Atkinson's inequality index can be derived by assuming a consumer evaluates income distributions from behind a veil of ignorance without knowing their own income. The consumer maximizes expected usefulness by integrating a utility function over the income distribution density. Atkinson's inequality index is defined as the relative cost of inequality and is the ratio of the average income to the equivalent certain income that yields the same total utility. Atkinson assumes the consumer has a constant relative risk aversion utility function, from which an expression for Atkinson's inequality index can be derived.

1. ( Every proof and expression should be done mathematically)
Atkinson's inequality index can be derived by assuming a consumer with preferences for the Von
Neumann-Morgenstern risk that 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 get in distribution. It therefore evaluates an income distribution with a density
fy(Y) By maximizing its expectation of usefulness. Suppose the well-being index SW(fy) Is the
value of this income distribution:
SW(fy)= integral ( 0 to Ymax) ( u(Y)*fy(Y)dy)
A relative inequality index can then be defined as the relative cost of inequality
I= (mu(y)))/mu(y)
Where mu(y) Is the average income in the distribution fy(Y) and is the equivalent income also
distributed, the parallel of the certain equivalent in risk theory. Is therefore implicitly defined by
U()= integral ( 0 to Ymax) ( u(Y)*fy(Y)dy)
Atkinson considers that this consumer has a utility function of the CRRA type:
U(Y)= (Y^(1))/1-
In which the risk aversion coefficient Is reinterpreted as a coefficient of aversion to inequality.
Give an expression for Atkinson's inequality index.