The later you pay, the higher the k.

1. Introduction

The age path (life-cycle timing) of a tax is an important yet often overlooked property of the tax because it affects the capital accumulation of the economy. By contrast, attention usually focuses on the incentive effect of a tax. For example, suppose a wage tax is replaced by a consumption tax. Neither tax reduces the net return to saving below the gross return, so a focus on the incentive effect might suggest that the capital accumulation of the economy would be unaffected by the replacement. But this reasoning neglects the fact that a consumption tax has a different age path from a wage tax; the individual pays less tax while working and more during retirement. It turns out that this postponement of tax over the life cycle tends to raise the capital per labor (k) in the economy.

Consider the proposal to terminate the taxation of capital income. This can be done in two different ways: convert the income tax to a wage (labor income) tax (exempt interest, dividends, and capital gains from the income tax) or convert the income tax to a consumption tax (tax consumption spending, not income). Does it matter which way? The impression is often given in the public finance literature that either way would have the same effect. But this ignores the fact that a consumption tax has a different age path from a wage tax.

Similarly, it has been shown that conversion of an income tax to a consumption tax raises the steady-state k of the economy regardless of the interest elasticity of saving and that the magnitude of the increase in k is unaffected by the magnitude of the saving elasticity (Seidman and Lewis 1999); but if the saving elasticity is irrelevant, what raises the steady-state k? This paper investigates the conjecture that "the later you pay, the higher the k" and demonstrates the importance of the age path of any tax for the capital accumulation of the economy.

Wage, income, consumption, and capital income taxes differ in their age paths as well as their incentive effects. Under a wage tax, a person pays tax while working, but not in retirement. Under an income tax, a person usually pays less annual tax during retirement than while working. But under a consumption tax, a person would pay as much annual tax whether retired or working, assuming the person saves enough to maintain consumption on retirement. Under a capital income (interest) tax, annual tax would rise and fall over the life cycle as the individual accumulates and decumulates wealth. This paper analyzes how the differing age path of each tax affects the capital accumulation of the economy.

The impact of the age path of a tax on the capital accumulation of the economy has been noted by other researchers (e.g., Diamond 1970; Atkinson and Stiglitz 1980; Bradford 1980; Summers 1981; Auerbach and Kotlikoff 1987; Ihori 1987, 1996; Gravelle 1994). Summers (1981, p. 538) shows that in a life-cycle growth model, a consumption tax generally achieves a higher steady-state capital per effective labor than a wage tax which raises the same revenue because "consumption taxation extracts revenue later in the individual's lifetime than does wage taxation and so causes more savings in the younger years." Ihori (1987, p. 386) writes that "the lump-sum approach shows clearly that even with the incentive effects ignored, the differing timing of tax payments would cause consumption, wage, and income taxes to achieve different intergeneration incidence during the transition process when tax rates are set to achieve identical tax revenue per worker." Auerbach and Kotlikoff (1987, p. 57) note the significance of the ag e path of a tax and find that "the consumption tax base generates significantly more long-run capital formation than either the wage tax or the income tax." Gravelle (1994, p. 42) recognizes this point, stating that "a switch to a consumption tax should increase the savings rate, while a switch to a wage tax has ambiguous effects."

Nevertheless, there continues to be a neglect of the importance of the age path of taxes in the public finance literature. For example, many public finance journal articles and textbooks continue to assert that a consumption tax is "equivalent" to a wage tax even though the two taxes have different age paths and consequently generate different steady-state k's for the economy.

Why this neglect? One reason is that some economists argue that the impact of the age path of a tax on capital accumulation can be offset by the appropriate government debt policy (Bradford 1980). This point is correct in theory. For example, although conversion from a wage tax to a consumption tax generally raises steady-state k when there is no change in debt policy, k can be kept the same with the appropriate change in debt policy (Seidman 1990).

In practice, however, when tax conversion occurs, it is doubtful that there will be any change in debt policy. For this reason, Summers (1981) analyzes tax conversion on the assumption that the budget remains balanced each year despite tax conversion--that is, there is no offsetting debt policy in response to conversion. When challenged by Bradford (1980), Summers makes this reply (p. 539): "David Bradford emphasizes the importance of this 'balanced budget' requirement. He argues that without such a requirement the government by running surpluses or deficits can control the level of the capital stock given any tax structure. The 'real world' feasibility of this sort of policy seems unclear."

We would go further than Summers and argue that there is no empirical evidence that proposals for tax conversion are accompanied by changes in debt policy that nullify the impact of tax conversion on capital accumulation. On the contrary, governments almost invariably discuss, debate, and set their debt policy--for example, a balanced budget rule--independently of tax policy. For example, in the debate in the U.S. Congress over fundamental tax reform in the 1990s, involving the pros and cons of a flat tax, a progressive consumption tax (Seidman 1997), and a national sales tax, there was never any discussion of altering government debt policy by legislators or even by economists testifying on the merits of each tax proposal. It is therefore important for economists to analyze the impact on capital accumulation of pure tax conversion--that is, tax conversion unaccompanied by offsetting debt policy.

A second reason for neglecting the impact of the age path of a tax on capital accumulation is the assumption that it is quantitatively unimportant. Indeed, if incentive effects were large in magnitude but timing effects were small, ignoring timing effects would be justified. But is this the case? A major purpose of this paper is to estimate the quantitative impact of the timing effect. We therefore utilize a model with numerical parameter values based on econometric studies. It turns out that the timing effect is in fact quantitatively important.

This paper builds specifically on the papers of Summers (1981) and Ihori (1987). Like Summers, we use a multiage life-cycle growth model, but unlike Summers, we compare conventional transactions-based taxes (wage, income, and consumption taxes) with lump-sum "age taxes" to isolate the impact of the timing effect. Like Ihori, we use lump-sum taxes, but unlike Ihori, we use a multiage model that is empirically calibrated, enabling us to estimate whether the timing effect is quantitatively important. We estimate that perhaps half the capital-stock gap between a consumption tax and an income tax is due to the timing effect.

In this paper, we investigate systematically the conjecture that the later you pay, the higher the k, first in a two-age (two-period) life-cycle model and then in a multiage (multiperiod) life-cycle model. The two-age model is simply the special case of the multiage model in which there are just two "ages"--work age and retirement age. We use the term "age" rather than "period" to emphasize the fact that it is time from birth, not calendar time, that is the focus of the model. The multiage model is empirically calibrated on the basis of recent econometric studies.

A central element of our analysis is to replace a transactions-based tax (income, wage, consumption, and capital income) with a corresponding lump-sum age tax that has the identical age path of tax payments over the life cycle. By doing so, we can isolate the impact that the pure timing effect has on capital accumulation and investigate its magnitude. It turns out, for an empirically plausible range of parameter values, that the quantitative impact can be considerable. Moreover, the timing effect often causes the impact of a tax on capital accumulation to be very different from what would be predicted from the incentive effect.

It is important to be precise about our assumptions concerning government spending. Like Summers (1981), we make the following assumptions. First, we assume that government spending consists entirely of government consumption purchases so that government spending has no direct impact on capital accumulation, thereby enabling us to clearly isolate the impact of taxes on capital accumulation. Second, we assume that the government annually balances its budget so that there is no government debt. As shown by Diamond (1965), this assumption matters for capital accumulation in a life-cycle growth model. Third, we assume that budget balance is adhered to regardless of the tax chosen to finance government spending. We share Summer's (1981) view that it is important to investigate what happens if debt is not varied when taxes change because such variation has virtually never been mentioned by legislators proposing fundamental tax reform. Fourth, in our multiage model, government spending per effective labor (which alw ays equals tax revenue per effective labor in each period) is the same regardless of the tax chosen to finance the government spending.

We begin with the two-age model and then proceed to the multiage model. For each model, we first examine lump-sum age taxes and then analyze the life-cycle timing effects of conventional transactions-based taxes.

2. The Two-Age Model

The two-age (work age and retirement age) model in this section utilizes a standard isoelastic intertemporal utility function and a Cobb-Douglas production function with no depreciation or technological progress. The production function is y = [mk.sup.[alpha]], where y is output per worker, m is a constant, k is capital per worker, and 0 < [alpha] < 1. We assume labor is fixed: Each worker who retires is replaced by one new worker, and each worker supplies one unit of labor; the number of workers equals the number of retirees. Each period, workers supply labor and retirees supply capital to firms. We assume that each factor is paid its marginal product where w is the wage per worker and r is the rental per unit of capital (the interest rate).

Our analysis throughout this paper focuses exclusively on the steady state of the model where k, y, w, and r remain constant over time. We assume that in this steady state, all expectations are realized so that there is no need to distinguish between expected and actual values. Each individual chooses ([C.sub.1], [C.sub.2]), consumption in the two ages of life (work and retirement), to maximize lifetime utility subject to a lifetime budget constraint. The utility function is isoelastic with an intertemporal elasticity of substitution [sigma] = 1/[gamma] and a subjective discount rate [rho]:

U = [([C.sup.1-[gamma].sub.1])/(1 - [gamma])] + [([C.sup.1-[gamma].sub.2] - 1)/(1 - [gamma])]/(1 + [rho]) if [gamma] [not equal to] 1,

U = ln [C.sub.1] + [ln [C.sub.2]/(1 + [rho])] if [gamma] = 1, (1)

where (1 + [rho]) > 0 and [gamma] > 0, so [sigma] = l/[gamma] > 0.

Age Taxes

Each person pays a lump-sum tax in each age of life. In the work age, each person pays lump-sum tax [T.sub.1] [greater than or equal to] 0 and in the retirement age lump-sum tax [T.sub.2] [greater than or equal to] 0. The government sets taxes to balance its cash-flow budget, so [T.sub.1] + [T.sub.2] = g, where g is government consumption purchases per worker. The individual's budget constraint is [C.sub.1] + [T.sub.1] + [([C.sub.2] + [T.sub.2])/(1 + r)] = w, or

[C.sub.1] + [[C.sub.2]/(1 + r)] = w - [T.sup.p], (2)

where [T.sub.p] [equivalent to] [T.sub.1] + [[T.sub.2]/(1 + r)], the present value of lifetime taxes; [T.sub.p] < w. The individual chooses ([C.sub.1], [C.sub.2]) to maximize Equation 1 subject to Equation 2; it can be shown (1) that the individual chooses ([C.sub.2]/[C.sub.1]) = [[(1 + r)/(1 + [rho])].sup.[sigma]] substituting for [C.sub.2] in Equation 2 yields

[C.sub.1] = [phi](w - [T.sub.p]), (3)

where [phi] [equivalent to] [phi][r, [sigma], [rho]] [equivalent to] 1/{1 + [[(1 + r).sup.[sigma]-1]/[(1 + [rho].sup.[sigma]]]}, so 0 < [phi] < 1. Because [S.sub.1] = w - [T.sub.1] - [C.sub.1], using Equation 3 we obtain

[S.sub.1] = (1 - [phi])(w - [T.sub.1])+[[phi][T.sub.2]/(1 + r)]. (4)

Then [C.sub.2] = [S.sub.1](1 + r) - [T.sub.2]. From Equation 4, the elasticity [epsilon] of [S.sub.1] with respect to r may be zero, positive or negative. For example, with no taxes ([T.sub.1] = [T.sub.2] = 0), if [sigma] = 1, then [epsilon] = 0 (because [partial][phi]/[partial]r = 0); if [sigma] > 1, then [epsilon] > 0 (because [partial][phi]/[partial]r < 0), and if [sigma] < 1, then [epsilon] < 0 (because [partial][phi]/[partial]r > 0).

Firms hire labor from workers and rent capital from retirees. Each period, each retiree supplies capital equal to [S.sub.1], the saving the retiree did last period as a worker. Because there is one worker for each retiree, k equals [S.sub.1], so from Equation 4,

k = (1 - [phi])(w - [T.sub.1]) + [[phi][T.sub.2]/(1 + r)]. (5)

Let [[theta].sub.1] [equivalent to] [T.sub.1]/w and [[theta].sub.2] [equivalent to] [T.sub.2]/w; note that ([T.sub.1], [T.sub.2]) is exogenous but that w is endogenous. Because each factor is paid its marginal product, (2) w = (1 - [alpha])[mk.sub.[alpha]] and r = [alpha]m/[k.sup.-1 - [alpha]], so k = [([alpha]m/r).sup.[1/(1 - [alpha])] and w = (1 - [alpha])m[([alpha]m/r).sup.[alpha]/(1-[alpha])], so [[theta].sub.1] = [[theta].sub.1][r, [T.sub.1]] and [[theta].sub.2] = [[theta].sub.2][r, [T.sub.2]]; note that r is endogenous. Substituting into Equation 5 for w and k yields an equation with one endogenous variable, r:

r = [[alpha]/(1 - [alpha])]/{(1 - [phi])(1 - [[theta].sub.1]) + [[phi][[theta].sub.2]/(1 + r)]}, (6)

where [phi]=[phi][r, [sigma], [rho]] [equivalent to] 1/{1 + [[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]]]}, [[theta].sub.1] = [[theta].sub.1][r, [T.sub.1]], [[theta].sub.2] = [[theta].sub.2][r, [T.sub.2]], and ([T.sub.1], [T.sub.2]) is exogenous. Given values for ([T.sub.1], [T.sub.2]) and the parameters ([sigma], [rho], [alpha]), r is obtained from Equation 6 by an iterative procedure. (3) Then k is obtained from k = [([alpha]m/r).sup.[1/(1-[alpha])]]. Setting [T.sub.1] = [T.sub.2] = 0 so [[theta].sub.1] = [[theta].sub.2] = 0 in Equation 6 gives the equation for [r.sub.o], the r in a no-tax (and no-government-spending) economy; [r.sub.o] = [[alpha]/(1 - [alpha])]/(1 - [phi][[r.sub.o]]). Imposing an age tax to finance government consumption purchases generates an r that may be greater than, equal to, or less than [r.sub.o]. In particular, if the tax ratio, [T.sub.2]/[T.sub.1], is set equal to the no-tax consumption ratio [([C.sub.2]/[C.sub.1]).sub.o], then the age tax r will be the same as [r.sub.o]. (4) Thus, if [T.sub.2]/[T.sub.1] is set equal to [([C.sub.2]/[C.sub.1]).sub.o], then the k under this age tax is the same as the k with no tax ([k.sub.o]); imposing an age tax with this particular ratio to finance government consumption purchases has no effect on the k of the economy.

Table 1 shows how k varies with alternative lump-sum age taxes. In the table, there are four alternative values of [sigma], ranging from 3.0 to 0.0625, and four values of [rho], ranging from -0.5 to 2.0 (note that in the two-age model, an "age" is roughly 30 years); [alpha] = 0.2 and m = 3.75 (this value of m is chosen so that, with no taxes and [sigma] = 1, [rho] = 1, then Equation 6 yields k = 1.000). In each row of Table 1, the sum [T.sub.1] + [T.sub.2] is held constant so that the balanced-budget requirement g = [T.sub.1] + [T.sub.2] is satisfied for a fixed g. Moving across any row from left to right, the ratio of retirement-age tax to total lifetime tax, [T.sub.2]/([T.sub.1] + [T.sub.2]), increases from 0% to 100%; thus, moving from left to right, the later the person pays. In every row in the table, moving from left to right, k increases. For example, with [sigma] = 1, [rho] = 1, when 0% is paid in retirement, k is 0.808, whereas when 100% is paid in retirement, k is 1.225. Thus, in every row of the ta ble, the later you pay, the higher the k. (5) Table 1 is based on [alpha] = 0.2, but the same result holds for [alpha] = 0.4, 0.6, and 0.8.

Conventional Transactions-Based Taxes

With transactions-based taxes, the individual's budget constraint is

[[C.sub.1]/(1 - [t.sub.c])] + [[C.sub.2]/(1 - [t.sub.c])]/(1 + [r.sub.n]) = (1 - [t.sub.w])w, (7)

where [t.sub.c], is the consumption tax rate, defined as [t.sub.c] [equivalent to] T/(C + T) so that C/(1 - [t.sub.c]) equals consumption plus consumption tax (6) and T = [t.sub.c]C/(1 - [t.sub.c]); [t.sub.w] is the tax rate on wage income; the net (after-tax) return [r.sub.n] [equivalent to] (1 -- [t.sub.r])r, where [t.sub.r] is the tax rate on interest income; with an income tax, [t.sub.y] = [t.sub.w] = [t.sub.r]. Then it can be shown (7) that the individual chooses ([C.sub.2]/[C.sub.1]) = [[(1 + [r.sub.n])/(l + [rho])].sup.[sigma]]; substituting for [C.sub.2] in Equation 7 yields

[C.sub.1]/(1 -- [t.sub.c]) = [phi](l -- [t.sub.w])w, (8)

where [phi] = [phi][[sigma], [rho], [r.sub.n]] [equivalent to] l/{1 + [[(1 + [r.sub.n]).sup.[sigma] - 1]/[(1 + [rho]).sup.[sigma]]]} so 0 < [phi] < 1. With a C Tax, [T.sub.1] = [t.sub.c][C.sub.1]/(1 - [t.sub.c]) = [t.sub.c][phi]w. Note that [phi] is a function of [r.sub.n]. Because [S.sub.1] = (1 - [t.sub.w])w - [[C.sub.1]/(1 - [t.sub.c])], using Equation 8 we obtain

[S.sub.1] = (1 -- [phi])(1 -- [t.sub.w])w. (9)

Then [C.sub.2]/(1 -- [t.sub.c]) = [S.sub.1](l + [r.sub.n]); and with a C Tax, [T.sub.2] [t.sub.c][C.sub.2]/(1 - [t.sub.c]) = [t.sub.c](1 - [phi])w(1 + r). Note that [t.sub.c] does not affect [S.sub.1], but [t.sub.r] does, through [phi] (except when [sigma] = 1, so [partial][phi]/[partial][r.sub.n] = 0). From Equation 9, the elasticity [epsilon] of [S.sub.1] with respect to [r.sub.n] may be zero, positive, or negative. If [sigma] = 1, then [epsilon] = 0 (because [partial][phi]/[partial][r.sub.n] = 0); if [sigma] > 1, then [epsilon] > 0 (because [partial][phi]/[partial][r.sub.n] < 0); and if [sigma] < 1, then [epsilon] < 0 (because [partial][phi]/[partial][r.sub.n] > 0). Because [kappa] = [S.sub.1],

[kappa] = (1 - [phi])(l - [t.sub.w])w. (10)

Recall that w = (1 -- [alpha])m[([alpha]m/r).sup.[alpha]/(1 - [alpha])] and [kappa] = [([alpha]m/r).sup.[1/(1 -- [alpha])]]; substituting into Equation 10 for w and [kappa] and replacing r by [r.sub.n]/(1 -- [t.sub.r]) yields an equation with one endogenous variable, [r.sub.n]:

[r.sub.n]/(1 -- [t.sub.r]) = [[alpha]/(1 -- [alpha])]/(1 -- [phi])(1 -- [t.sub.w]), (11)

where [phi] = [phi][r.sub.n]. Given values for the parameters, [r.sub.n] is obtained from Equation 11 by an iterative procedure. Then r = [r.sub.n]/(1 -- [t.sub.r]), and [kappa] is obtained from [kappa] = [([alpha]m/r).sup.[1/(1 -- [alpha])]].

Setting [t.sub.c] = [t.sub.w] = [t.sub.r] = 0 in Equation 11 gives [r.sub.o]. Because [t.sub.c] does not appear in Equation 11, just as it did not appear in Equation 9, the r with a consumption tax ([r.sub.c]) is the same as [r.sub.o]. Note that with a consumption tax, [T.sub.2]/[T.sub.1] = ([C.sub.2]/[C.sub.1]) Recall that when an age tax is set so that the tax ratio [T.sub.2]/[T.sub.1] = [([C.sub.2]/[C.sub.1]).sub.o], then the k with that particular age tax is also the same as [k.sub.o]. Summers (1981) says that a consumption tax has no effect on the steady-state k as long as the utility function is homothetic. The intuition is that it is evidently optimal for an individual to respond to a consumption tax by reducing consumption at all ages such that at each age, consumption plus tax is unchanged and saving is unchanged; hence, k is unchanged.

Table 2 presents values of k for four transactions-based taxes--wage (W Tax), income (Y Tax), consumption (C Tax), and interest (R Tax)--for the same combinations of [sigma] and [rho] as in Table 1. Note that because [t.sub.c] does not appear m Equation 11, the value of k under the C Tax is the same as the value of k with no tax, so the C Tax column is also the no-tax column. Although Table 2 is based on [alpha] = 0.2, the same qualitative patterns hold for [alpha] = 0.4, 0.6, and 0.8. The tax rates are set to achieve the same ratio of tax revenue to national income. In particular, under all taxes in Table 2, the ratio of revenue to income is 12%.

Table 2 also presents values of k for corresponding age taxes. For each transactions-based tax, the corresponding age tax is a lump-sum tax that has the same ([T.sub.1], [T.sub.2]) as the transactions-based tax. For each corresponding age tax, the steady-state r is obtained from Equation 6 by inserting the ([T.sub.1], [T.sub.2]) generated by the transactions-based tax; then k is obtained from k = [([alpha]m/r).sup.[1/(1-[alpha])]]. For example, corresponding to a Y Tax is an "Age Y Tax"--an age tax with the same ([T.sub.1], [T.sub.2]) that results from that Y Tax. It is shown in the Appendix that the k under the corresponding Age W Tax is the same as the k under a W Tax and that the k under the corresponding Age C Tax is the same as the k under a C Tax. Hence, in Table 2, there is only one column under the W Tax and one column under the C Tax.

How much of the difference in k between two transactions-based taxes is due to the difference in the life-cycle timing of the taxes? The difference in k between the two corresponding lump-sum age taxes we call the "timing effect." For example, consider the C Tax versus the Y Tax with [sigma] = 1 and [rho] = 1 in Table 2. The difference between the k's under these two transactions-based taxes is 1.000 - 0.852 = 0.148. The timing effect equals the difference between the k's under the two corresponding age taxes, 1.000 - 0.893 = 0.107. In this case, the timing effect accounts for 72% of the difference between the values of k under the two transactions-based taxes (0.107/0.148 = 72%).

We note several striking results from Table 2. In every row, the C Tax always achieves a higher k than the W Tax. For example, with [sigma] = 1, [rho] = 1, the C Tax k exceeds the W Tax k (1.000 vs. 0.816). Why is k higher under a C Tax than under a W Tax? After all, under both taxes, [r.sub.n] equals r: With a C Tax, saving is tax deductible when it occurs, and with a W Tax, saving is not deductible, but the income that will he earned from saving is exempt from tax. Neither tax drives a wedge between r and [r.sub.n]. So why the difference in k?

The answer is the timing effect. Life-cycle timing differs under the two taxes: [T.sub.2]/([T.sub.1] + [T.sub.2]) = 0 for the W Tax but is always positive for the C Tax. Recall that the k under an Age W Tax is the same as the k under a W Tax and that the k under an Age C Tax is the same as the k under a C Tax. Thus, "when you pay"--the timing effect--completely accounts for the difference in k under these two taxes.

Suppose an R Tax is replaced by a W Tax. It might be thought that this shift would raise the k of the economy. In fact, it usually reduces k in Table 2. Why? Because a switch to a W Tax means the person pays tax sooner in life, and this shift in timing tends to lower k. For example, with [sigma] = 1, [rho] = 1, the R Tax k exceeds the W Tax k (1.000 vs. 0.816). Only in a few cases where [sigma] is sufficiently high is the reverse true (e.g., for [sigma] = 3 and [rho] = 1).

Similarly, suppose a Y Tax is converted to a W Tax. Once again, it might be thought that this tax shift would raise the k of the economy because it removes the tax on capital income. Once again, this is not what usually happens in Table 2 because a switch to a labor income tax means that a person pays tax earlier in life, and the earlier you pay, the lower the k. For example, with [sigma] = 1, [rho] = 1, the Y Tax k exceeds the W Tax k (0.852 vs. 0.816). Only in a few cases where [sigma] is sufficiently high is the reverse true (e.g., for [sigma] = 3 and [rho] = 1).

Now suppose there is a switch from a C Tax to an R Tax. Under the C Tax, [r.sub.n] = r, whereas under the R Tax, [r.sub.n] < r because the income that will be earned from saving is taxed. As a result, one might expect this wedge between [r.sub.n] and r to reduce k, and this is indeed the case for [sigma] > 1. However, when [sigma] is less than 1 in Table 2, the k under the R Tax is greater, not less, than the k under the C Tax. Why? Because the person pays later with an R Tax than with a C Tax: [T.sub.2]/([T.sub.1] + [T.sub.2]) = 100% for the R Tax but is always less than 100% for the C Tax. So the timing effect favors the R Tax, and it dominates the wedge effect for [sigma] < 1. For [sigma] = 1, the two effects offset each other, and the C Tax k equals the R Tax k. (8)

Similarly, a shift from a Y Tax to an R Tax increases the wedge between r and [r.sub.n] because a higher tax rate is required to achieve the same ratio of revenue to national income. Does k therefore decrease? Usually not in Table 2 because a switch to an R Tax means the later the person pays. For example, with [sigma] = 1.0 and [rho] = 1.0, the R Tax k is 1.000, and the Y Tax k is 0.852. Only in a few cases where [sigma] is sufficiently high is the reverse true (e.g., for [sigma] of 3 and a [rho] of 1).

We now address a major tax-reform proposal: converting the Y Tax to a C Tax. Conversion would remove the wedge between r and [r.sub.n]. But what about the timing of tax over the life cycle? In contrast to the pairs of taxes examined previously, it is not obvious which tax is a greater postponer, but it can be shown that the C Tax is: [T.sub.2]/[T.sub.1] is always greater for a C Tax than for Y Tax. (9) Hence, the timing effect favors the C Tax. In fact, it can be shown that = [k.sub.c]/[k.sub.y] = [[1/(1 - [t.sub.y])].sup.1]/(1-[alpha])] > 0, where [k.sub.c] is the k under the C Tax and [k.sub.y] is the k under the Y Tax. (10)

The right column of Table 2 shows the percentage of the increase in k that is due to the timing effect (%DTT). The increase in k can be decomposed into three steps. First, we replace the Y Tax with an Age Y Tax. This replacement removes the wedge between r and [r.sub.n] while keeping the same timing of tax; for all rows in Table 2, k increases. Second, we replace the Age Y Tax with an Age C Tax. These two age taxes differ solely in timing; for all rows in Table 2, k again increases. Third, we replace the Age C Tax with a C Tax, but, as noted earlier, this has no effect on k. The right column gives the ratio of the increase in k from the timing effect (step 2) to the total increase in k from converting the Y Tax to the C Tax (steps 1-3). We will call this ratio the "percentage due to timing (%DTT)." For example, as noted previously, with [sigma] = 1 and [rho] = 1, the %DTT is 72% ([1.0 - 0.893]/l.0 - 0.852] 0.72). With [sigma] = 0.5 and [rho] = 1, the %DTI' is 85% ([1.066 - 0.933]/1.066 - 0.908]=0.85). With th e exception of the very high values for both [sigma] and [rho], the %DTT is over 60%, and for the smallest value of [sigma], the %DTT is over 90%.

To gain further insight on the impact of timing, consider the four age taxes that correspond to the four conventional taxes. For an Age W Tax, [T.sub.2] = 0, and for an Age R Tax, [T.sub.1] = 0. For an Age Y Tax and an Age C Tax, [T.sub.1] > 0 and [T.sub.2] > 0. As noted previously, it can be proved that [T.sub.2]/[T.sub.1] under a C Tax is greater than [T.sub.2]/[T.sub.1], under a Y Tax. Thus, the ranking of age taxes by timing of payments over the person's life cycle, from soonest to latest, is W, Y, C, and R. In each row, the ranking of age taxes, from smallest k to highest k, is exactly the same: W, Y, C, and R. Hence, in Table 2 for the four age taxes, the later in life the person pays, the higher the k of the economy. (11)

Finally, if the R Tax is replaced by an Age R Tax, the switch keeps the same timing of tax but removes the wedge between r and [r.sub.n]. In Table 2, each such switch increases k. (12)

3. The Multiage Model

The multiage model in this section is the same as the two-age model except that it has multiple ages and introduces labor growth at rate n and labor-augmenting technical progress at rate g; also, the production function is y = [k.sup.[alpha]] where k is capital per effective labor and y is output per effective labor. (13) In the steady state, total effective labor, capital, and output grow at the discrete period rate (n + g + ng), and k and y are constant. The wage per effective labor [w.sub.e] equals the marginal product of effective labor. In the steady state, [w.sub.e] is constant. In any year v, each worker of any age receives the same wage w, the marginal product of a worker in the economy in year v. Since [w.sub.e] is constant, w grows at the rate g, both over time and over each person's work life. Each person works from 1 to R and retires from R + 1 to J. Each person chooses a consumption path [C.sub.t] (and a saving path [S.sub.t]) from t = 1, ..., J to maximize lifetime utility subject to the person's lifetime budget constraint. The utility function is isoelastic with an intertemporal elasticity of substitution [sigma] = 1/[gamma] and a subjective discount rate [rho]:

U = [summation over (J/1)] [[C.sup.1-[gamma]] - 1)/(1 - [gamma])]/[(1 + [rho]).sup.t] if [gamma] [not equal to] 1,

U = [summation over (J/1)] 1n [C.sub.t]/[(1 + [rho]).sup.t] if [gamma] = 1, (12)

where (1 + [rho]) > 0 and [gamma] > 0, so the intertemporal elasticity of substitution [sigma] = 1/[gamma] > 0.

One advantage of our multiage model over our two-age model is that the parameters are calibrated empirically. We present four values of the intertemporal elasticity of substitution a, ranging from 1.00 to 0.10, based on the values used by other researchers. (14) We also consider three alternative values of p: 0.000, 0.0 15, and 0.030 (note that p is an annual p in the multiage model in contrast to the roughly 30-year p of the two-age model). We set [alpha] = 0.3, n 1%, and g 1%. Individuals work from age 21 through 65 and retire from 66 through 75 so that in contrast to the two-age model, the length of retirement is shorter than the length of work.

Age Taxes

An age tax levies a particular tax at a particular age. We assume that the tax at each age, [T.sub.t], grows at rate g over time, just as the wage grows at rate g over time. The lifetime budget constraint becomes

[summation over (J/1)] [C.sub.t]/[(1 + r]).sup.t] = [summation over (R/1)] [w.sub.t]/[(1 + r).sup.t] - [summation over (J/1)] [T.sub.t]/[(1 + r).sup.t], (13)

where [T.sub.t], is the lump-sum tax at age t. The individual chooses the consumption path [C.sub.t], to maximize Equation 12 subject to Equation 13. Aggregating over all age cohorts yields a ratio of aggregate consumption to aggregate labor income, C/wL. Aggregate saving S = wL + rK - C - T, where T is the aggregate age tax. In the steady state, the growth rate of capital, S/K, equals the growth rate of effective labor (n + g + ng). Using these equations and assuming that each factor is paid its marginal product (15) so that [w.sub.e] = (1 - [alpha])[k.sup.[alpha]] and r = [alpha]/[k.sup.[alpha]] (where km [equivalent to] K/E), we show (in the Appendix) that the steady-state r under an age tax is given implicitly by r = {[[alpha]/(1 - [alpha])](r - n - g - ng)}/E[r, [sigma], [rho], n, g, R, J, [theta]], (14)

where r appears on both sides of Equation 14 and is the only endogenous variable in that equation, E is a function given in the Appendix, and [theta] is the vector [[theta].sub.t] from t = 1, ... , J, where [[theta].sub.t] is the ratio of the lump-sum age tax for a person age t to the wage in the economy, so [[theta].sub.t] [equivalent to]/w. Both [T.sub.t] and w grow at rate g over time, so [[theta].sub.t] is constant over time. Note that since [[theta].sub.t] varies with w and w varies with r (because both are functions of k), [[theta].sub.t] is a function of r. Given values for the parameters in Equation 14, r is obtained by an iterative procedure. Then k is obtained from r = [alpha]/[k.sup.1-[alpha]].

Setting [[theta].sub.1] = [[theta].sub.2] = ... [[theta].sub.J] = 0 in Equation 14 gives [r.sub.o]. Imposing an age tax to finance government consumption purchases generates an r that may be greater than, equal to, or less than [r.sub.0]. (16)

Table 3 presents the steady-state k under two sets of age taxes: single age and life phase. Under the single-age tax, each person pays a tax at a single age of life: 30, 50, and 70. (17) Tax revenue per effective labor is made the same for all three single-age taxes. Looking across each row at the single-age taxes, we see that the later in life the person pays the tax, the higher the k. Three life-phase taxes are also presented in Table 3: a tax at each age of work ("work"), a tax at each age of life ("entire"), and a tax at each age of retirement ("retire"). (18) Tax revenue per effective labor is made the same for all three life-phase taxes. Looking across each row at the life-phase taxes, we observe the same result:

The later in life the person pays the tax, the higher the k.

Conventional Transactions-Based Taxes

With transactions-based taxes, the lifetime budget constraint becomes

[summation over (J/1)][[C.sub.t]/(1 - [t.sub.c])]/[(1 + [r.sub.n]).sup.t] = [summation over (R/1)](1 - [t.sub.w])[w.sub.t]/[(1 + [r.sub.n]).sup.t], (15)

so the individual chooses the consumption path [C.sub.t] to maximize Equation 12 subject to Equation 15. It is shown in the Appendix that the steady-state [r.sub.n] is given implicitly by

[r.sub.n]/(1 - [t.sub.r]) = {[[alpha]/(1 - [alpha])]([r.sub.n] - n - g - ng)}/(1 - [t.sub.w])F[[r.sub.n],[sigma],[rho],n,g,R,J], (16)

where F is a function given in the Appendix. (19) Note that [t.sub.c] does not appear in Equation 16, just as it did not appear in Equation 11 in the two-age model; hence, [k.sub.c] is the same as [k.sub.o]. (20)

Table 4 presents values of k for three (W, Y, and C) transactions-based taxes and corresponding age taxes for combinations of [sigma] and [rho]. For each transactions-based tax, the corresponding age tax is a lump-sum tax that has the same ([T.sub.1], [T.sub.2],..., [T.sub.J]) as the tax. For each corresponding age tax, the steady-state r is obtained from Equation 14 by inserting the ([T.sub.1], [T.sub.2], ... , [T.sub.J]) generated by the transactions-based tax; then k is obtained from r = [alpha]/[[k.sup.1-[alpha]]. For example, corresponding to a Y Tax is an Age Y Tax--an age tax with the same ([T.sub.1], [T.sub.2] ..., [T.sub.J]) that results from that Y Tax. In the Appendix, it is shown that (as in the two-age model) the k under an Age W Tax is the same as the k under a W Tax and that the k under an Age C Tax is the same as the k under a C Tax. Hence, in Table 4, there is only one column under the wage tax and one column under the C Tax. On the other hand, since the k under an Age Y Tax differs from the k under a Y Tax (just as in the two-age model), there are two columns under the income tax. In the table, the income tax rate [t.sub.r] = [t.sub.w] = [t.sub.y] = 30%, and all other taxes achieve the same revenue per effective labor as this income tax.

We comment on several striking results in Table 4. As in the two-age model, the C Tax always achieves a higher k than the W Tax. Neither tax drives a wedge between r and [r.sub.n]. But when you pay--the timing effect--differs: More tax is paid earlier in life with the W Tax than with the C Tax. In every row of Table 4, k is higher under a C Tax than under a W Tax. When you pay--the timing effect--completely accounts for the difference in k under the two taxes. Moreover, k under the C Tax is usually much larger than k under the W Tax. For example, as presented in the "C Tax k/W Tax k" column, with [sigma] = 0.25 and [rho] = 0.015, the ratio of the C Tax k to the W Tax k is 1.54; the ratios range from 1.27 ([sigma] = 1.0, [rho] = 0.030) to 1.79 ([sigma] = 0.1, [rho] = 0.000).

Consider switching from a Y Tax to a W Tax. Since a person pays more tax sooner under the W Tax, the timing effect tends to make k larger under the Y Tax. In Table 4, as in Table 2 (with the two-age model), the Age Y Tax always achieves a higher k than the Age W Tax. In Table 2, the Y Tax usually achieves a higher k than the W Tax. But in Table 4, the reverse is true: The W Tax usually achieves a higher k than the Y Tax (the exceptions in Table 4 are for [sigma] = 0.1 and p = 0.000 or [rho] = 0.015). Thus, the timing effect is still important in Table 4, as comparison of two age taxes shows. However, despite the timing effect, k is usually larger under the W Tax. For example, as presented in the "W Tax k/Y Tax k" column, with [sigma] = 0.25 and [rho] = 0.015, the ratio of the W Tax k to the Y Tax k is 1.08; the ratio ranges from 0.93 ([sigma] = 0.1, [rho] = 0.000) to 1.31 ([sigma] = 1.0, [rho] = 0.030). Other researchers have also found that the W Tax generally achieves a larger k than the Y Tax (Summers 1981; Auerbach and K otlikoff 1987). Apparently, in an empirically calibrated multiage model, the incentive effect usually outweighs the timing effect with the result that the W Tax usually achieves a higher k than the Y Tax.

We now reconsider a major tax reform proposal addressed previously in the two-age model: converting the Y Tax to a C Tax. As noted previously, conversion removes the wedge between r and [r.sub.n], and more tax is paid later in life because consumption rises smoothly on retirement, while income drops abruptly. It can be shown that the ratio [k.sub.c]/[k.sub.y] = [[1/(1 - [t.sub.y])].sup.1/(1-[alpha])] > 0, the same formula as in the two-age model. (21)

The "%DTT" column of Table 4 shows the percentage of the increase in k due to the timing effect. As in the two-age case, this increase in k can be decomposed into three steps. First, we replace the Y Tax with an Age Y Tax. This replacement removes the wedge between r and [r.sub.n], while keeping the same timing of tax; for all rows in Table 4, k increases. Second, we replace the Age Y Tax with an Age Y Tax. These two age taxes differ solely in timing; for all rows in Table 4, k again increases. Third, we replace the Age C Tax with a C Tax, but, as noted earlier, this has no effect on k. The "%DTT" column gives the ratio of the increase in k from the timing effect (step 2) to the total increase in k from converting the Y Tax to the C Tax (steps 1-3). For the combinations of [sigma],[rho] in the table, the percentage due to timing is substantial, ranging from 30% ([sigma] = 1.0, [rho] = 0.030) to 68% ([sigma] = 0.1, [rho] = 0.000).

4. Conclusion

This paper investigates the conjecture that "the later you pay, the higher the k" and illustrates the importance of the age path (life-cycle timing) of any tax for the accumulation of capital (k) in the economy. Income, consumption, and wage taxes differ in their age paths as well as their incentive effects. This paper studies how the differing age path of each tax affects the capital accumulation of the economy. We analyze two life-cycle (overlapping generations) models: two age and multiage. The multiage model is empirically calibrated on the basis of recent econometric studies. In both models, the production function is Cobb-Douglas, the utility function is isoelastic, the government annually balances its budget, there is no government debt, and tax revenue finances government consumption purchases. We ignore the possibility of offsetting changes in debt policy because, in practice, legislators do not alter debt policy when they choose among alternative taxes. In the two-age model, labor is fixed, and ther e is no technical progress, while in the multiage model, there is labor growth and technical progress. In the two-age model, the ratio of tax revenue (which always equals government spending in each period) to output is the same under all alternative taxes. In the multiage model, tax revenue per effective labor (which always equals government spending per effective labor in each period) is the same under all alternative taxes.

We investigate lump-sum age taxes in both models. In the two-age model, we fix total tax revenue and vary how much tax the person pays in the work age versus the retirement age of the life cycle. We find in every case for a wide variety of parameter values that the later the person pays tax, the higher the k of the economy. In the multiage model, we fix tax revenue per effective labor and consider two sets of age taxes: single age and life phase. Under the former, each person pays a tax at a single age of life; under the latter, each person pays a tax at each age throughout one phase of life (e.g., work or retirement). For a set of plausible parameter values, we find in every case that the later the person pays tax, the higher the k of the economy.

To analyze the timing effect of conventional transactions-based taxes in both models, we replace each tax with a corresponding lump-sum age tax that has the identical age path of tax payments over the life cycle. By doing so, we can isolate the impact that the pure timing effect has on capital accumulation. In both models, we find that the timing effect is very important in explaining the differential impact of alternative taxes on the capital accumulation of the economy. For example, we find the following:

(i) In both models, we find that the C Tax always achieves a higher k than the W Tax solely because of the timing effect. In the multiage model, for an empirically plausible range of parameter values, the C Tax k ranges from 27% to 79% higher than the W Tax k.

(ii) Surprisingly, in the two-age model, a Y Tax often yields a higher k than a W Tax. However, in the multiage model, the Y Tax usually yields a lower k than the C Tax despite the timing effect, as other researchers have found; for an empirically plausible range of parameter values, the W Tax k ranges from 31% higher to 7% lower than the Y Tax k.

(iii) Also surprisingly, in the two-age model, an R Tax often yields a higher k than a W Tax and yields a higher k than a C Tax when the intertemporal elasticity of substitution is less than one.

(iv) In both models, a C Tax always achieves a higher k than a Y Tax. In the two-age model, for some parameter values, the difference is due mainly to the timing effect. In the multiage model, for an empirically plausible range of parameter values, the percentage of the increase in k due to the timing effect is substantial, ranging from 30% to 68%. Hence, roughly half the gap between the C Tax k and Y Tax k is due to the timing effect.

Although the impact of the age path of a tax on the capital accumulation of the economy has been noted by several other researchers, most of the literature continues to focus on the incentive effect of taxes. This paper shows that the timing effect often causes the impact of a tax on capital accumulation to be very different from what would be predicted from the incentive effect.

Appendix

An expanded Appendix is available on our Web site at http://www.buec.udel.edu/seidmanl.sejexpandedappendix.html.

1. Two-Age Model

We show that the corresponding Age W Tax yields the same steady-state r (and hence the same k) as the W Tax. With a W Tax, r = [r.sub.n], so from text Equation 11, the r with the W Tax, [r.sub.w], is given by [r.sub.w] = [[alpha]/(l - [alpha])]/(1 - [phi][[r.sub.w]])(1 - [t.sub.w]). Note that [r.sub.n] in Equation 11 has been replaced by [r.sub.w] which can be done because [t.sub.r] = 0, so [r.sub.n] = [r.sub.w]. Now impose an age tax [T.sub.1] = [t.sub.w][w.sub.w], where [w.sub.w] is the wage under the W Tax and [T.sub.2] = 0. The r for this age tax is given by text Equation 6: r = [[alpha]/(1 - [alpha])]/(1 - [phi])(1 - [[theta].sub.1]). We now show that [r.sub.w], the r under the W Tax, satisfies Equation 6 so that the Age W Tax generates the same r as the W Tax. We test whether [r.sub.w] satisfies Equation 6: On the right side of Equation 6, we replace r by [r.sub.w]; r = [r.sub.w] implies that w = [w.sub.w] so that [[theta.sub.1] [equivalent to] [T.sub.1]/w = [T.sub.1]/[w.sub.w] = [t.sub.w], so we replace [[theta].sub.1] by [t.sub.w]. Then the right side of Equation 6 becomes [[alpha]/(1 - [alpha])]/(1 - [phi][[r.sub.w]])(1 - [t.sub.w]), which according to Equation 11 is equal to [r.sub.w]. Hence, [r.sub.w] does satisfy Equation 6, and the Age W Tax yields the same r as the W Tax.

In the expanded Appendix, we show that the corresponding Age C Tax yields the same steady-state r (and hence the same k) as the C Tax. We also show, by contrast, that the corresponding Age Y Tax does not yield the same steady-state r (and hence the same k) as the Y Tax because r [not equal to] [r.sub.n].

By an iterative procedure, we find different r's for the Y Tax and the corresponding Age Y Tax. We impose a lump-sum age tax that matches this Y Tax and compute r from Equation 6. Under the income tax steady state, a worker's tax [T.sub.1] equals [t.sub.y][w.sub.y], and a retiree's tax [T.sub.2] equals [t.sub.y][r.sub.y][k.sub.y]. To compute r from Equation 6, we need [[theta].sub.1] and [[theta].sub.2]; [[theta].sub.1] [equivalent to] [T.sub.1]/[w.sub.a], where [w.sub.a] is the age tax w and [[theta].sub.2] [equivalent to] [T.sub.2]/[w.sub.a]. Since we do not yet know [w.sub.a], for the initial iteration we compute [[theta].sub.t] by using the income tax [w.sub.y]; inserting this [[theta].sub.t] into Equation 6 yields an initial r; then we obtain an initial k from r = [alpha]/[k.sup.1-[alpha]] and a new [w.sub.a] = (1 - [alpha])[k.sup.[alpha]]. We recompute [[theta].sub.1] and [[theta].sub.2] using this [w.sub.a] and repeat this procedure until Equation 6 generates approximately the same [w.sub.a] used to co mpute [[theta].sub.1] and [[theta].sub.2].

2. Multiage Model

Let [M.sup.U.sub.L][X] [equivalent to] [summation over (U/L)] [X.sup.i] = ([X.sup.U+1] - [X.sup.L])/(X - 1) for X [not equal to] 1 and [M.sup.U.sub.L][X] = U + 1 - L for X = 1. The production function is y = [k.sup.[alpha]], where y [equivalent to] Y/E, k [equivalent to] K/E, and E is effective labor.

Age Taxes

We now derive text Equation 14, where E[r, [sigma], [rho], n, g, R, J, [theta]] [equivalent to] {[{[M.sup.R.sub.1][(1 + g)/(1 + r)] - [summation over (J/1)] [[theta].sub.t] [[(1 + g)/(1 + r)].sup.t]}[M.sup.J.sub.1][z]/{[M.sup.J.sub.1][[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]][M.sup.R.sub.1][1/(1 + n)]}] + {[[summation over (J/1)] [[theta].sub.t]/[(1 + n).sup.t]]/[M.sup.R.sub.1][1/(1 + n)]} - 1}, where z [equivalent to] [[(1 + r)/(1 + [rho])].sup.[sigma]]/(1 + n) (1 + g), and [[theta].sub.t] [equivalent to] [T.sub.t]/w = [T.sub.t][0][(1 + g).sup.v]/[w.sub.e][(1 + g).sup.v] = [T.sub.t][0]/[w.sub.e] = [T.sub.t][0]/(1 - [alpha])[([alpha]/r).sup.[alpha]/(1 - [alpha])], where [T.sub.t][0], the lump-sum age tax for a person age t in year 0, is a constant and [[theta].sub.t] = [[theta].sub.t][r] = [T.sub.t][0]/(1 - [alpha]) [([alpha]/r).sup.[alpha]/(1-[alpha])]. Each person chooses a path [C.sub.t] (t = 1,..., J) to maximize Equation 12 subject to Equation 13:

[U.sup.*] = [summation over (J/1)] [([C.sup.1-[gamma].sub.t] - 1)/(1 - [gamma])]/[(1 + [rho]).sup.t] + [lambda] {[summation over (R/1)][w.sub.t]/[(1 + r).sup.t] - [summation over (J/1)][T.sub.t]/[(1 + r).sup.t] - [summation over (J/1)] [C.sub.t]/[(1 + r).sup.t]}, [partial][U.sup.*]/[partial][C.sub.t] = [[C.sup.-[gamma].sub.t]/[(1 + [rho]).sup.t]] - [[lambda]/[(1 + r).sup.t]] = 0, so [C.sub.t] = [(1/[lambda]).sup.[sigma]] [[(1 + r)/(1 + [rho])].sup.[sigma]t] . [[theta].sub.t] [equivalent to] [T.sub.t]/w, so [T.sub.t] = [[theta].sub.t] [w.sub.0] [(1 + g).sup.t].

Substituting in Equation 13 for [C.sub.t] and [T.sub.t] yields [(1/[lambda]).sup.[sigma]] = [w.sub.0] {[M.sup.R.sub.l][(1 + g)/(1 + r)] - [summation over (J/1)] [[theta].sub.t][[(1 + g)/(1 + r)].sup.t]}/[M.sup.J.sub.l][[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]], so

[C.sub.t] = [w.sub.0]{[M.sup.R.sub.l][(1 + g)/(1 + r)] - [summation over (J/1)][[theta].sub.t][[[(1 + g)/(1 + r)].sup.t]} [[(1 + r)/1 + [rho])].sup.[sigma]t]/[M.sup.J.sub.l][[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]].

In year v for a person of age t, [w.sub.0] = [w.sub.e][(1 + g).sup.v-t], where [w.sub.e] is the wage per effective labor; in year v, [C.sub.t](v) = [w.sub.e][(1 + g).sup.v-t] {[M.sup.R.sub.l][(1 + g)/(1 + r)] - [summation over (J/1)] [[theta].sub.t] [[(1 + g)/(1 + r)].sup.t]}[[(1 + r)/(1 + [rho]).sup.[sigma]t]/[M.sup.J.sub.l][(1 + r)].sup.[sigma]-1/ - (1 + [rho]).sup.[sigma]]. In year v, the population age 0 is [(1 + n).sup.v], so that the population age t is [(1 + n).sup.v-t]. The consumption of the population age t in year v is [(1 + n).sup.v-t] [C.sub.t](v), so aggregate consumption in year v is C(v) = [(1 + n).sup.v] [w.sub.e][(1 + g).sup.v] [M.sup.R.sub.1)] [(1 + g)/(1 + r)] - [summation over (J/1)] [[theta].sub.t][[(1 + g)/(1 + r)].sup.t]} [M.sup.J.sub.1][z]/[M.sup.J.sub.1])[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]], where z [equivalent to][[(1 + r)/(1 + [rho])].sup.[sigma]]/(1 + n)(1 + g). In year v, aggregate labor in income w(v )L(v) equals [summation over (R/1)] [w.sub.e] [(1 + g).sup.v][(1 + n).sup.v-t], or w(v)L(v) = [w.sub.e][(1 + g).sup.v][(1 + n).sup.v] [M.sup.R.sub.1] [1/(1 + n)], so C(v)/w(v)L(v) equals

C/wL = {[M.sup.R.sub.1][(1 + g)/(1 + r)] - [summation over (J/1)][[theta].sub.t][[(1 + g)/(1 + r)].sup.t]} [M.sup.J.sub.1][z]/{[M.sup.J.sub.1][[(1 + r).sup.[sigma]-1] / [(1 + [rho]).sup.[sigma]]] [M.sup.R.sub.1][1/(1 + n)]}, (A1)

Which is independent of v. Aggregate saving is S [equivalent to] wL + rK - C - T, where T is the aggregate age tax; in the steady state. S/K = (n + g + ng), so (r - n - g - ng)K = C + T - wL, so

K/wL = [(C/wL) + (T/wL) - 1]/(r - n - g - ng). (A2)

We obtain T/wL as follows. The age tax of a person age t in year v is [T.sub.t](v) = [[theta].sub.t](v) = [[theta].sub.t][w.sub.e][(1 + g).sup.v]. The aggregate age tax T(v) = [summation over (J/1)] [(1 + n).sup.v-t][[theta].sub.t][w.sub.e][(1 + g).sup.v] = [w.sub.e] [(1 + g).sup.v][(1 + n).sup.v] [summation over (J/1)] [[theta].sub.t]/[(1 + n).sup.t] = {w(v)L(v)/[M.sup.R.sub.1][1/(1 + n)]}[summation over (J/1)] [[theta].sub.t]/[(1 + n).sup.t] because L(v) = [summation over (R/1)][(1 + n).sup.v-t] = [(1 + n).sup.v][M.sup.R.sub.1][1/(1 + n)], so [(1 + n).sup.v] = L(v)/[M.sup.R.sub.1][1/(1 + n)]. Dividing T(v) by w(v)L(v) yields

T/wL = [[summation over (j/1)][[theta].sub.t]/[(1 + n).sup.t]]/[M.sup.R.sub.1][1/(1 + n)], (A3)

which is independent of v. Let E be effective labor. K/wL = K/[w.sub.e]E = [k/[w.sub.e]. Since y = [k.sup.[alpha]], [w.sub.e] = [partial]Y/[partial]E = (1 - [alpha])[k.sup.[alpha]] and [partial]Y/[partial]r = [alpha]/[k.sup.1-[alpha]]. Thus

K/wL = [alpha]/(1 - [alpha])r. (A4)

Substituting Equations A1, A3, and A4 into Equation A2 yields Equation 14. Then r is obtained from Equation 14 by an iterative procedure, and k is obtained from r = [alpha]/[k.sup.1-[alpha]] and [w.sub.e] = (1 - [alpha])[k.sup.[alpha]].

In the expanded appendix, we show how to se [[theta].sub.t] in Equation 14 to obtain the numerical value of k for the particular age taxes presented in Table 3.

Conventional Transactions-Based Taxes

We now derive text Equation 16, where F[[r.sub.n], [sigma], [rho], n, g, R, J] [equivalent to] [{[M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n])][M.sup.J.sub.1][[(1 + [r.sub.n]).sup.[sigma]-1]/ [(1 + [rho]).sup.[sigma]]][M.sup.R.sub.1][1/(1 + n)]} - 1], where z [equivalent to] [(1 + [r.sub.n])/[(1 + [rho])].sup.[sigma]]/(1 + n)(1 + g). Each person chooses a path [C.sub.t](t = 1, ..., J) to maximize Equation 12 subject to Equation 15: [U.sup.*] = [summation over (J/1)][([C.sup.1-[gamma].sub.t] - 1)/(1 - [gamma])]/[(1 + [rho]).sup.t] + [lambda]{[summation over (R/1)](1 - [t.sub.w])[w.sub.t]/[(1 + [r.sub.n]).sup.t] - [summation over (J/1)][[C.sub.t]/(1 - [t.sub.c])]/[(1 + [r.sub.n]).sup.t]}, [partial][U.sup.*]/[partial][C.sub.t] = [C.sup.-[gamma].sub.t]/[(1 + [rho]).sup.t]] - ([[lambda]/(1 - [t.sub.c])[(1 + [r.sub.n]).sup.t]] = 0, so [C.sub.t]/(1 - [t.sub.c]) = [(1/[lambda]).sup.[sigma]] [(1 - [t.sub.c]).sup.[sigma]-1][[(1 + [r.sub.n])/[(1 + [rho])].sup.[sigma]t]. Substituting in Equation 15 for [C.sub.t]/(1 - [t.sub.c]), [(1/[lambda]).sup.[sigma]] = (1 - [t.sub.w])[w.sub.0][M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n])]/[(1 - [t.sub.c]).sup.[sigma]-1][M.sup.J.sub.1][[(1 + [r.sub.n]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]], so [C.sub.t]/(1 - [t.sub.c]) = (1 - [t.sub.w])[w.sub.0]{[M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n])]}[(1 + [r.sub.n])/[(1 + [rho])].sup.[sigma]t]/[M.sup.J.sub.1][[(1 + [r.sub.n]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]]]. In year v for a person of age t, [w.sub.0] = [w.sub.e][(1 + g).sup.v-t], where [w.sub.e] is the wage per effective labor, so in year v, [C.sub.t](v)/(1 - [t.sub.c]) = (1 - [t.sub.w])[w.sub.e][(1 + g).sup.v-t]{[M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n])]}[[(1 + [r.sub.n])/(1 + [rho])].sup.[sigma]t]/[M.sup.J.sub.1][[(1 + [r.sub.n]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]]]. The consumption of the population age t in year v is [(1 + n).sup.v-t][C.sub.t](v), so aggregate consumption plus consumption tax in year v is C(v)/(1 - [t.sub.c]) = [(1 + n).sup.v](1 - [t.sub.w])[w.sub.e][(1 + g).sup.v] [M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n])][M.sup.J.sub.1][Z]/[M.sup.J.sub.1][[(1 + [r.sub.n]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]]. In year v, aggregate net labor income (1 - [t.sub.w])w(v)L(v) equals [summation over (R/1)](1 - [t.sub.w])[w.sub.e][(1 + g).sup.v][(1 + n).sup.v-t], or (1 - [t.sub.w])w(v)L(v) = (1 - [t.sub.w])[w.sub.e][(1 + g).sup.v][(1 + n).sup.v][M.sup.R.sub.1][1/(1 + n)]; then [C(v)/(1 - [t.sub.c])]/(1 - [t.sub.w]w(v)L(v) turns out to be independent of v and equals

[C/(1 - [t.sub.c])]/(1 - [t.sub.w])wL = [M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n)][[M.sup.J.sub.1][z]/[M.sup.J.sub.1][[(1 + [r.sub.n]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]][M.sup.R.sub.1][1/(1 + n)]. (A5)

Aggregate saving is S [equivalent to] (1 - [t.sub.w])wL + [r.sub.n]K - [C/(1 - [t.sub.c])]; in the steady state, S/K = ( n + g + ng), so ([r.sub.n] - n - g - ng)K = [C/(1 - [t.sub.c])] (1 - [t.sub.w])wL, so

K/(1 - [t.sub.w])wL = [{[C/(1 - [t.sub.c])]/(1 - [t.sub.w])wL} - 1]/([r.sub.n] - n - g - ng). (A6)

Let E be effective labor. K/(1 - [t.sub.w])wL = K/(1 - [t.sub.w])[w.sub.e]E = k/(1 - [t.sub.w])[w.sub.e]; since [w.sub.e] = (1 - [alpha])[k.sup.[alpha]] and r = [alpha]/[k.sup.1-[alpha]],

K/(1 - [t.sub.w])wL = [alpha](l - [t.sub.r])/(1 - [t.sub.w])(1 - [alpha])[r.sub.n]. (A7)

Substituting Equations A5 and A7 into Equation A6 yields Equation 16. Then [r.sub.n] is obtained from Equation 16 by an iterative procedure.

Corresponding Age Taxes

We show that the corresponding Age W Tax yields the same steady-state r (and hence the same k) as the W Tax. With a W Tax, r = [r.sub.n], so from text Equation 16, the r with the W Tax, [r.sub.w] = [[[alpha]/(1 - [alpha])]([r.sub.w] - n - g - ng)]/(1 - [t.sub.w])F[[r.sub.w], [sigma], [rho], n, g, R, J], where F[[r.sub.w], [sigma], [rho], n, g, R, J] is given previously. Note that [r.sub.n] in Equation 16 has been replaced by [r.sub.w], which can be done because [t.sub.r] = 0, so [r.sub.n] = [r.sub.w]. Now impose an age tax [T.sub.t] = [t.sub.w][w.sub.ew][(1 + g).sup.t] for 1 [less than or equal to] t [less than or equal to] R and [T.sub.t] = for t [greater than or equal to] R + 1, where [w.sub.ew] is the wage per effective labor under the W Tax. The r for this age tax is given by text Equation 14:r = {[[alpha]/(1 - [alpha])](r - n - g - ng)}]/E[r, [sigma], [rho], n, g, R, J, [theta]], where E[r, [sigma], [rho], n, g, R, J, [theta]] is given previously. We now show that [r.sub.w] satisfies Equation [4 so that the Age W Tax generates the same r as the W Tax. We test whether [r.sub.w] satisfies Equation 14. On the right side of Equation 14 we replace r by [r.sub.w]; r = [r.sub.w] implies that w = [w.sub.w] so that [[theta].sub.t] [equivalent to] [T.sub.t]/w = [T.sub.t]/[w.sub.w] = [t.sub.w] for 1 [less than or equal to] t [less than or equal to] R and 0 for t [greater than or equal to] R + 1. Then the right side of Equation 14 becomes

{[[alpha]/(1 - [alpha])](r - n - g - ng)}/{[M.sup.R.sub.1][(1 + g)/(1 + r)] - [summation over (R/1)][t.sub.w][[(1 + g)/(1 + r)].sup.t]}

X [M.sup.J.sub.1][z]/[M.sup.J.sub.1][[(1 + r).sup.[sigma]-1]]/[(1 + [rho]).sup.[sigma]][M.sup.R.sub.1][1/(1 + n)]}] + {[[summation over (R/1)][t.sub.w]/[(1 + n)].sup.t]]/[M.sup.R.sub.1][1/(1 + n)]} - 1},

= {[[alpha]/(1 - [alpha])](r - n - g - ng)}/{[(1 - [t.sub.w])[M.sup.R.sub.1][(1 + g)/(1 + r)][M.sup.J.sub.1][z]/{[M.sup.J.sub.1][[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]][M.sup.R.sub.1][1/(1 + n)]}]

+ {[t.sub.w][M.sup.R.sub.1][1/(1 + n)]/[M.sup.R.sub.1][(1 + n)]} - 1},

= {[[alpha]/(1 - [alpha])](r - n - g - ng)}/(1 - [t.sub.w])[{[M.sup.R.sub.1][(1 + g)/(1 + r)][M.sup.J.sub.1][z]/{[M.sup.J.sub.1][[(1 + r).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]][M.sup.R.sub.1][1/(1 + n)]} - 1],

= {[[alpha]/(1 - [alpha])](r - n - g - ng)}/(1 - [t.sub.w])F[[r.sub.w], [sigma], [rho], n, g, R, J],

which according to Equation 16 is equal to [r.sub.w]. Hence, [r.sub.w] does satisfy Equation 14, and the Age W Tax yields the same r as the W Tax.

In the expanded Appendix, we show that the corresponding Age C Tax yields the same steady-state r (and hence the same k) as the C Tax. We also show, by contrast, that the corresponding Age Y Tax does not yield the same steady-state r (and hence the same k) as the Y Tax because r [not equal to] [r.sub.n].

By an iterative procedure, we find different r's for the Y Tax and the corresponding Age Y Tax. We impose a lump-sump age tax that matches this Y Tax path and compute r from Equation 14. Under the Y Tax steady state, a person's tax at age t equals [T.sub.t] = [t.sub.y] [Y.sub.t]. For a person age 0 in year 0, [w.sub.0] = [w.sub.e], so the Y tax path is [T.sub.t] = [t.sub.y] [Z[w.sub.e][(1 + g).sup.t]+r[A.sub.t]], where Z = 1 if t [less than or equal to] R and Z = 0 if t [greater than or equal to] R + 1. We generate [T.sub.t] as follows. [A.sub.1] = 0, [A.sub.t+1] = [A.sub.t] + [S.sub.t], [S.sub.t] = Z(1 - [t.sub.y])[w.sub.e][(1 + g).sup.t] + [r.sub.n][A.sub.t] - [C.sub.t], where [C.sub.t] = (1 - [t.sub.y])[w.sub.e]([M.sup.R.sub.1][(1 + g)/(1 + [r.sub.n])]][[(1 + [r.sub.n])/(1 + [rho])].sup.[sigma]t]/[M.sup.J.sub.l][[1 + [r.sub.n]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]]]. To compute r from Equation 14, we need the vector [theta], where [[theta].sub.t] = [T.sub.t]/[w.sub.ea][(1 + g).sup.t], [T.sub.t] is the lump-s um age tax, and [w.sub.ea] is the age tax [w.sub.e]. Since we do not yet know [w.sub.ea], for the initial iteration we compute [[theta].sub.t] by using the income tax [w.sub.ey], so initially [[theta].sub.t] = [T.sub.t]/[w.sub.ey][(1 + g).sup.t] inserting this [[theta].sub.t] into Equaltion 14 yields an initial r; then we obtain an initial k from r = [alpha]/[k.sup.1-[alpha]] and a new [w.sub.ea] = (1 - [alpha])[k.sup.[alpha]]. We recompute [[theta].sub.t] using this [w.sub.ea] and repeat this procedure until Equaltion 14 generates approximately the same [w.sub.ea] used to compute [[theta].sub.t] from [T.sub.t].

Table 1

k for Various Lump-Sum Age Taxes for the Two-Age Model (a)

                           [T.sub.2] asa Percentage of
                                    Total Tax
[sigma]   [rho]       0%       25%       50%       75%       100%

3.0       -0.5      2.928     3.109     3.287     3.464     3.636
           0.0      1.954     2.084     2.214     2.344     2.475
           1.0      0.764     0.822     0.882     0.945     1.010
           2.0      0.409     0.441     0.476     0.514     0.554
1.0       -0.5      1.923     2.080     2.235     2.390     2.542
           0.0      1.342     1.475     1.607     1.740     1.872
           1.0      0.808     0.911     1.014     1.119     1.225
           2.0      0.564     0.647     0.732     0.820     0.9 1
0.5       -0.5      1.446     1.598     1.746     1.893     2.039
           0.0      1.128     1.266     1.403     1.539     1.673
           1.0      0.830     0.953     1.076     1.198     1.320
           2.0      0.673     0.786     0.900     1.013     1.127
0.0625    -0.5      0.941     1.091     1.238     1.381     1.522
           0.0      0.898     1.048     1.193     1.335     1.473
           1.0      0.856     1.003     1.147     1.288     1.425
           2.0      0.831     0.978     1.121     1.260     1.397

(a)[alpha] = 0.2, m = 3.75.

Table 2

k for Various Taxes for the Two-Age Model (a)

                          W Tax                       Y Tax
                    ([t.sub.w] = 15%)     ([t.sub.w] = [t.sub.r] = 12%)
[sigma]   [rho]             k            k                 Age k

3.0       -0.5            2.953        3.065               3.088
           0.0            1.966        1.987               2.066
           1.0            0.767        0.739               0.812
           2.0            0.409        0.387               0.435
1.0       -0.5            1.941        2.027               2.059
           0.0            1.355        1.415               1.455
           1.0            0.816        0.852               0.893
           2.0            0.570        0.595               0.632
0.5       -0.5            1.463        1.554               1.575
           0.0            1.141        1.221               1.244
           1.0            0.841        0.908               0.933
           2.0            0.683        0.743               0.767
0.0625    -0.5            0.957        1.064               1.067
           0.0            0.914        1.020               1.023
           1.0            0.871        0.976               0.980
           2.0            0.847        0.951               0.954

                          C Tax                 R Tax
                    ([t.sub.c] = 12%)     ([t.sub.r] = 60%)
[sigma]   [rho]             k            k         Age k

3.0       -0.5            3.596        3.499       3.636
           0.0            2.331        1.994       2.470
           1.0            0.867        0.570       0.997
           2.0            0.454        0.264       0.542
1.0       -0.5            2.378        2.378       2.542
           0.0            1.660        1.660       1.872
           1.0            1.000        1.000       1.225
           2.0            0.698        0.698       0.910
0.5       -0.5            1.823        1.938       2.042
           0.0            1.433        1.562       1.678
           1.0            1.066        1.204       1.326
           2.0            0.872        1.011       1.134
0.0625    -0.5            1.248        1.517       1.533
           0.0            1.197        1.469       1.485
           1.0            1.145        1.421       1.437
           2.0            1.115        1.394       1.410


                            %DTT
[sigma]   [rho]     Y [right arrow] C tax

3.0       -0.5               96%
           0.0               77%
           1.0               43%
           2.0               28%
1.0       -0.5               91%
           0.0               84%
           1.0               72%
           2.0               64%
0.5       -0.5               92%
           0.0               89%
           1.0               85%
           2.0               81%
0.0625    -0.5               98%
           0.0               98%
           1.0               98%
           2.0               98%

(a)[alpha] = 0.2, m = 3.75. Percentage due to timing (%DTT) [equivalent
to] [{[DELTA] Age Tax k/[DELTA] Transactions-Based Tax k}.sup.*] 100.

Table 3

k for Various Single-Age and Life-Phase Taxes for the Multiage Model
(a)

[alpha]    p    Age 30  Age 50  Age 70   Work   Entire  Retire

1.0      0.000  17.102  20.115  22.285  18.099  18.310  19.748
         0.015  12.799  14.833  15.960  13.631  13.697  14.799
         0.030   9.659  11.035  11.625  10.232  10.364  11.084
0.5      0.000   9.620  11.838  12.920  10.527  10.725  11.875
         0.015   7.672   9.364  10.065   8.395   8.543   9.411
         0.030   6.177   7.454   7.878   6.717   6.827   7.454
0.25     0.000   4.770   6.392   6.819   5.441   5.569   6.406
         0.015   4.017   6.392   6.819   5.441   5.569   6.406
         0.030   3.417   4.434   4.613   3.809   3.881   4.378
0.1      0.000   1.549   2.127   2.165   1.659   1.705   2.015
         0.015   1.412   1.876   1.899   1.482   1.521   1.775
         0.030   1.293   1.668   1.681   1.334   1.366   1.578

(a)[alpha] = 0.3, n = 1%, g = 1%, Retire after age 65, end retirement at
age 75, [t.sub.y] = 30%.

Table 4

k for Various Taxes for the Multiage Model (a)

                    W Tax          Y Tax           C Tax
[sigma]     p         k         k        Age k       k

1.0       0.000     13.041    11.241    15.161    18.711
          0.015     10.416     8.411    11.856    14.000
          0.030      8.253     6.297     9.228    10.482
0.5       0.000      7.364     6.579     8.886    10.950
          0.015      6.128     5.218     7.274     8.686
          0.030      5.107     4.156     5.956     6.918
0.25      0.000      3.532     3.422     4.549     5.695
          0.015      3.066     2.840     3.852     4.727
          0.030      2.679     2.376     3.280     3.955
0.1       0.000      0.980     1.052     1.274     1.751
          0.015      0.908     0.936     1.151     1.559
          0.030      0.844     0.840     1.048     1.399

                    %DTT
[sigma]     p       Y [right arrow]C tax  C Tax k/W Tax k

1.0       0.000             47%                1.43
          0.015             38%                1.34
          0.030             30%                1.27
0.5       0.000             47%                1.49
          0.015             41%                1.42
          0.030             35%                1.35
0.25      0.000             50%                1.61
          0.015             46%                1.54
          0.030             43%                1.48
0.1       0.000             68%                1.79
          0.015             66%                1.75
          0.030             63%                1.66


[sigma]     p       W Tax k/Y Tax k

1.0       0.000          1.16
          0.015          1.24
          0.030          1.31
0.5       0.000          1.12
          0.015          1.17
          0.030          1.23
0.25      0.000          1.03
          0.015          1.08
          0.030          1.13
0.1       0.000          0.93
          0.015          0.97
          0.030          1.00

(a)[alpha] = 0.3, n = 1%, g = 1%, Retire after age 65, end retirement at
age 75, [t.sub.y] = 30%. Percentage due to timing (%DTT) = {[DELTA] Age
Tax k/[DELTA] Transactions-Based Tax k}*100.

Received January 2001; accepted May 2002.

(1.) [U.sup.*] = [([C.sup.1 - [gamma].sub.1] - 1)/(1 - [gamma])] [([C.sup.1 - [gamma].sub.2] - 1)/(1 - [gamma])]/(1 - [rho]) + [lambda]{w - [T.sub.p] - [C.sub.1] - [[C.sub.2]/(1 + r)]; then [partial][U.sub.*]/[partial][C.sub.1] = [C.sup.-[gamma].1] - [lambda] = 0, so [C.sup.-[gamma].sub.1] = [lambda], [partial][U.sub.*]/[partial][C.sub.2] = [C.sup.-[gamma].sub.2]/(1 + [rho]) - [lambda]/(1 + r) = 0, so [C.sup.-[gamma].2] = [lambda](1 + [rho])/(1 + r); hence, ([C.sub.2]/[C.sub.1]) = [[(1 + r)/(1 + [pho])].sup.[sigma]] where [sigma] = 1/[gamma].

(2.) y = [mk.sup.[alpha]] implies Y = [mK.sup.[alpha]][L.sup.1-[alpha]], where Y is output, K is capital, and L is labor, so MPK [equivalent to] [partial]Y/[partial]K [equivalent to] [alpha]m/[k.sup.1-[alpha]] and MPL [equivalent to] [partial]Y/[partial]L = (1 - [alpha])[mk.sup.[alpha]].

(3.) We construct a FORTRAN program to calculate r for the two-age (and later the multiage) model. The algorithm is as follows. We set r on the right-hand side of Equation 6 and then solve for the r on the left-hand side. Based on the sign of the difference of the two r's, we raise or lower the right-hand side r by a very small increment and then resolve Equation 6 for the left-hand side r. We continue to iterate until the difference between the r's disappears within a very small tolerance of five to six decimal places. The results are insensitive to a tighter tolerance. We checked and found that the results are not materially sensitive to the starting value of the right-hand-side r.

(4.) From footnote 1, [([C.sub.2]/[C.sub.1]).sub.o] = [[(1 + [r.sub.o])/(1 + [rho])].sup.[sigma]]. Suppose that an age tax ([T.sub.1], [T.sub.2]) is introduced such that [T.sub.2]/[T.sub.1] = [([C.sub.2]/[C.sub.1]).sub.o], so [[theta].sub.2]/[[theta].sub.1] [equivalent to] ([T.sub.2]/w)/([T.sub.1]/w) = [T.sub.2]/[T.sub.1] = [[(1 + [r.sub.o])/(1 + [rho])].sup.[sigma]]. Replacing [[theta].sub.2] by [[theta].sub.1][[(1 + [r.sub.o])/(1 + [rho])].sup.[sigma]] in Equation 6 yields r = [[alpha]/(1 - [alpha])]/{(1 - [phi])(1 - [[theta].sub.1]) + [[phi] [[theta].sub.1][[(1 + [r.sub.o])/(1 + [rho])].sup.[sigma]]/(1 + r)]}. We now show that r = [r.sub.o] satisfies this equation. If r = [r.sub.o], then if we replace [phi] by 1/{1 + [[(1 + [r.sub.o]).sup.[sigma]-1]/[(1 + [rho]).sup.[sigma]]]} in the second term in the denominator, that term reduces to (1 - [phi][[r.sub.o]])[[theta].sub.1], so the whole denominator reduces to 1 - [phi][[r.sub.o]]. But in the text we saw that [r.sub.o] = [[alpha]/(1 - [alpha])]/(1 - [phi][[ r.sub.o]]), so r = [r.sub.o] satisfies this equation, and the r for this age tax equals [r.sub.o]. A plausible conjecture is the following: If [T.sub.2]/[T.sub.1] > [([C.sub.2]/[C.sub.1]).sub.o], then k > [k.sub.o], whereas if [T.sub.2]/[T.sub.1] < [([C.sub.2]/[C.sub.1]).sub.o], then k < [k.sub.o]. We return to this conjecture later in this paper.

(5.) In each row, the constant sum [T.sub.1] + [T.sub.2] equals the sum under a consumption tax with a tax rate of 12% with the ([sigma], [rho]) for that row. Thus, the sum is constant across any given row but differs among rows because ([sigma], [rho]) differs.

(6.) Since [t.sub.c] [equivalent to] T/(C + T), T = [t.sub.c]C/(1 - [t.sub.c]), so C + T = C/(1 - [t.sub.c]).

(7.) [U.sup.*] = [[(C.sup.1 - [gamma].sub.1] -- 1)/(1 -- [gamma])] + [[(C.sup.1 - [gamma].sub.2] -- 1)/(1 -- [gamma])]/(1 + [rho]) + [lambda]{(1 -- [t.sub.w])w -- [C.sub.1]/(1 -- [t.sub.c])] -- [[C.sub.2]/(1 -- [t.sub.c])(1 + [r.sub.n])]}; then [partial][U.sup.*]/[partial][C.sub.1] = [C.sup.-[gamma].sub.1] -- [lambda]/(1 -- [t.sub.c]) = 0, so [C.sup.-[gamma].sub.1] = [lambda]/(1 -- [t.sub.c]), and [partial][U.sup.*][partial][C.sub.2] = [C.sup.- [gamma].sub.2]/(1 + [rho]) -- [lambda]/(1 -- [t.sub.c])(1 + [r.sub.n]) = 0, so [C.sup.- [gamma].sub.2] = [lambda](1 + [rho])/(1 -- [t.sub.c])(1 + [r.sub.n]); hence, ([C.sub.2]/[C.sub.1]) = [[(1 + [r.sub.n])/(1 + [rho)]].sup.[sigma], where [sigma] [equivalent to] 1/[gamma].

(8.) If [sigma] = 1, then [phi] = 1/{1 + [1/[(1 + [rho]).sup.[sigma]]]}, so [t.sub.r] has no effect on the right side of Equation 11; hence, the left side of Equation 11, r = [r.sub.n]/(1 - [t.sub.r]), is independent of [t.sub.r]. Thus, [k.sub.r] = [k.sub.c] = [k.sub.o].

(9.) Under a Y Tax, [T.sub.2]/[T.sub.1] = [alpha]/(1 - [alpha]), the ratio of the share of capital income to the share of labor income. Under a C Tax, [T.sub.2]/[T.sub.1] = [C.sub.2]/[C.sub.1]. With a C Tax or with no tax, [C.sub.2]/[C.sub.1] [[(1 + r)/(1 + [rho])].sup.[sigma]]; because the r with a C Tax is equal to the r with no tax, [T.sub.2]/ [T.sub.1] = [([C.sub.2]/[C.sub.1].sub.o]. It can be proved that [([C.sub.2]/[C.sub.1]).sub.o] > [alpha](l - [alpha]). The proof is as follows. With no tax, [C.sub.2] (1 + [r.sub.o])[S.sub.1] = (1 + [r.sub.o])(1 - [phi])w (using Equation 9), and from Equation 8, [C.sub.1] [phi]w, so [([C.sub.2]/[C.sub.1]).sub.o] = ( 1 + [r.sub.o])(1 - [phi])/[phi]. Since from Equation 11 [r.sub.o] = [alpha]/(1 - [alpha])(1 - [phi]), then [([C.sub.2]/[C.sub.1]).sub.o] = (1 - [phi] + [alpha][phi])/[phi](1 - [alpha]). Is (1 - [phi] + [alpha][phi]/[phi](1 - [alpha]) > [alpha]/(1 - [alpha])? It is if (1 - [phi]) > 0, which is true because o < [phi] < 1.

(10.) With a Y Tax, [t.sub.w] = [t.sub.r] = [t.sub.y]. From Equation 11, the [r.sub.n] with a Y Tax, [r.sub.ny], equals [r.sub.o] or [r.sub.c]. Since [r.sub.ny] = [r.sub.c], then the r with a Y Tax [r.sub.y]> [r.sub.c], so [k.sub.c] > [k.sub.y]. Since r = [alpha]m/[k.sup.1-[alpha]], then [r.sub.ny] = [r.sub.c] implies that (1 - [t.sub.y])[alpha]m/[k.sup.1-[alpha].sub.y] = [alpha]m/[k.sup.1-[alpha].sub.c], so [k.sub.c]/[k.sub.y] = [[1/(1 - [t.sub.y])].sup.1/(1 - [alpha])] > 0.

(11.) Recall our earlier conjecture concerning age taxes in the two-age model: If the age tax ([T.sub.1], [T.sub.2]) is such that [T.sub.2]/[T.sub.1] > [([C.sub.2]/[C.sub.1]).sub.o], then the k under this age tax is greater than [k.sub.o], whereas if [T.sub.2]/[T.sub.1] < [([C.sub.2]/[C.sub.1]).sub.o], then the k under this age tax is less than [k.sub.o]. This conjecture holds for all values in Table 2 (recall that the C Tax column gives the values of the no-tax k, [k.sub.o], because [k.sub.o], = [k.sub.c]). For an Age W Tax ([T.sub.2] = 0), [T.sub.2]/[T.sub.1] = 0 < [([C.sub.2]/[C.sub.1]).sub.o], and k under the Age W Tax is always less than [k.sub.o]. For an Age R Tax ([T.sub.1] = 0), [T.sub.2]/[T.sub.1] > [([C.sub.2]/[C.sub.1]).sub.o], and k under the Age R Tax is always greater than [k.sub.o]. For a Y Tax, as noted previously, [T.sub.2]/[T.sub.1] = [alpha]/(1 - [alpha]), and as proved previously, [alpha]/(l - [alpha]) < [([C.sub.2]/[C.sub.1]).sub.o], so for an Age Y Tax, [T.sub.2]/[T.sub.1] < [([C.sub.2]/[C.sub.1]).sub.o], and k under the Age Y Tax is always less than [k.sub.o].

(12.) This result is analogous to the result presented by Diamond (1970) in his study of the R Tax. Diamond introduces an interest income tax and returns the revenue to retirees with a lump-sum transfer; he finds that this tax/transfer reduces k. In our paper, we switch the financing of government consumption from an interest tax to a lump-sum tax on retirees; we find that this tax conversion increases k.

(13.) Recall that in the two-age model, the production function was y = [mk.sup.[alpha]], where k was capital per worker (k [equivalent to] K/L) and y was output per worker (y [equivalent to] Y/L).

(14.) 1.10 > [sigma] > 0.10, Hubbard and Judd (1986); [sigma] = 1.00, Beaudry and van wincoop (1996) and Laibson, Repetto, and Tobacman (1998); 1.00 [greater than or equal to] [sigma], Blanchard and Fischer (1993); 1.00 > [sigma] > 0.20, Evans (1983); 0.74 > [sigma] > 0.48, Attanasio and Weber (1993); [sigma] = 0.50. Carroll (1997); [sigma] = 0.33, Engen, Gale. and Scholz (1994), Hubbard, Skinner, and Zeldes (1995), and Engen and Gale (1996); 0.33 [greater than or equal to] [sigma], Gale (1998); [sigma] = 0.28, Camphelt and Mankiw (1989); [sigma] = 0.25, Anerbacls and Kotlikoff (1987), Auerbach (1996), Engen. Gravelle, and Smetrers (1997), Kotlikoff, Snsetters, and Watliser (1998); 0.20 > [sigma], Hall (1988); [sigma] = 0.18. Baraky et al. (1997). A review of empirical studies is given by Deaton (1992).

(15.) y = [k.sup.[alpha]] implies Y = [K.sup.[alpha]][E.sup.1-[alpha]], where Y is output, K is capital, and E is effective labor, so MPK [equivalent to] [partial]Y/[partial]K = [alpha]/[k.sup.1 - [alpha]] and MPE [equivalent to] [partial]Y/[partial]E = (1 - [alpha])[k.sup.[alpha]], where k [equivalent to] K/E.

(16.) [C.sub.o] = {[[(1 + [r.sub.o])/(1 + [rho])].sup.[sigma]] - 1) is the constant growth rate of consumption over the life cycle that the individual chooses in the no-tax/no-government-spending economy to maximize lifetime utility. Consider an age tax with a constant growth rate over the individual's lifetime, T, such that T = [C.sub.o]. Then substituting into Equation 14 shows that [r.sub.o], is also a solution for this age tax. Thus, if T = [C.sub.o], then the k under this age tax ([k.sub.a]) is the same as [k.sub.o]; imposing an age tax with this particular constant growth rate has no effect on the k of the economy. A plausible conjecture is the following: If the constant growth rate T > [C.sub.o], then [k.sub.a] > [k.sub.o], whereat if T < [C.sub.o], then [k.sub.a] < [k.sub.o]. We return to this conjecture later in this paper.

(17.) In Equation 14, [[theta].sub.t] is zero at all ages except at the taxed age. In each case, [[theta].sub.t] is set so that all three taxes yield the same total revenue per effective labor.

(18.) In Equation 14, under the work-phase tax, [[theta].sub.t], equals a fixed value for all t [less than or equal to] R and equals 0 for all t [greater than or equal to] R + 1. For the entire-phase tax, [[theta].sub.t] equals a fixed value for all t. Under the retire-phase tax, [[theta].sub.t], equals a fixed value for all t [greater than or equal to] R + 1 and equals 0 for all t [less than or equal to] R. In each case, [[theta].sub.t] is set to achieve the same tax revenue per effective labor.

(19.) This model is used in Seidman and Lewis (1999) to analyze the interest elasticity of saving.

(20.) This is similar to our earlier result in footnote 16 concerning an age tax with a constant growth rate over the individual's lifetime T: If T = [C.sub.o], then [k.sub.a] is the same as [k.sub.o] imposing an age tax with this particular constant growth rate has no effect on the k of the economy.

(21.) The proof is the same as in the two-age model (given in footnote 10) except it uses Equation 16 instead of Equation it.

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Laurence S. Seidman *

Kenneth A. Lewis +

* Department of Economics, University of Delaware, Newark, DE 19716, USA; E-mail [e-mail]; corresponding author.

+ Department of Economics, University of Delaware, Newark, DE 19716, USA; E-mail [e-mail].