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Frank Cowell: Microeconomics March 2007 Exercise 11.3 MICROECONOMICS Principles and Analysis Frank Cowell Ex 11.3(1): Question Frank Cowell: Microeconomics 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 The principal knows the agent’s type So maximises x y subject to where u = 0 for each individual type In the full-information solution the participation constraint binds there is no distortion Ex 11.3(1): full-information case Frank Cowell: Microeconomics Differentiate the binding participation constraint 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 Solution has MRS = MRT 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 make one person better off… …without making the other worse off Ex 11.3 (2): Question Frank Cowell: Microeconomics method: Derive the incentive-compatibility constraint Set up Lagrangean Solve using standard methods Compare with full-information values of x and y Ex 11.3 (2): “wrong” contract? Frank Cowell: Microeconomics 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 if he accepts the contract meant for him he gets utility but if he were to get a type-b contract he would get utility So a type a would prefer to take… a type-b contract rather than the efficient contract Ex 11.3 (2): incentive compatibility Frank Cowell: Microeconomics Given the uncertainty about lawyer’s type… …the firm wants to maximise expected profits This must take account of the “wrong-contract” problem just mentioned An a-type must be rewarded sufficiently… 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 Let p be the probability that the lawyer is of type a Expected profits are Structure of problem is as for previous exercises participation constraint for type b will be binding 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 The Lagrangean for the firm’s optimisation problem is: 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 Differentiate Lagrangean with respect to xa and set result to 0 yields m = pta Differentiate Lagrangean with respect to xb and set result to 0 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 Differentiate Lagrangean with respect to ya and set result to 0 Substitute for m: Rearranging we find 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 Differentiate Lagrangean with respect to yb and set result to 0 Substitute for l and m: Rearranging we find this is less than ¼[tb]2… …the full-information income for a b-type Ex 11.3 (2): optimal x Frank Cowell: Microeconomics Differentiate Lagrangean with respect to l and set result to 0 get the b-type’s binding participation constraint this yields which becomes Differentiate Lagrangean with respect to m 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 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 • x ^xb ^xa Ex 11.3: points to remember Frank Cowell: Microeconomics Standard “adverse-selection” results Full-information solution is fully exploitative Asymmetric information binding participation constraint for both types incentive-compatibility problem for a-types Second best solution binding participation constraint for b-type binding incentive-compatibility constraint for a- type no distortion at the top