4. Disneyland also o?ers a discount on admissions to residents of Southern California. (You show them your zip code at the gate.) What kind of price discrimination is this? What does this imply about the elasticity of demand for Disney attractions by Southern Californians? 25.4. This is third-degree price discrimination. Apparently the Disneyland administrators believe that residents of Southern California have more elastic demands than other visitors to their park.
26 Factor Markets
1. We saw that a monopolist never produced where the demand for output was inelastic. Will a monopsonist produce where a factor is inelastically supplied?
26.1. Sure. A monopsonist can produce at any level of supply elasticity.
2. In our example of the minimum wage, what would happen if the labor market was dominated by a monopsonist and the government set a wage that was above the competitive wage?
26.2. Since the demand for labor would exceed the supply at such a wage, we would presumably see unemployment.
3. In our examination of the upstream and downstream monopolists we de- rived expressions for the total output produced. What are the
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appropriate expressions for the equilibrium prices, p and k?
26.3. We ?nd the equilibrium prices by substituting into the demand functions. Since p = a?by, we can use the solution for y to ?nd p =
3a + c4
Since k = a?2bx, we can use the solution for x to ?nd k =
a + c2
.
27 Oligopoly
1. Suppose that we have two ?rms that face a linear demand curve p(Y )= a ? bY and have constant marginal costs, c, for each ?rm. Solve for the Cournot equilibrium output.
27.1. In equilibrium each ?rm will produce (a?c)/3b, so the total industry output is 2(a?c)/3b.
2. Consider a cartel in which each ?rm has identical and constant marginal costs. If the cartel maximizes total industry pro?ts, what does this imply about the division of output between the ?rms?
27.2. Nothing. Since all ?rms have the same marginal cost, it doesn’t matter which of them produces the output.
3. Can the leader ever get a lower pro?t in a Stackelberg equilibrium than he would get in the Cournot equilibrium?
27.3. No, because one of the choices open to the Stackelberg leader is to
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choose the level of output it would have in the Cournot equilibrium. So it always has to be able to do at least this well.
4. Suppose there are n identical ?rms in a Cournot equilibrium. Show that the absolute value of the elasticity of the market demand curve must be greater than 1/n. (Hint: in the case of a monopolist, n = 1, and this simply says that a monopolist operates at an elastic part of the demand curve. Apply the logic that we used to establish that fact to this problem.)
27.4. We know from the text that we must have p*1?1/n|ε|]=MC. Since MC > 0, and p>0, we must have 1 ? 1/n|ε| > 0. Rearranging this inequality gives the result.
5. Draw a set of reaction curves that result in an unstable equilibrium 27.5. Make f2( y1) steeper than f1( y2).
6. Do oligopolies produce an e?cient level of output?
27.6. In general, no. Only in the case of the Bertrand solution does price equal the marginal cost.
28 Game Theory
1. Consider the tit-for-tat strategy in the repeated prisoner’s dilemma. Suppose that one player makes a mistake and defects when he meant
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to cooperate. If both players continue to play tit for tat after that, what happens?
28.1. The second player will defect in response to the ?rst player’s (mistaken) defection. But then the ?rst player will defect in response to that, and each player will continue to defect in response to the other’s defection! This example shows that tit-for-tat may not be a very good strategy when players can make mistakes in either their actions or their perceptions of the other player’s actions.
2. Are dominant strategy equilibria always Nash equilibria? Are Nash equilibria always dominant strategy equilibria?
28.2. Yes and no. A player prefers to play a dominant strategy regardless of the strategy of the opponent (even if the opponent plays her own dominant strategy). Thus, if all of the players are using dominant strategies then it is the case that they are all playing a strategy that is optimal given the strategy of their opponents, and therefore a Nash equilibrium exists. How- ever, not all Nash equilibria are dominant strategy equilibria; for example, see Table 28.2.
3. Suppose your opponent is not playing her Nash equilibrium strategy. Should you play your Nash equilibrium strategy?
28.3. Not necessarily. We know that your Nash equilibrium strategy is
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the best thing for you to do as long as your opponent is playing her Nash equilibrium strategy, but if she is not then perhaps there is a better strategy for you to pursue.
4. We know that the single-shot prisoner’s dilemma game results in a dominant Nash equilibrium strategy that is Pareto ine?cient. Suppose we allow the two prisoners to retaliate after their respective prison terms. Formally, what aspect of the game would this a?ect? Could a Pareto e?cient outcome result?
28.4. Formally, if the prisoners are allowed to retaliate the payo?s in the game may change. This could result in a Pareto e?cient outcome for the game (for example, think of the case where the prisoners both agree that they will kill anyone who confesses, and assume death has a very low utility).
5. What is the dominant Nash equilibrium strategy for the repeated prisoner’s dilemma game when both players know that the game will end after one million repetitions? If you were going to run an experiment with human players for such a scenario, would you predict that players would use this strategy?
28.5. The dominant Nash equilibrium strategy is to defect in every round. This strategy is derived via the same backward induction process that
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