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microeconomics

Questions and Answers of Microeconomics

Verify the last sentence.
True or False: When a game tree is such that all information sets are single nodes, then subgame perfect Nash equilibrium is the same as perfect Bayesian Nash equilibrium.
Is every Bayesian Nash equilibrium also a perfect Bayesian Nash equilibrium? Is every perfect Bayesian Nash equilibrium also a Bayesian Nash equilibrium? Explain.
Suppose that, in the game in Graph 24.10, we had changed the receiver’s payoff from playing U when facing a sender of type II who plays left from 20 to 5 instead. Could there be a separating
Could the separating strategy 1L,R2 be part of an equilibrium in this case?
For the game in Graph 24.10, we have therefore found both a separating and a pooling equilibrium, but for the pooling equilibrium we needed to place a restriction on out-of-equilibrium beliefs. Do
How much higher a payoff would a type II sender get by switching her signal in this way?
In the previous section, we talked about subgame perfect strategies in ways that we cannot do here.What is different?
Suppose that we changed the –5 payoff in Graph 24.9 to 20. Demonstrate that this would imply that only the separating strategy 1L,R2 can survive in equilibrium
Suppose the –5 payoff in the lower right corner of the game tree were 0 instead. Would we still get the same subgame perfect equilibrium? Could 1R,R2 be part of a Nash equilibrium that is not
Determine for each of the three remaining sender pure strategies why the strategy cannot be part of a(subgame perfect) equilibrium.
Check that the payoffs listed in Graph 24.9 correspond to the payoffs in Graph 24.8.
Suppose that both bidders know how much each of them values the painting; that is, suppose the game was one of complete information. What would be the Nash equilibrium bidding behavior then? How does
What is the probability that Nature assigns a type greater than t to a player?What is the set of possible actions A for this game?
How would the outcome be different if the two games at the bottom of Graph 24.8 were the games in Table 24.5 and exercise 24a.7 (with player 2’s actions labeled L and R instead of U and D)?
True or False: Player 2 has four possible strategies while player 1 has two possible strategies.
How would you depict the complete information game from either of the payoff matrices in the graph if you had player 2 rather than player 1 at the top of the game tree?
True or False: If we depict a simultaneous move (complete information) game in a game tree, each player only has one information set.
Do you agree or disagree with the following statement: “Both complete and incomplete information simultaneous move games can be modeled as games in which Nature moves first, but Nature plays only
In what sense does the distinction between Nash and subgame perfect equilibrium illustrate how “off-the-equilibrium path” plans—that is, plans that are never executed in equilibrium—can be
If there are N players and T possible types, how many probabilities constitute my beliefs about the other players in the game?
Plot the best response functions to mixed strategies for the Prisoner’s Dilemma game and illustrate that there exists only a single, pure strategy equilibrium.
How might your answer to the previous exercise help explain why we see more cooperation in realworld Prisoner’s Dilemma games than we expect from the incentives contained in the game?
True or False: Whenever individuals find themselves in a Prisoner’s Dilemma game, there is profit to be made if someone can determine a way to commit players to change behavior.
If you model the decision about whether to be friendly to someone you run into as part of a Prisoner’s Dilemma, why might you expect people in small towns to be friendlier than people in big cities?
Why can’t the same type of “trigger strategy” sustain cooperation in a repeated Prisoner’s Dilemma that has a definitive end?
Would you playing “Cooperate always” also potentially be a best response for you to my trigger strategy? Would my trigger strategy then be a best response to your “Cooperate always” strategy?
Explain why this type of trigger strategy is “nice.”
True or False: If two players play “nice” strategies in the repeated Prisoner’s Dilemma, they will always cooperate with one another every time they meet
True or False: In an infinitely repeated Prisoner’s Dilemma game, every subgame of the sequential game is identical to the original game
Does the same logic hold for any repeated simultaneous game in which the simultaneous game has a single pure strategy Nash equilibrium? Put differently, does subgame perfection require that players
Why is this a Prisoner’s Dilemma game?
Why is this outcome inefficient from the perspective of the two players? Could it be efficient from the perspective of “society”?
In our example in Graph 24.4, we say that the subgame perfect equilibrium is not efficient from the perspective of the two players. Could it be efficient from the perspective of “society”?
Suppose the game had a third stage in which the existing firm gets a chance to reevaluate its price in the event that a new firm has entered the market. This would imply that the game tree in Graph
What are the Nash equilibria and the subgame perfect equilibria if player 2 rather than player 1 gets to move first in this version of the game?
True or False: In sequential move games, all pure strategy subgame perfect equilibria are pure strategy Nash equilibria, but not all pure strategy Nash equilibria are subgame perfect.
Is it also a Nash equilibrium for player 1 to play Right and player 2 to play (Left , Right)? If not, why was it a Nash equilibrium before when players were indifferent between coordinating on the
Can you find which strategies in the game depicted in Table 24.3 constitute a Nash equilibrium? (Hint: You should be able to find four combinations of strategies that constitute Nash equilibria.)
Suppose payoffs are as in exercise 24a.8 except that player 2’s payoff from playing Down is 10 less than before (regardless of what player 1 does). Is there a dominant strategy equilibrium? Is
Suppose both players’ payoffs are as in Table 24.5 except that player 1’s payoff when both players play Up is 20. Is there a dominant strategy equilibrium? Is there a unique Nash equilibrium? If
Suppose that player 2 has payoffs as in Table 24.4, while player 1 has payoffs as in Table 24.5. Write out this payoff matrix. Is there a dominant strategy equilibrium? Is there a unique Nash
True or False: If a simultaneous move game gives rise to a dominant strategy for a player, then that strategy is a best response for any strategy played by the other players.
are the two pure strategy Nash equilibria we have identified efficient?
Verify that the payoffs listed in Table 24.3 are consistent with those given in the game tree of Graph 24.2.
True or False: In simultaneous move games, the number of pure strategies available to a player is necessarily equal to the number of actions a player has available.
Suppose that for every player n in a game, the payoffs for player n depend on player n’s action as well as the sum of all the other players’ actions, but no single other player has, alone, a
In the last example (of employer and worker), which set of actions might be more appropriately modeled as continuous?
b. Will the single company make decisions different from that of the social planner in exercise 21.10? What does your answer depend on?
a. In exercise 21.10B(a), we discussed how a social planner’s cost function for each firm would differ from that of each individual firm. Review this logic. How does this apply when all the firms
B. Suppose, as in exercises 21.9 and 21.10, that each of the many firms around the lake has a cost function c1x2 5 bx 2 1 dX, where x is the firm’s output level and X is the total output by all
d. How would your answers to (b) and (c) change if the externality emitted by firms on the lake lowered rather than raised everyone’s marginal costs?
c. Suppose instead that by merging all the firms on the lake, the newly emerged firm will have obtained a monopoly in the output market for x. How would you now think about whether this merger is a
23.11 Policy Application: Regulating Market Power in the Commons: In exercises 21.9 and 21.10, we investigated the case of many firms emitting pollution into a lake. We assumed the only impact of
d. For what level of B is the monopolist’s output choice efficient?
c. What is the monopoly price?
b. Suppose the demand curve is equal to p1x2 5 A 2 ax. Determine the monopolist’s output level x M (assuming no price discrimination).
a. What is the marginal cost function for the monopolist? What is the social marginal cost function?
B. Suppose a monopolist faces the cost function c1x2 5 bx 2, but production of each unit of x causes pollution damage B.
f. Suppose that the production externality were positive instead of negative. True or False: In this case, the monopolist’s output level will be inefficiently low.
e. True or False: In the presence of negative production externalities, the per-unit tax that would cause the monopolist to behave efficiently might be positive or negative (that is, it might take
d. Illustrate a SMC curve with sufficient pollution costs such that the monopoly’s output choice becomes efficient.
c. Suppose that the monopolist pollutes in the process of producing, with the social marginal cost curve SMC therefore lying above the monopolist’s marginal cost curve. Does this change anything
b. Draw the marginal revenue curve and illustrate the monopolist’s profit-maximizing “supply point.”
23.10 Policy Application: Pollution and Monopolies: In Chapter 21, we discussed the externality from pollution-producing industries within a competitive market.A. Suppose now that the polluting firm
g. For what range of recurring fixed costs does the monopolist not produce in the absence of a subsidy from part (f) but produces in the presence of the subsidy? If recurring fixed costs are in this
f. How high a per-unit subsidy would the government have to introduce in order for the monopolist to produce the efficient output level?
e. What is the profit-maximizing output level if the monopolist can perfectly price discriminate?
B. Suppose demand and marginal costs are as specified in part A. Unless otherwise stated, assume no recurring fixed costs.a. Determine the monopolist’s optimal supply point (assuming no price
g. True or False: In the presence of distortions from market power, price distorting policies can be efficient.
23.9 Policy Application: Some Possible “Remedies” to the Monopoly Problem: At least when our focus is on efficiency, the core problem with monopolies emanates from the monopolist’s strategic
g. Suppose the government were to build and maintain the infrastructure needed to deliver electricity to people’s homes. It furthermore allows any electricity firm to use the infrastructure for a
f. Suppose B for the utility company is such that it cannot make a profit by behaving as you derived in (a) and suppose there are N households. Suggest a two-part tariff that will allow the utility
e. Suppose Microsoft and your local utility company share the same demand function. They also share the same cost function except for the fixed cost B. Given our description of the “problem”
B. We did not develop the basic mathematics of natural monopolies in the text and therefore use the remainder of this exercise to do so. Suppose demand for x is characterized by the demand curve p1x2
g. What would be the analogous government intervention in the software industry, and why might you think that this was not a very good idea there? (Hint: Think about innovation.)Could you think of a
f. Explain how the alternative of having the government lay and maintain the infrastructure on which electricity is delivered could address the same “problem.”
e. Now consider the “problem” in the utilities industry. How would setting a two-part tariff allow the utilities company to produce at zero profit? If properly structured, might its output level
d. In the case of Microsoft, how can the granting of a copyright on the software explain the existence of “the problem”? How much is Microsoft willing to pay for this copyright?
c. Put into words the “problem” in the two cases from a government’s perspective (assuming the government cares about efficiency).
23.8 Business and Policy Application: Two Natural Monopolies: Microsoft versus Utility Companies:We suggested in the text that there may be technological reasons for the barriers to entry required
B. Suppose the monopoly has marginal costs MC 5 x and faces the demand curve p 5 90 2 x as in exercise 23.1.a. If you have not already done so, calculate the profit-maximizing supply point 1x M, p M
i. Why can’t monopolists just use their market power to pass the entire tax on to the consumers?
h. By how much does deadweight loss increase as a result of the tax? (Assume that demand is equal to marginal willingness to pay.)
g. In terms of who pays the tax, does it matter which way the government imposes the per-unit tax on x?
f. What happens to the price paid by consumers (including the tax)? What happens to the price received by monopolists?
e. In your graph, illustrate the new marginal revenue curve and the impact of the consumption tax for the monopolist’s profit-maximizing output level.
d. Draw a new graph as in (a). Now suppose that the government instead imposes a per-unit tax t on consumption. Which curves in your graph are affected by this?
c. What happens to the price paid by consumers? What happens to the price that monopolists get to keep (given that they have to pay the tax)?
23.7 Business and Policy Application: Taxing Monopoly Output: Under perfect competition, we found that the economic incidence of a tax (that is, who ends up paying a tax) has nothing to do with
i. True or False: As the wage elasticity of labor supply increases, the monopsonist’s decision approaches what we would expect under perfect competition.
h. Can you write the MC side of the equation in terms of the wage elasticity of labor supply?
g. Consider the more general case of a monopsonist firm with production function f1,2 facing a labor supply curve of w1,2. Derive the MR 5 MC condition (which is the same as the condition that the
M does the monopsonist firm hire, and how does it compare to ,*?f. What wage w M does the firm pay, and how does it compare to w*?
B. Suppose that the firm’s production function is given by f1,2 5 A, a(with a , 1) and the labor supply curve is given by ws 1,2 5 b,.a. What is the efficient labor employment level ,* ? (Hint: You
j. Labor unions allow workers to create market power on the supply side of the labor market. Is there a potential efficiency case for the existence of labor unions in the presence of monopsony power
i. We gave the example of a modest-sized town with a dominant employer as a motivation for thinking about monopsonist firms in the labor market. As it becomes easier to move across cities, do you
h. Suppose the government sets a minimum wage of w* (as defined in (a)). Will this be efficiency enhancing?
g. After a monopolist decides how much to produce, she prices the output at the highest possible level at which all the product can be sold. Similarly, after a monopsonist decides how much to buy, he
f. Profit is maximized where MR 5 MC. Illustrate in your graph where marginal revenue crosses marginal cost. Will the firm hire more or fewer workers than a competitive market would (if it had the

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