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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?

gt6989b
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Mehdi Golafshan
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1 Answers1

0

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?

gt6989b
  • 54,422
  • We don't have the total utility. Our assumption those I wrote them. – Mehdi Golafshan Jan 15 '20 at 17:43
  • @MehdiGolafshan Of course you don't have it. But given what is in the hint, can you combine the components of total utility to actually write down the expression for it? – gt6989b Jan 15 '20 at 19:56
  • I asked from my friend, he told me u can do like this:

    first of all we should for every point $x$ of the segment find

    1. total pleasure if a person from point $x$ goes to the first end of the segment

    (this is $10 - p_1$ - distance to the end) 2) the same for going to the second end 3) compare these numbers, and zero

     – Mehdi Golafshan Jan 15 '20 at 23:28