Abstract
This paper demonstrates that the stream of uncertain income from human capital has systematic effects on the demand for risky physical capital assets. If labor supply is inelastic and real wages are known with certainty, then a labor income tax will reduce holdings of the risky physical asset. However, if labor income fluctuates randomly, a labor income tax may actually raise demand for the asset if human capital risk and physical capital risk are positively correlated. The idiosyncratic risk and nontradability of human capital also have implications for optimal taxation.
Economists have long been concerned with the effects of taxation on entrepreneurs’ risk-taking behavior. Since the seminal essay by Domar and Musgrave (1944), such issues have been analyzed in a pure portfolio choice framework. See Tobin (1958), Mossin (1968), Feldstein (1969, 1976), Stiglitz (1969), Sandmo (1969), and Dreze and Modigliani (1972). Presumably more risk taking leads to greater demand for risky assets and thus lowers the cost of risky capital. In the long run, such behavior helps to increase the economy’s capital stock and national income.
Although this literature has succeeded in dispelling the popular perception that taxing the returns on risky assets will necessarily reduce risk taking, it suffers from some serious limitations. Chief among them is that the earlier work largely ignores some important and closely related decisions, such as individuals’ saving-consumption and labor-leisure choices. Moreover, even if labor income is allowed, in addition to income from capital assets, the pure portfolio choice model usually takes such income as exogenously given (the labor supply is fixed) or assumes that individuals receive a certain stream of wage income.
Ignoring labor income fails to capture the crucial role of human capital in consumption and investment decisions, as human capital is often the more important form of wealth for most individuals.1 An extreme case is that in the absence of bequests the young are endowed with only human capital and no financial wealth.
To assume an exact stream of wage income is also unsatisfactory because in reality individuals cannot foresee the flow of their future labor income with certainty. For example, real wages are uncertain when the prices of consumption goods are uncertain. For some people, their otherwise smooth and continuous flow of labor income may contain jump components over the life cycle if they face a positive probability of temporary layoff or sudden loss of earning ability owing to health problems. Thus, like its physical counterpart, human capital is a risky asset. The risk of human capital income probably has important interactions with the risks of financial assets. Moreover, unlike physical capital, human capital cannot be traded for obvious moral hazard reasons. No claims on future wage income are actually traded on financial markets. This feature of human capital will undoubtedly affect society’s efficient risk pooling and risk sharing. Consequently, an analysis of the effects of taxation on risk taking should explicitly account for the riskiness and nontradabiiity of human capital.2
This paper recognizes the existence of risky, nontradable human capital income and attempts to offer an integrated treatment of individuals’ intertemporal consumption and labor supply as well as of their portfolio choices. The paper presents a basic model and first considers the simplest case with fixed labor supply and deterministic wage rate. Endogenous labor income is then introduced. The paper then turns to analyze the case when income from human capital follows geometric Brownian motion. Finally, it explores the implications of the risk and nontradabiiity of human capital for efficient taxation.
I. Basic Model
It is assumed that the uncertainty in the economy is generated by a two-dimensional stochastic process {dz,dq}, where dz and dq are standard Brownian motions and where dz dq = ηzq dt (ηzq being the instantaneous coefficient of correlation between dz and dq).
Individuals have two investment opportunities in this economy. One is investment in a risk-free asset with an instantaneous rate of return, r(t), and the other is investment in a risky capital asset, which may be viewed as a real production possibility. The consideration of this two-asset case proves convenient because it provides a well-defined measure of risk taking, given by the portfolio share of the single risky asset. Moreover, under certain conditions (such as with a constant consumption and investment opportunity set or with logarithmic utility functions), the continuous-time version of the Sharp-Lintner-Mossin capital asset pricing model (CAPM) obtains, and all individuals will hold the market portfolio, so that the many-asset case can be simplified as if the market were the only risky asset. The price of this risky asset can be modeled as an Ito process:
where α and σ are the instantaneous mean and standard deviation of the rate of return on the risky asset. With constant α and σ, equation (1) implies that the price of the risky asset is log-normally distributed. The flow of stochastic real wage income can be written as
Consider an individual who lives for T years and whose preference is described by a time-additive state-independent utility function, U[c(t), L(t)]. Assume that U[c(c), L(t)] exhibits the standard properties of monotonicity and concavity in consumption and leisure: that is, Uc > 0, UL > 0, Ucc < 0, and ULL< 0, where the subscripts on U denote the first-order and second-order partial derivatives with respect to c(t) and L(r), respectively.
At each point in time during the life cycle, the risk-averse individual chooses the rate of consumption c(t), the current leisure L(t), and a portfolio rule through which he or she can transfer wealth over time and across the states of nature.3 Let ω(t) denote the proportion of wealth to be invested in the risky asset. The general problem of intertemporal optimization is
where E0 denotes the expectation conditional on the information available at time 0. With income derived from human capital, dI, the budget constraint can be written as a stochastic differential equation: 4
Assume that labor income is subject to a proportional tax rate τ, with the government tax revenue being used to finance a public good that enters the utility function_ separately. If the individual consumes a fixed amount of leisure L and wage flow is deterministic (that is, dY—Y(t)dt), then the stream of after-tax labor income is dI = θ(1 - L)Y(t)dt, where θ = (1 - τ). Thus, the problem of equation (3) reduces to one with a single consumption good and with exogenous risk-free labor income. Defining the derived utility function of wealth as
and the Bellman equation as
one can write the first-order conditions for this problem:
where the subscripts on (U and J denote partial derivatives with respect to c, W, and t, respectively.
From equation (8), the share of the portfolio in the risky asset can be written as
In general, the optimal proportion of wealth held in the risky asset depends on the investor’s risk preference, wealth endowment, and age as well as on the parameters of the returns on traded assets a, r, and σ2. The individual’s investment in the risky asset is determined by his or her desire to attain a preferred risk-return trade-off in wealth. The more risk averse is the individual (the smaller the term—JW/[WJWW]), the smaller is his or her portfolio share in the risky asset. On the other hand, the larger the premium on the market, α - r, or the smaller the market risk, σ2, the larger is the proportion of his or her wealth that the individual will hold in the risky asset.
Note that a labor income tax affects the optimal portfolio demand by changing individuals’ risk-averse behavior and total wealth. To see this, consider the case of a utility function with constant relative risk aversion (CRRA) of consumption.5 Merton (1971) first derived explicit consumption and portfolio rules for the CRRA utility function:
where δ is the reciprocal of the relative risk aversion of consumption. Let WH denote the value of human capital; then,
In other words, the value of human capital is the present value of the lifetime flow of maximal labor income discounted at the risk-free market rate of interest. The crucial point here is that although human capital is not directly traded the individual can, in effect, sell his or her risk-free future labor income for present consumption by trading (shortselling) financial assets. Thus, equation (10) can be rendered
If there is only investment income (WH— 0), the individual will wish to maintain a constant share of the risky asset, ω*, in his or her portfolio over the life cycle. Then, the demand function for the risky asset is linear in the level of wealth. Thus, one obtains the usual separation theorem, which states that should be independent of the investor’s wealth and age for the case of the CRRA utility function. If the individual also receives labor income, he or she will treat his or her net worth of human capital as an addition to the current stock of wealth, and the individual’s portfolio share of the risky asset will be age dependent. The individual is willing to take more risk by investing more in the risky asset when he or she is young since the value of human capital is greatest in the early stage of life. As a person gets older, his or her ratio of WH to W will decline, and he or she will take a smaller position in the risky asset.
A labor income tax reduces the value of human capital and therefore decreases an individual’s holding of the risky asset. Fiscal policy is not neutral with respect to the individual’s risk-taking behavior. A current tax cut matched by a future tax increase tends to encourage investment in the risky asset. Since the government does not share the investment risk faced by private agents through a labor income tax, total (social) risk taking must also be reduced.
II. Interaction Between Labor Supply and Portfolio Choice
This section introduces the labor-leisure choice into the individual’s decision problem but continues to assume that income from human capital is nonstochastic.
The individual’s problem now is
From the Bellman equation.
the first-order conditions are
and
Combining equations (15) and (16) gives
Differentiate equation (17) with respect to τ and assume that U is separable in c and L; then,
Thus, an uncompensated increase in the tax on riskless labor income unambiguously discourages the individual’s labor supply when his or her preference is separable in the consumption good and leisure.
The optimal demand for the risky asset given by the first-order conditions takes the same form as equation (9). To derive a closed-form solution, a two-stage procedure is used. First, solve the following static optimization problem at each t:
Define U(C,t) = V[c*(C,t),L*(C,t)]. Because real wages are non-stochastic, there is no intertemporal uncertainty in the price of leisure relative to the consumption good. Therefore U(C,t) remains state independent. Then proceed to solve the following dynamic optimization problem:
Note that the leisure variable, L(t), no longer enters the budget constraint in equation (20). The problem now is formally identical to the one with a single consumption good. For a CRRA utility function, the optimal portfolio rule has the familiar form:
where WH is given by equation (11).
Note that ω** > ω*. For the same ratio of human wealth to current wealth, the individual takes more risk with an elastic labor supply than with a labor supply fixed at (1—L).6 Intuitively, the ability to adjust labor supply conditional on investment performance provides insurance for the individual’s investment risk. The effect of a proportional labor income tax, however, is to reduce private (and therefore social) risk taking, as in the case with fixed labor supply.
III. Uncertain Income from Human Capital
The presence of risky income from human capital complicates the problem in two ways. First, it introduces uncertainty into the relative price of leisure (denominated in the consumption good). As a result, the utility function becomes state dependent, with the state variable represented by the wage variable, such that U = U(C, Y, t), where C is redefined from equation (19) as the consumer’s aggregate expenditure, inclusive of the value of leisure measured in terms of the consumption good. Second, the presence of risky labor income has a wealth effect. It affects the dynamics of wealth accumulation, so that both the drift and the diffusion terms in the investor’s budget constraint are modified.
Assume that the stochastic behavior of wage income is as specified in equation (2): the wage income follows the geometric Brownian motion.7 Formally, the stochastic dynamic programming problem is
From the Bellman equation,
where σzq = σσyηzq is the covariance between the return on the risky asset and income from human capital. Then, the first-order conditions follow:
and
The optimal demand for the risky asset can now be derived:
The optimal demand for the risky asset consists of three terms. The first term, (-JW/JWW)(α - r)/σ2, is the usual speculative demand for the mean-variance maximizer.8 The second term,—θYσzq/σ2, is a labor income hedging demand. The last term, —QY(JYW/JWW)(σzq/σ2), is the state-variable hedging demand. These hedging terms arise from the nontradability of risky human capital. When the individual has human capital income as well as financial investment income, the risk of human capital income will play a role in determining his or her optimal portfolio behavior. In other words, the interaction between the human capital risk and the financial risk will affect the individual’s optimal holding of the risky asset. Compared with the case of riskless labor income in the previous sections, one can see that now, in addition to holding the risky market portfolio to attain the desired risk-return trade-off in wealth, the individual also uses the risky market portfolio to hedge against the unanticipated and possibly unfavorable changes in his or her labor income.
From the first-order condition (24),
and
If UCY— 0—that is, the direct utility function is state independent—then JWY/JWW = (∂C/∂Y)l(∂C/∂W) > 0, because the propensities to consume out of income and wealth are both positive. Then, the signs of both the hedging terms depend on the sign of the covariances between the return on the risky asset and wage income, σzq. If the return on the risky asset and that on wage income are negatively correlated (ηzq < 0), the existence of the risky human capital income produces a positive hedging effect on the demand for the risky asset. The same observation was made by Fischer (1975) in his exposition on demand for indexed bonds with a state-independent utility function.
Since
the effects of labor income taxation on risk taking will in general depend on the covariance of human capital risk and physical capital risk. The standard result under riskless real wage income no longer holds for the more general case of an uncertain income stream from human capital.
If wages and the return on investment in physical capital are negatively correlated, then a proportional labor income tax will reduce the hedging demand for the risky asset. Because part of the labor income risk is now shifted to the government,9 the individual then needs less of the risky asset in order to hedge the risk of nontradable human capital.
If the instantaneous rate of return on the risky financial asset is positively correlated with the unanticipated changes in wage income (ηzq > 0), then the individual’s hedging demand for the risky asset is negative: reverse hedging occurs. A labor income tax will actually encourage risk taking in this case.
If the return on the risky asset and that on wage income are uncorrected (ηzq > 0), then both the hedging terms in equation (26) will vanish. In this case, the individual’s labor income risk is idiosyncratic. Holding the risky financial asset cannot provide insurance against unanticipated changes in labor income. Although the idiosyncratic labor income risk will affect the individual’s risk aversion and consumption behavior, it does not affect his or her demand for the risky asset.10 Therefore, the hedging demand for the risky asset will be zero. In this case, the only mechanism for sharing and pooling individuals’ idiosyncratic labor income risk is the government’s taxation on labor income.
Which of these possibilities is empirically more relevant remains a subject for further research. Fama and Schwert (1977) study the role of human capital in the classical CAPM. They found that the relationships between the returns on human capital and the returns on various portfolios of traded assets are weak, so that the presence of human capital does not significantly affect the measurement of risk for traded capital assets. However, as the authors themselves acknowledged, their results cannot be viewed as definitive because of a number of measurement and estimation problems.
In addition to these hedging effects, the existence of labor income also produces a wealth effect. The risky stream of labor income, when appropriately capitalized, is treated as part of wealth in the consumption and portfolio demand functions.
IV. Optimal Taxation with Nontradable Risky Labor Income
The analysis above points to the importance of the interaction between the risks of financial assets and those of human capital income. The individual will invest more of his or her wealth in the risky capital asset if the return on the asset is negatively correlated with his or her labor income. In effect, the financial market can provide insurance against the labor income risk. This interaction between the individual’s nonhuman capital risk and human capital risk will, in addition to its effects on portfolio choice and risk-taking behavior, have implications for efficient income taxation, to which the discussion now turns.
Because human capital is nontradable, it clearly improves welfare to have some kind of social scheme that pools and shares the human capital risk when the private market fails to provide insurance instruments for individuals to hedge against the uncertain stream of labor income. The government, through its ability to tax, can play precisely such a role in diversifying the idiosyncratic human capital risk. Indeed, it is tempting to suggest a 100 percent wage tax combined with a lump-sum transfer. In reality, however, this type of public insurance program is unlikely to work because of its disincentive effect on labor supply. Therefore, one needs jointly to consider the insurance and efficiency aspects of labor income taxation.
Assume that a large number of individuals in the economy are identical in preferences, beliefs, initial endowments, and abilities, so tha tvariations tn labor income arise only from differences in “luck.” 11 In other words, the random fluctuations in labor income are caused by “idiosyncratic” risks that are uncorrelated among individuals. Formally, for each individual i, the uncertain stream of wage income is represented by equation (2), with dqi dz = ηqzdt, ∀i and dqi dj = 0,∀i ≠ j.
Consider a representative consumer who chooses his or her optimal consumption, labor supply, and portfolio share at each point in time, taking the tax rate on labor income, t, as given. The individual essentially faces the following problem:
One can write the following first-order conditions with respect to c, L, and w, respectively:
and
where θ = 1—τ, where J = J(W, t) is the derived utility function of wealth, and where σzq = σσy ηzq is the covariance between the individual’s financial risk and human capital risk.
How might the uncertainty about future wage income affect an individual’s labor supply decision? To see this, consider some simple cases. Substituting equation (33) into the first-order condition (32),
Suppose that the individual’s utility function is additively separable in the consumption good and leisure. If the individual’s human capital risk and financial risk are perfectly correlated (that is, |ηzq| = 1), then differentiating equation (34) with respect to t gives
Note that the term in the square bracket is positive. To see why, imagine there is a traded asset that replicates the income from a unit of human capital. At equilibrium, this traded asset is priced by the continuous-time CAPM:
where μy is the instantaneous rate of return on the traded asset. It is then easy to see that the square-bracketed term in equation (35) is positive when the rate of return on the riskless asset r is greater than zero. According to the standard assumptions about the utility function U[c(t), L(t)], it can be established from equation (35) that ∂L*/∂τ > 0. Therefore, just as in the case of risk-free labor income, an increase in the proportional wage tax will unambiguously reduce labor supply when the labor income risk and investment income risk are perfectly correlated. If, as a polar case, the human capital risk and the financial risk are uncorrelated (ηzq = 0), then differentiating equation (34) with respect to τ gives
Equation (37) clearly shows the possibility of a positive response of labor supply to an increase in the proportional wage tax. The greater the nondiversifiable labor income risk, the larger the share of current labor income in wealth, and the more risk-averse the individual, then the more likely it is that the right-hand side of equation (37) will turn positive and that the individual will increase his or her labor supply in the face of a higher wage tax. This surprising result arises because the government’s taxation of income provides insurance against the individual’s labor income risk.
Suppose that the government chooses a linear income tax schedule. At each point in time, the government taxes each individual’s labor income at the proportional rate ツ and makes a lump-sum income transfer dS(t) to each individual. The government faces the following intertemporal budget constraint:
where dG is the government revenue requirement. The government wishes to maximize the representative consumer’s lifetime utility given its budget constraint (38) so as to determine the optimal tax rate t*. In other words,
From the Bellman equation, the first-order condition with respect to τ is
The other two first-order conditions with respect to c and w are not presented here because they take the same forms as equations (31) and (33).
Using these first-order conditions from the individual’s maximization problem, one can, after some manipulations, obtain
Let F(τ) denote the left-hand side of equation (41). Obviously F depends on the interaction between the labor income risk and financial risk (ηzq) as well as on the labor income risk itself (σ2y).
Suppose that individuals’ labor income is risk free (σ2y = 0), then F(0) = 0. That is to say, the optimal taxation structure implies a zero tax rate on wages if tabor supply is elastic and if individuals do not face human capital risk; this is for the usual reason that, under certainty, one can have lump-sum taxation when all individuals are identical.
In general, a zero tax on wages cannot be optimal. Consider the case when the human capital risk and financial risk are uncorrelated (ηzq = 0). The expression of F becomes
Because evaluating F at τ = 0 gives
the first-order condition cannot be satisfied at a zero tax rate. In fact, if F(τ) = 0, then equation (41) yields
Therefore, if labor supply responds negatively to changes in the wage tax (∂L*/∂τ > 0), the optimal marginal tax rate should lie strictly between 0 and 1.
For the case of perfect correlation, it is straightforward to show that 0 < τ* < 1 if ηzq = +1 and that it may be optimal to have a wage subsidy, τ* < 0 if = -1. Therefore, even when individuals can diversify the risk of human capital by holding a risky financial capital asset, the optimal tax on their labor income is usually not zero.
These results closely parallel those obtained by Eaton and Rosen (1980), who use a static model of uncertainty with endogenous labor supply.12 They do not consider the interaction between investment risk and wage risk because the nonlabor income in their paper is assumed to be nonstochastic.
V. Conclusion
This paper has reexamined the effects of taxation on risk taking and on labor supply in a continuous-time life-cycle model. It has shown that the existence of income from human capital has systematic effects on individuals’ optimal portfolio choice and that the risk and nontradability of human capital have implications for an efficient income tax structure. Further work, especially empirical study, is needed to better understand how taxation affects individuals’ lifetime saving and portfolio decisions.
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Zuliu Hu was an Economist in the Research Department when this paper was written. He has since moved to the Fiscal Affairs Department. He holds a doctorate from Harvard University. The author is grateful to Robert Flood, Robert Merton, and Larry Summers for helpful comments.
Williams (1978) examines the effect of human capital on portfolio choice. In addition, Williams (1979) analyzes individuals’ decisions about their investment in human capital through their level of education.
In a similar model without human capital, Hamilton (1987) examines how taxation of risky financial assets, such as dividend and capital gains taxes, can affect individuals’ risk-taking behavior.
This paper abstracts from the individual’s retirement decision so that the individual has an active working life until the termination date T. The retirement problem has been extensively studied in the public finance and labor economics literature, see Diamond and Mirrlees (1978). One can similarly explore the problem of retirement with uncertainty in this continuous-time framework.
See Merton (1971) for a detailed derivation.
The constant relative risk aversion (CRRA) or isoelastic utility function belongs to the family of hyperbolic absolute risk-aversion utility functions.
In a closely related analysis, Bodie, Merton, and Samuelson (1992) show that the ability to vary labor supply ex post induces the individual to assume greater risks in his or her investment portfolio ex ante.
Merton (1971) solves a problem in which the wage income is given exogenously and is a Poisson process.
With continuous trading, the risk premium on the traded asset, α - r, is determined by Breeden’s (1979) consumption β model even with the existence of nontraded human capital, as long as both the asset prices and the consumption rate follow the Ito processes, as shown by Grossman and Shiller (1982).
Implicitly, it is assumed that the tax code contains full loss-offset provisions.
Although the idiosyncratic labor income risk itself does not affect the demand for the risky asset, the presence of human capital and labor income taxation will still produce a wealth effect and financial risk taking will be affected.
Stiglitz (1982) examines the problem of Pare to-efficient taxation when individuals differ in their productivities and the government has imperfect information about the “true” distribution of abilities across people. The assumption here is similar to that employed by Barsky, Mankiw, and Zeldes (1986), who examine the interaction between individual income uncertainty and income taxation in the face of a debt-financed tax cut.
Varian (1980) also examines the social insurance aspect of redistilbutive taxation in a static model of uncertainty. He does not explicitly consider the incentive effect of taxation on labor supply.