Question

Please show all your work.

Your firm has total costs equal to 40Q₁. Firm 2 has total costs of 50Q₂. The demand you face is P=4000-5Q. You are the Stackelberg leader and firm 2 is the

Stackelberg follower. What is your output and profits?

Answer 1

The price curve for the product is P = 4000 - 5(Q₁ + Q₂)

Hence, profit curve for firm 1 = F₁ = P*Q₁ - C(Q₁) = (4000 - 5 (Q₁ + Q₂))*Q₁ - 40Q₁ = 3660Q₁ - 5Q₁² - 5Q₁Q₂

Similarly, profit curve for firm 2 = F₂ = P*Q₂ - C(Q₂) = (4000 - 5 (Q₁ + Q₂)) - 50Q₂ = 3650Q₁ - 5Q₁² - 5Q₁Q₂

Hence, reaction curve for firm 1 and firm 2 are;

$Q_1 = \frac{\partial }{\partial Q1}F_{1}=3660-10{Q_{1}}-5Q_{2}$

$Q_2=\frac{\partial }{\partial Q2}F_{2}=3650-10{Q_{2}}-5Q_{1}$

Now, firm 1 is being the leader. Hence, substituting firm 2's reaction function in firm 1's profit function, we have;

$F_1=3660Q_1 - 5Q_1^2 - 5Q_1Q_2=3660Q_1 - 5Q_1^2 - 5Q_1\times \frac{(3650-5Q_1)}{11}=-\frac{30{Q_{1}}^{2}}{11}+\frac{22010Q_{1}}{11}$

Maximizinf F1, we have;

$\frac{\mathrm{d} }{\mathrm{d} Q_1}F_1==-\frac{60{Q_{1}}}{11}+\frac{22010}{11}=0$

Hence, Firm 1's output = 22010/60 = 366.83 = 367 say

Hence, Firm 1's profit is;

$F_1=-\frac{30{Q_{1}}^{2}}{11}+\frac{22010Q_{1}}{11}=\frac{24222005}{66}=367000$