Frank Cowell Microeconomics March 2007 Exercise 11 3

Frank Cowell: Microeconomics March 2007 Exercise 11. 3 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 11. 3(1): Question Frank Cowell: Microeconomics n n purpose: solution to an adverse selection problem method: find full-information solution from reservation utility levels. Then introduce incentive-compatibility constraint in order to find second-best solution

Ex 11. 3(1): participation constraint Frank Cowell: Microeconomics n n The principal knows the agent’s type So maximises x y subject to u u n where u = 0 for each individual type In the full-information solution u u the participation constraint binds there is no distortion

Ex 11. 3(1): full-information case Frank Cowell: Microeconomics n Differentiate the binding participation constraint u n n n to find the slope of the IC: Since there is no distortion this slope must equal 1 This implies Using the fact that u = u and substituting into the participation constraint:

Ex 11. 3(1): Full-information contracts Frank Cowell: Microeconomics u _b y §Space of (legal services, payment) §a-type’s reservation utility §b-type’s reservation utility §Contracts • y*a = 1 slope = 1 u _a y*b = ¼ 0 • x*b slope = 1 =½ x*a x =2

Ex 11. 3(1): FI contracts, assessment Frank Cowell: Microeconomics n Solution has MRS = MRT u u n since there is no distortion… …the allocation (x*a, y*a), (x*b, y*b) is efficient We cannot perturb the allocation so as to u u make one person better off… …without making the other worse off

Ex 11. 3 (2): Question Frank Cowell: Microeconomics method: n Derive the incentive-compatibility constraint n Set up Lagrangean n Solve using standard methods n Compare with full-information values of x and y

Ex 11. 3 (2): “wrong” contract? Frank Cowell: Microeconomics n n Now it is impossible to monitor the lawyer’s type Is it still viable to offer the efficient contracts (x*a, y*a) and (x * b , y * b )? Consider situation of a type-a lawyer u if he accepts the contract meant for him he gets utility u but if he were to get a type-b contract he would get utility So a type a would prefer to take… u u a type-b contract rather than the efficient contract

Ex 11. 3 (2): incentive compatibility Frank Cowell: Microeconomics n n Given the uncertainty about lawyer’s type… …the firm wants to maximise expected profits u n n This must take account of the “wrong-contract” problem just mentioned An a-type must be rewarded sufficiently… u n it is risk-neutral so that is not tempted to take a b-type contract The incentive-compatibility constraint for the a types

Ex 11. 3 (2): optimisation problem Frank Cowell: Microeconomics n Let p be the probability that the lawyer is of type a Expected profits are n Structure of problem is as for previous exercises n n u participation constraint for type b will be binding u incentive-compatibility constraint for type a will be binding This enables us to write down the Lagrangean…

Ex 11. 3 (2): Lagrangean Frank Cowell: Microeconomics n The Lagrangean for the firm’s optimisation problem is: u u u n where… l is the Lagrange multiplier for b’s participation constraint m is the Lagrange multiplier fora’s incentive-compatibility constraint Find the optimum by examining the FOCs…

Ex 11. 3 (2): Lagrange multipliers Frank Cowell: Microeconomics n n n Differentiate Lagrangean with respect to xa u and set result to 0 u yields m = pta Differentiate Lagrangean with respect to xb u and set result to 0 u using the value for m this yields l = tb Use these values of the Lagrange multiplier in the remaining FOCs

Ex 11. 3 (2): optimal payment, a-types Frank Cowell: Microeconomics n Differentiate Lagrangean with respect to ya u and set result to 0 n Substitute for m: n Rearranging we find u u u exactly as for the full-information case also MRS = 1, exactly as for the full-information case illustrates the “no distortion at the top” principle

Ex 11. 3 (2): optimal payment, b-types Frank Cowell: Microeconomics n Differentiate Lagrangean with respect to yb u and set result to 0 n Substitute for l and m: n Rearranging we find u u this is less than ¼[tb ]2 … …the full-information income for a b-type

Ex 11. 3 (2): optimal x Frank Cowell: Microeconomics n Differentiate Lagrangean with respect to l u and set result to 0 get the b-type’s binding participation constraint this yields u which becomes u u n Differentiate Lagrangean with respect to m u u u n and set result to 0 get the a-type’s binding incentive-compatibility constraint this yields These are less than values for full-information contracts u for both a-types and b-types

Ex 11. 3 (2): second-best solution Frank Cowell: Microeconomics §a-type’s reservation utility §b-type’s reservation utility §a-type’s full-info contract §b-type’s second-best contract §a-type’s second-best contract u _b y ^ya • • u _a ^yb 0 • ^xb ^xa x

Ex 11. 3: points to remember Frank Cowell: Microeconomics n n Standard “adverse-selection” results Full-information solution is fully exploitative u n Asymmetric information u n binding participation constraint for both types incentive-compatibility problem for a-types Second best solution u u u binding participation constraint for b-type binding incentive-compatibility constraint for a- type no distortion at the top