Inequality, Skills, Asset Choice, and Business Cycles

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Heejeong Kim, B.A., M.A.

Graduate Program in Economics

The Ohio State University

2017

Dissertation Committee:

Aubhik Khan, Advisor Julia K. Thomas Pok-sang Lam ⃝c Copyright by

Heejeong Kim

2017 Abstract

In these essays, I study the causes and implications of differences in household wealth, income, and consumption using quantitative general equilibrium settings that are consistent with household-level data. I explore frictions that are necessary to reconcile differences in household behavior, over age, income and wealth, with micro data. Moreover, within these environments, I evaluate the aggregate implications of differences across households for the long-run and recessions.

In the first chapter, “Inequality, Portfolio Choice, and the Great Recession”, I study the effect of heterogeneity, in both the level and composition of wealth, in a dynamic stochastic overlapping-generations economy where households face uninsurable unemployment, earnings, and liquidity risk. Households pay transactions costs when they adjust savings held in high expected return assets, which make such savings illiquid. They also hold liquid, lower return assets. I show that household-level disparities in liquidity are important for understanding differences in their behavior, as well as aggregate changes in consumption and investment, over the Great Re- cession. When I allow for a rise in both unemployment and disaster risk, reducing households’ expected income and the expected return on their illiquid savings, aggregate consumption and investment fall to levels seen in the recession. The response of aggregate consumption is sensitive to the behavior of wealth poor households with a high marginal utility of consumption. Facing a large possible fall in earnings, they

ii build precautionary savings in liquid assets. However, in a typical incomplete markets model with a single asset, all households would respond to the fall in the expected return on savings following a rise in disaster risk. The resulting substitution effect would offset much of the negative wealth effect on aggregate consumption. In contrast, when much of wealth is illiquid, many households do not respond to a fall in its expected return, substantially dampening the substitution effect. Moreover, wealthier households, more likely to adjust their portfolios, increase their shares of liquid assets. This results in a large fall in aggregate investment.

In the second chapter, “Skill Premia, Wage Risk, and the Distribution of Wealth”,

I challenge the conventional assumption of i.i.d wage shocks in a standard heterogeneous household economy with uninsurable idiosyncratic risk and studies the implications of unobserved heterogeneity, both in the mean and variance of wage processes, on aggregate wealth and income inequality and life-cycle profiles of earnings, income, and wealth by education groups. I document strong evidence of conditional means and variances of wage processes that rise with skills from the PSID using minimum distance estimation. The implications of these estimated skill-specific wage processes are studied in an incomplete-markets quantitative general equilibrium OLG model wherein households choose their education level. A discrete skill choice partly endogenizes earnings risk across households and introduces a channel through which households’ schooling decision affects their earnings as well as wealth. I show that, in contrast to a model with a common wage shock, the model with empirically consistent wage processes that differ by skills successfully explains much of aggregate inequality as well as life-cycle earnings, income, and wealth of skilled and unskilled households.

Furthermore, I ﬁnd that college education subsidies directed at the poor decrease

iii wealth inequality. However, these subsidies reduce the quality of college graduates and increase the quality of those left behind.

In the third chapter, “Segmented Asset Market and the Distribution of Wealth”, with Aubhik Khan, we study the effects of segmented asset markets on wealth distribution in a quantitative OLG model. Using the 2004 SCF data, we find significant heterogeneity in household portfolio choice across ages and wealth levels. First, 30 percent of the U.S. households hold high-yield assets defined as stocks, bonds, and mutual and hedge funds. Second, the probability of a household participating in high- yield asset markets is rising with age and wealth level. Lastly, wealthy households tend to hold more of their financial assets as high-yield assets. Solving for stationary equilibrium, we find that asset market segmentation is an important source of wealth inequality. Segmented asset markets lead to a substantial increase in wealth dispersion across households. Specifically, an alternative model without market segmentation generates a Gini coefficient for wealth that is approximately 7 to 10 percent lower. Second, we reproduce the empirical findings that households are more likely to hold high-yield assets if they are older and wealthier.

iv To my parents

v Acknowledgments

I would like to express my sincere appreciation to Aubhik Khan and Julia K.

Thomas for their constant guidance and support. Their presence brought me to the

Ohio State University, which has proven to be one of the greatest choices in my life. I have highly beneﬁted from their training and have gained a lot of knowledge about various economic theories and computational methods. Especially, I cannot express enough gratitude to my main advisor, Aubhik Khan, who has the attitude and substance of a great economist. He is a driven, dedicated and passionate re- searcher. Without his continuous support and guidance, this dissertation would have not been possible. I would also like to thank Pok-sang Lam, who repeatedly helped me throughout the program. His suggestions and advice considerably advanced the

ﬁrst and second chapters of my dissertation. Jonathan Heathcote deserves a special thanks for giving me an opportunity to visit the Federal Reserve Bank of Minneapolis.

I received invaluable comments and advice from him and Fabrizio Perri during my visit. Lastly, I want to express my love and gratitude to my family. Without their tremendous support and sacriﬁce, I would not be here. A huge thank you to Daeyeol, who is my brother and the best friend of my life, for his great care and support.

vi Vita

June 12, 1986 ...... Born - Seoul, Korea

2010 ...... B.A. Economics Sogang University 2012 ...... M.A. Economics The Ohio State University 2015 to present ...... Doctoral Candidate The Ohio State University

Fields of Study

Major Field: Economics

vii Table of Contents

Page

Abstract ...... ii

Dedication ...... v

Acknowledgments ...... vi

Vita ...... vii

List of Tables ...... xi

List of Figures ...... xiv

1. Inequality, Portfolio Choice and the Business Cycle ...... 1

1.1 Introduction ...... 1 1.2 Related Literature ...... 7 1.3 Impact of the Great Recession on heterogeneous households .... 9 1.4 Model ...... 14 1.4.1 Households ...... 15 1.4.2 Production and government ...... 20 1.4.3 Recursive equilibrium ...... 21 1.5 Calibration ...... 23 1.5.1 Numerical Overview ...... 29 1.6 Results ...... 32 1.6.1 Steady state ...... 32 1.6.2 Aggregate Dynamics ...... 35 1.6.3 Changes in the joint distribution ...... 51 1.7 Concluding Remarks ...... 57

viii 2. Skill Premia, Wage Risk, and the Distribution of Wealth ...... 59

2.1 Introduction ...... 59 2.2 Empirical analysis ...... 67 2.2.1 Estimation of skill-speciﬁc wage processes ...... 67 2.2.2 Wealth Inequality and College Education ...... 74 2.3 Economic Model ...... 77 2.3.1 Overview ...... 77 2.3.2 College Education Decision ...... 79 2.3.3 A Household in the Working Life ...... 81 2.3.4 A Household after Retirement ...... 81 2.3.5 Age- and Skill-varying Natural Debt Limits in OLG economy 82 2.3.6 Recursive Equilibrium ...... 83 2.4 Calibration ...... 85 2.4.1 College Education Cost and Borrowing Limits ...... 85 2.4.2 Initial Wealth Distribution ...... 87 2.4.3 Remaining Model Parameters ...... 87 2.5 Quantitative Results ...... 90 2.5.1 Common shock economy ...... 90 2.5.2 Aggregate inequality ...... 91 2.5.3 Life-cycle implications ...... 97 2.5.4 Importance of Education Choice ...... 106 2.5.5 Beneﬁts of College Education ...... 107 2.5.6 Implications of the endogenous labor supply ...... 109 2.6 Policy Experiments ...... 110 2.7 Concluding Remarks ...... 113

3. Segmented Asset Markets and the Distribution of Wealth ...... 117

3.1 Introduction ...... 117 3.2 Wealth Inequality and Household Portfolio Choice ...... 121 3.3 Economic Model ...... 127 3.3.1 Overview ...... 127 3.3.2 A Household in the High-Yield Asset Market ...... 129 3.3.3 A Household in the Low-Yield Asset Market ...... 130 3.3.4 Recursive Equilibrium ...... 131 3.4 Calibration ...... 134 3.4.1 Parameters ...... 134 3.4.2 Earning Shocks Estimation ...... 135 3.4.3 Idiosyncratic return risk for the high-yield asset ...... 140 3.5 Results ...... 141

ix 3.5.1 Cross-sectional distribution ...... 141 3.5.2 The implications of endogenous asset market segmentation . 146 3.5.3 Sensitive analysis to interest rate risk ...... 150 3.6 Concluding Remarks ...... 151

Bibliography ...... 153

Appendices 159

A. Appendix to Chapter 1 ...... 159

A.1 Data appendix ...... 159 A.1.1 2007-2009 SCF panel data ...... 159 A.1.2 PSID data ...... 160 A.2 Equilibrium Ex-dividend Price and Dividends ...... 168 A.3 Numerical Method ...... 169 A.3.1 Steady state ...... 169 A.3.2 Decision rules ...... 170 A.3.3 Aggregate Dynamics ...... 171 A.4 Additional Tables ...... 176

B. Appendix to Chapter 2 ...... 178

B.1 Data Appendix ...... 178 B.1.1 PSID data ...... 178 B.1.2 SCF data ...... 179 B.2 Numerical Method ...... 180 B.3 Additional Results on Policy Experiment ...... 181 B.4 Additional Figures and Tables ...... 183 B.4.1 Lorenz curve ...... 183 B.4.2 Life-cycle Proﬁle of Hours worked ...... 183

x List of Tables

Table Page

1.1 The distribution of wealth ...... 10

1.2 Growth rates of variables before the Great Recession ...... 11

1.3 Growth rates of variables during the Great Recession ...... 13

1.4 Changes in growth rates of variables ...... 14

1.5 Summary of parametrization ...... 31

1.6 Distributions of net worth, illiquid and liquid assets ...... 33

1.7 Business cycle statistics in a portfolio choice model (B ﬁxed) ..... 38

1.8 Business cycle statistics in a portfolio choice model (rf ﬁxed) ..... 38

1.9 Business cycle statistics 3 ...... 39

1.10 Peak-to-Trough declines: U.S. 2007 Recession and model ...... 44

1.11 Peak-to-Trough declines: U.S. 2007 Recession and model ...... 50

1.12 Fraction of liquid assets to total net worth ...... 50

1.13 Share of households who actively adjust their portfolios ...... 51

1.14 Share of adjustors in each quintile that disinvest in illiquid assets .. 52

1.15 Growth rates of variables 2005-2007 ...... 53

xi 1.16 Growth rates of variables 2007-2011 ...... 54

1.17 Changes in growth rates of variables ...... 55

1.18 Growth rates of variables with rising disaster risk ...... 56

1.19 Changes in growth rates of variables with rising disaster risk ..... 57

2.1 Minimum Distance Estimates in 2004 ...... 74

2.2 Wealth inequality in the U.S. economy ...... 75

2.3 Summary of parametrization ...... 89

2.4 A common wage shock process in 2004 ...... 90

2.5 Distribution of wealth ...... 91

2.6 Distribution of income ...... 94

2.7 The fraction of the skilled and unskilled in the top percentiles .... 96

2.8 The probability of being in the top percentiles ...... 97

2.9 Economies without educational choice ...... 106

2.10 Economies without the beneﬁt of college education ...... 108

2.11 An economy with exogenous labor supply ...... 110

2.12 Implications of educational subsidies ...... 112

3.1 Wealth Inequality in the U.S. Economy ...... 123

3.2 Portfolio Share of High-Yield Assets over Age ...... 126

3.3 Portfolio Share of High-Yield Assets over Wealth ...... 127

3.4 Earnings shock estimates ...... 139

3.5 Wealth distribution ...... 141

xii 3.6 Income distribution ...... 142

3.7 Importance of Endogenous Asset Market Segmentation ...... 147

3.8 Sensitive analysis to interest rate risk ...... 150

A.1 Summary statistics of data ...... 161

A.2 Composition of total expenditure in the BEA and PSID ...... 163

A.3 Spending as a fraction of total expenditure in the BEA and PSID .. 164

A.4 Share of earnings, income, expenditure and expenditure rates ..... 176

A.5 Single asset economy ...... 176

A.6 portfolio choice economy with disaster risk ...... 177

B.1 Distribution of wealth ...... 181

B.2 Distribution of income ...... 182

B.3 Implications of education subsidies ...... 182

B.4 Determinants of log of net worth (SCF data) ...... 184

xiii List of Figures

Figure Page

1.1 Labor market experience premium ...... 26

1.2 Borrowing limits ...... 28

1.3 Share of illiquid asset as a fraction of net worth over wealth deciles . 36

1.4 Share of illiquid asset as a fraction of net worth over age groups ... 37

1.5 Impulse response 1 ...... 41

1.6 Impulse response 2 ...... 43

1.7 Impulse response in a model with a constant return on liquid savings 46

1.8 Impulse response in a model with a constant stock of liquid asset .. 48

1.9 Impulse response in a single asset economy ...... 49

2.1 Male college wage premium ...... 69

2.2 Labor market experience ...... 70

2.3 Variance of persistent and transitory shock across skill groups .... 73

2.4 Life-cycle wealth accumulation ...... 76

2.5 Age- and skill-varying borrowing limits ...... 86

2.6 Distribution of wealth for skilled and unskilled households ...... 95

xiv 2.7 Life-cycle hourly wages for skilled and unskilled households ...... 98

2.8 Life-cycle earnings proﬁle for skilled and unskilled households .... 100

2.9 Life-cycle wealth accumulation for skilled and unskilled households .. 102

2.10 The model- implied life-cycle saving rates ...... 104

2.11 Life-cycle income for skilled and unskilled households ...... 105

2.12 College educated households given initial ability ...... 114

3.1 Share of Households holding High-yield Assets ...... 124

3.2 High-yield Assets over Wealth ...... 125

3.3 Labor Market Experience (PSID data 1968-2011) ...... 137

3.4 Distribution of Wealth ...... 143

3.5 Share of Households Holding High-Yield Assets over Age ...... 145

3.6 Share of Households Holding High-Yield Assets over Wealth ..... 146

3.7 Life-cycle saving rates and participation in high-yield asset market .. 148

A.1 Illiquid wealth share and risky wealth growth rate over wealth deciles 165

A.2 Illiquid wealth share and risky wealth growth rate over age groups .. 166

A.3 The composition of illiquid wealth in 2007 ...... 167

B.1 The Lorenz curve ...... 183

B.2 Life-cycle hours worked for skilled and unskilled households ...... 183

xv Chapter 1: Inequality, Portfolio Choice and the Business Cycle

1.1 Introduction

Models of the business cycle have typically assumed a single asset held by households. This includes recent heterogeneous agent models studying the effects of wealth inequality on the propagation of large recessions (Krueger et al., 2015). However, the data shows large differences in the composition of household wealth. Households not only make consumption-savings decisions but also choose how to allocate savings across assets of varying liquidity and returns. I quantify the importance of this channel for understanding business cycles and differences in their impact on rich and poor households. While these are natural questions to explore, the difficulty of solving a heterogeneous household model with multiple assets in dynamic stochastic general equilibrium has meant that there has been little work done in developing an answer.

Households’ portfolios of assets vary systematically with their age and wealth.1

In a large recession, reductions in earnings and increases in unemployment risk may lead to a reduction in the share of illiquid but productive assets. When accompanied by a rise in precautionary savings, this may help explain large declines in aggregate

1See Glover et al. 2016 and Khan and Kim 2015.

1 investment and consumption as well as a slow recovery. An evaluation of the quantitative importance of this channel is the goal of this paper. In particular, I examine how optimal portfolio choices, across liquid and illiquid assets that vary in their rates of return, aﬀect consumption and savings of diﬀerent households. Importantly, this heterogeneity in the composition of wealth, previously disregarded in business cycle models with uninsurable earnings risk, has a quantitatively important role in reproducing the large declines in aggregate quantities seen in the Great Recession.

While heterogeneity in the composition of wealth has recently received atten- tion in quantitative macroeconomic models, it has either been studied in endowment economies (Glover et al. 2016) or in partial equilibrium (Kaplan et al. 2015), making it difficult to quantitatively evaluate the importance of this channel for the business cycle. Moreover, in the absence of intra-generational inequality, it is hard to study the effects of changes in household portfolios on the cross-sectional distributions of net worth, income and consumption (Glover et al. 2016). Households of different ages hold different shares of illiquid wealth, which implies dissimilar impacts of asset price changes on their income and consumption. These differences are equally evident when examining households of different wealth. Thus, a realistic assessment of the effect of portfolio choice requires intra-generational differences that have been omitted in general equilibrium studies. To my knowledge, I am the first to explore a quantitative DSGE OLG framework where households, who face earnings, unemployment and liquidity risk, choose both their consumption and savings in low-yield liquid and high-yield illiquid assets. Liquidity risk arises through transaction costs of actively adjusting illiquid assets. It reinforces the response of households to a rise in earnings and unemployment risk.

2 Another contribution of this paper is to introduce disaster risk into a heterogeneous household economy and explore the role of a rise in the risk of economic disaster in the presence of multiple assets of varying liquidity. The rise in disaster risk lowers households’ expected income and wealth. De Nardi et al. (2012) argue that these are crucial determinants for understanding the fall in aggregate consumption during the Great Recession. In my model, a rise in disaster risk, alongside a fall in total factor productivity, generates a strong response in precautionary savings by all but the wealthiest households. This leads to a shift into liquid assets. Important in this mechanism, which matches well household level data over the Great Recession, is the assumption that economic disasters, when they happen, involve a relatively large fall in TFP which reduces households’ expected life-time earnings and the expected return to illiquid assets. This is in contrast to the existing implementation of disaster risk taken by Gourio (2012), who studies its asset pricing implications in a representative agent model. In the current work, there is a large recession when, following a fall in TFP, a rise in disaster risk makes households increase precautionary savings and reduce illiquid, productive assets.

Calibrating the model economy using household-level data, the distribution of transactions costs that gives rise to illiquidity in capital investment is quantitatively disciplined by reproducing the share of illiquid assets to net worth across households of diﬀerent levels of wealth and age. In deciding their savings in liquid and illiquid assets, which vary in their usefulness for consumption smoothing, households face borrowing limits that are a common percentage of age-speciﬁc natural debt limits that arise naturally in a life-cycle model with retirement. These age-varying borrowing limits are crucial in reconciling the micro-evidence showing a high share of illiquid assets for

3 the young and wealth poor. The model reproduces a share of liquid assets of about one-third, similar to the SCF. Additionally, allowing for heterogeneity in the return on savings across households reproduces a signiﬁcant fraction of the distributions of net worth, liquid assets, and illiquid assets seen in the data.

The economy, with assets that vary in their liquidity and return, exhibits an aggregate response to shocks that varies with the cyclicality of the return to liquid assets supplied by the government. When this supply is weakly countercyclical, the model can exhibit unusually large declines in investment and consumption over a recession. The steep fall in earnings, and rise in unemployment risk, that characterizes a recession compels households to smooth consumption by monetizing their illiquid assets. This substitution of liquid for illiquid assets leads to an unusually large decline in investment when compared to a single-asset economy.

A rise in the risk of a large economic disaster, during a recession, amplifies the effect of the fall in productivity and leads to larger declines in aggregate consumption and investment. The heightened risk of a further worsening in earnings exacerbates an already large negative wealth effect, driving a rise in precautionary savings and reducing consumption. Investment also plummets as the increased disaster risk lowers the expected return on capital. This results in a large portfolio adjustments into liquid assets, especially when the supply of liquid assets expands to maintain a constant safe real interest rate. When the government finances a constant return on liquid savings, the wealth composition heterogeneity model with a rise in disaster risk predicts peak- to-trough declines of 4.1 percent in consumption and 18 percent in investment, similar to the changes observed over the Great Recession. This is the result of a rise in the risk of a disaster that interacts with multiple assets of varying liquidity.

4 Importantly, in a single-asset economy without illiquid assets, a rise in disaster risk has little effect as increases in precautionary savings’ motives are largely offset by equilibrium changes in the return to capital. In the present work, households increase their precautionary savings and decrease their consumption in a large recession when they expect lower life-time earnings following a rise in disaster risk. (wealth effect)

However, in a single asset economy, all households respond to a fall in the return to savings which dampens the fall in consumption. (substitution effect) Thus, the substitution effect offsets the negative wealth effect, mitigating the effect of disaster risk. In contrast, when households hold both liquid and illiquid assets as in my model, a large fraction of non-adjusting households do not respond to a fall in the return to capital (illiquid assets). This sharply decreases the magnitude of the substitution effect, resulting in a large fall in aggregate consumption consistent with the data.

Heterogeneity in the composition of wealth, and the associated countercyclical substitution of liquid, low-yield assets for illiquid, productive investment in capital, leads to a slow recovery following a large recession. The half-life of aggregate consumption rises 1.4 times compared to that in a single asset economy. As the economy begins to recover, households that paid transactions costs to reduce their holdings of illiquid assets and smooth consumption against a fall in income are initially reluctant to reinvest in capital. Their shares of illiquid assets, while less than desired, remain in a range consistent with optimal adjustment in response to ﬁxed costs. These households’ tolerance for portfolio imbalances slows aggregate investment in capital and economic recovery.

Finally, I evaluate the models’ predictions for changes in the cross-sectional distribution of net worth, income, and consumption to those in the PSID. During ordinary

5 times, the portfolio choice economy successfully explains growth rates of net worth, earnings, income, and consumption that fall in wealth. These growth rates fall in a recession. More importantly, with countercyclical supply of liquid assets, savings rates rise in a large recession. In a typical model recession, spending rates would rise as households smooth their consumption. In contrast, household level data shows a fall in spending rates over the recent recession. My model economy suggests an importance of the interplay, between a rise in the probability of a large disaster and multiple assets of varying liquidity, for understanding this fall in spending rates in the

Great Recession. Thus, the model with heterogeneity in the liquidity, as well as the level of wealth, and uninsurable earnings, employment and liquidity risk, is consistent with an unusual characteristics of the household data over the recession.

The numerical method developed to solve the model may be of independent interest. As pointed out in Kaplan and Violante (2014), solving for a stochastic OLG economy with a bivariate cross-sectional distribution of assets is challenging using the Krusell-Smith algorithm as this approach involves a repeated long simulations to obtain an accurate parametric law of motion for the aggregate state. To resolve this diﬃculty, I extend the Backwards Induction method of Reiter (2002, 2010) to solve a life-cycle model with heterogeneity in household wealth, multiple assets, borrowing limits that vary in age, and ﬁxed transactions costs.

The remainder of the paper is organized as follows. Section 2 discusses the related literature. Section 3 documents the cross-sectional distribution of net worth, earnings, income, and consumption from the PSID. Section 4 presents the model economy.

Section 5 discusses the calibration. Section 6 presents quantitative results and Section

6 7 concludes.

1.2 Related Literature

This paper contributes to the literature that studies the impact of a recession on heterogeneous households. Krueger et al. (2016) is the ﬁrst to study changes in the joint distribution of earnings, income, and consumption during the Great Reces- sion in a heterogeneous households framework. They also explore the importance of cross-sectional wealth inequality for the aggregate dynamics of an incomplete-markets

Krusell-Smith economy with unemployment and earnings risk as well as permanent differences in households’ discount factors. They find that an economy with a more pronounced dispersion of wealth experiences a larger drop in consumption compared to an economy with less inequality as wealth-poor households sharply decrease their consumption in a recession. I further study changes in the joint distribution during the Great Recession by allowing differences in households not just at the level of wealth but also in the composition of their wealth. In addition, I explore how this endogenous portfolio channel alongside a rise in disaster risk amplifies the Great

Recession.

Guerrieri and Lorenzoni (2015) study the effects of a credit crunch during a recession in a heterogeneous-agent incomplete markets model with variable labor supply and idiosyncratic income shocks. They find that a credit crunch forces financially constrained households to repay their debt while it increases precautionary savings by unconstrained households, resulting in a drop in real interest rates. As financially unconstrained households respond to a fall in real interest rates by increasing their

7 consumption, the resulting aggregate output and consumption responses are modest.2

In their extended model with durable goods, output even increases by 0.4 percent following a credit crunch due to the high substitutability between bonds and durables. In contrast, liquidity risk embedded in illiquid wealth weakens substitutability between two assets in my model, generating a larger drop in consumption in a recession.

My work is also related to a body of work that studies the eﬀects of multiple assets on heterogeneous households’ behavior. Glover et al. (2014) study the intergenerational redistribution eﬀects of the Great Recession in a stochastic complete-markets

6-period OLG economy with i.i.d aggregate shocks. Studying financial markets characterized by two assets - a risk-free bond and a risky equity, they find that older cohorts are hurt more than younger cohorts as a result of a larger decline in asset prices compared to a decline in wages during the Great Recession. However, in the absence of cross-sectional heterogeneity in each cohort, they are unable to address effects of wealth inequality in the recession. Moreover, given fixed total stocks of capital and labor, all changes in the demand translate to security prices changes. On the contrary, wealth inequality arising from financial friction with rich heterogeneity across households, in my model, makes it possible to explore the impact of the Great

Recession across households with diﬀerent levels of wealth. Furthermore, my model economy possibly explain both price and quantity eﬀects of the asset demand during the Great Recession by allowing the aggregate capital to change.

Kaplan, Mitman, and Violante (2015) explore the consumption response to changes in house prices over the Great Recession. Their OLG economy has two assets, housing and liquid savings. However, by assuming an exogenous interest rate on liquid savings

2In their benchmark economy, a 10 percent drop in debt-to-GDP ratio only leads to one percent drop in output.

8 and exogenous labor supply, which is the only factor of production, the dynamics of consumption, investment, and GDP over the business cycle cannot be addressed by their framework.

My model economy is closely related to the framework of Kaplan and Violante

(2014) which studies the consumption responses to fiscal stimulus in an incomplete- markets life-cycle model with an optimal portfolio choice between a low-return liquid asset and a high-return illiquid asset involving a deterministic transaction cost to adjust a high-return illiquid asset. In contrast to a partial equilibrium analysis in their work, I solve the model in a general equilibrium to study implications of multiple assets that vary by liquidity and risk over the business cycle. In addition, I explore the effects of a disaster risk affecting households’ belief during the Great Recession.

1.3 Impact of the Great Recession on heterogeneous households

Understanding changes in the joint distribution of net worth, consumption, earnings, and income may be important for studying the Great Recession. In this section, using the 2005-2011 PSID data, I document the cross-sectional distributions of these variables before and during the Great Recession.34 Disaggregated wealth data is contained in the PSID since 2003, consisting of transaction accounts, stocks, bonds,

IRAs, business and farm equity, and debt. Consistent with the deﬁnition of net worth

3Krueger et al. (2016) also examined changes in the joint distribution of income, consumption, and net worth during the recent recession. 4The PSID consists of the SRC and SEO samples. Though SEO sample is designed to oversample the poor, I only used SRC sample to be consistent with sample selection for earnings estimation.

9 in the SCF, I deﬁne net worth as the sum of total assets minus total debt.5 Table 1 shows that the distribution of wealth in the PSID is well aligned with that in the SCF.

The PSID constructed wealth data explains more than 25 percent of total wealth held by the top 1 percent of households, consistent with existing ﬁndings from the SCF of a highly concentrated wealth distribution in the U.S. (Cagetti and De Nardi (2004),

Castenada et al. (2003), Khan and Kim (2015), and Kim (2016)). Recent waves of the PSID also provide detailed spending data. I aggregate this data to measure total expenditure, deﬁned as the sum of total spending on nondurable goods and services.6

Year Gini top 1% 5% 10% 50% 90% ≤ 0 2007 SCF 0.78 29.1 52.3 64.3 96.8 100 10.3 2007 PSID 0.76 25.8 47.9 62.1 96.3 101 10.2 2009 SCF 0.79 29.8 53.2 65.5 98.2 101 14.8 2009 PSID 0.79 28.5 50.1 64.6 98.1 101 14.3 Notes: Table 1 shows the wealth Gini coeﬃcient, the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the share of households with zero or negative asset holdings in the U.S. economy. For the PSID, drop three samples with wealth less than negative 99 million dollars.

Table 1.1: The distribution of wealth

Table 2 summarizes annualized growth rates of the average level of net worth, earnings, income, and consumption between 2005 and 2007 over 2005 wealth quintiles.

It also documents percentage points diﬀerence in expenditure rates which are deﬁned

5See Appendix A for details on sample selection and wealth categories. 6Items on expenditure surveyed in the PSID are listed in Appendix A. I also compare the PSID constructed expenditure data to BEA data in Appendix A.

10 as total spending as a fraction of income, over the sample period.7 Earnings are the sum of wages and salaries, bonuses, overtime, tips, commissions, and transfers.

Income is deﬁned as the sum of labor income, unemployment beneﬁts and income from assets.89

NW Quantile Net worth Earnings Income Expenditure Expen. rate Q1(poor) 172 8.1 7.3 0.6 -3.4 Q2 26 3.1 3.0 6.0 1.3 Q3 19 1.3 1.8 3.5 0.7 Q4 11 2.0 1.5 1.2 -0.1 Q5(wealthy) 9.2 2.2 5.0 4.0 -0.4 all 11 2.8 3.5 3.0 0.0 Notes : Table 2 shows annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID.

Table 1.2: Growth rates of variables across wealth quintiles before the Great Recession 2005-2007

The last row of Table 2 shows that aggregate net worth increases by 11 percent on average per year between 2005 and 2007 while aggregate earnings and income rise by

2.8 and 3.5 percent respectively. Aggregate consumption rises by 3 percent annually and the average consumption rate changes little over the sample period. Table 2 also reveals a few facts regarding the dynamics of each variable over wealth. First, it

7To study impact of the Great Recession for households with different levels of wealth, I keep households in each quintile fixed and calculate the annualized percentage change of the averages between two years following Krueger et al. (2016). 8Definitions of earnings and income are comparable to those in Krueger et al. (2016). The only difference is that I report taxable earnings and income while they report disposable income. Measures of disposable earnings and income are work in progress. 9Note that, in the PSID, income and earnings data are retrospective while reporting unit for consumption varies by items.

11 shows that growth rates of net worth, earnings, and income decline in wealth. This may be driven by mean-reversion which implies a higher probability of the rise in earnings or income for those who are already poor compared to rich. Upward bias in growth rates embedded in a small level of wealth may also contribute. Second, my PSID constructed spending data implies a non-monotonic change in consumption

(column 5). For instance, consumption grows the least for the bottom 20 percent of households and the most for households in the second quintile of the wealth distribution. Finally, the poorest 20 percent of households in 2005 experienced largest drop in their spending rates. Moreover, the middle distribution of households, who are between 21 to 60 percent of wealth distribution, increased their spending rates in

2007 relative to 2005.10

Table 3 summarizes annualized growth rates of net worth, earnings, income, and consumption between 2007 and 2011 to study how wealthy and poor households are aﬀected during the Great Recession. It reports the growth of the average of each category for the 2007 wealth quintiles. As seen in the last row of Table 3, in the aggregate, net worth fell by more than 4 percent while earnings and income exhibited little growth. Next, aggregate consumption fell by 1.4 percent and the expenditure rate fell by 0.1 percentage point.11

Across households of diﬀerent wealth levels, Table 3 shows net worth growth rates decreasing in wealth and becoming negative for the richest 40 percent of households.

Households in the fourth and ﬁfth quintiles experience an approximately 5 percent decline in their wealth while the poorest 20 percent of households saw more than 160

10Krueger et al. (2016) ﬁnd consumption growth rates falling in wealth as well as the largest fall in spending rates for households in the ﬁrst quintile over the same sample period. 11In comparison, Krueger et al. (2016) report a 1.6 percentage points drop in the expenditure rate.

12 NW Quantile Net worth Earnings Income Expenditure Expen. rate(pp) Q1(poor) 164 2.1 2.1 -0.2 -2.1 Q2 3.0 1.5 1.6 -2.0 -3.4 Q3 0.9 0.7 0.1 -3.3 -3.0 Q4 -4.7 0.4 0.3 -2.0 -2.3 Q5(wealthy) -5.3 -0.8 -0.3 0.7 0.9 all -4.1 0.1 0.1 -1.4 -0.1 Notes : Table 3 shows annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID.

Table 1.3: Growth rates of variables across wealth quintiles during the Great Reces- sion 2007-2011

percent rise in their wealth. I also observe rich households having lower growth rates of earnings and income relative to wealth poor households (third and fourth columns of Table 3). Next, consumption falls for all households but the richest group during the Great Recession. Lastly, I ﬁnd a fall in expenditure rates for the bottom 80 percent of households while those in the ﬁfth quintile slightly increase their spending rates. In other words, over the recession, all households increase their savings rates except for the richest 20 percent of households.12

To study severity of the Great Recession across the wealth distribution, Table

4 presents diﬀerences in the growth rates of variables between normal times (2005-

2007) and recession (2007-2011). For net worth, households with less wealth tend to experience a larger drop in their growth rates of wealth. Turning to earnings, while Table 3 shows declining only for the richest 20 percent of households over the

12Over 2007-2011, Krueger et al. (2016) also find net worth growth rates falling in wealth. How- ever, they find negative consumption growth rates for only those in fourth and fifth quintiles compared to the bottom 80 percent of households in my data. They also report rise in saving rates across all wealth quintiles.

13 recession, earnings growth slows down for all households, with the largest fall for the bottom 20 percent. Income growth rates for households in the top quintile not only turn negative (Table 3) but also fall the most compared to before the recession.

(column 4 of Table 4) Despite declines in consumption growth rates across all wealth quintiles, expenditure rates rise by 1.3 percentage points for households in the bottom and top quintiles. In contrast, remaining households raise their savings rates.13

NW Quantile Net worth Earnings Income Expenditure Expen. rate Q1(poor) -8.0 -6.0 -5.0 -0.8 1.3 Q2 -23 -1.6 -1.4 -8.0 -4.7 Q3 -18 -0.6 -1.7 -6.8 -3.7 Q4 -16 -1.6 -1.2 -3.2 -2.2 Q5(wealthy) -15 -3.0 -5.3 -3.3 1.3 all -15 -2.7 -3.4 -4.4 -0.1 Notes : Table 4 shows changes in annualized growth rates of the average net worth, earnings, income, and percentage points change in expenditure rates from the PSID.

Table 1.4: Changes in growth rates of variables between prior- and during the Great Recession

1.4 Model

In my model economy, there are four agents: households, an investment firm, a production firm, and a government. Households differ by age, productivity, employment status, and wealth. Household wealth is itself the sum of low-yield liquid assets and high-yield illiquid assets. Each period, a household chooses its consumption,

13On the contrary, Krueger et al. (2016) ﬁnd the largest drop in expenditure rate for households in the bottom quantile. This discrepancy may arise by the fact that they included the SEO sample which is representative of income poor households.

14 total saving, and asset portfolio. It has to pay an idiosyncratic transaction cost if

it chooses to cash in its illiquid wealth and re-balance its portfolio. Illiquid wealth

is capital and is held in an investment ﬁrm, which rents it to the production ﬁrm.

Government supplies the net quantity of the liquid asset. Interest payments on which

are ﬁnanced by tax revenue.

In this section, I ﬁrst describe households’ decision problems. Second, I discuss

the production economy and government. Lastly, I present the deﬁnition of recursive

equilibrium.

1.4.1 Households

Households live for a ﬁnite number of periods. They enter the labor market at

age j = 1, retire at age Jr = 35, and their last possible age is J = 60. While work-

ing, households face a stochastic idiosyncratic unemployment risk e(z) ∈ {0, pe(z), 1}

where z is the exogenous aggregate state which consists of a total factor productivity

shock η and disaster state d. This unemployment risk, which moves as a function of

the exogenous aggregate state, determines households’ working time in each period.14

Households can be partially (e = pe) or completely (e = 0) unemployed with probability πu(z) while they are full-time workers (e = 1) with probability 1 − πu(z). A fraction πp(z) of those not-fully-employed are working for a fraction pe(z) of a period and unemployed for the remaining time. In contrast, not fully employed households are laid oﬀ for the full period with probability 1−πp(z). Summarizing, the probability for each idiosyncratic employment status is as follows:

14Following Khan (2016), partial unemployment is introduced to match the mean and median duration of unemployment less than a model period of one year.

15 a full-time worker : 1 − πu(z)

a part-time worker : πu(z)πp(z)

unemployed : πu(z)(1 − πp(z))

Partially and fully unemployed workers receive unemployment beneﬁts from the gov-

ernment proportional to their possible earnings and unemployment duration in the

period; the replacement rate is θu. Households draw idiosyncratic productivity shocks ∈ { } ′ | ε each period, which follow a Markov chain ε ε1, ..., εnε , where P r(ε = εk ε = ∑ ≥ nε 15 εl) = πlk 0 and k=1 πlk = 1. These idiosyncratic productivity shocks alongside a labor market experience premium, l(j), determine the total eﬃciency units of labor for workers. After retirement, households receive social security beneﬁts proportional

− to their last earnings shock s(ϵJr 1).

As mentioned already, the economy has two assets - a high-yield illiquid asset,

a, and a low-yield liquid asset, b. Each unit of illiquid wealth pays a dividend d(z)

and has an ex-dividend price p(z). Adjusting illiquid assets to a value other than a

non-adjusted post-dividend balance involves an idiosyncratic ﬁxed adjustment cost, ξ,

denominated in units of output, and drawn from a time-invariant distribution, H(ξ).

Households paying these costs adjust illiquid wealth to a desired value; otherwise, the

after-tax dividend payments are re-invested in illiquid wealth.

Borrowing is only allowed in the liquid asset, which is supplied by the government

at a price q. Households face borrowing limits that are a ﬁxed fraction of their

15I combined persistent and transitory productivity shocks into one shock process for ease of notation.

16 16 age-varying natural debt limits, ϕbj. Note that ϕ is common across households of

diﬀerent age. Following Kim (2016), I derive age-varying natural debt limits bj in an

17 overlapping-generations economy. Given a natural debt limit for the next age bj+1 and a lowest possible future earnings xj+1, a natural debt limit for age j is deﬁned

as: ≥ − bj qbj+1 xj+1

This implies that the maximum debt a household can borrow is the discounted value

of the maximum debt it can borrow tomorrow and the lowest possible future earnings.

Age-speciﬁc natural debt limits are crucial for explaining more indebted younger gen-

erations as they allow households to borrow against future income. In the following,

ϕ < 1

Each period, a household, identiﬁed by its age j ∈ J = {1, ..., J}, illiquid asset,

a ∈ A ⊂ R+, liquid asset, b ∈ B ⊂ R, productivity, ε ∈ E, and working status

e(z) ∈ {0, pe(z), 1}, chooses consumption and savings. Savings is in two assets, and

adjusting the stock of illiquid wealth requires payment of a transactions cost. A

household adjusts its illiquid asset from the otherwise non-adjusted post-dividend

balance if it pays the current idiosyncratic cost, ξ ∈ Ξ. If a household does not pay its ﬁxed cost, it can only choose consumption and liquid savings in the current period. The distribution of households, µ, is over (j, a, b, e, ε) and evolves following

the mapping µ′ = Γ(z, µ) where z = (η, d) is a vector summarizing the exogenous

aggregate state.

16In a life-cycle model, the natural debt limit falls with age as households borrow against their future income. 17Given that borrowing is not allowed at the last possible age, I can solve natural debt limits backward by age.

17 I now present the problem solved by households. For the ease of notation, I

′ deﬁne transition probabilities for aggregate state vector z by P r(z = (η, d)g|z =

z ≥ (η, d)f ) = πfg 0. Each period, a working household has three idiosyncratic shocks; unemployment shock e, productivity shock ε, and portfolio adjustment cost ξ. If a

household pays its current ﬁxed cost ξ, it actively adjusts its illiquid wealth account

a′. After any adjustment decision, a household at age j with illiquid assets a, liquid

assets b, unemployment shock e, productivity shock εl, and adjustment cost ξ realizes

the value vj(a, b, e, εl, ξ; zf , µ) given the aggregate state (zf , µ). { } a n vj(a, b, e, εl, ξ; zf , µ) = max vj (a, b, e, εl, ξ; zf , µ), vj (a, b, e, εl; zf , µ) (1.1)

a n where vj represents the value of a household adjusting its risky illiquid assets and vj is ∫ e ξ the value of a non-adjusting household. Let vj (a, b, e, εl; z, µ) = 0 vj(a, b, e, εl, ξ; z, µ)H(dξ) be the expected value of a household before the idiosyncratic adjustment cost ξ is realized.

A worker receives labor income and an unemployed worker receives unemploy-

− ment beneﬁts. A retiree receives a proportional social security beneﬁts, s(ϵJr 1). An actively adjusting household, paying a resource adjustment cost, cashes in the total stock of wealth in both assets and re-balance its portfolio. Assuming Epstein-Zin preferences, I describe the optimization problem of a household who pays its adjustment cost to actively adjust its asset portfolios between high-yield illiquid and low-yield liquid assets.

a vj (a, b, e, εl, ξ; zf , µ) =  −  1 { } 1 σ 1−σ ∑nε ∑nz ∑1 1−γ  − 1−σ z e ′ ′ ′ ′ 1−γ  max (1 β)c + β πlk πfg πe(zg)vj+1(a , b , e , εk; zg, µ ) c, a′, b′ k=1 g=1 e=0

18 subject to

′ ′ c + q(zf , µ)b + p(zf , µ)a ≤ b + (p(zf , µ) + (1 − τa)d(zf , µ))a + x − ξ (1.2) { (1 − τ )w(z , µ)l(j)ε (e + (1 − e)θ ) if j < J x = n f l u r Jr−1 (1 − τn)s(ϵ ) otherwise ′ ≥ ′ ≥ ≥ b ϕbj, a 0, c 0

′ µ = Γ(zf , µ)

where bj is the age-varying natural debt limit. If a household does not pay its ﬁxed cost, it can only choose its stock of safe liquid wealth for next period, b′. Illiquid assets pay after-tax dividends which are re-invested.

n vj (a, b, e, εl; zf , µ) =  −  1 { } 1 σ 1−σ ∑nε ∑nz ∑1 1−γ  − 1−σ e ′ ′ ′ ′ 1−γ  max (1 β)c + β πlk πfg πe(zg)vj+1(a , b , e , εk; zg, µ ) c, b′ k=1 g=1 e=0

subject to

′ c + q(zf , µ)b ≤ b + x (1.3) { (1 − τ )w(z , µ)l(j)ε (e + (1 − e)θ ) if j < J x = n f l u r Jr−1 (1 − τn)s(ϵ ) otherwise ′ a = (1 + (1 − τa))d(zf , µ)a

′ ≥ ≥ b ϕbj, c 0

′ µ = Γ(zf , µ)

19 1.4.2 Production and government

The exogenous aggregate state z is summarized by a total factor productivity

shock η and disaster state d. Exogenous total factor productivity, η, follows a Markov

′ ∈ { } | ′ ≥ chain η η1, ..., ηnη with P r(η η) = πη,η 0. The economy can be either in a

normal (d = 1), high risk of disaster (d = 2), or disaster state (d = 3). A high risk of

disaster involves a higher probability of the disaster state next period. The disaster

state has an additional drop in TFP, determined by the parameter λ, conditional on the economy already experiencing a fall in TFP. 18 TFP is (1 − λ(d, η))η, where

λ(3, η = η1) = λ < 1 and λ(3, η) = 0 otherwise. I assume that d also follows a Markov chain d ∈ {1, 2, 3} with   p11 p12 p13 d   π = p21 p22 p23 p31 p32 p33

where pij is the probability of transiting from i state to j state.

An investment ﬁrm takes households’ supply of capital and rents capital to the

production ﬁrm at a rental rate rk(z, µ) where µ is the distribution of households.

The investment ﬁrm faces convex capital adjustment costs. ( ) k′ − k 2 Φ(k′, k) = k k

This adjustment cost introduces deviation in the price of capital from that of con-

sumption as it makes consumption and capital less than perfectly substitutable.

The production ﬁrm employs capital k and hires labor n to produce output

through a CRS production function y = (1 − λ(d, η))ηkαn1−α, where 0 < α < 1

18I will assume that the disaster state aﬀects TFP when the latter falls by more than one standard deviation from its mean. This will be the case only when η = η1.

20 and η > 0. Thus, the optimization problem of the production ﬁrm is ( ) max (1 − λ(d))ηkαn1−α − rk(z, µ)k − w(z, µ)n k,n

The government supplies a quantity of liquid assets, B, at a price q(z, µ). Labor

income is taxed at τn and dividend income at τa. Social security benefits and unemployment benefits are also taxed at τn. Government revenues are used to finance social security benefit payments, unemployment benefits payments, interest payments on debt, and government spending, G(z, µ) ≥ 0.19

1.4.3 Recursive equilibrium

Deﬁne the product space S = J × A × B × E × {0, pe, 1} for the distribution

of households. Given the Borel algebra S generated by the open subsets of S, µ :

S → [0, 1] is a probability measure over households. Households start with an initial

0 ∼ 2 wealth of zero and an initial labor productivity drawn from π logN(0, σπ). A recursive competitive equilibrium is a set of functions

(v, va, vn, ca, cn, ha, hn, ba, bn, χ, k, n, r, q, w)

such that:

(i) (v, va, vn) solves (1)−(3), and (ca, ha, ba) are the policy functions associated with

(2) for consumption, illiquid and liquid asset savings by a household that adjusts

its illiquid asset holdings. (cn, hn, bn) are the policy functions associated with

(3) for consumption and savings in illiquid and liquid assets by a non-adjusting

household. χ is the decision rule associated with (1), and χ = 1 when the ﬁxed

cost to adjust illiquid assets is paid.

19Every period, spending is chosen to balance the government budget.

21 (ii) The government budget is balanced ∫ ∫ ∑1 ∑J ∑nε − − G(z, µ)+Bs+ (1 τn)(s(εl1j≥Jr )+(1 e)θuwl(j)ε1j

(iii) Markets clear ∫ ∫ ∑1 ∑J ∑nε n(z, µ) = l(j)εleµ(j, da, db, e, εl) e=0 j=1 l=1 A B ∫ ∫ ∑1 ∑J ∑nε k(z, µ) = aµ(j, da, db, e, εl) e=0 j=1 l=1 A B ∫ ∫ ∑1 ∑J ∑nε B(z, µ) = bµ(j, da, db, e, εl) e=0 j=1 l=1 A B (iv) Prices are competitively determined

w(z, µ) = (1 − α)(1 − λ(d))ηkαn−α

rk(z, µ) = α(1 − λ(d))ηkα−1n1−α

p(z, µ) = 1 + Φ1(Gk(z, µ), k)

α−1 1−α d(z, µ) = α(1 − λ(d))ηk n − δ − Φ1(Gk(z, µ), k) − Φ2(Gk(z, µ), k)

where Gk(z, µ) is the aggregate law of motion for aggregate capital. Φ1 and Φ2

2021 are the derivatives of Φ with respect to Gk and k, respectively. ( ) ′ 2 20 ′ k −k Note that these equilibrium price function, using Φ(k , k) = k k imply p = 1 and d = αηkα−1n1−α − δ in steady state. 21Given the deﬁnition of recursive competitive equilibrium, I show the prices, p(z, µ) and d(z, µ) are consistent with equilibrium in Appendix B.

22 (v)

′ ′ µ (j + 1,A0,B0, e , εk) = (∫ ∫ ) ∑nε πe′ (z) πlk µ(j, da, db, e, εl)H(dξ) + µ(j, da, db, e, εl)H(dξ) ∀ j l=1 ∆1 ∆2

where

a a ∆1 = {(a, b, e, εl, ξ)|h (j, a, b, e, εl, ξ; z, µ) ∈ A0, b (j, a, b, e, εl, ξ; z, µ) ∈ B0 and χ(j, a, b, e, εl, ξ; z, µ) = 1}

n n and ∆2 = {(a, b, e, εl, ξ)|h (j, a, b, e, εl; z, µ) ∈ A0, b (j, a, b, e, εl; z, µ) ∈ B0 and χ(j, a, b, e, εl, ξ; z, µ) = 0},

(j, a, b, e, εl) ∈ S and ξ ∈ Ξ.

1.5 Calibration

In order to bring the model to data, I calibrate the model economy using household

data. Given its detailed wealth information, I use the 2007-2009 SCF panel data to

calibrate parameters directly aﬀecting asset holdings including the distribution of

adjustment cost. Parameters governing unemployment are calibrated to match the

mean and median unemployment duration in the CPS and the average unemployment

rate in the BLS. Lastly, I estimate the earnings shock process using the PSID.

In the SCF, I deﬁne illiquid wealth as stocks, business equity, net residential

property, net equity in non-residential real estate and net consumer durables.22 The remaining assets and debt are considered liquid following Kaplan et al. (2016). I calibrate β to match the capital (productive illiquid asset) to output ratio of 2.66.

Following Kaplan et al. (2016), I consider business equity, stocks and net equity in non-residential real estate as productive illiquid assets as well as 40 percent of

22Following Glover et al. (2016), money market mutual funds and quasi-liquid retirement accounts are included in stocks. In the absence of a housing market and collateralized borrowing, I include residential property net of mortgage debt following Kaplan and Violante (2014).

23 net housing and consumer durables.23 Given that a sampling unit in the SCF is a household, the total value of productive illiquid assets is divided by the average family size in 2007 to make it comparable to GDP per capita.24 The calibrated economy implies a 6.4 percent return on illiquid wealth in the steady state when a zero return on liquid wealth is targeted, resulting in a liquidity premium of 6.4 percent. The capital share of output is α = 0.36. Despite the share of liquid wealth to GDP not being a target, the calibrated economy explains a ratio of 29 percent, which is close to the 35 percent in the 2007 SCF.

I assume a model period of one-year. Households enter the labor market at age 25, retire at age 60, and live until age 84 with certainty. I assume Epstein-Zin preferences, allowing the elasticity of intertemporal substitution (EIS) to deviate from the inverse of the coeﬃcient of relative risk aversion. I set the coeﬃcient of relative risk aversion to 2 and the EIS is 1.5.25 Social security payments are paid based on the average of the highest 35 years of earning by Social Security Administration (SSA). In the model, calculating average earnings requires one more state variable, making computation more challenging. Given the high persistent earnings process, I proxy the history of earnings over a worker’s life-cycle using the level of earnings in the last working ∑ J −1 − r l(j) − period, s(ϵJr 1) = θ w j=1 εJr 1. Following Hosseini (2015), θ is chosen to match s Jr−1 s 23Kaplan et al. (2016) argue that a fraction of housing and consumer durables could be rented to business or used in production. I follow their approach. 24As GDP per capita is expressed in chain 2009 dollars, I adjust the value of productive illiquid wealth in 2009 dollars using CPI-IPUMS. 25Gourio (2012) shows that it is important to have the EIS greater than 1 in a model with a disaster risk to reproduce a countercyclical risk premia as seen in the data. He also shows that, in an economy with standard expected utility, investment is much less volatile than the data. As noted by Kaplan and Violante (2014), when regressing consumption growth on real interest rates, an estimate of the EIS is downward-biased because of measurement error and endogeneity issues. By contrast, the estimates derived using GMM are generally greater than 1. Similarly, Binsbergen et al. (2012) estimated a value of the EIS above 1 in a DSGE production economy in which households have Epstein-Zin preferences using maximum likelihood.

24 the replacement rate of 45 percent of average pre-tax earnings in the steady state.

Labor income is taxed at 27 percent (Domeij and Heathcote (2004)). I chose τa to

imply a 25 percent capital income tax rate in a steady state.26

I select parameters p, πu, πp to match mean and median unemployment durations

as well as the unemployment rate as in Khan (2016). The working period for a par-

tially employed household p is chosen to match the median unemployment duration

of 12 weeks between 1981 and 2016 in the CPS.27 The probability of a partial employment πp varies with a TFP shock, πp ∈ [πp − εp, πp + εp]. A steady state level of the

probability of a partial employment πp is chosen to match the mean unemployment

duration of 24 weeks between 1981 and 2016.28 The mean duration of unemployment

rises to 36 weeks after 2008 which is around 60 percent of a model period. I select εp

to match this rise in duration when the economy is in recession.29 The probability of

∈ − l h unemployment πu also varies with a TFP shock, πu [πu εu, πu + εu]. πu is chosen

l to match an unemployment rate of 5 percent in the steady state. εu is chosen to

h explain a 5 percent rise in the unemployment rate over the Great Recession while εu results in a 3 percent unemployment rate when the economy is in a boom.30 Lastly,

I choose 43.5 percent for a replacement rate of unemployment beneﬁts θu following

Nakajima (2012).

26Elenev et al. (2016) estimated a capital income tax rate of 20 percent from government corporate tax revenue as a share of GDP using BEA data. In Jermann and Quadrini (2012), the average tax rate is 35 percent. I chose a value in this range. 27Partial unemployment is introduced to handle the median unemployment duration less than a model period. As one year has 52 weeks, the median unemployment duration corresponds to 23 percent of a model period. 28 24 − − ∗ Note that the mean unemployment duration is calculated as 52 = πp(1 p) + (1 πp) 1. 29 36 − − − − ∗ Given p, this implies that 52 = (πp εp)(1 p) + (1 (πp εp)) 1 30Khan and Thomas (2008) measured a TFP drop by 2.18 percent in the recent recession. Thus, εu is chosen to imply 10 percent of unemployment rate when a TFP falls by 2.18 percent.

25 The idiosyncratic earnings shock ε consists of both a persistent and a transitory

component. The persistent shock follows an AR(1) process;

ε = φ + εv,

′ φ = ρεφ + εs,

∼ 2 ∼ 2 2 2 where εs N(0, σs ) and εv N(0, σv ). The values of ρε, σs , σv are the 2007 estimates in Khan and Kim (2016). They also estimated a labor market experience premium l(j) by running an OLS regression of the log of hourly wages on time dummies; an interaction term with education and time dummies; labor market experience approx- imated by age minus years of schooling minus 5; and experience-squared. Figure 1 illustrates the resulting labor market experience premium.

2 1.9 1.8 1.7 1.6

l(j) 1.5 1.4 1.3 1.2 1.1 1 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 Number of years in labor market

Figure 1.1: Labor market experience premium

26 The distribution of idiosyncratic fixed costs ζ ∈ [0, ζ] is assumed to be drawn from an uniform distribution. Households can borrow up to a fixed fraction ϕ of their age-varying natural debt limits, bj. I choose these parameters (ζ, ϕ) to match the share of households holding zero or negative net worth and the share of households holding positive illiquid assets but little liquid assets as seen in the 2007 SCF.These liquidity constrained households are comparable to poor and wealthy hand-to-mouth households respectively as defined in Kaplan and Violante (2014) and Kaplan et al. (2016).31 Kaplan and Violante (2016) point out that these households have a high response in consumption following a transitory income change.32 Thus, these parsimonious targets have the virtue of allowing for the mechanism that Kaplan and

Violante (2014) emphasize: a potentially important role for liquidity constrained households in driving a large aggregate consumption response.

Borrowing limits are important in determining the number of liquidity constrained households as these households have little liquid assets. Figure 2 shows the resulting age-speciﬁc borrowing limits which, in contrast to the standard age-invariant limit, allows more borrowing by younger cohorts. Thus, I calibrate parameters (ζ, ϕ) to match liquidity constrained households in the 2007 SCF. As seen in the next section, given this calibration strategy, the calibrated economy explains a signiﬁcant fraction of the distributions of the net worth as well as illiquid and liquid wealth. Furthermore,

31Kaplan and Violante (2014) defined wealthy hand-to-mouth (htm) households as those with liquid wealth less than half of their earnings but holding positive balances of illiquid wealth. They estimated a total fraction of hand-to-mouth households between 17.5 percent and 35 percent of households in the 2004 SCF, including both wealthy and poor households. Among htm households, they find 40 to 80 percent are wealthy. I instead define wealthy htm households as those with no liquid wealth but positive illiquid wealth. 32Kaplan and Violante (2014) showed that the average MPC for HTM households is around 40 percent while that for non-HTM households is only 7 percent.

27 the benchmark economy gives rise to the share of illiquid assets to net worth over wealth deciles and age cohorts without being a calibration target.

-0.1

-0.2

-0.3

-0.4

-0.5

negative liquid liquid wealth negative -0.6

-0.7 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82

age

Figure 1.2: Borrowing limits

′ Aggregate TFP is assumed to follow an AR(1) process in logs, log η = ρη log η +

∼ 2 εη where εη N(0, ση). Khan and Thomas (2013) estimated this process using Solow residuals calculated from data on real U.S. GDP, private capital, and total

33 employment hours. I use their estimates and discretize this process into nη = 3.

For the transition probabilities for disaster risk, I ﬁrst assume that the probability of staying in normal times is equal to the persistence of ordinary TFP shocks. Next, in order to make disaster a rare event, I set the probability of going to the disaster state from a normal state to a negligible value, p13 = 0. Next, I assume that the disaster

33Khan and Thomas (2013) estimated an AR(1) TFP shock process in a an yearly model with the same capital depreciation rate and the capital share in production function as my model economy. Though they assumed a DRS technology instead of a CRS technology, this diﬀerence only slightly aﬀects the standard deviation of a TFP shock process.

28 state is preceded by a rise in the risk of disaster, which itself has a higher probability of transiting into a disaster state. Imposing the following symmetry conditions, p11 = p33, p12 = p32, p13 = p31, p21 = p23, I only need to set one more parameter: the probability of staying in a high-risk state p22. Given that it is hard to pin down the transition probabilities in a high disaster risk state, I assume the following transition probability matrix:

  0.909 0.091 0 πd =  0.25 0.5 0.25  0 0.091 0.909 Disaster risk is usually deﬁned as wars or economic depressions which result in more than a 15 percent decline in real GDP per capita (Barro, 2006) and modelled as a shock to TFP or capital destruction. (Gourio, 2012). λ is chosen to match a drop in the expenditure rate for the bottom 20 percent of households in the Great Recession.34

1.5.1 Numerical Overview

I develop a two-stage approach to solve savings decisions with two assets. In the

ﬁrst stage, given a current ﬁxed cost, a household chooses whether or not to adjust its portfolio. If it adjusts, it chooses its savings in illiquid wealth, a′. In the second stage, given a′, I solve for the optimal choice of liquid wealth, b′, using the endogenous grid method (Carroll, 2006). To solve the model with aggregate uncertainty, I extend the Backward Induction Method of Reiter (2002, 2010). This involves generalizing the method to solve an OLG economy with bivariate cross-sectional distribution of continuous endogenous state variables. The Backward induction method of Reiter

34Results are not very sensitive to the transition probabilities parameters for disaster state. Details are available in next revision.

29 allows the distribution of households to vary in potentially rich ways as a function of an approximate aggregate state as it does not impose a parametric aggregate law of motion. Solving individual decision rules and the consistent aggregate law of motion simultaneously, it does not require repeated simulation, reducing computation time compared to Krusell and Smith (1998). Please see Appendix C for more details.

30 Parameters set externally Value α capital share of output(NIPA) 0.36 δ depreciation rate(NIPA) 0.069 τn, τa labor and capital income taxes 0.27, 0.25 (ρη, ση) TFP shock process (Khan and Thomas, 2013) (0.909, 0.014) σ coefficient of relative risk aversion 2.0 1 γ inverse of EIS 1.5 λ(d = 3) additional drop in tfp a in disaster state 0.1 θs replacement rate of avg pre-tax earnings for social security 0.45 θu replacement rate of avg pre-tax earnings for unemployment benefit 0.435 (σs, σv, ρε) earnings shock process (Khan and Kim, 2016) (0.2449, 0.3937, 0.9825) l(j) male hourly wage life-cycle profile see text Parameters calibrated Value moments to match data model β 0.937 capital to output ratio 2.66 2.69 pe 0.2287 median unemployment duration as a fraction of a model period 0.2287 0.2287 πd see text transition probability matrix for disaster state πe(z) see text unemployment rate 31 ζ 1.4 share of hhs holding positive illiquid asset but little liquid asset 0.32 0.32 ϕ 0.4 share of hhs holding zero or negative net worth 0.103 0.103

Table 1.5: Summary of parametrization 1.6 Results

I begin by discussing the steady state of the model economy. Next, I discuss the implications of wealth composition heterogeneity for business cycles driven by shocks to TFP and changes in disaster risk. Lastly, I evaluate predictions for changes in the joint household distribution of net worth, earnings, income and consumption.

1.6.1 Steady state

Table 6 summarizes distributions of net worth, illiquid asset, and liquid assets in the 2007 SCF and the steady state of the economy with heterogeneity in both the level and composition of net worth. Though it only targets the share of liquidity constrained households, the portfolio choice economy successfully reproduces much of the dispersion in wealth. As seen in the top panel, the quintile distributions of net worth in the benchmark (wealth composition heterogeneity) economy are broadly consistent with those in the 2007 SCF, explaining more than 70 percent of the total wealth held by the wealthiest 20 percent of households. The wealth Gini is within

9 percentage points of its empirical counterpart.35 As pointed out in Krueger et al.

(2016), reproducing enough dispersion in wealth in a model is important for explaining aggregate dynamics.36 A realistic skewness in the distribution of wealth is also crucial for studying changes in the cross-sectional distributions of consumption and income as the recent recession had disparate eﬀects on households with diﬀerent levels

35This is not driven by a high fraction of households near the borrowing limit as in Huggett (1996) but by the right tail of the distribution. The wealthiest 10 percent of households hold 57 percent of wealth while they hold 64 percent in the 2007 SCF. 36Krueger et al. (2016) show that the drop in aggregate consumption is 0.5 percentage points larger in an economy with more wealth inequality than in a representative agent economy.

32 of total wealth as shown in section 3.37

Net worth Q1 Q2 Q3 Q4 Q5 top 1% 5% 10% ≤ 0 Gini 2007 SCF -0.3 1.4 5.7 14.1 79.1 29.1 52.3 64.3 10.3 0.78 Benchmark 0.2 3.8 8.9 16.1 71.1 11.6 38.2 57.1 10.3 0.69 Single asset 1.1 4.9 9.7 16.7 67.6 9.9 34.5 52.3 2.16 0.64 Illiquid wealth Q1 Q2 Q3 Q4 Q5 top 1% 5% 10% Gini 2007 SCF 0.14 1.6 5.9 14.4 78.0 28.2 51.2 63.2 0.76 Benchmark 0.0 3.0 8.1 15.3 72.0 11.9 40.3 59.1 0.71 Liquid wealth Q1 Q2 Q3 Q4 Q5 top 1% 5% 10% Gini 2007 SCF -11.7 -0.53 0.92 7.8 103 47.1 76.2 90.5 0.92 Benchmark -8.3 -1.4 4.89 22.9 82.0 11.3 36.7 54.9 0.75 Notes : Table 6 shows the share of net worth, illiquid asset, and liquid asset across the wealth quintiles. It also reports the share held by the top 1, 5, and 10 households for net worth, illiquid asset, and liquid asset. Lastly, it reports the share of households with zero or negative net worth in the 2007 SCF and model and the Gini coeﬃcient for net worth, illiquid asset, and liquid asset.

Table 1.6: Distributions of net worth, illiquid and liquid assets

What explains differences between rich and poor households in the steady state of the benchmark economy? First, households with a favorable earnings shock have a propensity to smooth consumption by accumulating a higher level of wealth compared to others. Second, a household with a high level of earnings can afford portfolio adjustment costs and invest in illiquid assets to receive a higher return on savings which accelerates its wealth accumulation. Furthermore, fixed transaction costs realized to adjust illiquid asset holdings discourage households from investing in illiquid assets until they are sufficiently wealthy. This endogeneity in the composition of wealth

37While interesting, explaining the very right tail of the distribution is not of primary importance as the analysis largely focuses on the quintile distributions of wealth where the model matches the data fairly well.

33 is crucial for explaining asset positions before the recession and the subsequent heterogeneous responses during the recession as households with diﬀerent portfolios are dissimilarly aﬀected by changes in asset prices.

Table 6 also summarizes the distributions of high-yield illiquid assets and low-yield liquid assets in the 2007 SCF and in the portfolio choice economy. The distributions of illiquid assets are comparable to those in the data, reproducing a Gini of 0.71 and

72 percent of the illiquid assets held by the top 20 percent of households compared to 78 percent in the data. However, SCF shows more dispersion in liquid assets.

The liquid wealth Gini is around 0.91 and the top 20 percent of households hold all of net liquid assets in the economy. Borrowing by the bottom 20 percent of households represents approximately 12 percent of the total stock in the 2007 SCF.

Though the benchmark economy qualitatively matches the more skewed distribution of liquid relative to illiquid assets, the distribution of low-yield assets falls short of the data. The highly skewed empirical distribution may be driven by home equity loans that allow households to borrow against their illiquid wealth. As this home equity borrowing is considered liquid, in the model abstracting from collateralized borrowing leads to less inequality in liquid assets in the model compared to the data.

Importantly, age-varying borrowing limits that are a common percentage of age- consistent natural debt limits, alongside ﬁxed costs, give rise to empirically consistent average shares of illiquid assets across households of diﬀerent levels of wealth and age.

As seen in Figure 3, the benchmark economy predicts more than 80 percent of the total wealth held in illiquid assets for the top 80 percent of households and all of the wealth held as illiquid assets for the bottom 11-20 percent of households. The high ratio of illiquid assets to net worth, for the poorest 11-20 percent of households, is

34 the result of wealth-poor households borrowing in liquid asset markets. This drives a small level of wealth which biases up their share of illiquid assets to net worth.

Figure 4 shows the corresponding ﬁgure over age groups. Though the benchmark economy overestimates the average share of total wealth held in illiquid assets by the youngest cohort, it captures the share’s decline over age. Older cohorts tend to hold a higher fraction of safe liquid assets compared to younger cohorts as they have less time to recover from any fall in asset prices.38 In the benchmark economy, age-varying borrowing limits are critical in explaining the high share of illiquid assets held by the young. More borrowing by younger households also biases up their share of illiquid assets to net worth embedded in the small level of wealth.

1.6.2 Aggregate Dynamics

In this section, I ﬁrst present dynamic results for the model using TFP shocks.

Next, I introduce disaster risk to show how a rise in the probability of a severe economic recession, consistent with rational expectations, amplifies the effects of a shock to TFP. Throughout, I examine the model using two different assumptions about the supply of liquid assets. The first involves a constant stock of liquid assets (B fixed) while the other assumes a constant return (rf fixed). Thus, in the second case, the supply of liquid assets is perfectly elastic while, in the first, government policy holds the supply of such assets fixed.39 Note that, while the return on liquid savings is

38Glover et al.(2010) also find that younger cohorts hold less safe assets than old cohorts because of their large mortgage debt which is considered safe in their classification. I still find the high share of illiquid assets to total wealth by the young when measured using the net value of housing and consumer durables. 39Each of these assumptions about government policy over the business cycle serves as a benchmark. Over the 2007 recession, real interest rates on safe, liquid assets fell markedly while, at the

35 3

2.5

1.5

0.5

0 top 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 wealth deciles data model Notes : The average share of illiquid asset as a fraction of net worth over wealth deciles in the 2007 SCF (blue bars) and the benchmark economy (red bars).

Figure 1.3: Share of illiquid asset as a fraction of net worth over wealth deciles

held constant in equilibrium in a ﬁxed return model, this is consistent with general equilibrium characterized by the government policy which adjusts the supply of liquid assets to maintain the constant return.

Heterogeneity in the composition of wealth

In Table 7 and 8, I summarize business cycles moments from simulating two portfolio choice models.40 In both tables, we see that models with multiple assets of varying liquidity produce consumption approximately 40 percent as volatile as same time, a ﬂight to quality led to a large rise in their supply. In a future revision, I plan to address this by modelling a countercyclical supply of liquid assets that responds to a higher probability of disaster. The supply function describing government policy can be chosen to ensure a fall in the safe real interest rate consistent with the data. 40I simulate the model economies with an aggregate productivity shock for 2300 periods and drop the ﬁrst 300 periods.

36 2.5

1.5

0.5

0 25-34 35-44 45-54 55-64 65-74 over 75 age bin data model Notes : The average share of illiquid asset as a fraction of net worth over age group in the 2007 SCF (blue bars) and the benchmark economy (red bars).

Figure 1.4: Share of illiquid asset as a fraction of net worth over age groups

output and investment twice as volatile. Furthermore, the portfolio choice economy has a procyclical return on savings in liquid assets when their stock is ﬁxed (column

9 of Table 7). In contrast, when the return is rigid, the equilibrium stock becomes countercyclical (column 6 of Table 8). As a result, and somewhat surprisingly, the aggregate economy behaves similarly across the two cases where the supply of liquid assets is perfectly inelastic or elastic.

A rise in unemployment risk, during a recession, increases households’ demand for liquid assets. Households attempting to smooth consumption following a decline in their earnings increase their stock of liquid savings relative to illiquid assets. Given a ﬁxed stock of such assets, portfolio adjustments by households lower their return.

When instead the government choose to hold the return on liquid savings ﬁxed, this

37 portfolio adjustment drives a rise in liquid assets over a recession.41

x = YCIKBNE(r) rf w mean(x) 2.48 1.84 0.48 6.45 0.73 1.44 0.07 0.0 1.10 σx/σy (2.75) 0.38 1.95 0.22 n/a 0.86 0.27 0.02 0.29 corr(x, y) 1.0 0.95 0.99 -0.04 n/a 0.96 0.94 0.79 0.59 Notes : Table 7 presents means of GDP, consumption, investment in illiquid wealth, stock of capital, supply of liquid wealth, total hours worked, expected return on illiquid wealth, return on liquid savings and wage for the model simulated data. It also lists relative standard deviation to and correlation with GDP for each variable. I smooth series using a HP-ﬁlter with a smoothing parameter of 100.

Table 1.7: Business cycle statistics in a portfolio choice model (B ﬁxed)

x = YCIKBNE(r) rf w mean(x) 2.49 1.84 0.49 6.50 0.65 1.44 0.07 0.0 1.10 σx/σy (2.75) 0.38 2.0 0.22 0.26 0.86 0.56 n/a 0.29 corr(x, y) 1.0 0.96 0.98 -0.02 -0.46 0.96 0.93 n/a 0.59 Notes : Table 8 presents means of GDP, consumption, investment in illiquid wealth, stock of capital, supply of liquid wealth, total hours worked, expected return on illiquid wealth, return on liquid savings and wage for the model simulated data. It also lists relative standard deviation to and correlation with GDP for each variable. I smooth series using a HP-ﬁlter with a smoothing parameter of 100.

Table 1.8: Business cycle statistics in a portfolio choice model (rf ﬁxed)

41While these forces affect the composition of household savings, their aggregate effects in ordinary business cycles is small. As a result, fluctuations in GDP, consumption, and investment resemble those in a single asset economy subject to the same TFP and unemployment risk shocks. (Table D2)

38 Table 9 shows business cycle moments for the fraction of households who adjust their illiquid assets and for the share of liquid assets to the total stock of wealth in the economy. As mentioned above, during a recession, following a reduction in earnings and a rise in unemployment, more households increase their holdings of liquid assets to oﬀset the drop in their consumption, resulting in the counter-cyclical share of liquid assets across households. In the economy where the supply of such assets is allowed to vary, the share of liquid assets rises in a recession as their stock increases.

This results in a more variable share than when government policy holds the supply of such assets ﬁxed.

B fixed rf fixed B B x = adjusting pop K+B x = adjusting pop K+B mean(x) 0.144 0.102 mean(x) 0.147 0.091 σx/σy 0.279 0.192 σx/σy 0.272 0.309 corr(x, y) -0.485 0.046 corr(x, y) -0.424 -0.327 Notes : Table 9 presents means of the share of households who adjust their illiquid assets and the share of liquid assets to total wealth for the model simulated data. It also lists relative standard deviation to and correlation with GDP for each variable. I smooth series using a HP-filter with a smoothing parameter of 100.

Table 1.9: Business cycle statistics for the share of households adjusting illiquid assets and the share of liquid assets to total wealth

Now, I present impulse response functions for aggregate variables following a drop in TFP, accompanied by a rise in unemployment risk, in Figures 5 and 6 for a single asset economy as well as two heterogeneous composition of wealth economies. I assume that TFP falls for the ﬁrst two periods then starts to recover at the rate

39 implied by its persistence.42 Following a drop in TFP, output declines by 6 percent on impact both in a model with a single asset and in models with multiple assets of varying liquidity. Given a predetermined stock of capital, the fall in output is driven by the drop in TFP and aggregate labor supply. As labor supply is exogenous and all economies experience the same 5 percent rise in unemployment, the change in output on impact must be the same.

In Figure 5 (d), aggregate consumption exhibits a gradual, familiar, hump-shaped

response. Moreover, aggregate consumption falls more in economies with illiquid as-

sets compared to a single asset economy. For instance, in the absence of any change

in the aggregate stock of liquid assets, aggregate consumption decreases by approxi-

mately 2.3 percent while it only falls by 2 percent in the single asset case. The larger

fall in consumption is the result of illiquidity in wealth. The costs of adjusting illiquid

assets deters households’ ability to smooth consumption. As a result, consumption is

more volatile in response to shocks to TFP and employment.

Adding liquidity risk into a model produces a slow recovery which is one of the

characteristics of the Great Recession. In Figure 5 (d), the half-life of consumption in

the economy with a ﬁxed stock of liquid wealth is 1.4 times greater than that in a single

asset model. This slow recovery is the result of the equilibrium fall in the real return

to safe assets which depresses savings and consumption growth. Transactions costs

imply a range of portfolio shares over which households are unwilling to increase their

holdings of illiquid wealth. The slower growth in liquid assets for their non-investors

implies fewer households reaching a suﬃciently low share of illiquid wealth to invest

in capital.

42I choose 2.18 percent drop in a TFP to be consistent with the estimated decline in measured TFP over the Great Recession reported in Khan and Thomas (2013).

40 (a) total factor productivity (b) output 0 0 -0.5 -2

-1 -4 -1.5 -6 single asset economy

percent change percent -2 two asset economy (B fixed) percent change percent two asset economy (r_f fixed) -2.5 -8 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date date (c) hours worked (d) consumption 0 0 -1 -0.5 -2 -1 -3 -1.5 -4 -2 percent change percent -2.5 percent change percent -5 -6 -3 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date date

Notes : Figure 5 shows aggregate response to a 2.18 percent drop in TFP in a single asset economy (black solid line), in a heterogeneous asset economy with a ﬁxed quantity of liquid wealth (blue dashed line), and in a portfolio choice economy with a ﬁxed price of liquid wealth (red dotted line). The vertical axis measures the percent change of a variable from its mean.

Figure 1.5: Impulse response 1

41 Aggregate capital and investment fall more in portfolio choice economies following a drop in TFP compared to a single asset economy as seen in Figures 6 (a) and 6

(b). In a single asset economy, households can raise their buffer stock of savings only through capital, hastening the recovery. However, in an economy with multiple assets that bear different risk and liquidity, households, who invested in high-yield assets at the expense of liquidity, re-balance their portfolios toward liquid assets that offer value in smoothing consumption. This results in a larger drop in investment and capital compared to a single asset economy.

Table 10 summarizes peak-to-trough declines for each series in the Great Recession and in model economies. Following a 2.18 percent decline in TFP, an economy with one asset experiences 10 percent drop in investment and 2.4 percent drop in aggregate consumption compared to 19 percent and 4 percent, respectively, in data. As seen in the third and fourth rows of the Table 10, adding another asset that carries liquidity risk explains an additional 2 to 3 percentage points drop in aggregate investment.

Although aggregate consumption falls more by 0.2 percentage points in a ﬁxed liquid asset economy from those in a single asset economy, portfolio choice economies do not explain enough of the drop in aggregate consumption seen in data. In the following section, I introduce the disaster risk to study the ampliﬁcation mechanism of a rise in risk of a disastrous recession across households.

Disaster risk

In this section, I introduce aggregate disaster risk into the wealth composition heterogeneity economy to explore how a persistent rise in the risk of economic disaster interacts with diﬀerence in liquidity of household wealth. Figure 7 presents impulse

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Notes : Figure 6 shows aggregate response to a 2.18 percent drop in TFP in a single asset economy (black solid line), in a heterogeneous asset economy with a ﬁxed quantity of liquid wealth (blue dashed line), and in a portfolio choice economy with a ﬁxed price of liquid wealth (red dotted line). The vertical axis measures the percent change of a variable from its mean.

Figure 1.6: Impulse response 2

43 GDP I N C T F P data 5.59 18.98 6.03 4.08 2.18 single asset 5.92 10.00 5.58 2.45 2.18 two asset (B fixed) 6.06 12.06 5.58 2.61 2.18 two asset (rf fixed) 5.98 12.54 5.58 2.42 2.18 Notes : Table 10 shows peak-to-trough declines between 2008q4 and 2009q2 in the first row (Khan and Thomas, 2013). The second row is a single asset economy. The third row is a model with two assets, where the return on liquid wealth is fixed. The last row is a two asset model with a fixed stock of liquid assets.

Table 1.10: Peak-to-Trough declines: U.S. 2007 Recession and model

responses following a 2.18 percent drop in TFP and a rise in unemployment risk for the first two periods in a portfolio choice economy where the return on liquid savings is fixed over time while Figure 8 shows the corresponding figure in a portfolio choice model where the stock of liquid wealth is fixed. A rise in disaster risk raises households’ desire for liquidity. The solid line in Figure 7 shows that, when there is no change in disaster risk, the aggregate response of the economy following a TFP shock is close to those in Figures 5 and 6 where there was no disaster. This muted effect of aggregate disaster risk in ordinary times is the result of a rare probability of further deterioration in TFP and earnings.43

In contrast, a rise in the risk of a sharp fall in TFP, across households, implies a different response in the aggregate economy as seen in Figure 7. First, a rise in disaster risk qualitatively changes the familiar hump-shaped response of aggregate consumption in the wealth composition heterogeneity model with a fixed rate of return on liquid wealth. This is due to a change in the average substitution effect

43Recall that the likelihood of disaster in normal times is near zero.

44 when the future real rate of return to wealth is expected to be lower, following a rise in disaster risk, than the current average return. Over a typical TFP shock driven business cycle, the real return on assets rises over time following an initial drop. Over this period, the substitution effect leads to further reductions in consumption even as the negative wealth effect declines, generating a hump-shaped response in aggregate consumption. In contrast, when the real rate of return on liquid savings is rigid, the return to portfolios is expected to fall in the future with a rise in disaster risk. This fall in the expected real return on wealth changes the familiar hump-shaped response in consumption to a monotone one. Second, more cautious households drive a sharp initial drop in aggregate consumption. This larger drop in consumption is driven by the large negative wealth effect associated with the higher likelihood of economic disaster reducing TFP and earnings further. Lastly, aggregate investment experiences a large drop as the lower expected future return on capital causes a strong portfolio adjustment into liquid assets.44

As seen in Figure 8, if the stock of liquid assets is ﬁxed, the rise in precautionary savings through liquid assets by cautious households in a recession lowers the return on liquid savings (Figure 8 (d)). This fall in the return on liquid savings dampens the eﬀect of lower expected future average return on asset portfolios seen in Figure

7 as the current return on savings also falls more than before. While weakening the change in the substitution eﬀect from a rise in disaster risk, the rising probability

44Note that I model the disaster state as an additional drop in TFP that reduces earnings and the interest rate on savings. When I instead model disaster as a rise in capital depreciation which only aﬀects the return on savings, consumption rarely falls beyond that in the response to actual TFP. It is important to generate a rising risk of a fall in earning through an economic disaster rather than a lower return on savings to explain a sharp drop in both aggregate consumption and capital investment. Thus, a rise in disaster risk does not increase precautionary savings if disaster reduces the return on assets relative to earnings.

45 (a) total factor productivity (b) consumption 0 1 0 -0.5 -1 -1 -2 -1.5 -3 percet change percet percent change percent -2 -4 tfp shock tfp shock + disaster risk -2.5 -5 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date date

percet change percet 4 percent change percent -20 2 0 -25 -2 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date data

Notes : Figure 7 shows aggregate dynamics in the heterogeneous wealth composition economy following a 2.18 percent drop in TFP (black solid line) and a 2.18 percent drop in TFP alongside a rise in disaster risk (blue dashed line). The vertical axis measures percent changes of each variable from its simulation mean.

Figure 1.7: Impulse response in a model with a constant return on liquid savings (rf ﬁxed)

46 of disaster drives a far larger negative wealth eﬀect as it sharply reduces expected earnings compared to normal times. This drives a severe recession from an empirically consistent actual fall in TFP.

A rise in the risk of a further worsening of earnings has little implication for aggregate dynamics in a single asset economy. Figure 9 shows impulse responses for aggregate consumption and investment when all households save using the same asset. While aggregate consumption falls slightly more with a rise in disaster risk, on impact, we see that the aggregate dynamics of consumption and investment are close to those with only a shock to TFP. In a single asset economy, a rise in precautionary savings by households would lower the return to savings in equilibrium, oﬀsetting any aggregate eﬀect.

Table 11 summarizes peak-to-trough declines in both heterogeneous wealth composition economies with a rise in disaster risk. As I have mentioned, if the real rate of return on safe assets does not drop as much as predicted by a model with a ﬁxed stock, the rise in disaster risk drives a sharp substitution from illiquid capital to liquid assets, reproducing the drop in investment and consumption seen in the data.

Indeed, a portfolio choice economy with a rigid return on liquid savings explains the

4 percent drop in consumption as well as the 20 percent drop in investment over the

Great Recession.

The sharp simultaneous declines in investment and consumption, following a rise in the probability of an economic disaster, is driven by dissimilar responses from households with diﬀerent levels of wealth. Wealth poor households, with little liquid savings, face the risk of a sharp rise in the marginal utility of consumption following a further decline in earnings. They respond with portfolio adjustment selling illiquid

47 (a) total factor productivity (b) consumption 0 0.5 -0.5 0 -0.5 -1 -1 -1.5 -1.5 -2 percent change percent -2 change percent tfp shock -2.5 tfp shock + disaster risk -2.5 -3 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date date

percent change percent -10 -1 -12 change percentage -1.2 -14 -1.4 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date date

Notes : Figure 8 shows aggregate dynamics in the heterogeneous composition of wealth economy with disaster risk in a response to a 2.18 percent drop (black solid line) and to a 2.18 percent drop alongside with a rise in disaster risk (blue dashed line).

Figure 1.8: Impulse response in a model with a constant stock of liquid asset (B ﬁxed)

48 (a) consumption (b) investment 1 2 0 0 -2 -4 -1 -6 -8 -2 tfp shock percent change -10 percent change tfp shock + disaster risk -3 -12 1 11 21 31 41 51 61 71 81 91 1 11 21 31 41 51 61 71 81 91 date date

Notes : Figure 9 shows aggregate dynamics in a single economy with disaster risk in a response to a 2.18 percent drop (black solid line) and to a 2.18 percent drop alongside with a rise in disaster risk (blue dashed line).

Figure 1.9: Impulse response in a single asset economy

assets to pay oﬀ debt and build liquid savings. The fall in investment is the result of a general reduction in savings by wealthier households who, while roughly maintaining their share of illiquid assets, smooth consumption by reducing overall savings. Table

12 shows the fraction of liquid assets to total net worth over wealth quintiles in the steady state and over the ﬁrst date of this recession accompanied by a rise in disaster risk. As seen in the third row, higher risk of economic disaster boosts precautionary savings by the poorest 20 percent of households; their share of liquid savings rises from -127 percent to 18 percent when the return on liquid savings is ﬁxed.45

45While the increase in the share of liquid savings by the poorest 20 percent of households is amplified with a fall in the rate of return on liquid savings, a rise in disaster risk still boosts their precautionary savings compared to a recession without a rise in the probability of disaster. Indeed, without disaster risk, the start of a recession sees the share of liquid savings to net worth rise to 20 percent (fixed return model) and 13 percent (fixed stock model).

49 GDP I N C T F P data 5.59 18.98 6.03 4.08 2.18 two asset (B fixed) 5.71 12.83 5.58 2.82 2.18 two asset (rf fixed) 5.71 19.89 5.58 3.82 2.18 Table 11 shows peak-to-trough declines between 2008q4 and 2009q2 in the first row (Khan and Thomas, 2013). The second row is a model with two assets, where the return to liquid savings is fixed. The last row is a two asset model with a fixed stock of liquid asset.

Table 1.11: Peak-to-Trough declines: U.S. 2007 Recession and model

B ﬁxed steady state impact dates rf ﬁxed steady state impact dates t=0 t=1 t=2 t=0 t=1 t=2 all 0.096 0.100 0.101 all 0.096 0.106 0.108 Q1(poor) -1.27 0.26 0.34 Q1(poor) -1.27 0.18 0.33 Q2 0.28 0.27 0.25 Q2 0.28 0.26 0.25 Q3 0.17 0.17 0.17 Q3 0.17 0.16 0.17 Q4 0.14 0.14 0.14 Q4 0.15 0.14 0.14 Q5(wealthy) 0.07 0.07 0.07 Q5(wealthy) 0.07 0.07 0.08 Table 12 shows the share of liquid assets as a fraction of total wealth across wealth quintiles in the steady state and on impact dates following a TFP shock with a rise in disaster risk.

Table 1.12: Fraction of liquid assets to total net worth over the quintile distributions, B (K+B)

Table 13 and 14 summarize the shares of households who actively adjust their illiquid assets and the share of such adjustors, in each quintile, that actually disinvest illiquid assets, respectively. Comparing the share of adjustors in the fifth quintile to that in the first quintile, wealthy households are generally more willing to adjust their asset portfolios as they can afford frequent transaction costs. While the share of adjustors changes little over the business cycle, Table 14 shows that more adjusting

50 households monetize their illiquid assets in a recession. For instance, in the model with a rigid real return on liquid savings, the fraction of adjustors disinvesting in illiquid assets, in the recession, increases by 4 percentage points relative to the steady state. While poor households overall increase their liquid assets, a higher fraction of wealth poor adjustors increase their illiquid assets in a recession compared to the steady state as the price of illiquid assets falls.

B ﬁxed steady state impact dates rf ﬁxed steady state impact dates t=0 t=1 t=2 t=0 t=1 t=2 all 0.15 0.15 0.15 all 0.14 0.16 0.15 Q1(poor) 0.06 0.07 0.08 Q1(poor) 0.06 0.07 0.08 Q2 0.12 0.11 0.11 Q2 0.12 0.11 0.11 Q3 0.14 0.14 0.13 Q3 0.13 0.14 0.13 Q4 0.15 0.15 0.15 Q4 0.15 0.16 0.15 Q5(wealthy) 0.27 0.28 0.29 Q5(wealthy) 0.26 0.29 0.28 Table 13 shows the share of households who actively adjust their illiquid assets over wealth quintiles in the steady state and during impact dates.

Table 1.13: Share of households who actively adjust their portfolios across wealth quintiles

1.6.3 Changes in the joint distribution of net worth, income, and consumption before and during the Great Reces- sion

In this section, I examine the heterogeneous wealth composition economies for the dynamics of net worth, income, and consumption across households, comparing to the PSID before and during the Great Recession. I ﬁrst compare changes in the

51 B ﬁxed steady state impact dates rf ﬁxed steady state impact dates t=0 t=1 t=2 t=0 t=1 t=2 Q1(poor) 0.10 0.07 0.09 Q1(poor) 0.10 0.08 0.08 Q2 0.51 0.47 0.47 Q2 0.52 0.49 0.48 Q3 0.69 0.71 0.71 Q3 0.71 0.73 0.72 Q4 0.79 0.83 0.83 Q4 0.82 0.85 0.83 Q5(wealthy) 0.82 0.85 0.86 Q5(wealthy) 0.82 0.86 0.86

Table 1.14: Share of adjustors in each quintile that disinvest in illiquid assets

joint distribution in the data to those in heterogeneous wealth composition economies without disaster risk. Next, I discuss how disaster risk improves the models prediction on the dynamics of the cross-sectional distributions.

Recession without rising disaster risk

I simulate 60,000 households for several periods in (1) a heterogeneous composition wealth model with a constant return to liquid savings and (2) a model with a ﬁxed stock of liquid assets. Table 15 compares annualized growth rates of the average level of net worth, earnings, income, and consumption to those in the PSID between 2005 and 2007 over the 2005 wealth quintiles. In the last column of Table 15, I also report the percentage points changes in expenditure rates deﬁned as the fraction of total consumption to income. Note that growth rates of variables during normal times in data are comparable to those in a steady state in model economies.46

Table 15 shows that the model economies with multiple assets successfully explain some stylized facts seen in the PSID before the Great Recession. First, net worth

46In Table 12, the minor diﬀerence in the distribution of households drives slightly diﬀerent measures between the two model economies.

52 growth rates fall sharply in the level of wealth. For example, it explains the more

than 400 percent rise in net worth for the bottom 20 percent of households compared

to the small rise in net worth for the wealthiest 20 percent of households. Second,

growth rates of income and earnings in model economies are broadly consistent with

those in the PSID, generating the highest growth rates for the bottom 20 percent

of households. Falling growth rates of net worth, earnings, and income, in the level

of wealth, is mainly driven by high mean-reversion in earnings shocks. This mean-

reversion in shocks also leads to a higher growth in consumption for the bottom

compared to that for the top. However, as consumption rises less than income,

expenditure rates fall for all households but the top 20 percent.

Net worth Earning Income Expend. Exp. Rate (pp) Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) Q1(poor) 172 451 455 8.1 9.8 9.3 7.1 9.8 9.9 0.6 7.0 6.9 -3.4 -1.4 -1.5 Q2 26 10.9 11.2 3.1 5.8 5.4 3.0 5.8 5.8 6.0 3.0 3.1 1.3 -1.7 -1.6 Q3 19 2.9 3.0 1.3 2.5 2.4 1.8 2.5 2.5 3.5 0.8 0.9 0.7 -1.1 -1.0 Q4 11 1.3 1.4 2.0 -0.1 -0.5 1.5 -0.1 -0.1 1.2 -2.1 -2.0 -0.1 -1.4 -1.3 Q5(wealthy) 9.2 0.2 0.2 2.2 -2.1 -2.6 5.0 -2.1 -2.1 4.0 0.2 -0.2 -0.4 1.6 1.6 Notes : Table 15 shows annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID and in portfolio choice economies without disaster risk. (1) a model with ﬁxed return on liquid assets. (2) a model with ﬁxed stock of liquid assets.

Table 1.15: Growth rates of variables across wealth quintiles before the Great Reces- sion 2005-2007

Table 16 compares the corresponding growth rates between 2007 and 2011 over the 2007 wealth quintiles in the PSID to those in the model economies following a continuous drop in TFP, accumulating to an overall 2.18 percent across two periods.

The model economies successfully reproduce a concurrent slowdown in the growth of

53 net worth, earnings, and income compared to normal times (Table 15). Moreover, it

explains a fall in the level of net worth, earnings and income for those in the highest

quintile of the wealth distribution. The growth rates of the average consumption

across the quintiles of the wealth distribution decline in a recession, reproducing an

actual drop in the consumption for the wealthiest 60 percent of households. The

model economies also explain a rise in savings rates for households in the ﬁrst four

quintiles as well as a rise in consumption rate for the top 20 percent of households.

A rise in consumption rate for the top is due to the fact that wealthy households are

likely to be the old, who have less desire to save as they have a shorter remaining

lifetime compared to the young.

Net worth Earning Income Expend. Exp. Rate (pp) Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) Q1(poor) 164 420 438 2.1 6.0 6.0 2.1 7.9 7.9 -0.2 4.9 5.4 -2.1 -1.6 -1.3 Q2 3.0 9.9 10.1 1.5 2.7 2.9 1.6 4.1 4.2 -2.0 1.6 1.8 -3.4 -1.6 -1.4 Q3 0.9 1.8 2.0 0.7 0.1 0.2 0.1 1.1 1.0 -3.3 -0.6 -0.2 -3.0 -1.1 -0.8 Q4 -4.7 0.3 0.4 0.4 -2.3 -2.4 0.3 -1.4 -1.6 -2.0 -3.3 -3.0 -2.3 -1.4 -1.0 Q5(wealthy) -5.3 -0.8 -0.8 -0.8 -4.3 -4.2 -0.3 -3.4 -3.5 0.7 -0.8 -0.7 0.9 1.8 2.1 Notes : Table 16 shows annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID and in portfolio choice economies without disaster risk. (1) a model with ﬁxed return on liquid assets. (2) a model with ﬁxed stock of liquid assets.

Table 1.16: Growth rates of variables across wealth quintiles during the Great Reces- sion 2007-2011

To further evaluate the model’s predictions for the severity of the Great Recession across the wealth distribution, Table 17 documents the ﬁrst diﬀerence in growth rates between normal times and recession in model to the data. While not targeted, the two model economies explain a slowdown in the growth of net worth, earnings, income,

54 and consumption in a recession compared to a normal time. Most importantly, it

shows the largest fall in the growth rates of net worth and earnings for the bottom 20

percent of households. Although the model economies explain changes in consump-

tion rates across the wealth distribution in a recession (Table 16), it fails to explain

changes in the growth of expenditure rates between normal times and recession as

seen in the last column of Table 17. For example, the PSID data shows that a fall in

consumption rates for households in the second to fourth quintiles is pronounced in

the Great Recession compared to normal times while the model economies predicts a

less drop in consumption rate in a recession.

Net Worth Earning Income Expend. Exp. Rate (pp) Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) Q1(poor) -8.0 -31 -17 -6.0 -3.8 -3.3 -5.0 -1.9 -2.0 -0.8 -2.1 -1.5 1.3 -0.2 0.2 Q2 -23 -1.0 -1.1 -1.6 -3.1 -2.5 -1.4 -1.7 -1.6 -8.0 -1.4 -1.3 -4.7 0.1 0.2 Q3 -18 -1.1 -1.0 -0.6 -2.4 -2.2 -1.7 -1.5 -1.5 -6.8 -1.4 -1.1 -3.7 0.0 0.2 Q4 -16 -1.0 -1.0 -1.6 -2.2 -1.9 -1.2 -1.3 -1.5 -3.2 -1.2 -1.0 -2.2 0.0 0.3 Q5(wealthy) -15 -1.0 -1.0 -3.0 -2.2 -1.6 -5.3 -1.3 -1.4 -3.3 -1.0 -0.9 1.3 0.2 0.5 Notes : Table 17 shows changes in annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID and in portfolio choice economies without disaster risk. (1) a model with ﬁxed return on liquid assets. (2) a model with ﬁxed stock of liquid assets.

Table 1.17: Changes in growth rates of variables between prior- and during the Great Recession

With disaster risk

Table 18 reports the annualized growth rates of net worth, earnings, income and consumption in the 2007-2010 PSID as well as from the simulated heterogeneous

55 wealth composition economies with disaster risk.47 While growth rates in net worth,

earnings, and income are broadly similar to those in Table 16, the slowdown in con-

sumption growth in a recession is exacerbated by a rise in disaster risk, especially

when the return to liquid assets is ﬁxed. For example, in the model with ﬁxed re-

turns, consumption for the bottom 20 percent of households rises by 4.1 percent in

a recession with a rise in disaster risk compared to 4.9 percent without disaster risk

(Table 16). As explained earlier, this is driven by a higher expected return on savings

in the future compared to the current average return alongside with a strong negative

wealth eﬀect following a rise in disaster risk.

Net worth Earning Income Expend. Exp. Rate (pp) Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) Q1(poor) 164 439 470 2.1 7.3 7.4 2.1 7.8 7.8 -0.2 4.1 5.4 -2.1 -2.0 -1.2 Q2 3.0 9.7 10.6 1.5 3.7 3.9 1.6 4.3 4.0 -2.0 0.9 1.9 -3.4 -2.1 -1.3 Q3 0.9 1.8 2.2 0.7 1.0 1.0 0.1 1.2 0.7 -3.3 -1.0 -0.1 -3.0 -1.5 -0.5 Q4 -4.7 -0.1 0.5 0.4 -1.7 -1.6 0.3 -1.1 -1.8 -2.0 -3.8 -2.9 -2.3 -1.9 -0.8 Q5(wealthy) -5.3 -1.1 -0.8 -0.8 -3.9 -3.8 -0.3 -3.0 -3.7 0.7 -1.1 -0.7 0.9 1.4 2.2 Notes : Table 18 shows annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID and in portfolio choice economies with disaster risk. (1) a model with ﬁxed return on liquid assets. (2) a model with ﬁxed stock of liquid assets.

Table 1.18: Growth rates of variables across wealth quantiles during the Great Re- cession with rising disaster risk

Table 19 shows the annualized changes in the joint distribution between normal times and a recession. Involving a rise in disaster risk when the return on liquid savings is constant, this model economy predicts a more marked fall in consumption rates. This implies that a rise in disaster risk not only explains the aggregate response

47Note that the growth rates in normal times are same as those reported in Table 13.

56 of the economy seen in the data but also the micro-prediction of the economy. In

a typical recession with a single asset, households’ consumption smoothing desire

increases consumption rates. However, in the recent recession, consumption rates

sharply fall across households, which can be explained with a rise in the probability

of a further worsening of earnings with a rigid return on liquid savings. This suggests

the importance of disaster risk that ampliﬁes the eﬀects of a shock to TFP with

multiple assets of varying liquidity.

Net Worth earning Income Expend. Exp. Rate (pp) Quantile PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) PSID (1) (2) Q1(poor) -8.0 -12 15 -6.0 -2.5 -1.9 -5.0 -2.0 -2.1 -0.8 -2.9 -1.5 1.3 -0.6 0.3 Q2 -23 -1.2 -0.6 -1.6 -2.1 -1.5 -1.4 -1.5 -1.8 -8.0 -2.1 -1.2 -4.7 -0.4 0.3 Q3 -18 -1.1 -0.8 -0.6 -1.5 -1.4 -1.7 -1.3 -1.8 -6.8 -1.8 -1.0 -3.7 -0.4 0.5 Q4 -16 -1.4 -0.9 -1.6 -1.6 -1.1 -1.2 -1.0 -1.7 -3.2 -1.7 -0.9 -2.2 -0.5 0.5 Q5(wealthy) -15 -1.3 -1.0 -3.0 -1.8 -1.2 -5.3 -0.9 -1.6 -3.3 -1.3 -0.5 1.3 -0.2 0.6 Notes : Table 19 shows changes in annualized growth rates of the average net worth, earnings, and income and percentage points change in expenditure rates from the PSID and in portfolio choice economies with disaster risk. (1) a model with ﬁxed return on liquid assets. (2) a model with ﬁxed stock of liquid assets.

Table 1.19: Changes in growth rates of variables between prior- and during the Great Recession with rising disaster risk

1.7 Concluding Remarks

Solving an overlapping generations economy with uninsurable earnings, unemployment and liquidity risk in dynamic stochastic general equilibrium, I have examined the implications of household-level diﬀerences in the level and liquidity of wealth for aggregate dynamics and changes in the distribution of consumption, income and wealth across households. In particular, I quantify the role of household heterogeneity in the composition of wealth for understanding diﬀerences in their responses during the

57 Great Recession. This channel is ampliﬁed when households increase precautionary savings, and substitute liquid for illiquid assets, following a rise in disaster risk.

Liquidity varies across assets as higher returns involve investments associated with random transactions cost. Calibrating the distribution of liquidity costs to the real return on liquid and illiquid assets in the data, the model economy reproduces the empirical share of illiquid assets to net worth of households of diﬀerent levels of wealth and age. Moreover, the model explains much of the distribution in net worth, illiquid wealth, and liquid wealth seen in the data.

The implications of adding realistic heterogeneity not only in the level of wealth, but also in its liquidity, are important for an understanding of both aggregate and household-level changes over large recessions. In such recessions, a fall in earnings and a rise in unemployment risk force high-yield asset investors to monetize their wealth to smooth consumption, resulting in a sharp drop in investment. During the recovery, as

ﬁxed transaction costs lead households to tolerate lower than desired shares of illiquid assets in their portfolios, the economy with wealth composition heterogeneity recovers slowly. Most importantly, precautionary savings motives, following an increase in the probability of a large disaster, yield powerful changes in household behavior.

These changes in households’ consumption and expenditure rates are consistent with stylized facts seen in the joint distribution of consumption, income and net worth across households over the Great Recession.

In the next version of this paper, I will use the empirical cyclicality of liquidity premia to determine the cyclicality of the supply of liquid assets over the business cycle.

This will provide further information on the role of household portfolio substitutions, associated with an increase in precautionary savings, in large recessions.

58 Chapter 2: Skill Premia, Wage Risk, and the Distribution of Wealth

2.1 Introduction

Understanding unobserved heterogeneity in the variance of wages across households is crucial to explain (1) aggregate wealth and income and (2) cross-sectional diﬀerences in the age proﬁles of earnings, income, and wealth by education groups.

Although a large number of studies have found a rise in education wage diﬀerentials, both in means and variances, little work has been done to explore the implications of heterogeneous earnings risk on household behavior.48 This paper challenges the standard assumption of i.i.d wage shocks by allowing for the possibility that households with diﬀerent education levels face distinct wage risk, and endogenizes this wage risk through skill choice.

To evaluate the hypothesis that there exist conditional variances of wages that vary with skills, I estimate skill-speciﬁc wage processes from 1968-2011 PSID data. I

ﬁrst run an OLS regression to estimate time-varying college wage premia (between- group wage dispersion) and use minimum distance estimation method to estimate skill-speciﬁc wage risk, allowing for both persistent and transitory shocks (within- group wage dispersion). The estimated results indeed reject the typical assumption

48See Autor et al. (2008), Katz (1999), Juhn et al. (1993), Katz and Murphy (1992).

59 of i.i.d wage shocks across households and suggest time-varying heterogeneous wage risk that varies with skills. As is well known, the college wage premium increased rapidly starting in the 1980s. Importantly, overall wages have also become more dispersed over the same period, especially for college graduates, suggesting higher wage risk for the skilled.

I develop an incomplete-markets quantitative general equilibrium overlapping- generations model with endogenous skill choice and labor supply. I use this framework as a laboratory to examine the impact of distinct wage processes across skill levels.

College education decisions are made before the start of work and involve a resource education cost. Households have access to college loans. Having decided their skill choice, households with different education levels face skill-specific wage processes that account for both between- and within-group wage dispersion. For example, skilled workers benefit from a permanently higher wage - the college wage premium - and face a more volatile wage risk compared to unskilled households.

I set borrowing constraints for each age and skill group as a proportion of their natural debt limits. Given these age- and skill-varying borrowing constraints, the resource costs of college education are set to imply total spending on education equal to 134 percent of GDP per capita, consistent with the data. Moreover, the model economy gives rise to two-thirds of college graduates holding student loan debt as reported in the Project of Student Debt by NPSAS.

In my model economy, the saving behavior of skilled households is inﬂuenced by skill-speciﬁc wage processes through two channels. First, a more volatile wage shock for the skilled relative to the unskilled boosts their precautionary saving, and raises

60 their savings rates. Second, high levels of ex-post earnings by the skilled reduces savings rates through a strong income eﬀect. Note that the total level of savings remains high compared to unskilled households as a result of higher earnings. By deviating from the assumption of common wage risk to the entire population, the model partly endogenizes uninsurable risk through the discrete choice of skills. This introduces a channel through which households’ decisions aﬀect their wealth, in contrast, the primary force driving wealth inequality across households in a model with common wage risk is luck.

Using empirically consistent wage processes and an endogenous education choice, the model gives rise to sharply different implications of variable labor supply on wealth inequality. Specifically, endogenous labor supply increases wealth inequality in my model while it dampens inequality in a model without skill choice and skill-varying wage processes.49 This is first because, as households can work harder to pay back college loans, an intensive labor margin choice encourages productive households to pursue a college degree, driving more inequality. Second, skilled households, who face more volatile wage shocks, insure themselves from persistent downside wage risk by increasing hours worked while having a higher probability of high earnings compared to unskilled households. Thus, higher wage risk provides better opportunities to become wealthy for the skilled compared to the unskilled and increases the likelihood of more productive households becoming skilled given an intensive labor margin choice.

In an experiment with labor supply ﬁxed to the average level of hours worked at each skill level in the benchmark economy, the wealth Gini drops by 7 percentage points

49Castenada et al. (2003) points out the importance of modelling endogenous labor supply in a model with uninsurable idiosyncratic risk. Also, Pijoan-Mas (2006) shows that households make use of endogenous labor supply choice to smooth out consumption against idiosyncratic labor market risk.

61 and the share of wealth held by the wealthiest 10 percent of households drops by 10 percentage points from the benchmark economy.

My model explains much of the observed distribution of income and wealth in the data as well as asset accumulation behavior of skilled and unskilled workers despite the absence of strong saving motives such as entrepreneurship with collateral constraints (Cagetti and De Nardi,2004), preference shocks (Krusell and Smith, 1998), stochastic life cycles (Castenada et al.,2003), or intergenerational links (De Nardi,

2004). Moreover, the benchmark economy is broadly consistent with the fractions of skilled and unskilled households across wealth percentiles in the data and the probability of being in each wealth percentile for college and non-college graduates. Note that none of these moments are targeted in its calibration.50

I argue that the presence of skill-conditional variances for wages is vital in explaining both aggregate inequality and life-cycle facts across households. Indeed, in an alternative calibrated model with a common estimated wage shock, income inequality is exaggerated and the model fails to explain the disparity in wealth, earnings, and income across education groups. This common shock economy produces an income Gini of 0.61, compared to the corresponding empirical value of 0.49, and the share of income held by the richest 10 percent of households is 10 percentage points higher than that in data. More importantly, the common shock economy fails to explain the diﬀerence in life-cycle wealth accumulation of skilled and unskilled households. This is because common wage risk puts too much emphasis on luck to

50Castenada er al (2003) calibrate to the Gini indexes and points from the Lorenz curves of earnings and wealth. Cagetti and De Nardi (2006) also reproduce the skewed distribution of wealth by directly targeting the earnings and wealth Gini.

62 determine individual wealth, implying similar savings behavior, conditional on earnings, regardless of skill levels. In contrast, the benchmark economy endogenizes wage risk through skill choice, introducing an additional channel to determine households wealth accumulation.

Introducing an endogenous component for earnings, through skill choice, the model enriches the standard approach of i.i.d. wage shocks. However, a caveat is that it provides no distinction between heterogeneity and risk in the wage distribution. These two components affect households’ saving differently. A permanent rise in wages decreases saving rates through a strong income effect while a rise in wage risk increases saving rates by amplifying precautionary motives. Though I do not explicitly model heterogeneity in wages across college graduates, the estimated wage process has high persistence. Furthermore, wealth accumulation by skilled households is mainly driven by high ex-post earnings, implying a strong income effect, not by high precautionary savings.

To explore the role of endogenous educational choice and wage differentials across education groups on aggregate inequality, I conduct several experiments. In the first experiment, I abstract from skill differences, and assume a common wage process. Ab- stracting from between- and within-group wage differentials, this otherwise standard

Huggett economy with variable labor supply, naturally has less wealth inequality. By contrast, the increase in earnings risk for the unskilled, implied by the common wage shock, mitigates this eﬀect. As a result, the wealth Gini drops to 0.62 from 0.67 and the share of wealth held by the wealthiest 10 percent of households falls by 5 percentage points from the benchmark economy. In the second experiment, I lower the mean diﬀerence in wages across education groups. The decrease in the college wage

63 premium makes college education less attractive, resulting in 18 percent of households becoming skilled compared to 27 percent in the benchmark economy, and decreases wealth inequality. When I further remove the conditional variances of wages by skills, the wealth Gini drops to 0.60 from 0.67 in the benchmark economy and the wealthiest

10 percent of households hold 37 percent of total wealth compared to 48 percent.

Finally, I find that college education subsidies to the poorest 34 percent of households leads to overall welfare gains of 8 to 18 percent of life-time consumption when, depending on the size of the subsidy, compared to the benchmark economy. More- over, households becoming skilled under such policies are worse-off while households remaining unskilled in the policy economy are better-off. This is first because college education subsidies increase life-time earnings of households who have high marginal utility of consumption. Second, a rise in college-educated households in the policy economy dwindles wages for skilled workers, reducing the college wage premium.

Lastly, the composition of skilled households changes as low-ability households get subsidies for their college education. Furthermore, the equilibrium change in wages discourages high-ability households who would otherwise pursue a college degree from gaining skills, further aggravating the quality of skilled households. In sum, low-ability households crowd out high-ability households when education subsidies are targeting low earnings households.

While a large literature has documented the change in the U.S. wage structure, the macroeconomic implications of these changes have rarely been studied. Heathcote et al. (2010) explore the implications of rising wage inequality driven by skill-biased technological change for inequality in labor, earnings, and consumption. However, they abstract from the eﬀects of diﬀerent earnings risk by education groups, and its

64 implication for the wealth distribution, and model a small open economy. Assuming an utility cost of education instead of output costs, they also abstract from the eﬀect of liquidity constraints on college education decisions. In contrast, idiosyncratic resource costs of college education, and an initial asset distribution, alongside borrowing limits, determines college education in my model. Finally, I solve the model in general equilibrium.

The rise in within-group wage dispersion in the U.S. among college graduates is explored in Lemieux (2010) and Shin et al. (2016). Lemieux (2010) shows that much of the rise in wage dispersion is driven by highly educated workers. Shin et al. (2016) further explores the role of options in college education such as drop-out and underemployment, alongside rising wage dispersion, in explaining the weak rise of college graduation rates to the rising college wage premium since the 1980s in the

U.S. However, these papers do not consider the macroeconomic impact of distinct earnings risk across education groups.

This paper contributes to a body of work that studies the distribution of wealth.

In the U.S., the distribution of wealth is highly concentrated and skewed to the right.

For instance, excluding the self-employed, the U.S. wealth Gini coeﬃcient was around

0.77 in 2004, and the fraction of wealth held by the top 1 percent of U.S. households was around 29 percent.51 A large literature including Aiyagari (1994), Huggett (1996),

Krusell and Smith (1998) and Castaneda et al. (2003) has attempted to explain observed earnings and wealth inequality in the U.S. However, most of these models fail to generate both the signiﬁcant concentration of wealth in the top tail and a high

51Self-employed households are a small fraction of the overall population with, arguably, an additional savings motive. Cagetti and De Nardi (2003) show the eﬀect of entrepreneurship on the wealth distribution. However, excluding self-employed households, we still have a signiﬁcant amount of wealth inequality in 2004 SCF.

65 fraction of wealth-poor households in the bottom tail. For example, Huggett (1996)

produces the right U.S. wealth Gini coeﬃcient, mainly driven by a counterfactually

high variance for earning shocks, leading to a large fraction of households with zero

or negative asset holdings. Although Castenada et al. (2003) successfully reproduce

both tails of the wealth distribution and the U.S. wealth Gini coeﬃcient, their results

are derived in an environment characterized by a stochastic life-cycle. Stochastic

aging builds a strong precautionary saving motive, potentially important for wealth

accumulation.

My work is related to the entrepreneurship with collateral constraints model of

Quadrini (1998) and Cagetti and De Nardi (2006). Cagetti and De Nardi (2006) as-

sume that entrepreneurs borrow capital in loan markets with limited enforceability of

debt contracts. Households with proﬁtable entrepreneurial opportunities accumulate

large levels of wealth in order to increase their collateral and thus the level of capital

they are able to borrow. Assuming a two-stage life-cycle, with stochastic aging, and

allowing for altruism, Cagetti and De Nardi are able to reproduce the empirical Gini

coeﬃcient and the concentration of wealth amongst the top percentiles. These models

explore the eﬀect of occupational choice, in particular the choice to be an entrepreneur

or worker, on the distribution of wealth.52 Instead, I explain the distribution of wealth using a college education choice.

Hubbard et al. (1994) also introduce skill-varying earnings risk in a life-cycle

model to explore individual and aggregate wealth accumulation. In contrast to my

model, they abstract from education choice and variable labor supply. Moreover,

52They identify entrepreneurs in the data with the self-employed.

66 they fail to capture price feedback eﬀects on quantities by studying a partial equilibrium. Lastly, as they estimate earnings processes from the 1983-1987 PSID data, the assumed earnings risk for each education group is diﬀerent from my estimates.

Finally, Khan and Kim (2016) study the eﬀect of diﬀerent portfolio choices across households on the distribution of wealth, but only explain a wealth Gini of 0.71. Given a large empirical work showing a positive relationship between the return on savings and education, introducing portfolio choice with endogenous education decision may explain the remaining inequality not explained here.

The remainder of the paper is organized as follows. Section 2 summarizes the estimation of skill-speciﬁc wage processes as well as the U.S wealth distribution and college education in 2004. Section 3 presents the model economy. Section 4 discusses the calibration. Section 5 presents quantitative results Section 6 discusses policy experiments, and Section 7 concludes.

2.2 Empirical analysis

2.2.1 Estimation of skill-speciﬁc wage processes

An increase in education wage diﬀerentials, alongside a rise in the gender wage gap, is a major source of rising wage inequality seen in the U.S. (Juhn et al (1993),

Katz (1999), Katz and Murphy (1992)). Autor et al. (2008) show that this rising wage dispersion, since the 1980s, has been mainly driven by increases in upper-tail (90/50) inequality. They also ﬁnd a divergence in wages between high-skilled and low-skilled individuals. This asymmetric rise in wage dispersion across individuals suggests the existence of conditional variances of earnings shocks and brings into question the

67 standard assumption of i.i.d wage innovations used in the literature. Meghir and

Pistaferri (2004) also find heterogeneous variances for earnings across individuals, especially by education groups, and underline the importance of heteroscedasticity in earnings risk to explain individual behavior. Revisiting these findings, I allow for the possibility that households with different education levels face skill-specific wage processes, both in the mean and variance. Specifically, I estimate skill-specific wage processes using PSID data between 1968 and 2011.53 I first run an OLS regression to estimate time-varying college wage premia over the sample period (between-group wage dispersion). Next, I use minimum distance estimation to find wage shock process for each skill group, allowing for both persistent and transitory shocks (within-group wage dispersion).

In the following, wi,j,t,e represents the hourly wage of sample i where the head has age j and education level e in year t. I run the following regression of log hourly wage on time dummies; an interaction term with education, Di, and time dummies; labor market experience, θ; and experience-squared, θ2.54

2 log wi,j,t,e = βt,0 + βt,1Di + β3θ + β4θ + rbi,j,t,e

Figure 1 shows the estimated returns to college over the sample period. The college wage premium decreases until 1980, thereafter it increases rapidly until 1990. Figure

2 shows the estimated potential market experience function. Labor market experience increases hourly wages through the ﬁrst 29 years of work where wages rise around 80 percent relative to their initial level. Thereafter, hourly wages fall in experience until

53Generally, I follow the estimation strategy in Heathcote et al. (2010). 54Labor market experience is measured as age minus years of schooling minus 5. In years missing the variable for years of schooling, I proxy years of schooling for the individuals with a college degree as 16 and for the individuals without a college degree as 12.

68 retirement. Note that eﬃciency units of labor of households depend on both labor

2 market experience, l(j) = e(β3θ+β4θ ), and idiosyncratic wage (productivity) shocks.

Source: PSID data (1968-2011)

Figure 2.1: Male college wage premium

The regression residuals rbi,j,t,e are assumed to be the sum of idiosyncratic wage

55 shocks, ϵi,j,t,e, and measurement error, vei,j,t,e. Idiosyncratic shocks consist of both a

persistent component, η, and transitory component, ϵv. To be speciﬁc,

v ϵi,j,t,e = ηi,j,t,e + ϵi,j,t,e

e p ηi,j,t,e = ρ ηi,j−1,t−1,e + ϵi,j,t,e

55Following Heathcote et al. (2000), I use French (2002)’s estimate for the variance of measurement error in log hourly wages of 0.02.

69 2 1.9 1.8 1.7 1.6 1.5 l(j) 1.4 1.3 1.2 1.1 1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

experience

Source: PSID data (1968-2011)

Figure 2.2: Labor market experience

p ∼ 2 v ∼ 2 where ϵi,j,t,e N(0, σpt,e) and ϵi,j,t,e N(0, σvt,e). { 2 2 } I estimate within-group year-varying shock variances σpt,e, σvt,e , the persistence { e} 2 of the shock ρ , and the variance of the initial value for the persistent shock σπe , using minimum distance methods. These estimates are used as distinct idiosyncratic productivity shock processes, across skill groups, in the structural model. I use survey data from 1968 to 2011, but only estimate the variances through 2009 because of the

ﬁnite sample bias at the end of the sample period.56 I separate samples by education

56Given that the PSID has conducted a biennial survey starting from 1997, the estimation of annual shock processes there must confront the problem of observations missing for every other year. As Heathcote et al. (2010) point out, although the variance of the persistent shock for the missing years can be theoretically found using the available information from adjacent years, the resulting estimates are downward-biased because of insuﬃcient information. Therefore, I follow their approach and estimate variances for missing years by taking the weighted average of the two closest surrounding years.

70 groups, college graduates and non-college graduates, in order to estimate two separate

productivity shock processes. I assume that wage shocks between these two groups

are orthogonal to each other. I estimate L = 86 parameters for each education group.

The parameter vector is denoted by PL×1. From now on, I abstract from education variables for ease of notation.

The theoretical moment is deﬁned as

j P mt,t+n( ) = E(ri,j,tri,j+n,t+n)

which is the covariance between wages of individuals at age j in year t and t + n.57

To calculate empirical moments, I group individuals into 44 year and 26 overlapping

age groups. For example, the ﬁrst age group contains all observations between 25 and

34 years old and the second group contains those between 26 and 35 years old. The

empirical moment conditions are

j − j P mˆ t,t+n mt,t+n( ) = 0 ∑ j 1 Ij,t,n wherem ˆ = rbi,j,trbi,j+n,t+n and Ij,t,n is the number of observations of t,t+n Ij,t,n i=1 age j at year t existing n periods later.

The minimum distance estimator solves

min[ ˆm − m(P)]′[ ˆm − m(P)] P

where the vectors ˆm and m represent empirical and theoretical moments of dimension

8, 342 × 1. The identity matrix is used as the weighting matrix.

57The closed form of the theoretical moment is   { min(∑j−1,t) n  2 2(i−1) 2 2  1 if t = t + n E(ri,j,tri,j+n,t+n) = ρ ρ min(j − 1, t)σ + ρ σ + 1σ , 1 = π ηt−i+1 vt 0 otherwise i=0 .

71 Figures 3 displays the estimates of the variances of persistent and transitory wage

shocks for college and non-college graduates between 1968 and 2008. The overall wage

residual becomes more dispersed over time, especially for college educated workers

compared to non-college educated workers. Given that college graduates have higher

ex-ante earnings relative to non-college graduates, this is consistent with the ﬁnding

in Autor et al. (2008) which explains the rising wage inequality in the 1990s by

a persistent rise in upper-tail inequality summarized by the 90/50 log wage gaps.

Heathcote et al. (2010) also ﬁnd that dispersion in male hourly wages increases at

the top of the wage distribution after the 1990s.

The estimated persistence of the labor productivity shock, ρh, is 0.981 and the

2 variance of the initial persistent shock, σπ,h, is 0.108 for college graduates. The

l 2 corresponding values for non-college graduates are ρ = 0.986 and σπ,l = 0.131. We see that productivity shocks are persistent both for college and non-college graduates over the sample period. Further, the persistent shock variance increases to 0.04 in 2004 from 0.01 in 1983 for college graduates, while, over the same period, the persistent shock variance stays between 0.005 and 0.01 for non-college graduates.58

Table 1 shows the smoothed estimates of the skill-speciﬁc shock processes in 2004,

which are used to calibrate the benchmark model economy. Notice that the persistent

shock variance, 0.043, and transitory shock variance, 0.211, for skilled workers are

higher than the persistent shock variance of 0.006 and the transitory shock variance of 0.066 for unskilled workers. This implies more volatile wage risk faced by skilled workers compared to unskilled workers. Accounting for both skill premia and skill- speciﬁc wage dispersion, the estimated wage process for skilled workers ﬁrst-order

58This is in agreement with the ﬁnding in Shin et al. (2014) that college graduates face higher residual within-group earnings dispersion relative to the variance for non-college graduates.

72 (a) Persistent shock for college graduates (b) Transitory shock for college graduates

0.08 0.3000 0.07 0.2500 0.06 0.05 0.2000 0.04 0.1500 0.03 0.1000 0.02 0.01 0.0500 0.00 0.0000 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

year year Smoothed Estimates Estimates Smoothed Estimates Estimates

0.0250 0.1200

0.0200 0.1000 0.0800 0.0150 0.0600 0.0100 0.0400 0.0050 0.0200 0.0000 0.0000 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

year year Smoothed Estimates Estimates Smoothed Estimates Estimates Notes: Minimum distance estimates of the shocks for college and non-college graduates. Smoothed series are generated using a Hodrick-Prescott trend with a smoothing parameter of 100.

Figure 2.3: Variance of persistent and transitory shock for college and non-college graduates

stochastic dominates that for unskilled workers. As explained later, it is important

to introduce these realistic wage diﬀerentials between- and within- education groups

to explain distinct saving behavior by skill levels seen in the data.

I do not explore potentially important distinction in wage dispersion arising through

individual heterogeneity and risk. Chen (2008) ﬁnds that wage dispersion among

college graduates is mainly driven by a predictable component (heterogeneity) not

uncertainty (wage risk). These two components aﬀect households’ saving diﬀerently.

A permanent rise in wages decreases saving rates through a strong income eﬀect while

73 e 2 2 2 education ρ σpt,e σvt,e σπ,e skilled 0.981 0.043 0.211 0.108 unskilled 0.986 0.006 0.066 0.131 Notes : Estimates for the persistence of wage shock ρe, the variance of persistent and tran- 2 2 sitory shock, σpt,e, σvt,e,and the initial age 2 persistent shock variance, σπ,e for each skill group.

Table 2.1: Minimum Distance Estimates in 2004

a rise in wage risk increases precautionary savings. Although the model assumes that all wage dispersion arises through risk, the highly persistent estimated wage process leads to high ex-post earnings earned by skilled households, implying a strong income eﬀect.

2.2.2 Wealth Inequality and College Education

Table 2 summarizes the distribution of wealth in the U.S using the Survey of

Consumer Finances (SCF).59 For greater consistency with the PSID data used to estimate wage processes, I report wealth excluding self-employed households. It is worth noting that, even after excluding entrepreneurs, results from Table 2 show signiﬁcant wealth inequality among non self-employed households.60

59See section A.2 for the deﬁnition of wealth and elements of wealth. 60Cagetti and DeNardi (2006) emphasize entrepreneurial income as an important driver of wealth inequality in their model economy.

74 As seen in Table 2, the U.S. wealth distribution is highly concentrated and skewed to the right.61 In 2004, more than 28 percent of total wealth was held by the top 1 percent wealthiest households and approximately 64 percent of wealth in the economy was held by the top 10 percent of households. In contrast, the bottom 50 percent of households only held 3-4 percent of total wealth and more than 8 percent of households had zero or negative assets.62 Moreover, the wealth Gini coeﬃcient reaches to 0.77.63

Year 1% 5% 10% 50% 90% ≤ 0 Gini 2004 28.7 50.5 63.5 96.6 100 8.5 0.771 Notes : Table 2 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the wealth Gini coeﬃcient, the share of households with zero or negative asset holdings in the U.S. economy. (Source: 2004 SCF data)

Table 2.2: Wealth inequality in the U.S. economy

Figure 4 describes the average level of net worth of households with a college degree and those without a college degree over age. Though both skill groups have a relatively small amount of wealth at age 25, households with a college degree rapidly accumulate wealth, reaching 1.4 million dollars at age 61. In contrast, the level

61U.S. earnings inequality has risen over the last 30 years. However, there is no clear evidence of a corresponding trend in wealth inequality. See Heathcote, Perri, and Violante (2010). 62If we include self-employed households, the wealth distribution is more concentrated. The top 1 percent of households held around 33 percent of total wealth and the top 10 percent of households held 69 percent of total wealth. The wealth Gini coeﬃcient is 0.80 and the share of zero or negative asset holdings is 8.5 percent. 63To accurately measure the wealth Gini when some households have negative assets, I follow Chen, Tsaur, and Rhai (1982).

75 1.6 1.4

Millions 1.2 1 0.8 0.6 0.4 0.2 0 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82

Age skilled unskilled

Notes: I smoothed average level of wealth over age using a Hodrick-Prescott trend with a smoothing parameter of 100. (Source: 2004 SCF data)

Figure 2.4: Life-cycle wealth accumulation

of savings by households without a college degree at age 61 is less than $300, 000 dollars.64 A great disparity in wealth between skilled and unskilled households would at ﬁrst seem plausible. However, as seen in section 5, a standard incomplete-markets life-cycle model with a common wage shock process fails to explain these results.

64The importance of skills in the accumulation of wealth has become more pronounced in recent years. To show this, I ran two ordinary least squares regressions using the 1983 and 2004 SCF data. I regressed the log of real net worth on a college education dummy, the log of income, and age, race, and work status dummies. Table D.1 shows that the coefficient on the education dummy increases from 0.26 to 0.53 between 1983 and 2004 while coefficients and standard errors for other variables are similar across the two regressions. This increase in the coefficient on college education demonstrates that the tendency for skilled labor to hold a higher level of assets relative to unskilled labor has increased across the sample period.

76 Although a large literature documents a substantial dispersion of the U.S. wage structure since the 1980s, there is little work on the effects of a changing wage structure on the distribution of wealth. Though Heathcote et al. (2010) explore skill- premia and gender wage gaps as the driving force behind income and consumption inequality, their assumed wage structure does not allow for an asymmetric rise in wage dispersion by education. Moreover, by studying a small open economy, they abstract from its implications for distribution of wealth. In this paper, I study how differences in wage levels and risk between- and within- education groups affects individual wealth accumulation behavior and aggregate wealth inequality.65

2.3 Economic Model

2.3.1 Overview

In the model economy, there are three sets of agents: households, ﬁrms, and government. Household demographic structure involves J overlapping generations.

Each generation has a fraction of µj of the population and the total population is normalized to one. The time-invariant survival probability for a household from age ∑J j to j + 1 is ζj. At each date, a new cohort of measure µ1 = µj(1 − ζj) enters j=2 the economy. Households value consumption and leisure in each period and discount future utility by β. They are endowed with a unit of time in each period.

There are three life-cycle stages: education, work, and retirement. The economy consists of two types of labor, skilled workers and unskilled workers. Given a labor

65Campbell (2006) instead ﬁnds that college educated households have higher portfolio shares of stocks and stock market participation rate, suggesting higher returns on savings compared to non-college graduates.

77 productivity at initial working age j = 1 is known, a household gains skill by paying

66 a ﬁxed college education cost, κ ∈ (κl, κh), before he enters the labor market. This

cost is denominated in units of output and drawn from a time-invariant distribution

Q(κ) common across households. Households also have access to college loans which

allow them to borrow against future income. With the known initial productivity,

college loans encourage productive households to invest in a higher education, leading

more income and wealth dispersion between skilled and unskilled households.

Note that households who pay their ﬁxed cost permanently earn wage wh higher

than wl, the wage earned by unskilled households. Each period, a household is iden-

tiﬁed by age j, wealth a, idiosyncratic productivity shock, ε, and its education level.

During their working age, households earn both labor and capital income. After re-

tirement at age Jr = 35, they receive capital income and a lump-sum social security beneﬁt, b. Finally, their last possible age is J = 59. Households also receive acciden-

tal bequests, τr, from those that do not survive. Households are subject to borrowing

constraints as a fraction of age- and skill-varying natural debt limits.

A representative ﬁrm produces consumption and investment goods with aggre-

gate capital, K, and labor, N, through a strictly concave, constant returns to scale

production function F (K,N) = zKαN 1−α, where z > 0, and 0 < α < 1. Aggregate

labor N is itself constant elasticity of substitution function H(Hh,Hl) of two types

of eﬀective units of skilled and unskilled labor.

h l h θ−1 l θ−1 θ H(H ,H ) = [λ(H ) θ + (1 − λ)(H ) θ ] θ−1 (2.1)

66I assume that all households draw their initial labor productivity from the invariant distribution of the skilled.

78 Above, He is the aggregate labor input associated with education level e ∈ {h, l}, θ is

the elasticity of substitution between the two types of labor, and λ reﬂects skill-biased

demand shift. The rate of depreciation for capital is δ ∈ (0, 1).

Finally, the government levies taxes (τa, τn) on labor and capital income respectively. These revenues are used to ﬁnance social security payments to retirees. Let G

be government spending other than transfers to households. The government budget

is balanced.

2.3.2 College Education Decision

Before the initial age of working life, households complete an education decision.

They can choose either a college degree or high school diploma. Agents who pursue a college degree have to pay an idiosyncratic output cost κ which is drawn from the distribution Q(κ). However, they beneﬁt from a college wage premium which increases their lifetime earnings. Furthermore, the wage shock process diﬀers between skilled and unskilled workers.

Individuals decide to go to college if the value of pursuing a college degree is higher

than their value without a college degree. College education is possible only when

individuals have suﬃcient resources, including borrowing, to ﬁnance their education

costs. An output cost, unlike an utility cost, captures the fact that individuals may

be unable to go to college because of insuﬃcient income. This introduces a direct

role for borrowing constraints, and implicitly, college loans.

Let vh(1, a, ε, κ) be the value function of a household who pays an education cost

κ at age j = 1, with assets a and an idiosyncratic productivity shock ε. ve(j, a, ϵ)

79 represents the expected discounted value of a household, with education level e ∈

{l, h}, at age j with assets a and an idiosyncratic productivity shock ε. Given a and

κ, the household chooses consumption c, saving a′, and labor supply n. The problem

of a household who pays a ﬁxed education cost, κ, at age j = 1 is { } ∑Nϵ h h ′ v (1, a, εi, κ) = max u(c, 1 − n) + βζ1 πimv (2, a , εm) (2.2) c,a′,n m=1 subject to

′ h c + a = (1 + (1 − τa)r)a + (1 − τn)w l(1)εin + τr − κ

′ ≥ h ≥ ∈ a ϕa1 , c 0, n [0, 1]

e where aj is the natural debt limit for a household at age j with education level e. l(j) represents labor market experience.67

The problem of a household who does not pay a ﬁxed education cost is { } ∑Nϵ l l ′ v (1, a, εi) = max u(c, 1 − n) + βζj πimv (2, a , εm) (2.3) c,a′,n m=1 subject to

′ l c + a = (1 + (1 − τa)r)a + (1 − τn)w l(1)εin + τr

′ ≥ l ≥ ∈ a ϕa1, c 0, n [0, 1]

Note that this problem is same as that of a working household of any other age.

The optimal education decision, e(a, ε, κ), can be summarized as

{ h if vh(1, a, ε, κ) ≥ vl(1, a, ε) e(a, ε, κ) = (2.4) l otherwise − − h − ≥ h subject to m = (1 + (1 τa)r)a + (1 τn)w l(1)εin + τr κ ϕa1 . 67Following Heathcote, Storesletten, and Violante (2010), I assume a return to age as a proxy for labor market experience.

80 2.3.3 A Household in the Working Life

In this section, I describe the behavior of households during their working life, after the college education decision has been made. Let ve(j, a, ϵ) represent the expected discounted value of a household, with education level e ∈ {l, h}, at age j with assets a and an idiosyncratic productivity shock ε. Given a, the household chooses consumption c, saving a′, and labor supply n. The problem of a working household is

{ } ∑Nϵ e e ′ v (j, a, εi) = max u(c, 1 − n) + βζj πimv (j + 1, a , εm) (2.5) c,a′,n m=1

subject to

′ e c + a = (1 + (1 − τa)r)a + (1 − τn)w l(j)εin + τr

′ ≥ e ≥ ∈ a ϕaj, c 0, n [0, 1]

2.3.4 A Household after Retirement

Lastly, I describe the behavior of households after retirement. Let ve(j, a) be the

value function of a retiree, with education level e ∈ {l, h}, at age j with assets a. The

retiree chooses consumption, c, and saving, a′. The problem of a retiree is

e e ′ v (j, a) = max {u(c, 1) + βζjv (j + 1, a )} (2.6) c,a′

subject to

′ c + a ≤ (1 + (1 − τa)r)a + τr + b

81 ′ ≥ e ≥ a ϕaj, c 0

e where ϕaj is the borrowing limit and b is the social security beneﬁt.

2.3.5 Age- and Skill-varying Natural Debt Limits in OLG economy

e I derive natural debt limits, aj, in the model backward by age for each skill group. Note that borrowing is not allowed for the last age. Since labor supply is endogenous,

households can repay more debt by increasing their hours worked. This leads to

natural debt limits that are functions of the maximum possible labor supply. Given

e a natural debt limit, aj+1, for age j + 1, the maximum amount of debt a household can borrow at age j is deﬁned as:

1 ( ) e e − e aj = aj+1 xj+1 (2.7) (1 + (1 − τa)r)

e where xj+1 is the lowest possible labor earnings, given time endowment, or pension

e income at age j + 1 for a household with education level e. Speciﬁcally, xj+1 is the earnings from the lowest possible wage shock, conditional on maximum possible labor supply for workers and pension income for retirees.68 These natural debt limits, the

discounted value of the maximum debt households can borrow tomorrow and lowest

future possible earnings, allow households to borrow against their future income, im-

plying age-and skill-varying debt limits.

68By deﬁnition, natural debt limits depend on equilibrium prices.

82 2.3.6 Recursive Equilibrium

I deﬁne stationary recursive equilibrium. The distribution of households varies over age, wealth, labour productivity, and education level. Let J = {1,...,J} repre-

sent the set of indices for household age. Households’ wealth is a ∈ A = [a, ∞) and

their education level is described using e ∈ {l, h} where, as above, e = h if a house- { } hold is college educated and e = l otherwise. E = ε1, . . . , εNε deﬁnes the support

69 for household productivity shocks. Lastly, let Γ = {κl, . . . , κh} be the space of ﬁxed

education costs. The product space, S = J × A × E × {l, h} describes the space for

the distribution of households. Deﬁne S as the Borel algebra generated by the open

subsets of S. We deﬁne ψ : S → [0, 1] as a probability measure over households.

Households of age 1 begin with a0 drawn from χ(a0) and an initial productivity

0 ∼ 2 { }Nε 70 drawn from π logN(0, σπ), the invariant distribution for πim i,m=1. Let µ1 be the number of initial households with age j = 1. The distribution of non-college

educated households at this ﬁrst age is described by

∫ 0 ψ(1, A, εi, l) = πi µ1Q(dκ)χ(da0) {(a0,κ)|e(a0,ϵi,κ)=l}

The distribution of college educated households at initial age is described by ∫ 0 ψ(1, A, εi, h) = πi µ1Q(dκ)χ(da0) {(a0,κ)|e(a0,ϵi,κ)=h}

69I ﬁx the support for persistent shocks to that of the skilled and use the Tauchen algorithm to discretize the distribution implied by the variance of shocks for the unskilled onto this support. The initial value for the persistent shock is drawn from the same support. 70I assume a distribution for initial labor productivity using the value estimated for the skilled, 2 σπ,h, for both education groups.

83 In subsequent periods, j + 1, j = 1, ..., J − 1, the distribution of households with

each education level is given by the following. ∫ ∑Nε h ψ(j + 1, A, εl, h) = πim ζjψ(j, da, εi, h) { | h ∈ } i=1 (a,εi) g (j,a,εi) A ∫ ∑Nε l ψ(j + 1, A, εl, l) = πim ζjψ(j, da, εi, l) { | l ∈ } i=1 (a,εi) g (j,a,εi) A A recursive competitive equilibrium is a set of functions (vl, vh, gl, gh, nl, nh, cl, ch, e) and prices (wl, wh, r) such that:

i (vl, vh) solve (2), (3), (5) and (6). g : J×A×E×{l, h} → A is the associated op- ∫ h timal policy for saving. Note that g(1, A, E, h) = {κ|e(a,ε,κ)=h} g1 (A, E, κ)Q(dκ) h × × → where g1 : A E Γ A is the associated optimal policy for (2). n : J × A × E × {l, h} → [0, 1) is the associated optimal policy for labor supply,

c : J × A × E × {l, h} → R+ is the associated optimal policy for consumption,

and e : A × E × Γ → {l, h} is the education decision rule for paying the ﬁxed

cost to get a college degree.

ii Markets clear ∫ ∑J ∑Nε l l H = l(j)εin ψ(j, da, εi, l) j=1 i=1 A ∫ ∑J ∑Nε h h H = l(j)εin ψ(j, da, εi, h) j=1 i=1 A {∫ ∫ } ∑J ∑Nε K = g(j, a, εi, l)ψ(j, da, εi, l) + g(j, a, εi, h)ψ(j, da, εi, h) j=1 i=1 A A ∫ ∑Nε C + δK + G = F (K,H) − κψ(1, da, εi, h)Q(dκ), { | } i=1 (a,εi,κ) e(a,εi,κ)=h ∑ ∑ {∫ ∫ } J Nε where C = j=1 i=1 A c(j, a, εi, l)ψ(j, da, εi, l) + A c(j, a, εi, h)ψ(j, da, εi, h)

84 iii Government budget is balanced {∫ ∫ } ∑J ∑Nε h h l l τn(w H +w H )+τarK = G+b ψ(j, da, εi, l) + ψ(j, da, εi, h) A A j=Jr i=1

iv Prices are competitively determined

l h l w = D2F (K,H)D2H(H ,H )

h h l w = D2F (K,H)D1H(H ,H )

r = D1F (K,H) − δ

2.4 Calibration

2.4.1 College Education Cost and Borrowing Limits

The average annual college education cost in 2004, including tuition, fees, room

and board rates for full-time undergraduate students, is around $13, 993 in current

dollars.71 By assuming four years to complete college education, the total paid college

$13,993∗4 education cost as a share of GDP per capita in 2004 is $41,912 = 1.34. I assume that the college education cost is drawn from uniform distribution, κ ∼

U(κl, κu), where κl is set to 0. Households can borrow up to a ﬁxed fraction, ϕ, of their

e natural debt limits, aj. As skilled households can ﬁnance college education borrowing against future earnings, ϕ is a crucial parameter to determine both education costs

and college decisions. For these reasons, I choose the parameters (κu, ϕ) to match the

college completion rate of 27 percent for males between 25- and 59- years old in 2004

and the average college education cost as a share of GDP per capita.

71This includes both two-year and four-year institutions. The corresponding education cost for two-year institution is $7, 095 and $16, 510 for four-year institutions in National Center for Education Statistics.

85 The model-implied borrowing limits over age for each skill group in Figure 5 emphasize three points. Note that negative values for assets imply debt. First, skilled workers can borrow more than unskilled workers. This reflects a higher hourly wage received by the skilled compared to the unskilled. Second, labor market experience increases average wages, and thus workers’ ability to borrow. Once retired, both skilled and unskilled households borrow against fixed pension benefit, resulting in the same borrowing limits for both skill groups. Moreover, after retirement, households’ present discounted life-time earnings fall as they get older, and thus their ability to repay loans declines, their borrowing limits rise.

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 Age skilled unskilled

Figure 2.5: Age- and skill-varying borrowing limits

Importantly, while I have not targeted the share of student loan debt holders, the calibrated model economy gives rise to two-thirds of graduating seniors from four-year

86 institutions with student loan debt as reported in the Project on Student Debt by

National Postsecondary Student Aid Study (NPSAS).

2.4.2 Initial Wealth Distribution

The distribution of initial asset holdings, a0 ∈ [al, ah], is assumed to be drawn from a beta distribution, a0 ∼ Beta (β1, β2). I ﬁrst restrict the distribution curvature

parameters to be a reciprocal of each other, β2 = 1/β1 and ﬁx an upper bound of the

support, ah = 1. The remaining parameters, al and β1, are used to match 31.4 percent

of negative or zero asset holdings of households who are between 20 and 25 years old

and the zero mean of initial asset distribution. This distribution of assets, alongside

idiosyncratic education costs denominated in units of output, may determine the ef-

fect of liquidity constraints on households’ college education decisions. Speciﬁcally, it

can interact with borrowing constraints, discouraging some households who pursue a

college degree from gaining skills because of the insuﬃcient resources.72

2.4.3 Remaining Model Parameters

The benchmark model is calibrated to the 2004 U.S economy. Table 3 summarizes

the calibration strategy and parameter values. The model period is one year. House-

holds are assumed to enter the labor market at age 25 and retire at age 60, and their

last possible age is 84. Survival probabilities {ζj} are based on 2004 U.S. Life Tables

of the National Center for Health Statistics. The household period utility function is

72Heathcote et al. (2010) instead have an utility cost of college education, allowing individuals to choose college and gain skills independent of their resources.

87 CRRA.

c 1−γ (1 − ne)1−σ u(c , ne) = t + ψ t (2.8) t t 1 − γ 1 − σ

The relative risk aversion parameter, γ = 2.73 The parameter for leisure ψ is 2.6 to match average male hours worked of 30% of time endowment. Given this, σ = 4.3

matches a 0.48 Frisch elasticity of labor supply for men.74 I set β = 0.997 to replicate

the ratio of average wealth to average pre-tax earnings in 2004 SCF.75

I now turn to production, the parameter θ that governs the elasticity of substitution between skilled and unskilled labor is set to 1.67 following Krusell et al. (2000).

The capital share of output is α = 0.36 and the depreciation rate δ is 0.06. Following

Heathcote et al. (2000), λ is calibrated to match the observed college wage premium

of 1.64 in 2004.

Following Domeij and Heathcote (2004), the labor and capital taxes are set to

τn = 0.27 and τa = 0.40. The ﬁxed pension beneﬁt is set to a lump-sum payment

b that matches the ratio of maximum eligible individual beneﬁts to average pre-tax

individual earnings, amounting to 25.5 percent in 2004. This ratio is similar to the

ﬁnding in Heathcote et al. (2000) that a lump-sum social security beneﬁt around 24.5

percent of mean individual pre-tax earnings captures the redistribution of lifetime

earnings embedded in realistic social security system.

73Attanasio (1999) estimates this parameter between one and two.

74 1 (1−n) The Frisch elasticity of labor supply is ( σ ) n , where n is the average aggregate hours worked. 75Huggett and Ventura (2000) estimate a high discount factor, in their model, over 1. They point out that the value of the discount factor is an empirical issue in an overlapping generations model. OLG models do not have a theoretical restriction on the value of discount factor unlike inﬁnitely-lived horizon models.

88 Parameters set externally Value {ζj} Survival rates (2004 U.S. Life Tables) γ coefficient of relative risk aversion 2.0 α capital share of output(NIPA) 0.36 δ depreciation rate(NIPA) 0.06 θ elasticity of substitution between skilled and unskilled labor 1.67 τn, τa labor and capital income taxes 0.27, 0.4 l(j) male hourly wage life-cycle profile Parameters calibrated Value moments to match data model β 0.997 avg wealth to avg income ratio (2004 SCF) 5.2 5.2 ψ 2.6 average hours worked 0.33 0.33 σ 4.3 Frisch elasticity of male labor supply 0.48 0.48 b 0.07 ratio of maximum individual social security benefits to avg earnings 0.255 0.250 λ 0.51 wage premium to college education 1.64 1.64 κu 4.3 college enrollment rates 0.27 0.27 ϕ 0.47 average college education cost as a share of GDP per capita 1.34 1.34 β1 0.14 mean of asset holdings between 20 and 25 years old 0 0 89 al -0.003 share of households of age between 20 and 25 with zero or negative asset holdings 0.314 0.314

Table 2.3: Summary of parametrization 2.5 Quantitative Results

In this section, I present the results from an economy with skill-speciﬁc wage processes - the benchmark economy. To evaluate the importance of these skill-speciﬁc wage processes, I also calibrate an alternative economy with a common estimated wage shock process for both skill-groups and compare its results with the benchmark economy throughout the analysis.

2.5.1 Common shock economy

In an alternative common shock economy, there only exists between-group wage dispersion and both skilled and and unskilled households face the same productivity shock process. The estimates for a common wage shock process are reported in

Table 4. Importantly, this common wage shock process implies far greater volatility for unskilled households than the skill-speciﬁc wage shock process of the benchmark model. Indeed, unskilled households face a variance of persistent shock of 0.006 in the benchmark economy compared to 0.036 in the alternative economy.

2 2 2 ρ σpt σvt σπ 0.986 0.036 0.145 0.111 Note : Table 4 shows the persistence of wage shock ρ, the variance of persistent and tran- 2 2 sitory shocks, σpt , σvt , and the initial age persistent shock vari- 2 ance, σπ.

Table 2.4: A common wage shock process in 2004

90 2.5.2 Aggregate inequality

In Table 5, I summarize moments of the distribution of wealth and the share of households with a college degree in the 2004 SCF and the two economies - the benchmark economy and the common shock economy.

1% 5% 10% 50% 90% ≤ 0 Gini share 2004 SCF 28.7 50.5 63.5 96.6 100 8.5 0.77 0.27 Benchmark 11.9 32.8 47.9 94.6 101 12.3 0.67 0.27 Common shock 10.1 30.9 47.2 96.3 102 17.1 0.68 0.27 Note : Table 5 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the wealth Gini coeﬃcient, the share of households with zero or negative asset holdings, and the share of households with a college degree.

Table 2.5: Distribution of wealth

The benchmark economy implies 27 percent of households become skilled workers, receiving a higher hourly wage, as well as a more volatile wage shock process, relative to unskilled workers. This empirically consistent within- and between-group wage dispersion, alongside endogenous education choice, reproduces a signiﬁcant amount of the wealth inequality seen in the data. The wealth Gini coeﬃcient in the benchmark economy comes within 10 percentage points of the empirical wealth Gini and the share of wealth held by the top 10 percent of households is around 48 percent compared to

64 percent in the data. The top 1 percent of households hold 29 percent of wealth in the data compared to 12 percent in the model.76

76Although the model economy can not generate enough asset accumulation among the wealthiest households, it is surprising that such a large amount of inequality can be explained in the absence of strong saving motives arising from entrepreneurship with collateral constraints (Cagetti

91 The interaction of endogenous labor supply choices and skill-specific wage processes is key to understanding this success of the model. First, given access to college loans, an intensive labor margin choice allows productive households to become highly-educated. This is because productive but financially constrained households can increase hours worked during their working lives to repay their college loans. This selection effect increases the average earnings earned by skilled households. Earnings received by skilled households further rise through the persistence in their productivity shocks and a college wage premium. Second, the conditional variance of wage risk for the skilled implies a higher probability of both favorable and unfavorable shocks compared to unskilled households. This suggests that earnings differentials between skilled and unskilled households widen as well as that among the skilled.

Overall, skilled workers receive relatively high earnings and their accumulation of wealth increases inequality.

The distribution of wealth in the economy with a common wage shock is comparable to that in the benchmark economy as summarized in the last row of Table 5.

Speciﬁcally, the wealth Gini drops only by 1 point and the share of wealth held by the top 10 percent of households is close to that in the benchmark economy. However, wealth inequality in the common shock economy is driven by counter-factually high wage risk assumed for the unskilled. This assumed wage process increases unskilled households’ savings by two channels. First, it raises the probability of positive productivity shocks for the unskilled, indicative of high earnings. Second, an increase in labor market risk per se boosts their precautionary savings. Thus, the resulting dispersion in wealth in the economy with a common wage shock process is not just and De Nardi,2004), preference shocks (Krusell and Smith, 1998), stochastic life cycles (Castenada et al.,2003), or intergenerational links (De Nardi, 2004).

92 driven by skilled households but also unskilled households. While the diﬀerence in the implications for the distribution of wealth is minor between the benchmark economy and the common shock economy, as discussed in the next section, the common shock economy will not be consistent with the micro-predictions of the data on earnings, income, and wealth.

Figure 6 shows the distribution of wealth between education groups over the log of labor productivity and asset holdings in the benchmark economy. There is considerably more dispersion in productivity shocks for skilled households, a reflection of the higher wage risk they face. Both more volatile wage risk for skilled workers and college wage premia lead to higher inequality in the wealth distribution. Facing volatile wage risk, the precautionary saving motive of the skilled increases their savings following a positive productivity shock. However, a strong income effect arises over time following this rapid and significant accumulation of wealth, leading to eventual lower saving rates for the skilled compared to the unskilled. These two opposing forces together determine savings decisions of the skilled and the distribution of wealth in the economy.77

77See Figure D.1 for Lorenz curve.

93 Table 6 compares the distribution of income for the benchmark economy and the common shock economy to that from the 2004 SCF.

1% 5% 10% 50% 90% Gini 2004 SCF 12.8 26.2 37.2 81.8 98.8 0.49 Benchmark 5.5 18.4 31.0 87.1 100 0.52 Common shock 13.2 33.5 47.8 91.3 100 0.64 Notes : Table 6 shows the share of income held by the top 1, 5, 10, 50 and 90 richest households, the income Gini coeﬃcient.

Table 2.6: Distribution of income

Consistent with deﬁnition of income in 2004 SCF data, income in the model includes labor income, capital income, social security payments and transfers.78 Though the benchmark economy underpredicts skewness, the distributions of income in the benchmark economy are broadly consistent with those in the data, reproducing the income Gini of 0.52 and 31 percent of income held by the richest 10 percent of households compared to 37 percent in the 2004 SCF. In contrast, a common wage shock process overpredicts income inequality, generating an income Gini of 0.64 compared to 0.49 in data as seen in the last row of Table 6. This discrepancy in income inequality is a direct consequence of the amount of heterogeneity in wage risk across households. Speciﬁcally, with between- and within-group wage dispersions, in the benchmark economy, the fraction of households who face relatively volatile wage process drops to 27 percent of skilled households compared to the entire population in the common shock economy.

78See Appendix A.2 for the deﬁnition of income in the 2004 SCF.

94 Figure 2.6: Distribution of wealth for skilled and unskilled households 95 Table 7 and 8 summarize the over-identiﬁed predictions of the model. As shown in Table 7, the model is broadly consistent with the fractions of skilled and unskilled households across wealth percentiles in the data. For example, the model predicts that 74 percent of the top 10 percent of households are skilled as in the 2004 SCF.

The model also explains the high fraction of unskilled households between the 51 and

90 wealth percentile. Table 8 shows the probability of being in each wealth percentile for college and non-college graduates. Among college graduates, around 65 percent reach the top 50 percent of the wealth distribution compared to 67 percent in data.

Moreover, 18 percent of skilled households will be in the top 5 percent of the wealth distribution while the corresponding empirical value is 8 percent. Although these results are broadly consistent with data, the benchmark economy has over-emphasized the importance of college education choice on the life-time wealth accumulation, only allowing skilled households to be the wealthiest 5 percent of households.

2004 SCF 1% 5% 10% 50% 90% skilled households 0.85 0.78 0.74 0.50 0.38 unskilled households 0.15 0.22 0.26 0.50 0.62 Benchmark economy 1% 5% 10% 50% 90% skilled households 1.0 0.93 0.74 0.34 0.26 unskilled households 0.0 0.07 0.26 0.66 0.74

Table 2.7: The fraction of the skilled and unskilled in the top percentiles of the wealth distribution

96 2004 SCF 1% 5% 10% 50% 90% skilled households 0.02 0.11 0.20 0.67 0.93 unskilled households 0.00 0.02 0.04 0.40 0.89 Benchmark economy 1% 5% 10% 50% 90% skilled households 0.04 0.18 0.28 0.65 0.88 unskilled households 0.00 0.00 0.04 0.45 0.91

Table 2.8: The probability of being in the top percentiles of the wealth distribution

2.5.3 Life-cycle implications

To further evaluate the importance of heterogeneous mean and variance of wage

processes across households, I illustrate the role of educational choice and skill-varying

wage processes over life-cycle hours worked, wages, income, and wealth accumulation

by skill group. I simulate 40,000 households for each education group and over cohort

using the calibrated benchmark economy.

Ex-post wages Figure 7 (a) presents the actual level of wages realized for skilled and unskilled households during their working lives and Figure 7 (b) plots the age-speciﬁc ratio of skilled to unskilled wages in the benchmark economy.

As seen in Figure 7 (b), skilled workers receive ex-post skill premia around 1.75

during their working lives. Note that this value is higher than the estimated college

wage premium of 1.64. Given the endogenous education choice with a resource cost

of a college education, relatively high-ability households have larger net beneﬁts of

pursuing a college degree. Thus, wage diﬀerentials between education groups are

reinforced through endogenous skill choice in my economy. This selection eﬀect of

skilled households drives a wage gap between education groups seen in the bottom

97 (a) Skilled and unskilled households average wages

1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57

Age skilled wage unskilled wage

(b) Ratio of skilled to unskilled wages

2 1.95 1.9 1.85 1.8 1.75 1.7 1.65 1.6 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57

Age

Figure 2.7: Life-cycle hourly wages for skilled and unskilled households

98 panel.

Earnings Figure 8 compares the life-cycle earnings proﬁles across education groups from the benchmark economy (top panel) and from the common shock economy

(bottom panel) to those from the 2003 PSID data.79

For comparability between model and data, I calculate a factor which makes the

average earning of 50 year old unskilled households in the model equal to that in the

data, then multiply other series by the same factor. The benchmark economy with

skill-speciﬁc wage processes successfully reproduces a hump-shaped earnings proﬁle

for the skilled and a relatively ﬂat proﬁle for the unskilled.80 Given that hours worked

changes little over working lives, these age proﬁles for earnings are reﬂective of life-

cycle wages.81

The benchmark economy with shocks that diﬀer by skills predicts a diﬀerence in

earnings across education groups closer to the data than a model with the common

wage shock. In a common wage shock economy, the entire working population faces

the same earnings risk. This implies that if an unskilled household is lucky, he can

have a signiﬁcant amount of earnings without a college education, narrowing the gap

in earnings over education levels. It is worth noting that, in the common shock pro-

cess model, the primary determinant of wealth is luck. In contrast, in the model with

separate shock processes, wealth is partly endogenized by education choice. Indeed,

79Given the fact that the PSID data conducts a survey biennially since 1997, I used 2003 data for micro-evidence in this section. 80This is in agreement with the ﬁnding in Carroll and Summers (1991) that the average consumption-age and income-age proﬁles are more hump-shaped for college graduates than those for non-college graduates. 81See Figure D.2 in Appendix C for the life-cycle hours worked.

99 (a) Skill-speciﬁc wage shock

120 100 80 Thousands 60 40 20 0 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Age skilled (PSID) unskilled (PSID) skilled (separte shock) unskilled (separate shock)

(b) Common shock

100

Thousands 60

0 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Age skilled (PSID) unskilled (PSID) skilled (common shock) unskilled (common shock)

Notes : For the PSID data, smoothed series are generated using Hodrick-Prescott trend with smoothing parameter of 100.

Figure 2.8: Life-cycle earnings proﬁle for skilled and unskilled households

100 households, once skilled, face relatively volatile wages giving them a higher probabil-

ity of both favorable and unfavorable shocks.

Wealth Figure 9 depicts wealth over the life-cycle for skilled and unskilled households.

I normalize the model-implied wealth of 50 years old unskilled households to the level

in the data. As evident in the ﬁgure, the benchmark model economy successfully

reproduces the accumulation of wealth by skilled and unskilled workers seen in the

SCF (top panel) while it is hard to explain using a common wage shock (bottom

panel).

The conditional means and variances of wage processes that rise by skills, in the

benchmark economy, aﬀect skilled households’ saving decisions through two channels.

First, the volatile wage shock process faced by skilled households increases their pre-

cautionary saving, implying higher savings rates. Second, a relatively strong income

eﬀect that follows from high levels of ex-post earnings by skilled households reduces

their saving rates. However, the total level of savings remains high compared to un-

skilled households because of higher earnings. Nonetheless, the two opposing forces

result in lower saving rates by the skilled than those by the unskilled as illustrated

in Figure 10 (a).8283 In contrast, in the common shock economy, there exists no big discrepancy in wealth by skill levels as illustrated in Figure 9 (b). This implies similar

82The initial drop in savings rates by the skilled is also due to the college education cost payment. 83Absent a realistic social security system, households’ consumption after retirement is ﬁnanced out of their current asset holdings, and saving rates fall sharply after retirement. The drop in saving rates is large for college graduates as social security payments are small relative to their earnings while working.

101 (a) Skill-speciﬁc wage shock

1.6 1.4 1.2 Millions 1 0.8 0.6 0.4 0.2 0 -0.2 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82

Age skilled (SCF) unskilled (SCF) skilled (separate shock) unskilled (separate shock)

(b) Common shock

1.6 1.4 1.2 Millions 1 0.8 0.6 0.4 0.2 0 -0.2 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 Age skilled (SCF) unskilled (SCF) skilled (common shock) unskilled (common shock)

Notes : For the SCF data, smoothed series are generated using a Hodrick-Prescott trend with a smoothing parameter of 100.

Figure 2.9: Life-cycle wealth accumulation for skilled and unskilled households

102 saving rates between skilled and unskilled households given the same earnings risk

(Figure 10 (b)).84

In the absence of a mechanism explaining savings for retirees, such as a realis-

tic pension system, intergenerational bequest motive, or health shocks, my model

fails to match the persistent diﬀerence in wealth after retirement between skilled and

unskilled households. Thus, it may be useful to investigate after-retirement savings

motives to explain the remaining inequality in the data not captured in this paper.

Income Figure 11 plots the life-cycle proﬁles for total income including both labor and capital income. As before, I re-scale the model-simulated average income of the cohort of unskilled 50 years old to that in the data. It is worth noting that, in the benchmark economy, predicted life-cycle income paths of skilled and unskilled workers as well as the relative magnitude between the two education groups are similar to the

2004 SCF data while the common shock economy fails to match these. This conﬁrms the fact that income inequality is exaggerated in the common shock economy by assuming counter-factually high earnings risk for the entire population.

In summary, the analysis in this section highlights the importance of distinct wage

processes by skill level to understand wealth accumulation of skilled and unskilled

households. The standard assumption of i.i.d wage shock puts too much emphasis

on luck to determine individual and aggregate wealth, failing to match a disparity

in wealth across education groups. In contrast, endogenous skill choice, alongside

84Note that there is a more initial drop in saving rate by the skilled with a common wage shock compared to that in the benchmark economy. A model with a common wage shock has less incentives to be skilled compared to the benchmark economy. To induce the same number of skilled households in the common shock economy as in the data, I relaxed borrowing limits. This, in turn, reduces the initial savings rates of skilled workers.

103 (a) Skill-speciﬁc wage shock

1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 Age skilled unskilled

(b) Common shock

0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 Age skilled unskilled

Figure 2.10: The model- implied life-cycle saving rates for skilled and unskilled households

104 (a) Skill-speciﬁc wage shock

160 140 120

Thousands 100 80 60 40 20 0 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 Age skilled (SCF) unskilled (SCF) skilled (separate shock) unskilled (separate shock)

(b) Common shock

160 140 120

Thousands 100 80 60 40 20 0 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 Age skilled (SCF) unskilled (SCF) skilled (common shock) unskilled (common shock)

Notes : For the SCF data, smoothed series are generated using a Hodrick-Prescott trend with a smoothing parameter of 100.

Figure 2.11: Life-cycle income for skilled and unskilled households

105 skill-varying wage processes both in the means and variances, introduces a friction to explain diﬀerent saving behavior across households.

2.5.4 Importance of Education Choice

To further evaluate the importance of endogenous educational choice on wealth inequality, I study a standard incomplete markets life-cycle model as in Huggett

(1997), but with an endogenous labor supply decision. Note that the economy does not have an education decision. I assume a common estimated wage shock process across skill-groups summarized in Table 4.

As seen in Table 9, the Huggett economy decreases the wealth Gini to 0.62 from

0.67. Moreover, the share of wealth held by the top 10 percent of wealthiest households falls to 42 percent from 48 percent in the benchmark economy.

1% 5% 10% 50% 90% ≤ 0 Gini share 8.9 27.1 41.8 92.0 100 15.9 0.62 0.0 Notes : Table 9 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the wealth Gini coeﬃcient, the share of households with zero or negative asset holdings, and the share of households with a college degree.

Table 2.9: Economies without educational choice and with a common wage shock process

There are counteracting eﬀects at work in this model without a college decision.

First, eliminating between- and within-group wage dispersion decreases the diﬀerence

106 in wealth by education level, dampening inequality. However, as shown in Table 4, unskilled households face more volatile wage shocks than they do in the benchmark economy. This increased earnings risk for the unskilled itself increases wealth inequality by amplifying precautionary savings. In net, the ﬁrst eﬀect dominates, resulting in a drop of the wealth Gini by 6 percentage points. Therefore, endogenous skill choice plays a vital role in explaining inequality as skilled households endogenize earnings risk.

2.5.5 Beneﬁts of College Education

Absent empirically consistent within- and between- group wage dispersion, how much inequality can be explained by skill difference across households? To evaluate this, I first lower between-group wage dispersion. The second row in Table 10 shows the wealth distribution after decreasing the college wage premium to 1.1 from its empirical counterpart of 1.64, while holding other parameters unchanged.85 Note that the wealth Gini drops to 0.63 and the share of wealth held by the top 10 percent of households decreases by approximately 5 percentage points. This is because the decrease in the college wage premium discourages households from going to college, dropping the percent of skilled households in the economy to 18 percent compared to 27 percent in the benchmark model. With a college loan that has to be paid back during working age, discrete skill choice for the financially constrained depends on their labor income when they begin working. Thus, a drop in the college wage

85I fix wage for unskilled workers and interest rate to the steady state level. This is to see how the benefits of college education, both in means and variances of wage processes, affect college education decision, and thus wealth inequality.

107 premium makes college education less attractive to households, resulting in 18 percent of households obtaining skills.

I further remove within-group wage dispersion across education groups. The last row of Table 10 shows the results from a college wage premium equal to 1.1 and the unskilled wage shock process applied to both education groups. In this alternative economy, precautionary savings are weaker and skilled households are less likely to be earnings-rich relative to the benchmark economy. Consequently, the share of wealth held by the top 10 percent of households further decreases by 10 percentage points and the wealth Gini coeﬃcient drops to 0.60 from 0.67 in the benchmark economy.

These exercises clearly underline the importance of assuming empirically consistent wage diﬀerentials, both in means and variances, to examine the role of education choice on aggregate inequality.

1% 5% 10% 50% 90% ≤ 0 Gini share Benchmark 11.9 32.8 47.9 94.6 101 12.3 0.67 0.27 low college wage premium 9.76 27.8 42.5 93.4 101 12.6 0.63 0.18 low college wage premium 5.88 21.7 37.4 93.0 101 13.6 0.60 0.15 & unskilled wage shock Note : Table 10 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the wealth Gini coeﬃcient, and the share of households with a college degree.

Table 2.10: Economies without the beneﬁt of college education

108 2.5.6 Implications of the endogenous labor supply

In the literature, variable labor supply allows households to insure themselves from uninsurable idiosyncratic risk, dampening inequality.86 However, it is important to point out that, in a model with endogenous educational choice and empirically consistent wage processes for each skill group, variable labor supply drives more dispersion in wealth through two eﬀects. First, households who pursue a college degree can work harder to pay back college loans. Thus, variable labor supply encourages households to become skilled, driving more inequality. Second, households increase hours worked following a negative persistent productivity shock. Thus, skilled households can smooth earnings against downside risk from a volatile wage shock process by increasing their hours worked. However, a relatively volatile wage risk for the skilled per se implies a higher probability of drawing a positive productivity shock. As a result, a more volatile wage process with endogenous labor supply provides a better opportunity to earn higher income and eventually become wealthy for the skilled, driving inequality.

To see the impact of variable hours worked on inequality, I ﬁx labor supply for each skill group to the average level of hours worked by skill in the benchmark economy.

The skill premium and wage shock processes are assumed to be same as the benchmark economy. Table 11 shows that, without variable labor supply, none of households become skilled and the wealth Gini drops to 0.61 from 0.68 in the benchmark economy.

Moreover, the wealthiest 10 percent of households hold 38 percent of total wealth

86Castenada et al. (2003) points out the importance of modelling endogenous labor supply in a model with uninsurable idiosyncratic risk. Also, Pijoan-Mas (2006) shows that households make use of endogenous labor supply choice to smooth out consumption against idiosyncratic labor market risk.

109 compared to 48 percent in the benchmark economy. It is worth noting that, in contrast to the existing literature, variable labor supply leads to more dispersion in wealth in a model with a discrete choice and estimated skill-speciﬁc wage shock processes.

1% 5% 10% 50% 90% ≤ 0 Gini share 6.5 23.8 38.3 94.9 99.9 2.4 0.61 0.0 Note : Table 11 shows the share of wealth held by the top 1, 5, 10, 50and 90 wealthiest households, the wealth Gini coeﬃcient, the share of households with zero or negative asset holdings, and the share of households with a college degree.

Table 2.11: An economy with exogenous labor supply

2.6 Policy Experiments

In this section, I consider the eﬀects of college education subsidies on aggregate inequality and welfare using the calibrated benchmark economy. I denote χ as the associated welfare gain or loss, expressed in units of equivalent variation in lifetime consumption. In particular, χ measures the percentage change in lifetime consumption required to make households entering the benchmark economy indiﬀerent to those entering the policy economy.

Recall that ψ : S → [0, 1] is a probability measure over households where S =

J × A × E × {l, h}. Then, the welfare is measured as:

110 ∫ ∫ ∫ ∫ u ((1 + χ)ce(j, a, ε), ne(j, a, ε)) ψ(S) = J A E e ∫ ∫ ∫ ∫ u (ˆce(j, a, ε), nˆe(j, a, ε)) ψˆ(S) J A E e

wherec ˆe(j, a, ε),n ˆe(j, a, ε), ψˆ(S) denote decision rules and distribution in the policy

economy.

I design an educational subsidy policy such that households in the lowest x per-

cent earnings bracket receive a y percent subsidy for their college education costs from government.87 For example, the bottom row of Table 12 indicates the policy where

households in the bottom 34 percent of earnings receive a full college education cost

subsidy. Table 12 summarizes the implications of educational subsidies on welfare and

aggregate inequality.88 College education subsidies targeting low earnings households

decrease wealth Gini from 0.67 in the benchmark economy to 0.61 in the policy econ-

omy where bottom 34 percent of households receive a 100 percent subsidy. Moreover,

this increases the share of households acquired a college degree by 19 percent from

the benchmark economy.

Households entering the economy with the most generous college education sub-

sidy (bottom row of Table 12) experience welfare gains of 18 percent of life-time

consumption. This is mostly driven by unskilled households who enjoy a welfare gain

of more than 35 percent compared to those in the benchmark economy. Notably,

households becoming skilled with this education subsidy are worse-oﬀ, experiencing

welfare losses of 28 percent of consumption. The dramatic change in welfare is due to

87Government ﬁnances these subsidies by decreasing government expenditure. Given the fact that total subsidy payments are a small fraction of GDP in the model economy (0.5%), the implications of educational subsidies are robust when they are ﬁnanced by higher taxes. 88See Appendix C for detailed implications on the income and wealth distribution.

111 welfare wh x y %△wh %△wl share wealth Gini total skilled unskilled wl 12% 50% +2.25% −4.0% +2.44% 1.57 −2.36% +2.21% 0.28 0.66 12% 100% +8.11% −15.5% +11.0% 1.38 −8.71% +8.50% 0.34 0.64 34% 50% +7.91% −8.11% +5.96% 1.49 −5.27% +4.26% 0.30 0.65 34% 100% +17.5% −28.0% +35.8% 1.08 −19.0% +23.1% 0.46 0.61 Notes : Table 12 shows welfare changes for all households, skilled households, and unskilled households in the policy economy compared to the benchmark economy, college wage premium, percentage changes in wages from the benchmark economy, share of skilled households, and the wealth Gini coeﬃcient.

Table 2.12: Implications of educational subsidies

three forces. First, while these subsidies involve relatively small amounts of money to ﬁnance college education given to low ability households, they increase life-time earnings for these households who have high marginal utility of consumption, driving large welfare gains. Second, as subsidies encourage more households to acquire skills, wages for skilled workers fall while wages for unskilled workers rise, relative to the benchmark economy. Indeed, in the economy where 34 percent of households gets a full subsidy, skilled workers receive hourly wages 19 percent lower than the economy without subsidies. In contrast, wages for unskilled workers rise by 23 percent, resulting in college wage premium of 1.08 compared to 1.64 in the benchmark economy.

Lastly, the composition of households with a college degree changes with education subsidies. Figure 12 shows the share of households investing in a college education given their initial ability, in the benchmark economy and in two policy economies where the bottom 12 and 34 percent of households receive free college education respectively. Absent subsidies, high ability households are more likely to invest in a

112 higher education. For example, more than 60 percent of households in the highest ability group gains skills compared to less than 20 percent in the lowest ability group.

In contrast, in the policy economy, as low-ability households receive subsidies to gain skills, the average productivity of the skilled falls. Moreover, the drop in wages for skilled workers discourages high-ability households from going to college, further aggravating the quality of skilled workers. As seen in Figure 12, when subsidies are given to households in the bottom 34 percent of initial ability, only 20 percent of the highest ability group acquires a college degree compared to more than 60 percent in the benchmark economy. This suggests that low ability households crowd out high ability households in the economy with education subsidies. The net rise of 19 percentage points of skilled households, when 34 percent of households are fully sub- sidized, is the result of a 15 percentage points fall among skilled households who used to pay their education cost. Thus, given lower average productivity of the skilled, households pursuing a college degree in the policy economy expect lower average expected utility than those gaining skills in the benchmark economy. Likewise, the quality of unskilled households improves by educating those with low ability, increasing the expected utility of the remaining unskilled in the policy economy compared to benchmark economy.

2.7 Concluding Remarks

Exploring the eﬀect of education wage diﬀerentials, both in means and variances, on individual and aggregate wealth, I have developed an incomplete-markets quantitative overlapping-generations model characterized by a discrete college education

113 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.0 1.5 1.7 2.0 2.4 5.0 benchmark bottom 12% bottom 34% population

Notes : Share of skilled households given initial ability (bars). Total population of households for each ability group (red line).

Figure 2.12: College educated households given initial ability

114 choice involving resource costs, labor supply decision, and skill-varying life-cycle borrowing limits. The benchmark economy is calibrated to the 2004 U.S. data and gives rise to two-thirds of graduating seniors holding student loan debt as reported in the data.

The benchmark economy is broadly consistent with the distribution of wealth and income in the 2004 Survey of Consumer Finances. The model also reproduces the fraction of skilled and unskilled households across percentiles of the wealth distribution, and the probabilities of households reaching these levels of wealth. More importantly, the micro-predictions of the model on earnings, income, and wealth by education group successfully match the data while the standard common wage shock economy fails to explain these results.

The conditional means and variances of wages by education group, interacting with education choice, are essential to understanding distinct savings behavior of skilled and unskilled households. First, a relatively volatile wage risk, faced by skilled households, ampliﬁes their precautionary savings, and thus raise their saving rates.

However, this eﬀect is mitigated through lower saving rates driven by a strong income eﬀect as a result of high ex-post earnings. Second, the estimated wage risk for the skilled increases the probability of a favorable shock for skilled households, directly increasing their income and wealth, while labor supply choice allows them to self- insure against unfavorable shocks.

Extending the analysis, I find that college education subsidies directed at the poor make skilled households worse off on average while they make unskilled households better off. This is mainly driven by the composition of skilled households. Low- ability households acquire a college degree with education subsidies. However, a drop

115 in wages for skilled workers in the policy economy discourages relatively productive households from gaining skills, decreasing the average productivity of skilled households compared to the benchmark economy. This suggests that education subsidies worsen the quality of skilled workers while they improve that of unskilled households.

Broadly, this paper emphasizes the importance of understanding unobserved heterogeneity in the mean and variance of wages across households challenging the standard assumption of i.i.d wage risk. The presence of heterogeneous wage processes across households implies distinct saving behavior attributable to different earnings processes that vary with demographic characteristics, occupation, and education. My focus has been on the role of education-specific wage processes in explaining the distribution of income and wealth. More generally, the results suggest that different earnings process for different group of households is key to understanding income and wealth inequality.

116 Chapter 3: Segmented Asset Markets and the Distribution of Wealth

3.1 Introduction

Understanding the large levels of inequality observed in the distribution of wealth has been a primary focus of quantitative macroeconomic research in recent years.

Most studies have examined environments in which all households have access to the same assets. However, household data shows that households vary not only in the levels of their wealth, but also in the types of assets they hold. Wealthier households tend to save using assets that offer higher rates of return. Despite this evidence of household portfolio heterogeneity, little effort has been devoted to understanding how the distribution of wealth is shaped by differences in the return on savings across households.

We study a quantitative overlapping-generations model with segmented asset markets. Households diﬀer by age, wealth, income, and the types of assets they may hold.

They work in the first stage of their lives, then retire. Each period, households con- sume and borrow or save. Savings are invested in either of two assets which differ in their rates of return. Access to the higher return asset market requires the one time payment of a fixed cost. Households without current access have a discrete choice of whether or not to pay their current fixed cost to gain access to the high

117 yield asset market, or whether to continue saving using the low yield asset. There are three sources of risk in the model. First, individual income is risky and persistent.

Second, we allow for idiosyncratic uncertainty in the costs households face to access the high yield asset market. Third, households with access to high yield assets face an idiosyncratic interest rate shock on their returns.

Our benchmark model economy is calibrated to be consistent with household earnings from the PSID and data on household portfolio choice. We allow for an earnings process with both a persistent component and a transitory component. The estimated earnings process is very volatile, and earnings risk drives savings as households attempt to smooth consumption against fluctuations in income. We set the rate of return difference between the high and low return assets to be consistent with evidence on the long-run average difference in rates of return to low-yield assets and higher return assets.

Using the 2004 SCF data, we find 30 percent of households hold more than 1 percent of their total assets in the form of high return assets.89 Our model economy is consistent with this finding. Further, it reproduces the rising participation rate, seen in the data, in both age and age and wealth.90 Calibrating our model to 2004 earnings data, we find that asset market segmentation results in a 9 percentage point increase in the Gini coefficient for wealth as well as a rise in concentration for wealth amongst households in the top percentiles of the distribution. We also find that half of this rise is driven by endogeneity in asset market segmentation. When there is exogenous asset market segmentation, wealth Gini rises only by a 5 percentage

89Our ﬁndings are consistent with that of Kacperczyk, Nosal, and Stevens (2015), who compute a measure of similar high return asset market segmentation. 90The substantial heterogeneity in the share of household wealth invested in high return assets is consistent with the market participation evidence presented on Vissing-Jorgensen (2002).

118 point from the economy without asset market segmentation. This is because, in the economy with endogenous asset market segmentation, wealthy households are those who pay their fixed costs and benefit from high returns on their savings, driving further inequality. These results suggest that empirically plausible differences in the return to savings across households are an important source of wealth inequality.

The contribution of asset market segmentation to inequality hinges critically on our realistic quantitative overlapping generations model. In particular, households’ savings rates vary not only in response to the real interest rate and income risk, but also over age. Furthermore, the share of households using the high return asset rises over age and older households have lower savings rates. A greater proportion of older households having access to the high return asset market implies slower wealth growth among such households, as they have lower savings rates, than would arise if the average age of households in both asset market was similar. This reduces the impact of market segmentation on wealth inequality. This suggests the importance of a realistic demographic structure, and empirically consistent earnings process, for a quantitative evaluation of market frictions that may aﬀect wealth inequality.

In the U.S., the distribution of wealth is highly concentrated and skewed to the right. For instance, excluding the self-employed, the U.S. wealth Gini coeﬃcient was around 0.77 in 2004, and the fraction of wealth held by the top 1 percent of

U.S. households was around 29 percent. A large literature including Aiyagari (1994),

Huggett (1996), Krusell and Smith (1998) and Castaneda et al. (2003) has attempted to explain observed earnings and wealth inequality in the U.S. However, most of these models fail to generate both the signiﬁcant concentration of wealth in the top tail and a high fraction of wealth-poor households in the bottom tail. For example, Huggett

119 (1996) produces the right U.S. wealth Gini coefficient, mainly driven by a counterfactually high fraction of households with zero or negative asset holdings. Although Cas- tenada et al. (2003) successfully reproduce both tails of the wealth distribution and the U.S. wealth Gini coefficient in a modified stochastic neoclassical growth model, their results are derived in an environment characterized by a stochastic life-cycle and they choose the earnings of the highest income households. Stochastic aging builds a strong precautionary saving motive, potentially boosting wealth accumulation.

Our work is related to the entrepreneurship with loan market frictions models of

Quadrini (1998) and Cagetti and De Nardi (2006). These models focus on explaining wealth inequality through differences in earnings by entrepreneurs relative to workers. For example, Cagetti and De Nardi (2006) assume that entrepreneurs borrow capital in loan markets with limited enforceability of debt contracts. Households with profitable entrepreneurial opportunities accumulate large levels of wealth in order to increase their collateral and thus the level of capital they are able to borrow. Assum- ing a two-stage life-cycle, with stochastic aging, and allowing for altruism, Cagetti and De Nardi are able to reproduce the empirical Gini coefficient and the concentration of wealth amongst the top percentiles. They identify entrepreneurs in the data with the self-employed. Excluding the self-employed from earnings data, and from our measure of wealth, we focus on the importance of segmented asset markets on wealth inequality.

Our work is also related to Kaplan and Violante’s (2014) model where households must pay a ﬁxed cost each time they adjust their holdings of high return assets. In contrast, we assume a one time ﬁxed cost of access to high return assets, and focus on the distribution of wealth.

120 The remainder of the paper is organized as follows. Section 2 summarizes the U.S. wealth distribution in 1983 and 2004. Section 3 presents the model economy. Section

4 discusses our calibration and earning shock estimation. Section 5 presents results, and Section 6 concludes.

3.2 Wealth Inequality and Household Portfolio Choice

This section describes the stylized facts of the wealth distribution and household portfolio choice in the U.S using the Survey of Consumer Finances (SCF) data. SCF data is a household triennial data conducted by the Board of Governors of the Federal

Reserve System in cooperation with the Statistics of Income Division of the IRS since

1983. The SCF employs a dual frame sample design, one frame is multi-stage national area probability design which provides information on the characteristics of the population, and the other is a list sample to provide a disproportionate representation of wealthy households.91

Net worth in the survey is defined as total assets minus total debt. Total assets include financial assets and nonfinancial assets. Financial assets include current values and characteristics of deposits, cash accounts, securities traded on exchanges, mutual funds and hedge funds, annuities, cash-value of life insurance, tax-deferred retirement accounts, and loans made to other people. Nonfinancial assets include current values of principal residences, other real estate not owned by a business, corporate and non-corporate private businesses, and vehicles. Total debt includes the outstanding balances on credit cards, lines of credit and other revolving accounts, mortgages,

91Beginning with the 1989 survey data, the SCF methodology was substantially changed to include a multiple-imputation method implemented for missing variables.

121 installment loans for vehicles and education, loans against pensions and insurance policies, and money owed to a business owned at least in part by the family. We select households where the head of households’ age is between 25 and 85 years old.

Net worth is expressed in 2013 dollars. Full sample weights are used to calculate the wealth inequality measures.

The income process estimated using the PSID data selects households where income is not from self-employment. For consistency, we also measure wealth inequality excluding households who are self-employed. Table 1 summarizes the distribution of wealth in the U.S. As is well-known, the wealth distribution is highly concentrated and skewed to the right. Across years, more than 20 percent of total wealth was held by the top 1 percent wealthiest households and approximately 65 percent of wealth in the economy was held by the top 10 percent. The wealth Gini coeﬃcient is 0.771 in 2004 which is higher than the earnings Gini which is around 0.5.92 This is the well- known fact that wealth is more concentrated and unequally distributed than earnings in the U.S.93 The share of households with zero or negative asset holdings is around

8 percent in 2004.94

92To measure the wealth Gini when some households have negative assets, we follow Chen, Tsaur, and Rhai (1982). 93U.S. earnings inequality has risen over the last 30 years. However, there is no clear evidence of a corresponding trend in wealth inequality. See Heathcote, Perri, and Violante (2010). 94If we include self-employed households, the 2004 wealth distribution is more concentrated. The top 1 percent of households hold around 33 percent of total wealth and the top 10 percent of households hold 69 percent of total wealth. The wealth Gini coeﬃcient is 0.80 and the share of zero or negative asset holdings is 8.5 percent of the sample. However, the inclusion of self-employed households in the 1983 data changes the wealth inequality by little.

122 Year 1% 5% 10% 50% 90% ≤ 0 Gini 1983 36.6 57.6 68.1 96.7 100 8.7 0.789 2004 28.7 50.5 63.5 96.6 100 8.5 0.771 Table 1 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households and the wealth Gini coeﬃcient. Source: SCF data.

Table 3.1: Wealth Inequality in the U.S. Economy

Examining financial assets in the SCF, we define high-yield assets as stocks, bonds, money market mutual funds, and the share of annuities, trusts, and retirement accounts invested in stocks.95 We find that the share of households in 2004 with more than 1 percent of total assets held in high-yield assets is 30 percent. This definition of high-yield assets is comparable to the measure used by Kacperczyk, Nosal, and

Stevens (2015). Using the SCF data from 1989 to 2013, they also ﬁnd that 34 percent of households, on average, invest in stocks, bonds, or mutual funds, or use a brokerage account.

Figure 1 shows the share of households across 5-year age groups, holding more than

1 percent of their total assets as high yield assets. Examining households between 25 and 29 years old, only 15 percent hold high-yield assets. However, participation in high-yield asset markets increases with age until the retirement age of 64. In the 60-64 group, more than 40 percent of households hold high-yield assets. After retirement, the share of households with high-yield assets starts to decrease, falling to around 35 percent for households that are 80-85 years old. Figure 1 also shows a rising share of

95For annuity, trust, and retirement accounts, the SCF data provides a variable for the share of the amount invested in stocks. However, it does not provide the amount of these accounts invested in other kinds of potentially high-yield assets.

123 total ﬁnancial wealth invested in high-yield assets over age. For example, households aged 80 to 85 hold 20 percent of their total ﬁnancial wealth in high-yield assets compared to less than 10 percent for households aged 25 to 29.

0.45 0.25

0.4

0.35 0.2

0.3 0.15 0.25

0.2 0.1

YIELD ASSETS YIELD 0.15 FINANCIAL ASSET FINANCIAL

0.1 0.05 0.05 HIGH-YIELD ASSETS SHARE OF TOTAL TOTAL SHARE ASSETS OF HIGH-YIELD

SHARE HOUSEHOLDS OF HOLDING HIGH- 0 0 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-85

5-AGE BIN population share share of high-yield assets

Left axis: Share of households holding high-yield assets that are more than 1 percent of their total assets across age groups. (grey bars) Right axis: Share of high-yield assets as a proportion of total ﬁnancial asset across age groups. (blue line) Source: 2004 SCF data

Figure 3.1: Share of Households holding High-yield Assets

Figure 2 describes the share of households holding high-yield assets over wealth deciles as well as the share of high-yield assets as a fraction of total ﬁnancial assets.

Less than 10 percent of households in the ﬁrst wealth decile hold high-yield assets

124 compared to around 80 percent of households in the tenth wealth decile. Participa- tion in high-yield asset markets increases with the level of household wealth. Wealthy households invest more of their wealth in high-yield assets. For example, the top 10 percent wealthiest households hold more than 40 percent of their ﬁnancial assets in high return assets compared to less than 10 percent for the bottom 10 percent of households.

0.9 0.45

0.8 0.4

0.7 0.35

0.6 0.3

0.5 0.25

0.4 0.2

YIELD ASSETS YIELD 0.3 0.15 FINANCIAL ASSET ASSET FINANCIAL 0.2 0.1

0.1 0.05 HIGH-YIELD ASSETS SHARE OF TOTAL TOTAL OF SHARE ASSETS HIGH-YIELD

SHARE OF HOUSEHOLDS HOLDING HIGH- HOLDING HOUSEHOLDSOF SHARE 0 0 1 2 3 4 5 6 7 8 910 WEALTH DECILE population share share of high-yield assets

Left axis: Share of households holding high-yield assets exceeding 1 percent of total assets. Right axis: Share of high-yield assets as a proportion of total ﬁnancial assets. (Source: 2004 SCF data)

Figure 3.2: High-yield Assets over Wealth

125 Table 2 describes high asset holdings as a share of net worth over age instead of total ﬁnancial wealth as in the previous ﬁgures. There remains an increasing share of high return assets until retirement. We decompose these into stocks, corporate bonds and GSEs, government bonds and other assets such as hedge funds and balanced funds.96 Across age, high yield assets are predominantly stocks.97 As we might expect, older households, over 75, move to relatively safer assets, in particular government bonds. Nonetheless, their share of stocks remains high at 88 percent.

Age of high corporate & stocks govt. bond other the head wealth GSE bond all 6.67 91.80 1.36 3.28 3.56 25-34 1.69 93.06 0.68 2.21 4.04 35-44 6.34 92.67 0.42 2.08 4.82 45-54 6.86 93.32 1.22 1.41 4.06 55-64 9.95 92.86 1.30 3.30 2.54 65-74 9.16 87.76 4.00 5.46 2.78 75-85 8.09 88.39 1.22 7.90 2.48

Table 3.2: Portfolio Share of High-Yield Assets over Age

Table 3 shows the corresponding facts over wealth deciles. As Figure 2, there is still an increasing share of total wealth invested in high return assets. Moreover, high

96GSEs are bonds issued by government sponsored enterprises such as Fannie Mae and Freddie Mac. 97Note that we do not consider housing wealth as a high-yield asset. Once included, housing would become the biggest fraction of high-yield assets. However, the model economy abstracts from housing market and illiquidity in high-yield assets. Rather, high-yield assets are subject to idiosyncratic return risk which is closer to feature of stocks. See Kim (2017) for the implications of illiquidity in high-yield assets on the wealth distribution.

126 yield assets are overwhelmingly stocks. Throughout the section, we show that there

exists signiﬁcant heterogeneity in household portfolios.

Wealth high corporate & stocks govt. bond other Decile wealth GSE bond 1 -6.56 92.46 0.98 4.49 2.07 2 3.76 100.0 0.00 0.00 0.00 3 4.47 93.19 0.34 2.98 3.49 4 4.72 94.17 0.31 0.68 4.83 5 3.63 85.15 2.75 4.59 7.50 6 4.14 93.69 0.49 1.64 4.18 7 7.17 97.34 0.40 0.76 1.51 8 8.42 92.31 0.45 4.39 2.85 9 12.72 90.73 2.65 2.82 3.80 10 24.04 89.31 1.79 5.23 3.66

Table 3.3: Portfolio Share of High-Yield Assets over Wealth

3.3 Economic Model

3.3.1 Overview

In the model economy, there are four sets of agents: households, perfectly com-

petitive ﬁnancial intermediaries, ﬁrms, and a government. Household demographic

structure involves J overlapping generations. Each generation has a fraction of µj of the population and the total population is normalized to one. Households survive from age j to j+1 with survival probability ξj. Households value consumption in each period and discount future utility by β ∈ (0, 1). They are endowed with a unit of time in each period. Households enter the labor market at age j = 1, retire at age Jr = 40,

127 and J = 60 is the last age. During their working-life, an agent’s eﬃciency units of la-

bor depends on labor market experience, l(j), and idiosyncratic productivity shocks,

ϵ. After retirement, households receive social security beneﬁts proportional to their

− last earnings shock b(ϵJr 1).

The economy consists of two segmented ﬁnancial asset markets, a high-yield asset

market and a low-yield asset market. A household may gain permanent access to

the high-yield asset market by paying a ﬁxed cost to ﬁnancial intermediaries. At

the start of each period, any household is identified by age, j, assets, a, an idiosyncratic productivity shock, ϵ and its access to the high-yield asset market. For those without such access, they are also identified by a fixed access cost to financial intermediaries, ζ ∈ (0, ζ). This cost is denominated in units of output and drawn from a

time-invariant distribution H(ζ) common across households. Households in low-yield

asset market can pay their ﬁxed cost to access the high-yield asset market before

consumption and saving decisions. Once a household pays its ﬁxed cost ζ, it receives a higher expected return on their asset holdings, E(rh), but experiences idiosyncratic

r interest rate risk, ϵ , on this return. Otherwise, the return on asset remains rl, where rl < E(rh).

Consumption and investment goods are produced by a representative ﬁrm with

aggregate capital K and labor N, through a strictly concave, constant returns to scale

production function F (K,N). Aggregate capital K is the sum of the total capital

held by households in both low and high asset markets, K = qKl + Kh where Ke is

total capital held by households in asset market e ∈ {l, h}. Here, q is a technology pa-

rameter calibrated to match the diﬀerence in return on savings across asset markets.

The common rate of depreciation for these capital stocks is δ ∈ (0, 1). Finally, the

128 government receives tax revenue on labor, capital, and social security income. It also

collects accidental bequests from those who did not survive. Government revenue is

used to ﬁnance social security payments to retirees and its budget is balanced.

3.3.2 A Household in the High-Yield Asset Market

We now describe the behavior of households with existing access to the high-yield

asset market. Let vh(j, a, ϵ, ϵr) represent the expected discounted value of a household

at age j with assets a, an idiosyncratic productivity shock ϵ, and an idiosyncratic

interest rate shock ϵr. Note that households who have already paid the ﬁxed cost gain permanent access to the high-yield asset market. Given a, the household chooses

consumption c, saving a′, and labor supply, n. The problem of a working household is { } h r Nϵ Nr r h ′ r v (j, a, ϵi, ϵs) = max u(c) + βξjΣm=1πimΣt=1πstv (j + 1, a , ϵm, ϵt ) (3.1) c,a′,n subject to

′ ≤ − r − c + a (1 + (1 τa)rhϵs)a + (1 τn)wl(j)ϵin

a′ ≥ a, c ≥ 0, n ∈ [0, 1]

where a is the borrowing limit and l(j) is labor market experience.98 Capital income

and labor income are taxed at rates τa and τn respectively.

After retirement, the household does not work n = 0 and receives a social security

− beneﬁt payment b(ϵJr 1). The problem of a retiree is { } h Jr−1 r Nr r h ′ Jr−1 r v (j, a, ϵ , ϵs) = max u(c) + βξjΣt=1πstv (j + 1, a , ϵ , ϵt ) (3.2) c,a′

98Following Heathcote, Storesletten, and Violante (2010), we assume a return to age as a proxy for labor market experience.

129 subject to

′ − ≤ − r − Jr 1 c + a (1 + (1 τa)rhϵs)a + (1 τn)b(ϵ )

a′ ≥ a, c ≥ 0

3.3.3 A Household in the Low-Yield Asset Market

Households start their life without access to the high-yield market. At the start of each period, a household draws a ﬁxed ﬁnancial intermediary cost, ζ, and de- cides whether to pay this cost, and gain access to the higher return market, prior to consumption and saving decisions. Let vl(j, a, ϵ) represent the value function of a household in the low-yield asset market. ∫ ζ { } l l l v (j, a, ϵi) = max vl(j, a, ϵ), vh(j, a, ϵ, ζ) H(dζ) (3.3) 0

l l where vl is the value function of a household saving in the low return asset and vh is the value function of a household saving in the high return asset.

Savings in low-yield asset

l Given asset holdings a and idiosyncratic productivity shock ϵ, vl solves a working household’s problem { } ∑Nϵ l l ′ vl(j, a, ϵi) = max u(c) + βξj πi,lv (j + 1, a , ϵl) (3.4) c, a′, n l=1 subject to

′ c + a ≤ (1 + (1 − τa)rl)a + (1 − τn)wl(j)ϵn

a′ ≥ a, c ≥ 0, n ∈ [0, 1]

130 Savings in high-yield asset

l Let vh(j, a, ϵ, ζ) represent the value function of a household who pays a ﬁxed ﬁnancial intermediary cost ζ in the current period and saves using the high return

l asset. vh solves a working household’s problem { } l Nϵ Nr r h ′ r vh(j, a, ϵi, ζ) = max u(c) + βξjΣm=1πimΣt=1π0tv (j + 1, a , ϵm, ϵt ) (3.5) c, a′, n

subject to

′ c + a ≤ (1 + (1 − τa)rl)a + (1 − τn)wl(j)ϵn − ζ

a′ ≥ a, c ≥ 0, n ∈ [0, 1]

A retiree’s problem, in the low-yield asset market, is the analogue to (3.2) above.

3.3.4 Recursive Equilibrium

We define stationary recursive equilibrium. The distribution of households varies over age, wealth, labour productivity and access to the high-yield asset market. Let { } { } J = 1,...,J represent the set of indices for household age and E = ε1, . . . , εNε { } r r r define the support for household shocks to earnings. E = ε1, . . . , εNr define the support for an idiosyncratic interest rate shock and households wealth a ∈ A = [a, ∞).

Lastly, let Z = (0, ∞) be the space for ﬁxed costs drawn by households. The product space, Sl = J × A × E describe the space for the distribution of households in the low-yield asset market at the start of the period before the realisation of ﬁxed costs.

These costs of access to the high-yield asset market are relevant only for households in the low-yield asset market. Likewise, the product space, Sh = J × A × E × Er

131 describe the space for the distribution of households in the high-yield asset market.

Deﬁne Se as the Borel algebra generated by the open subsets of Se where e ∈ {l, h}.

We deﬁne ψe : Se → [0, 1] as a probability measure for households in each asset market e ∈ {l, h}.

Households of age 1 all begin with a0 ≥ 0 and an initial productivity drawn

0 ∼ 2 { }Nε from π log N(0, σπ), the invariant distribution for πi,j i,j=1. We assume all

such households are in the low-yield asset market. Let µ1 be the number of initial

households with age j = 1, their distribution is described by

l 0 ∈ ψ (1, A, εi) = πi µ1 iﬀ a0 A. (3.6)

In subsequent periods, j + 1, j = 1,...,J, the distribution of households in the

high asset market is given by the following. The ﬁrst term describes low-asset market

households that pay their ﬁxed cost draws, draw a labour productivity of εm at the

start of the next period, and save a level of wealth in the measurable set A. The

second term involves existing high asset market households of age j,

∫ ∑Nε h r l ψ (j + 1, A, εm, εs) = ξj πim ψ (j, da, εi) H (3.7)(dζ) | l ∈ i=1 {(a,εi,ζ) g (j,a,εi,ζ) A and χ(j,a,εi,ζ)=1} ∫ ∑Nε ∑Nr r h r +ξj πim πst ψ (j, da, εi, εt ). { r | h r ∈ } i=1 t=1 (a,εi,εt ) g (j,a,εi,εt ) A The distribution of households in the low return asset market of age j + 1, j =

1,...,J − 1, is given by

∫ ∑Nε l l ψ (j + 1, A, εm) = ξj πim ψ (j, da, εi) H (dζ) | l ∈ i=1 {(a,εi,ζ) g (j,a,εi,ζ) A and χ(j,a,εi,ζ)=0} (3.8)

132 ( ) l h l l l h l h A recursive competitive equilibrium is a set of functions v , v , vl, vh, g , g , c , c , χ and prices (rl, rh, w) such that: ( ) l h l l l × × × → i v , v , vl, vh solve (3.1)-(3.5), g : J A E Z A is the associated optimal policy for savings by a household that begins the period in the low

asset market, gh : J × A × E × Er → A is the associated optimal policy that

l attains the maximum in (3.1) and (3.2), c : J×A×E×Z → R+ is the associated

optimal policy for consumption by a household that begins the period in low

h r asset market, c : J × A × E × E → R+ is the associated optimal policy for

consumption by a household that begins the period in high asset market, and

χl : J × A × E × Z → {0, 1} is the decision rule for paying the ﬁxed cost of

access to the high-yield asset market.

ii Markets clear ∫ ∑J ∑Nϵ l Kl = aψ (j, da, εi) j=1 i=1 A ∫ ∑J ∑Nϵ ∑Nr h r Kh = aψ (j, da, εi, εt ) j=1 i=1 t=1 A

C + δK = F (K,N, ) ∑ ∑ ∑ {∫ ∑ ∫ } Jr−1 J Nϵ l l Nr h h r where N = j=1 µj and C = j=1 i=1 A c ψ (j, da, εi) + t=1 A c ψ (j, da, εi, εt ) . iii Government budget is balanced:   ∫ ∑J ∑Nϵ ∑Nr  r h r  τnwN + τa rlKl + rh εt aψ (j, da, εi, εt ) j=1 i=1 t=1 A {∫ ∫ } ∑Jr ∑Nϵ ∑Nr − l h r = (1 τn) b(εi) ψ (j, da, εi) + ψ (j, da, εi, εt ) j=1 i=1 A t=1 A iv Prices are competitively determined.

rl = qD1F (K,N) − δ

rh = D1F (K,N) − δ

w = D2F (K,N)

133 3.4 Calibration

3.4.1 Parameters

We set the length of a period to one year, and assume that households begin

working at age 25. Given this, we set J = 60 and assume that retirement is at age 65, implying Jr = 40. Survival probabilities {ξj} are based on 2004 U.S. Life Tables of the

National Center for Health Statistics. We assume that the period utility function is

iso-elastic and set the inverse of the elasticity of intertemporal substitution, σ = 1.5.

In the U.S., social security payments are paid based on the average of the highest

35 years of earnings. In the model, calculating average earnings requires one more

state variable. So, we proxy the history of earnings over a workers life-cycle using

J −1 − Σ r l(j) − the level of earnings in the last working period. Thus, b(ϵJr 1) = bw j=1 ϵJr 1, Jr−1 where b is the replacement rate. Here, b is chosen to match the replacement rate of 45 percent of average pre-tax earnings reported by social security administration.

Following Domeij and Heathcote (2004), labor and capital income tax rates are 27 and 40 percent respectively.

The subjective discount factor, β = 0.98, is chosen to match the capital to output

ratio of 3.0. We assume that the real interest rate paid on low return assets is zero

percent, broadly consistent with evidence from the Flow of Funds of the real return

to liquid assets such as checking and savings accounts and other components of M1,

rl = 0. Next, we chose the technological parameter q to imply the economy-wide

real interest rate of 5 percent when rl = 0. In the present analysis, borrowing is not

allowed, a = 0. This implies 4.2 percent of households holding zero or negative net

worth compared to 7.6 percent in the SCF.

134 In our present analysis, we assume that the distribution of fixed costs required to 1 access the high yield asset market follows a generalized beta distribution, Beta(γ, ), γ with support [0, ζ]. The parameters for distribution of financial intermediary costs are chosen to match the share of households investing in high-yield assets. This implies that roughly 1/3 of households save using high return assets. This is consistent with our findings from the SCF, reported in section 2, on the share of households holding more than 1 percent of their total assets as high yield assets. We interpret such households as having access to high yield assets, and we view the return on investment in the sector as representing the mean return for any portfolio that includes such assets.

3.4.2 Earning Shocks Estimation

We estimate labor market experience and the earning shock process using PSID

data between 1968 and 2011. The PSID data is a longitudinal survey of US individ-

uals and families conducted annually from 1968 to 1997, and biennially since 1997.

The original 1968 PSID sample combines the Survey Research Center (SRC) and

the Survey of Economic Opportunities (SEO) samples. We use the U.S. population

representative SRC sample.

Household earnings are deﬁned as the sum of the annual earnings of the head of

household and spouse. Total annual earnings include all income from wages, salaries,

bonuses, overtime, commissions, professional practice and the labor part of farm and

business income. We select households with no missing values for education and self-

employment status where: 1) the head of households’ age is between 25 and 59 years

135 old, 2) neither the head or spouse has positive labor income but zero annual hours

worked, 3) the hourly wage is not less than half of the minimum wage, and 4) income

is not from self-employment.99 Earnings are deﬂated with the CPI and expressed in

2013 dollars. After the sample selection, there are only 23 top-coded observations out of 42,127 observations.100 Following Autor and Katz (1999), we multiply all top-coded

observations by a factor of 1.5 times the top-coded thresholds.

Let yi,j,t be the earnings of household i with head at age j in year t. We run an

OLS regression of log earnings on time dummies; an interaction term with education,

2 101 edui, time dummies, labor market experience, θ; and experience-squared, θ . In

the absence of modeling education, the estimated earnings process may understate

the earnings dispersion in the economy. See Kim (2017) for an exploration of the role

of college wage premia on earnings and wealth inequality.

2 log yi,j,t = βt,0 + βt,1edui + β3θ + β4θ + rbi,j,t

Recall that eﬃciency units of labor depend on both labor market experience,

2 l(j) = e(β3θ+β4θ ), and idiosyncratic productivity shocks. Figure 3 shows the estimated potential market experience function. Labor market experience increases earnings through the ﬁrst 27 years when earnings rise 71 percent relative to their initial level.

Thereafter, total earnings fall in experience until retirement.

99We also estimated the earnings process using a sample that includes the income of the self- employed. The estimated earnings process changes a little (see below). 100All these top-coded observations are for head of households before 1983. No spouse’s earnings is top-coded. 101Labor market experience is measured as age minus years of schooling minus 5. In years missing the variable for years of schooling, we proxy years of schooling for the individuals with a college degree as 16 and for the individuals without a college degree as 12.

136 1.9

1.7

1.5

1.3

l(j) 1.1

0.9

0.7

0.5

0.3 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 1 3 5 7 9

experience

Figure 3.3: Labor Market Experience (PSID data 1968-2011)

The regression residuals rbi,j,t are assumed to be the sum of idiosyncratic productiv-

102 ity shocks, log ϵi,j,t, and measurement error, vei,j,t. Idiosyncratic productivity shocks consist of both a persistent component, log ϵp, and transitory component, log ϵv. To be speciﬁc,

p v log ϵi,j,t = log ϵi,j,t + log ϵi,j,t

p p log ϵi,j,t = ρ log ϵi,j−1,t−1 + ηi,j,t

∼ 2 v ∼ 2 where ηi,j,t N(0, σηt ) and log ϵi,j,t N(0, σvt ).

102Given the similarity in the estimates of earnings process to wage process estimated in Kim (2015), we use French (2002)’s estimate of a variance of measurement error in log hourly wages of 0.02.

137 Following Heathcote, Storesletten, and Violante (2010), we estimate year-varying

{ 2 2 } { } shock variances σηt , σvt , the persistence of the shock ρ , and the variance of per-

2 sistent shock for initial age σπ using minimum distance methods. We use survey data from 1968 to 2011, but only estimate the variances up to 2009 because of the ﬁnite sample bias at the end of a sample period. As Heathcote et al. (2010) point out, although the variance of the persistent shock for the missing years can be theoretically pinned down by the available information from adjacent years, the resulting estimates for missing years are downward-biased because of insuﬃcient information. Therefore, we follow their approach for the estimates in the missing years by taking the weighted average of the two adjacent years. We estimate a total of L = 86 parameters. The parameter vector is denoted by PL×1.

The theoretical moment is deﬁned as

j P mt,t+n( ) = E(ri,j,tri,j+n,t+n) which is the covariance between earnings of individuals at age j in year t and t + n.103 To calculate the empirical moments, we group individuals into 44 year and 26 overlapping age groups. For example, the ﬁrst age group contains all observations between 25 and 34 years old and second group contains those between 26 and 35 years old. The empirical moment conditions are

j − j P mˆ t,t+n mt,t+n( ) = 0

103The closed form of theoretical moment is { min(∑j−1,t) n 2 2(i−1) 2 2 1 if t = t + n E(ri,j,tri,j+n,t+n) = ρ [ρ min(j − 1, t)σ + ρ σ + 1σ ] , 1 = π ηt−i+1 vt 0 otherwise i=0 .

138 ∑ j 1 Ij,t,n wherem ˆ = rbi,j,trbi,j+n,t+n and Ij,t,n is the number of observations of t,t+n Ij,t,n i=1 age j at year t existing n periods later.

The minimum distance estimator solves the following problem

min[ ˆm − m(P)]′[ ˆm − m(P)] P

where ˆm and m are vectors of empirical moments and theoretical moments with dimension 8, 342 × 1. Note that we use the identity matrix as a weighting matrix.

Table 4 summarizes the resulting estimates for earning shock process. The esti-

mated persistence of earning shock, ρ, is 0.9825 and the age 1 variance of the persistent

2 earning shock, σπ, is 0.1546. The benchmark model economy is calibrated to 2004 with the smoothed estimates of the persistent shock variance 0.0574 and transitory

shock variance 0.1913.104

2 2 2 year ρ σpt σvt σπ 2004 0.983 0.057 0.191 0.155

Table 3.4: Earnings shock estimates

104We estimated earnings process with a sample that includes the income from self-employment. The estimated persistence of earnings is 0.9834 and the initial variance of persistent earning shock is 0.1553. The persistent shock variance estimate is 0.056 and transitory shock estimate is 0.1891. The estimates are essentially unchanged by the inclusion of income from self-employment.

139 3.4.3 Idiosyncratic return risk for the high-yield asset

We assume that an idiosyncratic interest rate shock, εr, follows two state Markov

r ∈ { r r } r r r 1 chain with the support ε εl , εh where εl < εh. Here, we restrict εl = r and the εh transition probability matrix is as follows:

r − r π11 1 π11 − r r 1 π22 π22

Towards the persistence and variability of this Markov Chain, we ﬁrst set its

r r persistence equal to the persistence of the earnings shock, π11 = π22 = 0.9825. In sensitivity analysis below, we show the results of the model are not sensitive to the persistence parameter. We then choose the variability of the shock to match an empirical estimate of the persistence of wealth growth rates of -0.47 from our constructed

PSID wealth data.

To measure the persistence of wealth growth rates, we ﬁrst construct a panel of wealth data using biennual waves of the PSID over 2003 to 2007. In the PSID, the following categories of wealth are available: 1) business and farm equity, 2) transaction accounts, 3) equity in real estate and vehicles, 4) stocks 5) bonds, 6) IRA, and 7) debt.105 Next, to measure the persistence of wealth growth rates, we run a regression of ln(at+2) − ln(at+1) on ln(at+1) − ln(at), where at is the level of wealth at time t.

105The PSID constructed wealth data is very similar to the wealth distribution in the SCF.

140 3.5 Results

3.5.1 Cross-sectional distribution

In Table 5 we compare several moments of the distribution of wealth for our benchmark model economy to those in the 2004 SCF.

1% 5% 10% 50% 90% ≤ 0 Gini 2004 SCF 28.7 50.5 63.5 96.6 100 8.5 0.77 Benchmark 15.8 40.5 56.5 94.4 100 4.2 0.71 Table 5 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households and the wealth Gini coeﬃcient.

Table 3.5: Wealth distribution

Our benchmark parameter set implies that, across all ages, 33 percent of households invest in the high-yield asset market. The higher return on wealth for such households increases inequality in the economy, and, as a result, the Gini coeﬃcient is 0.71, essentially close to the 2004 empirical counterpart. As another measure of inequality, the model reproduces a substantial fraction of the concentration of wealth seen among the top percentiles of households. The top 10 percent of households hold

56 percent of wealth in the model, compared to 64 percent in the 2004 SCF data.

The skewness in the distribution of wealth is less pronounced in the top percentile.

The top 1 percent of households hold 29 percent of wealth in the data, the model counterpart is 16 percent. Despite an absence of strong motives for wealth accumulation among the wealthiest households, as seen in the entrepreneurial borrowing

141 with collateral constraints model of Cagetti and De Nardi (2004) or the income risk

model of Castenada et al. (2003), our model economy is able to explain a substantial

fraction of wealth inequality.

Table 6 summarizes the income distribution both in the model economy and in the

2004 SCF.106 Though our earnings process is estimated from the PSID, the benchmark economy explains more than 85 percent of the income inequality in the SCF. Also, the model reproduces a level of skewness in the distribution of income consistent with the data.

1% 5% 10% 50% 90% Gini 2004 SCF 12.8 26.2 37.2 81.8 98.8 0.49 Benchmark 12.0 30.0 41.6 74.9 95.8 0.42 Table 6 shows the share of income held by the top 1, 5, 10, 50 and 90 richest households and the income Gini coeﬃcient.

Table 3.6: Income distribution

Overall, we see that the model economy generates an empirically reasonable dis-

tribution of income and explains a signiﬁcant portion of wealth inequality. This is

surprising given the diﬀerence in the population of wealthy households in the SCF,

which provides the data on wealth, and the scarcity of such households in the PSID,

which is the basis of our income process.107 Part of the reason for this success is our

106Income includes both labor and capital income. 107See Heathcote, Perri and Violante (2010) for a discussion of these issues.

142 exclusion of the self-employed both in our earnings estimation and in our calculation of moments from the wealth distribution.

Figure 3.4: Distribution of Wealth

Figures 4a and 4b show the distribution of wealth across households saving in the low return asset market and those saving in the high return asset market. While both sets of households have identical stochastic processes for earnings, the low return on savings (zero percent), in the former leads to compression in the distribution of

143 wealth. There is considerably more dispersion in wealth among households in the high-yield asset market where the effect of persistent earnings shocks on wealth are amplified through a higher rate of return on savings (five percent). Both distributions are concentrated in lower levels of earnings, a reflection of the considerable skewness in our log-normal earnings process.

We now examine asset market segmentation, and its relation to households wealth and age. As we reported in Figure 1, the SCF for 2004 shows the share of households holding high yield assets to be monotonically increasing over age until 65. In our model, Figure 5 captures this rise, and importantly, the largest share of households over age, around 0.40, is close to the data. It is important to remember these shares were not calibration targets. However, our model without reversibility of asset markets, does not exhibit the decline in participation after retirement seen in the data.

Also, the assumption that everyone starts their life-cycle in low-yield asset market underestimates participation by younger cohorts relative to the data.

As seen in Figure 5, participation in the high-return market increases with the years a households has been in the labour force. Initially, all households begin in the low asset market. Thereafter, as they accumulate wealth, the probability that they will pay any ﬁxed cost increases and thus their probability of having access to high returns on savings. Moreover, as households repeatedly sample ﬁxed cost draws over time, the probability of a relatively low cost that they will pay rises. Both forces lead to a rise in the fraction of households, as they gain experience, with access to the high return asset. Nonetheless, as households age, their lifetime compounded yield from higher returns decreases. Thus the threshold cost they are willing to pay for access, given any level of wealth, falls. As wealth accumulates slowly in the low return asset

144 0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-85

5-Age Bin Data Model

Figure 3.5: Share of Households Holding High-Yield Assets over Age

market, this leads to an eventual leveling oﬀ of the share of households in the high return market. Across ages, 1/3 of households save in the high return asset. This proportion is 0 for the youngest households, and levels oﬀ around 40 percent for the oldest.

Figure 6 displays the fraction of households in the high-yield asset market across levels of wealth.

Wealthier households tend to invest in the high yield asset market. First, higher levels of wealth make it more likely that a household will choose to pay any ﬁxed cost draw and save in the high-yield asset market. Second, for households already in this market, their higher returns to savings drive more rapid wealth accumulation compared to households saving using the low return asset.108 Given that the rising

108This is qualitatively consistent with the ﬁnding by Vissing-Jorgensen (2002) that participation in high return asset markets is more likely for wealthier households.

145 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1 2 3 4 5 6 7 8 910 Wealth Decile Data Model

Figure 3.6: Share of Households Holding High-Yield Assets over Wealth

participation by age and wealth are natural results of our environment, we emphasize that our economy is consistent with the cross-sectional distribution of households across low- and high-yield asset markets.

3.5.2 The implications of endogenous asset market segmentation

To see the importance of endogenous asset market segmentation, we consider several alternative economies. The ﬁrst exogenous segmentation economy has an initial share of households in the high asset market equal to the benchmark share.

With exogenous market segmentation, wealth inequality as well as the concentration of wealth drops. This is because exogenous market segmentation abstracts from the selection of households who move to the high asset markets and the diﬀerences, across

146 households, in age when they begin to invest in high yield assets. Both eﬀects lead

to less inequality, generating a wealth Gini of 0.67.

Asset market segmentation increases wealth inequality in our economy. We es-

tablish this by eliminating segmentation and setting a common return on savings for

all households. This return is equal to the wealth-weighted average return, 5.0, in our benchmark model. As seen in the last row of Table 7, there is a 9 percentage point reduction in the wealth Gini which falls to 0.62, and the share of wealth held by the top 10 percent of households decreases by 10 percentage points compared to the benchmark economy.

1% 5% 10% 50% 90% ≤ 0 Gini 2004 SCF 28.7 50.5 63.5 96.6 100 8.5 0.77 Benchmark 15.8 40.5 56.5 94.4 100 4.2 0.71 Exogenous seg 13.9 36.1 51.3 93.2 100 4.2 0.67 common market 11.7 31.5 46.0 90.0 99.8 2.3 0.62 Table 3 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the wealth Gini coeﬃcient, and the share of households in the high-yield asset market.

Table 3.7: Importance of Endogenous Asset Market Segmentation

While asset market segmentation, for our benchmark model calibrated to 2004

earnings data, increases inequality, there are several mechanisms in our quantitative

overlapping generations model, involving its demographic structure, that reduce its

impact. We begin our exploration of the determinants of the distribution of wealth

with a study of households’ savings behaviour over age and across sectors.

147 Figure 3.7: Life-cycle saving rates and participation in high-yield asset market

Deﬁning savings rates as the change in assets over cash on hand, Figure 7 shows average savings rates over years in the labour force, for households in the low return asset market and the high return asset market in the top panel and the participation rate in the bottom panel. The savings rate over working age declines monotonically for households. For example, savings rates at age 27 for households in the high return asset market is 12 percent falling to 2 percent in the last period in the labour force at age 59. Given positive savings rates before retirement, wealth is accumulated, and inequality generated, during years of employment. Over working ages, the average

148 savings rate for households in the low asset market is 2.0 percent; households in the high asset market have an average savings rate of 6.3 percent. Thus, households in the latter market respond to higher rates of return with higher savings rates.

The higher average savings rate, over working ages, of households using the high

yield asset is in response to higher rates of return on savings earned in that market.

In our alternative model without asset market segmentation, this responsiveness of

savings rate results in an average over working ages of 4.8 percent. Compared to

the 2.0 percent savings rate in the low return asset market of the segmented markets

model, the increased savings in the alternative model increases wealth accumulation

over households’ working lives, and drives inequality even in the absence of asset

market segmentation. As a result, the Gini coeﬃcient without market segmentation

remains high.

Second, these savings rates fall with age. Further, as seen in the bottom panel of

ﬁgure 7, households in the high return asset market tend to be older. As a result, the

life-cycle proﬁle of savings rates reduces the accumulation of wealth by households

with access to high yields on savings as the average age of households, in the high

asset market, is older. This is the second mechanism, arising from our model’s real-

istic demographic structure, which dampens the eﬀect of asset market segmentation

on wealth inequality. While high asset market households receive a higher return on

savings, their wealth grows more slowly than it would in the absence of savings rates

that decline with age.

149 3.5.3 Sensitive analysis to interest rate risk

We now turn to sensitivity analysis to different specifications for our interest rate shocks in the high return asset. First, the top panel of Table 8 shows that when interest rate shocks are twice and half of the mean return, the autocorrelation of biennual log difference in wealth is -0.47 while it is -0.471 in the data. (Benchmark case) This is for an interest rate shock process whose persistence is set to be the same as that of the persistent earnings shock. In contrast, this autocorrelation is too low when shocks are i.i.d. (No persistence) It is higher than the data (-0.42) when the variability of shocks is twice as large (High risk), and the autocorrelation is too low when there is no interest rate risk (Low risk). Thus, this is one reason to assume persistent interest rate shocks.

Persistence of biennial wealth growth rate Data Benchmark No persistence High risk No risk -0.471 -0.470 -0.556 -0.415 -0.563

Wealth distribution 1% 5% 10% 50% 90% ≤ 0 Gini Benchmark 15.8 40.5 56.5 94.4 100 4.2 0.71 No persistence 15.1 39.0 55.2 94.2 100 4.1 0.70 High risk 18.1 45.0 61.5 95.5 100 5.0 0.75 No risk 14.7 38.1 53.9 93.8 99.9 4.0 0.69

Table 3.8: Sensitive analysis to interest rate risk

In terms of the eﬀects of interest rate shocks on wealth inequality, in the bottom panel of table 8, we see that when there is no interest rate risk, the wealth Gini drops

150 by two percentage points and concentration of wealth also falls a little. Thus while interest rate shocks allow us to reproduce the autocorrelation of wealth growth rates

(top panel benchmark persistence of -0.47), they do not play a signiﬁcant role in generating inequality. When we consider i.i.d interest rate shocks instead of persistent shocks, there is little change in the distribution of wealth from the benchmark economy. Nonetheless, the autocorrelation of wealth growth rates, seen in the top panel, is signiﬁcantly more negative than the data.

If we double the variability of interest rate shocks, the wealth Gini increases to

0.75 and the concentration of wealth is also closer to the data. However, we calibrated the variability of interest rate shocks to match the persistence of biennial wealth growth rate from our constructed panel of wealth data, given our assumption that the persistence of these shocks was the same as that of the earnings process.

Table 8 indeed shows that the benchmark economy matches the persistence of biennial wealth growth rate of -0.47 compared to -0.42 for the model with more variable interest rate shocks.

3.6 Concluding Remarks

We develop a rich quantitative overlapping generations model where households face earnings and interest rate risk and asset markets are segmented. We estimate the stochastic process for earnings from the PSID, and evaluate the eﬀect of asset market segmentation on the distribution of wealth and income. Importantly, market segmentation is endogenous in our framework. Each period, households face a discrete choice of whether or not to pay a ﬁxed cost and gain an access to a high yield asset

151 market. This endogeneity has important implications for the eﬀect of asset market segmentation on wealth inequality. It implies that wealthier, older households tend to invest in the high yield asset market. A higher return on this asset leads wealthy households to accumulate their wealth fast, driving more dispersion in wealth. Indeed, eliminating market segmentation increases the wealth Gini gap between model and data by one and a half fold.

We measure asset market segmentation using SCF data on the fraction of households holding high return assets. Our model is consistent with our empirical ﬁndings that this share of households participating in high yield asset markets is rising with wealth and with age.

Given a large empirical work showing a positive relationship between the return on savings and education levels, in the next step of our research, we will introduce college educational choice and variable labour supply. Moreover, with separately estimated wage processes for non-college and college graduates including between- and within- group dispersion, college graduates have higher income to accumulate higher level of wealth relative to non-college households.

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158 Appendix A: Appendix to Chapter 1

A.1 Data appendix

A.1.1 2007-2009 SCF panel data

The SCF is household triennial data survey conducted by the Federal Reserve

Board. The SCF provides detailed information on assets and debts of households.

The SCF also employs a list sample design based on the IRS data to provide disproportionate sampling of wealthy households. Historically, the SCF had panel data from 1983 to 1989 and recently released two-year panel data for 2007 and 2009.

Net worth is deﬁned as total assets minus total debt. Total assets include both

financial and non-financial assets. Total financial assets include transaction accounts, certificates of deposit, directly held pooled investment funds, savings bonds, directly held stocks and bonds, cash value of life insurance, quasi-liquid retirement accounts, and other managed and miscellaneous financial assets. Total non-financial assets include vehicles, residential property, net equity in non-residential real estate, business equity, and other miscellaneous non-financial assets. Total debt consists of debt secured by residential property, credit card balances since last payment, installment loans, and other debts. I select households with a head who is between 25 and 85 years old and not self-employed. All the variables are deflated and expressed in 2013 dollars. Full sample weights are used.

159 A.1.2 PSID data

The PSID is a longitudinal survey of a sample of US individuals and families conducted annually from 1968 to 1977, and biennially since 1997. The original 1968

PSID sample combines the Survey Research Center (SRC) and the Survey of Eco- nomic Opportunities (SEO) samples. I use the U.S population representative SRC sample.

For greater consistency with the SCF, I select households with a head who is between 25 and 85 years old and drop the self-employed. I drop a sample observation if income is positive but annual hours worked is zero. If income is top-coded, I multiply it by a factor of 1.5 of the top-coded threshold following Katz (1999). All variables are deﬂated and expressed in 2013 dollars using IPUMS-CPI.

The PSID provides disaggregated data on wealth since 2003, consisting of business and farm equity, transaction accounts, equity in real estate and vehicles, stocks, bonds, IRA, and debt. As shown in Table 1, the distributions of wealth in the PSID are comparable to those in the SCF. Table A1 also summarizes averages in the PSID and SCF.

The PSID provides data on households’ consumption. In addition to food and housing, the PSID included items on transportation, health care, education, utilities, and child care since 1999. In 2005, additional items such as household furnishing and equipment, clothing and apparel, trips and vacations, and recreation and entertain- ment were added.

160 Data SCF PSID year 2007 2009 2007 2009 Sample size 13,085 13,085 481 481 Age of head 49 51 47 49 Total wealth 395,750 325,060 488,110 317,240 Labor income 55,477 54,820 65,655 64,023 Notes : All variables are expressed in 2013 dollars. For the PSID, drop three samples with wealth less than negative 99 million dollars. (Source: 2007-2009 PSID and SCF)

Table A.1: Summary statistics of data

I construct total expenditure in the PSID as the sum of nondurable goods and services.109 The PSID measures total spending on each item for the family. Since each item has a diﬀerent reporting time unit, I adjust it to an annual measure.

Reporting time unit varies by samples for food delivered, food eaten out, and food at home. Approximately half of respondents reported it weekly while the rest report it monthly or biweekly. I measure annual spending on these items based on individual reported time units. Following Krueger et al. (2015), I imputed the amount of rental services from home owners by multiplying the value of the main residence by

4 percent. Imputed rent and property taxes are included in expenditure on housing to be consistent with the BEA measure. Given that reported income is earned in the preceding year, there may exist time inconsistency between consumption and income.

In Table A2 and A3, I compare composition of total expenditure in the PSID to that in the BEA to comprehend the representativeness of the former micro data for

109Durable goods include motor vehicles and parts, furnishings and equipment, recreational goods and vehicles, and other durable goods.

161 macro aggregates.110 I used NIPA Table 2.3.6 Real Personal Consumption Expendi- tures in chain 2009 dollars (seasonally adjusted).111 Since the BEA measures current year consumption while the PSID reporting time unit varies by item, I take the average of the current and preceding years of the expenditure in the BEA to make the time unit closer to the expenditure measured in the PSID. Table A2 shows that the

PSID aggregate consumption accounts for around 50 percent of total expenditure in the BEA. In Table A3, spending on each item as a share of total expenditure in the

PSID is broadly comparable to that in the BEA. This suggests that, while micro data captures less expenditure than aggregate data in total, the dynamics in consumption at individual level may be explained by micro data. For instance, Krueger et al.

(2015) shows that the growth rates of the PSID aggregate closely follow BEA total spending, providing further support for using micro consumption data to measure individual growth rates.

110Total spending in the recent PSID is comparable to that in the CEX. Indeed, Charles et al. (2006) ﬁnd that the 2003 PSID covers 72 percent of total expenditure measured in CEX. 111Given the issue raised by chain-weighted dollars, I express PSID variables in 2009 dollars for Table A2 and A3.

162 2005 2007 2009 billions of 2009 dollars BEA PSID BEA PSID BEA PSID Nondurable goods 2,098 1,306 2,208 1,373 2,243 1,278 food 744 869 786 869 776 863 clothing 299 183 321 191 314 185 gasoline 299 254 297 313 284 230 other 753 n/a 814 n/a 821 n/a Services 6,255 3,008 6,592 3,006 6,679 3,059 housing and utilities 1,753 1,444 1,832 1,443 1,871 1,418 health care 1,466 335 1,544 344 1,613 418 transportation 333 268 337 229 306 258 recreation 360 106 385 110 383 135 food services 595 n/a 629 n/a 613 n/a ﬁnancial services and insurance 686 228 731 218 728 194 other services 856 626 895 662 891 638 total 8,353 4,283 8,812 4,363 8,873 4,326 Notes : Variables in the BEA are expressed in chain-weighted 2009 dollars. Variables in the PSID are expressed in 2009 dollars. Other services include chicle care, education, communication and vehicle services.

Table A.2: Composition of total expenditure in the BEA and PSID

163 2005 2007 2009 % of total expenditure BEA PSID BEA PSID BEA PSID Nondurable goods 25.1 30.5 25.1 31.5 25.3 29.5 food 8.9 20.3 8.9 19.9 8.8 20.0 clothing 3.6 4.3 3.6 4.4 3.5 4.3 gasoline 3.6 5.9 3.4 7.2 3.2 5.3 other 9.0 n/a 9.2 n/a 9.3 n/a Services 74.9 70.2 74.8 68.9 75.3 70.7 housing and utilities 21.0 33.7 20.8 33.1 21.1 32.8 health care 17.6 7.8 17.5 7.9 18.2 9.7 transportation 4.0 6.3 3.8 5.3 3.5 6.0 recreation 4.3 2.5 4.4 2.5 4.3 3.1 food services 7.1 n/a 7.1 n/a 6.9 n/a ﬁnancial services and insurance 8.2 5.3 8.3 5.0 8.2 4.5 other services 10.3 14.6 10.2 15.2 10.0 14.8 total(billions of dollars) 8,353 4,283 8,812 4,363 8,873 4,326 Notes : Variables in the BEA are expressed in chain-weighted 2009 dollars. Variables in the PSID are expressed in 2009 dollars. Other services include chicle care, education, communication and vehicle services.

Table A.3: Spending as a fraction of total expenditure in the BEA and PSID

164 5 40 4.5 30 4 3.5 20 3 10 2.5 0 2

1.5 -10 illiquid wealth share 07 share wealth illiquid

1 rates growth wealth illiquid -20 0.5 0 -30 top 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 2007 wealth deciles illiquid share 07 (as a fraction of asset) illiquid share 07 (as a fraction of wealth) illiquid wealth growth rates

Notes : The average illiquid asset share as a fraction of total asset (blue bars) and as a fraction of net worth (grey bars) in 2007. Exclude samples with negative net worth in 2007 for share measures or with net worth less than 0.1 percent of aggregate wealth. The growth rates of illiquid wealth between 2007 and 2009 (dashed line). Exclude samples with negative values in any sample period for growth rate measures. (Source: 2007-2009 SCF panel)

Figure A.1: Illiquid wealth share and risky wealth growth rate over 2007 wealth deciles

165 3 0

2.5 -5

2 -10 1.5 -15

illiquid share 07 share illiquid 1

-20 rates growh wealth illiquid 0.5

0 -25 25-34 35-44 45-54 55-64 65-74 over 75 2007 age bin illiquid share 07 (as a fraction of asset) illiquid share 07 (as a fraction of wealth) illiquid wealth growth rates

Figure A.2: Illiquid wealth share and risky wealth growth rate over 2007 age groups

166 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 top 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 2007 wealth deciles stocks business equity house residential property non-residential vehicle

Notes : The average share of each illiquid wealth as a fraction of total illiquid wealth. Exclude samples with negative or zero risky wealth. (Source: 2007-2009 SCF panel)

Figure A.3: The composition of illiquid wealth in 2007

167 A.2 Equilibrium Ex-dividend Price and Dividends

In this section, I show that deﬁnitions of ex-dividend price and dividend are con-

sistent with equilibrium and imply the aggregate resource constraint. Here, I simplify

the model by abstracting from government and liquid asset market.

Aggregate budget constraint for all households who are adjusting illiquid wealth

is as follows:

′ ≤ − ca + p(z, µ)ka wa(z, µ) + (p(z, µ) + d(z, µ))ka ζa

where xa is the sum of a variable x for all households who are adjusting and wa(z, µ) is the labor income for all active adjustors.

Using deﬁnitions of ex-dividend price and dividend, above aggregate budget con-

straint can be re-written as:

′ ′ ≤ α−1 1−α − − ′ − ca + (1 + Φ1(k , k))ka wa(z, µ) + (αηk n δ Φ2(k , k))ka ζa (A.1)

where k is the aggregate stock of capital.

Likewise, aggregate budget constraint for all households who are non-adjusting

illiquid wealth is:

cn ≤ wn(z, µ) (A.2)

′ α−1 1−α − − ′ − ′ kn = (1 + αηk n δ Φ1(k , k) Φ2(k , k))kn (A.3)

′ ′ ′ ′ Imposing xa + xb = x and Φ1(k , k)k + Φ2(k , k)k = Φ(k , k), equations (4)-(6)

imply the aggregate resource constraint

′ ′ c + k + Φ(k , k) ≤ y + (1 − δ)k − ζa

168 A.3 Numerical Method

A.3.1 Steady state

Stationary equilibria involve finite horizon dynamic programming problems with two endogenous state variables - illiquid wealth, a, and liquid wealth, b. The state space is six-dimensional: age, illiquid wealth, liquid wealth, unemployment shock, persistent and transitory earnings shocks. Solving the model with two endogenous state variables involves a significant amount of memory (RAM) to store decision rules as well as a lot of computation time. In solving the model, I combine unemployment shock, persistent and transitory shock processes into a single shock process ε, decreasing two dimension. More importantly, I develop a two-stage approach to solve savings decisions with two assets. In the first stage, given a current fixed cost, a household chooses whether or not to adjust its portfolio. If it adjusts, it chooses its savings in illiquid wealth, a′. In the second stage, given a′, I solve for the optimal choice of liquid wealth, b′, using endogenous grid method (Carroll, 2006).112

As the aggregate supply of liquid wealth is calibrated to match a rate of return of zero percent on liquid wealth, solving for stationary equilibria involves three prices

(w, p, d). Note that, in the absence of endogenous labor supply, wages are determined by the aggregate stock of capital. Moreover, p is ﬁxed to one given equilibrium price function as there is no aggregate capital adjustment cost in a steady state. Given the initial guess of prices, I compute decision rules and the distribution of households.

The latter is determined using a large grid over age, idiosyncratic shock, illiquid and liquid wealth. I use bilinear interpolation to place decision rules onto this grid.

112The prime denotes variables in the next period.

169 I update prices by bisecting for the aggregate capital, iterating through the above

steps, until prices converge.

A.3.2 Decision rules

I solve the household’s problem in two stages. Here, I abstract from aggregate

states for the ease of notation. In the ﬁrst stage, it has decided its illiquid wealth

′ for the next period , a . This may be (1 + (1 − τa)d)a if a household chooses not to

adjust, or it is the result of an active portfolio adjustment choice for a′ after paying

the ﬁxed cost ζ. Below, I describe the household’s problem at age j with illiquid asset

0 a, liquid wealth b, and productivity (working status) ε. Deﬁne vj as the intermediate value deﬁned over cash-on-hand, m, the future stock of illiquid wealth a′, and current

productivity.

The illiquid wealth problem { } 0 − ′ ′ 0 − vj (a, b, εi, ζ) = max max vj (m pa , a , εi) , vj (xi + b, (1 + (1 τa)d) a, εi) 0≤a′≤m (A.4)

subject to

m = xi(j, ε) + (p + (1 − τa)d) a + b − ζ

Note that if a household chooses to adjust illiquid wealth to a′, the remaining

cash-on-hand for consumption and liquid wealth in the second stage is m − pa′. How- ever, it is able to cash in its current stock of illiquid wealth, a. If a household does not pay its ﬁxed cost, it can not adjust the current stock of illiquid wealth and the cash available for liquid wealth and consumption is the sum of labor income if working or

170 pension beneﬁt if retired and the current stock of liquid wealth.

In the second stage, a household has already decided its illiquid wealth for the next period, a′. Given a′ and remaining cash-on-hand m, a household solves the problem below.

The consumption and liquid wealth problem ( ) 0 ′ e ′ ′ vj (m, a , εi) = max u (c) + βvj (a , b , εi) (A.5) b′

subject to

c + qb′ ≤ m

e where vj represents the expected value of a household at the beginning of the next period before ﬁxed cost is drawn.

A.3.3 Aggregate Dynamics

To solve the model with aggregate uncertainty, I extend the Backward induction method of Reiter (2002, 2010) to solve a stochastic overlapping-generations economy.

This involves generalizing the method to handle bivariate cross-sectional distributions of endogenous state variables. The Backward induction method of Reiter allows the distribution of households to vary in potentially rich ways as a function of an approximate aggregate state as it does not impose a parametric aggregate law of motion.

171 Moreover, this method does not involve repeated simulation, reducing computation time compared to Krusell and Smith (1998).113114

Backward induction method of Reiter (2002, 2010) selects a proxy distribution across a grid of approximate aggregate state based on distribution selection function

(DSF) which maps approximate aggregate states to cross-sectional distributions. A

DSF selects the proxy distribution that minimizes the distance to the reference distribution subject to moment consistency conditions.115 Solving for the DSF entails solving a large system of linear equations. With proxy distributions solved, backward induction simultaneously solves for households’ decision rules and an end-of-period distribution implying a future approximate aggregate state consistent with households’ expectations. This enforces consistency between individual behavior and the aggregate law of motion. Lastly, I simulate the model economy and weight simulated distributions using an inverse quadratic to update the reference distributions, and thus the DSF.

This paper contributes to the backward induction method of Reiter (2002, 2010) in two ways. First, I solve a stochastic OLG economy involving distributions deﬁned over 60 diﬀerent cohorts. The distribution of households over age and idiosyncratic shocks as well as two assets increases the dimension of the system of equations solved for proxy distributions making it intractable. To mitigate this problem, I aggregate

113Both Krusell and Smith (1998) and Reiter (2002, 2010) solve households’ decisions over approximate aggregate states which summarizes inﬁnite-dimensional cross-sectional distributions using a ﬁnite vector of moments. 114Krusell and Smith (1998) assume a parametric function to forecast an approximate aggregate state. Though Krusell and Smith (1998) update forecast rules based on realistic simulation-generated distributions, it is critical to have a long simulation to avoid sampling errors. These repeated long period simulations make it costly to solve a model with rich distribution of households using the Krusell and Smith’s method. 115I initially use the steady state distribution as a reference distribution.

172 full reference distributions over age and idiosyncratic shocks into a small subset of age and idiosyncratic type groups and calculate the weights mapping full distributions to aggregated ones before solving for proxy distributions. These weights keep the shape of proxy distributions over age and idiosyncratic shocks conditional on the level of wealth close to that of the original full distributions. Second, I solve the model with two endogenous state variables - illiquid and liquid wealth. To make the solution feasible, I solve the model with aggregate uncertainty over asset grids with a lower number of grid points than used for the steady state, both for decision rules and distributions. Having ﬁnished solving this model, I simulate the model economy over

ﬁner grids to have more accurate solution for households’ decisions.

I summarize the outline of the algorithm as follows: { } (1) Approximate the cross-sectional distribution in the aggregate state, z = z1, ..., znz , { } with a ﬁnite vector of statistics (moments) m = m1, ..., mnm . Here, I assume mi is ith-moment.

(2) Determine asset grids for decision rules and distributions with a lower number of { } { } grid points, A = a1, .., ana and B = b1, .., bnb , keeping bounds the same as those in the steady state. These grids are used until the backward induction is solved.

(3) Aggregate the full reference distribution, rµ(j, a, b, ε; z, m), across all age groups and a small subset of idiosyncratic types nε˜ ≤ nε, resulting in the reduced distribu- µ ∈ { } tion, r0 (a, b, ε˜; z, m) whereε ˜ ε˜1, ..., ε˜nε˜ , and calculate weights of this mapping.

µ → µ ω0(j, a, b, ε, ε˜; z, m): r0 (a, b, ε˜; z, m) r (j, a, b, ε; z, m)

µ (4) Choose a DSF which gives the proxy distribution, p0 (a, b, ε˜; z, m). I solve for

µ p0 (a, b, ε˜; z, m) as the solution to a problem that minimizes the distance to the reduced

173 µ distribution r0 (a, b, ε˜; z, m) while imposing that eachε ˜ sums to its reference value and moment consistency constraints. For each approximate aggregate state (z, m), a DSF solves :

∑na ∑nb ∑nε˜ µ − µ 2 min (p0 (ai, bk, ε˜l) r0 (ai, bk, ε˜l)) µ na, nb, nε˜ {p (ai,bk,ε˜l)} 0 i=1,k=1,l=1 i=1 k=1 l=1 subject to

∑na ∑nb ∑na ∑nb µ µ p0 (ai, bk, ε˜l) = r0 (ai, bk, ε˜l), l = 1, . . . , nε˜ (A.6) i=1 k=1 i=1 k=1

∑na ∑nb ∑nε˜ µ im a p0 (ai, bk, ε˜l)ai = mim , im = 1, . . . , nm (A.7) i=1 k=1 l=1

∑na ∑nb ∑nε˜ µ im b p0 (ai, bk, ε˜l)bk = mim , im = 1, . . . , nm (A.8) i=1 k=1 l=1

µ ≥ ∀ p0 (ai, bk, ε˜l) 0, i, k, l

where equation (A.6) represents type consistency conditions. Equations (A.7) and

(A.8) are moment consistency constraints for both assets. Lastly, probabilities should

be positive.

Ignoring non-negativity constraints for probabilities, the ﬁrst-order condition for

µ a b p0 (ai, bk, ε˜l), with λl, λim , and λim as Lagrange multipliers for (A.6), (A.7), and (A.8) respectively, is

∑nm ∑nm µ − µ − − a im − b im 2(p0 (ai, bk, ε˜l) r0 (ai, bk, ε˜l)) λl λim ai λim bk = 0 (A.9) im=1 im=1

{ µ } na, nb, nε˜ Finally, I solve a system of nε˜nanb+nε˜+2nm linear equations in ( p0 (ai, bk, ε˜l) i=1,k=1,l=1,

{ }nε˜ { a }nm { b }nm λl l=1, λim im=1, λim im=1).

174 As I ignored non-negativity constraints, the resulting solution to the system of

equations may have negative elements. If any of elements of solution are negative, I

set those elements equal to zero and reduce the system to the remaining elements. I

solve the reduced system iteratively until the solution has no negative elements.

(5) Using weights in (3), restore the full proxy distribution over age and idiosyncratic

shocks, pµ(j, a, b, ε; z, m).

(6) Simultaneously solve for households’ decision rules and an intrameporally consis-

′ tent future approximate aggregate m . Guess the aggregate law of motion Gk(z, m)

for approximate aggregate states. Given v(J + 1, a, b, ε; z, m) = 0, solve for decision

rules and value functions backwards by age over aggregate states. Compute the full

′ proxy distribution consistent end-of-period aggregate state m and update Gk(z, m).

Iterate until Gk(z, m) converges. Note that this solves for an aggregate law of motion

alongside households’ value functions.

(7) Given the value function solved by backward induction, simulate the model econ-

omy for T periods, then drop the ﬁrst T0 periods to develop new reference distribu-

′ tions. The simulation bisects for m . Let µt(j, a, b, ε) be the distribution of households

over the simulation period t = T0 + 1, ..., T .

I create new reference distributions as a weighted sum of µt, putting higher weights

for distributions generating moments, mt, closer to the vector of moments m. Deﬁne

index sets that group dates for the same exogenous aggregate state, z, I(z) = {t|zt = } z where z = z1, ..., znz . Let N(z) be the length of the vector I(z). The reference

distribution for each (z, m) is

1 ∑ δ (m, m ) rµ(j, a, b, ε; z, m) = 1 t µ (j, a, b, ε) N(z) δ(z, m) t t∈I(z)

175 where δ1(m0, m1) is deﬁned as the inverse of the Euclidian norm and δ(z, m) = ∑ t∈I(z) δ1(m, mt). (8) Iterate (3)-(7) to improve a DSF until no additional accuracy is achieved.

A.4 Additional Tables

% share of % Expenditure rate Earnings Disp income Expend. Earnings Disp income Quantile data model data model data model data model data model Q1 9.7 7.1 8.7 6.3 11.5 5.2 70.4 63.5 90.0 83.3 Q2 15.9 11.7 11.2 11.1 15.9 10.2 65.2 72.1 76.4 90.9 Q3 21.6 13.6 16.7 12.9 18.4 12.2 57.3 79.7 69.8 96.1 Q4 22.7 15.8 22.1 16.1 24.1 16.1 74.0 87.3 69.6 99.8 Q5 30.2 51.7 41.2 53.7 30.0 56.4 85.2 93.1 62.5 102

Table A.4: Share of earnings, income, expenditure and expenditure rates over wealth quantiles during normal times

x = YCIKBs NE(r) rf w mean(x) 2.66 2.14 0.54 7.81 n/a 1.44 0.05 n/a 1.18 σx/σy (2.75) 0.35 1.94 0.21 n/a 0.86 0.92 n/a 0.29 corr(x, y) 1.0 0.94 0.99 -0.05 n/a 0.96 0.95 n/a 0.59 Notes : Table A.5 presents means of GDP, consumption, investment in illiquid wealth, stock of capital, supply of liquid wealth, total hours worked, expected return on illiquid wealth, return on liquid savings and wage for the model simulated data. It also lists relative standard deviation to and correlation with GDP for each variable. I smooth series using a HP-ﬁlter with a smoothing parameter of 100.

Table A.5: Single asset economy

176 x = YCIKBs N p q w mean(x) 2.51 1.85 0.49 6.53 0.65 1.44 1.0 0.0 1.10 σx/σy (2.56) 0.39 1.99 0.23 0.27 0.83 0.27 0.0 0.30 corr(x, y) 1.0 0.97 0.98 0.02 -0.48 0.96 0.96 0.02 0.63 Notes : Table A.6 presents means of GDP, consumption, investment in illiquid wealth, stock of capital, supply of liquid wealth, total hours worked, price of illiquid wealth, and wage for the model simulated data. It also lists relative standard deviation to and correlation with GDP for each variable. I smooth series using a HP-ﬁlter with a smoothing parameter of 100.

Table A.6: portfolio choice economy with disaster risk

177 Appendix B: Appendix to Chapter 2

B.1 Data Appendix

B.1.1 PSID data

The PSID is a longitudinal survey of a sample of US individuals and families conducted annually from 1968 to 1997, and biennially since 1997. The original 1968

PSID sample combines the Survey Research Center (SRC) and the Survey of Eco- nomic Opportunities (SEO) samples. I use the U.S. population representative SRC sample.

Hourly wage is deﬁned as total annual earnings divided by total annual hours worked. Total annual earnings include all income from wages, salaries, bonuses, overtime, commissions, professional practice and the labor part of farm and business income. I select households with no missing values for education or self-employment status where: 1) the head of households’ age is between 25 and 59 years old, 2) the husband does not have positive labor income but zero annual hours worked, 3) the hourly wage is not less than half of the minimum wage, 4) income is not from self-employment. All the variables are deﬂated and expressed in 2013 dollars, using the CPI. After the sample selection, there are only 39 top-coded observations out of

178 52,211 observations.116 Following Autor and Katz (1999), I multiply all top-coded observations by a factor of 1.5 times the top-coded thresholds.

B.1.2 SCF data

SCF data is a household triennial data conducted by the Board of Governors of

the Federal Reserve System in cooperation with the Statistics of Income Division of

the IRS since 1983. The SCF employs a dual frame sample design, one frame is a

multi-stage national area probability design which provides information on the char-

acteristics of population, and the other is a list sample to provide a disproportionate

representation of wealthy households.117

Net worth, which is also reported as wealth, in the survey is deﬁned as total

assets minus total debt. Total assets include ﬁnancial assets and nonﬁnancial assets.

Financial assets include current values and characteristics of deposits, cash accounts,

securities traded on exchanges, mutual funds and hedge funds, annuities, cash-value

of life insurance, tax-deferred retirement accounts, and loans made to other people.

Nonﬁnancial assets include current values of principal residences, other real estate not

owned by a business, corporate and non-corporate private businesses, and vehicles.

Total debt includes the outstanding balances on credit cards, lines of credit and other

revolving accounts, mortgages, installment loans for vehicles and education, loans

against pensions and insurance policies, and money owed to a business owned at least

in part by the family. I select households where the head of households’ age is between

116All these top-coded observations are for head of households before 1983. 117Beginning with the 1989 survey data, the SCF methodology was substantially changed to include a multiple-imputation method implemented for missing variables.

179 25 and 84 years old. Net worth is expressed in 2013 dollars. Full sample weights are

used to calculate the wealth inequality measures.

Income includes wages and salaries, self-employment and farm income, tax-exempt

interest, taxable interest, dividends, returns from real estate, partnerships, subchap-

ter corporations, trusts and estates, realized capital gains and losses, payments from

unemployment insurance or workmen;s compensation, pension, social security, annu-

ity and disability payments, various types of welfare, alimony and child support, and

miscellaneous income. All of the variables are taxable income.

B.2 Numerical Method

Stationary equilibria are standard ﬁnite horizon dynamic programming problems.

The state space is ﬁve-dimensional: age, education, wealth, persistent and transitory

shocks. I determine decision rules for each skill group backward by age given a static

problem for the last age J. Each iteration of golden section search algorithm involves

bisection to solve for labor-leisure choice for workers. I allow age- and skill-varying

grids for the beginning period of assets starting with the borrowing limit for the

previous age. I log-spaced these asset grids to have ﬁner grids at low level of wealth

where the value functions have more curvature.

Solving for stationary equilibria involves three prices (wl, wh, r). Following Heath- cote et al. (2010), I use a ratio of the marginal products of skilled and unskilled labor to pin down λ which implies the observed college wage premium of 1.64:

( )− 1 wh (1 − λ) Hl θ 1.64 = = (B.1) wl λ Hh

This implies that I can solve the stationary equilibria only with two prices (wl, r)

subject to wh = 1.64wl. Given the initial guess of prices, I compute decision rules and

180 distribution. The distribution of households is determined using a large grid; weights are used to place decision rules onto this grid. I updated prices using Brodyen’s method which begins with the identity matrix as the initial guess of the Jacobian.

The Jacobian is updated using successive evaluations of objective function and its gradient. I iterate above steps until prices converge.

B.3 Additional Results on Policy Experiment

x y 1% 5% 10% 50% 90% ≤ 0 Gini share 2004 SCF 28.7 50.5 63.5 96.6 100 8.5 0.77 0.27 Benchmark 11.9 32.8 47.9 94.6 101 12.3 0.67 0.27 12% 50% 11.6 32.0 47.0 94.2 101 11.8 0.66 0.28 12% 75% 11.2 31.1 45.6 94.3 101 11.7 0.65 0.30 12% 100% 10.7 30.2 44.8 94.0 101 11.3 0.64 0.34 34% 50% 11.1 31.0 45.6 94.3 101 11.8 0.65 0.30 34% 75% 10.0 28.6 43.6 94.3 101 12.1 0.63 0.39 34% 100% 9.2 26.7 40.8 93.4 101 11.1 0.61 0.46 66% 50% 10.4 29.7 44.9 94.3 101 12.1 0.64 0.35 Notes : Table B.1 shows the share of wealth held by the top 1, 5, 10, 50 and 90 wealthiest households, the wealth Gini coeﬃcient, the share of households with zero or negative asset holdings, and the share of households with a college degree in 2004 SCF data, benchmark economy and an economy where households with the lowest x percent initial ability receive a y percent of subsidy for their college education costs.

Table B.1: Distribution of wealth

181 x y 1% 5% 10% 50% 90% Gini 2004 SCF 12.8 26.2 37.2 81.8 98.8 0.49 Benchmark 5.5 18.4 31.0 87.1 100 0.52 12% 50% 5.4 18.3 31.1 87.7 100 0.52 12% 75% 5.6 18.8 32.1 89.4 100 0.54 12% 75% 5.8 19.5 32.8 91.2 100 0.56 34% 50% 5.6 18.7 31.8 89.4 100 0.54 34% 75% 6.2 20.7 34.8 94.2 100 0.60 34% 100% 6.8 22.7 38.3 98.3 100 0.65 66% 50% 6.0 20.1 33.7 92.0 100 0.58 Notes : Table B.2 shows the share of income held by the top 1, 5, 10, 50 and 90 wealthiest households, the income Gini coeﬃcient in 2004 SCF data, benchmark economy and an economy where households with the lowest x percent initial ability receive a y percent of subsidy for their college education costs.

Table B.2: Distribution of income

wh x y total skilled unskilled %△wh %△wl r wl 12% 50% +2.25% −4.00% +2.44% 1.57 −2.36% +2.21% 2.60% 12% 75% +5.18% −10.1% +6.35% 1.47 −5.65% +5.02% 2.59% 12% 100% +8.11% −15.5% +11.0% 1.38 −8.71% +8.50% 2.54% 34% 50% +5.18% −9.08% +6.15% 1.47 −5.81% +5.28% 2.59% 34% 75% +11.6% −22.8% +19.2% 1.23 −14.2% +14.6% 2.57% 34% 100% +17.5% −28.0% +35.8% 1.08 −19.0% +23.1% 2.43% 66% 50% +9.08% −13.4% +10.8% 1.30 −11.7% +11.6% 2.56% Notes : Table B.3 shows welfare changes for a household, skilled household, and unskilled household entering policy economy where households with the lowest x percent initial ability receive a y percent of subsidy for their college education costs. It also shows college wage premium, percentage changes in wages from the benchmark economy and interest rate in the policy economy.

Table B.3: Implications of education subsidies

182 B.4 Additional Figures and Tables

B.4.1 Lorenz curve

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 top 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 2007 wealth deciles stocks business equity house residential property non-residential vehicle

Figure B.1: The Lorenz curve

B.4.2 Life-cycle Proﬁle of Hours worked

0.45

0.4

0.35

0.3

0.25

0.2 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57

Age skilled (PSID) unskilled (PSID) skilled (separate shock) unskilled (seaprate shock)

Figure B.2: Life-cycle hours worked for skilled and unskilled households

183 Year 1983 2004 Log of income 1.20∗∗∗ 1.15∗∗∗ (0.02) (0.02) Age 0.11∗∗∗ 0.09∗∗∗ (0.01) (0.01) Age squared −0.62∗∗∗ −0.49∗∗∗ (0.10) (0.10) College 0.24∗∗∗ 0.53∗∗∗ (0.06) (0.05) Race: white 0.36∗ 0.21∗ (0.21) (0.12) Race: black −0.30 −0.63∗∗∗ (0.23) (0.14) Race: hispanic −0.47∗ −0.29∗ (0.26) (0.15) Work status : working for other −0.10 −0.30∗∗ (0.13) (0.14) Work status : self-employed 0.91∗∗∗ 0.49∗∗∗ (0.15) (0.14) Work status : retired na 0.07 (0.36) (0.35) constant −5.58∗∗∗ −4.82∗∗∗ (0.36) (0.35) observations 3, 540 3, 992 R-square 0.65 0.78 College education dummy has a value of one if a household head has a college degree. Race dummies control for white, black, Hispanic and the other race. Work status dummies include working for some else, retired, and not working. Note: Standard errors in parentheses. * Significant at 10%; ** significant at 5%; *** significant at 1%. Work status-retired is not available in 1983. (Source: 1983 and 2004 SCF data)

Table B.4: Determinants of log of net worth (SCF data)

184