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2.3  Social Welfare and Inequality

Social welfare is regarded as an additive function over the equivalent income of households, , where . If the household itself is treated as the unit of analysis, where each household is assigned its own equivalent income, social welfare per household is simply:

(5)    

where is increasing and concave.

If the unit of analysis is the individual, so that the principle of anonymity (referred to alternatively in terms of compensation, or Pareto indifference) applies, social welfare per individual is given by:

(6)    

Finally, if the unit of analysis is the adult equivalent person, where the principle of transfers is satisfied, social welfare per equivalent adult is:

(7)    

Each of the three welfare measures is simply a weighted sum, over all households, of a function of the equivalent income of each household, . The only difference concerns the choice of the weights.[10]

The form of the additive welfare function discussed above is known to be consistent with the Atkinson inequality measure, . The Atkinson measure is defined as the proportional difference between the equally-distributed equivalent income, , and the arithmetic mean income, . Hence, is the living standard of a household which, if received by every ‘unit of analysis’ in the population, produces the same social welfare as the actual distribution, and:[11]

(8)    

Although this may be used with any form of , the most common form is:

(9)    

where is the degree of constant relative inequality aversion of a disinterested judge. For , the function becomes . In the case of the individual-based welfare function:

(10)    

The welfare function can be written in ‘abbreviated’ form in terms of the arithmetic mean and the measure of inequality, using . This has the convenient interpretation that welfare is equal to average income less a term, , which can be regarded as a measure of the ‘cost of inequality’.[12]

2.4  Poverty

The poverty measures examined here are based on the class introduced by Foster et al (1984). If the poverty line, below which individuals are judged to be in poverty, is , the poverty measure, , is defined, in terms of individuals, as:

(11)    

It would be possible to define poverty measures in terms of households or equivalent adults, but the present analysis concentrates on individuals. The measure corresponds to the widely used headcount measure of poverty, while is equal to multiplied by , where is the average of those below the poverty line. Hence depends on the average depth of poverty as well as the number of individuals below the poverty line. It can be shown that depends on the coefficient of variation of those in poverty, as well as and . In the case of poverty measures, there is no direct link with a social welfare function, as there is with the inequality measures.[13]

Notes

  • [10]In practice, microsimulation models assign a sample weight to each household so that appropriate population values can be obtained. The weights are often those provided by the statistical agency which collects the data, but they may also be modified for specific purposes. See Creedy and Tuckwell (2003) for an example of survey reweighting for microsimulation purposes in New Zealand.
  • [11]There is clearly potential for confusing the uses of the term ‘equivalent’ here, in terms of equally distributed equivalents and adult equivalent incomes.
  • [12]In this form  is the equally distributed equivalent income, though strictly abbreviated welfare per ‘person' is . However, the trade-off between equity and mean income is the same in each case. On abbreviated welfare functions, see Lambert (2001).
  • [13]For example, it would be convenient if social welfare could be expressed as average ‘income’, less the cost of inequality, less the cost of poverty. For further discussion and references, see Creedy (1997).
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