For the two firms in a duopolistic industry, T, = X_1 and r_2 = X_2 ; where and

Question

For the two firms in a duopolistic industry, T, = X_1 and r_2 = X_2 ; where and T_2 denote total cost for firms 1 and 2, respectively; while X. and X_2 denote the level of output for firms 1 and 2, respectively. The demand curve for the industry as a whole is given by the following equation: P = 100 - XY - X2 where P denotes the price of the industry's (homogeneous) good. Find the values of P, X_1 and X_2 in the Cournot equilibrium If the two firms then collude to form a cartel, calculate the resulting change in profits for each of them. Find the values of P, X_1 and X_2 in the Stackelberg equilibrium, assuming that firm 1 is the leader.

Explanation / Answer

(A)

Firm 1's reaction function:

P = 100 - X1 - X2

TR1 = P * X1 = 100X1 - X12 - X2X1

MR1 = 100 - 2X1 - X2

MR1 = MC1

100 - 2X1 - X2 = 2X1

4X1 = 100 - X2

X1 = 25 - 0.25X2 (firm 1's reaction function)

similarly, firm 2's reaction function is:

X2 = 25 - 0.25X1

Putting firm 2's reaction function in firm 1's reaction function, we get

X1 = 25 - 0.25X2 = 25 - 0.25( 25 - 0.25X1)

X1 = 25 - 6.25 + 0.0625X1

0.938X1 = 18.75

X1 = 19.99 , we can make it 30 units

Then, X2 = 25 - 0.25X1 = 25 - 0.25 * 30 = 17.5 units

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