Suppose in a Cournot duopoly that two firms, Firm 1 and Firm 2, face market demand $P = 50 -$ $Q$ and both have marginal cost, $M C = \$ 20$. Firm $1 ^ { \prime }$ s reaction function can be written as:
A) $Q _ { 1 } = 45 - 0.5 Q _ { 2 }$
B) $Q _ { 1 } = 25 - 0.5 Q _ { 2 }$
C) $Q _ { 1 } = 15 - 0.5 Q _ { 2 }$
D) $Q _ { 1 } = 10 - 0.5 Q _ { 2 }$

Question

Quizplus · Accepted Answer

To derive Firm 1's reaction function, we need to first find the total quantity demanded in the market as a function of both firms' output, Q. Solving for Q, we get:

Q = Q1 + Q2

Substituting in the demand equation, we get:

P = 50 - Q
Q = 50 - P

Setting marginal cost equal to marginal revenue (which is equal to price in a Cournot duopoly), we get:

20 = d/dQi (P*Qi + P*Qj)
20 = P + Qi*(dP/dQi) + Qj*(dP/dQj)
20 = (50-Q)*dQi/dQj
dQi/dQj = 20/(50-Q)

Now we can derive Firm 1's reaction function by solving for Q1 as a function of Q2:

20/(50-Q1-Q2) = 0.5
40 = 50 - Q1 - Q2
Q1 = 10 - 0.5Q2

Therefore, the answer is choice C.

Suppose in a Cournot Duopoly That Two Firms, Firm 1 P=50−P = 50 -P=50−

Suppose in a Cournot Duopoly That Two Firms, Firm 1 $P = 50 -$