# Taxation of Capital

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1665

## Abstract

Taxes on the income from capital have generated a large debate for two reasons. First, the contrast between the arguments for efficiency and equity seems to be particularly sharp here. Second, there exists a variety of views on the appropriate choice of an economic model and its parametric values that are relevant for policy. In this exposition, emphasis will be put on the dynamic aspects of a uniform tax on capital in general equilibrium. In most economies the tax on capital discriminates between different sectors (corporate capital, housing) and induces some static efficiency cost. This cost will be considered only very briefly in comparison with the dynamic efficiency cost.

Taxes on the income from capital have generated a large debate for two reasons. First, the contrast between the arguments for efficiency and equity seems to be particularly sharp here. Second, there exists a variety of views on the appropriate choice of an economic model and its parametric values that are relevant for policy. In this exposition, emphasis will be put on the dynamic aspects of a uniform tax on capital in general equilibrium. In most economies the tax on capital discriminates between different sectors (corporate capital, housing) and induces some static efficiency cost. This cost will be considered only very briefly in comparison with the dynamic efficiency cost.

## The Impact on Capital Accumulation

A dynamic framework is essential in the analysis of the taxation of capital income, and it is useful to recall that there are three generic models of capital accumulation. The first is the so called neoclassical model with an ad hoc specification of the saving function that depends on the flow of incomes and on the interest rate. Although there is little theoretical or empirical foundation for this form, a vague justification has been found in the argument that individuals may not optimize rationally over time, or that capital markets do not operate like standard intratemporal markets. However, the main value of this specification seems to be analytical expediency. In the second type of model, individuals optimize a life-time utility function with no bequest. The third model assumes that individuals care about the welfare of their next descendants as if they would be reincarnated in these descendants with the same utility function. A recursive argument implies that individuals act as if they would live forever.

Most dynamic studies on the taxation of capital income rely on one of these models, or a variation between these types. The models have different implications for the impact of the taxation of capital income on the level of capital accumulation and output, and for the method of evaluation of the tax on capital.

In the neoclassical model the tax reduces the net flow of saving (which is equal to the growth of capital on the balanced growth path at the natural rate), and has a negative impact on capital accumulation. The magnitude of the capital reduction is of course greater when the propensity to save from disposable income has a positive elasticity with respect to the rate of return, net of tax.

In the life cycle model, the capital stock behaves like the level of water in a bathtub. On the balanced growth path it is in equilibrium between the inflow of the savings of the younger generations and the outflow of the dissavings of the older generations. The impact of the tax depends on the elasticity of this equilibrium level with respect to the net rate of return to capital. The tax induces a decrease of the level of capital, if and only if the interest elasticity is positive (Diamond and Mirrlees 1971). A weaker condition is sufficient for a decrease of capital when there is a fixed factor of production such as land because the tax on capital induces an appreciation of land that diverts savings from capital (Chamley and Wright 1986).

When the utility function is additively separable, the value of the interest elasticity of the aggregate stock of capital that is generated by the life-cycle process in the steady state depends mainly on two parameters. The first is the short-run elasticity of saving with respect to the interest rate, which is proportional to the intertemporal elasticity of substitution of consumption. The second is the length of an individual’s horizon that determines the time span during which he can accumulate capital and consume it. When the length of an individual life tends to infinity, he has more time to save up to the point where the rate of return and the rate of time preference are equal. This implies that the elasticity of supply of the stock of capital with respect to the rate of return is larger (Summers 1981).

In the limit case where individuals have infinite lives, this elasticity of supply is infinite. The impact of the capital income tax is negative and its magnitude depends on the elasticity of the demand for capital by firms. The long-term impact of the tax may be large, but a steady state analysis may be misleading since it neglects the transition period.

## Dynamic Incidence

In the short-run, an increase of the tax rate on capital falls entirely on capital income. In the neoclassical model, the dynamic impact of the tax is a lower level of capital and of the wage rate in the long run, and a higher gross rate of return. In the example of an economy where all profits are saved and all wages consumed, a well-known result is that a tax on profit with transfer to workers, lowers the level of consumption of workers in the steady state. This occurs because the tax induces a shift away from the golden rule. For more general specifications, the incidence of this is shifted at least partially to labour income (Feldstein 1974).

The concept of factor incidence loses its meaning in an economy where the optimizing behaviour of agents is fully specified. In the life-cycle model for example, every individual goes through a worker and a capitalist phase. The proper evaluation method is the analysis of the welfare impact of the tax to which we now turn.

## Efficiency Cost

A first step in the computation of the welfare cost of the tax on capital is to assume that the economy is composed of a large number of identical individuals who are price takers. In the dynamic context, these individuals become families with an infinite horizon. The welfare cost of the tax is defined either with respect to lump-sum taxation, or as the differential welfare cost with respect to alternate forms of distortionary taxation.

The method of analysis is to consider a small variation of the tax rate combined with a change of lump-sum taxation or of other tax rates to keep the total revenues invariant. An important assumption is that the tax rate is constant over time. The efficiency cost is given by the difference between the levels of welfare (or its income equivalent), as measured on the dynamic path with and without the tax changes, respectively. For convenience, the original position the economy is in a steady state. An essential aspect of the method is that individuals have perfect foresight and optimize rationally over time. Other forms of expectations (such as myopic expectations or other types), may be convenient for large models but they lead to strange results. For example, a tax on capital income could correct myopic expectations and improve welfare if the level of the capital stock is lower than in the steady state.

When the tax rate is small and there are no other taxes, the welfare cost is of the second order with respect to revenues, a result that is well known for any tax. Note that the impact on the steady state level of output is of the first order. This illustrates how comparisons between steady states that ignore transitional effects can lead to large errors.

The effect of an increase of the tax rate on capital income has two components. In the short run, the level of consumption increases. In the long run the levels of capital, output and consumption are lower than in the initial steady state. The relative weights of these two effects depend on the difference between the growth rate of the economy and the discount rate. Only in the special case where these two rates are almost equal, is the transition component relatively insignificant. The comparison between steady states is then a proper evaluation method.

Consider first the case where the labour supply is fixed, and assume that there is no other tax in the original position. Two structural parameters are important for the value of the excess burden of the tax. First, there is a positive relation between the welfare cost and the intertemporal elasticity of substitution of the utility function. This effect is related to the transition path between steady states. It vanishes when the value of the growth rate tends to that of the discount rate.

Second, the welfare cost is positively related to the elasticity of substitution between capital and labour in the production function. For plausible values of this elasticity, the relation is almost linear. The smaller the elasticity, the larger is the fraction of the tax burden shifted to labour (which is a fixed factor here). When the elasticity is equal to zero, there is no distortion.

When the labour supply is elastic, the welfare cost is larger because the tax has a negative impact on the wage rate that affects the supply of labour. The marginal efficiency cost of the capital income tax is also larger when there are other taxes in the original position.

The differential welfare cost between the taxes on capital income and labour income is not always positive. A reduction of the tax rate on capital income that is maintained over time implies a lump sum transfer to the owners of the capital in place at the time of the tax reform. Therefore, a substitution of the capital income tax by the wage tax (at rates constant overtime), is not always efficient (Auerbach.et.al. 1984; Chamley 1985).

When the difference between the growth rate and the discount rate is small, the magnitude of the lump-sum transfer to the old capital is negligible with respect to the welfare cost of other taxes. In this case the differential welfare cost between the taxes on the incomes of capital and labour is positive. Its value depends on the interest elasticity of the demand for capital by firms and it is independent of the parameters of the utility function.

The measurement of the efficiency cost of the tax on capital income is more difficult when the population of individuals is heterogeneous, since it may involve implicit or explicit interpersonal comparisons of income and equity tissues. Auerbach et al. (1983, 1986), use a lump-sum redistributive authority to isolate the efficiency cost in an overlapping generation model.

Finally, the stylized model of dynamic general equilibrium is potentially a useful tool for the evaluation of a capital tax that is raised on a specific sector (such as the corporate tax), when agents optimize over time. Preliminary estimates indicate that for a production technology with constant returns to scale, the cost of the intrasectoral misallocation may be greater than the intertemporal welfare cost.

## Optimal Tax Rates and Redistribution

The standard method for the determination of a programme of efficient tax rates on capital income is to choose somewhat arbitrarily, an origin of time, and to analyse from that point on the standard second-best problem where the tax on capital income is one of the fiscal instruments.

The distortion induced by the tax on capital income increases with the interval between the moment of the announcement and the date at which the tax is actually raised because the supply elasticity of savings with respect to the rate of return increases also with time. This implies the disturbing property that the policy of second-best is time inconsistent. Since the tax on capital income has a very low efficiency cost at the beginning of the policy horizon, an arbitrary limit may have to be imposed on the tax rate for some initial interval of time so that the policy is defined.

An interesting result is that for fairly general assumptions, the long-run efficient value of the tax rate on capital income is equal to zero when the fiscal instruments are the taxes on the incomes of capital and labour, respectively. The two main assumptions are that a steady state exists in the long run, and that some individuals have an infinite horizon with an asymptotic rate of time preference equal to the social rate of time preference. The latter assumptions is satisfied when the individual’s utility function satisfies the axiom of Koopmans. It does not have to be separable between periods. The steady state is locally stable for additive utility functions and values of the tax rates that are not too large (Chamley 1986). The same result holds when the wage tax is replaced by an ad valorem consumption tax.

On the transition to the steady state the value of the tax rate on capital income is in general different from zero. For an economy where the government expenditures fluctuate, the debt has been considered as a useful instrument for tax smoothing and the minimization of the efficiency cost of raising revenues (Barro 1979). It is interesting to observe that when the efficiency cost of taxation is derived explicitly from price distortions, a tax on the income of capital (with a positive or negative rate), may perform the same function (Chamley 1980). Its role is to offset the intertemporal distortions that are caused by the variations of the tax rates on consumption and labour income that occur when the government budget has to be balanced in each period.

When individuals have finite lives and an operative bequest motive *à la* Barro, the standard Ramsey rules (Diamond and Mirrlees 1981), apply for the taxation of the savings that are used in life cycle consumption. However, the taxation of intergenerational transfers is suboptimal in the second-best (i.e. when the labour tax is an alternative to the capital income tax).

The result holds under a variety of assumptions about the heterogeneity of the population (Judd 1985), and casts some doubt on the redistributive value of capital taxation in the long run. This is in contrast to other studies that use an ad hoc specification of the processes of saving and income distribution (Stiglitz 1978). The result is not valid however when there are binding restrictions on negative bequests.

In a life-cycle framework the government that maximizes a social welfare function is chosen somewhat arbitrarily, as the representative of future generations. The level of the capital generated in the process of saving and dissaving for selfish life-cycle consumption, is not in general optimal when the welfare of future generations is taken into account in an intergenerational comparison.

An important issue here is whether the government can affect the level of capital directly through its saving or dissaving. If public saving is feasible, the efficient tax rates on the incomes of capital and labour on the dynamic path tend to values in the steady state that are determined by the Ramsey rules, and they depend on the price elasticities of the supply of labour and consumption at different instants (Pestieau 1974).

When there are binding restrictions on public saving or dissaving, they can be alleviated by exploiting the differences between the timings of the taxes on labour and capital, respectively. The capital income tax is levied later in life than the labour income tax. A government that is restricted from accumulating capital, could ‘entrust’ the young with some capital through a labour tax, and recover it later through the capital income tax in order to place it with the next generation and so on. The opposite policy can be used when the government wants to hide the public debt from the accountants. This ‘wealth carrying’ function of the tax system invalidates the standard formulate for efficient taxation (Atkinson and Sandmo 1980).

## Other Issues

The analysis has so far omitted the adjustment cost of investment and the international mobility of capital. The adjustment costs reduce the possibilities for intra- or intertemporal distortions, and the potential welfare gains of tax reform. The international mobility of capital has the opposite effect. One possible formulation of adjustment costs is the *q*-theory of investment that has been applied for the corporate tax by Summers (1981).

The issues of adjustment cost and international mobility have been integrated recently in an analytical model by Bovenberg (1986), who finds that the welfare cost of the capital income tax is significantly larger in open economies compared to closed economies, only when the degree of international capital is very high.

## See Also

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