Frank Cowell: Microeconomics Exercise 11.3 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

Frank Cowell: Microeconomics Ex 11.3(1): Question purpose: solution to an adverse selection problem 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 method: find full-information solution from reservation utility levels. Then introduce incentive-compatibility constraint in order to find second-best solution

Frank Cowell: Microeconomics Ex 11.3(1): participation constraint The principal knows the agent’s type The principal knows the agent’s type So maximises x  y subject to So maximises x  y subject to  where  = 0  for each individual type In the full-information solution In the full-information solution  the participation constraint binds  there is no distortion

Frank Cowell: Microeconomics Ex 11.3(1): full-information case Differentiate the binding participation constraint Differentiate the binding participation constraint  to find the slope of the IC: Since there is no distortion this slope must equal 1 Since there is no distortion this slope must equal 1 This implies This implies Using the fact that  =  and substituting into the participation constraint: Using the fact that  =  and substituting into the participation constraint:

Frank Cowell: Microeconomics Ex 11.3(1): Full-information contracts 0 y x slope = 1 x *a = 2x *b = ½ y *b = ¼ y *a = 1 bb __ aa   a-type’s reservation utility   b-type’s reservation utility   Space of (legal services, payment)   Contracts

Frank Cowell: Microeconomics Ex 11.3(1): FI contracts, assessment Solution has MRS = MRT 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 We cannot perturb the allocation so as to  make one person better off…  …without making the other worse off

Frank Cowell: Microeconomics Ex 11.3 (2): Question method: Derive the incentive-compatibility constraint Derive the incentive-compatibility constraint Set up Lagrangean Set up Lagrangean Solve using standard methods Solve using standard methods Compare with full-information values of x and y Compare with full-information values of x and y

Frank Cowell: Microeconomics Ex 11.3 (2): “wrong” contract? Now it is impossible to monitor the lawyer’s type 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 ) ? 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 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… So a type a would prefer to take…  a type-b contract  rather than the efficient contract

Frank Cowell: Microeconomics Ex 11.3 (2): incentive compatibility Given the uncertainty about lawyer’s type… Given the uncertainty about lawyer’s type… …the firm wants to maximise expected profits …the firm wants to maximise expected profits  it is risk-neutral This must take account of the “wrong-contract” problem just mentioned This must take account of the “wrong-contract” problem just mentioned An a-type must be rewarded sufficiently… An a-type must be rewarded sufficiently…  so that is not tempted to take a b-type contract The incentive-compatibility constraint for the a types The incentive-compatibility constraint for the a types

Frank Cowell: Microeconomics Ex 11.3 (2): optimisation problem Let  be the probability that the lawyer is of type a Let  be the probability that the lawyer is of type a Expected profits are Expected profits are Structure of problem is as for previous exercises 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… This enables us to write down the Lagrangean…

Frank Cowell: Microeconomics Ex 11.3 (2): Lagrangean The Lagrangean for the firm’s optimisation problem is: The Lagrangean for the firm’s optimisation problem is:  where…  is the Lagrange multiplier for b’s participation constraint   is the Lagrange multiplier fora’s incentive-compatibility constraint Find the optimum by examining the FOCs… Find the optimum by examining the FOCs…

Frank Cowell: Microeconomics Ex 11.3 (2): Lagrange multipliers Differentiate Lagrangean with respect to x a Differentiate Lagrangean with respect to x a  and set result to 0  yields  =  a Differentiate Lagrangean with respect to x b Differentiate Lagrangean with respect to x b  and set result to 0  using the value for  this yields =  b Use these values of the Lagrange multiplier in the remaining FOCs Use these values of the Lagrange multiplier in the remaining FOCs

Frank Cowell: Microeconomics Ex 11.3 (2): optimal payment, a-types Differentiate Lagrangean with respect to y a Differentiate Lagrangean with respect to y a  and set result to 0 Substitute for  : Substitute for  : Rearranging we find 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

Frank Cowell: Microeconomics Ex 11.3 (2): optimal payment, b-types Differentiate Lagrangean with respect to y b Differentiate Lagrangean with respect to y b  and set result to 0 Substitute for and  : Substitute for and  : Rearranging we find Rearranging we find  this is less than ¼[  b ] 2 …  …the full-information income for a b-type

Frank Cowell: Microeconomics Ex 11.3 (2): optimal x Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to  and set result to 0  get the b-type’s binding participation constraint  this yields  which becomes Differentiate Lagrangean with respect to  Differentiate Lagrangean with respect to   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 These are less than values for full-information contracts  for both a-types and b-types

Frank Cowell: Microeconomics Ex 11.3 (2): second-best solution 0 ^ xaxa y x ^ xbxb ^ ybyb bb __ aa ^ yaya   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

Frank Cowell: Microeconomics Ex 11.3: points to remember Standard “adverse-selection” results Standard “adverse-selection” results Full-information solution is fully exploitative Full-information solution is fully exploitative  binding participation constraint for both types Asymmetric information Asymmetric information  incentive-compatibility problem for a-types Second best solution Second best solution  binding participation constraint for b-type  binding incentive-compatibility constraint for a- type  no distortion at the top