Problem Set 7;Industrial Organization: Fall 2014;Due Date: Friday Dec. 2nd;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 game;i. 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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