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The Marginal Welfare Cost of Personal Income Taxation in New Zealand

5 Evaluating changes using a social welfare function

It is often desired to evaluate tax reforms in terms of their effects on specified demographic groups or for the population as a whole. The question then arises of the welfare metric and social welfare function to use. The complexities involved, particularly in the present context where there is considerable preference heterogeneity, are discussed by Creedy, Herault and Kalb (2011, pp. 17-18). One approach, used by Aaberge and Colombino (2008), and Ericson and Flood (2009) use a welfare metric in their social welfare function that is based on an independently estimated utility (or 'welfare') function which is considered to be the same for all individuals, although heterogeneity is recognised when dealing with labour supply behaviour. Hence, the utilities in their welfare metric may differ from the utility levels determining labour supply behaviour.[26]

The welfare metric investigated here is money metric utility per adult equivalent person, allowing fully for the fact that individuals have a probability distribution (conditional on being at observed hours in the pre-change or base tax structure). Money metric utility is the ‘cost', here expressed in terms of 'full income', of achieving a specified utility level at a given set of ‘reference prices'. The reference prices used are those applying in the 'pre reform' situation. While the difficulties associated with this metric are acknowledged, as Donaldson (1992, p. 89) stressed, ‘no methodology in applied welfare economics is perfect. Practical work is always limited by the availability of data and the problem of estimating the economic consequences of projects. Different evaluation procedures are, therefore, bound to be differentially useful in different situations'. The results presented here are based on the use of money metric utility per adult equivalent, using modified OECD scales and the individual as the unit of analysis.

The procedure is as follows. For each income unit, the initial money metric utility, M0, is obtained using pre-reform taxes as ‘reference prices': conveniently, this is equal to full income under the pre-reform system. Assuming that 80 hours is the maximum number of hours that can be worked per week, net income at 80 hours of work by all adult members of the income unit under pre-reform taxes represents full income for the income unit. Then, given the equivalent variation, EV, resulting from the reform, post-reform money metric utility is computed as M1 = M0 - EV for each set of error terms.[27] For each income unit, the adult equivalent size, s, is obtained and used to compute money metric utility per adult equivalent, mj,i for tax structure j and income unit, i. The probability distributions of m0,i and m1,i are used to make the social evaluations.

With the individual as the unit of analysis, each value of mj,i is weighted by the number of persons in the income unit. The welfare function is based on Atkinson's inequality measure, A(ε ), where ε is the degree of relative inequality aversion. The inequality measure is expressed as 1 minus the ratio of the equally distributed equivalent value to the arithmetic mean (Atkinson 1970). The equally distributed equivalent value is the value which, if obtained by everyone, gives the same social welfare as the actual distribution. Using an additive welfare function based on constant relative aversion, the equally distributed equivalent value is in general, for a set of values yi, (for i=1,…,n), equal to:

Results can be obtained for a range of inequality aversion parameters, ε . Finally, social welfare in each system is obtained using the abbreviated welfare function, associated with the Atkinson inequality measure, in general given by . This means that social welfare W is the equally distributed equivalent level of money metric utility.

Results are given in Table 8 where, as before, HES sample weights are used. Results are given for two values of the inequality aversion parameter. Table 8 also reports results using net income as the welfare metric, rather than money metric utility. In the former case, no value is attached to leisure (non-work) time. The percentage reductions in ‘social welfare' resulting from the tax rate increases for net income are almost double those for money metric utility. A high aversion to inequality implies a slightly larger percentage reduction in social welfare than a low aversion, when money metric utility is used. However, this is reversed when using net incomes. In the case of money metric utility, there is no change in inequality as a result of the uniform five percentage point change in all marginal tax rates.[28] The reduction in average money metric utility implies an unambiguous reduction in social welfare with this metric.

Table 8 - Social welfare function evaluations of income tax rate change
  Mean Atkinson's index Social welfare
Pre-reform net income 29,880 0.032 0.230 28,911 23,014
Pre-reform money metric 50,874 0.050 0.328 48,343 34,207
Post-reform net income 28,004 0.031 0.221 27,145 21,813
% change using net incomes -6.28 -3.13 -3.91 -6.11 -5.22
Post-ref money metric 49,208 0.050 0.328 46,758 33,071
% change using money metric -3.28 0 0 -3.28 -3.32

Note: Money metric utility and net income are measured in per adult equivalent terms. Social welfare is the equally distributed equivalent level of money metric utility, or net income: this is the abbreviated welfare function for the Atkinson inequality measure.

In the case of net incomes, which attach no value to changes in leisure, social welfare is necessarily much lower than for money metric utility. Bigger relative changes therefore result from the tax changes. Untaxed welfare benefits make a larger contribution to low-income groups. In this case measured inequality falls. There are small labour supply responses of single men and individuals in couples, but it has been seen that many single women and single parents display substantial reductions in labour supply, which imply that their net tax payments fall (after allowing for higher benefits). This again contributes to the reduction in the measured inequality of net incomes. However, this reduction in inequality of net incomes is not sufficient to outweigh the larger reduction in arithmetic mean welfare.


  • [26]Blundell and Shephard (2009) instead use utility (as in their labour supply specification) as the welfare metric, and adopt a social welfare function based on an iso-elastic transformation of utility.
  • [27]The equivalent variation divided by initial full income is thus the proportional change in money metric utility as a result of the tax change.
  • [28]The table gives inequality measures to three decimal places, but the Atkinson measures for money metric utility did change slightly at the sixth decimal point and was thus considered to have remained constant.
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