Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher

This article graphically illustrates the one-to-one duality mapping among the production function, the product supply equation, the derived factor demand equation, and the indirect profit function for the classical profit maximization problem. This pedagogical framework is then used to illustrate how empirical application of conventional duality theory can lead to distorted empirical results if the theory (e.g., Hotelling's lemma) does not apply because the firm is not a profit maximizer or because envelope results from the wrong optimization model are used. Although the presentation is in terms of profit maximization, the basic concepts can be extended to other maintained behavioral hypotheses such as cost minimization or utility maximization. Plausible reasons why a firm, even in a competitive market, may not behave according to the neoclassical maximization paradigm are briefly reviewed. Key words: duality theory, microeconomic theory applications, optimization. Applications of duality theory to empirical problems are widespread. Some of the claimed advantages of a dual approach are: (a) it opens up a richer class of operational functional forms, especially for multiproduct, multifactor production; (b) it brings theoretical coherence to the analysis, especially with respect to cross-commodity relationships, that is often lacking in nondual approaches; and (c) it is possible to obtain factor demand and product supply equations from an indirect profit function fit-ted to profit and price data without having empirical observations of quantities demand-ed or supplied (Pope 1982b; Lau and Yoto-poulos 1971, 1972; Young et al.). Most empirical studies have used duality theory associated with the conventional static, deterministic model of perfectly competitive firm behavior (e.g., Binswanger; Lau and Yo