Measuring the Elasticity of Tax Revenue: A Divisia Index Approach

Nurun N. Choudhry

doi:10.5089/9781451956528.024.A004

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Measuring the Elasticity of Tax Revenue: A Divisia Index Approach

Author: Mr. Nurun N. Choudhry

Publication Date:: 01 Jan 1979

eISBN:: 9781451956528

Language:: English

Keywords:: SP; money supply; output gap; tax function; revenue effect; Divisia index; tax yield; elasticity estimate; Tax elasticity; Consumption taxes; Income tax systems; Income and capital gains taxes

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The responsiveness of tax revenue to changes in national income, as measured in terms of elasticity, is of great interest to policymakers and researchers because elasticity is measured with reference to a given tax structure. However, since legislative changes in tax structure cause such elasticity to change over time, its direct measurement from historical revenue series is often difficult because the growth in revenue results both from the growth in the base caused by the increase in income and from discretionary tax changes.

Abstract

The responsiveness of tax revenue to changes in national income, as measured in terms of elasticity, is of great interest to policymakers and researchers because elasticity is measured with reference to a given tax structure. However, since legislative changes in tax structure cause such elasticity to change over time, its direct measurement from historical revenue series is often difficult because the growth in revenue results both from the growth in the base caused by the increase in income and from discretionary tax changes.

In estimating the built-in elasticity of a tax, historical revenue series must be adjusted to eliminate the effects on revenue of discretionary tax measures. The method of adjustment depends on the available data on such changes and on the type and frequency of those changes. A complete adjustment of historical revenue series is not possible in any of the major existing methods, such as (1) the proportional adjustment, (2) the constant rate structure, and (3) the dummy variable methods. ¹ Moreover, a great deal of information is needed for the adjustment of data, and it is often not available over a sufficient number of years. For instance, the proportional adjustment method requires use of budget estimates of tax yield owing to discretionary changes. Such data are difficult to obtain in many countries and, if available, they are of questionable reliability, as they may differ substantially from actual discretionary outturns. Similarly, the constant structure method, which requires the use of disaggregated data on taxes or effective tax rates and on the changing composition of the bases, places heavy demands on the availability of data. ² The dummy variable method, although it requires no adjustment of data, cannot be properly used if discretionary tax changes have been made frequently in the past.

This paper has two purposes. First, it proposes a method of estimating the elasticity of total tax yield that does not require the traditional adjustment of historical revenue to eliminate the effects of discretionary tax measures. Second, it applies this method to case studies of a few selected countries.

The proposed method of estimating the elasticity of tax revenue involves three steps. First, the effects of discretionary tax measures on revenue are estimated by an index that isolates the automatic growth of revenue from the total growth. Second, the buoyancy of tax revenue is estimated, with respect to gross domestic product (GDP) or any other aggregate income variable, by a standard regression technique. Finally the buoyancy estimate obtained in the second step is adjusted by a suitable transformation of the index of discretionary revenue, as estimated in the first step, in order to provide an estimate of the elasticity of tax yield.

The proposed method of measuring the revenue effects of discretionary changes is based on the principle of the Divisia index, which is widely used in measuring technical change. This method has two major limitations. First, despite a sound theoretical underpinning, in practical application, the Divisia index of discretionary tax change can underestimate (overestimate) the positive (negative) revenue effects of such measures. A second and related limitation is that, if discretionary changes produce very large revenue effects, this method does not give satisfactory results. ³ In this sense, this method does not replace the other existing methods. In fact, when full and reliable information about the discretionary effects is available, the proportional adjustment method should be preferred, since it can handle large or small discretionary changes without bias. The main advantage of the three-step Divisia index method is that it uses only historical data and requires no specific information on the revenue effects or on the frequency of past discretionary tax changes.

The conceptual framework required to use the Divisia index for measuring the effects of discretionary changes on revenue is discussed in Section I. This section also develops a methodology for estimating the elasticity of tax revenue by using such an index of discretionary change. Section II discusses the empirical results of the application of this methodology to the four case studies selected: the United States, the United Kingdom, Malaysia, and Kenya. In Section III, the estimates of elasticity, based on the disaggregative and aggregative versions of this methodology, are compared with those based on the constant structure method and the proportional adjustment method. A few concluding remarks are made in Section IV.

I. Methodology

1. conceptual framework

The method of estimating the elasticity of a tax system, as proposed in this section, was discovered through an intuitive appreciation that the characteristics of the effects of discretionary tax measures on tax yield are analogous to the effects of technical change on total productivity. The intuition is clear: discretionary tax measures produce changes in tax yield over and above those caused by the automatic growth in the tax bases, as technical change induces changes in total productivity over and above those that can be accounted for by increases in factor inputs.

At an aggregative level, it is generally assumed that there is a stable relationship between tax yield and the bases, just as it is assumed that a similar relationship exists between aggregate output and factor inputs. That is, analogous to the existence of an aggregate production function, there exists an aggregate tax function. In an aggregate production function, the growth in output maps the upward movement along the production curve (or surface) caused by increases in factor inputs. Total productivity is a convenient measure of such a movement relative to factor inputs. In much the same way, the growth in revenue maps the upward movement along the tax yield curve (or surface) caused by increases in the bases. Such movements along the tax yield curve can be represented by the elasticity of tax yield because it is an aggregative measure of the automatic growth in revenue relative to the growth in the base. The shape of an aggregate production function is determined by a given technology. The shape of an aggregate tax function must be determined by a given tax system, since it is the rate-base structure that determines tax yield. Clearly, if there are no technical changes or discretionary tax changes, then the given technology or tax system remains unaltered; hence, the shape of an aggregate production or tax function remains undisturbed.

Technical change is assumed to induce a shift in the production function because the given technology is altered. Analogously, therefore, a discretionary change must also induce a shift in the aggregate tax function, since it alters the given tax system. Thus, in the event of a discretionary tax change (a technical change), the change in tax yield (output) results not only from movement along the tax yield curve (production curve) caused by the growths in bases (factor inputs) but also from a shift in the curve caused by such a change.

More specifically, assume that there exists an aggregate tax function describing tax yield as a function of k bases (usually proxy bases for k categories of taxes). This is analogous to a production function showing the aggregate output obtainable from n factor inputs. Clearly, for a given tax structure and a given configuration of the bases, if there is no discretionary tax measure, there is no change in tax yield—just as, for any given level of factor inputs, the absence of technical change results in no change in output. Now, for any given set of bases, consider a discretionary tax action that alters tax rates and/or exemption levels (including allowances) of one or more categories of taxes. This action produces different revenues from those obtained without such an action. This difference or change in tax yield comes about only because of an induced shift (of either the slope or the intercept, or both) in the aggregate tax function caused by a discretionary tax action analogous to that caused by technical change in the production function, which produces a different output from that produced without technical change. ⁴

In a pioneering article, “Technical Change and the Aggregate Production Function,” Solow (1957) showed that, under certain conditions, the Divisia index is an appropriate index of factor inputs for the measurement of technical change. Such an index is derived from a weighted sum of growth rates of factor inputs, where the weights are the factors’ shares in output. The theoretical foundation of this index has received much attention in the literature, and it is now known that, under essentially the same conditions assumed by Solow about the aggregate production function, the Divisia index is “the best choice among index numbers.” ⁵

The index of technical change is defined as the ratio of an index of total productivity to an index of factor productivity, the latter measured by a Divisia index. This measure has an appealing intuitive interpretation: the percentage increase in total productivity caused by technical progress is equal to the percentage increase in output divided by the percentage increase in factor inputs. The appropriateness of the choice of this index of technical change is owing to its invariance property. ⁶ That is, if no technical change occurs, there is no change in the index, so that the growth in total productivity (or output) is entirely owing to increases in inputs. A change in the Divisia index, therefore, results in a measure of the change in total productivity that reflects any shifts in the production function induced by all sorts of factors that, taken together, are conveniently termed “technical change.”

Consider now the measure of the revenue effects of discretionary tax change. A Divisia index of discretionary tax change can be conceived as analogous to the index of technical change. Such an index should be equal to the percentage increase in total tax yield divided by the percentage increase in total tax yield owing to the automatic increase in the bases. Like the index of technical change, a change in this index should reflect the overall revenue effects of discretionary tax measures. Clearly, for such an index of discretionary tax change to be operational, it must be derived from an aggregate tax function analogous to a production function, and it must possess the invariance property. That is, if no discretionary tax measure exists, there is no discretionary revenue change, and the growth in tax yield is entirely owing to the growth in tax bases.

The necessary and sufficient conditions that guarantee the invariance property of the Divisia index are the following: ⁷

(a) There exists a well-defined continuously differentiable aggregate function, f(x₁(t), …, x_k(t)).

(b) The function f is linear homogeneous (that is, there are constant returns to scale).

There are valid reasons for treating condition (a) as a maintained hypothesis. For instance, if it is assumed that no aggregate tax function exists, it is meaningless to talk about the relationship between tax yield and the tax bases or about the relationship between tax yield and proxy bases such as national income, as well as about such concepts as the elasticity or the buoyancy of tax yield. Further, if it is assumed that no underlying aggregate tax function exists, there is a fundamental indeterminancy about tax yield and tax bases. The “continuously differentiable” character of an aggregate function ensures the regularity of such a function. Otherwise, erratic behavior of tax yield may occur. Moreover, the existence of a continuously differentiable aggregate tax function is central to the derivation of the Divisia index, even if it is not invariant. The crucial condition for the use of the Divisia index is therefore the second one. The requirement of linear homogeneity of f is restrictive. Even if we assume, a priori, that the aggregate tax function is homogeneous, it cannot be linear homogeneous. The presence of a progressive rate structure, as in income taxation, implies that increases in per capita income will produce more than proportional rises in revenues. In general, therefore, the aggregate tax function represents non-constant returns to scale.

Recently, Hulten has shown that the homogeneity assumption can be eliminated by a suitable transformation of the original Divisia index. ⁸ The removal of this restriction allows the application of the Divisia index to aggregate functions characterized by nonconstant returns to scale without violating its invariance property.

There is some justification for the homogeneity assumption of aggregate tax function, although it is not necessarily linear homogeneous, on historical and empirical grounds. The tax ratios (defined here as tax revenue divided by GDP), although they have moved upward in a large number of countries, have not done so erratically. The tax ratios for developing countries have increased somewhat faster than those for developed countries, but the average increase has been rather small. ⁹ These trends in tax ratios can be explained by writing aggregate revenue T as a homogeneous function of GDP (x)

\begin{matrix} T = a x^{μ} & (1) \end{matrix}

Notice that, with x rising through time, the tax ratio (T/x) remains constant or rises through time as the value of μ equals or exceeds unity.

The above aggregate tax function is homogeneous, and it has been extensively used in empirical studies for estimating the buoyancy or the elasticity of tax revenues. It can be shown that, under certain conditions, this aggregate tax function can be derived from an underlying function f—as set out in condition (a)—provided/is homogeneous. ¹⁰

For the purposes of this paper, it is assumed that f is homogeneous only, and not linear homogeneous. This is in keeping with the a priori notion of its properties. In theoretical investigations, such as production or consumption theories, many propositions are based on such theoretical underpinnings. ¹¹ With this perspective, the Divisia index of discretionary tax revenues is now derived.

2. derivation of the divisia index of discretionary tax revenues

Consider the continuously differentiable aggregate tax function at each instant of time

\begin{matrix} T (t) = f (x_{i} (t), …, x_{k} (t); t) & (2) \end{matrix}

where T denotes the aggregate tax yield, x_i denotes the proxy tax base for the k categories of taxes, and the time variable t is a proxy for discretionary tax measures. Analogous to the effect of technical change, the effects of discretionary tax changes at time t are obtained by taking the time derivative of f. Thus, by taking the logarithm of the tax function, differentiating with respect to time, and rearranging, equation (3) results:

\begin{matrix} \frac{f_{t} (t)}{f (t)} = \frac{\dot{T} (t)}{T (t)} - Σ_{1}^{k} \frac{f_{i} (t) x_{i} (t)}{f (t)} \frac{{\dot{x}}_{t} (t)}{x_{i} (t)} & (3) \end{matrix}

Setting $\frac{f_{i} (t) x_{i} (t)}{f (t)} = β_{i} (t)$ and $\frac{f_{t} (t)}{f (t)} = \frac{\dot{D} (t)}{D (t)}$ where D (t) is the Divisia index of discretionary tax change, equation (3) is rewritten as

\begin{matrix} \frac{\dot{D} (t)}{D (t)} = \frac{\dot{T} (t)}{T (t)} - Σ_{1}^{k} β_{i} (t) \frac{{\dot{x}}_{t} (t)}{x_{i} (t)} & (4) \end{matrix}

The apparent shift of the tax function f over time, $\frac{\dot{D} (t)}{D (t)}$ , is the growth of tax revenues owing to discretionary tax measures. Equation (4) can be integrated to get the index of discretionary tax revenue over the time interval [0,n]

\begin{matrix} \frac{D (n)}{D (0)} = [\frac{T (n)}{T (0)}] exp (- Σ_{1}^{k} \int_{0}^{n} β_{i} (t) \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} d t) & (5) \end{matrix}

Normalizing by setting D (0) = 1, D(n) can be viewed as the index of revenue growth owing solely to discretionary tax measures at time n.

As it stands, in empirical work, the computation of the index of discretionary revenue growth, as given by the right-hand side of equation (5), is a difficult task. This difficulty has been overcome in a recent article by Star and Hall (1976) in which they have shown that the fluctuating β_i(t) can be replaced by a constant ${\tilde{β}}_{i}$ , which is some form of weighted average of the β_i(t). This yields the following equation:

\begin{matrix} \int_{0}^{n} {\bar{β}}_{i} \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} d t = \int_{0}^{n} β_{i} (t) \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} d t & (6) \end{matrix}

By integrating the left-hand side of equation (6), the following equation is obtained:

\begin{matrix} {\tilde{β}}_{i} \log (\frac{x_{i} (n)}{x_{i} (0)}) = \int_{0}^{n} β_{i} (t) \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} d t & \begin{matrix} {(7)}^{12} \end{matrix} \end{matrix}

If the left-hand side of equation (7) is put into the right-hand side of equation (5), this yields

\begin{matrix} D (n) = \frac{T (n)}{T (0)} / Π_{i = 1}^{k} {[\frac{x_{i} (n)}{x_{i} (0)}]}^{{\tilde{β}}_{i}} & (8) \end{matrix}

Alternatively, in logarithmic form, equation (8) can be rewritten as

\begin{matrix} l o g D (n) = l o g (\frac{T (n)}{T (0)}) - Σ_{1}^{k} {\bar{β}}_{i} l o g (\frac{x_{i (n)}}{x_{i} (0)}) & (8^{'}) \end{matrix}

A few comments are in order here. First, the structure of the form of the Divisia index of discretionary tax revenues given by either equation (8) or equation (8’) is disaggregative and appealing. In equation (8), the index of total growth of tax revenues is divided by the index of automatic growth of tax revenues (as measured by $Π_{i = 1}^{k} {[\frac{x_{i} (n)}{x_{i} (0)}]}^{{\tilde{β}}_{i}}$ ). In equation (8’), the growth rate of the discretionary tax revenues is equal to the difference between the growth rates of total tax revenues and automatic tax revenues. The latter growth rate is a weighted sum of the growth rates of the (proxy) bases where the weight ${\tilde{β}}_{i}$ is obtained from equation (7). Second, the index D {n) does not require any adjustment of historical revenue data. This is so because the automatic growth of revenue is estimated directly from the data, unlike the existing methods of estimating the elasticity that “cleanse” discretionary tax changes from the data. This is the most important advantage of the Divisia index approach. Finally, the Divisia index D(n) is an exact index of discretionary tax revenues as derived from an underlying aggregate tax function such as equation (2). Therefore, it consolidates all the information, since it is an integral of the discretionary changes along the tax yield curve.

Theoretically, the index D(n) should provide an adequate estimate of discretionary revenue effects. In practice, there are some limitations because the discrete version of the β_i(t), which includes some discretionary revenue effects, is used. Ideally, β_i(t) equals $\frac{f_{i} (t) x_{i} (t)}{T (t)}$ . But f_i(t) unknown because the aggregate tax function cannot be estimated from unadjusted historical data. In the discrete version, therefore, the bias is upward (downward) when the discretionary changes produce positive (negative) revenue effects. The direction of bias in β_i(t) is picked up by the constant ${\tilde{β}}_{i}$ , which, in turn, causes bias in the estimate of automatic growth of tax revenue $Π_{i = 1}^{k} {[\frac{x_{i} (n)}{x_{i} (0)}]}^{{\tilde{β}}_{i}}$ in the same direction. Thus, it is likely that when discretionary changes produce positive (negative) revenue, the automatic growth of tax revenue will be overestimated (underestimated). It is also possible that when discretionary changes produce very large revenue effects, the automatic growth of tax revenue will be too high or too low, according to whether such changes produce positive or negative effects. ¹³

Despite these limitations, the index D(n) is expected to provide a reasonable measure of the effects of discretionary changes under normal circumstances. Also, when information on revenue effects of discretionary changes is not available, the index D(n) provides at least a rough indicator of the effects of such measures.

Estimation of the buoyancy of tax revenue from the underlying tax function is relatively simple. Earlier, it was assumed that the aggregate tax function T(t) = f (x₁(t), …, x_k(t); t) was homogeneous, although not necessarily linear homogeneous. When the degree of homogeneity of f is assumed to be r > 0, ¹⁴ it can be shown that if the growth rates of all the bases are equal to that of GDP (or any other aggregate income variable), then the tax function will have the form ¹⁵

\begin{matrix} T (t) = a x (t)^{r} D^{*} (t) = a x {(t)}^{μ} & (9) \end{matrix}

where x denotes GDP, D^* denotes an index of revenue growth owing to discretionary changes in the time interval [0,t], and μ denotes the buoyancy of tax yield. ¹⁶ The index D^* is a special case of the index D, and, for the time interval [0,n], it has the same form

\begin{matrix} D^{*} (n) = \frac{T (n)}{T (0)} / {[\frac{x (n)}{x (0)}]}^{\tilde{β} *} & (10) \end{matrix}

where ${\tilde{β}}^{*} = \frac{1}{n} \int_{0}^{n} β (t) \frac{ρ (t)}{\bar{ρ}} d t, ρ$ being the growth rate of GDP,

Also, from the relation (9), it follows that the index D^* for the time interval [0,n] can be written as

\begin{matrix} D^{*} (n) = {[\frac{x (n)}{x (0)}]}^{μ - r} & (11) \end{matrix}

Clearly, the index D^* is invariant. In fact, if there is no discretionary tax change in the time interval [0,n], the elasticity of the tax system r (which is also the degree of homogeneity of f) must equal the buoyancy μ of tax yield. But this implies that D^*(n) = 1.

It is now possible to estimate the elasticity r of tax yield from unadjusted historical data. First, the buoyancy μ is estimated from unadjusted historical revenue data for the time interval [0,n] by estimating the tax function T = ax^μ. Second, since the index D^* is derived from the underlying tax function/as the index D, the latter can be substituted in equation (11). This yields

\begin{matrix} D (n) = {[\frac{x (n)}{x (0)}]}^{μ - \hat{r}} & (12) \end{matrix}

Equation (12) is written in logarithmic form as follows:

\begin{matrix} \hat{r} = μ - \frac{l o g D (n)}{l o g [x (n) / x (0)]} & (13) \end{matrix}

Expression (13) now provides the estimate of the elasticity r that was sought. ¹⁷

The index D^* may also be computed from historical data on total tax yield and GDP to obtain an alternative estimate of the elasticity of tax revenue from equation (11) as follows:

\begin{matrix} {\hat{r}}^{*} = μ - \frac{l o g D^{*} (n)}{l o g [x (n) / x (0)]} & (14) \end{matrix}

In order to distinguish between these two estimates, $\hat{r}$ and ${\hat{r}}^{*}$ , it may be said that the estimate $\hat{r}$ is based on the disaggregative Divisia index method, while the estimate ${\hat{r}}^{*}$ is based on the aggregative Divisia index method. The focus here is on the estimate $\hat{r}$ , since it uses more information. If, however, the bases grow at the same rate as GDP or at rates proportional to the growth rate of GDP, then the estimates $\hat{r}$ and ${\hat{r}}^{*}$ are equal. ¹⁸

Although the direction of bias in the estimates of elasticity given in equations (13) and (14) is known, the extent of bias is unknown. Should this method of estimating elasticity still be used? If the choice is between the constant structure and the Divisia index methods of estimating elasticity, the Divisia index method should be used. The limitations of the constant structure method are discussed in some detail in Section III. If, however, the choice is between the Divisia index and the proportional adjustment methods and there is reliable data on the discretionary revenue effects, the proportional adjustment method should be used. ¹⁹

II. Empirical Results of Case Studies

The methodology for estimating the elasticity of tax revenues, as developed in the preceding section, has been applied to the United States, the United Kingdom, Malaysia, and Kenya. The selection of two developed and two developing countries has been made to represent a range of countries in which discretionary tax changes have occurred in most forms of taxation. In order to construct Divisia indices of the discretionary revenues, total tax revenue, comprising various categories of tax receipts in each country, was assumed to have the following underlying aggregate tax function:

\begin{matrix} T (t) = Σ_{i = 1}^{k} T_{i} (t) = f (x_{1} (t), …, x_{k} (t); t) & (15) \end{matrix}

where the proxy base x_i corresponds to the ith category of tax revenue, T_i. Five categories of federal tax revenues are considered for the United States, while four categories are considered for the other three countries. ²⁰

The major classification of tax revenues and corresponding proxy bases for the four countries are listed in Table 1. As the table shows, for the United States and the United Kingdom, federal/central direct taxes have been divided into three categories: individual, corporate, and social insurance and employment taxes, while the indirect taxes have been lumped into a consumption tax category. ²¹ In Malaysia and Kenya, individual and corporate income taxes are lumped into one category while indirect taxes are divided into three major categories according to their relative importance in the composition of tax revenue. The differences in the proxy bases between the two pairs of countries correspond to the differences in the classification of their tax revenue. As seen in Table 1, the United States and the United Kingdom have similar proxy bases corresponding to their similar classifications of tax revenues. On the other hand, the proxy bases for Malaysia and Kenya reflect the differences between the two countries in their classification of taxes. Some proxy bases, such as adjusted gross income of individuals/individual income before taxes and merchandise imports/exports, are as close to actual bases for individual income tax and import/export tax as data allow in these countries. Other proxy bases, such as private consumption and GDP at factor cost for consumption taxes and other indirect taxes, are rough indicators of the actual bases. ²² However, as long as the relative changes in the proxy bases approximately equal those of the unknown actual bases, the revenue effects of discretionary tax changes are captured reasonably well by a Divisia index.

Table 1.

Four Selected Countries: Categorization of Tax Revenues and Corresponding Proxy Variables

^¹Includes federal excise and customs taxes for the United States, and sales (value-added tax), excise, and customs taxes for the United Kingdom.

Table 1.

Four Selected Countries: Categorization of Tax Revenues and Corresponding Proxy Variables

Tax Categories	United States	United Kingdom	Malaysia	Kenya
Individual income tax	Adjusted gross income of individuals	Individual income before taxes	Chargeable income	Chargeable income
Corporate income tax	Net corporate income before taxes	Net corporate income before taxes	Chargeable income	Chargeable income
Consumption tax ¹	Private consumption	Private consumption	—	—
Import tax	—	—	Merchandise imports (c.i.f.)	Merchandise imports (c.i.f.)
Export tax	—	—	Merchandise exports (f.o.b.)	—
Excise	—	—	—	Monetary GDP at factor cost
Other indirect taxes	—	—	Private consumption	Monetary private consumption
Estate and gift tax	Personal income	—	—	—
Social insurance and employment tax	GDP at factor cost	GDP at factor cost	—	—

^¹Includes federal excise and customs taxes for the United States, and sales (value-added tax), excise, and customs taxes for the United Kingdom.

Table 1.

Four Selected Countries: Categorization of Tax Revenues and Corresponding Proxy Variables

Tax Categories	United States	United Kingdom	Malaysia	Kenya
Individual income tax	Adjusted gross income of individuals	Individual income before taxes	Chargeable income	Chargeable income
Corporate income tax	Net corporate income before taxes	Net corporate income before taxes	Chargeable income	Chargeable income
Consumption tax ¹	Private consumption	Private consumption	—	—
Import tax	—	—	Merchandise imports (c.i.f.)	Merchandise imports (c.i.f.)
Export tax	—	—	Merchandise exports (f.o.b.)	—
Excise	—	—	—	Monetary GDP at factor cost
Other indirect taxes	—	—	Private consumption	Monetary private consumption
Estate and gift tax	Personal income	—	—	—
Social insurance and employment tax	GDP at factor cost	GDP at factor cost	—	—

^¹Includes federal excise and customs taxes for the United States, and sales (value-added tax), excise, and customs taxes for the United Kingdom.

The revenue effects owing to all the discretionary tax measures taken during the period under review in each of the four countries have been estimated by Divisia indices based on the underlying aggregate tax function T (t) = f(x_i(t), …, x_k(t); t). The results, which are summarized in the first row of Table 2 (discretionary growth), express the overall effects of discretionary revenue changes in terms of the growth of total tax revenues. They indicate that, during the period 1955–75, the discretionary tax measures taken in the United States had very little overall effect on the growth of tax revenues, although such measures had significant effects on certain specific taxes. During almost the same period (1955–74), in the United Kingdom, the discretionary tax changes had the overall effect of reducing the growth of tax revenue by more than 0.4 per cent annually. For the periods 1961–73 and 1962–74 in Malaysia and Kenya, respectively, the discretionary tax measures produced additional revenue growth of about 1.0 per cent a year. These results, of course, conceal the year-to-year effects of such measures. For instance, in the United Kingdom, various budgets during the period 1955–72 had proposed tax measures that estimated positive changes in revenue in 8 out of the 18 years. The table also presents both total and automatic growth of revenue. Total growth of revenue is obtained from historical data, while automatic growth is obtained by subtracting the growth of discretionary revenue from total growth.

Table 2.

Four Selected Countries: Discretionary, Automatic, and Total Growth of Tax Revenues

(Annual growth in per cent)

Table 2.

Four Selected Countries: Discretionary, Automatic, and Total Growth of Tax Revenues

(Annual growth in per cent)

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Discretionary growth	0.02	–0.43	0.95	1.13
Automatic growth	7.08	9.39	9.00	11.05
Total growth	7.10	8.96	9.95	12.18

Table 2.

Four Selected Countries: Discretionary, Automatic, and Total Growth of Tax Revenues

(Annual growth in per cent)

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Discretionary growth	0.02	–0.43	0.95	1.13
Automatic growth	7.08	9.39	9.00	11.05
Total growth	7.10	8.96	9.95	12.18

According to the methodology described in this paper, the Divisia index of discretionary revenue growth has been utilized to adjust the estimated buoyancy of tax revenue in order to obtain the estimated elasticity. The estimates of buoyancy are obtained with respect to GDP, while the estimates of elasticity are obtained by adjusting buoyancy as given by formula (13) in Section I. The results for the four countries are presented in Table 3. Clearly, when the overall effect of discretionary tax changes is to increase revenue, the elasticity of tax revenue is expected to be smaller than buoyancy and vice versa when discretionary measures decrease revenue. Moreover, the greater the effect of discretionary measures, the larger the difference between buoyancy and elasticity. This is seen in Table 3. As mentioned above, discretionary tax measures had negligible revenue effects in the United States, with the result that the estimates of buoyancy and elasticity are almost identical. The observed differences between the values of estimated buoyancy and elasticity for the United Kingdom, Malaysia, and Kenya are also in accord with the direction and intensity of their discretionary revenue effects.

Table 3.

Four Selected Countries: Estimates of Buoyancy and Elasticity of Tax Revenue

Table 3.

Four Selected Countries: Estimates of Buoyancy and Elasticity of Tax Revenue

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Buoyancy	1.04	1.18	1.70	1.42
Elasticity	1.04	1.24	1.57	1.32

Table 3.

Four Selected Countries: Estimates of Buoyancy and Elasticity of Tax Revenue

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Buoyancy	1.04	1.18	1.70	1.42
Elasticity	1.04	1.24	1.57	1.32

The estimated elasticities (and buoyancies) reported in Table 3 differ considerably between and within the pairs of developed and developing countries. As can be seen, the estimates for the United States and the United Kingdom are smaller than those for Malaysia and Kenya. The estimate of elasticity ranges from a low of 1.04 for the United States to a high of 1.57 for Malaysia. The size of the elasticity of tax revenues is influenced by many factors, such as progressive elements in the tax system, distribution of income, and composition of bases. Discretionary tax measures and economic growth affect one or more of these factors and, hence, the size of the elasticity of tax revenue. The effects of such factors on the elasticity, in turn, are felt on the tax ratio (tax revenue relative to GDP). Thus, countries with a tax elasticity greater than unity must have a rising tax ratio through time. The larger the value of the elasticity or the buoyancy, the faster is the rise in the tax ratio. Data reveal that the tax ratio of Malaysia showed the fastest-rising trend, at an average annual rate of 2.1 per cent, followed by Kenya and the United Kingdom at annual rates of 1.5 per cent and almost 1.0 per cent, respectively. The tax ratio in the United States did not show an appreciable rising trend. ²³ The observed differences in the sizes of the elasticity and the buoyancy between and within the pairs of countries thus agree with the trends of their tax ratios.

In order to appreciate the contributions of the components of tax revenue in determining the size of the elasticity or the buoyancy of tax yield, the author has estimated the buoyancies of the components with respect to GDP ²⁴ (See Table 4.). Several observations can be made on the basis of the information in Table 4. First, there are substantial differences in the buoyancies between the components of tax revenue within each country. Second, although the components are not strictly comparable between the pairs of countries, the buoyancies of the components of tax revenue in Malaysia and Kenya are considerably larger than those of the United States and the United Kingdom. Third, in the United States, corporate taxes and consumption taxes (mainly federal excise and import taxes) are primarily responsible for lowering the overall buoyancy (and the elasticity) of tax revenue, since the estimated buoyancies of these components are well below unity. The largest contribution to the buoyancy of tax revenue has come from social security and employment taxes, with an estimated buoyancy of 1.73. Fourth, in the United Kingdom, the corporate income tax, with an estimated buoyancy of 0.56, has pulled down the overall buoyancy of an otherwise fairly buoyant tax system. Fifth, import and export taxes in Malaysia and the import tax in Kenya have estimated buoyancies of less than unity. All other components of tax revenue in these two countries are highly buoyant (In fact, other indirect taxes in Malaysia have the largest buoyancy of all other components among the four countries.).

Table 4.

Four Selected Countries: Estimates of Buoyancy for Different Categories of Taxes

Table 4.

Four Selected Countries: Estimates of Buoyancy for Different Categories of Taxes

Category of Tax	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Individual income tax	1.08	1.45	1.84	1.48
Corporate income tax	0.60	0.56	1.84	1.48
Consumption tax	0.53	1.11	—	—
Import tax	—	—	0.96	0.88
Export tax	—	—	0.83	—
Excise	—	—	—	1.31
Other indirect taxes	—	—	2.56	1.23
Social security and employment taxes	1.73	1.46	—	—

Table 4.

Four Selected Countries: Estimates of Buoyancy for Different Categories of Taxes

Category of Tax	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Individual income tax	1.08	1.45	1.84	1.48
Corporate income tax	0.60	0.56	1.84	1.48
Consumption tax	0.53	1.11	—	—
Import tax	—	—	0.96	0.88
Export tax	—	—	0.83	—
Excise	—	—	—	1.31
Other indirect taxes	—	—	2.56	1.23
Social security and employment taxes	1.73	1.46	—	—

The information presented in Table 4 may now be viewed in the context of discretionary tax measures taken in these countries. In the United States, the legal structure of individual income tax was virtually unchanged from 1955 to 1964. Thereafter, the Tax Reform Acts of 1964, 1968, and 1969 brought about important changes in the legal structure of income taxation by reducing the minimum tax rate and marginal tax rates across all income brackets. Other measures were also taken in 1970, 1971, and 1972 by the Revenue Act of 1969. The sharp increases in personal exemptions and standard deductions in 1972 were particularly important. The Tax Reform Acts also brought about important benefits relating to corporate profits taxation, particularly liberal increases in investment tax credits. Also, changes in depreciation provisions of the tax law, especially those made in 1971, and measures relating to foreign investments provided substantial tax relief to corporations. Except for the Excise Tax Reduction Act of 1965, which simplified and lessened the burden of these taxes, most of the other measures concerning excise taxes and import duties were of minor importance. The 1965 Excise Tax Reduction Act, however, had significant revenue-reducing effects. The losses in revenue from these sources were covered mainly by discretionary increases in social security tax payments. Although some revenue-increasing measures were taken in social security taxes between 1957 and 1965, the discretionary measures of 1967 led to important changes in social security tax laws that not only raised rates but also extended the base, by taxing certain earned income that had previously not been taxed, in order to meet rising future obligations. The sizes of the estimated buoyancies of the components of tax revenue indicate that these discretionary measures had substantial effects on the growth of revenues, although, on balance, the revenue-reducing measures in individual and, particularly, in corporate income taxes were offset by the revenue-increasing measures in social security and employment taxes. ²⁵

During the period 1955–74, discretionary tax changes have occurred frequently in the United Kingdom, particularly in individual taxation. On the whole, the discretionary measures applied to individual incomes, especially between 1955 and 1960 and between 1964 and 1972, lowered the minimum and marginal tax rates and increased personal exemptions and standard deductions. In 1974, however, individual income tax rates were raised. Major changes were made, though less frequently, in other forms of taxation, including several reforms in corporate taxation that provided substantial relief by increasing investment incentives through capital consumption and other allowances. During the period 1955–60, indirect taxes were not raised materially, but they were raised several times in later years, when changes included the introduction of the selective employment tax in 1966 and its subsequent abolition in 1973. The most important measure affecting indirect taxes was the introduction in 1973 of a value-added tax, which replaced the purchase tax. Several changes that were made in national insurance countributions (defined here as social security taxes) increased revenue. The estimated buoyancy of 1.45 for the individual income tax indicates that, on balance, the revenue-reducing measures did not impair the progressivity of the income tax structure. ²⁶ The low value of the buoyancy of the corporate tax at 0.54, however, indicates that the discretionary measures provided substantial tax relief to corporations even though corporate income before taxes rose at a faster rate than GDP. ²⁷ On the other hand, the estimated buoyancies of 1.11 and 1.46 for consumption (all indirect taxes) and social security taxes are fairly indicative of the revenue-increasing effects of discretionary changes.

Most of the discretionary measures relating to direct and indirect taxes in both Malaysia and Kenya produced additional revenues. In both countries, these measures not only resulted in periodic upward adjustments in rates but also broadened the bases, especially in indirect taxation, by extending coverage to many previously untaxed commodities. Substantial changes were made in the income tax system in both countries. In Malaysia, the Income Tax Ordinance was virtually unchanged until 1966. In 1967, a new ordinance raised the average statutory rates across all income brackets without making any change in the personal allowances and deductions. Subsequent amendments of the 1967 Ordinance made some minor changes and widened the tax base by adopting a world income concept in 1973. The 1967 Ordinance also imposed a 5 per cent development surcharge on all company income. ²⁸ Similar measures to increase income tax yield were also taken in Kenya between 1963 and 1973, particularly in 1965 and 1966, when important changes increased rates in top brackets and reduced personal exemptions and deductions in most income brackets. The 1965/66 amendments also raised the corporate tax rate from 37.5 per cent to 40 per cent. Discretionary changes were, however, more frequent in indirect taxes in both countries. The tariff schedule was periodically adjusted to increase the extent of differentiation, with the result that progressively higher rates were levied on luxury imports. Despite such progressivity in rates, specific rates continued to be maintained on a number of important items in both countries. Discretionary changes in other indirect taxes, particularly in taxes on domestic production, had sizable revenue-increasing effects. As may be seen in Table 4, the estimated buoyancies for income taxes of 1.84 and 1.48 for Malaysia and Kenya, respectively, indicate the revenue-increasing effects of such measures. ²⁹ The values of the buoyancies of import and export taxes, which are less than unity, mainly reflect the effects of specific rates in these countries. On the other hand, the revenue-increasing discretionary excise tax measures in Kenya and other discretionary indirect tax measures in Malaysia had very favorable effects on tax yield, as indicated by the high values of the estimated buoyancies.

By highlighting discretionary changes as applied to different taxes, the preceding discussion provides the reader with a better understanding of the contribution of these taxes to the buoyancy and elasticity of total revenue. Thus, in the United States and the United Kingdom, revenue-reducing discretionary changes in income taxation, particularly the taxation of company profits, have contributed to the low values of buoyancy and elasticity of tax revenue. In Malaysia and Kenya, revenue-increasing measures in both direct and indirect taxes have contributed to the relatively higher values of the buoyancy and elasticity of tax revenues.

III. Some Further Results

This section compares the estimates of elasticity in Section II with those obtained using three alternative methods. The first of these three, the aggregative Divisia index method, has already been discussed in Section I. The other two are the constant structure and the proportional adjustment methods. ³⁰ Elasticities of tax revenue based on the proportional method have been estimated for only the United Kingdom and Malaysia. ³¹ Table 5 presents estimates of elasticities based on different methods.

Table 5.

Four Selected Countries: A Comparison of Elasticities and Buoyancies of Tax Revenue Based on Alternative Methods

Table 5.

Four Selected Countries: A Comparison of Elasticities and Buoyancies of Tax Revenue Based on Alternative Methods

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Method used to compute elasticity
Disaggregative Divisia index	1.04	1.24	1.56	1.32
Aggregative Divisia index	1.05	1.27	1.51	1.26
Constant structure	1.01	1.02	1.19	1.00
Proportional adjustment	…	1.31	1.46	…
Buoyancy	1.04	1.18	1.70	1.42

Table 5.

Four Selected Countries: A Comparison of Elasticities and Buoyancies of Tax Revenue Based on Alternative Methods

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Method used to compute elasticity
Disaggregative Divisia index	1.04	1.24	1.56	1.32
Aggregative Divisia index	1.05	1.27	1.51	1.26
Constant structure	1.01	1.02	1.19	1.00
Proportional adjustment	…	1.31	1.46	…
Buoyancy	1.04	1.18	1.70	1.42

Several observations can be made on the basis of the information in Table 5. First, the elasticity estimates presented in Section II are very similar to those estimated using the aggregative Divisia index method. The fact that the aggregative method uses only the historical data on total tax revenue and GDP makes it a very attractive method for estimating an elasticity when time is a limiting factor. Second, the elasticity estimates based on the constant structure method are uniformly smaller than those based on any of the other three methods. Third, the elasticity estimates based on the proportional adjustment method are closer to those based on the first two Divisia index methods than to those based on the constant structure method. Finally, the constant structure method fails to detect the overall negative effects of discretionary changes in the United Kingdom, unlike the other three methods, since the elasticity estimate based on this method is smaller than the corresponding buoyancy estimate.

In the constant structure method, the simulated tax revenue is a linear combination of the various bases where the coefficients of linearity are the effective tax rates for a given reference year. The elasticity of tax revenue estimated by this method therefore depends on the combined variations of all the bases with respect to GDP. In other words, the estimate of elasticity is a function of the base-to-GDP elasticities. It can be shown that the elasticity estimate based on the constant structure method lies between the smallest and the largest values of the base-to-GDP elasticities. The exact value of such an estimate will be determined by a given reference year’s effective tax rates. Thus, differences in the tax structure between any two reference years will be reflected in the difference between the estimated values of elasticities with respect to the two given reference years. Two limitations should, however, be noted. First, even when discretionary changes produce overall negative (positive) revenue effects, the constant structure method does not guarantee that the elasticity estimate will be larger (smaller) than the corresponding buoyancy estimate. Second, there is a possibility that the elasticity estimated by the constant structure method will fail to detect the effects of discretionary changes. The first limitation arises because a tax system may have many progressive elements, so that even though the base-to-GDP elasticity may be low, the buoyancy of tax revenue with respect to GDP may be high. The second limitation is important. If all the bases grow at the same rate, so that the values of the base-to-GDP elasticities are all equal to λ, say, then, irrespective of the rate-base structure of a tax system and of reference points, the elasticity estimate based on the constant structure method will be λ. In particular, if λ = 1, implying that all the bases grow at the same rate as GDP, the elasticity estimated by the constant structure method will be unity. Under these circumstances, the constant structure method becomes totally inefficient as a method of estimating the elasticity of tax revenue. ³²

The above arguments are substantiated by the data given in Table 6, which reports the base-to-GDP elasticities for the components of tax revenue. As shown in Table 6, the variations in the base-to-GDP elasticities are considerably less for the United States and the United Kingdom than they are for Malaysia and Kenya. Also, the values of these base-to-GDP elasticities, with a few exceptions, do not deviate much from unity, particularly for the United States and to a lesser extent for the United Kingdom. Notice that for all four countries, elasticities based on the constant structure method (shown in Table 5) lie between the extreme values of the base-to-GDP elasticities. In fact, the elasticities estimated by the constant structure method are quite close to the unweighted average of the base-to-GDP elasticities given in Table 6.

Table 6.

Four Selected Countries: Base-to-GDP (Factor Cost) Elasticities

Table 6.

Four Selected Countries: Base-to-GDP (Factor Cost) Elasticities

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Individual income tax	1.03	1.08	1.63	0.80
Corporate income tax	0.98	1.10	1.63	0.80
Consumption tax	0.99	0.95	—	—
Import tax	—	—	0.99	1.18
Export tax	—	—	0.88	—
Excise	—	—	—	0.90
Other indirect taxes	—	—	1.06	1.00
Social security and employment taxes	1.04	1.00	—	—
Unweighted average	1.01	1.03	1.14	0.97

Table 6.

Four Selected Countries: Base-to-GDP (Factor Cost) Elasticities

	United States 1955–75	United Kingdom 1955–74	Malaysia 1961–73	Kenya 1962–74
Individual income tax	1.03	1.08	1.63	0.80
Corporate income tax	0.98	1.10	1.63	0.80
Consumption tax	0.99	0.95	—	—
Import tax	—	—	0.99	1.18
Export tax	—	—	0.88	—
Excise	—	—	—	0.90
Other indirect taxes	—	—	1.06	1.00
Social security and employment taxes	1.04	1.00	—	—
Unweighted average	1.01	1.03	1.14	0.97

The data in Table 6 can also be used to explain the small margin of difference between the elasticity estimates based on disaggregative and aggregative Divisia index methods. In Section I, it was stated that when all the bases grow either at the same rate as GDP or at rates that are proportional to the growth rate of GDP, the underlying aggregate homogeneous tax function T(t) =f(xi(t), x_k (t); t) can be written as T(t) = ax(t)^μ where x_i denotes the base and x denotes GDP. In Table 6, it can be seen that, with some exceptions, the base-to-GDP elasticities are contained within a ± 10 per cent deviation from unity. This implies that the deviations of growth rates of these bases from that of GDP are, on average, also contained within this band, which accounts for the “closeness” of the values of the two sets of elasticities. More importantly, however, it shows that when the variability among the tax-to-base elasticities is not expected to be large, the aggregative Divisia index method of estimating the elasticity of tax revenues saves a great deal of time, as it requires only data on total tax revenue and GDP time series.

To make a further comparison of these two methods of estimating elasticities and to see how discretionary changes have affected the responsiveness of the tax system over time in these four countries, the two sets of elasticities were estimated for different subperiods. Table 7 shows that the two sets of elasticity estimates for different subperiods are quite close for the United States, the United Kingdom, and Malaysia. For Kenya, the differences between the elasticity estimates are somewhat larger. Both methods appear to exhibit more or less similar movements in elasticity that are peculiar to each country, although the trends are relatively more pronounced in the disaggregative method. The results indicate that the elasticity of tax revenue has improved in Malaysia and, to a lesser extent, in the United States, while it has deteriorated marginally in the United Kingdom and, to a slightly greater extent, in Kenya. Aside from other factors that may have influenced the elasticity of tax revenue over time in these four countries, the discussion of discretionary changes at the end of Section II suggests that such changes have substantially affected the tax yield in these countries. The revenue-increasing measures in Malaysia and the revenue-decreasing measures in the United Kingdom must have played major roles in causing the upward trend in elasticity in the former country and the downward trend in the latter. In the United States, although the overall effects of discretionary revenue were negligible, the slight but gradual increase in elasticity could be attributed to the increase in progressivity of the individual income tax brought about by such measures despite reductions in rates and in standard deductions and allowances. ³³

Table 7.

Four Selected Countries: Comparison of Estimates of Elasticity of Tax Revenues Based on Disaggregative and Aggregative Divisia Index Methods for Different Periods¹

^¹DI denotes the disaggregated Divisia index method; ADI denotes the aggregated Divisia method.

Table 7.

Four Selected Countries: Comparison of Estimates of Elasticity of Tax Revenues Based on Disaggregative and Aggregative Divisia Index Methods for Different Periods¹

Period	United States		United Kingdom		Malaysia		Kenya
Period	DI	ADI	DI	ADI	DI	ADI	DI	ADI
1960	0.99	1.01	1.28	1.26	…	…	…	…
1965	1.01	1.04	1.26	1.25	1.46	1.47	1.44	1.27
1970	1.02	1.04	1.23	1.25	1.52	1.48	1.34	1.24
1973–75	1.04	1.04	1.24	1.27	1.57	1.51	1.32	1.26

^¹DI denotes the disaggregated Divisia index method; ADI denotes the aggregated Divisia method.

Table 7.

Four Selected Countries: Comparison of Estimates of Elasticity of Tax Revenues Based on Disaggregative and Aggregative Divisia Index Methods for Different Periods¹

Period	United States		United Kingdom		Malaysia		Kenya
Period	DI	ADI	DI	ADI	DI	ADI	DI	ADI
1960	0.99	1.01	1.28	1.26	…	…	…	…
1965	1.01	1.04	1.26	1.25	1.46	1.47	1.44	1.27
1970	1.02	1.04	1.23	1.25	1.52	1.48	1.34	1.24
1973–75	1.04	1.04	1.24	1.27	1.57	1.51	1.32	1.26

^¹DI denotes the disaggregated Divisia index method; ADI denotes the aggregated Divisia method.

IV. Concluding Remarks

In this paper, the characteristics of discretionary tax measures are considered as analogous to the characteristics of technical change. This analogy, and the assumption that the underlying aggregate tax function T(t) = f(x₁(t), …, x_k(t); t) is homogeneous, make it possible to use a Divisia index to measure the revenue effects of discretionary changes. The homogeneity assumption is justified on historical grounds, even though the tax function used in most research studies—T (t) = ax^μ(t), where x denotes GDP or any other aggregate income variable—implicitly assumes an underlying homogeneous tax function f. It has been shown that the elasticity of tax revenue can be estimated by adjusting the buoyancy by a simple transformation of a Divisia index of discretionary changes.

The method, however, has two limitations: (1) the Divisia index of discretionary tax change underestimates (overestimates) the positive (negative) revenue effects of such measures; and (2) if discretionary changes produce very large revenue effects, the method can produce unsatisfactory results.

The results of the application of the methodology (both disaggregative and aggregative Divisia index methods) to the United States, the United Kingdom, Malaysia, and Kenya are found to conform to the nature and direction of discretionary changes that have occurred in these countries during the periods under investigation. Also, in comparing these estimates of elasticity with those based on the constant structure and proportional adjustment methods, it is found that, while the proportional adjustment method gave estimates close to those of the Divisia index, the constant structure method uniformly gave the lowest estimates and also failed to detect the effects of discretionary changes in the case of the United Kingdom. The reasons for this failure, as well as for the general inadequacy of the constant structure method, were discussed in Section III. Finally, the movements of the elasticity of tax revenues for the four countries were traced by using both the disaggregative and the aggregative Divisia index methods, and it was found that both of these methods gave similar trends, although the trends were more pronounced for the disaggregative method.

In conclusion, it is believed that, despite its involved theoretical underpinning, the Divisia index methodology is simple, since it uses only unadjusted historical data. In fact, the aggregative Divisia index method requires only historical series on total tax revenue and GDP that are readily available for most countries.

APPENDIX

Mathematical Bases for Derivation of Divisia Index of Discretionary Tax Revenue and Properties of Constant Structure Method

In this Appendix, mathematical proofs are given for some of the statements made in the text.

1. derivation of an aggregate tax function: T = ax^μ

The aggregate tax function has been defined in the text as

\begin{matrix} T (t) = f (t) = f (x_{1} (t), …, x_{i} (t), …, x_{k} (t); t) & (16) \end{matrix}

where T denotes total tax yield and x_i denotes the (proxy) bases for the ith category of tax.

Given the aggregate tax function f, if the following two conditions hold: (1) the function f is homogeneous of degree r > 0, and (2) the (proxy) bases grow at the same rate as GDP (x), so that $\frac{{\dot{x}}_{i} (t)}{x_{i} (t)} = \frac{\dot{x} (t)}{x (t)}$ , then

\begin{matrix} T (t) = a x {(t)}^{r} D^{*} (t) = a x^{μ} (t) & (17) \end{matrix}

where D^* is an index of discretionary tax change.

Proof

The definition of the Divisia index D(t), as given in the text, yields:

\begin{matrix} \frac{\dot{D} (t)}{D (t)} = \frac{\dot{T} (t)}{T (t)} - Σ_{i = 1}^{k} \frac{f_{i} (t) x_{i} (t)}{f (t)} \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} & (18) \end{matrix}

By condition (1) above, the degree of homogeneity of f implies that

\begin{matrix} Σ_{i = 1}^{k} \frac{f_{i} (t) x_{i} (t)}{f (t)} = r & (19) \end{matrix}

Substituting equation (19) into equation (18), utilizing condition (2) mentioned previously, and writing D^* for D, the result is ³⁴

\begin{matrix} \frac{{\dot{D}}^{*} (t)}{D^{*} (t)} = \frac{\dot{T} (t)}{T (t)} - \frac{r \dot{x} (t)}{x (t)} & (20) \end{matrix}

If there has been no discretionary tax change in the period [0,t, then $\frac{{\dot{D}}^{*} (τ)}{D^{*} (τ)} = 0$ , for all τ in the time interval [0,t]. Therefore, the left-hand side of the differential equation (20) is zero and its solution is given by

\begin{matrix} T (t) = a x^{τ} (t) & (21) \end{matrix}

where $a = \frac{T (0)}{x {(0)}^{r}}$ . On the other hand, if there has been a discretionary tax change in the period [0,t], then $\frac{{\dot{D}}^{*} (τ)}{D^{* (τ)}} \neq 0$ for some τ in the time interval [0,t]. In this case, the differential equation (20) has the following solution:

\begin{matrix} T (t) = a x^{r} (t) \frac{D^{*} (t)}{D^{*} (0)} & (22) \end{matrix}

It follows from equations (21) and (22) that, when there has been no discretionary tax change in the period [0,t], then $\frac{D^{*} (t)}{D^{*} (0)} = 1$ . This implies the invariance of the index D^*, which is normalized by setting D^*(0) = 1.

Further, if there has been no discretionary tax change, the elasticity of tax yield r must equal the buoyancy μ of the yield. If, however, there have been discretionary tax changes, then the elasticity of tax revenue and the discretionary effects combined equal the buoyancy of the tax. Thus, the aggregate tax function f can be written as

\begin{matrix} T (t) = a x^{r} (t) D^{*} (t) = a x^{μ} (t) & (23) \end{matrix}

2. discrete formula for β_i(t)

The definition of β_i(t) in the text is given by

\begin{matrix} β_{i} (t) = \frac{f_{i} (t) x_{i} (t)}{f (t)}, i = 1, …, k & (\begin{matrix} 24 \end{matrix}) \end{matrix}

where $f_{i} (t) = \frac{\partial f (t)}{\partial x_{i} (t)} .$ At discrete points in time, β_i(t) can be computed as

\begin{matrix} β_{i} (t) = \frac{T_{i} (t) - T_{i} (t - 1)}{x_{i} (t) - x_{i} (t - 1)} . \frac{x_{i} (t)}{T (t)} & (25) \end{matrix}

Proof

The partial derivative $f_{i} (t) = \frac{\partial f (t)}{\partial x_{i} (t)}$ can be interpreted as the marginal revenue of the ith base. At discrete points in time, holding all the bases constant from t – Δ_t to t, the change in the ith base from x_i(t – Δt) to x_i(t) yields a change in total revenue that is given by the change in yield of the ith category of tax, T_i. Thus, the ith marginal revenue can be written

\begin{matrix} \partial f (t) = T_{i} (t) - T_{i} (t - Δ t) & (26) \end{matrix}

But $\partial f (t) = f (…, x_{i} (t), …) - f (…, x_{i} (t - Δ t), …)$

\begin{matrix} = Δ t \frac{Δ x_{i} (t)}{Δ t} \frac{\partial f (t)}{\partial x_{i} (t)} & (27) \end{matrix}

(by Taylor’s expansion, ignoring second- and higher-order terms). Equations (26) and (27) yield the following:

\begin{matrix} \frac{T_{i} (t) - T_{i} (t - Δ t)}{Δ t \cdot \frac{Δ x_{i} (t)}{Δ t}} = \frac{\partial f (t)}{\partial x_{i} (t)} = f_{i} (t) & (28) \end{matrix}

Now, for Δt = 1, we have $\frac{Δ x_{i} (t)}{Δ t} = x_{i} (t) - x_{i} (t - 1)$ . Therefore,

\begin{matrix} f_{i} (t) = \frac{T_{i} (t) - T_{i} (t - 1)}{x_{i} (t) - x_{i} (t - 1)} . & (29) \end{matrix}

Hence, substitution of equation (29) into equation (24) yields the discrete version of β_i (t) as given in equation (25).

3. computation of elasticity

An estimate of the elasticity of tax revenue, as given by equation (12) in the text, is

\begin{matrix} \hat{r} = μ - \frac{l o g D (n)}{l o g [x (n) / x (0)]} & (30) \end{matrix}

where μ = the buoyancy of tax revenue obtained by regressing log T(t) = α + μ log x(t) and

\begin{matrix} \log D (n) = l o g (\frac{T (n)}{T (0)}) - Σ_{i = 1}^{k} {\tilde{β}}_{i} l o g (\frac{x_{i} (n)}{x_{i} (0)}) & (31) \end{matrix}

Now, equation (7) in the text gives

\begin{matrix} {\tilde{β}}_{i} l o g (\frac{x_{i} (n)}{x_{i} (0)}) = \int_{0}^{n} β_{i} (t) \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} d t & (32) \end{matrix}

which, in discrete version, may be written as

\begin{matrix} {\tilde{β}}_{i} l o g (\frac{x_{i} (n)}{x_{i} (0)}) = Σ_{t = 1}^{n} β_{i} (t) \frac{x_{i} (t) - x_{i} (t - 1)}{x_{i} (t - 1)} & (33) \end{matrix}

Substitution of the expression for β_i(t), as given in equation (25), into equation (33) yields

\begin{matrix} {\tilde{β}}_{i} l o g (\frac{x_{i} (n)}{x_{i} (0)}) = Σ_{t = 1}^{n} [\frac{T_{i} (t) - T_{i} (t - 1)}{T (t)} \cdot \frac{x_{i (t)}}{x_{i (t - 1)}}] & (34) \end{matrix}

where T_i and x_i are defined as before.

Hence, substituting the right-hand side of equation (34) into equation (31) yields

\begin{matrix} l o g D (n) = l o g \frac{T (n)}{T (0)} - Σ_{i = 1}^{k} Σ_{t = 1}^{n} [\frac{T_{i} (t) - T_{i} (t - 1)}{T (t)} \cdot \frac{x_{i} (t)}{x_{i} (t - 1)}] & (35) \end{matrix}

Similarly, for the aggregative Divisia index D^*, it can be shown that

\begin{matrix} l o g D^{*} (n) = l o g \frac{T (n)}{T (0)} - Σ_{t = 1}^{n} [\frac{T (t) - T (t - 1)}{T (t)} \cdot \frac{x (t)}{x (t - 1)}] & (36) \end{matrix}

Substitutions of the computed values of log D and log D^* into equation (30) yield the estimates of elasticity, $\hat{r}$ and ${\hat{r}}^{*}$ .

4. some properties of the constant structure method

The methodology of constructing the constant rate series of tax revenue is described below.

Let $T (t) = Σ_{i = 1}^{k} T_{i} (t)$ = total tax revenue, comprising yields from k categories of taxes in period t; x_i(t) = the base for the ith category of taxes; and r = the reference year. The average effective tax rate for the ith category of taxes in the reference year is

\begin{matrix} t_{i} (r) = \frac{T_{i} (r)}{x_{i} (r)} & (37) \end{matrix}

so that ${\hat{T}}_{i} (t) = t_{i} (r) x_{i} (t)$ = the simulated tax yield of the ith category of taxes with respect to the reference year and

\begin{matrix} \hat{T} (t) = Σ_{i = 1}^{k} {\hat{T}}_{i} (t) = Σ_{i = 1}^{k} t_{i} (r) x_{i} (t) & (38) \end{matrix}

equals the simulated total revenue at period t.

The elasticity of tax revenue is obtained by the constant structure method as the slope of the regression equation

\begin{matrix} l o g \hat{T} (t) = a + λ l o g x (t) & (39) \end{matrix}

where x(t) = GDP in period t.

The normal equation for the ordinary least-squares estimate of λ is given (dropping the t variable) by

\begin{matrix} \hat{λ} = \frac{Σ_{t = 1}^{n} μ ν - n \bar{μ} \bar{ν}}{Σ_{t = 1}^{n} ν^{2} - n {\bar{ν}}^{2}} & (40) \end{matrix}

where $μ = l o g \hat{T}, ν = l o g x$ , and μ, ν are averages over the time series. Consider now the general case where the base-to-GDP elasticities are different from each other, implying that the bases grow at different rates. This can be represented by

\begin{matrix} x_{i} = a_{i} x^{λ i}; i = 1, …, k & (41) \end{matrix}

Utilizing these relations in equation (38) and, in turn, in equation (40), the result is

\begin{matrix} μ = l o g (Σ_{i = 1}^{k} b_{i} x^{λ i}); b_{i} = a_{i} t_{i} (r) & (42) \end{matrix}

so that

\begin{matrix} \hat{λ} = \frac{Σ_{t = 1}^{n} \log x \cdot log (Σ_{i = 1}^{k} b_{i} x^{λ i}) - n \bar{l o g x} \cdot \bar{l o g (Σ b_{i} x^{λ i})}}{Σ {(\log x)}^{2} - n {(\bar{\log x})}^{2}} & (43) \end{matrix}

In order to see the range of values within which $\hat{λ}$ lies, let λ_h and λ_s be, respectively, the largest and smallest of the base-to-GDP elasticities, λ_i—that is,

\begin{matrix} λ_{s} \leq λ_{i} \leq λ_{h} for all i = 1, …, k & (44) \end{matrix}

The inequality of system (44) implies

\begin{matrix} l o g (Σ b_{i} x^{λ s}) & \leq l o g (Σ b_{i} x^{λ s}) & \leq l o g (Σ b_{i} x^{λ h}) & (45) \end{matrix}

Substitution of $μ = l o g Σ b_{i} x^{λ s}$ into equation (40) yields

\begin{matrix} \frac{\overset{n}{Σ} μ ν - n \bar{μ} \bar{ν}}{Σ ν^{2} - n {\bar{ν}}^{2}} = \frac{\overset{n}{Σ} \log x \cdot \log (Σ b_{i} x^{λ s}) - n \bar{\log x} \cdot \bar{l o g (Σ b_{i} x^{λ s})}}{Σ {(\log x)}^{2} - n {(\bar{\log x})}^{2}} = λ_{s} & (46) \end{matrix}

since $l o g (Σ b_{i} x^{λ s}) = l o g [x^{λ s} (Σ b_{i})] = λ_{s} l o g x + l o g (Σ b_{i})$ .

Similarly, if $μ = l o g (Σ b_{i} x^{λ h})$ is substituted into equation (40), the result is:

\begin{matrix} \frac{\overset{n}{Σ} μ ν - n \bar{μ} \bar{ν}}{\overset{n}{Σ} ν^{2} - n {\bar{ν}}^{2}} = \frac{\overset{n}{Σ} \log x \cdot l o g (Σ b_{i} x^{λ h}) - n \bar{\log x} \cdot \bar{log (Σ b_{i} x^{λ h}})}{\overset{n}{Σ} {(\log x)}^{2} - n {\bar{(\log x)}}^{2}} & (\begin{matrix} 47 \end{matrix}) \\ = λ h \end{matrix}

It follows that the range within which $\hat{λ}$ lies in equation (43) is given by inequality system (45) and the relations (46) and (47)—that is,

\begin{matrix} λ_{s} \leq \hat{λ} \leq λ_{h} & (48) \end{matrix}

This inequality shows that elasticity estimated by the constant structure method must always lie between the largest and the smallest base-to-GDP elasticities irrespective of the reference year. Of particular importance is the case when all the bases grow at the same rate as GDP or at rates proportional to the growth rate of GDP. This implies that all the base-to-GDP elasticities are equal, so that λ_i = λ or 1 for i = 1, … k. In this situation, the elasticity $\hat{λ}$ estimated by the constant structure method equals either 1 or λ, depending on whether the bases grow at the same rate as GDP or at rates proportional to the growth rate of GDP.

Table 8.

Four Selected Countries: Estimates of Aggregate and Individual Tax and Base Equations

Sources: United States: Department of Commerce, Bureau of the Census, Historical Statistics of the United States: Colonial Times to 1970 (Washington, 1975), and Statistical Abstract of the United States: 1976 (Washington, 1976); Department of Commerce, Bureau of Economic Analysis, Survey of Current Business, various issues; Department of Commerce, United States Income and Ouput. United Kingdom: Central Statistical Office, National Income and Expenditure, 1964–74 (London, 1975); Chancellor of the Exchequer, Financial Statement and Budget Report, various volumes. Malaysia: Department of Inland Revenue, Annual Report, various issues. Kenya: East African Community, Income Tax Department, Report, various volumes; Ministry of Economic Planning and Development, Economic Survey, various volumes; and Ministry of Finance and Planning, Central Bureau of Statistics, Statistical Abstract, various volumes; United Nations, Statistical Office, Yearbook of International Trade Statistics, various volumes, and Statistical Yearbook, various volumes.

Table 8.

Four Selected Countries: Estimates of Aggregate and Individual Tax and Base Equations

Double Logarithmic Equation			Regressed on	Constant Term	Coefficient	R²	Durbin- Watson Statistic
Buoyancy method
	United States
		Total tax revenue	GDP	–1.888	1.040	0.988	2.001
		Individual income tax	GDP	–2.967	1.083	0.982	1.047
		Corporate profits tax	GDP	–0.594	0.595	0.852	1.551
		Consumption tax	GDP	–0.707	0.525	0.945	0.577
		Estate and gift tax	GDP	–6.966	1.215	0.933	0.788
		Social security and employment tax	GDP	–8.005	1.732	0.988	1.154
	United Kingdom
		Total tax revenue	GDP	–2.891	1.177	0.985	0.4%
		Individual income tax	GDP	–6.964	1.452	0.986	0.618
		Corporate profits tax	GDP	1.216	0.556	0.533	0.529
		Consumption tax	GDP	–2.969	1.110	0.972	0.331
		Social security and employment tax	GDP	–7.805	1.462	0.987	0.892
	Malaysia
		Total tax revenue	GDP	–7.828	1.696	0.966	0.820
		Income tax (individual and corporate)	GDP	–10.400	1.841	0.909	0.933
		Import tax	GDP	–2.466	0.964	0.915	1.050
		Export tax	GDP	–1.958	0.828	0.680	1.877
		Other indirect taxes	GDP	–17.155	2.560	0.848	0.605
	Kenya
		Total tax revenue	Monetary GDP	–4.176	1.422	0.993	2.643
		Income tax (individual and corporate)	Monetary GDP	–5.443	1.476	0.965	0.992
		Import tax	Monetary GDP	–2.130	0.881	0.927	1.466
		Excise	Monetary GDP	–5.385	1.307	0.922	0.823
		Other indirect taxes	Monetary GDP	–5.443	1.227	0.846	1.818
Constant structure elasticity (initial year)
		United States	GDP	–1.817	1.010	0.997	0.726
		United Kingdom	GDP	–1.316	1.017	0.999	0.999
		Malaysia	GDP	–3.620	1.191	0.8%	1.064
		Kenya	Monetary GDP	–1.876	1.002	0.988	1.590
Constant structure elasticity (end of year)
		United States	GDP	–1.680	1.012	0.998	0.766
		United Kingdom	GDP	–1.102	1.014	0.999	1.667
		Malaysia	GDP	–3.292	1.181	0.928	1.101
		Kenya	Monetary GDP	–1.033	0.912	0.991	1.475
Base equations
	United States
		Adjusted gross individual income	GDP	–0.553	1.031	0.999	1.252
		Net income of corporation (before tax)	GDP	–2.032	0.983	0.948	0.724
		Personal consumption	GDP	–0.310	0.993	0.998	0.777
		Personal income	GDP	–0.368	1.037	0.999	1.016
	United Kingdom
		Personal income (before tax)	GDP	–0.893	1.081	0.999	1.163
		Net corporate income (before tax)	GDP	–2.399	1.104	0.987	0.780
		Personal consumption	GDP	0.197	0.950	0.999	1.866
	Malaysia
		Chargeable income (individual and corporate)	GDP	–6.968	1.631	0.952	1.492
		Merchandise imports, c.i.f.	GDP	–0.615	0.985	0.614	0.975
		Merchandise exports, f.o.b.	GDP	–0.420	0.879	0.672	0.806
		Personal consumption	GDP	–0.853	1.059	0.981	1.491
	Kenya
		Chargeable income (individual and corporate)	Monetary GDP	–0.988	0.795	0.990	0.826
		Merchandise imports, c.i.f.	Monetary GDP	–2.062	1.179	0.970	1.775
		Monetary personal consumption	Monetary GDP	0.138	0.896	0.991	2.028

Table 8.

Four Selected Countries: Estimates of Aggregate and Individual Tax and Base Equations

Double Logarithmic Equation			Regressed on	Constant Term	Coefficient	R²	Durbin- Watson Statistic
Buoyancy method
	United States
		Total tax revenue	GDP	–1.888	1.040	0.988	2.001
		Individual income tax	GDP	–2.967	1.083	0.982	1.047
		Corporate profits tax	GDP	–0.594	0.595	0.852	1.551
		Consumption tax	GDP	–0.707	0.525	0.945	0.577
		Estate and gift tax	GDP	–6.966	1.215	0.933	0.788
		Social security and employment tax	GDP	–8.005	1.732	0.988	1.154
	United Kingdom
		Total tax revenue	GDP	–2.891	1.177	0.985	0.4%
		Individual income tax	GDP	–6.964	1.452	0.986	0.618
		Corporate profits tax	GDP	1.216	0.556	0.533	0.529
		Consumption tax	GDP	–2.969	1.110	0.972	0.331
		Social security and employment tax	GDP	–7.805	1.462	0.987	0.892
	Malaysia
		Total tax revenue	GDP	–7.828	1.696	0.966	0.820
		Income tax (individual and corporate)	GDP	–10.400	1.841	0.909	0.933
		Import tax	GDP	–2.466	0.964	0.915	1.050
		Export tax	GDP	–1.958	0.828	0.680	1.877
		Other indirect taxes	GDP	–17.155	2.560	0.848	0.605
	Kenya
		Total tax revenue	Monetary GDP	–4.176	1.422	0.993	2.643
		Income tax (individual and corporate)	Monetary GDP	–5.443	1.476	0.965	0.992
		Import tax	Monetary GDP	–2.130	0.881	0.927	1.466
		Excise	Monetary GDP	–5.385	1.307	0.922	0.823
		Other indirect taxes	Monetary GDP	–5.443	1.227	0.846	1.818
Constant structure elasticity (initial year)
		United States	GDP	–1.817	1.010	0.997	0.726
		United Kingdom	GDP	–1.316	1.017	0.999	0.999
		Malaysia	GDP	–3.620	1.191	0.8%	1.064
		Kenya	Monetary GDP	–1.876	1.002	0.988	1.590
Constant structure elasticity (end of year)
		United States	GDP	–1.680	1.012	0.998	0.766
		United Kingdom	GDP	–1.102	1.014	0.999	1.667
		Malaysia	GDP	–3.292	1.181	0.928	1.101
		Kenya	Monetary GDP	–1.033	0.912	0.991	1.475
Base equations
	United States
		Adjusted gross individual income	GDP	–0.553	1.031	0.999	1.252
		Net income of corporation (before tax)	GDP	–2.032	0.983	0.948	0.724
		Personal consumption	GDP	–0.310	0.993	0.998	0.777
		Personal income	GDP	–0.368	1.037	0.999	1.016
	United Kingdom
		Personal income (before tax)	GDP	–0.893	1.081	0.999	1.163
		Net corporate income (before tax)	GDP	–2.399	1.104	0.987	0.780
		Personal consumption	GDP	0.197	0.950	0.999	1.866
	Malaysia
		Chargeable income (individual and corporate)	GDP	–6.968	1.631	0.952	1.492
		Merchandise imports, c.i.f.	GDP	–0.615	0.985	0.614	0.975
		Merchandise exports, f.o.b.	GDP	–0.420	0.879	0.672	0.806
		Personal consumption	GDP	–0.853	1.059	0.981	1.491
	Kenya
		Chargeable income (individual and corporate)	Monetary GDP	–0.988	0.795	0.990	0.826
		Merchandise imports, c.i.f.	Monetary GDP	–2.062	1.179	0.970	1.775
		Monetary personal consumption	Monetary GDP	0.138	0.896	0.991	2.028

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Mr. Choudhry, economist in the Fiscal Analysis Division of the Fiscal Affairs Department, holds degrees in statistics and economics from the University of Dacca (Bangladesh) and the University of California at Berkeley.

For a discussion and an evaluation of these three major methods of adjustment of historical revenue for discretionary tax measures, see Chelliah and Chand (1974).

Tanzi (1969b) has developed a method of estimating the income tax elasticity of the United States that utilizes only cross-section data. Briefly, the method is based on the hypothesis that a time series of income tax revenue (T) and taxable income (TI) before and after tax, all on a per capita basis, can be simulated from data for these variables from different regions of a country. While the method is very appealing, its application is rather limited, as very few countries have such regional data. Recently, however, Tanzi’s method has been applied to estimate the elasticity of income tax in Denmark and Canada. See Andersen (1973) and Boucher (1977).

Some properties of the constant structure method limit its use. These are discussed in some detail in Section III, and the mathematical derivation of these properties is given in the Appendix.

These limitations are owing to the fact that the Divisia index is a line integral. In practical applications, especially with very large changes, the discrete version does not approximate the continuous function well.

Some aggregation problems ignored in the text discussion are worth mentioning. Both the aggregate production and tax functions assume homogeneous factor inputs and bases, respectively. An immediate consequence of this assumption is that the effects of compositional shifts in factor inputs and bases are not captured by the respective aggregate functions. As a result, the measurement of technical change or discretionary tax change can be biased. For example, consider a shift in, say, income distribution in favor of upper-income groups, with aggregate taxable income and the number of taxpayers unchanged. Clearly, even with no discretionary tax action, a change in revenue will take place, since the average income tax rate has increased. Similarly, a change in output occurs if there is a shift in the composition of labor between specific industry groups even when no technical change has taken place. Shifts in the composition of nonhomogeneous bases or factor inputs, such as those that often arise in reality, cannot be accounted for by a movement along the curve. If such shifts cause a change in tax yield or output, then this is perceived as a discretionary tax change or a technical change when, in fact, there was no such change.

See Hulten (1973)), p. 1017. The works of Solow (1957), Denison (1962), Kendrick (1961), Jorgenson and Griliches (1967), and Christensen and Jorgenson (1970) are important contributions to the logical foundations of the Divisia index.

Hulten has shown that the invariance property of the Divisia index is implied by the path independence property, which asserts that the value of the index depends on the current values of the variables in the production function and not on the historical path of the variables.

See Hulten (1973, p. 1018) for these necessary and sufficient conditions. A third condition required for the Divisia index of technical change, but not for that of discretionary tax change, states that the production function f has an associated input price vector that is unique up to a scalar multiplication. This implies that the ratio of marginal product between any two factors x_i and x_j equals their corresponding price ratio. That is, f_i/f_j = p_i/p_j. In the Divisia index of technical change, f_i and f_j are replaced by the factor prices p_i, and p_j. In the Divisia index of discretionary tax change, prices are not involved; hence, there is no need for the third condition.

Hulten demonstrated the elimination of the linear homogeneity assumption in the context of a production function characterized by nonconstant returns to scale. In fact, he also eliminated the use of the third condition referred to in footnote 7.

For a large number of developing countries, the average tax ratio has increased from 13.6 per cent for 1961–68 to 15.1 per cent for 1969–71. For more details, see Chelliah, Baas, and Kelly (1975).

See Section 1 in the Appendix for the derivation of the conditions.

The aggregation problem encountered in T = ax^μ is similar to that encountered in production or consumption functions.

¹²

Notice that equation (7) can be transformed as follows:

$\begin{matrix} {\tilde{β}}_{i} = \frac{1}{n} \int_{0}^{n} β_{i} (t) \frac{ρ_{i} (t)}{{\bar{ρ}}_{i}} & (7^{'}) \end{matrix}$

where $ρ_{i} (t) = \frac{{\dot{x}}_{i} (t)}{x_{i} (t)}$ and $n {\bar{ρ}}_{i} = \int_{0}^{n} \frac{{\dot{x}}_{i} (t)}{x_{i} (t)} d t = log (\frac{x_{i} (n)}{x_{i} (0)})$ . Thus, the constant ${\tilde{β}}_{i}$ s are the weighted average of the fluctuating β_i(t), where the weights are the ratios of the instantaneous rates of growth of the bases to their average rates of growth in the time interval [0,n]. Since historical data are often available on an annual basis, the relevant discrete version of equation (7) is given by

$\begin{matrix} {\tilde{β}}_{i} = \frac{1}{n} Σ_{t = 1}^{n} β_{i} (t) \frac{ρ_{i} (t)}{{\bar{ρ}}_{i}} & (7 ") \end{matrix}$

where $β_{i} (t) = \frac{T_{i} (t) - T_{i} (t - 1)}{x_{i} (t) - x_{i} (t - 1)} \cdot \frac{x_{i} (t)}{T (t)}$

See Section 2 of the Appendix for the derivation of the discrete version of β_i(f).

Even in measuring technical change, the same bias occurs in estimating automatic growth of factor productivity. This is because β_i(t), which is a factor share, can reflect the effects of technical change on factor productivity, which, in turn, bias (generally downward) the rate of technical change.

Usually, r is greater than, or equal to, unity. For the sake of generality, it is assumed that r > 0.

Also, if the bases grow proportionally with respect to GDP, the aggregate tax function will have the same form.

In practical application, the buoyancy μ is a function of the period [0, t].

Earlier in the paper, the possible sources of bias in the index of discretionary change were discussed. From expression (13), it follows that when discretionary changes increase revenue, the Divisia index method overestimates the size of the elasticity of tax revenue and vice versa when discretionary changes decrease revenue.

See Section 3 in the Appendix for the formulas for computing log D and log D ^* in order to obtain the elasticity estimates $\hat{r}$ and ${\hat{r}}^{*}$ .

Analytically, the Divisia index method is similar to the proportional adjustment method of estimating elasticity. Essentially, in adjusting total revenue of discretionary changes, the proportional adjustment method performs the following type of operation:

$\hat{T} (t) = \frac{T (t)}{D * * (t)}$

where D^**(t) is an index of discretionary change that is constructed as a chain index to adjust historical revenue T(t). The aggregate tax equation in the proportional adjustment method is given by

$\hat{T} (t) = a x^{r} (t)$

which is similar to

$\frac{T (t)}{D^{*} (t)} = a x^{r} (t)$

of the Divisia index method. The main difference between the two methods depends on how the two indices of discretionary change (D^* and D^**) are constructed. It has already been noted that D^* is biased in practice, since it is estimated residually from unadjusted historical data. In the proportional adjustment method, if the information on discretionary changes is reliable, then D^** may be a superior estimate of such changes. Consequently, the proportional adjustment method should be used under such circumstances.

The major categories of federal/central tax revenues considered for constructing Divisia indices of discretionary revenues are those shown in published government accounts for each country. Since estate and gift taxes, which are shown as a separate category for the United States, constitute less than 2 per cent of federal tax revenue, these taxes will not be discussed further in the text of this paper.

In the United Kingdom, social insurance and employment taxes are paid in the form of national insurance contributions by the taxpayers.

It is not easy to determine just what should be the proxy base for the estate and gift taxes for the United States. Private wealth may be a close proxy base for such taxes, but such data are not available. Similarly, GDP at factor cost may not be a close proxy base for the social insurance and employment tax. This tax is generally regressive with respect to individual or personal income. Of course, one could choose individual/personal income as a proxy base; however, since individual/personal income is almost unit elastic with respect to GDP at factor cost, the use of GDP as a proxy base is just as good under the circumstances.

Tax ratios in Malaysia and Kenya rose from 16.5 per cent of GDP in 1961 and 14.1 per cent in 1962 to 21.8 per cent in 1973 and 17.0 per cent in 1974, respectively. In the United Kingdom, the tax ratio rose with some fluctuations from 32.5 per cent of GDP in 1955 to 39.5 per cent in 1974. In the United States, after a sharp decline from 20.8 per cent of GDP in 1954 to 18.0 per cent in 1955, the tax ratio recovered in 1956 and thereafter remained fairly stable at around 20.5 per cent of GDP.

The elasticity of components of tax revenue cannot be estimated by the disaggregative Divisia method because it is assumed that each base represents its own tax category.

The effects of discretionary measures on tax yield can also be inferred from the movements in the shares of the components of tax revenue. For instance, despite the presence of progressive elements in individual income taxation, the share of the individual income tax remained virtually unchanged between 1955 and 1975 at about 45 per cent of tax revenue. The share of the corporate income tax steadily fell from 27.4 per cent in 1955 to 14.9 per cent in 1975 while that of the consumption tax (excises and import taxes) showed a similar declining trend, falling from 14.9 per cent to 7.4 per cent. On the other hand, the share of social security and employment taxes rose from 12.1 per cent in 1955 to 31.4 per cent in 1975, with sharp increases since 1967.

In a recent study of the U. K. tax system, it was found that the individual income tax had buoyancies of 1.02 and 1.45 for the periods 1950/51–1970/71 and 1960/61–1970/71, respectively, while the elasticity estimates based on the proportional adjustment methods were 1.37 and 1.53. The study also noted that “without these reductions [discretionary changes] the share of income tax in total revenue would have increased markedly” (See Baas and Dixon (1974), p. 14 and Table 5, p. 16.). The increase in the estimates of the elasticity and the buoyancy in the subperiod 1960/61–1970/71, compared with that for the whole period 1950/51–1970/71, indicates that, although the discretionary measures reduced revenue, they improved the progressivity of the individual income tax system.

The elasticity of corporate income before taxes with respect to GDP is found to be 1.10 (See Table 6.). The share of corporate taxes in total tax revenue declined from 18.0 per cent in 1955 to about 8 per cent in 1974.

See Choudhry (1975) on the effects of 1967 tax changes on elasticity in West Malaysia.

It should be mentioned that the smaller buoyancy of income tax in Kenya than in Malaysia reflects the differences in the responsiveness of the tax base to GDP rather than the differences in the progressivity of rate structures. In fact, the income tax system is more progressive in Kenya, but the buoyancy of the base with respect to GDP was found to be only 0.8, while in Malaysia the base-to-GDP buoyancy was found to be twice as high (See Table 6.).

For a description and evaluation of these two methods, see Bahl (1972).

The proportionally adjusted revenue data have been obtained from Baas and Dixon (1974), and Chand (1975).

Mathematical proofs of these statements are given in Section 4 of the Appendix.

In a recent study of the individual income tax utilizing cross-sectional data from each of the states in the United States, it was found that, despite revenue-reducing discretionary measures, the elasticity of individual income tax with respect to adjusted gross income gradually improved over time, from 1.37 in 1963 to 1.48 in 1972. See Tanzi (1976), p. 449, Table 4.

Instead of $\frac{{\dot{x}}_{i} (t)}{x_{i} (t)} = \frac{\dot{x} (t)}{x (t)}$ as given by condition (2), if it is assumed that the bases grow at a rate that is proportionate to the growth rate of GDP—that is, $\frac{{\dot{x}}_{i} (t)}{x_{i} (t)} = α \frac{\dot{x} (t)}{x (t)}$ for all i, then rα obtained in place of r in equation (20).

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IMF Staff papers: Volume 26 No. 1

Author: International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1979: Issue 001

Publisher:: International Monetary Fund

ISBN:: 9781451956528

ISSN:: 1020-7635

Pages:: 232

DOI:: https://doi.org/10.5089/9781451956528.024

Keywords
I. Methodology
- 1. conceptual framework
- 2. derivation of the divisia index of discretionary tax revenues
II. Empirical Results of Case Studies
III. Some Further Results
IV. Concluding Remarks
- APPENDIX
REFERENCES

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Table 8.

Four Selected Countries: Estimates of Aggregate and Individual Tax and Base Equations

Double Logarithmic Equation			Regressed on	Constant Term	Coefficient	R²	Durbin- Watson Statistic
Buoyancy method
	United States
		Total tax revenue	GDP	–1.888	1.040	0.988	2.001
		Individual income tax	GDP	–2.967	1.083	0.982	1.047
		Corporate profits tax	GDP	–0.594	0.595	0.852	1.551
		Consumption tax	GDP	–0.707	0.525	0.945	0.577
		Estate and gift tax	GDP	–6.966	1.215	0.933	0.788
		Social security and employment tax	GDP	–8.005	1.732	0.988	1.154
	United Kingdom
		Total tax revenue	GDP	–2.891	1.177	0.985	0.4%
		Individual income tax	GDP	–6.964	1.452	0.986	0.618
		Corporate profits tax	GDP	1.216	0.556	0.533	0.529
		Consumption tax	GDP	–2.969	1.110	0.972	0.331
		Social security and employment tax	GDP	–7.805	1.462	0.987	0.892
	Malaysia
		Total tax revenue	GDP	–7.828	1.696	0.966	0.820
		Income tax (individual and corporate)	GDP	–10.400	1.841	0.909	0.933
		Import tax	GDP	–2.466	0.964	0.915	1.050
		Export tax	GDP	–1.958	0.828	0.680	1.877
		Other indirect taxes	GDP	–17.155	2.560	0.848	0.605
	Kenya
		Total tax revenue	Monetary GDP	–4.176	1.422	0.993	2.643
		Income tax (individual and corporate)	Monetary GDP	–5.443	1.476	0.965	0.992
		Import tax	Monetary GDP	–2.130	0.881	0.927	1.466
		Excise	Monetary GDP	–5.385	1.307	0.922	0.823
		Other indirect taxes	Monetary GDP	–5.443	1.227	0.846	1.818
Constant structure elasticity (initial year)
		United States	GDP	–1.817	1.010	0.997	0.726
		United Kingdom	GDP	–1.316	1.017	0.999	0.999
		Malaysia	GDP	–3.620	1.191	0.8%	1.064
		Kenya	Monetary GDP	–1.876	1.002	0.988	1.590
Constant structure elasticity (end of year)
		United States	GDP	–1.680	1.012	0.998	0.766
		United Kingdom	GDP	–1.102	1.014	0.999	1.667
		Malaysia	GDP	–3.292	1.181	0.928	1.101
		Kenya	Monetary GDP	–1.033	0.912	0.991	1.475
Base equations
	United States
		Adjusted gross individual income	GDP	–0.553	1.031	0.999	1.252
		Net income of corporation (before tax)	GDP	–2.032	0.983	0.948	0.724
		Personal consumption	GDP	–0.310	0.993	0.998	0.777
		Personal income	GDP	–0.368	1.037	0.999	1.016
	United Kingdom
		Personal income (before tax)	GDP	–0.893	1.081	0.999	1.163
		Net corporate income (before tax)	GDP	–2.399	1.104	0.987	0.780
		Personal consumption	GDP	0.197	0.950	0.999	1.866
	Malaysia
		Chargeable income (individual and corporate)	GDP	–6.968	1.631	0.952	1.492
		Merchandise imports, c.i.f.	GDP	–0.615	0.985	0.614	0.975
		Merchandise exports, f.o.b.	GDP	–0.420	0.879	0.672	0.806
		Personal consumption	GDP	–0.853	1.059	0.981	1.491
	Kenya
		Chargeable income (individual and corporate)	Monetary GDP	–0.988	0.795	0.990	0.826
		Merchandise imports, c.i.f.	Monetary GDP	–2.062	1.179	0.970	1.775
		Monetary personal consumption	Monetary GDP	0.138	0.896	0.991	2.028

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Abstract

I. Methodology

1. conceptual framework

2. derivation of the divisia index of discretionary tax revenues

II. Empirical Results of Case Studies

III. Some Further Results

IV. Concluding Remarks

APPENDIX

1. derivation of an aggregate tax function: T = axμ

2. discrete formula for βi(t)

3. computation of elasticity

4. some properties of the constant structure method

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1. derivation of an aggregate tax function: T = ax^μ

2. discrete formula for β_i(t)