Taxation of Labor Income and the Demand for Risky Assets

Taxation of Labor Income and the Demand for Risky Assets Dougl~ W. Elmendorf Federal Reserve Board Miles S. Kimball University of Michigan June, 1996 We analyze the effectoflabor income risk on the joint saving/portfolio-composition problem. It is well known that when private insurance markets are incomplete, the insurance afforded by labor income taxes can reduce overall saving through the precautionary saving motive. This insurance may change the composition of saving as well, because the reduction in labor income risk may affect the amount of financial risk that an individual chooses to bear. We find that, given plausible restrictions on preferences, any change in taxes that reduces an individual’s labor income risk and does not make her worse offwill lead her to invest more in risky assets. This result holds even when labor income is statistically independent of the return to risky assets. We also find that the effect of labor income risk on financial risk-taking can be quantitatively important for realistic changes in tax rates. We would like to thank Louis Eeckhoudt, Benjamin Friedman, Roger Gordon, Greg Mankiw, and Andrew Samwick for helpful comments, and the National Science Foundation for financial support. The views expressed here are our own and not necessarily those of the Federal Reserve Board or its staff.

What isthe effect on saving ofa reduction in current taxes combined with an offsetting incre=e in taxes later in an individual’s life? In a world of perfect capital markets and no uncertainty, the individual should save the entire amount of the tax cut. In the real world, however, people face significant uncertainty about their future income, and because proportional or progressive income taxes reduce the variance of income, those taxes provide insurance against this uncertainty. An increase in future taxes increases this insurance and, through the precautionary saving motive, reduces an individual’s saving relative to the Ricardian benchmark just stated (see Chan, 1983, Barsky, Mankiw and Zeldes, 1986, and Kimball and Mankiw, 1989). The increased insurance provided by higher future taxes may have another effect on saving as well: it may change the composition ofsaving, because the reduction in labor income risk can affect the amount offinancial risk that an individual chooses to bear. In this paper we explore the effect of labor income taxes on the willingness to bear financial risk. We study a two-period life-cycle model in which individuals make two choices: how much to save in total, and how to divide that saving between a risky asset and a risk-free asset. We find that, given plausible restrictions on preferences, any change in taxes that reduces an individual’s labor income risk and does not make her worse off will lead her to invest more in the risky asset. This result holds even when labor income is statistically independent of the return to the risky asset, although not if the risky asset actually provides insurance for labor income risk. We also find that the effect oflabor income risk on financial risk-taking can be quantitatively important for realistic changes in tax rates. Consider again the deferral of labor income taxes with no change in the expected value within a person’s lifetime. In a neoclassical world with certain labor income, this tax reduction leaves national saving unchanged by raising private saving = much as public saving falls. It also has no effect on investment in the risky asset, because the future tax liability involves no risk and individuals want to offset that liability by holding more of the riskless =set. In a world with uncertain labor income, however, our analysis shows that deferring labor income taxes raises investment in the risky asset.1 In essence, individuals respond to a reduction in one risk by increasing their exposure 1 The Economic Report of the President (1996, p. 88) assertsour result without proof, as well asasserting the desirability of risk-taking and the importance of the adverse selection problem that we discuss below: “This income insurance[ofprogressive taxation] hasthedirect benefit of reducing theincome riskborne by individuals themselves, shifting it to society as a whole, but it also provides an indirect benefit. Because households will be willing to bear more risk if they have accessto income insurance, they will undertake investments (in both . financial and humancapital, including in~reasedlabor mobility) with greater risk and greater expected return. Aggregated over allindividuals, the effect of undertaking suchinvestments isahigher expected national income. Private markets will not offer suchincome insurancebecausethe inherentdifficultyof separating effort andluck from an individual’s abihty subjects private purveyors to adverse selection: those who expect poor outcomes wodd be more likely to purch~e the insurance. The income tax system, in contrast, applies to virtually all 1

to another risk.2 Surprisingly, the effect of this tax deferral on overall saving becomes unclear once we allow for changes in financial risk-taking. If the uncertainty of labor income were the only risk faced by an individual, then the standard analysis would apply: the individual would consume more and national saving would fall. But when the individual h= the opportunity to invest more in the risky asset, the additional uncertainty that this action creates will tend to decrease consumption and raise saving. In fact, we cannot rule out the possibility that this indirect precautionary effect might outweigh the direct precautionary effect and produce a net increase in saving. Our analytic results raise two questions. First, is the effect of taxes on labor income risk quantitatively important in people’s portfolio selections? We argue that it is likely to be important and present some illustrative calculations to that effect in the penultimate section of the paper. Second, is encouraging greater financial risk-taking a socially desirable or undesirable feature of labor income taxes? In their seminal paper on taxes and risk-taking, Domar and Musgrave (1944) write that “there is no question that increased risk-taking ... is highly desirable” (p. 391). They do not justify this claim, however, nor do most of the researchers who have followed them in work on this topic. A complete investigation of this issue lies well beyond the scope of this paper, although we can suggest several reasons why private markets mightgenerate too little risky investment.3 First and foremost is the lack of a complete market for human capital. Because human capital risk is undiversifiable for an individual but largely diversifiable for society as a whole, there is no presumption that individuals will undertake the socially optimal amount of risky investment in either human or physical capital. Indeed, we think there is some presumption that the optimal conditions can be approached more closely by diversifying idiosyncratic human capital risk through the tax system.4 Imperfections in the market for financial capital may inhibit risky investment as well. For example, entrepreneurs may be unable to diversify away the idiosyncratic risk of their projects economically active people, mitigating concerns with adverse selection.” 2 This restit is closely related to the fidings of Pratt and Zeckhauser (1987) and Kimball (1991) that, for a broad class of utility functions, when an agent is forced to accept one risk, the agent will be less willing to accept other independent risks. Kimball’s analysis is the more closely related to the analysis in this paper because (a) Kimball links the interaction between risksto the effect of the risks on expected marginal utility, which also governs consumption decisions, and (b) Kimball deals with differential changes in risk = well as with the discrete introductions of risk treated by Pratt and Zeckhauser. 3 We follow the literature in assuming that if individuals demand more risky =sets, more risky projects will be undertaken. For example, Feldstein (1983) asserts that “the net rates of return on capital in different usesare not generally equal but reflect the risk-return preferences of investors and their equilibrium portfolio . compositions” (p. 17). 4 To evaluate this presumption, one would need to explain why there is no private insurance against human capital risk. If the primary obstacle to private insuranceis moral hazard, there islittle reason to believe that the government can improve on the private market outcome. If adverse selection is an important obstacle, however, using the government’s coercive power of taxation may make possible a social gain. See footnote 1. 2

because adverse selection discourages the participation of outside investors. Since entrepreneurs’ labor income is highly correlated with the return to their financial capital, incre=ing the labor income tax rate is especially likely to increase their risky capital investment, as we show later. Third, the social return to risky investment will exceed the private return if there are technological spillovers or other positive externalities from such investment. For example, Shleifer and Vishny (1987) argue that aggregate demand externalities in an imperfectly competitive economy make the optimal amount of risky investment greater than the amount chosen (in the absence of taxes) by profit-maximizing firms. Fourth, capital income taxes at both the corporate and personal levels may have powerful effects on financial risk-taking. Unfortunately, there is little theoretical or empirical consensus on the direction or size of these effects, = shown by Sandmo’s (1985) survey. The paper is organized as follows. Section I discusses our work’s relationship to previous research. Section II presents the model, and Section III gives the main results. Section IV considers the quantitative significance of the results, and Section V concludes. I. Relationship to the Literature In its analysis of the effect of labor income taxes on the demand for risky assets, this paper bridges two lines of research. The first is concerned with the role of income taxes in providing insurance for risky labor income, and the resulting effect on the consumption/saving decision. The starting point for this research is the analysis ofthe consumption/saving decision under uncertainty, which began in earnest with Leland (1968), Sandmo (1970), Rothschild and Stiglitz (1971), and Dr&zeand Modigliani (1972). Recent contributions include Skinner (1988), Zeldes (1989), Kimball (1990a,b), Caballero (1990), Weil (1991), and Kimball and Weil (1991). Some research in this group examines the aggregate demand effects of tax cuts–for example, the papers mentioned in the introduction. Other research evaluates the welfare effects of redistributive taxation–for example, Eaton and Rosen (1980), Varian (1980), E~ley, Kiefer and Possen (1993), and Devereux and Smith (1994). All of this work, however, focuses on the effect of labor income taxes on total saving and investment, and says little about the possible effects of these taxes on portfolio composition. For example, Dr&zeand Modigliani discuss portfolio choices but determine only the conditions for their separability from saving decisions and conclude that perfect insurance markets for labor income are essential for separation to hold. Varian determines the optimal tax schedule as a balancing of the direct beneficial effect of social insurance on people’s utility and the detrimental effect ofsocial insurance on people’s saving. If it is appropriate to encourage investment in risky ~sets, then the 3

implicit insurance provided by taxes h= an additional benefit neglected by Varian. The second line of research is concerned with the role of capital taxes in providing insurance for financial risks and thus affecting the amount offinancial risk-taking in the economy. Domar and Musgrave (1944), and many following them, analyze optimal portfolio selection among a collection of fully marketable securities. Some of this research (summarized by Sandmo, 1985) includes the consumption/saving decision (Sandmo, 1969 and Ahsan, 1976) but does not allow for risky labor income. Friend and Blume (1975) discuss the role of human capital in their empirical study of the “market price of risk,” but they =sume a fixed amount of savings and do not address the role of taxes as insurance. Feldstein (1969) notes that “the optimal portfolio behavior for an individual is not independent of the uncertainty of his other income sources” (p. 762) but does not pursue the idea. Davies and Whalley (1991) analyze the effects of taxes on human and physical capital formation, but do not allow for uncertainty. The existing analyses of financial capital do not suffice as analyses of human capital because of fundamental differences between the two =ets. First, human capital can be acquired but not r=old–that is, human capital investment is irreversible except for a small amount of depreciation. Thus, the timing of decisions to invest in human capital is different from that for investing in most financial -ets. Second, the return to human capital depends on both unobservable effort and a large random element. This means that human capital risk is privately undiversifiable and uninsurable. Third, the random element in human capital returns is largely idiosyncratic. This provides the opportunity for the government to reduce each individual’s human capital risk without taking on additional risk itself.5 This paper isalso related to research on the possible “crowding in” ofinvestment bygovernment debt. Consider a reduction in taxes today accompanied by an offsetting increase in taxes in the future. Friedman (1978) argues that such a shift in the timing of taxes might reduce the cost of equity capital (though it would raise the cost of debt capital) and thus incre=e real investment.G Auerbach and Kotlikoff (1987) discuss crowding in that results from deferring capital income taxes. Temporarily lower taxes encourage individuals to save more, and ifthis effect exceeds the reduction in saving caused by the transfer of some of the tax burden to future generations, crowding in will occur. Our results show that crowding in of risky investment is in fact likely to occur, but for a 5 Merton [1983) shows that “a tax and transfer system not unlike the current social sec~ity system canreduce . or eliminate the economic inefficiencies” -that r--ult from the nonmarketability of human capital. Merton is concerned with theeffect of aggregatelabor income riskonfinancial risk-taking, not idiosyncratic labor income risk asin thispaper. 6 His discussion of fiancial crowding out and crowding in abstracts from the real crowding out which wodd occur in a fully-employed economy. 4

different re=n than those previously discussed. We return to this point in the Conclusion. II. The Joint Saving/Portfolio-Composition Problem in the Face of Labor Income Risk Setting. Our analysis uses a simple two-period life-cycle model with additively separable utility: U(c,c’) = u(c) +EV(C’), (1) where cisfirst-period consumption and c’issecond-period consumption. Both absolute risk aversion and the absolute strength of the precautionary saving motive decre=e with wealth. (We discuss these assumptions in more detail below.) We =sume that individuals earn a fixed amount of first-period labor income. That income combined with any initial wealth provides a fixed amount of wealth w to divide between firstperiod consumption and saving. That saving can be invested in two =ts—a risk-free bond with a real after-tax gross return of R, and a risky equity with a real after-tax ezcess return of 2 (i.e., borrowing $1 at the interest rate R to buy $1 of the risky security will yield on net, after taxes and repayment of the loan, the random amount $Z at the beginning of the second period).7 We -ume that individuals can freely borrow or lend through the riskless asset and can freely invest in (or short) the risky asset. Let ~ be the doZZarvaZueof the individual’s investment in the risky =set at the end of the first period (not to be confused with the share of the portfolio in the risky asset). Then at the beginning of the second period, the value of all the individual’s investments will be R(w – c)+ a~. We assume further that individuals hold risky human capital from which they earn income in the second period. The amount of human capital will be considered fixed as an approximation to the difference in timing between decisions about human capital investment (primarily the choice of occupation) and decisions about financial investment .8 Labor is supplied inelastically.g Following Barsky, Mankiw and Zeldes (1986), Kimball and Mankiw (1987), and Varian (1980), private insurance markets are assumed to be incomplete, leaving some amount of uninsured labor income risk. Some of this risk may be due to the possibility of disability, but probably a more 7 We are implicitly assuming that capital income taxes arelinear. . 8 See Kanbur (1981) and Driffill and Rose; (1983) on the choice of how muchhuman capital to hold, and Eaton and Rosen (1980) on the choice of the riskinessof humancapital. 9 Bodie, Merton and Samuelson (1992) study the effect of future labor supply flexibility (as opposed to labor income uncertainty) on the portfolio decisions of the young. 5

important source of uninsurable income risk is the possibility of doing worse than expected in one’s career. This risk is difficult to insure for both moral hazard re~ons (one might be tempted to expend less effort in advancing one’s career if failure is cushioned by insurance) and adverse selection reasons (those who have private information that they will do poorly in the future will be more likely to buy insurance than those who know they will do well). Providing insurance also entails marketing and administrative costs. For our purposes, the reasons for the absence of such insurance are not important. As long weconsider changes in tax parameters that are small enough that the amount of relevant private insurance remains at zero, we do not need to model explicitly why such insurance is unavailable.l” Thus, we model an individual’s second-period labor income as a random variable j with a fixed distribution. We want the joint distribution of labor income j and the excess return z to reflect both idiosyncratic income risk and the empirically observed positive correlation between aggregate labor income and the return on financial ~sets. 11 s. we ~sume that ~ is the sum of three components: a constant ~, a mean-zero random variable c independent of 2, plus a fraction ~ of Zitself. Formally: (Note that y is not the mean of j, but the mean of the portion of y that is uncorrelated with z.) As will be seen, our key conclusions hold even when ~ = O. The government redistributes labor income through a proportional tax on income above a certain level (yo) and a proportional rebate on income below that level. Thus, after-tax income in the second period is y.+(1– ~)(y – ye).12Note that any first-period labor income tax is effectively a lump-sum tax, because it interacts with neither uncertainty nor labor supply elasticity. And because there is Ricardian neutrality for changes in the timing of lumpsum taxes, any tax on first-period labor income can be treated as ifit were a lumpsum component ofsecond-period labor income taxes. 10 This strategy is~h=ed by the papers cited at thebeginning of theparagraph. Kaplow (1991) arWes forcef~Y that this approach is not adequate when the purpose of the study, as in Varian, is to judge the merits of government insurance. But our goal is not to determine whether taxes are an efficient solution to private insurance market failure; we simply note the absence of private insurance and the existence of government insurance, and study the effects of this situation on other features of the economic landscape. . 11 see B=ky, M*W and Zeldes (1986). - 12 Labor income taxes n~ not be line= aslong asthecontemplated changeinlabor income taXeSiSlinear, sinCe one could let urepresent after-tax labor income under the original tax pOliCYand thenlet 7 represent a linear surtax on what wasoriginally after-tax labor income. 6

Solution. Putting together everything above, maximizing (1) is equivalent to maximizing rn,:x u(c) + Ev(~(w – c)+ a~ + Y(I+ (1– T)(Y+ : + ~~ – Ye)). (3) To get to the heart of the mathematical structure of the problem, define x = Rw + yo, A= (1– ~), O= a + (1 – ~)~, and ~ = ~ – y. + Z. Then (3) is equivalent to ~~x u(c) + EU(Z – Rc + ~k + fl~). Y Define the pair of functions C(Z, A)and O(z, A)as the solution to (4)— (c(x, A),e(z, A))= arg ~~~ u(c) + EU(Z– Rc + Afi+ 02). (5) Then the solution to (3) is given by C*= C(RW + yo,1– r) (6) and O*= e(Rw+yo,1– T) – (1 – T)@. (7) Our goal is to analyze the effect of changes in the tax rate on c* and a“. Differentiating (7) (using subscripts for partial derivatives) reveals that da* — = –eA(Rw + yo,1– T)+ p. (8) dr Thus, the effect of incre~ing the tax rate r on the amount of risky investment is always more positive when ~ > 0 (there is a positive correlation between the returns on human capital and financial assets) than it is when ~ = O (the returns on human capital and financial assets are independent). In other words, when ~ > 0, any positive effect of labor income taxes on risky investment is enhanced. On the consumption side, (6) implies that dc” — = –CA(RW + yo, 1– ~), (9) dr so that ~ does not alter the effect of the tax rate r on consumption. To make progress in evaluating ~ and ~, wemust analyze the functions C(Z, A)and O(x, A). We begin by imposing some structure on the first and second-period utility functions u(.) and v(.). First, we ~ume that ~(.) and ~(.) are both monotonically increasing, strictly concave functions - (i.e., u’(o)>0, v’(”)>0, u“(”) <0 and ;“(o) < O). Second, we ~ume that d –v’’(z) <0. 9 (lo) K () v’(x) 7

that is, u(”)displays decre=ing absolute risk aversion. This is a standard assumption that h= a sound empirical b=is because it is necessary for risky investment to be a normal good (to vary positively with wealth). Decreasing absolute risk aversion also insures that v’” will be positivewhich implies a positive precautionary saving motive. Finally, we assume that d –v’” (z) <0, (11) K () v“(z) meaning that the precautionary saving motive decreases in strength with wealth. As shown in Kimball (1990a,b), –v’’’/v’’-or “absolute prudence’’-mewures the absolute strength of the precautionary saving motive, just as —v’t/v’me=ures the absolute strength ofrisk aversion. Therefore, the -umption in (11) is simply that the absolute strength of the precautionary saving motive is decreasing in wealth (“decreasing absolute prudence”). This condition is plausible a priori,13 and is not very restrictive for utility functions that already exhibit decreming absolute risk aversion, in the sense that almost all commonly used utility functions with decreasing absolute risk aversion also have decre=ing absolute prudence. 14 However, if one is trying, it is not difficult to construct a utility function that, over a certain range, satisfies (10) but not (11). III. The Effect of Labor Income Taxes on Saving and Portfolio Decisions We are now in a position to describe the effect of changes in the labor income tax rate on an individual’s total saving and on an individual’s saving in a risky financial asset. Wedo so by proving four propositions characterizing the functions C(Z, A)and ~(x, A). Recall that c is consumption in the first period; x is the nonstochastic part of second-period wealth; Ais 1 minus the future tax rate T;and Oequals the amount of explicit risky investment, o, plus the implicit investment in the risky asset through human capital, (1 – T)p. Because first-period wealth is held constant, the change in total saving equals the opposite of any change in first-period consumption. Inferring changes in risky saving from changes in Ois more 13 See the arguments in Kimball (1990b), one of which isthe following thought experiment: “Consider a college professor who has$10,000inthebank, andaRockefeller whoh= anet worth of $10,000,000,who have thesame preferences except for their differences in initial wealth. If each is forced to face a coin toss at the beginning of the next year, with $5,OOOto be gained or lost depending on the outcome, which one will do more extra saving to be ready for the possibility of losing? If one’s answeris that the college professor will do more extra saving, it arWes for decreasing absolute prudence.” More mechanically, Kimball (1990b) shows that absolute prudence isdecreasing aslong asthe wealth elasticity of risktolerance (which isalways equal to 1for constant relative risk aversion utility) does not increase too rapidly. 14 For example, all utility functions in the hyperbolic absolute risk aversion class that have (weakly) decreasing absolute riskaversion (suchw constant relative riskaversionorconstant absolute riskaversion utility functions) also have decreasing absolute prudence, and any mixture of utility functions that individually have decreasing absolute prudence also has decreasing absolute prudence. Quadratic utility has (weakly) decreasing absolute prudence but not the more basic property of decreasing absolute risk aversion. 8

complicated, however, because risky saving equals 8 minus (1 – ~)~. If the financial and human capital risks are uncormZated(~ = O)the extra term disappears, and risky investment is measured by 0. This is the principal c~e considered below. If the risks are positiueZycorrelated, our results are strengthened. In this c=e, a reduction in T lowers risky investment both by reducing O (as shown below) and by increasing the “after-tax beta” of human capital (1 – ~)~. If the risks are negatively correlated, a reduction in T might increwe risky investment (going against our story) because additional risky investment would be desirable to help insure against the increased human capital risk. Which of these three cases is most likely? For most people, ~ is probably close to zero. That is, their labor income risk is primarily idiosyncratic, and their financial risk is primarily aggregate.ls For entrepreneurs for whom the relevant risky investment is investment in their own company, ~ will be strongly positive. Employees of brokerage houses or of firms in procyclical industries may also have a substantially positive ~. Only for people with skills particularly appropriate for countercyclical industries (for example, bankruptcy lawyers) will ~ be negative. Since we consider the case of negative ~ atypical, but otherwise wish to be conservative, we concentrate on the case of@ = Oto obtain a re=nable lower bound for the effect we are interested in. We begin by considering the effects of an uncompensated change in future taxes; later we include the effects of an offsetting change in current taxes. Given the =umptions of monotonicity, concavity, decreasing absolute risk aversion and decre~ing absolute prudence, one can prove the following four propositions about C(Z, A) and 9(Z, A). Proofs can k found in Appendices A and B. Proposition 1says that both consumption and risky investment tncmase with wealth. Proposition 1: If u’(o)>0, u“(”) <0, v’(o)>0, v“(”) <0, and v(”) exhibits decreasing absolute risk aversion, then C.(Z, A)>0 (12) and ez(z, A)~ o. (13) Proving (12) requires only monotonicity and concavity. Proving (13) requires monotonicity, concavity and decre~ing absolute risk aversion. Neither (12) nor (13) depends on decreasing . absolute prudence. 15 ~- that oneof thejWtifications for the distinction between financial and humancapita isthe muchgreater difficdty in diversifying the latter. 9

Proposition 1 impli= that the wealth expansion path for c and Oobtained by holding Afixed and varying x is an upward-sloping line, as depicted in Figure 1. Proposition 2 implies that an increase in Ashifts the wealth expansion path downward—toward lower 6 for any given level of c. In words, Proposition 2 says that a decrease in the jutum tax mte shifts the consumer’s optimum toward less risky investment for any given level of consumption.16 Proposition 2: If u’(o)>0, U“(O)<0, v’(o)>0, V“(O)<0, and v(”) exhibits decreasing absoZute prudence, then ez(z, A)cA( A x ) , – CZ(X, A)eA(~ A) ,z o. (14) As shown in Figure 1, (CZ,~z) is the vector along the wealth expansion path, and (eZ, –CZ) is the downward perpendicular to that path. Therefore the expression in (14), e=C~ – C.e~, is the dot product of (C3, 0~) with the downward perpendicular to the wealth expansion path. Proposition 2 says that along with monotonicity and concavity, decreasing absolute prudence is enough to guarantee that the dot product is always positive. This means that an increase in A moves the point (c,8) at an acute angle to the downward perpendicular to the wealth expansion path, and thus shifts the wealth expansion path down. Note that the shift ofthe optimum toward lessrisky investment for any given levelofconsump tion does not mean that an individual will always undertake less risky investment. If a decrease in the future tax rate results in a large enough increase in consumption, the individual’s risky investment will increase as well. Consumption in turn is affected by two opposing forces—the increase in wealth due to the tax reduction tends to increue consumption, while the incre~ed need for precautionary saving due to the increase in risk tends to decrease consumption. The key here is the expected value of the individual’s stochastic second-period wealth, h = ~– uo+{, towhich the taxis applied. IfEkisvery large, then a reduction in the tax rate produces a large enough rise in expected after-tax income to override both the precautionary saving effect and the risk crowding effect, thus raising both consumption and investment in the risky asset. If E~ is somewhat smaller, then a reduction in the tax rate produces a large enough rise in expected after-tax income to override the precautionary saving effect and raise consumption, but not enough to override the risk crowding effect and raise risky investment. And even smaller values of Ek mean that a reduction in the tax rate lowers both consumption and risky investment. Proposition 16 The simple statement here is for P = O. If ~ > 0, the level of risky investment that goes along with any given level of consumption will fall even more with a reduction in the tax rate T. If ~ <0, the level of risky investment that goes along with any given level of consumption may rise with a reduction in the tax rate since risky investment wodd provide insurance for the additional human capital risk. 10

4 below characterizes the effect of tax= on consumption. Proposition 3 is closely related to Proposition 2. To explain the connection, it is helpful first to view Proposition 2 as saying that an increase in labor income risk and return that leaves precautionary saving unchanged still causes a reduction in the amount of an independent financial risk which is borne. In other words, the negative interaction between independent risks—termed “temperance” by Kimball (1992)—is stronger than the precautionary saving motive. This parallels Dr&zeand Modigliani’s (1972) finding that an increase in risk that leaves utility unchanged still causes an increase in the amount of precautionary saving. Kimball (1992) summarizes these results by writing that “just as decreasing absolute risk aversion implies that prudence is greater than risk aversion, decreasing absolute prudence implies that temperance is greater than prudence.” Ifdecreasing absolute prudence makes temperance stronger than prudence, and decreasing absolute risk aversion makes prudence stronger than risk aversion, then the combination ofdecreasing absolute prudence and decreasing absolute risk aversion should imply that temperance is stronger than risk aversion. In particular, Proposition 3 shows that, given decreasing absolute risk aversion and decreasing absolute prudence, even a compensated increase in independent lahr incomera”sk to which an individual is indiflemnt leads to a reduction in independent risky investment.17 A fortiori, any increase in independent labor income risk that is not compensated enough to make the individual indiflennt leads to a reduction in independent risky investment. Proposition 3: Ifu’(o)>0, u“(”) <0, v’(”) >0, V“(O)<0, v(”) exhibits decreasing absolute risk aversion and decreasing absolute prudence, and (15) then e~(z, A)<0. (16) This result isexactly what is required to analyze equation (8) above. An increase in Arepresents a decrease in the future tax rate and thus an increase in labor income risk. With no change in lumpsum taxes, it also represents an increase in wealth, and the combination of incre=ed risk and increased wealth may raise or lower utility. So, in what situations willcondition (15) hold? Clearly, 17 pratt and Z~aWer (1987) note that-under their assumptions, if a new insurance policy comes into the market, anvone who voluntarily purchases the POhCYwill do more of other risky investment. prop.osition 3 ,“ says that the combination of d~easing absolute prudence and decreasing absolute risk aversion is enough to guarantee that restit even when the consumption/saving decision isintegrated with the portfolio composition decision. 11

the precisenatureof preferencesplaysan importantrole. For a given change in taxes, someone with greater risk aversion is more likely to suffer a decline in utility and thus do less risky financial investment than someone with a greater tolerance for risk. But there is no straightforward way to characterize the restrictions on preferences that would be sufficient to guarantee condition (15) for any possible tax change. Therefore, we try instead to characterize the types of tax changes that would satisfy condition (15) for any preferences that meet our existing assumptions. We start with the ambiguous implication for utility of a decreme in the future tax rate combined with no change in lumpsum taxes. This means that a utility-compensating change in lumpsum taxes might be either an incre~e or a decrease. But an incre=e in lumpsum taxes large enough to leave expected tax payments unchanged would have no wealth eff’t and thus unambiguously lower utility. In other words, the set of inadequately compensated changes in labor income risk necessarily includes tax changes that are intertemporally revenue-neutral.18 What kinds of tax changes will be revenue-neutral? The answer depends on the interpretation given to the model’s risky financial =t. Consider first the C- where the risky financial =t embodiestheaggregate financial risk in the economy. Then a tax change is revenue-neutral if and only if yo = y, or equivalently, iff E ~ = E: = O. Tax revenue is not affected by idiosyncratic 19Tax revenue may appear to be a function labor income risk because of the law of large numbers. of the financial risk taken by individuals, except that the government can offset any change in its financial risk-bearing through other actions in the financial market. If the government does use the financial market to offset changes in its financial risk-bearing, or if taxpayers consider government risk-bearing m ifit were their own risk-bearing (as they should ifthe government eventually absorbs the financial risk it bears through stoch~tic lumpsum taxes), then the change in national financial risk-bearing is given directly by the change in 0. In this case, the defining characteristic of a revenue-neutral tax change is that it does not affect the expected present value of the part of labor income remaining after any implicit investment in the risky financial asset is removed. Now consider the c~e where the risky financial wet represents idiosyncratic projects for which private information prevents adequate diversification of returns. Then a small tax change is revenue-neutral if and only if Yo= Ej = y+ (~+ o*)E Z= y+ 8*EZ, 18 Remember that the model implies Ricardian equivalence for lump-sum taxes, aothe timing of lumpsum taxes is irrelevant. 19 Aggregate labor income risk will Still affect both individual incomes and government revenue. Because the government cannot insureindividuals against thisrisk through redistributive taxes, we do not focus on it here. 12

orquivalently,iff Ei = E [y– U()+ ~ = –O*EZ<0. In this second case, the effects on government revenue of both idiosyncratic labor income and financial risks are canceled out by the law of large numbers. The inequality –8*E 2 ~ Ofollows from the fact that the optimal exposure to a risk is always of the same sign as its expected value. Since in this case taxes help to diversify financial as well = nonfinancial risks, taxes are more valuable than in the first case where the risky asset represents aggregate financial risk. Thus, a revenue-neutral reduction in the tax rate is even less desirable to individuals here than in the first case,so that Proposition 3 can be applied. On either interpretation ofthe financial risky asset, therefore, the following corollary to Proposition 3 effectively guarantees that an intertempomlly revenue-neutml reduction in the future tax mte leads to a reduction infinancial risk-bearing. Corollary 3.1: If u’(”)>0, U“(O)<0, u’(s)>0, U“(O)<0, U(Oe)xhibits decreasing absolute risk aversion and decre~ing absolute prudence, and EL<0, then O~(x, A)~ O. Intertemporally revenue-neutral tax chang- include changes in the timing of income taxes like those discussed by Barsky, Mankiw and Zeldes (1986) and Kimball and Mankiw (1989). By Corollary 3.1, a postponement of labor income taxes—which appears here as an increase in T and a reduction in A-crowds in risky investment. To review, Proposition 3 says that a decre= in the future tax rate causes an individual to do less risky saving as long as the individual is not made better offby the tax change. It might appear that a similarly strong result could be derived about the effect ofsuch a tax change on consumption and total saving, but unfortunately this is not the case. This lack of a clear result is surprising, because the effect on the consumption/saving decision of an increase in labor income risk in the absence of an additional risky investment choice is settled by Leland (1968), Rothschild and Stiglitz (1971), and Drbze and Modigliani (1972). in that setting Leland (1968) shows that a single mean-zero risk leads to reduced consumption as long = u’” > 0. Rothschild and Stiglitz extend Leland’s result to mean-preserving spreads. Dr&zeand Modigliani show that any undesirable risk or undesirable increase in the scale of a risk leads to reduced consumption as long = absol~e risk aversion is decreasing. This result indicates that in the absence of an additional risky investment choice, a decrease in the labor income tax rate that does not make the individual better off will lead to less consumption and more saving. 13

What compiicatmtheanalysisof the precautionary saving effect in our model is the interaction of the labor income risk and the financial et risk. The direct effect of an imposed increase in human capital risk is still to incre~e saving through the bwic precautionary motive. But in the situation we model there is an indirect effect = well: the induced decrease in financial capital risk tends to decrease saving through the same precautionary motive. The induced decrease in financial capital risk is sometimes large enough to reduce the overall riskiness of the individual’s future income, and thus lead to a net reduction in precautionary saving. Thus, our main result about the eff-t of human capital risk on other risk-taking creates ambiguity about the effect of human capital risk on total saving. Appendix B provides a numerical counterexample to the idea that greater human capital risk must lead to less first-period consumption and more saving. The counterexample hm a utility function with decreasing absolute risk aversion and decreasing absolute prudence. For the parameter values chosen, even a mean-preserving scaling-up of human capital risk increases consumption and mdtices saving through a strong negative effect on financial risk-taking. Appendix 13also shows that anincreasein the meanof futurelaborincometo compensatefor the effectof the increased riskon utilitywouldreducesavingevenmore. Thus,withoutfurther-umptions, onecannotprove that a scaling-up of human capital risk raises saving. Yet, it seems appropriate to expect the direct effect to outweigh the indirect effect in most circumstances. Can we identify some circumstances in which this will be true? In particular, when can webe sure that a decrease in the future tax rate that does not make an individual better off willincre~e total saving? By a mathematical connection, this quation isequivalent to the question of when two ~ts whose quantities can be freely varied will both be normal goods. Proposition 4 says that an individual will save mom in response to an unpleasant tax change whenever human capital would be a normal good if its quantity could befreely varied. Proposition 4: Ifu’(”)>0, U“(O)<0, v’(o)>0, u“(”)<0 and – Rc+ @i+ Ai)) = o, (17) ;(U(C) + Ev(z IC,6)=[c(~,~),e(~,~)) then CJ(Z, A) ~ Oif and only if the optimal choice of A would haveapositivewealthel=ticity at (Z,A). Our two-periodmodel assumesthat individualscannotchangethe amountof humancapital that they hold. If they could vary this amount, however, the changes in human capital risk that result would be structurally equivalent to changes in (1 – ~), which equals A. Thus, condition 14

(17) says that the amount of human capital risk (captured by A) is optimal given the individual’s optimal choices of c and 0. Using notation defined in Appendix A, Proposition 4 says that if the agent is indifferent to a marginal change in A,then (d)~” sign(C~) = –sign ~ . (18) This result is exactly what is required to analyze equation (9) above. The economic logic underlying Proposition 4 is a link between two ~pects of individual behavior. One aspect of behavior—the one we are concerned about here-is the effect on saving of an expected-utilitypreserving incre~e in labor income risk when the amount of risky financial capital can be chosen as well. The other aspect of behavior—the one that is the basis for the proposition—is the effect of wealth on the optimal amount of risky human capital in a standard portfolio problem with another risky asset. Although these two effects may appear to be distinct, they are fundamentally the same. The first effect is the change in saving that results from an exogenous change in human capital risk, while the second effect is the change in human capital risk that results from an exogenous change in wealth (and thus, saving). In both c~es, the direct effect is a positive relationship: an increase in risk will tend to raise saving, and an increase in wealth and saving will encourage more risktaking. 20 But in both C=es there is an Offsetting, indirect effect that arises from the individual’s ability to vary another type of risk, namely financial risk.21 The intuition for this indirect effect in the first case is discussed above. The intuition in the second case is straightforward: the direct effect of extra saving on investment in the risky financial =set is positive, and the greater uncertainty about future income that this choice creates will discourage investment in other risky assets, like human capital. Thus, in both c~es the crucial issue is the complementarily (or substitutability) of human capital risk and saving in the face of endogenous adjustment of financial risk. This complementarily or substitutability is symmetric across these two c~es, which makes the condition in Proposition 4 a logical condition for the problem we are interested in. 22 First, suppose that the second-period utility function Proposition 4 h= two corollaries. displays constant relative risk aversion, perhaps with a displaced origin; that is, suppose that v 20 This relationship is guaranteed by the assumption of decreasing absolute risk aversion; see Kimball (1992). 21 P hasno effect on the wealth elasticity of human capital, since optimal adjustment of financial asset holdings cancels out any und~sired changes in financial risk-bearing implicit in changes in human capital holdings. Therefore, onecanassumewithout lossofgenerality thathumancapital andthefiancial assethaveindependent returns. 22 As an aside, there isalso oneimplausible set of -umptions that would guarantee normality of both risksand therefore that an undesirable increase in human capital risk would lead to more saving: the combination of increasing absolute prudence and decreasingabsolute risk aversion. Increasing abolute prudence, by implying 15

is in the hyperbolic absolute risk aversion cl= and has decreasing absolute risk aversion. Then, incre=es in wealth lead to equiproportionate incre~es in holdings of the risky assets, ensuring that both risky assets are normal. 23 Thus 7 by Proposition 4, constant reiative risk aversion implies that any undesimble increase in labor income risk leads to a reduction in first-period consumption and an increase in saving: Corollary 4.1: If u’(.) >0, U“(O)<0, v is of the form v(x) = ‘Z-~~~-7, defined for ~ > ~0, ~jt~ ‘y>0 (with v(z) = ln(z – x~) for ~ = 1), and ;(U(C) + EV(Z - Rc+02 + AL)) <0, (c,o)=(c(z,A),e(z, A)) then CA(Z,A)<0. Second, suppose that one of the two risks h= a two-point distribution. Given decre=ing absolute risk aversion, the other risk will be normal (a result that is new with this paper). Thus, by Proposition 4, if the risky financial asset has a two-point distribution, any undesirable increase in labor income risk leads to a reduction infirst-period consumption and an increase in saving: Corollary 4.2: If u’(o)>0, u“(”) <0, v’(”)>0, v“(”) <0, v(”) exhibits decreasing absolute risk aversion, 2 is a two-point risk, and In summary, it still appears somewhat likely that an undesirable increase in human capital risk will reduce first-period consumption and raise saving. Yet, if the financial risk has more than a two-point distribution and the utility function does not exhibit constant relative risk aversion, it is possible for an undesirable increase in human capital risk to raise first-period consumption and lower saving.24 complementarily between twoindependent assets,wotid guarantee that theoptimal quantities of the two assets wotid go upand down together and soguaranteepositive wealth elasticities. Even beyond thearguments given above for decreasingabsolute prudence, theseassumptionsareanimplausible combination: given monotonicity, concavity, and the Inada condition at infinity (u’(m)= O),globally increasing absolute prudence implies globally increasing absolute risk aversion (as one can see by reversing the direction of the proof in Kimball (1993, Appendix B)). 23 In t~s vein, Hmt (1975) shows that conditions stringent enough to guarantee that the mix of risky securities doesnotdepend onwealth, together withdecreasingabsolute riskaversion, guaranteeapositive wealth elmticity for every security. 2AIndeed,thenumefic~CouterexampleinAppendixB involves only a three-point distribution for the fiaIICid 16

IV. A Numerical Illustration Several of the results in the previous section indicate that the effect of human capital risk on financial risk-taking can be substantial. Proposition 2 shows that this effect is at le=t as strong as the effect of human capital risk on consumption, while Proposition 3 shows that the effect is at le~t M strong as the effect of human capital risk on utility. Thus, these results imply that changes in taxes that reduce labor income risk can have a noticeable effect on financial risk-taking. To provide more direct evidence on the magnitude of this effect, we present the following simple numerical illustration. We interpret our model as a life-cycle model, with each period lasting one generation. Let the utility function be u–– .5[ln(c~)+ Eln(tQ)], set the real interest rate to zero, and normalize initial resources to 2. The factor of.5 on the utility function means that a 1 percent increase in overall resources produces a .01 increase in utility. In the absence of human capital risk and any financial risk-taking opportunities, the individual would choose to consume 1 in each period, and would achieve a total utility of O. Now introduce a financial risk-taking opportunity. Suppose that borrowing one unit to invest in the risky asset has an equal chance of yielding .5 or -.25; that is, Z = .5 with probability one half and -.25 with probability one-half. It is e~y to calculate that in the absence of any human capital risk, the agent will continue to consume 1in the first period and invest 1in the risky asset. Consumption in the second period will be 1.5 or .75 with equal probability, and expected utility will be .03. Finally, weadd human capital risk, with a mean-zero symmetric two-point distribution, whose standard deviation after taxes is (1 – ~)av. Table 1shows optimal values of 0, c, and U for values of (1 – ~)au between Oand 1. Note that 0, c, and U are measured in comparable units, with a difference of .01 representing the effect of a 1 percent change in scale.25 Table 1 confirms the implication of our propositions that the effect of human capital risk on risky investment is greater risk, a small mean-zero human capital risk, and a utility function that is the sum of two logarithmic utilitv functions with different origins, so there is not much room to strengthen Corollaries 4.1 an~ 4.2. Note th~t a utility function that is the sum of two logarithmic utility functions codd arise as a reduced form from an underlying logarithmic utility function with a thirdbackgroundrisk. Thus, there isno way to extend Corollary 4.1to allow for sucha third (exogenous) risk. Corollary 4.2can readily be extended to sucha situation, since the financial risk wodd still have a two-point distribution, and decreasing absolute risk aversion isunaltered by a background risk. 25 This statement isalways true for ~, and istrue for Oand c when they are near 1. Taking logarithms of 8and c would make the comparison more exact. 17

than its effect on consumption, which is in turn greater than its effect on utility. (1 - T)o, e c u o 11 .03 .1 .99 .99 .03 .2 .95 .98 .02 .3 .89 .95 .00 .4 .82 .92 –.01 .5 .75 .88 –.04 .6 .68 .84 –.07 .7 .61 .79 –.10 .8 .54 .74 –.14 .9 .48 .69 –.19 1.0 .42 .63 –.24 If the standard deviation of future labor income is .5 (within the range studied by Barsky, Mankiw and Zeldes (1986)), then raising the future marginal tax rate from Oto 20 percent causes (1 - ~)oy to decline from .5 to .4. Table 1 shows that this 20 percentage point increase in the tax rate produces close to a 10 percent increase in risky investment (from .75 to .82). Thus, the semi-el~ticity of risky investment with respect to the future tax rate is roughly one-half. A 20 percentage point increase in the tax rate from 20 percent to 40 percent produces an 8 percent increase in risky investment (from .82 to .89), for asemi-el~ticity of roughly two-fifths. The size of this effect is sensitive to the amount ofhuman capital risk, ~one would expect. If the standard deviation of future labor income is .4, for example, then an increase in the future marginal tax rate from Oto 25~0produces an 8 percent increase in risky investment, for a semiel~ticity of roughly one-third. As human capital risk declines further, the semi-elasticity of risky investment with respect to the tax rate declines as well. There is also some direct empirical evidence that the effect ofhuman capital risk on financial risk-taking can be substantial. Guise, Jappelli, and Terlizzese (1994) study the portfolio choices of Italian households using the Survey of Income and Wealth. Their estimates suggest that the elimination of income uncertainty would increase the portfolio share of risky assets by 2 to 14 percentage points (Tables 5 and 7). V. Conclusion Individuals’ ability to earn labor income is often their most valuable asset; but this =set carries with it a large and mostly unmarketable risk. A decrease in current taxes combined with an offsetting future increase in proportional or progressive labor income taxes provides insurance 18

against this risk. (And because most labor income risk is idiosyncratic, individual uncertainty can be reduced with no increase in the uncertainty ofgovernment revenue.) Barsky, Mankiw and Zeldes (1986) show that the reduction in idiosyncratic labor income risk acts through the precautionary saving motive to reduce saving relative to a Ricardian benchmark. In this paper we show that the reduction in idiosyncratic labor income risk affects portfolio decisions as well. We analyze the effect of labor income risk on the joint saving/portfolio-composition decision in a two-period model. We show that, given plausible restrictions on preferences, any change in taxes that reduces an individual’s labor income risk and does not make her worse off will lead her to invest more in a risky security, even if its return is statistically independent of the labor income risk. A deferral of labor income taxes with no change in their expected present value is one such tax change. An additional curious result is that the effect oflabor income risk on portfolio composition can be so powerful that consequent indirect effects overturn the usual positive effect of labor income risk on overall saving that is familiar from the precautionary saving literature. Ruling Out this possibility requires relatively strict assumptions, such as constant relative risk aversion or a twopoint distribution for the return on the financial asset. This counterintuitive result turns out to be related to the difficulty of guaranteeing that a pair of independent risks will both be normal goods. One implication of our results concerns the effect of government debt issuance on risky investment. Suppose that the government reduces taxes today and raises taxes in the future, although not necessarily by a corresponding amount. The results in Section III imply that this increase in government debt willcrowd in risky investment whenever any ofthe followingconditions issatisfied: (1) the expected present value of taxes faced by an individual is unchanged—i.e., there is a change only in the timing of taxes (Corollary 3.1); (2) the incre=ed future taxes have a strong enough insurance effect that the policy raises expected utility (Proposition 3); or (3) the tax changes lead to higher current consumption (Proposition 2). Thus we concur with Friedman (1978) that crowding in will occur when the government reduces taxes now and pays off the debt with higher taxes in the future.26 But our results are based on 26 ~~el (1985) estimates that pol-tfo~o effects onrates of return are very small, but that crowding in Ofequity investment ismore likely than crowding out. We can foresee two waysin which anempirical analysis based on our approach would differ from Frankel’s. First, Frankel does not allow for the effects of future tax liabilities, which play an important role in our analysis. Second, Frankel constrains government debt to affect asset demands only through changes in the market portfolio, and therefore through the covariances of asset returns with the market portfolio. Inour approach, therisk aversion of theindirect utility function depends on expected future tax rates; therefore, changing those rates changes the market risk premium. 19

a neoclassical foundation of expected utility maximization in the absence of complete insurance markets. Moreover, individuals in our model are fully aware of their future tax liabilities, which allows us to show that financial crowding in can coexist with Ricardian equivalence of lumpsum tax rescheduling. Thus, our analysis establishes clear results about the effect of labor income risk on investment. in other risky assets,27but Cwts some doubt on the generality of previous results about the effect of labor income risk on total saving.28 27 It may be ~~rising that we Cm establish suchclearresultsabout labor income taxes and financial risk-taking when the literature on capital income taxes and financial risk-taking is replete with ambiguities. The main explanation for the difference is that individuals cannot trade away their risky human capital in the way that they can trade away risky financial assets. 28 One ~rection for fmther rese~ch is to extend our restits to models with more than two periods. Kimball (1990b) gives one idea of how this might be done. In a multiperiod model, the absolute risk aversion of the valuefunction isequal to theproduct of theabsolute riskaversion of the underlying period utility function and the marginal propensity to consumeout of wealth. Under conditions similar to those weassume,idiosyncratic labor income riskraisesthe absolute riskaversionof thevalue function both by raising themarginal propensity to consume out of wealth andby raising the absolute risk aversion of the underlying period utility function throughlowering consumption. Equivalently, Breeden (1986) finds that for continuous-time diffusion processes, theexpected rate ofreturn differential between riskyandrisklesssecurities shouldbe equal to theproduct ofan agent’s underlying risk aversion and the covariance of the rate of return differential with consumption growth. Grossman and Shiner (1982) show that this relationship can be aggregated: the market risk premium should be equal to the product of a weighted average of agents’ underlying risk aversions and the covariance between the rate of return differential and aggregate consumption growth. Idiosyncratic labor income risk raises the premium for holding risky assetsintwo ways: bylowering consumption, and therefore underlying riskaversion, and by raising the marginal propensity to consumeout of wealth and therefore the covariance of consumption with the returns on risky securities in which agentshave substantial positions. 20

Appendh A Proojs of Propositions 1-3 Five lemmas prepare the ground for proofs of Propositions 1–3. Define J(c, e;x, A)= u(c) + EV(Z– Rc + Ak+ 6Z). (Al) Then (c(z, A),e(z, A)) = arg ~~x J(c, 0;z, A). (A.2) ? Lemm~ 1–5characterize the function J. All of the results here assume that the risks 2 and ~ are nondegenerate and statistically independent of each other and that u’(0)>0, U“(O)<0, v’(”)>0, . ~“(.) < 0“ Lemmas 3 and 4 relYin addition Onv having decre=ing absolute risk aversion. Lemma 5 relies on v having decreasing absolute prudence, but does not rely on decre~ing absolute risk aversion. Subscripts denote partial derivatives. Lemma 1: Forall c, d, z and A, Jcc(c, e;x, A)= R2JZZ(C,6;X,~)+ U“(C), (A.3) Jcd(c, e;z, A)= –RJ=6(C,6;X,~), (A.4) JCZ(C6,;z, A)= –RJZ=(C,0;X,~), (A.5) J.}(C, 0;x, A)= –RJr~ (C,6;z, ~), (A.6) JOg<0 (A.7) and (A.8) Proof. Differentiate (A.1).~ Lemma 2: Forall c, 0, x and A, JCCJd8– J~d> R2[JZzJ6e– JjZ] >0, (A.9) (A.1O) (All)

and (A.12) Proof. Any sum ofconcave functions and any expectation over concave functions is concave. Since both u and v are jointly concave in all c, 0, x, Afor particular realizations of 2 and fi, the function J is jointly concave in all four arguments. The strict inequalities follow from U“(C)<0, v“(”) <0, (A.7), (A.8) and from the nondegeneracy and independence of 2 and i.2g= Lemma 3: For any random variable ~, the derived utility function O(z) = EV(2+ ~) (A.13) inherits decreasing absolute risk aversion from v. Proof. Both Nachman (1982) and Kihlstrom, Romer and Williams (1981) give proofs of Lemma 3. Since decre~ing absolute risk aversion is equivalent to convexity of ln(v’(z)), Lemma 3 is a consequence of Artin’s 1931Theorem that log-convexity is preserved under expectations. Marshall and Olkin (1979)give a particularly simple proof ofArtin’s theorem based on the fact that any sum of or expectation over positive semidefinite matrices is positive semidefinite. Decreasing absolute risk aversion of v implies convexity of in(v’(z + ~)) for each realization of ~, which is equivalent to positive semidefiniteness of the matrix V’(Z+ ~) V’(X+ ~+ 6) (A.14) U’(Z+ j +6) U’(Z+ q+ 26) “ [ 1 This implies positive definiteness of EVI(Z+ ~) EV’(X+ j+ 6) (A.15) [Ev’(x + ~+~) Ev’(z+ Q+ 2d) 1 and decreasing absolute risk aversion for 0(’).- Lemma 4: If O~ O, then J8Z(C,6;x, ~) ~ Owherever Jo(c, O;x,A) ~ O. Similarly, if A ~ O, then J~Z(c,0;x, A)~ Owherever J~(c, 6;x, ~) ~ O. Proof. By the symmetry between 8 and ~ in the definition of J (Al), proving one half of Lemma 4 is enough to prove both halves. Letting @= Afi– Rc in (A.13), ~o(c,6;x, ~) = E ZV’(X– Rc+ 62+ ~k) = E~ti’(X+ 8:) (A.16) 29 The Cauchy-Schw=tz inequalities which apply to the expanded versions of (A.g–A. 12) hold with equalitY o~Y when the random variables are perfectly correlated or when one of the random variables isdegenerate. 22

and J8Z(C,9;z,A) = EZV”(Z– Rc+ e: + ~k) = EZO”(z+ 02) (A.17) O“(Z+8Z)an because of the independence of 2 and ~. Decreasing absolute risk aversion of Omakes 0’(Z+6Z) increasing function of 2, so that O“(X+ 8Z) o“(x) 0’(2 + 02) – ti~(z) has the same sign m 2. Thus, Je(c,e;x, A)= EZO’(Z+ 0;) (A.18) implies (A.19) Lemma 5: For any c, Z, 6 ~ Oand A~ O, J8J(c, e; z,A)JCZ(C,0;x,A)– J@z(c, e; z,A)JAZ(C,6;x,A)~ o (A.20) Proof. Decreasing absolute prudence isequivalent to convexity of ln(–v”). Therefore, for any four quantities Z1,22, hl and h2, ln(–v’’(z+Ozl +Ahl)) +ln(-v’’(z+8z2 +Ah2)) ~ ln(–v’’(x+Ozl +Ah2)) +ln(–v’’(z+Oz2 +Ah1)) (A.21) when (Z2– Z1)(h2 – hl) z O,with the direction of the inequality in (A.21) reversed when (22—z1)(h2 – hl) ~ O.Exponentiating both sides of (A.21) and subtracting, the quantity always has the same sign as (Z2– Z1)(h2– hi). Thus, if 21, :2, 11 and ~2 are mutually independent random variables with il and 22having the same distribution as 5, and kl and fi2having the same distribution M fi, then 23

I Je~(c, e; x, A)JZZ(C, e; x, A)– Jdz(c, e; x, A) JAZ(C, e; x, A) (A.22) = {E2iv”(z + 85+ Ai)}{E V“(X + 82 + ~k)} - {E ZV”(X+ 82+ At)}{E kV”(Z + 8Z + ~fi)} 1 =zE(Z2– Z1)(h2– L~)[vfl(z +ez~+Ail)v”(z+ez2 +Atz) –v’’(z+ ezl+Ak2)v”(z +dE2+Ai~)] ~o.m Remark: Ifv exhibits increasing absolute prudence over the interval of interest, essentially the same proof can be used to show that Je~JZZ –JoZJ~Z <0. This converse to Lemma 5 allows one to establish the converseto Proposition 2.~ Proofof Proposition 1 The a,gent’sproblem can be rewritten as ~~x J(c, e;2,A). (A.23) ? The first-order conditions are Jc(c(z, A),e(x,A);x,A)= o (A.24) and J@(c(x, A),e(z,A);x,A) =0 (A25) Differentiating (A.24) and (A.25) with respect to xand~ and arranging the resultsin matrix form (A.26) Define J.. J.@ A= (A.27) JCe Joe “ 1 Then by Lemma 1, 1 Jg8 –JC6 A-l = (A.28) JCCJ4d-J:. [-Jce J.. 1 ~-~ JotI RJ6. = RJeZ R2JZZ+U” ‘ [ 1 24

where A = JCCJ6–e J:a >0 (A.29) by Lemma 2. Multiplying both sides of (A.26) on the left by A-l and restricting our attention for now to the left column of the result, (A.30) [::l=-A-l[~i:l –A-l Jee = [ RJ8Z R2J ‘ Z J’ Z ” + U,,1 [-::”1 = A-l R(JeeJZZ– JjZ) –u“Jez l [ 1 The effect of wealth on c given by Cz is positive by Lemma 2. The effect of wealth on Ogiven by @Zis positive by Lemma 4, in conjunction with the first-order condition (A.25).~ Proof of Proposition 2 Since the determinant of the product of two matrices is equal to the product of the determinants, (A.26) implies (A.31) and therefore (A.32) ~[J 8AJ.Z – Je=JJZ] A >— o, by Lemma 1, Lemma 5 and (A.29)., Remark: As noted in the remark to Lemma 5, incre~ing absolute prudence on any interval implies that Je~JZZ– J6ZJ~Z<0, and therefore that C~@Z– CZQ~ ~ O.Since it is always possible to choose z, ~and At in such a way that only the values of v on a small interval matter, decreasing absolute prudence is a necessary condition for (A.32) to hold for any x, 5 and ~~.~ Proof of Proposition 3 Calculating CJ and ~~ just as we calculated Cz and ~z (that is, by multiplying both sides of A.26 on the left by A-l), (A.33) [:tl=-A-l[~:l 25

Jge = –A-l [ RJo. R2J‘ZJ”Z+ U!’1[-:’”’1 ~_l R(JeeJxJ – JeZJg~) = [ R2(J$ZJZ~– Jz=Je~)– u“Je~1 “ Lemma 5 implies that R2(J@ZJz~– JzZJg~) <0. Moreover, since J.x <0, Lemma 5 implies that The first-order condition Je = O,together with Lemma 4, implies that Je. ~ O,and the assumption of Proposition 3 that J~ < 0, together with Lemma 4, implies that J~Z ~ O. Therefore, (A.34) guarantees that u“JO=J~Z -u’’Je~ < – <0. (A.35) J xx 26

Appendix B Numerical Counterezample and Proofs Related to the Eflect of Human C’apitalRisk on Consumption Numerical Counterexample: Increased Human Capital Risk Raising Current Consumption k’rom(A.33), CAh~ the same sign w JoeJZ~– JeZJe~= [E~2V”(z– Rc + ~k+ OZ)][EfiV”(Z– Rc + Ah + 02)] - [EZV”(Z- ~c + ~fi+ 82)][E~kV”(Z – Rc + ~k + 0:)]. The effect of a small mean-zero risk k that is independent of 2 can be analyzed by taking a secondorder Taylor expansion around ~ = Oand then taking the expectation over ~ to obtain the marginal distribution over 2. With the ~sumption Eh = Oeliminating the expectation of the first-order terms in the Taylor expansion, this yields J8@JZ~– JeZJ8~= [EZ2V”(X- Rc+ OZ)][o~Ev’”(Z - RC+ 8z)] - [E~V”(Z - Rc + 02)][o;E ~V’”(Z– Rc + 82)]+ O(C7:). Thus, for sufficiently small mean-zero ~, the effect of ~ on saving has the same sign as [E~2V”(Z – Rc + 8z)][EU’”(Z– Rc+ 82)]– [EZV”(Z– Rc+ 02)][E~V’”(Z– Rc + 02)]. Now,let v(x) = 31n(z) + 7A2ln(w + A3), where A is a large positive number. Let x– Rc = 5. It is always easy to choose a first-period utility function u and values for x and c to satisfy the first-order condition for optimal consumption. Let 2 equal -4 with probability .5– ~, -2 with probability .5 – ~, and A2 with probability ~. Choose ~ so that O= 1 becomes the optimal amount of 2 w h gets very small. That is, choose f to satisfy As A ~ +00, ~~ 1 in this equation. Then, straightforward but tedious calculations show that as ~ gets small and A ~ m, 27

I EZ2V’’(.)EV’’’(.)– E5v’’(.)EZV’’’(.) = ; >0. Thus, if k gets small enough and A gets big enough, CA becomes positive. Since C. is always positive, adding a positive constant to the mean-zero human capital risk ~ can only make CAmore strongly positive. n Proof of Proposition 4 From (A.33), sign(C~(x, A)) = sign(~~~~=~– (B.1) If Awere chosen optimally, the first-order condition JA(C,e;x, A)= o (B.2) would always hold. Allthree ofthe variables c, 8and J would be functions ofx alone. Differentiating (B.2) and the other two first-order conditions with respect to x would yield the matrix equation Define (B.4) By Cramer’s rule, d~” –1 ‘cc ‘Ce ‘=x JCe J$e JOZ (B.5) dx = det(B) JC~ Je~ JAx where the second line results from subtracting R times the third column from the first column of the determinant in the numerator on the first line. Since B is a negative definite 3 x 3 matrix, det(B) <0 and . d~” –sign(JOeJ~Z– Je~J6z) = –sign(C’~)., ‘tgn (-)dx = 28

Proof of Corollary 4.1 If (B.7) and (B.8) (B.9) and (B.1O) Therefore, ifthey were both freely chosen, the optimal values of Aand Owould be proportional to z – Rc – Zo,guaranteeing that both would increase with x.= Proof of Corollary 4.2 By (A.33), in order to prove the corollary, we must show that Je8J$J – JeZJg~<0 when z has a two point distribution. Given the fact that Jee < 0, this is equivalent to (B.11) where d6 JeZ — = —— (B.12) dx Jee is the optimal adjustment in 8 when x changes with Aremaining fixed. With ~ held fixed, ALis a fixed background risk, and the decre~ing absolute risk aversion of v is enough to guarantee that M > (). dz — Denote the two possible realizations of 2 as Zl, which occurs with probability p, and 22,which occurs with probability (1 – p). Then, the equation Je = O,which characterizes an optimal value of 0, becomes 29

pzlE [v’(z – Rc+ d~l+ At)]+ (1 – p)z2E[v’(z – Rc+ 622+ Afi)]= O, (B.13) where the expectations denoted by E here are only over h. (Note the use of independence between Z and k in expanding out the expectation over 2.) Clearly, (B.13) requires Z1 and 22 to be of opposite signs. Without loss of generality, let 22 >0>21. Taking the derivative of (B.13) with respect to Z, with Oadjusting to maintain this first-order condition, yields ‘zl(l+z’:)‘Ec’+v’’z’(lx+-Ak(1)l++z(2l-:p)E)z’’z2v2’’+(zA-Ri)lc=+ (B.14) Since Z2>0> z1, equation (B.14) implies that both (1+ zz~) and (1+ 21~) are of the same sign and therefore both positive. Turning to the quantity of interest, (B.11) can be expanded to “zJ+’’:p=(l+zl:)E~Av’’)’Rl’)cl sezl+ (B.15) ‘(1-p) (1+z2:)E’iv’’(z-Rc+ ez2+Ai)’ ~o Dividing (B.15) by the negative quantity -PZ1('l'+vZ''(12:-)RAck+)l`=z('1+-Ep''v(1''+(zz-A2R:ic))l+1ez (B.15) becomes 1 E [kV”(Z– Rc+ 8ZI + Ai)] + 1 E[~v”(z – Rc+ 022+ ~t)] < o (B.16) (-z1) E[v”(z – Rc + Ozl+ AL)] ~ E [v~l(~– Rc+ 8ZZ+ AL)] - In order to prove (B.16), use the stipulation that at the point in question, JA ~ O, or in expanded form, pE ~V’(Z– RC + 8Z~+ ~k) + (1– p)E kV’(Z– RC + 622i-~k) = 0. (B.17) Dividing (B.17) by the positive quantity 30

–pzl E [v’(z – Rc + Ozl + Ak)] = (1– p)E [v’(z – Rc + ez2 + AL)], (B.17) becomes 1 E [iv’(z – Bc+ Ozl+ Ak)] + 1E [kv’(z – Rc+ 8Z2+ ~~)] <0. (B.18) (-ZI) E[v’(x - Rc+8.z, + AZ)] ; EIvI(Z– Rc + Ozz+ AA)] Note that except for involving the first derivative of v instead of the second, (B.18) is of the same form as (B.16), which needs to be proven. To prove (B.16) from (B.18), we will show that each of the two terms in (B.16) is less than or equal to the corresponding term in (B.16). For z= 1,2, define E[kV”(2– RC + Ozi+ AL)] ~i= (B.19) E[v”(X – Rc+ Ozi + Ak)] “ Decreasing absolute risk aversion, which guarantees that –v” falls faster than v’ in proportional terms, implies that for any scalar realization h of the random variable k, (B.20) Taking expectations of both sides of (B.20) over random ~, E [hV”(X– RC + 6~i+ ~k)]– ~iE[V”(2– RC + dzi+ Jk)] (B.21) v“(X – RC+ Ozi+ Aqi) But, by the definition of vi, the right-hand side of (B.21) is less than or equal to zero. Thus, the left-hand side of (B.21) is also less than or equal to zero, implying Therefore, (B.18) implies (B.16).= Intuitively, the key to the proof is that with a twmpoint distribution for z, an increase in x with its consequent increase in 8 leads to an increase in z — RC + Ozi for both ~= 1 and ~= 2. This incre~e in x – Rc + Ozi leads to an incre~e in the marginal benefit of higher A relative to the expected marginal utility v’for both i = 1 and i = 2. Since the condition for optimal Oinsures that the importance of expected marginal utility for i = 1 and z= 2 is in a fixed ratio, this leads to an increase in the overall marginal benefit of A. 31

References Ahsan, Syed M., “Taxation in a Two-Period Temporal Model of Consumption and Portfolio Allocation,” Journal of Public Economics, 5 (April-May, 1976), 337-352. Auerbach, Alan J. and Laurence J. Kotlikoff, Dynamic FiscaZPoZicy (Cambridge: Cambridge University Press, 1987). 13arro,Robert J., “Are Government Bonds Net Wealth?” Journal of Political Economy 82 (November/December, 1974), 1095-1117. Barsky, Robert B., N. Gregory Mankiw, and Stephen P. Zeldes, “Ricardian Consumers with Keynesian Propensities,” American Economic Review, 76 (September, 1986), 676-691. Bodie, Zvi, Robert C. ~Merton,and William F. Samuelson, “Labor Supply Flexibility and Portfolio Choice in a Life-Cycle Model,” Journal of Economic Dynamics and Control, 16 (July-October, 1992), 427-449. Breeden, Dougl~ T., “Consumption, Production, Inflation and Interest Rates: A Synthesis,” Journal of Financial Economics, 16 (May, 1986), 3-39. Caballero, Ricardo, “Consumption Puzzles and Precautionary Savings,” Journal of Monetary Eco. nomics, 24 (January, 1990), 113–136. Chan, Louis Kuo Chi, “Uncertainty and Neutrality of Government Financing Policy,” Journal of Monetary Economics 11 (May, 1983): 351-372. Davies, James and John Whalley, “Taxes and Capital Formation: How Important is Human Capital?” ch. 6 in lvational Saving and Economic Performance, B. Douglas Bernheim and John B. Shoven, eds. (Chicago: University of Chicago Press, 1991). Devereux, Michael B., and Gregor W. Smith, “International Risk Sharing and Economic Growth,” International Economic Review 35 (August, 1994), 535-550. Domar, Evsey D. and Richard A. Musgrave, “Proportional Income Taxation and Risk-Taking,)’ Quarterly Journal of Economics 58 (May, 1944), 388-422. Dr&ze,Jacques H., and Franco Modigliani, “Consumption Decisions Under Uncertainty,” Journal of Economic Theory, 5 (December, 1972), 308–335. Driffill, E. J. and Harvey S. Rosen, “Taxation and Excess Burden: A Life-Cycle Perspective,” International Economic Review, 24 (October, 1983), 671-683. Easley, David, Nicholas M. Kiefer, and Uri M. Possen, “An Equilibrium Analysis of Fiscal Policy with Uncertainty and Incomplete Markets,” International Economic Review, 34 (November, 1993), 935-952. 32

Eaton, Jonathan, and Harvey S. Rosen, “Taxation, Human Capital, and Uncertainty,” American Economic Review, 70 (September, 1980), 705-715. Economic Report of the President (Washington, D.C., 1996). Feldstein, Martin, “The Effects of Taxation on Risk Taking,” Journal of Political Economy, 77 (Sept.-oct. 1969), 755-764. Feldstein, Martin, Capital Taxation (Cambridge: Harvard University Press, 1983). Feldstein, Martin, “The Optimal Level of Social Security Benefits,” QuarterZy Journal of Eco. nomics, 100 (May, 1985), 303–320. Frankel, Jeffrey A., “Portfolio Crowding-Out, Empirically Estimated,” Quarter/y JournaZof Economics, 100 (August, 1985), 1041–1066. Friedman, Benjamin M., “Crowding Out or Crowding In? Economic Consequences of Financing Government Deficits,” Brvokings Papers on Economic Activity, 3:1978, 593-641. Friend, Irwin, and Marshall E. Blume, “The Demand for Risky Assets,” American Economic Review,65 (December, 1975), 900–922. Grossman, Sanford J., and Robert J. Shiner, “Consumption Correlatedness and Risk Measurement in Economies with Non-Traded Assets and Heterogeneous Information,” Journa/ of Financial Economics, 10 (July, 1982), 195-210. Guise, Luigi, Tullio Jappelli and Daniele Terlizzese, “Income Risk, Borrowing Constraints and Portfolio Choicel” CEPR Discussion Paper # 888 (January, 1994). Hart, Oliver E., “Some Negative Results on the Existence of Comparative Statics Results in Portfolio Theory,” Review of Economic Studies, 42 (October, 1975), 615-621. Jaffe, Adam B., “Technological Opportunity and Spillovers ofR & D,” American Economic Review, 76 (December, 1986), 984-1001. Kanbur, S. M., “Risk Taking and Taxation: An Alternative Perspective,” Journal of Public ECOnomics, 15 (April, 1981), 163–184. Kaplow, Louis, “A Note on Taxation as Social Insurance for Uncertain Labor Income,” NBER Working Paper # 3708 (May, 1991). Kihlstrom, Richard E., David Romer, and Steve Williams, “Risk Aversion with Random Initial Wealth,” Econometric, 44 (June, 1981), 911-920. Kimball, Miles S., “Precautionary Saving in the Small and in the Large,” Econometric, 58 (January 1990a), 53–73. Kimball, Miles S., “Precautionary Saving and the Marginal Propensity to Consume,” NBER Work- 33

ing Paper # 3403 (1990b). Kimball, Miles S., “Standard Risk Aversion,” Econometric, 61 (May, 1993), 589-611. Kimball, Miles S., “Precautionary Motives for Holding Assets,” The New Palgrave D~ct~onary of Money and Finance, Peter Newman, Murray Milgate and John Eatwell, eds. (1992), Stockton Press, New York: 158-161. Kimball, Miles S., and N. Gregory Mankiw, “Precautionary Saving and the Timing of Taxes,” Journal of Political Economy, 97 (August, 1989): 863-879. Kimball, Miles S., and Phillipe Weil, “Precautionary Saving and Consumption Smoothing over Time and Possibilities,” NBER Working Paper # 3976 (1992). Leland, Hayne E., “Saving and Uncertainty: The Precautionary Demand for Saving,” Quarterly Journal of Economics, 82 (August, 1968), 465-473. MmDonald, Glenn M., “The Economics of Rising Stars,” American Economic Review, 78 (March, 1988), 155-166. Marshall, Albert W. and Ingram Olkin, Inequalities: Theory of Majorization and its Applications (New York: Academic Press, 1979). Merton, Robert C., “On the Role of Social Security as a Means for Efficient Risk Sharing in an Economy Where Human Capital Is Not Tradable,” Financial Aspects of the United States Pension System, Zvi Bodie and John Shoven, eds. (Chicago: University of Chicago Press, 1983). Nachman, David C., “Preservation of ‘More Risk Averse’ Under Expectations,” Journal of ECOnomic Theory, 28 (December, 1982), 361–368. Pratt, John W., “Risk Aversion in the Small and in the Large,” Econometric, 32 (January-April, 1964), 122-137. Pratt, John W., and Richard J. Zeckhauser, “Proper Risk Aversion,” Econometrica, 60 (January, 1987), 143-154. Rothschild, Michael and Joseph E. Stiglitz, “Increasing Risk II: Its Economic Consequences,” Journal of Economic Theory 3 (March, 1971), 66–84. Sandmo, Agnar, “Capital Risk, Consumption, and Portfolio Choice,” Econometric, 37 (October, 1969), 586-599. Sandmo, Agnar, “The Effect of Uncertainty on Saving Decisions,” Review of Economic studies, 37 (July, 1970), 353-360. Sandmo, Agnar, “The Effects of Taxation on Savings and Risk Taking,” Handbook of Public ECO- 34

nomics, A. J. Auerbach and M. Feldstein, eds. (Amsterdam: North-Holland, 1985). Shleifer, Andrei and Robert W. Vishny, “The Efficiency of Investment in the Presence of Aggregate Demand Spillovers,” Journal of Political Economy 96 (December, 1988), 1221-1231. Skinner, Jonathan, “Risky Income, Life Cycle Consumption, and Precautionary Savings,” JournaZ of Monetary Economics, 22 (September, 1988), 237–255. Varian, Hal R., “Redistributive Taxation as Social Insurance,” Journal of Public Economics, 14 (August, 1980), 49-68. Weil, Philippe, “Precautionary Savings and the Permanent Income Hypothesis,” Review of Economic Studies, 60 (April, 1993),367–383. Zeldes,StephenP., “Optimal Consumptionwith StochasticIncome: Deviations from Certainty Equivalence,” Quarterly Journal of Economics, 104 (1989), 275-298. 35

i (C(x,ko)e,(x,ko)) C(X,A1)@,(x,A*)) (cx,@x) c Figure 1