Problem%20Set%208%20-%20Oligopoly%20-%20Game%20Theory

Problem%20Set%208%20-%20Oligopoly%20-%20Game%20Theory -...

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 8 This problem set is due in section on Friday, November 20. Please collect your answers into a stapled packet, and write your own name and your TA’s name on the front of each packet, as well as your netID and the time of the section you usually attend. 0. (Aplia) Please do the assigned problems on the Aplia website. These problems are due at 11:45pm on Thursday, November 19. 1. (Monopolistic Competition / Game Theory: Hotelling’s Model of Spatial Competition) [ Some parts of this question are discussed intuitively in ch. 25.8‐25.10. Feel free to use what you learn from that part of the textbook in writing up your answer, but try the problem first with the textbook closed, and make sure you write up solutions in your own words. ] In 1929, economist and statistician Harold Hotelling proposed a “location model” of spatial competition that is now standard in a range of economic (and other social science) applications. The original storyline involves a boardwalk that stretches across the beach. Two ice cream vendors are deciding where to locate along a boardwalk in order to attract the most customers. Let’s model the boardwalk as the unit interval, [0, 1]. There are many consumers on the beach who would like to purchase ice cream from the vendors. To keep things simple, suppose that there is a continuum of consumers, one each at each point in the unit interval (to be formally correct, we say that the consumers are uniformly distributed on [0, 1]). Each consumer i’s utility function is: ui = { 0 if consumer i does not purchase ice cream if consumer i purchases ice cream, { v – p – di where v is consumer i’s reservation value for ice cream, p is the price of ice cream, and di is the transportation cost for the consumer to get to the vendor. Specifically, di is the distance between consumer i and the closest vendor. For example, if consumer i is located at 1/4 and the vendors are located at 1/3 and 2/3, then di = min{|1/3 – 1/4|, |2/3 – 1/4|} = 1/12. If a consumer is indifferent between purchasing from the two vendors, then the consumer purchases from either vendor with probability one‐half. For simplicity, suppose all consumers have the same valuation v = 3, and both vendors charge the same price p = 2. Each vendor’s cost of producing a unit of ice cream is c = 1, so each vendor j є {1, 2} earns profit πj = (p – c) (mass of consumers purchasing from j). For example, if the vendors are located at 1/3 and 2/3, then consumers located in the interval [0, ½) purchase from the first vendor, and consumers located in the interval (½, 1] purchase from the second vendor. (The consumer located at ½ is indifferent and could purchase from either vendor, but his decision will only have an infinitesimal effect on the mass purchasing from either vendor, so we can 1 safely ignore him.) Since the mass of consumers purchasing from each vendor is ½, each vendor earns profit πj = (2 – 1) (½) = ½. (a) Suppose the vendors locate at ¼ and ¾. Which consumers will purchase from each vendor? Draw a diagram of the unit interval, indicating the location of the vendors and the segment of the market captured by each vendor. (b) Because the consumers’ utility functions are denominated in money units (you can see this because price enters linearly), each ui is equal to i’s consumer’s surplus. Therefore, we can calculate consumers’ surplus by adding up all the consumers’ utilities. When the vendors are located at ¼ and ¾, explain why consumers’ surplus is equal to 4 ∫0 ¼ (1 – x) dx. What is the value of this expression? You do not need to prove it, but explain intuitively why consumers’ surplus is maximized when the vendors locate at ¼ and ¾ (as opposed to when the vendors locate at any other two points). (c) The vendors locating at ¼ and ¾ is not a Nash equilibrium of this game. Show that, holding fixed one vendor’s strategy, the other vendor has an incentive to deviate from his strategy. (d) Show that it is a Nash equilibrium for both vendors to locate at ½. In fact, this is the only Nash equilibrium. To see that, suppose that at least one vendor does not locate at ½. Show that a vendor who is not located at ½ can always increase his profit by moving slightly closer to ½. (e) Explain why the consumers’ surplus at the Nash equilibrium is equal to 2 ∫0½ (1 – x) dx. What is the value of this expression? Explain intuitively why the consumers are worse off at the Nash equilibrium than when the vendors locate at ¼ and ¾. (f) You do not need to prove it, but explain the logic for why there is no (pure strategy) Nash equilibrium when there are 3 vendors. When there are 4 vendors, explain why it is a Nash equilibrium for 2 of them to locate at 1/3 and 2 of them to locate at 2/3. (g) In addition to physical location decisions, the Hotelling model can be applied to a variety of settings where firms choose a continuous aspect of a product’s type. For example, consumers differ in how sweet they like their breakfast cereal to be, and cereal producers choose how sweet to make their cereal. How sweet a cereal is can be modeled as a point on the unit interval. There is a distribution of consumers’ ideal cereal types on [0, 1], and firms can be thought of as choosing a location on [0, 1]. The Hotelling model has also been influential in political science. Instead of locations along a beach, interpret [0, 1] as the political spectrum, where 0 represents the extreme left wing and 1 represents the 2 extreme right wing. Interpret the “vendors” as political candidates who choose how left/right wing to make their policy platforms. Interpret the “consumers” as voters who gain value v = 3 from voting (i.e., fulfilling their civic duty) but who must incur a cost p = 2 to vote (e.g., waking up early and going to the voting booth). The di term in the utility function captures the fact that voters prefer to vote for the candidate whose platform is closer to the voter’s own political opinions. In this political context, the prediction that when there are two political candidates, both will choose centrist platforms (locating at ½) is called the median voter theorem. Explain intuitively why the model predicts that two‐party democracies (like in the U.S., the U.K., and post‐WWII Germany) will have governments that adopt centrist policies, and many voters will feel that their political party’s policies are not liberal/conservative enough. Explain why the model predicts that multi‐party systems (like in Italy, Israel, and pre‐WWII Germany) will tend to have unstable governments that sometimes adopt extreme policies, but most voters will be satisfied that their party represents their political opinions relatively well. (h) Consider a two‐party democracy. Suppose that voters mostly care about how close the candidates’ political platforms are to their own political opinions, but their voting is also influenced a little bit by the candidates’ charisma. Explain (intuitively) why, in equilibrium, the vote share received by the candidates will be entirely determined by the candidates’ relative charisma. 2. (Oligopoly / Game Theory: An Entry Deterrence Game) [You can find solutions to some parts of this question in ch. 28.7‐28.8. Feel free to use what you learn from that part of the textbook in writing up your answer, but try the problem first with the textbook closed, and make sure you write up solutions in your own words. ] Being a monopolist (or more generally, having a lot of market power) is highly desirable, but maintaining a monopoly position requires barriers to entry. We have discussed several possible sources of barriers to entry, such as government licensing, exclusive ownership of a factor of production, and “natural monopoly” arising from cost structure relative to demand. Another barrier to entry is the threat of “anti‐competitive behavior”; if a competitor enters a monopolist’s market, the monopolist will lower prices so much as to drive the competitor out of business (and then raise prices again once the competitor is gone). This question explores a model of that kind of behavior. (a) There are two players, an Incumbent (the would‐be monopolist) and a (potential) Entrant. If the Entrant stays out of the Incumbent’s market, then the Incumbent gets payoff 9, and the Entrant gets payoff 1. However, if the Entrant enters the market, then the players’s payoffs depend on whether the Incumbents “fights” the Entrant (e.g., starts a price war) or “accommodates” the Entrant (e.g., they both charge competitive prices and split the market). If the Entrant enters and the Incumbent accommodates, the Entrant gets a payoff of 2, and the Incumbent gets a payoff of 1 (e.g., because the Incumbent faces a higher marginal cost of production if he increases production beyond his current level). However, if the Entrant enters and the Incumbent fights, then both earn a payoff of 0. Draw the 3 normal form of the game. Find the Nash equilibria of this game. (Hint: There are two pure strategy Nash equilibria and a continuum of mixed strategy Nash equilibria.) (b) When we have more information about the structure of a game, we can sometimes make more precise predictions than just Nash equilibrium. In this case, it seems reasonable to suppose that the Entrant makes his entry decision before the Incumbent decides whether to fight or to accommodate. Draw the extensive form of this game. Using backwards induction, show that the unique subgame perfect Nash equilibrium is that the Entrant enters and the Incumbent accommodates. Explain intuitively why this Nash equilibrium is subgame perfect and why the other Nash equilibria are not subgame perfect. (c) Now suppose that the Incumbent has the opportunity to invest in excess production capacity before the Entrant appears on the scene. This investment costs the Incumbent 2 units of payoff. However, if the Incumbent makes the investment, then his marginal cost of production is no longer higher in the event that he increases his output. Hence, if the Incumbent makes the investment, his payoff is 7 if the Entrant stays out and 2 if the Entrant enters, regardless of whether the Incumbent fights or accommodates (the Entrant’s payoffs are unchanged). Draw the extensive form for this larger game that includes the Incumbent’s choice of whether to make the investment. Using backwards induction, show that a pure‐strategy subgame perfect Nash equilibrium is that the Incumbent invests, and the Entrant stays out. Why is this an example of commitment? (d) The kind of anti‐competitive behavior we have been exploring was made illegal in the U.S. under the anti‐trust laws. Suppose you work for the Federal Trade Commission (FTC), the federal government agency in the U.S. that enforces anti‐trust laws. Suppose you observe a small firm enter a market that is dominated by a large firm, both firms lower prices, and the small firm goes out of business. Explain why it may be difficult to determine whether the large firm behaved anti‐competitively. What kind of data would you want to see to decide whether the large firm intentionally tried to drive the small firm out of business? Why do you think alleged violations of anti‐trust laws typically result in long court battles? 4 ...
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This note was uploaded on 12/28/2011 for the course ECON 3010 at Cornell University (Engineering School).

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