1. Consider Hotelling product dierentiation model we discussed in the class. Suppose that the length of the city is 1, transportation cost (for consumers) is 1 and marginal cost (of production) is also 1. Also suppose the cost of transportation is quadratic, that is if a consumer walks for x to get to a store that the transportation costs she pays is tx2 . p1 , and p2 are the prices that rms 1,2 charge respectively.Suppose that rm 1 is located at x1 = 0 and rm 2 is located at x2 = 1.a. What is the unique N.E. of this game?Now suppose that the length of the city is 2 and rm 1 is located at x1 = 0 and rm 2 is located at x2 = 2.b. As a function of prices, where will be the indierent consumer located?c. Find the demand functions for both rm in terms of prices.d. How does a rms demand curve respond to its price? What about the reaction to the other rm price? Interpret your answer.e. Write down the prot functions for the rms and nd rst order conditions.f. Find the unique N.E. of this game. Note that this is a symmetric game, so you can guess that prices are the same in equilibrium.g. what happened to the equilibrium prices as the city got bigger? What is your intuition?Now suppose that rm 1 is located at x1 = 0, and rm 2 is located at x2 = 1. The city is still of the length 2. Also assume that marginal costs are the same and equal to 1.h. Find the N.E. of this gamei. How do you compare equilibrium prices here to what you found if part (f )?j. If the only locations which are available are (x1 = 0, x2 = 2) and (x1 = 0, x2 = 1), what will be SPNE of the game in which they rst choose the location and then price? Note that rm 1 only has one option for location, x1 = 0.2. Consider Hoteling model of product dierentiation. In a city of the length 2 there are two rms which are located at the extremes. Transportation unit cost is 1 and it has the quadratic form.a. If marginal costs were the same and equal to 1 what would be N.E. of this game?Hint: Use the result in part (f ) in question 1.b. If marginal costs were the same and equal to 1 what would be N.E. of this game?Now suppose rms have dierent marginal costs. In particular, c1 = 1 and c2 = 2.c. Without doing any math where do you think the equilibrium prices should lie? Why?d. Find the equilibrium prices.e. How do you compare equilibrium prices in (d) to those in (a) and (b). Explain your intuition.
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