Derive the demand functions: Hotelling-style Model

Question

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.

Answer 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.