# 387hw3aa - by(2(3 after substituting c θ θ 1 = c 1 θ 2...

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Economics 387. Banking and Financial Intermediation. Spring 2002. Department of Economics, University of Texas Instructor: Dean Corbae, BRB 3.118, (o) 512-475-8530 Homework #3-Due 2/20/02 Consider the two period hidden information problem where there is a risk neutral principal and a risk averse agent studied in class based on R. Townsend (1982) with θ t { H,L } ,t =0 , 1 . max { c ( θ 0 ) ,c ( θ 0 1 ) } θ t { H,L } X θ 0 { H,L } π θ 0 " ¡ y c ( θ 0 )+ y θ 0 ¢ + β P θ 1 { H,L } π θ 1 ( y c ( θ 0 1 )+ y θ 1 · # (1) s.t. X θ 0 { H,L } π θ 0 u ( c ( θ 0 )) + β X θ 1 { H,L } π θ 1 u (( c ( θ 0 1 )) ω (2) u ( c ( θ 0 )) + β X θ 1 { H,L } π θ 1 u ( c ( θ 0 1 )) (3) u c ( e θ 0 )+ y θ 0 y e θ 0 · + β X θ 1 { H,L } π θ 1 u c ( e θ 0 1 ) · , θ 0 , e θ 0 u ( c ( θ 0 1 )) u c ( θ 0 , b θ 1 )+ y θ 1 y b θ 1 · , θ 0 1 , b θ 1 (4) 1. Show that under this formulation, the constraint set is not necessarily convex. Suggestion: Since (4) is binding (so that c ( θ 0 1 )canno tbe t

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Unformatted text preview: by (2)-(3) after substituting c ( θ , θ 1 ) = c 1 ( θ ) . 2. Transform (1)-(4) into a cost minimization problem over choice of cur-rent utility u θ and promised future utility w θ (as in Homework #1). Does the transformed problem necessarily have a convex constraint set? Under what conditions on utility functions is the constraint set for the transformed problem convex? Suggestion: Try CARA preferences. 1 3. Suppose the conditions which you stated in part 2 are not met, is there a way to convexify the constraint set? Suggestion: See Phelan, C. and R. Townsend (1991) &Computing Multi-Period, Information Constrained Optima±, Review of Economic Studies , Vol. 54, p. 599-617. 2...
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387hw3aa - by(2(3 after substituting c θ θ 1 = c 1 θ 2...

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