Problem Set 06 Solutions

Problem Set 06 Solutions - Econ 121 – Fall 2010 UC...

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Unformatted text preview: Econ 121 – Fall 2010 UC Berkeley Professor Cristian Santesteban Problem Set 7 ­ Solutions Due: Thursday, November 4 by 11:15am Problem 1 (10 points) Draw the profit‐possibility frontier (PPF) in the duopoly case when the firms have a) decreasing marginal cost and b) constant but unequal marginal costs. a) Decreasing marginal costs Π1 Π2 b) Constant but unequal marginal costs Π2 Π1 Problem 2 (5 points) Is it possible to make an argument that the fare dissemination activities of Airline Tariff Publishing Co. (ATP) were beneficial to consumers? What factors make this argument less than persuasive? The activities of the ATP could have been good for consumers in that the ATP provided consumers (or at least travel agents who then worked with consumers) with transparency and convenience in the form of a centralized database that consumers could use to compare prices and choose the best among competing offers. However, the use of first­ and last­ticket prices and footnote designators allowed airlines to secretly communicate with one another and agree on fare levels and the timing over which such fares would be in effect. As such, the ATP eliminated choice for consumers as airline prices mimicked each other and consumers ended up paying higher than competitive price levels. Problem 3 (5 points) “Price wars imply losses for all of the firms involved. The empirical observation of price wars is therefore a proof that firms do not behave rationally.” True or false? Explain. False. Price wars may be the outcome of rational punishment strategies in the presence of unobservable demand fluctuations. No firm cheats but the industry reverts to the Nash outcome for a set number of periods before returning to the collusive outcome. Problem 4 (50 points) Consider the Hotelling model with endogenous price competition. The utility of a consumer located at θ buying from a firm i located at θi charging price pi is given by: U (θ , θi, pi) = 20 – 2| θ ‐ θi| ‐ pi = 0 if she buys from firm i if she does not buy at all Transportation costs are linear. There are two firms, each located at one end of the linear city. Assume: M=1, MC=0. Given θ1 = 0 and θ2=1. a) Derive the best response functions for each firm. In order to find the BRF of each firm we first have to find the demand functions of each firm. Once we have the demand functions we can find the BRF after solving the firm’s profit maximization problem. To derive the demand function for firm 1, we find the marginal consumer. That is the consumer (specified by a specific location on the unit interval) that is indifferent between consuming between firms 1 and 2. All consumers to the left of the marginal consumer will buy from firm 1. All consumers to the right of the marginal consumer will buy from firm 2 The marginal consumer θj is given by: U(θj,0,P1)=U(θj,1,P2) 2θj + P1 = 2(1­θj) + P2 θj = ½ + (P2 – P1)/4 Firm 1’s demand function is given by: Q1 = M * θj = θj = ½ + P2/4 – P1/4 Firm 2’s demand function is given by: Q2 = M* (1­θj) = ½ + P1/4 – P2/4 Now we can solve each firm’s profit maximization problem to get the BRFs: Max P1 π = P1 * Q1 = P1 * (½ + P2/4 – P1/4) FOC: ½ + P2/4 – 2P1/4 = 0 P1 = 1 + P2/2 (this is firm 1’s BRF) By symmetry, P2 = 1+ P1/2 (this is firm 2’s BRF) b) Draw the best‐response functions and identify the Nash Equilibrium. In order to graph these, rewrite firm 2’s BRF as a function of P2. (Alternatively, you can rewrite firm 1’s BRF as a function of P1.) P1 = 2P2 – 2 (This is firm 2’s BRF rewritten as a function of P2) P1 c) Solve for the Nash Equilibrium prices. At the Nash Equilibrium, P1 = P2 = P* due to the symmetry of the problem. P* = 1 + P*/2 (from the BRF) Hence, P* = 2 d) Suppose now that instead of two independent firms, there is a monopolist who operates on both locations (i.e. who provides both varieties). If the monopolist wants to cover the whole market, what is the profit‐maximizing price for both goods? In order to cover the whole market, the monopolist must choose a price for each good such that all consumers get at least zero utility. A monopolist will maximize its profits if it sells both goods at the same price and gives utility equal to zero to the consumer located at 0.5. All other consumers will get utility greater than zero. So the consumer at 0.5 must get at least zero utility if it consumes either product 1 or product 2. U(1/2, 0, Pm) = U(1/2, 1, Pm) = 0 20 – 2*(0.5) – Pm = 0 Pm = 19 Nash Equilibrium P2 R2(P2) R1(P2) e) Would the monopolist cover the entire market if she can choose? (Hint: Suppose she does not cover the entire market, what is the optimal quantity supplied?) The monopolist would not cover the entire market if the gain from selling to less than the entire market at a price greater than 19 is bigger from the loss of sales from the lost customers. In order to answer that question we need to find the total demand for the monopolist’s products for P>19. The total demand is given by 2θ, where θ is the consumer who is indifferent between purchasing and not purchasing (i.e. getting utility 0) at a given Pm. So, we find θ by setting utility equal to 0 for any Pm: 20 ­ 2θ – Pm = 0 θ = 10 – Pm/2 So, we can then specify the monopolist’s total demand as follows: Qm = 1 for Pm<=19 Qm = 2θ = 20 – Pm for Pm>19 This is a kinked demand curve with a kinked marginal revenue curve. MR = 20 – 2Qm for Pm>19 MR = 0 for Pm<=19 The monopolist will maximize profits at the highest price where MR = 0; that is Pm=19 and Qm=1. So, yes, it is optimal for the monopolist to cover the entire market. Problem 5 (30 points) Consider the Hotelling linear city, with consumers of mass M uniformly distributed over the unit interval. Firms enter the market sequentially, and their location decisions are irreversible. Each firm offers one product variety. Prices are exogenously fixed at p − c = 1. a) If the fixed cost of entry is f = 2, what is the minimum market size (i.e. the lowest value of M) needed to sustain 10 firms in the free entry equilibrium? The minimum number of firms that can be sustained in a free entry equilibrium is Nmin = M/2f. So to sustain 10 firms and given f=2, we need M=40. b) Suppose the actual market size is M = 18. The government considers the following subsidy scheme: for each firm which enters, the government reimburses a fraction s (0, 1) of its entry cost. Which value of s is needed to have 9 firms entering in equilibrium? A subsidy of s reduces the entry cost from f to f’=f­s. So to sustain 9 firms for f=2 and M=18, we need to the following s: 9 = 18/[2(2­s)] or s = 1 ...
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  • Fall '07
  • Woroch
  • entire market, total demand, free entry equilibrium, marginal consumer, Nash Equilibrium  P2, unequal marginal costs

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