The Treatment of Indirect Taxation in Econometric Models: An Analytical Survey

Nurun N. Choudhry

doi:10.5089/9781451947557.024.A007

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The Treatment of Indirect Taxation in Econometric Models: An Analytical Survey

Author: Mr. Nurun N. Choudhry

Publication Date:: 01 Jan 1977

eISBN:: 9781451947557

ISBN:: 9781451947557

Language:: English

Keywords:: SP; income; indirect tax tax rate; econometric model; budget deficit; price level; aggregate demand; Consumption; Wages; Econometric models; Wage adjustments

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Recent literature on the simulation of existing econometric models of national economies directs little attention to the incidence of indirect taxation. This “inattention” is all the more conspicuous since men of affairs are paying closer attention to the effects of changes in indirect taxation. Such changes, particularly in an era of increasingly scarce natural resources, have a significant impact on prices, employment, output, and the balance of payments. Policymakers are therefore most interested in having at their disposal realistic econometric models that incorporate the effects of changes in indirect taxation.

Abstract

The ability to predict the consequences of changes in indirect taxation or, for that matter, of any policy action is critically dependent upon the specification of individual behavioral responses, as well as the overall specification of the real and financial sectors of the economy, in such econometric models. Of late, a number of national econometric models have been designed; however, a comparative review of the characteristics of such models with regard to the specification of indirect taxes has not been undertaken. Such a review would be useful to both policymakers and econometricians.

This survey of econometric models focuses attention on the specification of indirect taxes in econometric models and its implications for the effects of indirect taxes on prices. Section I briefly discusses, and provides examples to illustrate, the two major problems involved in specifying indirect taxes in econometric models. The general nature of the specifications of indirect tax functions and indirect tax rates in price formation equations in the selected econometric models is discussed in Section II. In Section III, the dynamic incidence of indirect taxation is investigated with the help of price and wage dynamics as incorporated in these models. Section IV consists of a few concluding remarks.

I. Problems of Specification of Indirect Taxes

Modern incidence theory makes a basic distinction between tax incidence on the supply side (or the sources of income) and incidence on the demand side (or the uses of income). When the impact of changes in indirect taxation is estimated using econometric models that reflect this kind of distinction, there is a definite improvement in both analytical rigor and quantitative results. An application of this theory is, however, extremely difficult. First, the myriad of interacting forces affecting the supply and demand schedules in different markets are difficult to quantify. Second, even if this difficulty is overcome, the theory of tax incidence has little to say about how indirect tax rate changes can be incorporated in the behavioral responses of the market mechanism.

Overcoming the first problem is relatively easy, at least in principle. Choosing a proper framework for the effects of indirect tax rate changes on prices will essentially depend on the use for which an econometric model is designed. At one extreme is the highly aggregative Keynesian model of income determination that is often extremely important qualitatively but seldom is acceptable to either econometricians or policymakers. At the other extreme is the logically consistent, but often unmanageable, general equilibrium analysis that incorporates even minor effects in its multimarket model of the economy, and consequently generates results that are difficult to assess from the standpoint of significance or reliability. Somewhere in between these limits lies the ideal model, which is both manageable and yields meaningful results on tax incidence.

The second problem is, perhaps, the more difficult one to tackle since it involves the element of time in an important way. Modern incidence theory, which deals with static incidence, can only attempt to explain the comparative-static effects of changes in equilibrium values. Policymakers, however, are interested not only in immediate effects but also in the time profile of effects. If, for instance, a policymaker is contemplating changes in a tax system to redistribute income in a desired direction, he is interested in the tax effects on income, employment, and prices—not only the effects in the immediate period after these tax changes are established but also the tax effects on saving, investment, and economic growth in later periods.

Broadly speaking, econometricians face at least two major problems in specifying indirect taxes in econometric models in order to measure the effects of incidence and shifting: (i) the general aggregation problem and (ii) the choice of an appropriate hypothesis to use in translating indirect tax rate changes into price changes in order to account for dynamic incidence. Both of these problems will be dealt with separately in this section.

1. the aggregation problem

The aggregation of indirect taxes is linked with the general problem of aggregation of production and consumption activities. The basic aggregation problem in formulating more flexible and more general functional forms to be used in econometric models is to maintain consistency of macrobehavioral relationships with regard to the aggregation of distinct commodities, whether in production or consumption. The meaning of “consistency” can be best explained by means of two simple examples, beginning with production activity.

Consider n commodities produced in n different firms, each with its own production function and each using specialized labor, so that wage rates differ between firms. The inverse demand function for labor for the ith firm, under the profit maximization assumption, is given by

\begin{matrix} w_{i} = p_{i} q_{i}^{'} (L_{i}), i = 1, \cdot \cdot \cdot n & (1) \end{matrix}

where W_i = nominal wage rate in the ith firm

P_i = producer price of the ith good

L_i = labor input in the production of ith good

$q_{i}^{'} = \frac{\partial q_{i}}{\partial L_{i}}$ = marginal productivity of labor.

Since the econometrician has to design a model containing only a limited number of equations, the n equations given in expression (1) have to be aggregated by means of a suitable price index P; a quantity index Q; an aggregate production function, Q(L), of the labor input index L; and a wage index W. Thus, the microequations given by (1) have to be transformed in the macrorelation

\begin{matrix} W = P Q^{'} (L) & (1') \end{matrix}

and, depending on the desired degree of disaggregation of production activities in an econometric model, the number of macrorelations of the type (1′) should correspond to the degree of disaggregation. The consistency requirement imposes the restriction that the total wage income in the aggregate (i.e., WL) equals the sum of wage income by firms (i.e., Σw_iL_i).¹

In econometric applications, the aggregate macrorelation (1′) has three variables: W, P, and L. Given any two of these variables, however defined (i.e., exogenously or endogenously), the remaining variable is obtained as a solution of (1′). A great majority of the econometric models surveyed in this paper tend to determine all these variables endogenously, in order to evaluate the connection between the various sectors in the models.²

Consider the aggregation of consumption activities. For the sake of simplicity, assume that the n commodities produced are consumed by households, and that indirect taxes (τ_i) are levied on consumption of these commodities on an ad valorem basis. Let C_d and C_s be the aggregate demand for, and supply of, consumer goods in nominal terms so that

C_{d} = \overset{n}{Σ} p_{i} (1 + τ_{i}) q_{i}

denotes consumers’ outlays on the n commodities, and

C_{s} = \overset{n}{Σ} p_{i} q_{i}

denotes the producers’ sales receipts; hence the difference

T_{I} = C_{d} - C_{s} = \overset{n}{Σ} τ_{i} p_{i} q_{i}

is the yield of indirect taxes levied at rates τi. Assume that the market prices of the consumer goods can be suitably aggregated into a single price index for consumer goods (P_c). The difference between the consumer price and the producer price of the ith commodity is the per unit yield of indirect taxation on that commodity. The consistency of aggregation of prices requires that there be a relationship between the consumer goods price index (P_c) and the producer price index (P). This can be written as

\begin{matrix} P_{c} = P (1 + τ) & (2) \end{matrix}

where τ is an idex of indirect tax rates (τ_i s). Further, since the production of q_is is already indexed by Q, and if the system is in static equilibrium, the yield of indirect taxes can be expressed as

\begin{matrix} T_{I} = τ P Q & (3) \end{matrix}

That (3) indeed holds true in static equilibrium can be shown as follows: The nominal outlay on consumption can be expressed as

\begin{matrix} C_{d} = P_{c} Q^{d} & (4) \end{matrix}

while the nominal sales revenues of producers is given by

\begin{matrix} C_{s} = P Q^{s} & (5) \end{matrix}

where Q^d and Q⁸ represent the demand for, and supply of, aggregated output. In equilibrium

\begin{matrix} Q^{d} = Q^{s} = Q & (6) \end{matrix}

Hence, the money yield of indirect taxes, which is the difference between C_d and C_s is given by (3).

Several observations about the above exposition of the nature of aggregation are in order. First, given the real demand for consumer goods (Q^d), it is relatively easy—at least in principle—to aggregate indirect taxes and study the results of their incidence. Second, if an econometric model is highly aggregative, such that there is only one aggregate consumption demand (Q^d), the differential impact of changes in individual indirect tax rates cannot be captured, since the aggregate index τ would behave as a general sales tax on all commodities. Clearly, if the policymaker is interested in the differential impacts of selective changes in indirect tax rates, there must be an appropriate degree of disaggregation of real consumption demands. Third, and related to the second observation above, is the issue of balanced budget incidence. In an econometric model with an aggregative consumption function, the financing of additional government expenditure by raising equivalent indirect tax revenue—for example, using sales tax A rather than sales tax B—would have the same balanced budget impact on private demand in both cases. In the real world, however, the balanced budget impact might be different in each case because of possible differences in the elasticity of consumption demand for different commodities, as well as possible differences in their supply elasticities. Fourth, in real economies, not all production is subjected to indirect taxation. The bulk of indirect taxation covers some consumer goods, while numerous other consumer goods are exempted. Accordingly, the indirect tax yield equation (3) should employ taxable commodities, rather than output Q. This could be done by some simple functional form, such as T_I = a + b(τPQ^d) or T_I = A(τPQ^d)^λ where λ is a parameter that reflects the actual base of indirect taxation. Fifth, the pure forms of equations (2) and (3) assume static equilibrium, while most econometric models incorporate dynamic elements to approximate reality. Hence, both these equations should have dynamic characteristics. Finally, in real situations, actual aggregate demand need not equal either actual production or potential output; therefore, there is an additional reason for modifying the pure form of the indirect tax yield equation (3).

A few words may be said about the general functional forms of production and consumption activities commonly employed in econometric models. In most models, the general functional forms—in particular, those used for consumption—recognize two important responses: substitution effects and income effects. Since these effects originate from microanalysis of consumer preferences, such functional forms generally embody implicit, often restrictive, assumptions regarding consumer preferences.

Most recent theoretical investigations of general functional forms point out that the conditions for perfect aggregation of producer and consumer demand models (based on microeconomic theory) are rather stringent.³ Although less restrictive theoretical functional forms have been developed, these have yet to be applied in econometric models.⁴ Consequently, econometric models make do with some of the more restrictive functional forms. However, this does not mean that econometric models apply ad hoc functional forms. A good many of the econometric models employ general functional forms that embody the concept of separability. This concept or idea allows commodities to be grouped in such a way that goods that interact closely (in the yielding of utility) are grouped together and treated as one commodity, while goods which are in different groups either do not interact or interact in only a general way. The degree of such grouping of commodities determines the extent of disaggregation that allows “reasonable” consistency in aggregation.⁵

As our interest in aggregation is primarily with regard to the incidence of indirect taxes, we should understand the implications of linear and nonlinear versions of the consumption functions (these versions include the permanent income hypothesis). Theoretical investigations of linear aggregation of consumer demand for individuals with identical tastes point out that macroresponse coefficients depend not only on the individual responses but also on the distribution of responses and incomes of consumers. This implies that linear aggregation of consumer demands do not, in general, satisfy the consistency requirement even if income distribution does not change.⁶ Most nonlinear aggregate consumption demand functions also do not satisfy the consistency requirement, as far as changes in income distribution are concerned. On balance, since consumers are not expected to have identical tastes and since income distribution may change in one way or another from time to time, it appears that aggregate consumer demand functions that satisfy, even approximately, the conditions for consistent aggregation do not exist.⁷

The preceding discussion facilitates an understanding of the problem of aggregation and its implications for the incidence of indirect taxes. For instance, if there is reason to believe that changes in certain indirect taxes may alter the consumption mix of consumers (such as increases in indirect taxes affecting consumption of luxury goods), or that there will be certain income distributional effects because of changes in the composition of output, it is likely that the existing general functional forms will bias the policy effects in such models. The extent of such bias and/or its direction is, of course, difficult to ascertain. One should at least be aware it exists and, if possible, anticipate its direction.

2. the problem of dynamic incidence

Static equilibrium analysis cannot deal with continuing effects of a given (indirect) tax policy, since most concepts of conventional economics, which hold rigorously in static general equilibrium, are not necessarily valid in a dynamic world where adjustment of prices and quantities does not take place instantaneously. In static analysis, the passing on of an indirect tax rate change to consumers alters the equilibrium price level. The chain, or the process, of price and quantity changes that involve time in an important way, however, is unknown. Also, most indirect taxes are, in the short or long run, treated as a part of business cost, and therefore passed on to consumers in the form of higher prices; but it is not known how fast this happens, or what the size of such forward shifting is during an inflationary period.

It may be argued that since static incidence analysis states that indirect taxes are, at least partially, passed on to consumers, a reduction in their rates ought to slow down inflation. A reduction in indirect taxation may well lower business costs, and therefore provide incentives to expand output. At the same time, it increases purchasing power, and therefore raises aggregate demand. The net result is unclear. If, however, the dynamics of inflation were such that it would eventually stop at some future date and prices would converge to a new, and higher, level, then it is possible that the new equilibrium price level would be lower than it would have been in the absence of an indirect tax reduction. It is, therefore, realistic to assume that a more satisfactory analysis of incidence of indirect taxes can be made in the context of macroeconomic theory, which attempts to integrate the dynamic elements of price and quantity adjustments.⁸

The dynamics of incidence of indirect taxes is, thus, an integral part of disequilibrium economics. Since much of the recent work in disequilibrium economics began because of the inadequacy of classical price dynamics in explaining movements in prices, wages, and output, a convenient approach to the problem of dynamic incidence will be to discuss it in the context of disequilibrium price and wage behavior.⁹

Most econometric models utilize some variants of the general theoretical structure of the markup hypothesis for their price formation equations. The basic premise of the markup hypothesis is that prices are set as a markup over “standard” or “normal” costs.¹⁰ The normal cost is a longer-run concept based on the underlying production function. Its strict use implies that temporary changes in costs arising from short-term variations in productivity are considered transient phenomena, and therefore need not be reflected in prices. The basic hypothesis can be modified so that temporary changes in costs may influence prices, though this influence need not be permanent.¹¹ The markup over cost may also be influenced by an excess or deficiency of aggregate demand relative to supply.¹²

The markup hypothesis of price behavior has several advantages in application to econometric models. First, although the basic hypothesis assumes an underlying production function, its modifications incorporate both long-run and short-run features. Second, it recognizes the importance of the cost of capital, which heavily influences the optimal pricing decision in the long run. Surprisingly enough, as we shall see, most econometric models omit this variable in their price-formation equations.¹³ Third, and related to the issue of optimal pricing, the markup hypothesis can easily lend itself to other concepts of long-run prices—such as the target-return pricing formula of Eckstein (1964).¹⁴ Fourth, the incorporation of indirect taxes can be carried out easily in the markup hypothesis, since such taxes can be considered as part of normal costs.¹⁵ Finally, the markup hypothesis does not require competitive conditions for its validity.¹⁶ This is an important factor, especially in view of the monopolistic elements in the real world.¹⁷ Therefore, this hypothesis is suitable for a satisfactory analysis of the shifting of indirect taxes.

In the markup hypothesis, the greatest influence on prices is the wage rate, because the latter constitutes the bulk of normal costs. Hence, variations in wage rates resulting from the inflation-real output (or inflation-unemployment) tradeoff, or other causes such as increases in wages through collective bargaining, will cause price changes. Further, since price adjustments do not take place instantaneously, the speed of adjustment of prices in response to changes in their determinants must have some dynamic pattern. Consequently, the nature of the dynamic incidence of indirect taxes depends not only on the basic price behavior hypothesis but also on how the price and wage adjustment equations incorporate dynamic elements. Wage adjustments, on the other hand, are believed to explain the behavior of unemployment, in the sense that there exists some relationship between the rate of change of money wages and the level of unemployment. In fact, in most econometric models, the latter hypothesis is an essential ingredient in capturing the dynamics of disequilibria among prices, wages, employment, and output.

In order to appreciate the nature of dynamic incidence of indirect taxes, the following examples of price and wage adjustment equations may be useful.

Assume that basic price behavior is determined by markup hypothesis, so that the target, or desired, price can be written as ¹⁸

\begin{matrix} P^{T} = \frac{λ K}{Q^{N}} + U L C^{N} + \frac{τ_{i} P}{1 + τ_{i}} & (7) \end{matrix}

where λ is the target rate of return on capital stock K; Q^N is the normal output; ULC^N is standard unit labor cost; τ_i is the rate of indirect taxation; and P is the market price of output.¹⁹ Without loss of generality (i.e., by properly choosing output and employment levels), we may write ULC^N = W, where W is the money wage rate.

Analogous to desired pricing, we assume that labor unions perceive a target or desired wage rate W^T, which is a function of market price P (a proxy for the cost of living index) and current money wage W. That is,

\begin{matrix} W^{T} = α P + β W & {(8)}^{20} \end{matrix}

If we postulate price and wage adjustments that depend on excess demand in the product and factor markets, as well as on the discrepancy between the target price and wage, respectively, we may write them as

\begin{matrix} \dot{P} / P = a_{1} X + a_{2} (\frac{P^{t} - P}{P}) + a_{3} \frac{\dot{W}}{W}, a_{i} > 0 & (9) \end{matrix}

and

\begin{matrix} \dot{W} / W = - b_{1} u + b_{2} (\frac{W^{t} - W}{W}), b_{i} > 0 & (10) \end{matrix}

where X is the real excess demand for output (i.e., $X = \frac{Q - Q^{N}}{Q^{N}},$ Q is the real actual demand) and $u = \frac{L^{N} - L}{L^{N}}$ is the unemployment rate.

Equations (9) and (10) generate the dynamics in the system, where producers attempt to obtain their price target P^T, and labor its desired wage rate W^T. If the desired price and wage are achieved, the dynamic system comes to rest and equilibrium is achieved (i.e., excess demand is zero so that Q = Q^N and the unemployment rate (u) is zero).²¹ This dynamic system, however, does not guarantee that the real wage offer by producers ${(\frac{W}{P})}_{1}$ equals the real wage claim by labor ${(\frac{W}{P})}_{2}$ when the system comes to rest. This can easily be shown by solving for $\frac{W}{P}$ from equations (9) and (10), by setting $\frac{\dot{P}}{P} = 0 = \frac{\dot{W}}{W},$ and utilizing equations (7) and (8). In fact, it is seen that

\begin{matrix} {(\frac{W}{P})}_{1} = 1 - \frac{τ_{i}}{1 + τ_{i}} - ρ & (11) \end{matrix}

and

\begin{matrix} {(\frac{W}{P})}_{2} = \frac{α}{1 - β} & (12) \end{matrix}

where $ρ = \frac{λ K}{P Q^{N}}$ is a constant in equilibrium.

Clearly, there is no reason to believe that the real wage rate equations (11) and (12) should be equal to each other. In order to achieve consistency (i.e., equality between $\begin{matrix} \begin{matrix} {(\frac{W}{P})}_{1} & a n d \end{matrix} & {\begin{matrix} (\frac{W}{P}) \end{matrix}}_{2} \end{matrix}$ ) in equilibrium, the model has to be closed by adding equations determining the demand for, and supply of, labor and output.

A few comments may be made here since a large number of econometric models utilize variants of this type of price and wage adjustment mechanism, which embodies a short-run Phillips curve relationship. First, indirect taxes can easily be incorporated into price formation behavior based on a modified markup hypothesis. The dynamic pattern of incidence, however, depends crucially on the nature of the price and wage adjustment equations.²² In this sense, without an explicit specification of dynamic elements in the price and wage adjustment equations, little can be said about the pattern of dynamic incidence on a priori grounds. Second, even if the dynamic elements are properly reflected in the price and wage adjustment equation, the dynamic incidence of indirect taxes can be biased. This is because the system may be inconsistent unless it is properly closed, so that demand and supply equations in the factor and product markets incorporate feedback from the disequilibrium behavior of prices, wages, and unemployment (see equations (11) and (12)). Third, as can be seen from the price and wage adjustment equations, the relationship between wages and unemployment can be shifted by means of changes in indirect taxes. Consequently, the equilibrium level of unemployment, if it exists, is dependent on the level of indirect taxes.²³ Fourth, changes in indirect tax rates do not necessarily get passed along to consumers via increased market prices, even though producers may try to pass their increased tax burden along fully by raising their target prices. The passing along may be only partial because of possible offsetting effects on prices, which may fall because of deficient aggregate demand. Finally, in periods of rapid wage increases caused by either rising expectations or other factors not reflected in price and wage adjustment behavior, changes in indirect taxes may have an imperceptible effect on prices. This is because the wage element has a heavy influence on prices.²⁴

The preceding discussion highlights the fact that the problem of dynamic incidence is an integral part of the problem involved in explaining the dynamic behavior of price and quantity adjustments. Further, it suggests that specification of the complexities involved in dynamic analysis is an extremely difficult task for a model builder.

II. Specification of Indirect Taxes in Selected Models

In this section, the specification of indirect taxes outlined in the preceding section is discussed with reference to the IS-LM structures (principally the IS structures) of 18 econometric models, 7 of which describe the U. S. economy. The remaining models describe the Canadian, Australian, German (Fed. Rep.), Japanese, British, Austrian, Finnish, and New Zealand economies. The 18 econometric models, of which the first 7 are models of the U. S. economy, are as follows: ²⁵

1. The Brookings Model (James S. Duesenberry, Gary Fromm, Lawrence R. Klein, and Edwin Kuh)
2. The Data Resources (DRI) Model (Otto Eckstein, Edward W. Green, and associates)
3. The Wharton Long-Term Annual and Industry Forecasting Model (Ross S. Preston)
4. The MIT-PENN-SSRC (MPS) Model (Albert Ando, Franco Modigliani, and Robert Rasche)
5. The Michigan Econometric (DHL-III) Model (Saul H. Hymans and Harold T. Shapiro)
6. The Bureau of Economic Analysis (BEA) Model (Albert A. Hirsch, Maurice Liebenberg, and George R. Green)
7. The Wharton Mark III Model (Michael D. McCarthy)
8. The RDX2 (Bank of Canada) Model (John F. Helliwell, Harold T. Shapiro, and associates)
9. The University of Toronto Quarterly (Toronto) Forecasting Model (Gregory V. Jump)
10. The TRACE Mark III R (University of Toronto) Annual Econometric Model of the Canadian Economy (John A. Sawyer)
11. The RBA1 (Reserve Bank of Australia) Model (W.E. Norton and J. F. Henderson)
12. An Econometric Model of the Australian (H-F) Economy (C. I. Higgins and V. W. Fitzgerald)
13. The Bonn Model of the German Economy (W. Krelle and G. Grisse)
14. The EPA Master Model (Quarterly Model of the Japanese Economy) (Economic Research Institute, Economic Planning Agency)
15 The London Business School Quarterly Econometric (LBS) Model (London Graduate School of Business Studies)
16. The Austria Link Model (Macroeconometric Model for the Austrian Economy) (Bernhard Boehm and Stefan Schleicher)
16. The Bank of Finland Model (On the Use of Two-Stage Least Squares with Principal Components: An Experiment with a Quarterly Model) (Juhani Hirvonen)
17. The New Zealand (Reserve Bank of New Zealand) Model (R. S. Deane).

All the models attempt to capture both the cyclical and trend characteristics of the economies they purport to represent. In other words, the models reflect stock-flow relations, lag structures, and nonlinearities which cause fluctuations when subjected to shocks. The extent of capturing these fluctuations varies greatly between models because of both their structural differences and their differences in the standardization of endogenous and exogenous variables. In addition, differences in the initial conditions cause considerable variation in the performance of such models.²⁶

Most of the econometric models, although they treat the underlying determinants of income and demand in varying degrees of aggregation, have placed less emphasis on industry production and output. The degree of disaggregation of the production sector, with the exceptions of the Brookings, the Wharton Long-Term, and the Wharton MK III models, is limited. This limitation is even more severe in most of the non-U. S. econometric models. In general, although output is primarily expenditure-determined, a majority of the models employ an aggregate production function, often of the Cobb-Douglas variety, to derive the demand for labor and capital. The latter demand, in the context of stock-adjustment theory, gives rise to investment demand in such models.

In varying degrees, the models have disaggregated the expenditure components of GNP. The extent of such disaggregation is relatively greater in the U. S. models, especially with regard to investment demand and inventory changes. The disaggregation of the fiscal sector, though quite substantial in such models as the DRI and the RDX2, is limited in most models. The financial sectors are more disparate. They range from the highly disaggregated financial sectors of the DRI, the MPS, the RDX2, and the Bonn models to the much less disaggregated financial sectors of the DHL-Ill, the BEA, the New Zealand, and the Japanese EPA models.

Despite differences in scale, purpose, and internal emphasis between the real and financial sectors, the econometric models can be seen as elaborate and complex variants of the simple Hicksian IS-LM framework. In this section, the focus is on the specification of indirect taxes in the consumption sector and on the price equations of the selected models.²⁷

1. indirect taxes in the consumption sector

In general, these econometric models have disaggregated the consumption demand of the private sector into a few major categories. Departure from the basic division of consumption among durables, non-durables, and services is found in a majority of the models.²⁸ At one extreme, with an aggregate consumption function, is the Bonn model; at the other extreme, the DRI and the Wharton Long-Term models have, respectively, 13 and 11 consumption demand categories. The theoretical underpinnings of real consumption demands, in all but the Bonn model,²⁹ adhere to some form of the permanent income hypothesis—through distributed lags on real disposable income or through lagged consumption variables. While this is true for most categories of consumption demand, in some models the treatment of consumption of non-durables (and of food consumption in the DRI) employs variants of the Keynesian absolute income hypothesis. In addition, while most models consider relative price effects, a few models also consider real balance effects, interest rate effects, and the effects of the past stock of durables. Some models also consider the effects of unemployment and population.³⁰

In Section I, it was pointed out that consistent aggregation of consumption in a general functional form does not generally account for shifts in income distribution. This is indeed true in these models. While the models disaggregate consumption demand in varying degrees, none has disaggregated disposable income, although a few models (such as the Brookings and the RDX2) have distinguished between wage and non-wage incomes. Moreover, although the underlying theory of consumption behavior is substantially the same in most models, actual specification of consumption functions differs materially. For instance, a large number of models employ a measure of permanent income that uses distributed lag weights on current and past disposable incomes, while others employ current disposable income and a lagged consumption term (sometimes even two-period lagged consumption, as in the RDX2).³¹ Also, inclusion of other variables (real balances, unemployment, etc.) in the consumption functions affects both the short-run and long-run marginal propensities to consume (MPCs).³² Further, although most models allow relative price effects, not all models allow the relative price effects in all individual disaggregated consumption functions.³³

Several implications for the effects of changes in indirect taxes follow from the general nature of consumption functions as found in the models. First, the inclusion of aggregate disposable income in consumption functions implies that changes in indirect taxes that cause shifts in income distribution cannot properly account for differential income effects, except to the extent that such changes affect aggregate disposable income. But the models can capture the differential price effects. Second, two different indirect tax measures which cause different shifts in income distribution, but cause equivalent changes in aggregate disposable income, have the same income effects; any differential effects they may have on aggregate consumption reflect the differential effects of relative prices. Third, a change in indirect taxes will have different effects, depending on the size of MPCs, since different specifications of consumption functions, although they embody the same underlying permanent income hypothesis, may give rise to differences in the MPCs between models. Fourth, although there are considerable differences in the MPCs between different categories of consumption within each model, differential impacts of changes in indirect taxes cannot be properly accounted for because shifts in income distribution are not reflected in aggregate disposable income (although relative price effects are). Finally, in models that either ignore relative price effects (such as the Bonn model) or do not incorporate these in all the disaggregated consumption demands (many models do this), the principal indicator of the pattern of shifting and incidence of indirect taxes is the income effect (via disposable income) of changes in such taxes.

Consistency in the aggregation of indirect taxation is also important, since the effects of rate changes are felt not only by the real sector; such rate changes produce feedback between the real and financial sectors via the government budget constraint and other macroaggregates, which are common to both of these sectors.³⁴ We now examine the indirect tax functions in the models.

The fact that indirect taxation covers not only most of consumption but also other components of gross national product (GNP) is reflected in most models. Except for three U.S. models (Wharton MK III, DHL-Ill, BEA) and three non-U.S. models (Bonn, EPA, Austria Link) where aggregate indirect tax yields are simply related to either gross domestic product or national income, the fiscal sector of most models disaggregates indirect taxes into consumption taxes and other categories (tariffs, state and local taxes), and attempts to specify the tax functions in terms of the relevant bases, rates, and other variables that affect their yields.³⁵

In most models, the stress on indirect taxation has been on consumption. In general, there are two types of functional forms preferred by most models, (1) T_I = A(τC)^λ or (2) T_I = a + b Σ τ_iC_i. The Brookings, MPS, BEA, and Finland models have the form (1) for individual categories of consumption, while most others have the form (2) or variants of it in which a broader definition of tax base that includes more than consumption is employed.³⁶ The RDX2 is the only model which employs the most consistent aggregation of indirect tax yield, in that it has $T_{I}^{i} = τ_{i} C_{i} a n d T_{I} = Σ T_{I}^{i} .$ .³⁷ For import taxes and other import fees and surcharges, most models relate their yields to taxable imports/total imports and GDP. In some models, either or both of these tax yields are treated exogenously.³⁸ It may be pointed out that, to the extent such indirect taxes are treated exogenously, the model cannot properly reflect the feedback between the real and financial sectors via the government budget constraint.³⁹ Also, the relevant indirect tax rates that are a suitable average index of effective rates over the class or category of goods in question are treated as fiscal policy instruments in a number of models. For consistency of aggregation, these rates must enter into price equations as, indeed, they do in some models.⁴⁰

The preceding discussion should suffice to highlight the nature of aggregation and its implications for the effects of indirect taxes in the selected econometric models, in particular, the probable importance of shifts in income distribution that may be caused by rate changes in the long run. Regarding the aggregation of indirect tax yields, this aspect of the problem is taken care of, in a limited sense, in most models. We now proceed to examine the incorporation of indirect taxes in price equations.

2. indirect taxes in price equations

There is considerable divergence in the manner of incorporating indirect tax rates in the models. The divergence between models arises mainly for three reasons: (i) differences in the level of aggregation of prices, (ii) differences in the mode of specifying the price formation equations, and (iii) differences in incorporating indirect tax rates, as well as defining such rates, in the price equations. It is beyond the scope of this paper to discuss in detail the differences in the price formation equations between models.⁴¹ However, a general overview of this topic is attempted in the ensuing discussion.

In the models, the price equations determine both an aggregate price index and a number of price indices for goods and services, commensurate with the level of disaggregation of the real sector.⁴² Although the models utilize some variants of the markup hypothesis, they differ in their approach to incorporation of the hypothesis in price equations.⁴³ Most models have employed either of the following two basic approaches: (i) the one-stage price determination approach, in which the prices of the goods and services are individually based on the markup hypothesis, and (ii) the two-stage price determination approach, in which some basic sectoral prices (such as the commonly employed price deflator for gross nonfarm business product) are determined by the markup hypothesis in the first stage. In the second stage, these basic prices are utilized as explanatory variables, among other relevant explanatory variables, to determine the deflators for final demand for goods and services. A few models combine these two basic approaches, in that deflators for some components of GDP (especially the consumer price index) are determined separately, while deflators for other components of GDP are determined by the two-stage approach.

The one-stage approach is utilized by seven non-U.S. models only, viz., the RDX2, RBA1, H-F, Bonn,⁴⁴ EPA, Austria Link, and New Zealand models. Those employing a mix of the two basic approaches are the MPS, Toronto, and LBS models. The remaining models employ the two-stage approach to price determination. Irrespective of the approach of price determination, the consistency criterion for price aggregation, as discussed in general terms in Section I, has been met by all the models. For instance, the aggregate consumer price index P_c is obtained by taking the weighted average of the price deflators for various categories of consumption.⁴⁵

It is difficult to ascertain the intrinsic superiority of either one of these approaches to price determination. It could be argued that the one-stage approach to price determination might be preferable, since it relates each price directly to its determinants; this is because unit labor cost is one of the principal determinants of prices and, since this variable is an aggregate index of interindustry unit labor costs, one-stage price determination might be more sensitive to changes in wages and productivity. On the other hand, it could be argued that price deflators for the components of GDP are, themselves, convenient aggregate indices of industry prices weighted by the fraction of industry outputs that go into a unit of final demand. Consequently, it might be preferable to determine the industry or sectoral prices first, which, in turn, could be utilized to explain the deflators of components of GDP. In other words, the two-stage approach to price determination might be more logical.⁴⁶

The case for the intrinsic superiority of the one-stage approach to price determination appears to be weak, in view of the logic of the two-stage approach. This is also reflected in the models, since a majority of them utilize the two-stage approach. In these models, although there are substantial differences in two-stage price formation, unit labor costs and demand pressures through capacity utilization are the main determinants of sectoral prices. Sometimes, changes in labor productivity and other cyclical factors also enter into these price equations. In the second stage, the price equations for components of GDP utilize sectoral prices as important determinants of their behavior.

The most interesting cases of two-stage price determination are found in the Brookings and Wharton Long-Term models. In these two models, in the first stage the sectoral or industry value-added prices are determined by the markup hypothesis. In the second stage, these prices are converted into consistent final demand price indices using the price-output conversion technique that utilizes an input-output framework. The only difference between the Brookings model and the Wharton Long-Term model is that the former employs a static Leontief input-output table, while the latter bases its input-output table on variable coefficients obtained from sectoral constant elasticity of substitution (CES) production functions.⁴⁷

With the exception of the EPA, the Finland, and the New Zealand models, indirect tax rates are incorporated in the price equations of all the above-mentioned models. As distinct from the level of price aggregation, which corresponds to the level of aggregation of the real sector, the level of aggregation of indirect tax rates follows neither the level of aggregation of indirect tax yield functions nor that of prices. As a result, in most models the components of final demand, such as imports (which are subjected to indirect taxation and for which there are customs receipt functions), do not often have the corresponding indirect tax rates specified in their price equations. Also, in models such as the Wharton MK III, some of the categories of consumption demand, such as services or “other nondurables,” do not incorporate indirect tax rates in their respective price equations. On the other hand, with the exception of a few models such as the Toronto model, import tariffs generally do not enter into the price equations of consumption categories.

In general, indirect taxes enter into the price equations of various categories of consumption, either as average rates in the GDP deflator or as commodity-group average rates. The typical form of the aggregate indirect tax rate, hereinafter called the A-type specification, is

\begin{matrix} P_{y} = \frac{P_{x} X + T_{1}}{X} = P_{x} +_{i} & (13) \end{matrix}

where P_y is the implicit GDP deflator, P_x is the GDP deflator at factor cost, τ = T_I/X is the aggregate average indirect tax rate, and X is the GDP at constant base-year prices.

The commodity-group average indirect tax rate specification, hereinafter called the B-type specification, has two alternative forms:

\begin{matrix} P_{i}^{m} (t) = λ_{i}^{m} (t, T) U L C (t) + γ_{i}^{m} (t, T *) τ_{i} (t) + ω_{i}^{m} (t) & (14) \end{matrix}

o r \begin{matrix} P_{i}^{f} (t) = \frac{P_{i}^{m} (t)}{l + τ_{i} (t)} = λ_{i}^{f} (t, T) U L C (t) + γ_{i}^{f} (t, T *) τ_{i} (t) + ω_{i}^{f} (t) & (14') \end{matrix}

where $P_{i}^{m}$ and $P_{i}^{f}$ are the ith sectoral market price and factor price, respectively. The symbols_λi and_iγ represent T and T*-period lag structures on unit labor cost ULC and indirect tax rate _τi, respectively. Other factors influencing prices, including lagged prices, are denoted by $w_{i}^{m}$ and $w_{i}^{f}$ . A variant of the B-type, which uses lagged prices as explanatory variables, has the form

\begin{matrix} P_{i} (t) = λ_{i} (t, T) U L C (t) + l_{i} (t, T^{*}) τ_{i} (t) P_{i} (t - 1) + w_{i} (t) & (14 ") \end{matrix}

where P_i could be either the market price or factor price, in which case $P_{i} = P_{i}^{f} = \frac{P_{i}^{m}}{1 + τ_{i}}, λ_{i}$ is a T-period lag structure, while l_i is a T*-period lag structure associated with the product variable τ_i (t) P_i(t - 1). The variable ω_i is as before, but excludes lagged prices.

The models which have the exact A-type specification of indirect taxes in price equations are the BEA, TRACE, and Austria Link models. The MPS model has a variant of the A-type equation, in that the implicit deflator for gross nonfarm business product, P_xbnf is defined as

P_{X B N F = (1 - λ)} \frac{P_{X B}}{1 - τ} + 100 λ,

where P_XB is the price deflator of gross nonfarm business product determined by the markup hypothesis, λ is the proportion of household product to gross nonfarm business product, and τ is the indirect tax rate defined as the ratio of indirect tax receipts to gross nonfarm business product.⁴⁸ With the exception of the EPA, Finland, and New Zealand models, the models under consideration have the B-type specification of indirect taxes, mostly the B(14) type, irrespective of whether they have one-stage or two-stage price formation equations.⁴⁹

The Toronto model, however, is an interesting case. It has variants of both A- and B(14)-type specifications. The private GDP deflator is a variant of the A type, but is based on the markup hypothesis, which also incorporates the unit price of imports, while consumer prices are based on the B(14) type, with both import prices plus tariffs and lagged values of private GDP deflators added to the markup hypothesis. While the Brookings,⁵⁰ Wharton Long-Term, and DRI models belong to the B(14)-type specification, the Wharton MK III is a particularly interesting case of B(14)-type specification. It first determines the manufacturing price by the markup hypothesis, and then utilizes this price in the price equations for various consumptions, along with commodity group indirect tax rates (defined as the tax yield over the commodity group divided by the value of the group consumption). In the DHL-III model, only changes in excise tax rates are specified in B(14)-type price equations for gross nonfarm business product. Later, this price variable is utilized in a double-logarithmic price equation for consumption goods. The only model with both B(14’)- and B(14”)-type specifications of indirect taxes in price equations is the RDX2 model.⁵¹

A few comments should be made about these specifications of indirect taxes, as well as the definition of such rates generally employed by the models. First, despite differences in the incorporation of indirect taxes, the incidence and shifting of such taxes in the models are, ceteris paribus, in the same direction as would be expected from the static theory of incidence.⁵² In both A-type and B-type (and their variants, as in the MPS and other models) specifications, a change in the indirect tax rate influences prices in the same direction. The only difference between these two types of specification is that a change in τ is fully passed on to the general price level in an A-type specification, while it is partially passed on for a corresponding change in τ in the B-type specification. In addition, to the extent that other prices of GNP components are functions of P_y (as, indeed, they are in the MPS, BEA, TRACE, and Austria Link models), there is relatively more quantity adjustment than price adjustment in models of the A type, especially if P_y is lagged in the other price equations, compared with models of the B type.

Second, in all the A-type models the response of P_x, the price deflator for gross nonfarm business product, to a change in τ is mainly through quantity adjustments. If this response is a weak one, the burden of adjustment falls mostly on the gross farm product deflator, which is determined residually in such models. In contrast, in the B(14)-type models prices respond to changes in the value of τ, both through quantity and other price adjustments. In other words, the relative price effects are well taken care of in the B(14)-type models.⁵³ Third, in a large number of models, of either the A type or the B type, the definition of τ_i is

τ_{i} = \frac{T_{i}}{B a s e}

where T_i is the indirect tax yield of the ith category. Base could be GDP or other relevant bases—consumption of durables, nondurables, and the like. Since both T_i and Base are endogenous variables in such models, the indirect tax rate so defined is also endogenous and is not a policy instrument. (This, of course, is important.) Thus, even though there need not be any discretionary indirect tax measures, there can be variation in τ_i because of nonproportional variations in T_i and Base.⁵⁴ As a result, endogenous variations in τ_i can cause changes in prices in such models.⁵⁵

The synoptic discussion of the models thus far suffices to highlight the nature of aggregation of the consumption sector and the incorporation of indirect taxes in the various models’ price equations. Despite differences between models in their internal emphasis on the aggregation and specification of indirect taxes, the incorporation of such taxes in most models appears to have the basic ingredients of the static incidence theory. With this perspective, it is worth investigating how the dynamics of price-wage movements in the models affect the dynamic pattern of incidence of such taxes.

III. Price-Wage Dynamics and Indirect Taxes in Models

The interconnections of indirect taxation with price and wage dynamics has already been discussed in Section I. Other things being equal, changes in indirect taxation initially tend to alter prices, which, in turn, set into motion interacting forces between prices and wages. The dynamics of such interacting forces determine price-wage movements over time. In this section, the nature of dynamic incidence in econometric models is investigated in the context of price-wage dynamics in the models.

The essence of price-wage dynamics, as found in most models, can be obtained by recasting equations (9) and (10) of Section 1.2 in a more familiar form. Assuming that an econometric model is properly closed, so that there is feedback between excess aggregate demand and unemployment (as is true of most models), these two equations, with slight modifications, can be combined into a semireduced form that yields the general Phillips curve equation.⁵⁶

\begin{matrix} f (\frac{Δ P}{P}, \frac{Δ W}{W}, u, π) = 0 & (15) \end{matrix}

where π denotes expected price inflation.

Since equation (15) is derived from a set of linear relations, it can be written as

\begin{matrix} \frac{Δ W}{W} = α_{0} + α_{1} \frac{Δ P}{P} + α_{2} u + α_{3} π & (16) \end{matrix}

where the α_i s are appropriate parameters such as α₁ > 0, α₂ < 0, and α₃ > 0. Further, if we assume that price expectations are formed on the basis of past rates of actual inflation, we may write ⁵⁷

π = Σ_{i = 1}^{T *} α_{3 i} {(\frac{Δ P}{P})}_{- i}, α_{3 i} \geq 0 f o r a l l i .

Equation (16) can now be put in a form found in most models,

\begin{matrix} \frac{Δ W}{W} = α_{0} + α_{1} (t, T *) \frac{Δ P (t)}{P} + α_{2} u & (17) \end{matrix}

where α₁ (t, T*) is a T*-period lag structure on the rate of actual price change.

None of the models have defined a target price or a target wage equation, as was done in the basic analytical framework in Section 1.2. However, the use in most models of lag structures in price equations suggests that an implicit partial adjustment of actual prices to desired prices is assumed.⁵⁸ Further, the use of an explicit lag structure in price equations eliminates the need for price adjustment equations. This, indeed, appears to have been the practice in most models. The Phillips curve form (17), as derived from the basic analytical framework, suggests that, even though target prices and wages, as well as inflationary expectations, are not explicit in the models, the incorporation of lag structures in a Phillips curve implies their existence via some partial adjustment mechanism (see footnote 58).

The transmission of the effects of indirect taxation in econometric models may be looked at in terms of their IS-LM structures and the Phillips curve equation. Broadly, we may trace the effect of a change in the indirect tax rate τ_i. An increase in τ_i, other things being equal, causes a rise in the market prices of consumer goods. The rise in consumer prices reduces disposable income and, therefore, reduces consumption and aggregate demand. The reduction in aggregate demand leads to a buildup of inventories, and eventually to cutbacks in investment and production. The simultaneous rise in consumer prices tends to push actual wages up. The wage push can, however, be dampened, offset, or even negated by an increase in unemployment owing to production cutbacks. A priori, the movement in actual wages will be determined by the relative, but offsetting, impacts of price changes and rising unemployment (i.e., the size of α_i and α_i in equation (16) or (17)). It is quite possible that an increase in τ_i may cause no rise in money wages if the existing level of unemployment is sufficiently high, even though the coefficient of the unemployment variable, as opposed to that of the price change variable, may be relatively smaller (in an absolute sense) in the Phillips curve equation.

At any rate, it is not clear, a priori, that an increase in the indirect tax rate will sustain the initial rise in prices, even when producers try to pass it on fully to consumers. Also, the effects of a change in indirect taxation work by means of the feedback between the real and monetary sectors. An increase in τ_i, other things being equal, initially tends to reduce the budget deficit, which, in turn, tends to push up interest rates. Rising interest rates cause business to invest less, and this aggravates the downward pressure on aggregate demand even more; feedback will be felt on output, prices, and wages as before. It is likely, however, that the eventual decline in output and aggregate demand will wipe out the initial reduction in the budget deficit, so that the upward shift in the LM curve will be relatively less than the downward shift of the IS curve. This implies that the feedback from the monetary sector counters, to some extent, the depressing effect of an increase in indirect taxation.

The dynamics of the incidence of indirect taxes in these models, in the context of a Phillips curve equation, may now be considered. Recall that the EPA, Finland, and New Zealand models do not incorporate indirect taxes in their price equations—in particular, the consumer price equations—although they do specify indirect tax functions. Of these three, the New Zealand model does not have a Phillips curve equation. As a result, the incidence of indirect taxes in these models is captured only to the extent that changes in such taxes have feedback from the monetary sector to the real sector via the government budget constraint.⁵⁹ The Phillips curve equation is also absent in two other models—the LBS and the Austria Link.⁶⁰ The dynamic feedback in these two models is, therefore, improperly specified despite the use of lag structures incorporating indirect tax rates in their price equations. The remaining models have Phillips curve equations, which are reproduced in Table 1.

Table 1.

The Phillips Curve in Models

As Table 1 shows, only a few models have employed variables other than those found in the Phillips curve equation (17). The Brookings model has employed a “rate of change of corporate profits before taxes” variable in its Phillips curve equation, while the MPS model has incorporated “a proportion of cash flow after taxes” variable. The EPA model, on the other hand, uses “corporate savings as a proportion of national income” in its equation. With the exception of the RBA1, Bonn, and Finland models, none of the models under discussion employs labor productivity or its rate of change in the respective Phillips curve equations.⁶¹ The only other models employing different variables are the Toronto and the H-F models. In the Toronto model, the use of employer and employee contribution rates for unemployment insurance, and another contribution rate for the Canada Pension Plan, in the Phillips curve equation act as added restraints on the downward movement of wage rates if these rates are increased in times of high unemployment. The more interesting case is the Australian H-F model, in which changes in award wage rates are first determined by the decisions of the National Wage Decision Board and Metal Trade Wage Decision Board.⁶² The changes in award wage rates are used in the Phillips curve equation to influence the movements of actual wage rates.

The price change variable in the Phillips curve equations of these models is the change in the consumer price index, which is a weighted average index of various categories of consumption. Most of the models, as discussed in Section II, incorporate commodity group indirect taxes in their respective consumer price deflators. A few models, however, employ an aggregate average indirect tax rate. The effects of a change in indirect tax rates in either of these two types of models are transmitted via the change in the consumer price index in the Phillips curve equation. The only difference is that models with an aggregative average indirect tax rate cannot compare the differential impacts of changes in individual commodity group indirect tax rates.

Table 1 reveals several interesting facts. First, although there is variety in the length of the lag structures associated with the unemployment variable in these equations, their modal length appears to be between two and four quarters in most models. The exceptions at the upper end are the RDX2 and RBA1 models that employ lag structures of 7 and 10 quarters, respectively, on their unemployment rates. Second, there is greater variety in the lag structure of the price variable in these equations. A number of models have lag structures or first differences extending as long as 12 quarters. Third, a majority of the models employ nonlinear Phillips curve equations, since the unemployment variable appears in reciprocal form.⁶³ Fourth, a majority of the models capture the cyclical effects of changes in the unemployment rate by taking the harmonic mean difference between average unemployment rates for two periods. Fifth, the Brookings and the Wharton MK III models employ smoothed-out average unemployment rates in their Phillips curve equations.

It is, indeed, difficult to ascertain the pattern of incidence of indirect taxes without knowledge of the dynamic multipliers. However, a priori, it appears that, on the basis of differences in the lag structures between models, and between the price change and unemployment variables in the Phillips curve equations within these models, the effects of changes in indirect taxes are felt for a longer period through the price change variables than through the unemployment variables (see Table 1, where the price variables have longer lags than the unemployment variables). Also, the coefficients associated with the price change variables appear, in general, to be relatively larger than those of the unemployment variables.⁶⁴

Interestingly enough, the coefficients of both price change and unemployment variables vary widely between models. Among the U. S. models, the estimates of the coefficients of price change variables range from 0.20 in the MPS model to 0.77 in the BEA model. The same estimates for the non-U. S. models range from 0.24, for the first four-quarter price change in the Toronto model, to 0.99, for the first one-quarter price change in the TRACE model. Similarly, estimates of the coefficients of unemployment variables show a wide divergence between U. S. and non-U. S. models.

Other things being equal, since the coefficients associated with the price change variable are significantly larger than those associated with the unemployment variable in all the Phillips curve equations presented in Table 1, it is expected that the positive effect of an increase in indirect taxes would outweigh the negative effect of a rise in unemployment. Therefore, while prices rise, wage rates may also rise with declining employment and output in these models.⁶⁵

Except for the Brookings and the LBS models, it has not been possible to obtain the dynamic multipliers of changes in indirect tax rates. The dynamic incidence multipliers for these two models are presented in Tables 2 and 3.

Table 2.

Dynamic Multipliers for an Excise Tax Cut in the Brookings Model¹

(1954 prices)

Source: Based on Fromm and Taubman (1968), pp. 66-64.

^¹The simulation of changes in excess tax rates from mid-1960 through 1972 are based on excise tax reductions enacted by the U. S. Congress in July 1965. The solution started with the equivalent 1960: III reduction of $2.7 billion in excise taxes.

Table 2.

Dynamic Multipliers for an Excise Tax Cut in the Brookings Model¹

(1954 prices)

Pass-along hypothesis	Effect on	1960: III	1961:II	1962:I	1962: IV
100 per cent	Gross domestic product (GDP)	0.925	1.370	2.222	2.370
80 per cent		0.703	1.148	1.888	2.074
50 per cent		0.592	0.925	1.592	1.777
100 per cent	Consumption	0.888	1.185	1.555	0.777
80 per cent		0.666	0.962	1.333	1.555
50 per cent		0.555	0.814	1.111	1.333
100 per cent	GDP deflator (per cent)	−0.010	−0.016	−0.014	−0.011
80 per cent		−0.008	−0.013	−0.012	−0.009
50 per cent		−0.007	−0.011	−0.010	−0.008
100 per cent	Unemployment rate (per cent)	−0.001	−0.002	−0.003	−0.004
80 per cent		0.000	−0.002	−0.002	−0.003
50 per cent		0.000	−0.001	−0.002	−0.003

Source: Based on Fromm and Taubman (1968), pp. 66-64.

Table 2.

Dynamic Multipliers for an Excise Tax Cut in the Brookings Model¹

(1954 prices)

Pass-along hypothesis	Effect on	1960: III	1961:II	1962:I	1962: IV
100 per cent	Gross domestic product (GDP)	0.925	1.370	2.222	2.370
80 per cent		0.703	1.148	1.888	2.074
50 per cent		0.592	0.925	1.592	1.777
100 per cent	Consumption	0.888	1.185	1.555	0.777
80 per cent		0.666	0.962	1.333	1.555
50 per cent		0.555	0.814	1.111	1.333
100 per cent	GDP deflator (per cent)	−0.010	−0.016	−0.014	−0.011
80 per cent		−0.008	−0.013	−0.012	−0.009
50 per cent		−0.007	−0.011	−0.010	−0.008
100 per cent	Unemployment rate (per cent)	−0.001	−0.002	−0.003	−0.004
80 per cent		0.000	−0.002	−0.002	−0.003
50 per cent		0.000	−0.001	−0.002	−0.003

Source: Based on Fromm and Taubman (1968), pp. 66-64.

Table 3.

DYNAMIC MULTIPLIERS FOR A PURCHASE TAX CUT IN THE LBS MODEL¹

(1972 prices)

Source: Ball and others (1975), p. 200.

^¹The solution of a sustained reduction in the purchase tax of £100 million per quarter started with the initial conditions of the first quarter of 1972.

Table 3.

DYNAMIC MULTIPLIERS FOR A PURCHASE TAX CUT IN THE LBS MODEL¹

(1972 prices)

Effect on	1972:I	1972:IV	1973:I	1973:IV	1974:I	1974:IV	1975:I	1975:IV
Gross domestic product (GDP)	0.300	0.720	0.850	0.970	0.930	0.820	0.740	0.580
Consumption	0.520	0.970	1.110	1.360	1.390	1.400	1.390	1.280
Unemployment rate (per cent)	−0.038	−0.162	−0.187	−0.211	−0.211	−0.208	−0.210	−0.209
Consumer price index (per cent)	−0.013	−0.014	−0.015	−0.014	−0.014	−0.013	−0.013	−0.013
GDP deflator (per cent)	0.000	−0.001	−0.001	0.001	0.002	0.003	0.004	0.005

Source: Ball and others (1975), p. 200.

^¹The solution of a sustained reduction in the purchase tax of £100 million per quarter started with the initial conditions of the first quarter of 1972.

Table 3.

DYNAMIC MULTIPLIERS FOR A PURCHASE TAX CUT IN THE LBS MODEL¹

(1972 prices)

Effect on	1972:I	1972:IV	1973:I	1973:IV	1974:I	1974:IV	1975:I	1975:IV
Gross domestic product (GDP)	0.300	0.720	0.850	0.970	0.930	0.820	0.740	0.580
Consumption	0.520	0.970	1.110	1.360	1.390	1.400	1.390	1.280
Unemployment rate (per cent)	−0.038	−0.162	−0.187	−0.211	−0.211	−0.208	−0.210	−0.209
Consumer price index (per cent)	−0.013	−0.014	−0.015	−0.014	−0.014	−0.013	−0.013	−0.013
GDP deflator (per cent)	0.000	−0.001	−0.001	0.001	0.002	0.003	0.004	0.005

Source: Ball and others (1975), p. 200.

^¹The solution of a sustained reduction in the purchase tax of £100 million per quarter started with the initial conditions of the first quarter of 1972.

Before discussing these dynamic incidence multipliers, a few points need to be made. First, as mentioned above, the Brookings model employs the markup hypothesis to determine the value-added prices that are utilized in an input-output framework to determine price deflators for GDP components. Fromm and Taubman (1968), in carrying out simulation exercises with an excise tax cut, employed the following mode of incorporating ad valorem excise taxes in price equations:

\begin{matrix} P v_{i} = (1 + τ_{i}) P v_{i} * & (18) \end{matrix}

where Pv_i* is the value-added price based on the markup hypothesis and Pv_i is the value-added price after the imposition of ad valorem tax rate τ_i. As it stands, equation (18) implies a 100 per cent passing along of the tax to consumers. In order to incorporate a different pass-along hypothesis, Fromm and Taubman assume that if firms retain, say, m per cent when τ_i is cut to τⁱ final demand prices fall from P_i to P_i so that

\begin{matrix} {\bar{P}}_{i} = 1 - m (\frac{τ_{i} - {\bar{τ}}_{i}}{1 + τ_{i}}) & (19) \end{matrix}

Second, in the LBS model, on the other hand, the purchase tax τ is incorporated in the price for consumer durables as

P c d = 0.0581 + 2.0757 (\frac{1}{4} Σ_{0}^{3} {U L C}_{- i}) + 0.0838 τ - 0.0045 T i m e + Σ_{1}^{3} α_{i} {D u m m y}_{i}

α_{i} = - 0.0054, 0.0004, - 0.0012.

In other words, in the LBS model, changes in indirect tax rates are partially passed on to consumers. Third, while the Brookings model has well-specified sectoral Phillips curve equations, the LBS model has none. Finally, the Brookings model has an implicit government budget constraint that assumes all deficits to be money financed; the LBS model has neither an explicit nor an implicit government budget constraint. It has been shown elsewhere⁶⁶ that fiscal policy effects in the Brookings model are reinforced by the monetary repercussions, so that the effect of an ad valorem-excise tax cut would be expected to be more pronounced in the Brookings model than in the LBS model. On the other hand, it is expected that the absence of a government budget constraint in the LBS model would bias the effects of a cut in the purchase tax.⁶⁷

The dynamic incidence multipliers, shown in Tables 2 and 3, exhibit several interesting features. First, the dynamic multipliers for GDP and consumption in the Brookings model are greater in magnitude than the corresponding dynamic multipliers in the LBS model.⁶⁸ Second, the indirect tax cut multipliers for consumption are smaller than those for GDP in the Brookings model, while it is the other way round in the LBS model.⁶⁹ Third, although the indirect tax cut multipliers for GDP and consumption increase up to the end of eight quarters in both models, they begin to decline in the LBS model thereafter.⁷⁰ Fourth, an examination of the Brookings model’s dynamic multipliers in Table 2 reveals that the effect of fully passing indirect taxes along to consumers is greater than the effect of partially passing them along (as seen in a comparison of multipliers for 100 per cent and 80 per cent pass-along hypotheses). However, the difference between the effects of passing along 80 per cent and 50 per cent of indirect taxes is relatively smaller than the difference between 100 per cent and 80 percent pass-along hypotheses. Fifth, in both models, cuts in indirect taxes substantially reduce the unemployment rate.⁷¹ Sixth, prices fall initially in both models, but economic expansion eventually pushes them back toward their initial levels. In the LBS model, the aggregate price level falls initially following a tax cut, but is pushed back to its original level by the end of eight quarters; however, economic expansion continues to exert upward pressure on prices.

The pattern of dynamic incidence, as shown in Tables 2 and 3, cannot be taken as the general pattern among models. The simulation periods in both the Brookings and LBS models were too short to permit this. Moreover, both these models have disparate IS-LM structures, as well as disparate wage-price dynamics. In fact, there is no Phillips curve equation in the LBS model. Also, the LBS model has lags of no more than five quarters in other structural equations, whereas the Brookings model has approximately eight-quarter lags in some of its structural equations other than the Phillips curve equation.

Even though most models have not simulated the effects of changes in indirect taxes, such simulations would not be of much use in a number of models that have endogenous indirect tax rates in their price equations. In fact, half the models discussed in this paper either do not incorporate indirect tax rates in their price equations (the EPA, Finland, and New Zealand models) or else incorporate them endogenously (the MPS, BEA, Wharton MK III, TRACE, Bonn, and Austria Link models). Of these nine models, the indirect tax functions in the Wharton MK III, TRACE, and Bonn models do not incorporate indirect tax rates exogenously, although they could be easily modified to do so.⁷² The other six models, however, incorporate indirect tax rates as policy instruments in their respective tax functions. The remaining nine models treat indirect tax rates as policy instruments in price equations, although they are not always incorporated in the indirect tax functions.⁷³

The EPA, Finland, and New Zealand models can simulate the effects of a change in indirect tax rates, since these rates are exogenous in their tax functions, even though their price equations do not incorporate them. In these models, an increase in indirect tax rates initially raises the indirect tax yield, which, in turn, reduces the budget deficit. The reduction in the budget deficit causes a reduction in the money supply to the extent that the amount of high-powered money declines. This causes an upward movement in the interest rate, which, in turn, tends to depress the level of investment (and, consequently, aggregate demand and, eventually, disposable income). At the same time, to the extent that unit labor costs and other determinants of price behavior change following a cutback in production, prices will also change in the same direction. In this way, these models respond to a change in indirect tax rates and can simulate the incidence effects.

As for the other six models that incorporate indirect tax rates as endogenous variables in their price equations, but employ statutory or other kinds of indirect tax rates as policy instruments in their tax functions, the response of these models to a change in indirect tax rates is quite similar to that of the above three models. The only difference in the latter six models is that the increase in prices may be more than it would have been if indirect tax rates were exogenous in the price equations.⁷⁴ In these six models, the increase in prices will reduce real disposable income, and hence consumption. At the same time, since investment is adversely affected (through the repercussions of budget deficits on the financial sector), this would lead to a further decline in aggregate demand and, in turn, cause output to decline. The decline in output may increase unit labor cost if wages do not fall sufficiently and/or labor productivity does not rise enough so that prices rise more than they would have if indirect tax rates were exogenous in the price equations.⁷⁵

What, in general, can be said about the incidence of indirect taxation in these models, even though the simulated dynamic multipliers (except for the Brookings and the LBS models) are not available? First, despite wide variety in the IS-LM structures and the methods of aggregation of indirect taxation in these models, the crucial element turns out to be the variety of treatment of indirect taxes in the price equations. Clearly, failure to incorporate indirect tax rates in the price equations results in, at best, improper incidence effects. On the other hand, when models such as the MPS, BEA, and Wharton MK III treat indirect tax rates endogenously in their price equations, they bias the incidence effects and may even cause irregularities in price movements. Second, although other models treat indirect tax rates exogenously in their price equations, a few of them do not incorporate these rates in their indirect tax functions. These latter models also bias the incidence effects of indirect taxation, since such models cannot properly reflect the interaction between the real and financial sectors because of bias in the indirect tax yield that affects the budget deficit or surplus (although the extent of this bias may be limited). Third, the simulated multipliers for the Brookings and the LBS models cannot be regarded as representative of other models, especially since the incidence effects on consumption differ so markedly between these two models. However, the simulation of incidence in the Brookings and LBS models indicate that the dynamic incidence of indirect taxation is likely to produce the effects predicted by static incidence theory, although the time path and size of the dynamic multipliers may vary among models. Finally, models (such as the Brookings, DRI, RDX2, and Toronto) that treat indirect taxes exogenously in their price equations should be considered better specified in this regard than models that fail to do so.

IV. General Conclusions

The following impressions emerge from this survey of the treatment of indirect taxation in econometric models: First, most econometric models are aggregative in their treatment of the consumption sector. Also, little effort has been made to capture the effects of periodic shifts in income distribution. As a result, the differential impact of indirect taxation is limited in such models.

Second, the substitution effects of price changes are comparable in magnitude to income effects, despite wide variations between models. This suggests that indirect tax measures, to the extent that they affect prices, have an important effect on aggregate demand.

Third, the disaggregation of indirect taxes in terms of their yields has been insufficient relative to the level of disaggregation of the consumption demands in most models. Consequently, the discretionary indirect tax measures are unsatisfactory.

Fourth, disaggregation of indirect tax rates in the price equations has been insufficient relative to the disaggregation of the consumption sector and its corresponding prices. Hence, few models can capture differential impacts of indirect taxation on broad commodity groups.

Fifth, the endogenous treatment of indirect tax rates in some models is a gross error, which not only assigns an improper role to this fiscal instrument but also causes unnecessary bias in the simulation results of such models. This is an important conclusion.

Sixth, the models reveal a wide variety of lag structures, not only between but also within models. Also, the Phillips curve equations have differences in lag structures between the price change variables and the unemployment rates. As a result, it is expected that the time path of incidence of indirect taxation will differ between models. This greatly affects the apparent efficiency of fiscal policy.

Seventh, in periods of rapid wage increases caused either by rising inflationary expectations or other factors not reflected in price-wage adjustment behavior, changes in indirect taxes may have negligible effects on prices. This is because the wage element has a very heavy influence on prices in most models.

Finally, since endogenous inflationary expectations can be represented by lag structures on actual price changes in the price-wage adjustment equations, most models have implicitly incorporated some form of inflationary expectations in their Phillips curve equations.

Three important lessons follow from the above impressions. First, econometricians should not be casual in their treatment of taxes, direct and indirect, in econometric models designed to study stabilization policies. The aggregation of the consumption sector, with its corresponding prices, must include appropriate aggregation of indirect tax rates, as well as indirect tax yield functions. Second, the form for price behavior should be made explicit, and the appropriate lag structures should be employed in wage-price dynamics to reflect feedback between aggregate demand and aggregate supply. Third, and most important, indirect tax rates must be treated exogenously in price equations, in order to properly trace the effects of discretionary indirect tax measures.

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Leijonhufvud, Axel (1973), “Effective Demand Failures,” Swedish Journal of Economics, Vol. 75 (March 1973), pp. 27–48.
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Leontief, Wassily W. (1936), “Composite Commodities and the Problem of Index Numbers,” Econometrica, Vol. 4 (January 1936), pp. 39–59.
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Leontief, Wassily W. (1947), “Introduction to a Theory of the Internal Structure of Functional Relationships,” Econometrica, Vol. 15

U.S. Models
1. The Brookings Model	$\frac{W_{i} - W_{- 4}^{i}}{W_{- 4}^{i}} = a_{0}^{i} + a_{1}^{i} \frac{1}{4} Σ_{i = - 3}^{0} {(\frac{P_{c} - P_{c - 4}}{P_{c - 4}})}_{i} + a_{2}^{i} \frac{1}{4} Σ_{i = - 3}^{0} τ_{d i}^{i} + a_{3}^{i} 1 / u * - a_{4}^{i} [\frac{W_{- 4}^{i} - W_{- 8}^{i}}{W_{- 8}^{i}}]$
	i = construction, durables, nondurables, wholesale and retail trade, regulated industries, and residual industries
	a₀ = (−0.04, 0.02, 0.04, 0.04, −0.04, −0.02)
	a₁ = (0.15, 0.88, 0.76, 0.07, 0.36, 0.52)
	a₂ = (2.33, 0.13, 0.04, 0.00, 0.53, 0.59)
	a₃= (0.0019, 0.0011, 0.0007, 0.0011, 0.0000 +, 0.0005)
	a₄ = (0.34, 0.43, 0.56, 0.56, 0.11, 0.50)
	u* =⅛ U_-4 + ¼ u_-3 + ¼ u_-2 + ⅛ u_-1 + ⅛ u
	τd = percentage changes in corporate profits before taxes, including inventory valuation
2. The DRI Model	Unavailable
3. The Wharton Long-Term Model	Unavailable
4. The MPS Model	$\frac{W - W_{- 2}}{W_{- 2}} = - 0.115 + 0.1961 (\frac{1}{u + u_{- 1}}) + 0.175 (\frac{Y P C C}{{Y P C C}_{- 1} + {Y P C C}_{- 2}})$
	(for private nonfarm business) $+ 0.4291 (\frac{P_{c - 2} - P_{c - 4}}{P_{c - 4}}) + 0.0023 (U T O - {U T O}_{- 2}) + {0.5279}_{ρ - 1}$
	where YPCC = cash flow of corporations after taxes
		UTO = old-age, survivors, and disability insurance contribution rate
		ρ = serial correlation term
	Alternate wage adjustment equation:
	$\frac{W - W_{- 2}}{W_{- 2}} = 0.002958 + 0.133161 (\frac{1}{u + u_{- 1}}) - 0.012810 (\frac{u - u_{- 2}}{u_{- 2}}) + 0.5703 (\frac{P_{c - 2} - P_{c - 4}}{P_{c - 4}}) + 0.03381 (U T O - {U T O}_{- 2}) + {0.05022}_{ρ - 1}$
5. The DHL-III Model	$\ln \frac{W}{W_{- 2}} = 0.012 + 4.18 \ln \frac{P_{c - 1}}{P_{c - 1}} + 0.52 (\frac{2}{u_{- 1} + u_{- 2}}) - 0.005 (\frac{u_{- 1} - u_{- 2}}{u_{- 2}}) - 0.516 [\ln \frac{{(W / P_{c})}_{- 2}}{{({W / P}_{c})}_{- 6}} - 0.032]$
6. The BEA Model	$\frac{W}{\frac{1}{4} Σ_{i = 1}^{4} W_{- i}} = - 0.443 + 0.769 \frac{P_{c - 1}}{\frac{1}{4} Σ_{2}^{5} P_{c - i}} + 0.00116 \frac{1}{Σ_{2}^{4} α_{i^{u -} i}} - 0.447 (u - \frac{1}{4} Σ_{1}^{4} u_{- 1}) - 0.0326 [\frac{1}{4} Σ_{0}^{3} {(\frac{u_{m}}{u})}_{- i}] + 0.678 (\frac{H}{\frac{1}{4} Σ_{1}^{4} H_{- i}})$
	i \|	0	1	2	3	4
	α \|	0.20	0.35	0.25	0.15	0.05

	u_m= unemployed males, aged 25-54 years
	H = average weekly hours for production workers in manufacturing
7. The Wharton Mark III Model	$\frac{W - W_{- 2}}{W_{- 1}} = 0.00336 + 0.1561 \frac{1}{u_{- 1}^{*}} + 0.3581 (\frac{P_{c} - P_{c - 2}}{P_{c - 1}}) + 0.4234 (\frac{W_{- 1} - W_{- 2}}{W_{- 2}})$
	u* = potential unemployment rate W = wage rate in manufacturing and mining (There are two similar wage adjustment equations for the regulated and commercial sectors.)
Non-U.S. Models
8. The RDX2 Model (Canada)	$\frac{W - W_{- 1}}{W_{- 1}} = λ_{i} (S_{i}) + {29.078}_{γ} - 0.0334 (\frac{{\frac{W}{P_{c}} + (\frac{W}{P_{c}})}_{- 1}}{2}) - 0.2386 {(\frac{Δ W}{W})}_{- 4} + 0.3335 {(\frac{Δ P_{c}}{P_{c}})}^{c} + 33.42 Σ {0.001}_{α - i (1 / u^{2}) - i}$
	S_i = dummy variable vector
	${(\frac{Δ P_{c}}{P_{c}})}^{c}$ = expected annual percentage change in consumer prices
	γ = labor efficiency factor
	i \|	0	1	2	3	4	5	6	7
	α \|	0.1196	0.1047	0.0897	0.0746	0.0598	0.0449	0.0299	0.0150
	Σ α_i = 0.5382
	(There is a wage adjustment equation for the construction sector as well.)
9. The Toronto Model (Canada)	$\frac{W - W_{- 4}}{W_{- 4}} = - 1.7466 + 26.7545 \frac{1}{\frac{1}{4} Σ_{0}^{3} u_{- i}} + 24.9316 [\frac{1}{\frac{1}{4} Σ_{1}^{3} u_{- i}} - \frac{1}{\frac{1}{4} Σ_{1}^{3} u_{- i - 4}}] + 24.22 (\frac{P_{c} - P_{c - 4}}{P_{c - 4}}) + 57.49 (\frac{P_{c - 4} - P_{c - 8}}{P_{c - 8}}) + 19.42 (\frac{P_{c - 8} - P_{c - 12}}{P_{c - 12}}) - 92.50 [\frac{({τ^{1}}_{4} + {τ^{2}}_{4}) - ({τ^{1}}_{- 4} + {τ^{2}}_{- 4})}{1 + {τ^{1}}_{- 4} + {τ^{2}}_{- 4}}]$
	τ¹ = employer and employee contribution rate for unemployment insurance τ² = employer and employee contribution rate for Canada Pension Plan
10. The TRACE Model (Canada)	$\frac{W - W_{- 1}}{W_{- 1}} = 0.985 \frac{P_{c} - P_{c - 1}}{P_{c - 1}} + 0.169 \frac{1}{u}$
	(here $\frac{P_{c} - P_{c - 1}}{P_{c - 1}}$ is used as a proxy for the expected rate of inflation)
11. The RBA1 Model (Australia)	$W = - 1.411 - 0.091 S_{1} - 0.040 S_{2} - 0.037 S_{3} + Σ_{0}^{11} λ_{i} P_{c - i} + Σ_{0}^{10} β_{i} u_{i} + Σ_{0}^{3} γ_{i} x_{- i}$
	S_i = dummy variable u = unemployed persons, in millions x = labor productivity in nonfarm business

	i \|	0	1	2	3	4	5	6	7	8	9	10	11
	λ \|	0.0577	0.1058	0.1443	0.1731	0.1924	0.2020	0.2020	0.1924	0.1731	0.1443	0.1058	0.0577
	β \|	−0.0220	−0.0410	−0.0560	−0.0670	−0.0740	−0.0780	−0.0740	−0.0670	−0.0560	−0.0410	−0.0220
	γ \|	0.12277	0.18415	0.18415	0.12277
12. Ths H-F Model (Australia)	Two-equation wage adjustment
	$\begin{matrix} (1) & \frac{W_{A} - W_{A - 1}}{W_{A - 1}} \end{matrix} = 0.003808 + 0.008009 Σ_{0}^{1} α_{i} {(W_{b d})}_{- i} + 0.00489 Σ_{0}^{4} β_{i} {(W_{m t})}_{- i} + 0.6474 Σ_{0}^{6} γ_{i} {(\frac{P_{c} - P_{c - 1}}{P_{c - 1}})}_{- i}$
	$\begin{matrix} (2) & \frac{W_{A} - W_{- 1}}{W_{- 1}} \end{matrix} = - 0.001329 + 1.2897 (\frac{0.88 W_{A} - W_{A - 1}}{{\bar{W}}_{- 1}}) + 0.12 (\frac{W_{A - 1} - W_{A - 2}}{{\bar{W}}_{- 2}}) + {0.002327}_{v} + 0.0108 (\frac{v}{v - 1}) - {0.428}_{ρ - 1}$
	i \|	0	1	2	3	4	5	6
	α \|	0.93	0.07
	β \|	0.59	0.30	0.10	0.01	0.00
	γ \|	0.00	0.48	0.30	0.15	0.06	0.01	0.00
	W_A = award wages (weighted average of four end-month wages to end-quarter wages with weights 1/6, 1/3, 1/3, 1/6; wages in dollars oer week)
	W_bd = National Wage Decision Board decision (per cent)
	W_mt = Metal Trade Wage Decision Board decision (per cent)
	W = average earnings of nonfarm wage and salary earners
	v = ratio of vacancies to persons registered for employment
	ρ = serial correlation term
13. The Bonn Model (Germany, Fed. Rep.)	$\begin{matrix} \frac{W - W_{- 1}}{W_{- 1}} \end{matrix} = 0.0325 + 0.0112 H W D S + 0.0389 \frac{X_{- 1} - X_{- 2}}{W_{- 1} - W_{- 2}}$
	HWDS = surplus demand for labor
	X = labor productivity
14. The EPA Model (Japan)	$\begin{matrix} \frac{W - W_{- 1}}{W_{- 1}} \end{matrix} = - 4.75285 + 0.466815 \frac{P_{c} - P_{c - 1}}{P_{c - 1}} - 0.0878642 \frac{1}{0.002 - \frac{1}{4} Σ_{1}^{4} u_{- i}} + 11.6074 V + 59.54 Z + 18.8193 Σ_{2}^{4} Z_{- i}$
	V = proportion of self-employed persons in the labor force
	Z = saving of corporations as a proportion of national income, adjusted for inventory valuation
15. The LBS Model (U.K.)	None
16. The Austria Link Model (Austria)	None
17. The Finland Model (Finland)	$\ln \frac{W}{W_{- 4}} = 0.769 \ln \frac{P_{c}}{P_{c - 4}} + 0.538 (\ln_{X} - \ln_{X - 4}) + 0.068 \frac{1}{Σ_{1}^{4} u_{- i}}$
	X = labor productivity (This equation is for competitive industries. Similar equations are specified for (noncompetitive) agriculture and forestry.)
18. The New Zealand Model (1974 version)	None

U.S. Models
1. The Brookings Model	$\frac{W_{i} - W_{- 4}^{i}}{W_{- 4}^{i}} = a_{0}^{i} + a_{1}^{i} \frac{1}{4} Σ_{i = - 3}^{0} {(\frac{P_{c} - P_{c - 4}}{P_{c - 4}})}_{i} + a_{2}^{i} \frac{1}{4} Σ_{i = - 3}^{0} τ_{d i}^{i} + a_{3}^{i} 1 / u * - a_{4}^{i} [\frac{W_{- 4}^{i} - W_{- 8}^{i}}{W_{- 8}^{i}}]$
	i = construction, durables, nondurables, wholesale and retail trade, regulated industries, and residual industries
	a₀ = (−0.04, 0.02, 0.04, 0.04, −0.04, −0.02)
	a₁ = (0.15, 0.88, 0.76, 0.07, 0.36, 0.52)
	a₂ = (2.33, 0.13, 0.04, 0.00, 0.53, 0.59)
	a₃= (0.0019, 0.0011, 0.0007, 0.0011, 0.0000 +, 0.0005)
	a₄ = (0.34, 0.43, 0.56, 0.56, 0.11, 0.50)
	u* =⅛ U_-4 + ¼ u_-3 + ¼ u_-2 + ⅛ u_-1 + ⅛ u
	τd = percentage changes in corporate profits before taxes, including inventory valuation
2. The DRI Model	Unavailable
3. The Wharton Long-Term Model	Unavailable
4. The MPS Model	$\frac{W - W_{- 2}}{W_{- 2}} = - 0.115 + 0.1961 (\frac{1}{u + u_{- 1}}) + 0.175 (\frac{Y P C C}{{Y P C C}_{- 1} + {Y P C C}_{- 2}})$
	(for private nonfarm business) $+ 0.4291 (\frac{P_{c - 2} - P_{c - 4}}{P_{c - 4}}) + 0.0023 (U T O - {U T O}_{- 2}) + {0.5279}_{ρ - 1}$
	where YPCC = cash flow of corporations after taxes
		UTO = old-age, survivors, and disability insurance contribution rate
		ρ = serial correlation term
	Alternate wage adjustment equation:
	$\frac{W - W_{- 2}}{W_{- 2}} = 0.002958 + 0.133161 (\frac{1}{u + u_{- 1}}) - 0.012810 (\frac{u - u_{- 2}}{u_{- 2}}) + 0.5703 (\frac{P_{c - 2} - P_{c - 4}}{P_{c - 4}}) + 0.03381 (U T O - {U T O}_{- 2}) + {0.05022}_{ρ - 1}$
5. The DHL-III Model	$\ln \frac{W}{W_{- 2}} = 0.012 + 4.18 \ln \frac{P_{c - 1}}{P_{c - 1}} + 0.52 (\frac{2}{u_{- 1} + u_{- 2}}) - 0.005 (\frac{u_{- 1} - u_{- 2}}{u_{- 2}}) - 0.516 [\ln \frac{{(W / P_{c})}_{- 2}}{{({W / P}_{c})}_{- 6}} - 0.032]$
6. The BEA Model	$\frac{W}{\frac{1}{4} Σ_{i = 1}^{4} W_{- i}} = - 0.443 + 0.769 \frac{P_{c - 1}}{\frac{1}{4} Σ_{2}^{5} P_{c - i}} + 0.00116 \frac{1}{Σ_{2}^{4} α_{i^{u -} i}} - 0.447 (u - \frac{1}{4} Σ_{1}^{4} u_{- 1}) - 0.0326 [\frac{1}{4} Σ_{0}^{3} {(\frac{u_{m}}{u})}_{- i}] + 0.678 (\frac{H}{\frac{1}{4} Σ_{1}^{4} H_{- i}})$
	i \|	0	1	2	3	4
	α \|	0.20	0.35	0.25	0.15	0.05

	u_m= unemployed males, aged 25-54 years
	H = average weekly hours for production workers in manufacturing
7. The Wharton Mark III Model	$\frac{W - W_{- 2}}{W_{- 1}} = 0.00336 + 0.1561 \frac{1}{u_{- 1}^{*}} + 0.3581 (\frac{P_{c} - P_{c - 2}}{P_{c - 1}}) + 0.4234 (\frac{W_{- 1} - W_{- 2}}{W_{- 2}})$
	u* = potential unemployment rate W = wage rate in manufacturing and mining (There are two similar wage adjustment equations for the regulated and commercial sectors.)
Non-U.S. Models
8. The RDX2 Model (Canada)	$\frac{W - W_{- 1}}{W_{- 1}} = λ_{i} (S_{i}) + {29.078}_{γ} - 0.0334 (\frac{{\frac{W}{P_{c}} + (\frac{W}{P_{c}})}_{- 1}}{2}) - 0.2386 {(\frac{Δ W}{W})}_{- 4} + 0.3335 {(\frac{Δ P_{c}}{P_{c}})}^{c} + 33.42 Σ {0.001}_{α - i (1 / u^{2}) - i}$
	S_i = dummy variable vector
	${(\frac{Δ P_{c}}{P_{c}})}^{c}$ = expected annual percentage change in consumer prices
	γ = labor efficiency factor
	i \|	0	1	2	3	4	5	6	7
	α \|	0.1196	0.1047	0.0897	0.0746	0.0598	0.0449	0.0299	0.0150
	Σ α_i = 0.5382
	(There is a wage adjustment equation for the construction sector as well.)
9. The Toronto Model (Canada)	$\frac{W - W_{- 4}}{W_{- 4}} = - 1.7466 + 26.7545 \frac{1}{\frac{1}{4} Σ_{0}^{3} u_{- i}} + 24.9316 [\frac{1}{\frac{1}{4} Σ_{1}^{3} u_{- i}} - \frac{1}{\frac{1}{4} Σ_{1}^{3} u_{- i - 4}}] + 24.22 (\frac{P_{c} - P_{c - 4}}{P_{c - 4}}) + 57.49 (\frac{P_{c - 4} - P_{c - 8}}{P_{c - 8}}) + 19.42 (\frac{P_{c - 8} - P_{c - 12}}{P_{c - 12}}) - 92.50 [\frac{({τ^{1}}_{4} + {τ^{2}}_{4}) - ({τ^{1}}_{- 4} + {τ^{2}}_{- 4})}{1 + {τ^{1}}_{- 4} + {τ^{2}}_{- 4}}]$
	τ¹ = employer and employee contribution rate for unemployment insurance τ² = employer and employee contribution rate for Canada Pension Plan
10. The TRACE Model (Canada)	$\frac{W - W_{- 1}}{W_{- 1}} = 0.985 \frac{P_{c} - P_{c - 1}}{P_{c - 1}} + 0.169 \frac{1}{u}$
	(here $\frac{P_{c} - P_{c - 1}}{P_{c - 1}}$ is used as a proxy for the expected rate of inflation)
11. The RBA1 Model (Australia)	$W = - 1.411 - 0.091 S_{1} - 0.040 S_{2} - 0.037 S_{3} + Σ_{0}^{11} λ_{i} P_{c - i} + Σ_{0}^{10} β_{i} u_{i} + Σ_{0}^{3} γ_{i} x_{- i}$
	S_i = dummy variable u = unemployed persons, in millions x = labor productivity in nonfarm business

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Abstract

I. Problems of Specification of Indirect Taxes

1. the aggregation problem

2. the problem of dynamic incidence

II. Specification of Indirect Taxes in Selected Models

1. indirect taxes in the consumption sector

2. indirect taxes in price equations

III. Price-Wage Dynamics and Indirect Taxes in Models

IV. General Conclusions

BIBLIOGRAPHY