Chapter 7 Review questions

Transcription

1 Chapter 7 Review questions 71 What is the Nash equilibrium in a dictator game? What about the trust game and ultimatum game? Be careful to distinguish sub game perfect Nash equilibria from other Nash equilibria The Nash equilibrium in a dictator game is for the proposer to keep all the money for himself The sub game perfect Nash equilibrium in an ultimatum game is for the receiver to accept any positive offer and so for the receiver to propose the smallest possible amount But, this is not the only Nash equilibrium Suppose the receiver threatens to reject any offer less than a split and the proposer offers a split This is a Nash equilibrium no one can gain from changing strategy given the strategy of the other There are many more similar Nash equilibria In a trust game a Nash equilibrium requires the proposer to not give any money back to the investor So, the only Nash equilibrium sees the investor keep the money for himself 72 Why might payoffs be lower in a public good game with punishment even if punishment increases contributions? Is it good to have the threat of punishment or not? We need to take into account the cost of punishment In a direct sense punishment is a deadweight loss because it lowers both the payoff of the punisher and person punished In an indirect sense punishment may be beneficial in increasing contributions Overall, therefore, punishment is only a good thing if the indirect benefit exceeds the direct cost This appears to be the case in western countries (see chapter eight for cultural differences) The reason is that the threat of punishment appears enough to change behavior no one actually has to do much punishing The costs of punishment are, thus, typically low It should be kept in mind, however, that punishment has the potential to escalate and become costly this is what we do observe in many non western countries

2 73 In section 715 I mentioned a study by Andreoni and Miller (2002) They asked subjects to play eleven different dictators game In each game the proposer was given a number of tokens, from 40 to 100 and asked to divide the tokens between them and a receiver Each token was worth $010 to $040 to the proposer and $010 to $040 to the receiver For example, a token might be worth $040 to the proposer and $010 to the receiver, or $020 to the proposer and $030 to the receiver How do you think the share given to receivers depended on the worth of tokens? How should it depend, to be consistent with utility maximization? Suppose I am given tokens and each token is worth $ to me and $ to you If I give you tokens then I end up with $ and you get $ Intuition would suggest proposers give more the higher is and the higher is relative to This is what we need in order that choice be consistent with utility maximization To see this in a bit more detail we can work through an example We could say that the price of me keeping a token for myself is 1/ the lower is the more costly for me to keep it Similarly the price of me giving a token to you is 1/ Suppose my utility function is a Cobb Douglas function, where is my relative gain, is your relative gain, and is a parameter that measures social preference Following standard arguments (see any microeconomics textbook) I maximize utility where where is marginal utility In this case that means So, I should set We next need the budget constraint which gives We can substitute for into the optimum to get Which simplifies to You should also get 1 1 These demand functions are as predicted The higher is and the more I keep for myself The higher is and the more I give to you

3 74 Should with equate material payoff with money? This is the convention in modeling social preferences, but can mislead In principle we should use the utility of money rather than money For example, if Miguel keeps $ in the ultimatum game then he keeps $ in utility If large amounts of money are at stake then this can make a difference To illustrate consider the ERC model looked at in the text and let be Miguel s payoff from keeping $ In the text we set 1 meaning money equals utility Suppose we set 1? We need to maximize utility with respect to At an internal optimum Rearranging we get If 1 this gives For instance, if 10 Miguel keeps all $10 and if 25 he keeps all $7 Solving for the optimum when 1 is not so easy (which is why it is simpler to equate money with utility) but we can look at some examples Suppose 05 Then we need 5 25 If 10 then a value of 6 works well enough So, Miguel gives away $4 If 25 then a value of 55 works well enough So, Miguel gives away $450 The key thing to take from this example is how the diminishing marginal utility of money, that we get when 05, means that Miguel gives away a lot more money In terms of intuition this is because Miguel has enough money that he will pay a high price for getting closer to the social reference point In general, the more money at stake the more we should expect people to focus on social outcomes than own monetary payoff

4 75 Find the general conditions such that punishment can work in a linear public good game in the ERC model and the Fehr Schmidt model We want to obtain a Nash equilibrium where everyone contributes $20 Suppose that Miguel deviates and contributes $ Anna punishes Miguel Specifically she pays $ in order to lower his payoff by $ If there are players in the game then the total amount contributed is 20 1 So, final monetary payoffs are where / In the ERC model Anna s relative share is Suppose we set 1/ the equal share Then which simplifies to 20 Note that if Anna punishes this much Miguel has no incentive to deviate But does Anna gain? If she does not punish her payoff is If she does punish her payoff is She will, therefore, punish if This will be satisfied for sufficiently small and large Now for the Fehr Schmidt model This time suppose Anna punishes Miguel enough that they both get the same material payoff So 20 meaning 20 1 Anna gets the same material payoff as Miguel and less than everyone else So, her payoff is If she did not punish her material payoff would be 20 less than Miguel So, overall

5 She is willing to punish if which gives or Why can punishment work in a linear public good game according to the Fehr Schmidt model if only one person is willing to punish, but in the ERC model only if two people are willing to punish? In the ERC model every person has an incentive to contribute a little bit less than $20 This is because a person does not mind being a little bit away from the social reference point (note the square term in the utility function) So, everyone needs the threat of punishment In the Fehr Schmift model a sufficiently inequality averse person would never want to contribute less than $20 (if everyone else is contributing $20) So, not everyone needs to be under the threat of punishment From a mathematical point of view this is an artifact of how I have set up the utility function the ERC model is quadratic and the Fehr Schmidt model linear But, there is a general point here Sometimes we only need one policeman of social norms because the policeman will play fair Sometimes we need someone checking up on the policeman 77 How can an intentions based model of fairness be adapted to take account of earned versus unearned payoff, and social norms? We simply need to change the definition of what is fair or kind For example, if money in an ultimatum game is earned so, the proposer earns the $10 he is asked to give away social norms would typically dictate it is not unkind for the proposer to keep the $10 An offer of $0 is not, therefore, considered unkind and the receiver may accept this offer Intention based models are very flexible to changes in the notion of what is fair or not But note that beliefs are still key here What does the proposer and receiver believe the other believes is a kind or unkind act?

6 78 In the intentions based model of section 731 the equitable payoff was derived looking only at Pareto efficient outcomes Why is this sensible? This will depend on the game But, in the ultimatum game it does make sense to restrict to Pareto efficient outcomes To see the distinction let us pick up the story in the text If Miguel believes Federica will accept offers of or more then he can offer $10 or $ and the outcome be Pareto efficient It is Pareto efficient because Federica will not reject The equitable payoff is then If we do not require a Pareto efficient outcome Miguel can offer $10 or $0 In this last case Federica will reject and so everyone loses out The equitable payoff is then 5 2 Notice that beliefs drop out of the story But, the very point of an intention based model is to capture beliefs So, the story is no longer as rich Specifically, we might want to say that offering $6 is not kindness if Miguel believes $ $6 He offers $6 because he believes anything else would be rejected 79 Is it better to be in a firm where you earn $50,000 and the average salary is $80,000 or in a firm where you earn $45,000 and the average salary is $30,000? First, let us look at this from the perspective of inequality aversion You would be earning $30,000 less than the average in one firm and $15,000 more than the average in the other This is likely to appeal to someone who is inequality averse Is it worth $5000 less salary? For some it might be Next think about of model of intentions Why are you getting so much less than the average in one firm and more in the other? You may or may not interpret this as fair depending on specific circumstances For example, if you are less skilled for the job then you may consider it fair you earn below the average If you suspect you earn less than the average because you are a women or black then you are clearly less likely to see it as fair

7 710 Why can loss aversion help explain why a cut in wages is worse than a rise in wages is good? How is this related to wage stickiness and habit formation? A cut in wages is clearly a loss But note the frame of reference: In principle we should take account of the rate of inflation and distinguish a nominal or real wage cut In reality, people seem more interested in the nominal wage A cut in the nominal wage is a very obvious loss If people are loss averse it is no wonder they do not like a nominal wage cut A cut in the real wage is a less obvious loss and people do not seem so bothered by it This is presumably because a nominal wage rise is interpreted as a gain even if it is a real wage cut If workers dislike nominal wage cuts then it is no surprise firms avoid them This causes wage stickiness Interestingly wage cuts are not uncommon in recessions Again, however, the framing is important to take account of In a recession the firm can say we are struggling, so it is either a wage cut or redundancy let s work together to get through the recession In this scenario a wage cut can be seen as fair and so does not generate as negative a reaction But, cutting wages in good times is likely to be a disaster The story I have given so far, and present in the text, is one of firms disliking a wage cut because it seems unfair it is an unkind act But, it is worth remembering that a wage cut has a more obvious direct effect of making a person poorer This is where habit formation is important If a person has become habituated to a particular standard of living they will be reluctant to lower that standard of living A wage cut is a direct threat to standard of living This can, unfortunately, turn a bad situation into a disaster as people in recessions cling to their old standard of living in the hope of a new job or a return to the pre recession wage and it does not come along in time to save from bankruptcy This can be connected to the reflection effect A person will risk all to not have a sure loss in standard of living 711 Should employers fine poorly performing workers, or give bonuses to well performing workers in order to provide the most effective incentives? Fines generate negative reciprocity This is probably not something that will succeed in the workplace Bonuses generate positive reciprocity This is far more likely to create a positive vibe through the workplace Note, however, the problem of reference dependence If bonuses become routine then getting one no longer creates the good feeling that induces positive reciprocity it was expected Bonuses, therefore, are probably best used sparingly and carefully This is arguably what the financial industry has lost sight of in recent years

8 712 What can we learn from the ultimatum game about the interaction between a buyer and seller of a good? Most natural is to think of the seller as the proposer and the buyer as receiver The seller sets a take it or leave it price This price dictates the share that he gets as profit and the consumer gets as surplus The buyer may reject an offer because it is unfair It is useful here to recall the distinction between transaction and acquisition utility If the price is high the buyer plans to keep a high share as profits Even though the acquisition utility is positive the buyer may reject such a deal because the transaction utility is low the deal is unfair Firms, therefore, need to be careful in how high a price they set and how much they take as profits We can also think of the buyer as proposer and the seller as receiver This is less common but does exist For example, in real estate it is typically the buyer that offers a price Or consider sport where a team offers a price to buy a player This time the seller may reject a deal as unfair even though the acquisition utility is positive