a. Reaction functions:
Q₁ = 1/4 * (200 - 3Q₂)
Q₂ = 1/4 * (200 - 3Q₁) + 28
b. Equilibrium Output:
Q₁ = 22
Q₂ = 32
c. Equilibrium Market Price:
P = 58
d. Profits in Equilibrium:
Π₁ = 396
Π₂ = 384
a. Reaction Functions:
Firm 1's profit maximization:
Profit (Π₁) = (200 - 3(Q₁ + Q₂))Q₁ - 26Q₁
Setting the derivative with respect to Q₁ to zero, we find the reaction function for firm 1:
Q₁ = 1/4 * (200 - 3Q₂)
Firm 2's profit maximization:
Profit (Π₂) = (200 - 3(Q₁ + Q₂))Q₂ - 32Q₂
Setting the derivative with respect to Q₂ to zero, we find the reaction function for firm 2:
Q₂ = 1/4 * (200 - 3Q₁) + 28
b. Equilibrium Output:
Setting the reaction functions equal to each other:
1/4 * (200 - 3Q₂) = Q₂
Solving for Q₂, we get Q₂ = 32.
Substituting Q₂ into the reaction function for firm 1:
Q₁ = 1/4 * (200 - 3 * 32)
Solving for Q₁, we get Q₁ = 22.
c. Equilibrium Market Price:
Substitute Q₁ and Q₂ into the inverse market demand function:
P = 200 - 3(Q₁ + Q₂)
P = 200 - 3(22 + 32)
P = 58
d. Profit in Equilibrium:
Substitute Q₁ and Q₂ into the profit functions for each firm:
Π₁ = (200 - 3(22 + 32)) * 22 - 26 * 22
Π₁ = 396
Π₂ = (200 - 3(22 + 32)) * 32 - 32 * 32
Π₂ = 384