FIRST DEGREE STOCHASTIC DOMINANCE FOR DISCONTINUOUS FUNCTIONS. L. P. Hansen, C. A. Holt, and D. Pe1ed. Discussion Paper No , September 1977

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1 FIRST DEGREE STOCHASTIC DOMINANCE FOR DISCONTINUOUS FUNCTIONS by L. P. Hnsen, C. A. Holt, nd D. Pe1ed Discussion Pper No , September 1977 Center for Economic Reserch Deprtment of Economics University of Minnesot Minnepolis, Minnesot 55455

2 FIRST DEGREE STOCHASTIC DOMINANCE FOR DISCONTINUOUS FUNCTIONS by L. P. Hnsen, C. A. Holt, nd D. Peled* Hnoch nd Levy [6J presented generl first degree stochstic dominnce theorem for the cse in which the Von Neumnn-Morgenstern utility function u(x) is non-decresing in x. In contrst to other ppers on this subject, Hnoch nd Levy did not ssume tht utility is differentible or even continuous. 1 Recently, Tesftsion [7J hs shown tht the proof of the Hnoch nd Levy first degree stochstic dominnce theorem ws incorrect. Tesftsivll then stted nd proved Himilr theorem for utility functions,h~h re non-decrehing nd continuous. The usefullness of the bove theorems extends beyond the rther si.nple comprison of expected utilities. For exmple, optimiztion problems yield first order conditions involving expecttions of derivtives of the objective function. In such problems, kinked py-off structure mkes it desirble to hve theorem tht pplies to discontinuous flillctions s well. Section I contins proof of severl first degree stochstic dominnce theorems for functions with discontinuities. Although the originl proof of the Hnoch nd Levy theorem ws deficient, it is shown in section II tht the theorem itself is correct subject to minor modifiction. An ppliction of these theorems is illustrted in section III in the context of simple inventory model. * We would like to thnk Kim Border, Clifford Hildreth, Jmes Jordn, Leonrd Shpiro, nd Christopher Sims for helpful comments.

3 -2- I. Some First Degree Stochstic Dominnce Theorems Following Fishburn [3J nd others, we will mke the distinction between strict nd non-strict dominnce. nd rndom vribles defined on n bstrct probbility spce (O,~, p). By definition, Xl stochsticlly domintes X 2 if p(x l > xl ~ p{x 2 > x} for ll x on the rel line R, where [Xi> x} ~ (w EO: Xi (w) > x}, i 2 1, 2. The dominnce is sid to be strict if P(XI > x} 3: P[X2 > x} Xo E R "rix E R, nd for some An ~quivlent definition of non-strict dominnce tht will be useful subsequently is given in the following lemm: Lemm 1. Let Xl nd X 2 be two rndom vribles. A necessry nd sufficient condition for Xl to stochsticlly dominte X 2 is tht Proof. () For necessity let x E R, nd consider sets of the form intege~. By hypothesis nd B ~ n B l' "rin. Il+ 00 I B = rx > x - -} where n n t 2 n P(An) ~ P(B n ) Therefore P( f1 n=l is positive nd by construction A --) A I n n+ An) = lim p(an) ~ lim P(Bn) = n-'co n-iq) p( n Bn). Then necessity follows by noting tht n A == {X ~ x} nd n==l n 1 n:l n Bn = (X2 ~ x} n:l 1 x + ;;-} (b) Sufficiency is proved in similr mnner. Let An = [Xl ~ nd sequences of sets, nd > P(XI > x} = p( u A ) ~l n by ssumption nd p(a )!! P(B ), \[n. n n re incresing Thus = lim P(A ) ~ lim p(b ) = ~J( ($"I U B) = P{X2 > x} n--ico n n...,p) n n=l n Q.E.D.

4 -3- The stndrd definition of the expected vlue of rndom vrible X is: EX where X+ (w) = = = J X+(w) dp(w) - J X-(w) ~(w) o o if X(w) > 0 otherwise, if X(w) < 0 otherwise EX is sid to exist s long s J x+ n both infinite. These integrls should Lebesgue integrls. d~) nd J X dp re not U be interpreted s generlized In the next two lemms, we show tht expected vlues of rndom vribles re ordered by first degree stochstic dominnce reltionships. Lemm 2. be rndom vribles such tht EX 1 nd EX 2 exist. If Xl stochsticlly domintes X 2 then EX I ~ EX 2 Proof. In order to prove this lemm, we mke use of well known result in probbility theory tht for non-negtive rndom vrible ("Xl 3 X, EX = J. P(X > x)dx. o J 00 P{Xi > x}dx, for i=l, 2 o X 2 it follows tht Ext ~ Exi CXlPLXi f < -x}dx, for i=l, 2. '0 which implies EX- - I ~ EX 2. Thus Then EXt~::: J (Xl P{X~ > x }dx o Since Xl Similrly, From Lemm 1 stochsticlly domintes EX~ ~ J OOr[X~ > x}dx = r{x l < -xl ~ P[X2 < -xl EX~ ~ EXi - EX 2 = EX 2 Q. E.D. 4

5 -4- The strict nlogue of Lemm 2 is: Lemm 2'. Let Xl nd X 2 be rndom vribles such tht t lest one of them hs finite expecttion. If Xl stochsticlly domintes X 2 in the strict sense,then EX I > EX 2 Proof. By hypothesis,there exists n x E R such tht P[XI > x} > P[X2 > x} Since P(X > yl is right continuous function of y, P(XI > y} > P(X2 > y} on some intervl [x' x + E 1, where E > Assume without loss of generlity tht c [, 00) nd From the - - proof of Lemm 3, EX I ~ EX 2 Hence EX I > EX 2. Q.E. D. Consider non-decresing function u nd two rndom vribles Xl nd X 2 It is not surprising tht dominnce reltionships between these two rndom vribles induce corresponding dominnce reltionships between the rndom vribles Lemm 3. Let u: R ~ R be non-decresing. If Xl stochsticlly domintes X 2, then u 0 Xl stochsticlly domintes u 0 X 2. Proof. Since u is non-decresing, it is Borel mesurble, nd therefore u 0 X nd u' 0 X 2 re rndom vribles. Let x E R. I cse (i) Suppose tht [z E R: u(z) > x} == if; Then [u o Xl > x} = [u o X 2 > x} = dj nd therefore p[u o Xl > x} 0.= 6 J!u 0 X 2 > x}.

6 -5- cse (ii) Suppose tht {z E R: u(z) > x} f 0' Let y = inf(z E R: u(z) > x},where y is possibly - m It follows tht (y, + cc) c {z E R: u(z) > x} c [y, +*'), so t1;lt [z E R: u(z) >x} is either (y, +(0) or [y, +=)., Consequently either () or (b) must hold: () (b) (u 0 X. > x} = {X. > y} for i = 1, [u 0 X. > x} = (X. ~ y) for i = 1, The result p(u 0 X > x} ~ p{u 0 X 2 > x} follows by hypothesis if () holds, nd by Lemm 1 if (b) holds. Q.E.D. A strict version of Lemm 3 cn be proved under stronger conditions on u. Lemm 3'. Let u: R _ R be non-decresing. Suppose tht Xl stochsticl1y domintes X 2 If there exists Xo E R such tht: (i) u(x) > u(x O ) for x > Xo nd (ii) P{Xl > x O } > ~J{X2 > x }, then u 0 Xl O stochsticlly domintes u 0 X in the strict sense. 2 Proof. By the monotonicity of u, Xi ~ Xo ~ u 0 Xi ~ u(x ) nd by O (i) Xi > Xo ~ u 0 Xi > u(x O ), for ~ 1, 2. Hence, (u 0 Xi > u(x O )} = [Xi> x O }, ~l, 2. Consequently by (ii) P[u 0 Xl > u(x )} > O p(u 0 X 2 > u(x )}. Then the conclusion follows from Lemm 3. O Q.E.D. theorem: Immedite ppliction of Lemm 2 nd Lemm 3 yields the following Theorem 1. Let u: R _ R be non-decresing. Suppose Xl stochsticlly domintes X 2 nd both nd exist. Then

7 -6- The strict version of theorem 1 follows from Lemm 21 nd 31~ Theorem II, Let u: R ~ R be non-decresing. Suppose tht Xl stochsticlly domintes X 2. If there exists n Xo e R such tht: nd if u 0 or hs finite expecttion, then > E(u 0 X 2 ) II. The Hnoch nd Levy First Degree Stochstic Dominnce Theorem Before we reconsider the originl Hnoch nd Levy theorem, brief discussion of integrtion my be useful. Hnoch nd Levy used Lebesgue-Stieltjes integrtion in their theorem nd proof. However, they encountered difficulties when they ttempted n integrtion by prts rgument. In order to void these difficulties, Tesftsion shifted to Riemnn-Stieltjes integrtion in her restricted theorem 5 nd proof. However, Riemnn-Stieltjes integrtion my not be pproprite for use in probbility theory. One of the resons for this is tht if X is rndom vrible with distribution function F, nd if u : R ~ R is Borel mesurble, then u 0 X is rndom vrible whose distribution function will be denoted by G. In this cse, it is desirble to hve n integrtion theory which yields the following equlities: E (u 0 X) = J x dg(x) R S u(x) df(x) R (1) provided tht ny of these integrls exist. If u nd F hve cmmon point of discontinuity, then the Riemnn-Stieltjes integrl on the right end of (1) does not exist even though f(x) = x my be Riemnn-Stieltjes integrble with respect to G. With Lebesgue-

8 -7- Stieltjes integrtion, (1) is stisfied. In this section ll integrls over the rel line re Lebesgue-Stieltjes integrls. Our modifiction of Theorem 1 in Hnoch nd Levy is: Theorem 2. Let U be the clss of non-decresing functions u: R... R, nd let F1 nd F2 be distribution functions. Then (I) nd (II) re equivlent: (II) s u(x) df 1 (x) for ny u E U wheneve~ both integrls exist, nd there is u* E U R for for some Xo E R which this inequlity is strict. Proof. Suppose tht (I) holds. For ny two distribution functions F1 nd F 2, there exist rndom vribles Xl nd X 2 defined on common probbility spce tht hve F1 nd F2 s their respective distribution functions. Then it follows from Theorem 1 tht: whenever the integrls on ech end exist. Then Next, let u*(x) = 1 if x E (x O ' + CD) nd zero elsewhere. Su*(x) dfl(x) R = = S u*(x) df 2 (x) R Conversely, suppose tht (II) holds. For ny y E R, let u (x) = 1 for x E (y, + CD ), zero elsewhere. Then y 1 - F1 (y) = J u (x) df 1 (x) RY J u (x) df 2 (x) R y =

9 -8- nd therefore, Fl(y) ~ F 2 (y). Since there is u* such tht it cnnot be the cse tht Fl(x) = F 2 (x) for ll x E R. Q.E.D. This theorem is essentilly the sme s the Hnoch nd Levy theorem, nd it does not impose the continuity conditions on u employed by Tesftsion. 6 III. An Appliction Stochstic dominnce theorems re often useful in the comprison of optimizing behvior under two different uncertinty regimes. In such optimiztion problems, the first order conditions cn contin expecttions of derivtives of the objective function. These derivtives will be discontinuous if the objective function is kinked. The following exmple illustrtes the ppliction of Theorems 1 nd l' in this type of problem. Consider n gent who is ble to sell units of homogeneous commodity for p dollrs per unit. The quntity demnded t this price is the reliztion of continuous rndom vrible x with density g(.) nd distribution function G( ). of g(.) is finite intervl [, bj. It is ssumed tht the support The discontinuity in this problem rises becuse the seller cn cquire the commodity for c dollrs t the beginning of the period, but the cquisition cost is n dollrs per unit obtined subsequent to the reliztion of the demnd x. It is ssumed tht ll demnd must be stisfied, nd tht n > p > c 7

10 -9- The seller's utility function u( ) is ssumed to be twice differentible with u' (.) > 0 nd u" (.) : O. Then the seller's expected utility s function of the initil inventory cquisition I is: I b J u(px - ci) g(x) dx + S u(px - TI(x - I) - ci) g(x) dx I for I Er, bj. The first order condition for determining the optiml inventory cn be written s n eqution in I: where co (I, x) b J cp(i, x) g(x)dx = 0 (2) tc u' (px - ci) if x ~ I - cj u' (px - TIX + TIl - ci) if x > I (3) It is strightforwrd to verify tht the following inequlities hold for ny density function h(-) with support [, bl b ~/I f S cp(i, x) hex) dx } < 0 for ll I E (,b), (4) Jb cp(, x) hex) dx > 0, nd b I ~(b, x) hex) dx < 0 Thus the optiml inventory level for the distribution G( ), denoted by I G, is uniquely determined by the first order condition (2). Now consider the effect of shift in the probbility distribution of x on the optiml inventory level. Specificlly, suppose tht the new distribution function of x is F(x) nd tht F(x): G(x) V x E [, bj. It is pprent from the concvity of u( ) tht ~(I, x) defined in (3) is non-decresing function of x with simple discontinuity t x = I Then the impliction of Theorem I is tht:

11 -10- j'bq:>(ig' x) f(x) dx ~ J' b (p(i G, x) g(x) dx = o (5) where the lst equlity follows from the first order condition (2). The optiml inventory for the distribution F denoted IF' is detennined by b f q:>(i F, x) f(x) dx It is pprent from (5), (6), nd the second order condition (4) o (6) tht If the se Her is risk verse (u" (.) < 0), then co (I, x) defined in (3) will be strictly incresing function of x. Thus it follows from Theorem I' tht IG < IF whenever F domintes G in the strict sense. In this exmple, the seller's utility is continuous function of the relized demnd x, the discontinuity ppers in the first order condition. In other problems, however, it my be the cse tht decision-mker's utility is itself discontinuous s function of some rndom vrible. For firm submitting seled bid for the right to exploit minerl lese, the mximum rivl bid cn be thought of s the reliztion of rndom vrible. If the firm's bid exceeds the mximum rivl bid, then the firm's profit would generlly be significntly lrger thn would be the cse if the lese is lost. Other discontinuities occur becuse contrcts commonly impose penlties if some mesure of performnce flls short of specified stndrd.

12 -11- Therefore, even though decision-mker's utility my be continuous function of monetry gin, utility my not be continuous function of the reliztion of rndom vrible tht ffects this monetry gin. In conclusion, the stochstic dominnce theorems in this pper my be useful in the nlysis of problems in which the objective function is kinked or discontinuous function of the reliztion of rndom vrible.

13 -12- NOTES 1. For exmple, Bw (lj nd Hdr nd Russell [4, 5J mke the differentibility ssumption. 2. If the distribution functions of Xl nd X 2 re Fl nd F2 respectively, then this definition is obviously equivlent to the requirement tht Fl(X) ~ F 2 (X) V x E R. 3. See Feller [2J, chpter 5, lemm Alterntively, lemm 2 cn be proved by showing tht if Xl stochsticlly domintes X 2 ' then there exists probbility spce (0*, "J*, p*) nd two rndom vribles X* nd X* tht 1 2 stisfy the following conditions: (1) Xi hs the sme distribution function s X:( ~ for i = 1, 2. (2) X* (w*) ~ 1 X* 2 (w*) V w* E n ie. In order to verify (1) nd (2), let Fl nd F2 denote the distribution functions of Xl nd X 2 respectively. Also let 0* be the open unit intervl, "J* the collection of Borel subsets of 0*, nd P* the Lebesgue mesure. For w i ( E 0*, define X~(w) - sup[x E R : F. (x) :=; w*} i = 1, 2. It follows tht ~ ~ the distribution function of X~ is F i Furthe rmore since ~ Xl ~ stochsticlly domintes X2 ' xt(w*) X; (w*) V w * E 0* 5. See [7J, Theorem 1*, p. 304.

14 -l3-6. As Tesftsion hs noted, Hnoch nd Levy were bit creless in the sttement of their theorem. Insted of using (II) in our Theorem 3, their theorem sttes tht I u(x) dfi (x) R i u(x) df 2 (X) ~ 0 R for ll u E U, with strict inequlity holding for some u*. However, the integrls on the left side my fil to exist. Even if both integrls exist, the left side my be of the fonn Condition (II) in our version of the theorem voids these technicl complictions. 7. An lterntive interprettion of this model is tht ll cquisitions must be mde t the beginning of the period. Then c is the cquisition cost nd IT - P is penlty incurred for ech unit of demnd tht is not stisfied. This penlty could reflect n expected reduction in future demnd.

15 -14- REFERENCES [lj Bw, V. "Optiml Rules for Ordering Uncertin Prospects," Journl of Finncil Economics, 2 (1975), [2] Feller, W. Modern Probbility Theory nd Its Applictions, Vol. II (Wiley, 1971). [3] Fishburn, P. "Convex Stochstic Dominnce with Continuous Distribution Functions," Journl of Economic Theory, 7 (1974), [41 Hdr, J. nd Russell, W. "Rules for Ordering Uncertin Prospects," Americn Economic Review, 59 (1969), [5J Hdr, J. nd Russell, W. "Stochstic Dominnce nd Diversifiction," Journl of Economic Theory, 3 (1971), [6J Hnoch, G. nd Levy, H. "The Efficiency Anlysis of Choices Involving Risk," Review of Economic Studies, 36 (1969), [7J Tesftsion, L "Stochstic Dominnce nd the Mximiztion of Expected Utility," Review of Economic Studies, 43 (1976),