# Efficiency Rents of Storage Plants in Peak-Load Pricing, I: Pumped Storage

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## Abstract

Programming methods are given for rental valuation of storage and conversion facilities in cyclical, continuous-time pricing problems, e.g. pumped storage of electricity. By identifying the points of differentiability of the short-run profit function, unique and separate marginal values can be imputed to the different capital inputs, even though these are perfect complements (which, with multiple outputs, does not entail fixed input proportions). The short-run (profit-maximising) operation problem has a dual that gives the shadow prices of the stored intermediate good. Formulae for the quasi-rents are derived in terms of the shadow prices. In particular, the unit reservoir rent is shown to be equal to the total positive variation of the shadow price over the cycle. This is unique if the (given) market price for the final good is continuous over time. The shadow prices also determine the optimal storage policy. It is also shown, by studying input requirement functions, that continuous-time problems of this kind can be incorporated into a model of general competitive equilibrium with the space of bounded functions as the commodity space.

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