Question

Consider a Hotelling linear city where four firms simultaneously decide their own locations 0 ≤ θi ≤ 1 (i = 1, 2, 3, 4) to sell their products. Suppose that one-sixteenth of consumers lives at a location x = 0, one-eighth of consumers lives at a location x = 1, and the remaining 13/16 of consumers live uniformly between x = 0 and x = 1.

1) In the Nash equilibrium, where is the location of the firm from which consumers who live at a location x = 0 will buy a product?
(a) x = 3/16.
(b) x = 1/16.
(c) x = 3/13.
(d) x = 1/4.
(e) x = 1/13.

2) In the Nash equilibrium, where is the location of the firm from which consumers who live at a location x = 1 will buy a product?
(a) x = 7/8.
(b) x = 15/16.
(c) x = 13/16.
(d) x = 11/13.
(e) x = 12/13.

3) In the Nash equilibrium, what is the maximum distance for which consumers must travel to buy a product from the closest firm?
(a) 4/13.
(b) 5/16.
(c) 3/16.
(d) 3/13.
(e) 1/8.

Answer 1

1) The firm where x=0 will be located at distance 1/16 as it will be located at the starting end of the locality because the firms are uniformly distributed and so are the people living in the locality as shown in the above figure.

2) And for the same reason the firm where x-1 will be located at distance 15/16 as the people who live at x=1 will be located at the ending of the locality

3) The maximum distance for which the consumers must travel to buy a product is 1/16+1/16=2/16=1/8, this is because the distance now doubles ie going to the firm and coming back from it.