Question

An industry has 3 firms (1, 2 and 3) in Cournot competition. They have no fixed costs, and their constant marginal costs are respectively\(c_1=\frac{9}{30} , 𝑐_2 =\frac{ 10}{30} , 𝑐_3 = \frac{11}{30} .\)
They face an industry inverse demand function 𝑃=1−𝑄, where 𝑃 is the market price and 𝑄 is the industry output (sum of outputs of the 3 firms). Suppose that 𝑄 𝑐 is the industry output under Cournot-Nash equilibrium. Then (𝑄 ^𝑐 )⁻¹ is equal to _______ (in integer).

Answer 1

Correct Answer: 2

Given:
Inverse demand: \( P = 1 - Q \), where \( Q = q_1 + q_2 + q_3 \).
Marginal costs: \[ c_1=\frac{9}{30},\quad c_2=\frac{10}{30},\quad c_3=\frac{11}{30}. \] 

Step 1 — Cournot best-response functions 
Firm \(i\)'s profit: \[ \pi_i = q_i(1 - q_1 - q_2 - q_3 - c_i). \] FOC: \[ 1 - q_i - Q_{-i} - c_i - q_i = 0 \] \[ \Rightarrow q_i = \frac{1 - c_i - Q_{-i}}{2}. \] 

Step 2 — Substitute the marginal costs
Compute \(1 - c_i\): \[ 1 - c_1 = 1 - \frac{9}{30} = \frac{21}{30} = 0.7, \] \[ 1 - c_2 = 1 - \frac{10}{30} = \frac{20}{30} = 0.666\dots, \] \[ 1 - c_3 = 1 - \frac{11}{30} = \frac{19}{30} = 0.6333\dots. \] Thus: \[ q_1 = \frac{0.7 - (q_2 + q_3)}{2}, \] \[ q_2 = \frac{0.6667 - (q_1 + q_3)}{2}, \] \[ q_3 = \frac{0.6333 - (q_1 + q_2)}{2}. \] 

Step 3 — Solve the system
Add the three best responses: \[ q_1 + q_2 + q_3 = \frac{0.7 + 0.6667 + 0.6333 - (q_1+q_2+q_3)\cdot 2}{2}. \] Let \(Q_c = q_1 + q_2 + q_3\). Then: \[ Q_c = \frac{2 - 2Q_c}{2} \] \[ Q_c = 1 - Q_c \] \[ \Rightarrow 2Q_c = 1 \] \[ \Rightarrow Q_c = 0.5. \] 

Step 4 — Compute \((Q_c)^{-1}\)
\[ (Q_c)^{-1} = \frac{1}{0.5} = 2. \] 

Final Answer: 2