Compatibility of Expected Utility and Approaches to Risk for a Class of Non Location-Scale Distributions

Compatibility of Expected Utility and Approaches to Risk for a Class of σμ / Non Location-Scale Distributions By Gerry Boyle and Denis Conniffe Abstract Proofs of compatibility of the expected utility and approaches to incorporating uncertainty in decision making exist for at least some utility functions and location-scale distributions. But there are severe constraints and it is desirable to investigate compatibility more widely. We do so for the class of distributions that are transformable to location-scale form by concave transformation and where the utility functions remain concave under transformation. The class is important, containing distributions such as the lognormal and Pareto, usually considered more appropriate for modelling income or wealth than those in the location-scale family. σμ / 1. INTRODUCTION As is well known, two frequently employed approaches to incorporating uncertainty about some key variable – often income or wealth - in analyses of the comparative statics of optimal decision-making are analysis and maximisation of expected utility. The former approach takes uncertainty as representable by the standard deviation of the variable. The decision maker is assumed to proceed by constrained maximisation of some function of the mean and standard deviation, which is increasing in the mean, decreasing in the standard deviation (the monotonicity conditions) and quasiconcave in σμ /  and  . The expected utility approach commences from a utility function which is monotonically increasing and concave in the variable. Uncertainty is embodied in a probability distribution for the variable and the decision maker evaluates outcomes in terms of expected utility. Elementary texts often remark that the approaches are equivalent for either a quadratic utility function or normality of distribution, although the latter case is really greatly constrained by being conditional on the existence of the expected utility. For other situations, the consistency of the two approaches has been debated at considerable length in the literature, which is so extensive that only publications directly relevant to the theme of this paper will be mentioned. Meyer (1987) and Sinn (1989) showed that equivalence of expected utility maximisation and analysis could be extended from the normal distribution to the location-scale family of distributions although again this is actually conditional on the existence of expected utilities. This is one of the key issues in discussing equivalence. Some of the most frequently advocated utility functions u(x) do not have expectations under normality, or some other location-scale distributions, but do under plausible distributions that are not in the location-scale family. For example, σμ / 2 xxu log)(  or , with  xxu )( 10    , do not have expectations under normality. Nor have they expectations under other location-scale distributions where x can be , such as the Gumbel. There are location-scale distributions for which x must be positive, such as the two parameter uniform and the two parameter exponential, but these are surely implausible as models for income or wealth. 0 However, the expectations do exist for several two parameter distributions that are not of locationscale form and that are plausible as models for income or wealth. The Pareto distribution was one of the first advocated (for example, Arnold, 1983) for that purpose1, but it is not in the location-scale family. The log normal has been frequently employed for income modelling, justified by both theoretical (for example, Aitchison and Brown, 1957) and empirical findings and, of course, it is not of location-scale type. Other non location-scale two parameter distributions and various multiparameter distributions have also been proposed (for example, Bandourian, McDonald and Jurley, 2003). Meyer’s insight that if an originally multiparameter non location-scale distribution is employed in circumstances where all parameters except the location and scale parameters are held constant it effectively becomes a location-scale distribution, does establish equivalence for some applied problems. But this situation, termed the ‘location scale condition’, must be of limited occurrence. So it is important to investigate if the equivalence between the expected utility function and approaches holds for the distributions for which the expectations exist, but that are not in the location-scale family. The approach in this paper is to investigate two parameter distributions of x that are not location-scale, but where the distribution of y = h(x) is location scale, where h(x) is a monotonically increasing and concave function of x. For example, if x is log normal, y = log x is normal and if x is Pareto, y = log x follows a two parameter exponential and this, like the normal, is a location-scale distribution. If we also require that our utility functions are concave when expressed as functions of y, we can show that equivalence will hold in the sense that σμ / ),())((   VxuE satisfies the monotonicity conditions throughout ),(   space and is quasiconcave in  and  in a region of that space. In our proofs we will exploit Meyer’s (1987) results for the location-scale case, although we will need to broaden his proof somewhat. The location-scale family is defined by the density 1 Nowadays it is recognised that the Pareto is unsuitable as a general income distribution, because it does not fit well at low incomes, although it does fit well to a population of higher income groups. But that might be the relevant population in contexts such as investment. 3          x f 1, where  is the location and  the scale parameter. The mean and standard deviation are 1 k     and 2 k    respectively, where and are constants. Meyer assumed 1 k2 k  and  equal to  and  , which would be true for a normal, but not for some other location-scale distributions, and the lower and upper variable bounds independent of parameters. These assumptions are not necessary and would, for example, prevent us from employing the two parameter exponential, where the lower bound depends on the location parameter. Our somewhat more general proof is given in Appendix A. 2. CONCAVE TRANSFORMATIONS TO LOCATION-SCALE DENSITIES As already mentioned, y = h(x) is monotonically increasing and concave and y has a location-scale distribution. The case y = x corresponds to x location-scale. Suppose the inverse function is . Clearly, any number of distributions of x with the required property can be found by obtaining the distributions of with h(x) any appropriate function and y following any of the location-scale distributions. But whether these distributions of x are good fits for income or wealth distributions is another matter. Some might be, even if they have not appeared in the literature, but probably many would not be. So, although much of the paper maintains the generality of y = h(x), our examples will use y = log x, where we know appropriate distributions exist. )()( 1ygyhx   )()( 1ygyhx   Suppose the utility function u(x) =u(g(y)) is also concave in y. Examples implying familiar utility functions are: bybyyyuxxbxxu 2/1,)(,1,)(loglog)( 22  10,)(,1,)(log)(    yyuxxxu 0,1)(,0,)( log     yx eyuxeAxu yyuxxxu log)(,1,loglog)(   . 4 Since y = h(x) has a location-scale distribution, we know that the expectation of u(y), , has ),( **  V * 2 the required properties for monotonicity and concavity with respect to the mean and standard deviation of the variable y. Of course, the expectation of u(x) with respect to the distribution of x must also be , since it amounts to just a change of variable in integration. However, we are interested in its properties with respect to  *  V),( **   and  and not and . We will investigate these in the next two sections. *  *  Since there are evidently concave utility functions of x that will not be concave in terms of y, we are restricting the class of utility functions somewhat. Some may be totally excluded, while others may require restriction of the range of the variable or parameters. For example, ,0,)( log  xexxu x  ,10    is concave in x, but with y = log x, y eyu  )( is not concave in y as its second derivative is positive. However, , x > 0. x exu   1)( with y = log x would give . y e eyu   1)( and then  yey eee y uy     1 2 2 . which is negative if the bracketed term is. So concavity requires 1  , or y > - log  . 3. THE MONOTONICITY CONDITIONS Since satisfies monotonicity conditions we know that ),( **  V *    V and *   V 2 Assuming V exists, of course. )(log)log(log yExE  will not exist if y is normal, that is, if x is log normal, but it will if y is two parameter exponential, that is, if x is Pareto. 5 are positive and negative respectively. Now                   * * * * VVV and                 * * * * VVV . So if    * and    * (1) are positive, as is plausible from y = h(x) being a monotonically increasing function of x, and    * and    * (2) are negative, as is plausible from y = h(x) being concave in x, the first derivatives of V with respect to  and  will be positive and negative respectively. The detailed proof is given in Appendix B. Of course, (1) and (2) imply the slope of an indifference curve        VV d d S/ is positive. For a first example, we take the case of x log normally distributed, so that y = log x is normally distributed and the utility function as u(x) = log x. This might seem a rather trivial utility function to choose, since then u(y) = y, which is on the limit of concavity, while and the slope of the indifference curve in space is zero. However, the case is important, both historically and because many textbooks on investment or portfolio theory (for example, Elton and Gruber, 1995, p.234 ) state that expected utility and analysis are equivalent for a log utility function given log normality. The claim is based on results in Elton and Gruber (1974), but the case had previously been analysed by Feldstein (1969) and commented on by others. *** ),(  V ),( **  μ σ/ Statistical textbooks write the parameters of the log normal as the mean and variance of the log of the variable, that is, of and . The mean and variance of the lognormal are *  *  6 2** 2 1    e and   1 2*2** 22     ee . Expressing in terms of *   and  gives         2 2 *1log 2 1 log    (3) and         2 2 2* 1log    . (4) Since is also *  ),(   V, differentiation of (3), as performed by Elton and Gruber (1974 ), gives )( 2 22 22        V and 22         V, which are positive and negative respectively. Utility is increased by increasing  for fixed  or decreasing  for fixed  . So if the set of possible values in ),(   space is bounded by a concave frontier the expected utility maximum will lie upon that frontier. The issue of precisely where on the frontier depends on how the slope of the indifference curve 22 2        d d S (5) changes along the curve and this will be returned to in the next Section. Of course, the signs of the derivatives of V with respect to  and  were already guaranteed by the general proof in Appendix B. In fact, for every utility function u(x) = u(g(y)), concave in y as well as x, with y = h(x) following a location-scale distribution, expected utility is compatible with analysis in this sense of the σμ / maximum occurring on the frontier. There are obviously very many possibilities, but keeping to lognormal x, one interesting case is . yx eAeAxAxu     log )( where 1  . In terms of y, this is the very frequently employed constant absolute risk aversion utility function. From the well known moment generating function of the normal distribution ),( **  V2*2* 2 1     eA . 7 It may be worth remembering that , which is not a function of in **** /   ddS *  accordance with constant absolute risk aversion. Substituting (3) and (4) into gives ),( **  V ),(   V=)/1log()1( 2 log 22       eA , (6) with    22 22 )2(          VA V,  22 )1(          VA V and 22 )2( )1(       S. (7) So S is a function of  and since  22 22 )2( )2(        S S. (8) This will be negative, showing decreasing absolute risk aversion, which is often considered plausible, if 22 )2(   . It could be positive for low mean income (or high  and variance), but as will be seen in the next section, quasiconcavity requirements are then infringed. Sinn (1989, p.152 ) presents a utility function of the form (6), although his derivation followed a quite different path. Assuming log normality is not essential, of course. Suppose x has a Pareto distribution     x x 1 1. The mean and variance of the distribution are 1     and )2()1( 2 2 2      The log of a Pareto has the (location-scale) two parameter exponential density ,, 1 )( )(   yeyf y     where   /1 and   log. The mean and variance of the exponential are and .   *22*   8 Taking u(x) = log x give as in the log normal case, but expressing in terms of s *** ),(  V*   and gives instead of (3) 2                            1111log),( 2 2 2 2 2 2 2 22 *            V . nd noting that Again, taking yx eAeAxAxu     log )( a              edye yy 1 )( 1 , we get                                   2 2 2 22 22 22 11 /11 /11 ),( AV stead of (6). Many other ),(   V incan be obtained corresponding to various u(x) and distributions of n the tractability of the integrals involved, mathematical expressions can metimes be difficult. 4. THE QUASICONCAVITY CONDITIONS x, although, depending o so We also want quasiconcavity of V with respect to  and  , so that the indifference curves are convex. The quasiconcavity condition is non-negativity of 3 2 2 2 2/2 222                                                                   VVVVVVVV . (9) While this will hold for low values of  for all utility functions and distributions, Appendix B shows it will not remain so over the whole of ),(   space for the class of utilities u(x) =u(g(y)) defined in Section 2. The quasiconcavity region depends on bo ies formation to locationscale and the behaviour of the indifference curve of ),( in ),( space. Seeking a single formula covering all transformations and utility functions leads to extremely unwieldy algebraic terms. th the propert of the trans f x log normal x and any u(x) = u(g(y)), concave in y as well as x, non negativity of (9) requires that **  V**  For the case o 9                                                        ** * 21 2 )( * 2* 2 * * 22 * * 3 * * 22 22          S d dSSS S, (10) with * * * * * * *       S S S d dS , be non-negative. This shows that quasiconcavity certainly holds in the interval 2 1 * *2            S, (11) when the elasticity of the indifference curve in space ),( **  * * log log  d Sd is equal to unity. When the elasticity is greater than unity the interval expands, since  can become somewhat larger, and when the elasticity is less than unity the interval contracts correspondingly. But it is difficult to make more clear-cut statements without considering specific utility functions. Similar conditions to (10) can be obtained for other distributions transformable to location-scale form, but in the case of any particular utility function, it is usually much easier to proceed by obtaining S as in the previous Section and then examining the positivity of .               S S S d dSS d dS This is what Feldstein (1969) did for the case of u(x) = log x with x log normal. Simple differentiation of (5) shows 322 2222 )2( )2)((       d dS so that convexity of the indifference curve requires 2/  . This has already been obtained more tediously in Appendix B and would also follow from (11). As mentioned in the previous section, Elton and Gruber (1974) also examined this case, but unlike Feldstein, did not think the lack of 16   dwwfkwu k V)()(...)(' 1 *1 2  and since is decreasing in x, negative values of )(' xu 1 kw  are being multiplied by larger values than are positive values. So, in view of (A1), the derivative of V with respect to *  is negative. Then the slope of an indifference curve * / ** * *        VV d d S is positive.    dwwfu V)((...)'' *2 2  is negative since is negative and )('' xu   dwwfkwu k V)()(...)('' 1 * 2 1 2 2 2 2  is negative for the same reason. The sign of   dwwfkwu k V)()(...)('' 1 ** 1 2 2  is unclear. It is positive if is positive and negative if is negative, but in either )(''' xu )(''' xu situation the Cauchy-Schwarz inequality implies 2 2 2 2 2 2 ** **                  VVV is non-negative. So V is a concave function of and , indifference curves are convex, and the *  *  equivalence of analysis to the expected utility approach is evident. While concavity is not σμ / essential for convexity of an indifference curve and that quasiconcavity will suffice, concavity implies quasiconcavity. Also, the main focus of this paper is to utilise these results to extend the equivalence to distributions that are not location-scale, but are transformable to that family. APPENDIX B EQUIVALENCE FOR TRANSFORMABLE DISTRIBUTIONS AND RESTRICTED U(X) Proving Monotonicity To show ),(   Vsatisfies monotonicity.we have to prove that the terms of (1) 17    * and    * are positive and that the terms of (2)    * and    * are negative. The distribution of x is not location-scale, but that of y=h(x) is. Now h(x) is assumed increasing and concave. The inverse transformation x=g(y) must be such that x=g(h(x)). Then .1 x h y g      So g(y) is increasing in y. Also 2 2 2 2 2 0x h y g x h y g                 and since h(x) is concave, its second derivative is negative and so the second derivative of g(y) is positive. That is, g(y) is increasing and convex. Clearly dy y fyg u l c c                 )( 1 or, with the same substitution as in Appendix A,     u l c c dwwf k kw g)())(**( 2 1  . Taking the limits of integration as understood and, for convenience, omitting arguments of g    ,)(' *dwwfg   which is positive since g(y) is increasing in y.   dwwfkwg k)()(' 1 *1 2   and since is increasing in y, negative values of )(' yg 1 kw  are being multiplied by smaller values than are positive values. So, in view of (A1), the derivative with respect to *  is positive. Also 18  dy y fyg u l c c                  2 2)( 1 or          u l c c dwwf k kw g)())(**( 2 2 1 2  . Then .)(')(2 * 2 * 2dwwfgg             Noting that, by definition, (B1) dwwfg )()(0   and remembering is increasing, negative values of 'g   g are being multiplied by smaller values than are positive values and so (B1) implies the derivative of  with respect to is positive. Again, *  .)()(')( 2 * 2 *1 2 2dwwfkwgg k            When and 1 kw    g the sign inside the integral is positive. When and 1 kw   g it is positive and when and 1 kw   g the sign inside the integral is also positive8. Then the fact that is increasing ensures the derivative of 'g  with respect to is positive. So the terms in the *  matrix M =                     ** **         are all positive. Now              d d                     ** **                   * *   d d and of course 8 Since  is the mean of a positive increasing convex function of y, it will be much larger than , which is zero for a normal or uniform, .5572 for a Gumbel and 1 for a two parameter exponential. 1 k 19            * *   d d                             ** ** .             d d So                             ** ** =1 M and evidently the diagonal elements are positive and the off diagonal elements negative. So    * and    * are positive and    * and    * are negative and ),(   Vsatisfies monotonicity. Investigating Quasiconcavity We need ),(   Vquasiconcave in  and  to ensure the indifference curve is convex. The condition for this is that 3 2 2 2 2/2 222                                                                   VVVVVVVV (B2) be positive. Now                                        * * * *2 2VVV or, working out terms 2 *2 *2 *2 * 2 * 2* 2** ** 2 2 * 2* 22                                                          VVVVV . Similarly 2 2   V is 20 2 *2 *2 *2 * 2 * 2* 2** ** 2 2 * 2* 22                                                          VVVVV and   V 2 is . 2 *2 * *2 * ** 2* 2**** ** 2** 2* 2                                                                                                     VV VVV Substituting these into the numerator of (B2) many terms cancel and the remaining terms give                                                     2 *2 * **** 2 *2 * 222 2     VVVVVVV 2 ****                          2 * * * *2 *2 *2 *2 *                                           VVVV 2 * * * *2 *2 *2 *2 *                                           VVVV                                                                     * * * * * * * * *2 * *2 * 2VVVVVV . The first term of this expression is the product of a squared term and the condition that is ),( **  V quasiconcave. But that condition must be true because Appendix A proved concave in ),( **  V *  and . So the first term is non-negative. However, it is unclear that the sum of all terms is non- *  negative for all (  ,  ) space. In the extreme situation of u(x) = h(x) = y, and all *** ),(  V second derivatives of V with respect to and are zero and the first term of the expression is zero. *  *  As might be expected the remaining terms reduce to                                                          2 * 2 **** 2 2 *222 2 *               . (B3) So we require , the expectation of the transformation, to be quasiconcave in *   and 21  . Now if we take x log normal and y = h(x) = log x, so that y is normal, (B3) can be shown by rather tedious evaluation of terms to be 2222 22 )( )2(     and this can be negative unless  2. Remaining with x lognormal, but any u(x) that is a concave function of y = log x, very laborious manipulation enables (B2) to be written                                                        ** * 21 2 )( * 2* 2 * * 22 * * 3 * * 22 22          S d dSSS S, (B4) where * * * * * * *       S S S d dS . Another formula obtained during the derivation of (B4) is                                                                    ** * )2( 21 2 1 * 2* 2 * * * 22 2 * * 2 * * 2 * * 22             SS SSS S S, (B5) which is employed in Section 5. Returning to (B4), the second term within chain brackets in (B4) could be positive, negative or zero depending on whether the elasticity of the slope along the indifference curve associated with ),( **  V is greater than, equal to, or less than, unity. The first term could be negative too, because *** ),(  V implies and all terms within the chain brackets vanish except , 0 *S22 2   implying, as before,  2 for positivity. So the region of quasiconcavity in ),(   space depends on both the quasiconcavity of the transformation expectation and the elasticity of the indifference curve in . ),( **  Corresponding conditions to (B4) can be obtained for other distributions transformable to locationscale form, but attempting to obtain a single formula applicable to all the relevant distributions and utility functions seems to result in almost intractable algebraic expressions. In any event, the 22 quasiconcavity region for particular cases is more easily obtained by direct examination of the convexity of the indifference curve in ),(   space via the positivity of       S S S d dS without explicit consideration of or its indifference curve. The examples in section 4 ),( **  V exemplify this. It may be worth remembering that for sufficiently small  , the expectation of any utility function can be written  )('' 2 1 )(),()( 2  uuVxuE  , so that )( )(' )(''     A R u u S , where is the Arrow-Pratt coefficient of absolute risk aversion. Then A R                    )( 1)( 2A A R R S S S d dS and this is positive, even if decreasing absolute risk aversion holds, if  is small enough. So every indifference curve commences with a convex region, the extent of which depends on the properties of the distribution and the utility function. REFERENCES Aitchison, J. and J. A. C. 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