# Utility Construction—Revisited

# Utility Construction—Revisited

The proposal that demand is governed by utility leads to the question of how to arrive at the utility, from evidence provided by the demand. For while demand is, in principle, directly observable, utility is not; it is just discussed, usually for its part in an explanation of demand. The question became a basis for mathematical questions making an enlargement of ‘mathematical economics’ where a host have contributed a variety of fragments. This chapter has to deal specifically with two construction topics, involving a demand function, or a finite demand correspondence. Pareto (1901) touched the utility construction question, dealing with a demand function, and remarked it could be done by solving certain partial differential equations. These referred to the inverse of the function. Volterra (1906) pointed out, in a review of Pareto, that the existence of a solution was not always assured but required certain ‘integrability conditions’, such as had been provided by a theorem of Frobenius. Thereafter the question became known as the ‘integrability problem’. Antonelli (1886) provided equivalent conditions, stated in the form of a symmetry. Slutsky (1915) did the same, by differentiation of first order Lagrange conditions, and brought into view the further necessary (in fact, more than necessary) second-order conditions. The approach gained a currency after rediscovery by Hicks and Allen (1934). Besides symmetry, the Slutsky matrix was required to have a negativity condition, intermediate between its being non-positive and negative definite and different from both. That this negativity requirement found by these discoverers had to be spurious is demonstrated by ‘The Case of the Vanishing Slutsky Matrix’ found by Afriat (1972), where there is a continuously differentiable demand function that has a utility but the Slutsky coefficients all vanish identically, as would be impossible under the negativity condition of Slutsky and the others. Lionel McKenzie (1957) made an elegant departure when he identified the Slutsky coefficient matrix with the matrix of second derivatives of a utility-cost function, so necessarily both symmetric and non-positive definite. The issue of sufficiency that had always remained completely without mention was taken up and settled by Afriat (1980). Afriat (1977, 1980) dealt with the same question when the demand function is replaced by a finite demand correspondence (1956, 1960), with the result now usually referred to as ‘Afriat's Theorem’.

*Keywords:*
choice, preference, utility, revealed preference, price index, price level, index numbers, Index Number Problem, integrability, Afriat's Theorem, new formula, consumption, demand, standard of living, Slutsky, Mckenzie, Hicks, Allen

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