Compute the entrepreneurs expected utility - what is the

Question

Dealing with asset substitution:-

Consider the fixed-investment model with a probability that the investment must be resold (redeployed) at an intermediate date because, say, it is learned that there is no demand for the product. The timing is summarized in Figure 7.16. An entrepreneur has cash A and wants to invest a fixed amount I > A into a project. The shortfall must be raised in a competitive capital market. The project yields R with probability p and 0 with probability 1 - p, provided that there is a demand for the product (which has probability x and is revealed at the intermediate stage; the final profit is always 0 if there is no demand, and so it is then optimal to liquidate at the intermediate stage).

Investors and entrepreneur are risk neutral, the latter is protected by limited liability, and the market rate of interest is 0.

(i) In a first step, ignore the possibility of asset substitution. The liquidation value is L = L₀, and the probability of success is pH if the entrepreneur works and p_L = p_H - ?_p if she shirks (in which case she obtains a private benefit B). Assume that the NPV of the project is positive if the entrepreneur works, and negative if she shirks. Assume that A ≥ A, where

• Interpret (1).
• Compute the entrepreneur's expected utility.
• What is the class of optimal contracts (or, at least, characterize the optimal contract for A = A)? (ii) Suppose now that, before the state of demand is realized, but after the investment is sunk, the entrepreneur can engage in asset substitution. She can reallocate funds between asset maintenance (value of L) and future profit (as characterized by the probability of success, say). More precisely, suppose that the entrepreneur chooses L and
• the probability of success is pH + τ(L) if the entrepreneur behaves and pL + τ(L) if she misbehaves;

• the function τ is decreasing and strictly concave;

• the entrepreneur secretly chooses L (multitasking). Consider contracts in which
• liquidation occurs if and only if there is no demand (hence, with probability x);
• the entrepreneur receives r_b(L) if the assets are liquidated, and Rb if they are not and the project is successful (and 0 if the project fails)

Interpret (2). Compute the minimum level of A such that the threat of (excessive) asset substitution is innocuous. Interpret the associated optimal contract. (Hint: what is the optimal asset maintenance (liquidation value)? Note that, in order to induce the entrepreneur to choose this value, in the case of liquidation you may pay r_b(L) = r_b if L is at the optimal level and 0 otherwise.)