(Hotelling model - 7 points) There is a single restaurant located at the center
ID: 1106750 • Letter: #
Question
(Hotelling model - 7 points) There is a single restaurant located at the center of a street of length 1 km. The restaurant has zero marginal costs. Consumers are uniformly distributed on the street (the interval [0,1]). Suppose that the transportation cost for each consumer is $1 per unit of distance (each kilometer of travel). The utility of a consumer who lives a units of distance from the restaurant is given by U = B a p, where p is the price of a meal and B is a constant. However, if the consumer does not eat at the restaurant her utility is U = 0.
(a) (2 points) Suppose that 0 < B < 1. Find the positions of the two consumers who are indifferent between eating at the restaurant and staying at home.
(b) (1 points) Express consumer demand for the restaurants as a function of p.
(c) (2 points) Solve for the price set by the firm and find its equilibrium quantity and profit.
(d) (2 points) What is the difference between horizontal and vertical product dif- ferentiation? Give a real-world example of each one.
Explanation / Answer
a) U = B - a - p
where, B is constant
a is units of distance
p is price of meal
If a consumer is indifferent between going to restaurant and staying at home then utility in both cases must be equal.
i.e,
B-a-p = 0
a = B-p
where a is customer distance from restaurant.
If 0<B<1
then, 0<a<1
hence, two consumers who are indifferent between eating at the restaurant and staying at home will be 'a' distance right of restarant and 'a' distance left of restaurant.
b) U = B-a-p
customer utility will be maximum if a = 0 and p = 0, then,
U = B
suppose 'B' is quantity of food when a = 0 and p = 0
then, consumer demand for the restaurants as a function of p
q = B - a - p
c) For maximisation,
As it is the only restaurant there then it must be charging monopolist price,
MR = MC
MC = 0 (given)
R = q * p
R = (B-a-p)*p
R = Bp-ap-p2
MR = B-a-2p
MR = MC
B-a-2p = o
P = (B-a)/2
q = B-a-p
q = B-a-(B-a)/2
q = B-a
profit = revenue - cost
profit = p*q - mc*q
profit = (B-a)2 / 2
d) In terms of quality, vertical differentiation refers to same type of product but varies considerably in terms of quality whereas, in horizontal differentiation, it is not easy to measure the qualitative difference between the product.
e.g: for vertical differentiation: jewelry, it is easy to differentiate between the real gold jewelry and gold plated jewelry.
Horizontal differentiation: salt, it is not at all easy to find the difference between the quality of brands of salt.
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