Unformatted text preview: Economics 3010
Fall 2009 Professor Daniel Benjamin
Cornell University Problem Set 8 Solutions 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: if consumer i does not purchase ice cream ui = { 0 { v – p – di if consumer i purchases ice cream, 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 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. 1 In order to maximize utility, consumers will go to the closest vendor. All customers left of ¼ are closest to vendor at A and will buy from him, while the customers right of ¾ will buy from vendor B. For the customers in between the two vendors, those who are left of the mid‐point between ¼ and ¾ will buy from A while those to the right will buy from B. Therefore, all the consumers between 0 and ½ (the arrow marked with A) will purchase from A and those between ½ and 1 (the arrow marked with B) will purchase from B. (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). When two vendors are located at ¼ and ¾, the minimum distance of travel is 0 and the maximum distance is ¼. Since 0 < di < ¼, consumers get higher utility when they buy ice cream than they don’t: ui = v – p – di = 2‐ 1 ‐ di = 1 ‐ di > 0 (1). Therefore all consumers in the unit line will purchase ice cream. For the continuum of the consumers distributed uniformly, the total surplus can be computed by integrating (1 ‐ di) (from (1) above) over the range of the distance 0 to ¼. For each of a quarter‐length block, the overall surplus is ∫0¼(1 – x) dx. The total surplus for the unit line is 4 ∫0 ¼ (1 – x) dx. The value of total surplus is: 4 ∫0 ¼ (1 – x) dx = 7/8 When there are two vendors and they are located at ¼ and ¾, the sum of the distances that consumers need to travel (note that here, the sum is represented as an integral since we have a continuum of consumers) is minimized. Hence, consumers’ surplus is maximized. 2 (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. Let’s fix B’s location. Given that B is at ¾, if A moves slightly toward B, A can take some of B’s market share. Therefore, A has an incentive to deviate from her current location. For example, if A moves from ¼ to A’ at 3/8 with B’s location fixed, A can increase its market share from ½ to 9/16 (the mid‐point between A’s position at 3/8 and B’s position at ¾) while B’s market share decreases to 7/16. (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. Let’s first show that A and B need to locate at the same point. Assume that A locates to the left of B. Then A could deviate to say the midpoint of A and B’s original locations, and end up with a strictly greater market share. This logic implies than in a Nash equilibrium, we necessarily have A’s location the same as B’s location. Now assume that even though A and B have the same location, this common location is not ½. For concreteness (and without loss of generality), assume this location is to the left of ½. Then B could locate to a point slightly to the right of A’s location, and B would then end up with more than ½ of the market share. This logic implies that in a Nash equilibrium, not only do we need A’s location to be the same as B’s location, we also need this common location to be 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. When both vendors are located at ½, consumers have to travel a longer distance. The minimum distance is 0 and the maximum is ½. The consumer surplus in each of the half‐length segment is ∫0 ½ (1 – x) dx and so the total surplus is 2∫0 ½ (1 – x) dx = ¾ Since consumers have to travel a longer distance (the average distance is ¼ now), the consumer surplus is smaller than when the vendors locate at ¼ and ¾ (the average distance was 1/8). 3 (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. The argument that for three vendors there is no Nash equilibrium is as follows: • If all three choose different positions, the ones in the extreme locations can move closer to the middle one and increase their market share. • If all three vendors choose the same position, then the three vendors split the market equally (one third each). Suppose that all three are located on the right (left) of the mid‐
point. Then, by slightly moving to the left (right), one player can attract all the consumers left to this new point and so increase its market share (to at least one half of the market).If they are all at the mid‐point, one player can capture a higher market share by slightly moving either to the left or the right (she’d then get a bit less than half of the market, which is still better than getting a third of the market). • If two choose the same position while the other chooses a different position then the lone vendor can move closer to the other two and capture more market. So no matter which location the three vendors are located at and no matter how they change their locations, they still have an incentive to deviate. Therefore, there is no Nash equilibrium. When there are 4 vendors and 2 of them are located at 1/3 and the other 2 of are located at 2/3, the vendors split the market equally. Fixing the location of three of the vendors, the remaining vendor cannot increase her market share by moving slightly to the left or to the right. Since there is no incentive to deviate from the current location, it is the Nash equilibrium. (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 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. 4 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. If we apply the Hotelling’s location model to two‐party democracies, the political spectrum can be described as below. Ranging from liberal to conservative, all the voters to the left of the mid‐point are Democrats while all to the right are Republicans. As we’ve seen in (d), in order to have more votes, the two parties will have to move toward each other. Hence, it is likely that both parties will adopt centrist policies. As we’ve seen in (f), there is no Nash equilibrium with three players. In order to gain more votes, the parties will constantly move around, which make the political system unstable. With more than three players, the equilibrium location pattern generally emerges and the location is more toward the end (as we see in the case of four players) instead of at the center (as in the two‐
player game). Therefore, the multi‐party system can be unstable (while the players move toward the equilibrium point), but some parties adopt more extreme policies. 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. In the two‐party system, the centrist policies tend to emerge. In equilibrium, two parties will be located exactly in the middle of the political spectrum. Then, voters will make their decisions completely based on the candidates’ relative charisma. 5 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 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.) The normal form of the game is the description of the players, actions, and the payoffs of the game in a matrix format. The game in this example can be written as below and we can identify two pure strategy Nash equilibria (PSNE): [Enter, Accommodate] (the entrant enters and the incumbent accommodates) and [Out, Fight] (the entrant stays out and the incumbent fights). Accommodate Fight Enter 2, 1 0, 0 Out 1, 9 1, 9 Entrant Incumbent 6 Now consider the mixed strategy equilibrium. The entrant mixes Enter and Out with probability p and (1‐p) and the incumbent mixes Accommodate and Fight with q and (1‐q). The table below summarizes the expected utility of each player when the other randomizes. Incumbent Fight (1‐q) Enter (p) Entrant Accommodate (q) 2, 1 0, 0 EUENT(Enter) = q*2+ (1‐q)*0 = 2q Out (1‐p) 1, 9 1, 9 EUENT(Out) = q*1+(1‐q)*1 = 1 • When the incumbent randomizes, the entrant’s strategy is as below: i)
ii)
iii) • EUINC(A) = p*1+(1‐p)*9 EUINC(F) = p*0+ (1‐p)*9 = 9 – 8p
= 9 – 9p If EUENT(Enter)>EUENT(Out) ( ⇔ 2q > 1, or q > ½), then the entrant will enter (p = 1). If EUENT(Enter)<EUENT(Out) ( ⇔ 2q < 1, or q < ½), then the entrant will stay out (p = 0). If EUENT(Enter)=EUENT(Out) ( ⇔ 2q = 1, or q = ½), the entrant is indifferent between entering and staying out, i.e. p can be anywhere between 0 and 1. When the entrant randomizes, the incumbent’s strategy is as below: i) ii) If EUINC(A) > EUINC(F), then the incumbent will accommodate. This happens when EUINC(A) > EUINC(F), i.e. 9 – 8p > 9 – 9p, or p > 0. This implies whenever entrant mixes with some positive probability p, the incumbent is sure to accommodate (q = 1). If EUINC(A) = EUINC(F), then the incumbent is indifferent. This happens when p = 0 and q can be anywhere between 0 and 1. 7 The best response curves show that the mixed strategy is for the incumbent to accommodate with probability with probability q, where (0 < q ≤ ½) and for the entrant to stay out with probability one. In the diagram, this is represented by the interval over which the two best response curves overlap. q
1 PSNE: p=1, q=1
[Enter, Accommodate] 1/2
Incumbent’s Best Response Mixed NE: p=0, 0<q≤1/2
[Out, A*q + F(1‐q), 0 < q ≤ ½] Entrant’s Best Response PSNE: p=0, q=0
[Out, Fight] 1 p The best response curves show the pure strategy Nash equilibrium as well, i.e. [Enter, Accommodate] and [Stay out, Fight] (two end points). 8 (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. The extensive form of the game is an illustration of the players, the rules, the outcomes, and the payoffs of the game in a game tree. The game tree below depicts the game in our example extensively. The first payoff represents the entrant’s and the second is the incumbent’s. (2, 1)
A
Incumbent
Enter F (0, 0) Entrant
Out
(1 ,9) A sub‐game is a part of the extensive form game which starts from a single node that contains all the successors of that node. There are two sub‐games in our example: the game itself and the sub‐game that starts from the incumbent’s node. A subgame perfect Nash equilibrium is a Nash equilibrium in every subgame. In the game above, if we examine the game from the final node where the incumbent moves, the dominant strategy is to accommodate (if the incumbent accommodates, his payoff is 1 whereas if he fights, the payoff is 0). Given that the incumbent will accommodate if the entrant enters, the entrant is better off with Enter instead of Stay Out (if the entrant enters, his payoff is 2 while the payoff is 1 if he stays out). Therefore, the sub‐game perfect equilibrium is for the entrant to Enter and the incumbent to accommodate. The incumbents’ threat of fighting in the event of new entry is not credible because doing so makes the incumbent worse off for sure than he accommodates. Hence, the threat is not credible, which makes [Fight, Stay Out] not a sub‐game perfect equilibrium. 9 (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 subgame perfect Nash equilibrium is that the Incumbent invests, and the Entrant stays out. Why is this an example of commitment? Sub1 (2, 2)
A
Incumbent
Enter F Entrant
Out Invest
Incumbent (0, 2) (1 ,7)
(2, 1) Sub2 A
Incumbent Not Invest Enter
Entrant F (0, 0) Out
(1 ,9) From the game tree, the subgame perfect Nash equilibrium can be explained intuitively as below: The incumbent can gain the highest payoff if he does not invest and the entrant stays out. But, once the incumbent chooses not to invest, it is best for the entrant to enter since given that the entrance enters, it is best for the incumbent to accommodate. Hence, [Not Invest, Stay out] is not feasible. On the other hand, if the incumbent chooses to invest, now the entrant is no longer sure that the incumbent will accommodate – the payoffs of accommodating and fighting are same for the incumbent. If the incumbent still chooses to accommodate, it is better for the entrant to enter, but there is a possibility for the incumbent to fight, which will make the entrant worse off than he stays out. Hence, in this case, the incumbent’s threat of fighting upon new entry is credible. Knowing this, the incumbent’s best response is to play “Fight” with a probability greater than ½ so that the expected payoff of entrant is smaller than 1 if he enters. This strategy will make the entrant stay out and the incumbent achieve the payoff of 7. Since the incumbent’s threat is credible, it is an example of commitment. 1 0 (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? It is difficult to tell whether the lower pricing is predatory or not. A low price may simply be a competitive behavior intended to maximize profit where competition is taken as given. A possible way to detect anti‐competitive behaviors is to look at the company’s cost. If the firm charges price below the short‐run average variable cost, this might indicate the predatory intent. Another way is to look at the incumbent’s price path. For instance, if the incumbent cuts its price around entry occurs, this might indicate anti‐competitive motivation. However, it is difficult to prove that the company’s aggressive business practice was indeed intended to remove the rival and moreover whether such conduct hurts consumer welfare. In the mean time, the competition may arise from new technology or from overseas. Therefore, alleged violations of anti‐trust law typically result in long court battles. For example, a case against IBM began in 1969 and was finally dropped in 1982. 1 1 ...
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 Economics, Microeconomics, Nash

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