A Recourse Certainty Equivalent for Under Uncertainty Decisions Aharon Ben-Tal Faculty of Industrial Engineering and Management Technion-Israel Institute of Technology Haifa, Israel and Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109-2117 Adi Ben-Israel Department of Mathematical Sciences The University of Delaware Newark, DE 19716 Technical Report 87-31 December 1987

A Recourse Certainty Equivalent for Decisions Under Uncertainty * Aharon Ben-Talt Adi Ben-Israel* 4 December 1987 Abstract We propose a new criterion for decision-making under uncertainty. The criterion is based on a certainty equivalent (CE) of a random variable Z, S((Z) = sup (z + Ezv(Z- z)} where v(.) is the decision maker's value-risk function. This CE is derived from considerations of stochastic optimization with recourse, and is called recourse certainty equivalent (RCE). We study (i) the properties of the RCE, (ii) the recoverability of v(.) from S,(.) (in terms of the rate of change in risk), (iii) comparison with the "classical CE" u-lEu(-) in expected utility (EU) theory, and Yaari's CE in his dual theory of choice under risk, (iv) relation to risk-aversion and (v) applications to models of production under price uncertainty, investment in risky and safe assets and insurance. In these models the RCE gives intuitively appealing answers for all risk-averse decision makers, without the pathologies inherent in the EU model, where the Arrow-Pratt indices are used to exclude certain risk averse utilities leading to implausible predictions. Key words: Stochastic optimization with recourse. Decision-making under uncertainty. Certainty equivalents. Risk aversion. Production under price uncertainty. Investment in risky and safe assets. Insurance.'Supported by the National Science Foundation Grant ECS-8604354 at the University of Delaware. t Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa, Israel, and Department of Industrial/Operations Engineering, The University of Michigan, Ann Arbor, MI 48109-2117. tDepartment of Mathematical Sciences, University of Delaware, Newark, DE 19716. 1

Abbreviations and special notation CE = certainty equivalent page 2 Ct = classical CE page 3 DM = decision maker, decision making D(T) = distribution functions with compact support page 25 D{xl,2, x23} page 22 EU = expected utility page 2 MPIR = mean preserving increase in risk page 22 MPSIR = mean preserving simple increase in risk page 22 Mu = u-mean CE page 3 RCE = recourse CE page 5 RV = random variable SP = stochastic program, stochastic programming SPwR = stochastic programming with recourse page 7 Sv = the RCE page 5 U = class of normalized utility functions page 10 [x,pl page 17 (a, p) page 18 Y! = Yaari's CE page 3 1 Introduction Decision making under uncertainty presupposes the ability to rank random variables, i.e. a complete order > on the space of RV's, with X > Y denotes X preferred to Y. If the preference order - is given in terms of a real valued function CE(.) on the space of RV's, X Y Y => CE(X) > CE(Y) for all RV'sX, Y we call CE(Z) a certainty equivalent (CE) of Z, corresponding to the preference >. In particular, a DM is indifferent between a RV Z and a constant1 z iff z = CE(Z). In the expected utility (EU) model, the DM is assumed to have a utility function u(.) which typically is strictly increasing (more is better) and concave. The DM's preference is then given by X > Y E Eu(X) > Eu(Y) (1.1) 1Regarded as a degenerate RV. 2

u- E u(X) > u-1 E u(Y), Accordingly we define the classical certainty equivalent (CCE) by C"(z) = u-Eu(Z) (1.2) Another CE, suggested by expected utility, is the u-mean CE Mu(.) defined implicitly by E u(Z - Mu(Z)) = 0 for all RV's Z (1.3) Mu does not induce the same order2 as the classical CE C,, i.e. the two CE's are not equivalent. Expected utility theory is "the major paradigm in decision making..., It has been used prescriptively in management science (especially decision analysis), predictively in finance and economics, descriptively by psychologists.... The EU model has consequently been the focus of much theoretical and empirical research... ", [32]. Empirical tests (e.g. [1], [2] and [17]) revealed systematic violations (also called "paradoxes") of the EU model axioms which were traced to the linearity in probabilities of the expected utility. Alternative theories of decisions under risky choices were proposed which avoid the said paradoxes, e.g. the prospect theory of Kahneman and Tversky [17], the local utility theory of Machina [22] and Yaari's dual theory [38]. In particular, Yaari's risk aversion is compatible with linearity in payments3. Given a monotone function f: [0,1] -- [0,1] with f(O) = 0 and f(1) = 1, Yaari's certainty equivalent Yf(.) is Yf(Z) = f(l-F(t))dt (1.4) where FZ is the cumulative distribution function of the RV Z. In particular, both Yaari's CE (1.4) and the u-mean CE (1.3) are shift additive in the sense that CE(Z + c) = CE(Z) + c for all RV Z and constant c (1.5) 2There are a utility tu() and RV's X, Y such that Eu(X) > Eu(Y) but Mu(X) < M,(Y). 3 "In studying the behavior of firms, linearity in payments may in fact be an appealing feature", [38, p. 96]. Indeed, a firm which divides the last dollar of its income as dividends, cannot be equated with the proverbial rich who value the marginal dollar at less than that. Yet both the firm, and the rich, can be risk averse. 3

In the EU model a risk-averse decision maker, i.e. one for whom a RV X is less desirable than a sure reward of EX, is characterized by a concave utility function u. The concavity of u also expresses the attitude towards wealth (decreasing marginal utility). Thus the DM's attitude towards wealth and his attitude towards risk are "forever bonded together", [38, p. 95]. Certain difficulties with the EU model are due to this fact. In Yaari's dual theory [38], and in the RCE model proposed here, the attitude towards wealth and the attitude towards risk are effectively separated. The above mentioned alternative theories, which lost much of the elegance, simplicity and tractability of EU, address the discrepancies between the EU model axidms and actual choices under risk as observed in psychological tests. In this paper we focus on the predictive usage of the EU model, which is dominant in economics and finance. Here too the EU model has a mixed record, giving valid predictions, as well as implausible ones. To be specific, we consider two models of economic behavior under uncertainty, a competitive firm under price uncertainty [31],[21] and investment in safe and in risky assets, [3],[9],[15]. For the competitive firm, the EU model yields the fundamental result, that optimal production under uncertainty is less than that under (comparable) certainty. It also gives a sensible condition (necessary and sufficient) for production to start, [31]. One would however expect that an increase in the selling price will result in increased production, but the EU model claims the opposite for certain risk-averse utilities. The dependence4 of the optimal output on the fixed cost is another source of difficulty. In the investment model, diversification is prescribed by the EU model under a natural condition. An impressive illustration of the predictive power of the EU model is the following result of Tobin, [36], which holds for all risk-averse utilities: "If a is the demand for risky investment when the return is a random variable X, then a/i + h is the demand when the return is the random variable (1 + h)X". In the investment model, when the rate of return of the safe asset increases, one would expect part of the investment capital to switch from the risky asset to the safe asset. However, the EU model allows the opposite behavior 4Called "paradoxical" in [21] and "seemingly paradoxical" in [31]. 4

for certain risk-averse utilities5. Also it was established empirically6 that the elasticity of demand for cash balance is > 17, but here again the EU model leaves open the possibility of elasticity < 1 for certain risk-averse utility functions. To avoid these pathologies (of the EU model), additional hypotheses are customarily imposed on the utility function u(.). These hypotheses are stated in terms of the Arrow-Pratt absolute risk-aversion index r(z) = - (z) (1.6) U'(z) and the Arrow-Pratt relative risk-aversion index R(z) = zr(z) (1.7) In the investment model, a typical postulate of Arrow [3] is: "r(.) is non-increasing" (1.8) The Arrow investment model [3] does not consider the effects of the rate of return p of the safe asset. This question was addressed in [15] where (p. 1068) it was shown that, even with r non-increasing, it is possible for p to increase and for investment money to shift from safe to risky assets! A sufficient condition to exclude this possibility is "r(-) is non-decreasing or R(.) < 1" (1.9) Both (1.8) and (1.9) are conditions on the 3rd derivative of the utility u, which may be difficult to check. Moreover, the only risk-averse utility function with r(.) both non-decreasing and non-increasing is the exponential utility. The CE advocated here is the recourse certainty equivalent (RCE) S,(Z):= sup {z + Ezv(Z- z)} (1.10) z where v(.) is the value-risk function of the DM, mapping the possible (yet to be realized) values z of a RV, into their values v(x) to the DM, at the time STo quote from [15]: "such optimal behavior appears to be unlikely". 6See references in [3, p. 103]. 7"Thus, the notion that security, in the particular form of cash balances, has a wealth elasticity of at least one, seems to be the only remaining explanation of the historical course of money holdings", [3, p. 104]. 5

of decision (before realization). We propose the RCE S, as a criterion for decision making (DM) under uncertainty i.e. for ranking RV's. The given value-risk function v induces a complete order > on RV's, X > Y: SV(X) > SV(Y) (1.11) in which case X is preferred over Y by a DM with a value-risk function v. Any new approach to DM under uncertainty should be measured against the standards of the classical expected utility (EU) approach. "What matters is whether the model offers higher predictive accuracy than competing models of similar complexity.... What counts is whether the theory... predicts behavior not used in the construction of the model", [32]. The advantages of the RCE approach, for predictive purposes, are demonstrated here by reexamining the above classical models of production and investment, and by studying a classical problem of optimal insurance coverage. In particular, the RCE approach (i) retains the successful predictions of the EU model (as listed above), (ii) does not require restrictive (thirdderivative) conditions on u (thus the conclusions are valid for the whole class of risk-averse utilities), and (iii) is mathematically tractable, comparable in simplicity and elegance to the EU model. We derive the RCE, using considerations of stochastic programming with recourse, in ~ 2. The main properties of the RCE are collected in Theorem 2.1. One such property is shift additivity, which holds for arbitrary value-risk functions v, S,(Z + c) = Sv(Z) + c for all RV Z and constant c. (1.12) Thus the RCE separates deterministic changes in wealth from the random variable which it evaluates. As mentioned above, shift additivity holds also in Yaari's dual theory of choice [38], and in the u-mean CE (1.3). In contrast, the classical CE (1.2) is shift additive only for linear and exponential utilities. For this reason, certain results (discussed in [4]), which in the EU model hold only for the exponential utility, hold in the RCE model for arbitrary utilities. Examples are the bridging of the gap between the buying and selling values of information, and the well known separation theorem in portfolio selection8. In Theorem 2.1 it is also shown that risk aversion in the sense of S,(Z) <EZ for allRV Z 8The proofs in [4] use only shift additivity. 6

is equivalent to9 v(x) x for all which, for normalized value-risk functions v (i.e. v(O) = 0 and v'(O) = 1) is a weaker requirement than the concavity of v(.). In Section 3 we discuss the recoverability of the value-risk function v(.) from the RCE Sv. It is shown that v(.) measures the rate of change in the RCE, when moving from a sure situation to a risky one (Theorem 3.1). Besides rendering a precise meaning of the value-risk function, this also suggests an empirical way to construct it from observed behavior. It is natural to ask, in the RCE model, what notion of risk-aversion corresponds to the concavity of v(.). The answer is given in Section 4, where we show that v(.) is concave iff the RCE Sv(-) exhibits risk-aversion in the sense of Rothschild and Stiglitz [28], i.e. aversion to mean preserving increase in risk. Certain functionals, associated with the RCE and useful in applications, are studied in Section 5. Section 6 is devoted to production under price uncertainty. The next two sections deal with investment in safe and in risky assets: The Arrow model [3] in Section 7, and a slight generalization in Section 8. An application of the RCE to the problem of optimal insurance coverage is discussed in Section 9. In the last section, ~ 10, we attempt to explain the success of the RCE theory in making plausible predictions with fewer assumptions than the EU theory. 2 The Recourse Certainty Equivalent A decision under uncertainty, as the name implies, is a decision made before the realization of the random variable in question. The consequences of this (apriori) decision depend on the (posteriori) realization. A rational decision maker must weigh these consequences according to their likelihood and value. This is the rationale for stochatic programming with recourse (SPwR). or two-stage stochastic programming, proposed in 1955 by G. Dantzig [10], see also Dantzig and Madansky [11] and Beale [5]. For illustration, consider the problem max {f(z): g(z) < Z } (2.1) 9We acknowledge the help of the referees in clarifying this point. 7

Here: z - decision variable, Z- budget, g(z) - the budget consumed by z, f(z) - the profit resulting from z. If Z is random then the apriori decision z may violate the (stochastic) constraint g(z) < Z (2.2) for some realizations of Z. In SPwR the optimal decision z' is determined by considering for each realization of Z a second stage decision y, consuming h(y) budget units and contributing v(y) to the profit. Thus z' is the optimal solution of max {f(z) + E (max {v(y): g(z) + h(y) < Z})} (2.3) The success of SPwR stems from the fact that it takes into account the trade off between greed (profit maximization) and caution (honoring the budget constraint). In those cases where z, y are scalars (e.g. levels of production), h(.) is monotone increasing ("more costs more") and v(.) is monotone increasing ("more is better"), we can rewrite (2.3) as max {f(z) + Ez (max {v(y): g(z) + y _ Z})} (2.4) where y, v correspond in (2.3) to h(y), v o h-1 respectively. If v is monotonely increasing then (2.4) is equivalent to: max {f(z) + E v(Z - g(z))} (2.5) in which y has been eliminated. The optimal value in (2.5) is the "SPwR value" of the SP (2.1). In this paper we use the SPwR paradigm to "evaluate" RV's. Our thesis is that assigning a value to a RV is in itself a decision problem. Thus, the "value" of a RV Z to a DM is the "most that he can make of it", i.e. value of Z = max {z: z < Z} (2.6) and we interpret (2.6) as the "SPwR value" which, by analogy with (2.5), is the RCE (1.10) sup {z + Ezv(Z- z)} z Z 8

Here v(x) is the current (before realization) value of the realized value z. We call v(.) the value-risk function. Remark 2.1 Another possible interpretation of (2.6) is max {z: u(z) < Ezu(Z)} (2.7) where u(.) is a utility function. For monotonely increasing u(.), the optimal value of (2.7) is then the classical certainty equivalent (1.2). The difficulties of modelling stochastic constraints by utility surrogates, such as the above or others, have been noted elsewhere, see [20]. In particular, the formulation (2.7) does not allow trade off between "greed" and "caution" as in (1.10). Remark 2.2 Note that Sv(Z) can be viewed as a temporal induced preference functional in the sense of Kreps and Porteus [19], see also [24], but unlike the multiperiod setting in the above references we have here a single period. However, in this single period there are two "periods", or time instants, induced by risk: The instant before the realization of the RV and the instant after. The time separation between these instants is irrelevant for our purposes. We list now several assumptions on v(.) which are reasonable, and useful for our purpose. Assumption 2.1 (vl) v(0) = 0 (v2) v(.) is strictly increasing (v3) v(z) < x for all x (v4) v(.) is strictly concave (v5) v is continuously differentiable Remark 2.3 By Assumption 2.1(vl),(v2) v(z) < 0 for x < 0, thus v(.) can also be viewed as a penalty function, penalizing violations of the constraint z<Z 9

Of particular interest is the following class of value-risk functions J=: v strictly increasing, strictly concave, continuously (2.8) U { v ~ differentiable, v(0) = 0, v'(0) = 1 which, for the purpose of comparison with the EU model, can be thought of as normalized utility functionsl~. The question of the attainment of the supremum in (1.10) is settled, for any v E U, in the following: Lemma 2.1 Let the RV Z have support [zmin, zma], with finite Zmin and zmax Then for any v E U the supremum in (1.10) is attained uniquely at some zs, Zmin I ZS < zmax, (2,9) which is the solution of E v'(Z - zs) =, (2.10) so that S,(Z) = zs + Ev(Z - zs) (2.11) Proof Note that Z - Zin > 0 with probability 1. Also v'(.) is decreasing since v is concave. Therefore E v'(Z - zmn) E v'(0) = 1 Similarly E v(Z - Zm2 ) > E v'() = 1 Since v' is continuous, the equationl1 Ev(Z-z) = 1 has a solution zs in [zmin, zm^], which is unique by the strict monotonicity of v'. Now zs is a stationary point of the function f(z)= z + Ev(Z- z) (2.12) which is concave since v E U, see (2.8). Therefore the supremum of (2.12) is attained at zs. 0 ~1For concave v the gradient inequality v(z) < v(O) + v'(O)z shows that all v E U satisfy (v3) of Assumption 2.1. 1 This equation is the necessary condition for maximum in (1.10). Differentiation "inside the expectation" is valid if e.g. v' is continuous and E v'() < oo, see [8, p. 99]. 10

Theorem 2.1 (Properties of the RCE) (a) Shift additivity. For any v: JR -, IR, any RV Z and any constant c Sv(Z + c) = S(Z)+ c (b) Consistency. Ifv satisfies (vl), (v3) then, for any constant c 12, SV(c) = c (2.13) (c) Subhomogeneity. lf v satisfies (vl) and (v4) then, for any RV Z, 1 Sv(AZ) is decreasing in A, A > 0 (d) Monotonicity. If v satisfies (v2) then, for any RV X and any nonnegative RV Y, S,(X + Y) > SV(X) (e) Risk aversion. v satisfies (v3) iff Sv(Z) < EZ for all RV's Z (2.14) (f) Concavity. Ifv E U then for any RV's Xo, X1 and 0 < a < 1, S,(aXc + (1 - a)Xo) > aS"(Xl) + (1 - a)Sv(Xo) (2.15) (g) 2nd order stochastic dominance. Let X, Y be RV's with compact supports. Then Sv(X) > S,(Y) for all v E U (2.16) if and only if E v(X) > Ev(Y) for allv E U (2.17) Proof. (a) For any function v: R — * IR, S,(Z+c) = sup z +Ev(Z+c- z)} z = c+ sup {( - c)+Ev(Z - ( - c))} =c + S(Z) 12Considered as a degenerate RV. 11

(b) For any constant c, Sv(c) = sup {z + v(c-z)} z < sup {z + (c - )} by(v3) C Conversely, S"(c) > {c + (c-c)} = c by (vl) (c) For any v:JR -, R and A > 0 define v\ by V,(z):= v(Ax), V: (2.18) Then S,,(Z) = Sv(AZ), for all RV Z, (2.19) as follows from, SV,(Z) = sup {z+ ^Ez(A(Z- )} 1 = suP{+Ezv(AZ-x)} (= Az) A ==,Sv(AZ) It therefore suffices to show that VA(z) is decreasing in A, A > 0 Indeed, let 0 < A1 < A2 By the concavity of v it follows, for all z, v(A2Z)- v(A1z) < v(Az) - V(0) A2 - A1 A and by (vl) V(A2Z) v(AlZ) A2 A1 12

(d) S(X + Y) = sup {z + Ev(X + Y- z)} z > sup {z + Ev(X - z)} by(v2) Z (e) If v satisfies (v3) then for any RV Z, S,(Z) = sup {z + Ev(Z- z)} < sup {z + E(Z - z)} = EZ Conversely, if for all RV's Z Sv(Z) < EZ then, for any RV Z and any constant z, z + Ev(Z-z) < EZ.Ev(Z - ) < E(Z - z).Ev(Z) < EZ proving (v3). (f) Let 0 < a < 1, and Xa = caX + (1 - a)Xo. Then by the concavity of v, for all zo, Zl, Ev(Xa - azi - (1 - a)zo) acEv(XI - z) + (1- a)Ev(Xo - zo) Adding az1 + (1 - a)zo to both sides, and supremizing jointly with respect to zt, ZO, we get Sv(Xo) > sup {a [zl + Ev(Xi - zI)] +(1 - a) [zo + Ev(Xo - zo)]} = aS(Xl) + (1- a)SV(Xo) (g) (2.17) = (2.16). Since each v E U is increasing, (2.17) implies z + Ev(X-z)> z+Ev(Y- z) Vz, andVvEU and (2.16) follows by taking suprema. (2.16) =: (2.17). Let zx, zy be points where the suprema defining S,(X) and Sv(Y) are attained, see Lemma 2.1. Then, for any v E U, Sv(X) = zx + Ev(X- zx) > zy + Ev(Y- zy), by (2.16) > z + Ev(Y- zx) 13

Therefore Ev(X - zx) > Ev(Y - zx) for all v E U, implying (2.17). 0 Remark 2.4 Theorem 2.1 lists properties which seem reasonable for any certainty equivalent. Property (b) is natural and requires no justification. The remaining properties will now be discussed one by one. (a) Note that shift additivity holds for all functions v, i.e. it is a generic property of the RCE. To explain shift additivity consider a decision-maker indifferent between a lottery Z and a sure amount S. If 1 Dollar is added to all the possible outcomes of the lottery, then an addition of 1 Dollar to S will keep the decision maker indifferent. (c) An important consequence (and the reason for the name "subhomogeneity") is S,(AZ) < ASv(Z), for all RV Z and A > 1 Thus indifference between the RV Z and its CE S,(Z) goes together with preference for AS,(Z) over the RV AZ, for A > 1. This is explained by E(AZ) = AEZ Var(AZ) = A2Var(Z) > AVar(Z) if A > 1 An interesting result, in view of (c) and (e), is that for v E U, lirm S,(AZ)= EZ (d) If v satisfies (vl) and (v2), and if the RV Z satisfies Z > Zmin with probability 1, then S (Z) > Zmin (2.20) This follows from part (d) by taking X = zmin (degenerate RV) and Y = Z - Zmin. (e) In the EU model, risk aversion is characterized by the concavity of the utility function. In the RCE model risk aversion is carried by the weaker property v(z) zX, Vz. We show in ~ 4 that concavity of v corresponds to strong risk aversion in the sense of Rothschild and Stiglitz, [28]. (f) The concavity of S,(.), for all u E U, expresses risk-aversion as aversion to variability. To gain insight consider the case of two independent RV's X1 and X0 with the same mean and variance. The mixed RV Xa = acX1+(1-ca)Xo 14

has the same mean, but a smaller variance. Concavity of S, means that the more centered RV Xa is preferred. The risk-aversion inequality (2.14) is implied by (f): Let Z, Z1, Z2,... be independent, identically distributed RV's. Then by (f), 1 a 1'= S,,( - ~ ) _ - sU(ZS ) l n i=l = Su(Z) As n - oo, (2.14) follows by the strong law of large numbers. In contrast, the classical CE u-'Eu(.) is not necessarily concave for all concave u. (g) In general, for a given u E U, E u(X) > Eu(Y) (2.21) does not imply Su(X) > S,(Y) (2.22) i.e. (2.21) and (2.22) may induce different orders on RV's, see [7]. Note however that in (2.16) and (2.17) the inequality holds for all u E U 13. This defines a partial order on RV's, the (2nd order) stochastic dominance, [16]. Example 2.1 (Exponential value-risk function) Here u(z) = 1 - e', V z, (2.23) and equation (2.10) becomes Ee-Z+z = 1, giving z$ = - log Ee-Z and the same value for the RCE S,(Z) = -log Ee-Z (2.24) A special feature of the exponential utility function (2.23) is that the classical CE (1.2) becomes u-lEu(Z)= - log Ee-Z showing that for the exponential function, the certainty equivalents (1.10) and (1.2) coincide. 13In which case Y is called riskier than X. 15

Example 2.2 (Quadratic value-risk function) Here14 u(z) = z - z2 z < (2.25) 2 and for a RV Z with zmx <. 1, EZ = A and variance a2, equation (2.10) gives zs = /s, and by (2.11) SU(Z) = - 2- a (2.26) Corollary 2.1 In both the exponential and quadratic value-risk functions n n S(E Zi) = E Su(Z,) (2.27) i=1 i=for independent RV's { Z1, Z2,..., Zn}15 0 Example 2.3 For the so-called hybrid model ([4],[33]) with exponential utility u and a normally distributed RV Z ~ N(IA, a2), 1 2 SU(z) = - 12 Example 2.4 (Piecewise linear value-risk function) Let () t t 0 O<a <1<3 (2.28) at, t>0 0< IfF is the cumulative distribution function of the RV Z, then the maximizing z in (1.10) is the l~-percentile of the distribution F of Z: z F 1-( and the RCE associated with (2.28) is Sv(Z) = P tdF(t) + a j tdF(t). 4The restriction xz 1 in (2.25) guarantees that t is increasing throughout its domain. S1The classical CE (1.2) is additive, for independent RV's, if u is exponential but not if u is quadratic. 16

The following result is stated for discrete RV's. Let X be a RV assuming finitely many values, Prob {X = x,} = pi (2.29) We denote X by X = [x,p], x= (xl, 2,....,Xn), p = (p,P2,...,i Pn) (2.30) The RCE of [x, p] is n Sv([x,p]) = max {z + v(xi - z)p} (2.31) i-=i We consider S,([x, p]) as a function of the arguments x and p. Theorem 2.2 (a) For any function v: R -- IR, and any x = (zl, 2,...,,n), the RCE S,([x,p]) is convex in p. (b) For v concave, and any probability vector p, the RCE Sv([x, p]) is concave in x. Proof. (a) A pointwise supremum of affine functions, see (2.31), is convex. (b) The supremand n + Pi V(X i- ) i=! is jointly concave in z and x. The supremum over z is concave in x, [27]. 0 We summarize, for a RV [x,p], the dependence on p and x, of the expected utility Eu(.) and 3 certainty equivalents. As a function As a function of p of x Eu, u concave linear concave u-WEu, u concave convex? Yf (1.4) convex linear SV __convex concave (if v is concave) 17

3 Recoverability and the Meaning of v(.) In ~ 2 we studied properties of S, induced by v. This section is devoted to the inverse problem, of recovering v from a given S,. The discussion is restricted to RCE's Sv defined by v E U. For these RCE's, we find v E U satisfying (1.10). Our results are stated in terms of an elementary RV X X f x, with probability p (3.1 = 0, with probability p = 1 - p ) which we denote (z, p). For this RV, Sv((, p)) = sup {z + pv(x - z) + pv(-z)} (3.2) z which we abbreviate So(z, p). Theorem 3.1 Ifv E U then v(z) = Sv(z,p) I[Po (3.3) Proof. For v E U the supremum in (3.2) is attained at z = z(x, p) satisfying the optimality condition (2.10) pv'(X - z(, p)) + pv'(-z(x, p)) = 1 (3.4) which, for p = 0 gives v'(-z(z, 0))= 1 and since v E U, Z(x, 0) =0 (3.5) Now, by the envelope theorem (appendix A), as, (pP) = v(x - z(, p)) - v(-z(, p)) (3.6) ap and (3.3) follows by substituting (3.5) and v(0) = 0 in (3.6). 0 To interpret this result consider an RCE maximizing individual who currently owns 0 $, and is offered the sum z with probability p. The resulting change in his RCE is A(X, p) = Sv(, p)- S(z,O) 18

E{(, p)} = px v(), p) pv()....... v(zx)...........-. t tangent to S(zx, ) at p = 0 slope = (z) = 8SV 0 1 Figure 3.1: Recovering v(z) from S,(z,p) and the rate of change is I. Theorem 3.1 says that this rate of change, for an infinitesimal change in risk (p -- 0) is precisely v(z), the value-risk function evaluated at z. Note that for a risk-neutral DM the added value A is E{(z,p)} = px. We illustrate this, for fixed z, in Fig. 3.1. The following theorem is a companion of Theorem 3.1. It says that the limiting rate of change ^(p is exactly the probability p of obtaining z. Theorem.2 Ifv E U then P = S,(zp) 3o (3.7) Proof. Substituting z = 0 in (3.4) gives v'(-z(O,p)) = 1 (3.8) By the envelope theorem (Appendix A) we get 8Sv(z p),p a: = pv'(z- Z,p)) 19

which, substituting z = 0 and (3.8) gives, as(o, p) zP) =p 9x p It is natural to ask, for any certainty equivalent CE(x, p), for the values CE(z, 0) the value risk function of CE CE(0,p) the probability risk function of CE We summarize theresults, in the following table, for the classical CE, the u-mean CE and the RCE. Certainty equivalent CE(x,p) -CE(x,0) ~CE(0,p) C, = u-xEu U(o)-(O) u()-u(o) P Mu u,(o) P S, v(z) p For the Yaari CE (1.4) Y(z.P) z[- f(p)], z > (3.9) We get: im Yf(x,p) - Yf(x,0) J f'(0), > 0 p.o+ p = xf'(l), O<0 and the two-sided derivatives li Y(x, p)- Yf(O, p) f)...,O+ X liUm Yf(x) - Yf(Op) - f() Remark 3.1 20

(a) The value-risk function for the EU model is thus precisely the normalized utility function UN(X):= (,(o) (UN(O) = 0, uN(0) = 1) This result suggests a new way to recover the utility function in the EU theory. (b) The probability-risk function (for a nonnegative RV) in Yaari's theory is thus precisely the function f(p) in terms of which Yf is uniquely defined. This is a new interpretation of f. (c) Note that the value-risk function corresponding to Yaari's CE is of the form (az, x_ 0 v() = z, >o 0 (3.10) with a = f'(O), j = f'(1). The convexity of f plus its normalization f(0)= 0, f() = 1 imply a< 1 < The function v in(3.10) is the source of the piecewise linear value-risk function in Exmaple 2.4. 4 Strong Risk Aversion In the EU model risk aversion is characterized by the concavity of the utility function, while in the RCE model it is equivalent to the weaker property (Theorem 2.1(e)) v(x) < z, V x. (4.1) It is natural to ask what corresponds, in the RCE model, to the concavity of v, i.e. vEU (4.2) The answer is given here in terms of a classical notion of risk-aversion due to Rothschild and Stiglitz [28], see also [12]. Definition 4.1 Let F, F be the c.d.f. of the RV's X, Y with support [a, b]. (a) If there is a c E [a, b] such that Fy(t) 2 FX(t), a t < c FX(t) > Fy(t), c < t < b 21

1 increasinf nreference P3 -/.______ 0 pi 1 Figure 4.1: Iso-EU and iso-mean lines in D{l, x2, x3} then Fy is said to differ from FX by a mean preserving simple increase in risk (MPSIR). (b) Fy is said to differ from FX by a mean preserving increase in risk (MPIR) if it differs from FX by a sequence of MPSIR's. Definition 4.2 An RCE maximizing DM with a value-risk function v exhibits strongrisk-aversion if FY differs from FX SV(Y) < SV(X) { by a MPIR This concept is best illustrated graphically as in [25]. Let Xl < X2 < 23 be fixed, and let D{zl,z2,23} denote the probability distributions over the values zx,x2,z3. Each p = (pi,p2,p3) E D{x(l,2,X3} can be represented by a point in the unit triangle in the (pt,p3)-plane as in Fig. 4.1, where P2 is determined by P2 = 1 - pi - p3. The dotted lines are loci of 22

distributions with same expectation (iso-mean lines) i.e. points (p, p3) such that pI xl + (1 - p - p3) 2 + p3 X3 = constant (4.3) As one moves in the unit triangle across the iso-mean lines, from the southeast (SE) corner to the northwest (NW) corner, the values of the mean (4.3) increase. Thus movement from the SE to the NW is in the preferred direction. The iso-mean lines are parallel with slope (i.e. Ap3/Apt) slope of iso-mean lines = - > 0 (4.4) X3-X 2 A movement along the iso-mean lines, in the NE direction corresponds to an MPIR as in Definition 4.1(b). Similarly, the solid lines in Fig. 4.1 represent iso expected utility curves which are parallel straight lines (due to the "linearity in probabilities" of the EU functional) with slope of iso-EU lines = u(z2) - (z > 0 (4.5) U(X3) - U(X2) The slope (4.5) is positive because of the monotonicity of u. For EU-maximizers strong risk-aversion corresponds to the iso-EU lines being steeper than the iso-mean lines, i.e. u(x2)-(X > - (4.6) U(Z3)- U(X2) X3 - X2 which holds for all z1 < z2 < X3 iff u is concave. Turning to the RCE functional, the iso-RCE curves are not straight lines (since the RCE functional is convex in the probabilities). For RCEmaximizers strong risk-aversion means that at each point (pl, p3), the slope of the iso-RCE curve (through that point) is steeper than the slope of the iso-mean line (given by (4.4), see Fig. 4.2. Let now p3 = p3(P1) be the representation of an iso-RCE curve. By the definition (1.10), p3 is solved from sup {z + pv(zl - )+ (1 -pi -pJ3)Cr3- Z)+ p3V(X3 - z)} = constant (4.7) z Then dIT z^ - zl strong risk-aversion < d3 > 2 - (4.8) dp; X3 - X2 23

1 3o 0 Pt 1 Figure 4.2: Iso-RCE curves and iso-mean lines in D{xi,,z2,3} Let the left side of (4.7) be written as a function s(p1,p2,P3) of the probabilities pi. Differentiating (4.7) with respect to pi we get 1s - 1 3 - (s2 - 3)p3 = 0, where si = - (4.9) api By the envelope theorem (Appendix A),Si = = u(X,- ) (4.10) where z' = z'(p, 3) is uniquely determined by (2.10) P1 V'( - Z*) + (1 - i - 3) V'(2 - z) + P3 V'(z3 - Z) = 1 Combining (4.9) and (4.10) we thus get, v(x22 - Z*) - v( - z') v(23 - Z) - V(Z2 - Z*) 24

and the Diamond-Stiglitz risk-aversion is, by (4.8) V(X - Z') - v(Z - Z') >2 - X (2 - z) - ( - " (4.11) v(x3 - ) - V(z2 - z) X3 - X2 (X3- Z) - (X2 - Z*) which holds for all xz < z2 < x3 iff v is concave. The above discussion can be generalized to a general RV X with distribution function F E D(T), where D(T):= {distribution functions with compact support T} (4.12) We note that the RCE S,(X) can be written as S,(X) = U(z,F)dF(x) (4.13) where U(, F) = z(F) + v(x - z(F)) (4.14) and the maximizing z(F) is obtained implicitly from (2.10) Jv'( - z(F))dF(z) = 1 Thus Sv, regarded as a function of F, S(X) = V(F) (4.15) is a generalized expected utility preference functional in the sense of Machina, [22]. By (4.13), U(z,F) is then the local utility function of Machina. We recall [22, Theorem 2] that for V(F) Frechet differentiable on D(T), the preference order induced by V is strongly risk-averse iff U(x, F) is concave in x for all F E D(T). Finally, by (4.14), the local utility U(., F) is concave for all F iff the risk-value function v(.) is concave. Remark 4.1 In the EU theory concavity of the utility u characterizes both risk aversion (CE(X) _ EX) and strong risk aversion, hence the two are equivalent in EU theory. In the RCE theory, risk aversion requires that v(x) < x while strong risk aversion requires the stronger property that v is concave. For non-EU theories this divergence between the two notions of riskaversion is not surprising. We note that in Yaari's dual theory Yf(X) EX requires f(t) < t, Vt whereas strong risk-aversion requires the convexity of f, [38, Theorem 2]16.'6The convexity of f, plus Yaari's normalization f(O) = 0, f(1) = 1 implies f(t) < t. 25

5 Functionals and Approximations Let Z = (Zi) be a RV in IR", with expectation p (vector) and covariance matrix S (if n = 1 then as above S = a2 ). For any vector y E IR, the inner product, n y.Z= EyiZi i=1 is a scalar RV. Given u E U, the corresponding RCE of y Z are taken as functionals in y, the RCE functional su(y):= Su(y Z), (5.1) We collect properties of the RCE functional in the following theorem, whose proof appears in Appendix B. Theorem 5.1 Let u E U be twice continuously differentiable, and let Z and s,(.) be as above. Then: (a) The functional s, is concave, and given by su(y) = zs(y) + Eu(y. Z - zs(y)) (5.2) where zs(y) is the unique solution z of Eu'(y Z - z)= (5.3) (b) Moreover, su(O) = o, VS(O)=,, V2s,(0) = u"(0)E (5.4) zs(O) = 0, Vs(O) = is (5.5) and if u is three times continuously differentiable, V2zs()'= U(O) I o0 (5.6) Theorem 5.1 can be used to obtain the following approximation of the functional s,(-) based on its Taylor expansion around y = 0. Corollary 5.1 If u is three times continuously differentiable then su(Y)= *.Y+TI u"(O)y. y+o( y II2) o (5.7) ~() ~. +~.(57 26

Remark 5.1 (a) In particular, for n = 1 and y = 1, it follows from (5.7) that the RCE has the following second-order approximation Su(Z) ft A + 2u"(O)a2 (5.8) 2 2 =, - (0)or2 where r(.) is the Arrow-Pratt risk-aversion index (1.6). (b) We also note that the approximation (5.7) is exact if (i) u is quadratic, or (ii) u is exponential, Z is normal. (c) By differentiating, and calculating the Taylor expansion of the classical CE (1.2) of y.Z, c(y) = u-lEu(y Z) (5.9) it follows that c,(y) is approximated by the right-hand side of (5.7). Thus we have cu(y) - s(y) = o(l1 y 112) (5.10) showing that the CE functionals (5.1) and (5.9) are close for small y. 6 Competitive Firm under Uncertainty The first application of the RCE is to the classical model studied by Sandmo [31], see also [21, ~5.2]. A firm sells its output q at a price P, which is a RV with a known distribution function and expected value EP =;. Let C(q) be the total cost of producing q, which consists of a fixed cost B and a variable cost c(q), C(q)= c(q)+ B The function c(.) is assumed normalized, increasing and strictly convex, c(O) = 0, c'(q) > 0, c"(q) > o Vq > 0 (6.1) The firm has a strictly concave utility function u, i.e. u'>0, u "<0 27

which is normalized so that u(0) = 0, u'(0) = 1. The objective is to maximize profit i(q) = qP - c(q)- B which is a RV. The classical CE (1.2) is used is Sandmo's analysis, so that the model studied is max u1 Eu(r(q)) q>O or equivalently, max Eu(r(q)) (6.2) q>O Here we analyze the same model using the RCE. For the sake of comparison with the EU model, we assume that the firm's value-risk function is u E U, i.e. is a utility. The objective of the firm is therefore max S,(ir(q)) (6.3) q>O Now maxSu(vr(q)) = max S(qP - c(q) - B) q>O q>O = max{Su(qP) - c(q)} - B by (1.12). We conclude: Proposition 6.1 The optimal production output q' is independent of the fixed cost B. D This result is in sharp contrast to the expected utility model (6.2) where the optimal output q depends on the fixed cost B: q increases [decreases] with B if the Arrow-Pratt index r(.) is an increasing [decreasing] function; the dependence is ambigious for utilities for which r(.) is not monotone. Note that the objective function in (6.3) is f(q) = su(q) - c(q) (6.4) where su(.) is the RCE functional (5.1). The function f is concave by Theorem 5.1 and the assumptions on c. Therefore, the optimal solution q' of (6.3) is positive if and only if f'(O) > 0. By (5.4) s'(0) = a, so q* > 0 if and only if p > c'(0) (6.5) 28

in agreement with the expected utility model (6.2). We assume from now on that A > c'(O) A central result in the theory of production under uncertainty is that, for the risk-averse firm (i.e. concave utility function), the optimal production under uncertainty is less than the corresponding optimal production qcer under certainty, that is for P a degenerate RV with value p. We will prove now that the same result holds for the model (6.3). First recall that the optimality condition for qcer is that marginal cost equals marginal revenue c'(qcer) = A (6.6) Proposition 6.2 q* < qcer for all u E U. Proof. The optimality condition for q' is 0 = f(q) = s (q) - c(q) (6.7) By Theorem 5.1 su(q) = z(q) + Eu(qP - z(q)) (6.8) where z(q) is a differentiable function, uniquely determined by the equation Eu'(qP- z(q)) = 1 (6.9) By the envelope theorem (Appendix A), su(q) = E{Pu'(qP- z(q))} (6.10) and the optimality condition (6.7) becomes EPu'(qP - z(q*)) = c'(q) (6.11) Multiplying (6.9) by j and subtracting from (6.11) we get E(P - A)u'(qP - z(q')) = c'(q') - i (6.12) or E{Zh(Z)} = c'(q) - 1 (6.13) where we denote Z:= P -, h(Z):= u'(q'Z + q' - z(q)) 29

Since u E U, it follows that h is positive and decreasing, and if can then be shown (see e.g. [21, p. 249]) that E{Zh(Z)} < h(O)EZ but EZ = E{P - it} = 0, and so, by (6.13), c'(q') < A and by using (6.6) c'(q') < c'(qcer) and since c' is increasing, q* < qcer 1 6.1 Effect of Profits Tax Suppose there is a proportional profits tax at rate 0 < t < 1, so that the profit after tax is r(q) = (1 - t)(qP - C(q)) As before, the firm seeks the optimal solution q* of (6.3), which here becomes max S,(r(q)) = max S,((1 - t)(qP - c(q) - B)) q>O q>o = max {S((1 - t)qP) - (1 - t)c(q)} - (1 - t)B q>0 which can be rewritten, using the RCE functional su(.) and omitting the constant (1 - t)B, max s((l - t)q)- (1- t)c(q) q>O Let the optimal solution be q = q(t). The optimality condition here is (1-t)((1 - t)q)- (1 - t)c'(q) = 0 giving the identity (in t),'((1 - t)q(t)) c'(q(t)) which, after differentiating (with respect to t), [(1 - t)(t - - (t)] s"((1 - t)q) = q'(t)c"(q) 30

and rearranging terms, gives'(t){c"(q)- (1 - t)((1 - t)q)} = -q(t)((1 - t)q) (6.14) The coefficient of q(t) is positive since c" > 0 and s,(.) is concave (Theorem 5.1(a)). The right-hand side of (6.14) is positive since q > 0, s" < 0. Therefore, by (6.14),'(t)> 0 and we proved: Proposition 6.3 A marginal increase in profit tax causes the firm to increase production. O In the classical expected utility case the effect of taxation depends on third-derivative assumptions; it can be predicted unambigiously17 only in one of the following cases: (a) r constant and R increasing, (b) r decreasing and R increasing, (c) r decreasing and R constant. In all these cases, the EU prediction agrees with our prediction in Proposition 6.3. 6.2 Effect of Price Increase If price were to increase from P to P + e (e fixed), then the corresponding optimal output q(e) is the solution of max { Su((P + c)q) - c(q)} = max { s,(q) + eq - c(q)} q>o Q:o The optimality condition for q(E) is s(q()) + e= c(q()) Differentiating with respect to e we get'(E)s"(4(e)) + 1 = ()c(()) hence b1 > (y) = cc"(q) s-"(q) > by the convexity of c and the concavity of su. We have so proved: 17See Katz's correction [18] to [31]. 31

Proposition 6.4 A marginal increase in selling price causes the firm to increase production. O This highly intuitive result is proved in the expected utility case only under the assumption that r(.) is non-increasing. 6.3 Effect of Ftitures Price Increase The RCE criterion was also applied to an extension [14] of Sandmo's model [31], dealing with a firm under price uncertainty and where a futures market exists for the firm's product. In [14, Proposition 5] it is shown that an increase in the current futures price causes a speculator or a hedger to increase sales, but not so for a partial hedger, unless constant absolute risk-aversion is assumed. This pathology is avoided in the RCE model, where the above three types of producers will all increase sales, [34]. 7 Investment in One Risky and in One Safe Assets: The Arrow Model Recall the classical model [3] of investment in a risky/safe pair of assets, concerning an individual with utility u E U and initial wealth A. The decision variable is the amount a to be invested in the risky asset, so that m = A - a is the amount invested in the safe asset (cash). The rate of return in the risky asset is a RV X. The final wealth of the individual is then Y= A - a + (1 + X)a= A + aX In [3] the model is analyzed via the maximal EU principle, so the optimal investment a* is the solution of max Eu(A + aX) (7.1) or equivalently max u-lEu(A+ aX) O<a<A Some of the important results in [3] are: (I1) a' > 0 if and only if EX > 0. 32

(12) a* increases with wealth (i.e. d. > 0) if the absolute risk aversion index r(.) is decreasing. (I3) The wealth elasticity of the demand for cash balance (investment in the safe asset) Em dm/dA -E:= m/A is at least one (7.2) EA m/A if the relative risk-aversion index u"(z) R(z) = -z u / is increasing (7.3) Arrow [3] postulated that reasonable utility functions should satisfy (7.3), since the empirical evidence for (7.2) is strong, see the references in [3, p. 103]. We analyze this investment problem using the RCE criterion, i.e. max Su(A + aX) (7.4) O<a<A where again we assume that the investor's value-risk function is u E U. The optimization problem (7.4) is, by (1.12), equivalent to max Su(aX)+ A O<a<A Let a* be the optimal solution. Using the RCE functional su(*), a* is in fact the solution of max su(a) (7.5) O<a<A Now, since su,() is concave a* > 0 if and only if s'(0) > 0 but by (5.4) 3'(0) = EX, and we recover the result (I1). Assuming (as in [3]) an inner optimal solution (diversification) 0 < a* < A (7.6) we conclude here, in contrast to (12), that da~ da* = ~ (7.7) dA 33

i.e. the optimal investment is independent of wealthl8. An immediate consequence of (7.7) is Em E >1 VuEU EA indeed Em A dm A d(A- a*) A da' A EA = mdA = A-a* dA A-a(' dA A-a' proving (7.2) for all risk-averse investors. Thus, in the RCE model, there is no need for the controversial postulate (7.3). The quadratic utility (2.33) 1 2 u(z)=z- z2 z<l1 2 violates both of Arrow's postulates (r decreasing, R increasing), and is consequently "banned" from the EU model. In the RCE model, on the other hand, a quadratic value-risk function is acceptable19. For this function the optimal investment a' is the optimal solution of max {3s(a) = /a - T2a2} O<a<A 2 where p = EX, a2 = Var(X). Therefore f /a2 if O< /a2<A a A if A/l2 > A showing that, for the full range of A values, a'(A) is non-decreasing, in agreement with (12). Moreover, if diversification is optimal, then Em A >1 EA A a2 Following [3] we consider the effects on optimal investment, of shifts in the RV X. Let h be the shift parameter, and assume that the shifted RV X(h) is a differentiable function of h, with X(O) = X. Examples are:'sHowever, initial wealth will in general determine when divesification will be optimal, i.e. when (7.6) will hold. 19Assuming 0 < X < t. 34

X(h) = X + h (additive shift), X(h) = (1 + h)X (multiplicative shift). For the shifted problem, the objective is max S,(aX(h)) (7.8) O<5<A Let a(h) be the optimal solution of (7.8), in particular a(0) = a*. Now S,(aX(h)) = ((a) + Eu(aX(h) - ((a)) (7.9) where ((a) is the unique solution of Eu'(aX(h) - ((a)) = 1 (7.10) The optimality condition for a(h) is d d{(a) + Eu(aX(h)- t(a))} = 0 which gives (using (7.10)) the following identities in h E{X(h)u'(a(h)X(h)- ((a(h)))} = 0 (7.11) E{u'(a(h)X(h)- ((a(h)))} 1 (7.12) Differentiating (7.11) with respect to h we get, denoting Z = aX(h) - t(a(h)), &(h)E{u"(Z}X(X-'(a(hW))} = E{X(h) [u'(Z) + a(h)X(h)u"(Z)]} (7.13) where a(h) = a(h) and similarly for X(h). The second order optimality condition for a(h), 2r Su(aX(h)) < 0, is here Eu"(Z)X(X -'(a(h))) > 0 hence, by (7.13), sign of a(h) = sign of E{X(h) [u'(Z) + aXu"(Z)]} exactly the same condition for the sign of a;(h) as in [3, p. 105, eq. (18)]. Therefore, the conclusions of the EU model are also valid for the RCE model. In particular. 35

Proposition 7.1 As a function of the shift parameter h, a(h) increases for additive shift, a(h) decreases for multiplicative shift. These results are illustrated for the quadratic value-risk function. There EX = Var(X) and h a(h) = a* + V( for an additive shift Var(X) a(h) = -— a* for a multiplicative shift (7.14) 1 +h In fact, (7.14) holds for arbitrary u E U, a result proved in [36] for the EU model. Proposition 7.2 If a* is the demand for the risky asset when the return is the RV X, then a(h) = a*/1 + h is the demand when the return is (1 + h)X. Proof The optimality condition for a* is E{u'(a'X - )X} = 0 (7.15) where ~* is the unique solution of Eu'(a'X- ) = 1 (7.16) The optimality conditions for a(h) are given by (7.11), (7.12). Now, for a(h)= = a, a(h)X(h) = a*X (7.17) and it follows, by comparing (7.12) with (7.16), that (a(h)) =' Substituting this in (7.11) and using (7.17), we see that (7.11) is equivalent to (7.16), and that a(h) = a*/1 + h indeed satisfies the optimality conditions (7.11), (7.12). C 36

8 Investment in a Risky/Safe Pair of Assets: An Extension We study the model discussed in [9] and [15], which is an extension of the model in Section 7. The analysis applies to a fixed time interval, say a year. An investor allocates a proportion 0 < k < 1 of his investment capital Wo to a risky asset, and proportion 1 - k of Wo to a safe asset where the total annual return per dollar invested is r > 1. The total annual return t per dollar invested in the risky asset, is a nonnegative RV. The investor's total annual return is kWot + (1 - k)Wor and for a utility function u, the optimal allocation k' is the solution of max Eu(kWot + (1 - k)Wor) (8.1) O<k<l The model of ~5, is a special case with Wo = A, t = 1 + X, kWo = a, r = 1. It is assumed in [9], [15] that u' > 0 and u" < 0, thus we assume without loss of generality that u E U. One of the main issues in [15] is the effect of an increase in the safe asset return r on the optimal allocation. The following are proved: (F1) An investor maximizing expected utility will diversify (invest a positive amount in each of the assets) if and only if Etu'(Wot) <'(Wt) < E(t) (8.2) Eu'(Wot) (F2) Given (8.2) he will increase the proportion invested in the safe asset when r increases if either (a) the absolute risk aversion index r(.) is non-decreasing, or (b) the relative risk aversion index R(.) is at most 1. The same model is now analyzed using the RCE approach, i.e. with the objective max S,(kWot + (1 - k)Wor) O<k<k where u denotes the investor's value risk function, assumed in U. Using (1.12) and the definition (5.1), the objective becomes max {(1 - k)Wor + su(Wok)}) (8.3) O<k< I 37

The following proposition, proved in Appendix C, gives the analogs of results (F1, (F2) in the RCE model. Proposition 8.1 (a) The RCE maximizing investor will diversify if and only if Etu'(Wot - r7) < r < E(t) (8.4) where 7 is the unique solution of Eu'(Wot - r) = 1 (8.5) (b) Given (8.4), he will increase the proportion invested in the safe asset when r increases, 0 Comparing part (b) with (F2), we see that plausible behavior (k' increases with r) holds in the RCE model for all u E U, but in the EU model only for a restricted class of utilities. We illustrate Proposition 8.1 in the case of the quadratic value-risk function (2.25). Here the optimal proportion invested in the risky asset is: 0 if r > E(t) k = {? if E(t) - WoC 2 < r < E(t) (8.6) 1 if E(t) - WOa2 > r where z2 is the variance of t. Thus k' is increasing in E(t), decreasing with a2 and decreasing with r (so that, the proportion 1- k* invested in the safe asset is increasing with safe asset return r). These are reasonable reactions of a risk-averse investor. We also see from (8.6) that k* decreases when the investment capital Wo increases. This result holds for arbitrary u 6 U, see the next proposition (proved in Appendix B). In the EU model, the effect of WO on k* depends on the relative risk-aversion index, see [9]. Proposition 8.2 If the investment capital increases, then the RCE-maximizing investor will increase the proportion invested in the safe asset. 0. Following the analysis in [3] and ~ 7, we consider now the elasticity of cashbalance (with respect to Wo). Here the cash balance (the amount invested in the safe asset) is m = (1 - k)Wo and the elasticity in question is Em. 38

Proposition 8.3 For every RCE-maximizing investor with u E U, Em>l >1 EWo - Proof. Em dm/dWo 1- k(Wo)- Wo do EWo m/Wo 1 - k-(Wo) hence Em dk EW > 1 if and only if (Wo) (8.7) EWo- dWo and the proof is completed by Proposition 8.2. 0 The equivalence in (8.7) shows that the empirically observed fact that Em/EWo > 1 can be explained only by the result established in Proposition 8.2 that dk*/dWo _ 0, a result which is not necessarily true for many utilities in the EU analysis. 9 Optimal Insurance Coverage Insurance models with two states of nature were studied in [13], [21] and the references therein. In this section we solve an insurance model with n states of nature, and give an explicit formula for the optimal allocation of the insurance budget, thus illustrating the analytic power of the RCE theory. 9.1 Description of the Model The elements of the model are: n states of nature p = (p,...,pn) their probabilities q, = premium for 1$ coverage in state i, qi > 0 B = insurance budget qi = qi/ E=l qj = normalized premium B = B/ jalt = normalized budget xi = income in state i x = (Zi,... zn) the decision variable 39

The budget constraint is n Eqi i = B (9.1) i=i We allow negative values for some zi's, i.e. we allow a person to "insure" and "gamble" at the same time, e.g. [13, p. 627]. For the RCE maximizer with value-risk function v, the optimal value of the insurance plan is r = max{ S([x,p]):. qi i = B} i-= n = %max max{z + Piv(zi-z)} (9.2) X, iiB i-=1 = S([x, p]) where x' = (xz) is the optimal insurance coverage. 9.2 The Solution Theorem 9.1 The optimal insurance coverage is qi=)-f;<^( ) (9.3) z? = B + E( qi ( J ) (9.3) Pi j=l Pi where; = (v')-! (9.4) Moreover, the optimal value of the insurance plan is r = s- E q (q) + E p iv(O(q)) (9.5) Pi Pi Proof. The problem (9.2) is maximizing a concave function subject to linear constraints. Since the Kuhn-Tucker conditions are necessary and sufficient P = min max L(x,z,A) (9.6) A X where L is the Lagrangian L(x, z, A) = z + E pi v(zi - z) + A(B - E qi, i) (9.7) 40

The optimal x*, z, A* satisfy 8L a = 1 - Pj'( - z*)=0 (9.8) 89L = piv'(z - z*)-Aqi= O, (i =,...,n) (9.9) OL = B- qs'x =O (9.10) From (9.9) and (9.8) we get 1 Eqi and consequently v'(z - ) = Pi Since v' is monotone decreasing (v is strictly concave) we write, using (9.4) xz - Z = (h), (i = 1,..., n) (9.11) Pi Multiplying (9.11) by qi and summing we get qi(x, - Z*) = Eqi(p) Pi..B-z' = E (i-. ) Pi *'* = B - Fqi )) (9.12) Pi which is compared with (9.11) to give (9.3). Finally, r = z' + 2piv(X - z') and (9.5) follows by (9.11) and (9.12). 0 In the above model, the price of insurance is actuarially fair if qi (i= 1,...,n) i.e. if the normalized premiums agree with the probabilities. For actuarially fair premiums we get from (9.3), using that v'(0) = 1 implies 4(1) = 0, x = B (i=,...,n) i.e. the individual is indifferent between the occurrence of states i = 1,..., n. 41

9.3 Special Case: Two States of Nature We translate the results of Theorem 9.1 to the special case of two states, as given in [13], [21, ~3]. Consider insurance against a single disaster. Specifically, let there be two states of nature: State Disaster Probability 1 occurs p 2 does not occur 1 p The final wealth is a RV f(.\ y + 3 with probability p (State 1) Xs) \W - W s with probability 1 - p (State 2) where W initial wealth a insurance coverage tr premium y income in disaster state In [21, ~3] this model is treated using the EU model max Eu(X(s)) S obtaining first order optimality conditions, comparative statics, and, in the case of exponential utility uA(x) =,(1- eA), (9.14) the explicit solution a=~s* 1=.... log(i (9.15) Ir + 1 A(r + 1) To apply Theorem 9.1 here we write the incomes in the two states and their probabilities x1 =y+s, pi =p 32 =W- rs, 2 = 1-p 42

We define the normalized premiums q,. (9.16) 1 q2:= 1-qI=l (9.17) The insurance budget (9.1) is implicit in this model. The budget B can be computed by qzt + q2z2 = ql (y + s) + q2 (W - s) = q + q2 W +s(q - q2r) but ql - q2r = 0 by (9.16) and (9.17), and therefore the budget is B = q, y + q2 W (9.18) Now, from (9.3), 2; = B+ (1 —gql) )- q2 ) = Cy+ w(+e [ plP)- P2) and therefore the optimal coverage is a 1 = xZ1-Y q=,[W -y +.)-_.)] _ ~+ 1 1+ [ + T (1 + )-)(1 - p) (9.19) Note that in this two state model, actuarlially fair insurance means 7r =, in which case s* = 1 (W - y) For the utility u\ of (9.14), we get by (9.4) 1logt =whih (u9),(t)h = orla. o which, substituted in (9.19), gives the formula (9.15) of 9*. 43

9.4 Related Work The RCE criterion was applied in [34] for studying the existence of optimal insurance contracts. Two fundamental results of Arrow [3] concerning * the optimality of 100% coverage (above deductibles) for a risk-averse buyer of insurance, and * the Pareto optimality of coinsurance for risk-averse insurer and buyer of insurance, were shown to hold as well in the RCE model. 10 Why Does the RCE Work? The models discussed above (~~ 6-9), give sufficient data for comparing the predictive powers of the RCE theory and the EU theory. We saw that the plausible predictions of EU are shared by RCE, and that the RCE criterion is a simpler and a more powerful analytical tool, e.g. ~ 9.2 where it gives an explicit solution for all risk-averse DM's, while in general the EU model can only provide comparative statics. Also the RCE predictions hold for all risk-averse DM's, while in the EU model risk-aversion does not suffice and, in order to avoid implausible predictions, restrictions (occassionally severe) must be imposed on the DM's subjective preference. The simplicity of the RCE criterion can be explained at the technical level. Shift additivity makes risky choices independent of constant factors (fixed costs, initial wealth), and by using the envelope theorem, comparative statics are free of certain ungainly derivatives. Such conveniences are in general unavailable to the EU maximizer. This however is not the whole story. The main advantage of the RCE theory, at the fundamental level of modelling choice under risk, is that its risk aversion is of the "right kind" from the start, without a need for qualifiers such as the Arrow-Pratt indices. Indeed, in the EU theory, behavior under uncertainty is analyzed in terms of the Arrow-Pratt indices r(.) and R(.). The typical postulates are (Al) r(w) = -OM is a non-increasing function of w (A2) R(w) = - wj, is a non-decreasing function of w 44

The economic literature contains several alternative formulations. In particular ([12, pp. 352-354] and [23, pp. 20-21]) (Al) is equivalent to (B1) If u(wl + cl) = Eu(wu + X) and u(w2 + c2) = Eu(w2 + X) for wl < w2, then cl < c2 and (A2) is equivalent to (B2) If u(wacl) = Eu(wtX) and u(w2c2) = E u(w2X) for uw < w2, then C1 > C2 Properties (B1), (B2) can be expressed directly in terms of the classical CE CU(X) = u-E u(X) Indeed, (B1) is equivalent to (Cl) Cu(X + w) - w is a non-decreasing function of w and (B2) is equivalent to (C2) I Cu(wX) is a non-increasing function of w Consider now the RCE Sv(X). The properties corresponding to (C1), (C2) are (S1) Sv(X + w) - w is a non-decreasing function of w (S2) S,,(wX) is a non-increasing function of w Now (S1) holds trivially, for any function v: ]R - R, by the shift additivity of the RCE, Theorem 2.1(a). In fact, Sv(X+w)-w is S,(X), a constant in w. Moreover, (S2) is the subhomogeneity property, proved in Theorem 2.1(c) for all v E U. Therefore, in the RCE theory the properties (S1) and (S2) hold for all value-risk function v E U, i.e. for all strongly risk-averse DM's. In the EU theory, risk-aversion coincides with strong risk-aversion (see ~ 4), but the properties (Al) and (A2) (which correspond to (S1) and (S2)) hold only for a restricted class of utilities. 45

References [1] M. Allais, "Le Comportement de 1'Homme Rational devant le Risque. Critique des Postulates et Axiomes de l'Ecole Americaine", Econometrica 21(1953), 503-546. [2] M. Allais and 0. Hagen (Editors), Expected Utility Hypotheses and the Allais Paradox, D. Reidel, Dordrecht, 1979. [3] K.J. Arrow, Essays on the Theory of Risk-Bearing, Markham, Chicago, 1971. [4] G. Bamberg and K. Spremann, "Implications of Constant Risk Aversion", Zeit. f.. Oper. Res. 25(1981), 205-224. [5] E.M. Beale, "On Minimizing a Convex Function subject to Linear Inequalities", J. Royal Statist. Soc. 17B(1955), 173-184. [6] A. Ben-Tal, "The Entropic Penalty Approach to Stochastic Programming", Math. Oper. Res. 10(1985), 263-279. [7] A. Ben-Tal and M. Teboulle, "Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming", Management Sci. 32(1986), 1445-1466. [8] N. Bourbaki, Elements de Mathematique. Fonctions d'une Variable Reelle, Vol. IX, Livre IV, Herman & Cie, Paris, 1958. [9] D. Cass and J.E. Stiglitz, "Risk Aversion and Wealth Effects on Portfolios with Many Assets", Rev. Econ. Stud. 39(1972), 331-354. [10] G.B. Dantzig, "Linear Programming under Uncertainty", Manag. Sci. 1(1955), 197-206. [11] G.B. Dantzig and A. Madansky, "On the Solution of Two-Stage Linear Programs under Uncertainty", Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 165-176, University of California, Berkeley, 1961. [12] P.A. Diamond and J.E. Stiglitz, "Increases in Risk and Risk Aversion", J. Econ. Th. 8(1974), 337-360. [13] I. Ehrlich and G.S. Becker, "Market Insurance, Self-Insurance, and SelfProtection", J. Polit. Econ. 80(1972), 623-648. 46

[14] G. Feder, R. Just and A. Schmitz, "Future Markets and the Theory of the Firm under Price Uncertainty", Quart. J. Econ. 86(1978), 317-328. [15] P.C. Fishburn and R.B. Porter, "Optimal Portfolios with One Safe and One Risky Asset; Effects of Change in rate of Return and Risk", Management Sci. 22(1976), 1064-1073. [161 J. Hadar and W. Russell, "Rules for Ordering Uncertain Prospects", Amer. Econ. Rev. 59(1969), 25-34. [17] D. Kahneman and A. Tversky, "Prospect Theory: An Analysis of Decision under Risk", Econometrics 47(1979), 263-291. [18] E. Katz, "Relative Risk Aversion in Comparative Statics", Amer. Econ. Rev. 73(1983), 452-453. [19] D.M. Kreps and E.L. Porteus, "Temporal von Neumann-Morgenstern and Induced Preferences", J. Econ. Th. 20(1979), 81-109. [20] I.H. LaValle, "Response to "Use of Sample Information in Stochastic Recourse and Chance-Constrained Programming Models": On the'Bayesability' of CCP's", Manag. Sci. 33(1987), 1224-1228. [21] S.A. Lippman and J.J. McCall, "The Economics of Uncertainty: Selected Topics and Probabilistic Methods", Chapter 6 in Volume 1 of Handbook of Mathematical Economics (K.J. Arrow and M.D. Intriligator, Editors), North-Holland, Amsterdam, 1981. [22] M.J. Machina, "'Expected Utility' Analysis without the Independence Axiom", Econometrica 50(1982), 277-323. [23] M.J. Machina, "The Economic Theory of Individual Behavior Towards Risk: Theory, Evidence and New Directions", Economic Series Tech. Report No. 433 (October 1973), Institute of Mathematical studies in the Social Sciences, Stanford University. [24] M.J. Machina, "Temporal Risk and the Nature of Induced Preferences", J. Econ. Th. 33(1984), 199-231. [25] M.J. Machina, "Choice under Uncertainty: Problems Solved and Unsolved", Econ. Perspectives 1(1987), 121-154. [26] J.W. Pratt, "Risk Aversion in the Small and in the Large", Econometrica 32(1964), 122-136. 47

[27] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. [28] M. Rothschild and J.E. Stiglitz, "Increasing Risk: I. A Definition", J. Econ. Th. 2(1970), 225-243. [29] M. Rothschild and J.E. Stiglitz, "Increasing Risk: II. Its Economic Consequences", J. Econ. Th. 3(1971), 66-84. [30] P.A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, Mass. 1947. [31] A. Sandmo, "On the Theory of the Competitive Firm under Price Uncertainty", Amer. Econ. Rev. 61(1971), 65-73. [32] P.J.H. Schoemaker, "The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations", J. Econ. Literature 20(1982), 529563. [33] J.K. Sengupta, Decision Models in Stochastic Programming, NorthHolland, Amsterdam, 1982. [34] S. Sharabany, Optimized Certainty Equivalent Criterion and its Applications to Economics under Uncertainty, M.Sc. thesis in Economics, Technion-Israel Institute of Technology, Haifa, Israel, June 1987 (Hebrew). [35] E. Silberberg, The Structure of Economics. A Mathematical Analysis, McGraw-Hill, New York, 1978. [36] J. Tobin, "Liquidity Preference as Behavior towards Risk", Rev. Econ. Stud. 25(1958), 65-86. [37] J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1947. [38] M.E. Yaari, "The Dual Theory of Choice under Risk", Econometrica 55(1987), 95-115. Appendix A. The Envelope Theorem This result is used repeatedly in this paper. For convenience we cite an elementary version here. See [35] and [30] for details and examples. 48

Theorem A.1 (The Envelope Theorem). Consider the unconstrained maximization maximize, y = f(z, q) Let z*(q) be the maximizer, for given q, and let y' = f(z'(q), q)= (q) Then e)n= af(z'(q), q) Appendix B. Proof of Theorem 5.1 (a) By (5.1) and (1.10), su(.) is the pointwise supremum of concave functionals, hence concave. The rest of (a) is proved as in Lemma 2.1. (b) For y = 0, (5.3) gives Eu'(-zs(O)) = 1 or u'(-zs(O)) = 1, proving that zs(O) = 0. From (5.2) it follows then that Su(0) = 0. Differentiating (5.3) with respect to y gives Eu"(y Z - zs(y))(Z - Vzs(y)) = 0 which at y = 0 becomes u"(O)(EZ - Vzs(O)) = 0 proving that Vzs(O) = Ai. Then, by differentiating (5.2) at y = 0 we get V^s(O) = 0. The expressions for V2zs(O) and V2s,(0) follow similarly by differentiating (5.3) and (5.2) twice at y = 0. O Appendix C. Results from Section 8 Proof of Proposition 8.1. (a) The objective function in (8.3) h(k) = (1 - k)Wor + s,(Wok) 49

is concave, by Theorem 5.1(a). Hence, the optimal solution x' is an inner solution, i.e. 0 < k* < 1 if and only if h'() > 0 and h'(1) <0 (C.1) Now h'(k) = -Wor + Wos'(Wok) (C.2) which becomes, upon substitution of the computed expression for as(.), h'(k) = -Wor + WoEtu'(Wokt - 7r(Wok)) (C.3) where 77(q) is the unique solution of Eu'(qt - )) = 1 (C.4) Therefore h'(O) = -Wor + WoE(t) h'(1) = -Wor + WoEtu'(Wo - 7(Wo)) and (C.1) is equivalent to (8.4). (b) Let k(r) be the optimal solution of (8.3) for given r, i.e. h'(k(r)) = 0, or using (C.3), -r + E{tu'(Wok(r)t - 7(Wok(r))} 0 Differentiating this identity (in r) with respect to r, we obtain -1 + E{tWo(k'(r)t - k'(r))7'(Wok(r))u"} = 0 or k'(r)WoEt(t - /')" = 1 (C.5) Now, the second order condition for the maximality of k(r) is 0 > h"(k) = WoE{tWo(t - r7')u"} (C.6) Therefore, k'(r) is multiplied in (C.5) by a negative number, and consequently k'(r) < 0 proving that k(r) [1 - k(r)], the proportion invested in the risky [safe] asset, is a decreasing [increasing] function of r, the safe asset return. o 50

Proof of Proposition 8.2. Let k = k(Wo) be the optimal solution of (8.3), i.e. h'(k(Wo)) = 0, or using (C.3) - r + E{tu'(Wok(Wo)t - r(Wok(Wo))} = 0 (C.7) Differentiating this identity (in Wo) we get Et [k(Wo) + Wok'(Wo)] [t - 7'(Wok(Wo))] u" = 0 or k'WoEt(t - t')u" = -EtkU" (C.8) By the second order optimality condition (C.6) it follows that, in (C.8), k' is multiplied by a negative number. Since the right hand side of (C.8) is positive (t, k > 0, u" < 0), it follows that k'(Wo)<O0 51

3