Comparing allocations under asymmetric information: Coase Theorem revisited

Transcription

1 Economics Letters 80 (2003) locate/ econbase Comparing allocations nder asymmetric information: Coase Theorem revisited Shingo Ishigro* Gradate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka , Japan Received 16 September 2002 accepted 10 December 2002 bstract This paper investigates the robstness of Coase Theorem nder asymmetric information. We identify the conditions nder which the same allocation is attained as an eqilibrim in a bilateral trade model no matter which informed or ninformed party has the bargaining power to make a contract offer Elsevier Science B.V. ll rights reserved. Keywords: symmetric information Coase Theorem Principal agent relationship Signaling games JEL classification: D80 D82 1. Introdction Coase Theorem has been often referred to as the reslt that allocative efficiency of a transaction among individals 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 particlar 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 eqilibrim allocation attained in the game (called P-game) in which an ninformed party (a principal) has the bargaining power to make a contract offer with that in the game (called -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 *Tel.: fax: address: ishigro@econ.osaka-.ac.jp (S. Ishigro) /03/$ see front matter 2003 Elsevier Science B.V. ll rights reserved. doi: / S (03)

2 68 S. Ishigro / Economics Letters 80 (2003) allocation (and hence the same ex ante efficiency) is implemented as an eqilibrim in both P-game 1 (screening game) and -game (signaling game). Or main finding is as follows: (i) the eqilibrim allocation in the P-game can be always attained as a perfect Bayesian eqilibrim (PBE) in the -game when there exist some bad types of agent who never contribte to the gains from trade as compared to stats qo. Ths in this case Coase 2 Theorem weakly holds nder asymmetric information. (ii) However, there exist no PBEs in the -game which attain the same eqilibrim allocation in the P-game when the following conditions are satisfied: firstly, the second best allocation is distorted at the worst type agent who yields the lowest vale to the principal. Secondly, no bnching (pooling) occrs at that type in the second best optimm as well. The conditions imposed in the standard mechanism design problems (so-called sorting condition and monotone hazard rate condition) cover the case for or second reslt (ii) to hold. 2. The model We will consider a bilateral trade model in which a risk netral principal contracts with a risk n netral agent. The principal has a qasi-linear tility fnction, V(a ) 2 t, where a [, R is an action (vector) chosen by the agent, t [ R a monetary transfer made from the principal to the agent, m and [ Q, R a type of agent, respectively. The agent has a qasi-linear tility fnction, t 2 c(a ), where c(a ) denotes the cost of choosing an action a. The reservation payoffs of both parties are normalized to zero. For simplicity, we will also assme that there exists an action a [ sch that V(a ) 5 c(a ) 5 0 for all [ Q. This means that there exists the contract, which specifies zero transfer and the action a, to garantee the reservation payoffs to the contracting parties. We also assme that c(a ) $ 0 for all a [ and all [ Q. The agent knows his type before contracting bt the 0 principal does not. Let P ( ) denote the prior belief held by the principal that agent is of type. n allocation is defined as a mapping m: Q D() which specifies for each type a probability distribtion on the action set where D() is the set of probability distribtions on. Note that we allow random allocation. Let also C 3 R denote the set of possible actions and transfers. We will 3 consider only non-random transfer schedle, denoted t: Q R. We will compare two different games depending on the bargaining powers to make a take-it-orleave-it contract offer. One is the game called the P-game which proceeds with the following timing: (1) the principal offers a contract C hm, tj, which consists of (possibly random) allocation mapping m and transfer schedle t. Let C( ) hm, t( )j also denote the incentive scheme designed for type agent where m [ D(). (2) The agent decides whether to accept this contract or not. (3) When 1 See Chade and Silvers (2002), Inderst (2001) and Jost (1996) for several isses in the informed principal models. lthogh Maskin and Tirole (1992) discss a similar problem (see their Proposition 12 and 13), their model differs from ors: they assme that more than two ninformed 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 or bilateral monopoly model. 2 We here se the term weakly becase there may exist other PBEs in the -game which do not attain the same allocation in the P-game. 3 Since the payoff fnctions of both the principal and agent are linear with respect to monetary transfer t, random transfer can be always replaced by nonrandom transfer withot loss of generality.

3 S. Ishigro / Economics Letters 80 (2003) accepted, the contract is exected. The other is the game called the -game which has the following timing: (1) the agent of type offers a contract C [ D() 3 R to the principal. The contract proposed by the agent also consists of (stochastic) action choice m [ D() and a transfer t [ R. (2) The principal decides whether to accept it or not. (3) When accepted, the contract is exected. In the P-game we can apply the revelation principle and se the direct revelation mechanism withot loss of generality. With slight abse of notation we define V(m ) E V(a )dm and c(m ) E c(a )dm. (1) In the first stage the principal chooses a contract C to maximize her expected payoff E[V(m ) 2 t( ) sbject to the standard incentive compatibility (IC) and individal rationality (IR) 4 constraints: [ arg max t( ˆ ) 2 c(m ) for all [ Q, (IC) û t( ) 2 c(m ) $ 0 for all [ Q. û Let C*( ) hm*, t*( )j be the optimal contract designed for type agent. Let m* also denote the eqilibrim allocation attained in the P-game. In the -game, the contract proposal made by the agent becomes a signal of sending his type in the -game, which is in contrast to the P-game. Let P( C) denote a posterior belief of the principal after a contract C hm, tj is offered by the agent. Then the principal decides to accept the contract C if and only if eq e [V(a ) 2 t dm dp ( C) $ 0. Let m(c) [ [0, 1 be a probability of the principal accepting the offered contract C, given her posterior belief P( C). Then in the first stage of the -game, agent of type will choose a contract C 5 hm, tj to maximize the expected payoff m(c)[t 2 c(m ). (IR) 3. Comparing allocations nder asymmetric information Now we will address the isse of whether or not the eqilibrim allocation in the P-game m* can be sstained as a PBE in the -game. We define the exclsion set, which is the set of agent s types who never contribte to the gains from trade as compared to the stats qo, as follows: ES h [ QV(a ) # 0 for all a ±aj. (2) Then we can show that the same allocation can be attained as an eqilibrim in both games: Coase Theorem is still valid even nder asymmetric information. 4 Note that in this definition we assme all types participate in the mechanism. However, no generality is lost by this condition becase when the participation of some type is not profitable for the principal she can always choose the stats qo action a and zero transfer t50 for that type, which garantee both herself and the agent the reservation payoffs, zero.

4 70 S. Ishigro / Economics Letters 80 (2003) Proposition 1. Sppose that ES ± 5. Then there exists a PBE in the -game which attains the eqilibrim allocation m* in P-game. Proof. Consider the following strategies and belief: the agent of type proposes the contract C*( ). The principal accepts this with certainty. The principal has the posterior belief P(? C*( )) when she is offered a contract C*( ) as follows: P( 9C*( )) 5 0 for all 9 [ Q\h 0 [ QC*( ) 5 C*( 0)j and P( 9C*( )). 0 for all other types. Moreover, the principal pts all positive weights on ES when she observes all other contracts C than hc*( )j, i.e. P(ESC) 5 1 for all C [ C hc [ D() 3 RC 5 C*( ) for some j. It is then obvios that the principal optimally accepts all contracts C [ C becase by definition V(m * ) 2 t*( ) $ 0 for any. Since C* satisfies (IC), type agent will also offer the contract C*( ) rather than C*( 9) for 9 ±, given the principal s above strategy. Sppose now that some type offers a contract C 5 hm, tj [ C. For sch deviation to be profitable, t 2 c(m ). 0 mst be satisfied. However, then V(m ) 2 t, 0 for all [ ES and all m [ D(). Given her belief that P(ESC) 5 1 for all C [ C, the principal will then reject sch deviation offer. Ths the agent cannot gain by the deviation. h The above reslt follows from the constrction of an off-the-path belief for observing other contracts C [ hc*( )j [Q in the -game. If ES 5 5, this is easily done by ptting all positive probabilities on the exclsion set ES when having observed sch deviation contracts. However, if this condition does not hold, Coase Theorem may not be extended to the case of asymmetric information. We now assme the existence of the worst type of agent, denoted, in the sense that V(a ) $V(a ) for all ± and a [. Let a ( ) also denote the first best action to maximize V(a ) 2 c(a ). Then we show the following. Proposition 2. Sppose that m * ± m* for any ± and also that m* does not pt mass one to a ( ). Then there exist no PBEs in the -game which attain the eqilibrim allocation m* in the P-game. Proof. Since m* does not pt mass one to a ( ), a ( ) ±a mst hold. This is becase, if a ( ) 5a 5 in the first best, then this mst be also the case in the second best, which contradicts the stated condition. Sppose, contrary to the claim, that there exists a PBE in which the allocation m* can be attained in the -game. Then type agent mst offer the contract ˆ C( ) hm*, ˆ t( )j in that PBE, which attains the allocation m*. In particlar there mst exist no ± sch that ˆ C( ) 5 ˆ C( ), becase m * ± m* by assmption. Now consider the deviation by type agent sch that C9 ha ( ), t9j is offered where t9 satisfies V(a ( ) ) 2 t9.v(m * ) 2 ˆ t( ), (3) t92c(a ( ) ). ˆ t( ) 2 c(m * ). (4) The right hand sides of (3) and (4) represent the eqilibrim payoffs of the principal and type agent, 5 Since V(m * ) 2 t*( ) #V(m* ) 2 c(m * ) #V(a ( ) ) 2 c(a ( ) ) 5V(a ) 2 c(a ) 5 0, we mst have m * (haj) 5 1 and t*( ) 5 0 de to the optimality of C*( ).

5 S. Ishigro / Economics Letters 80 (2003) respectively, when ˆ C( ) is offered. Note here that the principal s posterior belief P(? C( ˆ )) after ˆ C( ) was offered mst be P( C( ˆ )) 5 1 in the PBE becase ˆ C( ) ± ˆ C( ) for all ±. In fact sch transfer t9 exists becase m* does not pt mass one to a ( ) by assmption and hence this implies V(a ( ) ) 2 c(a ( ) ).V(m* ) 2 c(m * ). (5) Moreover, since V(m * ) 2 c(m * ) $ 0, the principal will accept the contract C9 with certainty: E[V(a ( ) )C9 2 t9 $V(a ( ) ) 2 t9.v(m* ) 2 ˆ t( ) $ 0 where E[? C9 denotes the expectation over Q conditional on the deviation offer C9. Here the first ineqality follows from the assmption that V(a ) $V(a ) for all ± and a [. This argment shows that the worst type agent will deviate from the eqilibrim contract C*( ) toc9, anticipating that the principal srely accepts sch contract, which is a contradiction. h Proposition 2 states that eqilibrim allocations of both games cannot coincide with each other when the first best action for the worst type is not implemented in the second best. This is becase if some distortion arises at the bottom (the worst type) in the allocation m* the worst type always has the incentive to offer the contract which incldes the first best action a ( ) in -game: sch contract offer attracts the principal and hence no PBEs exist for m* to be achieved. ll the conditions stated in Proposition 2 are satisfied in the standard mechanism design problems which assme so-called 6 Sorting Condition and monotone hazard rate condition. We ths conclde that whether or not Coase Theorem can be extended to the case of asymmetric information depends on whether or not there exist some inefficient types of agent who shold be exclded from transactions. References Chade, H., Silvers, R., Informed principal, moral hazard, and the vale of a more informative technology. Economics Letters 74, Inderst, R., Incentive schemes as a signaling device. Jornal of Economic Behavior and Organization 44, Jost, P.-J., On the role of commitment in a principal agent relationship with an informed principal. Jornal of Economic Theory 68, Maskin, E., Tirole, J., The principal agent relationship with an informed principal II: common vales. Econometrica 60, Roghly speaking this condition ensres that the indifference crves of different type agents cross each other only once in the plane of action and transfer.