: now we have a family of utility functions for wealth increments z indexed by initial wealth w.

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1 Lotteries with Money Payoffs, continued Fix u, let w denote wealth, and set u ( z) u( z w) : now we have a family of utility functions for wealth increments z indexed by initial wealth w. (a) Recall from last time that the coefficient of absolute risk aversion at z is A( z) : u"( z) / u '( z) Agent gets less risk averse as wealth increases iff she has decreasing absolute risk aversion. CARA (constant absolute risk aversion) utility u( z) ex( z), Az ( ) w. With CARA, the certain equivalent of a 2 N(, ) lottery is 2 /2. And the same formula gives an aroximation of the certain equivalent for small gambles under any- (continuous, concave) utility function:

2 U ( w c) U ( w x) f ( x) dx So if all realizations of for x are near Ex : then U ( w ) ( c ) U '( w ) 2 U ( w ) ( x ) U '( w ) ( x ) / 2 U "( w ) f ( x) dx (1 st order aroximation is just c ) 2 ( c ) U '( w ) ( / 2) U "( w ) c U w U w 2 ( / 2) "( ) / '( ) (remember here is small )

3 With CARA references the utility of a gamble that is equally likely to give g or l is.5 u( w l).5 u( w g).5ex ( w l).5ex ( w g).5ex( w) ex( l) ex( g) So certain equivalent c satisfies ex( ( w c)).5ex( w) ex( l) ex( g) c ln.5 ex( l) ex( g) / (so the certain equivalent is indeendent of w) Note that it doesn t much matter what g is unless l is small ( for examle for l 100, g l, and.1, the ex( g) term is negligible.) It is often argued that risk aversion is and should be decreasing in wealth.

4 Relative risk aversion measures attitudes towards lotteries that are roortional to wealth. Definition: The coefficient of relative risk aversion at wealth w is wu"( w) Rw ( ) :. u'( w) CARA utility u( z) ex( z) has relative risk aversion R( w) increasing in w. w, which is An agent with increasing relative risk aversion gets more averse to roortional risks as he gets wealthier. The coefficient of relative risk aversion measures the agent s risk remium (as share of wealth) for a small gamble that is roortional to her wealth: set U ( w(1 )) E[ U ((1 y) w)] and do local aroximations.

5 CRRA (constant relative risk aversion): u( w) ln w has R 1. 1 u( w) w,0 1 has R ; Aside: in static choice u"/ u ' matters for risk averion. Looking ahead, in dynamic models things like u"/ u ' or also influence resent vs. future tradeoffs and savings- this has led to interest in non-eu models in macro and finance

6 Risk aversion and asset allocation (Arrow [1965], Pratt [1964) MWG Examle 6.C.2: A risk averse agent divides ortfolio between safe asset with return of 1 and a risky asset with random return z, cdf F. Pick investment [0, w] to maximize U ( ) u( w ( z 1)) df. U concave objective due to risk aversion. first order condition U '( ) ( z 1) u '( w ( z 1)) df. So if Ez 1, U '(0) u '( w) ( z 1) df 0 and the otimum is 0. * If Ez 1, the otimal is strictly ositive. Note: both conclusions can fail with multile risky assets- the correlation structure matters.

7 Claim: Suose u 1 is more risk averse than u 2, that u1, u 2 are concave and * * differentiable, and that the otimal investment levels, satisfy the FOC with equality. Then * * Proof: To show this show that U ( ) U ( ) 0 for all [0,1]. ' ' 1 2 Because u 1 is more risk averse than u 2, u1 g u2 for some increasing concave g. Normalize u 1 so that g '( u2( w)) 1, then since g ' is decreasing, and u ( w ( z 1)) g ' u ( w ( z 1)) u ( w ( z 1)), ' ' ' ' u1( w ( z 1)) u2( w ( z 1)) ( z 1) 0. And since U1, U 2 are concave,. * * 1 2

8 CARA agent invests constant amount regardless of wealth, agent with decreasing absolute risk aversion invests more as wealth increases. Can also show that a CRRA agent invests constant share of wealth. Note that if the return on the safe asset increases, so does the investor s effective wealth. So she could invest less in the safe asset is her absolute risk aversion decreases fast enough. (Fishburn and Burr Porter, J Man Science [1976]). If the initial wealth w is stochastic, we need stronger condition than comaring risk aversion to conclude that agent 1 always invests less. Machina and Neilson Ema [1987] give a necessary and sufficient condition on the two utility functions.

9 Demand for Insurance Simlest model: Insurance against a urely monetary loss Initial wealth w, may lose 1 unit with robability. Can buy insurance against of the loss at cost of q. U ( ) u( w q (1 )) (1 ) u( w q). U '( ) (1 q) u '( w q (1 )) q(1 ) u '( w q) If q, U '(1) (1 q) u '( w q) q(1 ) u '( w q) 0, agent buys less than full insurance: otimal to choose a deductible. If q (actuarially fair insurance) the agent buys full insurance, 1. Then whether or not the loss occurs agent has utility u( w ) and marginal utility u '( w ).

10 But in a more general setting otimal urchase of actuarially fair insurance only equates the marginal utility in the various states. Insurance with State-Deendent Utility Some accidents and illnesses can change utility at each wealth level. Model this with state-deendent utility functions u, u. (see MWG 6E) So suose the agent s objective function is U ( ) u ( w q (1 )) (1 ) u ( w q) I Then when q if FOC holds the otimal urchase sets u '( w q (1 )) u '( w q), I H -this needn t equalize the utility levels -whether or not the agent buys full insurance e.g. whether 1. H H I

11 Risk Preference in the Lab Problem: Given the wealth of most lab subjects, they should be almost risk neutral to tyical lab gambles- but they re not. Holt and Laury AER [2002]: 2/3 of subjects exhibit non-trivial risk aversion to lotteries whose outcomes all range from [$0,$4]! Subjects asked to make 10 binary choices, 1 out of 10 aid (so same references as for a single choice if the indeendence axiom alies.)

12 Risk neutral agents choose risky at (5/10,5/10); same is true for for r [.15,15]. 1 u( x) x r 2/3 the subjects switched to risky at or after 5/10 so (if CRRA) their R>.15; average switch oint of 5.2 corresonds to CARA of.2.

13 Rabin Ema [2000]: (Under EU) aroximate risk neutrality holds not just for negligible stakes but for quite sizable and economically imortant stakes. Economists often invoke exected utility to exlain substantial (observed or osited) risk aversion over stakes where the theory actually redicts virtual risk neutrality. That is, an agent who rejects small gambles with ositive exected value over a range of wealth levels and has a concave utility function will reject very favorable large gambles. He shows this in a few related results, this one is the simlest to arahrase:

14 Corollary (Rabin [2000]) Suose that u is strictly increasing and weakly concave, and that there are g l 0 such that for all w,.5 u( w l).5 u( w g) u( w). Then for a function m (defined in the aer), for all integers k, for all m m( k),.5 u( w 2 kl).5 u( w mg) u( w), where ( ) mk can be imlausibly big or even infinite: The agent must reject very favorable gambles.

15 Intuition: If reject (1/2 chance -100, ½ chance 110), then.5 u( w 110).5 u( w 100) u( w) u( w 110) u( w) u( w) u( w 100) And since u is concave, 110 u '( w 110) u( w 110) u( w) u( w) u( w 100) 100 u '( w 100) So u'( w110) 10 : Marginal utility of wealth can t decrease too slowly. u'( w100) 11

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17 By same argument, if the agent rejects the gamble when w' w 210, then u '( w ) u '( w 320) 10 u '( w ) u '( w 110) 11. So u '( w 320) u '( w 320) u '( w 110) 10 u '( w 100) u '( w 110) u '( w 100) If can iterate 100 times, then because wealth is very low for high wealth , the marginal utility of So there is no value of g that would make the agent accet ½ chance of losing $1000.

18 This uses the reject at all w condition. But can get similar bounds if we only know the agent is risk averse and turns down the l,g gamble for all wealth levels less than some w: if.5 u( w 100).5 u( w 110) u( w) for all w 300,000 then at w 290,000 the agent refuses l=-1000,g= 718,000! Intuition: concavity says u '( x) u '(300,000) for all x 300,000. Nothing secial about ½- ½ bets, they are just used for convenience. What to make of this? Rabin suggests loss aversion as the key. But we also see small stakes risk aversion when gambles are entirely in the gains domain as in Holt and Laury.

19 An alternative exlanation combines the idea that the value of money comes when you send it and the idea of narrow bracketing based on mental accounts, as in Shefrin and Thaler Econ Inquiry [1988])..self-control is costly some mental accounts, those which are labelled wealth, are less temting than those labelled income. This leaves oen the question of how these brackets or accounts are set. Fudenberg-Levine AER [2006] roose that the mental account corresonds to daily or weekly consumtion exenditures, and argue that small stakes risk aversion comes from treating lab ayments as windfall gains.

20 Idea: Set mental account when not temted: ocket cash. Absent windfalls, either imossible or costly to send more than in mental account. Perfect foresight: imlement first-best consumtion by aroriate choice of cash=desired sending. After account is set, the cost of resisting temtation acts as tax on savings out of unlanned-for windfalls: lanned consumtion didn t take into account this cost. So if win $10 send it all, but if win $10,000 save some; thus high risk aversion for small winnings (as in exeriments) but non-crazy risk aversion for large winnings due to income smoothing. More on the FL model later if time ermits..

21 Stochastic Dominance Definition: For any ( ) let F be its cumulative distribution function (c.d.f): x F ( x) Pr[ z x] ( x) dx. F 1 : (0,1) be its quantile function F 1 ( u) inf x : F ( x) u. Definition: For q, ( ), first-order stochastically dominates q, written fosd q, if F ( x) F ( x) x. q Note: In the statistics literature this is simly called stochastic dominance and is said to be stochastically larger than q. (e.g. Mann and Whitney Ann. Math. Stat. [1947])

22 FOSD doesn t let us comare the lotteries realization by realization as they might be indeendent. But it does say we can reresent them with a air of erfectly correlated lotteries where one is always as large as the other: Fix q, ( ). For u (0,1) set these are the medians of and q. x F 1 ( u) and y F 1 ( u), e.g. if x.5 q Then F ( x) F ( x) x iff x( u) y( u) u (0,1), q and if we let u be uniformly distributed on (0,1) we have x ointwise at least as big as y. This may hel give intuition for the following result:

23 u :. Theorem: for every weakly increasing function fosd q iff ud udq Proof: (i) Suose there is y s.t. F ( y) F ( y). Then if u( x) 1( x y), ud 1 F ( y ) 1 F q( y ) udq. (ii) Conversely suose F ( x) F ( x) x. And to simlify assume the q q c.d.f. s are strictly increasing (no gas in the suort) and continuous (no atoms) on a common suort [a,b] and that u is continuous (can extend by aroximating the increasing function u with olynomials). Then we can integrate by arts:

24 b b b b u( x) df ( x) u( x) F ( x) F ( x) du( x) u( b) F ( x) du( x) a a a a b u( b) F ( x) du( x) u( x) df ( x). a q b a The characterization of FOSD as ranking the exected utility of lotteries alies for all non-decreasing utility functions. What if we add concave? Definition: If the suorts of and q lie in [a,b], we say second-order stochastically dominates q, q c c sosd q, if Fq( x) dx F( x) dxc [ a, b] a a. Theorem: Assume E Eq. Then on [a,b], b b udf a udf a q. sosd q iff for all non-decreasing concave u Proof omitted, use integration by arts.

25 The intuition comes from a related result: second-order stochastically dominates q only if we can write and q as the marginal distributions of random variables z and z q with zq z and E( z ) 0. Here lottery (a) is better than lottery (c) for any risk-averse utility function. Reading for next time: Strzalecki 5.8, , Neilson and Stowe J Risk Uncertainty [2002], Eer and Fehr-Duda Annual Review of Economics [2012], Bruhin, Eer and Fehr-Duda Ema [2015].