Stochastic estimation of firm technology, inefficiency, and productivity growth using shadow cost and distance functions

It is well-known that a firmʹs technology, allocative efficiency, technical efficiency, and productivity growth can be estimated using a shadow cost system, comprised of a shadow cost function and its share equations, whose arguments are outputs and shadow input prices (prices internal to the firm). We provide a dual characterization to estimate these measures using a shadow distance system. This system is comprised of a shadow distance function, expressed in terms of shadow input quantities and output quantities, plus the first-order conditions from the cost-minimization problem. An advantage of the distance system over the cost system is that we obtain direct estimates of input inefficiency with the former, but indirect estimates with the latter. We also show how to express cost function derivatives in terms of distance function derivatives, which allows calculation of returns to scale and price elasticities of demand from the estimated distance system. Using panel data on US electric utilities, we estimate both systems and find a strong similarity between their associated measures.