Shadow Pricing in Production Economics

This chapter is devoted to outlining production theoretical approaches to identifying shadow prices. Shadow prices have a long history in economics; they are perhaps most familiar from welfare economics and benefit-cost analysis. The focus here is narrower – shadow prices derived in a production theoretical framework. We seek to identify prices that are consistent with resource use or opportunity cost that would induce the decision-maker to choose the observed quantity vector. Market prices – if they exist – may be consistent with this condition, but there are many cases where they are not, often referred to as cases of market failure. We begin with the function representations of technology used to identify shadow prices, i.e., distance functions and their dual associated value functions. We emphasize the relationship between primal and dual spaces through their connection with calculus. Perhaps the most familiar example of our approach is Shephard’s lemma, which uses calculus and duality to find the optimal input quantities associated with the cost function, i.e., going from price space to quantity space. Here we adopt a dual Shephard’s lemma approach: begin with quantity space representations of technology, and use duality and calculus to find the associated dual support prices. These include shadow pricing of inputs or their characteristics, shadow prices of outputs, as well as pricing of inputs and outputs in a profit function and directional distance function framework. We include an appendix devoted to the choice of appropriate functional forms which accommodate the underlying structure of technology and the calculus. A more detailed theoretical development of the role of calculus in our approach to shadow pricing suggested to us by Robert Chambers is also included in an appendix.