Economics 2010a Fall 2003 Edward L. Glaeser Lecture 7 7. More on Uncertainty a. Prospect Theory, Loss Aversion b. Subjective Utility and Common Knowledge c. Risk Aversion d. First and Second Order Stochastic Dominance e. Asset Demand and Risk Aversion f. State Dependent Preferences g. Application: The Draft A great deal of attention has been recently paid to ideas like prospect theory and loss aversion (Kahneman and Tversky). Point # 1 of these studies– people are very sensitive to framing and current position. Rabin – people are risk averse over small bets– this is actually incompatible with standard measured levels of risk aversion. Point # 2: people are concave in the win domain (classically riskaverse), but convex in the loss domain. Subjective Probabilities and Common Knowledge Agreeing to disagree–the lemon juice out of your ear problem. Assume for a second that people were risk neutral–utility is always just cash. Some people say that you can get betting because of differences of opinion about probabilities. In other words, consider the situation where someone comes up to you and bets you 10 dollars that lemon juice is going to squirt out of your ear so you win if lemon juice doesn’t squirt out of your ear. You initially place probability p?.5, on this occuring. Certainly one might think that your expected gain from this bet is (1-p)10-p10 or (1-2p)10?0. Your opponent would offer the bet only if (2p’-1)10?0, so if p’?.5?p, it seems like you should have a horse race. But this can’t actually happen. To see this, let’s put some structure on learning. Remeber Bayes’ Rule P?B|A? ? P?B ? A?/P?A? This is always the key to understanding belief formations. So more information structure. The unconditional probability of a lemon squirting is p. If the lemon is squirting, then every person receives a positive signal with probability z. If the lemon is not squirting then every person recieves a positive signal with probability 1-z. The probability of lemon squirting conditional upon recieving a positive signal is zp/(zp?(1-z)(1-p)). There are actually four possible states of the world: Signal, Lemon Signal, No Lemon No Signal, Lemon No Signal, No Lemon (1) signal, no lemon, (2) signal, lemon, (3) no signal, no lemon, (4) no signal, lemon. The probabilties of each of these states are (1) (1-z)(1-p), (2) zp, (3) z(1-p), (4) (1-z)p If you haven’t received a signal– your probability assessment is (1-z)p/(z?p-2pz). If you have recieved a signal, your probability assessment is zp/(1-z-p?2pz). If there are two people, the situation is even harder– there are actually eight states of the world based on the signal that each person has recieved. In this gambling example we need to deal with two people: So we have the following states of the world: pz 2 ?1 ? p??1 ? z? 2 pz?1 ? z??1 ? p?z?1 ? z? pz?1 ? z??1 ? p?z?1 ? z? p?1 ? z? 2 ?1 ? p?z 2 If the other person has bet– what should you assume? So conditional upon you not getting a signal and the other person receiving a signal, the probability of lemon squirting is pz?1?z? z?1?z? or p. Both of you should make the same assessment and there is no room for betting. In generaly, you should never take a bet against someone whose utility function resembles your own. Risk Aversion– (New Topic) MWG Definition 6.C.1: A decision maker is a risk averter (or displays risk aversion) if for any lottery F(.) the degenerate lottery that yields the amount ?xdF?x? with certainty is at least as good as the lottery F(.) itself. If the decision-maker is always (for any F(.)) indifferent between these two lotteries, we say that he is risk neutral. Finally, we say that he is strictly risk averse if indifference holds only when the two lotteriest are the same (i.e. when F(.) is degenerate). It follows directly from the definition that the decision-maker is risk averse if and only if: ?u?x?dF?x? ? u ?xdF?x? This is Jensen’s inequality and it always hold is u(.) is concave. Definition 6.C.2: Given a utility function u(.): (i) the certainty equivalent of F(.) denoted c(F,u) is the amount of money at which the individual is indifferent bewteen the gamble F(.) and the certain amount c(F, u), i.e. u?c?F,u?? ? ?u?x?dF?x? (ii) For any fixed amoutn of money x and positive number ? the probability premium denoted by ??x,?,u? is the excess in winning probability over the odds that makes the individual indifferent between the certain outcome x and a gamble between the outcomes x ??and x ? ?. Proposition 6.C.1: Suppose a decision-maker is an expected utility maximizer with a utility function u(.) on amounts of money– then the following properties are equivalent: (1) the decision maker is risk averse (2) u(.) is concave (3) c(F,u)? ?xdF?x? for all F(.) (4) ??x,?,u? ? 0 for all x, ? MWG Definition 6.C.3: Given a twice differentiable utility function u(.), the arrow pratt coefficient of absolute risk aversion at x is defined as r A ?x,u? ? ? u ?? ?x? u ? ?x? The coefficient of relative risk aversion is ?x u ?? ?x? u ? ?x? Comparisons across individuals MWG Proposition 6.C.2: Given two utility function u 1 ?.? and u 2 ?.?, the following statements are equivalent: (1) r A ?x,u 2 ? ? r A ?x,u 1 ? for every x (2) there exists an increasing concave function ??.? such that u 2 ?.? ? ??u 1 ?.??, i.e. u 2 ?.? is a concave transformation of u 1 ?.? (3) c?F,u 2 ? ? c?F,u 1 ? for any F(.) (4) ??x,?,u 2 ? ? ??x,?,u 1 ? for any x and ? (5) whenever u 2 ?.? finds a lottery F(.) at least as good as a riskless outcome, x, then u 1 ?.? also finds that lottery at least as good as the riskless outcome. Stochastic Dominance MWG Definition 6.D.1: The distribution F(.) first order stochastically dominates G(.) if for every nondecreasing function u : ? ? ? we have ?u?x?dF?x? ? ?u?x?dG?x? Proposition 6.D.1: The distribution of monetary payoffs F(.) stochastically dominates the distribution G(.) if and only if F?x? ? G?x? for every X. Assume that F(.) stochastically dominates G(.) but F?x? ? G?x? , for some value of x denoted ? x ,i.e.F? ? x? ? G? ? x? Define the nondecreasing function u(x), where u(x)?1 for all x ? ? x, and u(x)?0 otherwise. We know ?u?x?dF?x? ? 1 ? F? ? x? and ?u?x?dG?x? ? 1 ? G? ? x? But if F? ? x? ? G? ? x? then ?u?x?dG?x? ? ?u?x?dF?x?, and that’s a contradiction. Now assume F? ? x? ? G? ? x? for all x, and prove stochastic dominance follows. ?u?x?dF?x? ? ?u?x?dG?x? ? ?u?x??dF?x? ? dG?x?? ? ?u?x?dG?x? ? ?u?x??dF?x? ? dG?x?? Let H(x)?F(x)-G(x), so we need to know if ?u?x?dH?x? ? 0 for all functions u(x). Integrate by parts to get ?u?x?dH?x? ? ?u?x?H?x?? 0 ? ??u ? ?x?H?x?dx H(0)?0 and lim x?? H?x? ? 0 so ?u?x?H?x?? 0 ? equals zero.The second term is negative if H(x)? 0 everywhere, so we’re done. Second Order Stochastic Dominance MWG Definition 6.D.2: For any two distributions F(.) and G(.) with the same mean, F(.) second order stochastically dominates (or is less risky than) G(.) if for every nondecreasing concave function u : ? ? ? ? we have ?u?x?dF?x? ? ?u?x?dG?x? Other definition: the variable y is a mean preserving spread of x, if y?x?z where ?zdH?z? ? 0 Proposition 6.D.2 Consider two distributions F(.) and G(.) with the same mean. Then the following statements are equivalent: (1) F(.) second-order stochastically dominates G(.). (2) G(.) is a mean preserving spread of F(.). If G(.) is a mean preserving spread of F(.), then ?u?x?dG?x? ? ? x ?? z u?x ? z?dH x ?z??dF?x? ? ? x u?? z ?x ? z?dH x ?z??dF?x? ? ?u?x?dF?x? But for all values of x ? z u?x ? z?dH x ?z? ? u?? z ?x ? z?dH x ?z?? by Jensen’s inequality. Asset Demand and Risk Aversion: We often talk about their being a risk-return frontier. One useful utility function for showing this is quadratic utility u?x? ? x ?.5?x 2 We use that fact that E?x ? E?x?? 2 ? E?x 2 ? ? ?E?x?? 2 to get that E?u?x?? ? E?x? ?.5??E?x ? E?x?? 2 ? ?E?x?? 2 ? ? E?x? ?.5??E?x?? 2 ?.5?Var?x? Of course, you’ve got this problem that x needs to be a lot less than ? to avoid utility falling with income. If you start with income y, and buy z units of an asset with price one, mean return 1?r and variance ? 2 , your income have a mean value of y?rz and a variance of z 2 ? 2 So E?u?x?? ? y ? rz ?.5??y ? rz? 2 ?.5?z 2 ? 2 In this case, the optimal choice of the risky asset gives us: r ? ??y ? rz?r ? ?z? 2 ? 0 or z ? r?1??y? ??? 2 ?r 2 ? What happens if there are multiple independent assets? Another way to think about it is that there is fixed amount of risky asset to be allocated– this implies an r, which is increasing in risk and decreasing in wealth. The pure mean-variance utility function is exponential, i.e. u?x? ? ?e ??x and if x is normally distributed, i.e. f?x? ? 1 ? 2 2? e ??x??? 2 2? 2 then E?u?x?? ? 1 ? 2 2? ??e ??x e ??x??? 2 2? 2 dx ? 1 ? 2 2? ??e ?2? 2 ?x??x??? 2 2? 2 dx ? 1 ? 2 2? ??e ? 2 ? 2 ?2?? 2 ?? x???? 2 ? 2 2? 2 dx ? ? e ???? ? 2 ? 2 2 1 ? 2 2? ?e ? x????? 2 ?? 2 2? 2 dx But 1 ? 2 2? ?e ? x????? 2 ?? 2 2? 2 dx ? 1,so E(U(x))??e ???? ? 2 ? 2 2 And maximizing this is just equivalent to maximizing ? ? ? 2 ? 2 State Contingent Preferences Here states differ not just by the amount of money you recieve in each state– but also by other circumstances, which can determine your utility from the payoff. Think of insuring against the loss of a loved one– you lose utility, sure, but does your marginal utility change. Consider a set of states S ? ?s 1 ,...s S ? The probabilities of each state is ? s The state contingent payout is ?x 1 ,...x S ? We can consider preferences, ?, over payoff vectors (i.e. over ?x 1 ,...x S ??. MWG Definition 6.E.4: The preference relation ? satisfies the sure thing axiom if for any subset of state E ? S whenever ?x 1 ,...x S ? and ?x 1 ? ,...x S ? ? differ only in the entries corresponding to E, the preference ordering ordering between ?x 1 ,...x S ? and ?x 1 ? ,...x S ? ? is independent of the the particular (common) payoffs for states not in E. Proposition 6. E. 2.: Suppose that there are at least three states, and that the preferences ? on ? ? S are continuous and satisfy the sure thing axiom, then ? admits an extended utility function representation. U?x 1 ,..x S ? ? ? s?1 S u s ?x s ? If u s ?x s ? ? u?x s ? , then preferences are not state contingent. The Draft (Bergstrom, Soldiers of Fortune) Suppose that society needs to hire "A" percent of its people to serve in the army and it finances this with taxes on all of society. What does the equilibrium look like? To get people to go into the army it must be that utilities are equalized across occupations. Let utility be U(C, I) where C is consumption (just the net wage) and I is an indicator function that takes on a value of 1 if the person is in the army. Let W A denote the army wage (which is to be chosen) and W denote the wage elsewhere. The Volunteer Army U?W ? T,0? ? U?W A ? T,1? As T solves T ? AW A , we can rewrite this as: U?W ? AW A ,0? ? U?W A ? AW A ,1? The draft army: here you can allocate people into the army– can you do better. You maximize: ?1 ? A?U?W ? T,0? ? AU?W A ? T,1? subject to T ? AW A , or ?1 ? A?U?W ? AW A ,0? ? AU?W A ? AW A ,1? The first order condition from this is: ?1 ? A?AU 1 ?W ? AW A ,0? ? ?1 ? A?AU 1 ?W A ? AW A ,1? or U 1 ?W ? AW A ,0? ? U 1 ?W A ? AW A ,1? Marginal utilities are equalized over space. Given that the free market equilibrium is to equalize total utilities and the optimal outcome is to equalize marginal utilities– there is a real difference. In which case does the volunteer overpay the army– in which case does the draft underpay the army. Some easy examples: U?C,I? ? U?C ? ZI? In this case, the volunteer army condition is: U?W ? AW A ? ? U?W A ? AW A ? Z? Or W ? AW A ? W A ? AW A ? Z Which also implies that U ? ?W ? AW A ? ? U ? ?W A ? AW A ? Z? And the draft and volunteer army yield the same utility. Separability gives us: U?C,I? ? U?C? ? ZI U?W ? AW A ? ? U?W A ? AW A ? ? Z If Z?0, then W A ? W and as long as U(.) is concave, the marginal utility of income is lower in the army than in the civilians, so the volunteer army underpays. If U(.) were convex obviously things would be reversed, or if cash and being in the army were strong complements. This is a nice example, but there are other cases– think about income transfers and local price levels. Assume two locations– distinguished by price level P and P’, and Amenity levels A and A’. Utility is U(Y/P,A)– income divided by local prices and amenities. Assume that there is a fixed amount of cash (K) to allocate between two locations and measure one of welfare recipients who have no other income. Assume first that there are q percent of the welfare recipients in the second neighborhood and that is fixed. State is trying to maximize utility or: qU? T P ,A? ? ?1 ? q?U T ? P ? ,A ? Subject to the constraint qT?(1-q)T’?K or qU? T P ,A? ? ?1 ? q?U K?qT ?1?q?P ? ,A ? The first order condition is: 1 P U 1 ? T P ,A? ? 1 P ? U 1 K?qT ?1?q?P ? ,A ? Point 1: Higher price areas have a lower marginal benefit from transfers because they are more expensive. Point 2: The key question is whether the amenities are complements or substitutes for income. Point 3: We haven’t even started to think about mobility responses to higher transfers.