DELEVERAGING DYNAMICS

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1 DELEVEAGING DYNAMICS JONATHAN GOLDBEG Abstract. How does the economy respond to a shift from loose credit for rms to tight credit? I develop a tractable model of deleveraging in which the scarcity of nancial assets interacts with tradeo between, on the one hand, taking advantage of investment opportunities, and on the other hand, the costs of risk management and holding liquid wealth. The model emphasizes (i) rms' dynamic accumulation of liquid wealth and (ii) heterogeneity in rms' ability to borrow. In the model, a decrease in the ability of rms to borrow leads to: increased capital misallocation and decreased total factor productivity (TFP); an increased wedge between the average marginal product of capital and the interest rate; and increased consumption riskiness. An endogenous decrease in the interest rate is shown to amplify these eects by discouraging retained earnings and risk management. I study the short- and long-run responses of the economy to temporary and permanent credit-crunch shocks, and characterize the conditions under which rms' dynamic balance-sheet adjustments lead to a decrease in TFP that is larger in the short run than in the long run. I present some preliminary numerical results suggesting that these amplication eects are large. Preliminary and Incomplete. Please do not circulate without permission. Date: February 202. Federal eserve Board. Any views expressed here are those of the author and need not represent the views of the Federal eserve Board or its sta. This paper is related to the rst chapter of my PhD dissertation. I am extremely grateful to my advisors George-Marios Angeletos, icardo Caballero and Guido Lorenzoni for their invaluable guidance and feedback. I also thank Vasia Panousi, Alp Simsek, Francisco Vazquez- Grande and various seminar participants for helpful comments. I am responsible for any errors. Contact: jonathan.goldberg@frb.gov.

2 DELEVEAGING DYNAMICS 2. Introduction A decline in rms' ability to borrow can lead to a reduction in rms' supply of nancial assets. The resulting scarcity of nancial assets can amplify the distortions caused by the decline in rms' ability to borrow. This scarcity can also lead to an increase in distortions for rms which did not experience a decline in their ability to borrow. To explore this idea formally, this paper builds a tractable dynamic model of the eects of a decrease in rms' ability to borrow. In the model, rms can only pledge to creditors a fraction of their future prots, and hence rms must nance part of their investment through retained earnings. However, the scarcity of nancial assets makes it expensive to retain earnings. A decline in rms' ability to borrow leads to greater scarcity of nancial assets and a decrease in the interest rate. This makes retained earnings more expensive, leading to greater investment distortions and decreased risk management. When there is heterogeneity in rms' ability to borrow, the endogenous decrease in the interest rate also increases the misallocation of capital across rms. By understanding how rms are aected by the scarcity of nancial assets and low interest rates, this model sheds light on how the economy responds to a shift from loose credit for rms to tight credit. The model shows that a credit crunch leads to an endogenous decrease in the interest rate, and that this decrease in the interest amplies the eects of the credit crunch on aggregate productivity and rm-level risk management. The model also shows how, as part of the same general-equilibrium dynamics, lower aggregate productivity and a reduced ability to manage rm-level risks leads to a larger decrease in the interest rate. The model shows that these general-equilibrium eects show up immediately in the short-run response of the economy to a credit crunch and also highlight the conditions under which these eects lead to a short-run overshooting in the level of investment distortions and consumption riskiness. These results underscore the importance of a general equilibrium framework for analyzing the economy's response to deterioration in the quality of the nancial system, and explain how the response to credit crunch in a closed economy will dier from the response if the interest rate is held constant, as in analyses that presume a small-open economy. In the model, rms, which are run by entrepreneurs, can only pledge a fraction of their future prots to creditors. As a result, a fraction of each rm's assets must be funded through retained earnings - which is expensive when the interest rate is low - and the rm may not be able to ooad all its idiosyncratic risks to diversied investors. This gives rise to a tradeo between, on the one hand, scale and, on the other hand, risk management and the cost of self-nancing through retained earnings. This tradeo is aected by the interest rate. A simple illustration of this tradeo comes from considering the long-run tightness of nancial constraints for an individual rm at an exogenous interest rate: if the interest rate is equal to the rate of time preference, the rm eventually retains enough earnings

3 DELEVEAGING DYNAMICS 3 to completely overcome the borrowing constraints. In contrast, if the interest rate is less than the rate of time preference, then in the long run, the rm will have binding nancial constraints and the entrepreneur will bear idiosyncratic risk. When there is a decrease in rms' ability to pledge their prots, the supply of nancial assets at a given interest rate decreases. This occurs for two reasons. The rst is a direct eect: conditional on the rms' capital choices, the maximum amount of nancial assets that rms can supply decreases. The second is an indirect eect: the decrease in rms' ability to borrow, holding rms' capital choices constant, reduces rms' ability to manage risks and increases the share of capital that must be nanced through retained earnings, a form of nancing which is expensive. Holding the interest rate constant, rms respond by choosing lower capital. Because the supply of nancial assets at a given interest rate decreases, the interest rate must decrease in order for the nancial market to clear. This endogenous decrease in the interest rate leads to an increase in the ratio between the marginal product of capital and the interest rate, an investment wedge which I call conditional under-investment. This is because only part of a rm's assets can be nanced at the market interest rate, and hence a rm's marginal product of capital does not vary one-for-one with the interest rate. The endogenous decrease in the interest rate also leads to a decrease in risk management. The lower interest rate leads entrepreneurs to increase their leverage to invest a greater share of their assets in physical capital, at the expense of a riskier consumption stream. In the baseline model with log utility, it can be shown analytically that the increase in investment distortions and the decrease in risk management occur immediately after the decrease in rms' ability to borrow and there is no over- (or under-) shooting. elaxing the assumption of log utility allows for richer dynamics, which I explore numerically. When there is heterogeneity in rms' ability to borrow, a credit crunch leads to an increase in misallocation, and this increase is amplied by the endogenous decrease in the interest rate. In the case of Cobb-Douglas production technology, this increase in misallocation takes the form of a decrease in aggregate measured total factor productivity. A lower interest rate generates increased misallocation because a rm's equilibrium marginal product of capital responds more sluggishly to a decrease in the interest rate if its ability to borrow is lower. Thus, a decrease in the interest rate causes high-pledgeability rms to expand capital and output to greater extent than equally productive rms with a lower ability to pledge future income. The model can also generate misallocation without heterogeneity in rms' ability to borrow, if productivity shocks are persistent. elated literature. My paper is related to three areas of the literature: (i) a growing set of papers on how the macroeconomy responds to changes in rms' ability to borrow; (ii) recent papers on the macroeconomic impact of uninsured idiosyncratic investment risk, including

4 DELEVEAGING DYNAMICS 4 Angeletos (2007) and Angeletos and Panousi (20); and (iii) papers, such as Holmstrom and Tirole (998), on the scarcity of savings instruments and rms' tradeo between scale and insurance. A number of papers have recently examined how the macroeconomy responds to changes in rms' ability to borrow. Important examples that focus on business-cycle frequencies include Jermann and Quadrini (20), Kahn and Thomas (200) and Buera and Moll (20). Other papers, such as Midrigan and Xu (200), Moll (20) and Buera, Kaboski and Shin (20), examine how steady-state outcomes vary with exogenous dierences in rms' ability to borrow, to shed light on dierences across countries with dierent levels of nancial development. Almost bridging these two literatures are papers, such as Buera and Shin (2008) and Buera and Shin (200), that examine the transition dynamics that ensue when there is an increase in rms' ability to borrow. My approach is complementary to these papers. One important dierence, however, is that my paper focuses on heterogeneity in rms' ability to borrow as the source of misallocation. In contrast, in most of these papers, the source of misallocation is a uniform restriction on rms' ability to borrow combined with persistent productivity shocks. In my paper, with heterogeneity in rms' ability to borrow, misallocation can occur even if there are no productivity shocks. In contrast, in papers that emphasize a uniform restriction on rms' ability to borrow, the extent of misallocation depends crucially on the characteristics of the shocks and, in the absence of shocks, (intensive-margin) misallocation often disappears quickly over time and is absent in the steady state (Banerjee and Moll 200). Another important dierence is the focus of my paper on endogenous changes to the interest rate caused by a decrease in rms' ability to borrow, and how these endogenous changes to the interest rate amplify the the eect of the shock on investment distortions and consumption riskiness. This focus is best highlighted by comparing my paper and Midrigan and Xu (200). Midrigan and Xu (200) assume a small open economy and hence in their paper the interest rate is exogenous. In counterfactual experiments exploring the eect of the elimination of borrowing constraints, the interest rate remains xed; Midrigan and Xu (200) nd that the eects on TFP of eliminating borrowing constraints in Colombia and South Korea are modest. In contrast, the central focus of this paper is the general The contribution of Kahn and Thomas (200) is to add xed costs of adjustment as additional frictions that amplify the misallocation caused by productivity shocks and a uniform restriction on rms' ability to borrow. Buera and Moll (20) study heterogeneity in recruitment costs and heterogeneity in investment costs, in addition to persistent productivity shocks, as sources of misallocation, in order to demonstrate that the impulse responses of measured aggregate wedges to a shock to rms' ability to borrow dier markedly across economies with dierent types of heterogeneity present. In Buera and Moll (20), the persistence of productivity shocks comes only from rms knowing their i.i.d. productivity shock before they make their investment and nancial decisions; they refer to this as heterogeneity in productivity. Nonetheless, the main intuition for why misallocation can occur in the economy with heterogeneity in productivity is the same as in other papers with persistent shocks, such as Moll (20) and Buera, Kaboski and Shin (20).

5 DELEVEAGING DYNAMICS 5 equilibrium change in the interest rate and its amplication eects. I am able to demonstrate and explain this amplication eect analytically and with some generality. My paper is also related to recent papers by Angeletos (2007) and Angeletos and Panousi (20) on the macroeconomic eects of uninsured idiosyncratic investment risk. These papers propose tractable frameworks for analyzing uninsured idiosyncratic investment risk, and use these frameworks to analyze the implications of this risk for aggregate saving and for the response of the global economy to nancial integration. This paper also studies unisured idiosyncratic investment risk. In particular, the moral-hazard problem that limits rms' ability to pledge future prots simultaneously limits the ability of owners of privatelyheld rms to ooad idiosyncratic investment risk. Hence, the credit-crunch shock that limits rms' ability to borrow is also a shock that limits their ability insure against idiosyncratic risk. Moreover, I analyze how endogenous changes in the interest rate caused by a decrease in rms' ability to borrow (and hedge risks) aects owners' ability to manage risks by making risk management more expensive. In my paper, nancial constraints arise endogenously from a single friction, the ability of rms to partially renege on their promises. There is no restriction on the state-contingency or the maturity structure of nancial contracts, except for the restrictions that arise due to the limited commitment problem faced by rms. Moreover, the share of idiosyncratic risk that can be ooaded to diversied investors depends, through the endogenous constraint, on the scale of investment and the dynamic nancing decisions of rms. In contrast, in Angeletos (2007) and Angeletos and Panousi (20), the share of idiosyncratic risk that can be ooaded is exogenous. Likewise, most papers on the macroeconomic eects of changes to rms' ability to borrow feature ad-hoc forms of market incompleteness; for example, Jermann and Quadrini (20) features ad-hoc costs of adjusting dividends, while many other papers in which the borrowing constraint can be micro-founded also include additional ad-hoc forms of market incompleteness, such as the absence of assets that are state contingent or with a maturity greater than one period. My paper builds methodologically on Angeletos (2007) and Angeletos and Panousi (20). In particular, I follow these papers in using constant returns to scale and homotheticity of preferences to deliver linear policies, which facilitates aggregation and, in the absence of persistent productivity shocks, makes the wealth distribution irrelevant. As a result, as in Angeletos (2007), the equilibrium here can be characterized with a low-dimensional, closedform recursive system of equations. In the model used here, this tractability permits the paper's main claims, including some claims about transition dynamics, to be demonstrated analytically. Although I allow for state-contingent assets with an arbitrary maturity structure, the model retains its tractability because restricting attention to one-period, statecontingent assets is shown in this setting to be without loss of generality.

6 DELEVEAGING DYNAMICS 6 In addition, my paper is related to Holmstrom and Tirole (998), which analyzes the scarcity of savings instruments and rms' tradeo between scale and insurance. This paper, and related contributions such as Holmstrom and Tirole (200) and ampini and Viswanathan (200), use a three-period model to analyze the eects of general-equilibrium limits to the availability of liquid assets. 2 In these papers, agents' initial liquid wealth and the external supply of assets play a crucial role, and there are no dynamics; in my paper, the evolution of agents' liquid wealth and the interest rate in response to a shock to rms' ability to borrow is endogenously determined. Lorenzoni and Guerrieri (200) studies the eects of a credit crunch. In particular, Lorenzoni and Guerrieri (200) study the response of an economy of Bewley-Aiyagari consumers to a permanent, unexpected tightening of their borrowing constraint. Like this paper, Lorenzoni and Guerrieri (200) emphasizes precautionary saving and the scarcity of liquid assets. One important dierence between these papers is that Lorenzoni and Guerrieri (200) focus on changes in the borrowing constraints of consumers, whereas this paper emphasizes shocks to the borrowing constraints of rms. Also, because my model is much more analytically tractable than a Bewley-Aiyagari economy, greater use can be made of analytical results here. 2. Model Time is discrete, indexed by t {0,,..., }. There is a continuum of rms, indexed by i. Each rm is run by an entrepreneur. There is only one good, used for both capital and consumption. Technology. In period t, rm i produces F (k i t, l i t, s i t), where k i t is capital, l i t is labor and s i t is the rm's idiosyncratic productivity. F is a neoclassical production technology. In period t, rm i has a history of idiosyncratic productivity shocks s i,t = {s i 0, s i,..., s i t} and invests k i (s i,t ) in the production technology. In period t+, the rm learns productivity s i t+. The rm then hires labor l i (s i,t+ ). The total output produced in period t + is: y i t+ = F (k i t, l i t+, s i t+). Period t + output net of labor costs is given by: π i t+ = y i t+ ω t+ l i t+. Firms take the wage ω t+ as given. Note that π i includes undepreciated capital. Productivity s i t is independently and identically distributed over i and t. Denote the cumulative distribution function over productivities s by P (s). I normalize the expected productivity to be equal to one. Corporate nance. 2 One exception to the three-period style of these papers is Farhi and Tirole (200), which uses an OLG model with Holmstrom and Tirole (998).rms to study bubbles.

7 DELEVEAGING DYNAMICS 7 The rm can access nancial markets by trading state-contingent promises that pay out conditional on the realization of productivity. At history h i t, the rm sells d i (s i,t+ ) Arrow- Debreu securities that represent a promise to pay one if state s i,t+ is realized. Because the shocks are idiosyncratic, in equilibrium, the unit price in period t is t+ P r(s t+ ), where t+ is the risk-free interest rate between periods t and t +. Although the promises are state-contingent, markets are incomplete because rms can renege on payment. In particular, if a rm reneges, the most that creditors can seize is a fraction θ j t of the rm's output net of labor costs. This setup nests no ability to borrow (θ = 0) and complete nancial markets (θ = ). When a rm reneges, the unmet portion of the rm's debt is erased. (When the rm is allowed to issue promises due in more than one period, all future promises by the rm are also erased). Given this, the rm will keep its promise to pay d i (s i,t+ ) if and only if: d i (s i,t+ ) θ t π i t+ By allowing state-contingent promises, I am able to focus exclusively on a single nancial friction, the possibility of reneging. In reality, entrepreneurs can make state-contingent promises in a variety of ways. For example, through default or renegotiation, putatively non-contingent debt becomes state-contingent. Note that θ t controls both the entrepreneur's ability to nance the rm's investments and his ability to hedge against idiosyncratic risk. The path of θ t is deterministic. Moreoever, I assume that lim t θ t exists. The supply of labor is inelastic and, in aggregate, equals L. I normalize L =. Preferences. I assume that entrepreneurs have time-separable constant relative risk aversion (CA) preferences: where β <. β t u(c i t) t=0 Budgets. An entrepreneur's budget constraint at state s t is: c i (s t ) + k i (s t ) t+ E [ d i (s t+ ) s t] π(k i (s t ), s t ) d i (s t ) Also, consumption and capital must not be negative: c i (h t ) > 0 and k i (h t ) > 0. The righthand-side of the equation is dened as the entrepreneur's period-t liquid wealth: w i π i t d i t. Note that liquid wealth, w i, is equal to cash on hand (or goods on hand, since this is a real economy with a single good), rather than actual wealth, which would reect the value of rm, which produces a stream of risky prots for the entrepreneur. I assume that workers are hand-to-mouth consumers, who live in nancial autarky and each period consume their wage.

8 DELEVEAGING DYNAMICS 8 Equilibrium. The initial conditions of the economy is given by the distribution of {k0, i d i 0} across rms. An equilibrium is a deterministic interest-rate and wage sequence { t, ω t } t=0, collections of state-contingent plans for entrepreneurs {c i t, kt, i d i t, lt} i t=0 and paths for aggregate levels of debt D t, consumption C t, capital K t, output Y t and labor L t such that: () the plans {c i t, kt, i d i t, lt} i t=0 maximize the utility of each entrepreneur; (2) the bond-market clears: D t = 0 for all t; (3) the labor-market clears: L t = L for all t; (4) aggregate quantities are determined by individual policies: D t = E[d i t], C t = E[c i t], Y t = E[yt], i and L t = E[n i t]. Note that the expectation in the market clearing conditions integrate across all histories and entrepreneurs. Because idiosyncratic uncertainty washes out in the aggregate, the path for aggregate debt D t owed in period t, aggregate capital K t chosen in period t, aggregate consumption C t and output Y t in period t is deterministic and given by D t = E[d i t], C t = E[c i t], Y t = E[yt]. i 3. Equilibrium characterization 3.. Partial equilibrium. To characterize individual behavior for given prices, the rst step is to recognize that, as in Angeletos (2007), the individual entrepreneur faces linear returns to private investment due to constant returns to scale in technology and the ability to adjust labor demand according to the realization of idiosyncratic productivity. Lemma. Individual labor demand lt+ i and output net of labor costs πi t+ capital kt, i decreasing in ω t and increasing in s i t. In particular: are linear in π i t+ = f(s i t+, ω t+ )k t and l i t+ = l(s i t+, ω t+ ) where f(s, ω) = max n [F (, n, s) ωn] and l(s,ω) = arg max n [F (, n, s) ωn]. The next step also builds on the same paper. Angeletos (2007) shows that with riskfree debt and no nancial constraint beyond the natural-solvency (no-ponzi) constraint, the combination of linear returns to private investment and CA preferences makes an entrepreneur's problem identical to the canonical portfolio choice problem. In my setting, with state-contingent debt and with an endogenous nancial constraint that will sometimes bind, I show that homothetic preferences and linear returns to private investment still make the problem tractable, because the nancial constraint is linear in output net of labor costs, which, as shown in Lemma, is linear in capital.

9 DELEVEAGING DYNAMICS 9 Lemma 2. Given prices, optimal consumption, investment and next-period state-contingent wealth for limited-pledgeability rms are linear in wealth: where (3.) τ c t = + (3.2) k t = w s,t = max (3.3) τ=t j=t g j c t c t+ h( t+, ω t+, θ t ) c t = c t w t k t = k t w t w s,t+ = w s,t w t { } (β t+ ) c ρ t, ( θ t ) c f(s; ω t+ ) k t t+ g t = g ( t+, ω t+, θ t ) h t t+ E and h t = h( t+, ω t+, θ t ) satises: [ { }] min f(s; ωt+ )h t (β t+ ) ρ, θt f(s; ωt+ )h t (3.4) [ [ { E f(s; ωt+ )] = ( θ t )E f(s; ω t+ ) max β (( θ t ) t+ f(s; ) }] ρ ω t+ )h t, 0 t+ To understand the form that the policy functions take, it is useful to examine the entrepreneur's problem, which can be written recursively as: (3.5) V (w t ; t) = subject to max u(w t + E[ k t,{w s,t+} f(s t+ ; ω t+ )k t w s,t+ ] k t ) + βe[v (w s,t+ ; t + )] t+ w s,t+ ( θ t ) f(s t+ ; ω t+ )k t for all s t+. The rst-order condition for capital is: [ (3.6) E f(st+ ; ω t+ )] = ( θ t ) t+ u (c t ) E[φ f(s s,t t+ ; ω t+ )] The rst-order condition for an individual rm's capital immediately implies that if the nancial constraints are binding for a positive measure of productivities for a positive measure of rms, then aggregate capital K t must be strictly less than the level of capital that maximizes expected prots next period, discounted at the market interest rate, less investment.

10 DELEVEAGING DYNAMICS 0 For given prices, the rst-order condition for wealth tomorrow in state s t+ and the envelope condition imply: (3.7) φ s,t = u (c t ) βu (c t+ (s t+ )). t+ Thus, next-period consumption will be equal across all next-period idiosyncratic states s for which the nancial constraint is not binding in the current period. Also, next-period consumption in these states will be lower than this-period consumption if β t+ <. Now consider any next-period idiosyncratic states s for which the nancial constraint is binding in the current period. Entrepreneurs would like to transfer wealth away from these nextperiod states, but the binding nancial constraint prevents them from doing so. Next- period consumption in these states will be higher than in any states for which the constraint is not binding. Combining the rst-order conditions for capital (3.6) and next-period wealth (3.7), one obtains (3.4). This highlights a trade-o faced by the entrepreneur between, on the one hand, insurance and the optimal inter-temporal allocation of consumption, and on the other hand, taking advantage of investment opportunities (scale). Conditional on a decision to be entirely self-nancing, the entrepreneur could (and would) perfectly insure his idiosyncratic risks; moreover, the entrepreneur would choose a path for next-period consumption that re- ected only his willingness to intertemporally substitute and the interest rate. In contrast, conditional on a decision to leverage his wealth as much as possible today, an entrepreneur's insurance would necessarily be incomplete. Moreover, for given prices, if tomorrow's idiosyncratic productivity is high, the entrepreneur's consumption path will be steeper than it would be under self-nancing or complete markets. Equation (3.2) shows how the entrepreneur's equilibrium wealth varies with the productivity realization. In particular, the next-period wealth function could be reproduced by a guaranteed payment to the entrepreneur plus a call options on ( θ) share of next-period c capital income, with a strike price of t c t+ (β t+ ) ρ. The nancial constraint in period t for state s t are binding when the call option is in the money for that state; that is, when productivity s t s t, where s is dened by ( θ) f(s t, ω t ) = ct c t+ (β t+ ) ρ. The remaining equations in this lemma are then determined from the budget constraint. In particular, g t is the ratio of the cost of the optimal amount of period-(t + ) consumption to the cost of the optimal amount of period-t consumption. Thus, if no nancial constraints bind in period t, then g t = t+ (β t+ ) ρ ; with no binding nancial constraints, constant elasticity of intertemporal substitution implies consumption in period t + should equal (β t+ ) ρ times period-t consumption. When nancial constraints do bind, the growth of consumption varies with idiosyncratic productivity and, for high productivities, is greater than (β t+ ) ρ ; this is reected in (3.3) and (3.).

11 DELEVEAGING DYNAMICS 3.2. General equilibrium: Characterization, existence and uniqueness. Because shocks are i.i.d. across rms, the market-clearing wage ω t = ω(k t ) is determined by L = K t l(s, ω(kt ))dp (s). Next, dene f(k t ) = f(s, ω(kt ))dp (s). We can then de- ne aggregate capital income in period t as Π(K t ) = K t f(kt ). By Lemma (2), consumption, investment and next-period state-contingent wealth are linear in individual wealth, making the distribution of wealth irrelevant for calculating aggregates, as in Angeletos (2007). This permits the following recursive characterization of general-equilibrium dynamics: Proposition 3. In general equilibrium, the aggregate dynamics satisify (3.4) and (3.8) (3.9) (3.0) (3.) (3.2) C t = c t Π (K t ) K t = k t Π (K t ) = + g t c t c t+ L = K t l(s, ω(k t ))dp (s) = c t + k t where g t = g ( t+, ω t+, θ t ) is given by (3.3). Equations (3.8) and (3.9) follow from the linearity of the optimal consumption and capital policies. Equation (3.0) comes from the rst order conditions of the rm's problem (3.5). Equations (3.) and (3.2) are, respectively, the labor-market clearing condition and the bond-market clearing conditions. Note that this system is recursive in (K t, c t ). In particular, (3.) can be used to eliminate ω t+ and (3.2) can be used to eliminate k t. Given (K t, c t ), equations (3.8) and (3.9) can be used to calculate C t and K t. Then (3.4) and (3.0) determine c t+ and t+. A steady state is a xed point of the dynamic system (3.8)-(3.). In steady-state, the capital stock K, the interest rate and the investment policy k solve: (3.3) K = kπ(k) (3.4) [ { E min f(s; K) k (β) ρ, θ f(s; }] K) k = 0 (3.5) [ ] E f(s; K) = ( θ)e [ { ( f(s; K) max β ( θ) f(s; ) }] ρ K) k, 0 For the remainder of the analysis, unless explicitly stated otherwise, I make the following assumptions. Assumption A. Agents have log utility: u(c) = log(c).

12 DELEVEAGING DYNAMICS 2 Proposition 4. There exists a unique equilibrium path for prices { t, ω t } t=0, aggregate quantities {K t, C t, Y t } t=0, and policies { k t, c t, w s,t } t=0, starting from any initial aggregate capital K 0 > 0 and converging to the unique steady-state (K,, k) determined by (3.3)- (3.5). Moreover, k t = β for all t and, if there exists a τ such that θ t = lim v θ v for all t τ, convergence is monotonic for t τ Properties of the steady state. Lemma 5. In the unique steady-state equilibrium, β. Moreover, in steady state, β < and nancial constraints bind for a positive measure of productivities if and only if there is a positive probability of productivity shocks s greater than ŝ(θ), where ŝ(θ) is dened by f(ŝ(θ); f ( β )) = β θ. For the remainder of the analysis, I make the following assumption, as in Angeletos (2007). Assumption A2. F (K, L, s) = F (sk, L, ). Assumption A2 implies that f(s; ω(k)) = sf K (K,, ). Under A2, Lemma 5 implies that nancial constraints bind in equilibrium and β < if and only if there is a positive probability of productivity shocks greater than or equal to θ times the mean productivity shock. 4. Effects of a decrease in firms' ability to borrow 4.. Steady state. The next results characterize the persistent eects of a decrease in rms' ability to borrow. By comparing across steady-states with dierent values of rms' ability to borrowθ, I am able to analyze the persistent eects of any long-lasting component of a shock to rms' ability to borrow. Understanding the persistent eects of a long-lasting shock is interesting in light of nancial-crisis episodes in which the recovery of nancial intermediation was very slow (ajan and Zingales (2003), einhart and einhart (200)) and in which the eects on output and total factor productivity were persistent (Cerra and Saxenna 2008, Kehoe and uhl 2009). Moreover, understanding how the steady-state of an economy responds to any permanent component of the decrease in θ is helpful for understanding how the economy responds to transitory shocks to θ, which is analyzed next. Proposition 6. Comparing steady-states for dierent values of rm's ability to borrow, a decrease in θ will lead to: (a) a (weak) decrease in the interest rate, ; (b) no change in aggregate capital, consumption or output; (c) a distribution over consumption paths that is riskier in the sense that entrepreneurs (weakly) prefer the distribution under the higher level of θ even though the distributions have

13 DELEVEAGING DYNAMICS 3 the same mean; moreover, there is a (weak) decrease in the productivity cuto s above which nancial constraints bind. The decrease in the interest rate, the increase in the riskiness of consumption, and the decrease in the cuto productivity at which nancial constraints bind will be strict if nancial constraints bind in the new steady state. In steady-state, the marginal product of capital equals the inverse of the discount factor, for any θ. To understand this result, consider the steady-state asset supply curve in this economy, dened as the amount of assets rms would supply in steady state per unit of wealth, conditional on rms' ability to borrow, θ, and an exogenously given interest rate. The per-unit-wealth supply curve, for a given θ, is the function mapping per-unit-wealth asset supply into the interest rate that would give rise to this supply. Because the prices of nancial assets are inversely proportional to the interest rate, the supply curve is downward sloping. When rms' ability to borrow decreases, this shifts the supply curve inward. As discussed below, the curve moves inward for two reasons: (i) mechanically, the rms' ability to borrow against a given amount of prots decreases; and (ii) prots per unit wealth, f k, decrease because investment is less attractive. The inward shift of the asset supply curve implies that the interest rate must decrease in order for the bond market to clear. This explains part (a) of the above proposition. Holding θ constant, the decrease in the interest rate will lead to an increase in aggregate capital, again, as will be shown below. It turns out that, with log utility, this increase in aggregate capital from lowering the interest rate (holding θ constant) exactly osets the decrease in aggregate capital from decreasing rms' ability to borrow θ (holding the interest rate constant). 3 This, together with the result that entrepreneurs' steady-state consumption share of wealth is β regardless of rms' ability to borrow, explains part (b) of the above proposition. If I were to relax the assumption of log utility and allow for CA utility with non-unitary relative risk aversion, it would no longer be the case that the increase in aggregate capital from lowering the interest rate (holding θ constant) exactly osets the decrease in aggregate capital from decreasing rms' ability to borrow θ (holding the interest rate constant); however, the two opposing eects from the exogenous change in rms' ability to borrow and the endogenous decrease in the interest rate would remain. 3 The intuition for this result is that, for any given interest rate and ability to borrow, entrepreneurs' steadystate consumption share of wealth will be β. In a closed economy, market clearing then implies that the share of wealth invested in capital will be β. Moreover, in a closed economy, the steady-state capital stock must equal the share of capital invested times aggregate prots (3.3). Thus, in a closed economy, the share of capital invested equals the average product of capital. Under constant returns to scale equals the expected marginal product of capital; that is, Π(K) = E[f]K. Thus, because the steady-state share of wealth invested is constant, it follows that the steady-state expected marginal product of capital is constant, and hence steady-state aggregate capital is constant. Moreover, since the consumption share of wealth is constant, aggregate consumption is constant.

14 DELEVEAGING DYNAMICS 4 Figure ( ) shows how a decrease in rms' ability to borrow aects rms' risk-management policy; it is drawn for the case at which nancial constraints bind before the decrease in θ. A decrease in rms' ability to borrow leads to a decrease in s, the productivity cuto at which nancial constraints start to bind. Mechanically, this is driven by two changes. First, because the interest rate is lower, rms' optimal choice for next-period wealth ignorning the nancial constraint is lower than before; this accounts for the decrease in the intercept of w s. Second, conditional on the amount of aggregate capital, rms can borrow less against next-period prots due to the decrease in θ; this accounts for the increase in the slope of w s. Since the w s policy chosen by rms' in the high-θ steady-state is feasible for rms' in the low-θ steady-state, and since the rm's problem is convex, it must be that entrepreneurs prefer the high-θ consumption path to the low-θ consumption path. This explains part (c) of the above proposition. Proposition 6 implies that a decrease in rms' ability to borrow leads to an increase in the steady-state ratio of the marginal product of capital to the interest rate; in particular, the former is unchanged and the later decreases. This ratio represents conditional underinvestment; that is, conditional on the interest rate, rms would like to invest more, but choose not to because of their nancial constraints. In particular, conditional on the interest rate, an individual rm could invest more, but due to the nancial constraints, this would require lowering consumption today in return for increasing consumption tomorrow in states of high productivity. In equilibrium, doing so is not attractive, because it would move consumption from to states of the world where the discounted marginal utility of consumption is lower than the marginal utility of consumption today. Proposition 6 implies that conditional under-investment increases when rms' ability to borrow decreases. These intuitions were based on some partial-equilibrium results that I will now make precise. In doing so, I will distinguish the eects of a decrease in rms' ability to borrow in a closed economy, and the eects of a decrease in rms' ability to borrow when the interest rate is held constant. The latter corresponds to the response of a small-open economy to a decrease in rms' ability to borrow. By decomposing the eects of a decrease in rms' ability to borrow into a partial-equilibrium component (decreasing θbut holding the interest rate constant) and a general-equilibrium component (holding θconstant but decreasing the interest rate), I can examine whether the endogenous decrease in the interest rate serves to amplify or dampen certain eects. Dene a small-open-economy equilibrium as equivalent to the closed-economy equilibrium dened in Section 2 above, except that the bond-market clearing condition is replaced with an exogenously given interest rate. Then, the aggregate dynamics in the small-open economy will satisfy

15 DELEVEAGING DYNAMICS 5 (4.) (4.2) (4.3) C t = c t (Π(K t ) D t ) K t = k t (Π(K t ) D t ) D t = E[ d t ](Π(K t ) D t ) and conditions (3.0), (3.) and (3.4) from Proposition 5, which characterized the aggregate dynamics in the closed economy. The steady-state formulations of (4.)-(4.3) are obtained by replacing aggregate quantities with their steady-state values. Lemma 7. If β <, there exists a unique steady-state small-open-economy equilibrium, and the measure of productivities for which nancial constraints bind is strictly positive. If β =, then there are multiple steady-state equilibria, but each features the same aggregate capital and perfect consumption insurance. To see why the set of productivities for which nancial constraints bind has positive measure if and only if β <, suppose, by contradiction, that β < and that nancial constraints did not bind for any productivity. Then next-period wealth will be lower than current wealth, regardless of productivity, and this is inconsistent with a steady state. This result is not specic to log utility. The next result, however, does require log utility, and facilitates the subsequent analysis. Lemma 8. In any (closed or small-open-economy) steady-state equilibrium, the consumption share of wealth c is β. It is useful here to dene the following measure of riskiness, which is weaker than secondorder stochastic dominance. Denition 9. A lottery over consumption paths {c A,t } is said to be riskier than an alternative lottery over consumption paths c B,t if β t u(c A,t ) < β t u(c B,t E[c A,t ] E[c B,t ] ) Proposition 0. Consider the eects 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) an increase in the ratio of the marginal product of capital to the interest rate if β < ; if β =, there will be no change in this ratio; (b) a decrease in aggregate capital, consumption and output if β < ; if β =, there will be no change in aggregate capital and output; (c) a decrease in E d, borrowing per unit wealth; (d) no change in the riskiness of consumption paths; the productivity cuto at which nancial constraints bind is unchanged.

16 DELEVEAGING DYNAMICS 6 When rms' ability to borrow decreases, rms are forced to nance a smaller share of their investment at the market interest rate. A larger share must be nanced out of retained earnings. However, when the market interest rate is low (i.e., β < ), retaining earnings is expensive: absent nancial constraints, entrepreneurs would prefer a consumption path that is declining over time. Hence, a decrease in rms' ability to borrow, holding the interest rate constant, serves to increase rms' eective cost of capital. This explains parts (a)-(c) of Proposition 0. To understand this argument more clearly, consider the special case of constant productivity (that is, no productivity shocks). With constant productivity, the rst-order condition (3.6) becomes: (4.4) f = θ + β( θ). That is, the marginal product of capital is equal to the weighted geometric average of the interest rate and the inverse of the discount factor. 4 An entrepreneur who invests an additional unit of capital can increase consumption today by θ f and consumption tomorrow by ( θ) f. In steady state, the marginal utility of consumption is constant; with discounting, this yields (4.4). Of course, if β <, the entrepreneur would prefer to move tomorrow's addtional consumption ( θ) f to today at the market interest rate, but the binding nancial constraint prevents it. 5 Part (d) of Proposition (0) is surprising. Although the decrease in the interest rate mechanically reduces the ability of the rm to manage risks, one obtains that risk management in the small-open economy steady state is unaected by a decrease in rms' ability to borrow. That is, the steady-state ratio of next-period wealth to today's wealth, w s, is unchanged. To see why, note that there are two ways w s = max{β, ( θ)fk} can change: the interest rate can fall, leading the entrepreneur to want a steeper downward-sloping consumption path in the absence of nancial constraints; or the minimum state-contingent ratio of tomorrow's wealth to today's wealth required by nancial constraints, ( θ) f k, can change. A decrease in θ, all else equal, would mechanically increase the later. However, because the interest rate is being held constant, this would be inconsistent with a steady state, as entrepreneurs' wealth, on average, would then be increasing over time. Thus, it must be that ( θ) f k does not change when θchanges, and hence risk management is unaected by a decrease in θin the small open economy. Therefore, the increase in consumption riskiness generated by 4 This is consistent with the result in Proposition 6 because in a closed economy with no productivity shocks, the general-equilibrium interest rate is equal to the inverse of the discount factor. 5 If entrepreneurs were risk neutral, then (4.4) would still apply. Moreover, with productivity shocks and risk neutrality, (4.4) would still apply, except that this equation would now give the expected marginal product of capital, rather than the certain marginal product of capital.

17 DELEVEAGING DYNAMICS 7 a decrease in rms' ability to borrow in the closed economy (Proposition 6) must entirely be driven by the endogenous decrease in the interest rate, as the next result makes clear. Proposition. Consider the eect of a decrease in the interest rate, holding θ constant. Comparing across steady-states, this will lead to: (a) an increase in the ratio of the marginal product of capital and the interest rate; (b) an increase in E d, borrowing per unit wealth; (c) a distribution of consumption paths that is riskier; the productivity cuto at which nancial constraints bind decreases. With a decrease in the interest rate, nancial constraints - as measured by conditional under-investment and the riskiness of consumption - tighten in steady state. This is because entrepreneurs can only nance part of their investment at the market interest rate; the rest must be nanced by retained earnings. A decrease in the interest rate means that retaining earnings are more expensive; in the absence of nancial constraints, an entrepreneur would have a steeper downward-sloping consumption path when the interest rate is lower. Hence, conditional under-investment increases. Consumption becomes riskier for two reasons. First, since the interest rate is lower, for any productivities for which the nancial constraint does not bind, the entrepreneur will choose an even lower value for w s than in the previous steady state. This serves to reduce w s for productivities below the new cuto s. Second, because the interest rate is lower, increasing scale is more attractive, even at the expensive of tighter nancial constraints. In particular, prots per unit of wealth, f k, increases, raising ws for productivities above the old cuto s Transition dynamics. The previous analysis raises questions about how the economy responds to a decrease in rms' ability to borrow not just in the long run, but at businesscycle horizons. Will it take a long time before the above-analyzed steady-state eects become present? In response to a permanent shock, will the short-run increase in distortions be larger or smaller than the long-run increase? How does the economy's response dier depending on the persistence of the nancial shock? When agents have log utility, it is possible to answer these questions analytically, and provide suggestive answers for other cases. Proposition 2. Suppose that at time t, the economy is in a steady state, with θ τ = θ for τ = {t, t,...}. Suppose that at date t, rms' ability to borrow falls unexpectedly and that the new path for rms' ability to borrow is given by ˆθ t. (a) There is a unique equilibrium path for the evolution of the economy. (b) At each point in time τ t, aggregate quantities and prices, the level of conditional under-investment, and risk-management policies w s,τ will correspond to the steady-state values for θ = ˆθ τ.

18 DELEVEAGING DYNAMICS 8 This result implies that, in response to an unexpected permanent decrease in rms' ability to borrow, the transition to the new steady state is instantaneous. At the time of the shock, the ratio of the marginal product of capital to the interest rate, or conditional under-investment, increases to its new steady-state level; the productivity cuto above which nancial constraints bind, s t, decreases immediately to its new steady-state value, and riskmanagement policy w s changes immediately to the new steady-state mapping; the interest rate immediately falls to its new steady-state level; and aggregate capital, consumption and output are unchanged. Similarly, if the decrease in rms' ability to borrow is temporary, with rms' ability to borrow falling sharply at time t and slowly recovering, conditional under-investment and consumption riskiness will increase on impact, and this amount of this increase will be the same as if the fall in rms' ability to borrow were permanent. Over time, as rms' ability to borrow recovers, conditional under-investment and consumption riskiness will decrease in line with the rebound in rms' borrowing ability. The instantaneous transition to the new steady state described in part (b) comes from the assumption of log utility. With log utility, there is no change in steady-state aggregate wealth, making the transition to a permanent shock immediate. If the assumption of unit relative risk aversion is weakened, there will be a change in steady-state wealth in response to a permanent shock, and the economy will transition to the new steady-state over time Maturity structure irrelevance. One advantage of this model of limited pledgeability is that restricting attention to one-period debt is without loss of generality. This true in the steady state and along the transition from any initial aggregate capital stock, and also for the response of an economy to an unexpected shock to rms' ability to borrow. Thus it is a single friction - entrepreneurs' ability to renege - that drives the results. In particular, consider an environment parameterized by n > in which entrepreneurs at history s t can trade state-contingent promises that pay out at history s t+τ, for τ {,.., n}. Denote by x(s t, s t+τ ) the amount of payment that the entrepreneur at history s t promises to pay at history s t+τ : The entrepreneurs budget constraint is (4.5) c(s t ) + k(s t ) f(k(s t ), s t ) x(s t, s t ) + n τ= τ v= t+v E t [x(s t, s t+i ) x(s t, s t+i )], where x(s t, s t+n ) = 0, since at t, the entrepreneur cannot trade promises with payos in period t + n. If the entrepreneur reneges at s t, this eliminates both his debt that is immediately due, x(s t, s t ), and his promises to repay in future periods, { x(s t, s t+i ) } n. As above, when i= the entrepreneur reneges, his creditors can seize only a fraction of his current period output, f(k, s). Hence, given the entrepreneurs budget constraint, the entrepreneur will not renege at s t if and only if

19 DELEVEAGING DYNAMICS 9 n (4.6) x(s t, s t [ ) + τ v= E t x(s t, s t+τ ) ] θf(k(s t ), s t ). t+v τ= The entrepreneur now chooses a collection { k(s t ), c(s t ), { x(s t, s t+i ) } n i=} to maximize utility subject to a collection of budget constraints (4.5) and nancial constraints (4.6) for each s t. Lemma 3. Consider a solution to the entrepreneurs' problem with n securities given by { k(s t ), c(s t ), { x(s t, s t+i ) } n i=}. There exists a policy of one-period positions d(s t ) promised at t to pay out at period t, n d(s t ) = x(s t, s t [ ) + τ v= E t x(s t, s t+τ ) ] t+v τ= that, together with the same policies for capital and consumption {k(s t ), c(s t )} satisfy the budget and nancial constraints of the problem with one-period promises and achieve the same utility. 5. Heterogeneity in Firms' Ability to Borrow The results in the previous section are suggestive of how misallocation might arise in this economy if there is heterogeneity in rms' ability to borrow, and how this misallocation might be amplied by an endogenous decrease in the interest rate generated by a fall in rms' ability to borrow. In this section, the model is extended to allow for heterogeneity in rms' ability to borrow. In particular, I assume that each rm belongs to one of two sectors that dier in their ability to pledge their future prots. Sectors are indexed by j. Firms in sector j can pledge θ j share of their future capital income. I assume that there are two sectors, with θ high > θ low. 6 I further assume that the labor market is segmented, with a competitive labor market for each sector. Assuming that labor is sector-specic is a convenient device to ensure that rms operate in each sector. If the labor-market were not segmented, only the θ high sector would operate; there would be no misallocation in equilibrium and θ low would not matter. Alternatively, in order for both sectors to operate in equilibrium, I could have assumed that sectors with a higher ability to borrow have suciently lower productivity, or that the sectors produce dierentiated goods that are suciently complementary in producing the nal good. These alternative assumptions would not alter the main qualitative results. I assume that workers inelastically supply L = units of labor to each sector. 6 The analysis could readily be extended to the case where each sector produces an intermediate good, and there is a competitive nal goods sector. However, this is omitted in this paper as the distribution of prices is not a focus here.