4. Consider two Cournot duopolists A and B who faces the demand function, and fo

Question

4. Consider two Cournot duopolists A and B who faces the demand function,

and for firm 2 the cost function is

(a) (5 points) Write down the best response functions for both firms

(b) (15 points) Solve for the equilibrium price and quantity in the market. [Note that the quantity in the market would be the sum of quantities chosen by the two firms.]

(c) (15 points) Now suppose the firm 1 is the Stackelberg leader, i.e, he chooses the quantity first. Solve for the equilibrium price and quantity in the market.

Explanation / Answer

a) Best response functions of firm 1 and 2:

Profit1 = (4-2(y1+y2))y1 - 2y1 = 4y1 -2y12 - 2y1y2 - 2y1 = -2y12 + 2y1 - 2y1y2

Profit2 = (4-2(y1+y2))y2 - y2 = 4y2 - 2y1y2 -2y22 - y2 = 3y2 - 2y1y2 -2y22?

dProfit1/dy1 = -4y1 + 2 - y2

f.o.c: dProfit1/dy1 = 0

-4y1 + 2 - y2 = 0

y1 = 0.5 - 0.25y2 This is best response function of 1.

Now, dProfit2/dy2 = 3 - 2y1 - 4y2

F.o.C: 3 - 2y1 - 4y2 = 0

y2 = 3/4 - 0.5y1 this is bwst response function of 2.

b) From best response functions

1.5 - 2y2 = 0.5 - 0.25y2

y2 = 0.571

and y1 = 0.357

and y = y1 + y2 = 0.928

and P = 4-2y = $2.14

c) 1being the leader will substitute the reaction function of 2 into its profit function

substitute  y2 = 3/4 - 0.5y1 into profit1

Profit1 = -2y12 + 2y1 - 1.5y1 - y12 = -3y12 + 0.5y1

F.O.C: dProfit1/dy1 = -6y1 + 0.5 = 0

y1 = 0.083

Now firm 2 will put this output into its own response function

y2 = 3/4 - 0.5(0.083) = 0.708

total output = 0.083 + 0.708 = 0.791

and P = $2.41

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