Question

A duopoly faces inverse demand \(p=120-Q\) (Rs.), with total output \(Q=q_A+q_B\). Firm A’s constant marginal cost is Rs.\ 20, which is exactly half of Firm B’s constant marginal cost. There are no fixed costs. If a Cournot–Nash equilibrium exists, \(Q\) is _____(in integer).

Hint:

For \(p=a-bQ\) with constant marginal costs \(c_i\), Cournot reactions are \(q_i=\dfrac{a-c_i-b q_j}{2b}\). With different costs, solve the two linear equations to get each firm’s output and sum for \(Q\).

Answer 1

Correct Answer: 60

Let \(a=120\), \(b=1\), \(c_A=20\), \(c_B=40\). Cournot best responses (linear demand): \[ q_A=\frac{a-c_A-b q_B}{2b}=\frac{120-20-q_B}{2}=50-\frac{q_B}{2}, \] \[ q_B=\frac{a-c_B-b q_A}{2b}=\frac{120-40-q_A}{2}=40-\frac{q_A}{2}. \] Solve simultaneously: substitute \(q_B\) into \(q_A\), \[ q_A=50-\frac{1}{2}\!\left(40-\frac{q_A}{2}\right) =50-20+\frac{q_A}{4} \Rightarrow \frac{3}{4}q_A=30 \Rightarrow q_A=40. \] Then \(q_B=40-\frac{q_A}{2}=40-20=20\).
Therefore \(Q=q_A+q_B=40+20=\mathbf{60}\).