Question

There are two firms in an oligopolistic industry competing in prices and selling a homogeneous product. Total cost of production for firm $i$ is \[ C_i(q_i) = 10q_i, \quad i = 1, 2; \] where $q_i$ is the quantity produced by firm $i$. Suppose firm $i$ sets price $p_i$ and firm $j$ sets price $p_j$. The market demand faced by firm $i$ is given by \[ q_i(p_i, p_j) = \begin{cases} 100 - p_i, & \text{if } p_i < p_j, \\ 0, & \text{if } p_i > p_j, \\ \dfrac{100 - p_i}{2}, & \text{if } p_i = p_j, \end{cases} \] for all $i, j = 1,2$ and $i \neq j$. Price can only take integer values in this market. Nash equilibrium/equilibria is/are given by

• A. $p_1 = 10, \ p_2 = 10$
• B. $p_1 = 12, \ p_2 = 12$
• C. $p_1 = 40, \ p_2 = 40$
• D. $p_1 = 11, \ p_2 = 11$

Hint:

In Bertrand competition with identical costs and homogeneous goods, Nash equilibrium price equals marginal cost.

Answer 1

The Correct Option is A, B, D

Step 1: Profit function of firm $i$.
If $p_i<p_j$, $\pi_i = (p_i - 10)(100 - p_i)$.
If $p_i = p_j$, $\pi_i = (p_i - 10)\frac{100 - p_i}{2}$.
If $p_i>p_j$, $\pi_i = 0.$
Step 2: Best response for firm $i$.
Firm $i$ chooses $p_i$ to maximize $\pi_i$. For $p_i<p_j$: \[ \frac{d\pi_i}{dp_i} = (100 - p_i) + (p_i - 10)(-1) = 90 - 2p_i. \] Set to 0: $p_i = 45.$ However, this yields $\pi_i = (45 - 10)(55) = 1925$ only if $p_i<p_j.$
Step 3: Check price undercutting logic (Bertrand model).
Since both firms have identical cost $c = 10$, equilibrium occurs where $p_1 = p_2 = c$. If any firm sets a slightly lower price, it captures the entire market; if price<10, profit<0. Thus, $p_1 = p_2 = 10$ is a Nash equilibrium: neither firm can increase profit by deviating.
Step 4: Conclusion.
Hence, equilibrium prices are $p_1 = p_2 = 10$.