Shadow prices and optimal cost in economic applications

TL;DR

This paper critically examines the limitations of shadow prices in economic optimization, demonstrating that they can misestimate marginal costs and are formulation-dependent, questioning their reliability in applications like social cost of carbon estimation.

Contribution

It presents an example showing shadow prices can misrepresent true marginal costs and highlights their dependence on problem formulation, challenging their conventional use.

Findings

Shadow prices may overestimate marginal costs in optimal control.

Cost estimation via shadow prices is formulation-dependent.

Shadow prices are not always relevant for estimating true optimal costs.

Abstract

Shadow prices are well understood and are widely used in economic applications. However, there are limits to where shadow prices can be applied assuming their natural interpretation and the fact that they reflect the first order optimality conditions (FOC). In this paper, we present a simple ad-hoc example demonstrating that marginal cost associated with exercising an optimal control may exceed the respective cost estimated from a ratio of shadow prices. Moreover, such cost estimation through shadow prices is arbitrary and depends on a particular (mathematically equivalent) formulation of the optimization problem. These facts render a ratio of shadow prices irrelevant to estimation of optimal marginal cost. The provided illustrative optimization problem links to a similar approach of calculating social cost of carbon (SCC) in the widely used dynamic integrated model of climate and the economy (DICE).