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Principles of Public Finance
The theoretical literature on tax reform discusses a desirable tax system that assures the required revenue when multiple taxes are available. First, let us compare labor income tax and interest income tax with the equal revenue requirement.
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The theory of optimal taxation is one of the oldest topics of public finance. Originally, the studies centered on the theory of optimal consumption taxation. Ramsey (
1927) published the first theoretical result, which proved to be the starting point of a great many studies on the subject and is well known today as the Ramsey tax rule, as explained in the main text of this chapter. This appendix examines the optimal combination of consumption taxes, labor income taxes, and capital income taxes in an overlappinggenerations growth model, based on Diamond (
1965). In the context of economic growth, we must also consider dynamic efficiency; namely, the golden rule. After investigating the first best solution, this appendix intends to clarify the relationship between the Ramsey rule and the golden rule when lump sum taxes are not available.
We apply the overlappinggenerations model of the advanced study from Chap.
5, in which every individual lives for two periods. We extend this basic model by incorporating several distortionary taxes and by allowing for endogenous labor supply; otherwise, a labor income tax becomes a lump sum tax.
An individual living in generation t has the following utility function:
where c
^{1} is the individual’s firstperiod consumption, c
^{2} is second period consumption, and x = (Z −
l) − Z = −
l is firstperiod net leisure. Z is the initial endowment of labor supply.
The individual’s consumption, saving, and labor supply programs are restricted by the following first and secondperiod budget constraints:
where
τ is the consumption tax rate,
γ is the tax rate on labor income, w is the real wage rate, s is the individual’s real saving, r is the real rate of interest, and
θ is the tax rate on capital income. T
_{t} ^{1} is the lump sum tax levied on the young in period t and T
_{t} ^{2} is the lump sum tax levied on the old in period t.
From Eqs. (
9.A2) and (
9.A3), the individual’s lifetime budget constraint reduces to
where
q
_{t} = (q
_{1t}, q
_{2t+1}, q
_{3t}) is the consumer price vector for generation t. Thus, we have the following relationships between consumer prices and tax rates:
The present value of a lifetime lump sum tax payment on the individual of generation t (T
_{t}) is given as
Equilibrium in the capital market is simply
where n is the rate of population growth and k is the capital/labor ratio.
The feasibility condition is in period t:
where g is the government’s expenditure per individual of the younger generation.
The government budget constraint in period t is given as
Equation (
9.A8) may be rewritten in terms of tax wedge:
where t
_{i} is a tax wedge and is given as
The tax wedge t
_{i} is the difference between the consumer price q
_{i} and the producer price. Note that labor income taxation means t
_{3} < 0 since x < 0. Capital income taxation (q
_{2} > 1/(1 + r)) means that the consumer price of c
^{2} is greater than the producer price (t
_{2} > 0). 1/q
_{2} − 1 is the aftertax net rate of the return on saving. Thus, for T
^{2} = 0, t
_{2} is given as
On the lefthand side, we multiply (1 + n) because t
_{2} is an effective tax rate on secondperiod consumption; hence, it is relevant for the older generation.
Observe that the government budget constraint, Eq. (
9.A8), is consistent with the production feasibility condition, Eq. (
9.A7). Namely, one of the three equations, Eqs. (
9.A4), (
9.A7), and (
9.A8), is not an independent equation that can be derived by the other two equations.
It is useful to formulate the optimal tax problem by using the dual approach. Let us write E(
q
_{t}, u
_{t}) for the minimum expenditure necessary to attain utility level u
_{t} when prices are
q
_{t} = (q
_{1t}, q
_{2t+1}, q
_{3t}). Then, we have
which implicitly defines the utility level of generation t as a function of consumer prices:
q
_{t} and lump sum taxes T
_{t}. Eq. (
9.A10) incorporates the lifetime budget constraint, Eq. (
9.A4).
However, as in the advanced study of Chap.
5, Appendix, the factor price frontier is written as
where
From Eqs. (
9.A3), (
9.A5.4), (
9.A6), (
9.A10) and (
9.A11), we can express the secondperiod budget constraint in terms of compensated demands as
E
_{i} denotes the partial derivatives for the expenditure function with respect to price q
_{i} (i = 1, 2, 3). Note that E
_{3} = x = −
l < 0. We call Eq. (
9.A12) the compensated capital accumulation equation.
The production feasibility condition, Eq. (
9.A7), is also rewritten in terms of compensated demands as
First of all, let us investigate the first best solution where two types of lump sum tax, T
^{1} and T
^{2}, are available. Then, the government does not have to impose any distortionary taxes:
\( \tau =\theta =\gamma =0 \). The government’s objective at time 0 is to choose taxes to maximize an intertemporal social welfare function, W, expressed as the sum of generational utilities discounted by the factor of social time preference,
β.
The associated Lagrange function is given as
where
λ
_{1t,}
λ
_{2t}, and
λ
_{3t} are Lagrange multipliers for the private budget constraint (
9.A10), the resource constraint (
9.A13), and the capital accumulation equation (
9.A12) respectively.
Differentiating the Lagrangian function, Eq. (
9.A14), with respect to T
_{t}, T
_{t} ^{1} , and r
_{t+1} respectively, we have
Considering the homogeneity condition, (
\( {\displaystyle {\sum}_{j=1}^3{q}_j{E}_{ij}=0} \)) and
\( \tau =\theta =\gamma =0 \), in the steady state, Eq. (9.A15.3) reduces to
This is the modified golden rule, which is the standard optimality condition of capital accumulation. The optimal levels of T
^{1} and T
^{2} are solved to satisfy the government budget constraint, Eq. (
9.A8), and the modified golden rule, Eq. (
9.A16). When two types of lump sum tax, T
^{1} and T
^{2}, are available, the government can attain the modified golden rule, Eq. (
9.A16), at the first best solution.
We are now ready to investigate normative aspects of distortionary tax policy. From this point on, we do not impose lump sum taxes:
\( {T}_t^1={T}_t^2=0 \). First, let us investigate the situation where all the consumer prices, q
_{1}, q
_{2}, and q
_{3}, are optimally chosen. In other words, we assume that the government can choose consumption taxes, wage income taxes, and capital income taxes optimally although lump sum taxes are not available.
The maximization problem may be solved in two stages. In the first stage, one can choose {r
_{t+1}} and {
q
_{t} = (q
_{1t}, q
_{2t+1}, q
_{3t})} (t = 0, 1, …) so as to maximize W. In the second stage, one can choose (
τ,
γ,
θ) to satisfy Eqs. (
9.A5.1), (
9.A5.2), and (
9.A5.3). Thus, our main concern here is with the first stage problem. The optimization problem is solved in terms of the consumer price vector. The actual tax rates affect the problem only through the consumer price vector.
In other words, the problem is to maximize
Equations (
9.A10) and (
9.A13) both have zero degree with respect to the q vector, but Eq. (
9.A12) does not. We consider the problem as follows. The maximum W is subject to Eqs. (
9.A10) and (
9.A13), and q
_{t} is uniquely determined to a proportionality. Then, Eq. (
9.A12) gives the level of q
_{2t+1}, which is consistent with the solution of our main problem. Thus, we obtain
Differentiating the Lagrangian function, (
9.A17), with respect to q
_{1t}, q
_{2t+1}, and q
_{3t}, we have
Differentiating with respect to r
_{t+1}, we obtain
In a steady state, Eq. (
9.A20) means
Considering Eq. (
9.A20) and the homogeneity condition (
\( {\displaystyle {\sum}_{j=1}^3{q}_j{E}_{ij}=0} \)), Eq. (
9.A19) in the steady state reduces to
or
Equation (
9.A21) is the modified Ramsey rule , an extension of the standard Ramsey rule for the intertemporal setting with β. Note that β appears in the secondperiod excess burden in Eq. (
9.A21). Hence, when all consumer prices are available, the optimality condition is given by the modified golden rule, Eq. (
9.A16), and the modified Ramsey rule, Eq. (
9.A21) (see Ihori
1981).
If
β = 1, Eq. (
9.A21) will be reduced to the standard Ramsey rule. In other words, if the government is concerned with steady state utility only, we have the standard Ramsey rule as well as the golden rule. The standard Ramsey rule describes the static efficiency point.
There is an important difference between our modified Ramsey rule and the standard static Ramsey rule even if we are only concerned with longrun welfare, ignoring transition (
β = 1). Our rule is derived under the assumption that all effective taxes are available in the sense that the government can choose all consumer prices (q
_{1}, q
_{2}, q
_{3}). This is because one cannot normalize
q in a dynamic system (unless lump sum taxes are available).
Atkinson and Sandmo (
1980) derived the standard Ramsey rule in the circumstance where debt policy is employed to achieve a desired intertemporal allocation. This rule (and hence the golden rule) is, however, also relevant to the second best solution where neither lump sum taxation nor debt policy is available. This is because changes in consumption taxes and labor income taxes have lump sum timing effects. Namely, an increase in consumption taxes with a reduction in labor income taxes is equivalent to an increase in lump sum taxes in the second period of life with a reduction in lump sum taxes in the first period of life. This tax timing effect is explained in the main text of the current chapter.
Assume for simplicity of interpretation that the crosssubstitution effects are zero (
\( \begin{array}{cc}\hfill {E}_{ij}=0\hfill & \hfill for\kern0.24em i\ne j\hfill \end{array} \)). Then, from Eq. (
9.A21), we have in the elasticity form
where e
_{i} is the effective tax rate (t
_{i}/q
_{i}) and
σ
_{ij} is compensated elasticity (q
_{j}E
_{ij}/E
_{i}).
If labor supply is completely inelastic (along the compensated supply curve), the optimal tax on secondperiod consumption is zero, while the tax on labor income is equivalent to a lump sum tax and could be set arbitrarily high. If, however, the demand for future consumption is inelastic, the argument is reversed, and future income is the ideal tax base from an efficient view.
In general, the optimal rate of effective tax, e
_{i}, depends upon the relative magnitudes of the elasticities. There is no particular reason to believe that the optimal rate should be the same for the three sources of the tax base. This interpretation carries over, with appropriate modifications, to the situation of nonzero crosselasticities.
Considering the homogeneity condition in elasticity terms,
\( {\displaystyle {\sum}_{j=1}^3{\sigma}_{ij}} \)= 0 (i = 1, 2, 3), Eq. (
9.A21) may be reduced to
If
\( {\sigma}_{13}={\sigma}_{23} \), Eq. (
9.A23) is reduced to
which implies
βe
_{2} = e
_{1.} Considering Eqs. (
9.A9.1), (
9.A9.2), and (
9.A16), we obtain q
_{1} = (1 + r)q
_{2}. Substituting Eqs. (
9.A5.1) and (
9.A5.2) into the above equation, we finally have
θ = 0. Thus, the optimal tax on interest income is zero.
\( {\sigma}_{13}={\sigma}_{23} \) is called the implicit separability condition. If this condition is satisfied, the government should not impose interest income tax.
In this economy, the government intends to realize two objectives: intertemporal efficiency and financing the public good, g. When two types of lump sum tax on the young, T
^{1}, and the old, T
^{2}, are available, the government can attain these two objectives at the same time. We call this the first best solution.
However, if the government cannot control the total amount of lump sum taxes, T, but can control the combination of T
^{1} and T
^{2}, it can realize intertemporal efficiency with the modified golden rule, but cannot finance the public good without imposing static efficiency costs. This situation is an example of the second best solution. Such an example is essentially the same as distortionary taxes with an ideal debt policy. The modified Ramsey rule is relevant as in the situation where all consumer prices are optimally chosen.
From Eq. (
9.A14), an increase in T
^{1} at given T
^{2} means
\( \frac{\partial W}{\partial {T}^1}+\frac{\partial W}{\partial T}={\lambda}_3\left({\lambda}_1+{\lambda}_3\right)={\lambda}_1 \). Thus,
λ
_{1} corresponds to the marginal benefit of lump sum transfer for each individual of the younger generation financed by distortionary taxes.
λ
_{1} is normally negative in the static model but may be positive in the present dynamic model. This is because an increase in disposable income in the first period of an individual’s life stimulates saving and capital accumulation, which may improve the dynamic efficiency of the economy.
λ
_{2} corresponds to the positive marginal benefit of a decrease in government revenues:
σ
_{22} < 0 and σ
_{33} > 0.
σ
_{23} > (<) 0 and
σ
_{32} < (>) 0 if c
^{2} and x are substitutes (complements). If γ and
θ are positive, e
_{2} > 0 and e
_{3} < 0. Hence,
σ
_{33} −
σ
_{23} > 0 implies that e
_{2} > 0.
σ
_{22} −
σ
_{32} < 0 implies that e
_{3} < 0 if
λ
_{1} is negative. This is exactly what one would expect from the static optimal efficiency point.
We have been concerned with the generality of the (modified) golden rule and the (modified) Ramsey rule in a growing economy. It has been shown that even with the second best solution, the golden rule and the Ramsey rule hold if all effective nonlump sum taxes are available. We have then shown that when consumption taxes are not available, the mixed Ramseygolden rule holds. Here, the optimal formulae include divergence from the golden rule at the third best solution.
A few studies have considered the optimal tax mix for an economy with heterogeneous individuals and distributional objectives. As pointed out before, if debt policy is chosen optimally, the intuition of the static results provides the correct guidance for tax policy in a dynamic economy. The standard separability result suggests that labor income taxes may be more efficient than capital income taxes, at least in some circumstances. Atkinson and Stiglitz (
1976) and Stiglitz (
1985) showed that if, with an optimal nonlinear income tax, the utility function is weakly separable between labor and all combined goods, there is no need to employ differential indirect taxation to achieve an optimum.
Further, Deaton (
1981) has shown that where there are many consumers, and only a linear income tax and proportional commodity taxes are allowed, weak separability between goods and leisure, together with linear Engel curves for goods, remove the need for differential commodity taxation. When applied directly to the taxation of saving, the optimal capital income tax rate may be reduced to zero. Atkinson and Stiglitz (
1976) suggested that the reason for the asymmetry between labor income and capital income is not because labor income is taxed in a nonlinear fashion, but because the difference between people is based on their wages and not the rates of return on saving.
Appendix
A of this chapter characterized tax structures that maximize the sum of generational utilities discounted by the social time preference in an overlappinggenerations growth model. Because the incentive effects are complicated and sensitive to parametric structure, theory alone cannot provide clearcut guidance to efficient dynamic tax structures. With the general model, the rates of tax are highly sensitive to the compensated elasticities and covariances. Unfortunately, we have little empirical data on some of these parameters.
At this stage, we have two alternatives. One is to address the quantitative issues of the incentive effects, using numerical simulation models in which agents live for many periods, as explained in the main text of this chapter. The other is to eliminate the incentive effects. It should be stressed that the impact on intergenerational incidence of converting an income tax to either a consumption or wage tax does not depend solely on the difference in such incentive effects on a representative person. Consumption taxes and labor income taxes are equivalent from the viewpoint of household budget constraint. Both taxes affect the relative price of consumption over time in the same way, as also explained in the main text of this chapter.
The present appendix thus employs the second approach that eliminates the incentive effects. Namely, within the framework of lump sum taxation, this advanced study, Appendix
B intends to analyze theoretically the effect of the timing of tax payments on the welfare of earlier generations during the transition process.
The rationale for this approach is not that we believe that such incentive effects of distortionary taxes are unimportant. Rather, the aim of this approach is to demonstrate that even if there are no incentive effects, different taxes generate different intergenerational incidence because consumers differ in their timing of payments of taxes. This is called the tax timing effect. The difference between consumption and labor income taxation is not the incentive effect. The tax reform concerning consumption and labor income taxation may well be evaluated within the framework of lump sum tax reform. It is useful to analyze the implications of lump sum tax reform for intergenerational incidence more fully.
Essentially, if the rate of interest is greater than the rate of population growth, the effect of consumption tax is to reduce the lifetime present value of taxation by postponing tax payments to later in life. This is called the tax postponement effect. Based on Ihori (
1987), we theoretically investigate under what circumstances the tax postponement effect is relevant and how the timing of tax payments affects intergenerational incidence.
The model is almost the same as in Appendix
A. For simplicity, it is assumed that labor supply is exogenous. We now incorporate lump sum taxes instead of distortionary taxes into the overlappinggenerations model of Appendix
A. Thus, a person born in period t has the following saving function:
Assuming consumption to be normal,
\( 0<{\mathrm{s}}_{\mathrm{w}}<1,\;0>{s}_{T^1}=\partial s/\partial {T}^1>1 \), and
\( 0<{s}_T=\partial s/\partial T<1 \). However, the sign of s
_{r} depends on the relative magnitude of income and substitution effects. For simplicity, s
_{t} is assumed to be independent of r
_{t+1}.
Hence, the economy may be summarized by the following equation, where T
_{t} ^{1} and T
_{t} are policy variables:
In order to analyze the welfare aspect of tax reform on each generation, it is useful to explore the dynamic properties of the economy. Under the stability condition, r will monotonously converge to the longrun equilibrium level, r
_{L}. This implies
at the steady state equilibrium.
The government budget constraint for period t is simply
The present value of lifetime lump sum tax payment on an individual of generation t (T
_{t}) is given as
From Eqs. (
9.B4) and (
9.B5), we have
Obviously,
\( {\mathrm{T}}_{\mathrm{t}}^1={\mathrm{T}}_{\mathrm{t}+1}^1={\mathrm{T}}^1 \) and
\( {\mathrm{T}}_{\mathrm{t}}^2={\mathrm{T}}_{\mathrm{t}+1}^2={\mathrm{T}}^2 \) when the tax structure is time invariant. T
_{t} ^{2} ,
\( {\mathrm{T}}_{\mathrm{t}+1}^2 \), and the third term appear only when the tax structure is time variant.
Suppose that the government changes the combination of lump sum taxes (T
^{1}, T
^{2}) in period j + 1. T
^{2} is raised and T
^{1} is reduced. This yields:
First, let us investigate the partial equilibrium effect of tax reform on the present value of the lifetime tax payment T. If r > n, postponing tax payments to later in life (
\( {T}^1\to {T}^2 \)) means a reduction of the lifetime present value of taxation. This is the socalled the tax postponement effect.
For future generations, j + 1 + i (i = 1, 2, …) for Eq. (
9.B5′) means that the present value of tax payments, T, decreases if and only if r > n. If r > n, this gives an extra benefit to the future generation. If r < n, the tax postponement effect is unfavorable for the future generation.
For the existing younger generation j + 1, the tax postponement effect works in the same way as with the future generation. The tax postponement effect is relevant to the steady state and the transition process. For the existing older generation j,
\( {\mathrm{T}}_{\mathrm{j}+1}^2 \) is increased, while T
_{j} ^{1} is not reduced. Thus, the lifetime present value of taxation T
_{j} is raised. This corresponds to the third term of Eq. (
9.B5′) and gives an extra burden to generation j. The result may be called the direct tax reform effect or the time horizon effect.
During the transition, the earlier generation may suffer significant reductions in welfare because of the tax reform. Note that this effect works irrespective of whether r is greater than n or not. In this sense, the effect should be distinguished from the tax postponement effect.
Let us investigate the impact of tax reform on savings. A reduction of T
^{1} directly increases an individual’s saving. However, if r > n, the decrease in T
^{1} reduces T and hence indirectly reduces her or his saving. However, considering (
9.B5′), we have
Hence, the direct effect of T
^{1} is always greater than the indirect effect of T
^{1}: An individual’s saving is raised irrespective of the sign of r – n. The lump sum tax reform (
\( {T}^1\to {T}^2 \)) increases the saving of the existing younger generation j + 1 and the future generation.
This may be called the (permanent) tax timing effect . This tax reform imposes a tax liability later in the life cycle. As a result, taxpayers tend to increase their saving early in the life cycle in order to meet the additional tax liability later in the life cycle.
The impact of this tax reform on generation j’s saving depends upon whether a member of generation j anticipates this tax reform in period j or not. If an individual of generation j does not anticipate the reform, her or his saving is unaffected by the tax reform. If she or he anticipates the reform, an increase in
\( {\mathrm{T}}_{\mathrm{j}+1}^2 \) raises T
_{j} and hence increases s
_{j}. This may be called the (temporary) tax timing effect.
Because of the tax reform, the saving function of future generations moves upward. Hence, the tax reform stimulates capital accumulation during the transition path. The new longrun equilibrium capital/labor ratio, k
_{L1}, is greater than the initial longrun equilibrium ratio, k
_{L0}. Thus, the tax reform (
\( {T}^1\to {T}^2 \)) stimulates capital accumulation in the long run.
If a member of generation j anticipates the tax reform, generation j’s saving is greater than the level indicated by the initial saving function. This leads to an extra initial capital endowment to generation j + 1. Thus, generation j’s extra saving stimulates capital accumulation during the earlier transition process. Note that this temporary tax timing effect disappears in the long run.
We now explore the welfare aspect of tax reform during the growth process. Let us examine the effect of tax reform on the utility of each generation j + i, u
_{j+1} (i = 0, 1, 2, …). If the tax reform is to increase T
^{2} and reduce T
^{1} from period j + 1 on, u
_{j} definitely decreases. This is because of the direct tax reform effect.
Moreover, if a member of generation j does not anticipate the tax reform, u
_{j} reduces further. The effect on the future generation j + i (i = 1, 2, …) depends upon the tax postponement effect and the temporary and permanent tax timing effect s. If r > n, the tax postponement effect is favorable for the future generation.
Let us investigate the welfare aspect of the tax timing effect. In order to analyze the welfare of each generation explicitly, it is useful to employ the expenditure function approach as in Appendix
A. The system is summarized by
where E[.] denotes the expenditure function and E
_{2}[.] denotes the compensated demand function for secondperiod consumption. Differentiating Eqs. (
9.B7) and (
9.B8) comprehensively, we have
where
\( {E}_u=\partial E/\partial {u}_t \),
\( {E}_{2u}=\partial {E}_2/\partial {u}_t \) and
\( {E}_{22}=\partial {E}_2/\partial \left(\frac{1}{1+{r}_{t+1}}\right) \). Hence,
where Δ is the determinant of the matrix of the lefthand side of Eq. (
9.B9). In addition, we have
Under the global stability condition, 0 < dr
_{t+1}/dr
_{t} < 1 at the steady state solution. Hence, Δ> 0. The sign of [.] in Eq. (
9.B10) is positive if the elasticity of substitution between labor and capital is large, which is consistent with the stability condition, Eq. (
9.B3). In such an instance, a higher capital endowment given to an individual’s generation makes her or his lifetime utility higher. An increase in k
_{t} raises w
_{t} and lowers r
_{t+1}. The former effect increases u
_{t}, while the latter effect decreases u
_{t}. If the elasticity of substitution is large, a decrease in r
_{t} raises w
_{t} significantly. The net effect is likely to increase u
_{t} under the stability condition.
Thus, on the transitional growth process where capital accumulation is monotonously increased, each generation’s lifetime utility is monotonously increased. Note that this favorable tax timing effect works, irrespective of the sign of r − n. Consequently, generation j’s extra saving is favorable for those future generations that are close to generation j. For distant future generations, generation j’s extra saving is unimportant. In this sense, the temporary tax timing effect is relevant only to future generations that are close to the present. The utility of distant generations depends upon whether longrun equilibrium is closer to the golden rule because of the tax reform than before. Hence, if r > n, the tax reform (
\( {T}^1\to {T}^2 \)) is favorable for distant future generations from the viewpoint of the tax postponement effect and the permanent tax timing effect.
Our analysis of tax reform and intergenerational incidence may be summarized in Table
9.B1, which shows that if r > n, tax reform has different impacts on the current older generation and the current younger and future generations. Namely, the tax reform (
\( {T}^1\to {T}^2 \)) harms the current older generation and benefits the future generation. However, the reverse tax reform (
\( {T}^1\leftarrow {T}^2 \)) benefits the existing older generation and harms the future generation. This is a tradeoff relationship between the current older generation’s welfare and the future generation’s welfare.
As is well known, if r > n, the growth path is efficient in the sense that no generation is better off unless some generations are worse off. In contrast, suppose the growth path is inefficient: r < n. Then, tax reform affects the welfare of the current older generation and the distant future generation in the same direction. However, even in this situation, if a member of the current older generation anticipates the tax reform, the temporary capital accumulation effect produces a tradeoff relationship between the current older generation and the future generation that is close to the present .
So far, we have considered the circumstance whereby taxes are lump sum. Our analysis suggests that the direct tax reform, the tax postponement, the temporal tax timing, and the permanent tax timing effects are important for the evaluation of tax reform. However, when taxes are distortionary, how would the results of this study be affected? With regard to the timing of tax payments, a wage tax corresponds to T
^{1} and a capital income tax corresponds to T
^{2}.
A consumption tax may be regarded as a combination of T
^{1} and T
^{2}. Among the three taxes, an individual pays wage taxes early in life. In this sense, converting a wage tax to a consumption tax is associated with the tax reform (
\( {T}^1\to {T}^2 \)). It should be stressed that the difference between consumption and labor income taxation is not the exemption from taxation of capital income or the incentive effect, but the different timing of tax payments. Thus, a tax reform concerning consumption and labor income taxation may well be evaluated within the framework of lump sum tax reform.
With regard to the income effect, the implications of distortionary tax reform are the same as in this study. For example, if the tax reform (
\( {T}^1\to {T}^2 \)) is desirable, then a capital income tax is better than a wage or consumption tax. However, a change in the tax rate on capital income also has an incentive effect. If the interest elasticity of saving is large, a reduction of the capital tax is desirable during the efficient growth process.
The lump sum tax reform model developed here should be regarded as a complement to the incentive analysis that has been used to compare income, wage, and consumption taxes. The standard incentive and simulation analyses are better suited to capturing the differing incentive effects of each tax.
The lump sum tax approach is better suited to exploring qualitatively the consequences of the differing timing of tax payments, an aspect of reality that has not been systematically analyzed in most of the literature that compares consumption, wages, and income tax. Even with the incentive effects ignored, the differing timing of tax payments causes consumption, wages, and income tax to achieve different intergenerational incidence during the transition process when tax rates are set to achieve identical tax revenue per worker.
Say whether the following is true or false and explain the reason.
A smaller tax rate is always more desirable than a larger tax rate in order to collect the same revenue.
In a twoperiod model, assume that the agent earns labor income in both periods. Show that the equivalence hypothesis between general consumption tax and labor income tax still holds.
Explain why the timing effect of taxation concerns only the income effect.
$$ {u}_t=u\left({c}_t^1,{c}_{t+1}^2,{x}_t\right), $$
(9.A1)
$$ \left(1+{\tau}_t\right){c}_t^1=\left(1{\gamma}_t\right){w}_t{l}_t{s}_t{T}_t^1\kern1em \mathrm{and} $$
(9.A2)
$$ \left(1+{\tau}_{t+1}\right){c}_{t+1}^2=\left[1+{r}_{t+1}\left(1{\theta}_{t+1}\right)\right]{s}_t{T}_{t+1}^2, $$
(9.A3)
$$ {q}_{1t}{c}_t^1+{q}_{2t+1}{c}_{t+1}^2+{q}_{3t}{x}_t+{T}_t=0, $$
(9.A4)
$$ {q}_{1t}=1+{\tau}_t, $$
(9.A5.1)
$$ {q}_{2t+1}=\frac{1+{\tau}_{t+1}}{1+{r}_{t+1}\left(1{\theta}_{t+1}\right)},\;\mathrm{and} $$
(9.A5.2)
$$ {q}_{3t}=\left(1{\gamma}_t\right){w}_t. $$
(9.A5.3)
$$ {T}_t={T}_t^1+\frac{T_{t+1}^2}{1+{r}_{t+1}\left(1{\theta}_{t+1}\right)}. $$
(9.A5.4)
$$ {s}_t=\left(1+n\right){k}_{t+1}{l}_{t+1}, $$
(9.A6)
$$ {c}_t^1+\frac{c_t^2}{1+n}+g+\left(1+n\right){k}_{t+1}{l}_{t+1}={w}_t{l}_t+{r}_t{k}_t{l}_t+{k}_t{l}_t, $$
(9.A7)
$$ {\tau}_t{c}_t^1+\frac{\tau_t{c}_t^2}{1+n}+{\theta}_t{r}_t{k}_t{l}_t+{\gamma}_t{w}_t{l}_t+{T}_t^1+\frac{T_t^2}{1+n}=g. $$
(9.A8)
$$ {t}_{1t}{c}_t^1+{t}_{2t}{c}_t^2+{t}_{3t}{x}_t+{T}_t^1+\frac{T_t^2}{1+n}=g, $$
(9.A8′)
$$ {t}_{1t}={\tau}_t={q}_{1t}1, $$
(9.A9.1)
$$ {t}_{2t}=\frac{\tau_t}{1+n}+\frac{\theta_t{r}_t{k}_t{l}_t}{c_t^2}=\frac{q_{2t}\left(1+{r}_t\right)1}{1+n}+\frac{\theta_t{r}_t{T}_t^2}{\left(1+n\right)\left[1+{r}_t\left(1{\theta}_t\right)\right]},\;\mathrm{and} $$
(9.A9.2)
$$ {t}_{3t}={\gamma}_t{w}_t={q}_{3t}{w}_t. $$
(9.A9.3)
$$ \left(1+n\right){t}_2=\frac{\left[\left(1+r\right)\frac{1}{q_2}\right]s}{c^2}={q}_2\left(1+r\right)1. $$
$$ E\left({q}_t,{u}_t\right)+{T}_t=0, $$
(9.A10)
$$ {\mathrm{w}}_{\mathrm{t}}=\mathrm{w}\left({\mathrm{r}}_{\mathrm{t}}\right), $$
(9.A11)
$$ {\mathrm{w}}^{\prime}\left({\mathrm{r}}_{\mathrm{t}}\right)={\mathrm{k}}_{\mathrm{t}}<0\ \mathrm{and}\ \mathrm{w}^{{\prime\prime} }>0. $$
$$ {q}_{2t+1}{E}_2\left({q}_t,{u}_t\right)+{T}_t{T}_t^1=\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right). $$
(9.A12)
$$ \begin{array}{l}{E}_1\left({q}_t,{u}_t\right)+\frac{E_2\left({q}_{t1},{u}_{t1}\right)}{1+n}+\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)+\\ {}\kern4em \left[w\left({r}_t\right)\left(1+{r}_t\right){w}^{\prime}\left({r}_t\right)\right]{E}_3\left({q}_t,{u}_t\right)+g=0.\end{array} $$
(9.A13)
$$ \begin{array}{l}W={\displaystyle {\sum}_{t=0}^{\infty }{\beta}^t\left\{{u}_t{\lambda}_{1t}\right[E\left({q}_t,{u}_t\right)}+{T}_t\left]{\lambda}_{2t}\right[{E}_1\left({q}_t,{u}_t\right)+\frac{E_2\left({q}_{t1},{u}_{t1}\right)}{1+n}+g\\ {}\kern2em +\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)+\left(w\left({r}_t\right)\left(1+{r}_t\right){w}^{\prime}\left({r}_t\right)\right){E}_3\left({q}_t,{u}_t\right)\Big]\\ {}\kern2em {\lambda}_{3t}\left[{q}_{2t+1}{E}_2\left({q}_t,{u}_t\right)+{T}_t{T}_t^1\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)\right]\Big\},\end{array} $$
(9.A14)
$$ \frac{\partial W}{\partial {T}_t}={\beta}^t\left({\lambda}_{1t}+{\lambda}_{3t}\right)=0, $$
(9.A15.1)
$$ \frac{\partial W}{\partial {T}_t^1}={\beta}^t{\lambda}_{3t}=0,\;\mathrm{and} $$
(9.A15.2)
$$ \frac{\partial W}{\partial {r}_{t+1}}=0 $$
(9.A15.3)
$$ 1+\mathrm{n}=\beta \left(1+\mathrm{r}\right) $$
(9.A16)
$$ \begin{array}{l}W={\displaystyle {\sum}_{t=0}^{\infty }{\beta}^t\Big\{{u}_t{\lambda}_{1t}E\left({q}_t,{u}_t\right)}{\lambda}_{2t}\Big[{E}_1\left({q}_t,{u}_t\right)+\frac{E_2\left({q}_{t1},{u}_{t1}\right)}{1+n}+g\\ {}\kern2em +\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)+\left(w\left({r}_t\right)\left(1+{r}_t\right){w}^{\prime}\left({r}_t\right)\right){E}_3\left({q}_t,{u}_t\right)\Big]\\ {}\kern2em {\lambda}_{3t}\left[{q}_{2t+1}{E}_2\left({q}_t,{u}_t\right)\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)\right]\Big\}.\end{array} $$
(9.A17)
$$ {\lambda}_{3t}=0. $$
(9.A18)
$$ \begin{array}{l}\frac{\partial W}{\partial {q}_j}={\beta}^t\Big\{{\lambda}_{1t}{E}_j\left({q}_t,{u}_t\right){\lambda}_{2t}{E}_{1j}\left({q}_t,{u}_t\right){\lambda}_{2t+1}\frac{E_{2j}\left({q}_t,{u}_t\right)\beta }{1+n}\\ {}\kern3em {\lambda}_{2t1}\;\frac{E_{3j}\left({q}_t,{u}_t\right)\left(1+n\right){w}^{\prime}\left({r}_t\right)}{\beta }{\lambda}_{2t}\left[w\left({r}_t\right)\left(1+{r}_t\right){w}^{\prime}\left({r}_t\right)\right]{E}_{3j}\left({q}_t,{u}_t\right)\Big\}=0\\ {}\kern3em \left(j=1,2,3\right).\end{array} $$
(9.A19)
$$ \begin{array}{l}\frac{\partial W}{\partial {r}_{t+1}}={\beta}^t\Big\{{\lambda}_{2t}\left(1+n\right)w^{{\prime\prime}}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)\\ {}\kern3em {\lambda}_{2t+1}\beta \left(1+{r}_{t+1}\right)w^{{\prime\prime}}\left({r}_{t+1}\right){E}_3\left({q}_{t+1},{u}_{t+1}\right)\Big\}=0.\end{array} $$
(9.A20)
$$ 1+\mathrm{n}=\beta \left(1+\mathrm{r}\right) $$
(9.A16)
$$ \begin{array}{l}{\lambda}_1{E}_j{\lambda}_2\left\{\left({q}_{1j}1\right){E}_{1j}+\left({q}_{2j}\frac{\beta }{1+n}\right){E}_{2j}+\right[\frac{\left(1+n\right){w}^{\prime }}{\beta }+{q}_{3j}w+\\ {}\kern3em \left(1+{r}_t\right){w}^{\prime}\left]{E}_{3j}\right\}=0\end{array} $$
$$ \frac{t_1{E}_{1i}+\beta {t}_2{E}_{2i}+{t}_3{E}_{3i}}{E_i}=\frac{\lambda_1}{\lambda_2}\;\left(\mathrm{i} = 1,\ 2,\ 3\right). $$
(9.A21)
$$ {e}_1{\sigma}_{11}=\beta {e}_2{\sigma}_{22}={e}_3{\sigma}_{33}, $$
(9.A22)
$$ {e}_1\left({\sigma}_{12}+{\sigma}_{13}+{\sigma}_{21}\right)+\beta {e}_2\left({\sigma}_{21}+{\sigma}_{23}+{\sigma}_{12}\right)={e}_3\left({\sigma}_{23}{\sigma}_{13}\right). $$
(9.A23)
$$ \left({\sigma}_{12}+{\sigma}_{13}+{\sigma}_{21}\right)\left(\beta {e}_2{e}_1\right)=0, $$
(9.A24)
$$ {s}_t=s\left({w}_t,{r}_{t+1},{T}_t^1,{T}_t\right). $$
(9.B1)
$$ s\left[w\left({r}_t\right),{T}_t^1,{T}_t\right]=\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right). $$
(9.B2)
$$ 0<\frac{{s}_w{w}^{\prime }}{\left(1+n\right)w^{{\prime\prime} }}<1 $$
(9.B3)
$$ {T}_t^1+\frac{T_t^2}{1+n}=g. $$
(9.B4)
$$ {T}_t={T}_t^1+\frac{T_{t+1}^2}{1+{r}_{t+1}}. $$
(9.B5)
$$ {T}_t=\frac{\left({r}_{t+1}n\right){T}_t^1}{1+{r}_{t+1}}+\frac{\left(1+n\right)g}{1+{r}_{t+1}}+\frac{T_{t+1}^2{T}_t^2}{1+{r}_{t+1}}. $$
(9.B5′)
$$ {\mathrm{T}}_{\mathrm{j}}^2<{\mathrm{T}}_{\mathrm{j}+1}^2={\mathrm{T}}_{\mathrm{j}+2}^2 = {\mathrm{T}}^2\;\mathrm{and}\kern0.24em {\mathrm{T}}_{\mathrm{j}}^1>{\mathrm{T}}_{\mathrm{j}+1}^1={\mathrm{T}}_{\mathrm{j}+2}^1={\mathrm{T}}^1 $$
$$ \frac{\partial s}{\partial {T}^1}={s}_{T^1}+\frac{\left(rn\right){s}_T}{1+r}<1+\frac{rn}{1+r}=\frac{1+n}{1+r} < 0. $$
(9.B6)
$$ E\left[\frac{1}{1+{r}_{t+1}},{u}_t\right]=w\left({r}_t\right){T}_t\kern1em \mathrm{and} $$
(9.B7)
$$ {E}_2\left[\frac{1}{1+{r}_{t+1}},{u}_t\right]=\left(1+n\right)\left(1+{r}_{t+1}\right){w}^{\prime}\left({r}_{t+1}\right){T}_{t+1}^2, $$
(9.B8)
$$ \left[\begin{array}{cc}\hfill {E}_u,\hfill & \hfill {E}_2\frac{1}{{\left(1+{r}_{t+1}\right)}^2}\hfill \\ {}\hfill {E}_{2u},\hfill & \hfill {E}_{22}\left[\frac{1}{{\left(1+{r}_{t+1}\right)}^2}\right]+\left(1+n\right)\left({w}^{\prime }+\left(1+{r}_{t+1}\right)w^{{\prime\prime}}\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill d{u}_t\hfill \\ {}\hfill d{r}_{t+1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {w}^{\prime}\hfill \\ {}\hfill 0\hfill \end{array}\right]d{r}_t, $$
(9.B9)
$$ \frac{d{u}_t}{d{r}_t}=\frac{1}{\Delta}{w}^{\prime}\left\{{E}_{22}\left[\frac{1}{{\left(1+{r}_{t+1}\right)}^2}\right]+\left(1+n\right)\left[{w}^{\prime }+\left(1+{r}_{t+1}\right)w^{{\prime\prime}}\right]\right\}, $$
(9.B10)
$$ \frac{d{r}_{t+1}}{d{r}_t}=\frac{E_{2u}{w}^{\prime }}{\Delta}. $$
(9.B11)
Table 9.B1
Tax reform and intergenerational incidence
Tax Reform

Current older generation

Near future generation

Distant future generation



\( {T}^1\to {T}^2 \)

r >
n

DTR(−)

TPP(+)

TPP(+)

TTT(+)

PTT(+)


r <
n

DTR(−)

TPP(−)

TPP(−)


TTT(+)

PTT(−)


\( {T}^1\leftarrow {T}^2 \)

r >
n

DTR(+)

TPP(−)

TPP(−)

TTT(−)

PTT(−)


r <
n

DTR(+)

TPP(+)

TPP(+)


TTT(−)

PTT(+)

9.1
Say whether the following is true or false and explain the reason.
A smaller tax rate is always more desirable than a larger tax rate in order to collect the same revenue.
9.2
In a twoperiod model, assume that the agent earns labor income in both periods. Show that the equivalence hypothesis between general consumption tax and labor income tax still holds.
9.3
Explain why the timing effect of taxation concerns only the income effect.
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 Titel
 Tax Reform
 DOI
 https://doi.org/10.1007/9789811023897_9
 Autor:

Toshihiro Ihori
 Verlag
 Springer Singapore
 Sequenznummer
 9
 Kapitelnummer
 9