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