Screening. Diego Moreno Universidad Carlos III de Madrid. Diego Moreno () Screening 1 / 1

Transcription

1 Screening Diego Moreno Universidad Carlos III de Madrid Diego Moreno () Screening 1 / 1

2 The Agency Problem with Adverse Selection A risk neutral principal wants to o er a menu of contracts to be o ered to an agent randomly drawn from a heterogenous population of agents. In the population of agents, a fraction q 2 (0; 1) is of type H, and the remaining fraction, 1 q, is of type L. Agents of type 2 fh; Lg are characterized by: A von N-M utility function u : R! R such that u(0) = 0; u 0 > 0 and u 00 0; representing his preferences. A real number u 0 specifying his reservation utility, and a function k c(e) describing his cost of e ort, where k > 0 and c : R +! R satis es c(0) = 0; c 0 > 0; and c 00 > 0. Thus, agents only di er in the value of the parameter k : Without loss of generality, let us assume that k L = 1; and k H = k > 1: Diego Moreno () Screening 2 / 1

3 The Principal s Problem The menu of contracts is designed in order to maximize expected pro ts and assure that every agent will accept one of the contracts o ered. Hence the principal s problem is: max q (EX (e H ) w H ) + (1 q) (EX (e L ) w L ) f(e H ;w H );(e L ;w L )g2r 4 + subject to: (PC H ; H ) u(w H ) kc(e H ) + u (PC L ; L ) u(w L ) c(e L ) + u (IC H ; H ) u(w H ) kc(e H ) u(w L ) kc(e L ) (IC L ; L ) u(w L ) c(e L ) u(w H ) c(e H ): Diego Moreno () Screening 3 / 1

4 This problem may be simpli ed by noticing that in an interior solution the participation constraint of the low cost type PC L is not binding, since u(w L ) c(e L ) u(w H ) c(e H ) (by IC L ) and hence it can be ignored. > u(w H ) kc(e H ) (because k > 1 and e H > 0) u (by PC H ), Diego Moreno () Screening 4 / 1

5 The Principal s Problem Suppressing the inequality PC L we may write the Lagrangian as: L() = q (EX (e H ) w H ) + (1 q) (EX (e L ) w L ) + H (u(w H ) kc(e H ) u) + H (u(w H ) kc(e H ) u(w L ) + kc(e L )) + L (u(w L ) c(e L ) u(w H ) + c(e H )) : Diego Moreno () Screening 5 / 1

6 The rst order conditions identifying an interior solution of the H = q (EX (e H )) 0 H kc 0 (e H ) H kc 0 (e H ) + L c 0 (e H ) H = q + H u 0 (w H ) + H u 0 (w H ) L u 0 (w H ) L = (1 q) (EX (e L )) 0 + H kc 0 (e L ) L c 0 (e L ) L = (1 q) H u 0 (w L ) + L u 0 (w L ) = 0; Diego Moreno () Screening 6 / 1

7 This system may be rewritten as q (EX (e H )) 0 c 0 (e H ) q u 0 (w H ) (1 q) (EX (e L)) 0 c 0 (e L ) 1 q u 0 (w L ) = H k ( L H k) (1) = H ( L H ) (2) = L k H (3) = L H : (4) Diego Moreno () Screening 7 / 1

8 In addition, the slackness conditions must hold. H (u(w H ) kc(e H ) u) = 0 (5) H (u(w H ) kc(e H ) u(w L ) + kc(e L )) = 0 (6) L (u(w L ) c(e L ) u(w H ) + c(e H )) = 0: (7) Since L > 0 by equation (4), then equation (7) implies that the constrain IC L is binding. Likewise, since H > 0 by equations (2) and (4), then equation (5) implies that the constrain PC H is binding. Diego Moreno () Screening 8 / 1

9 Let us show that in a solution to this system the incentive constraint of the incentive compatibility constrain of the high cost type, IC H ; is not bindig, that is u(w H ) kc(e H ) > u(w L ) + kc(e L ); and therefore the slackness condition (7) implies H = 0: Diego Moreno () Screening 9 / 1

10 In order to prove this, we rst show that in a solution e L e H. Since c(e L ) c(e H ) u(w L ) u(w H ) (by IC L ) k (c(e L ) c(e H )) (by IC H ), then This implies and hence (1 k) (c(e L ) c(e H )) 0: c(e L ) c(e H ); e L e H : Diego Moreno () Screening 10 / 1

11 Next we show that e L 6= e H : Suppose by way of contradiction that e L = e H : This implies that w L = w H for otherwise both types will choose the contract involving the largest wage (for identical e ort). Formally, the inequalities IC H and IC L imply 0 = c(e L ) c(e H ) u(w L ) u(w H ) k (c(e L ) c(e H )) = 0. Hence and therefore u(w L ) u(w H ) = 0; w L = w H : Diego Moreno () Screening 11 / 1

12 But if e L = e H and w L = w H ; then we can suppress the arguments in the system of rst order conditions, and write it as: (EX )0 q c 0 = H k ( L H k) (1) q u 0 = H ( L H ) (2) (EX )0 (1 q) c 0 = L k H (3) 1 q u 0 = L H : (4) Diego Moreno () Screening 12 / 1

13 Substituting H = 1=u 0 from (2) and (4) into equation (2) we get (1 q) H = L H L = (1 q) H + H : Substituting k H = (EX ) 0 =c 0 from (1) and (3) into equation (1) we get Since k > 1 and (1 Hence and since e L e H ; we have L = k[(1 q) L + H ]: q) H + H > 0 these two equations cannot hold. e L 6= e H ; e L > e H : Diego Moreno () Screening 13 / 1

14 Finally we show that IC H holds with strict inequality, and therefore H = 0 by the complementary slackness condition (6): Since L > 0; then IC L and e L > e H imply Then k > 1 implies u(w L ) u(w H ) = c(e L ) c(e H ) > 0: k(c(e L ) c(e H )) > c(e L ) c(e H ) = u(w L ) u(w H ); that is u(w H ) kc(e H )) > u(w L ) kc(e L ): Diego Moreno () Screening 14 / 1

15 Substituting H = 0 into the system of rst order conditions, we get q (EX (e H )) 0 c 0 = H k (e H ) L (1) q u 0 (w H ) = H L (2) (1 q) (EX (e L)) 0 c 0 = (e L ) L (3) 1 q u 0 (w L ) = L: (4) Diego Moreno () Screening 15 / 1

16 This system may be rewritten as (EX (e H )) 0 = kc0 (e H ) u 0 (w H ) + 1 q q (k 1) c0 (e H ) u 0 (w L ) (EX (e L )) 0 = c0 (e L ) u 0 (3; 4): (w L ) (1; 2) These two equations together with the two binding constrains u(w H ) = kc(e H ) + u (5) u(w L ) c(e L ) = u(w H ) c(e H ) (7) identify the optimal contract. Diego Moreno () Screening 16 / 1

17 Properties of the optimal menu: The contract o ered to the low cost type is optimal: by equation (3; 4), the Principal selects a contract on her demand of e ort from the low cost type. The contract o ered to the high cost type distorts the demand of e ort downward: the contract satisfying equation (1,2) is below the Principal s demand of e ort from the high cost type. This distortion makes the contract for the high types less attractive to the low type, which relaxes the incentive constraint for this type. As observed earlier, the low cost type captures a positive surplus which we can refer to as information rents. Diego Moreno () Screening 17 / 1

18 Exercise. EX (e) = 2e; and u(x) = x; c(e) = e 2 ; u = 0; k = 2; q = 1=2: Optimal contracts with complete information: E ort Supplies: w H = 2e 2 H ; w L = e 2 L : E ort Demands: 2 = 4e H ; i.e., e H = 1=2; 2 = 2e L ; i.e., e L = 1: Thus, the optimal contracts are (el ; w L ) = (1; 1); (e H ; w H ) = (1=2; 1=2): And the principal s expected pro t is E = 1 2 (2(1) 1) ! 1 2 = : Diego Moreno () Screening 18 / 1

19 With adverse selection the optimal menu of contracts solves, Solving the system we get 2 = 2e L 2 = 2 2e H w H = 2e 2 H w L e 2 L = w H e 2 H 2 (2 1) 2e H 1 (~e L ; ~w L ) = (1; 10=9); (~e H ; ~w H ) = (1=3; 2=9): The expected pro t is E~ = 1 2 2(1) = : Diego Moreno () Screening 19 / 1

20 With complete information the principal captures the entire surplus. Hence the social surplus is E = 3 4 : With adverse selection agents of type H capture some surplus. In this example, each L agent captures 1=16; and there is a fraction q = 1=2 of L agents in the population. Hence the social surplus is E~ ( 1 16 ) = < 3 4 : Adverse selection reduces the social surplus! Diego Moreno () Screening 20 / 1