Continuous entrepreneurial effort; liquidity needs:-

(i) An entrepreneur with initial cash A and protected by limited liability wants to invest in a fixed-size project with investment cost I > A. After the investment is made, the entrepreneur chooses the probability p of success (0 ≤ p ≤1) the disutility of effort is(The entrepreneur enjoys no private benefit in this model.)

In question (i) only, the profit isin the case of success and 0 in the case of failure. (We assume that R an expected rate of interest equal to 0 and everyone is risk neutral. Let R_b denote the entrepreneur's reward in the case of success. Solve for the optimal contract (R_b). Show that

(ii) Now introduce an intermediate liquidity shock ρ (for a now arbitrary level A of cash on hand). The cumulative distribution of ρ is F(ρ) on [0, ∞) and the density f (ρ). The effort decision is made after the value of ρ is realized and, of course, conditional on the choice of continuing (incurring reinvestment cost ρ). Suppose that the entrepreneur's stake in the case of continuation (Rb) is independent of ρ. Write the investors' breakeven condition. Write the optimization program yielding (Rb, ρ∗), where ρ∗ is the cutoff liquidity shock.