Question

The world price of oil depends on the world supply of oil. The supply schedule is as follows:

World Supply (barrels per day): 4 million 6 million 8 million

World price per barrel: $25 $15 $10

The country Iran had a marginal cost of $2 per barrel to extract oil. Iraq has a marginal cost of $4 per barrel. Their oil drilling capacities are that they each can produce a maximum of 4 million ball els per day, or a minimum of 2 million barrels per day. However, they can produce nothing in between because of high fixed costs.

a.) Assume that Iran and Iraq make their strategic decisions without knowing what the other has done. Compute the payoffs for each country and create a game table to fit the situation.

Answer 1

how much would CS want to build (that is, what is CS's best response function in

capacity)?

(b) If CS and LC each had to decide how much capacity to build without knowing

the other's capacity decision, what would the one-shot Nash equilibrium be in the

amount of capacity built?

Solution:

(a) If LC builds 100 units of capacity, then CS faces a residual demand of QCS = Q ¡ 100 =

1100 ¡ p. Its marginal revenue (contribution) is then MR CS = 1100 ¡ 2QCS. Equating

this marginal revenue with CS's capacity costs of 600 yields the optimal capacity for CS as

Q¤

CS = 250 units.

The generalization of this is to solve for CS's residual demand as a function of LC's

capacity QLC. That is, QCS = Q¡QLC = 1200¡QLC ¡p. CS's total revenue is then equal

to TR CS = pQCS = (1200 ¡ QLC + QCS)QCS and its marginal revenue can be obtained

by taking the derivative of TR CS with respect to QCS (treating QLC as a constant). This

yields MR CS = 1200 ¡ QLC ¡ 2QCS. Equating this marginal revenue to marginal cost and

solving for QCS yields QCS = 300 ¡ QLC=2 as CS's optimal capacity in response to any

capacity decision by LC.

(b) Since the two ¯rms are symmetric, LC's best response to CS is analogous to CS's best

response to LC, or QLC = 300 ¡ QCS=2. A Nash equilibrium requires that Q¤

LC = 300 ¡

Q¤

CS=2 and Q¤

CS = 300 ¡ Q¤

LC=2. Substituting Q¤

LC into Q¤

CS and solving for Q¤

CS yields

Q¤

CS = 200. Substituting this amount into the LC's best response function yields Q¤

LC =

200. At these capacities the market price is p = 1200 ¡ 200 ¡ 200 = 800. Each ¯rm's pro¯ts

are then (800 ¡ 600)(200) = $40; 000.

7.5¤ Consider a market for a homogeneous product with demand given by

Q = 37:5 ¡ P=4. There are two ¯rms, each with constant marginal cost equal to 40.

a) Determine output and price under a Cournot equilibrium.

b) Compute the e±ciency loss as a percentage of the e±ciency loss under monopoly.

Solution: (a) Duopolist i's pro¯t is given by

¼i = qip(Q) ¡ C(qi) = qi[150 ¡ 4(qi + qj )] ¡ 40qi;

where the term in the square brackets comes from the demand function. The ¯rst order

condition for pro¯t maximization is given by:

150 ¡ 4(qi + qj ) ¡ 4qi ¡ 40 = 0: (1)

By symmetry, we have qi = qj = 9:166. Also, p = 150 ¡ 8qi = 76:666.