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Research Report CCS 556

OPTIMIZED CERTAINTY EQUIVALENTS FOR DECISIONS UNDER UNCERTAINTY

11y

A. Den-Tal* A. Uen-ls r .ael1<>'<

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JAN 2 '3 't9S7 ' ' ,.

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Research Report CCS S5ü

OPTIMIZED CERTAINTY EQUIVALENTS FOR DECISIONS UNDER UNCERTAINTY

By

A. Ben-Tal* A. Ben-Israel**

December 1986

* 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 A8109-2117.

^Department of Mathematical Sciences, University of Delaware, Newark, DE 19716

This research was supported in part by National Science Foundation Grant ECS-8604354 at the University of Delaware, and by ONR Contract N0001A-82-K-0295, and National Science Foundation Grants SES-8A0813A and SES-8520806 with the Center for Cybernetic Studies at the University of Texas at Austin. Reproduction in whole or in part is permitted for any purpose of the United States Government.

CENTER FOR CYBERNETIC STUDIES

A. Charnes, Director College of Business Administration, 5.202

The University of Texas at Austin Austin, Texas 78712-1170

(512) A71-1821

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ABSTRACT

-■' A new approach Is proposed for some models of decision-making

under uncertainty, using optimized certainty-equivalents induced by

expected-utility. Applications to production, Investment and

inventory models demonstrate the advantages of the new approach.

n

KEY WORDS:

Expected Utility

Certainty Equivalent

Decision Making Under Uncertainty

Risk Aversion

Accession For

ifis GK\M id DTIC T/,3 H Unannou'- ".ri U Justific:.-..^ n

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1. INTRODUCTION

y Expected utility theory is rtthe major paradigm in decision

making.... It has been used prescriptively in management science (especially decision analysis), prediclively in finance and economics, descriptively by psychologists ... The expected utility (EU) model has consequently been the focus of much theoretical and empirical research ... ^ 115).

In spite of its power, elegance and success, considerable criticism has been directed at the EU theory, specially when used descriptively to model the decision processes underlying risky choice. In this context, evidence refuting the validity of the EU model axioms is relevant, e.g. (1), [ 11 ].

mmmMmmsmmmmmmmmmämsmmsmmm

A second usage, dominant in economics and finance, "is to view the EU model as predictive ... . What matters is whether the model oilers 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", [15]. In this sense the EU model has been, to a large extent, successful.

To be specific, we consider two models of economic behavior under uncertainty: A competitive firm under price uncertainty [14],[0] and investment in safe and in risky assets, [2],[()],[7].

For the competitive firm, the EU model yields the fundamental result, thai optimal production under uncertainty is loss Ihan that tinder (comparable) rertiüiily. Il also gives a sensible condition (necessary and siillicieiil) lor production to start. [11].

In the investment model, diversification is prescribed by the EU model under a natural condition. An interesting example of the predictive power of the EU model is the following result due to Tobin, [17]:

"If ö is the demand for risky investment when the return is a random variable A", then a/l+h is the demand when the return is the random variable (I+M-V".

The above conclusions hold for any risk-averse decision maker, i.e. one for whom a random variable .V is less desirable than a sure reward of /i.V. A fortunate aspect of the EU model is that risk-aversion is equivalent to the concavity of the utility function. Thus a fundamental altitude towards risk is characterized by a simple mathematical property.

Still, as a predictive tool the EU model is not without faults. It produces unnatural results, and it claims optimality for implausible behavior, even in the above production and investment models.

For a competitive firm under price uncertainty, [14], one would expect that an increase in the selling price will increase production. However, the EU

model claims the opposite for certain risk-averse utilities. The dependence ' of

msummmsM&mmm :<<s>w

Certainty equivalents Ben-Tal and Ben-Israel

the optimal output on the fixed cost is another source of difficulty.

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 does not preclude the oppo-

site behavior for certain utilities 2 . Also it was established empirically that the

elasticity of demand for cash balance is > 1 l , but here again the EU model leaves open the possiblity of elasticity < I for certain risk-averse utility functions.

To avoid these pathologies (of the ET model), additional hypotheses are customarily imposed on the utility function U. These hypotheses are stated in terms of the (absolute) Arrow-Pratt risk-aversion index

r(-) = - ^n '■•(--)

and the (relative) Arrow-Pratt risk-aversion index,

R'z)= z r{z)

In the investment model, a typical postulate is [2]

(l.l)

(1.2)

nr(z) is decreasing and R{z) is increasing"

Similar conditions are imposed on r() and li() in the production model, [II]. These monotonicity conditions on »"(•) and /?(•) are conditions on the first three derivatives of U. Additional conditions are placed on the magnitude of /?.

Some of these hypotheses are controversial in themselves 5 . Taken together they severely restrict the applicability of the EU model.

The problem (central to decision making under uncertainty), of selecting a "most desirable" one from a set of random variables (RV's), supposes an order on the space of RV's, allowing in particular the comparison between RV's and constants (degenerate RV's). In particular, if a decision maker is indifferent between a RV Z and a constant z, then for him 2 is a certainty equivalent

1 Called 'paradoxical' in [91, and "aeemingly paradoxical' in [14{. u To quote from [7], 'auch optimal behavior appear! to be unlikely'.

' See references in [2|, p 103.

* 'Thu*, the notion that aecurity, in the particular form of cash balances, hat a wealth «laaticity of at leait one, ieetni to be the only remaining explanation of the historical courae of money holding*', [2|, p. 104

" To quote from [2|, p. 07: 'The hypotheiii of increaaing relative ritk avenion if not to easily confrontable with intuitive evidence'.

KKSBS^^M*^^


(CE) of Z.

In the EU model, based on the classical theory of von Neumann and Mor- genstern, the decision criterion is maximal expected utility. For decision making purposes, the EU criterion coincides with the classical certainty equivalent €[•) defined, for a RV Z and a given utility U, by

V[C{Z))=EV{Z)

or

Indeed, by the monotonicity of V, EU{-) and C(-) induce the same order on RV's,

/•;r(.v) > KV{Y) &> r(A') > c(Y)

Other reasonable definitions for CE's, based on expected utility, are possible. For example, the CE C defined by

EU[Z-C) = Q (1.4)

In this paper we propose the use, as a decision criterion, the CE

SV{Z) = sup{z + EU(Z - z)} (1.6)

of thi! KV /, with respect to the utility V. We call Si(Z) an optimized certainty equivalent (OCE) of Z. It was introduced, in a difi'erent context, in (■!), [5]. Note that, unlike C{-), the CE's C and .s'r do not induce the same order as

EU.

The advantages of the OCE approach, for predictive purposes, are demon- strated here by reexamining the above classical models of production and investment. In particular, the OCE approach (i) retains the successful predictions of the EU model (as listed above), (ii) does not require restrictive (third-derivative) 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 * .

For problems where the objective is cost minimization, a natural OCE

* Not io for the CE defined by (1.4)

aawa^ft^sa^a«^^»^^ mmm&mmm


for a risk-averter is

Bp{Z) = mi{z + EP{Z - z)} (1.6)

where P^) is an increasing convex penalty function. This is not a departure from the classical theory, indeed, the von-Neumann Morgenstern principle of maximal expected utility could be alternatively developed as minimal expected penalty. Risk-aversion in this context means that a decision makpr, who has to pay a random amount /, prefers to pay the sure amount EZ. This is equivalent to acting on the basis of a convex penally function /*. We illustrate the OCE Bp{), by applying it to a classical inventory model, (10).

A central property of the OCR is shift additivity,

SV{Z + c) = .sVU) -f r (1.7)

for all Utilities V, llV's Z and constants c 7 . In contrast, the classical CK ('(■) is shift additive only for linear or exponential U, [3]. For this reason, certain results (discussed in [3]), which in the EU model hold only for the exponential utility, hold in the OCE 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 portfotoio selection 8 .

The motivation for the OCE's >'r and li^ is provided in (i'i. togellier with

basic properties. Associated functionals, useful in applications, are studied in §3. Section 4 is devoted to production under price uncertainty. The next two sec- tions deal with investment in safe and in risky assets: The Arrow model [2] in §"), and a slight generalization in §6. We conclude with an application of the ()("E Up to an inventory model, in §7.

2. OPTIMIZED CERTAINTY EQUIVALENTS.

Consider a transaction where the ownership of a future value of a random variable Z is about to change hands. A price z, acceptable to a buyer, must satisfy (in some sense) the stochastic constraint

7 Shift additivity alio holds for the CE given by (14), at well ai for th« CE in Yaari'i new axiomatic sys- tem lio|.

* The proofs in |3| use only shift additivity.

J*Mi3«^^


Z <Z (2-1)

From the seller's point of view, the minimal selling price of Z is therefore the value of the stochastic program

sup { r : c < Z } (2.2)

whore the meaning of the stochastic constraint (2.1) is left vague on purpose.

Similarly, a price z acceptable to a seller must satisfy the stochastic constraint

z > Z (2.3)

and the maximal buying price is the value of the stochastic program

inf ( z : z > Z } (2.1)

If / is a degenerate |{V. i.e. / assumes a known value // with probabilily 1. then (2.2) and (2.1) are dual linear programs, having /< as their common value.

One way to enforce the constraint (2.1) is to penalize its violation, following [4] we replace (2.2) by the (unconstrained) problem

sup { z + EU{Z - z)} (2.5)

where U( • ) is a penalty function. In particular, if U is a monotone increasing function with /"(()) = 0, then the term El'(Z - :) in (2.5) pennli/cs [rewards] values z which violate (satisfy] (2.1) in the mean.

Specifically, we can regard U{ • ) as a risk-averse utility function, thus adding concavity to the above properties. Then (2.5) represents a two-stage approach to the stochastic program (2.2), with payments z and the observed future value of Z — z, discounted by U( • ), [5].

Throughout this paper let U be the class of normalized utility func-

tions, 8

* The appropriate normalization for concave utilitiei, non-differentiable at 0, is

hm —L- t/(0) = 0, lim -^- = 1

mmmmmmmmmmmmmmiäsmm


{strictly concave, differentiable, increasing [ , .

functions U with (7(0) = 0 , ^(O) = ij

Given U £ \J, we define the sellers optimized certainty equivalent (SCE) of

Z by (1.5)

SV{Z) = sup{ : + EU{Z - z) }

S^{Z), called the "new certainty oquivalent" in [1], thus represents the miiiimal

selling price of Z.

In the constrained stochastic program (2.1) we similarly pen.'ili7,e the viola- tions of the constraint (2.3) by using a penalty function P( • ), selected from the class of normalized penalty functions

{strictly convex, dillerentiable, increasing | „.

functions H with /J(0) = 0 , F'(0) = ij

For any f G P, we rewrite (2.4) as

inf { z + EP{Z - z) ) (2.8)

and de.ine the buyers optimized certainty equivalent (BCE) of Z by (1.6)

Bp{Z) = inf{ z + EP(Z - z) }

Hp(Z) represents the maximal buying price of Z.

The transaction involving Z is made possible by the inequality

S(!{Z) < BpiZ) (2.9)

for all (/ G U, P 6 P, see Theorem 1(c) and (2.19). For such a pair {V, T} the interval

{SuiZ),BPiZ)\ (2.10)

contains all prices acceptable to both seller and buyer, hence all relevant CE's.

THEOREM 1 (Properties of SCE).

(a) Consistency. For any £/ £ U and a constant c 10

10 Comidertd u a degenerate RV.

JMasmfcoMfcrnft^


Su{c)=c (2.11)

(b) Shift additivity. For any RV Z, constant c and t/ £ U,

SuiZ + c) = SuiZ) + c (1.7)

(c) Risk aversion. If U Is an Increasing, normalized utility function, then

S(!{Z) < EZ for all RV's Z (2.12)

if and only if V is concave. Equality holds in (2.12) if and only if V is linear. (d) Lower bound. If /f is a RV, and Z > z^ with probability 1, then for any

v e u. ^mln < SuiZ) (2.13)

(o) Stochastic dominance. Let A", >' be RV's with compact supports. Then

Sr{X) > Sr{Y) for all (' G U (2.11)

if and only if

KL'(X) > Elr{Y) for all t' G U (2.15)

(f) Concavity. For any t/ € U, 0 < Q < 1 and RV's A:0 , A',

Su[aXl + (l-a)X0) > a ^(X.) + (l-«)5t^V0) (2.16)

(g) Exhaustion. Let Z be a nondegenerate RV with support \zmiQ , -max].

Then for any 2miD < z < EZ, there is f/ 6 U such that iV(^) — '•

['ROOF, (a) For L' 6 U, the gradient inequality implH's

V(:) < U{0) + zl"(0) = z for all ;

and therefore

Suie) = 8up{2 + V{c~z)} < sup{2 + (e-z)} = c. 1 f

AISO, Syic) > {C + ^(C-C)} = f .

(b) SyiZ + c) = sup{i + EU{Z + c-z)}

= c + sup{(2-c) -I- EU{Z - {z-c))} = c + SuiZ). z

(c) By Jensen's inequality, U is concave if and only if

EU[Z) < U{EZ) for all RV's Z

(with equality if and only if U is linear), which is equivalent to


2 + EU(Z-z) < z + U{EZ-z) for all z , and RV Z

and the result follows by taking suprema

SuiZ) = sup{z + EU{Z-z)} < sup{i + U{EZ-z)} z z

= S^EZ) = EZ by (a).

(d) Since U is monotone,

Su{Z) > snp{z + U{zmla - z)}

= :m-m . by (a).

(e) (2.15) => (2.1t). Since oacl» V £ U is increasing, (2.15) implies

z + EU{X-z) > z + EU{Y-z) for all z, and all U € U

and (2.14) follows by taking suprema. (2.14) => (2.15). Let zx , zy be points where the suprema defining -^/(A') and

SIJ{Y) are attained, see Lemma 1. Then, for any U E U,

SuiX) = zx + EU[X - zx) > zY + EU{Y - zY) , by (2.14)

> zx + i:v{\ - :.v)

Therefore

EV{X - zK) > EU(Y - zx) for all U £ U, implying (2.15).

(f) Let 0 < o < 1, and .Va = o.V, + (1 - o)A'0. Then by the concavity of f,

for all zQ , zx,

EU{Xa - azl - (I - a)2o) > aEU{Xx - ;,) + (1 - a)EU{X0 - z0)

Adding az^ + (1 — a)EU{XQ — ZQ) to both sides, and supremizing jointly with respect to Zy , zQ, we get

Su{Xa) > sup {a\zi + EU{Xl - zj] + (1 - a)[z0 + EU(X0 - z0)]}

= aSuiXy) + (1 - a)5(/(Xo)

(g) We define for any C/ £ U and a > 0 the function Ua by

timMmmMmmmmMmmMmmmmmmmmmmi


f'a(0= -f(«*). V* a

Then Ua € U. and the corresponding SCE is ll

^(^) = SUD{C+ ~U{a{Z - f\)} 0 n s

(2.17)

(2.18)

= —sup{c + U(aZ — :)}

= -SuiaZ) n

Sinro Sft [Z] is conlimious in o. if follows I'rom hcmnia 2 (Appendix R) thai t'cir

any

-min < Z < EZ

there is a > 0 such that SiJa[Z) = 2, proving the exhaustive property of tlu>

SCE. D

For the buyers OCE, an analogous theorem can be proved. In particular, 1110 analogs of purls (i^.fil) an»

EZ < BP[Z) < zmn (2.19)

for any 11V /, an upper bound zmAX and P £ P. The concavity in (f) is replaced

by convexity.

Theorem I lists properties which seem reasonable for any certainty equivalent. Properties (a) and (d) are natural and require no justification. The remaining properties will now be discussed one by one.

(b) To explain shift additivity consider a decision-maker indilferent 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 I Dollar to 5 will keep the decision maker indifferent.

(c) A decision maker is risk-averse (prefers EZ to Z) in our theory if and only if he is risk-averse in the EU theory, both risk-aversions equivalent to the

11 In the two-stage model of the SCE (2.18), Su can be interpretted ai the present value of the sum of the

payments i (at present) and 2 - i (in the future). Here a is the time discount factor

10

m mmmmm!mmmmmmmmmmmM®m>MtiM


concavity of the underlying utility U. (e) In general,

EU{X) > EU{Y)

does not imply

Sv{X) > Sr{Y)

(2.20)

(2.21)

i.e. for a given U, (2.20) and (2.21) may induce diflerent orders on the R.V.'s, see e.g. [5]. However, for the stronger preference of (2ad order) stochastic domi-

nance, with the inequality holding for all I' £ V t (2.11) and (2.15) are equivalent.

(f) The concavity of 5^(), for all U G U, expresses risk-aversion as aver-

sion to variability. To gain insight consider the case of two independent RV's with the same mean and variance. The mixed RV A'rt = «A'| + (I—a).V0 has

the same mean, but a smaller variance. Concavity of S(i means thai the more

centered RV Ara is preferred.

The risk-aversion inequality (2.12) is implied by (f): Let Z, Z{, ^2 , • ■ • be independent, identically distributed RV's. Then by (f).

= MZ)

As n —> oo, (2.12) follows by the strong law of large numbers.

In contrast, the classical CE U~lEU(-) is not necessarily concave for all concave U.

(g) This property simply means that any value between 2inln and EZ is

the certainty equivalent of some risk-averse decision maker.

The question of the attainment of the supremum in (1.5) is settled in the following:

LEMMA 1. Let the RV Z have support [2min , 2maxl> w'^ finite 2mjn and zmax. Then the supremum in (1.5) is attained at some Zg,

'" In which case Y is called Makler than X.

11

awiaaa^ata^^

Certainty equivalents

zmin < 25 < 'max'

which is the unique solution of

so that

EU'{Z - zs) = 1,

*V{Z) = :s + FA'{Z - zs)

Ben-Tal and Ben-Israel

(2.22)

(2.23)

(2.24)

PROOF. The function

H[Z) = z + El{Z - z) (2.25)

is concave, so its supromum is attained at a stationary point z^ satisfying (2.23)

(obtained by differetiating (2.25)). The uniqueness of cs- follows from the fact

that U' is strictly decreasing, implied by the strict concavity of U. From (2.23) it iilsü follows thai

and

proving (2.22).

''•(Cmln - *S) > 1

D

Similarly, the iiilimum in (1.0) is attained al some :/( in [:,„•„, . -„laxl- which is the unique solution of

KP'iZ - zB) = 1 (2.26)

so that

1>P{Z) = zB + EP{Z - zB) (2.27)

In the above discussion of Sy and Bp, the utility U and the penalty P are unre-

lated. In the following two examples we illustrate our results using pairs {P, V)

related by 13

P[z) = -(/(- 2 ) , \iz (2.28)

EXAMPLE 1. (Exponential utility). Here

11 Indeed (2.28) is a 1:1 correspondence between U and P. It also brings some symmetry into the discussion, since U and P (of (2.28)) penalize deviations in the same way. Another natural correspondence between U and P is /»A l/-'

12

Miijimm^^^


U{z) = 1 - e"* , \iz (2.29)

and equation (2.23) becomes Ee'2** = 1, giving 2S = ~ \ogEe~z and the same

value for the certainty equivalent

SyiZ) = - log Ee~z (2.30)

A special feature of the exponential utility function (2.29) is that the classical CE (1.3) becomes

U-lEU{Z) = - \ogEe-z

showing that for the exponential utility, the certainty equivalents (1.5) and (1.3) coincide.

The corresponding penalty, by (2.28). is

/'(;) = - l + ,--' (2.31)

giving the RCF,

HP{/) = [ogl-e2 (2.32)

EXAMPLE 2. (Quadratic utility). N Here

U{:) = z - -l- z1 z < 1 (2.33)

and for a RV / with :w^ < I. EZ ~ n and variance a2, e(pialion (2.23) gives

:s = n, and by (2.21)

S(!(Z) = /' - | *2 (2.31)

The corresponding penalty (2.28) is

Piz) = z + \ ^ (2.35)

with BCE

Bp{Z) = ti + j a2 (2.36)

Note that here Bp{Z) - SyiZ) = a1. We show in (3.12) that for small a2,

Bp{Z) — Sif{Z) is approximately linear in a2 for all t/ G U and P £ P.

COROLLARY 1. In both the exponential and quadratic utilities

11 The restriction < < I in (2.33) guarantees that U is increasing throughout its domain

13

fe^iimi^^^M^^


Su{yiZt)= rSuM) (2.37) 1=1 i=i

for independent RV's {Z^ Z?--,Zn}1*. □

EKWIPLE 3. For the so-called hybrid model ([3], [8], [16]). with exponciUial utility U and a normally distributed RV Z ~ N{//, <T

2),

^r(Z) = n - -a2

3. FUNCTIONALS AND APPROXIMATIONS.

Lot / = (#,) be a RV in It'1, with expectation ft (vector) and covariance

matrix D (if n = I then as above S = IT* ). For any vector y E li" , the inner product

n

y-z = Yl yizi i=i

is a scalar RV. Given U £ V and PGP, the corresponding CE's of yZ are taken as functional in y, the SCE functional

*(.'/) = Mr/). OU) and the BCE functional

b(y)= Upiy-Z). (3.2)

We collect properties of the SCF functional in the following theorem, whose proof appears in Appendix A. The analogous staements of the lUTl functional are omitted.

THEOREM 2. Let f/ £ U be twice continuously dilferentiable, and let Z and s() be as above. Then:

(a) The functional s is concave, and given by

s{y) = zs(y) + EU{yZ - zs{y)) (3.3)

where Zs{y) is the unique solution z of

The cUtiical CE (1.3) is additive, for independent RV's, if V is exponential but not if U is quadratic

14

^MS^MiM^iitei^^


EU'{yZ - z)= I (3.4)

(b) Moreover,

5(0) = 0 , Vs(0) = /i , V25(0) = (;"(0)E (3.5)

c,(0) = 0. VZS{0) = LI (3.6)

and if U is three times continuously differentiable,

v'-s,o, = »• (3-7)

D

Theorem 2. and the corresponding statements about the functional b(), can be used (o obtain the following approximations of the fmictionals «(•) . />(•) based on their Taylor expansions around y •— 0.

COROLLARY 2. If t/ , P are three times continuously differentiable then

*{y) = H'y + \u<<{0)y'Ly +o(||y||2) (3.8)

b{y) = ^y + \P"{o)yZy +o(||y||2) (3.9)

D

REMARKS

(a) In particular, for n = I and y = I, it follows from (3.8) and (3.9) that

$„(2) = h + ^U"(0y + o((r2) (3.10)

= „ - ir(0)a2+o(^)

where r() is the Arrow-Pratt risk-aversion index (l.l). Similarly,

BP{Z) = n+ |P"(0)a2 + o^2) (3.11)

and therefore

BP{Z) - SyiZ) = |(P"(0) - t/"(0))a2 + o^2) (3.12)

(b) We also note that the approximation (3.12) is exact if

16


(1) U is quadratic, or (ii) U is exponential, Z is normal.

(c) By differentiating, and calculating the Taylor expansion of the classical CE of y-z

ce{y)= U-[Er(rZ) (3.13)

it follows that ce{y) is approximated by the right-hand side of (3.12). Thus we have the unexpected result

ce{y)-s(y) = o(\\y\\-) (3.11)

showing that for small y the CE functionals (3.1) and (3.13) are practically the same.

I. COMPETITIVE FIRM UNDER UNCERTAINTY.

The first application of the OCE is to the classical model studied by Sandmo [14], see also [9, § 5.2], A firm sells its output q at at price P, which is a RV with a known distribution function and expected value EP = /i. 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) = e(q)+ /?

The function <•(•) is assumed normalized, increasing and strictly convex,

c(0) = 0, c'{q)>0< c"{q)>Q \iq > 0 (1.1)

The firm has a strictly concave utility function U, i.e.

U' > 0 , U" < 0

which is normalized so that U{Q) = 0 , (7'(0) = 1. The objective is to maximize profit

n{q) = qP - c{x) - B

which is a RV. The classical CE (1.3) is used is Sandmo's analysis, so that the model studied is

max U-lEU(n{q)) q >0

or equivalently,

16

&5ÖÖQM&S -™MM


max EU{ir{q)) q >0

(4.2)

Here we analyze the same model using the SCE (1.5). The objective of the firm is therefore

(1.3)

Now

max Suitiq)) q >0

max ^(/(^(fl)) = max ^uil^ ~ c(9) — ^) 9 > 0 ? > 0

= max {Sv{qP) - c{q)) - B q > 0

by (1.7). We conclude:

PROPOSITION I. The optimal production output q is independent of the fixed cost B. □

This result is in sharp contrast to the expected utility model (1.2) where the optimal output q depends on the fixed cost B: q increases (decreases] with B if the Arrov Pratt index r() is an increasing (decreasing] function; the dependence is iiinbisioiis for utilities for which r() is not monotone.

Note that the objective function in (4.3) is

f{q) = s{q)- c(q) (.1.4)

where «(•) is the SUE functional (3.1). The function / is concave by Theorem 2 and the assumptions on c. Therefore, the optimal solution q of (4.3) is positive if and only if /'(0) > 0. By (3.5) s*(0) = /<, so

q > 0 if and only if /i > c'(0) (4.5)

in agreement with the expected utility model (4.2). We assume from now on that

li > C'(0)

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 qctT under certainty, that is for P a degenerate RV with value /i. We will prove now that the same result holds for the model (4.3). First recall that the optimality condition

17

^^S^iÄK^m^^^^^^


for qcer is that marginal cost equals marginal revenue

C'ilcer) = /* (4-6)

» PROPOSITION 2. q < qctr for all U 6 U.

PROOF. The optimality condition for q is

0 = /'(/) = s'(/)-C'(/) (4.7)

Ry Theorem 2

s{q) = z{q) + EU{qP - z{q)) (4.8)

where z{q) is a differentiable function, uniquely determined by the equation

EW{qP - z(q)) = I (4.9)

Differentiating (4.8) with respect to q yields

s'{q) = z'(q) + E{[P - z'(q))U'(qP - z(q))}

= z'{q)[l - EU'{qP - Z(q))\ + EPU'{qP - z{q))

Using (4.9) we then get

5'(?)= E{PU'{qP - z{q))} (4.10)

Therefore the optimality condition (4.7) becomes

EPl'iu'P - :(<,')) = r■(q^) (1.11)

Multiplying (4.9) by /i and subtracting from (4.11) we got

E{P -n)U'(q,P-z{qt))=C'(q,)-fl (4.12)

or

E{Zh(Z)} = c^q,)-fl (4.13)

where we denote

Z .= P - n, h{Z) := U'(q'Z + q'n - z{q'))

Since C/ 6 U, it follows that h is positive and decreasing, and it can then be shown (see e.g. [9, p. 249]) that

E{Zh{Z)} < h{0)E{Z}

but E{Z} = E{P - n} =0, aud so, by (4.13),

c'iq') < H

and by using (4.6)

18

amimi&msMm&fii^^


C'(/) <cUef)

and since c* is increasing,

<?<</, cer D

KFFFCT OF PROFITS T.\X

Suppose there is a proportional profits tax at rate 0 < < < 1, so that the profit after tax is

*{q) = (1 - 0 (^ " C{q))

As before, the firm seeks the optimal solution q of (4.3), which here becomes

max S(t{n(q)) = max Sr{ (1 - /) {qP - r\q) - H)) q > 0 q > 0

= max {Sv[{i - t) qP) - (1 - t)c{q)} - (1 - i)B q > 0

which can be rewritten, using the SCE functional «(•) and omitting the constant

(1-00,

max s((l - t)q) - (I - t)c{q) q >0

Lei the optimal solution he if — q(t). The opiimality condition here is

(1 -/).s'((l - t)q) - (I - t)c'{q) = 0

giving the identity (in /),

.•((1 -t)q{t)) = c'(q(t))

which, after differentiating (with respect to t),

and rearranging terms, gives

q'{t){c"{q) - (1 - t)8"{{l - t)q)} = - 9(0*"((1 " t)q) (4.14)

The coefficient of q'{t) is positive since c" > 0 and «(•) is concave (Theorem 2(a)). The right-hand side of (4.14) is positive sina. 7 > 0 , s" < 0. Therefore, by (4.14),

q'{t) > 0

and we proved:

19

mmmtmmm&m mmmmmmmmmmmmmmM


PROPOSITION 3. An increase in profit tax causes the firm to increase produc-

tion, n

In the classical expected utility case the effect of taxation depends on third-derivative assumptions, and is undetermined if the relative risk-aversion index R{-) is not monotone.

EFFECT OF PRICE INCREASE.

If price were to increase from P to P + e (e fixed), then the corresponding optimal output q(f) is the solution of

max {.S>((P 4- f)q) - riq)] = max {*{<]) + tq - r{q)) q >0 q > 0

The optimality condition for q{t) is

*'{q{t)) + t =c'{q{i))

Differentiating with respect to e we get

f{<)s"{q{e))+l = q'{(y'{q{t))

hence

jr(0 = —i—1—- >o c"(<7) - s"(q)

by the convexity of c and the concavity of s. We have so proved:

PROPOSITION 4. An increase in selling price causes the firm to increase production. □

This highly intuitive result is proved in the expected utility case only under the assumption that r(-) is non-increasing.

5. INVESTMENT IN ONE RISKY AND IN ONE SAFE ASSETS: THE ARROW MODEL.

Recall the classical model [2] of investment in a risky/safe pair of assets, concerning an individual with utility U £ XJ 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).

20

wsa


The rate of return in the risky asset is a RV X.

The final wealth of the individual is then

Y = A-a+{l + X)a=A+aX

In [2] the model is analyzed via the maximal EU principle, so the optimal investment a is the solution of

or equivalently

max EU {A + a.V) (5.1) 0 < a < .4

max r(.4 4- «A') (5.2) 0 < « < A

Some of the important results in (2j are:

(Al) a* > 0 if and only if EX > 0.

(A2) o increases with wealth (i.e. —— > 0) if the absolute risk aversion index dA

r^) is decreasing.

(A3) The wealth elasticity of the demand for cash balance (investment in the safe asset)

Em dmldA • . i . /r ^ := L is at least one (5.2) £Vl ml A

if the relative risk-aversion index

fr"M R[z) = - z 77u~\ ls increasing (5.3)

Arrow (2j postulated that reasonable utility functions should satisfy (5.3), since the empirical evidence for (5.2) is strong, see the references in [2], p. 103.

We analyze this investment problem using the SCE criterion, i.e.

max SuiA + aX) (5.4) 0 < a < A

which, by (1,7) is equivalent to

21

^M^femd^Ma^^

tt.*l*J*l^l*ll^l"*.VWiWWW«lfllll.^4,"Ul1TW»l,**wvrn»»^.-»«*VB.imTi* mtrmumammtmncmmnuijwm


max Sij{aX) + A 0 < a < A

Let a be the optimal solution. Using the SCE functional «(•), a is in fact the

solution of

max s(a) (5.5) 0 < A < .4

Now, since s(-) is concave

(i* > 0 if and only if .s'(0) > 0

but by (3.5) s'(0) = EX, and we recover the result (Al).

Assuming (as in [2]) an inner optimal solution (diversification)

{) < a' < A (.->.())

we conclude here, in contrast to (A2), that

da ,. „. ■rfr = " (0')

i.e. the optimal investment is independent of wealth. 10

An immediate consequence of (5.7) is

Ew I:A

indeed

> i .- r G U

Em _ ^ii^» __ i! HA ZJL! = A (i _ da$ ) = A > I EA ~ m dA ' A - u' dA A - „ * <iA A -a'

proving (5.2) for all risk-averse utilities. Thus, in the SCE model, there is no need for the controversial postulate (5.3).

The quadratic utility (2.33)

1 ,2 U{z) = z - -^z* z < I

violates both of Arrow's postulates (r decreasing, R increasing), and is consequently "banned" from the EU model. In the SCE model, on the other hand, the

19 However, initial wealth will in general determine when diveiiflcation will be optimal, i.e. when (5.0) will hold.

22

mmmsMiiMm^Mm^^^^^^


quadratic utility is acceptable 17 . For the quadratic utility the optimal investment a is the optimal solution of

max {s{a) = pa - —a2 a2} 0 < a < A 2

where /* = EX , a2 = Var(X). Therefore

n/a2 if 0 <///a2 < 0 « (i = S

A if fi/a2 > 0

showing that, for the full range of A values, a (/t) is nondecreasing, in agreement with (A2). Moreover, if diversitication is optimal, then

Em __ A .

i:A ~ A - ft/a2

Following [2] we consider the ellects oti optimal investment, of shifts in the UV A'. Let h bt; the shift parameter, and assume that the shifted liV X(h) is a differentiable function of h, with A'(0) = X. Examples are

X(h) = X + h (additive shift), A'(/i) = (I + /i)Ar (multiplicative shift).

For the shifted problem, the objective is

max Sir(aX(h)) (5.8) 0 < a < . I

Let a(/j) be the optimal solution of (5.8), in particular «(0) = a . Now

%(«A'(M) = £(«) + h:i'{aX(h) ~ tin)) (5.9)

where £{a) is the unique solution of

EU'{aX(h) ~ ({a)) = 1 (5.10)

The optimality condition for a{h) is

-£{e(a) + EU{aX{h) - i{a)))

which gives (using (5.10)) the following identities in h

E{X(h)U'{a{h)X{h) - t{a(k))} = 0 (5.11)

" Auuming 0 < X < 1

28

s^^^^&a^^aaäa^^^^g^?^^


E{U'{a{h)X{h) - e(a(/»)) - ({a{h))} = 1 (5.12)

Differentiating (5.11) with respect to h we get, denoting Z = aX(h) — t{a{h)),

a{h)E{Uu{Z)X{X - r(a(/»))} + E{X{h)[U'{Z) + t{h)X{h)U"(Z)]} (5.13)

where <i(/i) := —-a(/i) and similarly for X{h). an

The second order optimality condition lor a[h), —7'sr("-M'')) ^ 0- is

hero

^/"(Z^X - fXM)) > o hence, by (5.13),

sign of h{h) = sign of E{X(h)\V'{Z) + nXV-(Z)\)

exactly the same condition for the sign of -—«(/») as in [2]. |). 105. eq/(18). Utt

Therefore, the conclusions of the EU model are also valid for the SC'E model. In particular:

PROPOSITION 5. As a function of the shift parameter h, a(/i) increases for additive shift, a(/i) decreases for multiplicative shift. D

Theses results are illustrated for the quadratic utility, '.'here

EX a =

Var(.V)

and

aih) = a* + ——TTT 'or un additive shift Var(A)

a[h) — -a for a multiplicative shift (5.14)

In fact, (5.14) holds for arbitrary (7 G U, a result proved in [17] for the EU model.

PROPOSITION £>. If a is the demand for the risky asset when the return is the RV AT, then a[h) ~ a /l+h is the demand when the return is (l + /i)Ar. PROOF. The optimality condition for a is

EiU'ia'X - t*)X} = 0 (5.15)

24

S&flS^BiB^^

f imjnpu-wHM.wi..

Certainty equivalents Ben-Tai and Ben-Israel

where ^ is the unique solution of

EU'{a'X - e*) = 1 (5.16)

The optimality conditions for a{h) are given by (5.11),(5.12). Now, for

fl(M=TL-a*(

a{li)X{h) = a'X (Ö.17)

and it follows, by comparing (5.12) with (5.16), that

e(«(M) = r Substituting this in (5.11) and using (5.17), we see that (5.11) is equivalent to (ri.lö), and that a{h) = « /l+/i indeed satisfies the optimality conditions (5.11),

(5.12). D

6. INVESTMENT IN A lllSKY/SAFC PAIR OF ASSETS: AN EXTENSION

We study the model discussed in [6] and [7], which is an extension of the model in §5. The analysis applies to a iixed time interval, say a year. An investor allocates a proportion 0 < A: < 1 of his investment capital \VQ to a risky

asset, and proportion l—k of W0 to a safe asset where the total annual return per dollar invested is r > 1. The total annual return / per dollar invested "m the risky asset, is a nonnegative KV. The investor's total annual return is

k\VQl + {l~k)\V0T

and for a utility function U, the optimal allocation k is the solution of

max t:U{kWQi + (l-Är)lV0r) (6.1) 0 < it < 1

The model of §5, is a special case with \V0 = A , t = l + X, kW0 = a , r = I.

It is assumed in [6], [7] that (7' > 0 and U" < 0, thus we assume without loss of generality that C/ € U.

One of the main issues in [7] is the effect of an increase in the safe asset return r on the optimal allocation. The following are proved: (Fl) An investor maximizing expected utility will diversify (invest a positive amount in each of the assets) if and only if

26

kt«M^^^^ig>iM<^^

ww IVFR


EtU'iWot) < T < E{t) (8.2)

Eb"{W0t)

(F2) Given (6.2) he will increase the proportion invested in the safe asset when

r increases if either (a) the absolute risk avursion index r(-) is nondecreasing, or (b) the relative risk aversion index /?(•) is at most I.

The same model is now analyzed using the SCE approach, i.e. with the objective

max 5'r(HvV + (l-A-)U„r) 0 < it < 1

Using (1.7) and the definilion (3.1), the objective becomes

max {(\-k)\VQr+ s(\Vük)) 0 < k < \

(6.:{)

The following proposition, proved in. Appendix (', gives the analogs of results (Fl). (F2) in the SCE model.

PROPOSITION 7. (a) The SCE maximizing investor will diversify if and only if

EtU'{W0t - n) < r < E{t) (6.4)

where rj is the unique solution of

/•nuv -//)=! (6.5) (b) Given (6.4), he will increase the proportion invested in the safe asset when r increases. □

Comparing part (b) with (F2), we see that plausible behavior (Ar increases with r) holds in the SCE model for all (/ G U, but in the EU model only for some U.

We illustrate Proposition 7 in the rase of the quadratic utility (2.33). Here the optimal proportion invested in the risky asset is:

k' =

0 if r > E{t)

Uli Wn<T2

if £(0 - WQa2 < r< E{t) (6.6)

1 if E(t) - W0(J2 > T

where a1 is the variance of t. Thus A; is increasing in E[t), decreasing with a2

and decreasing with r (so that, the proportion 1—A: invested in the safe asset is

26

^t^^at^^^:^^^

wwwviAiJunwuvia


increasing with safe asset return r). These are reasonable reactions of a risk-

averse investor.

We also see from (6.6) that A: decreases when the investment capital H'0

increases. This result holds for arbitrary (/ G U, see the next proposition (proved in Appendix C). In the EU model, the effect of Hr

0 on k depends on

the relative risk-aversion index, see [6].

PROPOSITION 8. If the investment capital increases, then the SCE-maximizing investor will increase the proportion invested in the sate asset. □

Following the analysis in [2] and §5, we consider now the elasticity of cash- balance (with respect to U'0). Here the cash balance (the amount invested in the

safe asset) is

m =(I-A-')U'o

and the elasticity in question is EW,

PROPOSITION 9. For every SCR-maximizing investor with U € U,

Jm_ > EW0 -

PUOOI".

dk*{WQ) 1 -k*{WQ)- »Ko—^

EW0 m/H'0 i - A-'(lV'0)

hence

-^- > I if and only if ... < 0 (ß.7) L VV 0 «IV 0

and the proof is completed by Proposition 8. □

The equivalence in (6.7) shows that the empirically observed fact that Em/EWQ > 1 can be explained only by the result established in Proposition 8

that dk ldW0 < 0, a. result which is not necessarily true for many utilities in the EU analysis.

27

iamas£^satt^&^^

niaiin


7. AN INVENTORY MODEL.

Consider a classical inventory problem, e.g. ([10], §2.5), where demand (for the item in question) occurs at a rate of d units per day. Orders are received immediately after they are placed with the supplier. Unsatisfied demand is backloggod until it can be satisfied. Holding cost is h per unit per day, shortage cost (penalty for unsatisfied demand) is p per unit per day. Material cost is r per unit, and there is a fixed transaction cost of k per order.

An ('i, s) policy is used (whenever stock level falls below .s. order up to

S). The aim is to minimize total cost per period, given by

TC = -^- + -^- 4- -^- + rrf (7.1) 2{S-s) 2{S~s) .S-.s v ;

and the optimal parameters of the policy are given by

P

The least-cost order quantity Q = S — s is then

S*= /M /ZPZ (7,2)

(7.3)

Q-^^/M^^± (7.4)

We now analyze this model under the assumption that, the demand D is a nonnegtaive RV. To compare our results with the deterministic case we assume that E{D) = d. We also denote the variance of D by a2. Since we have a cost, minimization problem, we use the BCE criterion, and so minimize the BCE of the total cost (7.1)

• n t ^ Ps2 kD ,. m mm Bp{—- + -7^ + + cD) SiS p{2{S-s) 2{S-s) S-s

which, jy (1.7), reduces to

mnins. ,)..= ^ + 1J^ + HJL- + c)} p.5)

where b{-) is the BCE functional of (3.2).

The first order necessary conditions for an optimal pair [S, s) are

28

i&ffiSijiiii^Sa^

■■■■■ H


(7.6) a/(5, 51 _ 85

1

2(5 - ^)2

B/(S, ^) _ Os

1

.c? . „^8^ ^ (2(5 - s)hS - (hS2 + pf) - 2kb'{-=^-^ + c)} = 0 S — s

(7.7)

{2(5 - s)pl + (/t52 + pr) + 2kb'{ _ _ f 0} = Ü 2(5 - sf 5 - .s

liy adding (7.G) and (7.7) we oblain the relation

I=-~S (7.8)

in analogy with (7.3). Substituting (7.H) in (7.6) wo got an equation for >':

' >(! + — ) P

Recall that fc^) is convex (proved analogously to the concavity of .?(•) in Theorem 2(a)). Therefore the function r(-) in (7.9) is increasing for 5 > 0. Moreover, b'{y) > 0 for all y > 0 since ^(O) = E{D) = d > 0. Therefore,

r(0) < 0 and r(oo) = oo

and equation (7.9) has a unique solution 5.

Comparing the solution (5, s") to the deterministic solution (5 , s ) (given in (7.2),(7.3)), we get:

PROPOSITION 10. For every BCE-maximizing decision-maker (i.e. for every l* ^ P) the optimal order quantity under uncertainty Q ;— S - J is larger than the optimal order quantity under certainty Q .

PROOF. By (7.9) with 0 := 1 + -, P

5^/ifl = 2kbt{~ + c) > 2kbt(0) = 2kd

hence

=■ ^ /Ikd /2kd /p+h _ g« 5> v"^ v~rv~r Using (7.8) and (7.3) we get a comparison for the other parameters

29

^gaamjiffi^tM^i^^^^

WJTPl im rnvwimnan

Certainty equtvalenta Ben-Tal and Ben-Israel

and finally

s = 6 < — —o = s P P

Q = S -s > S -s = Q D

The effect of changes in the cost parameters on i» can be determined from the optimality equation (7.9). These results are summarized in the Table 1, and compared to the analogous results in the deterministic case.

Increase in: Effect on S Effect on S * (stochastic demand) (deterministic demand)

Shortage cost p increase increase Holding cost h decrease decrease Transaction cost k increase increase Material cost c increase no effect

TABLE I. Effects of cost changes on policy parameters.

Thus we have the same effects of cost changes in the deterministic and stochastic cases, except for the material cost c: In the deterministic case the optimal policy is independent of c (see (7.2)), but in the stochastic case there is dependence, sec (7.9).

To illustrate the above results, consider the quadratic penally ('2.;ir)). Then the BCE functional (for the RV D) is

My) = ''y + y^V

(for the quadratic penalty, the approximation (3.9) is exact). The equation (7.9) determining the optimal i1 is here

PhO -2a2^ =2k{d +a'ic) es

(7.I0)

where tf = I + —. From (7.10) it follows that P

5 increases with the expected demand d, S increases with the demand variance a2.

Since Q = S — T = SO the same conclusions hold for the effects on the optimal order quantity Q. The result that Q increases with cr2 is intuitively clear: The risk-averse decision maker keeps a larger inventory to cope with increased

80

«^«^a^^^»«ai«i^^ MM

wvwrrswnmrwwm


demand fluctuations.

[1]

('»1

[6

[7

[8

[10

[H

(12

(13

(14

(15

REFERENCES

M. Alais and 0. Magen (Editors), Expected Utility Hypotheses and the Alais Paradox, D. Reidel, Dordrecht, Holland. 1979.

K.J. Arrow, Essays on the Theory of Risk-Bearing, Markham, Chicago, 1971.

G. Bamberg and K. Spremann, "Implications of constant risk aversion", Zeit. Oper. Res. 25 (1981), 205-221.

A. Ben-Tal, "The entropic penalty approach to stochastic programming", Math. Oper. Res. 10 (1985). 263-279.

A. Ben-Tal and M. Teboulle. "Expected utility, penalty functions, and dual- ity in slocliaslic nonlinear programming", Manayenienl Sri. 32 (19.%), 11'15-IU)6.

D. Cass and J.E. StigliU, "Risk aversion and wealth effects on portfolios with many assets". Rev. Econ. stud. 39 (1972), 331-354.

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 Sei. 22 (1976), 1064-1073.

J. Freund, "The introduction of risk into a programming model", Econome- /n>rt 24 (1956), 263-273.

S.A. Lippman and J..I. 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.

S.F. Love, Inventory Control, McGraw-Hill, New York, 1979.

M. Machina, ""Expected utility" analysis without the independence axiom", Econometrica 50 (1982), 277-323.

J.W. Pratt, "risk aversion in the small and in the large", Econometrica 32 (1964), 122-136.

R.T. Rockafellar, Convex Analysis, Princeton Unversity Press, Princeton, 1970.

A. Sandmo, "On the theory of the competitive firm under price uncertainty", Amer. Econ. Rev. 61 (1971), 65-73.

P.J.H. Schoemaker, "The expected utility model: Its variants, purposes, evidence and limitations", J. Econ. Literature XX (1982), 529-563.

31

testa^fta^^ft^ -*&-


[16] J.K. Sengupta, Decision Models in Stochastic Programming, North-Holland, amsterdam, 1982.

[17] J. Tobin, "Liquidity preference as behavior towards risk", Rev. Econ. Stud. (1958), 65-86.

[18] J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1947

[19] M.E. Yaari, "Risk aversion without diminishing marginal utility and the dual theory of choice under risk", Research Memorandum No. 65, Center for Research in Mathematical Economics and Game theory. The Hebrew Univer- sity, Jerusalem. February 1985.

APPENDIX A. PROOF OF THEOREM 2.

(a) By (3.1) and (1.9). s{-) is the pointwiso supremum of concave funclionals. hence concave. The rest of (a) is proved as in Lemma 2.3(a).

(b) For y = 0, (3.4) gives

EU'{ - zsm = 1

or U'{ - 25(0)) = I, proving that 25(0)) = 0. From (3.3) it follows then that

5(0) = 0.

DUlerentiaUng (3.4) with respect lo 1/ gives

EiJ-{rZ - 2s(y))^-v25(j/)) = o

which at j/ = 0 becomes

•'(()) [EZ ~ V;,(())) = 0

proving that Vz^O) = /i. Then, by differentiating (3.3) at y = 0 we get

V5(0) = 0.

The expressions for V2Z5(0) and V2«(0) follow similarly by differentiating

(3.4) and (3.3) twice at y = 0.

(c) and (d) are proved analogously. □

APPENDIX B. PROPERTIES OF 5^ .

LEMMA 2. Let t/ G U, a > 0, and let [/a be defined by (2.17). Then:

32

mmmimmmmmmmmmimmMm$MjmiM$m

mm


(a) Sy (Z) is monotone decreasing in a.

(b) WmSUa{Z) = n. a—Q

(c) If U is essentially smooth ([13], p. 251),

lim Sn{2) = -rain.

PROOF, (a) We prove that lrn is monotone decreasing in a. Indeed,

A.U{a2) = ±[azU'{az)- U{az)] act a*

< -^ - (mi a'

= 0 since /'(()) = 0.

The above inequality is the gradient inequality lor the concave function U. The monotonicity of Ua, as a function of a, is inherited by 5(/ of (2.18).

(b) By L'Hopital's rule, since ^'(0) = 1,

WmUjz) = 2 . \iz

from which (b) follows.

(<•) Since U is essentially smooth it can be shown that

lim Un{z) = - oo . \iz <() U -»00

from which (c) follows. □

APPENDIX C. RESULTS FROM §6.

PROOF OF PROPOSITION 7. (a) The objective function in (6.3)

h{k) r= (i_ib)^0r+ s{W0k)

is concave, by Theorem 2(a). Hence, the optimal solution x is an inner solution, i.e. 0 < fc < 1 if and only if


h'{0) > 0 and /»'(I) < 0 (Cl)

Now

h{k) = - W0T+ Wßs'iWok) (C.2)

which becomes, upon substitution of the computed expression for s'(-).

hs{k) = - W0T + WoEtWiW^kt - ti{WQk)) ((':.])

where T](q) is the unique solution of

EU(qt - //) = I (C.4)

Therefore

/.'(0) = - WQr+ \V0E{t)

/t'(l) = - W0T+ W0EtU(W0- r,{W0))

and (C.l) is equivalent to (6.4). (b) Let fc(r) be the optimal solution of (6.3) lor given r, i.e. Ir(k{r)) = Ü, or

using (C.3),

- r+ E{tU'{\V0k{T)t - //(UVfc(r))} = 0

Differentiating this identity (in r) with respect to r, we obtain

- I + E{tW0{k'{r)t - k'{r)V'{WQk{T)))U"} = 0

or

t(r)\V0Et{l~tr)V'- = I (C.5)

Now, the second order condition for the maximality of Ä;(r) is

0 > /»"(it) = W0E{lWQ(t - ,;')(/"} (CO)

Therefore, Ä:'(r) is multiplied in (C.5) by a negative number, and consoqucntly

^(r) < 0

proving that Ä;(r) [1 - k{r)], the proportion invested in the risky [safe] asset, is a decreasing (increasing] function of r, the safe asset return. □

PROOF OF PROPOSITION 8. Let k = A;(Vf0) be the optimal solution of (6.3), i.e. h'{k{ WQ)) = 0, or using (C.3)

- r + E{tU'(\V0k{WQ)t - v{WQk{W0)))} = 0 (C.7)

Differentiating this identity (in W0) we get

84

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pi>ia|»fi,frjwî'v<Tmw iw

Certainty equivalents Ben-Tal and Ben-Israel •

Et[k{W0) + W0k'{WQ)][t - ^(^(ô))]^' = 0

or

k'W0Et{t-ri')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

36

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IBIWHJWJI1«MI*11 wvw VWVWiVü I WM H'MVM WWnwwuwwwmwmmm'maimmmmmmmM mm*

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REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM

1. REPORT NUMBER

CCS 556

2. COVT ACCESSION NO. 1 RECRIENT'S CATALOG NUMBER

A/>*/7MS' 4. TiTLZ (»ml Sublllf) i TYPE OP REPORT « PERIOD COVERED

Optimized Certainty Equivalents for Decisions Under Uncertainty • PERFORMING ORG. REPORT NUMBER

1- AUTHORS)

A. Ben-Tal A. Ben-Israel

• . CONTRACT OR GRANT NUMBERf«) NSF Grant ECS-860435A ONR Contract N00014-82-K-0295 NSF Grants SES-840813A

SES-«s?n«nft 9. PERFORMINl} OHÜANUATIUN NAME AND ADDRESS

Center for Cybernetic Studies The University of Texas At Austin Austin, TX 78712

10 PROGRAM ELEMENT. PROJECT, TASK AREA A WORK UNIT NUMBERS

I I CONTROLLING OFFICE NAME AND ADDRESS

National Science Foundation

12 REPORT DATE

December 1986 II NUMBER OF F'Ar.t-S

38 "Hi MONlToniNd ACiENI'V NAMF * AODR ESSCK i/(//Br»n( from Cimlrnfllni) ()//lr«l IS. SECURITY CLASS. («I IM* r«tH<rl)

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Expected Utility, Certainty Equivalent, Decision Making Under Uncertainty, Risk Aversion

10. ABSTRACT (Canllnum an rrnut»» »Id» II n»a»»»mrr and HtnlHy by block nurnbtO

A new approach Is proposed for some mo<Jelt of decision-making under uncertainty, using optimized certainty-equivalents Induced by expected-utillty. Applications to production, investment and Inventory models demonstrate the advantages of the new approach.

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