# Optimization

An all-equity rm must raise a given amount I < ½ for a new project. The manager has the chance to modify the project once it has begun. The sequence is as follows:  Period 0. The rm borrows an amount B, raises an amount S in new equity, and invests I=B+S to begin the project. The face value of debt is D, and it must be repaid in period 2. The rate of interest is zero. The cost of issuing equity is c>0 per unit.  Period 1. The manager completes the project in one of two ways. The safe completion generates a certain cash ow of X=1 in period 2. The risky completion generates cash of X=θ with probability ½, and cash of X=0 with probability ½, where θ<2.  Period 2. Revenue is realized and all proceeds paid out. Assumption: When making the completion decision in period 1, the manager maximizes the wealth of shareholders. Assume: D < 1 and θ > 1 for simplicity. 1. Find the period 1 condition that determines whether the safe or risky project is adopted. 2. Find the face value of the debt, D, as a function of the amount borrowed, B. 3. Write down the rm’s optimization problem and characterize the optimal amount of debt as a function of θ and I. What might be a testable implication? 4. Prove that shareholders would be better o if the manager did not have the option to choose the risky project.

butwhy, you’re a master’s student, you should be able to solve this.