Microeconomic theorists have learned to take advantage of the symmetry afforded by reciprocal pairs of static optimization problems. Recall that the adjective reciprocal signifies that the second (or reciprocal) optimization problem reverses the roles of the original (or primal) problem's objective function and constraint function, and substitutes the minimization hypothesis for the maximization hypothesis. The classical economic example of this occurs in the archetype pair of reciprocal (but not dual) consumer problems: utility maximization and expenditure minimization.

A powerful advantage in working with reciprocal pairs of optimization problems is that one has a choice of which problem to analyze in order to extract the economic information, for the information in one problem can always be used to extract the information in the other. For example, in the modern proof of the negative semidefiniteness of the Slutsky matrix one first establishes the negative semidefiniteness of the substitution matrix, which comprises the first partial derivatives of the Hicksian demand functions with respect to the prices, by invoking the concavity of the expenditure function and the envelope theorem. Then one uses this result along with the Slutsky equation to establish the negative semidefiniteness of the Slutsky matrix. Thus the modern proof of the negative semidefiniteness of the Slutsky matrix works off the reciprocal expenditure minimization problem rather than the primal utility maximization problem, even though the theorem to be proven pertains to the utility maximization problem's solution.