Risk Aversion, Uninsurable Idiosyncratic Risk, and the Financial Accelerator

Transcription

1 Risk Aversion, Uninsurable Idiosyncratic Risk, and the Financial Accelerator Giacomo Candian HEC Montréal Mikhail Dmitriev Florida State University March 218 Abstract We study the role of uninsurable idiosyncratic investment risk and risk aversion for business cycle fluctuations. Relative to a real-businesscycle or New Keynesian model where idiosyncratic risk is fully hedged, the presence of uninsurable investment risk creates a risk premium on expected returns to capital. We show that this premium is less countercyclical than the more traditional external finance premium arising from agency costs, thus only mildly amplifying business cycles. When we combine non-diversified risk with agency frictions, the risk premium crowds out the external finance premium, significantly reducing the effect of the financial accelerator. We provide firm-level evidence supporting the model predictions about investment behavior. Our findings imply that financial innovations devised to complete markets may undesirably increase systemic fragility. JEL Classification: C68, D81, D82, E44, L26. Keywords: Risk Aversion; Uninsurable Idiosyncratic Risk; Financial Accelerator; Incomplete Markets. We thank Susanto Basu, Diego Comin, Fabio Ghironi, Peter Ireland, Robert King, 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 Banking 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. giacomo.candian@hec.ca. Web: mihail.dmitriev@gmail.com. Web:

2 1 Introduction According to Knight (1921), bearing risk is one of the defining features of entrepreneurship. Entrepreneurs are exposed to significant undiversified risk (Gentry and Hubbard, 24) and face extreme dispersion in equity returns (Moskowitz and Vissing-Jorgensen, 22). When entrepreneurial activity depends on external finance, the presence of this large undiversifiable risk may have important implications for macroeconomic dynamics. 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, 211; Liu and Wang, 214; Dmitriev and Hoddenbagh, 217), or present but coupled with the assumption of risk-neutral entrepreneurs (Bernanke, Gertler, and Gilchrist, 1999, BGG). Yet, 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, log-linear setup. 1 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 shocks, is 4 to 7 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 2 percent smaller when idiosyncratic risk is non-diversified. Following BGG, we consider an environment where entrepreneurs use their net worth and funds borrowed from households to invest in physical capital, whose returns are subject to both aggregate and idiosyncratic risk. 1 Log-linearization does not result in certainty equivalence because our steady state, while being deterministic in the aggregate sense, still features non-zero volatility of idiosyncratic productivity. In the steady state of our model, every entrepreneur is still exposed to significant idiosyncratic risk, which has a first-order effect. 2

3 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. When entrepreneurial net worth falls, leverage and defaults increase, resulting in a rise in the external finance premium, a subsequent fall in investment and asset prices, leading to further declines in net worth. Thus, the countercyclicality of external finance premium is a source of amplification of business cycle fluctuations. Introducing risk aversion gives rise to an additional risk premium relative to risk neutrality, as entrepreneurs require compensation for the volatility of their returns associated with uninsurable idiosyncratic risk. This risk premium is also countercyclical, as when net worth falls, a higher leverage implies a greater volatility of entrepreneurial returns. We show that this risk premium is less countercyclical than the external finance premium, and higher risk aversion leads to more stable business cycles relative to a world with agency problems but with risk-neutral borrowers. We begin our analysis of uninsurable idiosyncratic risk by studying the investment behavior of risk averse entrepreneurs that face no agency frictions. In this environment, entrepreneurs trade off the mean and variance of their returns. Their leverage is approximately proportional to excess capital returns (relative to the safe interest rate) and inversely proportional to their variance. For example, if excess returns to capital increase from 1 to 2 percentage points, leverage doubles. Thus, when returns to capital increase, capital purchases respond quite strongly, making the gap between capital returns and the safe rate of interest relatively stable in general equilibrium. This contrasts sharply with an environment with agency frictions and risk neutrality, where the rapidly rising cost of defaults prevents entrepreneurs from expanding their leverage when excess returns to capital rise, resulting in more volatile excess returns to capital over the business cycle. Finally, we combine risk aversion with agency frictions and find that this model also generates less amplification than the standard model with only 3

4 agency costs. This happens for two reasons. First, risk-averse entrepreneurs choose a lower leverage than their risk-neutral counterpart other things equal. This means that a given change in capital returns has a smaller effect on entrepreneurial net worth. Second, the gap between capital returns and the safe rate now reflects mostly the premium associated with risk aversion, which is less countercyclical than the external finance premium, thus further contributing to stabilizing the business cycle. 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 (212). 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 delivers the stabilizing effects of risk aversion. Methodologically, we are the first to our knowledge to incorporate riskaverse borrowers in a model of idiosyncratic, uninsurable investment risk and agency frictions, while keeping the analytical tractability of a log-linear framework. This task presents two key challenges. First, as shown by Gale and Hellwig (1985) and Tamayo (214), when entrepreneurs default in this environment, the optimal contract differs from debt; risk-neutral borrowers would surrender all their wealth, while risk-averse ones would retain some of their net worth. We embed these insights from contract theory in a stochastic general equilibrium model with optimal leverage choice. Second, because markets are incomplete, aggregation of individual histories becomes non-trivial, as is well known from the work of Krusell and Smith (1998). We address this challenge by modifying the overlapping-generation structure of entrepreneurs in BGG, allowing only newborn generations to work, so that future labor income does not affect entrepreneurial financial decisions. Under this assumption, every entrepreneur chooses the same leverage, and individual histories become irrelevant for aggregate dynamics. Non-diversified risk and risk aversion allow us to better match the micro evidence relative to the standard model with agency frictions. In the latter, the volatility of firms productivity is about thirty percent within a 4

5 quarter, being larger than the empirical value by about three times. In our calibration, we match the data on firm-specific volatility of TFP and sales (Castro, Clementi, and Lee, 21; Comin and Mulani, 26; Davis, Haltiwanger, Jarmin, and Miranda, 27; De Veirman and Levin, 214). 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 (25), and Angeletos (27). Our work differs from theirs, in that we study the interaction of non-diversified risk and agency friction. Covas (26) 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 (26) explore the role of risky investment for capital accumulation in an environment with asymmetric information. They find that the presence of investment risk leads to an under-accumulation of capital relative to an economy where idiosyncratic risk can be fully ensured. In our analysis, we also find under-accumulation of capital in steady-state in comparison to an economy where idiosyncratic risk is diversifiable. However, 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. Our findings are also related to Dmitriev and Hoddenbagh (217) and Carlstrom, Fuerst, and Paustian (216), who show that in models with agency frictions à 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 Hoddenbagh (217) and Carlstrom, Fuerst, and Paustian (216) suggest that insuring aggregate risk stabilizes the economy, we show that hedging idiosyncratic risk increases the economy s vulnerability to aggregate disturbances. The paper proceeds as follows. Section 2 derives the static optimal contract in partial equilibrium. Section 3 incorporates the resulting contract 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. 5

6 2 Static Optimal Contract 2.1 Risk Aversion and Non-Diversified Risk Without Agency Frictions 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 QKR k ω, where R k indicates aggregate returns to capital and log(ω) N ( 1 2 σ2 ω, σω) 2 the idiosyncratic return component that is specific to the entrepreneur with pdf φ(ω). 2 The utility is characterized by constant relative risk aversion, so the entrepreneur solves the following problem: [ 1 ρ max QKR k ω BR], (1) K,B 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 κ = ρσ 2 ω + o(σ 2 ω, ), (2) 2 To keep consumption non-negative we use a bounded normal distribution for log ω so that log ω [ 4σ ω, 4σ ω ] which covers of the distribution. 6

7 where = log( R k ), and κ = QK. Moreover, when we log-linearize the relationship κ = by allowing κ, σ R N ρσ 2 ω, and to change relative to the initial level κ ss, σ ω,ss, and ss, we obtain ˆκ = 1 ss ˆ 2ˆσω, (3) where ˆ = ss, ˆσ ω = ln(σ ω /σ ω,ss ), and ˆκ = ln(κ/κ ss ). 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 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 (214). 3 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 3 For earlier treatments of the contracting problem see Townsend (1979) and Gale and Hellwig (1985). 7

8 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 Ω V Ω. Following the literature, we assume that reports are always truthful so that s(ω) = ω for all ω Ω, which implies that the repayment function depends only on ω. 4 Definition 1. A 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, Ω V, where the lender chooses to verify the state of the world. The static problem in the presence of only idiosyncratic risk ω can be formulated as max K,R(ω) [QKRk (ω R(ω))] 1 ρ φ(ω)dω, (4) 1 ρ BR QKR k R(ω)φ(ω)dω µqkr ω Ω k ωφ(ω)dω, (5) V QK = B + N, (6) 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 4 See Tamayo (214) for details. 8

9 ω and ω, such that if ω < ω, R(ω) = ω ω if ω ω ω, R if ω > ω, where ω R ω ω, Ω V = [, ω). Proof. See the Appendix. The optimal contract is illustrated in Figure 1. 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. 5 Therefore, optimal risk-sharing requires 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. 6 Corollary 1. When ρ then ω, 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 complexity of the problem under risk aversion, the repayment function R(ω) is completely characterized by the thresholds (ω, ω) and by the non-default 5 Effectively, in the region ω (ω, ω) the borrower always receives ω. 6 In 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

10 Figure 1: Optimal contract with risk-averse entrepreneurs R.6 R(ω) ω ω ω repayment R. follows: L = This allows us to reformulate the contracting problem as (κr k ) 1 ρ g( ω, ω, max R) ( ) + λ κr k h( ω, ω, ω,ω, R,κ,λ 1 ρ R) (κ 1)R, where κ QK N, g( ω, ω, R) and h( ω, ω, R) are correspondingly: g( ω, ω, R) = ω h( ω, ω, R) = (1 µ) ω ω 1 ρ φ(ω)dω + ω 1 ρ φ(ω)dω + ω ω ωφ(ω)dω ω ω ω ω ω φ(ω)dω + R (ω R) 1 ρ φ(ω)dω, ω ω φ(ω)dω µ ωφ(ω)dω. The optimal κ, ω, ω and R are only functions of exogenous variables R k, R and parameters σ ω, 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. 1

11 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 = S 1 (, σ ω, µ, ρ). (8) We are interested in the decomposition of the excess returns to capital,, into the external finance premium 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: = S 2 (κ, σ ω, µ, ρ). (9) We interpret 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 σ ω under the optimal contract with monitoring costs µ. The external finance premium is simply defined as the capital wedge when risk aversion is set to zero: EF P = S 2 (k, σ ω, µ, ), (1) while the risk premium is defined as a residual component between the excess returns to capital and the external finance premium: RP = EF P = S 2 (κ, σ ω, µ, ρ) S 2 (κ, σ ω, µ, ) (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. 11

12 Figure 2: Optimal Leverage.7.6 ρ=.1, µ=.12, σ=.28 ρ=.1, µ=., σ=.28 ρ=., µ=.12, σ= ln(R k /R).4.3 risk premium.2.1 external finance premium ln κ Leverage Schedules. Figure 2 shows the relationship between the (annualized) excess returns to capital (R k /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 parameter, µ, is set to.12 and the standard deviation of idiosyncratic returns is σ ω =.28. We set ρ =.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.7, defaults start taking larger values, and monitoring costs become significant, thus preventing leverage 12

13 from expanding rapidly. For the standard calibrated value of the annual excess returns of.3 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 > ) 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 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 13

14 leverage and face smaller agency costs. Second, he will be more sensitive to changes in excess returns. However, 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 Keynesian model, where aggregate returns to capital and returns to lenders are determined 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, D t, and lends out the nominal amount B t to entrepreneurs. In equilibrium, deposits will equal loanable funds (D t = B t ). Households receive a predetermined real rate of return R t on their deposits. Each individual loan is subject to idiosyncratic and aggregate risk, but the financial 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 B t (j) from the representative lender, which he combines with his net worth N t (j) to purchase capital K t (j) at a unit price of Q t. That is, Q t K t (j) = N t (j) + B t (j). (12) 14

15 In period t+1, the entrepreneur receives an idiosyncratic shock, ω t+1 (j), that converts the raw capital K t (j) into effective units ω t+1 (j)k t (j), which are rented out to to perfectly competitive wholesalers at a rental rate R r t+1. The idiosyncratic shock ω t+1 (j) is drawn in an i.i.d. fashion from a log-normal distribution that obeys log(ω t+1 (j)) N ( 1 2 σ2 ω,t, σ 2 ω,t), so that the mean of ω is equal to 1. 7 After production takes place, the entrepreneur sells the undepreciated effective units of capital (1 δ)ω t+1 (j)k t (j) at price Q t+1, and settles his position with the lender, either by repaying the loan or by defaulting. In this way, an entrepreneur that draws an idiosyncratic shock ω t+1 enjoys a rate of return ω t+1 R k t+1 at time t + 1, where R k t+1 = Rr t+1 + Q t+1 (1 δ) Q t. (13) Thus, each entrepreneur in period t has access to a stochastic, constant returns technology ω t+1 (j)r k t+1. When the realization of ω t+1 (j) exceeds ω t+1, the entrepreneur is able to repay the loan at the contractual rate Z t+1. That is, B t Z t+1 = Q t K t R k t+1 R t+1. (14) When instead ω t+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 worth and supply inelastically one unit of labor in the aggregate at wage rate W e t. 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 lower net worth would realize 7 The timing is meant to capture the fact that the variance of ω t+1 is known at the time of the financial arrangement, t. 15

16 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 V e t (j) = (1 γ) s=1 γ s (C E t+s(j)) e 1 ρ t, (15) 1 ρ where C e t+s(j) is the entrepreneur j s consumption in case of his death, C e t (j) = N t (j), (16) defined as wealth accumulated from operating firms. The timeline for entrepreneurs is plotted in Figure 3. Figure 3: Timeline for Entrepreneurs t Rent capital K t 1 to wholesalers and receive return R k t Life/death of entrepreneur Take out new loan B t with lending rate Z t+1 t+1 Period t shocks are realized Pay off loan from period t 1 (B t 1Z t) or default Buy capital K t to rent in period t + 1 Period t + 1 shocks are realized 16

17 The dynamic problem can be formulated recursively as follows: max K t, R t+1, ω t+1,ω t+1 E t [ (κt R k,t+1 ) 1 ρ g( ω t+1, ω t+1, R t+1, σ ω,t )Ψ t+1 ], (17) 1 ρ [ s.t. Ψ t = 1 + γe t (κ t R k,t+1 ) 1 ρ g( ω t+1, ω t+1, R t+1, σ ω,t )Ψ t+1 ],, (18) βκ t R k,t+1 h( ω t+1, ω t+1, R t+1, σ ω,t ) = (κ t 1)R t. (19) As in BGG, R t is the safe rate known at time t. Lenders require to be paid R t on average, which implies that the contract must specify a triplet {ω t+1, ω t+1, R t+1 } contingent on R k t+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 with ν p > and ν σ <. ˆκ t = ν p (E t ˆRk t+1 R t ) + ν σˆσ ω,t, (2) Proof. Equation (2) 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 κ t 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 (2) that links leverage to the expected excess returns to capital. Note that equation (2) is identical in form to the one in BGG (equation (4.17) in their paper), once augmented with time-varying σ ω. The presence of risk aversion, however, changes the elasticity of leverage to the excess returns ν p 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 17

18 have that ν p ρ > ν σ ρ >. Our finding that the elasticity ν p increases with ρ is consistent with Figure 2 in the previous section. Moreover, a rise in σ ω, besides increasing defaults, mechanically raises the volatility of returns, to which risk-averse entrepreneurs optimally respond by cutting leverage more than their riskneutral counterpart. In terms of aggregation, the fact that each entrepreneur chooses the same leverage implies that κ t = Q t K t /N t. (21) Our assumptions about survival and new entry of entrepreneurs imply that aggregate net worth follows: N t+1 = γ ( Q t K t R k t+1 (Q t K t N t )R t µq t K t R k t+1 ωt+1 ωφ(ω)dω ) + W e t+1 + ɛ τ t. (22) The terms inside the brackets reflect the aggregate returns to capital to entrepreneurs, net of loan repayments and monitoring costs. The shock ɛ τ t represents an i.i.d. wealth transfer from the household to the entrepreneurs. Finally, aggregate entrepreneurial consumption is given by C e t = (1 τ e )(1 γ)(n e t W e t ). (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 nondiversified 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 { [ C 1 σ χh1+η ] } max E t β s t+s 1 σ t+s, (24) 1 + η s= 18

19 where C t is household consumption, and H t is household labor effort. The budget constraint of the representative household is C t = W t H t T t + Π t + R t 1 D t D t+1 + R n B t t 1 B t+1, (25) P t P t where W t is the real wage, T t is lump-sum taxes, Π t is lump-sum profits received from final goods firms owed by the household, D t are deposits in financial intermediaries (banks) that pay a real non-contingent gross interest rate R t 1 and B t are nominal bonds that pay a gross non-contingent interest rate R n t 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: { } C σ t = βr t E t C σ t+1 C σ t { C σ = βr n t+1 t E t π t+1, (26) }, (27) W t C σ t = χh η t. (28) We define the gross rate of inflation as π t+1 = P t+1 /P t. 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: ( 1 C t = The demand for retailer i s unique variety is ) ε c ε 1 ε 1 ε it di. (29) c it = ( pit P t ) ε C t, (3) 19

20 where p it is the price charged by retail firm i. The aggregate price index is defined as ( 1 P t = ) 1 p 1 ε 1 ε it. (31) 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 in a particular period t. Retailer i maximizes the following stream of real profits: max p it s= { ( ) θ s p it Pt+s w p ε E t Λ it t,s Y t+s}, (32) P t+s P t+s where Pt w is the wholesale goods price and Λ t,s β U C,t+s U C,t is the household s (i.e. shareholder s) stochastic discount factor. The first order condition with respect to the retailer s price p it is { ( ) p θ s ε [ E t Λ it p t,s Y it t+s ε P t+s P t+s ε 1 s= Pt+s w P t+s ] } =. (33) From this condition, it is clear that all retailers that are able to reset their prices in period t will choose the same price p it = Pt price level evolves according to i. The aggregate P t = [ θp 1 ε t 1 + (1 θ)(p t ) 1 ε] 1 1 ε. (34) Dividing the left and right hand side of (34) by the price level gives 1 = [ θπ ε 1 t 1 + (1 θ)(p t ) 1 ε] 1 1 ε, (35) where p t = P t /P t. Using the same logic, we can normalize (33) and obtain: p t = where p w t+s = P w t+s P t+s and p t+s = P t+s /P t. ε { s= θs E t 1 Λt,s (1/p t+s ) ε Y t+s p w t+s} ε 1 s= θs E t 1 {Λ t,s (1/p t+s ) 1 ε Y t+s }, (36) 2

21 3.5 Wholesalers Wholesale goods are produced by perfectly competitive firms and then sold to monopolistically competitive retailers who costlessly differentiate them. Wholesalers hire labor from households and entrepreneurs in a competitive labor market at real wage W t and W e t, and rent capital from entrepreneurs at rental rate R r t. 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 Y t = A t K α t 1(H t ) (1 α)ω (H e t ) (1 α)(1 Ω), (37) where A t denotes aggregate technology, K t is capital, H t is household labor, Ht e 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 Y t = A t K α t 1H (1 α)ω t. (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 P w t P t = p w t = 1 X t, (39) where X t is the variable markup charged by final goods producers. objective function for wholesalers is then given by The 1 max A t K H t,ht e,kt 1 X t 1(H α t ) (1 α)ω (H e t ) (1 α)(1 Ω) W t H t W e t H e t R r t K t 1. (4) t 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 r t = 1 X t α Y t K t 1, (41) W t = Ω (1 α) Y t, X t H t (42) W e (1 Ω) t = (1 α) Y t. X t Ht e (43) 21

22 Given that equilibrium entreprenerial labor in equilibrium is 1, we have W e t = 3.6 Capital Producers (1 Ω) X t (1 α)y t. (44) 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 K t = I t + (1 δ)k t 1 φ K 2 ( ) 2 It δ K t 1, (45) K t 1 where I t is investment in period t, δ is the rate of depreciation and φ K is a parameter that governs the magnitude of the adjustment cost. The capital producer s objective function is max I t K t Q t I t, (46) where Q t denotes the price of capital. The first order condition of the capital producer s optimization problem is 3.7 Goods Market Clearing The goods market clearing condition is ( ) 1 It = 1 φ K δ. (47) Q t K t 1 Y t = C t + I t + G t (48) where G t corresponds to government spending. Government spending is assumed to be proportional to output, so that G t /Y t is constant. We assume that aggregate monitoring costs are rebated lump sum to households so they do not enter goods market clearing. 22

23 3.8 Monetary Policy As in BGG, we assume that there is a central bank which conducts monetary policy by choosing the nominal interest rate R n t rule where ρ Rn according to the following log(r n t ) log(r n ) = ρ Rn( ) log(rt 1) n log(r) + ξπ t 1 + ɛ Rn t. (49) 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 ɛ 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. 3.9 Shocks Technology and idiosyncratic volatility follow an AR(1) process: log(a t ) =ρ A log(a t 1 ) + ɛ A t, (5) log(σ ω,t ) =(1 ρ σω ) log(σ ω,ss ) + ρ σω log(σ ω,t 1 ) + ɛ σω t. (51) where ɛ A, and ɛ σω denote exogenous shocks to technology, and idiosyncratic volatility, and σ ω,ss denote the steady state value for idiosyncratic volatility. Recall that σ 2 ω is the variance of idiosyncratic productivity, so that σ ω 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). 3.1 Equilibrium The nonlinear model has 24 endogenous variables and 24 equations. The endogenous variables are: R, R n, H, C, π, p, p w, X, Y, W, W e, I, Q, K, R k, N, k, ω, ω, R, Ψ, λ, A, σ ω, 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 optimal- 23

24 ity conditions (81), (82), (83), (84). The exogenous processes for technologyand idiosyncratic volatility follow (5) and (51), respectively. Nominal interest rate and wealth redistribution shocks are defined in (49) and (22), respectively. The model is log-linearized around the deterministic steady state and solved using Dynare. The log-linearized equations are reported in the Appendix. 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 =.99, the household s risk aversion parameter σ = 1 so that utility is logarithmic in consumption, and the elasticity of labor supply to 3 (η = 1/3). The share of capital in the Cobb-Douglas production function is α =.35, while the share of entrepreneurial labor 1 Ω is.1. Capital adjustment costs are φ k = 1, to generate an elasticity of the price of capital with respect to the investment-capital ratio of.25. Quarterly capital depreciation is δ =.25. The steady-state share of government expenditure in total output, G/Y, is set to.2. Regarding nominal rigidities, we set the Calvo parameter θ =.75, so that 25% of firms can reset their prices in each period so that the average length of time between price adjustments is four quarters. As our baseline, 24

25 we follow the BGG monetary policy rule and set the autoregressive parameter on the nominal interest rate to ρ Rn =.9 and the parameter on lagged inflation to ξ =.11. We set the persistence of the shocks to technology at ρ A =.99, 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 ρ Rn =.9. For our purposes, the most important part of the calibration regards the part of the model relating to financial frictions. Following Christiano, Motto, and Rostagno (214, CMR), we set the persistence of idiosyncratic volatility at ρ σω =.976 for all cases. Our risk-neutral calibration is similar to BGG and sets the monitoring costs µ, the probability of survival for entrepreneurs γ, and the standard deviation of idiosyncratic productivity σ ω to.12,.977 and.28, respectively. These values imply a steady-state leverage of 2, an annualized excess return to capital of 25 basis points, and an annualized default rate of 3.8 percent. This calibration is summarized in column 3 of Table 1 (Risk Neutral). In column 4 we report a calibration where we keep all structural parameters the same except risk aversion (Risk Averse 1). We increase the risk aversion coefficient to ρ =.1, consistently with our simulations of the static contract. As a result, (i) the steady-state leverage and defaults fall, while the steady-state excess returns to capital increase, and (ii) and the elasticity of leverage with respect to excess capital returns increases. To isolate the second channel, column 5 reports a different calibration for the risk-averse economy (Risk Averse 2) where an increase in risk aversion is coupled with the reduction in idiosyncratic risk, in order to obtain the same leverage and excess capital returns. Here our choice of σ ω is guided by the existing microeconomic evidence on firm productivity. Focusing on the cross-sectional volatility of sales growth, Castro, Clementi, and Lee (21) obtain a value for the firm-specific volatility of TFP between.4 and.12. Comin and Mulani (26), Davis, Haltiwanger, Jarmin, and Miranda (27) and a more recent study by De Veirman and Levin (214) report the volatility for the annual growth of sales to be between.24 and.3. Through the lens of our model, this range implies that a firm s standard deviation of idiosyncratic productivity, σ ω, is between.8 and.1. 8 We settle for a value 8 To obtain these values, we simulate our model in the steady state, where aggregate shocks are absent, but idiosyncratic shocks still affect firms. 25

26 of σ ω of.8 and subsequently choose a value for ρ =.5. The results reported in our simulations are robust to the choice of ρ and σ ω, as long as we select these two parameters to match the leverage and the average excess returns observed in the data. We also explore an alternative calibration, Risk Averse 3, which differs from Risk Averse 2 only in that it also lowers the monitoring costs to.21. This modification allows us to match, on top of leverage and excess returns, the default rate of the risk-neutral economy. 9 Table 1: Calibration Symbol Description Risk Neutral Risk Averse 1 Risk Averse 2 Risk Averse 3 A. Calibrated parameters ρ Risk aversion σ ω Std. Dev. idiosyncratic productivity γ Survival probability µ Monitoring costs B. Implied steady-state values κ Leverage ln(r k /R) Premium (%, annualized) Φ( ω) Default rate (%, annualized) C. Implied elasticities ν p Elasticity of leverage to returns ν p Predicted elasticity of leverage to returns - no agency costs ν σ Elasticity of leverage to id. risk ν σ Predicted elasticity of leverage to id. risk- no agency costs ˆκ t = ν p (E t ˆRk t+1 ˆR t ) + ν σˆσ ω,t 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 ν p and ν σ in equation (2) for the different calibrations, summarized in Panel C of Table 1. The Table shows that these elasticities are higher in absolute value for the risk-averse cases. How would a higher sensitivity of leverage to excess returns and to the volatility of idiosyncratic productivity affect business cycles? To predict the outcome it is helpful to think about the elasticity ν p in two extreme cases: the risk-neutral case with and without agency frictions. In a risk-neutral world 9 In Risk Averse 2, given our parametrization of risk aversion, the default cost is relatively high, so entrepreneurs choose a leverage of 2 only if they face very small chances of default. In Risk Averse 3, by lowering the default cost, we can get risk-averse entrepreneurs to default more frequently. 26

27 without financial frictions ν p. Even the smallest increase in expected capital returns makes entrepreneurs willing to hold an infinite amount of capital, owing to constant returns, so that in equilibrium returns to capital are equal to the safe rate. At the opposite end of the spectrum, when entrepreneurs are risk neutral and face agency frictions, ν p is small, reflecting the fact that even if capital returns rise, borrowing cannot increase much because marginal borrowing costs increase very quickly with leverage. In this case, large swings in excess returns are required to generate movements in leverage. Given that ν p in the risk-averse case is larger than in the riskneutral case with agency frictions, we should expect excess returns in the risk-averse case to still react to shocks (because financial frictions are still present), but more mildly than in the risk-neutral case. With smaller movements in the returns to capital and, therefore, the price of capital, we expect smaller fluctuations in net worth and less volatile business cycles. We use our static toy model with risk aversion and no agency costs to predict the elasticity of leverage to excess returns for the full model. For the case of Risk Averse 1, the specification with no agency costs implies ν p = 12.6, while the full contract delivers a slightly smaller elasticity ν p = This is intuitive, since Risk Averse 1 with positive agency costs should provide an intermediate elasticity between the risk-neutral case with only agency costs, and the case with positive risk aversion and no agency costs. As expected, with smaller leverage and defaults the results a closer to the case with positive risk aversion and no agency costs. We have a similar result for Risk Averse 2, where the toy model implies a higher sensitivity of leverage to excess returns ν p = 16, while the full model with agency costs and risk aversion gives ν p = The case Risk Averse 3 is the closest to the frictionless case with ν p = 181.8, which exceeds the value of 16 for the toy model with no agency costs. In Risk Averse 3 borrowers begin to use defaults as a tool for risk-sharing as monitoring costs approach zero. In contrast, our no-agency frictions toy model implies neither defaults nor risksharing. In this case, the full model with costly state verification with very low monitoring costs is closer to a world with risk neutrality and no agency costs than the toy model and delivers even higher elasticity ν p. 27