Chapter 5 Dynamic Contracting 5.1 Incomplete contracts In our earlier treatment of contracting problems, we assumed that the in- centive problem was generated by asymmetric information, either a problem of moral hazard (hidden actions) or adverse selection (hidden information). The incomplete contracts approach eschews asymmetric information because of its intractability and instead focuses on environments in which informa- tion is observable but not verifiable. “Observable” means the information is common knowledge among the contracting parties. “Non-verifiable” means that the information cannot be confirmed by an outside agency such as a court and hence cannot be made an explicit part of any written contract. A crucial aspect of the incomplete markets approach is that it allows for renegotiation, that is, agents can replace the pre-existing contract with a newcontractatanypointifitismutuallybeneficial. Renegotiation under incomplete information is analytically intractable, so this is another reason for avoiding informational asymmetries. 5.1.1 The holdup problem This example comes from Hart (1995). There are two firms, M1 and M2. Firm M2 produces a widget that is needed by firm M1 at a cost of C ? .There is no alternative supplier or purchaser. The gross return from the widget is R(i) if M1 makes a prior investment 1 2 CHAPTER 5. DYNAMIC CONTRACTING of i ≥ 0. Assume that R(0) >C ? ,R 0 (0) > 2, lim i→∞ R 0 (i) < 1,R 0 (i) > 0,R 00 (i) < 0,?i. Both parties are risk neutral and the interest rate is zero. Ignore issues of ownership. The first best maximizes total surplus: (i ? ,x ? ) ∈ argmax{(R(i)?i?C ? )x}, where x =0,1. If surplus is divided using symmetric Nash bargaining solution, invest- ment is sub-optimal (Grout, 1984). (??, ?x) ∈ argmax ? 1 2 (R(i)?i?C ? )x ? . Contractual solutions: ? If the type of widget produced can be specified in the contract, the first best can be achieved. ? If the level of investment can be specified in the contract, the first best can be achieved. Thereisalarge,finite number of states s =1,...,S.Awidgetoftypes is needed in state s,thatis,awidgetoftypes produces a return of R(i) in state s and nothing in other states. The cost of production is C ? for every type and state. The cost of writing a complete contract with a large number of states would be very high. The first best might be achieved as follows: M1 specifies thetypeofwidgetshewantsatdate1;ifM2 supplies that type, she receives p 1 ; if she fails to deliver, she receives p 0 .(Notethetypeofwidgetmustbe verifiable). To implement the first best, put p 0 ≥ 0 and p 1 ≥ p 0 + C ? .Since M1 gets the full marginal returns, investment will be optimal. Renegotiation causes problems with this setup. For example, suppose that M2 has the opportunity to make a take-it-or-leave-it counter o?er. Then M2 gets the surplus and again we have the holdup problem. In any case, when there is renegotiation, the outcome is independent of the contract. 5.1. INCOMPLETE CONTRACTS 3 5.1.2 Hart and Moore (1994) Another example of incomplete contracting is the Hart and Moore (1994) theory of debt. The returns to the asset are observable but not verifiable. Managers can run away with the cash but not the assets themselves. The only option open to the bondholders is to rest control of the asset from the manager. Suppose there are three dates t =1,2,3 and at the initial date the in- vestors purchase an asset (a machine) which they put in the control of a manager. The machine produces a random return y t at dates t =2,3 and canbeliquidatedforafixed amount L in period t =2(it has no scrap value at the last date). All uncertainty is resolved at date t =2in the sense that (y 2 ,y 3 ) become known for sure. At the first date, the investors and the manager write a contract which specifies, among other things, the payments to be made by the manager in each period. The contract cannot be made conditional on the earnings of the firm, as these are unverifiable; in fact, the only thing that can be verified is whether the manager has made a specified payment or not. If he has not, then the investors have the right to seize the asset and prevent the manager from using it. Let D be the minimum payment required at the second date. If the manager makes a payment of at least D at t =2, there is nothing the investors can do to stop him using the rest of the firm’s income for his own purposes after this. At the last date the manager will simply consume y 3 . There is nothing the investors can do to stop him. At the second to last date, the manager may lose control of the firm if he does not pay D to the investors, but he may get away with less. Clearly, he will never pay them more than D. If he only pays them L it will not be worthwhile seizing the asset. The exact division of the surplus is determined by a bargaining problem, but assuming the manager gets to make a take-it-or-leave-it o?er, he will clearly pay the minimum of D and L if he can. If y 2 < min{D,L} then the investors will seize the asset, even though this is ine?cient if L<y 2 + y 3 . If y 2 ≥ min{D,L} the manager will retain control, and his rents equal y 2 + y 3 ?min{D,L}. This story o?ers several lessons. First, the investors’ power is restricted to the threat of taking away control of the asset. Secondly, this threat is credible 4 CHAPTER 5. DYNAMIC CONTRACTING only if the manager cannot pay the investors as much as they would get by taking away his control of the asset. (Di?erent bargaining stories would lead to somewhat di?erent conclusions here). Thirdly, because the manager cannot commit to pay the investors in the future, the transfer of control may be ine?cient. 5.1.3 Aghion and Bolton (1992) Another example of incomplete contracts is provided by Aghion and Bolton (1992). Following Jensen (1986), Aghion and Bolton conclude that managers may use free cash flow to overinvest in order to capture private benefits. Debt financing is a method that can be used to restrain managers (the need to pay interest restricts cash flow) and to transfer control of the firm in certain states of nature. The essential assumption is that in states where cash flow is low, so that the manager cannot service the debt, it is optimal to restrain the manager (transfer control), whereas in states where cash flow is high and the manager can service the debt, it is optimal to leave control in the hands of the manager. 5.2 Renegotiation Renegotiation is typically regarded as a limitation on the ability of parties to write an e?cient contract. 5.2.1 Stiglitz and Weiss (1983) Credit markets are characterized by incomplete information, which gives rise to problems of adverse selection and moral hazard. Stiglitz and Weiss (1983) have argued that these problems are mitigated if lenders can threaten bor- rowers with punishment in the event of default or poor performance. For example, a firm that defaults on a bank loan may be refused credit in the fu- ture. We noted above that the threat of termination may improve incentives for making an e?ort. In analyzing the optimal use of threats, it is assumed that the lender can commit itself to a particular course of action in advance. Without commit- ment, past default should be regarded as a sunk cost. 5.2. RENEGOTIATION 5 Renegotiation thus creates a time-consistency problem. This is typical of contracting problems. However, when contracts are incomplete, renegotia- tion may fill a more beneficial role. 5.2.2 Aghion, Dewatripont and Rey (1994) The inability to specify all contingencies and what happens in those contin- gencies does not necessarily prevent achievement of the first best. Even if the state is not verifiable,messagesfromagentsmaybe.Ifagentsknowthe state and the contract is contingent on their messages, the outcome can in principlebemadecontingentonthestatethroughacommunicationgame. Here is an example from ADR in which the first best is achieved. Consider a two-person risk sharing problem in which there is a finite number of states. Without loss of generality we can assume that the states are numbered s =1,...,S. Descriptions of the state may be more complicated but we can replace those with a simpler message space. Let y s denote the income to be shared in state s. The optimal allocation requires agent 1 and 2 to receive x s and z s = y s ?x s in state s respectively. Notice that x s and y s ?x s are increasing in y s . Acontractc = {(x s ,z s )} specifies the amount that should be received by each agent in state s.Ifstates occurs, both agents announce numbers σ,σ 0 =1,...,S.Iftheyagree,agent1 receives x σ and agent 2 receives y s ?x σ . If they disagree, the agent who announces the state associated with the higher income becomes the residual claimant and the other agent receives theamountspecified in the contract Truth-telling is a Nash equilibrium and implements the first best. 5.2.3 Gale (1991) Repeated renegotiation can reduce the incompleteness of contracts enor- mously as the following example shows. The venture capitalist and the en- trepreneur share project revenue w. A complete contract is a pair of functions x(w) and y(w) that solve: max E[u(x)+λv(y)] s.t. x + y ≤ w a.s. A simple calculation shows that 0 <x 0 (w) < 1 and 0 <y 0 (w) < 1 for all values of w. 6 CHAPTER 5. DYNAMIC CONTRACTING The alternative to writing the complete contingent contract is for the ven- ture capitalist and the entrepreneur to write a much simpler contract, say a debt contract, and renegotiate the terms of the contract as more information becomes available. Suppose that time is divided into T periods t =1,...,T. The initial loan and investment are made before date 1 and the final outcome of the project is observed at date T. At each intervening date, some information about the eventual payo? arrives. Formally, we assume there is a sequence of random variables {w t } such that w t+1 = ? w t + a(h t ) with probability p(h t ) w t + b(h t ) with probability 1?p(h t ) , where a(h t ) >b(h t ) and 0 <p(h t ) < 1, h 1 = w 1 is a constant, w T = w,and the history h t =(w 1 ,...,w t ) is common knowledge at each date t. Let d t denote the face value of the debt chosen at date t and let m t denote the cumulative transfers made to the firm up to and including date t. The rules of the game are as follows: ? The firmandtheventurecapitalistareassumedtohavechosenan initial contract (d 0 ,m 0 ) before the first date. ? At each date t, there is a pre-existing contract (d t?1 ,m t?1 ).Thefirm proposes a new contract (d t ,m t ). ? The venture capitalist accepts or rejects the proposal. ? If the proposal is accepted, the venture capitalist makes a net transfer m t ? m t?1 to the firm and the firm’s debt is changed to d t .Ifthe proposal is rejected, nothing happens and the pre-existing contract at the next date will be (d t ,m t )=(d t?1 ,m t?1 ). ? At the final date t = T, there is no scope for renegotiation. The firm receives the payo? max{w T ?d T?1 ,0} + m T?1 and the venture capitalist receives the payo? min{d T?1 ,w T }?m T?1 . A subgame perfect equilibrium of the game achieves the first best risk sharing allocation. 5.2. RENEGOTIATION 7 5.2.4 The spanning condition The firm makes all the o?ers, so has all the bargaining power. The venture capitalist can guarantee that he will get at least V ? (d t?1 ,m t?1 |h t ) at date t,where(d t?1 ,m t?1 ) is the pre-existing contract and the information set is h t . In equilibrium, renegotiation of the contract (d t?1 ,m t?1 ) will give him exactly V ? (d t?1 ,m t?1 |h t ). Definition 1 If we can choose a contract (d t?1 ,m t?1 ) to satisfy V ? (d t?1 ,m t?1 |h t )=E[v(y(w T )|h t ] (5.1) for each date t and history h t , then we say that the spanning condition for implementation of the first-best risk-sharing allocation is satisfied. It turns out that this condition is su?cient as well as necessary for im- plementation of the first best. If the spanning condition is satisfied, then the bargaining game has a subgame perfect equilibrium that implements the first best. 5.2.5 Subgame Perfect Equilibrium Proposition 2 If the spanning condition is satisfied, there exists a Pareto- e?cient SPE of the renegotiation game, that is, a SPE that results in the implementation of first-best risk sharing. 5.2.6 An Example A parametric example illustrates the requirements of the theory and also al- lows us to see whether the spanning condition will be satisfied in a reasonable case. Suppose that both the firm and the venture capitalist have constant absolute risk aversion: u(x)=?e ?Ax v(y)=?e ?By and suppose that {w t } follows a random walk: w t+1 = ? w t + a w. pr. π w t + b w. pr. 1?π, 8 CHAPTER 5. DYNAMIC CONTRACTING for t =1,...,T?1,wherew 1 is a known constant. The first-order condition for e?cient risk sharing takes the form: Ae ?Ax(w) = Be ?By(w) which implies that x(w) is an a?ne function of y(w). Then the fact that x(w)+y(w) ≡ w implies that both x(w) and y(w) are a?ne functions of w. Suppose that y(w)=λw + μ,where0 <λ<1. Theninorderto implement the first-best, risk-sharing scheme, the debt contract adopted at date T ?1 when w T?1 is observed must satisfy: min{w T?1 + a,d T?1 }?m T?1 = λ(w T?1 + a)+μ min{w T?1 + b,d T?1 }?m T?1 = λ(w T?1 + b)+μ and since 0 <λ<1 this requires d T?1 ?m T?1 = λ(w T?1 + a)+μ w T?1 + b?m T?1 = λ(w T?1 + b)+μ or d T?1 = w T?1 + λa +(1?λ)b m T?1 =(1?λ)(w T?1 + b)+μ. So there are unique values of debt and transfers at date T?1 that implement the first best. Obviously, the adding-up condition implies that the same values of d T?1 and m T?1 will give the firm x(w). At dates t<T? 1, the problem is more complicated, because we have to choose d t and m t to give the venture capitalist the equilibrium status quo utility level rather than to give it a particular income level. As a result, the calculations are more complicated. However, the critical problem as we have seen is to ensure that the spanning conditions are satisfied. Recall that (d t ,m t ) can be chosen so that E[?exp{?B(min{w T ,d t }?m t )}|w t ]=E[?exp{?B(λw T + μ)}|w t ], which is equivalent to E[?exp{?Bmin{w T ,d t }}|w t ]=e ?B(μ+m t ) E[?exp{?B(λw T )}|w t ]. 5.2. RENEGOTIATION 9 We want to ensure that the analogous conditions hold at date t+1,thatis, E[?exp{?B(min{w T ,d t }?m t )}|w t + a]=E[?exp{?B(λw T + μ)}|w t + a] E[?exp{?B(min{w T ,d t }?m t )}|w t + b]=E[?exp{?B(λw T + μ)}|w t + b]. If we choose d = ∞ then min{w T ,d t } = w T and if we choose m t so that the ex ante expected utility of (d t ,m t ) is equal to the ex ante first-best expected utility, then μ + m t > 0 and e ?B(μ+m t ) < 1. Clearly, E[?e ?Bw T |w t + a]?E[?e ?Bw T |w t + b] >E[?e ?Bλw T |w t + a]?E[?e ?Bλw T |w t + b] >e ?B(μ?m t ) ? E[?e ?Bλw T |w t + a]?E[?e ?Bλw T |w t + b] ￠ so the ex ante condition implies that E[?e ?Bw T |w t + a] >e ?B(μ+m t ) E[?e ?Bλw T |w t + a] E[?e ?Bw T |w t + b] <e ?B(μ+m t ) E[?e ?Bλw T |w t + b] as required. That is, the spanning condition is satisfied, so there exists a level of debt 0 <d t < ∞ that equates the expected utility of the debt contract with the first-best expected utility in each of the information sets at date t +1. 5.2.7 Dewatripont and Maskin (1995) Dewatripont and Maskin (1995) have suggested that financial markets have an advantage over financial intermediaries in maintaining commitments to refuse further funding. The di?culty of renegotiating with a large number of bondholders increases the ability to commit. Although it is an interesting theoretical point, it is not clear how relevant Dewatripont and Maskin’s argument is in practice.(A possible exception is the case of sovereign debt, which lies outside the scope of the present dis- cussion). There are a couple of theoretical reasons why this may be so. In the first place, the incentive e?ects of terminations depend critically on the assumption that the borrower is restricted to dealing with one lender. If it is possible to switch to other sources of funds, whether they be intermediaries or markets, it may be impossible to prevent the extension of credit in the future anyway. Of course, the initial lender still has a claim on the borrower, and this will lead to some sort of bargaining problem. However, this is no di?erent from the renegotiation that would go on between a borrower and 10 CHAPTER 5. DYNAMIC CONTRACTING lender in any case. The point is that the lender cannot unilaterally prevent the financing of a positive NPV project if there is competition. A second reason why intermediaries may not find it di?cult to terminate a borrower with a bad history is asymmetric information. Renegotiation really only constrains the contracts when there is complete information about the borrower’s type. When the borrower’s type is unknown and default is taken to be a bad signal, it may be possible to find beliefs that support the termination of credit as a perfect Bayesian equilibrium. For example, suppose that there are two types of borrowers, good and bad. A good borrower can choose either a safe project that produces a certain return R or a risky project that produces a return H with probability π and 0 with probability 1 ?π. A bad borrower can only choose worthless projects that produce 0 in every state. The lender cannot observe the outcome but can observe whether the loan is repaid or not. If good borrowers are expected to choose safe projects, then a good borrower who chooses the risky project runs the risk of being confused with the bad borrowers and being excluded from the credit market forever, whether he deals with an intermediary or a competitive financial market. This threat is credible and will discourage good borrowers from choosing risky projects. Note that the argument does not depend on the prior probability of the bad type. For a formal model, see Diamond (1991) who analyzes reputation e?ects in credit markets. 5.3 Long-term and short-term contracts [To be completed] 5.4 References Aghion, P. and P. Bolton (1992). “An Incomplete Contracts Approach to Financial Contracting,” Review of Economic Studies 59, 473-494. Aghion, P., M. Dewatripont, and P. Rey (1994). “Renegotiation Design with Unverifiable Information” Econometrica 62, 257-82. Allen, F. and D. Gale (2000). Comparing Financial Systems.Cambridge, MA: MIT Press. Dewatripont, M.and E. Maskin (1995). “Credit and E?ciency in Central- ized and Decentralized Economies,” Review of Economic Studies 62, 541-55. 5.4. REFERENCES 11 Diamond, D. (1991). “Monitoring and Reputation: The Choice between Bank Loans and Directly Placed Debt,” Journal of Political Economy 99, 689-721. Gale, D. (1991). “Optimal Risk Sharing through Renegotiation of Simple Contracts,” Journal of Financial Intermediation 1, 283-306. Hart, O. (1995). Firms, Contracts, and Financial Structure (Clarendon Lectures in Economics), New York: Oxford University Press. Hart, O. and J. Moore (1994). “A Theory of Debt Based on the Inalien- ability of Human Capital,” Quarterly Journal of Economics 109, 841-879. Huberman, G. and C. Kahn (1988). “Limited Contract Enforcement and Strategic Renegotiation,” American Economic Review 78, 471-84. Stiglitz, J. and A. Weiss (1983). “Incentive E?ects of Terminations: Ap- plications to the Credit and Labor Markets,” American Economic Review 73, 912-27.