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: Write the market demand function.
\[ P=120-0.6Q \] Since
\[ Q=Q_1+Q_2 \] we can write
\[ P=120-0.6(Q_1+Q_2) \] \[ P=120-0.6Q_1-0.6Q_2 \]

Step 2: Write the profit function of firm 1.
Firm 1 revenue is
\[ TR_1=PQ_1 \] Therefore,
\[ TR_1=(120-0.6Q_1-0.6Q_2)Q_1 \]
Cost of firm 1 is
\[ C_1=6Q_1 \]
Hence, profit of firm 1 is
\[ \pi_1=(120-0.6Q_1-0.6Q_2)Q_1-6Q_1 \]
\[ \pi_1=114Q_1-0.6Q_1^2-0.6Q_1Q_2 \]

Step 3: Find firm 1's first-order condition.
Differentiate \(\pi_1\) with respect to \(Q_1\):
\[ \frac{\partial \pi_1}{\partial Q_1} = 114-1.2Q_1-0.6Q_2 \]
For profit maximization,
\[ 114-1.2Q_1-0.6Q_2=0 \]
Thus,
\[ 1.2Q_1+0.6Q_2=114 \]

Step 4: Write the profit function of firm 2.
Firm 2 revenue is
\[ TR_2=PQ_2 \] Therefore,
\[ TR_2=(120-0.6Q_1-0.6Q_2)Q_2 \]
Cost of firm 2 is
\[ C_2=0.5Q_2^2 \]
Hence, profit of firm 2 is
\[ \pi_2=(120-0.6Q_1-0.6Q_2)Q_2-0.5Q_2^2 \]
\[ \pi_2=120Q_2-0.6Q_1Q_2-0.6Q_2^2-0.5Q_2^2 \] \[ \pi_2=120Q_2-0.6Q_1Q_2-1.1Q_2^2 \]

Step 5: Find firm 2's first-order condition.
Differentiate \(\pi_2\) with respect to \(Q_2\):
\[ \frac{\partial \pi_2}{\partial Q_2} = 120-0.6Q_1-2.2Q_2 \]
For profit maximization,
\[ 120-0.6Q_1-2.2Q_2=0 \]
Thus,
\[ 0.6Q_1+2.2Q_2=120 \]

Step 6: Solve the two equations.
We have
\[ 1.2Q_1+0.6Q_2=114 \] and
\[ 0.6Q_1+2.2Q_2=120 \]
Solving these simultaneous equations gives
\[ Q_1=78.421 \] and
\[ Q_2=33.158 \]
Therefore, total output is
\[ Q=Q_1+Q_2 \] \[ Q=78.421+33.158 \] \[ Q=111.579 \]
Rounded off to two decimal places,
\[ Q=111.58 \]

Step 7: Final conclusion.
Hence, the total output produced by the duopolists is
\[ \boxed{111.58} \]