econ105_10s_ps6sol

econ105_10s_ps6sol - Problem Set 6 Solutions ECON105...

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Problem Set 6 Solutions ECON105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego 1. Work through the Green-Porter model of price wars for the following situations. a. First, consider the grim strategy where any price war lasts forever. That is, T = and V P = 0. What is the minimum for which collusion is achievable if = 3/4? Repeat with = 1/2 and = 1/4. If = 1/4 and = 5/6, will any firm cheat, and will there ever be a price war? When will cheat. b. Now assume V P = δ T V M , T < so price wars are of finite length. If = 1/4 and δ = 5/6, find the minimum T (shortest price war duration) required to prevent any firm from cheating. Do price wars ever occur in equilibrium? T=3 2. Consumers are uniformly distributed along a linear city, which runs from point 0 to point 1. A consumer has utility u j = ¾ - p j - z ij if she buys a product at price p j that is a distance of z ij away, and u i0 = 0 if she buys nothing. There is only one firm in the linear city, and this firm has zero marginal cost of production. Assume that a monopolist has two stores, one located at 0, the other at 1. Assume = 2. a. Will the monopolist serve part or all of the market? Given this, find the equilibrium price, quantity, and profits. p = 1/2, Q = 1, profits = 1/2 b. Are the locations of the two stores optimal from the monopolist’s point of view? Is it optimal from a social point of view? If the monopoly could relocate its stores, where would they go? (You need not solve any math for this part.) The locations of stores are either not socially optimum or monopolist’s optimum. Placing stores inside [0,1] interval will make both consumer and monopolist better off. c. Repeat part a. using the monopolist’s optimal choice of locations for its two plants. (You can use your results from a. without having to resolve anything.)
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stores at 1/4, 3/4; serve entire market, P= 11/16. Profits = 11/16. d. Assume now the monopolist has a fixed cost F of operating each store. Assuming stores can be opened and relocated costlessly, what is the socially optimal number of stores and where would a social planner locate them? Assuming stores can be opened and relocated costlessly, how many stores would the monopoly choose to have? The social optimum is to have (1/6F) 1/3 stores, the monopolist will choose to have (1/2F) 1/3 stores. e. (Easier practice question.) Repeat a. to d. using = 1. Repeat a, . p = 1/4, Q = 1, profits = ¼; repeat c, stores at 1/4, 3/4; serve entire market, P= 1/2. Profits = ½; The social optimum is to have (1/4F) 1/2 stores, the monopolist will choose to have (1/2F) 1/2 stores. See the t.a. for help. 3. A monopolist with marginal cost equal to zero has two stores, one located at the westernmost point and one located at the easternmost point of a circular city. Assume a consumer gets utility u = v p tz if she purchases one unit from a firm a distance of z away and zero otherwise. Let v be sufficiently large that all consumers are served in equilibrium. Finally, assume the monopolist has no fixed costs and cannot relocate her stores.
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