supose the following demand and cost function of duopoly firm
x=40-0.2p where x=x1+x2
c1=50+2x1+0.5x12
c2=100+10x2
drive riaction function
find cournot equilibrium quantity and price
calculate the equilubrium price and output of each firm assume firm 1 is sophisticated and firm 2 is follower firm?
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i have not a response
TC=20 P=10
Do it
x=40-0.2p, x=x1 +x2, c1 =50+2x1 +0.5x1², C2 =100+0.2P2
a, derived the isoproft function of each firm
b,derived the reaction function of each firm
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Student
answer
a) Isoprofit function of firm 1:
Profit (𝜋) = Total revenue (TR) – Total cost (TC)
Total revenue (TR) = Price (p) x Quantity (x1) = p*x1
Total cost (TC) = 50 + 2x1 + 0.5x1²
Profit (𝜋) = p*x1 - (50 + 2x1 + 0.5x1²)
Isoprofit function of firm 1:
𝜋1 = p*x1 - (50 + 2x1 + 0.5x1²)
Isoprofit function of firm 2:
Profit (𝜋) = Total revenue (TR) – Total cost (TC)
Total revenue (TR) = Price (p) x Quantity (x2) = p*x2
Total cost (TC) = 100 + 10x2
Profit (𝜋) = p*x2 - (100 + 10x2)
Isoprofit function of firm 2:
𝜋2 = p*x2 - (100 + 10x2)
b) Reaction function of firm 1:
For a given price (p), firm 1’s profit-maximizing output (x1) can be obtained by setting the derivative of 𝜋1 with respect to x1 equal to zero and solving for x1:
𝜋1 = p*(40-0.2p-x2) - (50 + 2*(40-0.2p-x2) + 0.5*(40-0.2p-x2)²)
Taking the derivative of 𝜋1 with respect to x1:
d𝜋1/dx1 = -p - 2 - (40-0.2p-x2)*0.5*(d/dx2)(40-0.2p-x2)
Setting d𝜋1/dx1 equal to zero and solving for x1:
d𝜋1/dx1 = -p - 2 - (40-0.2p-x2)*0.5*(-0.2) = 0
0.1(40-0.2p-x2) = -p - 2 - 4
0.1x1 = -p - 6 - x2
x1 = -10p - 60 - 10x2
Reaction function of firm 1:
x1 = -10p - 60 - 10x2
Reaction function of firm 2:
For a given price (p), firm 2’s profit-maximizing output (x2) can be obtained by setting the derivative of 𝜋2 with respect to x2 equal to zero and solving for x2:
𝜋2 = p*(40-0.2p-x1-x2) - (100 + 10x2)
Taking the derivative of 𝜋2 with respect to x2:
d𝜋2/dx2 = -p - 10 + (40-0.2p-x1-x2)*(-1)
Setting d𝜋2/dx2 equal to zero and solving for x2:
d𝜋2/dx2 = -p - 10 + 40 + 0.2p + x1 + x2 = 0
x2 = -p + 30 - 0.1x1
Substituting the reaction function of firm 1 into the reaction function of firm 2:
x2 = -p + 30 - 0.1(-10p - 60 - 10x2)
x2 = -1p + 24 - x2
2x2 = -1p + 24
x2 = -0.5p + 12
Reaction function of firm 2:
x2 = -0.5p + 12