<p>You are treating the absolute value wrong. You seem to have done it right the first time around, do it like that again. (Make a little drawing, think about which side of 10 and 50 the indifferent $X_i$ will be on, etc.)</p>

<p>Another thing: The demand function you derived for $Q_1$ only holds for certain ranges of $P_1$, $P_2$ and $P_3$, because if the prices of firms 1 and 2 are very high but $P_3$ is low, a customer indiffirent between firms 1 and 2 would go to firm 3.</p>

CC BY-SA 3.0
2015-11-19T10:39:36.513Z
2015-11-19T10:39:36.513Z
Giskard
<p>So I have this economics question that I have been trying for a while now and I can't seem to get the answer correctly. Below is the question and after I will show what I have so far. An explanation would help so much.</p>

<blockquote>
  <p>Consider the following Hotelling-style model. There are 3 Arms, each offering a single type of milk (always in 1 quart containers) that is horizontally differentiated along a single dimension, the percent of fat.</p>
  
  <p>Firm-1 offers non-fat milk. </p>
  
  <p>Firm-2 offers low fat milk, with 10% fat.   </p>
  
  <p>Firm-3 offers high fat milk, with 50% fat.</p>
  
  <p>There are 1,000 potential consumers. Some consumers hate fat and some love it. But the most fat any individual consumer would ideally have in their milk is 50%. In particular, assume that people's tastes for the ideal percent of fat is uniformly distributed from 0% to 50%. That is, the line starts at zero and goes to 50 (instead of 1 as we have done in the past).</p>
  
  <p>The utility that individual <em>i</em> obtains from purchasing a quart of milk from seller <em>j</em> is given by</p>
  
  <p>Uij = 5 - pj - 1/10*|Xi - Xj| </p>
  
  <p>Where Xi € [0,50] is individual <em>i's</em> ideal fat content and Xj € [0,50] is the fat content offered by firm <em>j</em>.</p>
  
  <p>For example, if a consumer is located at Xi = 2 were to purchase from firm 2, and firm 2 happened to set a price P2 = 3, they would obtain utility:</p>
  
  <p>Uij =5 - 3 - 1/10*|2 - 10| = 1.2</p>
  
  <ol>
  <li>Derive the demand functions for each of the three firms.</li>
  </ol>
</blockquote>

<p>So far I have been able to derive the demand function for firm 1 by first solving for Xi for the indifferent consumer between Firm 1 and Firm 2. </p>

<p>The line looks like this </p>

<blockquote>
  <p>|--x--|-------------|</p>
</blockquote>

<p>with the first line representing 0% fat, second 10%, and third 50%. The x represents the indifferent consumer between firm 1 and firm 2.</p>

<p>The math for the first firm is done by taking the Utility of the Indifferent consumer for firm 1 and setting it equal to the utility of the indifferent consumer for firm 2. </p>

<blockquote>
  <p>5-P1-1/10*|Xi-0|=5-P2-1/10*|Xi-10|</p>
</blockquote>

<p>by doing this, you end up with</p>

<blockquote>
  <p>Xi=5P2-5P1+5 with the Q1=(Xi-0)*1000/50</p>
</blockquote>

<p>Therefore the demand curve for firm 1 is</p>

<blockquote>
  <p>Q1=100P2-100P1-100</p>
</blockquote>

<p>The next step would be to be to solve for the demand curve for firm 3. My problem is when I set U2=U3 the Xis cancel. The math is below.</p>

<blockquote>
  <p>5-P2-1/10*|Xi-10|=5-P3-1/10*|Xi-50|</p>
</blockquote>

<p>This reduces to:</p>

<blockquote>
  <p>-P2-1+Xi/10 =-P3-5+Xi/10</p>
</blockquote>

<p>Finally this reduces to:</p>

<blockquote>
  <p>-P2+P3+4=0 </p>
</blockquote>

<p>The "Xi/10" on both sides cancel which does not make sense. Because of this result, I have no idea how to solve for the demand curve of Firm 3.  </p>

<microeconomics><mathematical-economics>
Derive the demand functions: Hotelling-style Model
THAT newbie
# Derive the demand functions: Hotelling-style Model

(Asked by: username_0 on 2015-11-19)

So I have this economics question that I have been trying for a while now and I can't seem to get the answer correctly. Below is the question and after I will show what I have so far. An explanation would help so much.

> Consider the following Hotelling-style model. There are 3 Arms, each offering a single type of milk (always in 1 quart containers) that is horizontally differentiated along a single dimension, the percent of fat.
>
> Firm-1 offers non-fat milk.
>
> Firm-2 offers low fat milk, with 10% fat.
>
> Firm-3 offers high fat milk, with 50% fat.
>
> There are 1,000 potential consumers. Some consumers hate fat and some love it. But the most fat any individual consumer would ideally have in their milk is 50%. In particular, assume that people's tastes for the ideal percent of fat is uniformly distributed from 0% to 50%. That is, the line starts at zero and goes to 50 (instead of 1 as we have done in the past).
>
> The utility that individual *i* obtains from purchasing a quart of milk from seller *j* is given by
>
> Uij = 5 - pj - 1/10\*|Xi - Xj|
>
> Where Xi € [0,50] is individual *i's* ideal fat content and Xj € [0,50] is the fat content offered by firm *j*.
>
> For example, if a consumer is located at Xi = 2 were to purchase from firm 2, and firm 2 happened to set a price P2 = 3, they would obtain utility:
>
> Uij =5 - 3 - 1/10\*|2 - 10| = 1.2
>
> 1. Derive the demand functions for each of the three firms.

So far I have been able to derive the demand function for firm 1 by first solving for Xi for the indifferent consumer between Firm 1 and Firm 2.

The line looks like this

> |--x--|-------------|

with the first line representing 0% fat, second 10%, and third 50%. The x represents the indifferent consumer between firm 1 and firm 2.

The math for the first firm is done by taking the Utility of the Indifferent consumer for firm 1 and setting it equal to the utility of the indifferent consumer for firm 2.

> 5-P1-1/10\*|Xi-0|=5-P2-1/10\*|Xi-10|

by doing this, you end up with

> Xi=5P2-5P1+5 with the Q1=(Xi-0)\*1000/50

Therefore the demand curve for firm 1 is

> Q1=100P2-100P1-100

The next step would be to be to solve for the demand curve for firm 3. My problem is when I set U2=U3 the Xis cancel. The math is below.

> 5-P2-1/10\*|Xi-10|=5-P3-1/10\*|Xi-50|

This reduces to:

> -P2-1+Xi/10 =-P3-5+Xi/10

Finally this reduces to:

> -P2+P3+4=0

The "Xi/10" on both sides cancel which does not make sense. Because of this result, I have no idea how to solve for the demand curve of Firm 3.

**Tags:** microeconomics, mathematical-economics

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## 1 answers

### Answer by username_1 on 2015-11-19. Score: 1

You are treating the absolute value wrong. You seem to have done it right the first time around, do it like that again. (Make a little drawing, think about which side of 10 and 50 the indifferent $X\_i$ will be on, etc.)

Another thing: The demand function you derived for $Q\_1$ only holds for certain ranges of $P\_1$, $P\_2$ and $P\_3$, because if the prices of firms 1 and 2 are very high but $P\_3$ is low, a customer indiffirent between firms 1 and 2 would go to firm 3.


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