# Sector-specific unemployment and corporate income tax incidence: a geometric exposition.

I. Introduction

Ever since Harberger's (3) work on the incidence of the corporate income tax, the general equilibrium analysis of tax incidence has been developed in the framework of fully employed factors.(1) Only recently, in the work of Atkinson and Stiglitz (1), Chang (2) and Miyagiwa (6) has the analysis been extended to include unemployment. In the first two studies unemployment is modelled as an economy-wide phenomenon arising from rigid wages in all sectors, while sector-specific rigid wage and sector-specific unemployment were considered in the last study. Miyagiwa (6), in particular, adapted the model of unemployment originally developed by Harris and Todaro (4) for describing the situation in many developing economies, which suffer from both interindustry wage-differentials and large-scale unemployment.(2)

The purpose of this paper is to examine the incidence of the corporate income tax under the assumption that rigid-wage and unemployment are specific to the corporate sector, as is assumed by Miyagiwa (6). The factor price frontier geometry for two-sector models developed in Mussa (7) and Woodland (10) is used. This paper utilizes the techniques of duality theory to provide an alternative demonstration of tax incidence analysis. This approach allows graphical illustrations of the results, with obvious pedagogical rewards.

II. The Model and Its Assumptions

Consider a model economy which produces two commodities in quantities |x.sub.1~ and |x.sub.2~ with homogeneous capital and labor (1 denotes the corporate sector and 2 the noncorporate sector). The two commodity markets are assumed to be perfectly competitive. The production functions for both sectors exhibit constant returns to scale with positive and diminishing marginal products. Capital and labor can move freely between the sectors but are fixed in supply. Capital is fully utilized, but labor is fully employed only in the noncorporate sector, where the real wage rate (|w.sub.2~) is competitively determined. Labor's unemployment in the corporate sector is due to an exogenously determined real wage (|w.sub.1~) which is rigidly set above the level required to clear labor markets. Hence, labor migrates from the noncorporate to the corporate sector, which causes unemployment.

With capital market equilibrium, the value of the marginal product of capital is the same in both sectors, so that

|r.sub.1~ = |r.sub.2~ = r. (1)

Labor reallocation is governed by the difference between the noncorporate wage and the expected present value of the earnings stream in the corporate sector, ||w.sup.e~.sub.1~, which equals |w.sub.1~ times the probability of finding (and keeping) a job in that sector. Let |Lambda~, the ratio of the unemployed (|L.sub.u~) to the employed (|L.sub.1~), be denoted |Lambda~ |equivalent to~ |L.sub.u~/|L.sub.1~; then the labor force in the corporate sector equals |L.sub.1~(1 + |Lambda~). The probability of finding (and keeping) a job there equals |L.sub.1~/(|L.sub.1~+|L.sub.u~) |equivalent to~ 1/(1 + |Lambda~).

Labor market equilibrium is attained when

(1 + |Lambda~)|w.sub.2~ = |w.sub.1~. (2)

Let |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ be the inelastically supplied endowments of capital and labor. Then

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

where |k.sub.i~ = |K.sub.i~/|L.sub.i~ is the capital-labor ratio in the ith sector. National income is then calculated as

|Mathematical Expression Omitted~.

Consider the unit cost functions, |c.sub.1~(|w.sub.1~,r) and |c.sub.2~(|w.sub.2~,r), which are the dual to the production functions in the corporate and the noncorporate sector. Perfect competition in each sector requires that in equilibrium,

|c.sub.1~(|w.sub.1~, r) = |p.sub.1~ (6)

and

|c.sub.2~(|w.sub.2~, r) = |p.sub.2~. (7)

These unit-cost functions are positive, concave, linearly homogeneous, and increasing functions of the relevant factor prices. Assuming that product prices in both sectors are fixed, equations (6) and (7) define the factor price frontiers for each sector. These frontiers are shown in Fig. 1.(3) The slope of each unit-cost curve equals |k.sub.i~(|equivalent to~ |K.sub.i~/|L.sub.i~), i = 1, 2. Under the assumption that the corporate sector is relatively capital intensive, |k.sub.1~ |is greater than~ (1 + |Lambda~)|k.sub.2~, the |Mathematical Expression Omitted~ curve cuts the |Mathematical Expression Omitted~ curve from above.(4) Since |w.sub.1~ |is greater than~ |w.sub.2~, the underemployment equilibrium points are to the left of where the curves intersect at point F, the full employment situation. In Fig. 1, |r.sup.0~ and ||w.sup.0~.sub.2~ are the initial reward to capital and noncorporate labor, respectively. In view of (2), the gap between the exogenously determined |w.sub.1~ and ||w.sup.0~.sub.2~ reflects the value of lambda which deals with unemployment in the corporate sector.

In view of (5), national income is determined by the straight line whose slope is equal to |Mathematical Expression Omitted~ in the (w,r) space given in Fig. 1. Line Y passes through point E, (||w.sub.2~.sup.0~,|r.sup.0~, which reflects an economy suffering from unemployment in the corporate sector.

III. Tax Incidence Analysis

It is assumed that the corporate income tax can be represented by a partial factor tax |t.sub.K~ denoting the ad valorem tax rate on the earnings of capital used in producing |x.sub.1~ which can be viewed as the composite corporate good. As in Harberger (3), the proceeds are returned to consumers as a lumpsum subsidy. An imposition of the corporate income tax raises the cost of capital in that sector from r to |r.sub.1~(1 + |t.sub.K~) = |r.sub.1~|T.sub.K~ so that

|r.sub.1~|T.sub.K~ = |r.sub.2~ = r|prime~ (8)

for capital market equilibrium.

Maintaining zero profit, hence, the |Mathematical Expression Omitted~ shifts, but not in a parallel fashion, inward to |c.sub.1~|prime~(|w.sub.1~,|r.sub.1~|T.sub.K~) = |p.sub.1~|prime~, which is shown in Fig. 2. It reduces the rental rate of capital to r|prime~ but raises the noncorporate wage from ||w.sup.0~.sub.2~ to |w.sub.2~|prime~. Since the gap between |w.sub.1~ and |w.sub.2~ is reduced, unemployment unambiguously falls. The economic explanation of this result is simple. Since |w.sub.1~ is held constant, the partial factor tax results in a factor substitution effect by which labor is substituted for capital, giving rise to a lower marginal product of labor in the corporate sector. As a result labor moves back to the noncorporate sector so that the rate of unemployment falls and the corporate sector becomes more capital intensive. Moreover, since the |Mathematical Expression Omitted~ curve is steeper at E|prime~ than it is at E, this partial factor tax also creates an output effect via the change in relative commodity prices by which the expanding production of noncorporate goods becomes more capital intensive as well. It creates an excess demand for the corporate output whereby the price rises from |Mathematical Expression Omitted~ to |p.sub.1~|prime~. With perfect mobility of capital, the common rental rate falls from |r.sup.0~ to r|prime~. This result is consistent in spirit with Miyagiwa's (6) Proposition 1 and 2 that capital bears more of the corporate tax burden than does labor, whether or not the corporate sector is relatively capital abundant. To establish our result via the dual approach we needed the Neary stability condition, |k.sub.1~ |is greater than~ (1 + |Lambda~)|k.sub.2~, and not the assumption of homothetic preferences which general equilibrium models usually include. This assumption is too limiting because it precludes any incidence effects operating through changes in relative commodity prices as well as changes in factor prices.

By utilizing this dual approach we can also analyze the effect of the corporate income tax on national income. The national income line should now be calculated as

|Mathematical Expression Omitted~

With the aid of (3) and (8), (9) can be rewritten as

|Mathematical Expression Omitted~

In Fig. 2, the new national income line, Y|prime~ (through E|prime~) is to the left of Y (through E), and this implies a decrease in national income. National income will fall because of the lower rental rate, and more of the labor force will be employed in the noncorporate sector at a lower wage rate relative to the corporate wage.

IV. Concluding Remarks

We have examined the implications of sector specific wage rigidity and unemployment for the incidence of the corporate income tax. The simple geometry used establishes the results and provides greater intuitive insights without the complicated mathematical manipulations required in earlier works. In addition, we have provided one significant new result, namely, that with the imposition of the corporate income tax, national income unambiguously declines even though it is coupled with a decline in the unemployment rate. The geometry presented in this paper can be applied to investigate other issues arising in the incidence literature. For example, the equivalence relation, as is demonstrated by Mussa (7), remains intact in spite of unemployment: a tax (subsidy) on an output is equivalent to an equal percentage tax (subsidy) on both inputs used in producing that output. It could also be utilized to study the poverty incidence of trade liberalization (a tariff reduction, perhaps) in developing countries.

Notes

1. For an exposition of Harberger's model and its extensions, see, for example, Atkinson and Stiglitz (1).

2. Their analysis was followed by a spate of related contributions including McCool (5) and Neary (8) among many others.

3. A similar figure is recently used by Wang (9) for demonstrating the backward incidence of pollution control.

4. The Neary stability condition, |k.sub.1~ |is greater than~ (1 + |Lambda~)|k.sub.2~, means that the corporate sector is capital-intensive relative to the noncorporate sector in the value sense as McCool (5) has pointed out.

References

Atkinson, A. B. and J. E. Stiglitz, Lectures on Public Economics (New York: McGraw-Hill, 1980).

Chang, C. H., "A General Disequilibrium Model of Tax Incidence." Journal of Public Economics 26 (February 1985): 123-133.

Harberger, A. C., "The Incidence of Corporate Income Tax." Journal of Political Economy 70 (June 1962): 215-240.

Harris, J.R. and M. Todaro, "Migration, Unemployment and Development: A Two-Sector Analysis." American Economic Review 60 (March 1970): 126-142.

McCool, T., "Wage Subsidies and Distortionary Taxes in a Mobile Capital Harris-Todaro Model." Economica 49 (February 1982): 69-80.

Miyagiwa, K., "Corporate Income Tax Incidence in the Presence of Sector-Specific Unemployment." Journal of Public Economics 37 (October 1988): 103-112.

Mussa, M., "The Two-Sector Model in Terms of Its Dual: A Geometric Exposition." Journal of International Economics 9 (November 1979): 513-526.

Neary, J. P., "On the Harris-Todaro Model with Intersectoral Capital Mobility," Economica 48 (August 1981): 219-234.

Wang, L.F.S., "Unemployment and the Backward Incidence of Pollution Control." Journal of Environmental Economics and Management 18 (May 1990): 292-298.

Woodland, A. D., "A Dual Approach to Equilibrium in the Production Sector in International Trade Theory," Canadian Journal of Economics 10 (February 1977): 50-68.

Leonard F. S. Wang Associate Professor of Economics, Indiana State University, and Professor of Economics, Department of Business Administration, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.

Ever since Harberger's (3) work on the incidence of the corporate income tax, the general equilibrium analysis of tax incidence has been developed in the framework of fully employed factors.(1) Only recently, in the work of Atkinson and Stiglitz (1), Chang (2) and Miyagiwa (6) has the analysis been extended to include unemployment. In the first two studies unemployment is modelled as an economy-wide phenomenon arising from rigid wages in all sectors, while sector-specific rigid wage and sector-specific unemployment were considered in the last study. Miyagiwa (6), in particular, adapted the model of unemployment originally developed by Harris and Todaro (4) for describing the situation in many developing economies, which suffer from both interindustry wage-differentials and large-scale unemployment.(2)

The purpose of this paper is to examine the incidence of the corporate income tax under the assumption that rigid-wage and unemployment are specific to the corporate sector, as is assumed by Miyagiwa (6). The factor price frontier geometry for two-sector models developed in Mussa (7) and Woodland (10) is used. This paper utilizes the techniques of duality theory to provide an alternative demonstration of tax incidence analysis. This approach allows graphical illustrations of the results, with obvious pedagogical rewards.

II. The Model and Its Assumptions

Consider a model economy which produces two commodities in quantities |x.sub.1~ and |x.sub.2~ with homogeneous capital and labor (1 denotes the corporate sector and 2 the noncorporate sector). The two commodity markets are assumed to be perfectly competitive. The production functions for both sectors exhibit constant returns to scale with positive and diminishing marginal products. Capital and labor can move freely between the sectors but are fixed in supply. Capital is fully utilized, but labor is fully employed only in the noncorporate sector, where the real wage rate (|w.sub.2~) is competitively determined. Labor's unemployment in the corporate sector is due to an exogenously determined real wage (|w.sub.1~) which is rigidly set above the level required to clear labor markets. Hence, labor migrates from the noncorporate to the corporate sector, which causes unemployment.

With capital market equilibrium, the value of the marginal product of capital is the same in both sectors, so that

|r.sub.1~ = |r.sub.2~ = r. (1)

Labor reallocation is governed by the difference between the noncorporate wage and the expected present value of the earnings stream in the corporate sector, ||w.sup.e~.sub.1~, which equals |w.sub.1~ times the probability of finding (and keeping) a job in that sector. Let |Lambda~, the ratio of the unemployed (|L.sub.u~) to the employed (|L.sub.1~), be denoted |Lambda~ |equivalent to~ |L.sub.u~/|L.sub.1~; then the labor force in the corporate sector equals |L.sub.1~(1 + |Lambda~). The probability of finding (and keeping) a job there equals |L.sub.1~/(|L.sub.1~+|L.sub.u~) |equivalent to~ 1/(1 + |Lambda~).

Labor market equilibrium is attained when

(1 + |Lambda~)|w.sub.2~ = |w.sub.1~. (2)

Let |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ be the inelastically supplied endowments of capital and labor. Then

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

where |k.sub.i~ = |K.sub.i~/|L.sub.i~ is the capital-labor ratio in the ith sector. National income is then calculated as

|Mathematical Expression Omitted~.

Consider the unit cost functions, |c.sub.1~(|w.sub.1~,r) and |c.sub.2~(|w.sub.2~,r), which are the dual to the production functions in the corporate and the noncorporate sector. Perfect competition in each sector requires that in equilibrium,

|c.sub.1~(|w.sub.1~, r) = |p.sub.1~ (6)

and

|c.sub.2~(|w.sub.2~, r) = |p.sub.2~. (7)

These unit-cost functions are positive, concave, linearly homogeneous, and increasing functions of the relevant factor prices. Assuming that product prices in both sectors are fixed, equations (6) and (7) define the factor price frontiers for each sector. These frontiers are shown in Fig. 1.(3) The slope of each unit-cost curve equals |k.sub.i~(|equivalent to~ |K.sub.i~/|L.sub.i~), i = 1, 2. Under the assumption that the corporate sector is relatively capital intensive, |k.sub.1~ |is greater than~ (1 + |Lambda~)|k.sub.2~, the |Mathematical Expression Omitted~ curve cuts the |Mathematical Expression Omitted~ curve from above.(4) Since |w.sub.1~ |is greater than~ |w.sub.2~, the underemployment equilibrium points are to the left of where the curves intersect at point F, the full employment situation. In Fig. 1, |r.sup.0~ and ||w.sup.0~.sub.2~ are the initial reward to capital and noncorporate labor, respectively. In view of (2), the gap between the exogenously determined |w.sub.1~ and ||w.sup.0~.sub.2~ reflects the value of lambda which deals with unemployment in the corporate sector.

In view of (5), national income is determined by the straight line whose slope is equal to |Mathematical Expression Omitted~ in the (w,r) space given in Fig. 1. Line Y passes through point E, (||w.sub.2~.sup.0~,|r.sup.0~, which reflects an economy suffering from unemployment in the corporate sector.

III. Tax Incidence Analysis

It is assumed that the corporate income tax can be represented by a partial factor tax |t.sub.K~ denoting the ad valorem tax rate on the earnings of capital used in producing |x.sub.1~ which can be viewed as the composite corporate good. As in Harberger (3), the proceeds are returned to consumers as a lumpsum subsidy. An imposition of the corporate income tax raises the cost of capital in that sector from r to |r.sub.1~(1 + |t.sub.K~) = |r.sub.1~|T.sub.K~ so that

|r.sub.1~|T.sub.K~ = |r.sub.2~ = r|prime~ (8)

for capital market equilibrium.

Maintaining zero profit, hence, the |Mathematical Expression Omitted~ shifts, but not in a parallel fashion, inward to |c.sub.1~|prime~(|w.sub.1~,|r.sub.1~|T.sub.K~) = |p.sub.1~|prime~, which is shown in Fig. 2. It reduces the rental rate of capital to r|prime~ but raises the noncorporate wage from ||w.sup.0~.sub.2~ to |w.sub.2~|prime~. Since the gap between |w.sub.1~ and |w.sub.2~ is reduced, unemployment unambiguously falls. The economic explanation of this result is simple. Since |w.sub.1~ is held constant, the partial factor tax results in a factor substitution effect by which labor is substituted for capital, giving rise to a lower marginal product of labor in the corporate sector. As a result labor moves back to the noncorporate sector so that the rate of unemployment falls and the corporate sector becomes more capital intensive. Moreover, since the |Mathematical Expression Omitted~ curve is steeper at E|prime~ than it is at E, this partial factor tax also creates an output effect via the change in relative commodity prices by which the expanding production of noncorporate goods becomes more capital intensive as well. It creates an excess demand for the corporate output whereby the price rises from |Mathematical Expression Omitted~ to |p.sub.1~|prime~. With perfect mobility of capital, the common rental rate falls from |r.sup.0~ to r|prime~. This result is consistent in spirit with Miyagiwa's (6) Proposition 1 and 2 that capital bears more of the corporate tax burden than does labor, whether or not the corporate sector is relatively capital abundant. To establish our result via the dual approach we needed the Neary stability condition, |k.sub.1~ |is greater than~ (1 + |Lambda~)|k.sub.2~, and not the assumption of homothetic preferences which general equilibrium models usually include. This assumption is too limiting because it precludes any incidence effects operating through changes in relative commodity prices as well as changes in factor prices.

By utilizing this dual approach we can also analyze the effect of the corporate income tax on national income. The national income line should now be calculated as

|Mathematical Expression Omitted~

With the aid of (3) and (8), (9) can be rewritten as

|Mathematical Expression Omitted~

In Fig. 2, the new national income line, Y|prime~ (through E|prime~) is to the left of Y (through E), and this implies a decrease in national income. National income will fall because of the lower rental rate, and more of the labor force will be employed in the noncorporate sector at a lower wage rate relative to the corporate wage.

IV. Concluding Remarks

We have examined the implications of sector specific wage rigidity and unemployment for the incidence of the corporate income tax. The simple geometry used establishes the results and provides greater intuitive insights without the complicated mathematical manipulations required in earlier works. In addition, we have provided one significant new result, namely, that with the imposition of the corporate income tax, national income unambiguously declines even though it is coupled with a decline in the unemployment rate. The geometry presented in this paper can be applied to investigate other issues arising in the incidence literature. For example, the equivalence relation, as is demonstrated by Mussa (7), remains intact in spite of unemployment: a tax (subsidy) on an output is equivalent to an equal percentage tax (subsidy) on both inputs used in producing that output. It could also be utilized to study the poverty incidence of trade liberalization (a tariff reduction, perhaps) in developing countries.

Notes

1. For an exposition of Harberger's model and its extensions, see, for example, Atkinson and Stiglitz (1).

2. Their analysis was followed by a spate of related contributions including McCool (5) and Neary (8) among many others.

3. A similar figure is recently used by Wang (9) for demonstrating the backward incidence of pollution control.

4. The Neary stability condition, |k.sub.1~ |is greater than~ (1 + |Lambda~)|k.sub.2~, means that the corporate sector is capital-intensive relative to the noncorporate sector in the value sense as McCool (5) has pointed out.

References

Atkinson, A. B. and J. E. Stiglitz, Lectures on Public Economics (New York: McGraw-Hill, 1980).

Chang, C. H., "A General Disequilibrium Model of Tax Incidence." Journal of Public Economics 26 (February 1985): 123-133.

Harberger, A. C., "The Incidence of Corporate Income Tax." Journal of Political Economy 70 (June 1962): 215-240.

Harris, J.R. and M. Todaro, "Migration, Unemployment and Development: A Two-Sector Analysis." American Economic Review 60 (March 1970): 126-142.

McCool, T., "Wage Subsidies and Distortionary Taxes in a Mobile Capital Harris-Todaro Model." Economica 49 (February 1982): 69-80.

Miyagiwa, K., "Corporate Income Tax Incidence in the Presence of Sector-Specific Unemployment." Journal of Public Economics 37 (October 1988): 103-112.

Mussa, M., "The Two-Sector Model in Terms of Its Dual: A Geometric Exposition." Journal of International Economics 9 (November 1979): 513-526.

Neary, J. P., "On the Harris-Todaro Model with Intersectoral Capital Mobility," Economica 48 (August 1981): 219-234.

Wang, L.F.S., "Unemployment and the Backward Incidence of Pollution Control." Journal of Environmental Economics and Management 18 (May 1990): 292-298.

Woodland, A. D., "A Dual Approach to Equilibrium in the Production Sector in International Trade Theory," Canadian Journal of Economics 10 (February 1977): 50-68.

Leonard F. S. Wang Associate Professor of Economics, Indiana State University, and Professor of Economics, Department of Business Administration, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.

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Author: | Wang, Leonard F.S. |
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Publication: | American Economist |

Date: | Mar 22, 1993 |

Words: | 1887 |

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