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
- 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
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.
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
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.
This reduces to:
Finally this reduces to:
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.