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

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

$q_i^{\prime }=\frac{\partial q_i}{\partial L_i}$ = 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

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

denotes consumers’ outlays on the n commodities, and

denotes the producers’ sales receipts; hence the difference

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

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

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

while the nominal sales revenues of producers is given by

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

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

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,

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

and

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 ${\left(\frac{W}{P}\right)}_1$ equals the real wage claim by labor ${\left(\frac{W}{P}\right)}_2$ 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

and

where $\rho =\frac{\lambda K}{P{Q}^N}$ 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 $\begin{array}{cc}\begin{array}{cc}{\left(\frac{W}{P}\right)}_1& and\end{array}& {\begin{array}{c}\left(\frac{W}{P}\right)\end{array}}_2\end{array}$) 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.

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 $T_I^i={\tau }_i{C}_{i}and{T}_I=\mathrm{\Sigma }{T}_I^i\mathrm{.}$.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

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:

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

where P_i could be either the market price or factor price, in which case $P_i=P_i^f=\frac{P_i^m}{1+{\tau }_i},{\lambda }_i$ 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

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

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.