Answer

Solution

1. 1

   Firm $i$'s best response: $q_i = \tfrac{1 - \kappa_i - q_j}{2}$.

2. 2

   Solving the simultaneous system,

3. 3
   $$q_i^* = \frac{1 - 2\kappa_i + \kappa_j}{3}.$$
4. 4

   The equilibrium price is $P^* = \tfrac{1 + \kappa_1 + \kappa_2}{3}$, so the equilibrium markup for firm 2 is

5. 5
   $$P^* - \kappa_2 = \frac{1 - 2\kappa_2 + \kappa_1}{3} = q_2^*.$$
6. 6

   Therefore,

7. 7
   $$\pi_2^* = (P^* - \kappa_2) q_2^* = \left(\frac{1 - 2\kappa_2 + \kappa_1}{3}\right)^2 = \frac{(1 - 2\kappa_2 + \kappa_1)^2}{9}.$$

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