# Bertrand game - Nash equilibrium

The quantity is limited to 300 but the monopoly quantity is equal to 400 and gives a monopoly price of 600. But if we plug the quantity of 300 into the demand function we get a price of 700.But I am confused.

Any help will be appreciated.Thank you.

Firm $i$'s profits $(\pi_i)$ as a function of its own price $(p_i)$ and the other firm's price $(p_j)$ are as follows :

\begin{eqnarray*} \pi_i(p_i, p_j) = \begin{cases} (p_i-200)\min(1000- p_i, 300) & \text{if } p_i < p_j \\ (p_i-200)\min\left(\frac{1000- p_i}{2}, 300\right) & \text{if } p_i = p_j \\ 0 & \text{if } p_i > p_j\end{cases} \end{eqnarray*}

, $i, j \in \{1,2\}$ and $i \neq j$.

We now find the best response correspondence of firm $i$ $(\text{BR}_i(p_j))$ by solving the following problem \begin{eqnarray*} \max_{0 \leq p_i \leq 1000} & \ \ \pi_i(p_i, p_j) \end{eqnarray*}

and we'll obtain

\begin{eqnarray*} \text{BR}_i(p_j) = \begin{cases} \{700\} & \text{if } p_j > 700 \\ \emptyset & \text{if } 400 < p_j \leq 700 \\ \{p_j\} & \text{if } 200 < p_j \leq 400 \\ \{p : p \ge 200\} & \text{if } p_j = 200 \\ \{p : p > p_j\} & \text{if } p_j < 200\end{cases} \end{eqnarray*}

$(p_1^*, p_2^*)$ is a Nash equilibrium of this game if it satisfy $p_1^* \in \text{BR}_1(p_2^*)$ and $p_2^* \in \text{BR}_2(p_1^*)$. This yields the following set of Nash equilibria :

$\{(p_1^*, p_2^*) : 200 \leq p_1^*= p_2^* \leq 400\}$

i.e., any action profile where both firms charge the same price, and that price lies in the interval $[200, 400]$ is a Nash equilibrium of the game.

• Thanh you so much Amit. I have one question why in the range of 400 to 700 the price that the firm chooses does not exist? – Stefanos Makridis Jun 12 '18 at 17:17
• When $400 < p_j \leq 700$, firm $i$’s profit increases as $p_i$ increases to $p_j$, but it drops abruptly at $p_j$. So there is no best response: firm $i$ wants to choose a price less than $p_j$, but is better off the closer that price is to $p_j$. For any price less than $p_j$ there is a higher price that is also less than $p_j$, so there is no best price. – Amit Jun 12 '18 at 17:33
• I see. Now I understand it. Thank you very much. – Stefanos Makridis Jun 12 '18 at 17:44
• Dear Amit I have an another question regarding Stackelberg setup. If you have time , I would appreciate to have a look. Thank you in advance. economics.stackexchange.com/questions/22432/stackelberg-setup – Stefanos Makridis Jun 12 '18 at 18:25
• why is 200 < pj ≤ 400 Nash equilibrium if the other one can keep lowing the price as in 400 < pj ≤ 700?? – Zulita Fernandez Dec 26 '18 at 21:29