Ice-cream seller on a beach, continuation)

Question

Unlike the problem from the trainer version, here the locations of the sellers are fixed but the prices are not. Suppose that a beach is a segment of length l and is uniformly occupied by bathers who wish to buy an ice-cream. The pleasure from eating an ice-cream is 10, the cost of walking a unit length towards the seller 1 ( the reversal walk is included). Additional ice-cream do not add pleasure. The sellers are situated on the edges of the segment and offer prices $p_1$ and $p_2$ ( in the units of pleasure). A bather compares total utility from buying an ice-cream at either seller and either chooses one of the sellers or stays at his place. The marginal costs of producing an ice-cream for the sellers are zero.

1. write down the utilities of a bather buying an ice-cream at each location. What areas buy ice-cream at the first seller? Second seller?Nowhere?

2. (Monopoly) suppose that $l=40$. what prices will the sellers offer?

3. (Competition) Suppose that $ l=3$. What prices will the sellers offer now?

4. Suppose that $l=9$. What prices will the sellers offer now? Is it possible that one of the sellers offers high monopoly prices instead of competing with the other seller?

Could u please hint me what should I do for that?

Answer 1

HINT

Note that if the bather has to walk more than $10$, he will never buy the ice cream, even for free, since walking is too much trouble.

To make this more formal, a bather at location $x$ has to walk $x$ to the first location (and $\ell - x$ to the second), so his utility must be:

• $10$ for eating the ice cream
• $x$ for walking
• $p_1$ for paying

What is the total utility? What is the same bather's utility for buying at the second location?