Consider a market that only includes two large firms. The (inverse) market demand is P = 100 – Q.3q2.Firm 1 has a cost function of C, = 2q1, and firm 2 has a cost function of C2Use a Cournot model to calculate the Nash equilibrium outputs q, and q2 of the two firms.and 92(a) Give each firm's profit as a function of(b) Compute the Nash equilibrium q, and q2.

Answered: Consider a market that only includes…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.5P

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Question

Consider a market that only includes two large firms. The (inverse) market demand is P = 100 – Q.
3q2.
Firm 1 has a cost function of C, = 2q1, and firm 2 has a cost function of C2
Use a Cournot model to calculate the Nash equilibrium outputs q, and q2 of the two firms.
and 92
(a) Give each firm's profit as a function of
(b) Compute the Nash equilibrium q, and q2.

Expert Solution

Step 1

Answer:

Given:

Inverse market demand function:

$P = 100 - Q W h e r e, Q = q_{1} + q_{2}$

$F i r m 1 \cos t f u n c t i o n : C_{1} = 2 q_{1} F i r m \cos t f u n c t i o n : C_{2} = 3 q_{2}$

(a).

Profit: it refers to the difference between the total revenue and total cost.

$T o t a l r e v e n u e = P r i c e \times Q u a n t i t y T o t a l r e v e n u e f o r f i r m 1 = P \times q_{1} = (100 - (q_{1} + q_{2})) \times q_{1} T o t a l r e v e n u e f o r f i r m 1 = 100 q_{1} - {q_{1}}^{2} - q_{1} q_{2} T o t a l \cos t = 2 q_{1} P r o f i t = T o t a l r e v e n u e - T o t a l \cos t P r o f i t o f f i r m 1 (π_{1}) = 100 q_{1} - {q_{1}}^{2} - q_{1} q_{2} - 2 q_{1} . . . . . . . . . . . . . . . e q u a t i o n 1 N o w, T o t a l r e v e n u e f o r f i r m 2 = P \times q_{2} = (100 - (q_{1} + q_{2})) \times q_{2} T o t a l r e v e n u e f o r f i r m 2 = 100 q_{2} - q_{1} q_{2} - {q_{2}}^{2} T o t a l \cos t = 3 q_{2} P r o f i t = T o t a l r e v e n u e - T o t a l \cos t P r o f i t o f f i r m 2 (π_{2}) = 100 q_{2} - q_{1} q_{2} - {q_{2}}^{2} - 3 q_{2} . . . . . . . . . . . . . . . . . e q u a t i o n 2$

Equation 1 1and equation 2 are the profit functions of firm 1 and firm 2 respectively.

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