Problem Set 2 Solutions

Problem Set 2 Solutions - Economics 3010 Fall 2010...

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Unformatted text preview: Economics 3010 Fall 2010 Professor Daniel Benjamin Cornell University Problem Set 2 Solutions 1. (Preferences: Money pumps and safety in markets) [The idea behind this question is taken directly from a recent theoretical research paper: Laibson, David I., and Leeat Yariv (2007), “Safety in markets: An impossibility theorem for Dutch books,” Caltech mimeo. If you are interested, feel free to read more details: ] (a) Naif is trying to decide what to buy for dessert. The possibilities are A (apple pie), B (brownies), or C (caramel ice cream). Naif has preferences over dessert and money. Some of her preferences between dessert and money are characterized by the following indifference relationships: A ~ $5 A ~ (B and $1) B ~ (C and $2) C ~ (A and $3) What does it mean if her preferences are complete? We first need to define the consumption set for this problem. The consumption set for a consumer is the set of all possible consumption bundles. Note that “possible” does not necessarily imply “affordable”. In this problem, we have two goods, dessert and money. There are three possibilities for dessert: {A,B,C}. For money, any positive amount is possible: R+ (this is notation for the set of strictly positive real numbers). So the consumption set in this problem is {A,B,C}x R+. We can now define completeness of preferences for allocations in this consumption set. Preferences are complete if for any two bundles X and Y from the consumption set, one of the following statements is true: (1) X is at least as good as Y, (2) Y is at least as good as X or (3) X is at least as good as Y and Y is at least as good as X. (b) Assume that Naif’s preferences are complete, and her preferences over money are monotonic. What can you conclude about her preferences between each possible pair of A, B, and C? What axiom do her preferences violate? Can you represent her preferences with a utility function? We are told that Naif’s preferences are complete and that her preference for money is monotonic. In part (a), we stated what it means for preferences to be complete. Now, what does it mean for preferences to be monotonic? If someone has a monotonic preference for some good, then she considers more of that good to be better; i.e. to make her better off. Now we are ready to answer the question. We will argue by contradiction. Assume that transitivity holds. We’re told that Naif is indifferent between A and {B and $1}. By monotonicity over money, A must then prefer A to B. We also know that she is indifferent between B and {C and $2}. By transitivity, she must then prefer A to {C and $2}. Applying again monotonicity over money, Naif must prefer A to C. Finally, Naif is indifferent between C and {A and $3}. By transitivity, Naif must prefer A to {A and $3}. But by monotonicity over money, we know that this last statement cannot be true. So we reached a contradiction and so Naif’s preferences do not satisfy transitivity. Transitivity is necessary for us to go from preferences to a utility function representation. So we can’t represent Naif’s preferences with a utility function. (c) A fellow named Arbitrageur realizes that he might be able to take advantage of Naif if he can get his hands on one each of A, B, and C. Suppose he can produce each at a cost of $4, so he goes ahead and pays $12 to produce an A, a B, and a C. Fill in the blanks to complete Arbitrageur’s strategy: (i) Sell A to Naif at a price of $4.99. (ii) Then offer to give __ to Naif if in exchange she gives him A and $2.99. (iii) Then offer to give B to Naif if in exchange she gives him __ and $1.99. (iv) Then offer to give A to Naif if in exchange she gives him B and $__. Would Naif agree to each of these steps? What is Arbitrageur’s profit (or loss) from steps (i)‐(iv), taking into account his costs of production? What if he kept repeating steps (ii) through (iv)? Arbitrageur (who we’ll now call Arbit for short) begins this process by producing an A, B, and a C for $12. In step 1, Arbit sells his A to Naif for $4.99. Naif accepts because she is indifferent between an A and $5. Before step 2 begins, Arbit has a B, a C, and $4.99 in revenue. He now offers his C to Naif for an A and $2.99. Naif will accept this offer because she is indifferent between a C and {A and $3}. We move on to step 3. Before it begins, Arbit has an A, a B, and $7.98 (4.99 + 2.99) in revenue. He now offers his B to Naif in exchange for a C and $1.99. Naif accepts this offer because she is indifferent between a B and {C and $2}. Heading into step 4 , Arbit has an A, a C, and $9.97 (4.99 + 2.99 + 1.99) in revenue. He now offers Naif an A in exchange for a B and $0.99. Naif accepts because she is indifferent between A and {B and $1}. So by the end of step 4, Arbit had a B, a C, and $10.96 in revenue. His total profit from steps 1 through 4 is ‐$1.04 (10.96 – 12). Note however that Arbit finishes these 4 steps with a B and a C. He could therefore go back to step 2 and keep on repeating steps 2 through 4 forever. His revenue will increase without bound, allowing profit to approach infinity. (d) Some economists have concluded that it is reasonable to assume people don’t have preferences like Naif’s because it is possible to take advantage of consumers like her (via a “money pump” like in part (c)). There are two versions of this argument. One version is that if somebody did have her preferences, that person would quickly go bankrupt and so wouldn’t have any effect on the economy. The second version is that if somebody did have her preferences, then we would see a lot of money pumps going on the real world; since we don’t see them, then nobody has those preferences. Choose one of these arguments, and explain it in a little more detail. Let’s begin by expanding on the first argument. If a person like Naif exists, then she would go bankrupt by the cycles mentioned in part (c). Without any money, Naif would not be able to make exchanges in the economy and, as a result, would have no influence on it. In other words, consumers like Naif would never participate in markets for long enough for them to have any noticeable effect on them. The second argument begins by saying that if people like Naif existed, then money pumps would be everywhere. Since we do not see money pumps in the real world, people like Naif don’t exist. This argument rests on the (reasonable?) supposition that whenever there are (infinite!) profits to be made, someone will surely take advantage of the opportunity. (e) So far, we have been assuming that Arbitrageur is a monopoly supplier of goods A, B, and C to Naif – that is, there is no one else who is competing with Arbitrageur to provide those goods. Now suppose instead there is a competitive supply of goods A, B, and C to Naif – i.e., there are an infinite number of suppliers competing with each other to sell their goods to Naif. Assume that each supplier can produce A, B, and C at a cost of $4 (the same as Arbitrageur). Explain why Arbitrageur can now charge Naif no more than $4 for A. Explain why Arbitrageur cannot charge Naif anything anymore to make trades between A, B, C. Suppose that Arbit decides to charge Naif more than $4 for an A. If he does this, then another producer can come in and undercut Arbit by a small amount and still make a profit. These undercuts continue to happen until price is pushed down to the marginal cost of $4. Using the same argument from above, we can conclude that Arbit will no longer be able to charge Naif anything to make trades between A, B, and C. If he offered Naif any trade of the form X in exchange for {Y and $m}, where X = A, B or C, Y = A, B or C and m>0, another arbitrageur would enter the market and undercut him. This is because it costs the same price ($4) to produce any of A, B or C (the logic is laid out more carefully in part (f) if this is not clear). (f) If Arbitrageur followed through with the sequence of trade from part (c) (but charging only as much as the market would allow), how much profit would he make? Explain why markets eliminate “money pumps” and thereby actually allow individuals like Naif to persist in having odd preferences. In a competitive market, Arbit can charge Naif only $4 for an A (Step 1). In Step 2, he can offer Naif a C for a maximum of A. To see that, assume Arbit offers Naif a C for {A and x} where x>0 chosen arbitrarily. Then since it costs as much to produce A and C, another arbitrageur would enter the market and undercut Arbit with the following offer to Naif: C in exchange for {A and y} where 0<y<x. As is evident, the only state of rest for this problem (or equilibrium) is for Arbit to offer Naif C in exchange for A. By the same logic, in Step 3 Arbit will offer Naif B in exchange for C. In Step 4, he will offer her A in exchange for B. So after going through steps 1 to 4 laid out in (c), we find that Arbit has revenue of $4 (from selling his A in step 1) and costs of $12 (the total cost of producing A, B, and C). His profit is therefore ‐$8. Repeating steps 2 through 4 doesn’t change anything because we have already argued that this can’t lead to more revenue. ...
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