Question

In a certain market, the elasticity of demand is -2. Three firms share the market. Firm A has a market share of 60%, Firm B has a market share of 30%, and Firm C has a market share of 10%. Recall that the Lerner index or price-cost margin is defined as the fraction of price that represents a markup over marginal cost: (P-MC)/P. Assume this market is an asymmetric Cournot oligopoly and compute the Lerner index for each firm based on its market share and the market demand elasticity.

a. Compute the Lerner index for Firm A.

b. Compute the Lerner index for Firm B.

c. Compute the Lerner index for Firm C.

d. Which firm has lowest Lerner index?

e. Which firm has lowest marginal cost?

Answer 1

Lerner index L = (P - MC)/P

At equilibrium MR = MC

P(1+1/e) = MC

1 + 1/e = MC/P

1 - MC/P = - 1/e

(P - MC)/P = - 1/e

So L =  (P - MC)/P = - 1/e

e = -2

L = - 1/(-2)

= 1/2

a)

L of firm A = 60% of 1/2

= (60/100)1/2

= 3/10

b) L of firm B = 30% of 1/2

= (30/100)1/2

= 3/20

c) L of firm C = 10% of 1/2

= (10/100)(1/2)

= 1/20

d) firm C has lowest lerner index

e) since Lerner index and MC are inversly related so Firm A has lowest MC because firm A has highest Lerner index among three firms