Therefore, social insurance through proportional income taxation is welfare enhancing as there is no aggregate risk. However, full insurance is impossible due ...
Optimal Taxation of Human Capital under Uncertainty reconsidered Bas Jacobs Erasmus University Rotterdam and CESifo Dirk Schindler
Hongyan Yang
Universit¨at Konstanz and CESifo
Universit¨at Konstanz
Februar 29, 2008 WORK IN PROGRESS DO NOT QUOTE Abstract In a model with ex-ante homogenous households, income risk and a general earnings function, we derive an optimal labor tax rate and optimal education subsidies. It turns out that the distinction of risk-increasing or risk-decreasing education does not have much effect on the optimal tax rate, but is crucial for optimal education subsidies. Moreover, an optimal education policy allows for better social insurance, as the tax-induced distortions in labor supply can be mitigated. This extends the analysis by Bovenberg and Jacobs (2005) to risky economies. Furthermore, we show that education policy itself has no insurance effect and is used solely for efficiency reasons and only if there is another policy instrument, which provides social insurance.
JEL Classification: H21, I2, J2 Keywords: labor taxation, human capital investment, education subsidies, idiosyncratic risk, risk properties
1
Extended Abstract and Paper Outline
Income risks as well as the risk of failure play an important role in educational investment and human capital accumulation. This has implications for the optimal design of tax and education policies, as has been shown in various contributions to the literature. See, for example, Eaton and Rosen (1980, AER), Hamilton (1987, IER), and Anderberg and Andersson (2003, JPubE). However, their findings differ, depending on the available policy instruments and depending on assumptions concerning how education influences risks and whether the labor supply decision is made before and after risks realize. The aim of the present paper is to provide a unifying framework, which integrates the previously studied approaches in order to characterize optimal taxation and education policies in risky economies. To that end we develop a one-shot model of human capital investments, labor supply and idiosyncratic risk. We assume additively separable preferences in labor and consumption. We furthermore extend the previous literature by employing a completely general earnings function that depends on human capital investment, labor supply and a random variable reflecting the uncertain state of nature. This general earnings function allows both for the possibility that education is a risky activity and the possibility that education hedges against labor market risk. Moreover, we separately study the cases where labor supply is decided before and after the realization of the income shock. Ex-ante homogenous households differ ex-post due to the realization of idiosyncratic risk in their ex-post income. Markets to insure earnings and/or human capital risks are missing. Therefore, social insurance through proportional income taxation is welfare enhancing as there is no aggregate risk. However, full insurance is impossible due to the endogeneity of labor supply, which causes a moral hazard problem. Income risk also affects education choices, depending on the shape of the earnings function. If education increases income risks, there will be underinvestment in education as individuals command a risk premium on their investment. In the case where education decreases income risks, individuals overinvest in education as education hedges against labor market risk. Therefore, education subsidies may seem potentially desirable so as to correct over- or under-investment in the presence of risk. 1
We fully characterize the optimal structure of tax and education policies, independently from the earnings function and the timing of labor supply. In particular, we will distinguish four cases: (i) education increases income risk and the labor supply decision is made before risk realizes (Anderberg and Anderson, 2003; Eaton and Rosen, 1980a, JPubE). (ii) education increases income risk and the labor supply decision is made after realization of risk (Eaton and Rosen, 1980b, AER; Hamilton, 1987). (iii) education and risk are negatively correlated and labor supply is determined under uncertainty (Anderberg and Anderson, 2003). (iv) education provides insurance against risk, but labor supply is determined after risk has realized. In doing so, our paper contributes in six major ways to the existing literature. First, we derive an explicit optimal tax rule. The optimal tax rate depends on the trade-off between insurance and incentives. The larger the gains of insurance – as measured by the normalized covariance between income and marginal utility of income – the higher is the tax rate. The larger are tax distortions in labor supply and education, the lower are tax rates. Further, by insuring labor market risks, the government is able to mitigate under- or over-investment in education by lowering risk-premia. Second, we derive an explicit rule for optimal education subsidies. We demonstrate that education subsidies are not used for insurance. This shows that the government does not aim to correct private over-investment or under-investment with education policy under laissez faire. Indeed, when there is no social insurance, governments cannot improve upon the market outcome by subsidizing education as this will upset the optimal private response to market risk. The fundamental reason is that only redistribution ex post can insure risks. As education subsidies are given ex ante, they do not insure risks, but distort education decisions. Therefore, the government does not use education policy in the absence of social insurance. These findings show that the first-best with full insurance is not the correct benchmark to judge the desirability of education subsidies, as in Anderberg and Andersen (2003). Third, we show that only with positive tax rates education subsides are employed in order to mitigate the labor tax distortions on labor and educational investment. Hence, education subsidies are used for efficiency reasons only. The 2
optimal subsidy increases in the tax rate. Accordingly, the combination of tax and education policies enables the government to provide better insurance via higher tax rates. This finding extends Bovenberg and Jacobs (2005, JPubE) to risky environments. Fourth, as education subsidies allow for more aggressive income insurance, the combination of tax and education policies results in less under- or over-investment. The adverse impact of missing insurance markets on educational investments is thereby reduced further, compared to the case where no social insurance cum education policy is employed. This additional role for education policies is a new finding and has not been identified before in the literature. Fifth, we explicitly derive the tax and subsidy elasticities of labor and education choices under risk by restricting the utility functions to have either constant relative risk aversion or constant absolute risk aversion and the earnings function to have a constant elasticity or constant semi-elasticity in education. Also this has not been done before and we demonstrate that risk-aversion parameters and the education elasticity of the earnings function are critical determinants of the optimal tax and education policy rules. Sixth, we show that the timing of labor supply decisions does not impact the structure of optimal tax and education policies, only the size of the labor supply elasticity. If labor supply is determined before risk realizes, taxation also has an insurance effect on labor supply. If individuals determine labor supply ex ante, the labor supply elasticity is lower compared to the case where households decide on labor supply after the realization of risk. Consequently, tax rates are higher in the former case. As a result, education subsidies are also higher when labor supply is determined ex ante rather than ex post.
2
Review of the Literature
Levhari and Weiss (1974) are the first, who examine the effect of risk on human capital accumulation and educational investment. Neglecting a labor supply decision, they assume a general earnings function Φ(e, ε), which depends on the level of human capital investment (education) e and a random factor ε. This factor captures various kinds of risk, which can be distinguished in uncertainty about 3
inputs and uncertainty about outputs.1 Uncertain inputs, i.e., contain imperfect knowledge about the own (learning) ability or about the quality of the school, in a nutshell, the risk of being successful in education. Uncertain outputs stand for uncertainty about market conditions and the productivity of the acquired human capital, accordingly, e.g. unemployment risk and stochastic wages. Depending on which kind of risk one focuses, this will have effects on the sign of the partial derivatives of the earning function. For input risk, e.g., the risk to fail at university, income risk should be increasing in the level of educational investment, hence Φeε > 0. If so, education is a risk-increasing activity. Instead, focusing on output risks, it can happen, that education decreases the exposure to risk. This is the case for, i.e., unemployment risk, which is significantly decreasing in education. Then, Φeε < 0 and education is said to be risk-decreasing. Levhari and Weiss (1974) show that educational investment is a device for selfinsurance. If education increases risk (Φeε > 0), households demand a positive risk premium in education πe > 0 and they will underinvest in order to reduce their exposure to risk. If education is instead a risk-reducing activity (Φeε < 0), overinvestment in education will insure against income risk and the households accept a negative risk premium πe < 0.2 If the risk under consideration is idiosyncratic, it can be fully diversified by pooling and it vanishes in aggregate. This is the case for a large part of human capital risk, e.g. for imperfect knowledge of abilities or quality of schools, accordingly especially for input risk. Hence, in principle, private insurance is possible at very low or even no costs and self-insurance by over- or underinvesting is inefficient from society’s point of view – although it is rational on individual level. However, in case of human capital, private insurance markets suffer from market failure and private insurance is not (or only to a very limited extend) available. As reasons are seen:3 (i) moral hazard. Any insurance changes the behavioral incentives, if private actions (i.e., learning effort) are not observable and create suboptimal behavior and inefficient insurance contracts, if any. (ii) adverse selection. If there is hidden information and the insured have an informational advantage 1
See Levhari and Weiss (1974), pp. 951. See Levhari and Weiss (1974), pp. 953. 3 See also Sinn (1996), Section 2, for a short discussion of the relevant literature. 2
4
(e.g., about their abilities), insurance companies can only offer average premiums. Then, the good risks will opt out and the insurance market can collapse. (iii) legal restrictions. Insuring human capital risk (from innate abilities or illness) may require contracts, which are settled very early in life, before the veil of ignorance is lifted. This would be only possible if (a) parents can sign binding contracts for their children, which have to settle for all their life, or if (b) the children themselves sign contracts, when they are very young. The former is not possible, yet, as long as slavery is precluded, and the latter is not possible as (very young) children are neither intellectually nor contractually (in a legal sense) capable to do so. (iv) crowding out. Existing social insurance crowds out private insurance markets, especially if mandatory social insurance is higher than the privately desired insurance coverage.4 Indeed, Sinn (1996, pp. 262) points out that crowing out cannot explain the non-existence of private insurance markets in this area, because the crowing out is far from perfect. As private markets are incomplete, social welfare can be increased by public intervention. Although this also suffers from moral hazard,5 the government overcome adverse selection and legal problems by providing mandatory social insurance against risks of lifetime careers. The most prominent insurance device in this respect is income taxation, which reduces the variance of incomes and redistributes resources between successful and unlucky individuals. Caring ex-ante for social risk insurance is moreover the same as having ex-post (in-)equality concerns, because for idiosyncratic risk any probability distribution of a shock converts into an income distribution, where each probability equals a relative frequency. Redistributing income and social insurance are just two sides of the same coin.6 The formal analysis of social insurance in case of human capital risk dates back to a seminal paper series by Eaton and Rosen (1980a,b), who have first extended the Levhari-Weiss approach for public policy instruments. The literature in this area can be distinguished along two lines: (i) Is education risk-increasing or risk-decreasing? (ii) Is labor supply chosen before or after risk has realized? Consequently, we are left with four cases: 4
See Andersson and Konrad (2003) for a formal analysis. See Wigger and von Weizs¨acker (2001) for a detailed analysis. 6 See also Sinn (1996), pp. 273. 5
5
Education risk-increasing L a b o r
after risk before risk
Eaton/Rosen (1980b) Hamilton (1987) Eaton/Rosen (1980a) Anderberg/Andersson (2003)
risk-decreasing — Anderberg/Andersson (2003)
All these papers deal with ex-ante homogenous households and idiosyncratic risk, and they all assume market failure in private insurance, accordingly missing insurance markets. In Eaton and Rosen (1980a) a human capital decision is neglected and an earnings function Φ(l, ε) = ε · w · l, exhibiting a multiplicative risk factor ε, is applied. Eaton and Rosen show that a proportional wage tax rate is preferable to lump-sum taxation, if labor supply l has to be chosen before risk realizes and households are therefore faced by income risk.7 The intuition is as follows: wage taxation provide insurance by reducing the variance in incomes and by returning tax revenue as deterministic lump-sum transfer. They state that for some levels of a proportional tax rate the insurance effect compensates induced distortions and increases welfare. Eaton and Rosen (1980b) extends this analysis. Now, they assume that labor supply is chosen after realization of risk, but households have to invest in human capital before the productivity shock reveals. Again, they use a multiplicative modeling of risk, where education e is risk-increasing. Costs of education are foregone earnings in a first period, w · e, and the intertemporal earnings function is given by Φ(e, l, ε) = ε · h(e) · l + (1 + r) · w · (1 − e). This setting fits well to input risk. Eaton and Rosen disclose some conditions, under which a wage tax increases human capital investment, and again they show that distorting labor taxation is welfare-enhancing, because the insurance effect dominates tax-induced distortions around t = 0.8 The analysis by Hamilton (1987) rests on the two-period model in Eaton and Rosen (1980b), assumes increasing educational risk as well, and reinforces the case 7
A similar argument is brought forward by Varian (1980), dealing with portfolio risk and allowing for non-linear taxation. 8 This analysis is extended to progressive labor taxation and the simultaneous use of education subsidies in Schindler and Yang (2007).
6
of a distorting labor tax at positive rate, but widens the focus to educational policy. Hamilton points out that, even though taxation provides some (social) insurance, there is still underinvestment in human capital, which is socially inefficient. Therefrom, he states that it would be preferable to give the government full control over human capital investment. Direct control of human capital is, however, unrealistic, hence, Hamilton proposes an interest tax as indirect tool for fostering educational investment. Interest taxation decreases the opportunity costs of human capital accumulation. A positive tax rate on interest, accompanying positive proportional wage taxation, is shown to be optimal under the (very) strong assumptions of (i) inelastic labor supply and (ii) either savings of zero or constant absolute risk aversion. Taken together, all these paper show that some social insurance is desirable – and that there still will be inefficient educational investment. However, neither of these papers really deal with optimal educational polices nor determines an optimal labor tax rate, balancing insurance and distortions. Anderberg and Andersson (2003) tackle the simultaneous design of tax and educational policies. They apply a model, where the wage w depends on a stochastic factor ε and on educational investment e and where education causes resource costs of e. The earnings function equals Φ(e, l, ε) = w(e, ε)·l−e and both risk-increasing and risk-decreasing education is considered. Whilst the households choose labor supply before risk realizes, the level of education is mandatorily set by the government – but costs are borne by the households. Anderberg and Andersson derive an optimal trade-off between insurance and tax-induced distortions in labor supply. In educational policy, they identify an insurance effect and an revenue-creation-effect. The latter effect can be divided into the subeffects of substitution, distributional and precautionary effect. Conditions under which each effect ceteris paribus calls for over-/underprovision of education are derived and the benchmark for evaluation is the First-best solution. It is shown that some of the effects may pull in different directions, but it seems that the main message is to choose overprovision if education is risk-decreasing and underprovision if exposure to risk is increasing in human capital. In fact, Anderberg and Andersson (2003) mix mitigation of distortions by use of educational policy and the self-insurance effect of households’ educational in7
vestment in deriving their creation-of-revenue effect. Consequently, their intuition and interpretation given is misleading, as will be discussed – among other aspects – in the following sections.
3
The Model
3.1
Preferences and assumptions
In a one-shot model, there is a continuum of ex-ante identical individuals, who have to decide on their educational investment and labor supply. They consume a homogenous good, whose price is normalized to 1, and their income is represented by an earnings function of the general form Φ = Φ(e, l, ε). This earnings function depends on education e and labor l and is subject to an idiosyncratic shock, measured by a random factor ε. Therefore, income and the returns to education are risky. We assume that, for any given value of ε, the marginal returns to education and to labor supply are positive, but decreasing, Φe , Φl > 0 and Φee , Φll < 0. Furthermore, the random variable ε is assumed to have a positive effect on income, Φε > 0. Human capital investment e is measured in pecuniary costs only and the government subsidizes education costs at a rate of s. Accordingly, the costs of an educational investment e accrued to the households are (1 − s)e. Income is taxed at a flat rate t and educational investment costs are assumed tax deductible. Moreover, the government grants a lump-sum transfer T , which can be seen as an earned-income tax credit. Taken together, the consumable income of a representative household is gross income Φ net of education costs and income tax plus the lump-sum transfer and, thus, consumption is given by9 c˜ = (1 − t) [Φ(ε, l, e) − (1 − s)e] + T.
(1)
For the timing structure and for the analysis to come, it is important, when decisions are made and how households’ choices and risk interact. 9
Henceforth, all variables, depending on the realization of ε and being stochastic, will be indicated by a tilde.
8
In any case, educational investment is chosen before risk realizes. However, human capital can both increase and decrease income risk. This circumstance is caught by the general form of the earnings function, which does not specify how educational investment interacts with income risks. In particular, we will distinguish two cases:10 (i) Educational investment itself causes and amplifies income risks. Technically, this is the case if Φeε > 0, thus if the shock affects marginal productivity positively as well. (ii) Educational investment hedges against income risks. This requires Φeε < 0, accordingly marginal productivity of learning is decreasing in the shock, whereas income is increasing. Labor supply instead can be chosen before or after realization of risk. If labor supply is chosen before the state of nature reveals, labor supply can be used as insurance device in order to stabilize income. Analogously to education, labor supply can thereby increase or decrease income risk. If labor is determined expost, there is no uncertainty concerning labor choice and income risks are not affected by labor supply. If labor supply is decided before risks realize, the timing structure of the model can be summarized as follows: Firstly, the government sets the proportional tax rate t, the subsidy rate s and the lump-sum transfer T . Secondly, the individuals observe these parameters and choose simultaneously their human capital investment e and their labor supply l. After that, (income) risk realizes, incomes are received and consumption takes place. If labor supply is chosen after realization of risk, we get instead a timing sequence in households’ decisions: They first decide on their educational investment, then risk realizes and after that, given their fortune in education, households choose their optimal labor supply – knowing for sure their individual wages.
3.2
Households optimization
The households derive utility from consumption and disutility from labor supply. We assume a von Neumann-Morgenstern expected utility function, being addi10
See the discussion in section 2 and Levhari and Weiss (1974), pp. 951.
9
tively separable in consumption and labor supply:11 U = E[u(c) − v(l)] = E[u(˜ c)] − v(l)
(2)
with u0 > 0, u00 < 0 and v 0 > 0, v 00 > 0. The subutility function of consumption is increasing and concave, whereas the disutility function of labor supply is increasing and convex. Furthermore, we impose Inada-conditions on both utility functions to avoid corner solutions. The household maximization problem can then be stated as max E [u(c)] − v(l)
(3)
subject to the budget constraint (1). Differentiation with respect to e and l leads to the first-order conditions E [u0 (.) (Φe (.) − (1 − s))] = 0,
(4)
(1 − t)E [u0 (.)Φl (.)] − v 0 (l) = 0.
(5)
For optimal educational investment the risk adjusted marginal return of education should equal its risk adjusted marginal costs. The second condition states that the risk adjusted return to labor (measured in additional consumption) should optimally balance marginal disutility of labor supply. To show the effects of risk properties on optimal education and labor supply we define: cov [u0 (.) , Φe (.)] πe ≡ − (6) E [u0 (.)] E [Φe (.)] πl ≡ −
cov [u0 (.) , Φl (.)] E [u0 (.)] E [Φl (.)]
(7)
πe is the negative normalized covariance between marginal utility of consumption and marginal return of human capital, therefore it mirrors the risk premium demanded in education. A positive risk premium implies that education enforces income risk, because πe > 0 corresponds to Φeε > 0 and educational investment 11
Note that only consumption is stochastic, while working time and therefore disutility of labor is deterministic.
10
increases risk. Marginal utility of consumption is decreasing in ε (as income is increasing in ε), but marginal productivity of education is increasing and the higher the educational level, the more exposure to risk there is. A negative risk premium, instead, mirrors a risk-reducing effect of education, due to Φeε < 0. πl is the negative normalized covariance between marginal utility of consumption and marginal return to labor, representing the risk premium in labor supply. Its interpretation is analogous to the risk premium in educational investment. Using the definitions of πl and πe , the first-order condition (4) can be rewritten as 1−s E [Φe (.)] = (8) 1 − πe If there is no income risk, condition (8) reduces to Φe (.) = 1 − s, the standard marginal return equal to marginal costs’ condition under certainty. If income is risky, however, expected marginal return of education can be either higher or lower than marginal costs, depending on the sign of the risk premium πe . In case that education increases risk, πe > 0, individuals require a positive risk-premium for their educational investment and therefore a higher expected marginal return than marginal costs. Hence, from a social point of view, riskaverse individuals invest too little in education. The risk premium turns in an additional cost factor and underinvestment can be seen as insurance device, acting as self-insurance in incomplete markets. If income risk decreases with education, πe < 0, marginal costs of education are, on the contrary, higher than marginal return, as risk aversion now acts as cost-subsidy. Then, risk-averse individuals invest too much in education in order to reduce their exposure to risk. In case of missing insurance markets, overinvestment in education then is the only (self-)insurance available. From this discussion it follows as well that (expected) marginal return is only equal to marginal costs of education, if either the households are risk neutral or if there are insurance devices (other than over-/underinvesting in education), providing full insurance. Only in these cases the risk premium will drop to zero. The first-order condition for labor supply (5) can be reformulated as v 0 (l) = (1 − t)(1 − πl )E [Φl (.)] E [u0 (.)] 11
(9)
in case labor supply is chosen before the shock. The marginal rate of substitution between consumption and labor must be equal to the risk adjusted net wage. Hereby, the emergence of risk and risk aversion acts as an additional tax on labor. If an increase in labor supply increases risk, the households demand a positive risk premium πl > 0 and, in order to decrease their risk, supply less labor than socially efficient. If labor supply decreases the exposure to risk, the risk premium turns negative and operate like a wage subsidy, leading to precautionary labor supply as self-insurance. Instead, if the labor supply decision is made after the shock, there is no uncertainty related to labor supply and the risk-premium for labor equals zero, πl = 0. We have v 0 (l) = (1 − t)Φl (.) . (10) u0 (c) Note that the value of ε is known with certainty and that the level of education e is already fixed, so that equation (10) is only a function of labor supply l.
3.3
Elasticities of education and labor supply
To be derived • explicit tax elasticities of education and labor supply • explicit subsidy elasticities of education and labor suppy Suggestion: Labor elasticities are higher if labor supply decision is made after risk has realized than in case of choosing labor supply under uncertainty.
3.4
Government
We assume a benevolent government, which maximizes social welfare and takes care for optimal (ex ante) social insurance and optimal (ex post) redistribution of incomes. The entire tax revenue from the wage tax is used to finance education subsidies and a lump-sum transfer, exogenous public expenditure is assumed to
12
be zero. As education costs are deductible from the labor tax base, the budget constraint of the government is given by t {E [Φ (.)] − (1 − s)e} = T + s · e
(11)
Since all income risk is idiosyncratic, tax revenue is deterministic according to the law of large numbers and revenue equals its expected value. Due to ex ante homogenous households the social welfare function is given by the indirect expected utility function of a representative household. The Lagrangian for maximization then is L ≡ E [u(˜ c∗ )] − v(l∗ ) + η [t (E [Φ (.)] − (1 − s)e) − se − T ] with η as the Lagrangian multiplier for the government budget constraint (11). Let the tax wedges on labor and education be defined as ∆l ≡ tE [Φl (.)] ,
(12)
∆e ≡ t (E [Φe (.)] − (1 − s)) − s.
(13)
∆l respectively ∆e measures the extent to which taxation and subsidization reduce the marginal return to labor supply respectively education. The first-order conditions for this maximization problem are: ∂L ∂E [u] ∂l ∂e = − η + η∆l + η∆e =0 ∂T ∂T ∂T ∂T ∂L ∂E [u] ∂l ∂e = + η (E [Φ (.)] − (1 − s)e) + η∆l + η∆e =0 ∂t ∂t ∂t ∂t ∂L ∂E [u] ∂l ∂e = − η(1 − t)e + η∆l + η∆e =0 ∂s ∂s ∂s ∂s
(14) (15) (16)
These are the central equations, which will be discussed in more detail in the following sections and from which optimal tax formulas will be derived.
13
4
Optimal Tax and Education Policies
In this section, we discuss the optimal policies for education and taxation. To do this, we first analyze optimal tax and education policies separately. Then, we derive the formula for choosing both policies simultaneously. After that, we discuss implications of our results under different assumptions concerning the earnings function and the timing of labor supply.
4.1
Optimal Lump-sum Transfer ∗
] Using Roy’s lemma ∂E[u = E [u0 ] the first-order condition for the optimal lump∂T sum transfer (14) becomes
∂L ∂l ∂e = E [u0 ] − η + η∆l + η∆e =0 ∂T ∂T ∂T We define b≡
u0 ∂l ∂e + ∆l + ∆e , η ∂T ∂T
as (net) social marginal valuation of income, including income effects on the tax base (see Atkinson and Stiglitz, 1980). Thus, the optimal lump-sum transfer is characterized by E [b] = 1
(17)
This expression states that, for social welfare maximization, expected social marginal value of a unit increase in lump-sum income should be equal to its resource costs. This corresponds to the standard condition of optimal income taxation (e.g. in Atkinson and Stiglitz, 1980). Notice that the law of large numbers implies that the probability distribution of income before shock is equal to the income distribution after risk realization. Therefore our result becomes b = 1 from an ex-post perspective.
4.2
Optimal Taxation
Analogously to Feldstein’s distributional characteristic (see e.g. Feldstein, 1972, or Atkinson and Stiglitz, 1980), we define the so-called insurance characteristic ξ 14
as the negative normalized covariance between gross income Φ and social marginal value of income b: cov [Φ, b] ξ≡− (18) E [Φ] E [b] The insurance characteristic ξ measures the magnitude of governmental concern about the variation of individual gross income, consequently about the effects of income risk. A positive insurance characteristic implies that lucky individuals, having high incomes after realization of risk, have a low welfare weight and individuals, having bad luck and earing lower income ex post, have a high weight in the welfare function. If so, income taxation is characterized by ex-post redistribution from lucky (‘rich’) to unlucky (‘poor’) individuals respectively by providing social income insurance from an ex-ante point of view. The latter is achieved via wage taxation by reducing the variance in net incomes. The insurance characteristic would be zero only if either the government is not concerned about income insurance (and redistribution), thus if all individuals have the same welfare weight no matter which income they have, or if there is no uncertainty and household income Φ is deterministic. After some reformulations (see the Appendix), the first-order condition for taxation (15) can be rewritten as t ξ (1 − πe ) ²et s =− + 1−t ²lt + πe ²et (²lt + πe ²et ) (1 − t) 1 − s
(19)
where ²lt and ²et are the income-weighted compensated tax elasticities of education and labor supply:12 E [Φl l] ∂l∗ 1 − t 0 and there is underinvestment in education – which is socially suboptimal. Instead, if there is overinvestment due to precautionary human capital accumulation and risk-decreasing education, consequently if πe < 0, the labor tax rate will be higher in order to insure and to decrease socially unproductive investment. Extending the model for an exogenously given subsidy rate s > 0, the optimal (1−πe )²et s tax rate is determined additionally by a second component (²lt +π · 1−s >0 e ²et )(1−t) if ²et < 0.14 Hence, a higher subsidy rate implies a higher tax rate: subsidizing education costs makes educational investment more attractive and mitigates tax distortions in education. Moreover, it fosters labor supply by increasing opportunity costs and therewith decreases distortions in labor supply as well. Therefore, education subsidies allow for a higher tax rate and for extended social insurance 13
However, the tax distortions in labor supply would still be higher than in standard models without endogenous human capital investment, as educational choice amplifies the labor elasticity. See, e.g., Jacobs (2005). 14 This term is positive as long as the optimal tax policy without educational subsidies leads to a positive tax rate, accordingly ²lt + πe ²et < 0 from (20) and ²et < 0, because πe ∈ [−1, 1] by definition.
16
of risks. Under certainty, the optimal tax rate is determined by t=−
²et s ²lt 1 − s
(21)
because ξ = 0 and πe = 0, then. Without exogenous government expenditures, the only rational for taxing income is to provide social insurance through an expost redistributive taxation and to correct over- or underinvestment in education that results from income risks and risk aversion. In a deterministic model with (ex-ante) homogenous households, however, there should be neither support for redistributive taxation nor for subsidies. Therefore, we should expect the optimal tax and subsidy rates to be zero in this special case – which will be shown in the following sections.
4.3
Optimal Education Policy
By rearranging the first-order condition (16) (see the Appendix), the optimal subsidy rate for a given tax policy can be characterized by s = 1−s
µ ²ls
²es
+ πe
1 − πe
¶ t.
(22)
In doing so, the income-weighted compensated subsidy elasticities of education and labor supply are defined as E [Φl l] ∂l∗ 1 − s R0 E [Φ] ∂s l E [Φe e] ∂e∗ 1 − s ≡ >0 E [Φ] ∂s e
²ls ≡
(23)
²es
(24)
The optimal subsidy is determined by two effects: Firstly, labor taxation distorts labor supply. These distortions can be mitigated by fostering (hampering) education via subsidies (tuition fees), as a higher (lower) stock of human capital increases the opportunity costs (forgone earnings) of leisure and stabilizes labor supply, if ²ls > 0 (²ls < 0). However, ceteris paribus, the subsidies create welfare-decreasing distortions in education itself. The optimal subsidy trades 17
these tax-weighted distortions off, which is mirrored in the first term in equation t ls (22), ²²es · 1−π . e The efficiency effects of education subsidies can be illuminated by looking at the tax wedge on education: ∆e t (E [Φe (.)] − (1 − s)) − s = =− 1−s 1−s
µ
²ls ²es
1 − πe
¶ t
(25)
Ceteris paribus, education is subsidized on a net basis, as long as higher education induces a higher labor supply, ²ls > 0 (and t > 0). Education should be taxed, instead, if an increase in education would decrease labor supply, ²ls < 0 – which seems not to be the normal case. In a nutshell, higher labor taxation ceteris paribus calls (in absolute values) for higher education subsidies in order to enhance efficiency and to mitigate distortions. This is in fact the ‘Siamese Twins’-intuition and extends the results by Bovenberg and Jacobs (2005) to risky economies. Secondly, education subsidies (respectively tuition fees s < 0) are used to mitigate risk-induced under- or overinvestment in education. This effect is represented πe by the second term in equation (22), 1−π ·t e The higher the labor tax rate is, the better social insurance is provided and the less self-insurance via educational investment is required. Hence, if education is risk-increasing and households are underinvesting (πe > 0), the inefficient low level of human capital ceteris paribus calls for educational subsidies. If education
decreases the exposure to risk (πe < 0), households tend to overinvest. This socially inefficient allocation can be improved by taxing education on a net basis. Again educational subsidies are used for efficiency reasons and not for (social) insurance. This gets very clear by recognizing from equation (22) that the optimal subsidy is not only increasing in the tax rate, but it is also equal to zero if there is no labor taxation (t = 0). In an economy where no social insurance via income taxation is provided, educational investment should not be subsidized/taxed. Consequently, we can sum up: Proposition 1. Education subsidies are only used in conjunction with labor taxes and are applied for efficiency reasons only. They do not provide any social insur18
ance. In case there is no labor taxation, the only insurance device for households is self-insurance by over- or underinvesting in education. This insurance is chosen optimally by the households, no governmental action is required. As subsidies are, moreover, state-independent and their ‘tax base’ is deterministic, they do not reduce the variance in income and therefore have no direct effect on the exposure to risk as well. Hence, the government can not improve market outcomes by subsidizing education in the absence of social insurance. Indeed, subsidizing education without income taxation would only distort the optimal responses of individuals to income risks.
4.4
Optimal Tax cum Education Policy
Combining the formulas for optimal tax and education policies (22) and (19), we receive the optimal tax rate, if the government can simultaneously employ optimal education subsidies: t∗ ξ (26) =− ls ∗ ²et 1−t ²lt − ²²es The optimal subsidy then results from substituting (26) into (22): s∗ = 1 − s∗
µ ²ls
²es
+ πe
1 − πe
¶ t∗
Equation (26) clearly indicates that the only reason for taxing income is to provide social insurance, in case there is no exogenous government expenditure (or if lump-sum taxes are not available). Hence, if there is no income or educational risk or if risk does not concern the government (which both is characterized by ξ = 0), there is no income taxation, when education subsidies are chosen endogenously. From (22), the optimal education subsidies are also zero, because subsidies are only used to alleviate tax distortions and to mitigate over- respectively underinvestment in education if there is a social insurance device available. Note that the risk premium in educational investment or over-/underinvestment in education does not enter equation (26) and has no direct impact on the optimal tax rate. Moreover, the insurance characteristics enters the formula for optimal 19
subsidies only indirectly via the optimal tax rate t∗ . These facts show that the role of education subsidies is really to mitigate tax distortions and inefficient resource allocation in education, whereas taxation is dedicated to insurance. Whilst all statements with respect to an optimal education subsidy in case of a given tax policy carry over to the simultaneous derivation of optimal tax and education policies, the interpretation of the optimal tax rate slightly changes. Of course, the optimal tax rate still increases in social risk and inequality aversion (ξ) and decreases in higher tax-induced distortions in labor supply (²lt ). If more education fosters labor supply (²ls > 0 and accordingly ²et < 0) and if this subsidy-induced effect is relatively strong compared to subsidy-induced distortions in educational investment (²ls /²es ), there is an offsetting effect on tax distortions, increasing the optimal tax rate. This effect is the higher, the more taxation effects education (²et ). The same effect on the tax rate is working, if labor and education are negatively related, as now ²ls < 0 and accordingly ²et > 0, but it is calling for tuition fees rather than education subsidies, then. Comparing the optimal tax policy cum education subsidies, equation (26), with the optimal tax rate in equation (20), where educational policy is absent, s = 0, taxation not only takes care for risk insurance in the latter case, but also has to tackle socially inefficient educational investment. Intuitively following Bovenberg and Jacobs (2005), one should expected that the optimal tax rate is higher, if the government has full-fledged instruments and can choose education subsidies. This intuition can be, however, misleading, if there is a trade-off between insurance and efficient investment in education and if inefficient educational investment calls for a higher tax rate than it would be chosen only in order to insure against risk. The intuition is correct, if the denominator in (20) is smaller than the denominator in (26), consequently if µ
²ls πe + ²es
¶ ²et ≤ 0.
(27)
Due to the symmetry of substitution effects in education and leisure, we have ²ls · ²et < 0 and can distinguish six cases (Tab. 1). Note that ²es > 0 in any case.
20
Tab. 1: Equation (27) is fulfilled, if
²ls < 0, ²¯et >¯ 0 ²ls = ²et = 0 ¯ ls ¯ πe > 0 |πe | > ¯ ²²es yes ¯ πe < 0
yes
yes
²ls > 0, ²et < 0 yes ¯ ¯ ¯ ls ¯ |πe | < ¯ ²²es ¯
We can conclude: Proposition 2. If labor supply and education are positively (negatively) related and there is underinvestment (overinvestment) in education, implementing an educational policy with either educational subsidies or tuition fees allows for fiercer income taxation and therefore for higher social insurance compared to using tax policy only. Proof. Proposition 2 follows directly from equation (27) and Tab. 1, because a positive (negative) relationship between labor and education implies ²ls>0 (²ls 0 (πe < 0). In the cases mentioned in Proposition 2 mitigating distortions via education subsidies and fighting against over-/underinvestment go in the same direction. If this is not the case, thus if, e.g., underinvestment (πe > 0) calls for education subsidies, but balancing tax distortions require tuition fees (²l s < 0), there is a trade-off and the optimal tax rate in the fully-fledged system is only higher if the ‘right’ effect dominates. Concerning the timing of the labor supply decision, we can note that the optimal tax structure does not depend in a qualitative sense on this issue, as the risk premium in labor supply does not enter any of the equation directly. However, the timing quantitatively has an impact on the tax rate. If labor supply is chosen after the realization of risk, labor supply loses its character as insurance device and the labor elasticities should increase. Thus, if labor supply is chosen after risk has realized, the tax-induced distortions are higher and the optimal tax rate should be lower than in case of a labor supply decision under uncertainty. This fits to the analysis of Eaton and Rosen (1980a), who show that the income-compensated substitution effect of the wage rate on labor supply is indeterminate (instead of being unequivocally negative), if there is wage risk. 21
5
Discussion of the Optimal Policies
To-do-list: • relate our results to previous literature • highlight effects of risk-increasing and risk-decreasing educational investment • discuss effect of timing • show that Anderberg/Andersson-intuition for under-/overprovision of education and comparison to First-best is misleading
6
Conclusions
In a model with a general earnings function and income risk, we derive an optimal labor tax rate and optimal education subsidies. It turns out that the distinction of risk-increasing or risk-decreasing education does not have much effects on the optimal tax rate, but is crucial for optimal education subsidies. Moreover, an optimal education policy allows for better social insurance, as the tax-induced distortions in labor supply can be mitigated. This extends the analysis by Bovenberg and Jacobs (2005) to risky economies. Furthermore, we show that education policy itself has no insurance effect and is used solely for efficiency reasons and only if there is another instrument, which provides social insurance. The flexibility of our model allows to summarize several earlier studies in this field. Our results allow to deepen, respectively to question, the intuitions given for their results. It turns out, e.g., that it is misleading to argue, as in Anderberg and Andersson (2003), educational policies should over- or underprovide education compared to a First-best solution. We argue that the First-best is the wrong reference point and that educational policy has to be evaluated against the laissezfaire allocation.
22
7 7.1
Appendix Optimal taxation
First-order condition for t can be written as ∂L ∂l ∂e = −E [u0 (Φ (.) − (1 − s)e)] + η (E [Φ (.)] − (1 − s)e) + η∆l + η∆e =0 ∂t ∂t ∂t ∗
] = −E [u0 (Φ (.) − (1 − s)e)] using Roy’s lemma ∂E[u ∂t Define X ≡ Φ (.) − (1 − s)e and using Slutsky equations
∂l ∂l∗ ∂l = − E (Φ (.) − (1 − s)e) ∂t ∂t ∂T ∂e ∂e∗ ∂e = − E (Φ (.) − (1 − s)e) ∂t ∂t ∂T the first order condition can be further reformulated as ∆l lE [Φ] E [Φl l] ∂l∗ 1 − t ∆e eE [Φ] E [Φe e] ∂e∗ 1 − t −E [(b − 1)X] + + =0 1 − t E [Φl l] E [Φ] ∂t l 1 − t E [Φe e] E [Φ] ∂t e Define the labor and education elasticities as follows. ²lt ²et
E [Φl l] ∂l∗ 1 − t ≡ E [Φ] ∂t l E [Φe e] ∂e∗ 1 − t ≡ E [Φ] ∂t e
(28)
There is premultiplication with the shares or elasticities of the earnings function. Define the insurance characteristic as ξ≡
E [(b − 1)X] −cov [Φ, b] =− E [Φ] E [b] E [Φ] E [b]
we can rewrite the first order condition as ξ+
∆l ∆e 1 1 ²lt + ²et = 0 1 − t E [Φl ] 1 − t E [Φe ]
Using the first order conditions of households, the net tax wedge on education ∆e 23
can be reformulated as ∆e =
E [Φe (.)] (−s + πe (s + t(1 − s))) 1−s
Substituting for ∆l and ∆e optimal taxation is given by ¸ ¸ · t πe (s + t(1 − s)) − s ξ+ ²lt + ²et = 0 1−t (1 − t)(1 − s) ·
or t ξ (²et − πe ²et ) s =− + (1 − t) ²lt + πe ²et (²lt + πe ²et ) (1 − t)(1 − s) If there is no education policy s = 0, optimal tax rate is t ξ = 1−t − (²lt + πe ²et )
7.2
Optimal education policy
Using Roy’s lemma written as
∂E[u∗ ] ∂s
= E [u0 ] (1 − t)e the first-order condition for s can be
∂l ∂e ∂L = E [u0 ] (1 − t)e − η(1 − t)e + η∆l + η∆e =0 ∂s ∂s ∂s Inserting for
∂l ∂s
and
∂e ∂s
the Slutsky equations ∂l ∂l∗ ∂l = + (1 − t)e ∂s ∂s ∂T ∂e ∂e∗ ∂e = + (1 − t)e ∂s ∂s ∂T
we have E [u0 ] ∂l ∂e ∆l ∂l∗ ∆e ∂e∗ + ∆l + ∆e −1+ + =0 η ∂T ∂T (1 − t)e ∂s (1 − t)e ∂s
24
with optimal lump-sum transfer b it reduces further to ∆l l
∂l∗ 1 − s ∂e∗ 1 − s + ∆e e =0 ∂s l ∂s e
because E[b] ≡ E[
u0 ∂l ∂e + ∆l + ∆e ]=1 η ∂T ∂T
Substituting for ∆e and ∆l the first order condition is given by t E [Φl (.)] l ∂l∗ 1 − s (−s + πe (s + t(1 − s))) E [Φe (.)] e ∂e∗ 1 − s + =0 (1 − t) E [Φ (.)] ∂s l (1 − s)(1 − t) E [Φ (.)] ∂s e or
t (πe (s + t(1 − s)) − s) ²ls = − ²es 1−t (1 − s)(1 − t)
where E [Φl (.)] l ∂l∗ 1 − s E [Φ (.)] ∂ (1 − s) l E [Φe (.)] e ∂e∗ 1 − s = E [Φ (.)] ∂ (1 − s) e
²ls = ²es
are the weighted compensated elasticities of labor and education w.r.t. subsidies. For any given tax rate the optimal education subsidy rate is therefore given by s = 1−s
µ ²ls
²es
+ πe
1 − πe
¶ t
References Anderberg, D., and F. Andersson, 2003. Investments in Human Capital, Wage Uncertainty, and Public Policy. Journal of Public Economics 87, 1521–1537. Andersson, F., and K. Konrad, 2003. Globalization and Risky Human-Capital Investment. International Tax and Public Finance 10, 211–228. Atkinson, A. B. and J. E. Stiglitz, 1980. Lectures on Public Economics. New York: McGraw-Hill. 25
Bovenberg, A. L. and Jacobs, B. 2005. Redistribution and education subsidies are Siamese twins. Journal of Public Economics 89, 2005-2035. Eaton, J., and H.S. Rosen, 1980a. Labor Supply, Uncertainty, and Efficient Taxation. Journal of Public Economics 14, 365–374. Eaton, J., and H.S. Rosen, 1980b. Taxation, Human Capital, and Uncertainty. American Economic Review 70, 705–715. Feldstein, M. S., 1972. Distributional Equity and the Optimal Structure of Public Prices. The American Economic Review Vol. 62, No. 1/2., pp. 32-36. Hamilton, J., 1987. Optimal Wage and Income Taxation with Wage Uncertainty. International Economic Review 28, 373–388. Jacobs, B., 2005. Optimal Income Taxation with Endogenous Human Capital. Journal of Public Economic Theory 7, 295 – 315. Levhari, D., and Y. Weiss, 1974. The Effect of Risk on the Investment in Human Capital. American Economic Review 64, 950–963. Schindler, D., and H. Yang, 2007. Risky Human Capital Formation Requires ‘Siamese Twins’. Research Group Heterogenous Labor Discussion Paper 07/10, University of Konstanz, Konstanz. Wigger, B.U., and R. von Weizs¨acker, 2001. Risk, Resources, and Education Public versus Private Financing of Higher Education. IMF Staff Papers 48, 547–560. Varian, H.R., 1980. Redistributive Taxation as Social Insurance. Journal of Public Economics 14, 49–68.
26
