Taxes, the speed of convergence, and implications for welfare effects of fiscal policy.

• Author: Russo, Benjamin

1. Introduction

   Economists' views on the relation between taxes and the long-run growth rate of per capita income have varied greatly as macroeconomic models have evolved. Neoclassical growth models leave no room for a relation between taxes and steady-state growth (Ramsey 1928; Solow 1956; Swan 1956; Cass 1965; Koopmans 1965). Tax policy can affect growth in early endogenous innovation models (Romer 1986, 1990; Segerstrom 1991; Grossman and Helpman 1991; Aghion and Howitt 1992). However, these models exhibit scale effects. (1) When scale effects are removed, the relation between taxes and steady-state growth tends to disappear (Jones 1995b; Segerstrom 1998; Young 1998). Tax policy can affect the growth rate in some versions of endogenous human capital models (King and Rebelo 1990; Rebelo 1991; Jones, Manuelli, and Rossi 1993). However, Lucas's (1990) simulations indicate that the tax effects are tiny. Stokey and Rebelo (1995) support Lucas's conclusion.

   The short run may be the only run in which a relation between taxes and per capita income growth might hold. Nevertheless, a short-run relation could have important lasting implications. Whether this is true depends on the economy's convergence speed. For example, a capital income tax cut may increase the saving rate, increasing growth. Even if diminishing returns dictate that faster growth is merely transitory, the temporary growth spurt shifts upward the long-run trajectories of capital per person and welfare. But the increase in saving requires lower consumption in the short run. IF the transition is slow, the short-run sacrifice could outweigh the long-run gain (Bernheim 1981; Judd 1987). The duration of the transition and, therefore, the size of the net welfare gain depend on the economy's convergence speed.

   The speed at which economies converge to their steady states has been analyzed extensively (Mankiw, Romer, and Weil 1992; Ortigueira and Santos 1997; Barro and Sala-i-Martin 1999). But even here, careful scrutiny suggests the absence of a relation between income taxes and growth. For example, income taxes do not affect the convergence speed in Barro, Mankiw, and Sala-i-Martin (1995) and Ortigueira and Santos (1997). Two points make it important to revisit the issue. First and foremost, as the example above suggests, convergence speed can alter the net welfare effects of fiscal policy. Second, though simple proportional income taxes do not affect the convergence speed, common features of realistic tax systems do. The purpose of this paper is to provide an example supporting both points.

   The particular example used here is the widely used depreciation deduction. However, the results are general: The analysis shows (i) that any tax parameter that affects required saving directly affects the convergence speed, and (ii) that any tax deduction causes the tax rate to affect convergence speed indirectly. Many features of realistic tax systems can affect required saving and, therefore, the economy's convergence speed. The accelerated depreciation allowances permitted by the U.S. tax code will have a larger effect than the example studied here. Tax benefits, such as deductions for charity, debt service, and amortization of goodwill and patents; investment tax credits for equipment and research and development (R&D); and the myriad other deductions in typical tax codes, as well as the personal income tax itself, affect required saving and can affect convergence speed.

   The next section of the paper incorporates a deduction for economic depreciation in the neoclassical model with exogenous saving (Solow model, hereafter) and explains the effect on the convergence speed. Section 3 explains the effect in the neoclassical model with endogenous saving (Ramsey model). Appendix C derives the same effect in a new growth model. The relation appears to be a robust feature of standard macroeconomic models. Section 4 compares the net welfare effects of income tax cuts with and without a depreciation deduction, under narrow and broad (human plus physical) capital. (2) The depreciation deduction reduces the size of net welfare effects. This is consistent with the fact that the deduction reduces the convergence speed, and suggests that the deduction reduces the size of the net welfare effects because it lengthens the transition, increasing short-run losses. Section 5 concludes the paper.

2. Taxes and Convergence Speed in the Solow Model

   Net saving is the difference between gross saving and the quantity of saving required to maintain the capital stock. In the Solow model with Cobb-Douglas production, a proportional income tax, and a depreciation deduction, net saving is

   k = s(1 - [tau])[k.sup.[alpha]] - [(1 - d[tau])[delta] + n + g]k, (1)

   where k is capital in effective labor units, k is k's time derivative, s is the exogenous after-tax saving rate, [tau] is the tax rate, [alpha] is capital's share of output, d is the proportion of depreciation deductible for tax purposes, [delta] is the rate of economic depreciation, n is the population growth rate, and g is the growth rate of labor effectiveness. The first term on the right-hand side of Equation 1 is gross saving. The absolute value of the second term is required saving. Except for the inclusion of the depreciation deduction, Equation 1 represents net saving in a standard Solow model with an income tax.

   The speed of convergence is the rate at which actual k approaches its steady-state value. Let [k.sup.*] be the steady-state capital stock. Near the steady state, convergence speed is determined by the negative of the coefficient on (k - [k.sup.*]) in the first-order Taylor expansion of Equation 1. (3)

   k [congruent to] {[alpha]s(1 - [tau])[k.sup.*[alpha]-1] - [(1 - d[tau])[delta] + n + g]}(k - [k.sup.*]). (2)

   The coefficient on (k - [k.sup.*]) is the response in net saving to a small change in k. The first term in the coefficient is the change in gross saving. The absolute value of the second term is the change in required saving. The convergence speed is determined by the difference in size of these two changes. If they were equal in size, net saving would not respond to k and the economy would not converge. Since the production function is concave, near the steady state the change in gross saving always is smaller than the change in required saving, net saving responds negatively to a small change in k, and the economy must converge.

   Setting Equation 1 equal to zero and solving for k gives [k.sup.*]:

   [k.sup.*] = [[s(1 - [tau])/(1 - d[tau])[delta] + n + g].sup.1/1-[alpha]]. (3)

   To get a standard form expression for the convergence speed, substitute Equation 3 into Equation 2 and take the negative of the coefficient on (k - [k.sup.*]):

   [beta] = (1 - [alpha])[(1 - d[tau])[delta] + n + g]. (4)

   Although the term s(l - [tau]) appears in Equations 1, 2, and 3, it drops out in the derivation of Equation 4. Barro and Sala-i-Martin (1999, p. 37) explain why the saving rate drops out: s does not affect convergence speed because it affects [k.sup.*] and the change in gross saving for given k in exactly offsetting ways. Likewise, the income tax per se affects [k.sup.*] and the change in gross saving in offsetting ways, so it does not directly affect the convergence speed. This was shown by Barro, Mankiw, and Sala-i-Martin (1995), and by Ortigueira and Santos (1997).

   Nevertheless, Equation 4 shows that the depreciation deduction, d, reduces the convergence speed. Equation 2 shows that d reduces the size of the change in required saving, which tends to reduce the size of the response in net saving and the convergence speed. On the other hand, Equation 3 shows that d has a positive effect on [k.sup.*]. This reduces the size of the change in gross saving, which tends to increase the size of the response in net saving and the convergence speed. Since the production function is concave, the latter effect must be smaller, so the convergence speed declines with d.

   Equation 4 also shows that the income tax rate enters through its relation with d. Although the income tax does not directly affect convergence speed, it does so indirectly, via the depreciation deduction. The larger the tax, the larger the amount of depreciation that is deductible, so an increase in the tax rate has the same effect as an increase in the deduction. Since the deduction causes the convergence speed to decline, the tax does also.

   Table 1 reports numerical values of [beta], under narrow and broad concepts of capital (Mankiw, Romer, and Weil 1992), with and without a depreciation deduction. The narrow concept of capital sets [alpha] to 0.3. The broad concept of capital sets [alpha] to 0.75. Also, n = 0.01, g = 0.02, and [delta] = 0.05. These values are used in Barro and Sala-i-Martin (1999). d is set equal to 0 in the first row. In the second row d 1.0 if [alpha] = 0.3, and d = 0.4 if [alpha] = 0.75. The smaller value of d is used under broad capital because this concept includes human capital, whose depreciation (unfortunately) is not deductible. (4) The tax rate is set equal to 35%. This is the statutory federal income tax rate on most corporate income.

3. Taxes and Convergence Speed in the Ramsey Model

   This section explains the effects of the depreciation deduction on convergence speed in the Ramsey model. After-tax profit of the representative firm is

   [pi] = (1 - [[tau].sub.C])([k.sup.[alpha]] - w - rk - (1 - d[[tau].sub.C])[delta]k, (5)

   where [[tau].sub.C] is a proportional tax on corporate cash flow, w is the wage rate, r is the rental price of capital, and all other symbols are defined as before. The interpretation of d, however, differs slightly from before. Here d represents only the corporate depreciation deduction, which is less than the total depreciation deduction. The total depreciation deduction is larger than d because the household effectively gets to deduct depreciation. This is clear in the...