Question

1. Consider a market of homogeneous products in which firms compete on price. Demand in this market is

given by

q(p) = 50 -10p

Consumers buy from the producer with the lowest price. If the prices of both firms are the same then they

purchase from E. There are both an incumbent firm M and a potential entrant E which can produce the good

at marginal costs 3 and 2 , respectively. Prior to entry, E must incur an entry cost equal to Ce is greater than or equal to 0 .

(a) Suppose that Ce = infinity . What are the equilibrium price, quantity, and surplus?

(b) Suppose that Ce = 0 . What are the equilibrium price, quantity, and surplus?

(c) What is the maximum value of Ce for which E does not make a loss if it enters?

(d) What is the maximum value of Ce for which it is optimal from a welfare perspective (i.e. total surplus)

for E to enter? (At the maximum value it is also optimal for E not to enter.)

2. Suppose that there is a single producer of a good and a single retailer. The producer’s marginal cost is 50

and the retailer’s marginal cost is the wholesale price w plus a unit retail cost equal to 50 . The producer

chooses the wholesale price and the retailer the retail price. The demand function is 200 - p .

(a) Write down the retailer’s profit function as a function of the retail price p and the wholesale price w .

(b) What is the optimal retail price choice as a function of the wholesale price?

(c) What is the corresponding quantity?

(d) What is the producer’s profit as a function of w ?

(e) What are hence the equilibrium values of w; p; q ?

(f) What are equilibrium producer and retailer pro ts (both separate and in aggregate)?

(g) Now suppose that the producer and retailer merge without a change in retail or production costs. Then

what would be the new equilibrium price, quantity, and profit?

(h) Provide a brief intuitive explanation for why the retail price is now less but profits are higher.

(i) If the producer and retailer are still separate rms, then how much of a $1 increase in the unit producer

cost gets passed through to the retailer and how much to the consumer?

(j) How would a $1 increase in the unit retail cost affect the wholesale and retail prices? Please explain.

3. Suppose that there is a nut manufacturer and a bolt manufacturer. Consumers need one of each. The cost

of producing a nut is 30 and the cost of producing a bolt is also 10 . Demand is

q(Pn; Pb) = 280 - 4pn - 4pb

(a) What are the equilibrium prices, quantities, and profits?

(b) What would be the corresponding numbers if the firms merged? You only need to set one combined price

for a nut–bolt pair.

(c) Explain why the price of a nut–bolt pair went down yet profits went up.

4. Consider a market for differentiated products with two producers. Each producer is tied to a single retailer.

Each producer firm sets a wholesale price. Then, the retailer chooses a retail price. The demand functions

are (

q1(p1, p2) = 100 - 3p1 + 2p2,

q2(p1, p2) = 100 + 2p1 - 3p2,

The production cost of each unit of either good is 56 . There are no retail costs and no fixed costs.

(a) For given wholesale prices w1;w2 , derive the optimal retail prices p1, p2 as a function of w1, w2 .

(b) Express the producers’ profit functions in terms of w1;w2 .

(c) What are the optimal wholesale prices?

(d) What are the optimal retail prices?

(e) What are retailer profits?

(f) What are producer profits?

(g) What would have been per firm profits absent a retail channel?

(h) What is the range of franchise fee amounts that would make both the producer and the retailer better

off with a retail channel?

Answer 1

1.

q(p) = 50-10p

or, p=(50-q)/10

Total Revenue(TR)=price * quantity

=[(50-q)/10]*q

Marginal revenue(MR) = dTR/dq

=5 - q/5

PART a : When Ce is infinity

E cannot enter. Because the entry costs are infinite, E will incur losses upon entry. So the entire market is captured by the incumbent(M).

Marginal cost (MC) = 3

At equilibrium, MR= MC

So, 5- q/5 = 3

or, quantity= 10

Then, p=(50-q)/10

= (50-10)/10

Price = 4

Surplus = Consumer surplus +Producer surplus

Consumer surplus(CS) =1/2* 10* (5-4)

=5

Producer surplus(PS) =Profits = (4-3)*10 = 10

Total Surplus = 10+5 = 15

Part B : When Ce is zero

Entrant captures the whole market.

Marginal cost (MC) = 2

At equilibrium, MR= MC

So, 5- q/5 = 2

or, quantity= 15

Then, p=(50-q)/10

= (50-15)/10

Price = 3.5

Surplus = Consumer surplus +Producer surplus

Consumer surplus(CS) =1/2* 15* (5-3.5)

=11.25

Producer surplus(PS) =Profits = (3.5-2)*15 = 22.5

Total Surplus = 11.25+22.5 = 33.75

PART C: MAXIMUM VALUE OF Ce

Incumbent's cost =3q

Entrant's cost = 2q +Ce

In order to not make losses,

2q+Ce < 3q

or, Ce<q