Contracts under Asymmetric Information 1 I Aristotle, economy - PDF document

Contracts under Asymmetric Information 1

I Aristotle, economy (oiko and nemo) and the idea of exchange values , subsequently adapted by Ricardo and Marx. Classical economists. An economy consists of a set of agents each of whom is characterized by her preferences and her initial endowments (resources). Walras and Pareto, neo-classical economists and the emergence of rigorous economics equi- librium, von-Neumann, Arrow, Debreu, McKen- zie, Nash, Aumann. Walrasian equilib- rium, competitive equilibrium and per- fect competition. Existence and Optimality of equilibrium. Book: Aliprantis et al. Uncertainty and the state contingent trade 2

model. II Asymmetric or Differential Informa- tion Economies. Walrasian Expectations equilibrium (WEE), and Rational Expectations Equilibrium (REE), Radner, Lucas, Prescott. What is the best possible contact we can reach when agents are asymmetrically informed? 1. Individual rationality (better off) 2. Efficient 3. Incentive Compatible 4. Existence 5. Implementable as a PBE of an extensive form graph (game tree). 3

Can we construct in a finite agent econ- omy a contact which has the above properties? NO Can we construct a second best con- tract? YES Can we construct an environment or framework where “first” best contracts can be reached? Yes, under perfect competition – negligible private information. Book: Glycopantis-Yannelis, Differential In- formation Economies , 2005. 4

1. Differential information economy (DIE) We define the notion of a finite-agent econ- omy with differential information for the case where the set of states of nature, Ω and the number of goods, l , per state are finite. I is a R l set of n players and I + will denote the set of positive real numbers. A differential information exchange econ- omy E is a set { ((Ω , F ) , X i , F i , u i , e i , q i ) : i = 1 , . . . , n } where 1. F is a σ -algebra generated by a partition of Ω; R l 2 I + is the set-valued func- 2. X i : Ω → tion giving the random consumption set 5

of Agent (Player) i, who is denoted by Pi; 3. F i is a partition of Ω generating a sub- σ - algebra of F , denoting the private infor- mation of Pi; R l 4. u i : Ω × I + → I R is the random utility function of Pi; for each ω ∈ Ω , u i ( ω, . ) is continuous, concave and monotone; R l 5. e i : Ω → I + is the random initial endow- ment of Pi, assumed to be F i -measurable, with e i ( ω ) ∈ X i ( ω ) for all ω ∈ Ω; 6. q i is an F -measurable probability function on Ω giving the prior of Pi. It is assumed that on all elements of F i the aggregate q i is strictly positive. If a common prior is assumed on F , it will be denoted by µ . We will refer to a function with domain Ω, 6

constant on elements of F i , as F i -measurable , although, strictly speaking, measurability is with respect to the σ -algebra generated by the partition. In the first period agents make contracts in the ex ante stage. In the interim stage, i.e., after they have received a signal 1 as to what is the event containing the realized state of na- ture, they consider the incentive compatibility of the contract. R l For any x i : Ω → I + , the ex ante expected utility of Pi is given by � v i ( x i ) = u i ( ω, x i ( ω )) q i ( ω ) . Ω Let G be a partition of (or σ -algebra on) Ω, belonging to Pi. For ω ∈ Ω denote by E G i ( ω ) 1 A s ignal to Pi is an F i -measurable function to all of the possible distinct observations specific to the player; that is, it induces the partition F i , and so gives the finest discrimination of states of nature directly available to Pi. 7

the element of G containing ω ; in the par- ticular case where G = F i denote this just by E i ( ω ). Pi’s conditional probability for the state of nature being ω ′ , given that it is actu- ally ω , is then  ω ′ / ∈ E G 0 : i ( ω )   ω ′ | E G � � q i i ( ω ) = q i ( ω ′ ) ω ′ ∈ E G : i ( ω ) .  E G � �  q i i ( ω ) The interim expected utility function of Pi, v i ( x |G ), is given by u i ( ω ′ , x i ( ω ′ )) q i ω ′ | E G � � � v i ( x |G )( ω ) = i ( ω ) , ω ′ which defines a G -measurable random variable. R l ) the space of all equiv- Denote by L 1 ( q i , I alence classes of F -measurable functions R l ; when a common prior µ is as- f i : Ω → I R l ) will be replaced by L 1 ( µ, I R l ). sumed L 1 ( q i , I 8

L X i is the set of all F i -measurable selections from the random consumption set of Agent i, i.e., R l ) : x i : Ω → L X i = { x i ∈ L 1 ( q i , I R l is F i -measurable and x i ( ω ) ∈ X i ( ω ) q i - a.e. } I n � and let L X = L X i . i =1 Also let R l ) : x i ( ω ) ∈ X i ( ω ) q i - a.e. } ¯ L X i = { x i ∈ L 1 ( q i , I n and let ¯ ¯ � L X = L X i . i =1 An element x = ( x 1 , . . . , x n ) ∈ ¯ L X will be called an allocation . For any subset of players ¯ S , an element ( y i ) i ∈ S ∈ � L X i will also be i ∈ S called an allocation, although strictly speaking 9

it is an allocation to S . In case there is only one good, we shall use the notation L 1 X i , L 1 X etc. When a common R l ) will be re- prior is also assumed L 1 ( q i , I R l ). placed by L 1 ( µ, I Finally, suppose we have a coalition S , with members denoted by i . Their pooled informa- i ∈ S F i will be denoted by F S 2 . We tion � assume that F I = F . Is it possible for agents to write incentive compatible and efficient or Pareto optimal contracts? Let us answer this question by considering a simple two agents example. Example 0.1 There are two Agents, 1 and 2, and three equally probable states of nature de- noted by a, b, c and one good per state denoted 2 The “join” W i ∈ S F i denotes the smallest σ -algebra containing all F i , for i ∈ S . 10

by x . The utility functions, initial endowments and private information sets are given as fol- lows: u 1 ( w, x 1 ) = √ x 1 , for w = a, d, c u 2 ( w, x 2 ) = √ x 2 , for w = a, b, c F 1 = {{ a, b } , { c }} e 1 ( a, b, c ) = (10 , 10 , 0) , e 2 ( a, b, c ) = (10 , 0 , 10) , F 2 = {{ a, c } , { b }} . Notice that a “fully”, pooled information, Pareto optimal, (i.e. a weak fine core outcome) is x 1 ( a, b, c ) = (10 , 5 , 5) x 2 ( a, b, c ) = (10 , 5 , 5) . (1) However, this outcome is not incentive com- patible because if the realized state of nature is a , then Agent 1 has an incentive to report that it is state c , (notice that Agent 2 cannot distin- guish state a from state c ) and become better 11

off. In particular, Agent 1 will keep her ini- tial endowment in the event { a, b } which is 10 units and receive another 5 units from Agent 2, in state c , (i.e., u 1 ( e 1 , ( a )+ x 1 ( c ) − e 1 ( c )) = u 1 (15) > u 1 ( x, ( a )) = 10) and becomes better off. Obviously Agent 2 is worse off. Similarly, Agent 2 has an incentive to report b when he observes { a, c } This example demonstrates that “ full or ex post Pareto optimality ” is not necessarily compatible with incentive compatibility . The following example will illustrate the role of the private information measurability of an allocation. Example 0.2 There are two Agents, 1 and 2, two goods denoted by x and y and two equally probable states denoted by { a, b } . The agents’ 12

characteristics are: u 1 ( w, x 1 , y 1 ) = √ x 1 y 1 , for w = a, b u 2 ( w, x 2 , y 2 ) = √ x 2 y 2 , for w = a, b e 1 ( a, b ) = ((10 , 0) , (10 , 0)) , F 1 = { a, b } F 2 = {{ a } , { b }} . e 2 ( a, b ) = ((10 , 8) , (0 , 10)) , The feasible allocation below is Pareto optimal (interim, ex post and ex ante). (( x 1 ( a ) , y 1 ( a )) , ( x 1 ( b ) , y 1 ( b ))) = ((5 , 2) , (5 , 5)) (( x 2 ( a ) , y 2 ( a )) , ( x 2 ( b ) , y 2 ( b ))) = ((15 , 6) , (5 , 5)) . (2) However, the allocation in (2) above is not incentive compatible because if b is the realized state of nature Agent 2 can report state a and become better off, i.e., u 2 ( e 2 ( b ) + ( x 2 ( a ) , y 2 ( a )) − e 2 ( a )) = u 2 ((0 , 10) + (15 , 6) − (10 , 8)) = u 2 (5 , 8) > u 2 ( x 2 ( b ) , y 2 ( b )) = u 2 (5 , 5) . 13

Notice that the allocation in (2) is not F 1 - measurable (i.e., measurable with respect to the private information of Agent 1). Hence, an individually rational, efficient (interim, ex ante, ex post) without the F i -measurability ( i = 1 , 2 ) condition need not be incentive compatible. Observe that one can restore the incentive compatibility simply by making the allocation in (2) above F i -measurable for each i, ( i = 1 , 2) . In particular, the F i -measurable alloca- tion below is incentive compatible, and private information ( F i -measurable) Pareto optimal. ( x 1 ( a ) , y 1 ( a )) , ( x 1 ( b ) , y 1 ( b )) = ((5 , 5) , (5 , 5)) ( x 2 ( a ) , y 2 ( a )) , ( x 2 ( b ) , y 2 ( b )) = ((15 , 3) , (5 , 5)) . The importance of the measurability condi- 14

tion in restoring incentive compatibility and of course guaranteeing the existence of an opti- mal contract is obvious in the above example and this approach was introduced by Yannelis (1991). Example 0.3 Consider a three person differ- ential information economy, with Agents 1, 2, 3, two goods denoted by x, y , and the three equal probable states are denoted by a, b, c . The agents’ utility functions, random initial endowments and private information sets are 15