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This gives characterizations of long-run producer optimum for a general convex technology. Each is either an optimization system or a differential system (i.e., a set of conditions put in terms of either the marginal optimal values or the optimal solutions to a primal-dual pair of programmes). One system is singled out because of its importance to applications. It is the Split SRP Optimization System, which serves as the preferred basis for the short-run approach because the technology of electricity generation is best described by production sets, and because short-run cost (SRC) minimization for each separate plant is easily split off as a subprogramme of short-run profit (SRP) maximization. The first differential system presented is the SRC/P Partial Differential System, which generalizes Boiteux’s original set of conditions. The L/SRC Partial Differential System has the same mathematical form but uses the LRC instead of the SRP. Equivalence of the two systems extends, to nondifferentiable costs, the Wong-Viner Envelope Theorem on equality of SRMC and LRMC. This is made possible by using the subdifferential as a generalized, multi-valued derivative, and by replacing input optimality with the stronger condition that the rental prices be equal to the profit-imputed values of the fixed inputs. The values must be based on increments to the operating profit (it is ineffective to try to value capacity increments by reductions in a nondifferentiable operating cost).

Such technologies describe the case of completely rigid industrial plants. Examples have already been encountered here in the context of electricity (Sect. 5.1): both thermal generation and pumped storage, though not hydro, are such techniques.

Once a pair of solutions (to a primal-dual programme pair) is found, a direct comparison of their values will show whether there is a duality gap. But there is also a method of checking for a gap at the outset—before solving the programmes. Namely, absence of a duality gap is equivalent to Type One semicontinuity of either optimal value, primal or dual (i.e., to semicontinuity of the primal value w.r.t. the primal parameters, or of the dual value w.r.t. the dual parameters). This well-known result—given in, e.g., [44, Theorem 15] and [36, 7.3.2]—is next stated for the SRP, LRC and SRC optimization programmes.

This illustrates the short-run approach to long-run equilibrium with Boiteux’s treatment of the simplest peak-load pricing problem—the problem of pricing the services of a homogeneous capacity which produces a nonstorable good with cyclic demands (such as electricity). A direct calculation of long-run equilibrium poses a fixed-point problem, but, with cross-price independent demands, short-run equilibrium can be determined instant-by-instant by the elementary method of intersecting the supply and demand curves. Long-run equilibrium can then be found by adjusting the capacity so that its unit cost equals its unit value defined as the unit operating profit.

This applies the model of Chap. 4 to the pricing, operation and investment problems of an Electricity Supply Industry with a technology that can include hydroelectric generation and pumped storage of energy, in addition to thermal generation. The application draws on the authors’ studies of operation and valuation of hydroelectric and pumped-storage plants (which recast Koopmans’s method of shadow pricing of hydroelectric resources). Those results are summarized and “fed into” the short-run approach. Thus the rudimentary one-station model of Chap. 2 is augmented to a continuous-time equilibrium model of electricity pricing with a diverse technology whose plant mix makes supply cross-price dependent, even in the short run (i.e., with the capacities fixed). Demand, too, is allowed to be cross-price dependent.

Building on Chap. 3, this sets up a framework for the short-run approach to long-run general-equilibrium pricing of a range of commodities with joint costs of production. Boiteux’s idea is thus developed into a frame for analysis of investment and pricing by an industry that supplies a range of commodities—such as a good differentiated over time, locations or events. In particular, it is shown that what is called the SRP Programme-Based System gives a full characterization of long-run market equilibrium. The question of methods for computing short-run market equilibria is only touched upon (the use of Walrasian tatonnement is suggested).

Koopmans's approach to hydroelectric storage is reformulated for competitive profit maximization, with electricity priced by time of use. Duality methods of linear programming are applied to the problems of operation and rental valuation of a hydro plant and its river. Both problems are approached by using time-dependent shadow pricing of water, and if the given market price for electricity ( p ) is a continuous function of time, then the shadow price function for water ( ψ ) is shown to be unique. The two prices ( ψ and p ) determine not only the optimal water storage policy but also the marginal values of the plant's capacities, defined as derivatives of the operating profit. In particular, the unit reservoir rent equals the total positive variation of the water value ( ψ ) over the cycle. Profit-imputed values of the river flow and of the hydro capacities (reservoir and turbine) are therefore definite – unlike Koopmans's cost-imputed values, derived from fuel savings in a mixed hydrothermal system. The marginal capacity values can be used to determine the optimum investment. The model applies also to other natural flows, e.g., water supply and geothermal or tidal energy.

With L∞ as the commodity space, the equilibrium price density is shown to be a continuous function of the commodity characteristics. The result is based on symmetry ideas from the Hardy-Littlewood-Pólya theory of rearrangements; and it includes, but is not limited to, the case of symmetric (rearrangement-invariant) production costs and additively separable consumer utility. For continuous-time peak-load pricing of, e.g., electricity, this allows the inclusion of storage and of cross-price dependent demands. In this context a continuously varying price has two uses. First, it excludes the demand jumps that arise from discontinuous switches from one price rate to another. Second, in the operation and valuation of hydroelectric and pumped-storage plants (studied elsewhere), price continuity guarantees that their capacities (viz., the reservoir and the converter), the energy stocks and, in the case of hydro also the river flows, have well-defined marginal values.

Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when $L^{\infty}$ is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production functions are Mackey continuous. A general form for production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. This gives a framework for settling Boiteux's conjecture on the shifting-peak problem. To make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges. The general equilibrium results are supplemented by results for prices supporting individual consumer or producer optima.

We present two ‘localisation’ techniques which facilitate verifications of the topological properties of sets, functions and correspondences needed in, e.g., economic equilibrium analysis with infinite-dimensional commodity spaces. The first uses the Krein-Smulian theorem, which shows that weak∗ upper semicontinuity of a concave function on a dual Banach space is equivalent to bounded weak∗ u.s. continuity. The second is based on the continuity of lattice operations: for a nondecreasing function on a topological vector lattice, we show that lower semicontinuity on a set bounded from below is equivalent to l.s. continuity on bounded subsets. In the case of L∞, we use convergence in measure to establish thatsequential semicontinuity, lower for the Mackey topology or upper for the weak∗ one, is equivalent to semicontinuity. This greatly simplifies some arguments; e.g., the Mackey continuity of a concave, nondecreasing integral functional on L+∞ becomes an immediate consequence of Lebesgue's Dominated Convergence Theorem. Other uses in mathematical economics are also discussed.

Assume thatL is a topological vector lattice andY is a closed subset ofL + ×R N, whereR N denotes theN-dimensional Euclidean space. It is shown that the setY−L + ×R + N is closed ifY has appropriate monotonicity properties. The result is applicable to the case ofL equal toL ∞ with the Mackey topology, τ(L ∞,L 1).

Norm-to-weak* continuity of excess demand as a function of prices is proved by using our two-topology variant of Berge’s Maximum Theorem. This improves significantly upon an earlier result that, with the extremely strong finite topology on the price space, is of limited interest, except as a vehicle for proving equilibrium existence. With the norm topology on the price space, our demand continuity result becomes useful in applications of equilibrium theory, especially to problems with continuous commodity spectra. Some auxiliary results are also given, including closedness of the total production set and additivity of the asymptotic cone operation. Both are needed in proving equilibrium existence by the use of the Debreu-Gale-Nikaido Lemma.

This is a new formal framework for the theory of competitive equilibrium and its applications. Our “short-run approach” means the calculation of long-run producer optima and general equilibria from the short-run solutions to the producer’s profit maximization programme and its dual. The marginal interpretation of the dual solution means that it can be used to value the capital and other fixed inputs, whose levels are then adjusted accordingly (where possible). But short-run profit can be a nondifferentiable function of the fixed quantities, and the short-run cost is nondifferentiable whenever there is a rigid capacity constraint. Nondifferentiability of the optimal value requires the introduction of nonsmooth calculus into equilibrium analysis, and subdifferential generalizations of smooth-calculus results of microeconomics are given, including the key Wong-Viner Envelope Theorem. This resolves long-standing discrepancies between “textbook theory” and industrial experience. The other tool employed to characterise long-run producer optima is a primal-dual pair of programmes. Both marginalist and programming characterizations of producer optima are given in a taxonomy of seventeen equivalent systems of conditions. When the technology is described by production sets, the most useful system for the short-run approach is that using the short-run profit programme and its dual. This programme pair is employed to set up a formal framework for long-run general-equilibrium pricing of a range of commodities with joint costs of production. This gives a practical method that finds the short-run general equilibrium en route to the long-run equilibrium, exploiting the operating policies and plant valuations that must be determined anyway. These critical short-run solutions have relatively simple forms that can greatly ease the fixed-point problem of solving for equilibrium, as is shown on an electricity pricing example. Applicable criteria are given for the existence of the short-run solutions and for the absence of a duality gap. The general analysis is spelt out for technologies with conditionally fixed coefficients, a concept extending that of the fixed-coefficients production function to the case of multiple outputs. The short-run approach is applied to the peak-load pricing of electricity generated by thermal, hydro and pumped-storage plants. This gives, for the first time, a sound method of valuing the fixed assets–in this case, river flows and the sites suitable for reservoirs.

The Wong-Viner Envelope Theorem on the equality of long-run and short-run marginal costs (LRMC and SRMC) is reformulated for convex but generally nondifferentiable cost functions. The marginal cost can be formalized as the multi-valued subdifferential a.k.a. the subgradient set but, in itself, this is insufficient to extend the result effectively, i.e., to identify suitable SRMCs as LRMCs. This goal is achieved by equating the profit-imputed values of the fixed inputs to their prices. Thus reformulated, the theorem is proved from a lemma on the sections of the joint subdifferential of a bivariate convex function. The new technique is linked to the Partial Inversion Rule of convex calculus.

The “short-run approach” calculates long-run producer optima and general equilibria by building on short-run solutions to the producer’s profit maximization problem and on profit-based valuation of the fixed inputs. We outline this method and illustrate it on an example of peak-load pricing.

We apply duality methods of linear and convex programming to the problems of operation and rental valuation of facilities for conversion and storage of cyclically priced goods, e.g. , energy. Both problems are approached by shadow-pricing the stock (which is a purely intermediate commodity); and if the given market price p for the final good is a continuous function of time, then the stock's shadow price function ? is shown to be unique (and continuous). Therefore, despite being perfect Allen-Hicks complements, the plant's capacities have definite and separate marginal values, which are expressed in terms of ? (and p). In particular, the unit reservoir rent equals the total positive variation of ? over the cycle. The optimal storage policy is also given in terms of ? and p). The marginal capacity values are used to determine the optimum investment. The framework can accommodate related storage problems (such as hydroelectric generation).

Using convex calculus, we extend the Wong-Viner Theorem to nondifferentiable costs by equating the capital inputs' rental prices to their profit-imputed marginal values. Thus extended, the short-run approach to LRMC pricing is applied to peak-load pricing with storage.

Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when L? is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production function are Mackey continuous. A general form of the production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. General equilibrium results are supplemented by those for prices supporting individual consumer or producer optima. Also, to make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges.

Duality methods of linear and convex programming are applied to impute definite marginal values to the fixed inputs of a hydroelectric plant from the operating profit. Our earlier analysis of pumped storage (of energy and other cyclically priced goods) is thus extended to valuation of an external inflow to the reservoir. Given a continuous time-of-use price for electricity, the profit-imputed hydro values are uniquely determined - unlike the corresponding values imputed from fuel savings for a mixed hydro-thermal system. In particular the water inflow is assigned a unique, time-dependent shadow price. The short-run profit is then differentiable in all the fixed inputs, so that unique and separate marginal values can be imputed to the reservoir and the turbine capacities (despite their perfect complementarity). The two rents can be expressed in terms of the shadow price for water (which determines the optimal storage policy). In particular, the unit reservoir rent equals the total positive variation of the shadow price over the cycle. Evaluation of profit-imputed rents is shown to be useful not only to a profit-maximising industry but also to a public utility aiming to price its outputs at long-run marginal cost and to optimise its capital stock on the basis of purely short-run calculations. In addition we verify the production set properties that are needed to incorporate such a storage problem into a continuous-time model of general competitive equilibrium with the space of bounded functions of time as the commodity space.

Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when L? is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production function are Mackey continuous. A general form of the production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. General equilibrium results are supplemented by those for prices supporting individual consumer or producer optima. Also, to make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges.

Let (A, %plane1D;49C;, μ) be a finite measure space, and let Ωµ, w+f denote the set of all nonnegative real-valued %plane1D;49C;-measurable functions on A weaklymajorized by a nonnegative function f, in the sense of Hardly, Littlewood and Pólya. For a nonatomic µ, the extreme points ofΩµ, w +f are shown to be the nonnegativefunctions obtained by taking a fraction (1−θ) of the largest values of and arranging them in any way on any subset of A of measure(1−θ), with values elsewhere set equal to zero. Topological properties of these extreme points are given.

I. Schur showed that a differentiable, real-valued, symmetric function defined on a Euclidean space is isotone in the majorisation order if and only if the vector of its partial derivatives at any point is arranged similarly to the vector of the coordinates of that point. This classical result is extended to the case of convex, possibly nonsmooth, symmetric functions defined on rearrangement-invariant function spaces. The result is applied to prove for convex symmetric functions that rearrangement and subdifferentiation commute. The extreme points of the subdifferentials of the support functions of (i) the set of all functions majorised by a given function, and (ii) the set of all nonnegative functions weakly majorised by a given nonnegative function, are characterised.

Let ( A , A , μ ) (A, \mathcal {A},\mu ) be a probability space, and let mes denote the Lebesgue measure on the Borel σ \sigma -algebra B \mathcal {B} in [ 0 , 1 ] [0,1] . The nondecreasing-rearrangement operator from the space L ∞ ( μ ) = L ∞ ( A , A , μ ) {L^\infty }(\mu ) = {L^\infty }(A, \mathcal {A}, \mu ) of real-valued essentially bounded functions into L ∞ = L ∞ ( [ 0 , 1 ] {L^\infty } = {L^\infty }([0,1] , B \mathcal {B} , mes) is shown to be uniformly continuous in the Mackey topologies τ ( L ∞ ( μ ) \tau ({L^\infty }(\mu ) , L 1 ( μ ) ) {L^1}(\mu )) and τ ( L ∞ , L 1 ) \tau ({L^\infty },{L^1}) on L ∞ ( μ ) {L^\infty }(\mu ) and L ∞ {L^\infty } , respectively.

We give variants on Berge's Maximum Theorem in which the lower and the upper semicontinuities of the preference relation are assumed for two different topologies on the action set, i.e., the set of actions availabe a priori to the decision-maker (e.g. a household with its consumption set). Two new uses are pointed to. One result, stated here without a detailed proof, is the norm-to-weak* continuity of consumer demand as a function of prices (a property pointed to in existing literature but without proof or precise formulation). This improves significantly upon an earlier demand continuity result which, with the extremally strong 'finite' topology on the price space, is of limited interest other than as a vehicle for an equilibrium existence proof. With the norm topology on the price space, our demand continuity result acquires an independent significance - particularly for practical implementations of the equilibrium solution. The second application referred to establishes the continuity of the optimal plan as a function of the decision-maker's information (represented by a field of events in a probability spcace of states).