# (p.128) A.2.2 From Welfare to Inequality

# (p.128) A.2.2 From Welfare to Inequality

The equivalent income function has the property of being *linearly homogeneous*—doubling all incomes doubles the equivalent income—whenever the original welfare function is *homothetic*.^{23} Intuitively, homotheticity makes the indifference curves ‘radial copies’ (blow‐outs or blow‐ins) of each other. In this case, the resulting inequality measure is definitely mean independent.^{24} Atkinson noted that additivity in the presence of homotheticity restricts consideration to the single‐parameter family:

^{25}

What happens if welfare is not homothetic? We lose the property of mean independence in the normative inequality measure, and this can thus introduce an ‘absolutist’ element in what is standardly thought of as being a *relative* concept (that of inequality). Blackorby and Donaldson (1978) present an alternative procedure which yields *relative inequality* measures for non‐homothetic social welfare functions.

Recall that the welfare indifference curves of non‐homothetic functions are not all radial copies of one another. Blackorby and Donaldson (1978) pick a ‘reference’ curve which is used to generate an ersatz (or an ‘as if’) welfare
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function that is linear homogeneous. Applying Atkinson's transformation then yields a relative measure or, more precisely, a different relative measure for *each* reference level of welfare. If the original social welfare function is homothetic, the ersatz welfare functions and relative inequality measures are all ‘reference free’ and we are back to the original Atkinson territory.

Blackorby and Donaldson's alternative transformation nicely extends Atkinson's line of analysis. The generalization is, however, achieved at some inescapable cost. As Blackorby and Donaldson have pointed out, the derived inequality index, while relative, need not be ‘normatively significant’. Away from the reference level, the ersatz and true welfare functions can disagree. Consequently, a particular redistribution of income can simultaneously lead to higher inequality *and* higher (true) welfare, breaking the inverse relationship (for a given mean income) underlying the classical Atkinson approach. The Blackorby‐Donaldson results identify the trade‐offs inherent in measuring ‘relative inequality’ through the social welfare approach when homotheticity cannot be presumed.

## Notes:

(23) A function is homothetic if it is an increasing transform of a linearly homogeneous function. Note that a welfare function and its equivalent income function are increasing transforms of one another.

(24)
If each income in *x* is doubled, *I* = (μ_{x} − *e* _{x})/μ_{x} is unchanged since both μ_{x} and *e* _{x} are doubled (where μ_{x} and *e* _{x} are the mean and equivalent incomes, respectively). It is interesting to note that Dalton's (1920) approach (*OEI‐1973*, p. 37) yields the same inequality measure as Atkinson's when welfare itself is linearly homogeneous.

(25)
As discussed in *OEI‐1973*, removing the requirement of additivity significantly broadens the range of welfare‐based relative inequality measures.