Question

Two firms compete by setting quantities simultaneously (Cournot Competition) in a market where demand is described by � = 100 − 2(�! + �"). The marginal cost of production for Firm 1 and Firm 2 is $6 and $10 respectively. a. Derive the reaction function of each firm. b. Compute the Cournot equilibrium quantities. Note: answers may be fractions.

Answer 1

(a).

(i). s1 = 1/4 ( 94 - 2s1), (ii). s2 = 1/4 (90 - 2s1).

(b). (i). s1 = 49/3, (ii). 43/3.

Step-by-step explanation:

We are given that;

J = 100 – 2(s1 + s2).

Therefore, J = 100 - 2s1 - 2s2.

(a).

(i).For firm one;

Js1 = 100s1 - 2s1^2 - 2s1s2. ---------------(1).

Differentiate the equation above to give;

d Js1/d s1 = 100 - 4s1 - 2s2. -------------(2).

The expression (2) above is known as Marginal revenue 1.

Recall that for profit maximization; Marginal revenue = marginal cost.

Hence, 6 = 100 - 4s1 - 2s2.

Therefore, we have;

94 = 4s1 + 2s2.

The reaction function of firm 1;

s1 = 1/4 ( 94 - 2s1).

(ii). For firm two;

Js2 = 100 - 2s1 - 4s2.

Recall that for profit maximization; Marginal revenue = marginal cost.

10 = 100 - 2s1 - 4s2.

The reaction function of firm two;

s2 = 1/4 (90 - 2s1).

(b).

(I). s1 = 1/4 ( 94 - 2s1).

s1 = 1/4 ( 94 - 2 × (1/4) 90 - 2s1).

s1 = 1/4 ( 94 - 45 + s1).

s1 = 1/4 ( 49 + s1).

Solve for s1.

s1 = 49/3.

(ii). s2 = 1/4 (90 - 2s1).

s2 = 1/4 (90 - 2 × 49/3).

s2 = 43/3