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1、 Copyright 2014 Pearson Education,Inc.CHAPTER 11 GAME THEORY AND ASYMMETRIC INFORMATION QUESTIONS 1.It is more of a problem for one-shot games.2.It is a variable sum game.3.It is more problematic for non-cooperative games.4.It is easier for an irrational person to make a credible threat.5.Because ex
2、plicit bargaining is illegal.6.Because firms need to agree on a joint outcome without being able to explicitly communicate with one another.7.Because if you can control the agenda you can create a tacit negotiation which has a solution that is favorable to you.8.The bounds on the war need to be qual
3、itative not quantitative.No price competition,not how much price competition,for example.9.Ace of spades.10.A focal point.11.Because different consumers will have different packets of information about the alternative versions of a good and as a result will have different preferences over identical
4、alternatives.12.No,asymmetric information implies that there is imperfect information and one side has a systematic advantage relative to the other side.Imperfect information simply means that the individual decision-making agents do not have all necessary information to make the decision.13.No.14.S
5、ellers have better information about the used cars they are selling than do buyers.15.As price increases,average quality increases in the used car market.If the increase in demand due to increased quality is larger than the reduction in demand due to the pure price effect then demand will be upward
6、sloping.16.The quality effect dominates.17.When it is differentially costly to provide.18.The job seeker has more information.Capital Budgeting 109 Copyright 2014 Pearson Education,Inc.19.Because it is differentially costly to obtain.20.No,education acts purely as a signal within Spences model.21.Ye
7、s because they are differentially costly to provide.22.The buyer has more information in this instance.23.Both act as a way of getting individuals to voluntarily separate themselves into groups according to their abilities or attributes.24.It is aimed at low risk individuals.25.Yes,the Lemons model
8、is an example of the adverse selection problem.26.Profit sharing and stock options.27.No because it is not differentially costly to provide.PROBLEMS 1.a.Any dominant strategy equilibrium is also a Nash equilibrium.This is discussed in Tables 11.1 and 11.2 b.A game may have a Nash equilibrium even if
9、 it does not have a dominant strategy equilibrium.This is discussed in Tables 11.1 and 11.2 c.Yes a firm can determine from the options available to the other firm that some strategies are more likely than others.If an opponent has a dominant strategy then it is likely that the firm will choose that
10、 strategy.As a result,the opponent can use that information to determine the optimal strategy.This is discussed in Table 11.2.d.No,a dominant strategy equilibrium means that one strategy is dominant for each player.Therefore there can be only one DSE and it only exists if all players have a dominant
11、 strategy.2.a.People will shop for the week based on the days that the grocery stores are open.Some new sales will occur due to being open an extra day but it may not make up for the added labor costs incurred.b.110 Game Theory and Asymmetric Information Copyright 2014 Pearson Education,Inc.c.Whites
12、 dominant strategy is to open Sundays.Grays dominant strategy is to open Sundays.Therefore,Open,Open is a dominant strategy equilibrium.This strategy is not,however profit maximizing as noted above.d.Since this game is played repeatedly,each firm is likely to be able to signal via its actions that i
13、t wishes to close on Sunday.Eventually both firms are likely to learn that this is the best strategy of course there is always the incentive to cheat on the bargain.It is worth noting that some states have“Blue Laws”that help firms achieve the Closed/Closed solution via government fiat.e.This is an
14、example of a prisoners dilemma because the dominant strategy equilibrium is dominated if each firm can be convinced to not cheat and remain closed on Sundays.3.a.The leader is Gray.Under all circumstances except when it is closed and White is open on Sunday,Gray has higher profits and in this instan
15、ce it has equal profit.b.Gray does not have a dominant strategy.If White chooses to remain open then Gray should also remain open.If White decides to close,then Gray should also close.Therefore,Grays choice depends on what White does Gray has no dominant strategy.c.Gray should stay open on Sundays.B
16、y analyzing what White is likely to do the appropriate strategy for Gray becomes clear.Regardless of Grays choice,staying open dominates for White.White is therefore likely to remain open.Based on the expectation that White will remain open,Gray should remain open as well.d.No this is not a prisoner
17、s dilemma.Consider the closed/closed solution as the starting point(since this is joint profit maximizing).Although White has an incentive to cheat and open,Gray does not.Therefore both firms do not have an incentive to cheat on the“dont confess/dont confess”outcome as is the case in a prisoners dil
18、emma.4.This is an example of a cooperative game.The fact that a better deal for you is certainly a worse deal for the seller you are purchasing from means you are playing a zero sum game,but the deals are explicitly negotiated therefore the game is cooperative.Explicit contracts are negotiated over
19、the attributes of the components as well as price and delivery date as well as over penalties if delivery is not completed in a timely manner.5.If negotiations cannot be explicitly agreed to(perhaps for legal reasons),then there is a greater need to have a recognizable outcome or a focal point solut
20、ion.The key skill is in formulating the problem so that the seemingly natural outcome is the one that is most advantageous to you.6.a.100%of cherry owners and 100%of lemon owners sell if P$8,000.If P$3,000.b.Since buyers are risk neutral they are willing to pay a weighted average of the market clear
21、ing prices in the good and bad segment.Pwtd=sPlemon+(1-s)Pcherry.With s=0.1,the weighted average price is$9,400=0.1$4,000+0.9$10,000.10%of cars will be lemons.c.With s=0.25,the weighted average price is$8,500=0.25$4,000+0.75$10,000.25%of cars will be lemons.Capital Budgeting 111 Copyright 2014 Pears
22、on Education,Inc.d.With s=0.5,the weighted average price is$7,000=0.5$4,000+0.5$10,000.100%of cars will be lemons and the price will decline to$4,000 because the weighted average price of$7,000 is below the reservation price of good used car owners.This is an example of the lemons problem.The asymme
23、tric information has caused an adverse selection problem in the used car market.e.The weighted average price must be at least as large as the good used car owners reservation price.We simply must solve for r in the equation:8000=s$4,000+(1-s)$10,000.8000=4000s+10000 -10000s.6000s=2000.s=1/3.When the
24、 portion of lemons is less than 1/3,there is no adverse selection problem in this used car market because both lemons and good used cars will be sold(at a weighted average price above the good used car owners reservation price).When the portion of lemons is greater than 1/3,there is an adverse selec
25、tion problem in the used car market because no good used car owner will be willing to sell.f.We would have the same result but simply at a different value of s.In fact,we already know the value of s from part(c).s=0.5 provides a weighted average price of$7,000 equal to the reservation price.If the p
26、ortion of lemons is less than 50%there would be no problem but if the portion of lemons is more than 50%the adverse selection problem would occur and all used cars sold would be bad.7.a.If all good cars are sold then all cars are sold so the fraction is 1.1/3 are lemons.b.If half of good used cars a
27、re sold then half are not.Since 1/3 of all cars are lemons,2/3 are good so half of these are sold.The other half are not sold.This means that 2/3 of all used cars are sold.The portion that is lemons is NOT 1/3 but rather 50%since all lemons are sold.c.If one quarter of good used cars are sold then 3
28、/4 are not.Since 1/3 of all cars are lemons,2/3 are good so 1/4 of these are sold.The other 3/4 are not sold.Total sold are therefore 1/3+2/3=1/3+1/6=.This means that half of all used cars are sold.The portion that is lemons is NOT 1/3 but rather 2/3 since all lemons are sold.d.The fraction sold is
29、1/3+G(2/3).The fraction that are lemons is:(1/3)/(1/3+G(2/3)=1/(1+2G).These equations work for all G between 0 and 1.e.No,the weighted average price in this instance is$8,333=1/3$5,000+2/3$10,000.The problem in this instance is that some of the good used car owners have a higher marginal valuation t
30、han$8,333.(In fact,given the uniform distribution assumed,2/9 do.)As a result,the adverse selection problem results.f.Given the full information prices and fraction of lemons sold derived in D we have:PB(G)=(1/(1+2G)$5,000+(2G/(1+2G)$10,000.112 Game Theory and Asymmetric Information Copyright 2014 P
31、earson Education,Inc.g.Inverse supply is a reservation price,PS(G).This can be set equal to PB(G)to obtain the market clearing price:(1/(1+2G)5000+(2G/(1+2G)10000=6000+3000G.First divide both sides by 1000 to make the numbers more manageable.(1/(1+2G)5+(2G/(1+2G)10=6+3G.Next multiply both sides by 1
32、+2G to remove the denominator on the left hand side:5+20G=(6+3G)(1+2G)Expanding we have:5+20G=6G2+15G+6.Placing all terms on the right hand side we have the following quadratic:0=6G2 5G+1.This may be factored as:6G2 5G+1=(3G 1)(2G 1).Therefore we have the solution:G=1/3 or G=.When G is between 1/3 a
33、nd the weighted average price is at least as high as marginal valuation(reservation price).The least distorted market occurs when G is as large as possible,therefore G=0.5 is the solution we will work with.Half the good used cars are sold,half are not.From part B we know that in this instance,the po
34、rtion of lemons is 50%.Therefore the market clearing price is$7,500.This same price may be derived from either formula,the inverse supply formula or the weighted average price formula.8.To have a solution in which some good used cars are sold,we require that the reservation price for a given level o
35、f G(the inverse supply equation)is at least as large as the weighted average price derived in part(f).The altered portion of lemons sold if G of the good used cars is sold is given by the new(d)equation,(d):The fraction sold is 1/2+G.The fraction that are lemons is therefore:(1/2)/(1/2+1/2 G)=1/(1+G
36、).Similarly,the fraction of sold cars that are good is:G/(1+G).(f)PB(G)=(1/(1+G)5000+(G/(1+G)10000.(g)The equality in(g)is to have a market price equal the inverse supply price:PB(G)=PS(G)if:(1/(1+G)5000+(G/(1+G)10000=6000+3000G.First divide both sides by 1000 to make the numbers more manageable.(1/
37、(1+G)5+(G/(1+G)10=6+3G.Next multiply both sides by 1+G to remove the denominator on the left hand side:5+10G=(6+3G)(1+G)Capital Budgeting 113 Copyright 2014 Pearson Education,Inc.Expanding we have:5+10G=3G2+9G+6.Placing all terms on the right hand side we have the following quadratic:0=3G2 G+1.This
38、has NO solution according to the quadratic formula.This is readily seen by noting that the“radical”part of the quadratic is a negative number.The radical part,B2 4AC,in this instance is 11.It remains to show that reservation price is always ABOVE weighted average price.The easiest way to show this i
39、s to simply check one value therefore check G=0.5.If G=0.5 then of the used cars are sold 2/3 of which are lemons and one third of which are cherries.The market price for this alternative is PB(0.5)=2/35000+1/310000=$6,667.The reservation price required to sell half the good used cars is according t
40、o PS(G)is:PS(0.5)=6000+0.53000=$7,500.The price demanded by good used car owners in order to have half of them be offered for sale is more than consumers would be willing to pay.The only solution for this market is for all used cars to be lemons.9.a.This is a constant elasticity demand function.The
41、price elasticity of demand is-2 and the quality elasticity of demand is+4.b.A(0)=0;A(5000)=100/3=33.3;A(10000)=50;A(15000)=60;A(20000)=66.6.c.Average quality increases at a decreasing rate as price increases.d.X(5000)=790;X(10000)=1000;X(15000)=922;and X(20000)=790.Each of these answers is obtained
42、using the CED function substituting the value of A(P)derived in part B in each instance.e.Since demand at a price of$10,000 is more than at$5,000 or$15,000 the demand curve must be backward bending somewhere between$5K and$15K.Formally,the price must be more than$5,000 and less than$15,000 at the po
43、int where it turns from being upward sloping to downward sloping(as price increases).f.Take the total derivative of demand and set equal to zero.dX/dP=X/P+X/AdA/dP.Note:The easy way to write derivatives with a CED function is to note that each partial is simply the exponent times X divided by the va
44、riable(in this instance P or A).This means we can write the derivative as:dX/dP=-2X/P+(4X/A)dA/dP.114 Game Theory and Asymmetric Information Copyright 2014 Pearson Education,Inc.But dA/dP=(P+10000)100 100P)/(P+10000)2=1,000,000/(P+10000)2,so:dX/dP=-2X/P+(4X/A)1,000,000/(P+10000)2 Substituting A(P)in
45、to the above we have:dX/dP=-2X/P+(4X(P+10000)/100P)1,000,000/(P+10000)2 Simplifying we obtain:dX/dP=-2X/P+(4X10,000)/(P+10000).We further simplify this by finding a common denominator for the two terms:dX/dP=(-2X(P+10000)+(4X10,000)/(P(P+10000).Further simplifying we obtain:dX/dP=(2X(-P-10000+20,000
46、)/(P(P+10000).dX/dP=(2X(10000 P)/(P(P+10000).Equation*We wish to set dX/dP=0.If a fraction equals zero,the numerator must equal zero.Since X 0,the other term in the numerator must equal zero:10000 P=0.This implies that dX/dP=0 when P=$10,000.We also see from Equation*that dX/dP 0 when P$10,000 and d
47、X/dP$10,000.g.This is most easily accomplished by programming A(P)and Q(P,A(P)into Excel.The graphs act exactly as expected.Average quality increases at a decreasing rate as price increases.Demand is backward bending:for prices below$10K,demand is upward sloping;for prices above$10K,demand is downwa
48、rd sloping;and at P=$10K,demand is vertical.Average quality,A(P)01020304050607080$0$5,000$10,000$15,000$20,000$25,000$30,000Price$5,000$10,000$15,000$20,000$25,000$30,00020040060080010001200 1020304050607080$0$5,000$10,000$15,000$20,000$25,000$30,000Demand,X(P)$0$5,000$10,000$15,000$20,000$25,000$30
49、,000020040060080010001200X 10.a.The firm should require 4 years of higher education.This is derived using Equation 21.4 that relates wage offers with the cost of receiving the signal:(H L)/CL e (H L)/CH.Using the numbers in this problem,we have:$14,000/$4,000=3.5 e$35,000 4$4,000.“no higher educatio
50、n wage”the net value of the high paying job.b.Education is a signal in this instance because it is differentially costly to obtain.Capital Budgeting 115 Copyright 2014 Pearson Education,Inc.c.In this model,there is no increase in productivity from education,per se.Educated workers are more productiv