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Contract design and costly verification games
Journal of Economic Behavior & Organization Vol. 34 (1998) 327±340
Contract design and costly verification games Chongwoo Choe* Department of Economics, La Trobe University, Bundoora, Victoria 3083, Australia Received 15 August 1995; accepted 3 March 1997
Abstract In this paper, the signalling subgame in costly verification models is studied in the context of the investor±entrepreneur contract without assuming the possibility of commitment to verification. It is shown that the game has a unique Perfect Bayesian Equilibrium in mixed strategies, implying that truth-telling and deterministic verification are not an equilibrium behavior. When the entire game starting from the stage of contract design is considered, it is shown that the contract designed by the informed entrepreneur results in less verification cost than the one designed by the uninformed investor. This could be taken as a normative criterion based upon which institutional arrangements as regards the distribution of bargaining initiative are to be designed. # 1998 Elsevier Science B.V. JEL classi®cation: D82 Keywords: Contract design; Costly veri®cation
1. Introduction How should a contract be designed when a party to a contract has ex post private information which is costly to verify? This is a question which has been asked by numerous economists over the past couple of decades. Such studies can be conveniently grouped into the following: Townsend (1979), Diamond (1984), Gale and Hellwig (1985) and Williamson (1987), among others, in the context of financial contract; Reinganum and Wilde (1985), Reinganum and Wilde (1986), Melumad and Mookherjee (1989), Bardsley (1994), Erard and Feinstein (1994), among others, in the context of tax auditing; Baron and Besanko (1984), Khalil (1992), among others, in the context of procurement contract. The analytical framework used to tackle the question is that of mechanism * Corresponding author. Tel.: +61 3 94792715; fax: +61 3 94791654; e-mail: [email protected] 0167-2681/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 7 - 2 6 8 1 ( 9 7 ) 0 0 0 5 5 - 3
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design (Harris and Townsend, 1981), which specifies a game to be played between the two parties. A general sequencing of the game thus defined is as follows: parties decide whether to accept a contract or not; when the contract is accepted by both parties, the outcome of the relation is realized (either with or without input from either party) which is observed only by one party, hence the private information; the privately-informed party reports the private information; upon receiving the report, the uninformed party decides whether to verify the report or not, based upon which payoffs to both parties are realized. Most studies so far have focused on the design of an incentive-compatible contract by relying, in one way or another, on the revelation principle. This amounts to assuming that the uninformed party can credibly commit to the verification of the report by the informed party, either on her own,1 or by employing a third party, say an auditor.2 While this commitment may elicit, if not always, a truthful report from the informed party, the very truth-telling vitiates the committed verification.3 The uninformed party must verify reports which she knows are truthful. This time inconsistency of verification is obviously due to the fact that the verification schedule is fixed ex ante, thereby terminating the whole game when the informed party makes a report. This does not seem to coincide with many practices in reality: debt covenants specify situations when creditors can exercise auditing, but the actual auditing is up to the creditors to decide; tax authorities do not announce auditing schedules to the public. Reinganum and Wilde (1986), Bardsley (1994) and Erard and Feinstein (1994) are notable exceptions in the sense that the commitment of the uninformed party is not presumed. Although all these studies establish the equilibrium in mixed strategies, they only direct attention to the costly auditing subgame. Khalil (1992) studied the entire game starting from the contract proposal without assuming the possibility of commitment, still obtaining the equilibrium in mixed strategies. However, by rendering to the uninformed party the bargaining initiative at the stage of contract proposal,4 Khalil does not explore other possibilities that may yield a less costly outcome. There is no reason a priori why we should let the uninformed party be the first mover in the bargaining game. This and other distinctions between Khalil's work and this paper are more fully discussed at the end of Section 2. The purpose of this paper is twofold. Firstly, I study the costly verification subgame without assuming the possibility of commitment. Since truth-telling vitiates verification and vice versa, we expect to obtain an equilibrium in mixed strategies. Based on the equilibrium of the subgame, I then look at the problem of contract design. Specifically, the question addressed is who should design the contract to minimize the verification cost. If the uninformed party designs the contract and assumes the bargaining initiative in an environment where commitment is not possible, one may expect that more resources 1 Examples include all the above-mentioned studies in the debtor±creditor contract, Reinganum and Wilde (1985), and Baron and Besanko (1984). 2 Examples are Baiman et al. (1987), and Melumad and Mookherjee (1989). 3 When the committed verification schedule is random as in Baiman et al. (1987), Townsend (1988), or Mookherjee and Png (1989), an additional problem arises in that it will be hard to see whether the uninformed party abides by the contractual terms. 4 The first stage bargaining game in this case starts with the uninformed party offering a contract on a take-itor-leave-it basis, which the informed party accepts or rejects.
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would be spent on verification than when the informed party designs the contract. In an attempt to extract as much surplus as possible, the uninformed party would set, not only the penalty for lying at its maximum level, but the reward for truth-telling at its minimum level. As long as the informed party is protected by limited liability and so the penalty for lying is bounded above, such a contract proposed by the uninformed party would make truth-telling less attractive, which in turn will trigger verification more often. If however, the informed party designs the contract, the reward for truth-telling would be set at its maximum level, making truth-telling more attractive, hence lessening verification. This could be taken as a normative criterion based upon which institutional arrangements as regards the distribution of bargaining initiative are to be designed. To this end, I study a simple model of investor±entrepreneur contract. While the terms `investor' and `entrepreneur' seem appealing in the current context, a slight modification of the current model can be used for the analysis of tax auditing.5 Without the possibility of commitment, the verification decision is treated as the strategy of the investor.6 Throughout the paper, the investor and the entrepreneur will be represented by female and male gender pronouns respectively. The rest of the paper is organized as follows. Section 2 describes the model and studies the costly verification subgame. In Section 3, the initial game of contract proposal is studied based on the result from Section 2, thereby tackling the problem of who should design the contract. Section 4 summarizes the main results of the paper and discusses possible extensions. The derivation of equilibria of the game in Section 2 is provided in the Appendix A. 2. Costly verification game and perfect Bayesian equilibrium The simplest of the costly verification games has two agents, to be called the entrepreneur and the investor, and one investment project. The entrepreneur alone can run the project but has to rely on the investor for external finance K, the necessary funds for the project. The exogenously given (net) rate of return on the investment is assumed to be zero. Thus the investor should be guaranteed at least K to take part in the contractual relation. The project has two possible monetary outcomes, xh, xl, xh>xl with publiclyknown probabilities p and 1ÿp respectively, and 0
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expected payoffs and, for simplicity, I will fix xl0. The reservation utility of the entrepreneur is assumed to be zero. To make the contractual relation meaningful, I make the following assumption which implies that the expected return from the project is large enough to cover the verification cost and the necessary funds for the investment. Assumption 1.
pxhcK.
In this section, the focus will be on the subgame that starts after the contract is signed. Based on the equilibrium of the subgame studied in this section, I turn in the next section to the entire game starting from the stage of contract proposal. Thus, this approach is equivalent to solving the entire game backwards. Given that verification is the strategy of the investor, the game proceeds in the following way: initially, Nature determines xh or xl; observing this, the entrepreneur makes a report of xh or xl; depending on the report, the investor decides whether to verify the report or not. Payoffs for each agent are determined by a contract which specifies the return to the investor contingent on the history: r : fxh ; xl g fxh ; xl g f0; 1g ! R where r(x, x0 , d) is the return to the investor when the actual outcome is x, the reported outcome is x0 and the verification decision is d2{0,1}, 0 for no verification and 1 for verification. For technical reasons, it will be assumed that whenever the entrepreneur issues a counterfactual report x0 6x, it costs him >0 which is the monetary equivalent of the non-pecuniary cost of falsification.7 is assumed to be sufficiently small. The entrepreneur's utility is the actual outcome less the return to the investor less the falsification cost, if any.8 Then r assumes eight possible values denoted by r1r(xh,xh,1), r2r(xh,xh,0), r3r(xh,xl,1), r4r(xh,xl,0), r5r(xl,xl,1), r6r(xl,xl,0), r7r(xl,xh,1) and r8r(xl,xh,0). In order to make the game definite, I impose the following restrictions on r: (a) limited liability: 0r[x,x0 ,d(x0 )]d(x0 )x[1ÿd(x0 )]min{x,x0 } for all x, x0 ; (b) the entrepreneur should be given an incentive not to lie; (c) the investor should find verification desirable only when a false report is detected; (d) individual rationality. The meaning of (a) is clear: with verification (d(x0 )1), the true outcome x becomes public, which should be the upper bound of the return, and without verification (d(x0 )0), the return cannot exceed min{x,x0 }. Restrictions (b) and (c) are in accord with the general spirit of costly verification games. (d) requires that the equilibrium expected payoffs to both agents be non-negative, for which we need a suitable definition of equilibrium. So I would neglect (d) for now and check it later for each equilibrium. Due to (a), we have xhrirj0, i1,2,3, j4,. . .,8. Restrictions (b) and (c) imply r3r2r1>0. Combining all these, we have the following assumption: 7 The cost of falsification may be real as in Lacker and Weinberg (1989). In the current context, it is an artificial device to reduce the number of equilibria. In particular, such a cost makes truth-telling more attractive to the entrepreneur if he is indifferent to truth-telling and lying without the cost. Moreover, one can show that the equilibrium with >0 leads to the same qualitative conclusion as the equilibrium when tends to zero. I thank an anonymous referee for suggesting this simplification. 8 In the case of verification, I assume that the investor pays the cost. That is, r is gross of any verification cost. In Gale and Hellwig (1985), there is also a private, non-pecuniary verification cost to the entrepreneur, which is not really essential for their result. So I assume away this cost.
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Fig. 1. Verification game.
Assumption 2.
xhr3r2r1>0rj, j4,. . .,8.
Now we have the following extensive form game: Fig. 1. A natural equilibrium concept I choose for this signalling game is that of Perfect Bayesian Equilibrium.9 Let I(x) be the probability that the investor verifies the given report of x, xxh, xl, a be the belief of the investor on a1 at the information set A, b, be the belief of the investor on b1 at the information set B, and E(x), be the probability that the entrepreneur reports xh given the observation of x, xxh, xl. Given the strategy of the entrepreneur, the belief of the investor can be computed using Bayes' rule: a b
pE xh ; pE xh 1 ÿ pE xl
p 1 ÿ E xh : p 1 ÿ E xh 1 ÿ p 1 ÿ E xl
A couple of quick observations simplify the task of solving the game. Note first that at any equilibria, the entrepreneur never reports xh when the actual outcome is xl, that is, E(xl)0. By reporting truthfully, the entrepreneur gets zero (expected) payoff, while making a counterfactual report of xh costs him . Given E(xl)0, the belief of the 9 One can also consider a Sequential Equilibrium. For this game, however, the set of Perfect Bayesian Equilibria and Sequential Equilibria coincide. See Fudenberg and Tirole (1991).
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investor simplifies to a1, bp(1ÿsE(xh))/p(1ÿE(xh))(1ÿp). For the investor, then, it is never in her interest to verify the report of xh, since she knows for sure that the actual outcome is xh given the reported outcome xh. Thus at any equilibrium, we must have I(xh)0. Costly verification by the investor arises only at the information set B. If the investor never verifies the report of xl, then the entrepreneur will always report xl as the cost of falsification is sufficiently small relative to the gains from such falsification. On the other hand, if the investor always verifies the report xl, then the entrepreneur will always report truthfully. This, as in other games of a similar nature such as inspection games, leads us to expect an equilibrium in mixed strategies. However, if the probability of a high outcome (p) is relatively low or the verification cost is relatively high, then the investor might verify the report only with small probability or might not even bother to verify the report at all simply because the expected gains from detecting a false report may not compensate the verification cost. In that case, the equilibrium report of the entrepreneur will be xl. This equilibrium would not satisfy the individual rationality for the investor. Sorting out this type of equilibria, the only meaningful equilibrium will indeed be in mixed strategies, the condition for whose existence is that the probability of a high outcome must be sufficiently high or the verification cost must be sufficiently low. The following proposition exhausts all possible equilibria, the derivation of which is in the Appendix A. Proposition 1.
The above game has three types of possible equilibria:
1. If pc/r3, then I(xh)0, I(xl)(r2ÿ)/r3, a1, bc/r3, E(xh)(pr3ÿc)/(pr3ÿpc), E(xl)0. As mentioned earlier, not all of the above equilibria satisfy individual rationality. At the type (1) equilibrium, the equilibrium expected payoff for the investor is ÿK since the entrepreneur always reports xl, which the investor never verifies, in which case the return to the investor is zero. This is because the probability of the outcome being xh is too low (or the verification cost is too high) so that the investor does not find it worthwhile to carry out costly verification. Thus it fails to satisfy individual rationality. So does the type (2) equilibrium for similar reasons. Thus it is expected that, for sufficiently large p, the type (3) equilibrium will satisfy individual rationality. Before I prove this last statement, the equilibrium expected payoffs for each agent at the type (3) equilibrium will be derived for future reference. For the investor, it is UI pE
xh r2
1 ÿ E
xh 1
xl
r3 ÿ c
1 ÿ pI
xl
ÿc ÿ K
r2 pr3 ÿ c ÿ K: r3 ÿ c
The expected payoff for the entrepreneur at the type (3) equilibrium is UE pE xh xh ÿ r2 1 ÿ E xh f1 xl xh ÿ r3 ÿ 1 ÿ I xl xh ÿ g pxh ÿ pr2 :
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Proposition 2. Suppose p[cr2K(r3 ÿc)]/r2r3 and Kc/r3 so that type (3) equilibrium exists. Note that 1>[cr2K(r3ÿc)]/r2r3>c/r3 since r3>c/p>c and K
10 For incentive-compatible contracts with more than two outcomes, non-truth-telling equilibria can exist. Consider for instance, the contract with the deterministic verification of Gale and Hellwig (1985). For any outcome higher than the threshold value for verification, the informed agent will report the minimum value, while for any outcome in the verification region, the informed agent is indifferent to truthful reporting and the reporting of any other value in the verification region. With a dichotomous outcome, however, this does not arise.
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to suppose that the uninformed principal, be it a tax authority or an investor, can dictate how much income or profit should be made. Finally, there is no reason a priori why we should focus on the game where the uninformed party has the initiative in designing the contract. In case commitment is possible, the problem of who designs the contract becomes irrelevant in so far as how much verification should be performed. The reason is that the optimal contract designed by the uninformed party, subject to the incentive compatibility and individual rationality for the informed party, leads to the same amount of verification as the optimal contract designed by the informed party, subject to the individual rationality for the uninformed party based on the (self-imposed) incentive compatibility for the informed party. In the absence of commitment, however, as the contract will affect the behavior of agents in the verification subgame, who designs the contract is likely to affect the amount of verification: hence the magnitude of welfare loss. This last point deserves further exploration, raising the general question of who should design the contract on efficiency grounds in the presence of asymmetric information. I turn to this problem in the next section within the simple setup of the present paper, leaving the exploration of the general question for future research. 3. Who should design the contract? The focus of this section is on the design of a contract that induces type (3) equilibrium in the verification subgame studied in Section 2. Since commitment to verification is not possible, who designs the initial contract will affect the behavior of the entrepreneur and the investor in a way different from what is prescribed in costly verification games with commitment. In the latter, whether the uninformed investor designs the contract or the informed entrepreneur designs the contract, resources spent on verification activity will be the same. This is because the incentive-compatible contract leads to the same equilibrium verification schedule, irrespective of who designs the contract, resulting in the same expected verification cost at the equilibrium. Thus the allocation of bargaining initiative ± namely, who designs the contract ± would not have any efficiency consequences, resulting only in (possibly) a different distribution of surplus between the two parties. With the possibility of commitment assumed away, the allocation of bargaining initiative does have an efficiency consequence. Consider, for instance, the contract designed by the uninformed investor. She has to choose the contract that optimally trades off her verification cost against the benefit from asking the entrepreneur to make a maximal payment whenever the outcome is xh. The latter will leave little rent for the entrepreneur, who then will find it increasingly tempting to make a false report since the limited liability will leave him, at worst wealth ÿ. This will lead the investor to spend more resources on verification than when the entrepreneur designs the contract. On the other hand, if the entrepreneur designs the contract, he can make his reward for truthtelling large enough, reducing his own incentive to lie, making verification by the investor less attractive than when the investor had the bargaining initiative. Thus one may expect that the contract designed by the informed entrepreneur will result in less welfare loss, or verification cost.
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Recall that at type (3) equilibrium, expected payoffs for agents were UI r2 pr3 ÿ c= r3 ÿ c ÿ K, UEpxhÿpr2. Expected verification cost at the equilibrium, denoted by M is Mc[p(1ÿE(xh)) I(xl)(1ÿp)I(xl)]c(r2ÿ)(1ÿp)/(r3ÿc). Finally, conditions used for the existence of type (3) equilibrium are collected: c C : xh r3 r2 > K; p > : r3 Since the condition p[cr2 K(r3ÿc)]/r2r3 is equivalent to the individual rationality for the investor, it will be replaced by UI0. Before we look at the problem of contract design, the following comparative statics provide useful information, whose proof is straightforward and is omitted. Proposition 3. (a) The expected verification cost decreases in r3, and p, and increases in r2 and c; (b) the expected payoff for the investor increases in r2 and r3; (c) the expected payoff for the entrepreneur decreases in r2. The intuition behind the above proposition is pretty clear. As r3 increases, or the penalty of lying increases, the investor is increasingly confident that the entrepreneur is telling the truth, hence less verification. The same is true if r2 decreases, or the reward for truth-telling increases. We now turn to the problem of designing an optimal contract as one which maximises the weighted expected utilities of the entrepreneur and the investor subject to given constraints. Let 2[0,1] be the welfare weight assigned to the expected payoffs of the entrepreneur, which may also be thought of as the proxy for the bargaining power of the entrepreneur. That is, the entrepreneur has full bargaining power if 1 and the investor does if 0. The optimal contract is a solution to Max r2 ;r3 UE 1 ÿ UI subject to C; UE 0; UI 0: Several observations simplify the solution to the above problem. First, the constraint (C) is sufficient for the individual rationality for the entrepreneur since r2xh, hence we can drop the constraint UE0 from the above problem. Second, note that the entrepreneur's expected payoff does not depend on r3, while the investor's expected payoff increases in r3. Thus in any optimal contract, we must have r3xh, implying the maximum penalty for lying, which is consistent with the costly verification literature. With r3xh, one can check that UI0 implies r2>K. Thus the constraints to the above problem are simplified to xhr2K(xhÿc)/(pxhÿc). The problem is now r2 pxh ÿ c K xh ÿ c ÿ K subject to xh r2 : Max r2 W pxh ÿ pr2 1ÿ xh ÿ c pxh ÿ c It is easy to check that dW( )/dr2 < 0 as > (pxhÿc)/(2pxhÿpcÿc). Two conclusions can be drawn immediately. If >(pxhÿc)/(2pxhÿpcÿc), or the bargaining power of the entrepreneur is large enough, then r2K(xhÿc)/(pxhÿc). That is, an optimal contract sets the reward for truth-telling to its maximum, leaving the investor with zero rent. On the other hand, if <(pxhÿc)/(2pxhÿpcÿc), then r2xh, leaving the entrepreneur with zero rent. A comparison of expected equilibrium verification costs (M), depending on the
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relative bargaining powers of the two agents, can then be made. Since M decreases in r2 as shown in Proposition 3, the expected equilibrium verification cost increases as the relative bargaining power shifts in favor of the investor. Intuition is pretty clear. If the investor designs the contract ( 0), she would maximize her rent by setting the maximum penalty for lying as well as the minimum reward for truth-telling. This will increase the probability of lying, leading to more verification than if the reward for truthtelling was larger. Nevertheless, what deters the entrepreneur from always falsifying the report is the falsification cost, albeit small. If the entrepreneur designs the contract ( 1), he can maximise his rent by imposing the maximum penalty for his lying and at the same time rewarding himself as much as possible for his honesty, which will make the investor less inclined to verify the report than when she designs the contract. Thus we have the following proposition: Proposition 4. The expected equilibrium verification cost is minimum if the entrepreneur designs the contract, and maximum if the investor designs the contract. What Proposition 4 is pointing at is the normative aspect of contract design. In the presence of private information which is costly to verify, the proposition sheds light on how the bargaining initiative should be allocated to minimize welfare loss, or verification cost. That is, the contract should be designed by the informed party. An additional element that would strengthen the validity of the proposition might be the cost of communicating private information. Nevertheless, it is to be noted that both the contract designed by the informed entrepreneur and the one designed by the uninformed investor are optimal given the fixed allocation of bargaining initiative unless some form of side payment is possible from the entrepreneur to the investor. As the bargaining initiative shifts from the investor to the entrepreneur, the one gets strictly worse-off while the other gets strictly better-off so that a Pareto comparison of the two arrangements is not possible. Should such a side payment be possible, however, then we should expect the contract designed by the entrepreneur to dominate the one designed by the investor. Suppose the initial bargaining initiative lies with the investor. The entrepreneur can always buy out the bargaining initiative of the investor with a side payment that would leave the investor at least as better-off as when she had the initiative. At the same time, the entrepreneur can make himself strictly better-off with the decreased verification cost. While the general message of Proposition 4 is unambiguous as regards the normative aspect of allocating bargaining initiative, the practice followed in reality does not appear to favor any single arrangement: (informed) firms issue debentures, not the investors; (uninformed) banks offer loan contracts, not the debtors. Admittedly, the key (and the only) feature of the present model, namely, the ex post private information, is just part of a whole list of elements that needs to be considered for more satisfactory analyses of institutional arrangements. Among them, for instance, is contracting cost. If firms can reduce contracting cost through gains from standardization by addressing many potential investors, the result of the paper will only strengthen the case for why firms issue debentures, not the other way around. The same argument may be used to establish the case for different arrangements. For example, the reason why uninformed banks offer
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loan contracts to many potential borrowers could be that the reduction in the contracting cost exceeds the increase in the monitoring cost. One may argue similarly for why tax authorities design tax schedules, not the taxpayers. Also, in the case of bank loans, as Wang and Williamson (1993) argue, the screening of ex ante private information may be of greater significance than the verification of ex post private information. It is by no means obvious if the inclusion of elements such as the presence of ex ante private information or contracting cost will lead to an unambiguous conclusion. 4. Summary and conclusions In this paper, the signalling subgame in costly verification models has been studied in the context of the entrepreneur±investor contract without assuming the possibility of commitment to verification. It was shown that the game has a unique Perfect Bayesian Equilibrium in mixed strategies, implying that truth-telling and deterministic verification are not equilibrium behavior, thereby questioning the popularity of debt contracts. Presumably, the widespread use of the debt contract is due to its simplicity, a formalization of which may well involve such elements as contracting costs. When the entire game starting from the contract design was considered, it was shown that the contract designed by the informed entrepreneur results in less verification cost than the one designed by the uninformed investor. This could be taken as a normative criterion based upon which institutional arrangements as regards the distribution of bargaining initiative are to be designed, although actual practice seems to provide mixed evidence. While the basic model in this paper is highly simplified, it can be extended to include such elements as collateral. Suppose the entrepreneur has collateral valued at >0 which will be confiscated by the investor if the actual return is xl or if it is xh but reported to be xl which is not verified. Provided that the collateral value is not large enough, one can solve the game again to find equilibria that are qualitatively the same as those obtained in the paper. As the payoffs for the entrepreneur with xxl now decrease uniformly by , the equilibrium strategy for the entrepreneur in that part of the game is still truth-telling. Given that, the equilibrium strategy for the investor upon receiving the report xh is not to verify the report. At type (3) equilibrium, we now have I(xl)(r2ÿ ÿ)/(r3ÿ ), E(xh)(pr3ÿp ÿc)/(pr3ÿp ÿpc) for which we need p>c/(r3ÿ ). Naturally the inclusion of collateral increases the probability of truth-telling and decreases the probability of verification. Based on this equilibrium, one can show that the results of Section 3 are still valid. As the investment technology of this paper is fixed with a positive net present value, the discussion of efficiency could be made based only on expected equilibrium verification costs. With more general models where levels of investment can also be chosen, it is needless to say that a proper welfare criterion should be net expected returns from the investment less the expected verification costs, rather than the expected verification costs alone. This would be a worthy extension of the present paper. Among a number of other extensions of the model, the exploration into whether the results of Section 3 will be robust under the environment with general adverse selection and moral hazard problems seems not only interesting but promising.
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Acknowledgements I would like to thank Peter Bardsley, seminar participants at La Trobe, New South Wales and Monash Universities, and participants at the Congress of the European Economic Association, Istanbul, August 1996, for helpful discussions. Two anonymous referees provided very helpful and constructive comments. Financial support of research grant from the School of Economics and Commerce, La Trobe University is gratefully acknowledged. Any remaining errors are my own. Appendix A Derivation of equilibria for the verification game Consider first the entrepreneur's problem. When the actual outcome is xh, the entrepreneur solves the following problem: MaxE xh E xh I xh xh ÿ r1 1 ÿ I xh xh ÿ r2 1 ÿ E xh I xl xh ÿ r3 ÿ 1 ÿ I xl xh ÿ : The solution to the above problem is E xh 1 if I xh r1 1 ÿ I xh r2 < I xl r3 ; 2 0; 1 if I xh r1 1 ÿ I xh r2 I xl r3 ; 0 if I xh r1 1 ÿ I xh r2 > I xl r3 :
(A)
When the actual outcome is xl, similar steps lead us to conclude that E(xl)0. As shown in Section 2, the beliefs of the investor then become a1, bp(1ÿE(xh))/ (p(1ÿE(xh))(1ÿp)). Next I look at the investor's problem. At information set A, the investor's problem is MaxI xh a I xh r1 ÿ c 1 ÿ I xh r2 1 ÿ a I xh ÿc 1 ÿ I xh r8 : Since a1 and r2r1>r1ÿc, we must have I(xh)0. The investor's problem at information set B is MaxI xl b I xl r3 ÿ c 1 ÿ b I xl ÿc: The solution to this problem is I xl 1 if b >
c c c ; 2 0; 1 if b ; 0 if b < : r3 r3 r3
(B)
Three cases in (B) will be considered separately for the derivation of equilibria. Case 1 b>c/r3: Then I(xl)1. We also know I(xh)0. From (A), then, we have I(xh)r1 (1ÿI(xh))r2r2<I(xl)r3r3 since r3 r2, leading us to conclude that E(xh)1. Then b0 which contradicts b>c/r3. Thus equilibrium does not exist in this case.
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Case 2 bc/r3: Then I(xl)2[0, 1]. I will consider the three cases in (A) separately utilizing the result, I(xh)0. Case 2.1 r2ÿ<I(xl)r3: Then E(xh)1, in which case b0 contradicting bc/r360. Thus equilibrium does not exist in this case. Case 2.2 r2ÿI(xl) r3 or I(xl)(r2ÿ)/r32(0,1): Then E(xh)2(0,1). From bc/ r3p(1ÿE(xh))/(p(1ÿE(xh))(1ÿp)), we obtain E(xh)(pr3ÿc)/(pr3ÿpc). For this probability to be meaningful, we need p>c/r3. Thus if p>c/r3, we have an equilibrium, I(xl)(r2ÿ)/r3, E(xh)(pr3ÿc)/(pr3ÿpc), I(xh)E(xl)0, a1, and bc/r3. Case 2.3 r2ÿ>I(xl)r3: Then E(xh)0, hence bp. Thus an equilibrium in this case is I(xl)<(r2ÿ)/r3, I(xh)E(xh)E(xl)0, a1, bpc/r3. Case 3 bI(xl)r30 in (A) since is sufficiently small. Thus E (x h )0 so that b p. In case p
References Baiman, S., Evans III, J.H., Noel, J., 1987. Optimal contracts with a utility-maximizing auditor, Journal of Accounting Research 25(2), 217±244. Bardsley, P., 1994. Tax compliance games with imperfect auditing, mimeo. La Trobe University. Baron, D.P., Besanko, D., 1984. Regulation, asymmetric information, and auditing, Rand Journal of Economics 15, 267±302. Diamond, D.W., 1984. Financial intermediation and delegated monitoring, Review of Economic Studies 51, 393±414. Erard, B., Feinstein, J.S., 1994. Honesty and evasion in the tax compliance game, RAND Journal of Economics 25(1), 1±19. Fudenberg, D., Tirole, J., 1991. Perfect Bayesian Equilibrium and sequential equilibrium, Journal of Economic Theory 53, 236±260. Gale, D., Hellwig, M., 1985. Incentive compatible debt contracts: The one-period problem, Review of Economic Studies 52, 647±663. Harris, M., Townsend, R.M., 1981. Resource allocation under asymmetric information, Econometrica 49, 33±64. Khalil, F., 1992. Auditing without commitment, Discussion Paper, 92-15, University of Washington. Kofman, F., LawarreÂe, J., 1993. Collusion in hierarchical agency, Econometrica 61(3), 629±656. Lacker, J.M., Weinberg, J.A., 1989. Optimal contracts under costly state falsification, Journal of Political Economy 97(6), 1345±1363. Laffont, J.-J., 1988. Hidden gaming in hierarchies: Facts and models, Economic Record 64, 295±306. Melumad, N.D., Mookherjee, D., 1989. Delegation as commitment: The case of income tax audits, RAND Journal of Economics 20(2), 139±163. Mookherjee, D., Png, I., 1989. Optimal auditing, insurance and redistribution, Quarterly Journal of Economics 104, 399±415. Reinganum, J.F., Wilde, L.L., 1985. Income tax compliance in a principal±agent framework, Journal of Public Economics 26, 1±18. Reinganum, J.F., Wilde, L.L., 1986. Equilibrium verification and reporting policies in a model of tax compliance, International Economic Review 27(3), 739±760.
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Tirole, J., 1986. Hierarchies and bureaucracies: On the role of collusion in organizations, Journal of Law, Economics, and Organization 2(2), 181±214. Townsend, R.M., 1979. Optimal contracts and competitive markets with costly state verification, Journal of Economic Theory 21, 265±293. Townsend, R.M., 1988. Information constrained insurance: The revelation principle extended, Journal of Monetary Economics 21, 411±450. Wang, C., Williamson, S.D., 1993. Adverse selection in credit markets with costly screening, mimeo., University of Iowa. Williamson, S.D., 1987. Costly verification, loan contracts, and equilibrium credit rationing, Quarterly Journal of Economics 102, 135±145.
Contract design and costly verification games Chongwoo Choe* Department of Economics, La Trobe University, Bundoora, Victoria 3083, Australia Received 15 August 1995; accepted 3 March 1997
Abstract In this paper, the signalling subgame in costly verification models is studied in the context of the investor±entrepreneur contract without assuming the possibility of commitment to verification. It is shown that the game has a unique Perfect Bayesian Equilibrium in mixed strategies, implying that truth-telling and deterministic verification are not an equilibrium behavior. When the entire game starting from the stage of contract design is considered, it is shown that the contract designed by the informed entrepreneur results in less verification cost than the one designed by the uninformed investor. This could be taken as a normative criterion based upon which institutional arrangements as regards the distribution of bargaining initiative are to be designed. # 1998 Elsevier Science B.V. JEL classi®cation: D82 Keywords: Contract design; Costly veri®cation
1. Introduction How should a contract be designed when a party to a contract has ex post private information which is costly to verify? This is a question which has been asked by numerous economists over the past couple of decades. Such studies can be conveniently grouped into the following: Townsend (1979), Diamond (1984), Gale and Hellwig (1985) and Williamson (1987), among others, in the context of financial contract; Reinganum and Wilde (1985), Reinganum and Wilde (1986), Melumad and Mookherjee (1989), Bardsley (1994), Erard and Feinstein (1994), among others, in the context of tax auditing; Baron and Besanko (1984), Khalil (1992), among others, in the context of procurement contract. The analytical framework used to tackle the question is that of mechanism * Corresponding author. Tel.: +61 3 94792715; fax: +61 3 94791654; e-mail: [email protected] 0167-2681/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 7 - 2 6 8 1 ( 9 7 ) 0 0 0 5 5 - 3
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design (Harris and Townsend, 1981), which specifies a game to be played between the two parties. A general sequencing of the game thus defined is as follows: parties decide whether to accept a contract or not; when the contract is accepted by both parties, the outcome of the relation is realized (either with or without input from either party) which is observed only by one party, hence the private information; the privately-informed party reports the private information; upon receiving the report, the uninformed party decides whether to verify the report or not, based upon which payoffs to both parties are realized. Most studies so far have focused on the design of an incentive-compatible contract by relying, in one way or another, on the revelation principle. This amounts to assuming that the uninformed party can credibly commit to the verification of the report by the informed party, either on her own,1 or by employing a third party, say an auditor.2 While this commitment may elicit, if not always, a truthful report from the informed party, the very truth-telling vitiates the committed verification.3 The uninformed party must verify reports which she knows are truthful. This time inconsistency of verification is obviously due to the fact that the verification schedule is fixed ex ante, thereby terminating the whole game when the informed party makes a report. This does not seem to coincide with many practices in reality: debt covenants specify situations when creditors can exercise auditing, but the actual auditing is up to the creditors to decide; tax authorities do not announce auditing schedules to the public. Reinganum and Wilde (1986), Bardsley (1994) and Erard and Feinstein (1994) are notable exceptions in the sense that the commitment of the uninformed party is not presumed. Although all these studies establish the equilibrium in mixed strategies, they only direct attention to the costly auditing subgame. Khalil (1992) studied the entire game starting from the contract proposal without assuming the possibility of commitment, still obtaining the equilibrium in mixed strategies. However, by rendering to the uninformed party the bargaining initiative at the stage of contract proposal,4 Khalil does not explore other possibilities that may yield a less costly outcome. There is no reason a priori why we should let the uninformed party be the first mover in the bargaining game. This and other distinctions between Khalil's work and this paper are more fully discussed at the end of Section 2. The purpose of this paper is twofold. Firstly, I study the costly verification subgame without assuming the possibility of commitment. Since truth-telling vitiates verification and vice versa, we expect to obtain an equilibrium in mixed strategies. Based on the equilibrium of the subgame, I then look at the problem of contract design. Specifically, the question addressed is who should design the contract to minimize the verification cost. If the uninformed party designs the contract and assumes the bargaining initiative in an environment where commitment is not possible, one may expect that more resources 1 Examples include all the above-mentioned studies in the debtor±creditor contract, Reinganum and Wilde (1985), and Baron and Besanko (1984). 2 Examples are Baiman et al. (1987), and Melumad and Mookherjee (1989). 3 When the committed verification schedule is random as in Baiman et al. (1987), Townsend (1988), or Mookherjee and Png (1989), an additional problem arises in that it will be hard to see whether the uninformed party abides by the contractual terms. 4 The first stage bargaining game in this case starts with the uninformed party offering a contract on a take-itor-leave-it basis, which the informed party accepts or rejects.
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would be spent on verification than when the informed party designs the contract. In an attempt to extract as much surplus as possible, the uninformed party would set, not only the penalty for lying at its maximum level, but the reward for truth-telling at its minimum level. As long as the informed party is protected by limited liability and so the penalty for lying is bounded above, such a contract proposed by the uninformed party would make truth-telling less attractive, which in turn will trigger verification more often. If however, the informed party designs the contract, the reward for truth-telling would be set at its maximum level, making truth-telling more attractive, hence lessening verification. This could be taken as a normative criterion based upon which institutional arrangements as regards the distribution of bargaining initiative are to be designed. To this end, I study a simple model of investor±entrepreneur contract. While the terms `investor' and `entrepreneur' seem appealing in the current context, a slight modification of the current model can be used for the analysis of tax auditing.5 Without the possibility of commitment, the verification decision is treated as the strategy of the investor.6 Throughout the paper, the investor and the entrepreneur will be represented by female and male gender pronouns respectively. The rest of the paper is organized as follows. Section 2 describes the model and studies the costly verification subgame. In Section 3, the initial game of contract proposal is studied based on the result from Section 2, thereby tackling the problem of who should design the contract. Section 4 summarizes the main results of the paper and discusses possible extensions. The derivation of equilibria of the game in Section 2 is provided in the Appendix A. 2. Costly verification game and perfect Bayesian equilibrium The simplest of the costly verification games has two agents, to be called the entrepreneur and the investor, and one investment project. The entrepreneur alone can run the project but has to rely on the investor for external finance K, the necessary funds for the project. The exogenously given (net) rate of return on the investment is assumed to be zero. Thus the investor should be guaranteed at least K to take part in the contractual relation. The project has two possible monetary outcomes, xh, xl, xh>xl with publiclyknown probabilities p and 1ÿp respectively, and 0
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expected payoffs and, for simplicity, I will fix xl0. The reservation utility of the entrepreneur is assumed to be zero. To make the contractual relation meaningful, I make the following assumption which implies that the expected return from the project is large enough to cover the verification cost and the necessary funds for the investment. Assumption 1.
pxhcK.
In this section, the focus will be on the subgame that starts after the contract is signed. Based on the equilibrium of the subgame studied in this section, I turn in the next section to the entire game starting from the stage of contract proposal. Thus, this approach is equivalent to solving the entire game backwards. Given that verification is the strategy of the investor, the game proceeds in the following way: initially, Nature determines xh or xl; observing this, the entrepreneur makes a report of xh or xl; depending on the report, the investor decides whether to verify the report or not. Payoffs for each agent are determined by a contract which specifies the return to the investor contingent on the history: r : fxh ; xl g fxh ; xl g f0; 1g ! R where r(x, x0 , d) is the return to the investor when the actual outcome is x, the reported outcome is x0 and the verification decision is d2{0,1}, 0 for no verification and 1 for verification. For technical reasons, it will be assumed that whenever the entrepreneur issues a counterfactual report x0 6x, it costs him >0 which is the monetary equivalent of the non-pecuniary cost of falsification.7 is assumed to be sufficiently small. The entrepreneur's utility is the actual outcome less the return to the investor less the falsification cost, if any.8 Then r assumes eight possible values denoted by r1r(xh,xh,1), r2r(xh,xh,0), r3r(xh,xl,1), r4r(xh,xl,0), r5r(xl,xl,1), r6r(xl,xl,0), r7r(xl,xh,1) and r8r(xl,xh,0). In order to make the game definite, I impose the following restrictions on r: (a) limited liability: 0r[x,x0 ,d(x0 )]d(x0 )x[1ÿd(x0 )]min{x,x0 } for all x, x0 ; (b) the entrepreneur should be given an incentive not to lie; (c) the investor should find verification desirable only when a false report is detected; (d) individual rationality. The meaning of (a) is clear: with verification (d(x0 )1), the true outcome x becomes public, which should be the upper bound of the return, and without verification (d(x0 )0), the return cannot exceed min{x,x0 }. Restrictions (b) and (c) are in accord with the general spirit of costly verification games. (d) requires that the equilibrium expected payoffs to both agents be non-negative, for which we need a suitable definition of equilibrium. So I would neglect (d) for now and check it later for each equilibrium. Due to (a), we have xhrirj0, i1,2,3, j4,. . .,8. Restrictions (b) and (c) imply r3r2r1>0. Combining all these, we have the following assumption: 7 The cost of falsification may be real as in Lacker and Weinberg (1989). In the current context, it is an artificial device to reduce the number of equilibria. In particular, such a cost makes truth-telling more attractive to the entrepreneur if he is indifferent to truth-telling and lying without the cost. Moreover, one can show that the equilibrium with >0 leads to the same qualitative conclusion as the equilibrium when tends to zero. I thank an anonymous referee for suggesting this simplification. 8 In the case of verification, I assume that the investor pays the cost. That is, r is gross of any verification cost. In Gale and Hellwig (1985), there is also a private, non-pecuniary verification cost to the entrepreneur, which is not really essential for their result. So I assume away this cost.
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Fig. 1. Verification game.
Assumption 2.
xhr3r2r1>0rj, j4,. . .,8.
Now we have the following extensive form game: Fig. 1. A natural equilibrium concept I choose for this signalling game is that of Perfect Bayesian Equilibrium.9 Let I(x) be the probability that the investor verifies the given report of x, xxh, xl, a be the belief of the investor on a1 at the information set A, b, be the belief of the investor on b1 at the information set B, and E(x), be the probability that the entrepreneur reports xh given the observation of x, xxh, xl. Given the strategy of the entrepreneur, the belief of the investor can be computed using Bayes' rule: a b
pE xh ; pE xh 1 ÿ pE xl
p 1 ÿ E xh : p 1 ÿ E xh 1 ÿ p 1 ÿ E xl
A couple of quick observations simplify the task of solving the game. Note first that at any equilibria, the entrepreneur never reports xh when the actual outcome is xl, that is, E(xl)0. By reporting truthfully, the entrepreneur gets zero (expected) payoff, while making a counterfactual report of xh costs him . Given E(xl)0, the belief of the 9 One can also consider a Sequential Equilibrium. For this game, however, the set of Perfect Bayesian Equilibria and Sequential Equilibria coincide. See Fudenberg and Tirole (1991).
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investor simplifies to a1, bp(1ÿsE(xh))/p(1ÿE(xh))(1ÿp). For the investor, then, it is never in her interest to verify the report of xh, since she knows for sure that the actual outcome is xh given the reported outcome xh. Thus at any equilibrium, we must have I(xh)0. Costly verification by the investor arises only at the information set B. If the investor never verifies the report of xl, then the entrepreneur will always report xl as the cost of falsification is sufficiently small relative to the gains from such falsification. On the other hand, if the investor always verifies the report xl, then the entrepreneur will always report truthfully. This, as in other games of a similar nature such as inspection games, leads us to expect an equilibrium in mixed strategies. However, if the probability of a high outcome (p) is relatively low or the verification cost is relatively high, then the investor might verify the report only with small probability or might not even bother to verify the report at all simply because the expected gains from detecting a false report may not compensate the verification cost. In that case, the equilibrium report of the entrepreneur will be xl. This equilibrium would not satisfy the individual rationality for the investor. Sorting out this type of equilibria, the only meaningful equilibrium will indeed be in mixed strategies, the condition for whose existence is that the probability of a high outcome must be sufficiently high or the verification cost must be sufficiently low. The following proposition exhausts all possible equilibria, the derivation of which is in the Appendix A. Proposition 1.
The above game has three types of possible equilibria:
1. If p
r2 pr3 ÿ c ÿ K: r3 ÿ c
The expected payoff for the entrepreneur at the type (3) equilibrium is UE pE xh xh ÿ r2 1 ÿ E xh f1 xl xh ÿ r3 ÿ 1 ÿ I xl xh ÿ g pxh ÿ pr2 :
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Proposition 2. Suppose p[cr2K(r3 ÿc)]/r2r3 and K
10 For incentive-compatible contracts with more than two outcomes, non-truth-telling equilibria can exist. Consider for instance, the contract with the deterministic verification of Gale and Hellwig (1985). For any outcome higher than the threshold value for verification, the informed agent will report the minimum value, while for any outcome in the verification region, the informed agent is indifferent to truthful reporting and the reporting of any other value in the verification region. With a dichotomous outcome, however, this does not arise.
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to suppose that the uninformed principal, be it a tax authority or an investor, can dictate how much income or profit should be made. Finally, there is no reason a priori why we should focus on the game where the uninformed party has the initiative in designing the contract. In case commitment is possible, the problem of who designs the contract becomes irrelevant in so far as how much verification should be performed. The reason is that the optimal contract designed by the uninformed party, subject to the incentive compatibility and individual rationality for the informed party, leads to the same amount of verification as the optimal contract designed by the informed party, subject to the individual rationality for the uninformed party based on the (self-imposed) incentive compatibility for the informed party. In the absence of commitment, however, as the contract will affect the behavior of agents in the verification subgame, who designs the contract is likely to affect the amount of verification: hence the magnitude of welfare loss. This last point deserves further exploration, raising the general question of who should design the contract on efficiency grounds in the presence of asymmetric information. I turn to this problem in the next section within the simple setup of the present paper, leaving the exploration of the general question for future research. 3. Who should design the contract? The focus of this section is on the design of a contract that induces type (3) equilibrium in the verification subgame studied in Section 2. Since commitment to verification is not possible, who designs the initial contract will affect the behavior of the entrepreneur and the investor in a way different from what is prescribed in costly verification games with commitment. In the latter, whether the uninformed investor designs the contract or the informed entrepreneur designs the contract, resources spent on verification activity will be the same. This is because the incentive-compatible contract leads to the same equilibrium verification schedule, irrespective of who designs the contract, resulting in the same expected verification cost at the equilibrium. Thus the allocation of bargaining initiative ± namely, who designs the contract ± would not have any efficiency consequences, resulting only in (possibly) a different distribution of surplus between the two parties. With the possibility of commitment assumed away, the allocation of bargaining initiative does have an efficiency consequence. Consider, for instance, the contract designed by the uninformed investor. She has to choose the contract that optimally trades off her verification cost against the benefit from asking the entrepreneur to make a maximal payment whenever the outcome is xh. The latter will leave little rent for the entrepreneur, who then will find it increasingly tempting to make a false report since the limited liability will leave him, at worst wealth ÿ. This will lead the investor to spend more resources on verification than when the entrepreneur designs the contract. On the other hand, if the entrepreneur designs the contract, he can make his reward for truthtelling large enough, reducing his own incentive to lie, making verification by the investor less attractive than when the investor had the bargaining initiative. Thus one may expect that the contract designed by the informed entrepreneur will result in less welfare loss, or verification cost.
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Recall that at type (3) equilibrium, expected payoffs for agents were UI r2 pr3 ÿ c= r3 ÿ c ÿ K, UEpxhÿpr2. Expected verification cost at the equilibrium, denoted by M is Mc[p(1ÿE(xh)) I(xl)(1ÿp)I(xl)]c(r2ÿ)(1ÿp)/(r3ÿc). Finally, conditions used for the existence of type (3) equilibrium are collected: c C : xh r3 r2 > K; p > : r3 Since the condition p[cr2 K(r3ÿc)]/r2r3 is equivalent to the individual rationality for the investor, it will be replaced by UI0. Before we look at the problem of contract design, the following comparative statics provide useful information, whose proof is straightforward and is omitted. Proposition 3. (a) The expected verification cost decreases in r3, and p, and increases in r2 and c; (b) the expected payoff for the investor increases in r2 and r3; (c) the expected payoff for the entrepreneur decreases in r2. The intuition behind the above proposition is pretty clear. As r3 increases, or the penalty of lying increases, the investor is increasingly confident that the entrepreneur is telling the truth, hence less verification. The same is true if r2 decreases, or the reward for truth-telling increases. We now turn to the problem of designing an optimal contract as one which maximises the weighted expected utilities of the entrepreneur and the investor subject to given constraints. Let 2[0,1] be the welfare weight assigned to the expected payoffs of the entrepreneur, which may also be thought of as the proxy for the bargaining power of the entrepreneur. That is, the entrepreneur has full bargaining power if 1 and the investor does if 0. The optimal contract is a solution to Max r2 ;r3 UE 1 ÿ UI subject to C; UE 0; UI 0: Several observations simplify the solution to the above problem. First, the constraint (C) is sufficient for the individual rationality for the entrepreneur since r2xh, hence we can drop the constraint UE0 from the above problem. Second, note that the entrepreneur's expected payoff does not depend on r3, while the investor's expected payoff increases in r3. Thus in any optimal contract, we must have r3xh, implying the maximum penalty for lying, which is consistent with the costly verification literature. With r3xh, one can check that UI0 implies r2>K. Thus the constraints to the above problem are simplified to xhr2K(xhÿc)/(pxhÿc). The problem is now r2 pxh ÿ c K xh ÿ c ÿ K subject to xh r2 : Max r2 W pxh ÿ pr2 1ÿ xh ÿ c pxh ÿ c It is easy to check that dW( )/dr2 < 0 as > (pxhÿc)/(2pxhÿpcÿc). Two conclusions can be drawn immediately. If >(pxhÿc)/(2pxhÿpcÿc), or the bargaining power of the entrepreneur is large enough, then r2K(xhÿc)/(pxhÿc). That is, an optimal contract sets the reward for truth-telling to its maximum, leaving the investor with zero rent. On the other hand, if <(pxhÿc)/(2pxhÿpcÿc), then r2xh, leaving the entrepreneur with zero rent. A comparison of expected equilibrium verification costs (M), depending on the
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relative bargaining powers of the two agents, can then be made. Since M decreases in r2 as shown in Proposition 3, the expected equilibrium verification cost increases as the relative bargaining power shifts in favor of the investor. Intuition is pretty clear. If the investor designs the contract ( 0), she would maximize her rent by setting the maximum penalty for lying as well as the minimum reward for truth-telling. This will increase the probability of lying, leading to more verification than if the reward for truthtelling was larger. Nevertheless, what deters the entrepreneur from always falsifying the report is the falsification cost, albeit small. If the entrepreneur designs the contract ( 1), he can maximise his rent by imposing the maximum penalty for his lying and at the same time rewarding himself as much as possible for his honesty, which will make the investor less inclined to verify the report than when she designs the contract. Thus we have the following proposition: Proposition 4. The expected equilibrium verification cost is minimum if the entrepreneur designs the contract, and maximum if the investor designs the contract. What Proposition 4 is pointing at is the normative aspect of contract design. In the presence of private information which is costly to verify, the proposition sheds light on how the bargaining initiative should be allocated to minimize welfare loss, or verification cost. That is, the contract should be designed by the informed party. An additional element that would strengthen the validity of the proposition might be the cost of communicating private information. Nevertheless, it is to be noted that both the contract designed by the informed entrepreneur and the one designed by the uninformed investor are optimal given the fixed allocation of bargaining initiative unless some form of side payment is possible from the entrepreneur to the investor. As the bargaining initiative shifts from the investor to the entrepreneur, the one gets strictly worse-off while the other gets strictly better-off so that a Pareto comparison of the two arrangements is not possible. Should such a side payment be possible, however, then we should expect the contract designed by the entrepreneur to dominate the one designed by the investor. Suppose the initial bargaining initiative lies with the investor. The entrepreneur can always buy out the bargaining initiative of the investor with a side payment that would leave the investor at least as better-off as when she had the initiative. At the same time, the entrepreneur can make himself strictly better-off with the decreased verification cost. While the general message of Proposition 4 is unambiguous as regards the normative aspect of allocating bargaining initiative, the practice followed in reality does not appear to favor any single arrangement: (informed) firms issue debentures, not the investors; (uninformed) banks offer loan contracts, not the debtors. Admittedly, the key (and the only) feature of the present model, namely, the ex post private information, is just part of a whole list of elements that needs to be considered for more satisfactory analyses of institutional arrangements. Among them, for instance, is contracting cost. If firms can reduce contracting cost through gains from standardization by addressing many potential investors, the result of the paper will only strengthen the case for why firms issue debentures, not the other way around. The same argument may be used to establish the case for different arrangements. For example, the reason why uninformed banks offer
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loan contracts to many potential borrowers could be that the reduction in the contracting cost exceeds the increase in the monitoring cost. One may argue similarly for why tax authorities design tax schedules, not the taxpayers. Also, in the case of bank loans, as Wang and Williamson (1993) argue, the screening of ex ante private information may be of greater significance than the verification of ex post private information. It is by no means obvious if the inclusion of elements such as the presence of ex ante private information or contracting cost will lead to an unambiguous conclusion. 4. Summary and conclusions In this paper, the signalling subgame in costly verification models has been studied in the context of the entrepreneur±investor contract without assuming the possibility of commitment to verification. It was shown that the game has a unique Perfect Bayesian Equilibrium in mixed strategies, implying that truth-telling and deterministic verification are not equilibrium behavior, thereby questioning the popularity of debt contracts. Presumably, the widespread use of the debt contract is due to its simplicity, a formalization of which may well involve such elements as contracting costs. When the entire game starting from the contract design was considered, it was shown that the contract designed by the informed entrepreneur results in less verification cost than the one designed by the uninformed investor. This could be taken as a normative criterion based upon which institutional arrangements as regards the distribution of bargaining initiative are to be designed, although actual practice seems to provide mixed evidence. While the basic model in this paper is highly simplified, it can be extended to include such elements as collateral. Suppose the entrepreneur has collateral valued at >0 which will be confiscated by the investor if the actual return is xl or if it is xh but reported to be xl which is not verified. Provided that the collateral value is not large enough, one can solve the game again to find equilibria that are qualitatively the same as those obtained in the paper. As the payoffs for the entrepreneur with xxl now decrease uniformly by , the equilibrium strategy for the entrepreneur in that part of the game is still truth-telling. Given that, the equilibrium strategy for the investor upon receiving the report xh is not to verify the report. At type (3) equilibrium, we now have I(xl)(r2ÿ ÿ)/(r3ÿ ), E(xh)(pr3ÿp ÿc)/(pr3ÿp ÿpc) for which we need p>c/(r3ÿ ). Naturally the inclusion of collateral increases the probability of truth-telling and decreases the probability of verification. Based on this equilibrium, one can show that the results of Section 3 are still valid. As the investment technology of this paper is fixed with a positive net present value, the discussion of efficiency could be made based only on expected equilibrium verification costs. With more general models where levels of investment can also be chosen, it is needless to say that a proper welfare criterion should be net expected returns from the investment less the expected verification costs, rather than the expected verification costs alone. This would be a worthy extension of the present paper. Among a number of other extensions of the model, the exploration into whether the results of Section 3 will be robust under the environment with general adverse selection and moral hazard problems seems not only interesting but promising.
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Acknowledgements I would like to thank Peter Bardsley, seminar participants at La Trobe, New South Wales and Monash Universities, and participants at the Congress of the European Economic Association, Istanbul, August 1996, for helpful discussions. Two anonymous referees provided very helpful and constructive comments. Financial support of research grant from the School of Economics and Commerce, La Trobe University is gratefully acknowledged. Any remaining errors are my own. Appendix A Derivation of equilibria for the verification game Consider first the entrepreneur's problem. When the actual outcome is xh, the entrepreneur solves the following problem: MaxE xh E xh I xh xh ÿ r1 1 ÿ I xh xh ÿ r2 1 ÿ E xh I xl xh ÿ r3 ÿ 1 ÿ I xl xh ÿ : The solution to the above problem is E xh 1 if I xh r1 1 ÿ I xh r2 < I xl r3 ; 2 0; 1 if I xh r1 1 ÿ I xh r2 I xl r3 ; 0 if I xh r1 1 ÿ I xh r2 > I xl r3 :
(A)
When the actual outcome is xl, similar steps lead us to conclude that E(xl)0. As shown in Section 2, the beliefs of the investor then become a1, bp(1ÿE(xh))/ (p(1ÿE(xh))(1ÿp)). Next I look at the investor's problem. At information set A, the investor's problem is MaxI xh a I xh r1 ÿ c 1 ÿ I xh r2 1 ÿ a I xh ÿc 1 ÿ I xh r8 : Since a1 and r2r1>r1ÿc, we must have I(xh)0. The investor's problem at information set B is MaxI xl b I xl r3 ÿ c 1 ÿ b I xl ÿc: The solution to this problem is I xl 1 if b >
c c c ; 2 0; 1 if b ; 0 if b < : r3 r3 r3
(B)
Three cases in (B) will be considered separately for the derivation of equilibria. Case 1 b>c/r3: Then I(xl)1. We also know I(xh)0. From (A), then, we have I(xh)r1 (1ÿI(xh))r2r2<I(xl)r3r3 since r3 r2, leading us to conclude that E(xh)1. Then b0 which contradicts b>c/r3. Thus equilibrium does not exist in this case.
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Case 2 bc/r3: Then I(xl)2[0, 1]. I will consider the three cases in (A) separately utilizing the result, I(xh)0. Case 2.1 r2ÿ<I(xl)r3: Then E(xh)1, in which case b0 contradicting bc/r360. Thus equilibrium does not exist in this case. Case 2.2 r2ÿI(xl) r3 or I(xl)(r2ÿ)/r32(0,1): Then E(xh)2(0,1). From bc/ r3p(1ÿE(xh))/(p(1ÿE(xh))(1ÿp)), we obtain E(xh)(pr3ÿc)/(pr3ÿpc). For this probability to be meaningful, we need p>c/r3. Thus if p>c/r3, we have an equilibrium, I(xl)(r2ÿ)/r3, E(xh)(pr3ÿc)/(pr3ÿpc), I(xh)E(xl)0, a1, and bc/r3. Case 2.3 r2ÿ>I(xl)r3: Then E(xh)0, hence bp. Thus an equilibrium in this case is I(xl)<(r2ÿ)/r3, I(xh)E(xh)E(xl)0, a1, bpc/r3. Case 3 b
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