I Introduction A DDITIVE and homogeneous production possibility frontiers have played an im-portant role in formulating statistical tests of the theory of production. In section II we characterize the class of production possibility frontiers that are homogeneous and additive. This class coincides with the class of frontiers with constant elasticities of substitution. Constancy of the elasticity of substitution has proved to be a fruitful point of departure for the analysis of production with one output and two factors of production, as in the pioneering study of capital-labor substitution by Arrow, Chenery, Minhas, and Solow.'For more than one product or more than two factors of production, constancy of elasticities of substitution and transformation is highly restrictive, as Uzawa and McFadden have demonstrated. 2 Our first objective is to develop tests of the theory of production that do not employ additivity and homogeneity as part of the maintained hypothesis. For this purpose we introduce new representations of the production possibility frontier in section III. Our ap-proach is to represent the production frontier by functions that are quadratic in the logarithms of the quantities of inputs and outputs. These functions provide a local second-order approximation to any production frontier. The resulting frontiers permit a greater variety of substitution and transformation patterns than frontiers based on constant elasticities of substitution and transformation. 3 A complete model of production includes the production possibility frontier and necessary conditions for producer equilibrium. Under constant returns to scale this model implies the existence of a price possibility frontier, defining the set of prices consistent with zero profits. 4 Necessary conditions for producer equilibrium, giving relative prices as a function of relative product and factor intensities, imply the existence of conditions determining relative product and factor intensities as a function of relative prices. The price possibility frontier and the conditions determining product and factor intensities are dual to the production possibility frontier and the necessary conditions for producer equilibrium. 5 Our second objective is to exploit the duality between prices and quantities in the theory of production. Our approach is to represent the price possibility frontier by functions that are quadratic in the logarithms of prices, paralleling our treatment of the production possibility frontier. These functions provide a local second-order approximation to any price fron-tier. The duality between direct and indirect utility functions employed in Houthakker's pathbreaking studies of consumer demand is analogous to the duality between production and price frontiers employed in our study of production.