1. a) Consider the principal-agent problem discussed in lecture, with the constraint that s = 0: The purpose of this problem is to derive the payoff (utility) possibility frontier. Data for the problem: Principal's utility is E(y - w), Agent's utility is Ew - a2=2, y = a + ; where E = 0:, w = by.

a) What effort level a does the Agent choose if she accepts a contract with s = 0 and bonus rate b? What utility level does the she derive from this the contract expressed as a function of b?

b) What utility does the Principal derive from this contract, also expressed as a function of b?

c) Now solve for b in terms of the Agent's utility u. Use that to express the Principal 's payoff u as a function of u.

d) Graph the payoff possibility set (u as a function of u). Label points on it corresponding to b = 0, b = 1=4, b = 1=2, b = 3=4, b = 1.

e) What is the optimal contract if we are only interested in the total value of the enterprise comprising Principal and Agent? What is optimal for the Principal, if the agent's outside option is worth (i) 0? (ii)1/8? (iii) 1/16? (iv) 9/32?