Interindustry flows and the incidence of the corporate income tax

Interindustry flows and the incidence of the corporate income tax

Journal of Public Economis INTERINDUSTRY 30 (1986) 359-368. North-Holland FLOWS AND THE INCIDENCE CORPORATE INCOME TAX Received April 1985, rev...

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30 (1986) 359-368.



Received April 1985, revised version



May 1986

This paper examines the empirical importance of interindustry flows in the incidence of the corporate income tax. I find that the omission of interindustry flows from general equilibrium tax incidence models can lead to serious errors, given their actual magnitude.

1. Introduction This paper explores the importance of interindustry flows in the incidence of the corporate income tax. Using a general equilibrium framework in the tradition of Harberger (1962) and transactions data for the U.S. economy, I find that the omission of interindustry flows can lead to dramatic over- and understatements of the impact of the corporate income tax on the functional distribution of income, depending on parameter values. Moreover, such omission cannot be justified by appealing to such things as separability or fixed factor ratios between value added and intermediate inputs in production; even under those assumptions, the incorporation of interindustry flows has a large impact on the results of the model. The model used here was developed by Bhatia (1982), who modified the original Harberger general equilibrium model of the incidence of the corporate income tax to account for interindustry flows. He derived analytical results regarding the nature of the incidence, whether the tax is borne disproportionately by capital or labor. In this paper I apply that model, with one modification, to data and various parameter values. I examine the magnitude of the incidence, and compare results with an analogous model without interindustry flows to see the degree of error caused by their omission. Bhatia considers a model that is identical to the original Harberger model save for its inclusion of interindustry flows.’ There are two sectors, the *I am grateful to Kul Bhatia and two referees for comments. The usual caveat applies. Support of the Centre of Policy Studies, Monash University, where some of this work was completed, is also gratefully acknowledged. ‘The notation used here is my own. Readers who desire greater detail on the model are referred to Bhatia’s paper or my working paper [Solow (198S)], available on request from the author. CMW-2727/86/$3.50


1986, Elsevier Science Publishers

B.V. (North-Holland)


J.L. Sokw,


heavily taxed or ‘corporate’ sector (X) and the lightly taxed or ‘noncorporate’ sector (Y). Each sector uses as inputs capital services (K,,K,), labor (L,, L,) and some of the output of the other sector ( Yx, X,, where the subscript denotes the purchasing sector). Consumers in the economy supply the capital and labor, and consume the remaining output of the two sectors (Xc, Yc). The prices of the two outputs and the two inputs are denoted by P,, P,, P, and P,, respectively. Following Harberger, it is assumed that both inputs are available in perfectly inelastic supply but are perfectly mobile across sectors, that both goods are produced under constant returns to scale, that consumers have identical, homothetic preferences, and that the government spends its revenue as consumers would or alternatively that all revenue is rebated to consumers in a lump-sum fashion. A perfectly competitive general equilibrium is assumed, and the incidence of the corporate income tax, treated as a marginal increase in a tax on capital in the corporate sector ( TKx) from an initially undistorted equilibrium, is examined. Incidence in this model is given by the effect of the tax on the rental/wage ratio, or Pi, where the asterisk denotes proportional change in the variable and P, is taken as numeraire. Bhatia derives an expression for Pi as a function of TKx that deoends on factor and consumption shares in the initial equilibrium, and on various elasticities of factor substitution and consumer demand. In order to evaluate tax incidence with this model, and to make a comparison of the models with and without interindustry flows, 1 thus require two sorts of information: an initial equilibrium and values for the elasticities. I will discuss each of these in turn, and then examine the difference that interindustry flows makes to tax incidence.

2. Initial equilibrium data To evaluate the shares and factor ratios that appear in the models, I require a transactions matrix aggregated into two sectors that are heavily and lightly taxed under the corporate income tax respectively. Such a matrix was arrived at by aggregating a 36-sector U.S. interindustry transactions matrix for 1974 that appears in Hudson et al. (1978). The aggregated matrix is given in table 1; the non-corporate sector is comprised of Agriculture, Table Interindustry


Supplying sector


Corporate Non-corporate Capital Labor

284.65 198.00 600.42

1 matrix

(1974 $ billion).




897.84 410.80

308.24 20 1.98

J.L. Solow, Interindustry flows


Forestry and Fisheries, Mining, Crude Petroleum and Natural Gas Extraction, Petroleum Refining, Services, Government Enterprise and a Miscellaneous category. All remaining production is included in the corporate sector. The corporate income tax rates used to differentiate the heavily and lightly taxed sectors come from Ballard et al. (1985), and are for 1973.’ The sectors were divided on the basis of being above or below the weighted average tax rate for all industry, but the division was quite readily apparent. Unfortunately, these are average, not marginal, tax rates; I am thus assuming that firms facing high average tax rates also were facing high marginal tax rates. The sectoral breakdown of the tax data matches that of the transactions data almost perfectly; the transactions data are disaggregated to a slightly liner level in some instances, but the aggregations that need to be done to get exact matching are obvious. For example, while the tax data have only one Mining sector, the transactions data have separate Metal Mining, Coal Mining, and Non-metallic except Fuel categories; the three disaggregated sectors were treated as facing the same tax rate, given by that in the tax data for the aggregated Mining sector. One other major difference between these sets of data is that the Hudson et al., data separate services from housing capita1 and consumer durables from the Finance, Insurance and Real Estate sector, while the Ballard et al., data distinguish between a Finance and Insurance sector, which is heavily taxed, and a Real Estate sector, which includes housing capital and is lightly taxed. In the aggregated matrix of table 1 the housing capital and consumer durables category has been included in the lightly taxed sector, and the Finance, Insurance and Real Estate sector has been included in the heavily taxed sector.

3. Parameter values Evaluation of the models also requires several parameter values; in particular, the income compensated elasticity of demand for corporate sector output, and parameters describing input substitution in both producing sectors. The latter consist of the elasticities of substitution between capital and labor in the two sectors (a” and 0’) in the Harberger model without interindustry flows, and three independent Allen elasticities of substitution for each sector of the model with interindustry flows3 Since estimates of these parameters are not readily available, and my interest is primarily in model ‘1 am grateful to John Shoven and Don Fullerton for providing these data. ‘With three inputs, there are actually nine Allen elasticities for each sector. However, symmetry of the Allen elasticities and the fact that the share-weighted sum of Allen elasticities with respect to any price must be zero means that three of the elasticities (two own-price and one cross-price) are sufftcient to generate the remaining six. For more on Allen elasticities, see Uzawa (1962).

J.L. Solow, lnterindustryflows


comparison, I derive incidence estimates using a range specifications. The income compensated elasticity of demand for corporate is evaluated from








where oD is the elasticity of substitution between corporate goods and noncorporate goods in the consumers’ utility functions. This causes some difficulty for model comparison. The problem stems from the calculations of the share of non-corporate output in total consumption. In the original Harberger model, output of each sector is equal to the value of capital and labor inputs into that sector under the assumption of constant returns to scale, and since the only use of output is final consumption, these give the share of non-corporate goods in total consumption. In the model with interindustry flows, total output of each sector is the sum of capital, labor and intermediate goods inputs into that sector under constant returns, and final consumption of sectoral output is the difference between total output and output used as intermediate inputs by the other sector. If the same capital and labor data are used in both models, the share of non-corporate goods in total consumption will differ. 4 Referring to eq. (l), it is clear that using the same income compensated demand elasticity implies different elasticities of substitution in utility, and using the same elasticity of substitution implies different elasticities of demand. I choose to use the same elasticity of substitution in demand, and allow the elasticity of demand for non-corporate goods to differ. I also modify the consumer demand equation used by Harberger and Bhatia to account for existing taxes in the initial equilibrium. Both assume that consumers’ demands for the two goods depend only on their relative prices, so that the demand for corporate sector output, written as a linear approximation in proportional changes, is given by XC*=&(P$-PP$).


Ballentine and Eris (1975) have pointed out that this specification implies that the initial equilibrium involves no taxes. They show that, in the face of existing taxes, the demand equation (2) should be written as

4Notice that in table 1 the total value of capital and labor inputs equals the total value of consumption, but neither sector’s value added equals its value of final consumption, nor are they even in the same ratio.

J.L. Solow, lnterindustry



M denotes the marginal T& is the existing differential


propensity to consume corporate goods, and tax on capital in the corporate sector. Since consumers’ preferences are homothetic, M is given by the share of corporate section goods in total consumption. T& is evaluated from the Ballard et al., tax data. It is given by the difference between the average corporate income tax rates for the two sectors, yielding a value of 0.1775.5 Experiments with other values show that the general results of the model are not particularly sensitive to this value. Input substitution in the original Harberger model is described by the elasticities of substitution in the two sectors, which can range from zero to infinity. The three-input production structures of the model with interindustry flows allow for much richer input substitution possibilities, including the possibility of input complementarity. For model comparisons, I could specify arbitrary Allen elasticities for the model with interindustry flows, but I would then have to calculate the corresponding capital/labor elasticities for the model without them. To make things easier, I make an additional assumption here; I assume that production in both sectors of the interindustry flow model can be represented by nested constant elasticity of substitution (CES) production functions, with capital and labor making up a CES subaggregate which then enters another CES function with the intermediate good. This assumption reduces the number of substitution parameters that have to be specified for each sector from three to two, these being the elasticities of substitution between capital and labor in the value added subaggregates (which I will denote.& and o;,) and the elasticities of substitution between value added and the intermediate inputs (denoted c& and G[,).” Once these are given, Allen partial elasticities of substitution can be calculated from this specification; for the purpose of comparing the two models, the natural thing to do is to let &=c? and ok,=&‘. The assumption of separability implies some restrictions on the model with interindustry flows. In particular, separability of this sort implies that the capital/labor ratio in each sector is a function of the after-tax rental/wage ratio only. This means that changes in the price of intermediate inputs relative to the numeraire cause no capital/labor substitution in either sector. These restrictions are the result of the assumption of separability only [for a straightforward discussion of separability of this sort, see Berndt and Wood (1979)]. The CES assumption is primarily for convenience; since the model is a linear approximation, the elasticities of substitution can always be ‘That part of the corporate income taxed faced by both sectors constitutes a general tax on capital income. It is well known that in a model such as this, with total inputs fixed, such a tax is non-distorting, and can be ignored. 6The reason that the number of parameters to be specified is reduced is that the nested CES production function is separable between the value added inputs and the intermediate good. This implies that the substitutability between capital and the intermediate good is the same as that between labor and the intermediate good.


J.L. Solow, lnterindustryflows

evaluated at the initial equilibrium, regardless of the functional form, which will be adequate for small changes in the tax. As the purpose here is to examine the empirical importance of interindustry flows, these results are intended to be illustrative; other assumptions about input substitutability will give different results. 4. Tax incidence results Given the inelastic factor supplies of the model, the nature of the functional incidence of the tax is given by the direction of the change in relative factor prices. If Pt is zero, capital and labor bear the burden of the tax in proportion to their initial shares of income. If Pz is negative (positive), the burden of the tax falls disproportionately on capital (labor) income. Ballentine and Eris (1975) have shown that capital bears the full burden of the tax (exclusive of the excess burden in the presence of existing taxes) if production in both sectors is Cobb-Douglas and on= - 1.0. This implies that if the elasticity of the rental/wage ratio with respect to the tax equals -0.44, capital bears the entire burden of the tax. It is theoretically possible for capital to bear more than the entire burden, too. Table 2 presents several sets of results from the model with interindustry flows using different parameter values. The results are grouped according to the assumed values of & and &. Corresponding results from the original Harberger model using these values for U* and 0’ are also given; these are rows 1, 8 and 11 of table 2. These results are meant to illustrate the effect of incorporating interindustry flows into the tax incidence analysis. They show a wide range of incidence possibilities; although the tax is disproportionately on capital income in all cases, the tax incidence runs from slightly more heavily on capital to more than entirely on capital. Of course, variations in elasticities of substitution of this magnitude represent considerable differences in specification.’

‘After this paper was completed, I became aware of another paper by Bhatia (1983) in which he applied data and parameter values to this model. Although the parameter choices do not correspond in all cases, our results are of similar order of magnitude; my results give tax elasticities that are about 25 percent smaller. These differences are attributable to two things. First, Bhatia does not incorporate existing taxes in the demand specification. Although this does not affect the qualitative results of the model, it does affect the magnitudes of the elasticities. Second, Bhatia uses the 1952-55 data from Harberger’s original paper, modified to account for interindustry flows, while I use data for 1974. (The reader who compares these papers should note that while u’s denote Allen elasticities in Bhatia’s paper, they denote ordinary elasticities of substitution in the CES functions in my paper. I derive Allen elasticities from these in order to calculate results.) My results do confirm Bhatia’s conclusions, in particular that assuming fixed factor ratios and/or inelastic demand causes errors if those assumptions are incorrect. If anything, my results suggest those errors are considerably larger than what Bhatia finds.


J.L. Solow, Interindustry~ows

Table 2 Tax elasticities

of relative

input prices under

alternative do=

I. uX=uY=

- 1.0

2. u~,,=“~,,=-1.0;0~,=~:,=-1.0


parameter uD=

specifications. -


0D= -2.0



- 0.29

- 0.48


- 0.40

3. I&=&=

- 1.0; r&=0:,=





4. 0;,=0;,=

- 1.0; &=&=








5. CKI. X-Y- OK,.- - 1.o;

u;y= c:,

6. c&=&=









7. u;,=0:,=

- 1.0: &,=


CT:,= -0.5





- 0.29




















8. ux=oy=

= 0.0


9. u~,.=o~,,=-o.5:uxyy=u:x=-1.0 IO. &=f&,= II. ?=uY=


12. fl;,.=0:,,= 13. &=&= 14. A;,= A;,=


-2.0 -2.0; -2.0; A&= -1.6,


- 1.0


1.0, Air= - 1.0, AfL= -0.8, A&=


Y sector = Cobb-Douglas.

Given the data underlying these estimates, ignoring interindustry flows causes considerable misstatement of the incidence of the corporate income tax. Depending on the assumed parameter values, the original Harberger model can either overstate or understate the results from the model with interindustry flows. And these differences can be considerable; among those cases where the elasticities of substitution between capital and labor in both sectors are - 1.0 (rows 1 through 7 of table 2), the largest overstatement by the Harberger model is 43 percent, which occurs when cn= -0.5 and a&= o:, = -2.0. The largest understatement is 44 percent, when on= -2.0 and o& = CJ:~ = 0.0. Moreover, the differences can be larger when other elasticities of substitution between capital and labor are used. Comparing rows 8 and 10 when on= -2.0, the Harberger model without interindustry flows understates the change in relative factor prices by 80 percent. It is possible to get overstatements of this magnitude with parameter values in these ranges as well, although none is shown here. Results using fixed factor ratios between value-added inputs and the intermediate inputs (rows 5, 10 and 13) are shown because this assumption is sometimes made in general equilibrium tax models as a first attempt at incorporating interindustry flows [see Melvin (1979) and Fullerton, Shoven

and Whalley (1978)]. These results suggest that such an assumption generally overstates the burden of the corporate income tax on capital if some degree of substitution between these inputs is possible, and that these overstatements can be large. For example, comparing rows 2 and 5 of table 2, assuming fixed factor ratios leads to an overstatement on the order of 25-30 percent. Moreover, it is not always the case that including interindustry flows and assuming no substitution gives more accurate results than ignoring interindustry flows if there is, in fact, a reasonable degree of substitutability. Notice that the. model with interindustry flows is considerably less sensitive to 0n than is the original Harberger model. This occurs because the corporate income tax has less effect on relative output prices on the presence of interindustry flows. In the Harberger model, the tax on capital in the corporate sector increases the cost of corporate sector output, which changes the relative prices of consumption goods. With interindustry flows that effect is mitigated, since the higher priced corporate sector goods are also used as an input in the non-corporate sector, causing an offsetting cost increase there. If the tax causes smaller changes in relative output prices, the willingness of consumers to substitute in response to those price changes becomes less important. Note also that the results of the two models correspond closely when in = & = afx. (The agreement is, in fact, exact when I$, =O.) This suggests the following set of conditions for the omission of interindustry flows to be approximately correct: that production in both sectors be separable between value-added and intermediate inputs, and that the elasticities of substitution between intermediate inputs and value-added be equal in the two sectors and also equal to the elasticity of substitution in demand. These are quite strong restrictions, however, and there is little reason a priori to expect them to hold. The nested-CES specification does not admit the possibility that inputs be complements in prod-ction. Yet there is some evidence to suggest that this is a real possibility. For example, Berndt and Wood (1975) found that capital and energy were complements in manufacturing. Their parameter estimates cannot be incorporated directly here, since they included four inputs in their analysis. However, given that in these data much of the non-corporate sector is energy-related and much of the corporate sector is manufacturing, it may be worthwhile to examine a case that is in the spirit of Berndt and Wood’s estimates. In row 14 of table 2 the Allen elasticities in the corporate sector between labor and capital and between labor and non-corporate goods are chosen (somewhat arbitrarily) to be unity, and the capital-non-corporate goods Allen elasticity is chosen to be - 1.0, indicating complementarity. The own-Allen elasticities are then derived from these and input shares SO as to maintain theoretical consistency. The non-corporate sector is assumed to have Cob&Douglas technology. In this case, the tax is borne dispropor-

J.L. Solow, Interindustryflows


tionately, although not entirely, by capital. It is not clear what the analogous version of the original Harberger model would be, as there is no unambiguous counterpart to the elasticity of substitution between capital and labor in the three-input case without separability.

5. Concluding remarks This paper has examined the empirical importance of interindustry flows in the incidence of the corporate income tax. Given their magnitude in the data, the omission of interindustry flows from general equilibrium tax incidence models can lead to dramatic errors. The inclusion of interindustry flows under the assumption of fixed factor ratios between intermediate goods and value-added, while appealing in that it accounts for product flows consistently, is not always an improvement, and can also lead to large errors if the assumption about factor substitutability is incorrect. The conditions under which the model without interindustry flows gives approximately correct answers involve strong restrictions and are not likely to hold. The results provided here are primarily meant to illustrate the importance of including interindustry flows in general equilibrium tax incidence models, rather than to measure the incidence of the corporate income tax. The reason is that the models considered here make other assumptions of the original Harberger analysis in order to focus on this particular issue. Many of these are unrealistic, and changing them will also change the tax incidence results. Moreover, Stiglitz (1973) has criticized the applicability of this sort of model to measurement of the incidence of the corporate income tax in particular, noting that the existence of debt financed capital and the deductibility of interest payments under the corporate income tax means the tax may not be on corporate capital at the margin. Nevertheless, small general equilibrium models are widely applied, and these results indicate that if the interindustry flows are an important feature of economic reality, their omission may lead to serious miscalculation.

References Ballard, C.L., D. Fullerton, J.B. Shoven and J. Whalley, 1985, Computation of general equilibrium for tax policy analysis (University of Chicago Press, Chicago). Ballentine, J.G. and I. Eris, 1975, On the general equilibrium analysis of tax incidence, Journal of Political Economy 83, 633-644. Berndt, E.R. and D.O. Wood, 1975, Technology, prices, and the derived demand for energy, Review of Economics and Statistics 57, 259-268. Berndt, E.R. and D.O. Wood, 1979, Engineering and economic interpretations of energy-capital complementarity, American Economic Review 69, 342-354. Bhatia, K.B., 1982, Intermediate goods and the theory of tax incidence, Public Finance 37, 318338. Bhatia, K.B., 1983, Tax effects, relative prices, and economic growth, in: D. Biehl, ed., Public finance and economic growth (Wayne State University Press, Detroit) 349-365.


J.L. Solow, Interindustryflows

Fullerton, D., J.B. Shoven and J. Whalley, 1978, General equilibrium analysis of U.S. taxation policy, in: Compendium of tax research 1978 (Department of the Treasury, Washington, D.C.). Harberger, A.C., 1962, The incidence of the corporation income tax, Journal of Political Economy 70, 2 15-240. Hudson, E., D. Jorgenson and D. O’Connor, 1978, The dynamic general equilibrium model log linear specification, Report to Applied Economics Division, Federal Preparedness Agency, General Services Administration. Melvin, J.R., 1979, Short-run price effects of the corporate income tax and implications for international trade, American Economic Review 69, 765774. Solow, J.L., 1985, Interindustry flows and the incidence of the corporate income tax, University of Iowa College of Business Working Paper No. 86-5. Stiglitz, J.E., 1973, Taxation, corporate financial policy, and the cost of capital, Journal of Public Economics 2, l-34. Uzawa, H., 1962, Production functions with constant elasticities of substitution, Review of Economic Studies 29, 291-299.