# A non-singular peaked Laffer curve: debunking the traditional Laffer curve.

Introduction

On a cool autumn evening in Washington in 1974, Art Laffer, 35, had one of those moments that end up defining someone for the rest of his life. Gerald Ford's chief of staff, Don Rumsfeld, and his deputy, Dick Cheney, things were different then-were sitting atop the Hotel Washington in the Two Continents lounge near the White House. Watergate and stagflation gripped the country. Ford wanted to WIN-Whip Inflation Now!--with a five-percent tax surcharge, which was supposed to re-ignite the American economy by taking big bites out of it. Today raising tax rates in a recession seems silly to almost everyone except Tom Daschle and the junior senator from New York. In the fall of 1974, Rumsfeld and Cheney were looking for alternatives. Happy to oblige was Laffer, who pointed to a mandala sketched on a cocktail napkin--two perpendicular lines and an arc--as the answer to the complex problems plaguing the nation. The Laffer Curve, one of the icons of supply-side economics, was born (American Spectator, Jan/Feb 2002).

It has been said that one of the great advantages of the Laffer curve is that you can explain it to a congressman in half an hour and he can talk about it for six months. But jokes aside, since its arrival on the public scene literally hundreds of journal articles have analyzed, dissected, rejected, accepted, objected, executed, and rehabilitated the Laffer curve. But all this literature has one thing in common; it implicitly assumes a single-peaked Laffer curve. In most of these articles the underlying assumption is that tax revenue approaches zero when the (average) tax rate is either zero or close to 1 (100%), and at some intermediate tax rates there is one peak point where tax revenue is at its maximum. Moreover, the approach taken by the literature to the aggregate (macro) Laffer curve is similar to that of the individualistic (micro) Laffer curve.

All journal articles and all text books in microeconomics, macroeconomics and public finance that we are aware of take the one peak approach (e.g., Borgas (2000), Stiglitz (1999), pp. 699-700, Rosen (1998), pp. 383-384). (1) Moreover, most authors do not even bother to distinguish between the micro and the macro Laffer curves. The aggregate (macro) Laffer curve is a vertical summation of the individualistic Laffer curves of all heterogeneous individuals in the society (in terms of hourly wage rate) at each tax rate. We will show that even if each individualistic Laffer curve has one peak point, the aggregate (macro) economic Laffer curve is likely to have multi (or at least dual) peaks.

This is based on several assumptions which reflect the wage distribution and labor supply curve in most western countries.

The assumptions are as follows:

1. The wage distribution demonstrates a very high degree of inequality. The distribution is one-tailed asymmetric with a narrow margin approaching very high wage rates while most of the population has a comparatively low wage rate. Parenthetically, Chinhui, Murphy and Piece (1993) claim that historically the wage rate inequality has always existed and is increasing over time, especially over the last several decades. Their data shows that from 1969 to 1989 the real wage of the median "income earner" remained stable, whereas the 10th percentile fell by about 20%. However, the real wage rate of the 90th percentile rose during this period by more than 15%. More recent data such as the distribution of wages in the U.S. in 1997 (see Borjas, 2000) clearly shows this one-tailed asymmetric distribution. This evidence strengthens our argument that wages exhibit a one-tailed distribution as well as a high degree of inequality.

2. Each individual who earns a given hourly wage rate, has a peak point of tax payment at some tax rate, which is different for different individuals. For the low wage earner the peak tax rate is relatively low, and for the high wage earner the peak tax rate is relatively high. Despite the above, a panel study by Martin Feldstein (1995), argues that for high-income individuals the current tax rates exceed the revenue maximizing rate. We do not intend to judge whether this claim is correct or not, as it is quite controversial, however one thing is clear: The peak points of revenue maximizing tax rates are different for different income groups. Even when the individuals are homogeneous in tastes and differ in wage rates, they are likely to have different peak points of revenue maximizing tax rates.

3. The individualistic supply curve of labor exhibits, at lower wage rates, (for most individuals in society) a positive relationship between labor supply and the wage rate. This demonstrates that the negative substitution effect of wage changes on leisure is dominant in comparison to the positive income effect. However, at relatively high wage rates the phenomenon of a backward bending labor supply occurs, indicating the dominancy of the income effect (see Link and Settle, [1981] who discuss a case of backward bending supply of married professional nurses).

Based on these three assumptions, we show that even if the individualistic Laffer curves are one-peaked, the aggregate Laffer curve may be (and based on U.S. income distribution data--is very likely to be) multi-peaked. The possibility of such a phenomenon is crucial for public finance theory, since a low tax rate peak implies a relatively heavy tax burden on lower wage earners, whereas the tax burden on the higher wage earners is relatively small. By adopting the higher tax rate associated with the second peak point, it is more likely that most of the tax rate is being imposed on the high wage earners. These considerations should of course be taken into account by policy makers when deciding on the desired tax rate.

A by-product of our paper is an introduction of specific utility functions with respect to consumption and leisure where leisure is a luxury good. The use of this simple utility function helps us to derive a backward-bending individualistic supply curve of labor in a manner that to the best of our knowledge is unique.

Finally after deriving the micro Laffer curves of individuals who differ in their wage rate and demonstrating how the multi-peaked aggregate macro Laffer curve is derived, we devote the last section to a discussion of some possible implications and conclusions.

The Case of Backward-Bending Labor Supply

Assume that each individual has an additive utility function, U, which is a positive function of the share of leisure, l, each day, i.e., 0 < l < 1, and daily consumption, C, that is measured in $ terms. The utility function that is maximized is as follows:

(1) U = 10C - [C.sup.2]/2 + 40l (2)

where the budget constraint is

(2) W(1 - t)[1 - l] = C (3)

The F.O.C. are M[U.sub.l]/M[U.sub.C] = W(1 - t),

thus

(3) 40/10 - C = W(1 - t)

or

(3') C = 10 - 40/W(1 - t)

From (2) and (3') we can derive the demand for leisure as:

(4) l = 1 - 10/W(1 - t) + 40/[[W(1 - t)].sup.2]

Because L + l = 1, we get the supply function of labor L as follows:

(5) L = 1 - l = 10/W(1 - t) - 40/[[W(1 - t)].sup.2]

The curve of L as a function of W(1-t) is introduced in Figure 1 with two regions.

[FIGURE 1 OMITTED]

From (5) we find that for a net wage rate per hour of W(1 - t) [less than or equal to] 4, L = 0. At a net wage rate of W(1 - t) = 8, L = 0.625 is at a maximum. (4)

For any increase in the net wage above 8, the daily labor supply is diminishing. (5)

The Shape of the Individual Laffer Curve

The Laffer curve shows the relationship between tax revenue, T, and the tax rate, t, for any level of basic gross wage rate W, and labor supply, L.

In our specific case where T = t x W x L, according to equation (5) above, we get

(6) T = t x W x [10/W(1 - t) - 40/[[W(1 - t)].sup.2] = 10 {(1-4/W) t - [t.sup.2]/[(1 - t).sup.2]

For t = 0 and t = 1 - 4/W there is no tax revenue.

For 0 < t < 1 - 4/W the tax revenue is positive.

For

(7) t = 1 - 4/W/1 + 4/W the tax revenue is at its maximum (peak point of the Laffer curve).

Because of the backward bending supply curve we can find another interesting characteristic.

At low tax rates an increase in t leads to a greater increase in labor supply. Therefore, the Laffer curve increases at an increasing rate. At some point the increase changes its form and continues to increase at a diminishing rate up to the peak point, then it starts diminishing until reaching zero tax revenue.

By taking the derivations of [d.sup.2]Tax [??]/d[t.sup.2] > 0 from (6) we can find that

if t < 1 - 8/W/1 + 4/W, the tax revenue is increasing at an increasing rate,

if 1 - 8/W/1 + 4/W < t < 1 - 4/W/1 + 4/W, the tax revenue is increasing by a diminishing rate.

At

(8) t = 1 - 8/W/1 + 4/W we find a saddle point.

In Table 1 we use a specific value of W = 40 to demonstrate tax revenue as a function of the tax rate t where the following results hold:

For t = 0 and t = 0.9, the tax revenue is zero.

For 0 < t < 0.8/1.1 = .7272727, tax revenue is increasing at an increasing rate.

For .7272727 < t < 0.8181 = 0.9/1.1, tax revenue is increasing at a diminishing rate.

At t = 0.8181 we reach the peak point where tax revenue is at its maximum of T = 20.25.

Based on (6), (7) and (8) above, we find that for a different wage group that earns W = 8 and has the same leisure-income preferences, at t = 1/3 tax revenue is at a maximum of 1.25, and there is no saddle point, i.e., the Laffer curve is increasing at a diminishing rate up to t=1/3 and approaches a tax revenue of zero at t = .5.

Now we turn to deriving the Laffer curve for the case of N identical individuals whose gross wage is 8(W = 8). The individual tax revenue is then multiplied by N and we get:

(6') Tax (N / W = 8) = N(5t - 10[t.sup.2])/[(1 - t).sup.2]

(6') is a simple vertical summation of the original equation (6) above for the case of W = 8.

This Laffer curve reaches its peak at t = 1/3 and zero value tax revenues at t = 0 and at t = 0.5.

Based on (6') of N identical individuals whose basic gross income is relatively low (W = 8) and assuming for simplicity only one individual whose gross income is significantly larger ([bar.W] = 40), we turn to deriving the aggregate Laffer curve. We have pointed out that studies show that this kind of one-tailed asymmetrical wage distribution is typical of the U.S., and by implication, perhaps also of many Western societies. We now turn to deriving the aggregate Laffer curve of society as a whole, i.e., the Laffer curve of N low wage individuals combined with (for the sake of simplicity) the one wealthy individual.

Again, we use the technique of vertical summation of (6') of N individuals with the Laffer equation whose income is [bar.W] = 40, as follows:

(6'') T(1/[bar.W] = 40) = 9t - 10[t.sup.2]/[(1 - t).sup.2]

TT, the total tax revenue derived by the summation of (6') and (6'') leads to

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

What is the shape of the TT curve? For the region where 0 < t < 1/3 the TT curve is definitely increasing.

In the region 1/3 < t < 0.5 the wealthy individual's Laffer curve is increasing, i.e., dT(1 / [bar.W] = 40)/dt > 0 but the value of this term might be smaller than the absolute value of dT(N / W = 8)/dt < 0.

Namely the tax revenue reduction of N poor individuals can be larger than the tax revenue increase of the wealthy individual especially for a relatively large N. In such a case the aggregate Laffer curve is diminishing with the tax rate in the region 1/3 < t < 0.5. However, we can reach a point where tax revenues from the N identical individuals approach zero (at t = 0.5), and still an increase in the tax rate leads to an increase in tax collection from the wealthy individual. This increase will continue until we reach the tax rate of t = 0.8. In this case it is clear that the Laffer curve has two peaks.

Now we derive the conditions under which the two peaks of the Laffer curve exist. From (6'') we get

(10) d[T(1 / W = 40)]/dt = (9 - 11t)/[(1 - t).sup.3]

From (6') we get:

(11) d[T(N / W = 8)]/dt = 5N(1 - 3t)/[(1 - t).sup.3] < 0 for t > 1/3

From (10) and (11) we can see that if 5N(3t - 1) > (9 - 11t) the Laffer curve is diminishing in the region 1/3 < t < 0.5.

At t = 9 + 5N/11 + 15N we reach the first peak point of the Laffer curve. For N = 1 the first peak point is closer to t = 0.05, while as N (the number of low wage individuals) increases the peak point will move toward t = 1/3.

Another peak point of curve TT can be obtained by taking the derivative of (9) with respect to t for the region where t > 0.5.

This second peak point of T revenue is obtained from equation (6''), i.e., at t = 0.818181.

Furthermore, we can find from (9) that for the case where N [approximately equal to] 13 at t [approximately equal to] 0.3 we reach total tax revenue of 20.25 where the tax burden is distributed between 13 low wage rate individuals and the wealthy individuals as follows:

(6') T(N = 13 x W = 8) = 13 (5 x 0.3596 - [10.0.3596.sup.2])/[(1 - 0.3596).sup.2] [approximately equal to] 15.75

(6'') T(1 / W = 40) = 9 x 0.3596 - 10 x [0.3596.sup.2]/[(1 - 0.3596).sup.2] [approximately equal to] 4.5

where 7/9 = 0.777 of tax revenues come from the low wage rate individuals, while in the case of t = 0.8181 all tax revenues come from the wealthy individual.

Implications and Conclusions

The shape of the individualistic Laffer curve is usually a curve with one peak point as illustrated by many economists.

The transformation from the individual curve to the aggregate Laffer curve does not lead necessarily to the same shape, and under certain conditions (conditions that appear to hold in many Western countries) it is more likely that the vertical summation of individualistic Laffer curves of different individuals will generate a curve with dual, multiple and even continuous regions of peak values of tax revenue. We obviously have not proven that this must occur. What we have shown is the conditions under which this is likely to occur, and these conditions (according to the previously discussed income distribution studies) appear to reflect today's actual empirical U.S. income distribution. At the very least this should point in the direction of the desirability of further research of the aggregate Laffer curve.

When a population group is characterized by homogeneity in tastes and preferences along with homogeneity in earning ability (as reflected in wage-per-hour differentials within the population group), it is clear that both the individual Laffer curve, and the derived aggregate Laffer curve would be single-peaked for tax rates ranging from 0 to 1.

However, in the event that earning power has a non-homogeneous distribution (and the empirical data points to a large majority concerning relatively little, with a slim minority earning relatively large sums) then the result will be a multi-peaked aggregate Laffer Curve, even if tastes and preferences are still assumed to remain homogeneous. This is because the Laffer peaks of each wage group are substantially different in both their peak values and in the tax rates at which those peaks are located, as the gap between the peak-tax rates of the various wage groups increases the likelihood of generating a multi-peaked macro Laffer curve dramatically increases. Therefore, in those Western countries (such as the U.S.) where the wage distribution is very diverse the probability of the Laffer curve being multi-peaked is high. This has serious practical implications to the policy maker. It is possible that at a given average tax rate the change in tax revenues resulting from a change in that tax rate could be ambiguous.

In the event that the Laffer curve is indeed multi-peaked, this may lead to the fascinating result that situations could arise where either a reduction or an increase in the tax rate at every marginal rate would yield an increase in tax revenue, which of course could never happen in the case of the traditional Laffer curve. If that were to occur, the issue would shift from tax revenues to the tax burden, i.e., on whom do we wish to impose the tax burden, on the middle income, the low-income or the wealthy. An analysis of this issue would consider a variety of factors, including political, psychological, and social issues and not necessarily the simple fiscal question of how to finance the government's budget. The issue of fairness also arises, if revenues raised from lower wage earners are used to finance public goods primarily consumed by higher wage earners.

Last but not least is the issue of how changes in the tax rate actually affect the tendency to work versus the opportunity to spend time on leisure. Again we can demonstrate that any change in the tax rate may encourage wealthy people to work more, while middle or low wage rate individuals may be affected differently.

Notes

(1.) As far as journal articles are concerned we have checked all major databases and have not found a single journal article that raises even the possibility that the aggregate Laffer curve might be multi-peaked. This holds true for earlier papers such as that of Charles E. Stuart in the JPE of 1981, up to and including recent papers such as that of Kent Matthews in the January 2003 issue of International Review of Applied Economics. None of these papers has suggested the possibility of a multi-peaked Laffer curve.

(2.) A more general case that demonstrates similar results can be U = [alpha]C - [beta][C.sup.2] + [gamma]l where [alpha][beta] and [gamma] are positive parameters.

(3.) Recent articles discuss more general models in which the use to which the tax revenue is put, is considered, i.e. where the tax revenues either finance the supply of a public good, or private good that is publicly provided, [see Gahvari (1998)], or the tax revenue generated is assumed to be allocated to the consumer as a transfer payment. For simplicity we ignore this issue and simply assume that the government acts like a firm that desires to maximize it revenues for its own benefit and the consumer perceives tax only as a burden on him/herself.

(4.) This we get by taking the derivative of (5) dL/d(W(1 - t)) = -10/[[W(1 - t)].sup.2] + 80/[[W(1 - t)].sup.3] = 0.

(5.) The backward bending of the labor supply curve may also exist for a specific value when marginal utility from leisure is increasing (and not necessarily constant) as we assume in equation (1) above. The proof can be provided by the author upon request.

References

Borjas, George J., "Labor Economics," McGraw Hill (2000)

Chinhui, Juhn, Kevin, M., Murphy, and Brooks Pierce, (1993), "Wage Inequality and the Rise in Returns to Skill," Journal of Political Economy, 101, June, pp. 410-42.

Feldstein, M., (1995), "The Effect of Marginal Tax Rates on Taxable Income: A Panel Study of the 1986 Tax Reform Act," June, pp. 551-72.

Gahvari, Firouz, (1988), "Does the Laffer Curve Ever Slope Down?" National Tax Journal, 41, 2, June, pp. 266-69.

Link, Charkes. Russel, Settle, (1981), "Wage Incentives and Married Professional Nurses: A Case of Backward-Bending Supply," Economic Inquiry, 19, January, pp. 144-56.

Matthews, K., (2003), "VAT Evasion and VAT Avoidance: Is There a European Laffer Curve for VAT?," International Review of Applied Economics, 17, January, pp. 105-14.

Rosen, H.S., (1998), Public Finance, Irwin McGraw Hill, Fifth Edition.

Sachs, D.J. and F. Larrain, (1993), Macroeconomics in the Global Economy, Prentice Hall, Inc., Englewood Cliffs, New Jersey.

Stiglitz, J.E., (1999), Economics of the Public Sector, W.W. Norton and Company, Third Edition.

Stuart, C.E., (1981), "Swedish Tax Rates, Labor Supply, and Tax Revenues," The Journal of Political Economy, 89, October, pp. 1020-39.

Uriel Spiegel * and Joseph Templeman **

* The Interdisciplinary Department of Social Sciences, Bar-Ilan University, Ramat-Gan, 52900, Israel and visiting Professor, University of Pennsylvania. Email: spiegeu@mail.biu.ac.il

** The College of Management, Rishon Letzion, 75190, Israel. Email: ytempeih@barak.net.il We wish to thank the referee for helpful and insightful comments

On a cool autumn evening in Washington in 1974, Art Laffer, 35, had one of those moments that end up defining someone for the rest of his life. Gerald Ford's chief of staff, Don Rumsfeld, and his deputy, Dick Cheney, things were different then-were sitting atop the Hotel Washington in the Two Continents lounge near the White House. Watergate and stagflation gripped the country. Ford wanted to WIN-Whip Inflation Now!--with a five-percent tax surcharge, which was supposed to re-ignite the American economy by taking big bites out of it. Today raising tax rates in a recession seems silly to almost everyone except Tom Daschle and the junior senator from New York. In the fall of 1974, Rumsfeld and Cheney were looking for alternatives. Happy to oblige was Laffer, who pointed to a mandala sketched on a cocktail napkin--two perpendicular lines and an arc--as the answer to the complex problems plaguing the nation. The Laffer Curve, one of the icons of supply-side economics, was born (American Spectator, Jan/Feb 2002).

It has been said that one of the great advantages of the Laffer curve is that you can explain it to a congressman in half an hour and he can talk about it for six months. But jokes aside, since its arrival on the public scene literally hundreds of journal articles have analyzed, dissected, rejected, accepted, objected, executed, and rehabilitated the Laffer curve. But all this literature has one thing in common; it implicitly assumes a single-peaked Laffer curve. In most of these articles the underlying assumption is that tax revenue approaches zero when the (average) tax rate is either zero or close to 1 (100%), and at some intermediate tax rates there is one peak point where tax revenue is at its maximum. Moreover, the approach taken by the literature to the aggregate (macro) Laffer curve is similar to that of the individualistic (micro) Laffer curve.

All journal articles and all text books in microeconomics, macroeconomics and public finance that we are aware of take the one peak approach (e.g., Borgas (2000), Stiglitz (1999), pp. 699-700, Rosen (1998), pp. 383-384). (1) Moreover, most authors do not even bother to distinguish between the micro and the macro Laffer curves. The aggregate (macro) Laffer curve is a vertical summation of the individualistic Laffer curves of all heterogeneous individuals in the society (in terms of hourly wage rate) at each tax rate. We will show that even if each individualistic Laffer curve has one peak point, the aggregate (macro) economic Laffer curve is likely to have multi (or at least dual) peaks.

This is based on several assumptions which reflect the wage distribution and labor supply curve in most western countries.

The assumptions are as follows:

1. The wage distribution demonstrates a very high degree of inequality. The distribution is one-tailed asymmetric with a narrow margin approaching very high wage rates while most of the population has a comparatively low wage rate. Parenthetically, Chinhui, Murphy and Piece (1993) claim that historically the wage rate inequality has always existed and is increasing over time, especially over the last several decades. Their data shows that from 1969 to 1989 the real wage of the median "income earner" remained stable, whereas the 10th percentile fell by about 20%. However, the real wage rate of the 90th percentile rose during this period by more than 15%. More recent data such as the distribution of wages in the U.S. in 1997 (see Borjas, 2000) clearly shows this one-tailed asymmetric distribution. This evidence strengthens our argument that wages exhibit a one-tailed distribution as well as a high degree of inequality.

2. Each individual who earns a given hourly wage rate, has a peak point of tax payment at some tax rate, which is different for different individuals. For the low wage earner the peak tax rate is relatively low, and for the high wage earner the peak tax rate is relatively high. Despite the above, a panel study by Martin Feldstein (1995), argues that for high-income individuals the current tax rates exceed the revenue maximizing rate. We do not intend to judge whether this claim is correct or not, as it is quite controversial, however one thing is clear: The peak points of revenue maximizing tax rates are different for different income groups. Even when the individuals are homogeneous in tastes and differ in wage rates, they are likely to have different peak points of revenue maximizing tax rates.

3. The individualistic supply curve of labor exhibits, at lower wage rates, (for most individuals in society) a positive relationship between labor supply and the wage rate. This demonstrates that the negative substitution effect of wage changes on leisure is dominant in comparison to the positive income effect. However, at relatively high wage rates the phenomenon of a backward bending labor supply occurs, indicating the dominancy of the income effect (see Link and Settle, [1981] who discuss a case of backward bending supply of married professional nurses).

Based on these three assumptions, we show that even if the individualistic Laffer curves are one-peaked, the aggregate Laffer curve may be (and based on U.S. income distribution data--is very likely to be) multi-peaked. The possibility of such a phenomenon is crucial for public finance theory, since a low tax rate peak implies a relatively heavy tax burden on lower wage earners, whereas the tax burden on the higher wage earners is relatively small. By adopting the higher tax rate associated with the second peak point, it is more likely that most of the tax rate is being imposed on the high wage earners. These considerations should of course be taken into account by policy makers when deciding on the desired tax rate.

A by-product of our paper is an introduction of specific utility functions with respect to consumption and leisure where leisure is a luxury good. The use of this simple utility function helps us to derive a backward-bending individualistic supply curve of labor in a manner that to the best of our knowledge is unique.

Finally after deriving the micro Laffer curves of individuals who differ in their wage rate and demonstrating how the multi-peaked aggregate macro Laffer curve is derived, we devote the last section to a discussion of some possible implications and conclusions.

The Case of Backward-Bending Labor Supply

Assume that each individual has an additive utility function, U, which is a positive function of the share of leisure, l, each day, i.e., 0 < l < 1, and daily consumption, C, that is measured in $ terms. The utility function that is maximized is as follows:

(1) U = 10C - [C.sup.2]/2 + 40l (2)

where the budget constraint is

(2) W(1 - t)[1 - l] = C (3)

The F.O.C. are M[U.sub.l]/M[U.sub.C] = W(1 - t),

thus

(3) 40/10 - C = W(1 - t)

or

(3') C = 10 - 40/W(1 - t)

From (2) and (3') we can derive the demand for leisure as:

(4) l = 1 - 10/W(1 - t) + 40/[[W(1 - t)].sup.2]

Because L + l = 1, we get the supply function of labor L as follows:

(5) L = 1 - l = 10/W(1 - t) - 40/[[W(1 - t)].sup.2]

The curve of L as a function of W(1-t) is introduced in Figure 1 with two regions.

[FIGURE 1 OMITTED]

From (5) we find that for a net wage rate per hour of W(1 - t) [less than or equal to] 4, L = 0. At a net wage rate of W(1 - t) = 8, L = 0.625 is at a maximum. (4)

For any increase in the net wage above 8, the daily labor supply is diminishing. (5)

The Shape of the Individual Laffer Curve

The Laffer curve shows the relationship between tax revenue, T, and the tax rate, t, for any level of basic gross wage rate W, and labor supply, L.

In our specific case where T = t x W x L, according to equation (5) above, we get

(6) T = t x W x [10/W(1 - t) - 40/[[W(1 - t)].sup.2] = 10 {(1-4/W) t - [t.sup.2]/[(1 - t).sup.2]

For t = 0 and t = 1 - 4/W there is no tax revenue.

For 0 < t < 1 - 4/W the tax revenue is positive.

For

(7) t = 1 - 4/W/1 + 4/W the tax revenue is at its maximum (peak point of the Laffer curve).

Because of the backward bending supply curve we can find another interesting characteristic.

At low tax rates an increase in t leads to a greater increase in labor supply. Therefore, the Laffer curve increases at an increasing rate. At some point the increase changes its form and continues to increase at a diminishing rate up to the peak point, then it starts diminishing until reaching zero tax revenue.

By taking the derivations of [d.sup.2]Tax [??]/d[t.sup.2] > 0 from (6) we can find that

if t < 1 - 8/W/1 + 4/W, the tax revenue is increasing at an increasing rate,

if 1 - 8/W/1 + 4/W < t < 1 - 4/W/1 + 4/W, the tax revenue is increasing by a diminishing rate.

At

(8) t = 1 - 8/W/1 + 4/W we find a saddle point.

In Table 1 we use a specific value of W = 40 to demonstrate tax revenue as a function of the tax rate t where the following results hold:

For t = 0 and t = 0.9, the tax revenue is zero.

For 0 < t < 0.8/1.1 = .7272727, tax revenue is increasing at an increasing rate.

For .7272727 < t < 0.8181 = 0.9/1.1, tax revenue is increasing at a diminishing rate.

At t = 0.8181 we reach the peak point where tax revenue is at its maximum of T = 20.25.

Based on (6), (7) and (8) above, we find that for a different wage group that earns W = 8 and has the same leisure-income preferences, at t = 1/3 tax revenue is at a maximum of 1.25, and there is no saddle point, i.e., the Laffer curve is increasing at a diminishing rate up to t=1/3 and approaches a tax revenue of zero at t = .5.

Now we turn to deriving the Laffer curve for the case of N identical individuals whose gross wage is 8(W = 8). The individual tax revenue is then multiplied by N and we get:

(6') Tax (N / W = 8) = N(5t - 10[t.sup.2])/[(1 - t).sup.2]

(6') is a simple vertical summation of the original equation (6) above for the case of W = 8.

This Laffer curve reaches its peak at t = 1/3 and zero value tax revenues at t = 0 and at t = 0.5.

Based on (6') of N identical individuals whose basic gross income is relatively low (W = 8) and assuming for simplicity only one individual whose gross income is significantly larger ([bar.W] = 40), we turn to deriving the aggregate Laffer curve. We have pointed out that studies show that this kind of one-tailed asymmetrical wage distribution is typical of the U.S., and by implication, perhaps also of many Western societies. We now turn to deriving the aggregate Laffer curve of society as a whole, i.e., the Laffer curve of N low wage individuals combined with (for the sake of simplicity) the one wealthy individual.

Again, we use the technique of vertical summation of (6') of N individuals with the Laffer equation whose income is [bar.W] = 40, as follows:

(6'') T(1/[bar.W] = 40) = 9t - 10[t.sup.2]/[(1 - t).sup.2]

TT, the total tax revenue derived by the summation of (6') and (6'') leads to

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

What is the shape of the TT curve? For the region where 0 < t < 1/3 the TT curve is definitely increasing.

In the region 1/3 < t < 0.5 the wealthy individual's Laffer curve is increasing, i.e., dT(1 / [bar.W] = 40)/dt > 0 but the value of this term might be smaller than the absolute value of dT(N / W = 8)/dt < 0.

Namely the tax revenue reduction of N poor individuals can be larger than the tax revenue increase of the wealthy individual especially for a relatively large N. In such a case the aggregate Laffer curve is diminishing with the tax rate in the region 1/3 < t < 0.5. However, we can reach a point where tax revenues from the N identical individuals approach zero (at t = 0.5), and still an increase in the tax rate leads to an increase in tax collection from the wealthy individual. This increase will continue until we reach the tax rate of t = 0.8. In this case it is clear that the Laffer curve has two peaks.

Now we derive the conditions under which the two peaks of the Laffer curve exist. From (6'') we get

(10) d[T(1 / W = 40)]/dt = (9 - 11t)/[(1 - t).sup.3]

From (6') we get:

(11) d[T(N / W = 8)]/dt = 5N(1 - 3t)/[(1 - t).sup.3] < 0 for t > 1/3

From (10) and (11) we can see that if 5N(3t - 1) > (9 - 11t) the Laffer curve is diminishing in the region 1/3 < t < 0.5.

At t = 9 + 5N/11 + 15N we reach the first peak point of the Laffer curve. For N = 1 the first peak point is closer to t = 0.05, while as N (the number of low wage individuals) increases the peak point will move toward t = 1/3.

Another peak point of curve TT can be obtained by taking the derivative of (9) with respect to t for the region where t > 0.5.

This second peak point of T revenue is obtained from equation (6''), i.e., at t = 0.818181.

Furthermore, we can find from (9) that for the case where N [approximately equal to] 13 at t [approximately equal to] 0.3 we reach total tax revenue of 20.25 where the tax burden is distributed between 13 low wage rate individuals and the wealthy individuals as follows:

(6') T(N = 13 x W = 8) = 13 (5 x 0.3596 - [10.0.3596.sup.2])/[(1 - 0.3596).sup.2] [approximately equal to] 15.75

(6'') T(1 / W = 40) = 9 x 0.3596 - 10 x [0.3596.sup.2]/[(1 - 0.3596).sup.2] [approximately equal to] 4.5

where 7/9 = 0.777 of tax revenues come from the low wage rate individuals, while in the case of t = 0.8181 all tax revenues come from the wealthy individual.

Implications and Conclusions

The shape of the individualistic Laffer curve is usually a curve with one peak point as illustrated by many economists.

The transformation from the individual curve to the aggregate Laffer curve does not lead necessarily to the same shape, and under certain conditions (conditions that appear to hold in many Western countries) it is more likely that the vertical summation of individualistic Laffer curves of different individuals will generate a curve with dual, multiple and even continuous regions of peak values of tax revenue. We obviously have not proven that this must occur. What we have shown is the conditions under which this is likely to occur, and these conditions (according to the previously discussed income distribution studies) appear to reflect today's actual empirical U.S. income distribution. At the very least this should point in the direction of the desirability of further research of the aggregate Laffer curve.

When a population group is characterized by homogeneity in tastes and preferences along with homogeneity in earning ability (as reflected in wage-per-hour differentials within the population group), it is clear that both the individual Laffer curve, and the derived aggregate Laffer curve would be single-peaked for tax rates ranging from 0 to 1.

However, in the event that earning power has a non-homogeneous distribution (and the empirical data points to a large majority concerning relatively little, with a slim minority earning relatively large sums) then the result will be a multi-peaked aggregate Laffer Curve, even if tastes and preferences are still assumed to remain homogeneous. This is because the Laffer peaks of each wage group are substantially different in both their peak values and in the tax rates at which those peaks are located, as the gap between the peak-tax rates of the various wage groups increases the likelihood of generating a multi-peaked macro Laffer curve dramatically increases. Therefore, in those Western countries (such as the U.S.) where the wage distribution is very diverse the probability of the Laffer curve being multi-peaked is high. This has serious practical implications to the policy maker. It is possible that at a given average tax rate the change in tax revenues resulting from a change in that tax rate could be ambiguous.

In the event that the Laffer curve is indeed multi-peaked, this may lead to the fascinating result that situations could arise where either a reduction or an increase in the tax rate at every marginal rate would yield an increase in tax revenue, which of course could never happen in the case of the traditional Laffer curve. If that were to occur, the issue would shift from tax revenues to the tax burden, i.e., on whom do we wish to impose the tax burden, on the middle income, the low-income or the wealthy. An analysis of this issue would consider a variety of factors, including political, psychological, and social issues and not necessarily the simple fiscal question of how to finance the government's budget. The issue of fairness also arises, if revenues raised from lower wage earners are used to finance public goods primarily consumed by higher wage earners.

Last but not least is the issue of how changes in the tax rate actually affect the tendency to work versus the opportunity to spend time on leisure. Again we can demonstrate that any change in the tax rate may encourage wealthy people to work more, while middle or low wage rate individuals may be affected differently.

TABLE 1 Laffer Curve Values for W = 40 t (tax rate) T (Tax revenue) 0.1 0.987654321 0.2 2.1875 0.3 3.673469 0.4 5.5555 0.5 8 0.6 11.12 0.7 15.555 0.72727 16.889 0.8 20 0.8181 20.25 0.83 20.10380 0.85 18.888 0.9 0

Notes

(1.) As far as journal articles are concerned we have checked all major databases and have not found a single journal article that raises even the possibility that the aggregate Laffer curve might be multi-peaked. This holds true for earlier papers such as that of Charles E. Stuart in the JPE of 1981, up to and including recent papers such as that of Kent Matthews in the January 2003 issue of International Review of Applied Economics. None of these papers has suggested the possibility of a multi-peaked Laffer curve.

(2.) A more general case that demonstrates similar results can be U = [alpha]C - [beta][C.sup.2] + [gamma]l where [alpha][beta] and [gamma] are positive parameters.

(3.) Recent articles discuss more general models in which the use to which the tax revenue is put, is considered, i.e. where the tax revenues either finance the supply of a public good, or private good that is publicly provided, [see Gahvari (1998)], or the tax revenue generated is assumed to be allocated to the consumer as a transfer payment. For simplicity we ignore this issue and simply assume that the government acts like a firm that desires to maximize it revenues for its own benefit and the consumer perceives tax only as a burden on him/herself.

(4.) This we get by taking the derivative of (5) dL/d(W(1 - t)) = -10/[[W(1 - t)].sup.2] + 80/[[W(1 - t)].sup.3] = 0.

(5.) The backward bending of the labor supply curve may also exist for a specific value when marginal utility from leisure is increasing (and not necessarily constant) as we assume in equation (1) above. The proof can be provided by the author upon request.

References

Borjas, George J., "Labor Economics," McGraw Hill (2000)

Chinhui, Juhn, Kevin, M., Murphy, and Brooks Pierce, (1993), "Wage Inequality and the Rise in Returns to Skill," Journal of Political Economy, 101, June, pp. 410-42.

Feldstein, M., (1995), "The Effect of Marginal Tax Rates on Taxable Income: A Panel Study of the 1986 Tax Reform Act," June, pp. 551-72.

Gahvari, Firouz, (1988), "Does the Laffer Curve Ever Slope Down?" National Tax Journal, 41, 2, June, pp. 266-69.

Link, Charkes. Russel, Settle, (1981), "Wage Incentives and Married Professional Nurses: A Case of Backward-Bending Supply," Economic Inquiry, 19, January, pp. 144-56.

Matthews, K., (2003), "VAT Evasion and VAT Avoidance: Is There a European Laffer Curve for VAT?," International Review of Applied Economics, 17, January, pp. 105-14.

Rosen, H.S., (1998), Public Finance, Irwin McGraw Hill, Fifth Edition.

Sachs, D.J. and F. Larrain, (1993), Macroeconomics in the Global Economy, Prentice Hall, Inc., Englewood Cliffs, New Jersey.

Stiglitz, J.E., (1999), Economics of the Public Sector, W.W. Norton and Company, Third Edition.

Stuart, C.E., (1981), "Swedish Tax Rates, Labor Supply, and Tax Revenues," The Journal of Political Economy, 89, October, pp. 1020-39.

Uriel Spiegel * and Joseph Templeman **

* The Interdisciplinary Department of Social Sciences, Bar-Ilan University, Ramat-Gan, 52900, Israel and visiting Professor, University of Pennsylvania. Email: spiegeu@mail.biu.ac.il

** The College of Management, Rishon Letzion, 75190, Israel. Email: ytempeih@barak.net.il We wish to thank the referee for helpful and insightful comments

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Author: | Spiegel, Uriel; Templeman, Joseph |
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Publication: | American Economist |

Geographic Code: | 1USA |

Date: | Sep 22, 2004 |

Words: | 3636 |

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