# There is a principal and an agent. The agent.s ability (or .) is either Low (a=0) or High (a=10). Only the agent knows his true type. The principal…

1. There is a principal and an agent. The agent.s ability (or .type.) is either Low (a=0) or

High (a=10). Only the agent knows his true type. The principal thinks Pr{a=0}=0.5.

Output is either q=0 or q=20. Output is surely q=20 if the agent has high ability.

However, if the agent has low ability, then Pr{q=20 | a=0}=0.8. The agent’s utility

function is U(w) = w^(1/2) and the principal’s profit is q-w: Reservation utilities: 2 for

the agent’s Low type, 4 for his High type, and 0 for the principal.

The principal offers the agent a choice: you can either work for a fixed wage w1 (which

doesn’t depend on output). Or you can choose a high-powered contract which pays w2

if q=20 but you’re paid nothing if q=0. (Of course, the agent is free to reject both

options and take his outside option instead).

What contract will the principal o.er in equilibrium? Derive numerical values for w1

and w2.

2. Joe owns and operates an hot dog stand in Central Park. The hot dog stand is worth

v to Joe. Joe, of course, knows v. Susan doesn’t know v. She thinks that v is equally

likely to be any number between 60 and 100 (i.e., a uniform distribution on [60; 100]).

But Susan knows she would be better at selling hot dogs than Joe (who is kind of

lazy). She thinks that if she buys the hot dog stand, then its value will increase by

20%. Thus, whatever is the true v, if the hot dog stand is worth v to Joe, then it

will be worth (1:2)v to Susan. Susan is now considering making a bid for the hot dog

stand. The bid will be .take-it-or-leave-it., i.e., Joe.s only choice is to either accept

or reject the bid. Let b denote the bid. If the bid is accepted, Susan.s payo. will be

(1:2)v-b and Joe gets b. If the bid is rejected, Susan’s payoff will be 0 and Joe gets

v. Susan doesn’t know the true v, so she has to consider her expected payoff

(a) What is the highest bid Susan can make without expecting to lose on the deal?

(That is, find the highest b such that Susan.s expected payoff. is non-negative.)

(b) What is Susan’s optimal bid? (That is, find the bid b which maximizes Susan’s

expected payoff)

1 Thus,