Answers: Assignment 6

Econ. 103, Fall 2002, Prof. Nancy Folbre

a) No player has a dominant strategy. A would prefer to buy a baseball ticket if B does so, but a movie ticket if B does so. Since the payoffs are symmetric, the same holds for B

b) There are two equilibria: BB and MM (movie-movie, and baseball-baseball). They are equilibria because they are mutual best responses, that is at BB, no player wants to unilaterally change his strategy and buy a movie ticket, and at MM, no player wants to unilaterally change his strategy and buy a baseball ticket. Note that this does not tell us anything about what will actually happen when they take their decisions simultaneously. It is very possible that one will buy a movie ticket and the other a baseball ticket. In other words, equilibria tell you that you stay there when you get there, but they don't tell you anything about what happens when you are not there (you need more information than a simple statement that a point is an equilibrium for this).

c) This game is not a prisoner's dilemma. For a game to be a prisoner's dilemma, it has to be the case that the only stable equilibrium, reached by a combination of dominant strategies, is inferior (for all players) to another outcome, which is not an equilibrium. To see this, recall that in the case depicted in the book, both prisoners would have been better off if both denied than having both confess (that is the aforementioned "other outcome"). However, common denial was not a possible outcome because confessing was a dominant strategy for both players, and the combination of those dominant strategies had them end up at the inferior confess-confess outcome.

d) The information given doesn't help, because B cannot be sure of A's choice.

e) same as above

10.3

1. If they all make their decisions individually, three llamas will be sent to the field. To see this consider the following: The first person to choose basically faces the option of buying a llama and making 22$ or buy the bond and make 15$, which will induce him to get the lama. Similarly, the second person has the choice of getting 18$ from a llama or 15$ from the bond, and the third between making 16$ from the llama or 15$ from the bond, so both of them will prefer to buy the llamas. However, the fourth person would only make 14$ if she sent a llama to the common, so she won't, and so on, so that when everybody has made a decision, three llamas are sent to the commons. The village income, in that case, is 3X16 + 3X15 = 93.
   
2. The socially optimal number of llamas is 1, because adding a second llama only brings 14$ to the community (2X18 – 1X22 = 14$), while investing that money would bring 15$. This is different from the actual number because now the negative effect that adding more llamas has on the owner of the llamas already present is taken into account. That is, the second individual only sees that he can make 18$ by buying a llama, not considering that he is reducing the amount gotten by the first husbander by 4$ (and 18-4=14 is the actual gain of the community from adding a second llama). Note that this is the externality in that setting: adding llamas imposes costs on the owners of existing llamas and these costs are not taken into consideration by the people who add the llamas (in other words, those costs are external to the investors' calculations).

The income of the community, if the socially optimal number of llamas is sent to the field is 1X22 + 5X15 = 97.

For this question, suppose the right is bought for one year, and then re-auctioned at the end of the year (other assumptions could be taken, but the answer would only differ quantitatively, not qualitatively). If a villager has a monopoly on grazing, this implies that she internalizes the externality mentioned above, in that she now considers the costs, as well as the benefits, of adding more llamas. Consequently, any villager buying the rights would only put one llama on the field, because to put one more she would have to borrow money (100$) at 15%, and would earn only 14$, as shown above. The income from putting one llama on the field is 22, and the profit, net of the next-best alternative, is 22-15=7. Now, since any single villager only has 100$, which is what the llama is going to cost, they have to borrow to buy the rights. The total amount any villager would be willing to repay at the end of the year is 7, because having to pay more would mean that they would be better off not buying the rights and investing their money instead.

c) Assume that the rights are sold forever and that the buyer will never have to repay the principal. In this case, the buyer will still put only one llama on the field; the amount of interest he is willing to pay each year is 22-15=7. At 15%, this means that any buyer would be ready to take on a loan of 46.67$ (46.67*0.15=7). For any loan higher than that, interest payments exceed 7$ and villagers are better off buying bonds, while a lower loan (which would mean interest payments lower than 7$) should not be possible as another villager would be ready to offer as much as 46.67 for the grazing rights. The rest of the problem is similar, given the same assumptions (such as the money is borrowed from outside, etc.) (try it, using the methodology above), and the income of the village each year is still 97$ (15*5+(46.67*0.15)+(22-7)=97).

 

A more realistic way of approaching the problem (that diverges from the text) would be the following: This 7$ is composed of the principal they borrowed plus the interest, so if they borrowed A, Ax(1.15)=7, or (7/(1.15))=A. Therefore, A = 6.08$, and this is they amount the rights would sell at, if the auction takes place in a perfectly competitive setting.

To evaluate the village income under that scheme, suppose that the auction proceeds are invested, and that the money was borrowed from outside the village, so that the right's buyer's repayment does not enter the village's income. The total income is:

• 15X5=75 from the people who did not buy the rights;
• 6.08X(1.15)=7 from the auction proceeds;
• 22-7=15 from the person who bought the rights and raised the llama.

Consequently, the income is the same as the social optimum: 97$.