## Measuring the Burden of High Taxes

## Table of Contents

## Notes

- Taxes collected for both purposes are also redistributed to rent-seekers - those who use the political process for gain they could not get in the free marketplace.
- Until the 1980s national production of goods and services, or gross national product (GNP), included some items that represented foreign wealth. This was altered to gross domestic product (GDP) to reflect more accurately wealth generated domestically. GNP and GDP figures differ slightly, but not enough to create a serious distortion.
- Calculated from data in Economic Report of the President, pp. 280, 372.
- Let the public goods in the previous period be designated as Gt-1 and the tax rate as . The balanced budget constraint means that G =Y, where Y is national output. The government collects taxes equal to Y, which leaves (1 - t)Y for private production. The production function that relates current national output to past output of public and private goods, then, is:

(1) Y_{t}= a(G_{t-1})^{b}[(1 - t)Y_{t- 1}]^{1-b}

= a(tY_{t-1})^{b}[(1 - t)Y_{t-1}]^{1-b}.

One might wish to recognize that not all government services are public goods and that some of the previous period's output is consumed. One can multiply G and (1 - t)Y by constants, such as a and b. This will make no difference to the results obtained in the paper. If one treats a and b as varying over time (e.g., the fraction of government expenditures devoted to public goods has declined relative to redistribution expenditures; saving has declined relative to consumption), we would have at^{b}and bt^{c}in the expression. The growth-maximizing tax rate remains unaffected. For further details about the development of this model, see Gerald W. Scully, "Taxation and Economic Growth in New Zealand," Pacific Economic Review, vol. 1, no. 2, September 1996, pp. 169-77; also see "The 'Growth Tax' in the United States," Public Choice, 85, 1995, pp. 71-90, and "What Is the Optimal Size of Government in the United States?" National Center for Policy Analysis, NCPA Policy Report No. 188, November 1994. - Dividing equation (1) in note 1 by Yt- 1 yields:

(2) Y_{t}/Y_{t-1}= 1 + g = at^{b}(1 - t)^{1-b}. - Differentiating g with respect to t in equation (2) in note 2 yields:

(3) dg/dt = at^{b-1}(1 - t)^{-b}(b - t).

The evaluation of this expression is quite simple. The first three terms of the expression -- a,t^{b-1}and (1 - t)^{-b}-- are positive. Designate the product of those terms as A. Thus:

(3') dg/dt = A(b - t).

The sign of dg/dt depends on whether the actual tax rate,t , is less than or greater than the growth-maximizing tax rate, b. - A constant, rather than a variable, growth rate.
- In the public finance literature, the "deadweight loss of taxation" arises from tax-induced distortions in allocative efficiency. This literature ignores technical inefficiency or output loss arising from taxation that causes actual output to be less than potential output. In this paper, deadweight loss is used in the latter sense.
- Two aspects of taxation cause deadweight loss. One is the gap between potential and actual output. The other arises from allocative inefficiency caused by the distortion of treating different income streams or assets differently for tax purposes. For example, interest deductibility of home mortgages induces more investment in housing than would otherwise have occurred. Deadweight loss from allocative inefficiency is not considered here.
- From the Economic Report of the President, pp. 280, 283, 371 (Washington, DC: USGPO, 1996); for years prior to 1959, Historical Statistics of the United States. For statistical reasons, the estimating equation (see note 4) is divided on both sides by one minus the tax rate and logarithms are taken.
- The explanatory variables are the tax rate and one minus the tax rate. But by definition these variables are not independent of each other. This high degree of intercorrelation may produce unreliable parameter estimates. More efficient estimates can be obtained by dividing both sides of the equation by one minus the tax rate. Thus we have a dependent variable that is one plus the growth rate divided by one minus the tax rate and an independent variable that is the tax rate divided by one minus the tax rate. The equation to be estimated is:

(1 + g)/(1 - t) = [a/(1 - t)][/(1 - t)]^{b}.

This relationship is nonlinear. It may be made linear for the purpose of estimation by taking the logarithms of both sides of the equation.

Estimation produces sensible results. The parameter estimate is highly significant and the statistical fit is good. Estimation produces a parameter estimate of the growth-maximizing tax rate of .2101. The regression is:

ln[(1 + g)/(1 - t)] = .5614 + .2101 ln[t/(1 - t)], R^{2}= .50.

(19.04) (6.74)

Various tests for heteroskedasticity and autocorrelation were conducted. Nothing of statistical significance was found. - The solution is straightforward. We have 1.327 = 1.753(.2101/.7899).2101. Multiplying [(1 + g)/(1 - t)] by [(1 - t)=.7899] we obtain [(1 + g) = 1.0484]. Subtracting 1 and multiplying by 100, we obtain 4.8 percent.
- The deadweight loss also can be calculated mathematically, as follows. Potential real GDP, at the growth-maximizing tax rate of 21 percent, grows at the rate of 4.85 percent per annum. The path of potential real GDP, Y*
_{t}, is given by Y*_{t}= Y0e^{g*t}, where g* is the maximum rate of economic growth. The actual path of real GDP is given by Y_{t}= Y_{0e}^{gt}. The difference between potential and actual real GDP at each time period is: Y_{0e}^{g*t}- Y_{0e}^{gt}. Integrating this difference over the interval from zero to infinity yields:

(1) (Y_{0e}^{g*t}/g*) - (Y_{0e}^{gt}/g) + [Y0(-g* + g)/g*g].

The path of real taxes is given by Tt = T_{0e}^{mt}, where m is the growth rate of real taxes, and is equal to 4.3 percent per annum. Integrating the tax function over the same interval yields:

(2) T_{0e}^{mt}/m - T_{0}/m.

Utilizing the initial values of Y_{0}and T_{0}and the respective growth rates and dividing (1) by (2) yields the average deadweight loss of $1.71. - The marginal deadweight loss is the change in the difference between potential and actual GDP over time divided by the change in real taxes. Given equations (1) and (2) in the above note, the marginal deadweight loss at time t is:

(Y_{0e}^{g*t}- Y_{0e}^{gt})/T_{0e}^{mt}. - The marginal deadweight loss is found by regressing the difference between real potential and actual GDP on real taxes. The result is:

DWLOSS = -1935.5 + 3.4409 REAL TAXES R^{2}= .93.

(11.40) (24.91) - V. Tanzi and L. Schuknecht, "The Growth of Government and the Reform of the State in Industrial Countries," IMF Working Paper (Washington, DC: International Monetary Fund, December 1995), p. 20.
- Ibid., p.18.
- Historical Statistics of the United States, vol. 1, p. 63.
- Ibid., p.58.
- Ibid., p.55.
- Statistical Abstract of the United States, 1995, p. 92.
- Assar Lindbeck, "Hazardous Welfare- State Dynamics," American Economic Review Papers and Proceedings, vol. 85, no. 2, May 1995, pp. 9-15.
- The reforms in New Zealand and the reduction in taxation there were precipitated by a foreign exchange crisis. New Zealand is the only OECD nation that has reduced the size and scope of government.