Risk Aversion, Uninsurable Idiosyncratic Risk, and the Financial Accelerator†

Giacomo Candian‡ Mikhail Dmitriev§ HEC Montr´eal Florida State University

May 2019

Abstract

We develop a tractable model to study jointly the role of non-diversifiable risk and financial frictions for business cycles. Non-diversifiable risk induces strong precautionary motives, which reduce the exposure of entrepreneurs to aggregate disturbances ex-ante, and make it easier to increase leverage ex-post. In general equilibrium, these precautionary motives dampen fluctuations in asset prices and risk premia, thus making the economy more resilient to financial shocks. We provide microeconomic evidence supporting the model’s predictions about firm behavior. Our results suggest that the origin and transmission of financial shocks lies with the well-diversified publicly-traded firms.

JEL Classiﬁcation: C68, D81, D82, E44, L26. Keywords: Risk Aversion; Uninsurable Idiosyncratic Risk; Financial Accelerator; Incomplete Markets.

†We thank the editor Jonathan Heathcote and two anonymous referees for constructive comments that improved the paper. Thanks also to Levent Altinoglu, Susanto Basu, Diego Comin, Fabio Ghironi, Peter Ireland, Robert King, Andrea Prestipino, Fabio Schiantarelli, Nicolas Vincent, and conference participants at the BC-BU Green Line Macro Meeting, Midwest Macro Meeting, Federal Reserve Bank of Cleveland, Federal Reserve Board, University of Washington, Florida State University, Macro Bank- ing and Finance Workshop, and CIRANO Workshop for useful suggestions. We are grateful to Vasia Panousi for sharing her dataset and for valuable insights. All remaining mistakes are our own. ‡Email: [email protected]. Web: https://sites.google.com/site/giacomocandian §Email: [email protected]. Web: http://www.mikhaildmitriev.org 1 Introduction

According to Knight(1921), bearing risk is one of the defining features of entrepreneurship. Entrepreneurs are exposed to significant undiversified risk (Gen- try and Hubbard, 2004) and face extreme dispersion in equity returns (Moskowitz and Vissing-Jorgensen, 2002). When entrepreneurial activity depends on external finance, the presence of this large undiversifiable risk may have important implications for macroeconomic dynamics. Yet, the theoretical literature on financial frictions has paid little attention to how entrepreneurs’ willingness to take on this risk affects the transmission of shocks over the business cycle. Indeed, general equilibrium models with credit market frictions assume that idiosyncratic risk is either absent (Kiyotaki and Moore, 1997, KM), fully diversified (Forlati and Lambertini, 2011; Liu and Wang, 2014; Dmitriev and Hoddenbagh, 2017), resolved at the beginning of the period so that there is no within period uncertainty (Bassetto et al., 2015) or present but coupled with the assumption of risk-neutral entrepreneurs (Bernanke, Gertler, and Gilchrist, 1999, BGG). In this paper, we fill this gap by introducing uninsurable idiosyncratic investment risk for risk-averse borrowers into a standard business cycle model. Our framework is flexible enough to study the consequences of uninsurable risk in an otherwise standard New Keynesian model, as well as to analyze the interaction of the former with agency frictions that are now a standard feature of medium-scale DSGE models, yet maintaining an analytically tractable model, log-linear setup. We show that non-diversified risk mitigates agency frictions and stabilizes business cycles. Indeed, in the presence of asymmetric information the response of output to financial shocks, such as risk and wealth redistribution shocks, is 35 to 80 percent smaller when entrepreneurs are risk averse than when they are risk neutral. Additionally, the responses of key macro variables to technology and monetary shocks are about 20 percent smaller when idiosyncratic risk is non-diversified. Our findings suggest that the privately-held entrepreneurial businesses and the less-diversified publicly-traded firms were unlikely to be the source or the amplifier of the Great Recession. Also, the less-diversified businesses and firms were not as negatively affected by financial shocks via the credit channel because of strong precautionary saving motives resulting from undiversifiable idiosyncratic risk. Hence, the origin and transmission of these shocks lies with the well-diversified publicly- traded firms. We consider an environment where entrepreneurs use their net worth and funds

2 borrowed from households to invest in physical capital, whose returns are subject to both aggregate and idiosyncratic risk. If there are no agency frictions and idiosyncratic risk is fully diversified, our model collapses to a standard New Keynesian framework, where capital returns are equal to the safe rate of interest up to a first- order approximation. In this model, the dynamics of net worth are irrelevant for economic outcomes, as entrepreneurs freely substitute equity with debt consistently with the Modigliani-Miller theorem. If lenders cannot verify the borrower’s return ex-post because of agency problems, they charge a premium on external finance as a compensation for the cost of defaults. This external finance premium drives a wedge between the returns to physical capital and safe rate of return. In bad times, when asset prices and net worth are low and expected returns to capital are high, agency costs prevent entrepreneurs from borrowing enough to support purchases of physical capital, as higher borrowing increases the risk of default. This leads to even lower asset prices and net worth, and even higher cost of defaults. As a result, the premium on external finance becomes highly volatile and countercyclical, thereby amplifying business cycle fluctuations. Introducing risk aversion creates an additional risk premium relative to risk neutrality, as entrepreneurs require compensation for the volatility of their returns associated with uninsurable idiosyncratic risk, and results in a lower leverage. This precautionary behavior helps in bad times for three reasons. First, lower leverage reduces losses from falling asset prices. Second, the external finance premium becomes less important for risk-averse borrowers. Third, risk-averse entrepreneurs have more flexibility to raise borrowing and invest in bad times if the risk premium increases. These precautionary motives support asset prices in recessions, preventing a further decline in net worth. The wedge between capital returns and the safe rate, now reflecting mostly the risk premium, becomes less countercyclical and less volatile, which stabilizes the business cycle. We found these results to be robust to all plausible values of risk aversion and sizes of idiosyncratic risk. We provide microeconomic evidence in support of our mechanism by studying the firm-level relationship between investment and capital returns in the presence of risk aversion, which we proxy using data on insider ownership, as in Panousi and Papanikolaou(2012). Indeed, when ownership is more diversified, entrepreneurs behave in a more risk-neutral way with respect to idiosyncratic risk. The evidence indicates that firms with higher insider ownership exhibit a stronger precautionary behavior and, importantly, a higher responsiveness of investment to future returns to capital. These findings corroborate the key channel at work in our model that

3 delivers the stabilizing effects of risk aversion. Methodologically, we are the first to our knowledge to incorporate risk-averse borrowers in a model of idiosyncratic, uninsurable investment risk and agency frictions, while keeping the analytical tractability of a model, that can be solved using standard perturbation techniques. A common challenge with incomplete-market models is that the wealth distribution—an infinite-dimensional object—is a rele- vant state variable for aggregate dynamics. We address this difficulty using over- lapping generations of entrepreneurs, where only new-borns work, so that future labor income does not affect financial decisions. Under these assumptions, the entrepreneurial problem is homothetic in net worth, which implies that aggregate quantities and prices in equilibrium are invariant to the wealth distribution. We solve the model using a two-step procedure that is similar in spirit to Re- iter(2009) and Winberry(2018). First, we compute for the steady state of the model, where there are no aggregate shocks but still idiosyncratic shocks. Notice- ably, this steady state does not display certainty equivalence because of the presence of risk-averse agents and idiosyncratic shocks. We then solve for the aggregate dynamics using first-order perturbation techniques around this steady state. We also use consider a second-order perturbation of the model to ensure the accuracy of the log-linear approximation and compute the welfare costs of business cycles. We build on the literature that studies the role of non-diversified risk and incomplete markets in business cycles, starting from Kimball(1993), Krusell and Smith (1998) and more recently Angeletos and Calvet(2005), and Angeletos(2007). Our work differs from theirs, in that we study the interaction of non-diversified risk and agency friction. Covas(2006) examines the link between precautionary motives and steady-state aggregate capital in an environment in which entrepreneurs face idiosyncratic investment risk and borrowing constraints. Meh and Quadrini(2006) study the long-run implications of risky investment for capital accumulation in an environment with asymmetric information. In contrast to these authors, our focus is on the implications of non-diversified entrepreneurial investment risk for the transmission of macroeconomic shocks over the business cycle. In a related paper, Bassetto et al.(2015) study the business cycles with entrepreneurs that are subject to financial constraints but are not exposed to idiosyncratic risk. They find that credit shocks have a very persistent effect on economic activity by eroding entrepreneurial net worth. Our results suggest that such effects are muted in the presence of non-diversified idiosyncratic risk, as strong precautionary motives limit entrepreneurial exposure to adverse shocks ex-ante and allow

4 entrepreneurs to increase their leverage ex-post. Our findings are also related to Dmitriev and Hoddenbagh(2017) and Carlstrom, Fuerst, and Paustian(2016), who show that in models with agency frictions a` la BGG, indexation of lenders’ repayments to aggregate variables stabilizes business cycle fluctuations. In their setup, entrepreneurs effectively buy insurance for aggregate risk from households in order to limit balance sheet movements. Differently, here we study a setting in which borrowers are unable to insure their consumption either from aggregate or from idiosyncratic risk. Thus, while Dmitriev and Hod- denbagh(2017) and Carlstrom, Fuerst, and Paustian(2016) suggest that insuring aggregate risk stabilizes the economy, we show that hedging idiosyncratic risk increases the economy’s vulnerability to aggregate disturbances and raises the welfare cost of business cycle fluctuations. The paper proceeds as follows. In Section 2, we conduct a partial-equilibrium analysis of non-diversifiable risk with and without agency frictions. Section 3 in- corporates the partial-equilibrium results into the general equilibrium framework. Section 4 contains our quantitative analysis. Section 5 conducts the empirical analysis using firms-level data, and Section 6 concludes.

2 Partial Equilibrium

2.1 Risk Aversion and Non-Diversiﬁed Risk Without Agency Fric- tions

In this section, we study the optimal contract between a risk-averse borrower (the entrepreneur) who invests in a project with idiosyncratic risk and a risk-neutral lender in the absence of agency frictions. The borrower never defaults on its debt and can freely borrow or save at an interest rate R. Lenders are risk neutral with respect to the idiosyncratic risk because, as will be true in the general equilibrium model developed below, they can diversify their lending activity across a large number of projects. An entrepreneur invests QK resources in a risky asset (capital), where K denotes the quantity of capital purchased and Q its relative price. He borrows B at a rate R and invests the borrowed funds along with his own net worth, N, so that QK = N + B. The return on the investment is QKRkω, where Rk indicates aggregate 1 2 2 returns to capital and log(ω) ∼ N (− 2 σω, σω) the idiosyncratic return component

5 that is speciﬁc to the entrepreneur with pdf φ(ω).1 The utility is characterized by constant relative risk aversion, so the entrepreneur solves the following problem:

R ∞[QKRkω − BR]1−ρφ(ω)dω max 0 , (1) K,B 1 − ρ s.t. QK = N + B.

If we assume that the variance of idiosyncratic returns and the wedge between the returns to physical capital and the safe rate are small, then we obtain the following results.

Lemma 1. Solving problem (1) gives the following relationship between leverage and the expected discounted returns to capital

4 κ = + o(σ2 , 4), 2 ω (2) ρσω

Rk QK where 4 = log( R ), and κ = N . Moreover, when we log-linearize the relationship 4 κ = 2 by allowing κ, σω, and 4 to change relative to the initial level κss, σω,ss, and ρσω 4ss , we obtain

1 ˆ κˆ = 4 − 2ˆσω, (3) 4ss

ˆ where 4 = 4 − 4ss, σˆω = ln(σω/σω,ss), and κˆ = ln(κ/κss). Proof. Equations (2) and (3) are obtained in the Appendix.

Lemma 1 describes the equilibrium relationship between leverage, risk aversion and the distribution of capital returns to a ﬁrst-order approximation. While the full solution potentially involves higher-order terms, the equations in (2) and (3) provide the intuition for the main mechanisms. Equation (2) shows that leverage grows proportionally with the excess returns to capital, and it is inversely proportional to the variance of the returns and the degree of risk aversion, which is a standard result of optimal portfolio analysis. A direct consequence of this result is that leverage is very sensitive to changes in excess returns to capital, as can be seen from equation (3). For example, if on average excess returns to capital are one percentage point, following an increase of these

1To keep consumption non-negative we use a bounded normal distribution for log ω so that log ω ∈ [−4σω, 4σω] which covers 0.99997 of the distribution.

6 returns to two percentage points leverage doubles. On the other hand, when we increase risk by one percentage point, leverage falls by two percent regardless of the size of the risk or degree of risk aversion.

2.2 Risk Aversion and Non-Diversiﬁed Risk with Agency Frictions. The General Case.

We now study the optimal contract between a risk-averse entrepreneur and a risk- neutral lender in the presence of agency frictions. The contract between the lender and borrower follows the traditional costly state verification (CSV) framework and resembles the optimal contract developed by Tamayo(2014). 2 The environment is similar to the previous section, except that now the lender cannot observe the realization of the idiosyncratic shock to the entrepreneurs, unless he pays monitoring costs µ which are a fixed percentage of total assets. In each state of the world ω ∈ Ω, the risk-averse entrepreneur chooses to report s(ω), and the report is verified in the verification set ΩV ⊂ Ω. Following Gale and Hellwig(1985), we focus only on incentive-compatible contracts, i.e., contracts where the borrower has no incentive to misreport the true realization of ω. This can be ensured by assuming arbitrarily large misreporting penalties.3 Incentive compatibility ensures that reports are always truthful, i.e. s(ω) = ω for all ω ∈ Ω, which implies that the repayment function depends only on ω.

Deﬁnition 1. An incentive-compatible contract under CSV is an amount of borrowed funds, B, a repayment function, R(ω), in the state of nature ω and a veriﬁcation set, ΩV , where the lender chooses to verify the state of the world.

2For earlier treatments of the contracting problem see Townsend(1979) and Gale and Hellwig (1985). 3Formally this means assuming the incentive compatibility constraint U(ω − R(s(ω), ω)) < U(ω − R˜(ω, ω)) for any s(ω), ω ∈ Ω, where U(·) is the utility function of the borrower, s(ω) is the report and ω is the true realization of the state. Imposing this constraint results in s(ω) = ω for any ω so in our subsequent analysis we drop the constraint and simply write R(ω) ≡ R˜(ω, ω) .

7 The static problem in the presence of only idiosyncratic risk ω can be formulated as

R ∞[QKRk(ω − R(ω))]1−ρφ(ω)dω max 0 , (4) K,R(ω) 1 − ρ Z ∞ Z BR ≤ QKRk R(ω)φ(ω)dω − µQKRk ωφ(ω)dω, (5) 0 ω∈ΩV QK = B + N, (6) 0 ≤ R(ω) ≤ ω. (7)

The ﬁrst equation is the expected utility of the entrepreneur from the investment return. The second equation is a participation constraint for the lender; it states that he should be paid on average the gross safe rate of return, R. The third equation just says that the entrepreneur uses the loan (B) and his own net worth (N) for acquiring capital. The ﬁnal inequality constraint states that repayments should be non-negative and cannot exceed the total value of assets. The following Proposition is a special case of Tamayo’s Theorem 1 case iii). Proposition 1. Under the optimal contract that solves the problem (4) subject to (5), (6), (7), the repayment function, R(ω), can be written as •∃ ω¯ and ω, such that  0 if ω < ω,  R(ω) = ω − ω if ω ≤ ω ≤ ω,¯  R¯ if ω > ω,¯ where ω¯ ≥ R¯ ≥ ω¯ − ω, ΩV = [0, ω¯).

Proof. See the Appendix. The optimal contract is illustrated in Figure1. When the lender monitors the borrower (ω ≤ ω¯), he does not seize all assets. If the borrower’s returns are very small (ω < ω), the lender does not receive any repayment. Conversely, if the borrower is a little more successful (ω < ω < ω¯), he keeps a ﬁxed amount ω of resources, while the lender seizes the rest. As in Townsend(1979)’s debt contract, when the borrower is not monitored, the lender receives a ﬂat payoff. The structure of the optimal contract in the defaulting region is the result of the borrower’s attempt to smooth his return across different states of the world.4 Therefore, optimal risk-sharing requires 4Effectively, in the region ω ∈ (ω, ω¯) the borrower always receives ω.

8 that the borrower be initially prioritized in the repayment. At the same time, the lender is indifferent to the structure of the repayment function, so long as his net payment covers the opportunity cost of his funds on average.5

Figure 1: Optimal contract with risk-averse entrepreneurs

0.9

0.8 ¯ 0.7 R

0.6 )

ω 0.5 R( 0.4

0.3

0.2 ω ω¯

0.1 45 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω

Corollary 1. When ρ → 0 then ω → 0, R¯ → ω¯ so that the optimal contract replicates the standard debt contract.

Corollary 1 states that when the borrower becomes risk neutral, the optimal contract converges to the debt contract of BGG. In this case, the repayment function is completely characterized by ω¯, as R¯ becomes equal to ω¯, and ω goes to zero. In other words, the debt contract of BGG is a special case of the richer risk-sharing agreement described in Proposition 1. An interesting implication of Proposition 1 is that, notwithstanding the complex- ity of the problem under risk aversion, the repayment function R(ω) is completely characterized by the thresholds (ω, ω¯) and by the non-default repayment R¯. This

5In this context, risk-sharing refers to the redistribution of wealth from non-defaulting to the defaulting entrepreneurs implemented through the ﬁnancial intermediary, subject to the asymmetry of information.

9 allows us to reformulate the contracting problem as follows:

(κRk)1−ρg(¯ω, ω, R¯) L = max + λ κRkh(¯ω, ω, R¯) − (κ − 1)R , ω,ω¯ ,R,κ,λ¯ 1 − ρ

QK ¯ ¯ where κ ≡ N , g(¯ω, ω, R) and h(¯ω, ω, R) are correspondingly:

Z ω Z ω¯ Z ∞ g(¯ω, ω, R¯) = ω1−ρφ(ω)dω + ω1−ρ φ(ω)dω + (ω − R¯)1−ρφ(ω)dω, 0 ω ω¯ Z ω¯ Z ω¯ Z ∞ Z ω h(¯ω, ω, R¯) = (1 − µ) ωφ(ω)dω − ω φ(ω)dω + R¯ φ(ω)dω − µ ωφ(ω)dω. ω ω ω¯ 0

The optimal κ, ω,¯ ω and R¯ are only functions of exogenous variables Rk,R and parameters σω, µ, ρ. The ﬁrst-order conditions for this problem are reported in the Appendix. The maximization problem cannot be solved analytically but we can gain some insights through numerical analysis. In particular, to separate the effect of risk aversion from that of agency frictions it is useful to consider the following decomposition.

Premia Decomposition. From solving the optimal contract, we obtain that the optimal leverage depends on the excess returns to capital, idiosyncratic volatility, monitoring costs, and the degree of risk aversion:

κ = S1(4, σω, µ, ρ). (8)

We are interested in the decomposition of the excess returns to capital, 4, into the external ﬁnance premium, generated by agency frictions, and the premium associated with risk aversion, which we will refer to as the risk premium. As there is a positive, monotonic relationship between optimal leverage and excess returns to capital, it is always possible to ﬁnd the following inverse relationship:

4 = S2(κ, σω, µ, ρ). (9)

We interpret 4 as the total premium on returns that is sufﬁcient to convince an entrepreneur with a risk aversion ρ to invest with a leverage of κ into a project with a variance of idiosyncratic returns σω under the optimal contract with monitoring costs µ.

10 The external ﬁnance premium is simply deﬁned as the capital wedge when risk aversion is set to zero:

4EFP = S2(k, σω, µ, 0), (10) while the risk premium is deﬁned as a residual component between the excess returns to capital and the external ﬁnance premium:

4RP = 4 − 4EFP = S2(κ, σω, µ, ρ) − S2(κ, σω, µ, 0) (11)

Thus, the risk premium corresponds to the additional premium for investment risk- averse agents require in order to have the same leverage, while facing the same project and the same lender as the risk-neutral agent.

Figure 2: Optimal Leverage

0.07 ρ=0.1, µ=0.12, σ=0.28 ρ=0.1, µ=0.00, σ=0.28 ρ=0.0, µ=0.12, σ=0.28 0.06

0.05

← risk premium 0.04 /R) k

4ln(R 0.03

0.02

0.01

← external finance premium

0 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ln κ

Leverage Schedules. Figure2 shows the relationship between the (annualized) excess returns to capital (Rk/R) and leverage κ in three different environments: risk neutrality with agency frictions (blue dashed line), risk aversion without agency frictions (pink dotted line) and risk aversion with agency frictions (green solid line). For the risk-neutral case, our parameterization follows BGG: the monitoring cost

11 parameter, µ, is set to 0.12 and the standard deviation of idiosyncratic returns is

σω = 0.28. We set ρ = 0.1 for the risk-averse case. Risk-neutral agents under zero excess returns to capital would prefer not to borrow at all, as borrowing has no benefits and extra costs in the form of monitoring costs. Thus, the dashed blue line begins at zero. As capital returns increase, leverage quickly builds up as the probability of defaults and expected monitoring costs are small. Therefore, we note that the dashed blue line is relatively flat for small values of the excess returns to capital. As capital returns grow and the level of leverage approaches the value of 2, or the logarithm of leverage approaches 0.7, defaults start taking larger values, and monitoring costs become significant, thus preventing leverage from expanding rapidly. For the standard calibrated value of the annual excess returns of 0.03 and a leverage equal to 2, which is equivalent to 6 percent of returns to net worth, we observe a low elasticity of leverage to capital returns, corresponding to a very volatile external finance premium that could significantly amplify the business cycle. The behavior of risk-averse agents that are not exposed to agency frictions is described by the dotted red line. Here entrepreneurs start borrowing (ln κ > 0) only when the capital returns significantly exceeds the safe rate, as risk-averse investors require to be compensated for taking risk. We also observe that for reasonable values of excess returns, risk-averse agents choose a smaller leverage than risk-neutral agents exposed to agency frictions. This is intuitive as risk-neutral agents invest more aggressively when agency frictions are small. As noted in Lemma 1, the leverage of risk-averse agents is quite sensitive to changes in excess returns to capital. This fast adjustment of leverage means that the risk premium associated with risk aversion will be relatively stable in general equilibrium, thus only weakly amplifying economic fluctuations. Finally, the green line combines the effect of risk aversion and agency frictions. We see that agents that are exposed to both types of frictions invest less than just risk-averse agents, or risk-neutral agents with only agency frictions. One can see that the green line is positioned to the left relative to the dotted red line or the dashed blue line. Intuitively, for the small values of leverage agency frictions are small, and risk-averse agents that are exposed to agency frictions behave similarly to the risk-averse entrepreneurs that are not exposed to agency frictions, making the risk premium predominant. However, as leverage increases and agency frictions build up, leverage increases more slowly in response to higher capital returns, and the solid green line becomes more vertical, becoming more similar to the risk-neutral

12 agents with agency frictions. How are these results sensitive to the degree of risk aversion? From equations (2) and (3), we know that risk aversion affects the level of leverage, but not its elasticity to the returns to capital. Hence, higher risk aversion would simply shift the pink line to the left, without affecting its slope. Under partial equilibrium we can compare the choices of risk-neutral and risk- averse entrepreneurs, facing identical projects with identical excess returns and risk size. First, a risk-averse entrepreneur will choose a smaller leverage and face smaller agency costs. Second, he will be more sensitive to changes in excess returns. How- ever, under general equilibrium systematically lower leverage will lead to lower aggregate capital investment and higher excess returns, which will partially offset the effect of partial equilibrium. We thus move on to the general equilibrium model to assess the overall effect of risk aversion for business cycle ﬂuctuations.

3 General Equilibrium

We now extend the contract to a dynamic setting in which entrepreneurs maximize their expected consumption path and embed it in a standard dynamic New Key- nesian model, where aggregate returns to capital and returns to lenders are deter- mined endogenously. There are six agents in our model: ﬁnancial intermediaries, entrepreneurs, households, capital producers, wholesalers, and retailers.

3.1 Financial Intermediaries

The representative lender costlessly intermediates funds between households and entrepreneurs. It takes nominal household deposits, Dt, and lends out the nominal amount Bt to entrepreneurs. In equilibrium, deposits will equal loanable funds

(Dt = Bt). Households receive a predetermined real rate of return Rt on their deposits. Each individual loan is subject to idiosyncratic and aggregate risk, but the ﬁ- nancial intermediary diversiﬁes his portfolio of loans across different entrepreneurs, so only aggregate risk remains.

3.2 Entrepreneurs

There is a continuum of entrepreneurs, indexed by j. At time t, the entrepreneur j obtains a loan Bt(j) from the representative lender, which he combines with his net

13 worth Nt(j) to purchase capital Kt(j) at a unit price of Qt. That is,

QtKt(j) = Nt(j) + Bt(j). (12)

In period t + 1, the entrepreneur receives an idiosyncratic shock, ωt+1(j), that con- verts the raw capital Kt(j) into effective units ωt+1(j)Kt(j), which are rented out r to to perfectly competitive wholesalers at a rental rate Rt+1. The idiosyncratic shock ωt+1(j) is drawn in an i.i.d. fashion from a log-normal distribution that obeys 1 2 2 6 log(ωt+1(j)) ∼ N (− 2 σω,t, σω,t), so that the mean of ω is equal to 1. After production takes place, the entrepreneur sells the undepreciated effective units of capital

(1 − δ)ωt+1(j)Kt(j) at price Qt+1, and settles his position with the lender, either by repaying the loan or by defaulting. In this way, an entrepreneur that draws an id- k iosyncratic shock ωt+1 enjoys a rate of return ωt+1Rt+1 at time t + 1, where

r k Rt+1 + Qt+1(1 − δ) Rt+1 = . (13) Qt

Thus, each entrepreneur in period t has access to a stochastic, constant returns tech- k nology ωt+1(j)Rt+1. When the realization of ωt+1(j) exceeds ω¯t+1, the entrepreneur is able to repay the loan at the contractual rate Zt+1. That is,

k ¯ BtZt+1 = QtKtRt+1Rt+1. (14)

When instead ωt+1(j) < ω¯t+1, the entrepreneur declares bankruptcy and is monitored by the lender. The residual assets are distributed according to the risk-sharing agreement between the entrepreneur and the ﬁnancial intermediary that solves problem (17)-(19) below. Following BGG, to prevent entrepreneurs from accumulating too much net worth and from being fully self-ﬁnanced, we assume that in each period a random fraction 1 − γ of them dies. At the same time, new entrepreneurs are born with zero net e worth and supply inelastically one unit of labor in the aggregate at wage rate Wt . It is well known, for instance from the work on incomplete markets by Krusell and Smith(1998), that if agents are risk averse and subject to uninsurable idiosyncratic risk, there is no simple way of aggregating individual histories, and one would need to keep track of the wealth distribution of all the entrepreneurs. Consider the case where entrepreneurs receive a wage income in every period. Entrepreneurs with

6 The timing is meant to capture the fact that the variance of ωt+1 is known at the time of the ﬁnancial arrangement, t.

14 lower net worth would realize that, even in the case of a very low draw of ω, they would be able to make up for their losses with their wages. Given their low net worth today, the variance of their net worth tomorrow is still relatively low even for a high leverage, therefore it will be optimal to choose a high leverage. In contrast, an entrepreneur with high net worth today would lose almost all of his wealth following the same draw of ω, so he would optimally choose a lower leverage. The issue of different leverages does not arise in BGG because entrepreneurs are risk neutral and thus are indifferent to the variance of their future wealth. To resolve the aggregation problem, we assume that entrepreneurs work only in the ﬁrst period of their lives and that they consume all their net worth only upon the event of death. If entrepreneurs survive, they reinvest all their proceeds. Since monitoring costs are proportional to the assets, entrepreneurial technology has constant returns to scale. Furthermore, the utility function is constant-relative-risk-aversion so that the entrepreneurial optimization problem is homothetic with respect to net worth. Therefore, all entrepreneurs choose the same leverage regardless of their individual histories, which allows us to track only aggregate net worth. Entrepreneur j’s value function is

∞ X (Ce (j))1−ρ V e(j) = (1 − γ) γs t+s , (15) t Et 1 − ρ s=1

e where Ct+s(j) is the entrepreneur j’s consumption in case of his death,

e Ct (j) = Nt(j), (16) deﬁned as wealth accumulated from operating ﬁrms. The timeline for entrepreneurs is plotted in Figure3.

Figure 3: Timeline for Entrepreneurs Take out Rent capital Kt−1 new loan to wholesalers Bt with and receive Life/death lending k t return Rt of entrepreneur rate Zt+1 t+1

Period t Pay off Buy capital Kt Period t + 1 shocks are loan from to rent in shocks are realized period t − 1 period t + 1 realized (Bt−1Zt) or default

The dynamic problem can be formulated recursively as follows:

15 k 1−ρ ¯ (κtRt+1) g(¯ωt+1, ωt+1, Rt+1, σω,t)Ψt+1 max t , (17) ¯ E Kt,Rt+1,ω¯t+1,ωt+1 1 − ρ k 1−ρ ¯ s.t. Ψt = 1 + γEt (κtRt+1) g(¯ωt+1, ωt+1, Rt+1, σω,t)Ψt+1 ,, (18)

k ¯ βκtRt+1h(¯ωt+1, ωt+1, Rt+1, σω,t) = (κt − 1)Rt. (19)

As in BGG, Rt is the safe rate known at time t. Lenders require to be paid Rt on ¯ average, which implies that the contract must specify a triplet {ωt+1, ω¯t+1, Rt+1} k contingent on Rt+1. This assumption about the repayment to the lenders makes entrepreneurs effectively bear the aggregate risk. The following Proposition summarizes the solution to the dynamic contracting problem.

Proposition 2. Solving problem (17)-(19) and log-linearizing the solution gives the following relationship between leverage, the expected discounted return to capital, and the the standard deviation of idiosyncratic productivity

ˆk κˆt = νp(EtRt+1 − Rt) + νσσˆω,t, (20) with νp > 0 and νσ < 0. Proof. Equation (20) is obtained in the Appendix.

Following our assumptions about entrepreneurial wage and consumption, all entrepreneurs choose the same leverage regardless of their net worth so that aggregate leverage κt will simply be equal to the leverage chosen by each entrepreneur. Moreover, to a first-order approximation, the complex financial agreement between borrowers and lenders simplifies to the single equation (20) that links leverage to the expected excess returns to capital. Note that equation (20) is identical in form to the one in BGG (equation (4.17) in their paper), once augmented with time-varying

σω. The presence of risk aversion, however, changes the elasticity of leverage to the excess returns νp and to the volatility of idiosyncratic productivity. In this sense, our framework fully nests the BGG framework, which allows us to compare the two models in a meaningful way. For all the calibrations considered we have that

∂ν ∂ν p > 0 σ > 0. ∂ρ ∂ρ

Our ﬁnding that the elasticity νp increases with ρ is consistent with Figure2 in the

16 previous section. Moreover, a rise in σω, besides increasing defaults, mechanically raises the volatility of returns, to which risk-averse entrepreneurs optimally respond by cutting leverage more than their risk-neutral counterpart. In terms of aggregation, the fact that each entrepreneur chooses the same leverage implies that

κt = QtKt/Nt. (21)

Our assumptions about survival and new entry of entrepreneurs imply that aggregate net worth follows:

k k R ω¯t+1 e τ Nt+1 = γ QtKtRt+1 − (QtKt − Nt)Rt − µQtKtRt+1 0 ωφ(ω)dω + Wt+1 + t . (22)

The terms inside the brackets reﬂect the aggregate returns to capital to entrepreneurs, τ net of loan repayments and monitoring costs. The shock t represents an i.i.d. wealth transfer from the household to the entrepreneurs. Finally, aggregate entrepreneurial consumption is given by

e e e e Ct = (1 − τ )(1 − γ)(Nt − Wt ). (23)

We let τ e → 1 so that entrepreneurs maximize their expected utility without introducing an additional distortion relative to a standard New Keynesian model with capital. Thus the models with agency frictions and non-diversiﬁed risk will differ from the New Keynesian model only because of the presence of the external ﬁnance premium and the risk premium.

3.3 Households

The representative household maximizes its utility by choosing the optimal path of consumption and labor

( ∞ ) X C1−σH H1+η max βs t+s − χ t+s , (24) Et 1 − σ 1 + η s=0 H where Ct is household consumption, and Ht is household labor effort. The budget constraint of the representative household is

n Bt Bt+1 Ct = WtHt − Tt + Πt + Rt−1Dt − Dt+1 + Rt−1 − , (25) Pt Pt

17 where Wt is the real wage, Tt is lump-sum taxes, Πt is lump-sum proﬁts received from

final goods firms owed by the household, Dt are deposits in financial intermediaries

(banks) that pay a real non-contingent gross interest rate Rt−1 and Bt are nominal n bonds that pay a gross non-contingent interest rate Rt−1. Households maximize their utility (24) subject to the budget constraint (25) with respect to consumption, labor, bonds, and deposits, yielding the following ﬁrst order conditions:

−σH −σH Ct = βRtEt Ct+1 , (26)

C−σH −σH n t+1 Ct = βRt Et , (27) πt+1

−σH η WtCt = χHt . (28)

We deﬁne the gross rate of inﬂation as πt+1 = Pt+1/Pt.

3.4 Retailers

The ﬁnal consumption good consists of a basket of intermediate retail goods, which are aggregated together in a CES fashion by the representative household:

ε Z 1 ε−1 ε−1 ε Ct = cit di . (29) 0

The demand for retailer i’s unique variety is

−ε pit cit = Ct, (30) Pt where pit is the price charged by retail ﬁrm i. The aggregate price index is deﬁned as 1 Z 1 1−ε 1−ε Pt = pit . (31) 0 Retailers costlessly differentiate the wholesale goods and sell them to households at a markup over marginal cost. They have price-setting power and are subject to Calvo (1983) price rigidities. With probability 1 − θ each retailer is able to change its price

18 in a particular period t. Retailer i maximizes the following stream of real proﬁts:

∞ ∗ w ∗ −ε X s pit − Pt+s pit max θ Et Λt,s Yt+s , (32) p∗ P P it s=0 t+s t+s

w UC,t+s where P is the wholesale goods price and Λt,s ≡ β is the household’s (i.e. t UC,t shareholder’s) stochastic discount factor. The ﬁrst order condition with respect to ∗ the retailer’s price pit is

∞ ( −ε ) X p∗ p∗ ε P w θs Λ it Y it − t+s = 0. (33) Et t,s P t+s P ε − 1 P s=0 t+s t+s t+s

From this condition, it is clear that all retailers that are able to reset their prices in ∗ ∗ period t will choose the same price pit = Pt ∀i. The aggregate price level evolves according to 1 1−ε ∗ 1−ε 1−ε Pt = θPt−1 + (1 − θ)(Pt ) . (34)

Dividing the left and right hand side of (34) by the price level gives

1 ε−1 ∗ 1−ε 1−ε 1 = θπt−1 + (1 − θ)(pt ) , (35)

∗ ∗ where pt = Pt /Pt. Using the same logic, we can normalize (33) and obtain:

P∞ s −ε w ε s=0 θ Et Λt,s(1/pt+s) Yt+spt+s p∗ = , t P∞ s 1−ε (36) ε − 1 s=0 θ Et {Λt,s(1/pt+s) Yt+s}

w w Pt+s where p = and pt+s = Pt+s/Pt. t+s Pt+s

3.5 Wholesalers

Wholesale goods are produced by perfectly competitive ﬁrms and then sold to mo- nopolistically competitive retailers who costlessly differentiate them. Wholesalers hire labor from households and entrepreneurs in a competitive labor market at real e r wage Wt and Wt , and rent capital from entrepreneurs at rental rate Rt . Note that capital purchased in period t is used in period t + 1. Following BGG, the production function of the representative wholesaler is given by

α (1−α)Ω e (1−α)(1−Ω) Yt = AtKt−1(Ht) (Ht ) , (37)

19 e where At denotes aggregate technology, Kt is capital, Ht is household labor, Ht is entrepreneurial labor, and Ω deﬁnes the relative importance of household labor and entrepreneurial labor in the production process. Entrepreneurs inelastically supply one unit of labor, so that the production function simpliﬁes to

α (1−α)Ω Yt = AtKt−1Ht . (38)

One can express the price of the wholesale good in terms of the price of the ﬁnal good. In this case, the price of the wholesale good will be

w Pt w 1 = pt = , (39) Pt Xt where Xt is the variable markup charged by ﬁnal goods producers. The objective function for wholesalers is then given by

1 max A Kα (H )(1−α)Ω(He)(1−α)(1−Ω) − W H − W eHe − RrK . (40) e t t−1 t t t t t t t t−1 Ht,Ht ,Kt−1 Xt

Here wages and the rental price of capital are in real terms. The ﬁrst order conditions with respect to capital, household labor and entrepreneurial labor are

r 1 Yt Rt = α , (41) Xt Kt−1 Ω Yt Wt = (1 − α) , (42) Xt Ht

e (1 − Ω) Yt Wt = (1 − α) e . (43) Xt Ht

Given that equilibrium entreprenerial labor in equilibrium is 1, we have

e (1 − Ω) Wt = (1 − α)Yt. (44) Xt

3.6 Capital Producers

While entrepreneurs hold capital between periods, perfectly competitive capital producers hold capital within a given period, and use available capital and ﬁnal goods to produce new capital. Capital production is subject to adjustment costs, according to 2 φK It Kt = It + (1 − δ)Kt−1 − − δ Kt−1, (45) 2 Kt−1

20 where It is investment in period t, δ is the rate of depreciation and φK is a parameter that governs the magnitude of the adjustment cost. The capital producer’s objective function is

max KtQt − It, (46) It where Qt denotes the price of capital. The ﬁrst order condition of the capital producer’s optimization problem is 1 It = 1 − φK − δ . (47) Qt Kt−1

3.7 Goods Market Clearing

The goods market clearing condition is

Yt = Ct + It + Gt (48)

where Gt corresponds to government spending. Government spending is assumed to be proportional to output, so that Gt/Yt is constant. We assume that aggregate monitoring costs are rebated lump sum to households so they do not enter goods market clearing.

3.8 Monetary Policy

As in BGG, we assume that there is a central bank which conducts monetary policy n by choosing the nominal interest rate Rt according to the following rule

n n Rn n Rn log(Rt ) − log(R ) = ρ log(Rt−1) − log(R) + ξπt−1 + t . (49) where ρRn and ξ determine the relative importance of the past interest rate and past inflation in the central bank’s interest rate rule. Shocks to the nominal interest rate are given by Rn . It should be noted that the interest rule in BGG differs from the conventional Taylor rule, where current inflation rather than past inflation is targeted.

21 3.9 Shocks

Technology and the standard devation of idiosyncratic productivity follow AR(1) processes:

A A log(At) =ρ log(At−1) + t , (50)

σω σω σω log(σω,t) =(1 − ρ ) log(σω) + ρ log(σω,t−1) + t . (51) where A, and σω denote exogenous shocks to technology, and idiosyncratic volatil- 2 ity, and σω denote the steady state value for idiosyncratic volatility. Recall that σω is the variance of idiosyncratic productivity, so that σω is the standard deviation of idiosyncratic productivity. Nominal interest rate shocks are deﬁned by the BGG Rule in (49). Wealth redistribution shocks are deﬁned in (22).

3.10 Equilibrium

The nonlinear model has 24 endogenous variables and 24 equations. The endogenous variables are: R, Rn, H, C, π, p∗, pw, X , Y , W , W e, I, Q, K, Rk, N, k, ω¯, ω, R¯,

Ψ, λ, A, σω, where the new variable λ corresponds to the Lagrange multiplier for the optimality conditions used in the Appendix. The equations defining these endogenous variables are: (26), (27), (28), (35), (36), (38), (39), (42), (44), (45), (47), (13), (22), (23), (21), (48), and financial contract participation (19), discounting condition (18) and optimality conditions (81), (82), (83), (84). The exogenous processes for technology and idiosyncratic volatility follow (50) and (51), respectively. Nominal interest rate and wealth redistribution shocks are defined in (49) and (22), respectively. The log-linearized system is reported in the Appendix. We solve the model using a method that is similar in spirit to Reiter(2009) and Winberry(2018). Their methods involve two steps. First, the steady state of the model in which there are no aggregate shocks but still idiosyncratic shocks is computed by taking a finite approximation of the infinitely-dimensional wealth distribution of individual agents. Noticeably, this steady state does not display certainty equivalence because of the presence of the idiosyncratic shocks. Second, they solve for the aggregate dynamics using perturbation techniques around this steady state.7 Our procedure involves the same two steps, with the difference that we do not need to approximate the wealth distribution in the steady state because the homotheticity

7Perturbation methods are appropriate because aggregate shocks are small.

22 of the entrepreneurial problem with respect to net worth makes individual wealth irrelevant for aggregate dynamics even in the fully non-linear model.

4 Quantitative Analysis

In section 3.2, we discussed the role of risk aversion in determining the elasticities of leverage with respect to the expected discounted returns to capital and to the standard deviation of idiosyncratic productivity. In particular, we have highlighted the fact that in partial equilibrium leverage becomes more responsive to the latter with higher risk aversion. While the partial equilibrium analysis suggests a higher sensitivity of leverage and, hence, a greater ampliﬁcation under risk aversion, the general equilibrium effect depends on the endogenous adjustment of prices and returns. In this section, we investigate quantitatively the general equilibrium effects of technology, monetary, idiosyncratic volatility, and wealth shocks for different coefﬁcients of risk aversion.

4.1 Calibration

Our baseline calibration largely follows BGG. We set the discount factor β = 0.99, the household’s intertemporal elasticity of substitution to 1 so that utility is logarithmic in consumption (σH = 1), and the elasticity of labor supply to 3 (η = 1/3). The share of capital in the Cobb-Douglas production function is α = 0.35, while the share of entrepreneurial labor 1 − Ω is 0.01. Quarterly capital depreciation is δ =

0.025. Capital adjustment costs are φk = 10, to generate an elasticity of the price of capital with respect to the investment-capital ratio of 0.25. The steady-state share of government expenditure in total output, G/Y , is set to 0.20. Regarding nominal rigidities, we set the Calvo parameter θ = 0.75, so that the average length of time between price adjustments is four quarters. As our baseline, we follow the BGG monetary policy rule and set the autoregressive parameter on the nominal interest rate to ρRn = 0.9 and the parameter on lagged inﬂation to ξ = 0.11. We set the persistence of the shocks to technology at ρA = 0.95, and keep the standard deviation at 1 percent. Following BGG, for monetary shocks we consider a 25 basis point shock (in annualized terms) to the nominal interest rate with persistence ρRn = 0.9. For our purposes, the most important part of the calibration regards the part of the model relating to ﬁnancial frictions, summarized in Table1. We map the

23 risk-neutral and risk-averse entrepreneurs to the corporate and non-corporate non- ﬁnancial sector of the US economy, respectively. Our risk-neutral calibration follows BGG and targets leverage, returns, and defaults. Speciﬁcally, we choose parameters to match the historical leverage of 2 obtained from the National Income and Product Accounts (NIPA) for the sample 1990-2010 and defaults of 3.8 for the delinquency rate on bank loans for the same period.8 For the returns, we target an equity premium of 5% or a level of returns to public equity of 9.2%, consistent with the estimates of Kartashova(2014) over the period 1990-2010. These returns to equity imply an annualized excess return to capital of 250 basis points.9 To match these targets we set the monitoring costs µ, the probability of survival for entrepreneurs

γ, and the steady-state standard deviation of idiosyncratic productivity σω to 0.12, 0.977 and 0.28, respectively. For the risk averse economy, we additionally target microeconomic evidence on the volatility of firm-specific productivity. Specifically, we still choose µ and γ to match returns and defaults. Given these two moments, there are different combinations of risk aversion, ρ, and firm-specific volatility, σω, that achieve the same leverage because both higher risk and higher risk aversion reduce the incentive to borrow. We choose the combination that matches the micro evidence on firm-specific productivity and investigate the sensitivity of our results in section 4.4. We consider three alternative parameterizations of the risk-averse economy. Be- cause risk aversion affects both the steady-state leverage and the elasticity to capital returns, we first consider a calibration, Risk Averse I, that isolate the effect on the elasticity while keeping leverage the same as in the risk-neutral economy. To choose

ρ and σω we look at the evidence from firm-level data. Using data on sales growth, Castro, Clementi, and Lee(2010) obtain a value for the firm-specific volatility of TFP between 0.04 and 0.12. Comin and Mulani(2006), Davis, Haltiwanger, Jarmin, and Miranda(2007) and a more recent study by De Veirman and Levin(2014) report the volatility for the annual growth of sales to be between 0.24 and 0.3. Through the lens of our model, this range implies that a firm’s standard deviation of idiosyncratic 10 productivity, σω, is between 0.08 and 0.1. We settle for a value of σω of 0.08, which requires a value of ρ = 0.5 to match a leverage of 2. Considering that households’ risk aversion is set to 1, we believe 0.5 is a reasonable value for entrepreneurial risk

8The data were obtained from the St. Louis Federal Reserve Bank’s online data base, FRED. The FRED mnenomic for the delinquency rate on commercial bank loans is DRALACBS 9With a leverage of 2, a 250bp excess return on assets implies a 5% excess return on net worth. 10To obtain these values, we simulate our model in the steady state, where aggregate shocks are absent, but idiosyncratic shocks still affect ﬁrms.

24 aversion and we stick to this number in all the calibrations. In our sensitivity analysis we show that the results are robust to any couple of ρ and σω that deliver the same leverage. The calibration Risk Averse 2 matches the leverage of 1.6 for the non-corporate non-financial sector from NIPA and estimates on the returns on private equity. For the latter, Moskowitz and Vissing-Jorgensen(2002) and Kartashova(2014) find that over the long run the historical returns to private and public equity are equal. Given that the leverage in the non-corporate sector is lower than in the corporate sector, we set excess returns to capital to 310bp, in order to generate the same returns to equity as in the risk-neutral economy. We increase the idiosyncratic volatility to 0.1 to generate the lower leverage. As expected, in order to match the benchmark values for default rates under risk aversion, it was necessary to establish a more favorable setting than under risk neutrality in terms of monitoring costs. In a shorter sample of 1990-2010, Kartashova(2014), finds that there is an equity premium on private equity over public equity, and estimates private returns to equity to be as high as 16.5%. We consider this scenario in Risk Averse 3 and match those returns to private equity by setting excess returns to capital to 7.1%, while keeping leverage at 1.6%.11 For all cases, we set the persistence of the standard deviation of idiosyncratic productivity to ρσω = 0.9706, following the estimates of Christiano, Motto, and Rostagno(2014, CMR).

Table 1: Calibration

Symbol Description Risk Neutral Risk Averse 1 Risk Averse 2 Risk Averse 3 A. Calibrated parameters ρ Risk aversion 0.0 0.5 0.5 0.5 σω Standard deviation idiosyncratic productivity 0.28 0.08 0.10 0.15 γ Survival probability 0.977 0.976 0.979 0.961 µ Monitoring costs 0.120 0.120 0.026 0.065 B. Implied steady-state values κ Leverage 2 2 1.6 1.6 ln(Rk/R) Premium (%, annualized) 2.5 2.5 3.1 7.1 Φ(¯ω) Default rate (%, annualized) 3.8 0.0 3.8 3.8 C. Implied elasticities νp Elasticity of leverage to returns 21.7 157.7 147.4 61.4 νp Predicted elasticity of leverage to returns - no agency costs ∞ 160 130 56 νσ Elasticity of leverage to id. risk -0.69 -1.95 -2.00 -1.94 νσ Predicted elasticity of leverage to id. risk- no agency costs 0 -2 -2 -2

ˆk ˆ κˆt = νp(EtRt+1 − Rt) + νσσˆω,t

11This choice generates an 11.5% excess return over the safe rate. With a average Federal Funds rate of 4.3 over the sample we obtain a return of about 16% for private equity.

25 4.2 Leverage, Capital Returns and Ampliﬁcation

For all calibrations we consider, entrepreneurial risk aversion alters the way in which the economy reacts to shocks. This different sensitivity is captured by the different values of the two elasticities νp and νσ in equation (20) for the different calibrations. We summarize these elasticities in Panel C of Table 1. First, let us consider the case where agents are risk-neutral and there are no agency costs. In this case the elasticity with respect to risk, νσ = 0, as borrowing is frictionless and risk fully diversified. Frictionless borrowing coupled with risk neutrality also means that the elasticity with respect to excess returns, νp, is infinite: even the smallest increase in excess capital returns would make entrepreneurs be willing to purchase an infinite amount of capital, owing to constant returns. This implies that in equilibrium excess returns are zero.

Second, consider the case of risk-neutral agents with agency costs. Here, νσ is negative, as extra risk increases the likelihood of defaults, exacerbates the asymmetric information problems between lenders and borrowers, and reduces the optimal leverage. Also, the sensitivity of leverage to excess capital returns drop from ∞ to 21.7, as increasing leverage in response to higher excess capital returns becomes more difﬁcult because it raises the likelihood of default. Agency frictions will thus have a strong ampliﬁcation effect in general equilibrium, as borrowers cannot use leverage to stabilize prices and net worth. Third, consider the case of risk-averse agents with no agency costs, which we have discussed in section 2.1. As in equation (2), the elasticity of leverage to risk is equal to -2 following from mean-variance optimization. This partial equilibrium effect of higher risk is stronger for risk-averse agents than for risk-neutral agents with agency costs. The elasticity νp varies from 56 to 160 for risk-averse agents facing no agency costs, which means that these agents can strongly increase their leverage in response to higher excess returns. This is intuitive, as entrepreneurial choices are constrained only by their tolerance for risk, and risk-averse borrowers are ready to invest if the returns become attractive enough. The ability to increase leverage strongly in response to changes in returns will have a stabilizing effect on prices, net worth, and on the business cycle overall. Finally, adding agency costs for risk-averse agents does not materially affect their decision making. The precautionary motives dominate, as the sensitivity of leverage to risk and excess returns are almost the same when we compare risk-averse agents with or without agency costs.

26 4.3 Simulations

In this section, we simulate our models and study the impulse responses of key macroeconomic variables to different shocks, comparing the case of risk aversion and the case of risk neutrality relative to a frictionless model. The frictionless economy is a New Keynesian model without agency frictions where idiosyncratic risk is fully diversiﬁed. It features the steady-state parameter values of the risk-neutral economy, except that the elasticity of leverage to excess capital returns νp → ∞ and the elasticity of leverage to the standard deviation of idiosyncratic productivity 12 νσ = 0.

4.3.1 Wealth and Risk Shocks

Figure7 shows the impulse responses to a wealth shock that transfers in a lump- sum fashion 1% of the initial net worth of entrepreneurs to households. A drop in wealth reduces the net worth of entrepreneurs and increases their leverage. In the frictionless economy, this wealth transfer does not affect real economic outcomes, as entrepreneur costlessly substitute their own equity with debt. With risk-neutral borrowers and agency frictions, this substitution leads to higher defaults and an increase in the external finance premium. Consequently, entrepreneurs reduce their demand for capital, putting downward pressure on the price of capital and reinforc- ing the fall in net worth, investment, and output through the financial accelerator mechanism described in BGG. For the cases of positive risk aversion, lower net worth has a small effect on agency frictions and, therefore, the external finance premium remains almost flat. On the other hand, the initial increase in leverage due to the lower net worth leads to a larger exposure to risk. In partial equilibrium, the larger exposure to risk would make entrepreneurs cut back on their investment demand, potentially leading to a severe recession. However, in general equilibrium the risk premium rises to com- pensate entrepreneurs for the additional risk. Given the high sensitivity of leverage to excess returns to capital for risk-averse entrepreneurs, we observe that even a modest increase in the risk premium is sufficient to motivate borrowers to invest and stabilize the economy. Quantitatively, the impact response of output falls from

12The frictionless economy can achieve the same steady state as the risk-neutral economy with agency frictions by setting a tax rate on the purchases of physical capital by entrepreneurs equal to the steady-state external ﬁnance premium. The tax revenue is rebated back to entrepreneurs in a lump-sum fashion, except the portion that corresponds to the steady-state monitoring costs, which is rebated to households.

27 0.32% in the risk-neutral economy to about 0.06% in the risk-averse cases. Figure8 shows the impulse responses to a risk shock. In the frictionless world, this shock does not affect economic variables. In the risk-neutral economy, a spike in the volatility of idiosyncratic productivity risk raises the probability of a low realization of ω, thus increasing the external finance premium required to cover the higher costs of default. Entrepreneurs respond by borrowing less and by reducing the quantity demanded of capital goods. This leads a fall in the price of capital and net worth, setting in motion the financial accelerator. The impact response of output to risk shocks is substantially muted when entrepreneurs are risk averse. On the one hand, the increase in the idiosyncratic volatility does not have a large effect on defaults or overall monitoring costs, and consequently on the external finance premium. On the other hand, the increase in idiosyncratic risk has a negative effect on the borrower’s utility and leads to an increase in the risk premium. Similarly to the wealth shock, even a moderate increase in the risk premium is sufficient to incentivize borrowers to invest into physical capital, thereby limiting the decline in the price of capital, net worth and output relative to the risk-neutral case. Indeed, under risk aversion output falls by 0.05% in Risk Averse 2 and 0.08% in Risk Averse 3 in response to a one-standard-deviation risk shock, compared to 0.12% under risk neutrality. The dampening effect of risk aversion is more pronounced for a wealth shock than for a risk shock. This is intuitive because, while risk aversion makes entrepreneur more sensitive to rises in average returns, it makes them also more fearful of their volatility. Thus, other things equal, entrepreneurs are less willing to invest when expected return rise as a result of an increase in risk than they are when returns rise as a consequence of wealth redistribution.

4.3.2 Technology and Monetary Shocks

Figure9 plots the impulse responses to a technology shock. The shock immediately stimulates the demand for capital, leading to an investment boom. The increase in investment raises asset prices, which boost net worth and reduce leverage. In the frictionless world, the change in net worth leads to a substitution of debt with equity and has no other effect on the economy. For risk-neutral borrowers, lower leverage leads to a large decline in the external ﬁnance premium and sets in motion the ﬁnancial accelerator. For risk-averse borrowers, lower leverage is instead mostly associated with a decline in the risk premium. As above, risk-averse borrowers have a higher sensitivity of leverage to the excess returns to capital, thus the latter are more

28 stable, resulting in less volatile asset prices and in a milder amplification compared to the risk-neutral case. One appealing feature of general equilibrium models with costly state verification and risk-neutral borrowers is that they amplify monetary shocks and make the responses of macro variables more persistent, thanks to the endogenous dynamics in net worth. Figure 10 shows the impulse responses of the two models with varying degrees of risk aversion with respect to a 25 basis-point shock to the interest rate. The model with higher risk aversion displays an output response that is twenty percent smaller on impact vis-a-vis the risk-neutral case and is practically identical to the frictionless economy. As for the technology shock, the desire to stabilize excess returns to capital results in more moderate business cycle fluctuations under risk aversion.

4.3.3 Welfare

In our discussion about the dynamic responses in the different economies we put the emphasis on the volatility of output following financial shocks across different models. Here we focus on the consequences of the volatility of output as well as other macroeconomic variables for the welfare of the household.13 The welfare measures we report are computed based on a second-order approximation to the nonlinear equilibrium conditions of the models. Specifically, we calculate the perpetual increase in consumption that would make the household in a given environment as well off as in a deterministic economy without aggregate shocks. The shocks sizes are chosen to generate a standard deviation of output of 0.03 in the risk-neutral economy. We follow closely the computational strategy outlined in Schmitt-Grohé and Uribe(2005). Table 2 summarizes our results. First, we find that in the risk neutral economy wealth shocks and risk shocks produce a welfare cost that correspond to a perpetual flow of consumption of 0.042 percent and 0.173 percent, respectively. As a refer- ence point, the business cycle costs of technology shocks that we calculate for the corresponding frictionless economy is 0.056 percent, comparable with the original estimates of Lucas(1987). Second, these losses are significantly lower in a model

13The model has two agents: households and entrepreneurs. Our preferred welfare measure fo- cuses on the household for three reasons. First, entrepreneurial consumption account for a trivial portion of aggregate consumption. Second, entrepreneurs preferences change across different speci- ﬁcations of the model, which would make welfare comparison not meaningful. Third, even in absence of preference changes, the choice of Pareto weights would not be obvious.

29 with risk-averse entrepreneurs: 0.020 percent for wealth shocks and 0.027 for risk shocks in our Risk Averse 2 benchmark. That is, these shocks are half as costly or 4.5 times less costly, depending on the type of shock. If we interpret the risk averse economy as an economy where idiosyncratic risk is not diversifiable and the risk neutral economy as an economy where entrepreneurs can insure idiosyncratic shocks by trading state-contingent securities, these findings suggest that “completing” the markets entails a welfare cost that can range from half the size to 2.5 times the typically estimated costs of business cycle fluctuations. Our findings also suggests that in sectors where idiosyncratic risk is non-diversified, risk aversion can act as a self-insurance mechanism that reduces exposure to the adverse consequences of financial shocks. On the contrary, in sectors where idiosyncratic risk is more diversifies, such as the non-financial corporate sector, that resemble more our risk-neutral economy, macroprudential regulation may be useful to prevent excess leveraging ex-ante, which magnifies the impact of financial shocks. It should also be noted that, if on the one hand, the presence of risk aversion and uninsurable risk makes financial shocks less harmful, on the other hand, it can lead to higher steady-state premia on risky capital, which implies a lower steady- state level of capital and consumption (Angeletos, 2007). Thus, in implementing measures against excess leveraging, macroprudential policy will have to trade off the benefits of reduced consumption and employment volatility need with the costs of a lower level of average consumption. A full quantitative analysis of these trade- offs is well beyond the scope of this paper and we leave it as an open question for future research.

Table 2: Welfare Analysis

Risk Neutral Risk Averse 1 Risk Averse 2 Risk Averse 3 Frictionless

Welfare Cost of Business Cycle (%)

Technology Shock 0.067 0.046 0.060 0.072 0.056 Monetary Shock 0.039 0.018 0.014 0.016 0.018 Risk Shock 0.173 0.037 0.027 0.069 0.000 Wealth Shock 0.042 0.017 0.020 0.013 0.000

Notes : The entries represent the welfare losses for the household value function relative to the deterministic steady state. To turn these utility ﬂows into consumption equivalents, we divide this utility difference by the steady state marginal utility of consumption and express them as a fraction of aggregate consumption. The welfare gain thus represents the perpetual ﬂow increase in consumption that equates lifetime utility under the two environments. For example, 0.039 is a perpetual increase in annual consumption of 0.039 percent (or a one-time increase of 3.9 percent).

30 4.4 Sensitivity Analysis

The above section demonstrated that risk aversion dampens the effects of the financial accelerator under some specific calibrations. As a first check, we explore the effects of changing risk aversion leaving the other parameters unchanged at the values set in the case Risk Averse 2. As varying risk aversion changes the steady state of the model, to conduct this exercise we solve about 100 non-linear systems of equations and the corresponding DSGE models.

Figure 4: Sensitivity Analysis I

3 5 1 Risk Shock Wealth Shock 2.8 0.9 4.5

2.6 0.8 4 2.4 0.7

3.5 2.2 0.6

2 3 0.5 Leverage 1.8 0.4 2.5

1.6 0.3 2 Relative Output Response on Impact

1.4 Excess Returns to Capital (annualized %) 0.2

1.5 1.2 0.1

1 1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Note: The ﬁgure shows the effects of changing risk aversion (x-axis) on the steady state leverage, returns to capital, and on the response of output to ﬁnancial shocks. The response of output is depicted as a fraction of the response of output in the risk-neutral economy of calibrated as in Table 1.

Each point in Figure4 corresponds to one DSGE model for a particular value of ρ. The left panel shows that with higher risk aversion entrepreneurs reduce their capital investment, taking on less leverage. In general equilibrium a lower amount of physical capital leads to higher rates of return, shown in the middle panel. These two panels show that varying entrepreneurial risk aversion between 0 and 1 produces realistic values of leverage and excess capital returns. For all these values, the right panel shows that risk aversion mutes the response of output to ﬁnancial shocks. For example, for ρ equal to 0.5, a risk shock produces a response of output in the risk-averse economy that is only 40% as large as in the risk-neutral economy. This relative response can vary between 30% and 45% depending on the steady state

31 considered. For wealth shocks, the stabilizing effect of risk aversion is larger, 82% when ρ = 0.5, ranging between 13% and 20%. Considering that the extent of the stabilizing effect can vary, how do we choose a particular value of risk aversion, given that we do not observe preferences directly? In section 4.1, we resolved this issue by choosing a value of the volatility of idiosyncratic productivity, σω, that matched the micro evidence and we subsequently backed out ρ to keep matching leverage. However, we recognize that idiosyncratic ﬁrm volatility is notoriously difﬁcult to measure. In Figure5 we investigate to what extent our particular choice of σω and ρ affects our results by considering all possible combinations of these two parameters that achieve the same leverage.

Figure 5: Sensitivity Analysis II

0.4 1 Risk Shock 0.9 Wealth Shock 0.35

0.8 0.3 0.7

0.25 0.6

0.2 0.5

0.4 0.15

0.3 0.1

Relative Output Response on Impact 0.2

0.05 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Note: The ﬁgure shows the effects of changing risk aversion (x-axis) when leverage, returns, and defaults are kept the same. The response of output in the right panel is depicted as a fraction of the response of output in the risk-neutral economy of calibrated as in Table 1.

Like in Figure4, each point corresponds to one DSGE model for a particular value of ρ. The left panel shows that as risk aversion increases we need to lower the quantity 14 of risk (captured by σω) to keep leverage the same. On the right panel, we can see that as soon as risk aversion reaches the very low value of 0.1 the relative response of output flattens out at the values found in Section 4.3.1 for the Risk Averse 2 case. These findings confirm that the stabilizing effects of risk aversion are robust to

14 In practice, changing ρ and σω changes also average returns and defaults. In our experiment we are adjusting µ and γ as well to keep matching those targets. The changes to these parameters are tiny.

32 changes in parameter values as long as we match measurable and realistic targets.

4.5 Two-sector Model

To highlight the effects of risk aversion and uninsurable risk in the clearest way, we have so far focused on two stark cases where all entrepreneurs are either risk neutral or risk averse. In both cases, all entrepreneurs differed only by their net worth but otherwise chose the same leverage, which implied that the realizations of idiosyncratic shocks and individual wealths were irrelevant for aggregate dynamics. In practice, of course, there is considerable heterogeneity among firms along many dimensions. For example, there are sectors where idiosyncratic risk is more diversified, e.g. the non-financial corporate sector, and sectors where individual entrepreneurs are more exposed to non-diversified risk. To see how heterogeneity affects our results, we add to our model the assumption that there are two types of entrepreneurs with different preferences. Heterogeneity of preferences across different groups of entrepreneurs implies that wealth redistribution across sectors will matter of aggregate dynamics. One key advantage of the tractability of our framework and its simple aggregation is that extending it to multiple sectors is quite straightforward and not computationally more intensive. In the first sector, entrepreneurs are risk-neutral and are calibrated to match the data for the corporate sector. The risk-averse borrowers populate the non-corporate sector. They have lower leverage and observe higher expected returns to physical capital. Each sector produces an intermediate good with labor and a sector-specific capital. The intermediate products are then used to produce a single wholesale good via a CES aggregator. Retailers combine wholesale goods into a single final good that can be consumed or converted to a sector-specific capital subject to capital adjustment costs. Labor is fully mobile so that wages are equal across sectors. We give a precise mathematical description of the two-sector model in the Appendix. Figure6 demonstrates the effect of a positive wealth shock in the heterogeneous sector model. The shock redistributes 1% of wealth from the households to the entrepreneurs. In the top panels, the black line corresponds to the one-sector model, where all entrepreneurs are risk neutral. The blue (red) lines reflects the response with heterogeneous sectors with a low (high) elasticity of substitution,ζ, between sectors of 0.5 (2). We find that regardless of the elasticity of substitution, the model that includes a sector with risk-averse entrepreneurs generates roughly half of the response of aggregate output and investment following a wealth shock compared to

33 the one-sector risk-neutral model. While the elasticity of substitution has little effect on aggregate dynamics, it strongly affects the sectoral ones, as can be seen in the bottom panels. As expected, for both low and high substitution, the sector with risk- neutral agents generates stronger volatility of the investment. However, while with low substitution the volatility for investment is almost 5 times larger for the risk- neutral agents relative to risk-averse borrowers, with high elasticity of substitution this relative difference is only 2.5 times.

Figure 6: Two-Sector Model: Impulse Responses to a Wealth Shock

Aggregate Output Aggregate Investment 0.35 2 1 Sector - Risk Neutral 1 Sector - Risk Neutral 2 Sectors - Low Substitution 1.8 2 Sectors - Low Substitution 0.3 2 Sectors - High Substitution 2 Sectors - High Substitution 1.6

0.25 1.4

1.2 0.2 1

Percent 0.15 Percent 0.8

0.1 0.6 0.4 0.05 0.2

0 0 4 8 12 16 20 4 8 12 16 20

Sectoral Investment - Low Substitution Sectoral Investment - High Substitution 2 2 Risk Neutral Sector Risk Neutral Sector 1.8 Risk Averse Sector 1.8 Risk Averse Sector 1.6 1.6

1.4 1.4

1.2 1.2

1 1 Percent Percent 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 4 8 12 16 20 4 8 12 16 20

Note: This ﬁgure shows the impulse responses to a 1% wealth redistribution shock in the 2 sector and 1 sector model. In the two-sector economy, the sectors are calibrated following the parameterization Risk Neutral and Risk Averse 2 of Table 1. For low and high substitutability we set ζ = 0.5 and ζ = 2, respectively.

34 5 A Test Using Firm-Level Data

Our main theoretical results follow from the observation that in the presence of uninsurable risk, more risk-averse firms leverage themselves less in steady state, and use this spare borrowing capacity to invest more aggressively when expected returns to capital rise. In this section, we attempt to test for the presence of this relationship in the data. Specifically, we look at the firm-level relationship between investment and expected returns for varying degrees of risk aversion. A key challenge in testing the prediction of our theory is to find an empirical measure of risk aversion. Following Panousi and Papanikolaou(2012), we proxy for risk aversion using data on insider ownership, under the assumption that a higher degree of managerial ownership is a sign of a less diversified business, which would result in a more risk-averse behavior. Indeed, in our model with positive risk aversion, entrepreneurs bear all the risk associated with their business, which would correspond to a case where ownership is extremely concentrated. In contrast, in the BGG model, entrepreneurs have linear preferences and maximize expected equity value, and therefore do not take risk into consideration. This scenario is akin to a case in which firm owners are fully diversified.

5.1 Data Description

Our dataset comes from Panousi and Papanikolaou(2012). The sample of firms includes all publicly traded firms from Compustat, except firms from the financial sector (SIC code 6000-6999), utilities (SIC code 4900-4949) and government-regulated industries (SIC code > 9000). Firms with missing values for investment, cash flows, size, leverage, stock returns and negative book values of capital were also dropped from the original dataset.15 The data on managerial ownership, which spans from 1988 to 2005, was obtained from the Thomson Financial Institutional Holdings database. The available measure of insider ownership in year t is the reported yearly holdings of a firm’s shares held by firm officers at the end of that year or at the latest filing date, as a fraction of the shares outstanding in the firm. After merging these data with Com- pustat, removing missing observations and zero ownership values, we are left with a sample of 32,444 observations for 5,415 firms at an annual frequency from 1988 to 2005. 15The dataset is described in greater details in Panousi and Papanikolaou(2012).

35 We divide firms into quintiles according to their level of insider ownership. Table 3 reports time-series averages of the median firm characteristics within ownership quintiles. Insider ownership varies substantially across the sample, from a median of 0.06% in the bottom quintile to a median of 17.43% in the top quintile. As the Table documents, firms with higher insider ownership have higher profitability, as measured by Tobin’s Q and cashflow to capital, and tend to invest more. In line with our model’s predictions, these firms also exhibit a more pronounced precautionary behavior: they have lower leverage, hold more cash and pay fewer or no dividends.

Table 3: Summary Statistics

Level of Insider Ownership Low 2 3 4 High Insider Ownership 0.06 0.32 1.01 3.40 17.43 Investment to capital 8.76 9.41 10.57 12.18 12.47 Tobin’s Q 1.64 1.80 1.93 2.27 2.156 Cashﬂow to capital 6.58 6.64 7.47 8.71 8.38 Market Capitalization 52.32 22.67 10.80 8.06 6.15 Assets 57.37 21.81 10.18 6.24 4.40 Cash to Book Assets 6.58 6.64 7.47 8.71 8.38 Physical Capital to Book Assets 63.47 58.13 54.11 47.36 46.61 Dividend to Cash Flow 3.69 2.14 0.74 0.00 0.00 Book Debt to Book Assets 21.96 20.92 19.39 17.29 16.88

Notes : The table reports time-series average of ﬁrm characteristics sorted by the level of insider ownership. Except for Tobin’s Q, all characteristics are in percent. Firm market capitalization and book assets at time t are scaled by the average at time t across ﬁrms.

5.2 Regression Analysis

Next, we test whether investment rates for ﬁrms with higher insider ownership exhibit stronger responsiveness to excess returns to capital or investment opportunities compared to investment for ﬁrms with lower insider ownership. We frame our analysis using a variation of a typical investment regression:

X 0 (I/K)i,t = β0 + β1Xi,t + βjXi,t × INSDi,j,t + Zi,tγ + ηi + gt + vi,t (52) j∈{2,3,4,H}

In equation (52), Xi,t is ﬁrm i’s expected return on capital investment, which we proxy with ﬁrm i’s current sales-to-capital ratio (Y/K)i,t, as in Gilchrist, Sim, and

Zakrajˇsek(2014). The ﬁrm ﬁxed effect ηi controls for unobserved heterogeneity in

36 firm investment opportunities. The time fixed effect gt controls for the state of the macroeconomy, including the safe interest rate Rt. The matrix Zi,t is a set of control variables. The term Xi,t × INSDi,j,t represents the interaction of expected returns with our proxy for risk aversion, managerial ownership. This allows us to test for the key prediction of our model: firms with higher insider ownership should be more responsive to investment opportunities, implying that the coefficients on the interaction terms should be positive. We have excluded the term Xi,t × INSDL from the regression so that β1 is interpretable as the slope coefficient for firms in the bottom quintile, while βj represents the difference in the slopes between firms in the j−th and bottom quintiles. Following Eberly, Rebelo, and Vincent(2008), we use a semi- log specification as it provides a better fit to our data, given the positive skewness in the sales-to-capital ratio and Tobin’s Q. Table4 reports our regression results. Column 1 shows that the general rela-

Table 4: Investment Regressions

Dependent variable: (I/K)i,t (1) (2) (3) (4) (5) log(Y/K)i,t 0.121 0.162 0.154 0.115 0.086 (17.19) (9.72) (9.74) (8.15) (6.94)

log(Y/K)i,t × INSD2 0.011 0.010 0.016 0.017 (0.50) (0.47) (0.75) (0.90)

log(Y/K)i,t × INSD3 0.049 0.044 0.040 0.022 (1.85) (1.72) (1.60) (0.94)

log(Y/K)i,t × INSD4 0.114 0.109 0.095 0.075 (3.87) (3.70) (3.24) (2.72)

log(Y/K)i,t × INSDH 0.104 0.096 0.085 0.046 (3.83) (3.57) (3.22) (1.99)

Observations 32,444 32,444 32,444 32,444 32,444 R2 0.66 0.77 0.77 0.78 0.79 Fixed effects F F F, T F, T F, T Controls Q, K No No Q Q, K

Notes: F and T stand for firm and time fixed effects, respectively. Q and K denote a firm’s ˆ 1 PN Tobin’s Q (log Qi,t−1) and size (log Ki,t−1 = log(Ki,t−1/ N i=1 Ki,t−1), respectively. Time and firm fixed effects are interacted with the quintile dummies. Standard errors are clus- tered at the firm level, and t-statistics are reported in parenthesis. tionship between a firm’s sales and investment rate is positive and significant, con-

37 trolling for firm and time fixed effects as well as firm size and Tobin’s Q.16 Columns 2-3 introduce the interactions between a firm’s sales and insider ownership quintiles, and sequentially control for firm and time fixed effect. Consistent with the theory, the coefficients on these terms, and in particular on the one capturing the different slope between the top and bottom quintiles, log(Y/K)i,t × INSDH , are positive and highly statistically and economically significant. The estimates in column 3 suggest that a rise in a firm’s sales-to-capital ratio by 10 percent causes the investment rate to increase by 1.54 percentage points for firms with low insider ownership, and by 2.50=1.54+0.96 percentage points for the group with the highest insider ownership. Thus, the effect of an increase in the sales-to-capital ratio on investment is 62 percent larger for firms with high insider ownership. The magnitude of this differen- tial effect does not change once we control for Tobin’s Q and a firm’s size in columns 4-5. The robustness of our results has been confirmed along a number of dimensions: we have used Tobin’s Q as an alternative proxy for investment opportunities; we have included additional controls in the regression such as the lagged equity-to- assets ratio and the lagged cashflow-to-capital ratio. We have also controlled for lagged investment, which is known to be an important predictor of current investment that could eliminate the importance of other explanatory variables (Gilchrist and Himmelberg, 1995; Eberly, Rebelo, and Vincent, 2012). In our case, the presence of this additional regressor does not substantially increase the R2, which is already high given the presence of fixed effects. Overall, results remain statistically significant and qualitatively similar across these robustness checks.17 Our empirical findings seem to validate the theoretical channel predicted by our model. They suggest that firms with higher insider ownership exhibit a stronger precautionary behavior, hold more cash and invest more when investment opportunities appear on the horizon. Our model also predicts that in general equilibrium these firms that are more exposed to idiosyncratic risk act as a stabilizing force on the business cycle. Taken together, these results imply that privately-held entrepreneurial businesses and/or the less diversified publicly-traded firms could not have been the source or the amplifier of the Great Recession shock. Indeed, these firms do not appear to be as much directly affected by the credit channel because of the strong precautionary motives arising from undiversifiable idiosyncratic risk. Rather, the origin and transmission of financial shocks lies with the well-diversified

16Firm size is measured as a ﬁrm’s capital stock scaled by total capital. 17These results are available upon request.

38 publicly-traded ﬁrms.

6 Concluding Remarks

In this paper, we studied the role of uninsurable idiosyncratic investment risk and risk aversion for business cycle fluctuations. We provide an analytically tractable model that allows studying the interaction of non-insurable risk and asymmetric information problems between lenders and borrowers. This interaction is important because, while financial frictions constrain the availability of credit to borrowers, uninsurable idiosyncratic risk affects the borrower’s willingness to invest. We demonstrate that risk aversion creates strong precautionary motives that have a stabilizing effect on business cycles. Our findings suggest that the privately-held entrepreneurial businesses and/or the less diversified publicly traded firms were unlikely the source or the amplifier of the Great Recession. Also, the less-diversified businesses and firms were not as negatively affected by the financial shocks via the credit channel because of strong precautionary saving motives resulting from undiversifiable idiosyncratic risk. Hence, the origin and transmission of these shocks lies with the well-diversified publicly- traded firms. Small and medium-sized corporations, as well as non-corporate businesses, play an important role in the US economy and are exposed to asymmetric information problems and non-diversified risks. For instance, Davis, Haltiwanger, Jarmin, Mi- randa, Foote, and Nagypal(2006) find that more than two-thirds of non-farm business employment is accounted for by privately held firms. Our framework provides the intuition and point of departure for studying further these sectors in more serious multi-sector business cycle models.

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42 Figure 7: Impulse Response to Wealth Shock Consumption Price of Capital Frictionless Excess Returns to Capital 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 0 0 0

-0.1 -0.1 -0.2 -0.3 -0.4 -0.5 0.05 0.06 0.04 0.02

-0.05 Percent Percent Percent Risk Averse 3 Risk Averse 2 Leverage Investment External Finance Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 Risk Averse 1 1 0 0 0 -1 -2

1.5 0.5 0.5

-0.5 -1.5

Percent 0.06 0.04 0.02 Percent Percent Risk neutral Output Networth Risk Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 -3 10 0 0 5 0

-1 -2 20 15 10

-0.1 -0.2 -0.3 -0.5 -1.5 Percent

Percent Percent

Note: All impulse responses are plotted as percent deviations from steady state.

43 Figure 8: Impulse Response to Risk Shock Consumption Price of Capital Frictionless Excess Returns to Capital 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 0 0 0

-0.1 -0.2 0.02 0.01 0.05 0.04 0.03 0.02 0.01

-0.01 -0.02 -0.03 -0.05 -0.15 Percent

Percent Percent Risk Averse 3 Risk Averse 2 Leverage Investment External Finance Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 Risk Averse 1 0 0 0

0.1

-0.2 -0.4 -0.6 -0.1 0.15 0.05 0.04 0.03 0.02 0.01

Percent -0.05 -0.15 Percent Percent Risk neutral Output Networth Risk Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 0 0 0

0.1

-0.1 -0.1 -0.2 -0.3 -0.4 0.03 0.02 0.01

-0.02 -0.04 -0.06 -0.08 -0.12 Percent Percent Percent

Note: All impulse responses are plotted as percent deviations from steady state.

44 Figure 9: Impulse Response to Technology Shocks Consumption Price of Capital Frictionless Excess Returns to Capital 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 -3 10

8 6 4 2 0 1 0 0

-2

0.8 0.6 0.4 0.2 0.4 0.3 0.2 0.1 Percent

Percent Percent Risk Averse 3 Risk Averse 2 Leverage Investment External Finance Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 -3 10 Risk Averse 1

1 0 0 8 6 4 2 0

-2

0.2 0.1 Percent 1.5 0.5

Percent 0.25 0.15 0.05

-0.05 Percent Risk neutral Output Networth Risk Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 -3 10

1 0 0 3 2 1 0

0.8 0.6 0.4 0.2 0.5 0.4 0.3 0.2 0.1 Percent

-0.1 Percent Percent

Note: All impulse responses are plotted as percent deviations from steady state.

45 Figure 10: Impulse Response to Monetary Shocks Consumption Price of Capital Frictionless Excess Returns to Capital 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 0 0 0

0.5 0.4 0.3 0.2 0.1 0.6 0.4 0.2

Percent

Percent -0.01 -0.02 -0.03

-0.005 -0.015 -0.025 Percent Risk Averse 3 Risk Averse 2 Leverage Investment External Finance Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 Risk Averse 1 2 1 0 0

2.5 1.5 0.5

-0.5 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7

Percent Percent -0.01 -0.02 -0.03 Percent Risk neutral Output Networth Risk Premium 4 8 12 16 20 4 8 12 16 20 4 8 12 16 20 -3 10

0 1 0 1 0

-1 -2 -3 -4 -5

0.6 0.5 0.4 0.3 0.2 0.1 1.2 0.8 0.6 0.4 0.2 Percent

Percent Percent

Note: All impulse responses are plotted as percent deviations from steady state.

46 Appendix (for online publication)

A. Proof of Lemma 1

The entrepreneur is maximizing

1−ρ max QKωRk − BR , (53) K E using B = QK − N, then we can reformulate it

1−ρ max NR + QK(ωRk − R) (54) K E

Now since NR does not affect the optimal choice, we can divide everything by NR and obtain

Rk 1−ρ max E 1 + k(ω − 1) (55) k R

Rk Rk Denote log(ω R ) = x and ω R − 1 = y. Now we can use the relationship

Rk 1 y = ω − 1 = ex − 1 = x + x2 + o(x2) (56) R 2

Let us rewrite this expression as

1−ρ max E 1 + ky (57) k

Rk Taking a second-order Taylor approximation around y = 0, where y = k(ω R −1) we obtain ρ(1 − ρ) 2 2 2 max E 1 + (1 − ρ)ky − k y + o(y ) (58) k 2 Now we will assume that variance of ωRk is small, and the mean E(ωRk) is close to R. Taking the ﬁrst-order condition with respect to k yields 2 2 E (1 − ρ)y − ρ(1 − ρ)ky + o(y ) = 0 (59) so we can express k as y k = E + o(y2) (60) ρEy2

47 x2 2 We now substitute the expression y = x + 2 + o(x ) to obtain

x2 E(x + ) k = 2 + o(x2) (61) ρEx2

We would like to get the expression for leverage and omit small terms of higher 2 orders. We will consider σω, 4 to have small values of the ﬁrst order. On the 2 2 4 other hand, variables like 4 , σ 4, σω will be considered to be of higher orders and dropped out from the expression or be o(x).

σ4 log2 ω = (log ω)2 − ( log ω)2 + (log ω)2 = σ2 + ω = σ2 + o(x). (62) E E E E ω 4 ω

Then

2 2 Ex = E(log ω + 4) = 2 2 2 E(log ω) + 2E(log ω)E4 + E(4 ) = σω + o(xt+1) (63)

Now we obtain 4 + o(x) 4 k = E + o(x2) = E + o(x) 2 2 (64) ρσω + o(x) ρσω If we focus on the expression

4 κ = E , (65) ρσ2 then the elasticity of leverage with respect to excess returns and risk are given by:

d log k 1 νk = = (66) dE4 4 d log k ν = = −2 (67) σ d log σ

Note that neither of the elasticities depends on the risk-aversion parameter or the size of risk.

A. Proof of Proposition 1

The proof follows Tamayo(2014). First, note that when the report is not veriﬁed (ω∈ / ΩV ) the repayment function must only depend on the report ω˜, i.e. we have ∗ R(˜ω). Therefore, the entrepreneur will choose ω = arg minω˜ R(˜ω) so the contract

48 may as well set R(˜ω) = R¯. Second, under the optimal contract, in the verification region R(ω) ≤ R¯ because otherwise the contract would not be incentive compatible. Specifically, the entrepreneur would prefer to misreport ω∈ / ΩV and pay R¯. Finally, it can also be shown that ΩV must be a lower interval (for the proof see Lemma 3 in Tamayo(2014)). These findings can be summarized by saying that the optimal repayment function follows:  R(ω) ≤ R,¯ if ω ≤ ω¯ R(ω) = (68) R,¯ if ω > ω¯

Now let us rewrite the contracting problem using the above results as

1−ρ 1−ρ k k R ω¯ R 1−ρ R ∞ R ¯ 1−ρ max 0 κ R [ω − R(ω)] dΦ(ω) + ω¯ κ R [ω − R] dΦ(ω) (69) Rk Z ω¯ s.t. κ R(ω)dΦ(ω) + R[1 − Φ(¯ω)] − µΦ(¯ω) ≥ (κ − 1) (70) R 0 R¯ ≤ ω¯ (71) R(ω) ≤ ω ∀ω ≤ ω¯ (72) R(ω) ≥ 0 ∀ω ≤ ω¯ (73)

QK where we have plugged in the constraint (6), used the deﬁnition of leverage κ = N and rescaled the objective function and constraints by the exogenous parameters N and R. Assign the multipliers λ, ξ, γ1(ω) and γ2(ω) to the constraints. The Lagrangian reads:

Z ω¯ Rk 1−ρ Z ∞ Rk 1−ρ max κ [ω − R(ω)]1−ρdΦ(ω) + κ [ω − R¯]1−ρdΦ(ω)+ 0 R ω¯ R Rk Z ω¯ λ κ R(ω)dΦ(ω) + R[1 − Φ(¯ω)] − µΦ(¯ω) − (κ − 1) + R 0 Z ω¯ Z ω¯ ¯ ξ(¯ω − R) + γ1(ω)(ω − R(ω))φ(ω)dω + γ2(ω)(R(ω))φ(ω)dω 0 0

The ﬁrst order necessary conditions with respect to R(ω), R,¯ ω¯ after appropriate

49 rescaling of the multipliers can be written as:18

1−ρ RK −ρ RK −γ1(ω)φ(ω) − κ R {[ω − R(ω)] φ(ω) + λκ R φ(ω) + γ2(ω)φ(ω) = 0 ∀ω ≤ ω¯ (74) RK 1−ρ Z ∞ RK − ξ − κ [ω − R¯]−ρdΦ(ω) + λκ [1 − Φ(¯ω)] = 0 (75) R ω¯ R 1−ρ 1−ρ K K K ξ R 1−ρ R ¯ 1−ρ R ¯ − φ(¯ω) − κ R [¯ω − R(¯ω)] + κ R [¯ω − R] − λκ R [R(¯ω) − R − µ] = 0 (76) and the complementary slackness conditions:

Rk Z ω¯ 0 = λ κ R¯(ω)dΦ(ω) + R[1 − Φ(¯ω)] − µΦ(¯ω) − (κ − 1) (77) R 0 0 = ξ[¯ω − R¯] (78)

0 = γ1(ω)[ω − R(ω)] (79)

0 = γ2(ω)R(ω) (80)

Suppose that γ1(ω) > 0 for all ω < ω¯. Then it must be that γ2(ω) = 0, from the complementary slackness conditions. Then equation (74) would imply that λ > −ρ RK −ρ κ R (0) which is not possible. Hence it must be true that γ1(ω) = 0 for all ω ≤ ω¯ and a standard debt contract is not optimal. We know from (74) that −ρ γ1(ω) = 0 ⇐⇒ (ω − R(ω)) ≥ λ. Now there are two possible cases. Suppose −1/ρ R γ2(ω) = 0 for all ω ≤ ω¯. Then the contract speciﬁes that R(ω) = ω − λ RK κ . By complementary slackness it should be the case that R(ω) > 0 for all ω, which is not possible because if ω = 0, R(ω) > 0 would not be feasible. Then it must be the −1/ρ R case that γ2(ω) > 0 for some ω which implies R(ω) = 0 and ω ≤ λ RK κ for the same ω. Hence there is a lower interval where R(ω) = 0. Call the upper bound of −1/ρ R this interval ω ≡ λ RK κ . Therefore R(ω) = 0 if ω ≤ ω and R(ω) = ω − ω if ω ≤ ω ≤ ω.¯ 18We do not need the ﬁrst-order condition with respect to κ to prove the proposition.

50 B. Proof of Proposition 2

The Lagrangian is

(κ Rk )1−ρg(¯ω , ω , R¯ , σ )Ψ L = t t+1 t+1 t+1 t+1 ω,t t+1 + Et 1 − ρ k ¯ λt+1 κtRt+1h(¯ωt+1, ωt+1, Rt+1, σω,t) − (κt − 1)Rt

The ﬁrst order conditions are ∂L k 1−ρ = Et (κtRt+1) gt+1Ψt+1 − λt+1Rt = 0 (81) ∂kt ∂L (κ Rk )1−ρg Ψ = t t+1 ω,t¯ +1 t+1 + λ κ Rk h = 0 (82) ∂ω¯ 1 − ρ t+1 t t+1 ω,t¯ +1 ∂L (κ Rk )1−ρg Ψ = t t+1 ω,t+1 t+1 + λ κ Rk h = 0 (83) ∂ω 1 − ρ t+1 t t+1 ω,t+1 k 1−ρ ∂L (κtRt+1) gR,t¯ +1Ψt+1 k = + λ κ R h ¯ = 0 (84) ∂R¯ 1 − ρ t+1 t t+1 R,t+1

∂L Now we can express λt+1 from ∂ω¯ = 0

k 1−ρ (κtRt+1) gω,t¯ +1Ψt+1 1 λt+1 = − k (85) 1 − ρ κtRt+1hω,t¯ +1

Now we plug this condition into the three other equations and obtain

k 1−ρ ∂L k 1−ρ (κtRt+1) gω,t¯ +1Ψt+1 1 = Et (κtRt+1) gt+1Ψt+1 + k Rt = 0 ∂kt 1 − ρ κtRt+1hω,t¯ +1 k 1−ρ k 1−ρ ∂L (κtRt+1) gω,t+1Ψt+1 (κtRt+1) gω,t¯ +1Ψt+1 1 k = − k κtRt+1hω,t+1 = 0 ∂ω 1 − ρ 1 − ρ κtRt+1hω,t¯ +1 k 1−ρ k 1−ρ ∂L (κtRt+1) gR,t¯ +1Ψt+1 (κtRt+1) gω,t¯ +1Ψt+1 1 k = − κ R h ¯ = 0 ¯ k t t+1 R,t+1 ∂R 1 − ρ 1 − ρ κtRt+1hω,t¯ +1 we can transform this system to ∂L k 1−ρ gω,t¯ +1 = Et (Rt+1) Ψt+1 gt+1 + k Rt = 0 (86) ∂kt (1 − ρ)κtRt+1hω,t¯ +1 ∂L hω,t+1 = gω,t+1 − gω,t¯ +1 = 0 (87) ∂ω hω,t¯ +1 ∂L gω,t¯ +1 = gR,t¯ +1 − hR,t¯ +1 = 0 (88) ∂R¯ hω,t¯ +1

51 ˆ Since in the equation (86)Ψt+1 and Rk,t+1 enter as multiplicative terms and the gω,t¯ +1 ˆ term gt+1 + k Rt is equal to zero in the steady state, Ψt+1 and Rk,t+1 (1−ρ)κtRt+1hω,t¯ +1 have no effect in the first order approximation. Therefore, to find the approximate solution it is sufficient to consider the following system:

k ¯ κtRt+1h(¯ωt+1, ωt+1, Rt+1, σω,t) = (κt − 1)Rt (89) gω,t¯ +1 Et gt+1 + k Rt = 0 (90) (1 − ρ)κtRt+1hω,t¯ +1 g g ¯ ω,t¯ +1 = R,t+1 (91) hω,t¯ +1 hR,t¯ +1 g g ω,t¯ +1 = ω,t+1 (92) hω,t¯ +1 hω,t+1

We can substitute kt and obtain gω¯ Rt+1 Et (1−ρ)Rk,t+1hω,t+1 1 = (93) Rk,t+1 Etgt+1 1 − ht+1 Rt g g ¯ ω,t¯ +1 = R,t+1 (94) hω,t¯ +1 hR,t¯ +1 g g ω,t¯ +1 = ω,t+1 (95) hω,t¯ +1 hω,t+1

Whenever the gradient of this system has full rank at the steady state, we will be ˆ ¯ˆ ˆ ˆ able to ﬁnd an approximate solution of ω¯t+1, ωˆt+1, Rt+1 as functions of EtRk,t+1 − Rt, ˆ ˆ Rk,t+1 − Rt and σˆω,t. Using this fact and log-linearizing equation (89) will give us

ˆ ˆ ˆ kt = νp(EtRk,t+1 − Rt) + νσσˆω,t (96)

C. Log-linear Model

The log-linear model has 17 equations and 17 variables, because algebraic manipu- lations with the Calvo model allow to replace (35), (36) and (39) with (100), and drop p∗ and pw, while simplifying the ﬁnancial contract allows to replace (18), (19),

52 (81), (82), (83), (84) with (110) and drop ω,¯ ω, R,¯ Ψ. The equations are:

ˆ ˆ ˆ − σH EtCt+1 − Ct + Rt = 0, (97) ˆn ˆ Rt = Rt + Etπˆt+1, (98) ˆ ˆ ˆ ˆ ˆ Yt − Ht − Xt − σH Ct = ηHt, (99) (1 − θ)(1 − θβ) πˆ = − Xˆ + β πˆ , (100) t θ t Et t+1 ˆ ˆ ˆ ˆ Yt = At + αKt−1 + (1 − α)(1 − Ω)Ht, (101) ˆ ˆ ˆ Kt = δIt + (1 − δ)Kt−1, (102) ˆ ˆ ˆ Qt = δφK (It − Kt−1), (103) ˆk ˆ ˆ ˆ ˆ ˆ Rt+1 = (1 − )(Yt+1 − Kt − Xt+1) + Qt+1 − Qt, (104) G C I 1 − Yˆ = Cˆ + Iˆ, (105) Y t Y t Y t ˆ ˆ ˆ ˆ m m ˆ φt = Qt−1 + Kt−1 − Nt−1 + νσ σˆω,t−1 + νp (Et−1Rk,t − Rt−1), (106) ˆ ˆ ˆ ˆ W e ˆ e N−W e ˆ τ Nt = γ κRk(ˆκt−1 + Rk,t) − κRκˆt−1 − (κ − 1)RRt−1 − φφt + N Wt + N Nt−1 + t , (107) ˆ ˆ ˆ κˆt = Kt + Qt − Nt, (108) ˆ e ˆ ˆ Wt = Yt − Xt, (109) ˆk ˆ κˆt = νp(EtRt+1 − Rt) + νσσˆω,t, (110) ˆ A ˆ A At = ρ At−1 + t , (111) ˆn Rn ˆn Y ˆ Rn Rt = ρ Rt−1 + ξπˆt + ρ Yt + t , (112)

σω σω σˆω,t = ρ σˆω,t−1 + t . (113)

53 D. Partial versus General Equilibrium Effect

Figure 11: Partial versus General Effect

Partial Equilibrium General Equilibrium -0.6 -0.005

-0.01 -0.8

-0.015 -1 -0.02

-1.2 -0.025

-1.4 -0.03 Response of Capital of Response Response of Capital of Response -0.035 -1.6 -0.04

-1.8 -0.045 Risk Shock Wealth Shock -2 -0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Note: The figure shows the effects of changing risk aversion (x-axis) on the partial and general equilibrium response of capital to financial shocks. The rest of the model parameters follow Risk Averse 2 and the steady-state is being kept fixed by adjusting σω as in Figure5.

Figure 10 demonstrates the partial and general equilibrium effects of risk and wealth shocks on purchases of physical capital for different coefﬁcients of risk-aversion. The results from general equilibrium were obtained from the impulse responses of the model, while partial equilibrium effects of shocks were constructed under the assumption of excess returns to capital being ﬁxed at their steady state values. Each point on the general equilibrium dashed line requires running a DSGE model for the corresponding value of the risk aversion, computing the impulse response for purchases of physical capital on impact. Every point of the partial equilibrium response requires recomputing the steady state derivatives for each value of the risk aversion. The response of capital purchases under partial equilibrium is more volatile than in general equilibrium by about two orders of magnitude. This is a consequence of the fact that the demand for physical capital in general equilibrium is equal to the supply of physical capital stock, which is almost vertical in the short run. Thus, the

54 endogenous movement of prices plays a major role in capital dynamics by dampening partial equilibrium effects. Recall that entrepreneurs with the same preferences choose the same leverage, as the entrepreneurial problem is homothetic with respect to net worth. Therefore, the decrease of net worth by one percent in partial equilibrium leads to a decline of physical capital purchases by one percent as well. In general equilibrium, there are two additional channels in addition to the partial equilibrium response. First, excess returns to capital increase, Rk/R, because the lower capital purchases reduces temporarily the price of capital. Higher expected returns to capital lead to the rise in leverage and higher purchases of physical capital, having a stabilizing effect. We refer to this mechanism as a leverage channel. Sec- ond, a decrease in the price of physical capital leads to lower net worth, especially since the entrepreneurs have a leverage greater than one. This tends to have an amplifying effect on business cycles. We refer to this mechanism as a net worth channel. For low coefficients of risk aversion, the general equilibrium effect is relatively large because of the net worth channel. As the coefficient of risk-aversion increases, the general equilibrium effect becomes smaller because of the higher sensitivity or risk averse entrepreneurs to high excess returns to capital. Under the partial equilibrium, the effect of a risk shock for risk-neutral agents leads to an increase in defaults, which translate into higher interest rate on loans, and, consequently, reducing leverage. For the agents with higher risk-aversion, we observe a larger reduction of leverage in partial equilibrium, because in addition to higher defaults, the risk shock directly increases the volatility of returns, from which the risk averse borrower tries to back away. In general equilibrium, precautionary motives allow entrepreneurs with higher risk-aversion to increase their leverage more when they face higher excess returns to capital. In other words, for risk-averse agents leverage channel is strong enough to support the price of capital and prevent the fall of net worth, which instead dominates in the case of risk neutral agents. Finally, it is worthwhile noticing that the responses of capital purchases for both shocks in general equilibrium flatten out as soon as risk aversion reaches 0.1, which shows that our results are not sensitive to a particular value of this parameter.

55 E. Two-Sector Model

The complete system with a two-sector model is described by the following system of equations:

α (1−α)Ω Y1,t = AtK1,t−1H1,t , (114) α (1−α)Ω Y2,t = AtK2,t−1H2,t , (115)

K1,t = I1,t + (1 − δ)K1,t−1, (116)

K2,t = I2,t + (1 − δ)K2,t−1, (117) 1 I1,t = 1 − φK − δ , (118) Q1,t K1,t 1 I2,t = 1 − φK − δ , (119) Q2,t K2,t

1 Y1,t+1 ζ−1 αY1,t+1 a + Q1,t+1(1 − δ) k Xt+1 Yt+1 K1,t+1 R1,t+1 = , (120) Q1t 1 Y2,t+1 ζ−1 αY2,t+1 a + Q2,t+1(1 − δ) k Xt+1 Yt+1 K2,t+1 R2,t+1 = , (121) Q2t

φ1,t = Q1,t−1K1,t−1g(EtR1k,t,Rt,R1,kt, σ1ω,t, ρ1), (122)

φ2,t = Q2,t−1K2,t−1g(EtR2k,t,Rt,R2,kt, σ2ω,t, ρ2), (123)

N1,t = γ(K1,t(Rk,1t − Rt − φ1,t) + N1,tR) + We,1t, (124)

N2,t = γ(K2,t(Rk,2t − Rt − φ2,t) + N2,tR) + We,2t, (125)

K1,tQ1,t κ1,t = , (126) N1,t K2,tQ2,t κ2,t = , (127) N2,t ζ−1 Y1,t Y1,t We,1t = a(1 − Ω)(1 − α) , (128) Xt Yt ζ−1 Y2,t Y2,t We,2t = (1 − a)(1 − Ω)(1 − α) , (129) Xt Yt

κ1,t = f(EtR1k,t+1,Rt, σ1ω,t, ρ1), (130)

κ2,t = f(EtR2k,t+1,Rt, σ2ω,t, ρ2), (131)

σω σω σ1ω log(σ1ω,t) = (1 − ρ ) log(σ1ω,ss) + ρ log(σ1ω,t−1) + t (132)

σω σω σ2ω log(σ2ω,t) = (1 − ρ ) log(σ2ω,ss) + ρ log(σ2ω,t−1) + t (133)

56 −σH −σH Ct = βRtEtCt+1 , (134) C−σH −σH n t+1 Ct = βRt Et , (135) πt+1 Y Y ζ−1 η 1,t 1,t −σH Ht = (1 − α)Ω a Ct (136) H1,tXt Yt Y Y ζ−1 η 2,t 2,t −σH Ht = (1 − α)Ω (1 − a) Ct (137) H2,tXt Yt

(1 − gy) Yt = Ct + It, (138) 1 ε−1 ∗ 1−ε 1−ε 1 = θπt−1 + (1 − θ)(pt ) , (139) ε V p∗ = t (140) t ˜ ε − 1 Vt

−σH −1 Vt = Ct YtXt + βθEt(Vt+1πt+1) (141)

˜ −σH ˜ −1 Vt = Ct Yt + βθEt(Vt+1πt+1 ) (142) A A log(At) = ρ log(At−1) + t , (143) n n Rn n Rn log(Rt ) − log(R ) = ρ (log(Rt−1) − log R) + ξπt−1 + t , (144) ζ ζ ζ Yt = aY1,t + (1 − a)Y2,t (145)

It = I1,t + I2,t (146)

Equations (145) and (146) reflect how sector-specific output and investment are combined to produce corresponding aggregates. The parameter ζ character- izes the elasticity of substitution between sectors, and the parameter a corresponds to the weight of the sector 1 in the production of aggregate output. Equations (134)-(146) describe the behavior of aggregate variables, while equations (114)- (133) correspond to sector-specific relationships. All other parameters, except a and ζ, are identical to the one-sector model. Overall, we have 20 sector-specific k variables Yi,Ki,Ii,Hi,Qi,Ri ,Ni, φi, κi, σiω with i = 1, 2 and 13 aggregate variables ∗ ˜ e C,R,Rn, π, p ,V, V, X ,H,W ,Y,I,A.