A Note on Beneficial Changes in Random Variables

Transcription

1 The Geneva Papers on Risk and Insurance Theory, 17: (1992) 1992 The Geneva Association A Note on eneficial Changes in Random Variables JOSEF HADAR TAE KUN SEO Department of Economics, Southern ethodist University, Dallas, TX Abstract This paper is an extension of Jack eyer's paper titled "eneficial Changes in Random Variables Under ultiple Sources of Risk and Their Comparative Statics" published in the June 1992 issue of this journal. The extension consists of showing which of the sufficient conditions in eyer's Theorems 1 and 3 are also necessary, and which are not. In addition, conditions are provided which are necessary and sufficient for general beneficial changes to imply a decrease in the demand for insurance. Key words: Demand for Insurance, Risk Aversion, First-Degree Stochastic Dominance Shifts, ean-preserving Shifts In the June 1992 issue of this journal, Jack eyer has analyzed the effects on the demand for insurance of beneficial changes in the distribution of losses, given that there are two sources of risk. eyer's results are presented in four theorems, each of which states sufficient conditions for the demand for insurance not to increase following a beneficial shift. Two of eyer's theorems allow for general FSD and PC shifts in distributions, while in the other two theorems the shifts are induced by particular transformations of the original random variable, and hence are of a more restrictive type. In this Note we extend eyer's work in the following way: (i) We show that in the case of general FSD shifts, eyer's condition given in his Theorem 1, Ra(z) -< 1, is also necessary, where RR denotes the relative risk aversion function. (ii) For the case of general PC shifts, we show that the three inequalities in eyer's Theorem are not necessary. Similarly, the two weaker inequalities provided by Dionne and Gollier [1992] in their Theorem 3' are not necessary either. (iii) For general PC shifts we provide a condition that is necessary and sufficient. As the reader will see, even with the aid of a lemma, the necessity proofs, unlike those of the sufficiency, are non-trivial and somewhat lengthy. In the derivation of our results we adopt the insurance model of eyer in which the decision maker chooses an optimal coverage so as to maximize expected utility of terminal wealth. The latter is defined by Z = Y + - X + g(x - P), where Z is terminal wealth, Y denotes the uninsurable asset, -< y -<, denotes the size of the insurable asset, X is the random loss, -< x -<, P is the total premium payment, < P <, and 8 denotes the co-insurance rate; that is, the fraction of that is insured. (We denote the random variables Z, X, Y by

2 172 JOSEF HADAR AND TAE KUN SEO capital Fetters, and their values by the corresponding lower case letters.) Like eyer, we assume that the decision maker is risk averse with a thrice differentiable utility function u. The derivatives of u are denoted by u', u", u", respectively. It is also assumed that the decision maker's maximization problem has an interior solution. efore presenting our results, we state a lemma which simplifies the proof of our theorems; the proof of the lemma, which is a bit lengthy, is given in the Appendix, z Lemma: Let the variables and parameters satisfy y >--O, > O, <-x <-, O < P <, <8< I. We define the following functions: +(x; y,, P,8) = u'(y + - x + 8(x - P))(x - P), qb(v) = u'(y + v)v, where v = - x + 8(x - P). (i) Assume that u' >- O, u" <- O. Then given any y,, P, 8, the function + is non-decreasing in x if and only if, for any y, the function + is non-decreasing in v. (ii) Assume that u' >- O, u" <- O, u" >- O. Then given any y,, P, 8, the function + is convex in x if and only if, for any y, the function d~ is concave in v. Proof." See the Appendix. The decision maker under consideration maximizes expected utility which can be written as U(8;F) = for u(y + - x + 8(x- P))dF(x)dG(y), where F is the loss distribution function of the insurable asset, and G is the loss distribution function of the other asset; the two assets are assumed to be independently distributed. The optimal value of 8 when the loss distribution is given by F ~ is denoted by 8i. Now let (8) denote the difference between the derivatives of expected utility with respect to g under the two distributions F and FI; formally, O(g) = Ua(g; F ~) - U~(8; F ) = of of - P)d[F'(x) - F (x)]dg(y). (1) u'(z)(x - P) = u'(z)[1 - RR(z)] - u"(z)(y + - P). For the We note that -x case in which the loss distribution undergoes a general "beneficial" FSD shift, that is, F~(x) >- F (x) for all x, eyer's condition for a decrease in 8 is RR(Z) --< 1

3 ENEFICIAL CHANGES IN RANDO VARIALES 173 for all z. This condition clearly implies (in conjunction with risk aversion) that the function u'(z)(x - P) is non-decreasing in x for all 8. Then it follows from the basic FSD theorem, Fishburn and Vickson [1978], that the inner integral in (1) above is non-positive, and so is (8), for all 8. Evaluating (8) at 8 = 8 gives (8) -- UdS; F ~) -, which implies, since U is concave in 8, that G -< 8. This is eyer's. Theorem 1: We now show that the condition RR -- 1 is also necessary. 3 1": Assume that u' ->, u" -<. Suppose that 8~ -< 8 whenever F FSD F j. Then the agent's relative risk aversion function RR must satisfy RR(z) --< 1 for all z. Theorem Proof." Assume that 81 -< 8 for any F FSD F 1, and any G,, P, where X ~and Y, i =, 1, are stochastically independent. Then we claim that for any y,, P, 8, the function ~ (as defined in the Lemma) is non-decreasing for all x. Suppose this is not the case. Then there exist Yo, o, Po, 8o such that +x(x; Yo, o, Po, 8o) < for x over a non-degenerate interval IC[, o]. Since +x(x; y, o, Po, 8o) is continuous in x and y, there exist intervals I~ and I2 Such that I~ = (sl, s2), <s~<s2'<o, I2 = ( Y o - e, Yo + e ), < y o e < y o + e <, and t~x(x; y, o, Po, 8) < for all x in 1~ and y in 12. (2) The first step in the proof is to show that there exist stochastically independent random variables X and yo such that expected utility is maximized at 8 = 8. Let y be uniformly distributed on I2, with G Odenoting its distribution function. We now define the following: f P f +(x; Y, o, Po, 8o)dxdG (Y) < O, f fo [3 = +(x; y, o, Po, ~o)dxdg (Y) > O, ~/ = [3po + (o O~ - Po). Consider the random variable X with density fo and distribution function F where -yfo(x) = [3, f o r -< x < Po, e~ 1, for P -< x -< o,, elsewhere.

4 174 JOSEF HADAR AND TAE KUN SEO Then it follows that F is continuous and strictly increasing in [, ]. We now have of,j *(x; y' o, Po, go)df (x)dg (Y) = -o~/o P ff o fo +~o +(x; y, o, Po, go)df (x)dg (Y) ~(x; y, o, Po, go)df (x)dg (Y) o (_~) + 1([3) O. (3) Hence, if = o, P = Po < o, and X and yo as defined above, then expected utility is maximized at g = g. Now choose any random variable X l such that X ~and yo are independent, and X ~ has the following distribution function F': (i) F 1 is continuous on [, o], (ii) Fl(x) > F (x), for sl < x < s2, and Fl(x) = F (x), elsewhere. It is clear that F FSD F t. Differentiating the expression for expected utility, and evaluating this derivative at g = go, we obtain fro ffo U~(go; F ~) = = tb(x; y, o, Po, go)d[fl(x) - F (x)]dg (Y), by (3) t" -- +(x; y, o, Po, go)dfl(x)dg (Y), ~szc / /,(x; d - FO(x oo,, d s1 t" = Y + ~ (. s2 / Y -- e d / Sl +x(x; Y' o, Po, go)[f (x) - Fl(x)]dG (Y) >, d where the last equation has been obtained from integration by parts of the inner integral, and the inequality follows from (2) and F (x) < FJ(x) on I~. The inequality implies that expected utility is maximized at some g = g~ > go when X ~and yo as defined above, and F FSD F ~. This contradicts the assumption of the theorem, and therefore the function +(x; y,, P, g) must be non-decreasing in x for all y,, P, g. Then the L e m m a implies that the function +(v) = u'(y + v)v is non-

5 ENEFICIAL CHANGES IN RANDO VARIALES 175 decreasing in v for any y->. This, in turn, is equivalent to the condition RR(Z) --< 1 for all z. [] We now turn to the case of mean-preserving contractions (PC) which e y e r addresses in his T h e o r e m 3. In that theorem, e y e r shows that the demand for insurance will not increase if the risk-averse decision maker's preferences are characterized by non-increasing absolute risk aversion, non-decreasing relative risk aversion, and RR(Z) for all z. As it turns out, none of these conditions is necessary; similarly, the conditions in Theorem 3' of Dionne and Gollier, which involve the measure of partial risk aversion, are also not necessary. Our next theorem states a condition which is necessary and sufficient for the demand for insurance not to increase following a PC in the loss distribution. Theorem 3": Assume that u' >- O, u" <<- O, u'" >- O. Then 81 <- 8ofor any F 1 PC F if and only if, for any y >- O, the function u'(y + v)v is concave in v. Proof." Sufficiency Rewriting equation (1), after replacing u'(z)(x - P) by +(x; y,, P, 8), we have (8) = of of +(x; y,, P, 8)d[Fl(x) - F (x)]dg(y). (1 ') According to the Lemma, the concavity of +(v) = u'(y + v)v implies that + is convex in x for any y,, P, 8. Then it follows from the basic SSD theorem with equal means, Fishburn and Vickson [1978], that the integral in (1') is non-negative for all 8. Evaluating the integral at 8 = 8, we have (8) = U~(8; F I) -<. Then it follows from the concavity of U that 81 -< 8. Necessity Assume that 8, -< 8 for any F 1 PC F and any, P, G such that X ~ and Y, i =, 1, are stochastically independent. Then we claim that for any y,, P, 8, the function + is convex in x. Suppose this is not the case. Then by the continuity of +xx (which is implied by the continuity of u") there exist Y, o, P, such that t~ x(x; Yo, o, Po, 8) < for x over some non-degenerate interval IC[, o]. Since +xx is continuous in x and y, there exist intervals I~ and 12 such that I1 = ( s 1, s2), < I2 = (Yo - si < s2 < o, s, Yo + s), < Yo - s < Yo + s <, and ~x (X, y,, P, 8) < for all x in Il and y in 12. (2')

6 176 JOSEF HADAR AND TAE KUN SEO Following the same procedure as in the proof of Theorem 1", we construct stochastically independent random variables X and y with distribution functions F and G, respectively, satisfying the following conditions (the explicit construction is omitted to save space): (i) F is continuous and strictly increasing on [, ], (ii) G o is distributed uniformly on I2, (iii) expected utility is maximized at = 8 when the random variables are X and y. We now choose a random variable X 1 with distribution function F I with the following properties: (i) X ~ and y are stochastically independent, (ii) F r is con- tinuous on [, ], (iii) -/e[fl(x)- F (x)]dx <, for sl < ~ < s2, and Sl" ] S 2 sfl [F~(x) - F (x)]dx =, (iv) Fl(x) = F (x), for x not in 11. The above properties imply that F 1 PC F. To simplify notation, we shall write tb(x; y) -- tb(x; y,, P, 8). Now we get the following equations: U~(~, F 1) = Of Of O j(x; y)dflc A)dG(y), : of of ljj(x;y)d[f l(x) - F (x)]dg (y), since 8 o is the maximizer for F. Integrating the inner integral by parts twice, and invoking (2'), we have J ('y + E ' ~ ~'X U~(~; F 1) = y-~j qsxx(x; Y) [Fl(t) - F (t)]dtdxdg (Y) >. sl The last inequality and the concavity of U imply that expected utility is maximized at some g = gl > g, even though F 1 PC F. Since this contradicts the assumption of the theorem, it follows that, for any y,, P, g, the function qs(x; y,, P, 8) is convex in x. y the Lemma this implies that for any y, the function +(v) = u'(y + v)v is concave in v. [] To relate this theorem to Theorem 3 of eyer, note that, given u" ->, our necessary and sufficient condition "for any y ->, +"(v) -< for all v" is equivalent to "+"(v) -< for all v and y =," which, in turn, is equivalent to RA(X)[RR(X ) -- 1] --< RR(X), for all x. (4) eyer's conditions, it will be recalled, are R A > (risk aversion), RA--<, (DARA), RR --< 1, and RR >-- (increasing relative risk aversion). These conditions are clearly sufficient (even without the DARA condition) since they imply that the left-hand side of inequality in (4) is non-positive, and the right-hand side is non-negative. To illustrate that none of the last three inequalities in eyer's sufficient con-

7 E N E F I C I A L C H A N G E S IN R A N D O V A R I A L E S 177 dition is necessary, we shall provide examples of three utility functions that satisfy the n e c e s s a r y and sufficient condition in (4), yet each one violates one of e y e r ' s inequalities. For the purpose of this illustration, we assume that = = 1/2, so that z = y + - x + g(x - P) < 1. For the utility functions given below, which also satisfy u'" >-, the following statements can be shown to hold: u~(z) = - e u2(z) = U3(Z ) = % 1 <c<2, (e~zz ~ d z, o OLZ Z 2, v i o l a t e s R R_< l; ~ >, [3 -> 3, a[3 -< 1/3, violates R'R --> ; c~ > 1, violates R' A _<. We should add that the w e a k e r sufficient inequalities provided by Dionne and Gollier in their T h e o r e m 3' are not n e c e s s a r y either; these are R~ >- and Rp -< 1, where Rp, the partial risk aversion function, is defined as Rp = RA(Zl + z2)z 2, with zl = y + - P, and z 2 = (8-1)(x - P). That the latter inequalities are not necessary follows from the observation that the utility function ul a b o v e violates Rp -< 1, and the example u2 violates R~, ->. The main conclusions of this N o t e are the following: (i) In the case of F S D shifts, e y e r ' s condition, R R _< 1 for all z = zj + z2, is also necessary, and so is the condition of Dionne and Gollier, Rp -< 1 for all z~ and z2, since the two inequalities are equivalent. (ii) For the case of PC shifts, the three inequalities of eyer, and the weaker inequalities of Dionne and Gollier, are not necessary. In lieu of these inequalities, we have provided a condition that is necessary and sufficient for the desired result. (iii) Our condition is valid for the case of a single source of uncertainty as well as for two s o u r c e s? While this has been shown in this Note for the case in which the two random variables are independently distributed, we can, in fact, assert that these results also extend to the case of a general interdependence in which the conditional distribution functions undergo either FSD or PC shifts; i.e., for Fl(xly) FSD F (x[y) for all y, and for F~(xly) PC F (x]y) for all y. If in equation (1), the term [F~(x) - F (x)] is replaced by [F(xly) - F (xly)], e y e r ' s result in T h e o r e m 1 will still hold, and if the same substitution is p e r f o r m e d in equation (1 '), it will not affect the result of our Theorem 3". Since independence is a special case of interdependence, there is no need to modify the necessity proofs. Finally, it should be pointed out that the results in this N o t e do not in general carry o v e r to insurance models with a deductible provision. Such models are quite a bit more complicated, and at this time, unambiguous results are available only for particular shifts in the loss distribution. See Schlesinger [1981] and E e c k h o u d t, Gollier and Schlesinger [1991].

8 178 JOSEF HADAR AND TAE KUN SEO Appendix Proof of Lemma From the definitions of the functions + and qb one can see that +x = qb'(v) + (P - )u"(y + v), and ~ ~ = (8-1)[d)"(v) + (P - )u'"(y + v)], (A.1) (A.2) wherev = - x + ~(x - P). Part (i) From (A.1) it follows that +x -> if+' -. To show the converse, assume that +x -> for all x and all y,, P, ~. Then we claim that, for any y, we must have +'(v) -> for all v. Assume this is not the case, that is, assume that there exist Yo and Vo such that +'(Vo) = u'(yo + Vo) + vou"(yo + Vo) <. (Note that vo >, since otherwise +'(vo) ->.) Now choose Xo, o, Po, ~o, Po < o, such that Vo = o - Xo + ~o(xo - Po). (A.3) We now define the function -q(; P) = - xo + ~o(xo - P). Then we have ~(o; o) < ~q(o; Po) = vo, and -q(; ) = ( - Xo)(1 - ~o) --+ co as --+ co. Therefore there exists ~, ~ > o such that ~(I; 1) = 1 - Xo + ~o(xo - 1) = vo. (A.4) Let P~ be any real number satisfying i > P~ > max{~ +'(Vo) 2u"(yo + Vo)' Po}. (Note that u"(yo + vo)<, since otherwise +'(vo)>.) Then we have < (P~ - u"(yo + vo) < -(1/2)+'(Vo). Furthermore, we see that "q(1; P > -q(~; = vo = "q(o; Po) > "q(o; Pl)- Hence there exists z such that o < 2 < 1, and ~](2; Pl) = Vo. Then at Yo, 2, PI, 8 o we have tbx = <b'(vo) + (P1-2)u"(Yo + Vo)< ~b'(vo) + (P1 - u"(yo + Vo)< ~b'(vo) - (1/2)+'(Vo) <, which is a contradiction. Hence for any y, we must have +'(v) -> for all v. Part (ii) From (A.2) we can see that q~xx -> if d)" -<. To prove the converse, assume that d)"(vo) > for some vo and Yo. (Note that v o >, since otherwise +"(vo) -<.) For any Po, choose xo, o, 8o, and 1 as in (A.3) and (A.4) above. Then define P2 to be any real number such that +"(Vo) ~ > P2 > max{1 2u'"(yo + Vo)' Po}.

9 ENEFICIAL CHANGES IN RANDO VARIALES 179 (Note that u'"(yo + v)>, since otherwise +"(v)-<.) Hence we have > (P2 - u'"(y + v) > - (1/2)qb"(v), and this implies -q(~; P2) > vi(1; = v = ~l(; P) > "q(; Pz). Therefore there exists 3 such that < ~ < I, and ~q(3; Pz) = v. Then at x, 3, P2, ~ we have *x~ = (g - 1)[+"(v) + (P2-3)u'(Y + v)] < (g - 1)[+"(v) + (P2 - I) u'"(y + v)] < (g - 1)[+"(v) - (1/2)+"(v)] <, which proves the converse. Notes I. This implies that the premium payment is unfair; that is, E[X] < P. Another aspect of the premium payment which is critical to this analysis is that the premium remains unchanged when the loss distribution shifts. This was also pointed out by Dionne and Gollier [1992], who argue that in certain circumstances this is not an unreasonable assumption. 2. This lemma is quite similar to the lemmas we proved in two earlier papers, Hadar and Seo [199, 1992]. However, since the form and properties of the objective function in this paper are different from those in the earlier papers, the proof of the present lemma is also different. 3. We denote our theorems by "double primes" to indicate that they relate to the similarly numbered theorem of eyer, and to distinguish them from the corresponding theorems in Dionne and Gollier which are denoted by a single prime. 4. For an insurance model with only one source of uncertainty, the sufficiency and necessity of our conditions has been shown in Hadar and Seo [1991]. References DIONNE, G. and C. GOLLIER [1992]: "Comparative Statics Under ultiple Sources of Risk with Applications to Insurance Demand," The Geneva Papers on Risk and Insurance Theory, 17 (June 1992), EECKHOUDT, L., C. GOLLIER and H. SCHLESINGER [1991]: "Increases in Risk and Deductible Insurance," Journal of Economic Theory, 55 (December 1991), FISHURN, P.C. and R.G. VICKSON [1978]: "Theoretical Foundations of Stochastic Dominance," in G.A. Whitmore and C. Findlay (Eds.), Stochastic Dominance, D.C. Heath and Company. HADAR, J. and T.K. SEO [199]: "The Effects of Shifts in a Return Distribution on Optimal Portfolios," International Economic" Review, 31 (August 199), HADAR, J. and T.K. SEO [1991]: "Changes in Risk and Insurance," Working Paper #9126, Southern ethodist University. HADAR+ J. and T.K. SEO [1992]: '+General Changes in Uncertainty," Southern Economic Journal, 58 (January 1992), EYER, J. [I 992]: "eneficial Changes in Random Variables Under ultiple Sources of Risk and Their Comparative Statics," The Geneva Papers on Risk and Insurance Theory, 17 (June 1992), SCHLESINGER+ H. [1981]: '+The Optimal Level of Deductibility in Insurance Contracts," The Journal of Risk and Insurance, 48 (September 1981 ),