INTRODUCTION
There are two ways of introducing second-best government policy in a competitive economy. The first is when the government has to raise an exogenously specified amount of revenue without recourse to lump-sum taxation. The second-best tax system then minimizes the distortions.1
In the second-best, the government designs an optimal dynamic tax formula at date zero, and remains precommitted to it (i.e., the government chooses all future consumer prices at date zero). If lump-sum taxation is allowed, the analysis reduces to the first-best.
Although Chamley (1986) takes the first view of second-best taxation where the government is just revenue raising, Judd (1985) views the government as redistributive. The zero capital-income tax is a result robust to several generalizations. Atkeson et al. (1999) provides a survey of the robustness of the zero capital taxation result using the primal approach. See also Renström (1999) for a survey of the dynamic optimal tax literature using the dual approach.
In this paper, we focus our attention on analyzing the second-best labor income taxation in the Judd-Chamley model. The goal of our paper is to find conditions on preferences that give rise to positive labor taxation.3
Chamley (1985) analyzes second-best wage taxation in the absence of a capital tax and suggests that the exact tax policy depends on the utility function. We explore this issue and show that the functional form of the utility function is important in determining the sign of the labor-income tax.
There is a literature looking at this issue from various perspectives. Lucas and Stokey (1983), and Ayagiri et al. (2002) look at optimal labor taxation in the absence of capital. For our purposes, it is important to include physical capital. One may then explore various optimal-tax outcomes, including the possibility of zero tax on both capital and labor. Jones, Manuelli, and Rossi (1993) address the issue of optimal labor taxation including physical and either human capital or productive government spending. However, their analysis is mostly based on simulation with specific functional forms, and does not admit a closed form solution with a fairly general preference structure.
Jones, Manuelli, and Rossi (1997) include human capital in addition to physical capital, so that the labor tax has an intertemporal distortion. They show that there are certain cases when the labor tax is zero in steady state. See also Reinhorn (2006) for a clarification of those results, in particular regarding interior solutions of a model with human capital in addition to physical.
Hansen (1985) and Rogerson (1988) establish that indivisible labor explains aggregate fluctuations better than models with divisible labor. A subsequent literature explored further into the business cycle implications of indivisible labor [see Greenwood and Huffman (1988), Hansen and Sargent (1988)]. Mulligan (1999) points out that the tax implications are different for these two classes of models.
A related paper, Basu, Marsiliani, and Renström (2004), derives necessary and sufficient conditions on preferences for the capital tax to become zero in finite time in an indivisible labor setting. Those preferences are special cases of our HARA class derived in Section 4 of this paper.
Why is the issue of optimal taxation in an environment with indivisible labor worth exploring? With indivisible labor, we can unravel a rich set of preferences for which labor is taxable. We accomplish this by establishing a connection between the household's demand for unemployment insurance and the normality of leisure. When leisure is normal, in order to raise the tax base, the fiscal authority would tax labor to induce the household to work harder. When labor supply is indivisible, and the leisure is normal, the individual buys insurance to equate the utility gain from not working to the utility cost of the insurance purchase. The novelty of our approach is that we further derive a HARA class of preferences (i.e., preferences with Hyperbolic Absolute Risk Aversion) with nonseparable leisure for which this insurance is a linear function of after-tax wage income. Because positive insurance demand and normality of leisure are inextricably connected, one can then answer the tax policy question posed in this paper: What class of preferences would warrant positive labor taxation? For those preferences, we can derive the time path of the optimal labor tax. In addition, we can compare the optimal time paths of labor income taxes between divisible and indivisible labor economies, which turn out to be very different between these two scenarios. This happens because labour supply elasticities are different between these two models.
There is also a pedagogical reason for exploring the optimal labor taxation in an indivisible labor scenario. There is a literature on optimal taxation arguing that labor should be taxed when market is incomplete in the sense that there is neither an insurance market nor a set of contingent-claims markets (where the household can diversify income risk). The government then uses corrective taxes for the missing markets.8
This has been explored in the literature on risk sharing. There, even without a government revenue constraint, a labor tax may be levied. The labor tax then corrects for a market failure (a missing insurance market). In a two-period setting, Hamilton (1987) demonstrates that the optimal wage tax is positive if the second period labor income is uncertain. The issue is indirectly dealt in the macroeconomics literature concerning debt nonneutrality. Chan (1983) and Barsky et al. (1986), as well as Basu (1996), examine the debt nonneutrality hypothesis when future income is uncertain. Although none of these models explicitly deals with the labor-supply decision, there is one common result: taxing future income at a flat rate would be welfare improving if markets were unable to fully insure households from future income risk.
As expected, we find that the second-best capital-income tax is zero at the steady state.
We also characterize the short-run dynamics of the labor income tax. We find that when leisure is normal, the optimal labor tax is increasing over time as long as capital is taxed. When leisure is inferior, and labor is subsidized, the subsidy is increasing over time as long as capital is taxed. We also look at two special cases of the HARA class, iso-elastic utility and negative exponential utility. We find that for widely used iso-elastic preferences, the optimal labor income tax is positive both in the short and long run. For negative exponential utility, where leisure is neutral, the optimal labor-income tax is zero in the short and long run when the government can use debt as an instrument. Optimal taxation of labor is, therefore, generally positive except for some special types of preferences.
The paper is organized as follows. The following section presents the model. Section 3 derives the optimal-tax implications in a second-best world. Section 4 derives a class of HARA preferences, and presents examples of the time path of the optimal labor tax. Section 5 compares the time paths of optimal taxes between divisible and indivisible labor economies. Section 6 briefly discusses the case when public debt is zero. Section 7 concludes.
AN ECONOMY WITH INDIVISIBLE LABOR
Individual Economic Behavior
Following Hansen (1985) and Rogerson (1988), we consider an economy where labor supply is indivisible. The consumption set is restricted so that that the individuals can work either full time, h0, or not at all. Households have access to an insurance market where they can buy insurance. In each period, the household member engages in an employment lottery, choosing the probability of working, α(t), and the probability of not working, (1−α(t)). This makes her wage income uncertain. She has access to an insurance market where she buys unemployment insurance, y(t).The household's consumption (cs(t)) and asset accumulation (
are thus potentially contingent on whether the household works (s = 1) or not (s = 2). There is no intrinsic uncertainty, which means that preferences and technology are nonstochastic. The household thus solves the following maximization problem:
where a(t) equals the sum of outstanding public debt, b(t), and the capital stock, k(t), that earn the after-tax interest at rate ρ(t) =(1−τK(t))r(t), and ω(t) =(1−τL(t))w(t) is the after tax wage; r(t) and w(t) are the rental- and wage-rates, respectively, τK(t) and τL(t) are the proportional tax rates on capital- and labor-income, respectively, and p(α(t)) is the competitive price of insurance.10
The optimization problem laid out in our paper resembles Hansen (1985). We make it explicit that the agent starts with the same asset but potentially can transit to two possible states: work or no-work. In equilibrium, they will, however, have the same asset equation via insuring themselves against income risk. Alternatively, we can transform the problem into a single state variable by defining a=αa1(t)+(1−α)a2(t), and get exactly the same first-order conditions.
The household randomizes the labor-supply decision in this setting by choosing a probability of work α(t). A realistic description of this arrangement is that the representative household consists of a family of N members. In each period, the household decides the proportion, α(t), of members working. The labor supply is then α(t)h0N. The household can buy insurance on a competitive market to diversify the income uncertainty arising from (1 − α(t))N of its members not working. After choosing the probability of work, α(t) the household is precommitted to it, and cannot renege. The insurance company then charges the actuarially fair premium, 1 − α. This rules out adverse selection in the model. The household then realizes that, when choosing work probability, the insurance premium is a linear function of the probability.
Substituting the zero-profit condition into (2) and (3), the current value Hamiltonian of the representative household can be written as
The first-order conditions are (subscripts denoting partial derivatives)
Using (6), (7), and (8) it follows that
which means perfect insurance. In other words, by buying insurance, the individual equalizes the marginal utilities across states. Equation (9) gives the optimal time paths of state-contingent consumption, c1(q(t)) and c2(q(t)), as functions of the co-state variable q(t). However, this does not necessarily imply that household will equalize consumption across states. For consumption equalization, one requires an additional restriction on the preferences that the utility function is additively separable between consumption and leisure, meaning uc(1−h)=0. It turns out that without any such restriction on the preferences, the household will not choose to have full consumption insurance as in Hansen (1985).12
Note that if the utility function is nonseparable, then the household optimally chooses not to have full consumption insurance. One can easily set up a social planning problem and demonstrate that no other consumption allocation dominates the allocation governed by (12).
An individual's asset accumulation is thus independent of her employment history. This implies that individuals starting with the same a0 will have the same a(t) at all t, regardless of their employment history. Substituting (10a) in (2) and (3) gives
Notice now that the household chooses full insurance if the optimal consumption bundles are such that c1(q(t)) = c2(q(t)). In the absence of any restriction on y(t), the household can choose to have positive, negative or zero insurance.13
One needs to be careful about the non-negativity constraint on consumption while thinking about negative unemployment benefit. y(t) can be negative as long as c2(t) is nonnegative. We assume interior solutions, meaning c2(t) > 0.
which upon the use of (11) can be rewritten as
The household chooses to buy a positive insurance, y(t), if the utility gain from not working balances the utility cost of the insurance purchase.
Production
There is large number of competitive firms in the economy each operating under the following constant returns-to-scale technology:
where
and
.
The Government
The government taxes labor and capital income to finance an exogenously specified sequence of public spending, g(t), the use of which is not explicitly modeled. It adjusts two tax rates, τL(t) and τK(t), continuously. The government is assumed to borrow and lend freely at the market rate of interest, r(t). The government's budget constraint is, therefore, given by
with b(0)=b0.
Equilibrium
The equilibrium is characterized by the following conditions:
- (a) Facing w(t), r(t), τL(t), τK(t), the household chooses optimal sequences of c(t), a(t), α(t),y(t) that solves the problem stated in (1), (2) and (3).14
We assume no-Ponzi games.
- (b) Given an exogenous steam of government spending, g(t), the government precommits to a tax sequence, τL(t) and τK(t), and a debt sequence, b(t), that satisfies the government budget constraint (14).
- (c) Goods, labor, rental markets clear meaning
Notice that the equilibrium level of employment, h(t) (≡α(t)h0) is determined by the time path of the probability of work, α(t). The equilibrium time path of α(t) can be determined in two steps. First, using (12) one determines the market clearing after tax wage, ω(t) as a function of q(t). Define that equilibrium wage function as
Next, using (16) and (18), one can characterize the path of α(t) as a function of k(t), q(t), and τL(t) as follows:
PROPOSITION 1. In equilibrium, y(t)>0, if and only if
(leisure is normal at date t); y(t)<0, if and only if
(leisure is inferior at date t).
Proof. See Appendix A.
Positive insurance demand and normality of leisure are inextricably connected.15
In a divisible labor economy, Renström (1999) finds that noninferiority of leisure is sufficient for the wage-income tax to be positive. In the indivisible-labor setting, normality is both necessary and sufficient for a positive labor-income tax.
The insurance market is added to obtain insights about the relationship between normality of leisure and the optimal tax, as well as deriving preference structures for which leisure is normal. One can alternatively construct an environment with contingent-claims markets as in Shell and Wright (1993), and derive the same equilibrium allocation as a sunspot equilibrium. A proof of this equivalence is available from the authors on request.
SECOND-BEST OPTIMAL TAXATION
We now solve for the optimal tax problem for the government for this economy with indivisible labor. The government solves a Ramsey problem for precommitted tax sequences, τL(t) and τK(t), that maximize the household's utility functional (1) subject to its own budget constraint (14), the economy wide resource constraint (15), the first-order optimality conditions (9), (10b), and (12), and a no-confiscation constraint on capital income as follows:17
No such confiscation constraint is relevant for labor income taxation because if labor income is confiscated by the government it is optimal for the household to set α(t) = 0, which means no production. By contrast, the capital income can be confiscated and the government can eventually own all the capital to run production.
Using (16) and (17), and the CRS property of the production function, the government's budget constraint, (14), can be rewritten as
We may write the government's current value Hamiltonian as follows (ignoring the time indices from now on):
In principle, the government faces the states, k, b, and q, and chooses the controls ρ and τL. For algebraic convenience, we pose the government's problem as follows. The government chooses the controls ρ and α. Then using the equilibrium sequence of α as in (19), one can determine the optimal labor tax, τL.18
It is straightforward to verify that for given k and q, α′(τL) < 0 and hence α(·) can be inverted with respect to τL.
The Optimal Capital-Income Tax
One may now establish, as in Judd (1985) and Chamley (1986), that the optimal capital-income tax is zero in the steady state, also in our economy with indivisible labor. This can be verified as follows. At steady state the individual's marginal utility of consumption is constant over time, implying, by (10b), that θ = ρ. This in turn implies, by (27), that μ is constant in the steady state. With μ being constant (and q constant), equation (23) implies that also λ is constant in the steady state. Setting the time derivative of λ to zero in (26), and using θ=ρ gives (ρ−f1)(λ−μ)=0, which holds if and only if f1=ρ because (λ−μ)>0. We summarize this result in terms of the following proposition.
PROPOSITION 2. If the indivisible labor economy converges to a steady state under the second-best tax program, then in the steady state the optimal capital-income tax is zero, τK=0.
The Optimal Labor-Income Tax
Our primary interest in this paper is to explore carefully the optimal labor-tax implications. The problem is complicated by the fact that in an economy with indivisible labor and lottery, the consumer has the option to transfer consumption not only across dates but also across states. Whether the government should tax wageQ1 income in the second best depends crucially on the household's risk preference, which will determine its demand for insurance. The following proposition states a key result about the relationship between the household's demand for insurance and the second-best wage taxation.
PROPOSITION 3. Under the second-best tax program, from the date when the no-confiscation constraint (20) on capital ceases to bind, the optimal wage-income tax is positive (negative) if the household's demand for insurance, y, is positive (negative).
Proof. See Appendix B.
Thus, there is a direct link between the sign of y and the sign of the labor-income tax. The question arises whether the optimal labor-income tax is zero when y=0, the case when leisure is neutral. This cannot be directly inferred from Proposition~3, because it is based on an equilibrium wage equation subject to the condition that y is nonzero.19
To see this clearly note that the proof of Proposition 3 rests on the fact that
, which holds when y is nonzero.
PROPOSITION 4. If the preferences of the consumer are such that the consumer chooses to buy zero insurance (meaning y=0), the optimal labor-income tax is zero at all dates.
Proof. Plugging (12′) into (23) and using (11), one obtains
It immediately follows that when y=0, f2=ω, which means τL=0.
Because Proposition 1 gave us a one-to-one relationship between normality/neutrality/inferiority of leisure and positive/zero/negative insurance, we can conclude from Propositions 3 and 4 that there is a one-to-one relationship between normality/neutrality/inferiority of leisure and the sign of the optimal labor tax. That is, the labor tax should be positive/zero/negative if leisure in normal/neutral/inferior. Notice that this is not just a steady-state result but holds at least from date when the nonconfiscation constraint does not bind.
The issue arises whether for normal/inferior leisure, the labor-income tax/subsidy rises or falls over time. It is difficult to characterize the exact time path without fully specifying the preferences. In the next section, we derive a broad class of HARA preferences for which it is possible to characterize the exact short run path of the labor tax.
DYNAMICS OF THE LABOR TAX: A PARAMETRIC EXAMPLE
Consider a class of preferences for which insurance demand is proportional to the wage income as follows:
where π is a parameter that can be either positive, negative, or zero. It turns out that the parameter π pins down alternative preference structures for which leisure may be normal, neutral or inferior.
Full Insurance
The benchmark case of full insurance arises π=0. We have the following result.
PROPOSITION 5. Necessary and sufficient for π=0 is that the utility function u (c, 1−h0) is additively separable in leisure: uc(1−h)=0.
Proof. If π = 0, from (11) and (30) it follows that c1 = c2. Next, note that when c1 = c2, the only way marginal utilities can be equalized in (9) is by setting uc(1−h) = 0. This proves necessity. Next we prove the sufficiency. If uc(1−h) = 0, the = from (9) it follows that c1 = c2, which on plugging in (11) and (30) gives π = 0.
The additive separable leisure in the utility function is widely used in the literature. When the utility function is additively separable in leisure, it implies that leisure is a normal good. In this case, evidently labor is taxed as per Proposition~2. However, the converse is not true. For normality of leisure, it is not necessary that uc(1−h)v=0. We now derive a wide range of preferences involving nonseparable leisure that belongs to a family of preferences known as the Hyperbolic Absolute Risk Aversion (HARA) class. Such a family of preferences can be derived when π ≠ 0.
HARA Preferences
We shall establish now that for π ≠ 0, the derived HARA class can encompass various possibilities: (a) 0 < π < 1, partial insurance; (b) π=1 no insurance; (c) π > 1, negative insurance, and (d) π < 0, over-insurance.
Plugging (30) into (11) to eliminate ω, we get
Plugging (31) into (12′) gives
where
and u2=u(c2(q), 1). Equation (32) holds for all preferences for which π≠0.20
Note that π = 0 cannot be nested under this case, which means additively separable utility function does not belong to our derived class of preferences.
LEMMA 1. It is necessary that the class of preferences for which π ≠ 0, satisfies the following condition:
where
and
denote the derivatives of u and c in state s={1, 2} w.r.t. c and q, respectively.
Proof. Define (32) as the implicit function J(q, π) = 0. Using the implicit function theorem,
For (34) to be zero, it is necessary that either Jq=0 or Jπ=∞. However, Jπ=∞ would mean
q being the marginal utility of consumption cannot be infinity at the optimum. Thus, for (35) to hold, π must equal zero, which violates the restriction that π≠0. Thus, it is necessary that Jq=0, which yields (33).
LEMMA 2. For Jq = 0 ∀ q, it is necessary that the utility function is of the following Hyperbolic Absolute Risk Aversion (HARA) form:
for s={1, 2}. Bs and Ds are constants (possibly dependent of h), with B>0, and A is a constant independent of h.
Proof. Taking the derivative of (9) w.r.t. q gives c
. Plugging this into (33), we get
Because preferences are state independent (i.e., expected utility), the left- and right-hand sides must equal the same constant, (say−A), that is,
for s={1, 2}. This means that A and π cannot be state dependent (i.e., dependent of h). Integrating (38) twice (see, Appendix C for derivation) yields (36).
We next show that for the class of HARA preferences (36), the constant D is state independent (i.e., independent of leisure).
LEMMA 3. For (32) and (36) to hold, it is necessary that that D1=D2.
Proof. Take the derivative of (36)
Then, by using (9), we find cs as functions of q:
Inserting (40) into (36) gives
Plugging (40) and (41) into (32), it follows that D1=D2.
Because in the preference class (36), only Bs and cs can be state-dependent, it must be the case that Bs is a function of leisure. Using this insight, define Bs = ϕh, where h can take two possible states 0 and h0. Our next task is to characterize the precise restrictions on π, which generates leisure as a “good” (with positive marginal utility) in the utility function. We are ready to state a key proposition.
PROPOSITION 6. Necessary and sufficient for insurance to be a constant fraction of the after-tax wage income is that the preferences belong to the following class:
where A and D are constants. ϕh is a function of leisure as follows:
- (a) If π>0, then ϕh=ϕ(h), with ϕ(h0)>ϕ(0).
- (b) If π<0, then ϕh=ϕ(1−h), with
Proof. Necessity of (42) follows from Lemma 2 and 3. Sufficiency can be checked by evaluating equation (32) for the class in (42). Finally, conditions (a) and (b) imply that utility of working is less than of not working, everything else equal.
The Case of Zero Labor-Income Taxation
In the next step we derive a subclass of preferences from the HARA class for which labor should not be taxed at the second-best optimum. This is a special case when leisure is neutral.
PROPOSITION 7. If the utility function is of the following exponential class
where D is a constant, and} ϕh = ϕ(h), with ϕ(h0) > ϕ(0), then the household self-insures meaning y = 0}, and the optimal labor-income tax is zero at all dates.
Proof. Recall from Proposition 3 that when the individual takes no insurance, the optimal wage tax is zero. In the present context, y = 0 if π = 1 (see equation 30). Take limits of (42) as π → 1 (which requires setting A equal to unity) and apply l'Hôpital's rule to obtain (43).
Proposition 7 has an important implication. It basically provides a special case where self-insurance could be optimal. Even if the insurance market is present, households choose to self-insure (meaning the demand for insurance, y = 0) in this special environment by using saving and labor supply as instruments.21
In general, savings alone would not equalize marginal utilities across states. To see this clearly, note that in the absence of an insurance market, there is no equation (8), and hence (9) will not hold generally. Thus, savings alone cannot equalize marginal utilities across states. If the insurance market is dropped, the model reduces to an incomplete market scenario where the intertemporal marginal rates of substitution in consumption (which is governed by the two co-state variables q1 and q2 in our model) will not be equalized across states because of the emergence of uninsurable income risk. This uninsurable risk arises because a fraction of the households do not work.
Why Does Normality of Leisure Mean a Positive Tax on Labor?
We have found that whether labor should be taxed or not depends on whether the individual demands positive insurance or not, which in turn depends on whether leisure is normal or inferior. A relationship thus exists between normality of leisure and the insurance demand via the individual's preferences. To see this connection clearly, differentiate (19) with respect to τL(t), (keeping k(t) constant), to obtain the following useful decomposition of a change in tax rate, τL(t) on work effort, α(t):
The first term in (44) represents the compensated labor-supply response when the tax rate changes (the substitution effect). If q(t) is held constant, it follows from (18) that ω(t) is also constant and it is straightforward to verify from (16) that the first term is: f2/[(1−τL)f22·h0], and is negative. This substitution effect thus captures the distortionary effect of the wage-income tax. As far as this substitution effect is concerned, a higher labor tax lowers labor supply and thus lowers the tax base.
The second term in (44) reflects the income effect of a change in the wage tax rate, which works through the effect of τL(t) on α(t) via its effect on q(t). When τL(t) is higher, it lowers the permanent income of the household, thus lowering consumption in both states for given α(t). The marginal utility of consumption, q(t), thus rises, which means ∂q(t)/∂τL(t) > 0. A loss of permanent income (represented by higher q(t)) would boost the labor supply, α(t), if leisure is a normal good. Next, verify that (44) can be rewritten as:22
To see this, insert (30) into (A.5) and use (16) to obtain ∂α(t)/∂q(t) = −(1−π)f2/f22q(t) h0.
The normality of leisure (π < 1) thus makes the income effect of a higher wage tax positive. This positive income effect tends to increase the tax base (w(t)α(t)h0) when τL(t) rises, which countervails the distortionary effect of a higher τL(t). The government can thus tax labor more in those economies. By contrast, if π > 1, leisure is inferior. The income effect is then negative, reinforcing the distortionary effect of wage taxation. In this case, the government should subsidize labor.
Should Labor Be Taxed When the Utility Function Is Isoelastic in Consumption?
Consider now a special case of the HARA class, equation (42), when A = 0. We have the following lemma concerning the elasticity of interstate substitution (call it σ hereafter).
LEMMA 4. For a specific class of HARA utility functions with} A = 0, the elasticity of interstate substitution (σ) is given by 1−π.
Proof. By definition,
Using (42) gives σ=1−π.
The utility function thus becomes isoelastic in consumption, and the elasticity of interstate substitution in consumption is uniquely characterized by π. Furthermore, we must have π < 1 when A = 0, otherwise positive marginal utility, uc > 0, is violated. This immediately implies that leisure is normal when the utility function is isoelastic in consumption, and thus labor must be taxed.
To summarize, the HARA class of preferences (42) embraces a variety of utility functions, and thus covers a wide range of optimal taxation schemes. These cases include: (a) isoelastic utility in consumption (A = 0), in which case the optimal labor tax is positive, (b) negative exponential utility function (π = 1), where the optimal labor tax is zero, and (c) quadratic when π = 2 (where
, in which case the optimal labor tax is negative (i.e., a subsidy).
Transitional Dynamics of the Wage-Income Tax in the HARA Case
We shall now analyze the transitional dynamics of labor income tax for the range of preferences discussed earlier. Using (29) one obtains
then taking the time derivative of (46) using (10b), (26), and (27), one obtains
Appendix C outlines the steps in deriving the second equality. Based on the second equality in (47), and the fact that q−μ>0, and λ−μ>0, we have the following proposition.
PROPOSITION 8. If π < 1 labor is taxed (at least from the date at which the nonconfiscation constraint does not bind), and the wage tax is increasing as long as capital is taxed. If π = 1 labor is always untaxed. If π > 1 labor is subsidized (at least from the date at which the nonconfiscation constraint does not bind), and the subsidy is increasing as long as capital is taxed.23
When π=1, it turns out that the nonconfiscation constraint is binding for a period t ∈ [0, T*), and capital income is taxed at 100%. From T* and onward, capital is untaxed, and the economy is at its steady state level. T* is a function of the present value of the stream of gt, t ∈ [0, ∞) discounted at the rate θ.
COMPARING THE LABOR TAX IN INDIVISIBLE AND DIVISIBLE LABOR ECONOMIES
In this section, we compare the optimal tax paths of the indivisible and divisible labor economies. We establish here in terms of an example that the optimal labor tax paths could be very different between these two economies because of different labor supply elasticities. The current value Hamiltonian for the household for the divisible labor economy is
The first-order conditions are
Equations (49) through (50) form a system of equations such that c and h can be solved in terms of q and ω. For our purpose we are interested in the compensated labor supply response with respect to a change in ω holding q constant. We get
Denoting
as the compensated labor supply, and assuming an additively separable utility function, U(c, 1 − h) = V(c) + L(1 − h), Renström (1999) shows that for this divisible labor economy, the time path of labor tax is given by
In order to make a valid comparison between divisible and indivisible labor economies, we need to assume the same preference structures for both. Recall from Proposition 5 that for the indivisible laboreconomy π=0 in (30) if and only if the utility function is additively separable. Substituting (30), with π=0, into (29) gives
Comparing (46a) and (53), we see that the time paths of labor taxes are very different for these two economies because of the presence of the labor-supply elasticity term η on the right-hand side of (53).
Hansen (1985) shows that the equilibrium allocation of an indivisible labor economy can be replicated in a divisible labor economy with a utility function linear in leisure. From (53), this immediately means that η is infinite for such a divisible labor economy. The optimal labor tax path described in (53) thus reduces to the indivisible labor case (46a) in such a case. Except this special case, the optimal paths for τL are generally different between these two economies.24
One suspects that in general the optimal labor tax is lower in an indivisible labor economy because of a higher labor supply elasticity. In order to make such a quantitative comparison, we need to solve the differential equations in λ, q and μ in both economies. Although differential equations look the same for both economies, the solution will be different. The reason is that the initial values of these co-state variables are different for these two economies (except the special case when η is infinite), and also the time from which the nonconfiscation constraint does not bind also will differ. The details of these derivations are available from the authors on request.
NO DEBT SCENARIO
Until now, we assumed that the government intertemporally balances the budget by using public debt as an instrument. How would the optimal tax formula change when public debt is disallowed meaning b = 0? In the absence of public debt, the government is left only with two policy instruments, namely, wage tax and capital tax. The zero capital tax still remains socially optimal in steady state, which means that labor has to be always taxed to finance the exogenous stream of government spending in the steady state. In certain cases, we may even have a corner solution for the labor supply where the labor supply hits the upper bound (α=1).
The intuition for the possibility of corner solution goes as follows. To sustain such a zero capital tax regime, the household may have to stretch the labor supply to the upper bound (a corner solution) to finance the government spending. This happens particularly when the household values leisure as an inferior or neutral good. As the economy-wide capital stock grows, the household does not then mind working harder to finance the government spending just to sustain a socially optimal zero capital tax regime. Thus labor supply hits the corner when leisure is nonnormal. This kind of a corner solution does not arise when debt is allowed as an instrument because the benevolent government can always float debt to alleviate the strain on laborsupply.25
The details of the derivation of the zero debt case are available from the authors on request.
CONCLUSION
In this paper, we address the issue of optimal wage taxation in a dynamic complete-markets setting. We find that the second-best labor tax depends crucially on the degree of complementarity between consumption and leisure. If leisure is a not neutral, there is scope for government intervention in the form of labor taxation or subsidy in a complete market environment. We investigated this question in a fairly general setting with preferences allowing nonseparability between consumption and leisure. Our conclusion is that labor should be taxed if leisure is normal. The optimal wage tax could be zero or negative only for very special scenarios where leisure is nonnormal, and the government has access to public debt as a policy instrument. If the government is restricted to balance the budget period by period, the wage subsidy result ceases to hold. Thus, a normative theory of taxation generally implies that labor should be taxed in the long run. We also find that the optimal labor tax paths are generally different between the divisible and indivisible labor economies. The results obtained here provide useful guidance in designing optimal tax policy in a dynamic environment. A useful extension of this paper would be to investigate similar issues in a third-best environment where the government may not necessarily commit to a specific tax design.
APPENDIX A
Proof of Proposition 1. First note that leisure in an indivisible labor economy is normal if
, which means the household chooses a lower probability of work when its wealth is higher. Next observe that
Using (10a), one obtains
Next note by the application of Envelope property of the value function J(a0) in (1) that J′(a0)=q0. By strict concavity of the value function (J″(a0)<0), it follows that
. Thus from (A.2), it follows that
. From (A.1), it means that
Using (12) and (16), define the following implicit function (time indices suppressed)
Using the implicit function theorem and (9) one obtains
From (A.3) and (A.5), it follows that
is negative, zero, or positive if and only y(t) is positive, negative, or zero.
APPENDIX B
Proof of Proposition 2. Plugging (12) into (23) and using (11), one obtains
Premultiply (24) by q and exploiting the fact that Ω′(q)=[c1−c2−ωh0]/qh0, one obtains
Next note that
Plugging (B.3) into (B.2) gives
Next note that
Plugging (27) into (B.5)
Using (2), (3), and (10a), the household's budget constraint can be rewritten as
which after plugging into (B.6) gives
Next noting that a=b+k, rewrite (25) as
Taking the time derivative of (B.9), one obtains
Using (B.4) and (B.8) in (B.10) one obtains
Chamley (1986) shows that the confiscation constraint, (20) cannot be binding forever. In our case, if it is binding forever, consumption falls to zero in both states. Suppose that it ceases to bind at date t1. Because ν is the multiplier associated with the confiscation constraint, (20), this implies that
for t≥t1. Plugging (B.12) into (B.11) and simplifying terms, we get
for t≥t1.
Plugging (B.13) into (B.1), one obtains
Because λ > 0 and μ < 0 and
, and
(by concavity of u), it follows that the sign of (f2−ω) is the same as the sign of y. Hence, the labor income tax is positive (negative) when y >(<) 0.
APPENDIX C
Derivation of equation (36): Inverting (38) gives
for s={1, 2}. Or equivalently
Integrating both sides with respect to cs gives
where Ms is any constant, possibly dependent on s.
Taking exponents of both sides gives
where Bs=exp(Ms), and consequently is positive.
Integrating both sides with respect to cs finally gives
where Ds is any constant, possibly dependent on s.
Derivation of equation (46). Note that
Next plug in (10b), (26), and (27) into the right-hand side of (C.6) to obtain (47).
We would like to thank Satyajit Ghosh, Gary Hansen, Barthalow Moore, William T. Smith, and two anonymous referees for useful comments. We also thank workshop participants at the University of Sydney, University of Warwick, and the 2002 meetings of the Society for Economic Dynamics and the 2003 European Meeting of the Econometric Society. Basu gratefully acknowledges the research support given by Tilburg University and NWO, and Renström gratefully acknowledges financial support from the European Commission (TMR Grant FMBI983151). M. Armellini and K. Cowton are acknowledged for secretarial support.
