<p>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 :</p>

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

<p>, $i, j \in \{1,2\}$ and $i \neq j$.</p>

<p>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} &amp; \ \ \pi_i(p_i, p_j) \end{eqnarray*}</p>

<p>and we'll obtain </p>

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

<p>$(p_1^*, p_2^*)$ 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>

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

<p>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.</p>

CC BY-SA 4.0
2018-06-12T17:17:59.520Z
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
CC BY-SA 4.0
2018-06-12T17:33:56.203Z
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
CC BY-SA 4.0
2018-06-12T17:44:38.827Z
I see. Now I understand it. Thank you very much.
Stefanos Makridis
CC BY-SA 4.0
2018-06-12T18:25:27.393Z
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. https://economics.stackexchange.com/questions/22432/stackelberg-setup
Stefanos Makridis
CC BY-SA 4.0
2018-12-26T21:29:24.097Z
why is 200 < pj ≤ 400 Nash equilibrium if the other one can keep lowing the price as in 400 < pj ≤ 700??
Zulita Fernandez
CC BY-SA 4.0
2018-12-27T17:53:05.467Z
@ZulitaFernandez There are three main possible results from a firm's price: they could get all the demand, half the demand, or none. Going from no demand to half is always good, but since each firm can produce only $300$ units, going from half to all benefits the first only if half the demand is less than $300$, i.e. total demand is less than $600$, which happens when the price is $400$. Once the price falls below $400$, getting all the market confers no benefit, so once a firm gets half the market, they have no incentive to lower their price further.
Acccumulation
CC BY-SA 4.0
2018-06-12T13:52:54.123Z
2018-06-12T14:01:43.180Z
2018-06-12T14:01:43.180Z
Amit
<p><a href="https://i.sstatic.net/RA6nv.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/RA6nv.jpg" alt="Exercise"></a></p>

<p>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. <a href="https://i.sstatic.net/GYJ8c.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GYJ8c.jpg" alt="Picture 2 "></a></p>

<p>Any help will be appreciated.Thank you. </p>

<microeconomics><game-theory><self-study>
Bertrand game - Nash equilibrium
Stefanos Makridis
# Bertrand game - Nash equilibrium

(Asked by: username_0 on 2018-06-12)

[![Exercise](https://i.sstatic.net/RA6nv.jpg)](https://i.sstatic.net/RA6nv.jpg)

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. [![Picture 2 ](https://i.sstatic.net/GYJ8c.jpg)](https://i.sstatic.net/GYJ8c.jpg)

Any help will be appreciated.Thank you.

**Tags:** microeconomics, game-theory, self-study

---

## 1 answers

### Answer by username_1 on 2018-06-12. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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.  
> **username_0**: I see. Now I understand it. Thank you very much.  
> **username_0**: 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. https://economics.stackexchange.com/questions/22432/stackelberg-setup  
> **username_2**: why is 200 < pj ≤ 400 Nash equilibrium if the other one can keep lowing the price as in 400 < pj ≤ 700??  
> **username_3**: @ZulitaFernandez There are three main possible results from a firm's price: they could get all the demand, half the demand, or none. Going from no demand to half is always good, but since each firm can produce only $300$ units, going from half to all benefits the first only if half the demand is less than $300$, i.e. total demand is less than $600$, which happens when the price is $400$. Once the price falls below $400$, getting all the market confers no benefit, so once a firm gets half the market, they have no incentive to lower their price further.  

---

