Suppose the market has two firms, and market demand is p = 200 - 4y. The cost functions for all firms are C(y_i)= 600 + 30y_i. The equilibrium number of firms this market can support under Cournot is: A)2. B)5. C)3. D)4.

Question

Quizplus · Accepted Answer

In order to find the equilibrium number of firms in a Cournot model, we need to first find the best response function for each firm. The best response function shows how much output each firm will produce given the output of the other firm. To find the best response function, we set the derivative of profit with respect to output equal to zero:

π1 = (p - c1(y1 + y2))y1
π1 = (200 - 4(y1 + y2) - (600 + 30y1))y1
π1 = (200 - 4y1 - 4y2 - 600 - 30y1)y1
π1 = (600 - 34y1 - 4y2)y1

Taking the derivative with respect to y1 and setting it equal to zero, we get:

dπ1/dy1 = 600 - 68y1 - 4y2 = 0
68y1 = 600 - 4y2
y1 = (600 - 4y2)/68

This is the best response function for firm 1. We can find the best response function for firm 2 in a similar way:

π2 = (p - c2(y1 + y2))y2
π2 = (200 - 4(y1 + y2) - (600 + 30y2))y2
π2 = (200 - 4y1 - 4y2 - 600 - 30y2)y2
π2 = (200 - 34y2 - 4y1)y2

Taking the derivative with respect to y2 and setting it equal to zero, we get:

dπ2/dy2 = 200 - 4y1 - 68y2 = 0
68y2 = 200 - 4y1
y2 = (200 - 4y1)/68

Now we can substitute y2 into the first equation to get the reaction function for firm 1, and then substitute that into the second equation to get the reaction function for firm 2. We can then solve for the Cournot equilibrium by setting y1 = y2:

y1 = (600 - 4(200 - 4y1))/68
y1 = (408 - 4y1)/17
17y1 = 408 - 4y1
21y1 = 408
y1 = 408/21

y2 = (200 - 4(408/21))/68
y2 = 552/357

Therefore, the equilibrium number of firms is 2.

Suppose the Market Has Two Firms, and Market Demand Is