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.