First-Order Taylor Series Approximations and Cost Functions
完整刻画了与齐次柯布-道格拉斯生产函数对偶的成本函数，发现连续成本函数具有一阶泰勒级数近似解释当且仅当它是该生产函数的对偶，即成本函数在价格和产出上均需齐次。
This note gives a complete characterization of the cost function which is the dual of a homogeneous Cobb-Douglas production function.! The main result is that a continuous cost function has a first-order Taylor series approximation interpretation if and only if it is the dual of a homogeneous Cobb-Douglas production function. In other words, all Taylor series expansions of cost functions, when expanded jointly in prices and output, will collapse to the Cobb-Douglas form when terms of secondand higher-order are dropped. This is surprising, for it means not only that the cost function is homogeneous in prices as dictated by economic theory, but that in order to have a first-order Taylor series approximation interpretation it must be homogeneous in output as well. The analysis was motivated by the work of Barnett and others who have demonstrated that the second-order Taylor series expansions used to estimate cost, indirect utility, and profit functions frequently fail to satisfy certain regularity properties.2 The objective of this paper was to identify the functional forms of all continuous functions which provide a first-order Taylor series approximation to an arbitrary cost function at a point. While one would not expect to find any acceptable first-order approximations for cost functions since they cannot model arbitrary substitution possibilities, that expectation is rigorously confirmed here. The analysis is an extension of Eichhorn (1974, 1978) and Stehling (1975), and is similar to Fare and Sung (1986).