Debt Contracts with Short-Term Commitment

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1 Debt Contracts with Short-Term Commitment Natalia Kovrijnykh Department of Economics University of Chicago February 13, 2008 Abstract This paper analyzes the role of short-term commitment by the lender in a dynamic relationship where the borrower cannot be legally forced to make repayments. I show that short-term commitment can decrease social welfare compared to both the full and no-commitment cases considered by most of the literature. I show that the size of investment is positively related to the borrower s income. In addition, both underinvestment and overinvestment can occur in equilibrium. I also introduce the borrower s outside option and do comparative statics with respect to it. I show that the social welfare is non-monotonic in the borrower s outside option. If the borrower s outside option is interpreted as a measure of competitiveness of the credit market, this implies that an increase in the strength of competition has an ambiguous e ect on welfare. Furthermore, numerical results suggest that as the outside option of the borrower increases, the renegotiation-proof equilibria converge to the Markov equilibrium, where the agents strategies depend only on the borrower s liquidity. That is, the welfare gain from using complicated history-dependent strategies instead of simple Markov strategies is small when the borrower s outside option is high. 1 Introduction Consider a situation where an agent, the borrower, can operate a stochastic production technology, but does not have access to the necessary capital input. Another agent, the lender, has enough I would like to thank Fernando Alvarez, Gadi Barlevy, Marco Bassetto, Je rey Campbell, William Fuchs, Veronica Guerrieri, Anil Kashyap, Vijay Krishna, Robert Lucas, Alessandro Pavan, Robert Shimer, Hugo Sonnenschein, Nancy Stokey, Balázs Szentes, Harald Uhlig, and Marcelo Veracierto for helpful discussions and criticism. I am also grateful for comments by seminar participants at the University of Chicago, Federal Reserve Bank of Chicago, University of Minnesota Carlson School of Management, University of Houston, Texas A&M, Arizona State University, and University of Pennsylvania Wharton. 1

2 capital, but cannot operate the technology himself. The agents enter into repeated relationship, where they jointly generate and share surplus. In such a dynamic relationship, the lack of commitment can cause ine ciencies for the following reason. If the lender expects not to be repaid in the future, he might not invest into the borrower s technology. Similarly, the borrower might not repay the lender if she expects the lender not to invest in the future. If the parties cannot credibly commit to future decisions, such pessimistic expectations can be self-ful lling and lead to ine ciencies. The goal of this paper is to analyze how the amount of commitment power a ects investment and welfare. An example of a bilateral relationship described above is the interaction between a foreign investor (the lender) and the host country (the borrower). Indeed, the seminal paper by Thomas and Worrall (1994), among others, analyzes lending contracts in the context of foreign direct investment. In this case, commitment problems are particularly important because international contracts are hard to enforce in general. Another example is a close long-term relationship between a bank and a rm. 1 The lack of commitment on the side of the borrower can arise because, for example, the rm can use accounting tricks to divert cash ows. (See, e.g. Hart and Moore 1998, and Albuquerque and Hopenhayn 2004.) Most of the studies on dynamic lending with limited commitment on the side of the borrower make one of the two extreme assumptions about the lender s ability to commit: They either assume that the lender has full commitment power or cannot commit at all. Full commitment means that, at the beginning of the relationship, the lender is able to commit to investment decisions for all future periods and all possible histories. In the no-commitment case, all contracts must be selfenforcing. Interestingly, the common prediction of models with either of these assumptions is that, in optimal subgame perfect equilibria, the socially e cient outcome can be achieved over time if the agents are patient enough. However, the reasons for e ciency are quite di erent in the two cases. If the lender has full commitment power, e ciency can be achieved because he can commit to e cient investment decisions in the future. The lender is willing to do so for the following reason. The borrower, knowing that the lender will invest e ciently in the future, compensates the lender for past investments by large payments whenever she has enough liquidity. In contrast, if neither parties can commit, autarky becomes an equilibrium. In the optimal contract, any deviation from the equilibrium behavior is punished by entering into autarky forever. If agents are su ciently 1 The literature provides many explanations for the existence of such close relationships. For example, a bank might internalize informational problems and serve as a corporate monitor of its client rms see, e.g., Diamond (1984) and Hoshi, Kashyap, and Scharfstein (1991). 2

3 patient, this punishment is strong enough to induce the agents to make e cient decisions. Therefore, it is just natural to ask: What happens if the lender does not have full commitment power, but can at least partially commit to future decisions? And perhaps most importantly: How does the introduction of such an intermediate assumption a ect social welfare? Indeed, my main objective in this paper is to investigate predictions of a model with a partial, short-term commitment structure, where the lender is able to commit to one-period contracts. I model short-term commitment in the following way. Before the borrower decides how much to repay to the lender, the lender provides her with a menu specifying the amount of investment as a function of the borrower s repayment. If the borrower makes a certain amount of repayment, the lender invests the amount that corresponds to the borrower s payment in the menu. That is, the lender is able to credibly commit to such a menu. 2 This way of modeling short-term commitment explicitly allows new contracts being o ered in each period. Indeed, in lender-borrower relationships we often observe parties negotiating the terms of a contract over and over again. Hence, in a model of such a relationship, the explicit assumption of the possibility of signing new contracts seems realistic. I show that introducing short-term commitment creates a nontrivial incentive problem on the side of the lender that has not been previously analyzed in the literature. When the lender can fully commit, his future incentives are irrelevant. Without commitment, the payo to the lender who deviates from the optimal contract (i.e., violates its terms), is his autarkic payo. 3 Hence the lender has incentives to follow the equilibrium contract as long as his payo exceeds this value. Allowing the lender to commit to one-period contracts increases the lowest payo that the lender can receive in equilibrium (which can be used as a punishment for a deviation). The reason is that the lender can now credible promise investment to the borrower, and the borrower is willing to pay for it. This allows the lender to generate positive surplus. Therefore, since introducing short-term commitment increases the payo to the lender from deviating, it makes it harder to discipline the lender in equilibrium. The above discussion leads to a somewhat surprising prediction that, in environments where short-term contracts can be enforced, the social welfare might be lower than in environments where no contracts can be enforced. While the majority of papers on lending with limited commitment 2 This essentially means that the lender is able to commit to one-period contracts. It would not be hard to generalize this model so that the commitment is over a nite number of periods. 3 Even if one requires renegotiation-proofness, as I do in this paper, so that autarky is not considered a credible punishment, the lowest payo to the lender in renegotiation-proof equilibria is still his autarkic value. See Thomas and Worrall (1994). 3

4 focuses on the borrower s incentives, this paper shows that the possibility of writing short-term contracts makes it crucial to analyze the incentives of the lender. I characterize renegotiation-proof equilibria in this model. The main consequences of shortterm commitment are the following. First, the amount of investment is increasing in the borrower s liquidity. This result is consistent with empirical ndings in the corporate nance literature. 4 Second, both underinvestment and overinvestment can occur in equilibrium. On the one hand, if the borrower does not have enough liquidity to compensate the lender, the lender underinvests. On the other hand, when the borrower s income is high, the lender overinvests. The reason is that overinvestment increases the next period s expected income, and hence the next period s expected investment. Such a contract is attractive to the borrower, and she is willing to pay for it. That is, the lender uses overinvestment as a way of committing himself to high investments in the future. Thus, the short-term commitment causes a positive relationship between the current income and investment, which, in turn, induces overinvestment. Third, investment uctuates over time. The reason is that the borrower s production technology is stochastic, and therefore, her liquidity also varies stochastically across di erent periods. These ndings are in contrast to the predictions of most models with either the full- or no-commitment assumptions, where only underinvestment occurs until the time when the maximum level of investment (which is often the e cient amount) is reached. In a model where there is one lender and one borrower, an important question arises: Would competition among lenders improve e ciency? In order to address this question, I introduce the outside option of the borrower, which is the payo that she receives after her contractual relationship with the lender comes to an end. This outside option can be interpreted, for example, as the value from an alternative use of the production technology, and perhaps most importantly, the payo from searching for another lender. Hence one can consider the borrower s outside option as a measure of competitiveness of the credit market. One of the most interesting observations is that the social welfare is non-monotonic in the borrower s outside option: it decreases rst and then increases. The reason for this result is that the e ect of an increase in the borrower s outside option on the investment can be negative or positive depending on the level of the borrower s income. For high income levels, the borrower s 4 See Hubbard (1998) for a survey. Many of existing papers explain liquidity sensitivity of investment using models of asymmetric information. This paper provides an alternative explanation - the lack of commitment. Close bank- rm relationships are often suggested to mitigate the information problems and hence the associated liquidity problems, see, e.g., Hoshi, Kashyap, and Scharfstein (1991). However, Fohlin (1998) nds that relationship banking provides no consistent lessening of rms liquidity sensitivity. 4

5 participation constraint binds, and the borrower with a higher outside option would prefer to reject the lender s old contract. Therefore, the lender is forced to invest more at high income levels. But this observation has another consequence: it is now more costly for the lender to extract the borrower s income. Therefore, when the income is low so that the borrower s participation constraint does not bind and the lender only takes into account his own future gains, he has less incentives to invest. I show that the rst, positive e ect dominates when the borrower s outside option is su ciently high, and the second, negative e ect dominates when the outside option is low. An implication of this result is that the strength of competition in the credit market has an ambiguous e ect on social welfare. More precisely, if the competition is weak in the credit market, then a marginal increase in the strength of the competition can actually lead to a decrease in social welfare. However, if the credit market is competitive enough, making the competition even stronger increases e ciency. This nding is in line with empirical predictions that bank competition can have both positive and negative economic e ects. 5 The nonmonotonicity result of my model is in contrast to predictions of similar models with full or no commitment by the lender, which suggest that the social surplus is decreasing in the borrower s outside option. 6 Furthermore, I show with numerical computations that as the outside option of the borrower increases, the set of renegotiation-proof equilibria converges to the Markov equilibrium. (In a Markov equilibrium, the agents strategies are allowed to depend only on the current income but not on the history of the play.) That is, the welfare gain from using complicated history-dependent strategies instead of simple Markov strategies is small when the borrower s outside option is high. A possible interpretation of this result is the following. In this model, the renegotiation-proof equilibria have the property that the previous history can be summarized by the value to the lender, which can be interpreted as debt. The next period s debt can be viewed as being part of a contract, indicating to the parties which equilibrium will be played in the next period. Since contracts are complete, the next period s debt can be conditioned on the next period s output. In the case of a Markov equilibrium, the current income of the borrower fully determines which strategies are played in a given period. Therefore specifying the level of debt in a contract is irrelevant, as is the possibility of making the level of debt conditional on output. Hence, this paper provides a possible explanation for why we often observe debt contracts that do not specify 5 See, e.g., Petersen and Rajan (1994, 1995) and Cetorelli and Gambera (2001). Boot (2000) and Cetorelli (2001) provide reviews of the literature. 6 See, e.g., Albuquerque and Hopenhayn (2004) and Opp (2007). 5

6 state-contingent repayment schedules, even when this possibility appears feasible. Related Literature. The papers that are most related to mine are Thomas and Worrall (1994), Albuquerque and Hopenhayn (2004), and Kovrijnykh and Szentes (2007). Thomas and Worrall (1994) consider optimal contracts between one lender and one borrower in the context of foreign direct investment. They assume no commitment on either side. Albuquerque and Hopenhayn (2004) study a relationship between a bank and an entrepreneur, where the borrower cannot commit, but the lender has full commitment power. 7 The main prediction of these papers is that while there is underinvestment initially, borrowing constraints cease to bind eventually, and the rst-best investment is made from that point on. In Thomas and Worrall the economy converges to the steady state over time. In contrast, in my model there is no convergence in general, and both underinvestment and overinvestment can occur in equilibrium. In addition, Albuquerque and Hopenhayn predict that the value of the project is decreasing in the outside option of the entrepreneur, in contrast to the prediction of my model that this relationship is non-monotonic. Kovrijnykh and Szentes (2007) consider a framework similar to mine, with two lenders simultaneously making o ers to the borrower. Furthermore, contracts are incomplete: the next-period s debt cannot be conditioned on the next-period s income of the borrower. In contrast, in my model only one lender makes o ers to the borrower, and contracts are complete in the sense that they can be history-dependent. Doyle and Van Wijnbergen (1994) study a repeated bargaining model of foreign direct investment, where, similar to my model, negotiation over the output transfer occurs in every period. Bulow and Rogo (1989) present a constant recontracting model of sovereign debt with a nite horizon, where agents engage in Rubinstein bargaining and o ers are made in each period. Output is exogenous in both of these papers. The o ers only specify how the surplus is shared, and the models make no predictions about the investment process and associated ine ciencies. In contrast, in my model negotiations over both the investment and the output transfers take place. The following papers on foreign direct investment are also related. Schnitzer (1999) analyzes an in nite horizon model where all investment is made in period zero and is then sunk. She shows that the risk of expropriation may cause underinvestment if the outside option of the investor is too low, and overinvestment if it is too high. Janeba (2002) studies a model where a country cannot commit to a long-run tax policy, and a multinational rm cannot commit to invest in only one country. The author shows that the lack of commitment may induce the rm to invest in 7 The authors assume that there are many lenders initially, but they do not play a strategic role, as the borrower writes a long-term contract with only one of them. 6

7 a high-cost (but more credible) country. Alternatively, the rm might invest an optimal amount in a low-cost (but low-credibility) country, but hold excess capacity in the high-cost country for strategic purposes. In my model the lender invests only into one country, but similarly to Janeba I am interested in how incentives of the lender are a ected by the limited commitment on the side of the borrower. A recent paper by Opp (2007) considers a framework similar to Thomas and Worrall (1994), where the government has access to a relatively ine cient autarkic technology, and the government is more impatient relative to the foreign rm. The author nds that the steady-state payo to the government is non-monotonic in the autarkic productivity. The steadystate investment, and hence the total steady-state surplus in his model are non-increasing in the autarkic productivity as long as the equilibrium is not the autarky. The autarkic productivity can be interpreted as the outside option of the borrower. That is, Opp s result is in contrast to the prediction of my model where the total surplus is non-monotone in the borrower s outside option even while autarky is not an equilibrium. This work is also related to the following papers on corporate lending. Sigouin (2003) considers a relationship between a risk-neutral creditor and a risk-averse entrepreneur, where assets and capital serve as physical collateral. The author shows that both underinvestment and overinvestment can occur in equilibrium, in line with my predictions. However, in his model overinvestment occurs because it facilitates risk sharing and because capital plays a role of collateral. In contrast, in my model agents are risk neutral and debt is unsecured. The reason why overinvestment occurs is because it increases the next period s liquidity and thus increases the lender incentives to invest in the next period. Clementi and Hopenhayn (2006) analyze a borrower-lender relationship with asymmetric information where the lender does not observe either the use of funds or the output. The optimal contracts have a feature that the economy eventually converges to one of two absorbing states: A sequence of bad shocks results in the liquidation, while a sequence of good shocks leads to the rst best. 8 Atkeson (1991) studies optimal sovereign lending contracts with moral hazard between a borrower and overlapping generations of short-lived lenders. In Atkeson s model the lenders do not observe whether the borrower invests or consumes borrowed funds. To provide incentives to the borrower, the optimal contract must specify a fall in consumption and investment for the lowest realizations of output. My model produces a similar result investment is increasing in output, and investment is ine ciently small if the output is small. 8 Also see Quadrini (2004) who shows that the liquidation of the rm can arise as an outcome of a renegotiationproof contract. 7

8 2 Model Production and Preferences There are two risk neutral agents, a lender and a borrower. The time horizon is in nite, time is discrete, and agents discount the future according to the discount factor 2 (0; 1). The borrower can operate a stochastic technology that transforms capital goods into consumption goods. If the amount of capital investment is K, then the output in the next period, in terms of consumption goods, is F (K) = sf (K) ; where s is the realization of a random shock. The function f is strictly increasing, strictly concave, and satis es the Inada conditions. The shock is distributed according to the continuous cumulative distribution function G, which has a strictly positive density g on [0; 1]. The goods are perishable and completely depreciate every period. The lender has enough capital to invest in production in every period. In addition, the lender can instantaneously transform one unit of capital good into one unit of consumption good and vice versa. This means the lender is indi erent between the two goods. 9 maximize the discounted present value of expected consumption. Timing and Contracts Each agent s goal is to The lender can commit to one-period contracts only. A typical contract is a pair (C; K) 2 R 2 +, where C is the consumption of the borrower, and K is the investment in the production technology. By o ering such a contract, the lender commits to invest K if the borrower makes a repayment of I C to the lender. More generally, a contract can be a probability mixture of these pairs. 10 The timing is as follows. At the beginning of a period, the output realization is I, which is simultaneously observed by the agents. The lender o ers a set of contracts to the borrower. The borrower either accepts a contract from this set or rejects all contracts. accepts a contract (C; K), she gives I invests K units of capital in the production technology. 11 If the borrower C units of consumption good to the lender, and the lender The borrower consumes C, the lender consumes I C K, and the period ends. If the borrower rejects all contracts, the relationship is automatically terminated and the game ends. 12 The payo s upon termination of the relationship 9 Having two di erent goods ensures the borrower cannot invest into the technology herself. This assumption is also used by Thomas and Worrall (1994) and Kovrijnykh and Szentes (2007). 10 Since the borrower is risk neutral, C can always be assumed to be deterministic even though contracts can be random. 11 If the borrower accepts a random contract, then the outcome of the lottery is rst observed. 12 The assumption that the game ends if the borrower rejects all o ers means that the lender can commit to terminate the relationship. I also consider an extension of this model, which can be found at where I relax this assumption. Instead I assume that after re- 8

9 are zero to the lender and I + to the borrower, where 0 is the value of the borrower s outside option. To see how this game with short-term commitment is di erent from a game where the lender has no commitment power, consider the following setup, which is similar to the one analyzed in Thomas and Worrall (1994). At the beginning of each period, rst the lender chooses the level of investment. Production takes place and the income is realized and observed by both agents. Then the borrower decides how much to repay to the lender from the current output. The agents consume and the period ends. In this setup, in each period each agent chooses his action to maximize the present value of his utility at that point of time. In particular, the investment choice maximizes the lender s payo after the borrower has made the repayment. This is not the case with the short-term commitment structure that I consider. After the borrower makes the repayment I C, it might not be in the lender s interest ex-post to invest the amount K speci ed in the contract. The lender does invest K because he committed to it previously. In other words, without commitment the lender chooses a strategy according to his reaction function, while with the short-term commitment the lender commits to a reaction function. Also notice that, in the model without commitment, nothing links the current output to future investment except for the equilibrium behavior of the agents. This is not the case with the shortterm commitment. The reason is that the amount of the borrower s liquidity limits the repayment that the lender can extract from her, and therefore it a ects the level of o ered investment. I will show that in equilibrium investment is increasing in the liquidity of the borrower. The First-Best Investment. Since both agents are risk neutral, the rst-best investment maximizes K + R 1 0 sf (K) dg (s). The solution, KF B, is de ned by the following rst-order condition: 1 = f 0 K F B R 1 0 sdg (s). Let SF B denote the rst-best social surplus if the borrower s income is zero. That is, S F B = [ K F B + f K F B R 1 0 sdg (s)]= (1 ). When the borrower s income is I, the rst-best surplus equals S F B + I. Equilibrium Concept The goal of this paper is to analyze renegotiation-proof equilibria in the game described above. The concept of renegotiation-proofness requires that the Pareto frontier corresponding to optimal subgame perfect equilibria has the property that the o -equilibrium continuation payo s themselves must lie of the constrained Pareto frontier. The idea behind this equilibrium concept is that jecting the lender s o er, the borrower can choose between taking the outside option and staying in the relationship with the current lender. This model is harder to analyze analytically, but the main qualitative results remain the same. For su ciently high values of the borrower s outside option the solutions to the two problems coincide. 9

10 even if a player deviates from the equilibrium behavior, the agents can renegotiate their implicit contract and can reach a constrained e cient agreement. Renegotiation-proofness is usually de- ned for repeated games. The game that I consider is not repeated for the following reason. The income I a ects the game played in a given period, in particular, it a ects the value to the borrower from taking her outside option. In addition, the distribution of income in each period is determined by the investment made in the previous period. However, this game still has a repeated-game-like structure, with the income parametrizing the changing physical environment. This simple structure allows me to generalize the de nition of renegotiation-proofness given for repeated games to the considered environment. This extension of the de nition is one of the contributions of this paper. 13 Let (I) be a set of subgame perfect equilibria for income I, and = f (I) ji 0g be a set of subgame perfect equilibria for all income levels. Denote P (I) = fp (I) j p (I) is the payo pro le corresponding to (I) 2 (I)g the corresponding set of payo s for income I. The collection of these sets for all income levels is denoted by P = fp (I) ji 0g. De nition 1 P is weakly renegotiation-proof if (a) 8I, 8p 2 P 0 2 P (I) such that p 0 Pareto dominates p (that is, p 0 p), 14 and (b) 8I, 8p 2 P (I), p is generated using continuation payo s from P. De nition 2 P is renegotiation-proof if P is weakly renegotiation-proof, and there is no other weakly renegotiation-proof set of payo s P 0, P 0 6= P, that Pareto dominates P, i.e., there is no P 0 such that 8I, 8p 2 P (I) 9p 0 2 P 0 (I) such that p 0 p. As we will see later, a weakly renegotiation-proof set of payo s will be a xed point of an operator, corresponding to a recursive problem. Before setting up this recursive problem, I de ne Markov equilibrium. I am interested in Markov equilibria for several reasons. First, as I will show below, any Markov equilibrium is weakly renegotiation-proof. This allows me to prove existence of a renegotiation-proof equilibrium. Second, I could prove certain results analytically for Markov equilibria and only show numerically for renegotiation-proof equilibria. And nally, I 13 I extend the de nition of weak renegotiation-proofness (internal consistency) given in Bernheim and Ray (1989), and Farrell and Maskin (1989). Also see Ray (1994). The additional requirement for renegotiation-proofness extends the corresponding condition given in Van Damme (1991). 14 Some de nitions use the notion of weak Pareto dominance here, i.e., they impose p 0 p (p 0 i > p i, i = 1; 2) instead of p 0 p. It can be shown that in this model the two requirements lead to the same results. That is, I am not imposing a stronger requirement by using Pareto dominance instead of weak Pareto dominance. 10

11 show with numerical computations that as the outside option of the borrower increases, the set of renegotiation-proof equilibrium payo s converges to the set of Markov equilibrium payo s. De nition 3 A subgame perfect equilibrium is Markov if in each period the strategies of the players depend only on the current income realization, but not on the previous history of the play. 3 Bellman Treatment In this section, I formulate recursive problems corresponding to Markov and renegotiation-proof equilibria. Markov Equilibria Fix a Markov equilibrium, and let L M (I) and B M (I) denote the values to the lender and to the borrower at income I, respectively. The lender solves the following maximization problem: ^L M (I) = max C;K I C EK + ELM (sf (K)) (1a) s.t. C + EB M (sf (K)) I +, (1b) and n L M (I) = max I 0 ;I 00 ;2[0;1] max 0; ^L o n M (I 0 ) + (1 ) max 0; ^L o M (I 00 ) s.t. I 0 + (1 ) I 00 = I. (2) The choice variables are consumption of the borrower and investment. 15 The maximization is subject to the participation constraint of the borrower. It requires that the equilibrium payo of the borrower at income I is at least as high as her outside option, I +. Problem (2) incorporates the possibility that if the value ^L M (I) is less than zero, then the lender prefers to terminate the relationship (by o ering a contract that the borrower would reject) and receive a payo of zero. n Hence at income I, the payo to the lender is at least max 0; ^L o M (I). The lender can achieve a possibly higher payo by using lotteries. The payo frontier for a given income level I is a singleton: P (I) = (B M (I) ; L M (I)). Hence by the de nition of weak renegotiation-proofness, we have the following result: Claim 1 Any Markov equilibrium is weakly renegotiation-proof. Although a Markov equilibrium is weakly renegotiation-proof, it is not renegotiation-proof in general. Furthermore, in a Markov equilibrium the lender is never constrained to deliver more to 15 I allow the use of lotteries, hence the expectations in the continuation values are taken both with respect to the randomness in the contract and the shock. 11

12 the borrower than her outside option. That is, the lender always has the full bargaining power. This is not the case in the problem that I will consider below. Weakly Renegotiation-Proof Equilibria The problem of nding a set of weakly renegotiation-proof payo s can be written recursively, with the income and the value to one of the agents as state variables, and the continuation values as control variables. Let L (I; B) denote the value to the lender if the income is I and the value to the borrower is at least B. Essentially, the arti cial state variable B summarizes the previous history of the play. The maximization problem of the lender is as follows: L (I; B) = max C;K;B (:) I C EK + EL (sf (K) ; B (sf (K))) s.t. C + EB (sf (K)) B, B (I 0 ) B min (I 0 ) for all I 0, L (I 0 ; B (I 0 )) d (I 0 ) for all I 0, (3) where B min (I) = sup fb j B 2 arg max BI+ L (I; B)g corresponds to the lowest payo to the borrower on the frontier for income I. The value d (I) is the lender s value from the most pro table deviation when the borrower s income is I. In problem (3), B I + and L (I; B) d (I), for otherwise one of the agents would deviate. The value to the deviating lender, d (I), is de ned by the following two problems: 2 ^d (I) = min B 4 max C;K I C EK + EL sf (K) ; B d (sf (K)) 3 5 d (:) s.t. C + EB d (sf (K)) I + s.t. B d (I 0 ) B min (I 0 ) for all I 0, L I 0 ; B d (I 0 ) d (I 0 ) for all I 0, (4) and n d (I) = max I 0 ;I 00 ;2[0;1] max 0; ^d o n (I 0 ) + (1 ) max 0; ^d o (I 00 ) s.t. I 0 + (1 ) I 00 = I. Below, I describe the above problems in details, and later establish any weakly renegotiation-proof set of payo s corresponds to a solution to these problems. Consider problem (3) rst. The choice variable of the lender is an o er that he makes, (C; K). The continuation values B (I 0 ) are not part of the lender s strategy. They specify which equilibrium is played in the next period if the income realization is I 0, and the lender has not deviated. Given B, the maximization with respect to B (:) ensures that problem (3) generates the constrained Pareto e cient outcome. The rst constraint in problem (3) is the promise-keeping constraint, which guarantees that the value that the borrower receives is indeed at least B. The second and the third constraints 12 (5)

13 determine which continuation payo s can be feasible in equilibrium. 16 The second constraint is the next period s participation constraint of the borrower. It says that the borrower prefers to play her equilibrium strategy and receive the payo of B (I 0 ) to taking her outside option. The value B min (I 0 ) appears on the right-hand side of the constraint instead of I 0 + for the following reason. For some values of income I, if the borrower is only promised her outside option, B = I +, the promise-keeping constraint might not bind, and hence the actual lowest payo that the borrower receives in equilibrium for income I exceeds her outside option: B min (I) > I +. Incorporating B min (I 0 ) into the second constraint guarantees that B (I 0 ) indeed corresponds to the actual payo that the borrower receives at income I 0. The third constraint in problem (3) is the next period s participation constraint of the lender. It ensures that the lender prefers playing his equilibrium strategy to deviating. I now turn to the characterization of the function d, the value to the lender from the most pro table deviation. Suppose the income of the borrower is I. Instead of making an equilibrium o er and delivering a payo of at least B to the borrower, as in problem (3), the lender can make a di erent o er. The borrower will accept it as long as it delivers her a payo of at least I +. The agents expect that if the lender deviates, the continuation payo of the borrower at income I 0 will be B d (I 0 ). The lender chooses (C; K) to maximize his payo subject to the participation constraint of the borrower, taking B d (:) as given. Again, the second and third constraints give restrictions on which B d (:) can be supported as the equilibrium continuation payo s, so that neither of the agents deviates. To satisfy renegotiaion-proofness, the continuation payo of the lender for income I 0 is again given by the value function L, that is, the continuation payo s lie on the same frontier where the equilibrium payo s lie even after the lender has deviated. The value to the lender is minimized with respect to B d (:) because the lower the value to the lender from the deviation, the higher welfare it is possible to achieve. That is, the lender who contemplates a deviation expects that, as a punishment, the continuation equilibria will be chosen in the least favorable way to him. Problem (5) is an analog of problem (2). Claim 2 Any weakly renegotiation-proof equilibrium corresponds to a solution to the recursive problem described by (3) (5). Notice that for a xed B, the value to the lender given by problem (3) depends on I in an additively separable way, and the choice of the lender in this problem only depends on B. Hence 16 If K is deterministic, the second and the third constraints can be written as B (sf (K)) B min (sf (K)) and L (sf (K) ; B (sf (K))) d (sf (K)), for all s 2 [0; 1]. The form in which the constraints are written in problem (3) means that they have to be satis ed for each pairs of realizations of the random variables K and s. 13

14 I can rewrite problem (3) using B as the only state variable. Using this observation, de ne N (B) = L (I; B) I to be the value to the lender minus income, when the payo to the borrower is at least B. I want to rewrite problems (3) and (4) in terms the function N instead of L. The lowest payo to the borrower for income I, B min (I), can be written as max fi + ; B 0 g, where B 0 = sup fb j B 2 arg max B N (B)g. If the promise-keeping constraint in problem (3) does not bind at, then B 0 would be the largest B at which this constraint does not bind. Then problems (3) and (4) can be written as N (B) = max C;K;B (:) C EK + E [sf (K) + N (B (sf (K)))] s.t. C + EB (sf (K)) B, B (I 0 ) max fi 0 + ; B 0 g ; for all I 0, (6) I 0 + N (B (I 0 )) d (I 0 ) for all I 0, and ^d (I) = min B d (:) 2 4 max C;K I C EK + E sf (K) + N B d (sf (K)) s.t. C + EB d (sf (K)) I +, s.t. B d (I 0 ) max fi 0 + ; B 0 g for all I 0, 3 5 (7) I 0 + N B d (I 0 ) d (I 0 ) for all I 0. Problems (5), (6) and (7) de ne the operator T in the following way. For a pair of functions (N; d), the value generated by (6) is T 1 (N; d), and the value generated by (5) and (7) is T 2 (N; d). Then T (N; d) = (T 1 (N; d) ; T 2 (N; d)). Any xed point of the operator T corresponds to a set of weakly renegotiation-proof payo frontiers. However, not any xed point corresponds to a set of payo s which is renegotiation-proof. The reason is that renegotiation-proofness also requires that there is no other xed point that generates payo frontiers that are Pareto dominating. I showed in Claim 1 that any Markov equilibrium is weakly renegotiation-proof. It is easy to verify that a Markov equilibrium corresponds to a xed point of the operator T. De ne N M (I + ) = L M (I) I, the value to the lender minus income in a Markov equilibrium. Then N M ; L M is a xed point of T such that B min (I) = B max (I) = B M (I) and I+N (I + ) = d (I) = L M (I) for all I. Then B d (I 0 ) = B (I 0 ) = B M (I 0 ) for all I 0 and I + T 1 (N; d) (I + ) = T 2 (N; d) (I) = L M (I) for all I. Although a Markov equilibrium is weakly renegotiation-proof, it is not renegotiation-proof in general. A xed point corresponding to a renegotiation-proof frontier satis es N (B) N M (B) for all B. In this model there is a payo frontier for each income level. I rewrote the problem in such a way that this family of frontiers is summarized by two functions, N and d. Let (N; d) be a 14

15 xed point of T corresponding to a set of renegotiation-proof payo frontiers P. I will refer to the function N as the aggregate renegotiation-proof frontier, because it comprises in itself all the frontiers for di erent income levels. For convenience, de ne N d (I + ) = d (I) I. Figure 1 shows graphically how the set of payo frontiers P can be obtained from the functions N and N d. The functions N and N d are depicted by black solid and dashed curves, respectively. Recall that the frontier for income I is P (I) = f(b; I + N (B)) jb 2 [B min (I) ; B max (I)]g. The lowest equilibrium payo to the borrower is B 0. Consider, e.g., an income level I such that I + > B 0. Then B min (I) = I + is the lowest payo to the borrower on the renegotiation-proof frontier for income I. The graph shows how to nd the highest payo to the borrower for income I, B max (I). First draw the vertical line through B min (I). Its intersection with N d is marked with a white dot. Draw a horizontal line through this intersection until it crosses the curve N to obtain B max (I): N d (I + ) = N (B max (I)). The two black dots on the curve N correspond to the payo pairs (B min (I) ; N (B min (I))) and (B max (I) ; N (B max (I))). In order to obtain P (I), the portion of the N-curve in between the two black dots should be shifted vertically by I. The grey curves show frontiers for di erent income levels. Since L (I; B) = I + N (B), these frontiers have the same slope along any vertical line. The set of such frontiers for all income levels gives us the set P. N Renegotiationproof frontier for income I, P(I) P N(B) I N d (B) First-best frontier, S FB B 0 B 0 B min (I) B max (I) B Figure 1. Obtaining P from N and N d. In most papers, the lowest equilibrium payo to the lender is his outside option. It is either exogenous or is only in uenced by physical variables that are determined in equilibrium, for example, the amount of capital, as in Sigouin (2003). In this paper the lender s lowest equilibrium payo is not his outside option, which is zero, but his value from the best deviation while keeping 15

16 the relationship in place. This value is endogenously determined and is in uenced by, and in turn, in uences the whole equilibrium structure. 4 Analysis and Results I start by describing properties of Markov equilibria in the next subsection. Then I will analyze properties of weakly renegotiation-proof equilibria. 4.1 Properties of Markov Equilibria Consider problem (1) introduced in Section 3. Since random o ers are allowed, the lender s value function is concave, and any concave function is di erentiable everywhere but on at most a countable set. Using concavity and the fact that she shock is continuous, I will use the rst-order approach to solve the problem. For simplicity of the exposition, I will assume in most of the analysis that is low enough so that L M (I) > 0 for all I, and hence L M (I) = ^L M (I) for all I. Let M and M be the Lagrange multipliers on the constraints (1b) and C 0, respectively. The rst-order condition for C is 1 + M + M = 0. The Envelope condition for I is L M0 (I) = 1 M = M 2 [0; 1]. That is, L M0 (I) > 0 as long as C 0 binds. Hence we have Lemma 1 The function L M is concave and L M0 2 [0; 1]. Lemma 2 describes the borrower s value function. It says that either the borrower s participation constraint binds and she receives her outside option, or she receives the same value as when her income is zero. Lemma 2 B M (I) = max I + ; B M (0). Investment. The rst-order condition with respect to K is 1 = f 0 (K) Es L M0 (sf (K)) + M B M0 (sf (K)). (8) Let K M (I) denote the lender s choice of investment in problem (1) if only deterministic o ers were allowed. Furthermore, de ne I = inf I j L M0 (I) = 0 = sup I j C M (I) = 0, the highest income level for which the borrower s consumption, C M, is zero. The following proposition characterizes investment in a Markov equilibrium. 16

17 8 >< Proposition 1 (i) K M (I) >: = K M (0) for I B M (0) strictly increasing for I 2 B M (0) ; I = K M I for I I. (ii) No random o ers are used, that is, investment K M (I) is made at I with probability one. (iii) K M (0) < K F B. (iv) K M I > K F B. The above proposition says that if the participation constraint of the borrower does not bind, then the lowest level of investment is made, which is strictly below the rst-best level, K F B. On the other hand, if the borrower consumes, then the highest level of investment is made, which exceeds K F B. Otherwise, the investment strictly increases in the borrower s income. What causes ine ciencies in this environment is that the borrower controls the output and can choose not to repay to the lender. If the income is low today and the lender does not have to deliver more than the outside option to the borrower, then he underinvests. This is because the borrower cannot commit to repay to the lender tomorrow for his current investment. Without an up-front compensation, the lender is only willing to invest an amount that makes the surplus from the future relationship worth it. A higher investment can only be induced by a higher up-front payment from the current output. On the other hand, when the borrower s income is su ciently high, investment exceeds the rst-best level. This increases the expected output, and hence the expected investment in the next period. That is, the lender uses overinvestment as a way of committing himself to high investments in the next period. Such a contract is attractive to the borrower, and she is willing to pay for it. 17 Existence. First I show that a Markov equilibrium exists, and that it is not autarky. Notice that the function B M (I) appears in the constraint of problem (1), and this function is a ected by the function L M (I) (see the proof of Lemma 2). As a result, the monotonicity property of Blackwell s su cient conditions fails to hold. Therefore, the usual dynamic programming techniques cannot be used to show existence and uniqueness of the xed point. I de ne an operator, T M, that maps L M into T M L M, and apply Schauder s xed-point theorem to establish the existence of a Markov equilibrium. By Claim 1, any Markov equilibrium is weakly renegotiation-proof. Hence part (i) of the following proposition shows that a weakly renegotiation-proof equilibrium exists. Then it is straightforward to show that a renegotiation-proof equilibrium also exists, see part (iii). 17 If the borrower had access to a storage technology with the rate of return equal, then K M I > K F B would still hold as long as <, where = 1= 1 is the rate of time preference. If = then the rst-best level of investment is indeterminate, and if > then K F B = 1. 17

18 Proposition 2 (i) A Markov equilibrium exists. (ii) If < S F B =, then autarky is not a Markov equilibrium. (iii) A renegotiation-proof equilibrium exists. Complete Information Case. Suppose there is no uncertainty. Assume, for example, that s t = R 1 0 sdg (s) a is constant in all periods. It is straightforward to show that parts (i) (iii) of Proposition 1 still apply. Let fk t g t1 be a sequence of investment levels over time along a Markov equilibrium path if there is no uncertainty. That is, K t+1 = K M (af (K t )) for t 1, and K 1 = K M (I 1 ), where I 1 is some initial level of income of the borrower. Since K M (0) < K F B, if I 1 is low enough, then there is underinvestment initially. Claim 3 If there is no uncertainty, then K M I = maxi K M (I) = K F B and 9 such that for t, K t = K F B. Notice the di erences and similarities between the investment decisions in the Markov equilibria with and without uncertainty. In both cases the commitment problem on the side of the borrower causes underinvestment. If the output is deterministic, then over time the borrower accumulates enough liquidity to buy the rst-best level of investment in each period. The borrower s future participation constraints seize to bind and the rst-best surplus is achieved from that point on. On the contrary, if the output is uncertain, the liquidity problem is never eliminated. 4.2 Properties of Weakly Renegotiation-Proof Equilibria In this subsection I will characterize some properties of weakly renegotiation-proof equilibria. Consider problem (6). Let and be the Lagrange multipliers on the promise-keeping constraint and C 0, respectively. The rst-order condition with respect to C is 1++ = 0. The Envelope condition with respect to B is N 0 (B) = = 1. Since 0, we have N 0 (B) 2 [0; 1]. Furthermore, N 0 (B) < 1 whenever C 0 binds. De ne B = inffb j N 0 (B) = 1g = supfb j C (B) = 0g, the highest value to the borrower such that the borrower s consumption is zero. The aggregate payo frontier, N, is downward sloping because an increase in B is costly to the lender. The above results are summarized in Lemma 3. Concavity follows because random o ers are allowed. Lemma 3 The function N is concave and N 0 2 [0; 1]. Continuation Values and Bargaining Power. The following two claims describe optimal choices of the continuation values B and B d. 18

19 Claim 4 If the promised value to the borrower is B, then the following choice of continuation values B is optimal: 8 B I 0 ; B >< = >: B, if B min (I 0 ) B B max (I 0 ), B max (I 0 ), if B > B max (I 0 ), B min (I 0 ), if B < B min (I 0 ). Since N is concave, it is optimal to set B (I 0 ) equal to B whenever possible. For I 0 at which the borrower would deviate at B, B (I 0 ) is set to B min (I 0 ). Similarly, for I 0 at which the lender would deviate at B, B (I 0 ) is set to B max (I 0 ). The value B min (I 0 ) corresponds to the point on the frontier for income I 0 where the lender extracts all the surplus, or, in other words, has the full bargaining power. Similarly, B max (I 0 ) corresponds to the point on the frontier where the borrower extracts all the surplus. Hence Claim 4 shows how the bargaining power of the agents changes endogenously over time depending on the state of the economy. Lemma 4 The function d is concave and d 0 (I) 2 [0; 1]. Claim 5 (i) If I and are low enough (so that d 0 (I) = 1), then B d (I 0 ) = B max (I 0 ) for all I 0. (ii) If I is high enough (so that d 0 (I) = 0), then B d (I 0 ) = B min (I 0 ) for all I 0. The result of Claim 5 is based on the fact that N (B) is decreasing in B and B+N (B) is increasing in B. The value to the lender can be written as max K K + E[sf (K) + N(B d (sf (K)))] + minfi; EB d (sf (K)) g, where the last term is the repayment R = I C. For I low enough the value to the lender equals max K I K + E[sf (K) + N(B d (sf (K)))], which is minimized if B d (sf (K)) is the highest possible for each s, so that N(B d (sf (K))) is the lowest possible for each s. On the other hand, if income is very high, the value to the lender is max K K + E[sf (K) + N(B d (sf (K))) + B d (sf (K))], which is minimized if B d (sf (K)) is the lowest possible for each s, so that N(B d (sf (K))) + B d (sf (K)) is the lowest possible for each s. It is worth comparing Claims 4 and 5. For example, for a high enough current income I, B (I 0 ; I +) = B max (I 0 ) for all I 0, while B d (I 0 ; I) = B min (I 0 ). This means that when the income is high, the lender making the equilibrium o er expects that in the next period the borrower will have the full bargaining power. On the other hand, if the lender deviates, he has the full bargaining power in the next period. Investment. Let g (s) (sf (K)) and g (s) (sf (K)) be the Lagrange multipliers on the participation constraints of the borrower, B (sf (K)) maxfsf (K) + ; B 0 g, and the lender, 19