Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between nd b. Thus [, b] is the mximl support of ll the probbility distributions we will consider. (In prctice my be zero if there is limited libility, but could become negtive; the upside potentil could be unbounded. Thus we my wnt to llow = nd b =. This rises some technicl complictions which we ignore.) Thus we hve new feture: W cn be continuous rndom vrible, nd lottery is then probbility distribution over [, b]. Consider n expected-utility mximizer with utilityof-consequences function u(w ), evluting prticulr lottery with cumultive distribution function F (W ) nd density function f(w ). The expected utility is u(l) = u(w ) f(w ) dw. (If you find integrls difficult, try writing out the sums for discrete distribution tking on vlues W < W <... < W n, probbilities f i, nd cumultive probbilities F i = i j= f i. But ctully tht is lgebriclly more messy.) Once the u-function is known, the expected utility cn be evluted for ny lottery, nd ny pir of lotteries cn be compred. But we re often interested in mking broder comprisons tht re vlid for very generl clsses of utility functions, tht is, in sttements such s ny person whose u-function stisfies such nd such property will prefer lottery L to lottery L. There re two especilly importnt questions of this kind: [] Wht kind of reltionship must exist between lotteries L nd L, or equivlently, between their CDFs F nd F, to ensure tht everyone, regrdless of his ttitude to risk, will prefer L to L (so long s he likes more welth to less). In other words, to ensure tht for ll incresing utility functions u(w ), u(l ) u(w ) f (W ) dw > u(w ) f (W ) dw u(l ). The reltion of F to F tht ensures this property is clled first-order stochstic dominnce (FOSD). [] Wht kind of reltionship must exist between lotteries L nd L, or equivlently, between their CDFs F nd F, to ensure tht every risk-verse person will prefer L to L. In other words, to ensure tht for ll concve utility functions u(w ), u(l ) u(w ) f (W ) dw > u(w ) f (W ) dw u(l ).
The reltion of F to F tht ensures this property is clled second-order stochstic dominnce (SOSD). Comments: () For rbitrry L nd L, neither my dominte the other in either of these senses. In other words, both of these dominnce concepts re prtil orderings of lotteries, not complete orderings. () Your first thought my be tht first-order stochstic dominnce should correspond to hving higher men, nd second-order stochstic dominnce should correspond to hving lower vrince, but neither of these intuitions is correct. You should be ble to see fter moment s thought tht higher men is not going to suffice for FOSD: F my hve higher men but so much higher vrince thn F tht someone with n incresing but concve utility function will like L less thn L. But the point bout vrince nd SOSD is more subtle. We will see counterexmples soon. First-Order Stochstic Dominnce To find the mthemticl concept of FOSD, begin by observing tht for ny distribution function F nd its density function f, nd for ny utility function u u(w ) f(w ) dw = [ u(w ) F (W ) ] b u (W ) F (W ) dw = u(b) u() 0 = u(b) u (W ) F (W ) dw u (W ) F (W ) dw integrting by prts Therefore compring expected utility under two different distributions (if they hve different supports [, b], tke the biggest nd define the other density outside its support to be 0): u(w ) f (W ) dw u(w ) f (W ) dw = = u (W ) F (W ) dw + u (W ) [ F (W ) F (W ) ] dw. u (W ) F (W ) dw We wnt this to be positive for ll incresing functions u, tht is, for ll functions with u (W ) > 0 for ll W [, b]. Tht will obviously be true if F (W ) > F (W ) for ll W (, b). We cnnot void equlity t the endpoints nd b becuse F () = F () = 0 nd F (b) = F (b) =. But tht does not hurt the result. We cn lso llow equlities t isolted points within the intervl. Agin I leve more rigorous proofs to more dvnced (grdute) courses. But the converse is lso true: if F (W ) < F (W ) for ny rnges of W (, b), then we will be ble to construct function u for which u (W ) is very positive nd significnt, sy equl to, in these rnges, nd still positive but very smll, sy some ɛ, everywhere else. By choosing ɛ smll enough, we cn then mke the integrl on the right hnd side negtive. This counterexmple shows tht F (W ) < F (W ) for ll W (, b) is lso necessry for u(l ) > u(l ) to be true for ll incresing u functions.
So we hve the desired result if, nd only if, F (W ) > F (W ) for ll W (, b). We might mke this our definition of FOSD, nd stte the result proved in the bove rgument s theorem: Definition : The distribution F is first-order stochstic dominnt over F if nd only if F (W ) < F (W ) for ll W (, b). Theorem : Every expected-utility mximizer with n incresing utility function of welth prefers the lottery L with distribution F to L with distribution F if nd only if F is FOSD over F. Figure shows two such cumultive distribution functions. I hve tken = 0 nd b = for convenience. Since both distributions strt out t 0 when W = = 0, nd then F (W ) becomes less thn F (W ), it must be the cse tht F (W ) is fltter thn F (W ) for smll W. Since the slope of the cumultive distribution function is the probbility density function, the density f (W ) must be less thn the density f (W ) for smll W. Conversely, both distributions climb to t W = b =, but F (W ) climbs from smller vlues thn F (W ), it must be the cse tht F (W ) is steeper thn F (W ) for W close to. Tht is, the density f (W ) is higher thn the density f (W ) for W close to. Figure shows the density functions. F(x) F (x) F (x) x Figure : FOSD: CDF comprison f(x) f (x) f (x) x Figure : FOSD: Density comprison The density comprison cn be restted s sying: f (W ) cn be obtined from f (W ) 3
by shifting some probbility weight from lower to higher vlues of W. Tht is n intuition for why nyone who prefers more welth to less prefers the lottery L to L. For your informtion, the functions re f (W ) = W ( W ), f (W ) = W ( W ), F (W ) = 4 W 3 3 W 4, F (W ) = 6 W 8 W 3 + 3 W 4. Second-Order Stochstic Dominnce The procedure is similr except tht we hve to integrte by prts twice. Write S(W ) = W F (w) dw. This is the integrl of the cumultive distribution function, so its vlue t ny point W is the re under the F (w) curve for w going from to W. And by the fundmentl theorem of the clculus, the cumultive distribution function is its derivtive: F (W ) = S (W ) for ll W. Cll the S function the super-cumultive distribution function for short. Note tht S() = 0. Now crry out the process of integrtion by prts twice: u(w ) f(w ) dw = [ u(w ) F (W ) ] b u (W ) F (W ) dw Therefore = u(b) u() 0 = u(b) u (W ) F (W ) dw u (W ) F (W ) dw = u(b) [ u (W ) S(W ) ] b + u (W ) S(W ) dw = u(b) u (b) S(b) + u(w ) f (W ) dw = u (b) [S (b) S (b)] + u (W ) S(W ) dw u(w ) f (W ) dw u (W ) [S (W ) S (W )] dw integrting by prts int. by prts gin We wnt our dominnce definition to be such tht every risk-verse person prefers L nd every risk-loving person prefers L. Therefore we wnt the expression on the right hnd side of the lst line to be positive for ll concve functions u, tht is, for ll functions with u (W ) < 0 for ll W [, b]; we lso wnt the sme expression to be negtive for ll convex functions, tht is, for ll functions with u (W ) > 0. And therefore, on the borderline, we wnt risk-neutrl person to be indifferent between L nd L. Tht is, the whole expression on the right hnd side of the lst line should be 4
zero if utility function stisfies u (W ) = 0 for ll W (, b). But this condition reduces the integrl on the right hnd side to zero. So the first term u (b) [S (b) S (b)] must be zero s well. Since u (b) 0, it must be tht S (b) = S (b). Wht does this condition signify? For ny distribution F, we hve S(b) = F (W ) dw = F (W ) dw = [ F (W ) W ] b f(w ) W dw = F (b) b F () = b E[W ] f(w ) W dw int. by prts Therefore, if the two distributions hve S (b) = S (b), then the expected vlues of W under the two distributions should lso be equl, which we cn write s E (W ) = E (W ). In other words, the men welth should be the sme under the two lotteries. This is very nturl condition to require when we wnt preference between them to depend only on the ttitudes towrd risk. So we re left with u(w ) f (W ) dw u(w ) f (W ) dw = u (W ) [S (W ) S (W )] dw. Proceeding exctly s the cse of first-order stochstic dominnce, we see tht the right hnd side will be positive for ll utility functions stisfying u < 0 if, nd only if, S (W ) < S (W ) for ll W (, b), nd S (b) = S (b). So we hve our definition of second-order stochstic dominnce (SOSD), nd theorem: Definition : The distribution F is second-order stochstic dominnt over F if nd only if S (W ) < S (W ) for ll W (, b) nd S (b) = S (b), where S(W ) = W F (w) dw. Reclling tht the end-point equlity of the S-functions imposes equl mens, nd F is going to be riskier thn F, we lso sy tht F is men-preserving spred of F. Theorem : Every expected-utility mximizer with concve utility function of welth prefers the lottery L with distribution F to L with distribution F if nd only if F is SOSD over F. Actully the result is even more generl. In the whole rgument we nowhere used the sign of u. So ll we needed ws concve function, whether it be incresing or decresing. So we hve ctully proved more generl Theorem : The inequlity u(w ) f (W ) dw > u(w ) f (W ) dw holds for every concve function u if, nd only if, F is SOSD over F. 5
Figure 3 shows such comprison of two S functions. But S functions re not immeditely menble to intuition, so we wnt to see how the underlying F or f functions compre. Figures 4 nd 5 show them. 0.5 S(x) S (x) S (x) x Figure 3: SOSD: S-function compriison F(x) F (x) F (x) F (x) F (x) x Figure 4: SOSD: CDF comprison f(x) f (x) f (x) x Figure 5: SOSD: density comprison For your informtion, the functions re f (W ) = 30 W ( W ), f (W ) = 6 W ( W ), F (W ) = 0 W 3 5 W 4 + 6 W 5, F (W ) = 3 W W 3, S (W ) =.5 W 4 3 W 5 + W 6, S (W ) = W 3 0.5 W 4. 6
We cn now interpret the differences between the two distributions. It helps to compre the simpler but somewht similr nlysis in the cse of first-order stochstic dominnce. There the cumultive F (W ) ws entirely below F (W ), nd the densities f i (W ) were the derivtives (slopes) of the cumultives F i (W ); therefore f (W ) ws rightwrd shift of f (W ). (See Figures nd.) Here, the super-cumultive S (W ) is entirely below S (W ), nd the cumultives (CDFs) F i (W ) re the derivtives (slopes) of the super-cumultives S i (W ); therefore F (W ) is rightwrd shift of F (W ). But the F i (W ) re incresing functions, unlike the densities f i (W ); contrst Figures 4 nd. Thus F (W ) strts out fltter thn F (W ) (is smller for smller W vlues), then crosses it (is lrger for lrger W vlues), but must fltten out gin s the two pproch the common vlue for W = b. In other words, F (W ) is fltter for smll nd lrge vlues of W, nd is steeper in the middle. The density functions f i re the slopes of the cumultives F i ; therefore the comprison of densities is tht f (W ) is smller thn f (W ) for smll nd lrge vlues of W, nd is lrger in the middle. This is shown in Figure 5. In other words, the density f (W ) hs higher pek, nd smller tils, thn the density f (W ). The distribution of the outcomes of lottery L is more spred out thn tht of the outcomes of L. Tht lso explins why the switch from f (W ) to f (W ) is clled men-preserving (becuse S (b) = S (b) ) spred. And it is quite is nturl reflection of the ide we wnt to cpture, nmely tht ll risk-verse persons prefer L to L, or tht L is in some uniform sense riskier thn L. Being men-preserving spred imples hving higher vrince, but not conversely. To prove the impliction, just consider the function φ(w ) = ( W W ) where W is the expecttion of W, the sme under the two distributions (remember menpreserving). The function φ is convex. If F is SOSD over F, our generl Theorem gives φ(w ) f (W ) dw < φ(w ) f (W ) dw. But ech side is just the vrince of W under the respective distribution. Actully these pictures show only the simplest wy in which the condition in the theorem cn be fulfilled. Figure 3 remins the true story, but the Figures 4 nd 5 derived from it cn hve more complicted shpes. In more generl Figure 4, the grphs of F (W ) nd F (W ) cn intersect multiple times. F (W ) must still strt out fltter nd end up fltter, so the number of intersections must be odd. And becuse F (W ) strts out fltter, the re under it (its integrl S (W ) ) strts out smller thn the corresponding re for F (W ). In subsequent intervls where F (W ) > F (W ), the gp between S (W ) nd S (W ) strts to close. (And in other intervls where F (W ) < F (W ) gin, the gp widens gin.) But the requirement of SOSD is tht the gp should never close entirely until we rech the right hnd end-point b. These figures re difficult to drw; see Fig. 3.7 on p. of the book by Hirshleifer nd Riley tht is cited s occsionl reding in the syllbus. The ides generted by Figure 5 cn be formulted into lterntive (nd equivlent) definitions of SOSD: 7
Definition (): Let W i denote the rndom vrible following distribution F i, for i =,, nd E (W ) = E (W ). Then F is sid to be SOSD over F if there exists rndom vrible z with zero expecttion conditionl on ny given vlue of W, such tht w hs the sme distribution s w + z, or in other words, w equls W plus some dded pure uncertinty or noise. Definition (b): Of two distributions yielding equl expected vlues, F is sid to be SOSD over F if it is possible to get from F to F by sequence of opertions which shift pirs of probbility weights on either side of the men frther wy, while leving the men unchnged. The dding noise ide is simple nd we will use it occsionlly. We won t need the detils of how to construct bsic men-preserving spreds so I omit them; if you wnt to, you cn find them in the Rothschild-Stiglitz rticle pp. 33 37, reprinted in the Dimond- Rothschild book pp. 03 07. But it is very importnt to understnd how nd why higher vrince does not ensure SOSD. Since we wnt to show tht it is possible to hve n instnce where L hs higher vrince thn L but risk-verse person prefers L to L, so F is not SOSD over F, we need only construct n exmple. There re vrious exmples of this kind in the literture. Here is reltively simple nd comprehensible one: The two lotteries, defined by their vectors of welth consequences nd probbilities, re: L = (0,, 4, 6; /4, /4, /4, /4) L = (0., 3, 5.9; /3, /3, /3) It is esy to clculte tht the mens nd vrinces re L : Men = 3, Vrince = 5.000 L : Men = 3, Vrince = 5.607 Tke the utility function u(w ) = { W if W 3 3 + W if W > 3 Figure 6 shows the two lotteries nd Figure 7 shows the utility function: /4 /3 L L 0 0. 3 4 5.9 6 Figure 6: Lotteries in exmple for SOSD lower vrince This gives expected utilities: EU(L ) = 5.000, EU(L ) = 5.033 8
9 u u(w) 6 3 3 6 W Figure 7: Utility in exmple for SOSD neq low vrince so the person prefers L despite the fct tht it hs higher vrince thn L nd the sme men. Note tht the utility function is stright line in the regions below nd bove W = 3, but hs kink t 3. In other words, the concvity (risk-version) rises solely becuse the person dislikes downside risk more thn he vlues upside potentil. This is not unusul; in fct it is one spect of ctul behvior emphsized by mny critics of the usul smooth expected utility, nd we will pick up on this ide lter. But you cn get nother exmple by mking very slight chnge to the u function to mke it smooth nd strictly concve everywhere; you just need rpid chnge of slope in smll intervl round W = 3. Now consider the expected bsolute devitions, tht is, E( W W ) for the two distributions: for L it is, nd for L it is.933. Therefore L hs lrger bsolute devition, even though it hs smller vrince. And our utility function, which hs constnt slope below the men nd nother smller constnt slope bove the men, cn be expressed s if the person dislikes bsolute devition. We cn write or Therefore u(w ) = {.5 +.5 W 0.5 (3 W ) if W 3,.5 +.5 W 0.5 (W 3) if W > 3, u(w ) =.5 +.5 W 0.5 W 3. EU(W ) =.5 +.5 E(W ) 0.5 AbsDev(W ) where AbsDev(W ) denotes the bsolute devition of W. So it is nturl tht the lotteries re rnked by bsolute devition rther thn by vrince. And once you understnd this ide, you cn get similr counterexmples by constructing utility functions tht involve some other mesure of centrl tendency thn the vrince. 9
And how does this relte to the definition of SOSD? For tht, we need to compre the super-cumultive functions for the two distributions. A little work shows yields Figure 6. I hve shown S thicker, nd hve shown the points 0. nd 5.9 out of scle for clrity of ppernce. We see tht the two super-cumultives S nd S cross, so F is not SOSD over F. 3 S S 0 0. 3 4 5.9 Figure 8: SOSD vs. vrince comprison 6 Another wy to look t this is tht SOSD is kind of comprehensive requirement tht requires every conceivble mesure of centrl tendency under one distribution to be smller thn tht under the other. The vrince is only one such mesure; therefore it being smller does not ensure tht the person prefers tht distribution if he ctully cres bout some other mesure. 0