A Note on the Consumer Surplus Path-of-Integration Problem

Abstract

It is well known that Marshallian consumer surplus1 is generally dependent on the path of integration. Silberberg (1972) emphasized this dependency by demonstrating that Marshallian consumer surplus is unbounded above and below for arbitrary paths of integration. Despite the approximation results of Willig (1976), because of the path-dependency problem and the lack of a legitimate a priori restriction on the admissible paths of integration, one could maintain that Marshallian consumer surplus is potentially unreliable and generally invalid as an index of preference.2 The purpose of this paper is to elucidate a rigorous solution to the Marshallian consumer surplus path-of-integration problem. The family of expenditure function indices (which includes the Hicksian variations of consumer surplus) is examined and shown to be equivalent to Marshallian consumer surplus with certain restrictions on the admissible paths of integration; essentially, the implicit restriction guarantees the order-preserving property of these indices.3 The logical conclusion is that any a priori restriction on the admissible paths of integration that yields an order-preserving index is legitimate. In addition to the family of expenditure function indices, "monotonic variations" satisfy this principle (Zajac, 1979; Stahl, 1980).