12varianbergstrom - Chapter 12 N AME Uncertainty...

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Chapter 12 NAME Uncertainty Introduction. In Chapter 11, you learned some tricks that allow you to use techniques you already know for studying intertemporal choice. Here you will learn some similar tricks, so that you can use the same methods to study risk taking, insurance, and gambling. One of these new tricks is similar to the trick of treating commodi- ties at different dates as different commodities. This time, we invent new commodities, which we call contingent commodities . If either of two events A or B could happen, then we define one contingent commodity as consumption if A happens and another contingent commodity as con- sumption if B happens. The second trick is to find a budget constraint that correctly specifies the set of contingent commodity bundles that a consumer can afford. This chapter presents one other new idea, and that is the notion of von Neumann-Morgenstern utility. A consumer’s willingness to take various gambles and his willingness to buy insurance will be determined by how he feels about various combinations of contingent commodities. Often it is reasonable to assume that these preferences can be expressed by a utility function that takes the special form known as von Neumann- Morgenstern utility . The assumption that utility takes this form is called the expected utility hypothesis . If there are two events, 1 and 2 with probabilities π 1 and π 2 , and if the contingent consumptions are c 1 and c 2 , then the von Neumann-Morgenstern utility function has the special functional form, U ( c 1 , c 2 ) = π 1 u ( c 1 ) + π 2 u ( c 2 ). The consumer’s behavior is determined by maximizing this utility function subject to his budget constraint. Example: You are thinking of betting on whether the Cincinnati Reds will make it to the World Series this year. A local gambler will bet with you at odds of 10 to 1 against the Reds. You think the probability that the Reds will make it to the World Series is π = . 2. If you don’t bet, you are certain to have $1,000 to spend on consumption goods. Your behavior satisfies the expected utility hypothesis and your von Neumann- Morgenstern utility function is π 1 c 1 + π 2 c 2 . The contingent commodities are dollars if the Reds make the World Series and dollars if the Reds don’t make the World Series . Let c W be your consumption contingent on the Reds making the World Series and c NW be your consumption contingent on their not making the Series. Betting on the Reds at odds of 10 to 1 means that if you bet $ x on the Reds, then if the Reds make it to the Series, you make a net gain of $10 x , but if they don’t, you have a net loss of $ x . Since you had $1,000 before betting, if you bet $ x on the Reds and they made it to the Series, you would have c W = 1 , 000 + 10 x to spend on consumption. If you bet $ x on the Reds and they didn’t make it to the Series, you would lose $ x ,
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162 UNCERTAINTY (Ch. 12) and you would have c NW = 1 , 000 x . By increasing the amount $ x that you bet, you can make c W larger and c NW smaller. (You could also bet against the Reds at the same odds. If you bet $ x
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