I. Introduction

This paper studies the effects of taxation on economic performance, factor allocations and capital flows, and derives implications for the optimal taxation of factors of production (human and physical capital) in an open economy. In particular, it stresses the impact of factor income taxation on the private sector’s decisions to accumulate physical capital and improve labor efficiency through human capital accumulation. It builds on three related strands of literature: the literature on endogenous economic growth, the literature on the effects of international taxation on capital flows, and the literature on dynamic optimal taxation. The positive analysis in this paper examines the effects of labor and capital income taxation on long—run growth; the normative analysis addresses the question of the optimal path of taxes on labor income, capital income and the income from foreign assets.

Following the seminal work of Romer (1986) the literature on economic growth has experienced a revival. Theoretical studies have developed models in which the rate of growth of an economy is determined endogenously, instead than by exogenous factors such as preferences, population growth and technical progress.^1/ For example, Lucas (1988) considered human capital as an additional engine of economic growth, together with physical capital. This literature suggests that distortions (such as non lump—sum taxation) will affect the rate of growth of income, consumption and capital accumulation in an endogenous growth set-up, while they have only an effect on the level of such variables in a neoclassical exogenous growth model.

Traditionally, the normative analysis of optimal taxation of factor incomes was developed in neoclassical models of exogenous growth. One of the most remarkable results of this literature was obtained by Chamley (1985, 1986) and Judd (1985). These authors considered an infinite-horizon exogenous growth models with a representative agent deriving utility from consumption of final goods and leisure time, and showed that, in such models, the optimal tax rate on capital income is zero in the long run. Given an exogenous stream of public expenditures, the optimal tax plan consists in taxing capital income heavily in the short run, since the supply of capital is relatively inelastic; in the long run, however, capital income tax rates discourage capital accumulation: expenditures should be financed only with taxes on labor income since labor/leisure (i.e., the individual’s time endowment) is the only factor in fixed supply. This asymmetry between the optimal long run taxation of capital income (zero) and labor income (positive) is quite surprising; one should then consider whether the result is robust and under which conditions it may or may not hold. In particular, it is important to assess whether the Chamley-Judd result hold in models of endogenous growth where both human and physical capital are engines of accumulation and growth.

In such a context, we study the role of the technology for human capital accumulation; the nature of the “leisure” activity; and international capital markets in determining the effects of taxation on long-run economic growth. We also determine the optimal taxation of human capital, physical capital and foreign assets. The key characteristics of our model are the following. First, we consider a general set-up where both physical and human capital can enter in the production of new human capital. ^2/ We are therefore able to analyze whether and how the direct inclusion of physical capital inputs in the production of human capital affects the results about the effects of taxation of growth and the optimal long run taxation of factor incomes. Second, we study the implications of alternative specifications of leisure production for the optimal factor taxation results. Our specification is quite general and includes as subcases the conventional “raw time” specification (leisure is the fraction of time that is spent away from work and study), the “quality time” model (leisure is human capital times the fraction of the time endowment that is not spent working or studying), “home production” (leisure is a non-market good produced with human and physical capital) and the case of no leisure. Third, we develop an open economy model that allows us to discuss the optimal taxation factor incomes (including the income from foreign assets) in a context of international capital mobility.

Our main results, summarized in table 1, are the following:

Table 1.

Optimal Taxation of Human and Physical Capital

w = Balanced Growth Rate

Table 1.

Optimal Taxation of Human and Physical Capital

LEISURE → HUMAN CAPITAL ↓	No Leisure η = 0	Quality Time L = (1-u-z)H	Home Production L=[(1-u-z)H]1-γ * [(1-v-x)K]γ
H = B z H (β = 0)	$\begin{array}{cc}\mathrm{w}\ne \mathrm{f}& ({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}& \left({\mathrm{\tau }}^{\mathrm{k}}\right)\end{array}$	$\begin{array}{c}\mathrm{w}\ne \mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$	$\begin{array}{c}\mathrm{w}\ne \mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$
	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}=0\\ {\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}>0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}=0\\ {\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}=0\\ {\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$
H = B (zH)1-β (XK)β	$\begin{array}{c}\mathrm{w}=\mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$	$\begin{array}{c}\mathrm{w}=\mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$	$\begin{array}{c}\mathrm{w}=\mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$
	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}={\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}={\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}={\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$

w = Balanced Growth Rate

View Table

Table 1.

Optimal Taxation of Human and Physical Capital

LEISURE → HUMAN CAPITAL ↓	No Leisure η = 0	Quality Time L = (1-u-z)H	Home Production L=[(1-u-z)H]1-γ * [(1-v-x)K]γ
H = B z H (β = 0)	$\begin{array}{cc}\mathrm{w}\ne \mathrm{f}& ({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}& \left({\mathrm{\tau }}^{\mathrm{k}}\right)\end{array}$	$\begin{array}{c}\mathrm{w}\ne \mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$	$\begin{array}{c}\mathrm{w}\ne \mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$
	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}=0\\ {\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}>0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}=0\\ {\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}=0\\ {\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$
H = B (zH)1-β (XK)β	$\begin{array}{c}\mathrm{w}=\mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$	$\begin{array}{c}\mathrm{w}=\mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$	$\begin{array}{c}\mathrm{w}=\mathrm{f}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\\ \frac{\mathrm{K}}{\mathrm{H}}=\mathrm{g}({\mathrm{\tau }}^{\mathrm{K}},{\mathrm{\tau }}^{\mathrm{H}})\end{array}$
	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}={\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}={\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$	$\begin{array}{c}{\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{K}}={\mathrm{\tau }}_{\mathrm{\infty }}^{\mathrm{H}}=0\end{array}$

w = Balanced Growth Rate

1. The steady-state growth rate of the economy in models with no leisure is qualitatively similar to that in models where the leisure activity is modeled as “quality time” or “home production”. This is because in the last two cases leisure is a non-market activity produced with constant returns to scale to reproducible factors. Leisure can therefore be reinterpreted as a non-market consumption good; consequently, the model is substantially equivalent to one in which there is no leisure.

2. Under the three specifications for leisure described above (no leisure, quality time and home production), the human capital accumulation function has important implications for the dependence of growth rates on factor income taxes. In particular, if human capital is produced with both human and physical capital (with CRS in the two inputs), the steady state growth rate of the economy and the real rate of interest depend on both labor and capital income tax rates. In this case, a zero long-run taxation of both capital and labor income will be optimal. However, if human capital accumulation uses human capital only (with CRS), the steady state growth rate of the economy will not depend on either factor income tax rate.

3. When human capital is produced only with human capital, long-run growth does not depend on tax rates but the optimal long run taxation of labor and capital will depend on the model of leisure considered. If leisure is modeled as “quality time” or “home production”, the growth rate of the economy does not depend on factor taxes but the steady state physical to capital ratio depends on both tax rates. Since tax rates on both factors create such an intertemporal distortion, the optimal long run tax on both human and physical capital will be zero in these cases. In the model without leisure, the tax on physical capital affects the steady state physical to human capital ratio but the tax on human capital does not. In this sense, the tax on labor is lump-sum and it is therefore optimal to tax only labor in the long run while the tax on physical capital is zero.

4. Under a residence-based taxation system, the tax on net foreign assets can be derived residually once the tax rates on domestic factor incomes are set. Specifically, the tax rate on labor affects the equilibrium real after-tax rate of return on physical capital. Consequently, the tax rate on foreign assets will depend on both the tax rate on capital and on labor income. The optimal long—run tax on foreign assets is shown to be zero whenever the optimal tax on physical capital is zero.

5. If the government has to balance its budget in every period because of borrowing and lending restrictions, capital and labor income should be taxed at the same positive rate in the long run whenever it is optimal to have zero long run taxation of labor and capital income with an unconstrained government. ^3/

6. When the leisure activity is modeled as “raw time”, the balanced growth rate of the economy depends on both labor and capital income tax rates regardless of the way the human accumulation technology is specified. This dependence of the growth rate on both tax rates implies the optimal long run tax on both human and physical capital will be zero in the “raw time” model. ^4/ In summary, our results imply that the optimal long run tax on both capital and labor income is zero (or symmetric if borrowing is not allowed) under very general conditions regarding the production of human capital and the specification of the leisure activity. The only case in which the long run tax on capital is zero while the one on labor is positive is that of a model without leisure and with human capital produced only with human capital.

The rest of the paper is organized as follows. In Section 2 we discuss the existing literature on taxation, growth and international capital markets. In section 3 we present our general setup, and in Section 4 we solve for the competitive market equilibrium. In Section 5 we discuss the conditions under which the steady-state growth rate of the economy and factor allocations depend on the tax rates on capital and labor income. Section 6 presents the solution to the government’s optimal taxation problem. Section 7 briefly discusses some policy implications, and Section 8 concludes.

II. A Review of the Literature

The modern literature, on the optimal taxation of factors of production is based on the seminal work of Frank Ramsey (1927). Ramsey studied the problem faced by a government that needs to raise a given amount of revenue by using commodity taxation. The government would like to raise revenue as “efficiently” as possible, but needs to take into account that the behavior of the private sector is going to be influenced by the tax system in place. Formally, the problem is solved by determining the optimal behavior of private agents for a given tax system, and then choosing the tax system that maximizes private agents’ welfare, subject to the constraints imposed by private agents’ behavior and by the government’s revenue needs.

Building on Ramsey’s work, Chamley (1985, 1986) and Judd (1985) showed that in neoclassical models of exogenous growth the optimal long-run tax on capital is zero, while the optimal long-run tax on labor income is positive; capital should be taxed heavily in the short run, when it is in relatively inelastic supply. These results about the asymmetric long-run taxation of labor and capital may be significantly modified in models in which both human and physical capital are engines of endogenous growth. In this regard, Lucas (1990) presented a model of endogenous growth and showed that it is optimal not to tax capital income in the long run even when human capital accumulation is an additional source of long-run growth. He also showed that when the time devoted to human capital accumulation is exogenous, the Chamley-Judd result is obtained again—all long-run taxation should fall on labor income. However, he does not derive the implications of his model for the optimal long run taxation of labor income when the accumulation of human capital is endogenous.

Recently, a number of authors have started to address this issue. ^5/ Jones, Manuelli and Rossi (1993a, 1993b) extend the specification of Lucas (1990) by modeling human capital as a non-market good and by assuming that a flow of final market goods, in addition to effective human capital, enters in the production of human capital. They show that, if human capital is accumulated with constant returns to its reproducible inputs (human capital and market goods), both capital and labor income taxes should be zero in the long run. Bull (1993a) argues that this result is obtained also in a two-sector model in which human capital can be “produced” using physical capital, human capital and intermediate goods as inputs, and/or accumulated through learning-by-doing in the final goods sector. Since government expenditure is positive, the implementation of this tax plan requires high short-run taxation on both factors, in order to accumulate government assets that will finance long-run government spending.

Our model contributes to this literature by studying the role of the technology for human capital accumulation, the nature of “leisure” and international capital markets in determining the effects of taxation on long-run economic growth.

Regarding the first issue, it is clear that the presence of human capital as a reproducible factor modifies the traditional Chamley—Judd results. If human and physical capital were symmetric goods, both perfectly substitutable with consumption and accumulated through savings, the impact of capital and labor taxation would then be similar (Bull 1993a; the first model in Jones, Manuelli and Rossi 1993a). In this case human capital is just a second capital good that is reproducible with the same technology as physical capital. Assuming that human and physical capital are perfectly symmetric is, however, restrictive. Human capital differs from physical capital in at least three dimensions: (1) it is not substitutable with consumption; (2) it is a non-market good; and (3) its accumulation depends on a production function with inputs possibly different from those entering in the production of final goods and physical capital.

Concerning point (1), while most growth models specify physical capital as being perfectly substitutable with consumption (final goods can either be consumed or accumulated in physical capital), it is more realistic to assume chat human capital cannot be consumed (we can consume cars instead of using them to produce final goods but we cannot “consume our brain”).

Concerning point (2), human capital accumulation should be thought as a non-market activity whose inputs are not subject to factor income taxation. Specifically, while the labor income deriving from the time spent in the production of final goods can be taxed, the time input (and implicit labor income) used in production of human capital is usually not taxed. Similarly, any implicit income of physical capital goods entering in the production of human capital cannot be taxed either (for example, a capital good such as a computer used in the production of final goods earns an income that is taxable but the same computer used for increasing one’s own human capital earns an implicit income that is not taxable).

Concerning point (3), the production technology for human capital accumulation may use different inputs and/or have different capital intensity relative to the production technology for final goods. For example, Lucas (1988, 1990) assumes that human capital is a non—market good whose accumulation requires only human capital (or effective labor, i.e., a time fraction of human capital) as its input. While Lucas assumes that physical capital inputs do not enter in the production of human capital, Rebelo (1991) and Mulligan and Sala—i-Martin (1993) consider two-sector models where human capital is produced using both human and physical capital, with factor intensities that differ from those for the production of final goods/physical capital. Alternatively, Jones, Manuelli and Rossi (1993a, 1993b) consider models where human capital is a non-market good that is produced with effective human capital and a flow of market goods, but no direct physical capital input. ^6/

The second issue regards the role of leisure specification and its implications for the optimal factor taxation. In all the recent models of optimal factor taxation (Lucas 1990, Jones, Manuelli and Rossi 1993a, b and Bull 1993a, b), leisure is considered as a non-market good that requires the use of “raw time” only. An older literature, however, considered leisure as a more complex non-market activity requiring the use of both human and physical capital inputs, in addition to raw time. For example, in Becker (1965) and Heckman (1976) leisure is modeled as “quality time”, defined as human capital times the fraction of the time endowment that is not spent working or accumulating human capital. ^7/ Extending this idea of leisure as being quality time, Greenwood and Hercowitz (1991) model leisure as a form of “home production” that uses effective labor and effective physical capital in its production. ^8/ The idea that leisure might be a physical capital intensive activity makes sense since most forms of leisure require the use of capital goods (think of video—stereo entertainment, sporting equipment and so on). Why should the specification of leisure matter? The answer is that when leisure consists of time off work and education only, it cannot be “increased” alongside consumption. If, however, it is an activity that uses reproducible factors, such as human and physical capital, in addition to time, it can be increased together with consumption; at the same time, the decisions to accumulate human and physical capital will take into account their impact on the future enjoyment of leisure as well.

Finally, international capital markets are important because domestic investors will consider the option of investing abroad (and vice versa for foreign investors). Consequently the rate of return that domestic residents can obtain on foreign assets will affect their decision whether to invest in physical capital domestically or buy foreign assets. Such a decision is clearly affected by the tax rates on domestic and foreign capital income. More subtly, it may be influenced by the tax rate on human capital as well should the latter affect the domestic rate of return on capital.

The literature on taxation and international capital flows has been developed in a large number of studies, such as Frenkel, Razin and Sadka (1990) and Razin and Slemrod (1990). These studies have stressed the importance of the principle governing taxation of foreign assets and liabilities held by agents of different countries; residence versus source-based taxation of foreign assets will have very different implications for the world allocation of savings and investment. A number of recent papers consider the positive and some normative effects of taxation on growth in open economy models of endogenous growth. Rebelo (1992) surveys the literature on endogenous economic growth in open economies; Buiter and Kletzer (1991) consider the effects of residence-based taxes on savings in a two-country OLG endogenous growth model; Correia (1992) addresses the issue of optimal taxation of capital income in an open economy while Razin and Yuen (1993a, b) consider a two-country model with human and physical capital and endogenous fertility choice. These contributions do not, however, address the issue of the optimal relative taxation of physical and human capital in models of endogenous growth.

The literature on the effects of taxation on economic growth has examined a number of other interesting issues that are not addressed in our model. If government expenditure is endogenous and productive—for example, when it enters in the production function for final or capital goods—the long run optimal tax on capital might not be zero. This issue has been addressed by Barro (1990), Jones, Manuelli and Rossi (1993a, 1993b), Judd (1990), Zhu (1991) and Corsetti (1992). If there are externalities in the production of final goods, as in Romer (1987) or in the production of human capital, as in Lucas (1988), the optimal tax plan might require subsidies to the activities with positive externalities (see Yuen (1991)). When the rate of time preference is endogenous, Zee (1994) shows that the growth effects of a tax on capital income in a standard Ak model are ambiguous. Judd (1990), King (1990), Chari, Christiano and Kehoe (1991a), Zhu (1991) and Corsetti (1992) study optimal taxation in stochastic settings. In the presence of rents generated by factors in fixed supply (such as labor in models without leisure) it may be optimal to tax capital in the long run if there are limits on the taxation of rents (Jones, Manuelli and Rossi 1993b). Finally, the effects of indirect taxation are examined, in Bull (1993a) and Jones, Manuelli and Rossi (1993b) among others.

III. The Model

We consider a two-sector open economy: a final goods sector that produces consumption goods and physical capital, and an education sector that produces human capital. ^9/ The economy is small, and takes the world real interest rate as given. Physical capital is perfectly mobile across countries, while labor (human capital) is immobile.

IV. The Competitive Equilibrium

The representative consumer takes the paths of τ^K, τ^H and τ^F as given and chooses the paths of C, K, H, F, B, u, v, x, z to maximize (5) subject to (2) and (6). This case includes as its subcases three models of leisure: the “home production” model (η>0 and 0<γ<1); the “quality time” model (η>0 and γ=0) and the “no leisure” model (η=0). ^11/

We can define non-human wealth W_t = K_t + B_t, + F_t and re-write (6) as follows:

Next, we observe that since domestic bonds and foreign assets are perfect substitutes for the consumer, they should yield the same after—tax rate of return:

This also implies that the consumer’s maximization problem can only determine aggregate holdings of domestic and foreign bonds (B_t + F_t); the amount of domestic bonds being held is determined by the supply from the government.

The first-order conditions with respect to C, W, K, H, v, x, u and z respectively can be expressed as follows:

The remaining two FOCs are the constraints (2) and (6). Equation (13) states that the shadow price of consumption (physical capital) must equal the marginal utility of consumption in every period. Equation (14) is the FOC for capital accumulation: the rate of change of the shadow price of consumption must equal the marginal product of capital net of tax, which must also equal the rate of return on government bonds. Equation (15) is the corresponding FOC for human capital accumulation, relating the change in the shadow price of human capital to its marginal rate of return. Equation (16) and (18) describe the optimal allocation of physical and human capital respectively between production of market goods and “home production”. Finally, conditions (17) and (19) describe the optimal allocation of physical and human capital between the “education” sector and home production.

The transversality conditions are:

From (12) and (14) it is straightforward to obtain:

That is, under residence-based taxation the interest rate on domestic bonds must equal the net after-tax rate of return on capital and the after-tax rate of return on foreign assets. This result is clearly an implication of the assumption that in the absence of uncertainty domestic bonds, domestic capital and foreign assets are perfect substitutes.

Using equations (9), (10) and (16)-(19) we can express the sectoral allocation of factors as a function of technology parameters and taxes:

According to (22), when the tax on labor income (capital income) rises (falls), the capital/labor ratio in the sector producing goods rises with respect to the capital/labor ratio in the sector producing human capital and with respect to the capital/labor ratio in the home production sector. It is interesting to note that changes in tax rates do not cause changes in the relative capital intensity between the education sector and the home production sector, since both these sectors are not directly taxed.

This economy will exhibit a balanced growth path, along which consumption, physical capital and human capital grow at the same rate, while factor allocations (u, v, x and z) remain constant.^12/ Log-differentiating (13) and using the fact that factor allocations are constant along the balanced growth path, we obtain:

where time subscripts have been omitted. Along the balanced growth path, the shadow prices of physical and human capital must decline at the same rate. Equating (13) and (15) and using (9), (10), (17) and (19) we can determine the physical to human capital ratios in the three sectors along the balanced growth path:

where the terms D_i are constants involving the technology parameters α, β, A and B, reported in the Appendix. The ratio of “market” consumption to leisure can be determined in an analogous fashion, and is given by:

Clearly, higher factor income tax rates will tend to shift consumption from “market” goods to “home production”.

Using (10), (13), (23) and (24) we obtain the balanced growth rate of the economy (equation 26) and the steady state net real interest rate (equation 27):

From equations (24), (25) and (26) it is clear that both tax rates on domestic factor incomes will in general distort the allocation of factors between sectors and reduce the rate of growth of the economy.

When will the economy reach the balanced growth path? In a closed economy, the economy has to accumulate physical and human capital until it reaches the capital-labor ratio that is associated with balanced growth. In an open economy, however, it is possible to augment (reduce) the domestic capital stock instantaneously by borrowing (lending) on international capital markets. This is what will happen, and therefore the balanced growth path will be reached immediately. In other words, if tax rates do not change over time a small open economy will exhibit no transitional dynamics.

V. Taxation and Long-Run Growth

We will now discuss the main results of the balanced growth solution of our model. In particular, we analyze the conditions under which the balanced growth solutions of the model depend on the tax rates on labor (τ^H), capital income (τ^K) and income from net foreign assets (τ^F).

First, note that the “quality time” model of leisure (γ=0), the “home production” model of leisure (γ > 0) and the model with no leisure (η = 0) do not qualitatively differ from each other since the equilibrium after-tax real interest is identical in the three models (it does not depend on either γ or η). Moreover, the growth rate in the “quality time” and “home production” models is equal and its ratio relative to the growth rate in the “no leisure” model is a constant ([θ / (θ-η(1-θ))]) that depends only on parameters of the utility function θ and η (see equation 26). ^13/

This qualitative similarity of “quality time” and “home production” models of leisure with the case of no leisure results from the fact that leisure is modeled as a non–market activity produced with constant returns to scale to reproducible factors—either human capital only (when γ=0) or both human and physical capital (when γ > 0). Therefore, leisure can be reinterpreted in these two cases as a non–market consumption good and the model is substantially equivalent to one in which there is no leisure.

Second, note that the assumption of perfect capital mobility and the hypothes is of residence-based taxation imply that the after-tax return on all investments (domestic capital, domestic bonds and foreign assets) are equal in every period (see equation 21). In the absence of taxation of foreign asset income, perfect capital mobility should lead to the equalization of domestic real rates of return with the world interest rate and the equalization of domestic and world growth rates. In fact, assume that in the absence of taxation of domestic capital and labor income, the domestic real interest rate is equal to the world rate (r=r^*); this would be the case if the technology and preference parameters in the world economy were equal to those in the domestic economy and if there were no distortionary taxes in the world economy. This would also imply that, in the absence of domestic distortionary taxes on capital and labor, the home growth rate of consumption and GNP would be equal to the world growth rate. ^14/

Consider now the effects of residence-based taxation. This form of taxation allows the after-tax return earned by domestic residents to differ from the world interest rate: the net after-tax return on domestic capital will be equal to the world real interest rate net of the tax on domestic residents’ income from foreign assets. This divergence between domestic and world interest rates under residence—based taxation implies that domestic and world growth rates will also differ (see Rebelo 1991, 1992 on this point).

Next, the solution (27) for the steady state net after-tax domestic real interest rate shows that, in general, this real return will depend both on the tax on domestic capital income and on labor income. Given the equality between this real return and the after-tax return on foreign assets, it follows that:

Proposition 1 In the case of β > 0, the residence taxation principle implies that, for any given tax on labor and capital income, there is a unique feasible value for the tax on foreign assets so that the returns to domestic and foreign investments are equalized. Such a tax rate is equal to:

where r is defined in equation (27).

Proof See equations (12) and (21). ‖

According to this proposition, the equilibrium after—tax real interest rate is determined once the tax rate on capital income and labor income are chosen. Therefore, there is only one value for the tax on foreign assets’ income such that the returns to domestic and foreign investments are equal.

Proposition 2 In the case of β > 0, since the equilibrium real interest rate is a negative function of the labor income tax, the equilibrium tax on foreign assets will be a positive function of the tax on labor income.

Proof See equations (21) and (27). ‖

Intuitively, since an increase in the tax on labor income reduces the after tax real interest rate r, a higher tax on foreign asset income is required to maintain the parity between domestic and foreign returns.

Proposition 3 When physical capital does not enter in the production of human capital (β = 0), the long-run growth rate and the equilibrium real interest rate are independent of the tax rates on capital and labor income regardless of how the leisure activity is modeled. In this case:

Proof See equations (26) and (27). ‖

The intuition for the result is simple. If human capital is produced with human capital only, an increase in the labor tax rate will reduce the return to current work effort but it will also reduce the return to human capital accumulation (and the return to the leisure activity) by the same amount. Therefore, the fraction of time spent working, studying and producing leisure will not be affected by a change in the labor tax rate and the rate of growth of the economy and the real interest rate will be unaffected as well.

It is known that equation (30) represents the steady state growth rate of economies à la Lucas (1988) where human capital accumulation is CRS in human capital only $(\dot{H}=BzH)$ and there is no leisure (η = 0) (see Barro and Sala-i-Martin 1994). Our analysis generalizes that result by showing that, in the Lucas case of β=0, a qualitatively similar steady state solution is obtained when we consider economies where leisure is introduced and modeled as “quality time” or “home production”. Specifically, the growth rate will be independent of the technological parameters for the production of final goods and physical capital (α and A) while the net real interest rate will be independent of preferences and equal to the productivity level of human capital minus depreciation (B-δ).

Proposition 4 When β = 0, the steady state physical to human capital ratio in the production of final goods is independent of the tax rate on labor income but a negative function of the tax rate on capital income:

Proof See equation (24). ‖

The explanation for the independence of the capital ratio from the labor tax is the same as the one given in Proposition 3. A change in the capital income tax rate will instead reduce the return to physical capital accumulation while not affecting the return to human capital accumulation; therefore, the physical to human capital ratio in the production of final goods will fall.

Proposition 5 When β = 0, the steady state physical to human capital ratio in the economy is independent of the tax rate on labor income but a negative function of the tax rate on capital income in the model with no leisure; in the models where leisure is “quality time” or “home production” such a capital ratio depends on both the tax on human and physical capital.

Proof From (22), the equilibrium capital/labor ratio in the economy is given by

From Proposition 1, we know that when β = 0, the fraction z of time spent accumulating human capital is independent of both tax rates. When γ > 0, we can use the equality between the first and the third term in (22) to express v as a function of u and both tax rates. Using the economy’s resource constraint (4), we can establish that u, v and v/u are a function of both tax rates. Since from (23) we know that vK/uH depends only on τ^K, it follows that K/H is a function of both tax rates. When γ=0, v = 1; the resource constraint (4) establishes that u is a function of both tax rates. From equation (28) it follows that K/H is also a function of both tax rates. Finally, when there is no leisure z = 1 - u; since z is independent of both tax rates (Proposition 1), so is u. In this case (32) establishes that K/H depends only on the tax rate on physical capital. ‖

The explanation for the above proposition is the following. As discussed above, when there is no leisure in the model, the tax on labor income does not affect the fraction of time spent working (u) and studying (1-u) as long as β = 0. Then, since v = 1 in this case, equation (31) shows that the equilibrium physical to human capital ratio in the economy will also be independent of τ^H but dependent on τ^K. In the specifications where leisure is modeled as “quality time” or “home production”, the tax on labor income does not affect the fraction of time spent studying (z) (and therefore does not affect growth) but it affects the allocation of time between work (u) and leisure activities (1 - u - z). In particular, an increase in the labor tax reduces the fraction of time spent working and increases the fraction of time spent in the leisure activity. Therefore, the equilibrium human and physical capital in the economy will be affected by the labor tax. The above proposition is important for the derivation of the optimal taxation of factors. In fact, when β = 0 and there is no leisure, the labor tax does not create any intertemporal distortion: it does not affect either the growth rate of the economy nor the capital labor ratio in the economy. Conversely, if leisure is modeled as “quality time” or “home production”, the labor tax does not affect growth but it creates an intertemporal distortion since the economy wide K/H is affected. A tax on capital income is always distortionary when β = 0 because it affects the physical to human capital ratio regardless of whether there is leisure or not in the model.

Proposition 6 When β = 0, the equilibrium tax on foreign asset income will not depend on the tax rates on labor and capital income. In this case:

Proof See equation (27). ‖

The intuition for the proposition is clear. When β = 0, the real aftertax interest rate is a constant, B - δ, that does not depend on either the tax rate on capital or labor income. Therefore, the unique value of the tax on foreign asset income that equalizes returns to domestic and foreign investments will be independent of the choice of the tax on capital and labor.

Proposition 7 When physical capital enters in the production of human capital (β > 0) the steady state growth rate of the economy will negatively depend on the tax rate on both capital income and labor income. Moreover, the steady state physical to human capital ratios in the final goods and human capital sectors will also depend on both factor tax rates.

Proof See equations (24) and (26). ‖

The intuition for the proposition is easier to present for the case of no leisure, but is the same in the equivalent cases of leisure as “quality time” or “home production”. We showed above that when β = 0, the return to and the cost of human capital accumulation (i.e. the net of tax wage) are affected in the same proportion by a change in labor taxes, leaving the time allocation decision unchanged. In other terms, since the cost of human capital accumulation is effectively tax-deductible, labor income taxation does not affect the incentive to accumulate human capital. ^15/ However, if physical capital is also used in the production of new human capital (β > 0), the return to human capital is reduced more than its cost. In particular, the cost of physical capital inputs used in the production of human capital is not reduced by the labor income tax since these inputs are not tax deductible. More generally, as suggested by Trostel (1993), if any other inputs in addition to human capital enter in the production of human capital, its return will be reduced by more than its cost. Therefore human capital accumulation will be reduced by an increase in the labor tax rate.

The above results imply that, for the three specifications of the leisure activity considered so far (the “quality time”, the “home production” and the “no leisure” models), the specification of the human capital accumulation function has important implications for the dependence of growth rates on factor income taxes. In particular, if human capital is produced with both human and physical capital (with CRS in the two inputs), the steady state growth rate of the economy depends on both factor tax rates. However, if human capital accumulation uses human capital only (with CRS), the steady state growth rate of the economy will not depend on either factor income tax rate. Moreover, in this case the steady state ratio between effective human and physical capital will depend on the tax on capital income but not on the labor income tax.

The result that the dependence of the growth rate on factor tax rates has to do with whether physical capital enters in the production of human capital (i.e. on whether β is positive or zero) holds both when leisure does not enter the utility function and when it is produced with constant returns in reproducible factors. What happens in the case in which leisure is not produced with constant returns to reproducible factors? One such case—leisure modeled as a “raw time” activity—is the one most studied in the literature.^16/ In this case “raw time” is a non-reproducible factor that is constrained by the agent’s total time endowment. In Milesi-Ferretti and Roubini (1994) we show that, if leisure is modeled as “raw time”—or, more generally, as an activity not produced with CRS in reproducible factors—the balanced growth rate of the economy will always depend on the tax rates on capital and labor income regardless of whether physical capital inputs enter or not in the production of human capital (i.e., regardless of whether β is positive or zero). ^17/

As discussed above, residence-based taxation allows the after-tax return earned by domestic residents to differ from the world interest rate. This divergence between domestic and world interest rates under residence-based taxation implies that domestic and world growth rates will also differ. Residence—based taxation therefore implies that the long-run equilibrium and growth rate of an open economy will be equal to that of a closed economy. The main difference relative to the closed economy case is the absence of transitional dynamics in an open economy. Starting from an initial equilibrium without international capital flows, the opening of international capital markets will imply that, given the initial conditions (the initial stock of physical and human capital), the country will borrow or lend from the rest of the world so as to instantaneously change the stock of domestic capital to the steady state desired ratio of physical and human capital. In the presence of perfect capital mobility, the economy will therefore jump to the long run allocation of resources and grow along the balanced growth path; while in a closed economy the transition to the balanced growth path will take time.

VI. Optimal Taxation Analysis

So far, we have discussed the conditions under which the growth rate of the economy and the capital ratios in the various sectors will depend on the tax rates on labor and capital income. We turn now to the analysis of the conditions under which it will be optimal to have a zero long run taxation of a factor of production.

VII. Policy Implications

What are the policy implications of our results for the actual conduct of tax policy ? In recent years, the question of the optimal degree of taxation of capital income has been hotly debated in academic and policy circles. For example, in many industrial countries the issue of capital gains taxation has attracted a lot of attention. The results of Chamley (1985, 1986) and Judd (1985), derived in neoclassical exogenous growth models, provided a theoretical rationale for the widely held view that the taxation of capital income should be kept at a minimum, because such taxes can reduce capital accumulation. Our analysis suggests that while the taxation of capital income is distortionary and can have negative growth consequences when the growth rate of the economy is endogenous, a similar effect is caused by the taxation of labor income when human capital is an additional engine of economic growth. Such an effect of labor taxation is absent in exogenous growth models, where the labor factor cannot be accumulated in the form of human capital.

In this sense, the analysis suggests that distortionary taxes have in general negative growth effects and should be kept at a minimum in the long run. What are the short-run implications? Formally, in the absence of constraints on the ability of the government to borrow and lend, the optimal taxation problem yields a solution that involves initially high taxes on both human and physical capital. This allows the government to accumulate a sufficient quantity of assets to finance government expenditure in the long run without any further recourse to taxation. In practice, this is not a realistic solution, for a number of reasons.

The first practical problem is that the government is unable to commit to a given path of taxes from now on to the foreseeable future. Therefore, this optimal taxation scheme will be subject to time—consistency problems. In particular, in every period the government will have an incentive to tax more heavily existing capital, while refraining from taxing investment. ^21/

A second problem is that in practice government expenditure is not exogenous, and high short—run rates of taxation may lead to more spending, rather than to the accumulation of assets to finance long-run expenditure.

A third problem is that in practice the ability of the government to borrow and lend is likely to be restricted. In the limit case in which the government has to balance its budget in every period, the model suggests that physical and human capital should be taxed in a similar fashion. In this sense, the important message of the paper is not that taxation of labor and capital income should be high in the short run and zero in the long run; it is rather that human and physical capital should be taxed similarly if they both contribute to accumulation and long-run growth.

With regard to the taxation of foreign assets, endogenous growth models suggest that differential taxation of domestic and foreign asset income is feasible only with a residence-based taxation scheme, provided, of course, that foreign asset income can effectively be recorded and taxed. With source-based taxation, return differentials would imply unlimited amounts of capital inflows or outflows. Under residence-based taxation, the tax on net foreign asset income should be set at the level that equates domestic and foreign post-tax returns on capital. If zero taxation of domestic capital is optimal and feasible, zero taxation of foreign assets would also ensue.

In this paper we have not considered indirect taxes. The relative merits of indirect versus direct taxation have been widely discussed in the literature; in the framework of neoclassical growth models, it has been suggested that consumption taxation is superior to direct factor income taxation. However, the results of Jones, Manuelli and Rossi (1993b) and Bull (1993a) suggest that in the long run all taxes, including a consumption tax, ought to be zero. The reason is that when growth is endogenous even a distortionary consumption tax affects long-run growth. It would, however, be interesting to consider the implications for indirect taxation of restrictions on the government’s ability to borrow and lend.

Another interesting policy aspect that is captured with the “home production” specification is the notion that factors of production in the “market” sector may be elastically supplied even in the short run. In practice, the “home production” sector may be reinterpreted as the “informal” sector of the economy, where capital and labor income are not taxed. The analysis suggests that “excessive” taxation of incomes in the formal sector will lead agents to transfer capital and labor resources to the informal sector in order to avoid taxation.

VIII. Concluding Remarks

In this paper we have considered the role of the human capital accumulation technology and of the nature of “leisure” activity in determining the optimal taxation of labor, capital and foreign assets in a small, open economy. Traditional optimal taxation analyses in exogenous growth models stressed that the tax rate on capital income should be zero in the long run, while the one on labor should be positive. We have shown that in endogenous growth models this result obtains only under restrictive specifications of human capital accumulation and leisure production. Under more general specifications, capital and labor income should be taxed similarly. Another implication of this paper is that the labor tax rate will affect the real after-tax rate of return on domestic capital. Therefore, in the presence of free capital mobility, equalization of tax rates on domestic and foreign capital may not be sufficient to prevent capital flight.

A general but unrealistic feature of the optimal taxation solution is the accumulation of budget surpluses in the short run to finance government expenditure without recourse to distortionary taxation in the long run. This result is due to the fact that reproducible factors are supplied relatively inelastically in the short run but elastically in the long run. Future research should re-examine the issue of dynamic optimal taxation subject to a realistic set of restrictions on government’s behavior.