Suppose we have two individuals with the same utility function, defined over their incomes: U_A = U(I_A) and U_B = U(I_B). This utility function is increasing in income, but is subject to diminishing marginal utility. That is, U’(I) > 0 and U"(I) < 0. There is a fixed amount of income, I, to be distributed between persons A and B and we wish to distribute this income between them so as to maximize the combined utility U = U_A + U_B.

To begin, we note that B’s income is the amount left over after A gets I_A. That is, I_B = I -I_A. With this substitution, we wish to maximize U = U(I_A) + U(I - I_A) with respect to I_A. Take the first derivative, and equate to zero: = U’(I_A) -U’(I -I_A) = 0. As U’’(I) is strictly negative, this condition can hold only if I_A = I -I_A, or I_A = ½I. That is, maximum combined utility is achieved only when each individual gets exactly half the combined available income.