Sep 18, 2017

# (RANDOM PRIVATE BENEFITS). CONSIDER THE VARIABLE-INVESTMENT MODEL: AN ENTREPRENEUR INITIALLY HA

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(random private benefits). Consider the variable-investment model: an entrepreneur initially has cash A. For investment I, the project yields RI in the case of success and 0 in the case of failure. The probability of success is equal to pH ∈ (0, 1) if the entrepreneur works and pL = 0 if the entrepreneur shirks. The entrepreneur obtains private benefit BI when shirking and 0 when working. The per unit private benefit B is unknown to all ex ante and is drawn from (common knowledge) uniform distribution F:

with density f (B)ˆ = 1/R. The entrepreneur borrows I − A and pays back R1 = r1 in the case of success. The timing is described in Figure 3.10. (i) For a given contract (1, r1), what is the threshold B∗, i.e., the value of the private per-unit benefit above which the entrepreneur shirks? (ii) For a given B∗ (or equivalently r1, which determines B∗), what is the debt capacity? For which value of B∗ (or r1) is this debt capacity highest? (iii) Determine the entrepreneur’s expected utility for a given B∗. Show that the contract that is optimal for the entrepreneur (subject to the investors breaking even) satisfies

Interpret this result. (iv) Suppose now that the private benefit B is observable and verifiable. Determine the optimal contract between the entrepreneur and the investors (note that the reimbursement can now be made contingent on the level of private benefits

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