1) On a remote island there are two hotels. Each hotel has 200 rooms.Suppose tourists visit the island only on 2 occasions: a summer holiday and a winter holiday. The marginal cost of hosting guests in a room is constant and equal to mc=40 per room in both hotels and both seasons. Assume there is no cost if a room remains empty. In the summer

holiday the total demand for hotels in the island is P = 700-Q. In the winter total demand is P = 100-Q. In each season, the two firms compete according to Bertrand competition with capacity constraints. Consumers who take vacations in one season never take a vacation in the other season so there is no effect of price in one season on demand in the other season.

a)In the winter what is the Bertrand equilibrium? What would the market price be?

b)In the summer could p=mc be an equilibrium? Explain. Suggest equilibrium prices.

Verify that these suggested price constitute an equilibrium (that is, show that no firm has an incentive to deviate from the price you proposed).

2)Consider a location model of differentiated products where the set of possible products is the line segment [0,1] and consumers are uniformly distributed along the line segment.Transportation costs in this model are equal to td, where d= |x - x*|is the distance between the consumer's ideal variety and the variety she purchases. If a consumer with ideal variety x* purchases variety x at price p, then her utility is

u = s - p - t( |x - x*| ).

If the consumer does not purchase the good her utility is u = 0. A consumer buys the good only if her utility from buying is at least zero.

a)Suppose the market is served by a monopolist selling variety x = . For given s and t, find the demand the monopoly faces as a function of p (remember, consumers want to buy the good only if their utility is non-negative). Make sure that as part of your answer you specify (i) for which range of prices no consumer would want to buy the good so that demand is zero [hint: even the consumer located at x=0.5 does not buy]; (ii) for which range of prices all consumers would

want to buy the good [hint: even the consumer located at x=0 wants to buy the good]; (iii) For the range of prices such that some consumers buy, and some do not, state how demand depends on the price. [Hint: what is the location of a consumer who is indifferent between buying and not buying? Consumers who are closer, will but].

b)Explain in words why a monopolist selling variety x = is better off the smaller is t.

c)Assume now that the market is served by 2 firms located at 0 and 1. Each firm has a marginal cost of production c. Assume all the consumers buy, find the equilibrium. [Hint: This is what we did in class.]

d)Explain in words why a (when all the market is served) a duopolist selling variety x = 0 and facing a rival who sells variety x = 1 may be better off the bigger t.

3)Consider a location model of differentiated products where the set of possible products is the line segment [0,1] and consumers are uniformly distributed along the line segment.Assume that the transportation costs is quadratic td2, where d= |x - x*|is the distance between the consumer's ideal variety and the variety she purchases. If a consumer with ideal variety x* purchases variety x at price p, then her utility is

u = s - p - t( x - x*)2.

If the consumer does not purchase the good her utility is u = 0. A consumer buys the good only if her utility from buying is at least zero. Suppose the market is served by 2 firms located at 0 and 1. Each firm has a marginal cost of production c. Assume all the consumers buy, find the equilibrium. [Hint: Follow the steps we did in class, just notice that here the transportation cost is quadratic.]

4. In a local fair, there are two snack vendors (firms) 1,2 each selling one food item. The firms engage in simultaneous price competition. Consumers view the two goods as differentiated. The marginal cost of production for each firm is c=0.Demand for the two goods is given by the following system:

q1=100-3p1+p2

q2=100-3p2+p1

a)Derive each firm's best response function; illustrate the two functions on the same graph. Are prices strategic substitutes for strategic complements?

b)Find the equilibrium prices and the profits of the firms. What quantity does each firm sell?

c)Suppose vendor 1 is not free to choose his price but instead he needs to follow the directions of his boss who made him commit to charge a price p1=21 no matter what the other vender does. Vendor 2 maximizes his own profit given this fixed strategy of vendor 1. What price will vendor 2 set? Is it higher or lower than the equilibrium price you found in part b? Is it higher or lower than p1=21?

d)Given the price p2you found in part c, is p1=21 a best response for firm 1 if it were able to choose its price freely?