Question

1. In the Principal Agent experiment with asymmetric information, expected payoffs for the risky project are as follows (where RS equals revenue if successful, RU equals revenue if unsuccessful, w equals wage, b equals bonus, p equals the probability of success, e_H equals the cost of high effort and e_L equals the cost of low effort):

Boss (if worker chooses high effort):              p*(RS) + (1-p)*(RU) - w - p*b

Boss (if worker chooses low effort):               RU - w

Worker (if worker chooses high effort):          w + p*b - e_H

Worker (if worker chooses low effort):           w - e_L

Suppose:

RS = 800

RU = 200

p = 0.5

e_H = 100

e_L = 50

a. (15 pts.) What minimum bonus will induce the worker to exert high effort?

b. (15 pts.) At the minimum bonus amount, will the boss’s payoff be higher under high effort than under low effort? Show work to get credit.

c. (10 pts.) Bonus: What range of bonus amount will both induce high effort from the worker and ensure the boss is better off than under low effort? Show all work to get credit.

Answer 1

(a) The worker will put higher effort if:

the payoff from high effort >= payoff from low effort

=>  w + p*b - e_H >=  w - e_L

=> 0.5*b - 100 >= -50

=> 0.5b >= 50

=> b >= 50/0.5 = 100

Thus, the minimum bonus required is 100

(b) Boss's payoff with high effort = p*(RS) + (1-p)*(RU) - w - p*b

= 0.5*800 + 0.5*200 - w - 0.5*100 = 450 - w

Boss;s payoff with low effort = RU - w = 200 - w

SInce 450 - w > 200 - w , boss's payoff is higher under high effort.

(c) For the boss ot have high payoof in low effort we have,

200 - w > 0.5*800 + 0.5*200 - w - 0.5*b

=> 200 - w > 500 - w - 0.5*b

=> 0.5b > 300

=> b > 300/0.5 = 600