UTILITY FUNCTIONS WHOSE PARAMETERS DEPEND ON INITIAL WEALTH

Transcription

1 # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA Bulletin of Economic Research 55:4, 2003, UTILITY FUNCTIONS WHOSE PARAMETERS DEPEND ON INITIAL WEALTH Christian S. Pedersen and S. E. Satchell Oliver, Wyman and Company Faculty of Economics and Politics, Cambridge University ABSTRACT Conventional one-period utility functions in Economics assume that initial wealth only enters preferences through the definition of final wealth. Consequently, those utility functions most utilized (i.e., exponential and quadratic) have implausible risk characteristics. The authors characterize a new class of utility function whose risk parameters depend upon initial wealth and obtain several desirable results. In particular, investors with quadratic and exponential utility functions can have decreasing risk aversion, and risky assets in a quadratic utility multi-asset environment do not have to be inferior as implied by the traditional framework. Keywords: utility functions, risk, initial wealth JEL classification numbers: D81, G11 Correspondence: Christian S. Pedersen, Oliver, Wyman and Company, 1 Neal Street, Covent Garden, London WC2H 9PU, UK. cpedersen@owc.com. Christian S. Pedersen is also an Honorary Associate of the Financial Econometrics Project, Department of Applied Economics, Cambridge University. The opinions expressed in this article are those of the authors alone and are completely independent of Oliver, Wyman and Company. 357

2 358 BULLETIN OF ECONOMIC RESEARCH I. INTRODUCTION The purpose of this paper is to present a new class of utility function, which has the feature that the parameters constituting part of their specification depend explicitly upon initial wealth. That utility functions could have such a property was recognized as early as 1951 by Mosteller and Nogee (1951), who comment upon their experimental investigation into choice under uncertainty: One possible criticism could be that the subject changes his utility curve with these changes in capital, so that each decision he makes depends on the amount of money he has on hand at that particular moment. Their results provided the motivation for the Consistency Axiom used by Pfanzagl (1959) in his derivation of a theory of measurable utility, and amongst other applications led to the literature on wealth-dependent preference switching, originating with Bell (1988). A strong motivation for this class is the well-known fact that the few tractable utility functions available to economists typically have undesirable risk characteristics for example, the exponential utility function implies constant absolute risk aversion whilst quadratic utility exhibits increasing absolute risk aversion. Arrow (1982) has shown that in an optimal choice over two assets, one riskless and the other risky, investors with such risk characteristics would keep constant or decrease their holdings in the risky asset as wealth increases. This has long been seen as unrealistic and could easily contaminate the many applications of these utility functions to solving decision problems. Further, in a multiasset environment, quadratic utility in its most conventional form implies that risky goods held by investors are inferior i.e., as your wealth increases, your holding in risk assets does not increase. Our suggested utility functions eliminate these problems by specifying extra relationships between initial wealth and the coefficients that measure risk, without departing from the expected utility framework. Although more general theories of choice under uncertainty exist, this particular approach has not, to the best of our knowledge, been explicitly formulated before. Indeed, within a more general theoretical framework, the antecedents of these ideas can be linked to several other initiatives. One may relate the concept of initial-wealth dependent utility to developments in economic theory, which consider aggregation issues by discussing the shape of the wealth distribution across agents and model jointly utility and wealth (see, for instance, Grandmont, 1992 and Hildenbrand, 1983). In addition, applied economists who use expected utility to evaluate performance have been using these ideas for some time (see, for instance, McCulloch and Rossi, 1990 and West, Edison and Cho, 1993). Our approach in this paper does not focus on equilibrium issues, however. In Section II, we present further details of where these ideas have appeared in the literature, define our class of utility function and give a definition of the associated measure of risk-aversion. In

3 UTILITY FUNCTIONS AND INITIAL WEALTH 359 Section III, we specialize to the class of utility functions with hyperbolic absolute risk aversion (HARA) and present conditions under which exponential and quadratic utility have been decreasing absolute risk aversion. We then illustrate the consequences of this for basic investment decisions under risk, generalizing to a multi-asset environment using the general quadratic utility functions. We reserve Section IV for our conclusions. II. UTILITY FUNCTIONS WHOSE PARAMETERS DEPEND UPON INITIAL WEALTH In conventional theory, utility U () is defined exclusively in terms of terminal wealth, ew, which in turn is defined as: ew ¼ W 0 1 þ ex ð1þ for initial wealth W 0 and the rate of return, ex, on a portfolio or simple lottery. Choice under uncertainty is then analysed by maximizing the expectation of Uð ewþ subject to the appropriate budget constraints. In such a framework, W 0 only enters the utility function via ew, although there is no underlying theoretical reason why it should not be allowed to enter separately. We thus present our general class of utility functions. Definition 1. The class of utility functions whose parameters depend on initial wealth, W 0, is given by U W 0 ; ew ; ð2þ where the first component refers to when initial wealth enters the utility function independently of ew. To interpret such a utility function, we find the following ideas helpful. Expected utility theory defines preferences over a set of gambles for a given wealth level W 0. It is clearly possible to conceptualize what one s preferences over the same set of gambles would be if wealth were changed. If we now vary wealth over some index set, then UðW 0 ; ewþ represents a family of utility functions defined over that set. Comparative statics that involved changing attitudes to risk as W 0 vary must presumably have such a model in mind our definition adds no complication except to make the functional dependence explicit through other channels than final wealth, which in turn makes for more flexible modelling of comparative statics. Expected utility theory tells us that, in a context where initial wealth is fixed, it is only final wealth that our preferences are based on. To the authors knowledge, it has not been established anywhere in literature

4 360 BULLETIN OF ECONOMIC RESEARCH that the underlying utility function which forms the basis for the expected utility function has to be independent of endowments, since this preserves linearity in probabilities. Indeed, Mas-Colell et al. (1995), page 185 (top), remark: Although the axioms (of expected utility) yield the expected utility representation they place no restrictions whatsoever on the form of the utility function. Utility functions of the form given by (2) have already appeared in the Economics literature although the authors are not aware of any formal treatment of such until the present. For instance, Grauer (1981) defines a generalized logarithmic utility function U W 0 ; ew ¼ ln ew W 0 2 which forms the basis for an asset pricing model. A further case in Cho, Edison and West (1993) who use a generalized quadratic utility function is discussed later. Functions in (2) also arise naturally when working with target-based preferences, where utility functions defined on returns take different forms above and below a preset threshold (see, for instance, Fishburn, 1977 and Holthausen, 1981). In Satchell (1996) and Pedersen (1999 and 2000), such piecewise utility functions are taken from the Management Science literature and applied to asset pricing. This involves transforming utility functions defined on returns using (1), implying that constant returns targets are transformed to wealth targets that vary with initial wealth, which locates the resulting utility function defined on wealth in the general class (2). This notion of transforming from utility of return to utility of wealth is discussed in more detail in Ingersoll (1987, pp ). An interesting area of possible application of these functions is Portfolio Turnpike Theory (see Ross, 1974 and Huberman and Ross 1983). Portfolio Turnpike Theory is the study of multi-period investment strategies that maximize final period expected utility of wealth and where the strategy is simplified so that the impact of changing wealth through the investment periods is not considered; in Ross (1974) the strategies are defined by constant portfolio weights for the multiple holding periods. It may well be possible to extend existing results to look at certain turnpike problems where the wealth variation changes the risk characteristics of the utility function. Further, Morrison (1998) uses examples of our general class of utility functions when analysing the endowment effect in the willingness to pay and willingness to accept. Indeed, one much-researched related area, which can trivially be extended using this framework is that of Two Fund Money Separation (TFMS), i.e., the property that agents optimally choose a mix of a safe asset and the same bundle of risky assets. This is the cornerstone of several equilibrium theories in Economics. In a fundamental result, Cass and Stiglitz (1970) showed that only the hyperbolic absolute risk aversion functions (HARA) see Eeckhoudt and Gollier (1995) for a detailed description of these have the TFMS property. However, their analysis makes no joint assumptions on initial,

5 UTILITY FUNCTIONS AND INITIAL WEALTH 361 wealth and preferences and so the set of feasible utility functions for TFMS is enlarged when moving to the class (2). This is further explored and discussed in Pedersen (1999). We do not claim that the cited works are the only ones in the literature which use members of our new class of utility functions, nor that (2) is the only way to incorporate additional wealth effects into the existing frameworks. In this paper, we focus on how the definition of absolute risk aversion should be amended and the obvious consequences for optimal decision-making under risk leaving the numerous other applications and possible alternative approaches for future research. II.1. Definition of absolute risk aversion For our utility function (2) we will refer to: R A (W 0 ) ¼ U 0;2(W 0; W 0 ) U 0;1 (W 0 ; W 0 ) ð3þ where U i,j denotes differentiating i times with respect to the first argument and j times with respect to the second argument, as the Conventional Arrow-Pratt Co-efficient of Absolute Risk Aversion (CARA). However, unlike in the conventional framework, this is no longer interpretable as a quantity proportional to the risk premium, ðw 0 ; exþ, the amount of wealth an individual is willing to forego to avoid a mean-zero additive gamble ex, for utility functions in (2). In this case, one should solve for the risk premium using the equation: h i U W 0 (W 0 ; ex); W 0 (W 0 ; ex) ¼ E U(W 0 ; W 0 þ ex) ð4þ where E denotes the expectations operator. In (4), the risk premium enters the left-hand side twice as we forego initial wealth but only once on the right-hand side, since the gamble taken only affects terminal wealth. We thus get the following theorem. (Since all theorems and lemmas in this paper involve standard calculations, we omit proofs, which are available upon request from the authors.) Theorem 1. With utility given as in (2), the risk premium for a small gamble ex ð0; 2 Þ is given by: (W 0 ; ex) ¼ U 0;2 (W 0; W 0 ) U 1;0 (W 0 ; W 0 ) þ U 0;1 (W 0 ; W 0 ) : ð5þ It is immediately obvious that when the first component is zero or, more generally, W 0 does not affect utility independently of ew, equation (5)

6 362 BULLETIN OF ECONOMIC RESEARCH reduces to equation (3) as U 1,0 (W 0, W 0 ) ¼ 0. From (5), it is immediate that the expression corresponding to (3) is: R G A (W 0) ¼ U 0;2 (W 0; W 0 ) U 1;0 (W 0 ; W 0 ) þ U 0;1 (W 0 ; W 0 ) ð6þ which we name the Generalized Arrow-Pratt measure of Absolute Risk Aversion (GARA). It can further be shown that this generalist measure of risk aversion is invariant to affine transformations of the utility function, and so has the same desirable properties as CARA. We make further assumptions about UðW 0 ; ewþ starting with the following standard properties. Assumption 1: U 1,0 (W 0, W 0 ) þ U 0,1 (W 0, W 0 ) > 0. Assumption 2: U 0,1 (W 0, W 0 ) > 0. Assumption 3: U 0,2 (W 0, W 0 ) < 0. The first two assumptions imply that an increase in initial wealth increases utility and that the overall contribution to this increase (including that via final wealth) is strictly positive. It is an interesting consideration that an initial wealth increase could in fact decrease the component which does not include final wealth (i.e., we could have U 1,0 (W 0, W 0 ) < 0, which is the assumption that drives some of our later results). The third assumption is the standard concavity assumption which, combined with the second assumption, asserts that conventional absolute risk aversion holds. We are now in position to use (4) to identify exact conditions under which the risk premium is decreasing in initial wealth. We get the following result. Theorem 2. When utility is given by (2) and Assumption 1 holds, the risk premium decreases in initial wealth if and only if E U 0;1 (W 0 ; W 0 þ ex) þ U 1;0 (W 0; W 0 þ ex) > U 1;0 W 0 (W 0 ; ex); W 0 (W 0; ex) ð7þ þ U 0;1 W 0 (W 0 ; ex); W 0 (W 0; ex) Note that in the case of U 1,0 (.,.) ¼ 0 the result collapses to the traditional characterization in the conventional framework. Also, for utility functions which are three times differentiable in both arguments, (7) can be interpreted as imposing certain conditions on the third partial derivatives. In conventional theory, decreasing risk aversion implies that the third derivative is positive, as can be seen easily from (3). The changes in the general approximation (5) with changes in wealth will similarly

7 UTILITY FUNCTIONS AND INITIAL WEALTH 363 imply restrictions on the third partial derivatives of our general utility function (2). III. CHOICE BETWEEN RISKY ASSETS We now turn to more specific functions in order to illustrate some of the advantages of our new framework. In particular, we will examine the aspects of choice under uncertainty when choosing optimal portfolios made of a riskless and one or more risky assets. In order to illustrate more specifically the power of our new framework, we start by introducing the general quadratic and exponential utility functions. Definition 2. The general quadratic utility function is defined by: U W 0 ; ew ¼ ew b(w 0 ) ew 2 ; and the general exponential utility function is defined by: U W 0 ; ew ¼ e b(w 0) W e : ð8þ ð9þ In both cases, b(w 0 ) > 0 and b(w 0 ) is at least twice differentiable. Clearly, there is no restriction as to how W 0 enters the function. However, for our illustrative purposes, and to ensure mathematical tractability, we find the chosen parameterizations useful. As mentioned in the Introduction, functions of this type have appeared in the literature. For instance, Cho, Edison and West (1993, p. 36), use quadratic utility with constant absolute risk aversion equal to one. This would imply that: 2bðW 0 Þ ¼ 1 1 2bðW 0 ÞW 0 in our notation, which solves for: 1 bw ð 0 Þ ¼ 2(W 0 þ 1) so that, implicitly, b is a function of W 0, and their utility function is thus located in our general class (2). Although we move to a more general framework the quadratic is still consistent with mean-variance preferences. Further, one major criticism of the quadratic utility function in the conventional function is that it displays increasing risk aversion; similarly the exponential utility function

8 364 BULLETIN OF ECONOMIC RESEARCH has constant absolute risk aversion. The following result shows that the generalized exponential and quadratic functions can give both decreasing general absolute risk aversion (DGARA) and decreasing conventional absolute risk aversion (DCARA). Theorem 3. The general quadratic (8) has DGARA iff and DCARA iff 2W 0 bw ð 0 Þb 0 ðw 0 ÞþW0 2 bw ð 0Þb 00 ðw 0 ÞþW 0 b 0 ðw 0 Þ W0 2 ð b0 ðw 0 ÞÞ 2 þb 0 ðw 0 Þ bw ð 0 Þþ2ðbW ð 0 ÞÞ 2 < 0 ð10þ The general exponential (9) has DGARA iff and DCARA iff b 0 ðw 0 Þþ2ðbW ð 0 ÞÞ 2 < 0: ð11þ 2b 0 ðw 0 ÞW 0 þ 2ðb 0 ðw 0 ÞÞ 2 bw ð 0 Þ bw ð 0 Þb 00 ðw 0 Þ < 0 ð12þ b 0 ðw 0 Þ < 0: ð13þ The interpretation of this result for the quadratic case is that an increase in initial wealth needs to decrease the risk-return trade-off sufficiently to off-set the tendencies of agents with quadratic utility to prefer riskless assets as we approach the bliss point bw ¼ 1 2bðW 0 Þ. Another interpretation is that an increase in initial wealth moves the bliss point by decreasing b(w 0 ); moving the bliss point further away makes risky assets more attractive, which implies a decreasing aversion to risk. Taken together, these considerations make a convincing case for b 0 (W 0 ) < 0. We next consider as a specialization of Theorem 3 a power function representation for b (W 0 ) in both the extended exponential and extended quadratic functions. Again, the choice of specific function is, of course, free, but this particular choice enables us to adequately demonstrate the effect of our constructions without unnecessary algebra. Indeed, it allows us to derive more specific conditions in which desirable risk characteristics are attainable, and explore how these characteristics vary in wealth. Corollary 1. Suppose that Then (10) is satisfied if <0 and bw ð 0 Þ ¼ W0 : ð14þ

9 UTILITY FUNCTIONS AND INITIAL WEALTH 365 ð1 ÞW 0 >(þ2)w0 þ1 ; while (11) is satisfied if <0 and W 0 > 2 ðþ1 1 Þ : Also (12) is satisfied if <0 and W 0 > ð 1Þ while (13) is satisfied if <0. Hence, the selection of a power representation for b(w 0 ) allows decreasing risk aversion in both the new and conventional sense and in both exponential and quadratic utility without forcing an overly complex parametric relationship between wealth and utility. As a natural extension of this, we now turn to optimal asset allocations in a world with a riskless and a risky asset. 2 ; III.1. One riskless and one risky asset There are only a small number of utility functions which admit closed form solutions for optimal asset allocations when making non - trivial distributional assumptions. In this section we examine how our new general exponential and quadratic functions ((8) and (9), respectively) can produce more appropriate results and comparative statics with respect to initial wealth. We consider an individual with utility function UðW 0 ; ewþ facing an investment decision where she needs to allocate an optimal amount,, of initial wealth W 0, to a risky asset with rate of return, er, and the remaining amount, W 0, is invested in a safe asset with fixed rate of return, denoted by r f. Clearly, final wealth will then be given by ew ¼ W 0 1 þ r f þ er rf ; and the optimization problem becomes max U W 0; W 0 1 þ r f þ er rf : ð15þ Before moving to specific cases we present a general result which, for our framework, corresponds to the standard result in portfolio theory proved in Arrow (1982) and Huang and Litzenberger (1988, Chp. 2) that with

10 366 BULLETIN OF ECONOMIC RESEARCH decreasing risk aversion, the fraction allocated to the risky asset decreases in initial wealth. Theorem 4. If the general utility function (2) exhibits DCARA and E U 1;1 (W 0 ; W) f er r f > 0 > 0 If (2) exhibits ICARA and E U 1;1 (W 0 ; W) f er r f < 0 < 0 Once more, one can easily confirm that these conditions reduce to the conventional result in Arrow (1982) when we remove the independent initial wealth term. We now use this result to demonstrate how asset allocations with the general quadratic and exponential functions differ from the conventional set-up. Theorem 5. For the general quadratic utility function in (8), the optimal allocation in risky assets is a ¼ (1 2cb(W 0)) 2b(W 0 )( 2 þ 2 ) ; ð16þ where c ¼ W 0 (1 þ r f ), ¼ Eðer r f Þ > 0 and 2 ¼ Varðer r f Þ. > 0 iff b 0 ðw 0 Þ < 4 1þ r f ð bw0 ð ÞÞ 2 : 0 This shows that under certain circumstances we can have an increase in the allocation to equity as we get wealthier when using quadratic utility. This corresponds to the earlier results on decreasing risk aversion. We get the following similar result for general exponential utility. Theorem 6. For the general exponential utility function (9) the optimal fraction invested in equity, a, increases in initial wealth if and only if

11 UTILITY FUNCTIONS AND INITIAL WEALTH ðw 0 bw ð 0 ÞÞ < 0: 0 Consequently the exponential utility function will no longer produce wealth-independent optimal asset allocations, which has been its main shortcoming in earlier theory. Since it is immediate that this generalization is at no extra cost in terms of computational or algebraic complexity, Theorem 6 essentially merges one of the most fundamental properties of investor behaviour with the exponential utility representation without incurring undue complications. Given the popularity of the exponential utility in Economics, precisely due to the mathematical tractability, we consider the generalized exponential utility function (9) one which merits the attention of theorists and practitioners alike. III.2. Asset demand with several risky assets We now consider the case where an investor is choosing between several risky assets and, in particular, illustrate how our new framework can overcome the common problem that optimally selected assets with quadratic utility are inferior that is, as initial wealth increases, your holding in the risky assets decreases. We shall thus focus on the generalized quadratic function introduced earlier, but a parallel analysis could be performed for the exponential. Suppose then that an investor maximizes U W 0 ; ew ¼ ew 1 2 b(w 0) ew 2 ð19þ where b(w 0 ) > 0and ew is restricted so that Assumptions 1, 2 and 3 above are satisfied (similar to the restriction U 0 ð ewþ > 0 in the traditional case). We also assume there are (n þ 1) assets of which the first n are risky and asset ðn þ 1Þ is assumed riskless. The vector ¼ð 1 ; 2 ;...; n Þ denotes the expected returns on the risky assets and the covariance matrix of the assets is denoted by ¼ () ij.asusual, is assumed positive definite, and and e ¼ ð1; 1;...; 1Þare assumed linearly independent. If x i denotes the real value of current holdings in asset i, andr f the return on the riskless asset, then W 0 ¼ P nþ1 i¼1 x i, and final wealth is defined as: ew ¼ Xnþ1 x i (1 þ i ) i¼1 ¼ Xnþ1 i¼1 x i ( i r f ) ¼ W 0 (1 þ r f ) þ x 0 ( r f e) ð20þ

12 368 BULLETIN OF ECONOMIC RESEARCH in matrix format. Consequently, by defining ¼ r f e, we can write expected utility EðUðW 0 ; ewþþ as: W 0 1 þ r f þ x bw ð 0Þ W0 2 1 þ r 2þ2W0 f 1 þ r f x 0 þ x 0 0 x þ x 0 x ð21þ and so the first-order conditions with respect to the choice of weights on the risky assets U W 0 ; ew ¼ 0 ) ð1 bðw 0 ÞW 0 ð1 þ r f ÞÞ ¼ bðw 0 Þð 0 þ which imply that the optimal allocation bx in the risky assets are: 1 bx ¼ 0 1 þ bw ð 0 Þ W 0 1 þ r f : ð22þ Using the formula for partitioned inverses, this can be written as: 1 bw ð 0 Þ W 0 1 þ r f bx ¼ 1 þ : ð23þ Thus, assuming that, and b(w 0 ) are independent of current prices, it is clear b0 ðw 0 Þ 1 þ r bw ð ¼ 0 Þ 2 f 1 : 0 1 þ 0 1 Now, consider the standard case when b(w 0 ) ¼ b > 0. Since, in that case, you need 1 b W 0ð1 þ r f Þ > 0 to ensure U 0 ð ewþ > 0, the sign of bx 0 are the same in fact determined by the sign of 1 so i i that as you get wealthier, you reduce the absolute holding in all risky assets and increase your holdings in the safe asset. In particular, in a situation where all risky assets are held long, a wealth increase causes you to decrease your risk profile by selling all risky assets! In other words, all risky assets are inferior goods. This is clearly at odds with fundamental observations and basic assumptions in Finance. In the general case, however, this can be avoided by an appropriate choice of b(w 0 ). More specifically, any function satisfying the relationship b0 ðw 0 Þ ð1þr bðw 0 Þ 2 f Þ > 0 and the assumption that bx i is positive for all i, would ensure that all risky goods are normal.

13 UTILITY FUNCTIONS AND INITIAL WEALTH 369 The result above is merely illustrative of the usefulness of our approach. Hamada and Royama (1971) provide an alternative solution to the problem, but they do not include a riskless asset and use expected rates of return as prices and expected wealth as wealth. Hence, in their model, inferior is defined relative to changes with respect to expected wealth and not, as in our model, by changes in initial wealth, consistent with the traditional characterizations in microeconomics. We further add that our framework can also be used to examine conditions under which assets are substitutes or complements, as defined in the original microeconomic sense. This links our research to the correlation between assets (clearly, you expect positively correlated assets to be good substitutes), which could potentially be used to examine the empirical usefulness of our, and the traditional, approaches. IV. CONCLUSION This paper has considered how to extend utility functions to incorporate initial wealth into the risk parameters. Whilst the mathematics of the extensions are straightforward, the conclusions that follow seem to be helpful in eliminating some of the unrealistic features of the small number of tractable utility functions available to students of decision making under uncertainty. Our work is not exhaustive; the list of previous examples of the use of initial-wealth dependent utility functions in the literature together with the new examples presented in the paper does not cover all useful applications. For instance, in a multi-period setting our utility functions are related to those used in works on habit formation (initially popularized by Pollak, 1970; see more recent references in Alessie and Lusardi, 1997), where utility functions are defined over not only current consumption but also the fraction of wealth not consumed in the previous period. This is also a feature of the household technology preference structure of Hansen and Sargent (1996), who make the bliss-points of their multi-period quadratic utility functions depend upon past consumption. A further link (pointed out to us by an anonymous referee), which expands on an earlier observation in Section II, is to the endowment effect in the contingent claims valuation literature (see, for instance Thaler, 1980, Knetsch, 1989 and Kahnemann, Knetsch et al. 1990). We hope that since our framework still yields closed-form solutions to the corresponding optimization problems in the traditional framework, extensions made immediate by using our new class of utility functions will enable researchers to resolve possible anomalies with regards to changes in wealth in these and other areas of Economics.

14 370 BULLETIN OF ECONOMIC RESEARCH REFERENCES Alessie, R. and Lusardi, A. (1997). Consumption, saving and habit formation, Economics Letters, vol. 55, pp Arrow, K. (1982). Risk perception in psychology and economics, Economic Inquiry, vol. 20, pp Bell, D. (1988). One-switch utility functions and a measure of risk, Management Science, vol. 34, pp Cass, D. and Stiglitz, J. (1970). The structure of investor preferences and asset returns, and separability in portfolio allocation: A contribution to the theory of mutual funds, Journal of Economics Theory, vol. 2, pp Cho, D., Edison, H. and West, K. (1993). A utility-based comparison of some models of exchange rate volatility, International Economics, vol. 35, pp Eeckhoudt, L. and Gollier, C. (1995). Risk: Evaluation, management and sharing. Prentice Hall, Harvester Wheatleaf. Fishburn, P. (1977). Mean-risk analysis with risk associated with below-target returns, American Economic Review, vol. 67, pp Grandmont, J. (1992). Transformations of the commodity space, behavioral heterogeneity, and the aggregation problem, Journal of Economic Theory, vol. 57, pp Grauer, R. (1981). Investment policy implications of the capital asset pricing model, Journal of Finance, vol. 36, pp Hamada, K. and Royama, (1971). Substitution and complementarity in the choice of risky assets, in: Hesler, D. D. and Tobin, J. (eds. ) Risk Aversion and Portfolio Choice, Wiley and Sons, New York. Hansen, L. and Sargent, D. (1996). Recursive linear models of dynamic economics, mimeo. Hildenbrand, W. (1983). On the law of demand, Econometrica, vol. 51, pp Holthausen, D. (1981). A risk-return model with risk and return measured as deviations from a target return, American Economic Review, vol. 71, pp Huang, C. and Litzenberger, R. (1988). Foundations for Financial Economics, London: Elsevier Science Publishing Company Inc. Huberman, G. and Ross, S. (1983). Portfolio turnpike theorems, risk aversion and regularly varying utility functions, Econometrica, vol. 51, Ingersoll, J. (1987). Theory of Financial Decision Making, RowmanandLittlefield Publishers Inc. Kahnemann, D., Knetsch, J. L. and Thaler, R. H. (1990). Experimental tests of the endowment effect and the coarse theorem, Journal of Political Economy, vol. 98(6), pp Knetsch, J. L. (1989). The endowment effect and evidence of non-reversible indifference curves, American Economic Review, vol. 79(5), pp Mas Colell, A., Whinston, M. and Green, J (1995). Macroeconomic Theory, Oxford University Press: Oxford. McVulloch, R. and Ross, P. E. (1990). Posterior, predictive, and utility-based approaches to testing the arbitrage pricing theory, Journal of Financial Economics, vol. 28, pp

15 UTILITY FUNCTIONS AND INITIAL WEALTH 371 Morrison, G. C. (1998). Understanding the disparity between WTP and WTA: Endowment effect, substitutability, or imprecise preferences? Economics Letters, vol. 59(2), pp Mosteller, F. and Mogee, P. (1951). An experimental measurement of utility, Journal of Political Economy, vol. 59, pp Pedersen, C. (1999). Topics in risk in finance, PhD thesis, Trinity College, University of Cambridge, England. Pedersen, C. (2000). Separating risk and return in the CAPM: A general utilitybased approach, European Journal of Operational Research, (forthcoming) Pfanzagl, J. (1959). A general theory of measurement applications to utility, Naval Research Logistics Quarterly, vol. 6, pp Pollak, R. A. (1970). Habit formation and dynamic demand functions, Journal of Political Economy, vol. 78, pp Ross, S. (1974). Portfolio turnpike theorems for constant policies, Journal of Financial Economics, vol. 1, pp Satchell, S. (1996). Lower partial moment capital asset pricing model: A re-examination, Discussion Paper 20, IFR Birkbeck College. Thaler, R. (1980). Towards a positive theory of consumer choice. Journal of Economic Behaviour and Organisation, 1, pp

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