Question

Let us consider a duopoly market with the following market demand and firm-specific cost functions:

Hint:

In a Cournot-type duopoly problem, write each firm’s profit function separately, take the first-order condition with respect to its own output, and solve the reaction equations simultaneously.

Answer 1

Correct Answer: 111.58

Step 1: Set up the demand.
With $P=120-0.6Q$ and $Q=Q_1+Q_2$,
\[ P=120-0.6Q_1-0.6Q_2 \]

Step 2: Firm 1's reaction.
Profit is $\pi_1=(P)Q_1-6Q_1$. Setting its derivative in $Q_1$ to zero,
\[ 1.2Q_1+0.6Q_2=114 \]

Step 3: Firm 2's reaction.
Profit is $\pi_2=(P)Q_2-0.5Q_2^2$. Setting its derivative in $Q_2$ to zero,
\[ 0.6Q_1+2.2Q_2=120 \]

Step 4: Solve the two lines.
Solving gives $Q_1\approx78.42$ and $Q_2\approx33.16$.

Step 5: Add the outputs.
\[ Q=78.42+33.16=111.58 \]
\[ \boxed{111.58} \]