Question

Two firms face the following demand curve: P = 50 – 5Q, where Q = Q1 + Q2. The firms cost functions are C1 (Q1) = 20 + 10Q1 for firm 1 and C2 (Q2) = 10 + 12Q2 for firm 2.
Suppose both firms have entered the industry. What is the joint profit maximizing level of output

Answer 1

Joint profit maximizing level of output means that we need to maximize sum of profits of both firms, i.e. act like a cartel:

$\pi=\pi1+\pi2 \to max(Q1, Q2)$

Profit of each firm equials earnings minus costs:

$\pi1 = Q1*P(Q1+Q2) - C1(Q1)$

$\pi2= Q2*P(Q1+Q2) - C2(Q2)$

This gives us equation for total profit:

$\pi = Q1∗P(Q1+Q2)−C(Q1) +Q2∗P(Q1+Q2)−C(Q2)\\\\=P(Q1+Q2)*(Q1+Q2) - C(Q1)- C(Q2)$

In order to maximize $\pi$ we need to find its partial derivatives with respect to $Q1$ and $Q2$:

$$\frac{\partial \pi }{\partial Q1}= P(Q1+Q2)+(Q1+Q2)* \frac{\partial P}{\partial (Q1+Q2)}*\frac{\partial (Q1+Q2)}{\partial Q1} -\frac{\partial C1}{\partial Q1}$$

$\frac{\partial \pi }{\partial Q2}= P(Q1+Q2)+(Q1+Q2)* \frac{\partial P}{\partial (Q1+Q2)}*\frac{\partial (Q1+Q2)}{\partial Q2} -\frac{\partial C2}{\partial Q2}$

By plugging in given formulas for cost functions and demand curves we get:

$\frac{\partial \pi }{\partial Q1}= 50 - 5*(Q1+Q2)+(Q1+Q2)*(-5)*1-10 = \\\\=40-10*(Q1+Q2)$

$\frac{\partial \pi }{\partial Q2}= 50 - 5*(Q1+Q2)+(Q1+Q2)*(-5)*1-12 = \\\\=38-10*(Q1+Q2)$

Technically, we should calculate such $Q1$ and $Q2$ that both derivatives mentioned above would be equial to zero. In our case it's impossible, as firms have diffrent marginal costs. In such situation cartel will decide to produce only via the first firm (with minimum marginal costs), so profit maximization will effectively be maximazing $\pi1$ given that $Q2=0$: $\pi1 = Q1*P(Q1) - C1(Q1) = Q1*(50-5*Q1) - 20 -10Q1=\\\\=50Q1-5*(Q1)^2-20-10Q1=\\\\=-5*(Q1)^2+40*Q1-20$

Joint profit maximizing output is 4