Redistribution and the Income of the Median Voter

Redistribution and the Income of the Median VoterAuthor(s): Phillip NelsonSource: Public Choice, Vol. 98, No. 1/2 (Jan., 1999), pp. 187-194Published by: SpringerStable URL: http://www.jstor.org/stable/30024474 .

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Public Choice 98: 187-194, 1999.

a 1999 Kluwer Academic Publishers. Printed in the Netherlands. 187

Redistribution and the income of the median voter *

PHILLIP NELSON Department of Economics, Binghamton University, Binghamton, NY 13902-6000, U.S.A.

Accepted 9 December 1996

Abstract. The income of the median voter has been measured by median income. This measure fails to consider the income distribution of both voters and number of adults per family. Proper measures of the income of the median voter change standard results. This income is no longer less than mean income; its ratio to mean income is only slightly related to the ratio of median to mean income.

Many of the uses of the median voter model depend on the relationship between mean income and the income of the median voter. The ratio of the two plays a central role both theoretically and in econometric estimation. In both, median family income has been used to measure the family income of the median voter. These are not the same concepts. This paper measures the error produced by the confusion of the two. I show that the measurement error is so large that it creates dramatic differences in the results of the theoretical literature and has some important econometric consequences as well.

This measurement problem is particularly serious in the analysis of redistribution - one of the more important applications of the median voter model. Economists have long been interested in the determinants of redistribution in a democracy. Some have focused on externalities. For example, higher income to the poor provides an insurance policy to others who might become poor later (Bishop, Formby, and Smith, 1991; Overbye, 1995). Or higher income to the poor may be a bribe to reduce anti-social behavior such as crime and riots (Piven, 1971; Hirshleifer, 1994). Others (Nelson, 1994) have claimed that voting decisions are made in terms of group rather than individual narrow self-interest. Often, however, economists have assumed that each voter votes his own self-interest excluding externalities. I call this the voting power hypothesis.

Meltzer and Richard (1981) and (1983) explore the voting power hypothesis using a uni-dimensional median voter model. In the uni-dimensional model the amount of redistribution is the decision variable. Since there is only one dimension, the pattern of redistribution is necessarily given if the amount

* Kenneth Greene made many useful suggestions.



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of redistribution is variable. Meltzer and Richard (1981, 1983) assume that each unit will be given a flat amount financed by a proportional tax on income. The median voter, as determined by the person who pays the median amount of net taxes, then determines the amount of that redistribution. The level of redistribution will be such that the median voter is indifferent to any change in its amount.

The median voter model with a constant marginal tax rate generates redistribution from rich to poor only because the median voter is assumed to have lower than mean income. If there were no labor supply responses, those with mean income would be indifferent to such redistribution because they would pay as much in taxes as they received in transfers. Given the dead- weight losses incurred because of supply responses, such redistribution hurts those with mean income. A voter could be indifferent to marginal increases in redistribution only if he had less than mean income. In order for the median voter model to generate redistribution without externalities the median voter must have less than mean income. If the median voter model is an appropriate model of voter behavior, it is important to know whether the income of the median voter is or is not less than mean income.

The rationale for the assumption that the median voter's income is less than mean income is the positively skewed income distribution. But that rationale is not sufficient. The relevant median comes from a different income distribution than that appropriate for the relevant mean. Voting power is determined by the median of those who vote. Since taxes are collected from all units whether they vote or not, the appropriate mean is the mean income of all units. The poor are less likely to vote than the rich. As a result, the median income of those who vote increases relative to the mean income of everybody.

Another less obvious process also operates. Not surprisingly, family income rather than individual income is the income determinant of voting behavior.' There are more adults over 18 in those families with higher family income than those with lower family income, so higher income families have more potential votes.

When both of these considerations are taken into account over the period 1972-89 the family income of the median voter weighted by the number of adults per family is slightly higher than mean family income for all families rather than being substantially lower. This result is based on data from the National Opinion Research Center (1994). My procedures are discussed in the Appendix. Table 1 reports my results.

Look at the geometric mean of the ratio of median family income in a year to mean family income for that year over this period. It is .8229. Next, calculate median income for a year by restricting the sample to families whose



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Table 1. The ratio of various median family incomes to mean family incomea

Year Over-all Median for Weighted median voters voter medianb

1972 .812 .962 1.017 1973 .864 .943 1.004 1974 .869 .972 1.045 1975 .866 .950 1.033 1976 .849 .934 1.019 1977 .787 .892 .985 1978 .816 .931 1.029 1980 .824 .951 1.070 1982 .771 .875 1.022 1983 .821 .913 .991 1984 .831 .926 1.033 1985 .826 .927 1.011 1986 .808 .926 1.051 1987 .805 .891 .945 1988 .824 .916 1.051 1989 .801 .885 1.005 Geometric .823 .924 1.019

mean

aWhile the median varies by column, the mean re- mains the same - the mean family income for all persons. bWeighted by the number of adults per family.

respondent voted in the relevant presidential election and continue to use the income for all families to calculate the mean. The geometric mean for the new ratio over this period is .9241. Now re-calculate this median for each year by weighting each voting family, as determined by whether the respondent from that family voted, by the number of adults in the family. Again, keep the mean the same: the mean income in a year for all families. The geometric mean of the new ratio over my years 1972-1989 is 1.0191. Given 16 yearly observations (data is not available for every year), this ratio is significantly different than I at the 5% level. (t = 2.51) However, for all practical purposes the income of the median voter is virtually identical to the mean income of all families.

There are at least two reasonable objections to these results. First, in terms of a simple self-interest model permanent income rather than actual income should be the key to voting decisions. Part of the reason for both the smaller voting probabilities and the fewer adults per family of the poor as measured by current family income is their relative youth. To handle this problem I



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Table 2. The ratio of various median family incomes to mean family income by age groupsa

Age Over-all Median for Weighted median voters voter medianb

< 30 years .793 .965 1.077 30 to 60 .922 .961 1.014 > 60 years .676 .773 .860

aThe median of medians 1972-89. b Weighted by the number of adults per family.

focus on the age group where current income most closely corresponds to permanent income: the age group from 30-60 years for respondents (see Table 2). The ratio of median family income for that group to mean family income for that group is .9217. Limiting the median to family income for voters for that group and keeping the mean the same as previously, that ratio is .9614. Changing the median by weighting voting families by the number of adults and, again, keeping the mean the same, the ratio is 1.0138.2

Another possible objection centers around the definition of income. The relevant income for the median voter model is before tax, before transfer income. If the mean and median of this income were equal, there would be no redistribution from voter power given a constant marginal tax rate. However, the family income measure used by NORC is before tax, after transfer income. One suspects that the former will have a larger variance than the latter, since the poor get more transfers than the rich. Hence, given the skewness of the income distribution, the difference between mean and median income probably will be larger in the before tax, before transfer case, so it is less likely that the adjustments we have made will make the two approximately equal. However, if the median voter model is the appropriate model of voter power, transfers and taxes will never be so high as to produce an equality between mean after transfer, after tax income and the appropriately defined after transfer, after tax median income, much less an equality between the two for after transfer, before tax income.3 When the after transfer, after tax mean equals its median, the median voter directly neither gains nor loses from an additional dollar of redistribution. But he incurs an indirect cost: the dead- weight loss from redistribution. So the median voter has pushed redistribution too far if voting power were the sole explanation for redistribution.

Since redistribution is affected by other votes than those involving Presi- dents, the exclusive focus on presidential votes - a limitation of the NORC data - might produce problems as well. However, what is known about voter participation rates strongly suggest that the discrepancy between family in-



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come of the median voter and median family income will be even greater for non-presidential elections. Voter participation rates are lower for non- presidential elections. It is generally accepted that Republicans benefit from lower participation rates. This makes sense if the proportion of high-income adults that vote is less sensitive to changes in the participation rate than this proportion for low-income adults. However, my results do not confirm this hypothesis. Over the period 1972-1989 there was virtually no correlation between voting participation rates for presidential elections by year and the ratio of the income of the median voter to median income. r = -.074. If ones priors are to be believed, the adjustments in the medians for presidential elections understate the required adjustment for all elections. If one is convinced by the correlation, it would appear that the presidential election results are an unbiased estimate of the appropriate median for all elections.

The preceding analysis assumed a constant marginal tax rate. One can always find a tax rate schedule such that the median voter would be better off with redistribution, ignoring externalities. But are those the tax and transfer rates that are actually observed? One of the problems with the actual estimates is how to allocate by income the beneficial effects of government expenditures that are not direct transfers. Neither Fullerton and Rogers (1993) and Browning and Johnson (1984) (called FR and BJ from now on) try. In fact, FJ ignore all non-cash transfers, such as Medicare, even if they can be easily allocated. In consequence, both have total tax payments far exceeding transfers. Two simple options for allocating these transfers are, first, that they are constant per family or, second, that they are directly proportional to before tax dollars. For those with mean income the two rules produce the same estimated transfers: the average of these transfers. If one ignores supply responses to taxes, those with mean income break-even with proportional taxes. Using the BJ estimates of the actual net taxes, families at the mean lose .036 of their income, again ignoring supply responses. For the FR estimates that loss is .0001 of mean income. Given efficiency losses those with mean income are certainly worse off given the best estimates of the actual tax pattern. Since the relevant median income is approximately equal or slightly in excess of mean family income, these results imply that those with this median income are worse off given the actual pattern of net taxes. Given the median voter model, redistribution cannot be explained without externalities.4

The relationship of the income of the median voter to mean income also plays an important role in the literature on public demand for a private good, (for example, Spann, 1974; Bergstrom, 1973). It is argued that, in spite of the dead weight loss generated, the median voter gains from such public



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purchases. Their analysis depends upon the income of the median voter not being greater than mean income. I find that the income of the median voter is slightly greater than mean income - about 2% greater. Though that difference is statistically significant, it probably is not economically significant. This result is somewhat less supportive to the Bergstrom, Spann position than the previously accepted view that the income of the median voter was significantly less than mean income.

The ratio of mean income to the income of the median income has played an important role in other non-redistributive theories of government expenditures (for example, Borcherding, 1985). These studies, then, estimate this role by using the ratio of median to mean income. In these cases the important question raised by these studies is not whether the income of the median voter is greater or less than mean income, but how much measurement error is generated by the substitution of median income for the income of the median voter. I can estimate this measurement error for the particular case of the ratio of median to mean income by year for the United States over the period 1972- 1989 by correlating the logarithm of the ratio of median income to mean income with the logarithm of this ratio for the income of the median voter. It is a paltry .324 and not statistically significant. Only 10% of the variance in the appropriate measure is explained by variation in the actual measure. This result is not just attributable to random errors in measurement. If that were the case, then, B = 1 in the equation:

logRA = a + flogR + g (1)

where R = the ratio of median to mean family income, RA = this ratio with the median the median of voters weighted by the number of adults. However, our estimated B is significantly less than 1 at the 5% level. (b = .348), implying that / is negatively correlated with log R.

I conjecture that one of the reasons for this negative correlation is the comparative impact of the growth in single adult families on R and on u. An increase in single adult families will tend to reduce R because these single parents are concentrated in low-income groups. However, /

would increase with the growth of single adult families because of the decreased weight to low-income families generated by a relative reduction in their family size. This surmise is confirmed by the correlation of the logarithm of the number of adults per family by year with the logarithm of the ratio of median income for voters weighted by adults per family with the unweighted median for that group. It is -.718, significant at the 1% level. Furthermore, the adults variable is positively correlated with the logarithm of R: r = .577. No wonder R works so poorly in predicting RA.



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Substantially more than half the measurement error is generated by the appropriate weighting of families by the number of adults. The correlation of the logarithm of the ratios using median income with the logarithm of the ratios using median income of voting families without weighting these families by the number of adults is .710, implying a 50% measurement error. At the very least, studies using R should correct in part for its measurement error by including the number of adults per family as an additional variable.

It must be emphasised, however, that R'S poor performance in measuring RA for my data might not generalise to cross-section studies. For recent time series the number of adults per family is quite important in determining the crucial ratios. For cross-sectional data or earlier time-series that may or may not be the case.

Notes

1. In some unpublished regressions using the other variables employed by Nelson (1994), family income was significantly related to "Republicanism" but individual income was not, nor was family size, so that family income per capita would not be an appropriate income measure either.

2. See Table 2 for these results by age groups. This median is less than the mean only for the age group > 60 years, and this is a group for which income has only the roughest correspondence to permanent income.

3. An even stronger statement can be made in terms of after transfer, before tax income. Taxes reduce the variance of before tax income. If the means and appropriately defined median income are equal for before tax, after transfer income, the median will be greater than the mean for after tax, after transfer income.

4. Mine has not been the first study that has maintained that the median voter model cannot explain redistribution without externalities. Using a quite different approach, Bishop, Formby, and Smith (1991) made that claim, but their work is vitiated by their failure to consider the value of increased leisure in estimating the returns to the median voter after government redistribution.

References

Bergstrom, T. (1973). A note on efficient taxation. Journal of Political Economy 81:187-191. Bishop, J., Formby, J. and Smith, W. (1991). Incomplete information, income redistribution,

and risk averse voter behavior. Public Choice 68: 41-55. Borcherding, T.E. (1985). The causes of government expenditure growth: A survey of the U.S.

evidence. Journal of Public Economics 28: 359-382. Browning, E. and Johnson, W. (1984). The trade-off between equality and efficiency. Journal

of Political Economy 92: 175-203. Fullerton, D. and Rogers, D. (1993). Who bears the lifetime tax burden? Washington: The

Brookings Institution. Hirshleifer, J. (1994). The dark side of the force. Economic Inquiry 32: 1-10.



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Meltzer, A. and Richard, S. (1981). A rational theory of the size of government. Journal of Political Economy 89: 914-927.

Meltzer, A. and Richard, S. (1983). Tests of a rational theory of the size of government. Public Choice 41: 403-417.

Miller, H. (1971). Rich man, poor man. New York: Crowell. National Opinion Research Center (1954). General social surveys 1972-1992. Chicago:

Chicago University Press. Nelson, P. (1994). Voting and imitation. Economic Inquiry 32: 92-102. Overbye, E. (1995). Explaining welfare spending. Public Choice 83: 313-335. Piven, E (1971). Regulating the poor. New York: Pantheon. Spann, R. (1974). Collective consumption of private goods. Public Choice 62: 63-73.

Appendix

I calculated means and medians for the whole period 1972-89 covered by the NORC data. In NORC family income is given by class intervals. In my calculations I used as my estimates of income the mid-point of the reported class intervals with two exceptions: (1) I estimated the mean of the open-ended class for each year by assuming a Pareto curve for that year and following the procedures of Miller (1971). (2) I assumed that observations for the median class were equally distributed over that class. To make the observations from the various years comparable I took the ratio of my various median incomes to mean income for each year.

For the medians by age group I used a somewhat less accurate but unbiased procedure to avoid the calculation burdens produced by the deficiencies in the SAS program. I assumed that even in the median class all observations were concentrated at the class mid-point. I, then, calculated the median of the ratios of median to mean income by years. I was able to compare the results of this procedure to the more accurate procedure above for the three medians for the over-all distributions. Errors ranged from 1 to 4%, clearly not enough to change any of our conclusions.

NORC asks respondents whether they voted for president in the last presidential election. I used this as a measure for whether all adults in the family voted or not. Since NORC tries to choose at random adult respondents for any family, little bias should be produced by assuming that all adults in the family behaved as did the respondent.



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Redistribution and the Income of the Median Voter

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