A. Individuals

In each country i, the representative member of the generation born at time t plans his consumption so as to maximize the utility function³

where $c_{1t}^i$ and $L_t^i$ denote the first-period consumption and constant labor supply, respectively, with $L_t^i=L^i;c_{2t+1}^i$ and $G_{t+1}^i$ denote the second-period consumption and public good provision, and it is assumed that the functions u (∙) and m (∙) are strictly increasing, strictly quasi-concave and twice continuously differentiable, with $\underset{c_1^i\to 0}{\lim }{u}_{c_1}^{\prime }(c_1^i,c_2^i)=\underset{c_2^i\to 0}{\lim }{u}_{c_2}^{\prime }(c_1^i,c_2^i)=+\infty$, and $\underset{G^i\to 0}{\lim }m\prime \left(G^i\right)=+\infty$, with public goods entering in a separable manner. The assumptions on public goods are clearly strong. The additive separability of the utility function is unfortunately restrictive, as is the assumption that utility derived from public goods is independent of the circumstances of the recipients.⁴

The representative individual maximizes his utility subject to the lifetime constraints

where i ≠ j. The first constraint states that the first-period wage income, $w_t^i{L}^i$, can be consumed or saved. In turn, saving $S_t^i$ is invested either home, $S_t^{ii}$, or abroad, $S_t^{ij}$. The last constraint requires that, in the second period, the individual consumes all his wealth, including principal and net returns, given tax rates and the after-tax domestic returns $r_{t+1}^i$ and $r_{t+1}^j$. The individual is supposed to have perfect foresight regarding $r_{t+1}^i$ and $r_{t+1}^j$, and the future tax rates on capital income both at home and abroad.

Capital is internationally mobile and, in equilibrium, it is allocated across countries so as to equate national after-residence tax returns ${\beta }_{t+1}^{ii}$ and ${\beta }_{t+1}^{ij}$ (with i, j = 1, 2 and i ≠ j) which implies that $r_{t+1}^1=r_{t+1}^2$; thus, in equilibrium

where R_t+1 is the economy-wide return to capital before residence tax and depends on both countries’ source taxes. As far as their asset allocation is concerned, private investors respond to differences in national source-based taxes ${\sigma }_{t+1}^i$, but not to changes in the residence tax rates ${\rho }_{t+1}^i$. In fact, the burden of residence taxation is independent of where savings are invested, and has no effect on the international location of capital.

In equilibrium, the individual is, therefore, indifferent to investing at home $(S_t^{ii})$ or abroad $(S_t^{ij})$, caring only about the level of total saving. We can then consolidate previous constraints into a single present-value budget constraint

The solution to the consumer’s problem of maximizing utility subject to (2) is straightforward. Applying the standard method of constrained optimization, the first order condition for a maximum is

where the conditions on the utility function u ensure that $c_{1t}^i>0$ and $c_{2t+1}^i>0.$.

This also gives us the saving behavior of the single generation in each country i,

where, differentiating with respect to the variables $w_t^i$ and ${\beta }_{t+1}^i$ we obtain $0<{\xi }_{w_t^i}^{i\prime }<1,$, while the sign of ${\xi }_{{\beta }_{t+1}^i}^{i\prime }$ is undetermined. In this model, saving is an increasing function of wage-income, but the effect of the net rate of return ${\beta }_{t+1}^i$ is ambiguous. An increase in the interest rate (or a decrease in the tax rates) leads individuals to shift consumption from the first to the second period (substitution effect), but it also makes it possible to increase consumption in both periods (income effect). The overall impact of these effects depends on the elasticity of substitution between consumption in the first and second period of life. If this elasticity is greater (smaller) than one (where one is assumed to be the expenditure elasticity), then the substitution (income) effect dominates, and an increase in the rate of return ${\beta }_{t+1}^i$, or a lower tax rate, leads to an increase (decrease) in saving.

It may be noticed that the behavior of generation t depends on both domestic and foreign taxes. It directly depends on the domestic residence tax rate ${\rho }_{t+1}^i$ and through the worldwide return to capital R, it also depends on the source tax rate at home and abroad $({\sigma }_{t+1}^i,{\sigma }_{t+1}^j).$.

At times, it may be convenient to work with homothetic utility functions. In this case, the optimal conditions take, in fact, a particularly simple form with saving being linear in the wage income:

If we further restrict the attention to the special case with Cobb-Douglas utility functions, the factor sⁱ (∙) becomes a constant (see Appendix I). In what follows, these simple examples will be useful in discussing a number of tax-related issues.

B. Firms

In each country i, firms act competitively, hiring local labor and renting capital to produce a single homogeneous output according to the aggregate net production technology $F_i\left(A^i,K_t^i,L^i\right).$ This is a twice continuously differentiable function with decreasing returns in capital and labor, and a local productivity factor Aⁱ.⁵

To model continual endogenous growth, we assume that firms in each country accumulate new capital and inadvertently contribute to the productivity of capital locally invested by others. In this sense, production spillovers exist that are national in scope, and the productivity factor may be defined as ${\overline{A}}^i{K}_t^i$. This represents uncompensated spillovers of knowledge or ideas from one producer to another within the same country as for example in Arrow (1962).

In an integrated world economy countries have access, however, to a larger set of knowledge and technologies than they have in a non-integrated environment. In particular, the stock of capital available abroad affects productivity at home, and international mobility of this capital implies transfers of technology and ideas, so that technology spills over from one country to others. To capture these cross-border technological externalities, the productivity factor may be redefined as $A^i(K_t^i,K_t^j),$, where i ≠ j (Bertola, 1993). With this formulation, capital abroad affects productivity at home and cross-country capital flows imply international transfers of technology and knowledge. This mechanism of growth is a very simple extension of the learning by doing model originally proposed by Arrow (1962), and later studied by Romer (1986). In addition, however, it takes into account the fact that domestic production and investment cannot be independent when countries are economically integrated.⁶

The existence of international growth externalities has recently received widespread attention. Coe and Helpman (1995), for example, analyzed the effects of R&D spending in OECD countries and concluded that each country’s total factor productivity significantly depends on foreign R&D capital. Using a special version of the IMF’s MULTIMOD, these authors (and Bayoumi, 1999) calculated that⁷

“When all industrial countries raise R&D spending by an amount equivalent to ∙½ of 1% of GDP, the long-run US output gain is 50% higher than in the case when only US R&D spending rises.” (Bayoumi et al, 1999, p. 425)

Turning to the details of the model, in each country i the production function $F_i\left(A^i,K_t^i,L^i\right).$ is assumed to be linearly homogeneous in the domestic stock of capital $K_t^i$ and in the productivity factor Aⁱ. In turn, the productivity factor is linearly homogeneous in $K_t^i$ and $K_t^j$, and at the limit Aⁱ→ 0 as $K_t^i\to 0$ or $K_t^j\to 0,$ so that positive productivity requires a positive level of capital in both countries. The assumption that the productivity factor Aⁱ is linearly homogeneous in capital stocks is the natural counterpart of the typical AK-model of growth that has been so widely used in single country models. In these models, the linear-homogeneity assumption captures the productivity spillover in national context. In a two-country context, however, it would not be reasonable to assume that the foreign capital simply adds to the domestic effect, thereby generating increasing returns to capital. Rather, the assumption here is that domestic and foreign capital combine to have the same impact on productivity as assumed in a single country model.

Under these assumptions, we can write the output in each country i as

where the constant supply of labor Lⁱ has been omitted, and ${\alpha }_t=\frac{K_t^j}{K_t^i}$, with i ≠ j. We then have a sort of open-economy version of an AK-model where endogenous growth is sustained by a positive external effect of both domestic and foreign capital stocks on total factor productivity. It is worth noting that in this model, the return to capital is constant in the aggregate, and this is a sufficient condition for having economic growth; however, decreasing returns still prevail when domestic capital is considered separately. In addition, domestic production depends on how capital is located between the two countries.

In this context, profit-maximizing competitive firms pay factors their marginal productivity

where $F_{i{k}_t^i}^{\prime }{|}_{A^i}$ is the marginal productivity of capital for the single firm which takes the productivity factor Aⁱ as exogenously given, and the dependence of the ${\overline{r}}^i$ and ωⁱ functions derives from the property of linear homogeneity of the production function.

It may be noticed that at each time the interests of capital owners (the old people) do not necessarily coincide with those of wage earners (the young). Increases in the relative allocation of domestic capital (the ratio α_t) rise, in fact, the private marginal product of capital and have ambiguous effects on that of labor, possibly decreasing it, as

where $H_{i\alpha t}^{\prime }\left({\alpha }_t\right)$ is positive, so that the sign of ${\omega }_{\alpha t}^{i\prime }\left({\alpha }_t\right)$ is ambiguous. This creates a potential intergenerational conflict that is of great importance in examining the intertemporal effects of capital taxes.

At times, it is convenient to work with the more restrictive class of Cobb-Douglas production functions

and with a multiplicative form for the productivity spillovers

where μ + μ* = 1 – ε and μ, μ*, μ_i > 0 in both countries so to satisfy previous assumptions.

In this case, the national product $Y_t^i,$ the private return to capital ${\overline{r}}_t^i$, and the wage rate $w_t^i$ take a very simple form, all being functions of the international allocation of capital ${\alpha }_t=\frac{K_t^j}{K_t^i}$,

In this simplified case, the private rate of return is not a function of domestic capital $K_t^i$ other than through α_t, whereas domestic product and wage rise proportionally with $K_t^i$.

C. Public Sector

As seen earlier, in each country i the government provides an amount of local public goods $G_{t+1}^i$ to each generation t and finances these expenditures by levying distortionary taxes on the residents’ income from saving and on income from domestically invested capital $({\rho }_{t+1}^i,{\sigma }_{t+1}^i)$. If one unit of output can be transformed to one unit of public good, each government i’s revenue constraint may be then written as

where variables are per capita as there is no population growth and population has been normalized to one. The first term on the right-hand side of this expression denotes revenue from the source-based tax ${\sigma }_{t+1}^i,$, and the other two terms indicate revenue from taxes on the return to national savings at time t; these savings are invested, respectively, at home and abroad, and yield a return at time t + 1.

As the reader may have already noticed, government debt has been ignored. In this model with overlapping generations, debt policy would, in fact, be equivalent to levying differential lump-sum taxes, and our purpose is to explore decentralized government behavior in the absence of these taxes.⁸

D. Market Equilibria and Accumulation

At this point, we can complete the building blocks of this two-country model of growth, and determine the equilibria in the national factor markets and in the world capital market, as well as derive the law of capital accumulation for this economy.

A. Momentary Equilibria and International Allocation of Capital

The effect of domestic taxes on the international allocation of capital has been the subject of much attention in the literature on capital taxation, and it is from this issue that we start our discussion. To examine this effect, we need to introduce the concept of single-period or momentary equilibrium. This is defined as an international allocation of capital that, given tax rates, implies equilibrium in both countries’ national markets. It may then be characterized by a set of prices $\left\{{\omega }_t^i,r_t^i\right\}$ and consumption-savings allocations such that, in each country, firms maximize profits, agents maximize utility, and governments satisfy their budget constraints. The exact equilibrium clearly depends on the existing national fiscal policies $\left\{{\rho }_t^i,{\sigma }_t^i,G_t^i\right\},$, and on the aggregate stock of capital, ${\overline{K}}_t$, inherited at the beginning of the period. At each time t, the momentary equilibrium can, therefore, be fully described as a function of the current national policies and the inherited stock of capital.

The momentary equilibrium can be used to determine the international allocation of capital at each instant and the dependence of tax rates. In particular, in equilibrium the aggregate stock of capital ${\overline{K}}_t$ must be located between the two countries, so that ${\overline{K}}_t=K_t^i+K_t^j$ and $\frac{K_t^j}{K_t^i}=\alpha ({\sigma }_t^i,{\sigma }_t^j)$ (as derived from the arbitrage condition (8)) with i, j = 1, 2 and i ≠ j. These conditions provide a system of equations that can be solved for the equilibrium amount of capital in each country i, $K_t^i$, as a function of the vector of source-based taxes $({\sigma }_t^i,{\sigma }_t^j)$, given the stock of capital, ${\overline{K}}_t$. At each time t, we then have that

At each instant, the equilibrium allocations of capital $(K_t^i,K_t^j)$ depends on the source-based tax rates ${\sigma }_t^i$ and ${\sigma }_t^j$ but not on residence taxation. Residence taxes do not, in fact, discriminate residents’ investment across different countries. Changes in residence taxes do, of course, affect the individual saving behavior and the future inherited stock of capital, but at each instant they leave the momentary equilibrium, and the international allocation of the existing stock of capital, unchanged.

What does this tell us about the effect of domestic taxes on the allocation of capital? Differentiating previous condition (8) using equations (6) for ${\overline{r}}^i({\alpha }_{t+1}),$, with i, j = 1, 2 and i ≠ j, and evaluating the expression for the simplest case of symmetric countries, we obtain that at each time t,

as from the properties of production functions, ${\left(F_{i{k}_t^i}^{\prime }{|}_{A^i}\right)}_k^{\prime }<0$

In equilibrium, the amount of capital in each country is a declining function of the domestic source tax, but an increasing function of the other country’s tax rate. In this setting, each country should then be concerned that increases in domestic source taxes lead to an outflow of capital to foreign countries. It is this externality that has received much attention in the literature on international capital taxation.

To get a better understanding of the international effect of domestic taxes, it may be useful to work out a specific example with Cobb-Douglas production functions. In this case, the equilibrium allocation of capital in each country i takes a very simple form.¹¹

In this simple example, the international allocation of capital and the effect of taxes can be illustrated in figure 1. In this figure, both countries’ net rates of return are plotted together as a function of the variable $\chi =\frac{K^1}{\overline{K}}$, where 0 ≤ χ ≤ 1 The capital market is in equilibrium when $r_t^1=r_t^2$, that is at the intersection point I where condition (8) holds.

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Momentary Equilibrium

Citation: IMF Working Papers 2004, 091; 10.5089/9781451851540.001.A001

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As shown in the figure, any increase in the domestic source tax rate (e.g., in country 1) shifts the net-return curve of the country downwards (the dashed line). This leads to an international reallocation of capital with a lower domestic capital stock and a higher level of capital in the other country (i.e. a smaller χ). In equilibrium, the international interest rate also falls as a result of the higher source-based tax rate. This cross-country capital flow effect can be interpreted as a one-period or static fiscal externality of domestic policy and has been often discussed in the literature on international capital taxation.

This example brings out an important point: that the capital relocation effect is a pure static or single-period effect and does not depend on capital accumulation and economic growth. To illustrate this point, consider the case with Cobb-Douglas production functions with no externalities and no economic growth (so that μ = μ* = 0 and 0 < ε < 1). As in the previous case, an international equilibrium exists. In this equilibrium, changes in domestic tax rates lead, again, to a reallocation of capital in favor of the tax-reducing country.¹²

It is not, of course, suggested that economic growth is unimportant; however, an important distinction exists between the effect of domestic taxes on the international allocation of the current capital, which is static, and the impact on economic growth, which is dynamic. It is the static relocation effect which has received much attention in the debate on capital taxation, but this is not the only effect we should consider. A tax change also affects the growth process and the future stock of capital at any instant. Indeed, the static analysis cannot be simply repeated generation after generation, and it is to this dynamic effect that we now turn.

B. Intertemporal Equilibria, Economic Growth, and Welfare

Much of the political rhetoric about capital tax reforms has centered round the effects on economic growth; but, how exactly do domestic taxes influence growth in an integrated economy? To consider this, we need to introduce the concept of intertemporal equilibrium. This can be defined as that sequence of momentary equilibria in which national consumption profiles steadily grow from generation to generation, given the sequence of prevailing fiscal policies ${\{{\rho }_t^i,{\sigma }_t^i,G_t^i\}}_{t=0}^{\infty }$ in each country.

To simplify the analysis, let us focus the attention on the growth rate of domestic product, and let us assume for convenience that national policies are constant. In this case, we can use the property of linear homogeneity of production technologies and write each country’s product as a function of the capital ratio α and of a single country’s stock of capital; we have, for example, that

where, the inelastic supply of labor has been omitted, and the capital ratio $\alpha ({\sigma }_t^i,{\sigma }_t^j)$ is constant because of constant tax policies. The domestic product in the two countries thus grows at a common rate. However, it should be noticed that the claims on that product, i.e. the national income, may grow at different rates as these depend on domestic savings. For example, one country may save more than the other, so that it will come to own an increasing share of the entire world product. Thus, domestic product and national income do not necessarily grow at the same rate. From previous expressions, it also follows that domestic tax policies may affect both the world-wide growth rate and the distribution of production across countries by changing the ratio α because growth and distribution depend on how capital is internationally allocated; however, tax policy cannot generate differences in the countries’ rates of growth of domestic product.

At this point, we can examine the effect of domestic taxes on the rate of economic growth in this two-country economy. To this purpose, we can focus on a single country since the other is characterized by identical dynamics. In particular, previous accumulation equations (10) give us the law of motion of capital in each country i,

where τⁱ = (ρⁱ, σⁱ) and τⁱ = (ρⁱ, σⁱ), and we have dropped the time index for the tax rates as these are constant. This equation describes a relationship between the domestic stock of capital $K_{t+1}^i$ and$K_t^i$ at different times, given the vectors of tax rates τⁱ and τ^j and the fact that the inherited stock of capital in the other country can be written in term of the domestic stock of capital as $K_t^j=\alpha K_t^i$ Therefore, with well behaved utility functions, the dynamics can, eventually, be expressed as

where g_i = g(τⁱ, τ^j) is constant (for an example see appendix I).

Equation (11) gives us the rate of economic growth in each country, and in the entire economy, as a function of domestic and foreign tax rates. Thus, the growth rate in each country is not function of domestic fiscal decisions only: the tax rates of the other country also affect the domestic growth rate, much in the same way as an externality.¹³ In fact, domestic taxes create an intertemporal or “growth” externality. The main interest of this model rests precisely in these intertemporal transmission effects. Unfortunately, previous equations do not allow us to directly examine the growth effects of domestic taxes since the functional forms of the problem are unknown.¹⁴

To get a better understanding of the effects of domestic taxes on international growth, it is useful to work out a specific example. We may consider for simplicity countries that are symmetric but for their tax policy, use the Cobb-Douglas technologies described above, and introduce homothetic utility functions.¹⁵ For this class of utility functions, the individual optimality conditions take, in fact, a very simple form with the saving in each country depending on the stock of domestic capital (equation (5)),

where ${\beta }_{t+1}^i=(1-{\rho }^i)R_{t+1}$. Substituting these expressions in (10), $K_{t+1}^i=\frac{S_t^i+S_t^j}{1+a_{t+1}}$, we obtain a very simple expression for the rate of economic growth

where, for convenience, the terms η_i, η_j have been normalized to be equal to one, and Lⁱ = L^j = L.¹⁶

Armed with this simple example, we can now examine the international effects of national taxation on economic growth, and we start considering the impact of residence taxation. From equation (12), we can observe that residence taxes have no effects on the international allocation of capital α as α (σⁱ, σ^j), but they do affect the rate of growth through changes in the saving propensity factor s$\left({\beta }_{t+1}^i\right)$. In particular, an increase in residence taxes has a zero, positive or negative impact on the national (as well as foreign and international) rate of growth, depending on whether the elasticity of substitution at home is equal, smaller or greater than one (where one is the value of the expenditure elasticity for homothetic utility functions). A clear distinction exists between the static effect of residence taxes on the international allocation of capital and the dynamic effect on economic growth.

We conclude that residence taxes do not affect the international allocation of capital, but they may raise or lower capital accumulation; in fact, they are not necessarily harmful to economic growth. Higher residence taxes change the saving behavior of generations and, if the elasticity of substitution is, for example, smaller than one, then they eventually boost economic growth. This result has long been known in the theoretical literature on saving behavior and a large volume of empirical work suggests that elasticities of substitution are indeed generally less than unity. Thus, the empirical possibility exists that increases in residence taxation do lead to faster growth, while leaving the international allocation of capital unchanged.¹⁷

The effect of changes in the source-based tax rates, σⁱ, are more complex to examine. As clear from previous equation (12), source taxes affect the growth through two channels. They influence the propensity to save s (∙), as well as the international allocation of capital, α (∙), thus leading to ambiguous effects on the rate of growth.

Some definite conclusions can be, however, reached if we restrict further the attention to the special case of Cobb-Douglas utility functions. In this case, s (∙) becomes a constant, and national growth rates take a very simple form,

where $a^{\mu *}={\left(\frac{(1-{\sigma }_t^j)}{(1-{\sigma }_t^i)}\right)}^{\frac{1}{2}}$ with i, j = 1, 2 and i ≠ j (see appendix I). Under this assumption, the rate of growth g_i is thus independent of the residence-based tax rates ρⁱ, but it depends on source taxation through the ratio α (σⁱ, σ^j).

In this special case, there is a national tax (or subsidy) rate σ^i* which, given the parameters of the model, maximizes the rate of economic growth.¹⁸ Clearly, there is a non-monotonic relation between source-based taxes and the rate of economic growth, and this takes the form of a Laffer curve-type relation. This relation is the result of the special cross-country growth mechanism $A^i(K_t^i,K_t^j)$. With this mechanism, only one international allocation of capital exists that maximizes growth externalities, and this can only be achieved with specific national source tax rates.

This special case brings out the important point that lower source taxes increase the stock of domestic capital but they do not necessarily raise the rate of economic growth. Growth depends, in fact, on how capital is allocated across countries. For example, any change away from the growth maximizing tax rate, even lower taxes or higher subsidies, increases the domestic share of the existing stock of world capital, but reduces the rate of growth both at home and abroad. Clearly, an important distinction exists between the static effect of domestic source taxes on the international allocation of capital and the dynamic effect on the rate of growth, and these effects may well differ.

This simple example also serves to show that the effect of a tax change on domestic product may be different from the effect on national income, namely welfare. This can be seen with a simple example. Suppose that the despite the assumption of symmetric countries, country i levies a lower source tax than country j. In this case, an increase in country i’s source tax would raise the rate of growth of domestic product but, under specific parameterization of the model, it will also lower domestic saving, hence reducing the national income.¹⁹ Clearly, the supposed advantage of a tax change varies with the performance criterion we adopt, in this case domestic product and welfare (national income). Considerable circumspection is then necessary in assessing the effect of a tax policy in a dynamic model.

C. Comparing Predictions

The predictions of the model described here may be compared with those others have reached in infinite-life Ramsey models. In their analysis of public finance and economic growth, for example, Barro and Sala-I-Martin consider a Ramsey model with different mechanisms of growth, and conclude:

“Putting the results together, the implication is that the models predict positively correlated movements in r (the net rate of return) and γ (the rate of economic growth).” (Barro and Sala-I-Martin, 1992, p. 655)

thus, capital taxes reduce net returns and the rate of growth. They examine a closed economy, but the same result applies to the case of open economies with internationally mobile capital. Lejour and Verbon (1997), for example, use a Ramsey accumulation model to investigate the issue of tax competition in a two-country world, and observe

“So, a higher domestic source-based capital-income tax will generate a lower growth rate of wealth, …” (Lejour and Verbon, 1997, p. 485)

Thus, the effect of domestic taxes on the international allocation of capital and on economic growth coincide.

The main reason these conclusions are different from ours is that they draw on infinite-lived Ramsey models of accumulation. In these models, time preferences limit opportunities for cross-country tax and growth dependences.²⁰ Suppose, for example, that each national economy consisted of Ramsey infinite-lived optimizing individuals who maximize the infinite stream of their utility discounted by their rates of time preference. In this case, the long-run equilibrium stock of capital in each country is given by the so-called modified “golden rule” (see for example in Blanchard and Fisher, 1989, p. 57). This requires that, in the stationary state, the net-of-tax interest rate in each country equals the rate of time preference of individuals or, in terms of previous notation, that in each country i,

where $\overline{\theta }$ is the individual rate of time preference which is assumed to be constant and equal in the two countries.²¹ From this rule, it is clear that in Ramsey-like models, domestic taxes have no long-run international effect. In these models, changes in the domestic source-based tax ${\sigma }_t^i$ affect the net-of-tax return to home capital, and cause capital to move across borders, but this is a short-term effect. The long-run equilibrium level of capital abroad is given, in fact, by the modified golden rule that remains unchanged; thus, the equilibrium long-run level of capital is not affected by changes in the foreign tax rates. The residence-based tax rate ${\rho }_t^i$ does not influence the international allocation of capital either, since the burden of residence taxation is independent of where savings are invested. It follows that in the Ramsey-type models domestic tax policies have no long-run international effect.²²