Credit-Crunch Dynamics with Uninsured Investment Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Credit-crunch dynamics with uninsured investment risk Jonathan E. Goldberg 2013-47 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Credit-crunch dynamics with uninsured investment risk Jonathan Goldberg∗ April 9, 2013 Abstract I study the effects of credit tightening in an economy with uninsured idiosyncratic investment risk. In the model, entrepreneurs require an equity premium because collateralconstraintslimitinsurance. Aftercollateralconstraintstighten, theequitypremiumandtheriskinessofconsumptionriseandtherisk-freeinterestratefalls. Ishow that, both immediately after the shock and in the long run, the equity premium and the riskiness of consumption increase more than they would if the risk-free rate were constant. Indeed,thelong-runincreaseintheriskinessofconsumptiongrowthispurely ageneral-equilibriumeffect: iftherisk-freeratewereconstant(asinasmallopeneconomy), an endogenous decrease in risk-taking by entrepreneurs would, in the long run, completely offset the decrease in their ability to diversify. I also show that the credit shock leads to a decrease in aggregate capital if the elasticity of intertemporal substitution is sufficiently high. Finally, I show that, due to a general-equilibrium effect, thereisno“overshooting” intheequitypremium: inresponsetoapermanentdecrease infirms’abilitytopledgetheirfutureincome,theequitypremiumimmediatelyjumps to its new steady-state level and remains constant thereafter, even as aggregate capital adjusts over time. However, if idiosyncratic uncertainty is sufficiently low, credit tightening has no short- or long-run effects on aggregate capital, the equity premium, ortheriskinessofconsumption. Thusmypaperhighlightshowinvestmentriskaffects the economy’s response to a credit crunch. Keywords: Incomplete markets, idiosyncratic risk, business cycles, risk-free rates JEL codes: D52, E44, G11, O16 ∗Federal Reserve Board. I thank Vasia Panousi, Alp Simsek, Francisco Vazquez-Grande and various seminarparticipantsforhelpfulcomments. IamparticularlygratefultoGeorge-MariosAngeletos,Ricardo Caballero, Guido Lorenzoni and Iván Werning for their advice and encouragement. Any views expressed here are those of the author and need not represent the views of the Federal Reserve Board or its staff. Contact: jonathan.goldberg@frb.gov 1

1 Introduction This paper studies the short- and long-run effects of a shift from loose credit for firms to tight credit in an economy with uninsured investment risk. I focus on a general-equilibrium link between firms: financial claims issued by one firm can (directly or indirectly) be the asset of another firm. The debtor firm seeks to finance an investment. The creditor firm (which owns the debt perhaps in the form of a bank deposit) holds the other firm’s debt in ordertofinancefutureinvestmentsorasabufferagainstabadshocktoitsownprofitability. When a firm faces tighter credit conditions, it seeks to decrease its borrowing: in part because it can no longer borrow as much against a given investment and in part because, with tighter credit, some investments are not worth pursuing and hence do not generate collateral. If many firms face tighter credit conditions, they will all seek to decrease their borrowing at the same time. Hence there will be fewer stores of value for other firms to hold. To explore this linkage between firms and how it affects the economy’s response to a credit crunch, I develop a tractable dynamic, general-equilibrium model in which collateral constraints limit the ability of entrepreneurs to hedge idiosyncratic risk. The model illustrates the interaction between idiosyncratic investment risk and limited commitment. In contrast, many papers on uninsured investment risk, including Angeletos (2007), Angeletos and Panousi (2009), Angeletos and Panousi (2011), and Panousi (2012), abstract from borrowing constraints, while many papers on shocks to collateral constraints, including Buera and Moll (2012) and Midrigan and Xu (2012), abstract from uninsured investment risk. Thus, the paper’s main contribution is to characterize the short- and long-run effects of credittighteninginageneral-equilibriummodelwithuninsuredinvestmentrisk. Iemphasize the role of an endogenous decrease in the risk-free rate in determining the effects of credit tightening on capital accumulation, consumption riskiness, and the equity premium. Preview of the model and results. In the model, firms are run by entrepreneurs and the output of each firm is subject to an idiosyncratic shock. Entrepreneurs can issue state-contingent assets of any maturity, but the set of financial contracts available to an entrepreneur is constrained because, in the event of default, creditors’ recovery is limited. This limited-enforcement problem gives rise to collateral constraints. Due to these collateral constraints, taking advantage of investment opportunities requires forgoing insurance. Thus, entrepreneurs require a (private) equity premium. The shock that I study takes the form of a decrease in creditors’ recovery in the event of default. I begin by studying the economy’s long-run response. When collateral constraints tighten, entrepreneurs are forced to retain a greater share of their profits. As each entrepreneur shifts to self-financing, he reduces the quantity of financial assets that can be ownedbyotherentrepreneurstoinsureagainsttheirbadidiosyncraticshocks. Thishasim- 2

portantgeneral-equilibriumeffects: therisk-freeratefallsandconsumptiongrowthbecomes riskier. The equity premium rises, for two reasons: first, credit tightening reduces the share of (risky) profits that entrepreneurs can sell to diversified investors; and, second, because therisk-freeratefalls,hedgingismoreexpensive. Correspondingly,therisk-adjustedreturn to saving falls. This gives rise to an income effect and a substitution effect, and thus aggregatecapitaldecreasesifandonlyiftheelasticityofintertemporalsubstitutionissufficiently large. Ishowthatiftherisk-freeratewereconstant,asinasmallopeneconomy,theeconomy’s responsewouldbequalitativelydifferent. Inparticular,becausetherisk-freeratewouldnot fall as entrepreneurs shift to self-financing, the decrease in firms’ ability to borrow would lead entrepreneurs to scale back risk-taking. This endogenous decrease in risk-taking would be so large that, in the long run, the riskiness of consumption growth would be unchanged –eventhoughcredittighteningmechanicallyreducestheshareofprofitsthatentrepreneurs can sell to diversified investors. The equity premium would still rise, because, for a given investment, an entrepreneur has to bear more risk. However, because entrepreneurs would accumulate safe financial assets, they would be better hedged. Hence, the equity premium would not rise as much as it does when the risk-free rate is endogenous. This highlights the general-equilibrium effects of the endogenous decrease in the riskfree rate after a shock to collateral constraints. In the long run, the increase in the equity premiumishigherthanitwouldbeiftheinterestratewereconstant. Moreover,theincrease in the riskiness of consumption growth can be accounted for by the decrease in the risk-free rate. This analysis of the long-run response to credit tightening begs the question of how the economy responds in the short run. I show that there is no “overshooting” in the equity premium: inresponsetoapermanentdecreaseinfirms’abilitytopledgetheirfutureincome, the equity premium immediately jumps to its new steady-state level and remains constant thereafter, even as aggregate capital adjusts over time. One corollary of this result is that if idiosyncratic uncertainty is sufficiently low, credit tightening has no short- or long-run effectsonaggregatecapital,theequitypremium,ortheriskinessofconsumption. Ialsoshow that the absence of overshooting is a general-equilibrium effect. Finally, in the case where theelasticityofintertemporalsubstitutionisequaltoone,Ishowthat,intheshortrun,the increaseintheequitypremiumandtheriskinessofconsumptioncausedbycredittightening are larger than they would be if the risk-free rate were unaffected by credit tightening. 2 Related literature Mypaperbuildsontheliteratureonuninsuredinvestmentrisk,includingAngeletos(2007), Angeletos and Panousi (2009), Angeletos and Panousi (2011), and Panousi (2012). These 3

papers study how aggregate capital is affected by idiosyncratic investment risk using models in which entrepreneurs can trade only a risk-free bond. These papers ignore borrowing constraints(exceptforthe“natural” borrowingconstraintthatconsumptionneedalwaysbe positive) and, like Aiyagari (1994) and Guerrieri and Lorenzoni (2011), which use similar models of uninsured labor-income risk, ignore the micro-foundations for incomplete markets. In contrast, I explicitly incorporate a limited-enforcement problem that gives rise to collateral constraints. Even though my paper features collateral constraints and state-contingent debt, some of the economic forces at work in Angeletos (2007) are similar and one of my results – that aggregate capital decreases when firms’ ability to pledge their income decreases if and only if the elasticity of intertemporal substitution is greater than one – appears very similar to the main result of Angeletos (2007); in Angeletos (2007), aggregate capital is lower without state-contingentdebtthanwithcompletemarketsifandonlyiftheelasticityofsubstitution is above a related threshold. However, it is not the case that my model nests Angeletos (2007), or vice versa. In my model, even when firms cannot pledge any future profits, the zero correlation between idiosyncratic productivity and the payment to outside investors is an endogenous, general-equilibrium outcome: in partial equilibrium, the entrepreneurs in my model could hedge, even when they cannot pledge any future income, by receiving, for example,aminimumsalaryfrominvestorsregardlessoftherealizedshock. Thus,evenwhen firmscannotpledgeanyfutureprofits,thecorrelationbetweenidiosyncraticproductivityand the payment to outside investors is zero only because no entrepreneur can supply the assets that other entrepreneurs can use to hedge; that is, the risk-free rate must be sufficiently lowsuchthatentrepreneursarewillingtoforgoinsurance. Incontrast, inAngeletos(2007), the correlation between idiosyncratic productivity and the payment to outside investors is exogenous. One advantage of my approach is that, in my paper, idiosyncratic productivity riskandtheleveloffinancialdevelopmentcanvaryindependently;incontrast,inAngeletos and Panousi (2011), idiosyncratic productivity risk and financial development are one and the same. TwopapersthatemphasizeidiosyncraticriskandborrowingconstraintsareCovas(2006) and Guerrieri and Lorenzoni (2011). Covas (2006) studies the effects of idiosyncratic investment risk on steady-state aggregate capital in a calibrated Bewley-type model of entrepreneurs with an ad-hoc (as opposed to “natural”) borrowing constraint. In the paper’s simulations, Covas (2006) often finds that, with uninsured investment risk and borrowing constraints, steady-state aggregate capital is higher than under complete markets. Covas (2006) attributes these results, in part, to precautionary saving. That is, uninsured investment risk and borrowing constraints increase the hedging demand for the safe asset, depressing the risk-free rate and thus making the risky entrepreneurial investment more attractive. However, as highlighted in my paper, this decrease in the risk-free rate also 4

increases the cost of hedging, making risky entrepreneurial investment less attractive. In Covas (2006), it is not clear why the former effect tends to dominate the latter effect. Guerrieri and Lorenzoni (2011), like my paper, studies the short-run effects of a shock to borrowing constraints in the presence of idiosyncratic risk. Unlike my paper, Guerrieri and Lorenzoni(2011)focusonuninsuredlaborincomeriskinaBewley-typemodel, ratherthan uninsured investment income risk in a model with collateral constraints on investment. In Guerrieri and Lorenzoni (2011), as in my paper, a credit crunch leads to a decrease in the risk-free rate, providing an example of a shock that can lead to a negative real risk-free rate and potentially a binding zero-lower-bound on the nominal risk-free rate. Eggertsson and Krugman (2012) explore a similar point in a simple, tractable model of consumers who borrow and lend due to heterogeneity in impatience to consume. This paper also contributes to a growing literature that studies the short-run macroeconomicresponsetoshocksthatlimitfirms’abilitytopledgetheirprofitstooutsideinvestors. Some papers in this literature, including Jermann and Quadrini (2011), Kahn and Thomas (2010) and Buera and Moll (2012), study shocks to parameters, such as outside investors’ recovery rate in bankruptcy, that directly enter into firms’ collateral constraints. Other papers, including Christiano, Motto and Rostagno (2010), Gilchrist, Sim and Zakrajsek (2010), and Arellano, Bai and Kehoe (2010), study shocks to idiosyncratic and aggregate uncertainty in the presence of financial constraints. Jermann and Quadrini (2011), like this paper, examines the short-run effects of credit tightening. However, their results rely on two particular assumptions: (i) an ad-hoc cost of changing the payout to equityholders; and (ii) tax benefits of debt. If there are no ad-hoc costs to changing dividends and no tax benefits of debt, shocks to firms’ ability to borrow wouldnotaffecttherealeconomy. Inmypaper,incontrast,therearenofinancialconstraints beyond those that arise endogenously from the firms’ limited enforcement problem, and entrepreneurs’ financial decisions are driven by the investment risk they face. Kahn and Thomas (2011), in a calibrated DSGE model, study how fixed costs of capital adjustment affecttheeconomy’sresponsetoadecreaseinfirms’abilitytoborrow. InKahnandThomas (2011), idiosyncratic investment risk is important, but for a different reason than in my paper. In Kahn and Thomas (2011), equityholders are fully diversified, and idiosyncratic uncertaintymattersonlybecauserespondingtoshocksmaytriggerfixedcostsofadjustment. Inmypaper,asinAngeletos(2007),idiosyncraticuncertaintymattersbecauseentrepreneurs areundiversifiedownersoftheequityoftheirfirms. Also,likeJermannandQuadrini(2011), Kahn and Thomas (2011) features ad-hoc financial frictions, such as the inability of widely held firms to issue equity. Buera and Moll (2012), like this paper, analytically characterizes the short-run response to a collateral-constraints shock and emphasizes the effect of endogenous changes to the risk-free rate. However, the firm’s problem and the economy’s response to a credit crunch 5

in Buera and Moll (2012) are different than in my paper. Buera and Moll (2012) assume thateachfirm’sproductivityisknownwithcertaintyatthetimeofinvestmentandfinancial contracting; financial frictions limit the transfer of wealth from low-productivity firms to high-productivityfirmsandthereisnouninsuredinvestmentrisk. Incontrast,inmymodel, eachfirm’sproductivityisunknownatthetimeofinvestmentandfinancialcontracting,but the distribution of next-period productivity is the same for each firm. As a result, in my model,financialfrictionsdonotgiverisetomisallocationacrossfirms,butthereisuninsured investment risk. Unsurprisingly, the economy’s response to a collateral-constraints shock in my paper is qualitatively different than the economy’s response in Buera and Moll (2012). For example, Buera and Moll (2012) focus on the investment wedge, defined as the tax on investment in a representative-agent economy that would make the Euler equation hold, as in Chari, Kehoe and McGrattan (2007).1 One of Buera and Moll (2012)’s main results is that, with idiosyncratic productivity shocks, the investment wedge is equal to zero and is unaffected bychangesinfirms’abilitytoborrow. AlthoughBueraandMoll(2012)’sinvestmentwedge is not the focus of my paper, I find that, comparing across steady states, Buera and Moll (2012)’s entrepreneurial investment wedge is generally non-zero, has a sign that depends on the elasticity of intertemporal substitution, and increases in absolute value when firms’ ability to borrow decreases. A number of calibration papers in this literature also abstract from uninsured idiosyncratic investment risk by assuming that tomorrow’s productivity is revealed before today’s investment and financial-contracting decisions. Midrigan and Xu (2012) study a shock to collateral constraints in a calibration that matches Korean establishment-level data. Buera and Shin (2010) study the effects of capital liberalization and a decrease in idiosyncratic taxes in a calibration that matches six emerging-market growth-acceleration episodes. Bassetto, Cagetti and DeNardi (2010) study a variety of financial shocks in a calibration that matches the U.S. firm-size distribution. Like my paper, these papers study both short-run dynamics and long-run effects. It bears noting that, although my paper differs in important ways from these papers, many of these papers also emphasize the general-equilibrium effects of endogenous changes to the risk-free rate. For example, in the literature on collateral constraints, Midrigan and Xu (2012) find that steady-state TFP losses due to financial frictions are about 50 percent larger in a closed economy than in a small open economy. Similarly, Buera and Shin (2010) find that reducing idiosyncratic distortions leads to a much larger increase in TFP when accompaniedbyacapital-accountliberalizationthatresultsinanexogenousincreaseinthe interest rate. In the literature on uninsured investment risk, Panousi (2012) shows that the 1More specifically, they define the entrepreneurial investment wedge as the tax on investment in a representative-agent economy that would make the Euler equation hold for the aggregate consumption offirmowners. 6

effect of capital taxation on capital accumulation is theoretically ambiguous, a surprising result that emerges because capital taxation leads to an increase in the risk-free rate and this in turn leads to a decrease in the equity premium. 3 Model Time is discrete, indexed by t ∈ {0,...,∞}. There is a continuum of infinitely-lived entrepreneurs, indexed by i. Each entrepreneur runs a firm. There is only one good, used for both capital and consumption. Technology. In period t, firm i has a history of idiosyncratic productivity shocks si,t = {si,si,...,si} and invests ki in the production technology. In period t+1, the firm learns 0 1 t t productivitysi . Thefirmthenhireslaborli . Thetotaloutputproducedinperiodt+1 t+1 t+1 is: yi =F(ki,li ,si ). t+1 t t+1 t+1 F isaneoclassicalproductiontechnology. Periodt+1outputnetoflaborcostsisgivenby: πi =yi −ω li . t+1 t+1 t+1 t+1 Firmstakethewageω asgiven. Isuppressthedependenceofindividualvariablesonthe t+1 history of idiosyncratic shocks. Productivity si is independently and identically distributed over i and t. t Denote the cumulative distribution function over productivities s by P(s). I normalize the expected productivity to be equal to one. Financial markets. The firm can access financial markets by trading state-contingent promisesthatpayoutconditionalontherealizationofproductivity. Athistorysi,t,thefirm sellsaportfolioofArrow-Debreusecuritiesthatrepresentapromisetopaydi nextperiod. t+1 Note that the promised payment di may depend on the realization of si . Because the t+1 t+1 shocks are idiosyncratic, the proceeds from this sale, in equilibrium, are 1 E (cid:2) di (cid:3) , where Rt t t+1 R is the risk-free interest rate between periods t and t+1 and the expectation is taken t with respect to productivity si . t+1 Althoughthepromisesarestate-contingent,marketsareincompletebecauseentrepreneurs can renege on payment. In particular, if an entrepreneur reneges, the most that creditors can seize is a fraction θ of the firm’s output net of labor costs. This setup nests no ability t to borrow (θ = 0) and complete financial markets (θ = 1). When a firm reneges, the t t unmet portion of the firm’s debt is erased. (When the firm is allowed to issue promises due in more than one period, all future promises by the firm are also erased). Given this, the 7

firms repay in period t+1 if and only if: di ≤θ πi t+1 t t+1 By allowing state-contingent promises, I am able to focus exclusively on a single financial friction, the possibility of reneging.2 Note that θ controls both the entrepreneur’s ability t to finance the firm’s investments and his ability to hedge against idiosyncratic risk. The path of θ is deterministic but potentially time varying. Moreover, there exists a t τ <∞ such that θ =θ if t≥τ. t Preferences. I assume that entrepreneurs have Epstein-Zin preferences with constant elasticity of intertemporal substitution and constant relative risk aversion. That is, associatedwithastochasticconsumptionstream{ci}∞ isastochasticutilitystreamthatsatisfies t t=0 the following recursion: ui =U(ci)+βU (cid:0)CE (cid:2) U−1(ui ) (cid:3)(cid:1) t t t t+1 where β < 1 and CE (ui ) = Υ−1(cid:0) E (cid:2) Υ(ui ) (cid:3)(cid:1) is the certainty-equivalent of ui cont t+1 t t+1 t+1 ditional on period-t information. Υ and U are given by: c1−γ c1−1 (cid:15) Υ(c)= and U(c)= . 1−γ 1− 1 (cid:15) Note that γ > 0 denotes the coefficient of relative risk aversion and (cid:15) > 0 denotes the elasticityofintertemporalsubstitution. AsinAngeletos(2007),allowing(cid:15)γ ∈(0,∞)permits the forces that operate in this economy to be more clearly elucidated while also nesting the frequently made assumption that (cid:15)γ =1. The supply of labor is inelastic and, in aggregate, equals L. I normalize L=1. Budgets. An entrepreneur’s budget constraint at state stis: ci+ki− 1 E (cid:2) di (cid:3) ≤πi−di t t R t t+1 t t t Consumption and capital must not be negative: ci > 0 and ki > 0. The right-hand-side t t of the equation is defined as the entrepreneur’s period-t liquid wealth: wi ≡ πi−di. Note t t t that liquid wealth, wi, is equal to “cash on hand” (or “goods on hand,” since this is a real t 2It is without loss of generality to restrict attention to one-period promises. To see this, note that, if firms traded promises due in up to n>1 periods in the future, a firm could replicate the cash flows from any set of multi-period promises by making a one-period promise in period-t (due in period-t+1) with a face value equal to the total expected value of the period-t+1 through period-t+n promises, discounted to period t+1 using the spot-rate curve derived from {Rτ}t τ + = n t+ − 1 1. The entrepreneur would not renege on this set of one-period promises if the entrepreneur would not renege on the original set of multi-period promises. Rampini and Viswanathan (2010) make a similar point in a partial-equilibrium setting with a constantrisk-freerateandaconstantabilityoffirmstoborrowagainstcapital. 8

economy with a single good), rather than actual wealth, which would reflect the value of firm. I assume that workers are hand-to-mouth consumers, who live in financial autarky and each period consume their wage. Equilibrium. Theinitialconditionoftheeconomyisgivenbythedistributionof{ki ,di} −1 0 across firms. An equilibrium is a deterministic interest-rate and wage sequence {R ,ω }∞ , t t t=0 collections of state-contingent plans for entrepreneurs {ci,ki,di,li}∞ and paths for aggret t t t t=0 gate levels of debt D , consumption C , capital K , output Y and labor L such that: t t t t t 1. the plans {ci,ki,di,li}∞ maximize the utility of each entrepreneur; t t t t t=0 2. the bond-market clears: D =0 for all t; t 3. the labor-market clears: L =L for all t; t ´ ´ 4. aggregate quantities are determined by individual policies: C = ci, K = ki, ´ ´ t i t t i t D = di, and L = li. t i t t i t Notethatthepathforpricesandaggregatequantitiesisdeterministicbecauseidiosyncratic uncertainty washes out in the aggregate. 4 Equilibrium characterization 4.1 Partial equilibrium In the model, profits and labor demand for each entrepreneur are linear in the physical capital owned by the entrepreneur, due to constant returns to scale in technology and the ability to adjust labor demand according to the realization of idiosyncratic productivity: πi =f˜(si ,ω )ki and li =l(si ,ω )ki t+1 t+1 t+1 t t+1 t+1 t+1 t where f˜(s,ω)=max [F(1,z,s)−ωz] and l(s,ω)=argmax [F(1,z,s)−ωz]. z z Thus, the firm’s problem can be written recursively as: V(wi;t)= max U(wi−xi)+βUΥ−1E [ΥU−1V(wi ;t+1)] (4.1) t t t t t+1 ki,xi,{wi } t t t+1 subject to (cid:18) (cid:19) 1 1 E [wi ]= E [f˜(si ,ω )]−1 ki+xi (4.2) R t t+1 R t t+1 t+1 t t t t and, for each si , t+1 9

wi ≥(1−θ )f˜(si ,ω )ki (4.3) t+1 t t+1 t+1 t In period t, the entrepreneur chooses how much to consume, wi −xi, and its capital t t ki. These choices determine, through the budget constraint (4.2), the expected value of t next-period wealth. The firm also chooses a mapping from next-period productivity to next-period wealth, wi , that must satisfy the budget constraint (4.2) and the limitedt+1 enforcement constraints (4.3). The (private) equity premium is 1 E [f˜(si ,ω )]−1. When this equity premium is Rt t t+1 t+1 positive, the firm faces a three-way tradeoff between insurance, the timing of consumption, and taking advantage of its investment opportunity. On the one hand, increasing capital ki t increases expected next-period wealth, as shown in (4.2). On the other hand, for any state of the world si,t+1 for which (4.3) is binding, an increase in capital requires an increase in theamountofnext-periodperiodwealththatmustbeallocatedtothatstateattheexpense of either current consumption or the next-period wealth allocated to other states. If U−1V is weakly concave, which I will verify below, then the period-t+1 wealth level {wi } will feature some constant level of wealth for every state si below a threshold t+1 t+1 si∗ , and for every state si greater than this threshold, wealth will be determined by t+1 t+1 (4.3). That is, wi will be equal to the greater of a fixed salary or the minimum wealth t+1 level consistent with repayment. I denote the fixed salary by ni, so that t (cid:110) (cid:111) wi =max ni,(1−θ )f˜(si ,ω )ki . (4.4) t+1 t t t+1 t+1 t Using this intuition, firms’ optimal decisions for given prices can be characterized in closed form. Lemma 1. Given prices, optimal consumption ci, capital ki, and minimum payoff ni are t t t linear in wealth wi: t ci t = c˜ wi t t ki t = k˜ =(1−c˜)κ (4.5) wi t t t t ni t = n˜ =(1−c˜)η wi t t t t where  −1 ∞ τ (cid:88)(cid:89) c˜ t = 1+ β(cid:15)ρ(cid:15) j −1  (4.6) τ=tj=t (cid:104) (cid:110) (cid:111)(cid:105) ρ = CE max η ,(1−θ )f˜(s ,ω )κ (4.7) t t t t t+1 t+1 t 10

and (cid:104) (cid:110) (cid:111)(cid:105) {η ,κ } = argmaxCE max η,(1−θ )f˜(s ,ω )κ t t t t t+1 t+1 η,κ subject to (4.8) 1 (cid:110) (cid:111) (cid:18) 1 (cid:19) E[max η,(1−θ )f˜(s ,ω )κ ] = E[f˜(s ,ω )]−1 κ+1. R t t+1 t+1 R t+1 t+1 t t Lemma 1 shows that one can think about the entrepreneur’s decision of how much to consume as equivalent to that of an investor choosing how to consume and how much to invest in a risky asset with a risk-adjusted return ρ . This risk-adjusted return ρ reflects t t theoptimaltradeoffbetweentheexpectedvalueandriskinessofnext-periodwealth. Dueto thehomotheticityofpreferences,thelinearityofprofitsinphysicalcapital,andthelinearity ofthelimited-enforcementproblem,theentrepreneur’schoicesofhowmuchtoconsumeand how to divide savings among physical capital and insurance are linear in the entrepreneur’s wealth. Thus, we can write (4.4) as: wi (cid:110) (cid:111) t+1 =w˜ =max n˜ ,(1−θ )f˜(si ,ω )k˜ (4.9) wi t+1 t t t+1 t+1 t t Equation (4.9) shows how the limited-enforcement problem interferes with insurance and the optimal timing of consumption. In period t, the financial constraint (4.3) binds for all productivity realizations si above a threshold si∗ , which is uniquely determined by: t+1 t+1 n˜ =(1−θ )f˜(si∗ ,ω )k˜ . t t t+1 t+1 t For productivity realizations below the threshold si∗ , there is full insurance: that is, t+1 ci = c˜ n˜ wi for all si ≤ si∗ . Moreover, for these low productivity realizations, t+1 t+1 t t t+1 t+1 the consumption growth rate is undistorted in the following sense: an entrepreneur at history si,t is indifferent between (i) an additional unit of consumption next period in states with si ≤si∗ ; or (ii) 1 P(si∗ ) units of consumption in period t.3 t+1 t+1 Rt t+1 In contrast, for productivity realizations for which the financial constraint does bind, insurance is limited: that is, ci is strictly increasing in si for si > si∗ . Moreover, t+1 t+1 t+1 t+1 for these high productivity realizations, the consumption growth rate is “too high”: the entrepreneur would prefer 1 (cid:0) 1−P (cid:0) si∗ (cid:1)(cid:1) units of consumption in period t rather than Rt t+1 an additional unit of consumption next period in states with si >si∗ . t+1 t+1 Correspondingly,whenfinancialconstraintsbindwithpositiveprobability,entrepreneurs requireanexpectedreturnonphysicalcapitalgreaterthantherisk-freerate. Thefirst-order 3In the case of standard expected utility, for productivity realizations below the threshold si∗ , the t+1 consumptiongrowthrateofentrepreneuriisequalto(βRt)(cid:15),reflectingtheusualtradeoffbetweenconsuming Radditionalgoodsnextperiodoroneadditionalgoodtoday. 11

conditions of (4.8) imply that the equity premium is: (cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21) 1 1 w˜ E [f˜(si ,ω )]−1= E (1−θ )f˜(si ,ω ) 1−Υ(cid:48) t+1 (4.10) R t t+1 t+1 R t t t+1 t+1 n˜ t t t Consider an entrepreneur in period t with history si,t. At the margin, increasing capital ki t byoneunitrequiresanentrepreneurtohold(1−θ )f˜(si ,ω )unitsofadditionalwealth t t+1 t+1 next period in any state with si ≥ si∗ in order to satisfy the financial constraint for t+1 t+1 that state. To do so, the entrepreneur must hold less wealth in states with si < si∗ .4 t+1 t+1 However, period-t+1 consumption, ci , is already higher in states with si ≥ si∗ than t+1 t+1 t+1 (cid:16) (cid:17) in states with with si <si∗ . Thus, Υ(cid:48) w˜t+1 <1 for si >s∗ , and hence the equity t+1 t+1 n˜t t+1 t+1 premium is positive if financial constraints bind (i.e., if P(si∗ )<1). t+1 4.2 General equilibrium Because shocks are i.i.d. across firms, the period-t market-clearing wage ω = ω(K ) is t t−1 determined by L = K E (cid:2) l(si,ω(K )) (cid:3) . We can then define aggregate capital int−1 t−1 t t−1 (cid:104) (cid:105) comeinperiodtasΠ(K )=K E f˜(si,ω(K )) .ByLemma1,anentrepreneur’s t−1 t−1 t−1 t t−1 consumption, investment and minimum payoff are linear in his wealth, making the distribution of wealth irrelevant for calculating aggregates. This permits the following recursive characterization of general-equilibrium dynamics: Proposition 2. In equilibrium, the aggregate dynamics satisfy: C = c˜Π(K ) (4.11) t t t−1 K = k˜ Π(K ) (4.12) t t t−1 1 1 = 1+ β(cid:15)ρ(cid:15)−1 (4.13) c˜ c˜ t t t+1 1 = c˜ +k˜ (4.14) t t where k˜ is given by (4.5) and ρ is given by (4.7) . t t Equation (4.13) would hold in any model for an agent with Epstein-Zin preferences facing a sequence of risk-adjusted returns {ρ }∞ ; in this paper, this sequence of riskt t=0 adjusted returns comes from the optimal allocation of savings to capital (which earns a return subject to idiosyncratic risk) and financial assets (which can partially insure that idiosyncratic risk). Note that this system is recursive in (K ,c˜). In particular, given c˜, the markett−1 t t clearingcondition(4.14)canbeusedtodeterminek˜ . Next,equations(4.11)and(4.12)can t 4Theentrepreneurisindifferentbetweenadecreaseinperiod-tconsumptionandadecrease(withequal expectedvalue)inconsumptioninanystatessi,t+1 withsi ≤si∗ . t+1 t+1 12

be used to calculate C and K . Then the first-order conditions of (4.8) determine η and t t t R , which in turn imply ρ . Finally, (4.13) determines c˜ . t t t+1 A steady state is a fixed point (K,c˜) of the dynamic system (4.11)-(4.14) with K > 0 and c˜>0. 5 Steady state analysis I make the following assumption, as in Angeletos (2007). Assumption 1. F(K,L,s)=F(sK,L,1). Assumption 1 implies that f˜(s;ω(K)) = sF (K,1,1). An example of technology that K satisfies Assumption 1 is Cobb-Douglas technology. Proposition 3. (a) There exists (cid:15) ∈ [0,1) such that a steady state exists if and only if (cid:15)>(cid:15). (b) If a steady state exists, it is unique. In steady state w˜ = max{n˜,(1−θ)s} where n˜ satisfies: E[w˜]=E[max{n˜,(1−θ)s}]=1. (5.1) Aggregate capital is then determined by: 1 F K = β (CE[w˜]) 1 (cid:15) −1 (5.2) and the interest rate is determined by: (CE[w˜]) 1 (cid:15) βR= ≤1. (5.3) (cid:0)CE[w˜] (cid:1)γ n˜ Also, n˜ <1 and βR<1 if and only if P( 1 )<1. 1−θ Proposition 3 shows that, in steady state, entrepreneurs bear idiosyncratic risk and the interest rate is less than entrepreneurs’ discount rate if and only if there is a positive probability of a sufficiently high idiosyncratic productivity.5 To see why, note that when entrepreneurs borrow, they must keep at least (1 − θ) share of profits in order to credibly promise repayment. This means that, even when the economy is in a steady state, an entrepreneur’s wealth and consumption will grow after a sufficiently high realization of idiosyncraticproductivity. However,growthinwealthandconsumptionafterahighidiosyncraticshockisonlyconsistentwithasteadystateifentrepreneurs’wealthandconsumption 5Notethatifθ>0,thereisauniquen˜ thatsatisfies(5.1). Ifθ=0,theremaybemorethanonen˜ that satisfies (5.1); however, if n˜(cid:48) and n˜(cid:48)(cid:48) both satisfy (5.1), it will be the case that w˜(cid:48) = max{n˜(cid:48),(1−θ)s} = w˜(cid:48)(cid:48)=max{n˜(cid:48)(cid:48),(1−θ)s}=salmosteverywhere. SeetheproofofProposition3formoredetail. 13

decrease after a low idiosyncratic shock. Hence, if there is a positive probability of a sufficiently high productivity shock, the interest rate must be less than entrepreneurs’ discount rate to encourage entrepreneurs to hedge less against low idiosyncratic shocks. Hereafter, I will confine attention to economies which have a unique steady state. This is guaranteed by Assumption 2. Assumption 2. If P( 1 ) < 1, then (cid:15) > (cid:15) ≡ (1+ logβ )−1, where n˜ is 1−θ logCE t[max{n˜,(1−θ)s}] determined by E[max{n˜,(1−θ)s}]=1. Thethreshold(cid:15)inAssumption2isuniquelydeterminedbypreferenceparametersβ and γ, the distribution of idiosyncratic productivity P(s), and the ability of firms to borrow θ. The threshold (cid:15) does not depend on the production function F. It can be shown that (cid:15)<1 and that (cid:15) is increasing in β and γ and decreasing in θ. 5.1 Effect of a credit crunch The characterization of the economy’s steady state in Proposition 3 immediately leads to the following result. Proposition 4. Comparing steady-states for different values of firms’ ability to borrow, a decrease in θ will lead to: (a) a decrease in the minimum payoff n˜ and an increase in the riskiness of consumption; that is, the growth rate of idiosyncratic consumption with θ =θ second-order stochastically H dominates the growth rate of idiosyncratic consumption with θ =θ if and only if θ >θ ; L H L (b) a decrease in the interest rate, R; (c) an increase in the (private) equity premium, FK(K); R (d)adecreaseinaggregatecapitalifandonlyiftheelasticityofintertemporalsubstitution is greater than one. These changes will be strict if financial constraints bind in the new steady state. When there is a decrease in θ, firms are forced to retain a greater share of profits in states of high idiosyncratic productivity. For this to be consistent with steady state condition (5.1), entrepreneurs must insure less against low productivity realizations; and (5.3) implies that a decrease in the interest rate is required if entrepreneurs are to save less against low-productivity states. More specifically, consider the steady states associated with θ and θ , with θ < θ , L H L H andsupposeP( 1 )<1,sothatfinancialconstraintsbindwithpositiveprobabilityinthe 1−θL θ steady state. Denote the productivity threshold above which financial constraints bind L in the θ steady state by si∗. Then there will be a cutoff productivity sˆ∈ (si∗,∞) such L θL θL that: for any productivity si < sˆ, the consumption growth rate in the θ steady state is L lowerthantheconsumptiongrowthrateintheθ steadystate; foranyproductivitysi >sˆ, H 14

the opposite holds. Hence, it follows that a decrease in θ leads to a simple increase in risk in the distribution of consumption growth. According to (5.2), this increase in the riskiness of consumption leads to a decrease in aggregate capital if and only if the elasticity of intertemporal substitution is greater than one. This is because credit tightening leads to a decrease in the risk-adjusted return to saving, giving rise to opposing income and substitution effects; when the EIS is higher, the substitution effect is stronger. Using Assumption 1 and (4.10), the equity premium can be written as: F (K ) (cid:20) (cid:20) (cid:18) (cid:18) w˜ (cid:19)(cid:19)(cid:21)(cid:21)−1 K t = 1−E (1−θ )s 1−Υ(cid:48) t+1 (5.4) R t t n˜ t t Thus, in the general-equilibrium steady state: F (cid:20) (cid:20) (cid:26) (cid:18) (cid:18) (1−θ)s (cid:19)(cid:19)(cid:27)(cid:21)(cid:21)−1 K = 1−E max 0,(1−θ)s 1−Υ(cid:48) (5.5) R n˜ With a shift from θ to θ , for each additional unit of investment, an entrepreneur has H L to retain a greater share of the upside risk. Moreover, bearing the risk associated with entrepreneurial investment is even less attractive in the θ steady state than in the θ L H steady state, because, in the θ steady state, the interest rate is lower and it is more L expensive to hedge. Hence, as (5.5) shows, a decrease in θ leads to an increase in the equity premium. In the remainder of this section, I will examine how the economy would react differently to a decrease in θ if the risk-free rate were held constant, as in a small open economy. 5.2 Understanding the endogenous decrease in the interest rate and its effects ToobtainaclearerunderstandingofProposition4,defineasmall-open-economyequilibrium asequivalenttotheclosed-economyequilibriumdefinedabove,exceptthatthebond-market clearing condition is replaced with an exogenously given interest rate. Using the budget constraint (4.2) and Lemma 1, define the optimal debt policy as: di (cid:110) (cid:111) t+1 =d˜ =min f˜(si ,ω )k˜ −n˜ ,θ f˜(si ,ω )k˜ (5.6) wi t+1 t+1 t+1 t t t t+1 t+1 t t The aggregate dynamics in the small-open economy will satisfy:     C c˜ t t  K = k˜ (Π(K )−D ). (5.7)  t   t  t−1 t D E [d˜ ] t+1 t t+1 15

as well as the optimality condition (4.13) and the first-order conditions of (4.8). A smallopen-economy steady state is a fixed point of this system with c˜>0 and K >0. Lemma 5. Suppose βR<1 and θ <1. (a) There exists (cid:15) ∈ (0,1) such that a small-open-economy steady state exists if and soe only if (cid:15)>(cid:15) . soe (b) If a steady state exists, it is unique and entrepreneurs are not perfectly insured. Intuitively,if βR<1 andfinancialconstraintsdidnotbind,then,overtime,firmswould be decumulating wealth, which is inconsistent with a steady state.6 As in the analysis of the closed economy, I hereafter confine attention to economies that have a unique steady state. The next result shows how a small-open economy would respond to a decrease in θ. Proposition 6. Suppose βR < 1. Consider the effects of a decrease in θ, holding the interest rate constant, as in a small open economy. Comparing across steady-states, this will lead to: (a) no change in the riskiness of consumption growth; the steady-state consumption (cid:110) (cid:111) growth rate w˜ =max n˜,(1−θ)sF k˜ is unchanged; K (b) a decrease in net leverage, E[d˜], and expected profits per unit wealth, F k˜; K (c) an increase in the equity premium (and hence a decrease in aggregate capital). Even though a decrease in θ mechanically implies that a smaller share of risky future profits can be sold to diversified investors, Proposition 6 states that a decrease in θ leads to no change in the distribution of steady-state consumption growth. The riskiness of consumption growth is unchanged because a decrease in θ endogenously leads to a decrease in risk-taking that exactly offsets the exogenous decrease in entrepreneurs’ ability to hedge their risks. To see this, consider the steady-state version of the partial-equilibrium result (5.3): n˜ =(βR)γ 1CE[w˜]1− (cid:15) 1 γ (5.8) Clearly, with a constant interest rate, no change in the riskiness of consumption after a decrease in firms’ ability to pledge is consistent with (5.8): the certainty-equivalent of the wealth growthrate isunchanged; the interestrate isunchanged; andtheminimumpayoff n˜ isunchanged. Intuitively,itistheinterestratethatcontrolshowcostlyitisforentrepreneurs to hedge idiosyncratic risk. Theresponseoftheequitypremiumtoadecreaseinfirms’abilitytoborrow,holdingthe interest rate constant, can then be seen from (5.4). Because the riskiness of consumption is unchanged, Υ(cid:48)(cid:0)w˜(cid:1) is unchanged: shifting one unit of consumption from low-productivity n˜ 6IfβR=1,thentherearemultiplesteady-stateequilibria,buteachfeaturesthesameaggregatecapital andperfectconsumptioninsurance. 16

states to high-productivity states is equally unattractive before and after the decrease in firms’ ability to borrow. Nonetheless, mechanically, for each unit of physical capital that an entrepreneur holds, more consumption must be shifted from low-productivity states to high-productivity states. Correspondingly, the equity premium increases. Hence, aggregate capital decreases unambiguously. Proposition 6 explains why the risk-free rate falls in a closed economy in response to a decrease in θ. Consider a small, open economy with βR < 1 and suppose that, in the initial steady state, its current account was in balance. In this small open economy, a decrease in θ leads to a persistent current-account surplus. The decrease in leverage, E[d˜] = E[min{sF k˜−n˜,θsF k˜}], has a mechanical component and an endogenous one. K K Mechanically, the decrease in θ means that, for high realizations of next-period productivity, the share of next-period profits pledged must decrease, in order to satisfy the limitedenforcement constraint. In addition, the decrease in θ leads to an endogenous decrease in expected profits per unit wealth, F k˜. Thus, a decrease in θ in the small open economy K leads to a decrease in net leverage, which explains why, in the closed economy, the interest rate must fall in response to a decrease in θ. ThenextpropositionisthecomplementtoProposition6: thatis,Proposition7examines how a small-open economy is affected by changes in the risk-free rate. Proposition 7. Consider the effect of a decrease in the interest rate, holding θ constant. Comparing across steady-states, this will lead to: (a) a decrease in the minimum payoff n˜ and an increase in the riskiness of consumption growth; (b) an increase in net leverage, E[d˜], and expected profits per unit wealth, F k˜; K (c) an increase in the equity premium. In the small open economy, a decrease in the interest rate, holding all other exogenous variablesconstant,leadstoanincreaseintheriskinessofconsumptiongrowth. Theintuition is clearest with standard expected utility, which features (cid:15)γ = 1. With standard expected utility, the steady-state consumption growth rate for low productivity realizations equals (βR )(cid:15), reflecting the usual tradeoff between consuming R additional goods next period t or one additional good today. Thus, a decrease in the risk-free rate leads to a decrease in insurance against low productivity realizations. In order for this to be consistent with steadystate, expectedprofitsperunitwealthmustincrease. Correspondingly, netleverage, E[d˜]=min{sF k˜−n˜,θsF k˜] increases as well. K K As before, the response of the equity premium to a decrease in the interest rate, holding all other exogenous variables constant, can be seen from (5.4). Because the riskiness of consumption increases, Υ(cid:48)(cid:0)w˜(cid:1) decreases: shifting one unit of consumption from lown˜ productivity states to high-productivity states is more painful than before the decrease in the interest rate. Thus, even though there is no change in the amount, per unit of physical 17

capital, of wealth that must be shifted from low-productivity states to high-productivity states, reducing insurance is more painful because of the decrease in the risk-free rate. 6 Transition dynamics In this section, I study how the economy responds in the short run to the decrease in firms’ ability to borrow. In the preceding analysis of the long-run response, I showed that a decrease in firms’ ability to borrow leads to an increase in the equity premium and the riskiness of consumption. I also showed that steady-state aggregate capital may increase or decrease in response to a decrease in firms’ ability to borrow, depending on the elasticity of intertemporal substitution. These results raise a number of questions about the shortrun dynamics. For example, one might conjecture that if entrepreneurs accumulate wealth over time in response to credit tightening, then there will be overshooting in the equity premium: the equity premium would shoot up at first, but this initial increase would be partially reversed over time. 6.1 No overshooting of the equity premium Thus, in this section, I fully characterize the transition dynamics of the equity premium and the riskiness of consumption for any path for {θ }. It turns out that, due to a generalt equilibrium effect, the equity premium and one measure of consumption riskiness do not display any over- or under-shooting in response to a decrease in firms’ ability to borrow. In this section, I explicitly denote the dependence of steady-state variables on the share of profits that firms are able to pledge in the long run; for example, K(θ) is steady state aggregate capital when lim θ =θ. t→ t Proposition 8. Atanydatet, theequitypremiumandthecoefficientofvariationofperiodt+1 consumption are the same as in the steady state with firms’ ability to borrow equal to θ . More specifically, t F(K ) F(K(θ )) t = t R R(θ ) t t ci and the cumulative distribution function of t+1 is the same as the c.d.f. of w˜(θ ). Et [ci t+1 ] t Proposition 8 implies that, in response to a permanent decrease in firms’ ability to borrow, the equity premium and the the coefficient of variation of next-period consumption immediately jump to their new steady-state levels; there is no overshooting. During the transition, aggregate capital, K , the minimum payoff, n˜ , and expected profits, F (K )k˜ , t t K t t may be changing, but the equity premium and the coefficient of variation of next-period consumption will remain constant after their initial jump to their new steady-state levels. 18

Another implication of Proposition 8 is that, at time t, the future path for firms’ ability to borrow, {θ }∞ , does not affect the period-t equity premium or the coefficient of τ τ=t+1 variationofperiod-t+1consumption, althoughitmayaffectperiod-taggregatecapitaland consumption. The absence of overshooting described in Proposition 8 is a general-equilibrium effect. The equity premium, given in (5.4), is greater when idiosyncratic consumption is riskier; ci thatis,when t+1 ismorerisky. Thus,theequitypremiumcanovershoot(orundershoot) Et [ci t+1 ] after a permanent shock only if the riskiness of consumption varies with aggregate capital. However, in the closed economy studied here, this is not possible. Consider a shock to aggregate capital, all else equal: if in response to the shock, entrepreneurs sought to take more risk by increasing leverage, this would lead to an increase in the supply of financial assets, leading entrepreneurs to hedge more and thus contradicting the premise that the aggregate wealth shock leads to more risk taking. 6.2 Understanding the role of the interest rate in the short run It is interesting to consider the short-run dynamics of the risk-free rate and whether the equity premium and consumption riskiness increase more in the short-run than they would if the interest-rate were fixed. I address this questions for the special case in which the elasticity of intertemporal substitution is equal to one. With an EIS equal to one, the path for aggregate capital is unaffected by the path for collateral constraints and idiosyncratic uncertainty. Proposition 9. Suppose that (cid:15)=1. (a) The path for aggregate quantities {K ,C }∞ is unique and converges monotonically t t t=0 to the unique steady state, with K =F−1(1) and C =(1−β)Π(K). K β (b) Compare two economies, H and L, that are identical in their initial conditions and exogenous parameters, except that θH > θL. Then RH > RL and the period-τ equity pre- τ τ τ τ mium in economy L is greater than it would be if the period-τ risk-free rate in economy L were exogenous and equal to RH. τ Proposition9showsthat,withtheEISequaltoone,adecreaseinfirms’abilitytopledge leads, on impact, to an endogenous decrease in the risk-free rate and that this decrease in the risk-free rate amplifies the concurrent increase in the equity premium. And because the equity premium depends on the riskiness of idiosyncratic consumption growth, Proposition 9impliesthattheendogenousdecreaseintherisk-freeratealsoleadstogreaterrisk-taking.7 7Morespecifically,thedistributionof ci τ , + H 1 representsasimpleincreaseinriskrelativetothedistri- Eτ[ci τ , + H 1 ] ci,L ci,L butionof τ+1 ,asimpliedbyProposition8. Moreover,thedistributionof τ+1 representsasimple Eτ[ci τ , + L 1 ] Eτ[ci τ , + L 1 ] increaseinriskrelativetothedistributionthatwouldoccurineconomyLiftheperiod-τ risk-freeratewere exogenousandequaltoRH. τ 19

Thereasonthatadecreaseintherisk-freerateinperiod-τ leadstoalargerequitypremium intheshort-runissimilartothereasonthatadecreaseinthelong-runrisk-freerateleadsto alargerequitypremiuminthelongrun. Whentherisk-freerateislow,hedgingisexpensive; thus entrepreneurs choose to hedge less and correspondingly they demand a higher equity premium. 7 Conclusion Anumberofrecentpapershavestudiedtheshort-andlong-runeffectsofshockstocollateral constraintsinmodelsthatabstractfromuninsuredinvestmentrisk. Atthesametime,other papershavestudieduninsuredinvestmentriskwhileignoringborrowingconstraints. Inthis paper,Icharacterizedtheeconomy’sresponsetoashocktocollateralconstraintsinamodel with uninsured risk. In the model, credit tightening leads to an increase in the equity premium and the riskiness of consumption. I showed that, both immediately after the shock and in the long run, the equity premium and the riskiness of consumption increase morethantheywouldiftherisk-freeratewereconstant. Indeed,thelong-runincreaseinthe riskiness of consumption growth was shown to be a purely general-equilibrium effect: if the risk-free rate were constant (as in a small open economy), an endogenous decrease in risktakingbyentrepreneurswould,inthelongrun,completelyoffsetthedecreaseintheirability to diversify. I also showed that the credit shock leads to a decrease in aggregate capital if theelasticityofintertemporalsubstitutionissufficientlyhigh. Finally,Idemonstratedthat, due to a general-equilibrium effect, there is no “overshooting” in the equity premium: in response to a permanent decrease in firms’ ability to pledge their future income, the equity premium immediately jumps to its new steady-state level and remains constant thereafter, even as aggregate capital adjusts over time. Because the results highlight how changes in the risk-free rate affect risk-taking and the equity premium, it would be useful to extend the model to understand the implications of uninsured investment risk for monetary policy and the optimal level of government debt. 20

References Aiyagari, SRao.1994.“Uninsuredidiosyncraticriskandaggregatesaving.” TheQuarterly Journal of Economics, 109(3): 659–684. Angeletos, George-Marios.2007.“Uninsuredidiosyncraticinvestmentriskandaggregate saving.” Review of Economic Dynamics, 10(1): 1–30. Angeletos, George-Marios, and Vasia Panousi. 2009. “Revisiting the supply side effects of government spending.” Journal of Monetary Economics, 56(2): 137–153. Angeletos, George-Marios, and Vasia Panousi. 2011. “Financial integration, entrepreneurialriskandglobaldynamics.” Journal of Economic Theory,146(3):863–896. Arellano, Cristina, Yan Bai, and Patrick Kehoe. 2010. “Financial Markets and Fluctuations in Uncertainty.” Federal Reserve Bank of Minneapolis Working Paper. Bassetto, Marco, Marco Cagetti, and Mariacristina De Nardi. 2010. “Credit Crunches and Credit Allocation in a Model of Entrepreneurship.” Chicago Fed mimeo. Buera, Francisco J, and Benjamin Moll. 2012. “Aggregate implications of a credit crunch.” Mimeo, UCLA. Buera,FranciscoJ,andYongseokShin.2010.“Financialfrictionsandthepersistenceof history: A quantitative exploration.” National Bureau of Economic Research Working Paper No. 16400. Chari, VV, Patrick J Kehoe, and Ellen R McGrattan. 2007. “Business Cylce Accounting.” Econometrica, 781–836. Christiano, Lawrence, Roberto Motto, and Massimo Rostagno. 2010. “Financial factors in economic fluctuations.” ECB Working Paper. Covas, Francisco. 2006. “Uninsured idiosyncratic production risk with borrowing constraints.” Journal of Economic Dynamics and Control, 30(11): 2167–2190. Eggertsson, Gauti B, and Paul Krugman. 2012. “Debt, Deleveraging, and the Liquidity Trap: A Fisher-Minsky-Koo Approach.” The Quarterly Journal of Economics, 127(3): 1469–1513. Gilchrist, Simon, Jae W Sim, and Egon Zakrajšek. 2010. “Uncertainty, Financial Frictions, and Investment Dynamics.” Mimeo, BU. Guerrieri, Veronica, andGuidoLorenzoni.2011.“Creditcrises,precautionarysavings, and the liquidity trap.” National Bureau of Economic Research Working Paper No. 17583. 21

Jermann, Urban, and Vincenzo Quadrini. 2010. “Macroeconomic effects of financial shocks.” Mimeo, Wharton. Khan, Aubhik, and Julia K Thomas. 2011. “Credit shocks and aggregate fluctuations inaneconomywithproductionheterogeneity.” NationalBureauofEconomicResearch Working Paper No. 17311. Midrigan, Virgiliu, and Daniel Yi Xu. 2012. “Finance and misallocation: Evidence from plant-level data.” Mimeo, NYU. Panousi,Vasia.2012.“Capitaltaxationwithentrepreneurialrisk.” Mimeo,FederalReserve Board. Rampini, Adriano A, and S Viswanathan. 2010. “Collateral, risk management, and the distribution of debt capacity.” The Journal of Finance, 65(6): 2293–2322. 22

Appendix A: Proofs Proof of Lemma 1 Define B(xi;t)= max Υ−1E [ΥU−1V(wi ;t+1)] (7.1) t t t+1 ki,{wi } t ,t+1 subject to (4.2) and (4.3). Using (7.1), we can write (4.1) as: V(wi;t)=maxU(wi−xi)+βU(B(xi;t)). (7.2) t t t t xi t Conjecture the following solution: V(wi;t) = U(a wi) (7.3) t t t ci t = c˜ wi t t ki t = k˜ wi t t ni t = n˜ wi t t wi (cid:110) (cid:111) t+1 = max n˜ ,(1−θ )f˜(si ,ω )k˜ wi t t t+1 t+1 t t where a , c˜, k˜ , and n˜ are time-varying but deterministic coefficients to be determined. t t t t From (7.3), we have that B(xi;t)=maxai CE (max(ni,(1−θ )f˜(si ;ω )ki)) (7.4) t t+1 t t t t+1 t+1 t ki,ni t t subject to E [max(ni,(1−θ )f˜(si ;ω )ki]=E [f˜(si ;ω )]ki+R (xi−ki) (7.5) t t t t+1 t+1 t t t+1 t+1 t t t t Dividingthefirst-orderconditionwithrespecttoki bythefirst-orderconditionwithrespect t to ni, one obtains: t ni  E t (cid:104) max (cid:110) 0,(1−θ t )f˜(si t+1 ;ω t+1 ) (cid:111)(cid:105) − (cid:16) E t [f˜(si t+1 ;ω t+1 )]−R t (cid:17)1/γ k t i t = E t (cid:104) max (cid:110) 0,((1−θ t )f˜(si t+1 ;ω t+1 ))1−γ (cid:111)(cid:105)  (7.6) where si∗ is defined by ni =(1−θ )f˜(si∗ ,ω )ki. t+1 t t t+1 t+1 t This,togetherwiththelinearityof(7.5),impliesthatthesolutionsniandkito(7.4)-(7.5) t t are linear in xi and that B(xi;t) is linear in xi. In particular, we have B(xi;t)=a ρ xi, t t t t t+1 t t 23

where ρ is given by (4.7). Moreover, re-arranging (7.6), one obtains (4.10). t Substituting for B(x ;t) into (7.2), one obtains: t V(wi;t)=U(wi−xi)+βU(a ρ xi). t t t t+1 t t Due to homotheticity, the solution of this problem will be of the form xi =(1−c˜)wi. t t t The envelope condition is: 1−1 −1 a (cid:15) =c˜ (cid:15). (7.7) t t Takingthefirst-ordercondition,usingtheenvelopecondition,andrearranging,oneobtains: 1 1 =1+ β(cid:15)ρ(cid:15)−1. (7.8) c˜ c˜ t t t+1 Forward iteration of (7.8) implies (4.6). Finally, to verify (7.3), one must confirm that U(a wi)=U(c˜wi)+βU(CE (cid:0) a w˜ wi(cid:1) ). t t t t t t+1 t+1 t Substituting from ρ = 1 CE [w˜ ], where ρ is defined as in 4.7, one has t 1−c˜t t t+1 t U(a wi)=U(c˜wi)+βU(a wi(1−c˜)ρ ). t t t t t+1 t t t Dividing both sides by U(wi) and using the envelope condition (7.7), one obtains (7.8).♦ t Proof of Proposition 3 From assumption 1, one obtains Π(K )=F (K )K . Setting K =K =K in (4.12), t K t t t t−1 one obtains the steady-state condition: F (K)k˜ =1. (7.9) K Hence, in steady state, w˜ = max{n˜,(1−θ)s} and (5.1) holds. If θ > 0, there is a unique n˜ ∈ (0,1] that satisfies (5.1). If θ = 0, there may be more than one n˜ that satisfies (5.1). However, consider w˜(cid:48) = max{n˜(cid:48),(1−θ)s} and w˜(cid:48)(cid:48) = max{n˜(cid:48)(cid:48),(1−θ)s}, where n˜(cid:48) and n˜(cid:48)(cid:48) both satisfy (5.1). If θ =0, then w˜(cid:48) =w˜(cid:48)(cid:48) =s almost everywhere. Using (4.7) and (7.8), one can write: c˜ 1−c˜ t =βCE t [w˜ t+1 ]1−1 (cid:15) ( c˜ t )1 (cid:15) (7.10) t+1 Then, setting c˜ =c˜ =c˜in (7.10), one obtains t t+1 c˜=1−βCE[w˜]1−1 (cid:15) (7.11) 24

From (7.11), observe that c˜= 1−β > 0 if n˜ = 1. If n˜ < 1, then CE[w˜] ∈ (0,1). Hence, if n˜ <1, there exists (cid:15)=(1+ logβ )−1 <1 such that c˜>0 if (cid:15)>(cid:15). logCE[w˜] Using (4.14), (7.9) and (7.11), one obtains (5.2). Next, using (4.2) , (7.6) and (7.11) , one obtains (5.3). Note that si∗ = n˜ . If P( 1 )<1, then: n˜ <1; P(si∗)≤P( 1 )<1; 1−θ 1−θ 1−θ and, from (5.3), βR < 1. If P( 1 ) = 1, then: n˜ = 1; P(si∗) = P( 1 ) = 1; and from, 1−θ 1−θ (5.3), βR=1.♦ Proof of Proposition 4 I will compare the steady states corresponding to two different values for firms’ ability to borrow: θ and θ <θ . H L H Suppose that P( 1 ) < 1. Denoting explicitly the dependence of steady-state policies 1−θL on θ and using (5.1), one obtains that n˜(θ ) > n˜(θ ). This result, together with (5.5), H L implies that FK(θH) < FK(θL). R(θH) R(θL) Next, I will show that a decrease in θ generates a mean-preserving spread in w˜. Note that, in steady state, w˜ is the growth rate of idiosyncratic wealth and the growth rate of idiosyncratic consumption. Denote by G(x) the cumulative distribution function of w˜(θ)= max{n˜(θ),(1−θ)s}. Observe that E[w˜(θ )]=E[w˜(θ )]=1. Furthermore, H L   =0 if x<n˜(θ )  L  G(x;θ )−G(x;θ )= >0 if x∈(n˜(θ ),n˜(θ )) . L H L H ≤0 if x>n˜(θ )  H Thus,adecreaseinθgeneratesamean-preservingspreadinw˜. ThisimpliesthatCE[w˜(θ )]> H CE[w˜(θ )]. Then, from (5.2), we have that K(θ ) < K(θ ) if and only if (cid:15) > 1. Finally, L L H (5.3) and n˜(θ )>n˜(θ ) imply that R(θ )>R(θ ).♦ H L H L Proof of Lemma 5 In this proof, I will assume that s is a continuous random variable and that P(s) has positive support over (0,∞). However, the more general proof is very similar. In a small open economy, n˜ and F k˜ are uniquely determined by the steady-state con- K ditionE[w˜]=E[max{n˜,(1−θ)sF k˜}]=1and(5.3), whichisapartial-equilibriumsteady- K state condition. Define a function z(x) by E[max{z(x),sx}]=1, where the domain of x is [0,1]. Observe that: z(0) = 1; z(1) = 0; z is continuous; and z is strictly decreasing in x. Write (5.3) as: (CE[max{z(x),sx}]) 1 (cid:15) βR= ≡H(x) (7.12) (cid:16) (cid:17)γ CE[max{1,s x }] z(x) Note that: H(0)=1; lim H(x)=0; and H is strictly decreasing in x. Thus, associated x→1 with any BR < 1 there is a unique value for F k˜ and a unique value for n˜ < 1 that K 25

satisfy E[max{n˜,(1−θ)sF k˜}]=1 and (5.3). Then (4.10) determines the unique value of K aggregate capital. From (7.11), observe that c˜> 0 if (cid:15) > (cid:15) = (1+ logβ )−1, where w˜ = max{n˜,(1− logCE t[w˜] θ)sF k˜}.♦ K Proof of Proposition 6 From (7.12), one observes that x and z(x) do not depend on θ. Hence, a decrease in θ leads to a decrease in F k˜ and has no effect on n˜. (5.4) implies that a decrease in θ leads K to an increase in FK.♦ R Proof of Proposition 7 From (7.12), we see that a decrease in R leads to an increase in x and thus a decrease in z(x). Hence, a decrease in R leads to an increase in F k˜ and a decrease in n˜. (5.4) implies K thatadecreaseinRleadstoanincreasein F R K.Becausen˜ decreasesandF K k˜ increases,one obtainsthatadecreaseinRleadstoasimpleincreaseinriskinw˜ =max{n˜,(1−θ)sF k˜}.♦ K Proof of Proposition 8 As in the main text, I explicitly denote the dependence of steady-state variables on the share of profits that firms are able to pledge in the long run; for example, K(θ) is steady state aggregate capital when lim θ =θ. t→ t Financial market clearing implies: E[d˜]=E[min{sF (K )k˜ −n˜ ,θ sF (K )k˜ }]=0 t K t t t t K t t whichdeterminesauniquevaluefor FK(Kt)k˜ t,with FK(Kt)k˜ t = FK(K(θt))k˜(θt) = 1 .Then, n˜t n˜t n˜(θt) n˜(θt) from 5.4, we have FK(Kt) = FK(K(θt)). Rt+1 R(θt) Next, observe that ci n˜ t+1 = max{ t ,(1−θ )s} E[ci ] F (K )k˜ t t+1 K t t = max{n˜(θ ),(1−θ )s}=w˜(θ ) t t t ci andhencethedistributionof t+1 (conditionalonperiod-tinformationanduncondition- Et[ci t+1 ] ally) is equal to the distribution of w˜(θ ).♦ t Proof of Proposition 9 ProofofPart(a). From(4.6)andtheassumptionthat(cid:15)=1,oneobservesthatc˜ =1−β t for all t. Then, from (4.14), one obtains that k˜ = β. Moreover, from (4.12), we have that t steady state capital satisfies F (K) = 1 and that the growth rate of aggregate capital is K β given by Kt+1 =βF (K ), so that convergence is monotonic. Kt K t 26

Proof of Part (b). Propositions 4 and 8 imply that FK(K τ L) > FK(K τ H). From part (a), RL RH τ τ we have that KL =KH and hence RH >RL. τ τ τ τ Finally, consider an economy in which firms’ ability to borrow is given by (cid:8) θL(cid:9)∞ but t t=0 theperiod-τ interestrateisexogenousandequaltoRH. Suppose,bycontradiction,thatthe τ period-τ equitypremiuminthishypotheticaleconomyweregreaterthantheperiod-τ equity premium in economy L. Then, because RH >RL, we have that period-τ aggregate capital τ τ in this economy is less than period-τ aggregate capital in economy L. Because aggregate capital in period-τ −1 is identical in the hypothetical economy and in economy L, this implies that period-τ net leverage in the hypothetical economy is less than period-τ net leverage in economy L, which is given by E[d˜L ]=0. Equation (5.4) then implies that the τ+1 period-τ equity premium in the hypothetical economy is greater than the period-τ equity premium in economy L, a contradiction.♦ 27