Doing the Exercise 1 and 2.
20utility-11-5-19-H%20(1).pdf a complete setting. Missing was a description of preferences, of what people want. Without some such description, we can neither describe what they do if they have opportunities to engage in risk-sharing arrangements nor can we describe what is best for them, best in senses we will define. The rest of this chapter is devoted to such a description. The description we use at least until our discussion of anomalies is widely used, but is somewhat controversial. It is called expected-utility preferences (or von Neumann-Morgenstern expected utility). 1 Expected utility In almost everything we do, it is enough to have one underlying good say, rice. If you are uncomfortable with that label, you could call it something else: food or even income or wealth. Usually, in economics we think of there being many goods. However, for our focus on uncertainty, it is helpful to deal with the case of one underlying good, rice, and with probability distributions of consumption of that good. In all the symmetric and asymmetric information examples above, at the point where people take actions, they have to be thinking about the conse- quences of those actions for the distribution they will get. In any situation in which people are choosing the amount of insurance to buy (auto or home or something else) or are choosing whether or not to make a particular gamble (the purchase of a lottery ticket), they should be thinking about the distri- bution of wealth implied by each amount of insurance that they might buy or each gamble that they undertake. We will almost always be working with distributions which have a finite set of possible consumption levels, each level is represented by a nonnega- tive real number and we use S to denote the number of possible outcomes. A distribution is then represented by two lists, a list of possible outcomes and their associated probabilities. For now, we denote the set of possible consumption levels for a person by a list (C1, C2, ..., cs), where c, is consump- tion in outcome s, (At times, we will denote the list by c.) We denote the associated list of probabilities by (71 , 72, ..., Is), where n, is the probability that outcome s occurs. (At times, we denote that vector by 7). If the per- son has this distribution, then they get consumption c. with probability T.. In everything we do, we treat the probabilities as given and known. Such probabilities are often called objective as opposed to subjective.Outility-11-5-19-H%20(1).pdf speaking, affine) transformation. A consequence is that we can always arbi- trarily name two values of the u function provided they are consistent with u being strictly increasing. For example, if we replace consumption by wealth in terms of $'s, we can always say that u(1k) = 0 and u(10k) = 1, where k stands for thousand. We can use these values and think about how we would construct the function u for a person based on how they answer questions about various gambles. 1.1 Eliciting a person's u function Here is a question we could ask a person to elicit aspects of the person's utility function u. As mentioned, we fix u(1k) = 0 and u(10k) = 1. To elicit u(5k), we ask for what magnitude of w is the person indifferent between having 5k for sure and having 1/ with probability 7, and 10k with probability (1 -71). Suppose the person answers 71 = .3. If we interpret that answer in terms of expected utility, then we have the equation 4(56) - 3u(1k) 71(106)Outility-11-5-19-H%20(1).pdf of stage (ii), everyone sees the realized amount of rainfall. Here is another symmetric-information setting we study. Again, there are N rice farmers. But now there is a simple version of what is called idiosyncratic uncertainty. Each farmer will realize a crop whose size is either w1 or w2, where w2 > w1. For each farmer there is an independent coin tossing: it comes up tails with probability m1 6 [0, 1] and heads with probability 12 = 1 - m1. We use 7 = (71, 72) to denote the distribution. When it comes up heads, the farmer realizes w2; otherwise, the farmer realizes wj. At stage 2, everyone sees what each farmer realized and based on those realizations any deals made at stage (i) are carried out. Such symmetric-information settings can be fit into the standard model of what is called competitive general equilibrium-the complete-economy version of demand-supply models. The extension of that model to settings with uncertainty is called the Arrow-Debreu model, which was first formulated in the 1950's. In section I, we study how that is done. One of the applications of that model that we study is called arbitrage-pricing theory (APT). In part II, we consider three kinds of asymmetric information: moral haz- ard, adverse selection, and ex post asymmetric information about outcomes. The second farmer example above can be generalized to become a moral hazard model in the following way. Assume that rather than a common for all N farmers there are, say, two possible T's, 7" and 7 with 12 > 72 (that is, a 7 farmer has a higher probability of obtaining w2, the higher crop size, than a 7 farmer). Moreover, assume that each farmer chooses a costly effort level that determines whether they become 7 or w farmers. Finally, assume that each farmer knows their own effort level, but no one else does. That same example can be generalized to be an adverse-selection model in the following way. Rather than a common 7 for all N farmers, there are, again, two possible T's, 7" and 7 with 7. > 75. Now, at the start, each farmer realizes either all or w. Each farmer knows their own 7, but no one else does. Finally, that same example becomes a setting with ex post asymmetric information about outcomes if each farmer sees his crop realization, but others can see it only at some cost. We conclude with a brief discussion of what seem like anomalies behavior that is inconsistent with the models studied. Relative to those models, there seems to be too much stock trading and too much gambling. We describe ex- planations that have been offered. explanations that are generally consideredtility-11-5-19-H%20(1).pdf Now let (C1, C2, ..., Cs) and (71, 72, ..., Ts) = T be one distribution and let (C1, C2, .... Cs) and (71, 72, ..., T's) - 7', be another distribution. (The assump- tion that the list of outcomes is the same is without loss of generality because we permit some of the probabilities to be zero. It also allows us to describe a distribution solely by the vector of probabilities.) The expected utility assumption (or hypothesis) is that, for some utility function, u a strictly increasing function whose domain is the set of outcomes and whose range is the set of real numbers the person strictly prefers 7 to w' (written 7') if [ Till(G) > [ TU(G). (1) 1=1 Notice that the function u also describes the utility that the person gets from sure or certain consumption. (Also, the person is indifferent between 7 and 7' (written 7 ~ 7') if strict inequality in (1) is replaced by equal- ity.) The left side of (1) is called the expected utility of the distribution 7 and the right side is called the expected utility of the distribution '. (This kind of expected utility with objective probabilities is called von Neumann- Morgenstern expected utility because they introduced it in the book they wrote that initiated the field of game theory.) Notice that each side of in- equality (1) is a real number. Two different utility functions, say, u and v, can give the same preference ordering over distributions. In particular, if v(x) = au(x) + b for some numbers a, b with a > 0, then u and v imply the same preference ordering over distributions. Exercise 1. Let " and T' be two different consumption distributions for the person and let v(x) = au(x) + b with a > 0. Show that if >_, mu(c,) > Er, Tu(c), then E_, TV(C.) > >, To(C.). (Hint: Start with _ Tiv(G.) and produce a string of inequalities as follows. Replace v(c.) by au(c.) + b. Then, use the assumption in the "if" clause. Finally, replace au(c;) + b by An implication of this result and the analogous results for indifference is usually stated as saying that u is unique up to a positive linear (strictly speaking, affine) transformation. A consequence is that we can always arbi- trarily name two values of the u function provided they are consistent with utility-11-5-19-H%20(1).pdf Here is a question we could ask a person to elicit aspects of the person's utility function u. As mentioned, we fix u(1k) = 0 and u(10k) = 1. To elicit u(5k), we ask for what magnitude of 71 is the person indifferent between having 5k for sure and having 1/ with probability m and 10k with probability (1 -71). Suppose the person answers 71 - .3. If we interpret that answer in terms of expected utility, then we have the equation u(5k) = 3u(1k) + .Tu(10k) = .7 where the second equality comes from the normalization, u(1k) = 0 and u(10k) = 1. Therefore, we have u(5k) = .7. Having determined that u(5k) = .7, we could ask other questions. For example, we could ask, for what magnitude of 71 is the person indifferent between having 3k for sure and having 1k with probability m, and 5k with probability (1 - 71). Suppose the person answers, 71 = .2. If we interpret that answer in terms of expected utility, then we have the equation u(3k) = 2u(1k) + .8u(5k) = (.8)(.7) =.56, where the second equality comes from the normalization, u(1k) = 0, and the previous result that u(5k) = .7. Notice that we are eliciting the person's attitude toward risk. There are no right or wrong answers to these answers. Exercise 2. Put yourself in the position of the above thought experiment and pose and answer 4 questions that permit you to find your u function for 4 different magnitudes of r between 1k and 10k. Use some software to sketch those points of your u function. Exercise 3. Extend the previous exercise to some magnitudes of a > 10k and some magnitudes of r w1. For each farmer there is an independent coin tossing: it comes up tails with probability 71 E [0, 1] and heads with probability 72 - 1 - 71. We use 7 = (71, 72) to denote the distribution. When it comes up heads, the farmer realizes w2; otherwise, the farmer realizes w1. At stage 2, everyone sees what each farmer realized and based on those realizations any deals made at stage (i) are carried out. Such symmetric-information settings can be fit into the standard model of what is called competitive general equilibrium the complete-economy version of demand-supply models. The extension of that model to settings with uncertainty is called the Arrow-Debreu model, which was first formulated in the 1950's. In section I, we study how that is done. One of the applications of that model that we study is called arbitrage-pricing theory (APT). In part II, we consider three kinds of asymmetric information: moral haz- ard, adverse selection, and ex post asymmetric information about outcomes. The second farmer example above can be generalized to become a moral hazard model in the following way. Assume that rather than a common for all N farmers there are, say, two possible T's, 7 and 7 with my > 12 (that is, a w farmer has a higher probability of obtaining w2, the higher crop size, than a w farmer). Moreover, assume that each farmer chooses a costly effort level that determines whether they become 7 or n farmers. Finally, assume that each farmer knows their own effort level, but no one else does. That same example can be generalized to be an adverse-selection model in the following way. Rather than a common 7 for all N farmers, there are, again, two possible 7's, TH and a with my > m. Now, at the start, each farmer realizes either 7" or 74. Each farmer knows their own 7, but no one else does. Finally, that same example becomes a setting with ex post asymmetric information about outcomes if each farmer sees his crop realization, but others can see it only at some cost. We conclude with a brief discussion of what seem like anomalies behavior that is inconsistent with the models studied. Relative to those models, there seems to be too much stock trading and too much gambling. We describe ex- planations that have been offered, explanations that are generally considered part of what has come to be called behavioral economics. Although we said that our example of rice farmers was a setting, it was not
