Econ 8601 Fall 2018 Take-Home Final. You can choose when to work on it, but please work only one

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1 Econ 8601 Fall 2018 Take-Home Final Please work alone. day on it. You can choose when to work on it, but please work only one Question 1. Perfect Price Discrimination. Consider the Logit Model of Product Differentiation from homework 1. There are firms indexed by andan outside good labeled by 0. Each firm 1 has constant marginal cost equal to.there is a unit measure consumers. Let index an individual consumer and suppose the utility of consumer from purchasing good and paying price has utility = + for =12 = 0 for good 0. A consumer is therefore summarized by his or her vector of draws ε =( 0 1 ) since consumers are otherwise the same. Assume the are drawn type 1 extreme value which delivers the logit choice probabilities. We allow for perfect price discrimination. Each firm observes the entire vector of draws ε for each consumer and can set prices contingent on ε (i.e. can set a price specific to individual.). The firms compete in a Bertrand fashion for each individual consumer. (a)takeasgiventhe firms in the industry and calculate the equilibrium of price competition when perfect price discrimination is feasible. Derive formulas for the market shares of each firm. (b) A potential entrant is considering entry into this industry. If it comes in, it will be firm = +1, and consumers will all get a new logit draw +1 for this firm. The firm has given values of +1 and +1. If the firm enters, it pays a fixed cost +1. Derive a condition determining whether the firm enters. Discuss the connection between the private incentive for entry and the social incentive. That is, how does the total of producers and consumers surplus change if the firm enters? 1

2 Question 2 A monopolist faces a constant elasticity demand curve, = for 1. On the production side, the firm uses inputs of up to different suppliers, that are indexed =1. There is a fixed cost of 0 to obtain inputs from, witha constant marginal cost of for supplier. Let be an indicator variable equal to one if supplier is chosen and let = : such that =1 ª be the set of chosen suppliers. The production function for output given the choice of suppliers and the input levels of the chosen supplier = Ã X! ( 1) (1 ) ( 1) Standard calculating show that the minimum variable cost given supplier choice and the input price vector is given by (a) Suppose. increased. ( ) = " X 1 # 1 1 Discuss comparative statics with supplier choice as is (b) Now index firms by. Suppose firms vary in and in fixed cost = + whre the are i.i.d. log normal. Suppose and are known and assume. Suppose you observe the output of each firm and the input levels from each supplier. Sketch a strategy for estimating the the and the distribution of the. 2

3 Question 3 Consider the following model of the widget industry. There are two potential firms =12. There are a continuum of consumers indexed by. If both firms enter the industry, then consumers make a discrete choice between buying one unit of product from firm 1, or one unit from firm 2, or buying neither. Refer to the choice to buy from neither as the outside good and index it by =0. Following the standard discrete choice logit setup, each consumer draws i.i.d. type 1 extreme value utility shocks ( ) for each alternative. Consumers also care about the price of the product of firm. Finally, let be total quantity sold of firm product across all consumers and let = be the industry quantity. Consumers care about total industry quantity. Let be the utility of consumer for option. It is given by 0 = 0 1 = = Then =1if =max{ }. Assuming a unit measure of consumers, and letting () be the density of, totalsalesoffirm are Z = ( )( 0 )( 1 )( 2 ) Thepresenceofaggregate purchases in consumer utility indicates a spillover. For 0, the spillover is positive, e.g. the consumer likes buying a widget more when others are buying a widget. If 0, the spillover is negative. Note the spillover is independent of which firm other consumers buy from. That is, consumers care whether their neighbors are buying a widget instead of the outside good, but not about the brand their neighbors buy. 3

4 The two firms have constant marginal cost. Firm has a random type 1 extreme value profit shockfromentry and from staying out. If firm enters and the price and quantity are and, it s profit fromentryis =( ) +, and its profit fromstayingoutis =. Timing is as follows. There are three stages. In stage 1, firms 1 and 2 privately observe their profit shocks and simultaneously make their decisions about whether to enter or stay out. In stage 2, prices are set. A firm that enters sees whether or not its rival also entered in stage 1. If both firms entered they simultaneously set 1 and 2. If firm entered and its rival did not, firm knows it has a monopoly when setting. In stage 3, consumers observe which firms have entered and see prices and simultaneously make their purchase decisions. Note that if firm does not enter we treat this as = intheconsumerchoiceproblem,and =0and =0. (a) Define and characterize an equilibrium of the above model. (b)showthatif 0, the equilibrium is unique. homework as needed.) (You can refer to results from (c) Simplify by changing the environment so that there is price regulation in stage 2. Specifically, if firm enters its price is set at =. Discuss the possibility of multiple equilibria if 0. (d). Suppose there exists a cross-section of markets that are indexed by. Assume that the various markets differ in two respects but are otherwise identical. First, markets vary in, the constant term in consumer utility, that is now indexed by market. Second, the potential entrants in each market get different profit shock draws,, thatare across (and ). Suppose a researcher observes 4

5 data for the cross-section of markets on entry, prices and quantities 1 and 2. Discuss strategies for estimating the model parameters. Discuss potential threats to identification in the context of this data generating process. Discuss how your answer might differ if the entry stage were removed from the model, and there were exogenously two firmsineachmarket. 5