Comparing Allocations under Asymmetric Information: Coase Theorem Revisited

Comparing Allocations under Asymmetric Information: Coase Theorem Revisited Shingo Ishiguro Graduate School of Economics, Osaka University 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan August 2002 Abstract This paper investigates the robustness of Coase Theorem under asymmetric information. We identify the conditions under which the same allocation is attained as an equilibrium of a bilateral trade no matter which informed or uninformed party has the bargaining power to make a contract offer. Keywords: Asymmetric Information, Coase Theorem, Principal Agent Relationship, Signaling Games JEL Classification Numbers: D80, D82 Corresponding Author. Tel: +81-6-6850-5220, Fax: +81-6-6850-5256. E-mail Address: ishiguro@econ.osaka-u.ac.jp 1

1 Introduction Coase Theorem has been often referred to as the result that allocative efficiency of a transaction among individuals with quasi linear preferences is not affected by their bargaining powers. In this paper we investigate whether or not Coase Theorem can be extended to the environments with asymmetric information. In particular we consider a bilateral trade model where an agent has a private information (his type ) which affects both his own and the principal s benefits. Then we compare the equilibrium allocation attained in the game (called P game) in which an uninformed party (a principal) has the bargaining power to make a contract offer with that in the game (called A game) in which an informed party (an agent) has the bargaining power to do so. We say that Coase Theorem holds in this environment if the same allocation (and hence the same ex ante efficiency) is implemented as an equilibrium in both P game and A game. Since the informed party (the agent) makes a contract offer in A game, this game is known as the informed principal problem (See Maskin and Tirole (1992) for a general treatment on this issue.) Recently several papers have also examined the informed principal problem in various contractual settings. 1 However, our main focus is to compare equilibrium allocations between signaling and screening models and identify the conditions under which these coincide with each other, while the existing papers have paid little attention to such issues. 2 Our main finding is as follows: (i) The equilibrium allocation in P game can be always attained as a perfect Bayesian equilibrium (PBE) in A game when there exist some bad types of agent who never contributes to the increase of the gains from trade as compared to status quo. Put differently, the attainable efficiency (ex ante total payoffs of the principal and agent) is same in both P game and A game. Thus in this case Coase Theorem weakly holds even under asymmetric information. 3 (ii) However, there exist no PBEs in A game which attain the same equilibrium allocation in P game when the following conditions are satisfied: First, the second best allocation is distorted at the worst type agent who yields the lowest value to the principal. Second, no bunching (pooling) occurs at that type in the second best optimum as well. We also show that the former condition is satisfied under fairly general conditions (for example when the type set is finite and action is a continuous variable.) The conditions imposed in 1 See for example Chade and Silvers (2002), Inderst (2001, 2002) and Jost (1996). 2 Although Maskin and Tirole (1992) discuss the similar problem (see their Proposition 12 and 13), their model differs from ours: They assume that more than two uninformed parties compete to offer contracts in the screening model. This Bertrand competition essentially gives the informed party the bargaining power even in the screening model, which is in contrast to our bilateral monopoly model. 3 We here use the term weakly because there may exist other PBEs in A game which does not attain the same allocation in P game. 2

the standard mechanism design problems (so called sorting condition and monotone hazard rate condition) also results in the downward distortion and no bunching, which covers the case for our second result (ii) to hold. Therefore, we conclude that whether or not Coase Theorem holds even under asymmetric information depends on whether or not there exist some inefficient types who should be excluded from transaction. 2 The Model We will consider a bilateral trade model in which a risk neutral principal contracts with a risk neutral agent for implementing some profitable project. The principal has a quasi linear utility function, V (a; θ) t, where a A R n is an action (vector) chosen by the agent, t R a monetary transfer made from the principal to the agent, and θ Θ R m a type of agent respectively. Note that we allow both multidimensional action choices and types. V (a; θ) is assumed to be strictly concave with respect to a. The agent also has a quasi linear utility function, t ψ(a; θ), where ψ(a; θ) denotes the cost of choosing an action a. ψ(a; θ) is assumed to be strictly convex with respect to a. The reservation payoffs of both parties are normalized to zero. For simplicity, we will also assume that there exists an action a A such that V (a; θ) =ψ(a; θ) = 0 for all θ Θ. This means that there exists the contract, which specifies zero transfer and the action a, to guarantee the reservation payoffs to the contracting parties. 4 The action a can be thought of as the status quo action in that it gives the parties the status quo payoffs, zero. The agent knows his type θ before contracting but the principal does not. Let P 0 (θ) denote the prior belief held by the principal that agent is of type θ. An allocation is defined as a mapping µ :Θ (A) which specifies for each type a probability distribution on the action set A where (A) is the set of probability distributions on A. Note that we allow random allocation. We will compare two different games depending on the bargaining powers to make a take it or leave it contract offer. One is the game called P game in which the principal (uninformed party) proposes a contract and the other is the game called A game in which the agent (informed party) does so. In P game we can apply the revelation principle and use the direct revelation mechanism without loss of generality. Let C A R denote the set of possible actions and transfers. Since the payoff functions of both the principal and agent are linear with respect to monetary transfer t, random transfer can be always replaced by nonrandom 4 More generally, we may use an alternative formulation by adding the null contract, which specifies nothing and hence ensures the reservation payoffs, to the set of possible contracts. To save notation, we will not adopt such approach. 3

transfer without loss of generality. 5 Let t : Θ R be a nonrandom transfer schedule. We will then consider stochastic contracts only on the part of action choice a. P game has the following timing: P game: 1. The principal offers a contract C {µ, t}, which consists of (possibly random) allocation mapping µ and transfer schedule t. Let C(θ) {µ θ,t(θ)} also denote the incentive scheme designed for type θ agent where µ θ (A). 2. The agent decides whether to accept this contract or not. 3. When accepted it, the contract is executed. On the other hand, A game proceeds with the following timing: A game: 1. The agent of type θ offers a contract C (A) R to the principal. The contract proposed by the agent also consists of (stochastic) action choice µ (A) and a transfer t R. 2. The principal decides whether to accept it or not. 3. When accepted it, the contract is executed. We first consider the full information case that the principal also knows about θ at the contracting stage. Moreover, under our assumptions, random allocation is not optimal in this first best case. Then so called Coase Theorem holds, i.e., the first best allocation a FB (θ) does not depend on which party has the bargaining power to make a contract offer. a FB (θ) is defined as a FB (θ) arg max V (a; θ) ψ(a; θ). (1) a A Note that a FB (θ) is uniquely determined. Let Π FB (θ) V (a FB (θ); θ) ψ(a FB (θ); θ) denote the first best payoff. 3 Equilibrium Allocations 3.1 Equilibrium in P Game As we noted, the equilibrium allocation in P game can be attained by using a direct revelation mechanism C(θ) ={µ θ,t(θ)} θ Θ. With slight abuse of notation, we define V (µ θ ; θ) V (a; θ)dµ θ (2) A 5 Indeed a random transfer can be replaced by its expected value without affecting the results. 4

and ψ(µ θ ; θ) A ψ(a; θ)dµ θ. (3) The optimization program the principal should solve at the first stage is given as follows: (P) max E[V (µ θ ; θ) t(θ)] C subject to θ arg max t(ˆθ) ψ(µˆθ; θ) for all θ Θ (I C) ˆθ t(θ) ψ(µ θ ; θ) 0 for all θ Θ (IR) where expectation E[ ] is taken over Θ. (IC) means the incentive compatibility constraint that type θ agent is induced to tell the truth. (IR) means the individual rationality constraint that each type obtains at least the reservation payoff, zero. 6 Let C (θ) {µ θ,t (θ)} be the optimal contract designed for type θ agent. Let µ also denote the equilibrium allocation attained in P game. 3.2 Equilibrium in A Game The contract proposal made by the agent becomes a signal of sending his type in A game, which is in contrast to P game. Thus we will confine our attention to the perfect Bayesian equilibrium (PBE) as follows: (i) Sequential Rationality: Each type of agent optimally offers a contract C to maximize his expected payoff. The principal optimally makes the acceptance decision, given the strategy taken by the agent and her belief about the agent s type. (ii) Belief Consistency: The principal s belief is updated according to the Bayes rule, given the above agent s strategy, whenever it is possible. Let P (θ C) denote a posterior belief of the principal after a contract C {µ, t} is offered by the agent. Then the principal chooses to accept the contract C if and only if [V (a; θ) t]dµdp (θ C) 0. (4) Θ A Let m(c) [0, 1] be a probability of the principal accepting the offered contract C, given her posterior belief P (θ C). The agent of type θ will then choose a contract C = {µ, t} to maximize the expected payoff m(c)[t ψ(µ; θ)]. (5) 6 Note that in this definition we assume all types participate in the mechanism. However, no generality is lost by this condition because when the participation of some type is not profitable for the principal she can always choose the status quo action a and zero transfer t = 0 for that type, which guarantee both herself and the agent the reservation payoffs, zero. 5

4 Comparing Allocations under Asymmetric Information Now we will address the issue of whether or not the equilibrium allocation in P game µ can be sustained as a PBE in A game as well. We define the exclusion set as ES {θ Θ V (a; θ) ψ(a; θ) < 0 for all a a}. (6) ES is the set of agent s types who never contribute to the increase of the gains from trade as compared to the status quo. Of course, if ES, we have a FB (θ) =a for all θ ES. Then we can show the following result. Proposition 1. Suppose that ES. Then there exists a PBE in A game which attains the equilibrium allocation µ in P game. Proof. Consider the following strategies and belief: The agent of type θ proposes the contract C (θ). The principal accepts this with certainty. The principal has the posterior belief P ( C (θ)) when she is offered a contract C (θ) as follows: P (θ C (θ)) = 0 for all θ Θ \{θ Θ C (θ) =C (θ )} and P (θ C (θ)) > 0 for all other types. Moreover, the principal puts all positive weights on ES when she observes all other contracts C than {C (θ)}, i.e., P (ES C) = 1 for all C/ C {C C C = C (θ) for some θ}. It is obvious that the principal optimally accepts all contracts C C because by definition of optimal contract C we have V (µ θ ; θ) t (θ) 0 for any θ. Since C satisfies (IC), type θ agent will also offer the contract C (θ) rather than C (θ ) for θ θ, given the principal s above strategy. Suppose now that some type θ offers a contract C = {µ, t} / C. For such deviation to be profitable, t ψ(µ; θ) > 0 must be satisfied. However then V (µ; θ) t< V (µ; θ) ψ(µ; θ) < 0 for all θ ES. Given her belief that P (ES C) =1 for all C / C, the principal will then reject such deviation offer. Thus the agent cannot gain by the deviation. Q.E.D. Proposition 1 states that Coase Theorem weakly holds under asymmetric information in the sense that the same allocation can be an equilibrium, no matter how bargaining power to make a contract offer is allocated to an informed party or an uninformed party. Thus the same ex ante efficiency can be attained in both P game and A game. The intuition for this result to hold is simple: The original (IC) means that each type does not mimic the other types. Thus in order to sustain µ as a PBE in A game it is sufficient to construct off the path belief for observing other contracts C/ {C (θ)} θ Θ. If ES, this is easily done by putting all positive probabilities on the exclusion set ES when having observed such deviation contracts. 6

The condition stated in Proposition 1 means that some inefficient types are excluded from transaction. If this condition does not hold, Coase Theorem may not be extended to the cases of asymmetric information. Such issue will be addressed in the following proposition. To this end, we assume the existence of the worst type of agent, denoted θ, which satisfies V (a; θ) V (a; θ) for all θ θ and a A. This condition is satisfied when V (a; ) is monotone with respect to θ. Proposition 2. Suppose that µ θ µ θ for any θ θ and also that µ θ does not put mass one to a FB (θ). Then there exist no PBEs in A game which attain the equilibrium allocation µ in P game. Proof. Since µ θ does not put mass one to afb (θ), a FB (θ) a must hold. This is because, if a FB (θ) =a in the first best, then this must be also the case in the second best, 7 which contradicts the stated condition. Suppose, contrary to the claim, that there exists a PBE in which the allocation µ can be attained in A game. Then type θ agent must offer the contract Ĉ(θ) {µ θ, ˆt(θ)} in that PBE, which attains the allocation µ. In particular there must exist no θ θ such that Ĉ(θ) =Ĉ(θ), because µ θ µ θ by assumption. Now consider the deviation by type θ agent such that C {a FB (θ),t } is offered where t satisfies V (a FB (θ); θ) t >V(µ θ; θ) ˆt(θ) (7) t ψ(a FB (θ); θ) > ˆt(θ) ψ(µ θ; θ) (8) where the right hand sides of (7) and (8) represent the equilibrium payoffs of the principal and type θ agent respectively when Ĉ(θ) is offered. Note here that the principal s posterior belief P ( Ĉ(θ)) after Ĉ(θ) was offered must be P (θ Ĉ(θ)) = 1 in the PBE because Ĉ(θ) Ĉ(θ) for all θ θ. In fact such transfer t exists because µ θ does not put mass one to afb (θ) by assumption and hence this implies V (a FB (θ); θ) ψ(a FB (θ); θ) >V(µ θ; θ) ψ(µ θ; θ). (9) Moreover, since V (µ θ ; θ) ψ(µ θ ; θ) 0, the principal will accept the contract C with certainty: E[V (a FB (θ); θ) C ] t V (a FB (θ); θ) t >V(µ θ; θ) ψ(µ θ; θ) 0 7 Since V (µ θ; θ) t (θ) V (µ θ; θ) ψ(µ θ; θ) V (a FB (θ); θ) ψ(a FB (θ); θ) =V (a; θ) ψ(a; θ) = 0, we must have µ θ({a}) = 1 and t (θ) = 0 due to the optimality of C (θ). 7

where E[ C ] denotes the expectation over Θ conditional on the deviation offer C. Here the first inequality follows from the assumption that V (a; θ) V (a; θ) for all θ θ and a A. This argument shows that the worst type agent θ will deviate from the equilibrium contract C (θ) toc, anticipating that the principal surely accepts such contract, which is a contradiction. Q.E.D. Proposition 2 states that Coase Theorem may not be extended to the case of asymmetric information when the first best action for the worst type is not implemented in the second best. This is because if some distortion arises at the bottom (the worst type) in the allocation µ the worst type always has the incentive to offer the contract which includes the first best action a FB (θ) in A game: Such contract offer attracts the principal and hence no PBEs exist for µ to be achieved. All the conditions stated in Proposition 2 are satisfied in the standard mechanism design problems which assume so called Sorting Condition 8 and monotone hazard rate condition are satisfied. These conditions result in the monotonicity of action choice and its downward distortion except for the most efficient type. However, even when these standard conditions are not met, we will show that some distortion always arises at the worst type when the following certain conditions hold. Assumption 1. (i) Θ is finite. (ii) P 0 (θ) > 0 for some θ θ. (iii) A is a closed interval in R. (iv) V (a; θ) and ψ(a; θ) are differentiable with respect to a A Assumption 2. (i) ψ(a; θ) ψ(a; θ) for all θ θ and a A. (ii) ψ a (a FB (θ); θ) >ψ a (a FB (θ); θ) for θ θ. Assumption 1 (i) and (ii) say that the type set is finite and prior belief puts some positive weight on other types than the worst one. The remaining conditions of Assumption 1 mean that we restrict attention to the smooth environment. Assumption 2 simply means that type θ agent is the worst in the sense that his action cost is the highest as well as he gives the principal the lowest value. Assumption 2 (ii) is weaker than the standard sorting condition, which requires ψ a is monotone with θ for all a. Proposition 3. Suppose that a FB (θ) a and Assumption 1 2 are satisfied. Then the optimal contract which solves the problem (P) does not induce the worst type θ agent to choose the first best action a FB (θ) with certainty. 8 Roughly speaking this condition ensures that the indifference curves of different type agents cross each other only once in the plane of action and transfer. 8

Proof. Suppose contrary to the claim that µ θ puts mass one to afb (θ). Since the following inequalities must be satisfied: t (θ) ψ(µ θ; θ) t (θ) ψ(µ θ; θ) t (θ) ψ(µ θ; θ) 0 due to (IC), Assumption 2 (i) and (IR), the relevant (IR) is only for the worst type θ. Thus (IR) must be binding at θ because of the optimality of µ θ, i.e., t (θ) =ψ(a FB (θ); θ). Now consider the change of the optimal contract as follows: Pick a contract C {a,t } such that t ψ(a ; θ) =t (θ) ψ(a FB (θ); θ) = 0, with a a FB (θ) ɛ for small ɛ>0, and offer such contract to the agent who reported he is of the worst type θ. Moreover, the contracts offered to other types θ θ are modified as C ɛ (θ) {µ θ,t (θ) αɛ} where α>0 and ɛ>0 is sufficiently small (See below for the specification of α). Note that all types θ θ do not mimic the other types θ θ because transfer is only modified for those types by subtracting the same constant αɛ. The worst type also does not have the incentive to mimic other types because t ψ(a ; θ) = 0 t (θ) ψ(µ θ; θ) > t (θ) αɛ ψ(µ θ ; θ) where the first inequality follows from the original (IC). Then we will check that every type θ θ also has no incentive to mimic the worst type θ. This constraint can be written by t (θ) αɛ ψ(µ θ ; θ) t ψ(a ; θ) = ψ(a ; θ) ψ(a ; θ) = ψ(a FB (θ) ɛ; θ) ψ(a FB (θ) ɛ; θ). Let L(ɛ) and R(ɛ) denote the first left hand and last right hand expressions of the above inequalities respectively. By the original (IC), we have L(0) R(0). Moreover, L (ɛ) = α and R (0) = ψ a (a FB (θ); θ) ψ a (a FB (θ); θ) < 0 by Assumption 2 (ii). Thus the above incentive compatibility constraint will be satisfied by taking small ɛ and α<min {ψ a(a FB (θ); θ) ψ a (a FB (θ); θ)}. θ θ Finally we show that the above modified contract improves the principal s expected payoff, which contradicts the optimality of {C (θ)} θ Θ. Under the modified contract, the principal obtains the following expected payoff Π(ɛ) θ θ P 0 (θ)[v (µ θ ; θ) t (θ)+αɛ]+p 0 (θ)[v (a FB (θ) ɛ; θ) ψ(a FB (θ) ɛ; θ)]. 9

Differentiating this with respect to ɛ and taking ɛ +0, we obtain Π (0) = α θ θ P 0 (θ) P 0 (θ)[v a (a FB (θ); θ) ψ a (a FB (θ); θ)]. Since the second term in the above expression becomes zero due to the optimality of a FB (θ) and its first order condition, we have Π (0) = α θ θ P 0 (θ) > 0 by Assumption 1 (ii), which is the desired contradiction. Q.E.D. Combining Proposition 2 and 3, we conclude that the equilibrium allocation µ in P game cannot be attained as a PBE in A game under fairly general conditions (Assumption 1 and 2), when the worst type is productive in the sense that a FB (θ) a and no bunching occurs at that type in the second best optimum. References [1] Chade, H., Silvers, R., 2002. Informed Principal, Moral Hazard, and the Value of a More Informative Technology, Economics Letters 74, 291 300. [2] Inderst, R., 2002. Contractual Signaling in a Market Environment, Games and Economic Behavior 40, 77 98. [3] Inderst, R., 2001. Incentive Schemes As a Signaling Device, Journal of Economic Behavior and Organization 44, 455 465. [4] Jost, P-J., 1996. On the Role of Commitment in a Principal Agent Relationship with an Informed Principal, Journal of Economic Theory 68, 519 530. [5] Maskin, E., Tirole, J., 1992. The Principal Agent Relationship with an Informed Principal II: Common Values, Econometrica 60, 1 42. 10