Transformations and the Income Tax

My interest in contingency table problems and the interactions of contingency tables goes back a long time. Early on I had been aware of how the problem of restructuring a graduated income-tax paralleled the non-interactive modelling of a complex contingency table, but I did not pursue the issue then. I do so now. Income tax problems are indeed quite complicated and it is'not my purpose to go into these questions in detail, to bring out inequities or loopholes, or to note marginal situations such as ones in which one might effect a substantial tax savings simply by slightly overstating one's income (I didn't believe it either, until I was shown). My primary interest is in considering how to make an equitable adjustment in an income-tax structure, whether to increase or to decrease total tax revenue, granted that the tax structure one begins with is a fair one. The correspondence of the tax restructuring problem and the contingency table problem arises in the following way. Suppose the tax rate in the i'th bracket of the original rate table is Pi, in the restructured table Pi'. If the restructured rates are equitable then in some sense the difference between Pi' and Pi is constant for all i, i.e., we have chosen a fair transformation such that f(Pi') f(Pi) = constant for all i. A contingency table situation obeying such a rule would be one displaying no interaction in the scaling defined by f(P). Relative to the contingency table problem I recognized that difficulties could arise with certain scaling transformations, that is with certain selections of f(P), as I discussed in an unpublished presentation at the 1956 annual statistical meetings in Montreal. It was unreasonable, I indicated, to consider a scaling like f (P) = P since a constant addition might not be appropriate for every P. Thus if P1 = 0.20, P1' = 0.60, P2 = 0.80, what is P2'? It cannot be 1.20, since that is outside the 0-1 range for a binomial parameter. On similar grounds it would be improper to employ f(P) = log P, for which f(P') f(P) = log (P'/P), and so depends on the ratio of the P values. For the example shown the calculated P2' value would be 2.40, again outside the 0-1 range. The same kind of difficulty would attend use of f(P) = log(1 P), albeit a proper value for P2' = 0.90 would result in the present example. The difficulty would arise in situations like P1 = 0.80, P1' = 0.40, P2 = 0.20, for which P2' would be -1.40. Scalings like these which I considered unreasonable for the contingency table situation seem not to have a fairness property which would be needed if they are to be used for restructuring a tax table. The fairness property I have in mind is that scaling should give acceptable and reasonable results both when employed to increase taxes or to decrease taxes. Where the tax rate is 90% (the maximum rate today is 70%, but in the past it has been as high as 91%) it could be pleasant to benefit from a flat reduction equal to 10% of the tax-but the same kind of rule could be near-confiscatory in situations where to increase the total tax take by 10%, a flat percentage increase was made. Thus across-the-board percentage slashes or increases in the tax are not fair ways to restructure a tax table. A flat change in the tax rate, as for example 10 percentage points, can produce similar unfair results with an increase from 90% to 100% being the reverse of a decrease from 90% to 80%. Use of the f (P) = log(1 P) scaling does avoid near-confiscatory tax rates, save in the instance where they are somewhat near-confiscatory for everyone. With this scaling it is the relative change in the net income after taxes which matters. But here a maximum tax rate of 90% would be reduced but moderately to 87.5%, where a minimum tax rate, say of 20%, is reduced to zero-in both cases there is a relative increase of 25% in net income. (The change in the tax structure made fully effective in 1965 approximated a flat slash of 16% of the tax on joint taxable incomes up to $100,000, and single taxable incomes up to $50,000. For marginal incomes above those maxima the percentage reduction in tax was still greater, attaining 23.1% with the peak tax rate dropping from 91% to 70%; this yielded a net income after taxes 3.33 times as great for that part of one's income taxable at the maximum rate. The justification for what might seem to be an unfair tax restructuring could be that the initial tax table was an unfair one imposed under wartime conditions and in need of correction. Yet another basis could be the recognition that excessively high marginal tax rates are in practice ineffective. Most individuals who might be subject to them are adept in the use of taxavoidance devices and do not pay the maximum rate. Lowering the maximum rate could reduce the incentive for tax avoidance and is fairer for those who do not pursue tax avoidance with dedication.) The common difficulty of the several scalings illustrated above is that they do not have the unlimited range -x to +x, P itself being limited to the 0-1 range, while log P and log (1 P) are limited to the range -ox to 0. A scaling avoiding such limitation is provided by any cumulative distribution defined over -x to +x, three of which I illustrate below. Table I shows for each of these three the altered rates apply* Biostatistics Center, The George Washington Univ., 7979 Old Georgetown Rd., Bethesda, MD 20014. Work done with the hopefully tolerant support of the Public Health Service under Grant CA-15686 from the National Cancer Institute.