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Problem Set 4
Q1. A retailer faces downstream demand of P=240-2Q and has one supplier who has 0
marginal cost. The retailer’s only cost is what it pays the supplier. What will happen in
this market in terms of price, quantity and profits? What would happen if there were two
retailers who competed according to the Cournot model? What would happen if one of
the supplier became vertical integrated and only sold its output through its own retailer?
The Case of one Supplier and one Retailer:
The MR of the retailer would be MR=240-4Q. Since they will set MC which is just the
price of the supplier equal MR, we have that demand for the supplier is given by P=2404Q. Therefore the supplier’s MR=240-8Q. as the suppliers MC=0 we have MR=MC
implies 240-8Q=0 or Q=30. The supplier’s price is 120 and the retailers’ price is 180.
The supplier’s profits would be 120*30=3600 and the retailer’s profits would be (180120)*30=1800.
The Case of Two Retailers and one supplier:
Retailer 1 will have demand given by P=240-2q1-2q2 so it will have MR=240-4q1-2q2.
The retailer will set MR=MC so we will have Ps=240-4q1-2q2, where Ps is the supplier’s
price. By symmetry we know that q1=q2 so we have 240-Ps=6q1 or q1= 40-Ps/6. So the
quantity the supplier will sell is Q= 80-Ps/3. We can rewrite this as 240-3Q=Ps. So
marginal revenue for the supplier is MR=240-6Q. Since MC=0 we have the optimal
quantity is 40, at a price of 120. Each firm will buy 20 units and the market price given
by the original demand curve is 160. The supplier’s profit is 4800 and each retailer
makes (160-120)*20=800.
The vertically integrated firm:
MR=240-4Q (of the consumers) and MC=0 (for the producer) so the optimal quantity is
60 and the price is 120. The profits would be 7200.
Q2. A worker has preferences given by u(w,e) = w - .5e2 where w denotes wealth and e
denotes effort. This worker has an outside offer that will earn her a utility of u*=10.
Sales revenue is increasing in the worker’s effort, TR= 20e, so the marginal benefit of
effort is 20. The only cost to this firm is the wage it must pay to the worker. To get the
worker to be willing to accept the job and exert a given level of effort, what does her
wage offer have to be (that is what is the individual rationality constraint)? What is the
expression for the firm’s profit? The marginal cost of effort is just e (the cost of effort is
.5e2 so the derivative is just e). Assuming that effort is observable, what level of effort
should the firm specify in its contract, what wage should it pay and what profit will the
firm earn?
The worker must get w - .5e2 > 10 or w > 10+ .5e2. The firm’s profits would be 20ew=20e-10-.5e2. Taking the first derivative and setting it equal to 0 we find that the
optimal effort is e=20. The wage would need to be 10+.5(20)2= 210. The firm’s profits
would be 20*20-210=190.
Q3. The value of a worker’s output is 10e and the cost to the worker of exerting effort is
e2. Supposing that effort is unobservable and that the worker has an outside offer with a
payoff of 5, what contract can the employer write to earn the maximum profit?
Since the effort is unobservable, the worker will always choose 0 effort for any flat wage.
The key is that worker needs to internalize the benefits of the effort which can be
achieved by selling the worker the right to their output (this is just an externality
problem). The worker will accept a contract that yields a payoff of at least 5. Therefore
it must be that 10e – e2 – fee > 5. Once the worker accepts the contract they will select e
to maximize 10e – e2 – fee which is where the first derivative equals 0. That is where 10
– 2e – 0 = 0 or e=5. The largest fee that the firm can charge for the right to the output is
where 10(5) – (5)2 – fee = 5 or fee = 20.
Q4. The value of a worker’s output is (8/3)e1/3 and the cost to the worker of exerting
effort is e. Supposing that effort is unobservable and that the worker has an outside offer
with a payoff of 3, what contract can the employer write to earn the maximum profit?
This is similar to problem 3. It must be that (8/3)e1/3 – e – fee > 3. The optimal effort is
where (1/3)*(8/3)e-2/3 – 1 = 0 or (8/9)e-2/3 = 1 or e-2/3 = 9/8 or e2/3 = 8/9 or e = (8/9)3/2
=0.838. So the fee should equal (8/3)e1/3 – e – 3 = -1.3. This means that the firm would
have to pay the worker for the franchise. The firm would be better of not paying
anything and earning a profit of 0 and hence there should be no production or contract in
this case. (To be honest, I mistyped the problem which was not supposed to be this ugly
but it is a good mistake in that it brought up a situation that otherwise you might not have
thought about occurring, namely one in which shutting down is optimal.)