Economics 2010a Fall 2002 Edward L. Glaeser Lecture 6 6. Choice Under Uncertainty a. Representing Uncertainty: Lotteries and Compound Lotteries b. Axioms of Expected Utility c. The Expected Utility Theory d. Empirical Challenges to Expected Utility Theory—the Paradox Business e. Application: Crime and punishment In lecture 4, we discussed utility over time, and thought about a utility function of the form u?c 1 ,c 2 ,..c T ? that was defined over future periods of consumption. We were particularly interested in the time-separable form of this utility function u?c 1 ,c 2 ,..c T ? ? ? t?1 T ? t u?c t ? and we even assumed that ? t ? ? t Uncertainty has many similarities. Think of consumption is different "states of nature" rather than time periods. You can imagine a two states world (it rains, it shines), or an 11 states world (the sum of two dices come up anything from 2-12) or an almost unimaginably large state space in space which actually live. Then the utility function can be thought of as being defined over states of the world: u?c 1 ,c 2 ,..c S ? where c s is consumption in state s. In fact– this is the most important point– if you define all forms of consumption as "state contingent consumption" – good x in state s, then the MWG discussion in chapters 1-3 can handle uncertainty with no changes whatsoever. The state-space analogue to time separability is state separability of u?c 1 ,c 2 ,..c S ? ? ? s?1 S u i ?c s ? In some cases, we prefer the even more extreme assumption that: u?c 1 ,c 2 ,..c S ? ? ? s?1 S p s u?c s ? where p s is a constant. We can assume that ? s?1 S p s ? 1, so that these constant terms can be interpreted as "subjective probabilities" (i.e. weights that add to one). In the case that the value of p s equals the objective probability of state s occurring (for each state s) , we call this an expected utility model. We will derive this in a second more formally– but if we believe that assets span– i.e. there are enough assets so that you can actually by and sell goods in each state of the world, then consumers maximize: ? s?1 S p s u?c s ? ?? w ? ? s?1 S k s c s where k s indicates the cost of consumption in state s, which leads to first order conditions: ? s u ? ?c s ? ? ?p s or u ? ?c z ? u ? ?c s ? ? k z p s k s p z The ratio of the marginal utility of consumption in the different states equals the ratio of the prices divided by the ratio of the probabilities. Note that for these first order conditions to make sense u?.? must be concave. In particular, if the ratio of the probabilities equals the ratio of the prices– this would be true if all bets were fair– consumption levels are equal across states. But more generally– economics tells you to equalize marginal utilities of consumption not total utilities. Back to MWG and the more formal treatment MWG Definition 6.B.1: Asimplelottery L is a list L ? ?p 1 ,...p N ? with p n ? 0 for all n and ? n?1 N p n ? 1 where p n is interpreted as the probability of an outcome n occurring. A simple lottery is a point in the N ? 1 dimensional simplex, i.e. the set ?? p ? ? ? N : ? n?1 N p n ? 1 MWG Definition 6.B.2: Given K simple lotteries L k ? ?p 1 k ,...p N k ?, k ? 1,2,...,K, and probabilities ? k ? 0 with ? k ? k ? 1,thecompound lottery ?L 1 ,...L K ;? 1 ,...? K ? is the risky alternative that yields the simple lottery L k with probability ? k for k ? 1,...K. For any compound lottery, ?L 1 ,...L K ;? 1 ,...? K ? , the corresponding reduced lottery is the simple lottery: L k ? ? k ? k p 1 k , ? k ? k p 2 k ,.. ? k ? k p N k The decision-maker is assumed to have preferences, denoted by ?, over lotteries. For this to make sense– assume that each state n, which occurs with probability p n k in lottery k has a fixed payoff x n . The set of alternatives (i.e. the equivalent of X in chapter 1) is denoted ? is the set of all simple lotteries over outcomes or consequences C. We will assume that the decision-maker has rational preferences (i.e. transitive and complete) over ?. Assuming continuity (defined in MWG 6.B.3) means that just as before there exists a utility function that will rank lotteries. Axioms of Expected Utility MWG Definition 6.B.4: The preference relation ? on the space of simple lotteries ? satisfies the independence axiom: if for all L,L ? ,L ?? ? ? and ? ? ?0,1? we have L ? L ? if and only if ?L? ?1 ? ??L ?? ? ?L ? ? ?1 ? ??L ?? . The ranking is immune to adding on extra lotteries. Sometimes I think of this as saying that your utility from getting a higher probability of state j is independent of your probability of state k. Definition 6.B.5: The utility function U : ? ? ? has an expected utility form if there is an assignment of numbers ?u 1 ,...u N ? to the N outcomes such that for every simple lottery L ? ?p 1 ,.p N ? ? ? we have U?L? ? ? j?1 N u j p j A utility function U : ? ? ? with the expected utility form is called a von Neumann-Morganstern expected utility function. Proposition 6.B.1: A utility function U : ? ? ? has an expected utility form if and only if it is linear, i.e. if and only if it satisfies the property that U ? k?1 K ? k L k ? ? k?1 K ? k U?L k ? for any K lotteries, k ? 1,...,K and weights ? k ? 0 with ? k ? k ? 1 Proof: First, if the utility function has the expected probability form then ? k?1 K ? k U?L k ? ? ? k?1 K ? k? n?1 N u n p n k ? ? n?1 N ? k?1 K ? k p n k u n ? U ? k?1 K ? k L k so linearity holds. More surprisingly, if U ? k?1 K ? k L k ? ? k?1 K ? k U?L k ? then we can write L k ? ?p 1 k ,...p N k ? ? ? n?1 N p n L n where L n places a probability of 1 on state n and zero on all other states. We have then U?L k ? ? U ? n?1 N p n L n ? ? n?1 N p n U?L n ? ? ? n?1 N p n u n which has the expected utility form. MWG Proposition 6.B.3: Suppose that the rational preference relation ? on the space of lotteries ? satisfies the continuity and independence axioms. Then ? admits a utility representation of the expected utility form. That is, we can assign a number u n to each outcome n ? 1,...,N in such a manner that for any two lotteries L ? ?p 1 ,.p N ? and L ? ? ?p 1 ? ,.p N ? ? we have L ? L ? if and only if ? j?1 N u j p j ? ? j?1 N u j p j ? The MWG Proof proceeds in four steps: Step 1: We assume that there are best and worst lotteries in ?, denoted L and L,so that for all L ? ? , we know that L ? L ? L. Furthermore if L ? L all lotteries are equivalent, so U?L? can just be represented by a constant and trivially, the proposition holds, so we consider only the case where L ? L. Step 2: If L ? L ? and ? ? ?0,1? then L ? ?L? ?1 ? ??L ? ? L ? This follows from the independence axiom. Directly using the axiom tells us that: L ? ?L? ?1 ? ??L ? ?L? ?1 ? ??L ? ? ?L ? ? ?1 ? ??L ? ? L ? Now to make the preferences strict. Assume that: ?L? ?1 ? ??L ? ? ?L? ?1 ? ??L ? L, but then ?L? ?1 ? ??L ? ? ?L? ?1 ? ??L Now, because the independence axiom is an if and only if condition, it follows that L ? ? L, we have a contradiction. Likewise, if ?L? ?1 ? ??L ? ? ?L ? ? ?1 ? ??L ? ? L ? then L ? ? L and again a contradiction results. Step 3: Let ?,? ? ?0,1?. Then ?L ??1 ? ??L ? ?L ??1 ? ??L if and only if ? ? ? Part a: If ? ? ? (which also means that ? ? 1) show that ?L? ?1 ? ??L ? ?L? ?1 ? ??L Let ? ? ??? 1?? ? 1 and note that ?L ??1 ? ??L ? ?L ??1 ? ????L? ?1 ? ??L? By step 2 we know that L ? ?L? ?1 ? ??L so if we use step 2 again we know that ?L? ?1 ? ????L? ?1 ? ??L? ? ?L? ?1 ? ??L, which means that ?L? ?1 ? ??L ? ?L? ?1 ? ??L Part b: If ?L ??1 ? ??L ? ?L? ?1 ? ??L show that ? ? ? Assume the contrary, i.e. ? ? ? , in that case (using the conclusion of part a immediately above) we conclude that ?L? ?1 ? ??L ? ?L? ?1 ? ??L and this is a contradiction. Assume that ? ? ? , in that case ?L? ?1 ? ??L ? ?L? ?1 ? ??L and that also contradicts ?L? ?1 ? ??L ? ?L? ?1 ? ??L Step 4: For any L ? ? there is a unique ? L such that L ? ? L L? ?1 ? ? L ?L From step 3, we know the sets ?? ? ?0,1? : ?L? ?1 ? ??L ? L? and?? ? ?0,1? : ?L? ?1 ? ??L ? L? cannot share any elements, so there must exist some value ? ? L for which if ? ? ? ? L , then ?L? ?1 ? ??L ? L and if ? ? ? ? L then ?L? ?1 ? ??L ? L. Consider the sequence of ? ? s : ? ? L ? 1? ? ? L n that sequence converges to ? ? L and there must be in the set ?? ? ?0,1? : ?L? ?1 ? ??L ? L? since that set is closed. Likewise the sequence ? ? L ? 1? ? ? L n converges to ? ? L and must be in the set ? ? ?0,1? : ?L? ?1 ? ??L ? L . Since ? ? L is in both ? ? ?0,1? : ?L? ?1 ? ??L ? L and ? ? ?0,1? : ?L? ?1 ? ??L ? L , it follows that L ? ? L L? ?1 ? ? L ?L To prove uniqueness– assume the contrary, i.e. there exists both an ? L and ? L ? and note that if they are not equal one must be greater than the other, and using step three this implies that one must give strictly greater utility, a contradiction. Step 5: The utility function U : ? ? ? that assigns U?L? ? ? L represents the preference relation ? Consider any two lotteries L and L ? which are assigned ? L and ? L ?.IfL ? L ? , it follows that ? L L? ?1 ? ? L ?L ? ? L ?L? ?1 ? ? L ??L which from step 3 implies that ? L ? ? L ? If ? L ? ? L ? it follows (again from step 3) that ? L L? ?1 ? ? L ?L ? ? L ?L? ?1 ? ? L ??L,or L ? L ? . Step 6: The utility function U?L? ? ? L is linear, i.e., U??L? ?1 ? ??L ? ? ? ?U?L? ? ?1 ? ??U?L ? ? for all L,L ? and ? ? ?0,1?. We know that L ? ? U?L ? ?L? ?1 ? U?L ? ??L and L ? U?L?L? ?1 ? U?L??L Using the independence axiom twice we find: ?L? ?1 ? ??L ? ? ??U?L?L? ?1 ? U?L??L? ? ?1 ? ??U?L ? ?L? ?1 ? U?L ? ??L ? L??U?L? ? ?1 ? ??U?L ? ?? ?L?1 ? ?U?L? ? ?1 ? ??U?L ? ?? But this means that U??L? ?1 ? ??L ? ? ? ?U?L? ? ?1 ? ??U?L ? ? and we’ve got linearity which from Proposition 6.B.1 gives us the expected utility form. Challenges to Expected Utility Theory– The Paradox Game The Allais Paradox What would you prefer: $500,000 for sure or a 10% chance of $2.5 million, an 89% change of $500,000 and a 1% chance of zero. What would you prefer: an 11% chance of $500,000 and an 89% chance of zero or a 10% chance of $2.5 million and a 90% chance of zero. Most people answers to this violate the independence axiom (and expected utility theory). What do we make of this? Machina’s Paradox: Three outcomes: (1) stay home, (2) watch a movie about Venice, (3) go to Venice. Assume that you prefer the lottery that puts probability 1 on outcome (3), to the lottery that puts probability 1 on outcome (2) to the lottery that puts probability 1 on outcome (1). In some cases, people would prefer a lottery which puts a 99 % chance on (3) and a 1% chance on (1) to a lottery that puts a 99% chance on (3) and a 1% chance on (2). The idea here is that regret seems to be important in some cases. The Ellsberg Paradox – Risk vs. Uncertainy You get 1,000 dollars if a red ball gets picked. Option A: choose urn number 1, which has know to have 50 red balls and 50 black balls. Option B: choose urn number 2, which has only black and red balls, but in unknown proportions. People generally choose option A. But let’s say that you get paid 1,000 if a black ball was chosen. Which urn would you choose? This is a bit of a paradox and relates to Knightian ideas of risk and uncertainty. Crime and Punishment– An Application (Becker, 1968) The reason that we have probabilities involved in that we always assume that criminals get caught with some probability (denote it ?), and if they get caught, they pay a fine equal to F. The benefit of a crime is B (if the criminal is not caught) Two ways to proceed. First, we can assume that each person commits only one crime (if any). As such, the decision-maker commits a crime whenever ?1 ? ??U?W?B? ??U?W ? F? ? U?W? (Note I am assuming that you don’t get to benefit from the crime if you get caught). Imagine that there is a distribution of B throughout the population characterized by a cumulative distribution function G?B?, and other than B everyone is identical, and assume that there exists some B for which ?1 ? ??U?W?B? ??U?W ? F? ? U?W? holds. Furthermore, assume that g?B? ? 0 (i.e. the density is strictly positive) everywhere. Claim: there exists a value of B, denoted B ? , at which all individuals are indifferent between committing crimes and not committing crimes. All individuals with B ? B ? will commit crimes and all individuals with B ? B ? will not commit crimes. The fact that ?1 ? ??U?W?B? ??U?W ? F? ? U?W? starts out negative, ends positive and is monotonic and continuous in B guarantees this claim. Thus, B ? is defined so that ?1 ? ??U?W?B ? ? ??U?W ? F? ? U?W? The share of the population that is criminal equals 1 ? G?B ? ?. Claim: The value of B ? is increasing with F, and ?, so the number of criminals is falling with F, and ?. Totally differentiate: ?1 ? ??U?W?B ? ? ??U?W ? F? ? U?W? To get: ?B ? ?? ? U?W?B ? ? ? U?W ? F? ?1 ? ??U ? ?W?B ? ? ? 0 ?B ? ?F ? ?U ? ?W ? F? ?1 ? ??U ? ?W?B ? ? ? 0 Also ?B ? ?W ? U ? ?W? ? ?1 ? ??U ? ?W?B ? ? ? ?U ? ?W ? F? ?1 ? ??U ? ?W?B ? ? If F ? fW – i.e. punishment is lost time in the labor force, or a tax that is proportional to labor, then we have: ?1 ? ??U?W?B ? ? ??U?W?1 ? f?? ? U?W? and ?B ? ?W ? U ? ?W???1???U ? ?W?B ? ????1?f?U ? ?W?1?f?? ?1???U ? ?W?B ? ? This is always positive if the coefficient of relative risk aversion satisfies ?X U ?? ?X? U ? ?X? ? 1 because aU ? ?ax? is falling with a, and that implies that U ? ?W? ? ?1 ? f?U ? ?W?1 ? f?? . Becker Claim # 1: If catching criminals is expensive but punishing them is free (say through crimes), then you should drive the punishment probability to zero and increase the punishment to infinity. This is particularly obvious in the linear case, where B ? solves: ?1 ? ???W?B ? ? ???W ? F? ? W or B ? ? ?F 1?? Becker Claim # 2: There is no reason to expect that criminals will stop being criminals after they go to jail, if it was optimal to rob before they go to jail is will probability be optimal afterwards as well. Solving for the optimal punishment. Assume that there is a social cost C per crime. Assume that it is costly to try to catch people, and there is just a function ???? that captures this cost. Assume that the social cost per unit of punishment is ? times the number of people who are punished times the size of punishment F. Then the social welfare problem (if we exclude the welfare of the criminals) is to find ? and F that minimize: ???? ? ?1 ? G?B ? ???C???F?. This yields a first order condition for ?: ? ? ??? ? ?1 ? G?B ? ???F ? g?B ? ? ?B ? ?? ?C???F? ? and a first order condition for F ?1 ? G?B ? ???? ? g?B ? ? ?B ? ?F ?C???F? ? 0 This can be rewritten: ??F ? Fg?B ? ? 1 ? G?B ? ? ?B ? ?F ?C???F? and using the notation ?? Fg?B ? ? 1?G?B ? ? ?B ? ?F We get Log?F? ? Log?C? ?Log ? 1 ?? ? Log??? ? Log??? Thus the optimal fine is a function of: (1) social damage of the crime, (2) the elasticity of the crime with respect to punishment, (3) the probability of arrest and (4) the costliness of punishing the criminal. To solve for the optimal probability of arrest, we combine the first order conditions to get ?? ? ??? ? ? ?B ? ?? ? F ?B ? ?F g?B ? ??C???F? or using the formulas ?B ? ?? ? U?W?B ? ??U?W?F? ?1???U ? ?W?B ? ? and ?B ? ?F ? ?U ? ?W?F? ?1???U ? ?W?B ? ? we get ?? ? ??? ? U?W?B ? ??U?W?F??FU ? ?W?F? ?1???U ? ?W?B ? ? ?g?B ? ??C? Note that if B ? is too close to zero, then the term in brackets may not be positive– this would mean that the marginal benefit of increasing the probability of catching a criminal is negative– how could that happen? One condition that we sometimes us to guarantee an interior solution in these cases is the Inada condition: lim x?0 f ? ?x? ? ?, which guarantees a positive probability of arrest.