Question

Suppose that the demand function facing two duopolistic firms under Cournot's model is Q = 250 - (10/25)p, where Q = Q1 + Q2, and the cost functions are C1 = 10Q and C2 = 20Q2.

a) Derive the reaction function of each duopolist.
A) Q1 = (1000 - 2.5p)/3
B) Q2 = (1000 - 2.5p)/3
C) Q1 = (100 - 0.25p)/3
D) Q2 = (100 - 0.25p)/3

Answer 1

The reaction function of each duopolist is Q1 = (p - 10)/10 and Q2 = (p - 20)/10.

Step-by-step explanation:

The reaction function of each duopolist can be derived by maximizing their profits. The profit function for each firm can be calculated by subtracting the cost function from the revenue function, which is the price multiplied by the quantity (Q). To find the reaction function for each duopolist, we need to differentiate the profit function with respect to Q and set it equal to zero.

For firm 1:

Profit = (p - 10)Q1

dProfit/dQ1 = p - 10 - 10Q1 = 0

Q1 = (p - 10)/10

For firm 2:

Profit = (p - 20)Q2

dProfit/dQ2 = p - 20 - 10Q2 = 0

Q2 = (p - 20)/10

Therefore, the reaction function of each duopolist is:

Q1 = (p - 10)/10

Q2 = (p - 20)/10