Abstract
This outlines the book’s content: it is a new formal framework for the theory of perfectly competitive equilibrium and its industrial applications. Based in good part on ideas of Boiteux and Koopmans, the “short-run approach” is a scheme for calculating long-run producer optimum and market equilibrium by building on short-run solutions to the producer’s profit maximization problem, in which capital inputs and natural resources are treated as fixed. These inputs are valued at their marginal contributions to the operating profit. Since short-run profit is a concave but generally nondifferentiable function of the fixed inputs, their marginal values are defined as the generally nonunique supergradient vectors. Also, they usually have to be obtained as solutions to the dual programme of fixed-input valuation. The key property of the dual solution is therefore its marginal interpretation, but this requires the use of a generalized, multi-valued derivative of a convex function—the subdifferential—because an optimal-value functions are commonly nondifferentiable. Applied to the peak-load pricing of electricity generated by thermal, hydro and pumped-storage plants, the approach gives a sound and practical method of valuing the fixed assets (the river flows and the geological sites suitable for reservoirs).
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- 1.
When carried out by iterations, the calculations might also be seen as modelling the real processes of price and quantity adjustments.
- 2.
The theory of differentiable convex functions is, of course, included in subdifferential calculus as a special case. Furthermore, the subgradient concept can also be used to prove, rather than assume, that a convex function is differentiable—by showing that it has a unique subgradient. This method is used in [21, 23], [27, Section 9] and [30, Section 9].
- 3.
From Sect. 3.2 on, short-run cost minimization is split off as a subprogramme, whose solution is denoted by \(\check{v}\left (y,k,w\right )\). In these terms, \(\hat{v}\left (\,p,k,w\right ) =\check{ v}\left (\hat{y}\left (\,p,k,w\right ),k,w\right )\).
- 4.
A short-run approach to equilibrium might also be based on short-run cost minimization, in which not only the capital inputs (k) but also the outputs (y) are kept fixed and are shadow-priced in the dual problem, but such cost-based calculations are usually much more complicated than those using profit maximization: see Sect. 4.1
- 5.
Boiteux’s work is also presented by Drèze [15, pp. 10–16], but the short-run character of the approach is more evident from the original [9, 3.2–3.3] because Boiteux discusses the short-run equilibrium first, before using it as part of the long-run equilibrium system. When Drèze mentions short-run equilibrium on its own, it is only as an afterthought [15, p. 16].
- 6.
This is not an unacceptable condition, but some capacities can of course be zero in long-run equilibrium. The long-run model meets the usual adequacy assumption, as does the short-run model with positive capacities, and so existence of an equilibrium follows from Bewley’s result [7, Theorem 1], which is amplified in [31, Section 3] and [29] by a proof based on continuity of demand in prices.
- 7.
As is well known, this process does not always converge, but there are other iterative methods.
- 8.
In general, this is an inclusion rather than an equality: see (4.2.19).
- 9.
By contrast, SRC minimization for a system of plants can be difficult because it involves allocating the system’s given output among the plants. Its complexity shows in, e.g., the case of a hydro-thermal electricity-generating system studied by Koopmans [35]. The decentralized approach taken here (Chaps. 4 and 5 with their references) avoids having to deal directly with the formidable problem of minimizing the entire system’s cost: see the Comments with Formulae (4.1.3) and (4.1.4).
- 10.
This shows how mistaken is the widespread but unexamined view that nondifferentiabilities of convex functions are of little consequence: the very points which, in a sense, are exceptional a priori turn out to be the rule rather than the exception in equilibrium. Also, it is only on finite-dimensional spaces that convex functions are “generically smooth” or, more precisely, twice differentiable almost everywhere with respect to the Lebesgue measure (Alexandroff’s Theorem). On an infinite-dimensional space, a convex function can be nondifferentiable everywhere.
- 11.
Capital inputs are called independent if the SRP function (\(\Pi _{\mathrm{SR}}\)) is linear in the capital-input bundle \(k = \left (k_{1},k_{2},\ldots \right )\); an example is the multi-station technology of thermal electricity generation. Such a technology in effect separates into a number of production techniques with a single capital input each, and so Boiteux’s analysis applies readily: to ensure that a short-run equilibrium is a long-run equilibrium, it suffices to require cost recovery for each production technique θ with k θ > 0, although one must also remember to check that any unutilized production technique (one with k θ = 0) is unprofitable at the equilibrium prices (e.g., that \(r_{\theta } \geq \int \left (\,p\left (t\right ) - w_{\theta }\right )\mathrm{d}t\) for any unbuilt type θ of thermal station, with unit capital cost r θ and unit fuel cost w θ ).
- 12.
To distinguish the two quite different meanings of the word “Lagrangian”, it shall be occasionally expanded into either “Lagrange function” (in the multiplier method of optimization) or “Lagrange integrand” (in the calculus of variations only).
- 13.
Without involving \(\Pi _{\mathrm{SR}}\), the inclusion (∂ y C LR ⊆ ∂ y C SR) can be improved only by making it more precise but no more useful: \(\partial _{y}C_{\mathrm{SR}}\left (y,k\right )\) can be shown to equal the union of \(\partial _{y}C_{\mathrm{LR}}\left (y,r\right )\) over \(r \in -\partial _{k}C_{\mathrm{SR}}\left (y,k\right )\), i.e., over all those fixed-input price systems r for which k is an optimal fixed-input bundle for the output bundle y (given also the suppressed variable-input price system w): see (3.9.11).
- 14.
- 15.
See [21] and [23] for examples of an SRP programme in which the output space is \(L^{\infty }\left [0,T\right ]\) and a “singular” price term places the price system outside the predual \(L^{1}\left [0,T\right ]\), but it is the timing of the singularity, and not just its presence, that determines whether the programme is soluble or not.
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Horsley, A., Wrobel, A.J. (2016). Introduction. In: The Short-Run Approach to Long-Run Equilibrium in Competitive Markets. Lecture Notes in Economics and Mathematical Systems, vol 684. Springer, Cham. https://doi.org/10.1007/978-3-319-33398-4_1
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