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microeconomics

Questions and Answers of Microeconomics

g. What happens to the equilibrium probability of a crime being reported as N increases?
f. For what value of y have we identified a Bayesian Nash equilibrium?
e. What is the condition for individual n to not report the crime if xn , y? What is the condition for individual n to report the crime when xn $ y?
d. What is the expected payoff of not reporting the crime for individual n whose value is xn?What is the expected payoff of reporting the crime for this individual?
c. Consider now whether there exists a Bayesian Nash equilibrium in which each player n plays the strategy of reporting the crime if and only if xn is greater than or equal to some critical value y.
b. From here on out, suppose that P1x2 5 x/b. Does what you concluded in (a) hold?
a. What is P102? What is P1b2?
B. Suppose from here on out that everyone values the reporting of crime differently, with person n’s value of having a crime reported denoted xn. Assume that everyone still faces the same cost c of
j. If the cost of reporting the crime differed across individuals (but is always less than x), would the set of pure Nash equilibria be any different? Without working it out, can you guess how the
i. True or False: If the reporting of crimes is governed by such mixed strategy behavior, it is advantageous for few people to observe a crime, whereas if the reporting of crime is governed by pure
h. What is the probability that a crime will be reported in this mixed strategy equilibrium? (Hint:From your work in part (f), you should be able to conclude that the probability that no one else
g. Using your answers to (d) through (f), derive d as a function ofc, x, and N such that it is a mixed strategy equilibrium for everyone to call with probabilityd. What happens to this probability
f. What is the expected payoff from not calling when everyone calls with probability d? (Hint:The probability that one person does not call is 11 2 d2, and the probability that 1N 2 12 people don’t
e. What is the payoff from calling when everyone calls with probability d , 1?
d. Next, suppose each person calls with probability d , 1. In order for this to be a mixed strategy equilibrium, what has to be the relationship between the expected payoff from not calling and the
c. There are many pure strategy Nash equilibria in this game. What do all of them have in common?
b. Is there a pure strategy Nash equilibrium in which more than one person reports the crime?
a. Each person then has to simultaneously decide whether or not to pick up the phone to report the crime. Is there a pure strategy Nash equilibrium in which no one reports the crime?
A. Begin by assuming that everyone places a value x . c on the crime being reported, and if the crime goes unreported, everyone’s payoff is 0. _Thus, the payoff to me if you report a crime is x,
24.16 Everyday and Policy Application: Reporting a Crime: Most of us would like to live in a world where crimes are reported and dealt with, but we’d prefer to have others bear the burden of
e. In which equilibrium—the one in part (c) or the one in part (d)—do the equilibrium beliefs of the incumbent seem more plausible?
d. Next, suppose that the payoffs for (P, G2 changed to (4, 2) and the payoffs for (U, G) changed to (3, 2) (with the remaining payoff pairs remaining as they were in A(f)). Do you get the same pure
c. What combinations of strategies and (incumbent) beliefs constitute a pure strategy perfect Nash equilibrium? (Be careful: In equilibrium, it should not be the case that the incumbent’s beliefs
b. For what range of d is it a best response for the incumbent to play G? For what range is it a best response to play F?
a. Suppose that the incumbent believes that a challenger who issues a challenge is prepared with probability d and not prepared with probability 11 2 d2. What is the incumbent’s expected payoff
B. Consider the game you ended with in part A(f).
f. Can you still use the logic of subgame perfection to arrive at a prediction of what the equilibrium will be if the incumbent cannot tell whether the challenger is prepared or not as you did in
14, 22, and the payoffs for 1U, F2 changed to (0, 3) (with the other two payoff pairs remaining the same). Assuming again that the incumbent fully observes both whether he is being challenged and
e. Next, suppose that the payoffs for (P, G2 changed to (3, 2), the payoffs for 1U, G2 changed to
d. Next, suppose that the incumbent only observes whether or not the challenger is engaging in the challenge (or staying out) but does not observe whether the challenger is prepared or not.Can you
24.15 Everyday, Business, and Policy Application: To Fight or Not to Fight: In many situations, we are confronted with the decision of whether to challenge someone who is currently engaged in a
i. What happens to the pooling wage relative to the highest possible wage in a separating equilibrium as d approaches 1? Does this make sense?
h. Could there be an education level e that high productivity workers get in a separating equilibrium and that all workers get in a pooling equilibrium?
g. What levels of education e could in fact occur in such a perfect Bayesian pooling equilibrium?Assuming e falls in this range, specify the pooling perfect Bayesian Nash equilibrium, including the
f. Assuming that every job applicant is type 2 with probability d and type 1 with probability 11 2 d2, what wage offers will firms make in stage 2?
e. Next, suppose instead that the equilibrium is a pooling equilibrium; that is, an equilibrium in which all workers get the same level of education e and firms therefore cannot infer anything about
d. Given your answers so far, what values could e take in this separating equilibrium? Assuming e falls in this range, specify the separating perfect Bayesian Nash equilibrium, including the
c. What wages will the competing firms offer to the two types of workers? State their complete strategies and the beliefs that support these.
b. Suppose that there is a separating equilibrium in which type 2 workers get education e that differs from the education level type 1 workers get, and thus firms can identify the productivity level
a. Will workers behave any differently in stage 3 than they did in part A of the exercise?
B. Now suppose that employers cannot tell the productivity level of workers directly; all they know is the fraction d of workers that have high productivity and the education level e of job
f. True or False: If education does not contribute to worker productivity and firms can directly observe the productivity level of job applicants, workers will not expend effort to get education, at
e. What level of e will the two worker types then get in any subgame perfect equilibrium?
d. Would the wages offered by the two employers be any different if the employers moved in sequence, with employer 2 being able to observe the wage offer from employer 1 before the worker chooses an
c. Note that we have assumed that worker productivity is not influenced by the level of education e chosen by a worker in stage 1. Is there any way that the level of e can then have any impact on the
b. Given that firms know what will happen in stage 3, what wage will they offer to each of the two types in the simultaneous move game of stage 2 (assuming that they best respond to one
A. Suppose first that worker productivity is directly observable by employers; that is, firms can tell who is a type 1 and who is a type 2 worker by just looking at them.a. Solving this game
24.14 Everyday, Business, and Policy Application: Education as a Signal: In Chapter 22, we briefly discussed the signaling role of education; that is, the fact that part of the reason many people
f. All else being equal, where would you expect the most screaming per child: in a single-parent household, a two-parent household, or in a commune?
e. What if the child lives in a two-parent household? What if the child is raised in a commune where everyone takes care of everyone’s children?
d. Children can be like terrorists: screaming insanely to get their way and implicitly suggesting that they will stop screaming if parents give in. In each instance, it is tempting to just give them
c. If you had to guess, do you think small countries or large countries are more likely to negotiate with pirates and terrorists?
b. Suppose that only a single country is targeted by terrorists. Does the Prisoner’s Dilemma still apply?
Can you use the logic of the Prisoner’s Dilemma to explain why so many countries negotiate even though they say they don’t? (Assume pirates cannot tell who owns a ship before they board it.)
A. Often, countries have an explicit policy that “we do not negotiate with terrorists,” but still we often discover after the fact that a country (or a company that owns a shipping vessel) paid a
24.13 Policy Application: Negotiating with Pirates and Terrorists (and Children): While we often think of pirates as a thing of the past, piracy in international waters has been on the rise.
i. What is the highest that w can get in order for the firm to best respond to workers (who play the strategy in (b)) by playing the strategy in (a)? Combining this with your answer to (h), how must
. How much of a premium above the market wage w* does this imply the worker requires in order to not shirk? How does this premium change with the cost of effort e?How does it change with the
. Combining this with your answer to parts (e) and (f), explain why the following equation must hold:Ps 5 w 1 d cgPs 1 11 2 g2 w*1 2 d d . (24.11)Derive from this the value of Ps as a function ofd,
g. Suppose that the worker’s expected payoff from always shirking is Ps. If the worker does not get caught the first day he shirks, he starts the second day exactly under the same conditions as he
f. Suppose that the worker gets unlucky and is caught shirking the first time and that he therefore will not be employed at a wage other than the market wage w* starting on day 2. In that case, what
e. Suppose the firm offers w 5 w. Notice that the only way the firm can ever know that the worker shirked is if its payoff on a given day is 12w2 rather than (x 2 w), and we have assumed that this
d. Use this to determine the present discounted value Pe of the game (as a function of w,e, and d)for the worker assuming it is optimal for the worker to exert effort when working for the firm.
c. Suppose everyone values a dollar next period at d , 1 this period. Suppose further that Pe is the present discounted value of all payoffs for the worker assuming that firms always offer w 5 w and
b. Consider the following strategy for the worker: Accept any offer w so long as w $ w*; reject offers otherwise. Furthermore, exert effort e upon accepting an offer so long as all previous offers
a. Consider the following strategy for the firm: Offer w 5 w . w* on the first day; then offer w 5 w again every day so long as all previous days have yielded a payoff of 1x 2 w); otherwise offer w
B. The problem in the game defined in part A is that we are not adequately capturing the fact that firms and workers do not typically interact just once if a worker is hired by a firm. Suppose, then,
f. The subgame perfect equilibrium you just derived is inefficient. Why? What is the underlying reason for this inefficiency?
e. Suppose w* is related to g, x, and e such that it is efficient for workers to be hired by the firm only if they don’t shirk, that is, if the conditions you derived in (c) and (d) hold. What is
d. Suppose the worker exerts effort e if hired by the firm. Since e is a cost for the worker, how must w* be related to 1x 2 e2 in order for it to be efficient for non-shirking workers to be hired by
c. How must w* be related to g and x in order for it to be efficient for the worker not to be employed by the firm if the worker shirks?
24.12* Business Application: Carrots and Sticks: Efficiency Wages and the Threat of Firing Workers: In our treatment of labor demand earlier in the text, we assumed that firms could observe the
b. What is the perfect Bayesian equilibrium of this game in the context of concepts discussed in Chapter 23? Explain.
a. Can you model the price-setting decision by the monopolist as a game of incomplete information?
B. Next, suppose that the monopolist is unable to observe the consumer type but knows that a fraction r in the population are low demand types and a fraction 11 2 r2 are high demand types. Assume
e. Next, suppose that the monopolist cannot charge a fixed fee but only a per-unit price, but he can set different per-unit prices for different consumer types. What is the subgame perfect
d. How is this analysis similar to the game in exercise 24.5?
c. True or False: First degree price discrimination emerges in the subgame perfect equilibrium but not in other Nash equilibria of the game.
24.11 Business Application: Monopoly and Price Discrimination: In Chapter 23, we discussed first, second, and third degree price discrimination by a monopolist. Such pricing decisions are strategic
e. Why do you think I called the lowest winning bid the “market price”? Can you think of several ways in which the allocation of students to faculty might have become more efficient as a result
d. Would it surprise you to discover that for the rest of my term as chair, I never again heard complaints that we had a “TA shortage”? Why or why not?
c. In my annual e-mail to the faculty at the beginning of the auction for rights to match with students, I included the following line: “For those of you who are not game theorists, please note
b. I replaced the system with the following: Aside from some key assignments of graduate students as TAs to large courses, I no longer assigned any students to faculty. Instead, I asked the faculty
a. Under this system, faculty complained perpetually of a “teaching assistant shortage.” Why do you think this was?
B. This part provides a real-world example of how an auction of the type analyzed in part A can be used.When I became Department Chair in our economics department at Duke, the chair was annually
g. True or False: The outcome of the sealed bid second price auction is approximately equivalent to the outcome of the sequential (first price) auction.
f. In equilibrium, approximately what price will the winner of the sequential auction pay?
e. Now consider a sequential first price auction in which an auctioneer keeps increasing the price of x in small increments and any potential bidder signals the auctioneer whether she is willing to
d. Suppose that players are not actually sure about the marginal willingness to pay of all the other players, only about their own. Can you think of why the Nash equilibrium in which all players bid
c. Can you think of another Nash equilibrium to this auction?
b. Suppose individual j has the highest marginal willingness to pay. Is it a Nash equilibrium for all players other than j to bid zero and player j to bid vj?
a. Is it a Nash equilibrium in this auction for each player i to bid vi?
vA. Suppose there are n different bidders who have different marginal willingness to pay for the item x.Player i’s marginal willingness to pay for x is denoted vi. Suppose initially that this is a
24.10 Everyday and Business Application: Auctions: Many items are sold not in markets but in auctions where bidders do not know how much others value the object that is up for bid. We will analyze a
f. Does the first mover have an advantage in this infinitely repeated bargaining game? If so, why do you think this is the case?
e. Use this insight to derive how much player 1 offers in period 1 of the infinitely repeated game.Will player 2 accept?
d. Given your answers, why must it be the case that x 5 100 2 d1100 2 dx2?
c. In part A of the exercise, you should have concluded that when the game was set to artificially end in period 3 with payoffs x and 1100 2 x2, player 1 ends up offering x1 5 100 2 d1100 2 dx2 in
b. Suppose that, in the game beginning in period 3, it is part of an equilibrium for player 1 to offer x and player 2 to accept it at the beginning of that game. Given your answer to (a), is it also
a. True or False: The game that begins in period 3 (assuming that period is reached) is identical to the game beginning in period 1.

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