<p>This is a version of <strong>Hotelling's Model of Spatial Differentiation</strong> </p>

<p>The easiest solution to this is to find an "X" in-between [0,2] such that the customer is indifferent between going to Firm A or Firm B. </p>

<p>This will happen when 
<span class="math-container">$ P(A) + Distance(X-A) - Utility = P(B) + Distance(B-X) - Utility $</span></p>

<p>Then demand [0, X] would go to A, and demand [X, 2] will go to B. Then using Best Response Functions for this Simultaneous game we can compute prices, quantity, and profits.</p>

<p>I have attached my solution below. For more information, you can see -  <a href="https://books.google.co.in/books?id=DXcJAAAAQBAJ&amp;pg=PA544&amp;lpg=PA544&amp;dq=snyder%20nicholson%20hotelling&amp;source=bl&amp;ots=RWmQxfSq2W&amp;sig=ACfU3U3WC_orI1OuFUPLrukZ7N5PNDyOPA&amp;hl=en&amp;sa=X&amp;ved=2ahUKEwj7ofWa2ZLpAhXRzTgGHQBtCIcQ6AEwBHoECAsQAQ" rel="nofollow noreferrer">SNYDER NICHOLSON HOTELLING'S BEACH EXAMPLE</a>
<a href="https://i.sstatic.net/5nabb.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5nabb.jpg" alt="enter image description here"></a>
<a href="https://i.sstatic.net/BU0WU.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BU0WU.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2020-05-01T12:48:35.887Z
2020-05-01T12:48:35.887Z
Poorvi
<p>Consider a town with consumers represented by a closed interval $[0,2]$ with the consumers spread continuously and uniformly. There are two stores, $A$ and $B$ who sell the same product at $p_A$ and $p_B$ at no cost. </p>

<p>A consumer has a gross utility of 4 from having the product which is reduced by price paid and travel costs, which is determined as distance traveled. Each consumer only buys 1 unit of the product, and if the consumer does not buy from either store, he has a utility of 0. For example, if a consumer is located at $x=1.5$ and buys from store B at $p_B=1$ which is located at $x_B=2$, then his utility is $4-1-|2-1.5| = 2.5$. </p>

<p>Suppose that $A$ is located at $x_A=0$ and $B$ is located at $x_B=2$. Also, suppose stores much charge at $p \le 4$.</p>

<p>Suppose stores are profit maximizers. What are the equilibrium prices, quantities, and profits for both stores?</p>

<hr>

<p>My initial thoughts are that, in this scenario, the profit functions for each store is denoted as:</p>

<p>$$ \pi_A=p_A+p_A(\frac{p_B-p_A}{2}) $$</p>

<p>$$ \pi_B=p_B-p_B(\frac{p_B-p_A}{2}) $$</p>

<p>We can take the first order condition for each, to get the maximizing profit. </p>

<p>$$ \frac{\partial \pi_A}{p_A} = 1 + \frac{p_B}{2} - p_A = 0 \implies p_A = \frac{p_B}{2} + 1 $$</p>

<p>$$ \frac{\partial \pi_B}{p_B} = 1 - p_B + \frac{p_A}{2} = 0 \implies p_B = 1 + \frac{p_A}{2} $$</p>

<p>By substituting one into the other, we have that $p_A=p_B=2$, $q_A=q_B=1$ and $\pi_A=\pi_B=2$. </p>

<p>I'm not very sure about this solution or the direction I took to solve it. Any advice here would be appreciated on what other direction I should try.</p>

CC BY-SA 3.0
2018-04-29T04:06:12.967Z
Your solution looks correct to me.
Herr K.
<game-theory><nash-equilibrium>
Nash equilibrium for Bertrand Model with Spatial Differentiation
Felix Jen
# Nash equilibrium for Bertrand Model with Spatial Differentiation

(Asked by: username_0 on 2018-04-29)

Consider a town with consumers represented by a closed interval $[0,2]$ with the consumers spread continuously and uniformly. There are two stores, $A$ and $B$ who sell the same product at $p\_A$ and $p\_B$ at no cost.

A consumer has a gross utility of 4 from having the product which is reduced by price paid and travel costs, which is determined as distance traveled. Each consumer only buys 1 unit of the product, and if the consumer does not buy from either store, he has a utility of 0. For example, if a consumer is located at $x=1.5$ and buys from store B at $p\_B=1$ which is located at $x\_B=2$, then his utility is $4-1-|2-1.5| = 2.5$.

Suppose that $A$ is located at $x\_A=0$ and $B$ is located at $x\_B=2$. Also, suppose stores much charge at $p \le 4$.

Suppose stores are profit maximizers. What are the equilibrium prices, quantities, and profits for both stores?

---

My initial thoughts are that, in this scenario, the profit functions for each store is denoted as:

$$ \pi\_A=p\_A+p\_A(\frac{p\_B-p\_A}{2}) $$

$$ \pi\_B=p\_B-p\_B(\frac{p\_B-p\_A}{2}) $$

We can take the first order condition for each, to get the maximizing profit.

$$ \frac{\partial \pi\_A}{p\_A} = 1 + \frac{p\_B}{2} - p\_A = 0 \implies p\_A = \frac{p\_B}{2} + 1 $$

$$ \frac{\partial \pi\_B}{p\_B} = 1 - p\_B + \frac{p\_A}{2} = 0 \implies p\_B = 1 + \frac{p\_A}{2} $$

By substituting one into the other, we have that $p\_A=p\_B=2$, $q\_A=q\_B=1$ and $\pi\_A=\pi\_B=2$.

I'm not very sure about this solution or the direction I took to solve it. Any advice here would be appreciated on what other direction I should try.

**Tags:** game-theory, nash-equilibrium

> **Comments:**
> **username_1**: Your solution looks correct to me.  

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## 1 answers

### Answer by username_2 on 2020-05-01. Score: 1

This is a version of **Hotelling's Model of Spatial Differentiation**

The easiest solution to this is to find an "X" in-between [0,2] such that the customer is indifferent between going to Firm A or Firm B.

This will happen when
$ P(A) + Distance(X-A) - Utility = P(B) + Distance(B-X) - Utility $

Then demand [0, X] would go to A, and demand [X, 2] will go to B. Then using Best Response Functions for this Simultaneous game we can compute prices, quantity, and profits.

I have attached my solution below. For more information, you can see - [SNYDER NICHOLSON HOTELLING'S BEACH EXAMPLE](https://books.google.co.in/books?id=DXcJAAAAQBAJ&pg=PA544&lpg=PA544&dq=snyder%20nicholson%20hotelling&source=bl&ots=RWmQxfSq2W&sig=ACfU3U3WC_orI1OuFUPLrukZ7N5PNDyOPA&hl=en&sa=X&ved=2ahUKEwj7ofWa2ZLpAhXRzTgGHQBtCIcQ6AEwBHoECAsQAQ)
[![enter image description here](https://i.sstatic.net/5nabb.jpg)](https://i.sstatic.net/5nabb.jpg)
[![enter image description here](https://i.sstatic.net/BU0WU.jpg)](https://i.sstatic.net/BU0WU.jpg)


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