# Derive the demand functions: Hotelling-style Model

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.

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.