finalmacro2sp05answers - Econ 387L Macro II Spring 2005...

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Econ 387L: Macro II Spring 2005, University of Texas Instructor: Dean Corbae Final Exam 1. Consider the following risk sharing problem. An insurer has a very large endowment x and a household has state contingent income y θ , θ { H, L } with y L < y H and prob ( θ = H ) = π (0 , 1) . The insurer is risk neutral and the household has a strictly increasing and concave utility function u ( c ) with u 0 (0) = . a. (10 points) Formulate the planner’s problem of maximizing the house- hold’s expected utility subject to ex-ante participation of the insurer under the assumption that household income is observable and if the insurer chooses not to participate, she simply consumes her endowment. Characterize the equilib- rium allocation of consumption. (Hint: Write the problem in terms of transfers between the household and insurer and assume that the insurer’s endowment is so large that her post transfer consumption is never negative). Answer: Let p θ denote the payment (if positive) or receipt (if negative) from the household to the insurer. Under full information, the planner’s problem is max πu ( y H p H ) + (1 π ) u ( y L p L ) s.t.πp H + (1 π ) p L 0 y θ p θ 0 where the resource constraint is satis fi ed by construction. The fi rst order condi- tions imply u 0 ( y H p H ) = u 0 ( y L p L ) . Thus we have 2 equations in 2 unknowns: y H p H = y L p L πp H + (1 π ) p L = 0 which yields p H = (1 π )( y H y L ) > 0 and p L = π ( y H y L ) < 0 . Hence the planner transfers from the household when it has high income and to the household when it has low income. b. (10 points) Formulate the planner’s problem of maximizing the house- hold’s expected utility subject to ex-ante participation of the insurance company under the assumption that household income is unobservable and the planner must implement an allocation that satis fi es incentive compatibility. Character- ize the equilibrium allocation of consumption. Is the allocation from part a implementable? What is? Answer: Two incentive compatibility constraints must be added to the above problem: u ( y H p H ) u ( y H p L ) ⇐⇒ 0 p H p L u ( y L p L ) u ( y L p H ) ⇐⇒ p H p L 0 The only way for both constraints to be satis fi ed is if p H = p L or the pay- ment/transfer is independent of message. If p < 0 , the insurer’s participation 1

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constraint is violated and if p > 0 , the planner can make the household better o ff by setting p = 0. Thus, there can be no insurance in a static setting. c. (20 points) Now suppose the insurer and household are in a repeated, two period setting with private information. In particular, assume that the income shocks to the household are iid across time and that it report its income to the insurer in both periods. Formulate the planner’s problem. Can the planner
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