A Unified Approach to Measuring Poverty and Inequality
Generalized mean of order
Figure 2.6: Generalized Means and Parameter
`
xN WGM(x; α)
WE(x) WA(x) WG(x) WH(x) x1 –∞
–2
1 –1 0 Parameter
2
∞
General Means as Welfare Measures The transfer principle ensures that the general means may be interpreted as social welfare measures. Actually, the general means for a < 1 are commonly interpreted as measures of social welfare. This form of welfare function was considered by Atkinson (1970), who then defined a helpful transformation of the function called the equally distributed equivalent income (ede). The utility function that Atkinson assumed to obtain his particular ede was 1 1 U(x n ) = (x n ) a for a < 1 and a ≠ 0 and U(x ) = ln x for a = 0 for all n. n n a a The ede represents the level of income x ede, which, if received by all people in a society, yields the same welfare level as that of the original income distribution. Thus, like the general mean itself, the value of ede depends on the parameter a, and for vector x, the ede of order a is EDE(x; a) = WGM(x; a). Sen Mean The usual mean can be reinterpreted as the expected value of a single income drawn randomly from the population. Now, suppose that instead of a single income, we were to draw two incomes randomly from the population (with replacement). If we then evaluated the pair in terms of the lower of the two incomes, this would lead to the Sen mean, which is defined as the expectation of the minimum of two randomly drawn incomes.6 These two random incomes are drawn with replacement, which means that these two incomes may belong to the same person in a society. If every income in distribution x
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