Relative Risk Aversion: What Do We Know?
Preliminary Draft Please Do Not Quote
Donald J. Meyer Department of Economics Western Michigan University Kalamazoo, MI 49008 [email protected]
Jack Meyer Department of Economics Michigan State University East Lansing, MI 48824 [email protected]
Abstract: The magnitude and slope of the relative risk aversion measure depends on the variable used as the argument of the utility function, and on the particular definition or measure used for that variable. Furthermore, the various relative risk aversion measures that result from these different variables are related to one another in a manner determined by the relationship between the variables themselves. These facts are used to make adjustments to several estimates of relative risk aversion so that those estimates can be directly compared with one another. After adjustment, the significant variation in the reported estimated values for relative risk aversion for a representative agent is substantially reduced.
1 Relative Risk Aversion: What Do We Know?
1. Introduction When economists model choices made by expected utility maximizing decision makers, a variety of variables are used to represent the possible outcome of the decision being represented. Wealth, consumption or profit are most often used, although a number of other variables, including income, return, or rate of return are sometimes selected. In addition to using different variables as the argument of the utility function, the various studies also frequently define or measure the chosen variable in different ways. Profit for agricultural producers, for instance, might or might not include the value of unpaid labor provided by the farm family. Similarly, wealth usually, but not always, excludes the value of human capital, and the return to a portfolio can be calculated before or after taxes are paid on investment income. These variations in the way the outcome of a risky choice is defined or measured can affect the measure of relative risk aversion determined for the representative decision maker, changing both its magnitude and its slope. Failure to fully recognize this fact has contributed significantly to the confusion that prevails concerning what is known about relative risk aversion and risk preferences in general. There are instances where the results from one study are viewed as contradicting those from another, when, after adjusting for differences in outcome variables, the results support rather than contradict one another. The opposite also occurs; evidence concerning the slope and level of relative risk aversion for one outcome variable is sometimes used to support that same assumption for an entirely different outcome variable, when in fact, the two relative risk
2 aversion measures cannot be the same and must differ in a predictable way. Even after forty years of published research concerning measures of relative risk aversion, this confusion is viewed by some as so severe that the existing body of evidence is dismissed in favor of introspection.1 The variation in the choice of outcome variable and the way it is measured has also lead to the scattered and disconnected nature of the information that is available concerning risk attitudes. The risk attitudes of decision makers such as those of agricultural producers, or consumers, or stock market investors, are each studied separately. Most often, information concerning risk preferences obtained for one group is not used when specifying the risk taking characteristics of other groups, and when it is, the specification often fails to adjust appropriately for the variation in the outcome variables. The purpose of this study is to help remedy this situation by carefully reviewing a small portion of the information that others have provided concerning relative risk aversion measures for expected utility maximizing decision makers. In each of the studies examined the reported measure of relative risk aversion is taken as it is given, but special attention is paid to how the outcome variable is defined or measured, and the effect that this has on the measure of relative risk aversion that is obtained. This paper has two parts. The first part provides a framework for comparing and contrasting information concerning relative risk aversion from the various theoretical and empirical studies with their differing outcome variables. Since a single set of risk preferences can be represented by many different relative risk aversion measures, one for each different
3 outcome variable, a reference outcome variable is suggested, and its relative risk aversion measure is chosen to be the common scale for measuring relative risk aversion. After this first step, the second part of the analysis begins the task of gathering, and translating to this common scale, the information concerning risk aversion acquired during the past forty years. This task is a lengthy one, and only a very small portion is accomplished in this particular study. The papers chosen for review are among the more prominent studies for each of three important outcome variables. One paper for each variable is reviewed in considerable detail, while the others are discussed briefly. Information obtained from theoretical analysis as well as empirical evidence is examined. The outcome variables considered are wealth, consumption and profit. The risk aversion information for utility from wealth is derived mainly from examination of portfolio decisions, while that for utility from consumption comes from a consumption survey and from asset pricing studies. The risk attitudes associated with utility from profit are based on decisions made by agricultural producers. The main focus and contribution of this paper is the translation of known information concerning relative risk aversion to a common measurement scale. An important and perhaps unexpected secondary contribution, however, is the finding that the evidence concerning relative risk aversion for various decision makers, and for a very wide variety of outcome variables, is more unified and consistent than casual inspection would indicate. Both the slopes and magnitudes of relative risk aversion measures are less different across decision makers, and across a wide variety of decisions, than the unadjusted measures seem to indicate.
4 The paper is organized as follows. First, in the next section, the framework of discussion and the notation used in the paper are given. The focus is on the measure of relative risk aversion for a decision maker. A particular reference variable and the associated measurement scale are suggested as being natural ones to use when discussing information concerning relative risk aversion. Following this discussion are three separate sections reviewing information concerning relative risk aversion for wealth, for consumption, and for profit, respectively. Finally, the paper concludes with a brief summary of what is known concerning relative risk aversion, both level and slope, for individuals making risky economic decisions.
2. Notation and Framework of Analysis For forty years now, the measures of risk aversion defined by Pratt and Arrow have been used to describe the risk attitudes of expected utility maximizing decision makers. These measures apply most directly to a utility function that has a single argument, and this is the only case considered here. The expected utility maximizing decision maker is assumed to make choices, and these choices then combine with the realizations of random parameters to yield a single dimension random outcome such as wealth or consumption or profit over which utility is defined. For the purposes of this review, the decision model need not be specified more precisely than this. Initially the discussion uses X and Y to represent two different random outcome variables, or perhaps the same outcome measured or defined in two different ways. For a particular decision maker, u(X) and v(Y) denote the utility function whose expected value is maximized when the outcome variable is X or Y, respectively. Even though
5 these two utility functions are associated with, and represent a single set of risk preferences, they are likely to differ due to the different outcomes being considered. Given utility function u(X) or v(Y), the definitions of Pratt and Arrow specify two primary risk aversion measures employed in economic analysis. Au(X) = -u''(X)/u'(X) denotes the measure of absolute risk aversion, and Ru(X) = Au(X)·X = -u''(X)·X/u'(X) denotes the measure of relative risk aversion for u(X). Similar definitions give Av(Y) and Rv(Y) for v(Y). Obviously, if either the absolute or the relative risk aversion measure is specified, the other is easily determined. Thus, one can choose to present risk preference information for any particular utility function in terms of either its absolute or its relative risk aversion measure. The discussion here focuses on relative risk aversion because this measure is unaffected by a change in the unit of measure for the outcome. Relative risk aversion is the elasticity of marginal utility, making its magnitude more comparable across the various studies. Although X and Y are different outcome variables, often these outcome measures are related to one another. If so, the nature of the relationship between X and Y determines how the risk aversion properties of u(X) relate to those for v(Y). In the various studies reviewed here, the observed relationship between Y and X stems from two sources. In some cases, the values for Y and X are linked within an economic model. For example, in a multi-period consumption model with random return on saved wealth where Y and X represent consumption and wealth, respectively, Y is the optimal level of consumption for any realized value for wealth X. Similarly, consumption for agricultural producers can depend on the profit or net income earned from risky agricultural production decisions.
6 A second, and more frequently observed source of a relationship between outcome variables, results from using different definitions or measures for a particular outcome. The differences between the definitions or measures typically involve including certain components in one definition or measure, and not in the other. This may be due to data limitations or for other reasons. In this case, the variables X and Y are discussed as if they represent the same outcome, but careful examination indicates that there are differences. For example, Y and X can each refer to the return to a decision to invest in risky and riskless assets, but Y is that return after taxes are paid on investment income, while X represents return before taxes. The specific nature of the tax process then determines the relationship between Y and X. Similarly, the definition or measure of wealth can include or exclude the value of human capital, and assumptions concerning the relationship between the values of human capital and other forms of wealth determine how Y and X are related. One of the main tasks in reviewing the literature describing risk preferences is to determine how the outcome variable used in a particular study is related to other outcome variables used elsewhere so that the various reported findings can be appropriately compared and contrasted. The easiest case to deal with is when outcome variables Y and X are deterministically related; that is, where Y = Y(X), and each realization of the random outcome Y is determined by the corresponding realization of X. This implies that the utility functions u(X) and v(Y) must satisfy u(X) = v(Y(X)). More complex analysis results when the relationship between Y and X is not completely deterministic. For the deterministic relationship Y = Y(X), the connection between the risk aversion measures for u(X) and v(Y) is a simple calculation. u(X) = v(Y(X)) implies that
7 u'(X) = v'(Y)·Y' and u''(X) = v''(Y)·(Y')2 + v'(Y)·Y''. Together these imply that Au(X) and Av(Y) are related by Au(X) = Av(Y)·Y' - Y''/Y'. Notice that the second term, -Y''/Y', is the equivalent of the absolute risk aversion for, and is a measure of the concavity of, the relationship Y(X). Similarly, the relative risk aversion measures satisfy Ru(X) = [Rv(Y)][Y'·X/Y] - [Y''·X/Y'], where the last term is the equivalent of the relative risk aversion measure for the relationship Y = Y(X). One special form for Y = Y(X) is particularly simple and is frequently observed, and hence warrants particular notation. This is the case where Y and X are linearly related, that is, Y = a + bX. The risk aversion measures then satisfy Au(X) = b·Av(Y) and Ru(X) = [bX/Y]Rv(Y) = [bX/(a + bX)]Rv(Y). Notice that even though the function relating Y and X is linear and thus has no curvature, when the intercept in this relationship is not zero, it is still the case that the magnitude and the slope of relative risk aversion for u(X) and v(Y) are different. When the intercept is positive, the magnitude of Rv(Y) is necessarily larger than that for Ru(X), and Rv(Y) can be negatively sloped even when Ru(X) is positively sloped. It is the case that the elasticity of the slope for Ru(X) is necessarily larger than that for Rv(Y) whenever v(Y) displays decreasing absolute risk aversion. A proof of this is provided in the Appendix. The case Y = a + bX arises both as an equilibrium restriction in an economic model, and also from measurement or definitional disparities. For instance, for certain utility functions, optimal consumption is a linear function of beginning of period wealth, with a positive intercept reflecting the fact that consumption decisions are made to ensure that consumption is less variable than wealth.2 This smoothing of consumption implies that for any set of risk preferences, the relative risk aversion for utility from consumption
8 is necessarily larger than for utility from wealth, and the findings that the first is decreasing and the second is increasing, are consistent findings. The linear relationship Y = a + bX can also result from measurement differences. Specifically this occurs when components are included with Y and are not included with X, and these excluded components are either constant or are proportional to X. When Y is more inclusively defined or measured than is X, it is the case that Rv(Y) is always larger than Ru(X), and the elasticity of Rv(Y) is smaller than that for Ru(X) under decreasing absolute risk aversion (DARA). A second type of relationship between Y and X that is sometimes observed is not completely deterministic. There can be random components that are included with Y but not X so that Y = X + Z, where Z is random. When Z and X are independently distributed the relationship between u(X) and v(Y) is given by u(X) = EZ v(Y) = EZ v(X + Z). If Z has a zero mean, Z is referred to as background risk. The implications of background risk for the risk aversion properties of u(X) and v(Y) are discussed in the literature. In the analysis here, the more usual situation is where Z has a positive support rather than a zero mean. For this case, the fact that Z is random does not alter the primary finding that including more components when measuring or defining a variable has the effect of increasing the level of relative risk aversion and decreasing its elasticity, assuming DARA.3 To compare studies with two different or differently measured outcome variables, only the relationship between Y and X is needed. To summarize and examine information from a larger number of studies, however, it is convenient to pick a particular variable as a reference X and to relate all the Yi from the various studies to this reference
9 outcome variable. Doing this allows risk aversion information to be converted to a common measurement scale. The question then is which X to choose when defining this common scale? In a formal sense, it does not matter, but in a practical sense, familiarity is an important consideration. For this reason, wealth is chosen as the reference outcome variable. Furthermore, the definition of wealth that is selected includes with wealth the value of all assets, and only those assets, whose quantities can be freely adjusted. This type of wealth is referred to as Arrow-Pratt (A-P) wealth. A-P wealth is chosen as the reference outcome variable for several reasons. The main reason, however, is that it is the variable and definition discussed extensively by Arrow and by Pratt, and hence goes back to the earliest discussion of measures of relative risk aversion. This particular definition of wealth also continues to be that employed in most theoretical analysis of portfolio behavior, and can be linked to other measures of wealth employed in empirical studies. With the selection of a reference variable, it is now appropriate to alter the general notation to reflect this. Rather than X, W is used as the argument of utility u(·), and Ru(W) denotes the relative risk aversion measure for u(W). Ru(W) is referred to as the Arrow-Pratt (A-P) measure of relative risk aversion. In the analysis, all estimates of relative risk aversion for any other outcome variable are adjusted so that they provide information concerning this specific A-P measure of relative risk aversion. With this framework in place, the task of reviewing the available information concerning measures of relative risk aversion can begin.
3. Relative Risk Aversion for Wealth As indicated earlier, Arrow and Pratt provide definitions of measures of absolute and relative risk aversion. These definitions are formulated for an outcome variable or
10 argument of utility referred to as wealth, and this wealth is assumed to be freely allocable to the various available assets. This section reviews a portion of the information concerning relative risk aversion for wealth that has accumulated since those definitions were introduced. The review begins with information relying primarily on theoretical analysis. One way to determine a risk aversion measure’s properties is to ask which properties lead to comparative static predictions that are consistent with observation, and which do not. This approach to the question, initiated by Arrow, has been used extensively in the literature describing the various properties of risk aversion measures since that time. For instance, Arrow argued that absolute risk aversion for wealth, Au(W), is a decreasing rather than an increasing function. Pratt also placed particular emphasis on this property of decreasing absolute risk aversion. Decreasing absolute risk aversion (DARA) is supported in various ways, including analysis of the portfolio model with a single risky and riskless asset. It is well known that under DARA, a wealth increase leads to an increased holding of the risky asset, while increasing absolute risk aversion (IARA) leads to the opposite conclusion. This result, and evidence of how portfolio composition changes as wealth changes, supports DARA and rejects IARA. As a consequence, today DARA is a routinely assumed and accepted as a property of u(W).4 Arrow also presented theoretical arguments concerning the slope and magnitude of relative risk aversion for u(W). His hypothesis, that Ru(W) is increasing (IRRA) and approximately one, is less supported than is DARA by the ensuing research, but still is a commonly used reference point when discussing relative risk aversion for wealth and
11 often other outcome variables as well. Using the same portfolio model, Arrow shows that as wealth increases, the proportion of wealth invested in the risky asset declines under IRRA, increases under decreasing relative risk aversion (DRRA), and is unchanged under constant relative risk aversion (CRRA). Arrow cites evidence concerning the wealth elasticity of cash balance or money demand, which he interprets as demand for the riskless asset, in support of IRRA. As Friend and Blume indicated many years ago, the support for IRRA is less than persuasive and that still is the case today. Arrow’s argument that Ru(W) should be near one is based on a limiting argument. He notes that for utility to be bounded from below as wealth approaches zero, and to be bounded from above as wealth become infinite, it must be that relative risk aversion approaches a number less than one as W goes to zero and a number greater than one as W becomes infinite. It is important to recognize that this argument applies to any outcome variable and thus applies to all relative risk aversion measures. Requiring that Ru(W) be near one appears to have little support. Nonetheless, Arrow’s conclusion that it is broadly permissible to assume that Ru(W) increases with W, although theory does not rule out some fluctuation, and that if for simplicity reasons CRRA is to be assumed, then Ru(W) = 1 is the appropriate assumption, has significantly influenced assumptions made concerning relative risk aversion for wealth and the other outcome variables. Cass and Stiglitz employ a more general portfolio model with many risky assets, and determine which aspects of Arrow’s comparative static findings continue to hold. They are able to show that while IRRA is not sufficient to predict the effect of wealth changes on the exact composition of the risky portion of the portfolio, IRRA does lead to a similar conclusion concerning the relative share of the portfolio allocated to the riskless
12 asset. Cass and Stiglitz maintain the assumption that all wealth is freely allocable across the various assets. Another important comparative static finding associated with relative risk aversion is presented by Hirshleifer and Riley (H-R) who show that in a state preference model satisfying the axioms of expected utility, if relative risk aversion is less than one, the price elasticity of demand for any state contingent claim is greater than one, and thus all contingent claims are gross substitutes rather than gross complements. Furthermore, when relative risk aversion is greater than one, state contingent claim are gross complements and own price elasticity is less than one. This comparative static result is interpreted as lending some support for the assumption that Ru(W) is less than one. The state preference model used in the H-R analysis assumes that the initial endowment of wealth can be freely used to purchase the various state contingent claims, an assumption consistent with that made for assets in the portfolio models of Arrow or Cass and Stiglitz. As a final statement pertaining to theoretical findings concerning relative risk aversion for wealth, it is noted that there exists a rather large body of work by a number of researchers, including Hadar and Seo, who show that IRRA and Ru(W) ≤ 1 lead to “sensible” comparative static findings. Hadar and Seo demonstrate, for instance, that a first degree stochastic dominant (FSD) improvement in the return to the risky asset, always leads to an increased allocation to that asset when the decision maker has A-P relative risk aversion less than one, a seemingly sensible result, while the opposite can occur if Ru(W) is greater than one. Other similarly “sensible” comparative static findings can be listed as support for the assumption of Ru(W) ≤ 1 and IRRA for u(W). Little or no support of this type for the opposite assumptions exists.
13 As a reminder, the aforementioned theoretical support is for the A-P relative risk aversion measure, Ru(W), where the measure of wealth, W, includes the value of all assets that can be freely reallocated and only the value of those assets. The portfolio model used to develop this theoretical support could be modified to include other forms of wealth whose value cannot be reallocated. Making such a change alters and complicates the link between the effect of wealth increases and IRRA or DRRA. This particular point is revisited in the review of the empirical analysis of Friend and Blume. Direct empirical evidence concerning relative risk aversion for wealth is provided in an early and prominent empirical investigation by Friend and Blume (F-B), and this is a study chosen for detailed review. In their analysis, data concerning the composition of the asset portfolios of 2100 households are used to determine how allocation to risky and riskless assets varies with household wealth. The proportion invested in risky assets and how this proportion changes with wealth is interpreted as providing cross section evidence concerning the magnitude, and the increasing or decreasing nature, of relative risk aversion for utility for wealth for a representative decision maker. When carrying out their empirical analysis, F-B present several different estimates of the relative risk aversion measure for a representative decision maker, each estimate associated with a different measure of wealth. Three of these estimates and their associated measures of wealth are discussed here. Two different decisions by F-B lead to wealth outcome variables which are different from A-P wealth. First, they assume that households recognize the taxation of income from investing, and therefore F-B estimate relative risk aversion for expected utility from wealth after taxes are paid on investment income. Second, F-B sometimes choose to include, and other times exclude the values of
14 housing and human capital when measuring wealth. Determining the effect of these decisions on the measure of relative risk aversion is an important consideration when interpreting their findings. Since F-B’s explicit modeling of the taxation of investment income affects each of their estimates in the same way, this issue is discussed first. When an asset earns a rate of return and that rate of return is taxed, after tax income is likely the variable of main concern to the investor. F-B explicitly recognize this point and alter the simple portfolio model of Arrow to include this feature. Putting this into the notation and terminology from section 2, F-B examine the risk taking properties of utility for outcome variable Wt, v(Wt), where Wt is after tax wealth. How the risk aversion properties for this utility function relate to those for u(W) is determined once the relationship between Wt and W is identified. When all investment income is taxed at a constant rate t, after tax wealth to the decision maker is given by Wt = W0 + W0(1 - t)(α·r + (1 - α)ρ), where W0 is initial wealth, 0 ≤ t < 1 is the tax rate, r and ρ are the rate of return to the risky and riskless assets, respectively, and α is the proportion of wealth allocated to the risky asset.5 Arrow-Pratt wealth is this same expression with t = 0. Thus, Wt = t·W0 + (1 - t)W. W and Wt are linearly related, and the intercept in this relationship is positive and reflects the fact that it is the income from wealth, not wealth itself, which is taxed. Inserting the intercept and slope values into the general derivation given in section 2, indicates that the relative risk aversion measures for these two outcome variables satisfy Ru(W) = [W(1 - t)/Wt]Rv(Wt) = [(Wt - t·W0)/Wt]Rv(Wt).
15 To convert the estimates of Rv(Wt) reported by F-B to the reference A-P relative risk aversion scale, Ru(W), information concerning the variable adjustment factor [(Wt - t·W0)/Wt] is needed. It is clear that this adjustment factor is always less than one. Two lines of reasoning argue that this factor is approximately (1 - t). First, at Wt = W0, the adjustment factor is exactly (1 - t). Second, letting rp = (α·r + (1 - α)ρ) denote the portfolio rate of return, then the adjustment factor is equal to (1 + rp - t - rp·t)/(1 + rp - rp·t), and a Taylor series expansion of this shows that for small values for rp, the adjustment factor is approximately (1 – t). When Wt rather than W is used as the outcome variable, the slope of relative risk aversion is also affected, but for portfolio rate of return values of 10% this effect is quite small and is ignored here. When the tax rate itself varies with wealth level, this too can alter the estimated value for relative risk aversion, and also appears to be a second order effect and is ignored. As a consequence, to adjust the F-B estimates to Ru(W), we simply multiply by (1 - t), concluding that to a first approximation, F-B’s use of after tax wealth biases their estimates of relative risk aversion upward by 1/(1 - t).6 Average tax rates during the time period of the F-B study are approximately 50% for the high wealth categories, and marginal tax rates are even higher. Thus, the reported relative risk aversion levels for those wealth categories are reduced significantly when converted to an estimate of Ru(W). The exact reductions for the three wealth measures and for all wealth levels are reported in Table 1 given later in this section. Before presenting this table, however, the three wealth measures themselves are described in more detail.
16 The first wealth measure used by F-B includes the value of several categories of financial assets and is reported in their table 1. This measure, denoted here as W1, contains mainly assets who value is quite easily reallocated to other assets. If W1 completely measures the conceptual W of Arrow and Pratt, then the only adjustment necessary to convert Rv(W1) to Ru(W) is the tax adjustment just described. It is possible, however, that W1 fails to include the value of all freely allocable assets, or includes assets which are not freely allocable. F-B suggest that the value of housing, which they purposely exclude from W1, may be an important asset to consider even though it is more difficult to reallocate than are many of the assets included with W1. Thus, F-B include housing as a risky asset in their second (and in the third) measure of wealth, denoted here as W2, and reported in their table 2. Since all wealth categories have some housing equity7 on average, F-B’s treatment of housing equity as a risky investment implies that the average proportion of wealth held in risky assets is increased for all investors. This necessarily leads to lower estimates of relative risk aversion. In this portfolio allocation framework, omitting any risky asset leads to an overestimate of relative risk aversion while omission of a riskless asset leads to an underestimate. In addition to reducing relative risk aversion, including the value of the risky housing asset also changes the slope of the relative risk aversion measure because in the data analyzed, housing is a much larger share of wealth for low wealth categories than for those with higher wealth, and hence the reduction in relative risk aversion due to the inclusion of housing equity is most pronounced at low wealth levels. At the highest wealth levels, housing is such a small portion of wealth that the effect of including it is nearly zero. Our Table 1 gives the F-B estimates of relative risk
17 aversion for utility from W1 and W2 and also scales these estimates by (1 - t) to adjust for taxes. In a third measure of wealth, W3, along with the equity value of housing, F-B also include the value of human capital. They assume, however, that human capital is not a usual risky or riskless asset since its value cannot easily be reallocated to other assets. They use a Capital Asset Pricing Model approach to take into account the portion of the riskiness of the return on human capital that cannot be diversified away. Thus, the riskiness of human capital impacts the allocation of other wealth among risky and riskless assets, even though human capital itself cannot be reallocated. One way to represent this simply is to assume that human capital’s return is like that of a fixed portfolio of the risky and riskless assets, that is, human capital earns a risky rate of return rh = (β·r + (1 - β)ρ), where β is the correlation with return on risky assets, and rh is the return to human capital. When human capital is included in the portfolio allocation model in this way, its effect on estimated risk aversion depends critically on the relative magnitudes of desired level for α and the given level for β. When β and α are equal, the inclusion of human capital has no effect on the choice of α because the riskiness of human capital is just what would have been chosen had the decision maker been able to freely reallocate that wealth among the risky and riskless assets. On the other hand, when the chosen α is less than β, the chosen α is lower than it would have been without human capital in order to compensate for the higher than desired risk level associated with human capital. The reverse occurs when the chosen α is larger than β. In this case the riskiness associated with human capital is less than the desired level, so to compensate, the investor chooses a portfolio of financial assets that is
18 riskier than consideration of only the measure of relative risk aversion would imply. The consequence of this is that including human capital with financial assets, reduces the estimate of relative risk aversion when β > α, and increases the estimate when β < α. When rh = (β·r + (1 - β)ρ), both the return on human capital and on the portfolio of financial assets are linear functions of r. This implies that W3, the measure of wealth including human capital, is a linear transformation of W2, the measure that excludes it. Straightforward but tedious algebra detail how the slope and intercept of this relationship depends on α and β, and on the relative sizes of financial wealth W2, and human capital H. The scale factor W2/[(1 - β/α)H + W2] transforms Rv(W3) in Rv(W2). As must be the case, no adjustment is necessary when α and β are equal or if H is zero. The magnitude of the adjustment is greater the larger H is relative to W2. The scale factor is greater or less than one depending on whether β > α or β < α. F-B report the data needed to calculate this scale factor and these values along with the tax adjustment are used in Table 1 to transform the F-B estimate of Rv(W3) into Rv(W2) and then into Ru(W), the A-P reference relative risk aversion scale. In summary, Table 1 reports three of F-B’s estimates of relative risk aversion for a representative decision maker. These estimates are based on the same data, and differ because each uses a different measure of wealth. The unadjusted estimates are quite different from one another, and it is not clear which is best estimate by any particular criterion. The analysis here has taken these estimates as they are given and using the information provided concerning the various wealth measures, has adjusted the estimates to provide three alternative estimates of Ru(W), the A-P relative aversion measure. While
19 the adjusted estimates clearly are still not the same as one another, they are more similar to one another than before adjustment. There are several reasons for the remaining variation in the estimates of Ru(W). First, logically it cannot be the case that all three wealth measures include exactly the complete set of assets that can be freely reallocated since the three measures include different assets. Second, F-B define five wealth categories at the outset and maintain these categories even as the wealth measure changes. Since investor wealth increases with the inclusion of additional assets, a large number of investors change wealth category as the wealth measure is changed. For example, there are 523 households with wealth in the 1-10 thousand range when the measure is W1, but only 114 remain in this category for W3, where housing and human capital are included. Because the average of the relative risk aversion levels for these households is used to determine the representative relative risk aversion level for that wealth group, this procedure necessarily averages across a different set of households whenever the wealth measure is altered. All of this prevents the adjustment to a common scale from being exact. Which of the various estimates of relative risk aversion for the group of investors surveyed is the best in a statistical, procedural and measurement sense is unclear. All estimates of Ru(W) decline more with wealth than do the unadjusted estimates, and even though the adjusted estimated magnitudes are still greater than one, the deviation from one is considerably smaller than reported by Friend and Blume.
4. Relative Risk Aversion for Consumption In this section, a portion of the literature presenting information for relative risk aversion for utility from consumption is reviewed. As was the case for wealth, different
20 measures of consumption have been used and this must be taken into account. In addition to adjusting for measurement differences, however, it is also necessary to link the various consumption measures to A-P wealth. Most often, the studies that estimate relative risk aversion for consumption do not themselves provide such a link. Thus, other sources for this information are used, and a brief discussion of a way to relate consumption and A-P wealth is included. Theory provides less guidance concerning the slope or magnitude of relative risk aversion for consumption than it does for wealth. Arrow’s bounded utility argument, which applies to all relative risk aversion measures, indicates a nearness to one in the limit, but other than this, no particular assumption concerning relative risk aversion for consumption, Rv(C), is supported by comparative static analysis. Rothschild and Stiglitz do point out the importance of both the slope and magnitude of Rv(C) when they show that an increase in the riskiness of the return on saved wealth leads to more consumption now and less later whenever Rv(C) is nondecreasing and less than one, and that the opposite occurs when Rv(C) is nonincreasing and greater than one. It is also the case that the most common assumption made in multi-period consumption models where utility is additive over time is that Rv(C) is constant. This may be due, however, to the convenience of the power function when analyzing such models. Several papers present a limited amount of empirical evidence concerning relative risk aversion for consumption although in most cases measuring risk aversion is not the primary focus of the research. Some of this work is discussed later in the section. The work that reviewed in most detail is that by Barsky, Juster, Kimball and Shapiro (BJKS) who analyze the survey responses of 11,707 individuals to a sequence of hypothetical
21 questions concerning gambles over lifetime income. These responses, and several assumptions concerning risk preferences, allow individuals to be placed in one of four categories defined by the magnitude of their relative risk aversion. Although the survey questions themselves are expressed as gambles over lifetime income, BJKS interpret the relative risk aversion information that is obtained as for permanent consumption. Thus, BJKS determine a range for the magnitude of Rv(C) for each individual where C is interpreted as permanent consumption. After determining the approximate magnitude of Rv(C) for each individual, BJKS ask how relative risk aversion level varies with a wide range of demographic variables. Unfortunately for the purposes here, individuals do not report permanent consumption or lifetime income. Instead, two related variables, current wealth and current income, are discussed. BJKS group individuals into five equal size categories defined by either current wealth or current income, and find that average relative risk tolerance, the inverse of relative risk aversion, neither increases nor decreases significantly as the wealth or income level associated with the category changes. Average risk tolerances for the various levels of current wealth (income) range from .2318 to .2601 (.2310 to .2556). Risk tolerance decreases at first, but then increases slightly, as either current wealth or current income increases. To convert this relatively constant risk tolerance finding into information concerning Rv(C), requires at least two additional assumptions. First, it is assumed that grouping individuals by permanent consumption rather than by current income or current wealth would not affect the findings significantly. The fact that both current income and current wealth lead to similar findings lends some support for this assumption. Second,
22 since only average risk tolerance estimates are reported, these averages must be converted to average relative risk aversion levels. This would be very simple if the inverse of the average were the average of the inverse, but Jensen’s inequality tells us that it is just a lower bound. Based on the relationships reported by BJKS in their first table where they report both average risk tolerance and average relative risk aversion grouped by question response, it appears that average relative risk aversion can be as large as double the inverse of average risk tolerance. Thus, our interpretation of the BJKS finding is that Rv(C) is nearly constant with a magnitude of somewhere between four and eight. BJKS do not model the relationship between permanent consumption and A-P wealth so other sources are used for that information. Ogaki and Zhang do not provide an estimate for the magnitude of relative risk aversion for consumption, but do provide information concerning the slope of this function. Using a utility function of the form u(CF) = [(CF - x)1-α - 1]/(1 - α), whose relative risk aversion measure is αCF/(CF - x), they develop a procedure for estimating the value for x. It is the case that when x = 0, relative risk aversion is constant, and otherwise is decreasing or increasing depending on whether x is positive or negative, respectively. In their analysis, CF represents food consumption for households in Pakistan and India. Ogaki and Zhang obtain large positive and significant estimates of the value for x, and therefore conclude that relative risk aversion for food consumption for these individuals is decreasing. To get a sense for the rate of decrease that they find, Ogaki and Zhang’s point estimate for x is approximately equal to 2/3 the average level of CF in their sample. This implies that Rv(CF) has an elasticity equal to -2 at the average level of food
23 consumption. Ogaki and Zhang do not relate food consumption CF to A-P wealth in their analysis. The macroeconomics literature dealing with asset pricing, the equity premium puzzle and the habit formation utility function has also determined approximate values for relative risk aversion for consumption. It does this by requiring risk aversion levels to be consistent with observed asset prices and interest rates. These models also directly relate consumption to A-P wealth within an optimization model. A prominent paper in this literature is one by Constantinides, who assumes that the consumer’s marginal utility from current consumption depends in part on past consumption levels. A simple model that leads to this habit level of consumption is formulated. Utility for consumption takes the form v(C) = (C - x)1-α /(1 - α) at each point in time, where x is the habit level of consumption whose value depends on past consumption levels. The relative risk aversion measure for this utility function v(C) is αC/(C - x), and is identical in form to that of Ogaki and Zhang. Since the proposed values for x are positive and a large part of average consumption, this form for utility for consumption is severely DRRA. Using annual data for aggregate consumption, and the return on risky and riskless assets for 1889-1978, Constantinides examines various possible parameter values for the process leading to the habit level of consumption, x(t). He reports six pairs of values that are not rejected by the observed data. In his model, the optimal and the habit levels of consumption are random and vary with time, and thus the information given concerns their mean values. Included in that information are the mean values for Rv(C) and Ru(W). Constantinides assumes that wealth can be freely allocated across assets and thus W is interpreted as A-P wealth. No other wealth or sources of income are included in his
24 model. In the data used in the calibration, consumption is aggregate consumption and is treated here as the same as permanent consumption for the representative consumer. The six parameterizations of the habit formation model that are consistent with the observed consumption and rate of return data imply mean values for Rv(C) ranging from 11 to 17. The corresponding mean values for Ru(W) range from 7.03 to 2.78. It is the case that Rv(C) is between 1.5 and 6 times larger than Ru(W) depending on which of the six particular forms of the model is considered. It is also the case, that the habit level of consumption, x, on average is approximately 80% of consumption at the mode and this is true for all six of the parameterizations. This later fact implies that the elasticity of Rv(C) at the mode is minus four, implying an even faster rate of decrease than that determined by Ogaki and Zhang. Consumption and A-P wealth can be related to one another in at least two different ways. One is to specify a model where the optimal level of consumption depends on wealth. The Constantinides model that was just reviewed, as well as those of Kimball and Mankiw, and Meyer and Meyer each do this. Although the utility functions and other aspects differ across these models, in each case, the optimal level of consumption is demonstrated to be a linear function of wealth, C = a + b·W with a positive intercept. Partly for this reason, and also for simplicity, it is assumed here that the link between consumption and wealth takes this linear form. This implies that the adjustment to use when converting information concerning Rv(C) to the reference scale involves multiplying Rv(C) by [b·W/(a + b·W)]. To determine the exact properties of this conversion factor, estimates of the relative sizes of the constant, a, and bW are needed. In the Constantinides’ analysis, the
25 habit level of consumption is approximately 80% at the mode, and thus b·W represents the remaining 20%. This implies that the conversion factor is .2 at that point. National income account data also indicates that this is a reasonable number. Disposable personal income, for instance, is approximately 80% from compensation to labor and 20% from rate of return on wealth in the form of return to capital and equity. (Dornbusch and Fisher, Kuznets). Assuming that C = a + b·W, and that at the mean, b·W is 20% of consumption implies that Ru(W) = Rv(C)/5 at the mean. In addition, the linear relationship between C and W implies that the elasticity of Ru(W), εRu(W) is related to the elasticity of Rv(C), εRv(C), by the equation εRu(W) = [b·W/(a + b·W)] εRv(C) + [a/(a + b·W)]. Thus, at the mean, εRu(W) = .2 εRv(C) + .8. This implies that the Constantinides finding that εRv(C) = -4, is equivalent to εRu(W) = 0. Thus, very severe DRRA for utility for consumption and CRRA for utility for A-P wealth are consistent findings in that analysis. Using this same adjustment factor, the BJKS finding that Rv(C) ranges from 4 to 8 and is relatively constant leads to a value for Ru(W) ranging from .8 to 1.6 with a positive elasticity equal to +.8 or larger. Thus, their relative risk aversion estimate implies that Ru(W) satisfies IRRA and has a magnitude near one. To convert the Ogaki and Zhang finding of DRRA for relative risk aversion for food consumption to information concerning Ru(W) requires a link between food consumption CF and A-P wealth. When CF is proportional to C or a linear function of C with a positive constant term, the Ogaki and Zhang elasticity of minus two for Rv(CF) is also consistent with IRRA for u(W). In summary, the link between consumption and A-P wealth suggests that Rv(C) is significantly larger than Ru(W). Moreover, elasticity of Ru(W) is also larger than that for
26 Rv(C), and so much so that the evidence of a strongly declining Rv(C) can be associated with a relatively flat or even increasing Ru(W).
5. Relative Risk Aversion for Profit In this section, information concerning relative risk aversion for profit is discussed. This discussion is facilitated by first providing a simple model relating profit and A-P wealth to one another. This model is specifically formulated for agricultural producers who provide a portion of their labor and capital, and their entrepreneurial talent to the production process. Although the discussion is phrased in terms of agricultural producers, any decision maker who similarly derives income from several sources can be discussed within the same modeling framework. National income data, which attributes income to its various sources, can be discussed within this framework as well. Assume that income to an agricultural producer comes from three sources, compensation for labor, the profit of the producer as an entrepreneur, and the rate of return earned by the producer’s wealth. Labor income could be from the farm or from off-farm employment, and rate of return on wealth could be from the equity in the agricultural enterprise or wealth invested elsewhere. To establish notation, let I denote total income and therefore I = Y + π + r·W, where Y, π and r·W represent labor income, profit, and rate of return on wealth, respectively. It is assumed that wealth can be reallocated among assets and is A-P wealth. The value of human capital and value of the entrepreneurial talent are not part of this A-P wealth. These three components of income are taken to have their usual economic meaning. Income attributable to labor or to capital is the opportunity cost of that input, and profit is economic profit. Because it is often the case that income attributable to labor and/or capital is misreported and in some cases not
27 even paid, an important aspect of interpreting results concerning estimates of relative risk aversion for profit involves determining exactly what is included with the measure of profit that is used. The link between profit and A-P wealth comes primarily from the above identity, but this link is further strengthened by linking consumption and income. National income account data suggests that in the aggregate, personal disposable income and consumption are nearly the same. Therefore, the assumption here follows that of BJKS, and consumption is assumed equal to income. Thus, the identity C = Y + π + r·W, is used to connect relative risk aversion information for wealth, consumption and profit to one another. Be reminded that C is a very broad measure of consumption equal to total disposable income, and W is A-P wealth. When C = Y + π + r·W represents the relationship between C, π and W, one can determine how the relative risk aversion measure for either Y, π or W relates to that for C, assuming the levels of the other two components are held fixed.8 Doing this implies that relative risk aversion for profit is equal to that for consumption multiplied by the factor [π/(Y + π + r·W)]. Similarly, the relative risk aversion measure for wealth is equal to that for consumption multiplied by [r·W/(Y + π + r·W)]. Combining these expressions implies that the relative risk aversion measures for A-P wealth and profit satisfy Ru(W) = Rv(π)[r·W/π]. Thus, the relative sizes of Y, π and r·W become critical information when relating relative risk aversion for profit to that of either A-P wealth or consumption. The empirical paper estimating levels of relative risk aversion for agricultural producers that is selected for careful review is that by Saha, Shumway and Talpaz (SST) who use firm level data for fifteen wheat farms in Kansas to jointly estimate risk aversion
28 and a stochastic production function. This paper is chosen in part because of the care with which it describes the data used to estimate relative risk aversion. The variable used by SST as the argument of utility is referred to as wealth although what is actually measured is net or disposable income. This variable includes farm revenues and off-farm income and subtracts from these revenues the costs of agricultural production. Due to data limitations, no labor costs are subtracted from farm revenues. Thus, assuming that capital costs have been accounted for correctly, the argument of utility in our notation includes economic profit π and also labor income Y. It does not include rate of return earned by wealth. Thus, our perception is that the estimate of relative risk aversion reported by SST for these agricultural producers is for variable (Y + π) and it, therefore is adjusted to the reference scale Ru(W), by multiplying by [r·W/(Y + π)]. Aggregate agricultural statistics indicate that r·W and π are approximately equal in size, and that Y is larger than either. As a consequence this adjustment factor is a fraction, less than one half, and the SST estimate of relative risk aversion of five is comparable to the previously reviewed estimates for Ru(W). Another way to evaluate the information concerning relative risk aversion supplied by SST estimate is to ask what it implies concerning relative risk aversion for consumption. To convert their estimate to information concerning Rv(C) involves multiplying by [(Y + π + r·W)/(Y + π)]. This number is necessarily larger than one, and agricultural statistics indicate that in the aggregate, it is approximately 1.5. Thus, the SST estimate implies an estimate of relative risk aversion for consumption similar to those reported in that section.
29 Theory also provides some information concerning relative risk aversion for profit. Sandmo shows that the effect of an increase in a proportional tax on profit is to increase (decrease) output if relative risk aversion for profit is increasing (decreasing). He does not offer evidence supporting either DRRA or IRRA. Hadar and Seo show that a first degree stochastic dominant increase in output price leads a firm to increase output level if Rv(π) ≤ 1, but that with Rv(π) > 1, this need not be the case. This is viewed as support for Rv(π) ≤ 1. This evidence is for economic profit.
6. Summary and Conclusions This review attempts to consolidate and standardize the information that is available concerning the risk attitudes of representative decision makers. Information concerning relative risk aversion for different outcome variables and various types of decision makers is used. When appropriately adjusted for differences in outcome variables, the reported relative risk aversion measures are quite similar across decision makers, much more similar than the unadjusted information indicates. The data reviewed here suggests that for A-P wealth, a narrowly defined wealth measure, Ru(W) is near, but larger than one and increasing. Evidence for this comes from estimates for relative risk aversion for wealth using various wealth measures, and also from relative risk aversion estimates for consumption and profit. This same evidence implies that for consumption, when broadly defined as disposable income, Rv(C) is much larger, at least five at the mean, and likely even as large as ten. Furthermore, relative risk aversion for consumption is strongly decreasing so that values observed away from the mean can vary significantly. It is especially important to recognize that for consumption levels below the mean, Rv(C) could be 20, 30 or 50 as some have suggested. Finally,
30 relative risk aversion for economic profit is like to be somewhere in between Ru(W) and Rv(C), but most similar to Ru(W).
31 TABLE 1 W2
W1 Wealth Level
32 Appendix Theorem 1: When C = a + bW with a > 0 and v(C) satisfies DARA, then the elasticity of Ru(W) is larger than that for Rv(C).
Proof: When C = a + bW, then Ru(W) = (bW/C)Rv(C). Also, DARA for v(C) implies that Av(C) = Rv(C)/C is falling, and this implies that Rv'(C) ≤ Rv(C)/C. Differentiating implies that Ru'(W) = Rv(C)(bC - b2W)/C2 + Rv'(C)(b2W/C). Substituting the inequality implies that Ru'(W) ≤ Rv'(C)(bC - b2W)/C + Rv'(C)(b2W/C) when a ≥ 0. This is equal to Ru'(W) ≤ Rv'(C)b, which when converted to elasticities gives [Ru'(W)W/Ru(W)] ≤ [Rv'(C)C/Rv(C)] .
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1. In 2001, Feldstein and Ranguelova, for instance, point to the widely varying evidence concerning relative risk aversion for consumption, and instead use a thought experiment to allow readers to choose the appropriate assumption to make concerning its magnitude. 2. Gollier refers to this as time diversification. 3. Pratt discusses the risk aversion properties of convex combinations of utility functions. 4. Note that even though DARA specifies the slope of Au(W), the assumption also limits the rate at which relative risk aversion, Ru(W), can increase. DARA does not limit the rate of decrease of Ru(W). 5. For this altered portfolio model, the relationship between IRRA and DRRA is no longer that presented by Arrow. Now, it is possible that DRRA is associated with a smaller share of initial wealth being allocated to the risky asset as wealth increases. 6. One way to think of this is to observe that by imposing taxes on income from investing, the taxing authority shares the risk of investing with the investor. As a consequence the investor is willing to assume more risk, and evidence concerning the magnitude of the risk assumed no longer implies the same level of risk aversion that is implied in an investment setting without taxes. 7. Friend and Blume report and use both the equity and the gross value of housing as measures of housing, but argue that the equity value is the better measure. 8. This is what is done in Kimball and Mankiw and Meyer and Meyer, for instance, where either Y or W are fixed, and a relationship between C and the other component of income is determined. Profit is assumed to be zero in both cases.