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387hw2sp07 - an optimal debt contract where in exchange for...

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Economics 387. Dynamic Contracts. Spring 2007. Department of Economics, University of Texas Instructor: Dean Corbae, BRB 3.134A, (o) 512-475-8530 email: [email protected], web: www.eco.utexas.edu/˜corbae Homework #2 - Due 2/13/07 Consider the following variant of the costly state veri fi cation problem by R. Townsend (1979) studied in class. The main di ff erence between homework and class is that both the principal and agent are risk neutral. Suppose there is a risk neutral entrepreneur who has a project that yields a random return x [0 , x ] drawn from distribution function F ( x ) at the end of the period if an investment is made at the beginning of the period and yields zero otherwise. There is a risk neutral lender who has one unit of an investment good, access to a storage technology which yields r at the end of the period, and access to a veri fi cation technology which takes κ units of e ff ort in order to observe the return on the entrepreneur’s project. You are to characterize
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Unformatted text preview: an optimal debt contract where in exchange for 1 unit of investment good lent to the entrepreneur at the beginning of the period, the lender receives return B ( x ) ∈ [0 , x ] if the entrepreneur reports e x ∈ S ⊂ [0 , x ] and the lender veri f es the report while the lender receives R ( e x ) if the entrepreneur reports e x ∈ S c and the lender does not verify. 1. Prove that R ( e x ) must be a constant (say R ) on e x ∈ S c . 2. State the programming problem which maximizes the entrepreneur’s utility subject to participation by the lender. 3. Prove that the veri f cation region S = { e x : e x ∈ [0 , x ∗ ] } (i.e. a lower interval). 4. Prove that B ( x ) = x < R. Note that (1)-(4) implies that the optimal contract has all the essential features of a debt contract where veri f cation can be interpreted as bankruptcy. 1...
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