We can derive the Atkinson inequality index by assuming a consumer with preferences in the face of the Von Neumann-Morgenstern type risk that is placed behind the veil of ignorance and asked to estimate distributions of income. Since he is behind the veil of ignorance, this consumer does not know what income he will get in the distribution. It therefore evaluates an income distribution having a densityfY(y) by maximizing its expected utility. Suppose that the welfare index SW(fY ) is the value of this distribution of income:

SW(fY)=0ymaxu(y)fY(y)dy

A relative inequality index can then be defined as the relative cost of inequality I=YY;

where Y is the average income in the distribution fY(y)and is the equivalent income also distributed, the parallel of the certainty equivalent in the theory of risk. is therefore implicitly defined by u()=0ymaxu(y)fY(y)dy

Atkinson considers that this consumer has a utility function of the CRRA type:

u(y)=1y1;

in which the coefficient of risk aversion, is reinterpreted as a coefficient aversion to inequality. Give an expression for the inequality index of Atkinson