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a) Generate the reaction functions for Cournot duopolists
a) Generate the reaction functions for Cournot duopolists. Both firms face the demand p(Y)=90-2Y where Y is the sum of firm outputs. Each have the same cost function c(y)=4y where y is the individual firm's output.
b) Solve the reaction functions simultaneously to find the Cournot equilibrium. What is the equilibrium price and quantity?
Expert Solution
a)
Demand, p(Y)=90-2Y
Let the output produced by firm 1 be Y_1.
Let the output produced by firm 2 be Y_2.
Firm 1:
Revenue, R_1 =P*Y_1={90-2(Y_1+Y_2)}*Y_1
Marginal revenue, MR_1=90-2(Y_1+Y_2)-2Y_1
=90-4Y_1-2Y_2
Cost of firm 1: c(y)=4Y_1
Marginal cost, MC=4
At equilibrium, MR_1=MC
90-4Y_1-2Y_2=4
4Y_1+2Y_2=86
The reaction function of firm 1 is: (86-2*Y_2)/4
R_1(Y_2)= (86-2*Y_2)/4
Similarly, firm 2 reaction function R_2(Y_1)=(86-2*Y_1)/4
b)
Reaction function of both firms:
4Y_1+2Y_2=86
4Y_2+2Y_1=86
Solving both equations Y_1=Y_2=14.33
Price, P=61.34
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