A Political Economy Model with Occupational Choice

Higher Taxation for Fairer Redistribution? A Political Economy Model with Occupational Choice

Anne P. Villamil, Xiaobing Wang and Ning Xue ∗

Abstract

This paper explores the relationship between inequality and taxation in a po-

litical economy model where agents with diﬀerent wealth choose to be workers or

entrepreneurs. Theoretical models usually predict a positive relationship between

inequality and taxation, while the results of empirical studies are mixed. By in-

troducing occupational choice, we show that more inequality may cause it to be

positive or negative depending on the elasticity of investment to net-of-tax rate and

the elasticity of labor supply to net-of-tax rate. In this framework, agents vote for

their preferred tax rate based on their own income (wage income, investment proﬁts

and transfers) and the occupational choice of others. Tax policy aﬀects the economy

through two eﬀects. A higher tax rate discourages entrepreneurial investment and

thus affects the tax revenue collected which reflects the traditional “efficiency and

equity trade-off”. In addition to this direct effect, it indirectly affects the income

of agents by reducing the equilibrium wage as fewer entrepreneurs reduces labor

demand.

Keywords: Redistribution, Taxation, Inequality, Occupational choice, Majority

Voting

JEL Classiﬁcation: H21; I31; D63

∗Villamil: University of Iowa, Iowa City, IA, USA. Email: [email protected]; Wang: Depart- ment of Economics, the University of Manchester, Oxford Road, Manchester M13 9PL. Email: Xiaob- [email protected]; Xue: Department of Economics and Related Studies, the University of York, Email: [email protected].

1 1 Introduction

In many developed countries, inequality has been high and even increasing. Under democracy, majority voting enables agents to vote for higher taxation and higher transfers. Generally speaking, these measures would reduce inequality. However why do not elec- torates in these democracies choose higher tax rates and higher transfers? In addition to the many mechanisms put forward in the literature, this paper explains this puzzle in a political economy framework through occupational choice and general equilibrium eﬀect. Theoretically, under well-functioning democracies, public policies are decided by the median voter (Persson and Tabellini, 2000). Because the median voter often receives more transfers than the tax paid, thus is the beneﬁciary of income redistribution, higher inequality should result in a larger demand for redistribution. Meltzer and Richard (1981) developed a political economy model where the demand for redistribution is positively related to the inequality. However, this seemingly reasonable hypothesis has not been supported by empirical evidences. A number of empirical studies, such as Perotti (1996), Rodriguez (1999), Georgiadis and Manning (2012), showed mixed results in terms of both over time and cross-countries studies. Over time, it is observed that with an increase in inequality, the level of tax and redistribution have been reduced in some countries. It is also observed that some countries with higher inequality have lower taxation and lower level of redistribution, such as the US as compared to most European countries (Alesina and Angeletos, 2005). Although the total amount of tax and redistribution has increased in some countries (Besley and Persson, 2009; Hao et al., 2018), the increase in taxation and redistribution seems unable to curb increasing inequality. In recent decades, both income and wealth inequality have increased in most developed countries (Piketty, 2014; Alvaredo et al., 2018). Figure 1 compares the national income share of top 1% and bottom 50% in the US from 1980 to 2014, and shows that income inequality has been increasing.

2 Figure 1: Top 1% and Bottom 50% national income shares of the US from 1980 to 2014

Source from: World Wealth & income database

This raises a question: why do the majority who decide the policy not choose higher taxation and higher redistribution? A number of papers have discussed this question and provided explanations from different perspectives (Larcinse, 2007; Benabou and Ok, 2001; Alesina and Glaeser, 2004; Alesina and Angeletos, 2005; Bethencourt and Kunze, 2015). They focus mostly on models where heterogeneous agents vote for their preferred tax rate depending on their pre-tax incomes which are exogenously given. Many optimal taxation theories argue from the perspective that tax distort incentives and behaviors (Mirrlees, 1971; Meltzer and Richard, 1981; Saez, 2001; Ngai and Pissarides, 2011). 1 This paper aims to explain the puzzle considering the general equilibrium effect of taxation in a political economy framework. We consider the effect of tax on the investment decision of entrepreneurs in a model where agents could choose whether to be an

1It is argued that agents would response to the tax rate by changing their behaviors in the following ways: 1) a tax on labor income may reduce labor input. For example, agents with high level of ability would work fewer hours if the marginal tax rate is very high; 2) a tax on capital would lower returns to capital and thus reduce saving rate and the willingness to invest; 3) other types of tax may also affect agents’ behavior such as people’s allocation of work time among different sectors, a firm’s choice of using more advanced technology or less productive technology to produce.

3 entrepreneur or a worker. In an economy with heterogeneous agents, each agent is endowed with a bequest obtained from the last generation. An agent chooses whether to be an entrepreneur or a worker based on the future incomes of her occupational choice. A worker sells her labor to the firm and gets the equilibrium wage, lends her bequest and gets interest from the world market. An entrepreneur earns profit by running the firm and pays a proportional tax. If the after-tax income of being an entrepreneur is higher than the income she could obtain as a worker, she will choose to be an entrepreneur. Oth- erwise, she will choose to be a worker. The income of the two occupations will be affected differently by the tax rate leading to various preferences to tax rate among agents. For a given level of output, higher tax rate increases the amount of tax revenue and transfers and thus improves equality. However, tax distorts incentives and behavior, and the level of output. Higher tax rate on capital may distort incentives for agents to become entrepreneurs. Thus it reduces the tax base, and ends up with lower tax revenue and lower transfer. Furthermore, having fewer entrepreneurs reduces the aggregate labor demand and results in lower wage for workers. Thus a higher tax rate on capital not only hurts the profitability of entrepreneurs, but also the wage of workers. Therefore, for a worker who is the median voter, she faces two opposite effects from a tax policy. Her chosen tax rate will optimize her total income which consists of wage income and transfer income. So there is an optimal tax rate for the median voter, where the chosen tax rate may be lower than the one maximizes only the transfer payment. Under this optimal tax and redistribution policy, the median voter will have lower transfer but enjoys higher wage. When there is an increase in the inequality of initial wealth (a decrease in median voter’s bequest), in equilibrium, the change in the tax rate depends on which effect dominates. This paper closely follows the literature in occupational choice (Meh, 2005 and 2007; Antunes et al., 2008; Bohacek and Zubricky, 2012). The effects of tax rate on the occupational choice have been examined in macroeconomic models but little attention has been paid to the the determination of tax rate in a political economy framework where people vote for the tax rate considering their own occupational choice. This paper contributes to the literature in the following ways. This is the first paper

4 that brings the general equilibrium effect of taxation on wage earning into the political economy framework. Meltzer and Richard (1981) provide a general equilibrium model of a labor economy, where the share of income redistribution is determined by majority rule. In their model and many other political economy models concerning redistribution , an increase in inequality in terms of an increase in mean income relative to the income of the decisive voter increases the size of redistribution. However, this classic result is contrary to the current economic reality, where voters are not voting for higher redistribution while inequalities have been increasing in many democratic countries (Picketty, 2014). We argue that the conventional literature only considers the partial equilibrium effect of taxation, where a higher tax rate may mean higher tax collection and redistribution. However, it may also hurt agents’ wage earnings through the general equilibrium effect. This paper shows that whether an increase in inequality lead to higher or lower demand for redistribution depends on the elasticity of investment to net-of-tax rate and the elasticity of labor supply to net-of-tax rate. Secondly, it is the first study that inte- grates occupational choice into a political economy model and studies how the tax rate and occupational choice are related and endogenously determined.

This paper is organized as follows: section 2 reviews the recent literature; section 3 introduces the macroeconomic model with occupational choice where the occupation and income of each agent are identiﬁed; section 4 shows how the tax rate is determined in a political economy framework; section 5 presents the simulation results; section 6 discusses the relationship between inequality and preference to redistribution. section 7 concludes.

5 2 Literature Review

Conventional theoretical literature in political economy shows that there is a positive relationship between income distribution and redistribution. The most influential study is the paper by Meltzer and Richard in 1981. They analyzed individuals’ preferences to redistribution in a theoretical model and showed that under the median voter theorem, more inequality (which is measured by a lower ratio of median income to mean income) will lead to a higher tax rate as a poorer median voter prefers more transfers and is more likely to choose a higher tax rate. This hypothesis has been tested by a number of empirical studies and the results are mixed. Whereas some studies find evidence to the above argument (Meltzer and Richard, 1983), some studies find no evident relationship between income distribution and redistribution (Rodriguez, 1999; Georgiadis and Manning, 2012). Rodriguez (1999) tested the relationship between the income distribution skewness and redistribution of the US from 1947 to 1991 and failed to find evidence to support the hypothesis. Georgiadis and Manning (2012) found that the demand for redistribution in the UK decreases when the inequality increases. Studies concerning this relationship have provided some explanations to this puzzle from different perspectives ( Larcinse, 2007; Hirschman and Rothschild, 1973; Benabou and Ok, 2001; Alesina and Glaeser, 2004; Alesina and Angeletos, 2005; Bethencourt and Kunze, 2015). Larcines (2007) argued that individuals with higher income are more likely to vote which is always neglected in the conventional literature. Therefore, the pivotal voter will be richer than the median voter among the whole population and thus will vote for a relatively lower tax rate. Another explanation is Prospects of Upward Mobility hypothesis which is firstly proposed by Hirschman and Rothschild in 1973 and developed by Benabou and Ok in 2001. They argued that when voting on public policy, individuals will consider not only their current income but also their expected income. From this hypothesis, individuals would like to vote for a lower tax rate because of the prospect that they or their offspring may move up the income ladder. There are also explanations which take ideological factors such as individual experience and perception

6 of fairness into consideration. Preferences to redistribution depend on income and some ideological factors as well (Piketty, 1995; Alesina and Glaeser, 2004; Alesina and Angele- tos, 2005). Individuals will be more risk-averse if they have experienced income decrease or misfortune in the past and will thus vote for a higher tax rate. And individuals are more likely to accept inequality brought by effort rather than luck. If the differences in income are brought by luck, they will vote for a relatively higher tax rate. Bethencourt and Kunze (2015) provide an explanation from the perspective of tax avoidance. The basic idea is that an increase in tax rate will lead to more tax avoidance which will shrink the tax base. At the same time, individuals will choose to increase the tax enforcement to improve the effect of this increase in tax rate. However, tax enforcement is costly so an increase in the tax rate leads to two possible equilibria which depend on the elasticity of tax rate and tax enforcement. These studies attempted to explain this puzzle in different scenarios and focused on models where the pre-tax incomes of individuals are exogenously given. Optimal taxation theory literature suggests that tax often distorts people’s incentives, and thus affects the pre-tax income of agents. The fundamental framework of optimal taxation is based on the equity-efficiency tradeoff built by Mirrlees (1971) where the social planner makes taxation decisions taking into consideration the effect of the tax on individuals’ labor supply. Based on Mirrlees (1971), recent literature on optimal taxation (Feldstein, 1995; Saez, 2001; Gruber and Saez, 2002; Mankiw and Weinzierl, 2010; Alesina et al., 2011) has examined optimal tax policy design using the elasticity method. The result varies with different social welfare functions and behavioral response elasticities. Specifically, Saez (2001) studies the partial equilibrium effect that the labor supply of an agent can be affected by a change in taxation. An increase in the tax rate could have two effects. The first effect is called the mechanism effect that describes the change of tax revenue if no behavior responses are considered. The second effect is the behavioral response which is the change of pretax income with the change of tax rate. Whilst in this paper, in addition to the two effects established in optimal taxation theory, an indirect

7 effect of tax on the income of agents is considered. A tax change will affect the decision of an agent as to whether he chooses to be an entrepreneur or a worker. It subsequently affects the aggregate labor supply and labor demand and thus the equilibrium wage. The negative effect of taxation such as the corporate taxation on wages has been supported in empirical studies. In the assessment of welfare effect of local corporate tax cuts in the US, Serrato and Zidar (2016) estimate that workers bear about 30% of the tax incidence. A reduction of federal corporate tax rate in the US from 35% to 21% is estimated to increase workers’ average annual wage by $4000 (CEA, 2017). Fuest et al. (2018) evaluate the incidence of corporate tax in Germany and find that workers bear 51% of the tax burden. Through taxation on occupational choice between entrepreneurs and workers, this paper shows the redistribution of income through indirect, general equilibrium effects. The effect of tax rate on the wage has received increased attention only recently to study its general equilibrium effect. Scheuer (2014) examined the Pareto optimal tax policy in a private information economy with endogenous firm formation and demonstrated that when the same tax policy is applied to profits and labor incomes, the optimal Pareto tax depends on the general equilibrium effect. Sachs et al. (2016) study the incidence and the optimal design of nonlinear income taxes in a general equilibrium Mirrleesian economy. However, policy on taxation and government spending in western democracies such as OECD countries are arguably decided by a political process and should be studied in a political framework. The focus of Scheuer (2014) is on the design of optimal tax policies on profits and labor income, while the focus of this paper is political economy of income distribution and redistribution and their implications for inequality and social welfare. Specifically, we study the implications of tax policy on occupational choice through the general equilibrium effect. Moreover, the literature focuses mostly on the labor supply response to taxation and the incomes of agents are labor incomes. Empirical evidence has shown that a large proportion of the richest individuals are self-employed (Kuhn and Ríos-Rull, 2016; De

8 Nardi and Fella, 2017). By incorporating occupation choice, 1) we are able to examine both the response of labor demand and supply and thus the effect of tax rate on the equilibrium wage rate; 2) it is possible to consider different sources of income when analyzing the effect of policy on individual’s preference to tax rate. A large and growing body of literature has emphasized the importance of occupational choice in macroeconomic studies ( Meh, 2005 and 2007; Bohacek and Zubricky, 2012) and agents with different occupations will behave differently to changes in policies (Kitao, 2008). These studies focus mainly on the effect of tax scheme reforms on entrepreneurship and hence the whole economy but do not consider the effect of occupational choice on agents’ preference to tax, leaving the question of how the tax rate is determined unanswered. The effect of an increase in tax rate has been examined in the literature in supply-side economies such as the Laffer curve and discussions of the elasticity of taxable income (Feldstein, 1995; Gruber and Saez, 2002; Trabandt and Uhlig, 2011). The Laffer curve describes the relationship between the average tax rate and tax revenues with the idea that an increase in the tax rate will lead to an increase in the total tax revenue first and then a decrease in the total tax revenue. A number of studies estimate the elasticity of taxable income to marginal tax rate but the estimated elasticity varies . The only paper that studies the determination of tax rate where agents are able to make occupational choice is Boadway et al. (1991). They investigated the optimal income tax when income from entrepreneurship is considered. Agents with different levels of abilities choose whether to be a worker or an entrepreneur and the final tax rate is determined by the social planner by maximizing the total utilities of the society. Our study is different in the following two ways. Firstly, we study the tax rate in a political economy framework where the tax rate is determined by majority voting. In democratic countries, public policy is determined by majority voting rather than the social planner. Secondly, we consider another essential production factor, the capital which plays an important role in agents’ occupational choice by affecting loans borrowed etc. We examine the effect of heterogeneity in capital endowments on income distribution

9 which is emphasized in the literature (Atkinson, 2003; Piketty, 2014; Stiglitz, 2012).

3 The model

In this section, we will present a macroeconomic model with occupational choice. Every agent has an initial endowment which is called a bequest from the previous period, one unit of labor and is of the same ability. She can either be a worker, deposit the bequest in the bank and earn the wage and interest or an entrepreneur, invest the bequest in the firm and earn the profit. There are no financial frictions in the model. For simplicity, we assume that there is one firm using the capital of all entrepreneurs. 2 Each entrepreneur invests in the firm in the form of shares and gets income from the profit of production. Government levies a proportional tax on the income of entrepreneurs and labour income of workers. The tax revenue will be used to provide transfers which will be part of agents’ incomes. Each agent will make occupational choices first according to her bequest level and then choose her optimal tax rate. Suppose the entrepreneurs borrow loans at the world interest rate, r , which is given in the model. The timing of events is as follows. (i) At the beginning of every period, each agent is endowed with an amount of bequest. (ii) Each agent makes occupational choice taking factor prices and tax rate as exoge- nous3. (iii)With the given occupational choice and wealth distribution, the tax rate is determined by majority voting.4 (iv) Investment and production take place. Agents obtain incomes. Tax is paid and transfers are made. (v) Each agent decides the amount of bequest left to the next generation and the

2In the economy, there are a number of heterogeneous firms. As the aggregate levels of the economy are needed later to work out the aggregate labor demand, we aggregate the firms at the beginning for simplicity in mathematical calculation and assume each entrepreneur gets a share of the profits of the aggregate firm which depends on her bequest. This aggregation will not affect our results. 3As a single agent cannot affect the factor price and tax rate, she takes the prices of factors and the policy as given when making decisions. 4In fact, (ii) and (iii) happen simultaneously. That is, individuals make their occupational choice with foresight of factor prices and tax rate. And voters for tax rate are assumed to have full knowledge of the wealth distribution and the results of occupational choice.

10 amount of consumption to maximize her utility. The distribution of initial wealth endowments of the next generation is determined.

3.1 Agents

The economy consists of a number of heterogeneous agents and ﬁrms. Suppose the entrepreneurs borrow loans at the world interest rate, r , which is given in the model. Agents are employed either as workers or entrepreneurs.

5 i min max At the beginning of period t , agent i inherits a bequest of bt[bt , bt ] , drawn

i from a cumulative distribution function F (bt) and derives utility from consumption ct

i and bequest left to the next generation, which is denoted as bt+1. The utility function of agent i is supposed to be

i i i i i Ut = U(ct, bt+1) = ln(ct) + γln(bt+1) (1)

where γ measures the weight of bequest left to the next generation in the utility function. Agent i will maximize her utility subject to her budget constraint.

i i i i ct + bt+1 = It + xt (2)

i where It denotes the after-tax factor income of agent i which refers to income from

i labor and capital, and xt is the transfer received by agent i which will be discussed later. At optimal, consumption and bequest level will be a proportion of the total income.

1 (ci)∗ = (Ii + xi) (3) t 1 + γ t t

γ (bi )∗ = (Ii + xi) (4) t+1 1 + γ t t

5We focus on a one-period static model in the paper. The current period is denoted by period t which is involved to distinguish periods as there are bequests which are the transmission over periods.

11 The indirect utility of agent i is thus

i ∗ i i (Ut ) = (1 + γ)ln(It + xt) + γln(γ) − (1 + γ)ln(1 + γ) (5)

3.2 Demand of labor and capital

The production function is supposed to be

E α1 α2 β Yt = A(Bt ) (Kt) Lt (6)

E where Bt is the investment from entrepreneurs which is the total amount of the

bequests of entrepreneurs, Kt is the loans borrowed, Lt is the input of labour and α1 +

6 α2 + β = 1. The ﬁrm solves the following problem taking the capital and wage as given.

E E α1 α2 β Π(Bt , wt) = A(Bt ) (Kt) Lt − wtLt (7)

where wt is the competitive wage. Labor demand of the ﬁrm is thus

E α1 α2 A(B ) (Kt) β 1 E t 1−β Lt(Bt , wt) = ( ) (8) wt

E Equation (8) indicates that more investment from entrepreneurs ( a larger Bt ) will lead to higher labor demand. Substituting equation (8) into (7), the ﬁrm’s proﬁt for a given level of capital is

1 β β E E α1 α2 1−β 1−β Π(Bt , wt) = (1 − β)(A(Bt ) (Kt) ) ( ) (9) wt

6 E Equation (5) implies that the self-owned capital Bt and the loans Kt contribute to the production E differently. Entrepreneurs not only invest their bequest Bt but also exert effort in the process. For example, they need to make investment decisions and monitor the workers. As argued in Banerjee and Newman (1993), with one unit of labor, the entrepreneurs can monitor the actions of workers. Therefore, E the return of the self-owned capital Bt is different to that of the loans Kt.

12 The ﬁrm solves the following problem

E max[Π(Bt , wt) − (1 + r)Kt] (10)

∗ The optimal amount of loan,Kt , is

1 β E α1 β (A(B ) ) 1−β ( ) 1−β α2 1−β ∗ t wt K = ( ) 1−α2−β (11) t 1 + r

The proﬁt of the ﬁrm, which is also the total income of entrepreneurs,Πt , is therefore

β β 1 α α2 ∗ E ∗ α α 2 α E Πt = Π (Bt , wt) − (1 + r)Kt = α1( ) 1 (A) 1 ( ) 1 Bt (12) wt 1 + r

Equation (12) implies that proﬁt increases with the investment from entrepreneurs and decreases with the competitive wage and the world interest rate.

i The pretax income of an entrepreneur with a bequest level of bt is

i β α i bt β 1 α2 2 i α1 α1 α1 Πt = E (Πt) = (1 − α2 − β)( ) (A) ( ) bt (13) Bt wt 1 + r

Let τt denotes the tax rate on the income of entrepreneurs and labour income of

i Ei workers. The after-tax income of an entrepreneur with a bequest level of bt , It , and a j W j worker j with bt , It are

β β 1 α α2 Ei α α 2 α i It = (1 − τt)α1( ) 1 (A) 1 ( ) 1 bt (14) wt 1 + r

W j j It = (1 − τt)wt + (1 + r)bt (15)

3.3 Supply of labor and capital

We assume that labor is only supplied by workers. An agent chooses to be an entrepreneur if the income she could get as an entrepreneur is at least as high as that she could get if she was a worker. We will show that there exists a benchmark amount of capital endowment

i (bequest), bt, so that agent i will become an entrepreneur if bt ≥ bt. Otherwise, she will

13 choose to be a worker. Therefore, the supply of labor is the total population whose bequest is smaller than

bt. As the distribution of bequests is supposed to be F (bt), labor supply is F (bt)Nt where

Nt is total population in the economy. We assume that there is a world capital market, where interest rate, r, is exogenously given. An entrepreneur can borrow from the world capital market and pay the world interest rate r. A worker can deposit her wealth in this capital market and obtain the world interest rate r.

3.4 The Equilibrium (Occupational Choice)

Deﬁnition. The equilibrium in this economy is deﬁned by the wage rate wt, the interest

rate r, the tax rate τt, in which there exists an agent who is indiﬀerent to being a worker and an entrepreneur and the labor market clears.

i Lemma. There exists a bt that for a given tax rate τt, agent i with bt will become an

i i entrepreneur if bt ≥ bt and a worker if bt < bt.

Proof. For agent i to be an entrepreneur, she will choose to invest in the ﬁrm if and only if her total income of being an entrepreneur is not less than the total income of being a worker.

Ei i W i i It + xt ≥ It + xt (16)

Substituting equation (14) and (15) into (16),

β β 1 α α2 α α 2 α i i (1 − τt)α1( ) 1 (A) 1 ( ) 1 bt ≥ (1 − τt)wt + (1 + r)bt (17) wt 1 + r

That is,

i (1 − τt)wt bt ≥ β 1 α (18) β α 2 [(1 − τ )α ( ) α1 (A) α1 ( 2 ) α1 − (1 + r)] t 1 wt 1+r

Let bt be the bequest level with which the income of being an entrepreneur is the same to that of being a worker.

14 (1 − τt)wt bt = β 1 α (19) β α 2 [(1 − τ )α ( ) α1 (A) α1 ( 2 ) α1 − (1 + r)] t 1 wt 1+r

Ei i W i i At bt, It + xt = It + xt. Therefore, the aggregate labor supply is the amount of workers–agents whose be-

bt quest level is less than bt which is F (bt)Nt = Nt 0 dF (bt). The total investment from max ´ E bt max entrepreneurs is B = Nt btdF (bt) where b is the largest amount of bequest t bt t ´ inherited. From equation (8) and (11) , the aggregate labor demand is

1 β 1−α2 α α2 α E α 2 α Lt = (A) 1 Bt ( ) 1 ( ) 1 (20) wt 1 + r

Labor market equilibrium requires that

bt 1 β 1−α2 α α2 α E α 2 α (A) 1 Bt ( ) 1 ( ) 1 = Nt dF (bt) (21) wt 1 + r ˆ 0

From equation (19) and (21),

1 α α α 2 BE 1 1−α2 2 1−α2 t 1−α2 (1 − τt)A ( ) β( W ) 1+r Nt bt = 1 α β (22) α 2 BE 1−α2 2 1−α2 t α2−1 (1 − τt)α1(A) ( ) ( W ) − (1 + r) 1+r Nt and

α E α 1 α2 2 Bt 1 1−α2 1−α2 1−α2 wt = β(A) ( ) [ W ] (23) 1 + r Nt

W bt where Nt = Nt 0 dF (bt). 1 α β ´ α 2 BE 1−α2 2 1−α2 t α2−1 We assume that (1 − τ)α1(A) ( ) ( W ) > (1 + r) to ensure a positive 1+r Nt 1 α β α 2 BE 1−α2 2 1−α2 t α2−1 threshold level of bequest. If τt is large enough that (1−τt)α1(A) ( ) ( W ) − 1+r Nt

(1 + r) < 0, bt ≤ 0 and there will be no entrepreneurs and thus no production. The tax base and income of all individuals will be zero which is the case discussed in most literature such as the Laﬀer curve eﬀect.

15 At bt, the labor market clears. Agents whose bequest is smaller than bt will be workers and agents whose bequest is larger than bt will be entrepreneurs. The equilibrium level of aggregate output, total proﬁt in the economy are

α E β 1 α2 2 Bt E 1−α2 1−α2 α2−1 Yt = A ( ) ( W ) Bt (24) 1 + r Nt

α E β 1 α2 2 Bt E 1−α2 1−α2 α2−1 Πt = α1A ( ) ( W ) Bt (25) 1 + r Nt

j The total income of worker j with bequest bt is

α E α W j j 1 α2 2 Bt 1 j j 1−α2 1−α2 1−α2 It + xt = (1 − τt)βA ( ) ( W ) + (1 + r)bt + xt (26) 1 + r Nt

i The total income of entrepreneur i with bequest bt is

α E β Ei i 1 α2 2 Bt i i 1−α2 1−α2 α2−1 It + xt = (1 − τt)α1A ( ) ( W ) bt + xt (27) 1 + r Nt

Proposition 1. A higher tax rate will lower the pretax labour income of workers but increase the pretax income of entrepreneurs.

Proof. From equation (22),

α α 1 2 BE 1 1−α2 α2 1−α2 t 1−α2 A ( ) β( W ) (1 + r) dbt 1+r Nt = E > 0 dτt Bt 1 α2 E β 1 α2 E α1+α2 d( W ) βα1(1−τ) B B N 2 1−α2 α2 1−α2 t α2−1 1−α2 α2 1−α2 t 1−α2 t M − A ( ) {M( W ) + (1 − τt)A ( ) β( W ) } 1+r 1−α2 Nt 1+r Nt db (28)

1 β α BE 2 α2 1−α2 t α2−1 where M = (1 − τt)α1(A( ) ) ( W ) − (1 + r) > 0. 1+r Nt BE E W d( t ) W dBt E dNt W Nt −Bt Nt dbt dbt dbt As = W 2 < 0, > 0, which indicates that an increase in the dbt (Nt ) dτt tax rate will lead to an increase in the bequest threshold to be an entrepreneur. For the agent who is indiﬀerent to being a worker and an entrepreneur, an increase in the tax rate decreases her income of being an entrepreneur. She will then choose to be a worker. The ﬁrm will get less investment from the entrepreneurs resulting in less loans borrowed and less labor demand.

16 From equation (23),

BE E t 1 α2 α1+α2−1 d( W ) dwt α2 βα1 Bt Nt dbt 1−α2 1−α2 1−α2 = (A) ( ) ( W ) (29) dτt 1 + r 1 − α2 Nt dbt dτt

E Bt d( W ) As Nt < 0 and dbt > 0 as discussed above, dwt < 0. dbt dτt dτt 1 α β α 2 BE i i 1−α2 2 1−α2 t α2−1 i The pretax income of an entrepreneur with bt is Πt = α1A ( ) ( W ) bt. 1+r Nt E E E Bt Bt Bt d( ) d( ) 1 α β−α +1 d( ) NW NW db dΠi α 2 BE 2 β NW t t t t 1−α2 2 1−α2 t α2−1 t As = < 0, = α1A ( ) ( W ) > 0. dτt dbt dτt dτt 1+r Nt α2−1 dτt

The intuition of proposition 1 is straightforward. dwt < 0 implies that there is a dτt negative relationship between the tax rate and the competitive wage while there is a positive relationship between the factor income of an entrepreneur and the tax rate as

i indicated by dΠt > 0. A higher tax rate will improve the bequest threshold resulting in dτt E less entrepreneurs and more workers. From equation (20), a decrease in Bt caused by less entrepreneurs will lead to a decrease in aggregate labor demand. As the number of workers also increases, there will be an excess of labor supply. Therefore, the competitive wage rate must decrease to make the labor market clear. As laborers are cheaper, each entrepreneur obtains a higher profit. The pre-tax return of capital for an entrepreneur increases. Therefore, a higher tax rate increases the pre-tax income of an entrepreneur. Overall, an increase in the tax rate will have a negative effect on the factor income of workers and a positive effect on the factor income of entrepreneurs. At the same time, the number of workers increases. Therefore, the pre-tax income inequality between the two types of occupations increases.

3.5 Government

The government levies a proportional tax on the factor income of entrepreneurs. Let Tt be the tax revenue in period t.

α E β 1 α2 2 Bt E 1−α2 1−α2 α2−1 Tt = τt(α1 + β)A ( ) ( W ) Bt (30) 1 + r Nt

17 The redistribution is designed in a way that the rich receives no more transfers than the poor. We suppose the tax revenue is redistributed to agents according to their bequest level and agents with less bequest receive more or the same transfers with those with more bequest. Under this condition, there are many ways to redistribute the tax revenue as transfers. For example, the transfers could be distributed evenly to each agent. It is also possible to design a rule that agent with the least amount of bequest receives the most amount of transfers while agent with the most amount of bequest receives the least

amount of transfers. Let Bt be the total amount of bequests inherited. The proportion of

max tax revenues received by the poorest agent could be bt while that for the richest agent Bt min is bt . The ways the tax revenue is distributed will not aﬀect our mechanism as long as Bt i the distribution rule depends only on the initial bequest owned. Let ηt be the proportion i i dηt of tax revenue received by agent with bt as transfers and i ≤ 0. The transfer received dbt by agent i is

i i xt = ηtθtTt (31)

where θt measures the eﬃciency of tax system and θt(0, 1). θtTt measures the amount of taxation that is actually collected while (1−θt)Tt is the collection cost wasted.

If everyone receives the same amount of transfer, xi = θtTt . t Nt As the government budget is balanced,

Nt i θtTt = xtdi (32) ˆ0 and

Nt i ηtdi = 1 (33) ˆ0

Proposition 2. Given the initial wealth distribution F (bt),

G (i) there exists a τt below which an increase in the tax rate will be good for eﬃciency.

T (ii) there exists a τt below which an increase in the tax rate will improve the tax

G T revenue. And τt <τt .

G G T (iii) if τt [0, τt ), both the eﬃciency and equality would be improved. If τt [τt , τt ),

18 T there exists a tradeoff between efficiency and equality. If τt [τt , 1), the efficiency and equality in the economy could be improved if τt decreases.

Proof. From equation (24),

dY 1 α α2 α1+α2−1 β+α2−1 1 db t 1−α 2 1−α E 1−α W 1−α E W t = A 2 ( ) 2 (Bt ) 2 (Nt ) 2 (βBt − α1Nt bt)f(bt) (34) dτt 1 + r 1 − α2 dτt

As dbt > 0, a small τ leads to a low b and thus a large BE and small number of dτt t t t W E W workers, Nt (βBt − α1Nt bt) is positive when τt is relatively small. As τt increases,

E W there will be less entrepreneurs and more workers, (βBt − α1Nt bt) is negative when τt is relatively large.

Equation (34) implies that dYt will be positive given a smaller τ and negative when dτt t τ is large. Let τ G be the tax rate which satisfies dYt = 0. At τ G, the aggregate output t t dτt t is maximized. The effect of an increase in the tax rate on the aggregate output in the economy is ambiguous. On the one hand, an increase in the tax rate results in an increase in the threshold level of bequest which decreases the number of entrepreneurs in the economy as we have shown above. Having less entrepreneurs leads to less investment and thus less output. On the other hand, more workers in the economy leads to a decrease in the labor cost (lower wage) and thus lead to more employment and more output. The overall effect depends on which effect dominates which is related to the distribution of bequest at the beginning of the period and values of parameters. When the tax rate is relatively

low, the threshold of bequest, bt, is relatively low. An increase in the tax rate will lead

E to an increase in the labor supply and decrease in Bt . The positive eﬀect brought by the increase in the labor supply on the output could make up for the negative eﬀect of a

E decrease in Bt . As the positive eﬀect dominates, the output increases with the tax rate.

When the tax rate is high and the threshold of bequest, bt, is relatively high, the negative

E eﬀect will dominates. An increase in the tax rate will result in a large decrease in Bt and a small increase in the labor supply. Therefore, the output decreases when there is

19 an increase in the tax rate. Similarly, from equation (30),

α α β 1 α2 2 E 1 W T = τ (α + β)A 1−α2 ( ) 1−α2 (B ) 1−α2 (N ) 1−α2 (35) t t 1 1 + r t t

The eﬀect of tax rate on the tax base consists of three parts. First of all, given the

same tax base, a higher tax rate τt will raise the tax revenue.We call this the mechanical eﬀect in the spirit of Saez (2001). Secondly, as a higher tax rate raises the bequest

E threshold to become an entrepreneur, there will be less entrepreneurs and thus less Bt resulting in a lower tax base. Thirdly, the increased bequest threshold bt will lead to an increased labor force which could compensate the loss in tax base. The first two effects are standard in the study of optimal taxation where a higher tax rate would reduce the tax base by dismotivating labor supply. In our model, as we consider both the labor supply and demand, a higher tax rate has two possible effect on the tax base and also on the production of the economy. Therefore, a higher tax rate has a mechanical effect and also affect the tax base through behavioral responses. As discussed above, the tax base increases with the tax rate when it is relatively low and decreases when it is relatively high, which implies that the tax revenue follows the Laffer curve effect.

T Suppose τt is the tax rate at which the tax revenue is maximized. Due to the

T G G T mechanical eﬀect of tax rate, τt > τt . The speciﬁc values of τt and τt depend on the distribution of the initial wealth in the economy. We will not discuss their values in details.

In his influential book, Okun (1975) argued that either equality or efficiency would be sacrificed when the other is to be achieved or valued more highly. More recent attention on the tradeoff discovers that equality and efficiency could be positively related in the long run. For example, in the sustainable growth, equality could enhance efficiency. The findings of our study suggest that the relationship between equality and efficiency depends.

20 G If τt (0, τt ), an increase in τt would improve the output level in the economy and also the tax revenue. Therefore, both the efficiency and equality in the economy could be enhanced by choosing a higher tax rate. The economy would suffer from the losses in both efficiency and equality otherwise. As the tax rate is determined by majority voting and the decisive voter cares only about her own utility, the tax rate is relatively low and

G T there will be a loss in efficiency and equality. If τt [τt , τt ), there exists a tradeoff between efficiency and equality which depends on the preference of the median voter in

T the economy. If τt [τt , 1) , the eﬃciency and equality in the economy could be improved

if τt decreases.

4 Government policy under majority voting

From Section 3, agents with bequest smaller than bt will be workers and agents whose bequests are larger than bt will be entrepreneurs. The total income of each agent is identiﬁed taking the tax rate as given. This section will show how the ﬁnal tax rate is determined in a political economy equilibrium.

i From section 3, the total income of entrepreneur i with bt is

α E β Ei i 1 α2 2 Bt i i E 1−α2 1−α2 α2−1 It + xt = A ( ) ( W ) [(1 − τt)α1bt + ηtθtτt(α1 + β)Bt ] (36) 1 + r Nt

j and the total income worker j with bt is

α E α W j j 1 α2 2 Bt 1 W j 1−α2 1−α2 1−α2 It + xt = A ( ) ( W ) [(1 − τt)β + θtητt(α1 + β)Nt ] + (1 + r)bt (37) 1 + r Nt

Equation (36) and (37) show that the income of the two occupations will be aﬀected diﬀerently by the tax rate leading to various preferences to tax rate among agents. In democracies where the policy is determined by majority voting, each agent will vote for her most preferred tax rate. We can show that there exists a Condorcet winner in our framework, as the preferences of agents could be ranked.

Proposition 3. A Condorcet winner always exists and coincides with the most preferred

21 m∗ tax rate of the agent with median wealth, τt .

Proof. See Appendix.

We assume that the agent with median level of bequest is a worker which is consistent with the case in real life where there are always more workers than entrepreneurs and the median agent is always a worker.

W m m m Let It , xt and ηt be the after-tax factor income, transfers and proportion of tax

m revenue received by agent with median level of bequest, bt respectively. Her total income is

W m m m m It + xt = (1 − τt)wt + (1 + r)bt + ηt θtTt (38)

m∗ Let τt be her most preferred tax rate.

m∗ m m τt argmax {(1 − τt)wt + (1 + r)bt + ηt θtTt} (39)

m∗ Equation (39) implies that the median voter chooses the optimal tax rate, τt , after recognizing the cost and benefit of the change in the tax rate. Consistent with the literature, we show that the preferences for redistribution depend on their wealth levels and, in democracies where the policy is determined by majority voting, the final policy is the most preferred policy of the median voter who is relatively poorer and always a worker in our framework. The tax rate affects the total income of median voter through two channels. A higher tax rate will enhance the mechanical effect. However, the increase in the tax rate reduces the number of entrepreneurs and hence the labor demand. The tax revenue and the transfer received will be directly affected. At the same time, the number of workers increases. The competitive wage will decrease to make the labor market clear. Therefore, the overall effect of an increase in the tax rate is indeterminate. The median voter

j ∂xt considers not only the direct eﬀect of the taxation, m∗ but also its indirect eﬀect on ∂(τt )

∂wt the equilibrium wage, m∗ . The tax rate depends on these two eﬀects. ∂(τt ) In the equilibrium, the utility of the median voter is maximized. Once the voting result of tax rate is realized, other variables such as the occupational choice of each

22 agent, their total incomes, the output and transfers are determined. As discussed above, the efficiency and equality trade-off depends on the tax rate chosen. In this political-economic equilibrium, as the tax rate is chosen by the median voter, the voted tax rate may not satisfy the objective of other agents. This microeconomic decision made by the median voter has profound implications on efficiency and equality for this economy. In countries where the policy is determined by the social planner, the tax rate would

m∗ be different to τt , generating different economic outcomes in terms of the efficiency and equality of the economy. The social planner will choose the optimal tax rate that maximizes its social welfare function. For example, if the social planner cares about

G eﬃciency, the ﬁnal tax rate would be τt . If the social planner considers only the tax

revenue, the tax rate would be τT . If the social planner cares about both eﬃciency and

G T equality, the ﬁnal tax rate will be somewhere between τt and τt . The following example compares the tax choice of the median voter with that of an output-maximizing social planner.

m∗ G If τt = τt , the choice of median voter is consistent with that of the social planner. If

m∗ G m∗ G τt (0, τt )[ or τt [τt , 1) ], as discussed in proposition 2, an increase (or a decrease)

m∗ in τt would improve the output level in the economy and also the tax revenue. As the tax rate is determined by majority voting and the decisive voter cares only about her own utility, the economy would suffer from losses of efficiency when compared to the social planner case and there exists a trade-off between efficiency and equality which depends on the preference of the median voter in the economy. The model could be extended to study the evolution of inequality and economic development which are determined endogenously in every period. In the literature of inequality evolution, Kuznets (1995) proposes the Kuznets curve which describes an inverted U-shaped relationship between inequality and economic development from 1931 to 1948. He argues that in the first period of development, as the advanced technology is in the hands of a small proportion of the population, benefit from the increased productivity is mainly reaped by a few people. As a result, inequality increases in the first period of in-

23 dustrialization. As the technology develops, the increased productivity will bring benefit to more and more people decreasing the inequality. Piketty (2014) provided an updated Kuznets curve and extended the analysis by including a third period of development when the inequality increases again after the 1970s due to globalization and development of financial sector. In his book, Piketty (2014) highlights the role of capital in the rising inequality. Capital is very unevenly distributed and is owned by a few people at the beginning of 19th century. Due to shocks such as World Wars, inequality decreased after the 1920s. However, from the 1970s, inequality increased to the level before the World Wars. He argued that the dramatic rise in inequality is a result of higher growth rate of capital returns than that of the economy. As the rate of capital return keeps increasing faster than the wage income and capital is unevenly distributed, the capital owners gain much more income and accumulate more wealth resulting in large wealth and income inequality. Even if inequality increased dramatically, the tax rate voted by the relatively poor decisive voter has not increased much. Our model provides a new explanation for increased inequality but relatively low tax rate. In our model, the rich are the entrepreneurs who earn income by investing their capital in firms. Others supply labor and earn their income from wage and deposit paid at the level of world interest rate. For the entrepreneurs, the rate of return from capital is higher than the wage of being a worker. A higher tax rate decreases the number of entrepreneurs and thus the labor demand. At the same time, it increases the tax revenue and thus amount of transfers. The decisive voter who is the agent with median level of wealth and a worker, prefers a relative low tax rate and chooses not to appropriate the rich because she also cares about her own income (wage) rather than the transfers only. When looking at the aggregate level, the decisive voter puts more weight on efficiency than equality. As the tax rate is relatively low, the entrepreneurs is able to get more profits by investing their capital and workers getting a higher wage even though it would result in a higher level of wealth and income inequality.

24 5 Simulation results

From the above propositions, the optimal tax rate for the median voter depends on the distribution of initial wealth at the beginning of the period. We simulate the model to match it with the data. This section presents the numerical results. In order to ﬁnd out the voted tax rate of the economy, values of parameters and the distribution of the initial wealth must be assigned. The baseline economy are calibrated to match some key statistics of the US economy. The model period is 1 year.

Table 1: Parameter values

Parameters Values Descriptions β 0.55 Labor income share α1 0.285 Self-equity income share α2 0.165 Capital income share r 0.02 Interest rate γ 0.06 Proportion of income left as bequest θ 0.8 Eﬃciency of tax system

Table 1 lists the parameter values used here. According to the literature, for the US economy, the labor share is around 0.67 and the capital share is around 0.33. The income

E from Bt can be understood as the income from their own capital which is bequest and the entrepreneur skills. In this model, we don’t pay entrepreneurs for their raw labor and we don’t consider their entrepreneurial skills. However, when matching data, the income generated by entrepreneurship should be included under β. Therefore, α1 is the sum of share of entrepreneur skills and that of self-owned wealth. As β measures only the share of raw labor, we set β at 0.55 (Antunes et al., 2008). To match the data that labor share is 0.67, the share of entrepreneur skills is 0.12. In the production, even though there are two types of capital involved, self-owned wealth inherited and the loans borrowed from the capital market, they contribute to the production in the same way. Therefore, the share of self-owned wealth and that of loans,

α2 are the same and α2 = 0.165. The value of α1 is thus α1 = 0.12 + 0.165 = 0.285. We set the interest rate to be 2% and γ is set to be 0.06 so that the fraction of total income left as bequest is 0.06 (Antunes et al., 2008).

25 The eﬃciency of the tax system is set at 0.8. The value of A is calibrated so that the percentage of entrepreneurs is around 5%. We set the value of A at 10000. The distribution of initial wealth of agents must be determined to pin down the model. Empirical studies have shown that the distribution of wealth and earnings is highly right skewed (Castaneda et al., 2003; Kuhn and Ríos-Rull, 2016). In the simulation, the distribution of the initial wealth is assumed to be lognormal distributed and is set to closely match the data of wealth inequality and distribution from Survey of Consumer Finances (SCF) 2013 as much as possible. Let µ and σ be the mean and standard deviation of wealth distribution. We set the value of µ be 11.307 and σ be 1.934. Table 2.1 compares the US data and statistics generated by the baseline model. As measured by Kuhn and Ríos-Rull (2016) based on the SCF, the wealth gini of US in 2013 is 0.85 and the mean-to-median ratio is 6.49. According to Kuhn and Ríos-Rull (2016), the percentages of wealth held by the top 1% , 5% and 10% are about 35.5%, 62.9% and 75.0% respectively.

Table 2: Basic statistics of US and the model

Wealth Mean-to- Percentage 5% 10% % of en- Gini median wealth in the trepreneurs ratio top 1% US data 0.85 6.49 35.5 62.9 75.0 5% Model results 0.83 6.49 36.0 62.8 75.3 3.33% with universal transfer

We simulate the model in the case where the tax revenues are redistributed evenly. Suppose there are N agents in the economy. A universal transfer is considered so that the

i Tt amount of transfers received by each agent is xt = N . Every agent gets the same amount of transfers. The optimal tax rate chosen by the median voter is 0.42. The percentage of entrepreneurs in the total population is 3.33%. As we have discussed above, a higher tax rate has two eﬀects on the income of the median voter. On one hand, it increases the tax burden of the entrepreneurs and thus there will be less entrepreneurs which decreases the wage rate. On the othre hand, it would increase the amount of transfers paid to

26 the median voter as the tax revenue may increase. Instead of choosing a very high tax rate or a zero tax rate, the optimal tax rate chosen by the median voter is 0.42 which is consistent with the data. The ratio of general government expenditure over GDP in 2013 is 0.39. Besides labor force participation choice (occupation choice), a worker may also consider how to allocate time to paid work and other nonpaid activities such as leisure. With regard to labor-leisure trade-oﬀ, as entrepreneurs do not supply labor, there is no impact to them. For workers endowed with higher levels of wealth and thus have higher income, they would prefer to supply less labor as a result. A worker with lower endowment would want to supply more labor and she would prefer a lower tax rate (which would increase labor demand and hence wage). It might yield diﬀerent tax rate preferred by the median vote depending on the wealth distribution and utility function, but it will not change the mechanism discussed in our paper.

6 Inequality and preference to redistribution

m∗ From proposition 3, there exists a wining tax rate τt which maximizes the total income of the median voter. Substituting equation (23) and (30) into equation (38),

α E α W m m 1 α2 2 Bt 1 m W m 1−α2 1−α2 1−α2 It +xt = A ( ) ( W ) [(1−τt)β +ηt θtτt(α1 +β)Nt ]+(1+r)bt (40) 1 + r Nt

W m m m∗ d(It +xt ) The optimal tax rate τt satisﬁes m∗ = 0. dτt E Bt d( W ) E W Nt (1−τt) dBt (1−τt) dNt (1−τt) Let E = > 0, E = > 0 and W = < B d(1−τ ) BE B ,1−τ d(1−τ ) E N ,1−τ d(1−τ ) W ( W ),1−τ t t t Bt t Nt N ( W ) Nt E Bt 0. E is the elasticity of to (1 − τ ) which measures how the ratio between ( B ),1−τ N W t NW t investment of entrepreneurs and labour supply react to the the change in the net-of-tax rate, 1 − τt. BE ,1−τ measures the elasticity of investment to the net-of-tax rate. N W ,1−τ measures the elasticity of labour supply to net-of-tax rate.

27 Diﬀerentiating equation (40) with respect to τt and rearranging,

β m∗ θ(α1 + β) − m [1 − α2 + α1 BE ] τ ηt ( ),1−τ t = NW (41) m∗ W β (1 − τt ) θ(α1 + β)N {( ) BE + E } t α2−1 ( ),1−τ B ,1−τ NW

Equation (41) is the formula for ﬁnal tax rate decided by majority voting. It depends

m on E , E and η . ( B ),1−τ B ,1−τ t NW 1 α α α 2 BE 1 1−α2 2 1−α2 t 1−α2 As shown above, wt = β(A) ( ) [ W ] . Tax rate aﬀects the equilibrium 1+r Nt E Bt wage rate through . Therefore, E measures the elasticity of equilibrium wage N W ( B ),1−τ t NW rate with respect to the net-of-tax rate. Firstly, let’s consider the case where the eﬀect of tax rate on equilibrium wage is not E Bt d( W ) Nt considered. That is, = 0 and thus BE = 0. dτt t ( W ),1−τ Nt

BE m∗ t τ 1 β W t = − Nt (42) m∗ m E (1 − τt ) BE ,1−τ η θ(α1 + β)BE ,1−τ Bt

E dBt (1−τt) m m m∗ As BE ,1−τ = E > 0, a larger ηt (a smaller bt ) will increase τt . That is, when d(1−τt) Bt there is an increase in income inequality caused by a poorer median voter, the median voter will prefer a higher tax rate which is consistent to the result of Meltzer and Richard (1981). If the eﬀect of tax rate on equilibrium wage is considered, we obtain

m ∗∗ Proposition 4. When there is an increase in inequality, for the new tax rate (τt ) ,

m ∗∗ m∗ i) (τt ) > τt if (α1BE ,1−τ + βN W ,1−τ ) ≥ 0

m ∗∗ m∗ ii)(τt ) < τt if (α1BE ,1−τ + βN W ,1−τ ) < 0

Proof. From equation (41), when the eﬀect of tax rate on equilibrium wage is considered, E Bt d( W ) Nt (1−τt) E = > 0 and the optimal tax rate becomes B d(1−τ ) BE ( W ),1−τ t t N ( W ) Nt

β m∗ θ(α1 + β) − m [1 − α2 + α1 BE ] τ ηt ( ),1−τ t = NW (43) m∗ W β (1 − τt ) θ(α1 + β)N {( ) BE + E } t α2−1 ( ),1−τ B ,1−τ NW

28 −β[1−α2+α1 E ] ( B ),1−τ m∗ m NW The relationship between τt and ηt depends on the sign of W β . θ(α1+β)Nt {( ) E + E } α2−1 ( B ),1−τ B ,1−τ NW β As BE > 0, the sign depends on ( ) BE + E . ( ),1−τ α2−1 ( ),1−τ B ,1−τ NW NW As β 1 ( ) E + E = (α E + β W ) (44) ( B ),1−τ B ,1−τ 1 B ,1−τ N ,1−τ α2 − 1 NW 1 − α2

the most preferred tax rate of the median voter satisﬁes

β (1 − α + α E ) m∗ ηm 2 1 ( B ),1−τ τt 1 t NW m∗ = β − W (45) (1 − τt ) ( ) BE + E θ(α1 + β)Nt (α1BE ,1−τ + βN W ,1−τ ) α2−1 ( ),1−τ B ,1−τ NW

β m∗ From equation (45), if (α1 E + β W ) > 0, ( ) BE + E > 0, τ B ,1−τ N ,1−τ α2−1 ( ),1−τ B ,1−τ t NW m and ηt are positively related. That is, a larger inequality caused by a poorer median voter will result in a higher tax rate.

β m∗ m If (α1 E +β W ) < 0, ( ) BE + E < 0, τ and η are negatively B ,1−τ N ,1−τ α2−1 ( ),1−τ B ,1−τ t t NW related. That is, a larger inequality caused by a poorer median voter will result in a lower tax rate.

A larger inequality in the distribution of initial wealth which is caused by a decrease in the bequest of the median voter, will generate two possible results depending on the elasticity of investment to net-of-tax rate and the elasticity of labour supply to net-of-

m m tax rate. As a smaller bt will lead to a larger ηt resulting in a larger transfer, the median voter’s preference to the tax rate increases as she will get more transfers. At the same time, an increase in the tax rate would decrease the wage as we discussed above. Therefore, an increase in the tax rate will increase the transfers but decrease the wage. The overall eﬀect of a change in the tax rate thus depends on the elasticity of the wage and the taxable income which are based on the initial wealth distribution at the

beginning of period t, F (bt). If the positive effect outweighs the negative effect of tax rate, more inequality in the distribution of initial wealth leads to a higher tax rate which is consistent with Meltzer and Richard (1981). However, if the negative effect dominates, the tax rate will not increase with the rise in the inequality in the distribution of initial

29 wealth. This tax rate results in a new wealth distribution which will be evolved in the following periods.

7 Conclusion

This paper investigates the relationship between inequality and prefererne for taxation in democratic countries through the general equilibrium effect of tax rate by incorporating occupational choice into a political economy model. Existing literature has shown that the decisive vote will choose a higher tax rate when inequality increases which is not consistent with the empirical studies. A few explanations have been discussed to explain this puzzle from different perspectives. This paper connects three strands of literature in public finance, public choice, and occupational choice, to solve the puzzle that, following majority voting, agents do not always vote for a higher tax rate and higher level of redistribution in an economy with high and increasing inequality. This is due to the general equilibrium effects (and production distortions) that an increase in tax rate would reduce the incentives for agents to become entrepreneurs, which subsequently decreases the aggregate labor demand and reduce wage rates for workers. Different from the conventional public finance literature, which emphasizes the direct effect of tax policy on welfare, this paper focuses on the indirect welfare consequences of tax policy, through general equilibrium effects on wages. Specifically, this paper presents a political economy model with occupational choice where agents with different capital endowments hold different preferences to redistribution. At equilibrium, the final tax rate is determined by the agent with the median level of bequest who is relatively poor and is a worker. The results show that as a higher tax rate could not only affect the redistribution but also decrease the competitive wage which is part of a worker’s income, the decisive voter who is usually a worker would not choose a very high tax rate to expropriate the rich. An increase in inequality would lead to two possible results of redistribution depending on the elasticity of investment to net-of-tax rate and the elasticity of labour supply to net-of-tax rate. This tax rate which is deter-

30 mined at the microeconomic level has profound implications on the economy. As the tax rate is determined by the median voter who cares only about her own utility, there would be losses in eﬃciency and equality for the economy compared with the case where the tax rate is chosen by the social planner. In the present paper, the tax revenue is redistributed as the transfers and the redistribution rule is not fully examined. It would be interesting to study the optimal tax rate when the tax revenue is used in diﬀerent ways. For example, the tax revenue could be spend either on the provision of some utility-enhancing public goods such as health (which increases agents income/utility), or productive public goods which improves the productivity of the private sector such as infrastructure. How the tax revenue is spent is important in the analysis of the determination of the tax rate (Rogerson, 2007) and will be an extension for future research.

31 Appendix A

Proof of Proposition 3

Gans and Smart (1996) showed that the preferences to tax rate are single-crossing, if they satisfy the Spence-Mirrlees condition that the marginal rate of substitution can be ordered by the initial bequest. In below, we prove that in our model, the marginal rate of substitution can be ordered by the initial bequest. From the utility function in equation (1) and the budget constraint stated in equation

j (2), the indirect utility for worker j with bequest bt is

W j ∗ j j (Ut ) = (1 + γ)ln[(1 − τt)wt + (1 + r)bt + ηt θtTt] + γln(γ) − (1 + γ)ln(1 + γ) (46)

The Spence-Mirrlees condition requires that the marginal rate of substitution between

j τt and Tt to vary monotonically with individual’s bequest bt .

W j d[(1−τt)wt] ∂Ut /∂τt dτt MRS(τt,Tt) = − W j = − j (47) ∂Ut /∂Tt ηt θt

Substituting equation (23) into equation (47),

1 α β α 2 BE BE α β BE 1−α2 2 1−α2 t α2−1 1 t A ( ) ( W ) [β( W ) + BE ( W )] 1+r N N 1−α2 ( ),1−τ N t NW t MRS(τt,Tt) = j (48) ηt θt

E Bt d( W ) Nt (1−τt) where E = > 0. B d(1−τ ) BE ( W ),1−τ t t N ( W ) Nt i Similarly, for entrepreneur i with bt,

i Ei ∗ bt i (Ut ) = (1 + γ)ln[(1 − τt) E (Π) + ηtθtTt] + γln(γ) − (1 + γ)ln(1 + γ) Bt

32 The marginal rate of substitution is

i bt d[(1−τt) E (Π)] Ei Bt ∂Ut /∂τt dτt MRS(τt,Tt) = − Ei = − i (49) ∂Ut /∂Tt ηtθt

Substituting equation (25) into equation (49),

1 α β α 2 BE β 1−α2 2 1−α2 t α2−1 i i A ( ) ( W ) {α1b + α1b BE } 1+r N t t α2−1 ( ),1−τ t NW MRS(τt,Tt) = i (50) ηtθt

For the agent who is indiﬀerent to being a worker and an entrepreneur,

W E It + xt = It + xt (51)

Substituting equation (26) and (27) into equation (51),

α E α α E β 1 α2 2 Bt 1 1 α2 2 Bt 1−α2 1−α2 1−α2 1−α2 1−α2 α2−1 (1−τt)βA ( ) ( W ) +(1+r)bt+xt = (1−τt)α1A ( ) ( W ) bt+xt 1 + r Nt 1 + r Nt (52)

Diﬀerentiating equation (52) with respect to τt and rearranging,

E E Bt Bt E E d( W ) E d( W ) Bt Bt α1 Nt Bt β Nt β( W )[−( W ) + (1 − τt) ] = −α1bt[( W ) + (1 − τt) (53) Nt Nt 1 − α2 dτt Nt α2 − 1 dτt

E Bt d( W ) Nt (1−τt) Substituting E = into equation (53), B d(1−τ ) BE ( W ),1−τ t t N ( W ) Nt

E E β Bt α1β Bt α b + α b E = β + ( ) E (54) 1 t 1 t ( B ),1−τ W W ( B ),1−τ α2 − 1 NW Nt 1 − α2 Nt NW

i From equation (57), it is straightforward that for bt > bt,

E E i i β Bt α1β Bt α b + α b E > β + ( ) E (55) 1 t 1 t ( B ),1−τ W W ( B ),1−τ α2 − 1 NW Nt 1 − α2 Nt NW

Equation (48), (50), (55) show that 1) for both workers and entrepreneurs, the

33 j j i dηt marginal rate of substitution is negatively related to ηt and ηt respectively. As j < 0 dbt i dηt and i < 0, for both workers and entrepreneurs, the marginal rate of substitution is dbt positively related to their bequest level. 2) As the bequest of entrepreneurs is larger than bt, the marginal rate of substitution of an entrepreneur is higher than that of a worker. Therefore, the marginal rate of substitution of the whole population can be ordered by the initial bequest. As the preferences to tax rate are single-crossing, a Condorcet winner always exists and coincides with the most preferred tax rate of the agent with median level of bequest.

34 References

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