A shadow price of a resource is its value to the decision maker. Shadow prices frequently equal market prices, but there are many instances in which they differ from each other. Methods for measuring shadow prices are closely related in the constrained optimization literature of mathematical programming, and in the econometrics literature. Within the constrained theoretical optimization framework, the shadow price (often called a “dual value”) is the change in the objective function of the optimal solution, which is the result of a small change in the value of a constraint. Each constraint has an associated shadow price, which is represented by a Lagrange multiplier. This can be illustrated by considering the Lagrange multiplier associated with revenue maximization based on the sale of a commodity x at price p, where the firm faces the constraint that the amount available to sell cannot exceed X. The optimal solution is for the firm to sell X units of x, and the shadow price of x is p. If one considers this same problem, but subject to a production function constraint, namely x = f (L ), where L is the quantity of labor employed, the intuitive result is that the shadow price of a unit of output equals its marginal cost.
If the individual is assumed to maximize utility, then the interpretation of a shadow price becomes less intuitively obvious. Let us assume that the individual wishes to maximize the utility derived from leisure and consumption of good x, subject to two constraints: (1) money stock plus earned income must equal or exceed his expenses, and (2) leisure plus work hours cannot exceed total available hours. In this case, each constraint will be binding (since neither money nor time will be wasted), and each shadow price has an interesting interpretation. The shadow price of time equals the marginal utility of leisure. The shadow price of money (expressed in “utils” per dollar) equals the marginal utility of consumption divided by the price of the good x. This outcome shows that all scarce resources have a positive price, even if it is only expressed in terms of utility. That is, even if a scarce good or input is not sold directly in a market, it still has a shadow, or implicit, price. In a decentralized market, the ratio of market prices equals the ratio of shadow prices, and shadow prices are expressed in terms of utility, while actual prices are expressed in terms of dollars. When the Lagrangian takes the value of zero, its associated constraint is non-binding. With more than one constraint, additional Lagrangian multipliers are used, each with a shadow price interpretation for that resource. In economics, it is assumed that both goods and time are scarce, so that zero shadow prices do not occur.
In many cases, the shadow price differs from the actual price for a good or an input. In the industrial organization literature, this could be due to rate-of-return regulation as illustrated by Scott Atkinson and Robert Halvorsen (1984). Further, monopoly power, monopsony power, labor union regulations, distortionary taxes, sticky prices, and export/import regulations are just some of the many constraints that can cause divergences between actual and shadow prices. In the case of rate-of-return regulation for electric utilities, the shadow price of capital exceeds the actual price of capital, causing the utility to over-invest in capital. (Färe and Primont  and Mas-Colell et al.  provide many more examples of constrained cost minimization, utility maximization, profit maximization, and revenue maximization in a mathematical programming framework. In particular, Färe and Primont stress the dual relationship between various constrained optimization models.)
Econometricians have focused on estimating shadow prices (sometimes called virtual prices) for inputs and outputs that are not marketed, so that actual prices are not observed, and for inputs and outputs where shadow prices differ from actual observed prices. Robert Halvorsen and Tim Smith (1991) employ a restricted cost function to estimate the time path of the shadow price of an unextracted resource, whose actual price is not observable because of the prevalence of vertical integration in the industry. The basic technique involves taking partial derivatives of the restricted cost function with respect to the quasi-fixed input, and then employing Hotelling’s lemma.
Hedonic prices have also been estimated as shadow prices of attributes whose market prices are not available due to bundling. Hedonic regressions have a reduced form: The equilibrium price of a commodity is regressed on its attributes, which are typically not available in an unbundled form, and therefore are not separately priced. Examples are automobile price as a function of automobile attributes and housing price as a function of housing attributes. (For examples of these regressions see Atkinson and Halvorsen  and Atkinson and Crocker .) Hedonic prices have also been calculated to determine the value of human life, as shown by W. KipViscusi (2004).
Atkinson and Jeffrey Dorfman estimate the shadow price for tradable sulfur dioxide emission permits by calculating partial derivatives of the distance function with respect to sulfur dioxide emissions. Following Rolf Färe and Daniel Primont (1995), one divides the partial derivative of the distance function with respect to a good by its derivative with respect to sulfur dioxide, and then equates this to the ratio of the market price of this good to the unknown shadow price of sulfur dioxide. One then solves for the latter. The estimated shadow price can then be usefully compared to the market price for traded permits, as this market is quite thin.
A large additional literature has developed that maintains the hypothesis that shadow prices to the firm may differ from actual or market prices in instances where actual prices are observed. Shadow cost, shadow distance, and shadow profit systems have been developed and estimated. With shadow-cost systems, if ratios of estimated shadow prices differ from ratios of actual prices, then the firm is allocatively inefficient. Atkinson and Halvorsen (1984) and Atkinson and Christopher Cornwell (1994) assume shadow-cost minimization and estimate shadow-cost functions to compute the allocative efficiency of electric utilities and airlines, respectively. Panel data facilitates the estimation process, as shown by Atkinson and Cornwell. Atkinson and Primont (2002) extend this analysis to an estimation of a shadow-distance system, which employs the first-order conditions for cost minimization. A shadow-distance system estimates the dual to a shadow cost system by computing shadow quantities rather than shadow prices. After assuming shadow-profit maximization by firms, a shadow-profit system was devised by Atkinson and Halvorsen (1980) and Atkinson and Cornwell (1998). This system is used to estimate the shadow prices of inputs and outputs, where the latter are measures of monopoly power. Shadow prices for inputs and outputs have also been estimated to compute the extent of monopoly and monoposony power by Atkinson and Joe Kerkvliet (1989).
Attempts have been made to translate shadow-price estimates into the potential savings from eliminating allocative inefficiency. (See, in particular, Kumbhakar .) However, Atkinson and Dorfman (2006) show that any such decomposition is not unique, unless at least one input’s shadow price can be related to its actual price via a factor of proportionality.
Atkinson, Scott E., and Christopher Cornwell. 1994. Parametric Measurement of Technical and Allocative Inefficiency with Panel Data. International Economic Review 35: 231-245.
Atkinson, Scott E., and Christopher Cornwell. 1998. Profit versus Cost Frontier Estimation of Price and Technical Inefficiency: A Parametric Approach with Panel Data. Southern Economic Journal 64 (3): 753-764.
Atkinson, Scott E., and Thomas Crocker. 1987. A Bayesian Approach to Assessing the Robustness of Hedonic Property Value Studies. Journal of Applied Econometrics 2 (1): 27-45.
Atkinson, Scott E., and Jeffrey Dorfman. 2006. Chasing Absolute Cost Savings in a World of Relative Inefficiency. Working paper. Department of Economics, University of Georgia.
Atkinson, Scott E., and Robert Halvorsen. 1984. Parametric Efficiency Tests, Economies of Scale, and Input Demand in U.S. Electric Power Generation. International Economic Review 25 (3): 647-662.
Atkinson, Scott E., and Robert Halvorsen. 1990. The Valuation of Risks to Life: Evidence from the Market for Automobiles. Review of Economics and Statistics 72: 133-136.
Atkinson, Scott E., and Joe Kerkvliet. 1989. Dual Measures of Monopoly and Monopsony Power: An Application to Regulated Electric Utilities. Review of Economics and Statistics 71 (2): 250-257.
Atkinson, Scott E., and Daniel Primont. 2002. Measuring Productivity Growth, Technical Efficiency, Allocative Efficiency, and Returns to Scale Using Distance Functions. Journal of Econometrics 108: 203-225.
Färe, Rolf, and Daniel Primont. 1995. Multi-Output Production and Duality: Theory and Applications. Boston: Kluwer Academic Publishers.
Halvorsen, Robert, and Tim R. Smith. 1991. A Test of the Theory of Exhaustible Resources. Quarterly Journal of Economics 106 (1): 123-140.
Kumbhakar, Subal C. 1997. Modelling Allocative Inefficiency in a Translog Cost Function and Cost Share Equations: An Exact Relationship. Journal of Econometrics 76: 351-356.
Mas-Collel, Andreu, Michael D. Whinston, and Jerry R. Green. 1995. Microeconomic Theory. Oxford: Oxford University Press.
Viscusi, W. Kip. 2004. The Value of Life: Estimates with Risks by Occupation and Industry. Economic Inquiry 42 (1): 29-48.
Scott E. Atkinson