1. Consider a market with inverse demand p = 11 ? Q, where Q is the sum of outputproduced by all firms in the market. Fi...

1. Consider a market with inverse demand p = 11 ? Q, where Q is the sum of output

produced by all firms in the market. Firms have identical cost functions c (q) = 2 + q.

(a) Suppose there is a single firm in this market. What price and quantity will the

firm choose? What will be consumer surplus? What will be the firm’s profit?

(b) Suppose there are n firms in the market that compete by simultaneously choosing

quantities. What quantities, q c(n) will they choose in equilibrium? (Hint:

remember that all firms are identical). How do these quantities change with the

number of firms? What is the total quantity Qc

(n) produced in equilibrium and

how does it change with the number of firms in the market? What is the total

profit ? in the industry and how does it change with n? What is the consumer

surplus and how does it change with n? Discuss with subtlety and insight.

(c) Suppose now that the number of firms is determined endogenously. What will be

the free-entry equilibrium number of firms? What is the sum of consumer surplus

and industry profits in this equilibrium?

(d) What is the efficient number of firms in the market? What is the sum of consumer

surplus and industry profits?

(e) Provide some intuition for the failure of free entry to lead to an efficient equilibrium.

(f) Would eliminating the fixed cost associated with entry eliminate the discrepancy

between the free entry equilibrium and the social optimum? Explain

2. Consider a Cournot triopoly where, prior to any possible merger, firms have the same

(constant) marginal cost and same fixed cost: Ci (qi) = 30qi + F for firms i = 1, 2, 3.

Market demand for the homogeneous product is described by the linear (inverse) demand

function: P (Q) = 150 ? Q where Q = q1 + q2 + q3 is industry output.

(a) Express the profit of each (symmetric) triopolist as a function of F.

Suppose now that firms 1 and 2 merge, with the resulting industry being a Cournot

duopoly. Through the merger, the merged firms are able to eliminate duplication of

overhead costs so that the post-merger cost function of the merged firm is given by:

C12 (q12) = 30q12 + F12 where q12 is the post-merger output of the merged entity, with

F12 (b) Compute the post-merger profit of the merged firm.

(c) Compare profits in (a) and (b) and decide how much savings in overhead cost is

necessary for the two firms to find it profitable to merge.

1

Now suppose that, instead of the previous setting, the two merging firms realize savings

in variable cost. To keep matters simple, assume away any fixed costs before or after the

merger: F = 0. Let post-merger costs facing the merged firm be: C12 (q12) = 30cq12

where c merger.

(d) Compute the post-merger profits of the merged firm. [Note: the market becomes

an asymmetric-cost Cournot duopoly.]

(e) Compare profits in (d) with pre-merger profits to determine how large the (variable)

cost savings must be for the merger to be profitable.

(f) Compute Cournot equilibrium market price before and after this merger, and decide

how large the cost savings must be for the merger to benefit consumers.

3. Consider a homogeneous good industry with two firms i = 1, 2. Market demand is

given by

P = 100 ? Q

where P is the market price and Q is aggregate output. Firms compete in quantities.

Costs of production are zero.

(a) Suppose firm 1 chooses output q1 first. Firm 2 then observes q1 and chooses its

quantity q2. Determine the equilibrium quantities and price. What are equilibrium

profits?

(b) Using firm 2’s strategy derived in part (a), show that firm 2’s profits are declining

in q1.

(c) Now consider the following variation of the game: after firm 1 has chosen q1, firm

2 observes q1 and has to decide whether to enter the industry or not. If firm 2

enters, it has to pay a fixed fee F and then chooses its quantity q2. If firm 2 does

not enter, q2 = 0 and its profits are zero. Which output q1 does firm 1 have to set

in order to deter firm 2’s entry?

(d) Assuming that firm 1 sets output to deter entry as in (c), how does price vary

with F? Explain (You may assume throughout the remainder of this problem

that the quantity necessary to deter entry is larger than the monopoly quantity).

(e) Determine the lowest level of F so that firm 1 wants to deter entry.

4. Aike (Brand A) and Beebok (Brand B) are leading brand names of fitness shoes. The

direct demand functions facing each producer are given by

qA (pA, pB) = 180 ? 2pA + pB

qB (pA, pB) = 120 ? 2pB + pA

Assume zero production costs (cA = cB = 0) and answer the following questions.

(a) Derive the price best-response function of firm A as a function of the price set by

firm B. Show your derivations.

2

(b) Derive the price best-response function of firm B as a function of the price set by

firm A. Show your derivations.

(c) Solve for the Nash-Bertrand equilibrium prices,

p

b

A, pb

B

. Then, compute the

equilibrium output levels

q

b

A, qb

B

, the equilibrium profits

?

b

A, ?b

B

and aggregate

industry profits ?

b

A + ?

b

B.

(d) Suppose now that the two producers hold secret meetings in which they discuss

fixing the price of shoes to a uniform (brand-independent) level of p = pA = pB.

Compute the price p which maximizes joint industry profits, ?A + ?B. Then,

compute aggregate industry profits and compare them with the aggregate industry

profits made under Bertrand competition which you computed in part (c).

(e) Suppose now that the two firms merge. However, they decide to keep selling

the two brands separately and charge, possibly, different prices. Compute the

prices pA and pB which maximize joint industry profits, ?A + ?B. Then, compute

aggregate industry profits and compare them with the aggregate industry profits

under Bertrand competition which you have already computed under separate

ownership.

5. Consider a linear “Hotelling” city inhabited by two types of consumers, male (m) and

female (f), each distributed uniformly in the interval [0, 1]. There are two nightclubs

located at the extremes of the interval, that compete to attract customers. They both

have the same marginal cost, c, of supplying each type of customer. Let p

l

i be the price

charged by the club in location l to each type i = m or f. A consumer of type i who

is located in x ? [0, 1] gets utility

U

0

i

(x) = u ? p

0

i ? tx

for going to club 0, or

U

1

i

(x) = u ? p

1

i ? t(1 ? x)

for going to club 1, where t is the marginal transportation cost (same for everyone).

The value of clubbing (u) is so high that all consumers visit one or the other venue.

(a) Calculate the demand function for each of the clubs.

(b) Compute the equilibrium prices for each club.

Now suppose that male consumers enjoy an additional benefit ?nl

, when going to club

l, where nl

is the number of women going to club l, and ? is a constant. Women’s

preferences remain the same as before. Assume that 2t > ?.

(c) Calculate the demand function for each club.

(d) Calculate the equilibrium prices. (Hint: assume a symmetric equilibrium, i.e.,

p

0

m = p

1

m = pm and p

0

f = p

1

f = pf .)

(e) Interpret your results in parts (c) and (d). How do prices depend on ? and

t? Explain also what is the role of the condition 2t > ? . Without doing any

calculation, could you argue what would happen if this condition was not satisfied?