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Exercise 11.3

This exercise explores the adverse selection problem and the introduction of incentive compatibility constraints to find the second-best solution. It discusses full-information contracts, the optimisation problem, and the optimal payments for each type.

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Exercise 11.3

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  1. Exercise 11.3 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

  2. Ex 11.3(1): Question • 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

  3. Ex 11.3(1): participation constraint • 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

  4. Ex 11.3(1): full-information case • 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 = uand substituting into the participation constraint:

  5. ub _ ua _ slope = 1 slope = 1 Ex 11.3(1): Full-information contracts • Space of (legal services, payment) • a-type’s reservation utility y • b-type’s reservation utility • Contracts • y*a = 1 • y*b = ¼ x 0 x*b = ½ x*a = 2

  6. Ex 11.3(1): FI contracts, assessment • 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

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

  8. Ex 11.3 (2): “wrong” contract? • 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

  9. Ex 11.3 (2): incentive compatibility • Given the uncertainty about lawyer’s type… • …the firm wants to maximise expected profits • it is risk-neutral • This must take account of the “wrong-contract” problem just mentioned • 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

  10. Ex 11.3 (2): optimisation problem • 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…

  11. Ex 11.3 (2): Lagrangean • The Lagrangean for the firm’s optimisation problem is: • where… • l is the Lagrange multiplier for b’sparticipation constraint • m is the Lagrange multiplier fora’sincentive-compatibility constraint • Find the optimum by examining the FOCs…

  12. Ex 11.3 (2): Lagrange multipliers • 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

  13. Ex 11.3 (2): optimal payment, a-types • 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

  14. Ex 11.3 (2): optimal payment, b-types • 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

  15. Ex 11.3 (2): optimal x • 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

  16. ub _ ua _ ^ ^ ^ ^ xb yb ya xa Ex 11.3 (2): second-best solution • a-type’s reservation utility • b-type’s reservation utility y • a-type’s full-info contract • b-type’s second-best contract • a-type’s second-best contract • • • x 0

  17. Ex 11.3: points to remember • Standard “adverse-selection” results • Full-information solution is fully exploitative • binding participation constraint for both types • Asymmetric information • 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

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