Two independent ice cream vendors own stands at either end o

Two independent ice cream vendors own stands at either end of a 2 mile long beach. Everyday there are 200 beach-goers who come to the beach and distribute themselves uniformly along the water. Every beach-goer wants exactly one ice cream during the day, and values the ice cream from both stands at $10. All of the beach-goers would rather be sunbathing or in the water, so they have a disutility to walking on the beach of $0:50 per mile. Early\'s Ice Cream, the rm at location 0, is an early riser and always posts her price rst. Cali Creamery, at location 2, is more laid back and posts her price just before the beach opens (the beach requires all prices be posted by the time the beach opens). Both stands get their ice cream from the same place and pay $2 per unit.

1. Each individual is also referenced by a location x on the beach between 0 and 2. What are the utilities of purchasing from Early\'s and Cali for the person at location x = 1:50, given that Early\'s names price pe and Cali names price pc? What are the utilities for each individual as a function of their location on the beach, x?

Solution

Let Ue(x) and Uc(x) be the utility of the individual at x when she buy the ice cream from Early’s Ice Cream and Cali Creamery, respectively. \'

Then,Ue(x) = 10-Pe-x

and Uc(x) = 10-Pc-(2-x) = 8 -Pc+x

1.a) Given that If x=1.5 then

Ue(1.5) = 10-Pe-1.5 = 8.5-Pe

Uc(1.5) = 8+1.5-Pc = 9.5-Pc

1.b) Since each consumer maximizes his or her utility to decide which store to go to,the utility of the consumer at location x is

U(x) = max{Ue(x),Uc(x)}

U(x) =max{10-Pe-x,8-Pc+x}.