The Atkinson inequality index can be derived by assuming a consumer with Von Neumann-Morgenstern risk preferences who is placed behind the veil of ignorance and asked to evaluate income distributions. Since he is behind the veil of ignorance, this consumer does not know what income he will obtain in the distribution. He therefore evaluates an income distribution with density f_Y (y) by maximizing his utility expectation. Suppose that the welfare index SW(f_Y ) is the value of this income distribution:

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

A relative inequality index can then be defined as the relative cost of inequality

I=Y/Y

whereY is the average income in the f_Y (y) distribution and is the equally distributed equivalent income, the parallel of the certainty equivalent in risk theory. is thus implicitly defined by

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

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

u(y)=y1/1

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