Abstract
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).
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Notes
- 1.
Even if the objective were nonlinear, it could always be replaced by a linear one with an extra scalar variable, subject to an extra nonlinear constraint: as is noted in [12, p. 48], minimization of \(f\left (y\right )\) over y is equivalent to minimization of ϱ over y and ϱ subject to: \(\varrho \geq f\left (y\right )\), in addition to any original constraints on y.
- 2.
More generally, this is so whenever the optimand separates into a function of \(\left (r,k\right )\) plus terms independent of r and k.
- 3.
Formally, this follows from the definitional conjugacy relationship ( 3.1.14) between \(\Pi _{\mathrm{SR}}\) and \(\Pi _{\mathrm{LR}}\) (as functions of k and r, respectively) by using the first-order condition ( B.5.5) and the Inversion Rule ( B.6.2) of Appendix B.
- 4.
This is equivalent to joint convexity of the constrained minimand (which is the sum of the minimand and of the 0-∞ indicator function of the constraint set). In [44] it is called “the minimand” for brevity.
- 5.
A linear change of variables makes it a saddle function: \(4f \cdot y = \left (\,f + y\right ) \cdot \left (\,f + y\right ) -\left (\,f - y\right ) \cdot \left (\,f - y\right )\) is convex in f + y and concave in f − y.
- 6.
In this as in other contexts, it can be convenient to think of extrinsic perturbations either as (i) complementing the intrinsic perturbations (which are increments to the fixed inputs) by varying some aspects of the technology (such as nonnegativity constraints), or as (ii) replacing the intrinsic perturbations with finer, more varied increments (to the fixed inputs). For example, the time-constant capacity k θ in ( 5.2.3) is an intrinsic primal parameter. The corresponding perturbation is a constant increment Δ k θ , which can be refined to a time-varying increment \(\varDelta k_{\theta }\left (\cdot \right )\). The perturbation \(\varDelta k_{\theta }\left (\cdot \right )\) is complemented by the increment \(\varDelta n_{\theta }\left (\cdot \right )\) to the zero floor for the output rate \(y_{\theta }\left (\cdot \right )\) in ( 5.2.3). The same goes for all the occurrences of Δ k and Δ n in the context of pumped storage and hydro (where Δ ζ is another complementary extrinsic perturbation, of the balance constraint ( 5.2.15) or ( 5.2.35)).
- 7.
The dual constraint A T σ = p must be changed to A T σ ≥ p if y ≥ 0 is adjoined as another primal constraint (in which case the primal LP can be interpreted as, e.g., revenue maximization—given a resource bundle s, an output-price system p and a Leontief technology defined by an input-coefficient matrix A).
- 8.
Without a proof of value differentiability, the Generalized Envelope Theorem is given also in, e.g., [47, 1.F.b].
- 9.
These are the Lagrangian Saddle-Point Conditions (\(0 \in \partial _{\sigma }\mathcal{L}\) and \(0 \in \hat{\partial }_{y}\mathcal{L}\)) for the present LP case.
- 10.
In this case, equivalence of the Kuhn-Tucker Conditions and the FFE Conditions can be seen directly, but it holds always (since, by the general theory of CPs, each system is equivalent to the conjunction of primal and dual optimality together with absence of a duality gap).
- 11.
The standard dual to the ordinary CP of maximizing a concave function \(f\left (y\right )\) over y subject to \(G\left (y\right ) \leq s\) (where G 1, G 2, etc., are convex functions) is to minimize \(\sup _{y}\mathcal{L}\left (y,\sigma \right ):=\sup _{y}\left (\,f\left (y\right ) +\sigma \cdot \left (s - G\left (y\right )\right )\right )\) over σ ≥ 0 (the standard dual variables, which are the Lagrange multipliers for the primal constraints): see, e.g., [44, (5.1)]. And \(\sup _{y}\mathcal{L}\) (the Lagrangian’s supremum over the primal variables) cannot be evaluated without assuming a specific form for f and G (the primal objective and constraint functions).
- 12.
Since the minimand \(\left \langle w\,\vert \,v\right \rangle\) is not jointly convex in w and v, w cannot serve as a primal parameter (it will turn out to be a dual parameter).
- 13.
As the notation indicates, \(\overline{\Pi }\) and C are thought of mainly as dual expressions for \(\Pi \) and C (although duality of programmes is fully symmetric).
- 14.
A similar remark applies to the full and reduced shadow-pricing programmes, ( 3.3.4) for \(\left (\,p,r\right )\) and the one in ( 3.4.7) for p alone. Taken as the primal parameterized by w, each has the same dual, viz., the SRC programme ( 3.1.10)–( 3.1.11). And both of the other vector data (y and k) are dual parameters for the full programme ( 3.3.4). But the datum k is neither a dual nor a primal parameter for the reduced programme in ( 3.4.7).
- 15.
These arguments use the subprogramme and duality concepts to view \(\Pi _{\mathrm{SR}}\) in two ways: (i) as the value of a subprogramme, and (ii) as the primal value—and thus arrive in two ways at the FIV programme for r in ( 3.2.4). In detail: since \(\Pi _{\mathrm{SR}}\) is the value of the subprogramme of LRP maximization obtained by fixing k, its (concave) conjugate w.r.t. k is \(-\Pi _{\mathrm{LR}}\) as a function of r. It follows—by ( B.5.5) and ( B.6.2)—that k solves the “conjugacy programme” in ( 3.2.1) if and only if: r solves the “reverse conjugacy programme” in ( 3.2.4) and ( 3.2.5) holds. Also by using the conjugacy between \(\Pi _{\mathrm{SR}}\) and \(\Pi _{\mathrm{LR}}\), the same programme for r can be derived as the dual of SRP maximization parameterized by k. This is done in Proposition 3.10.1 (which additionally identifies p and w as the dual parameters): it is the formal foundation of duality for CPs that the dual minimand is the primal parameter times the dual variable, minus the (concave) conjugate of the primal maximum value (as a function of the primal parameter)—and here, the sum is \(\left \langle r\,\vert \,k\right \rangle + \Pi _{\mathrm{LR}}\left (r\right )\). See, e.g., [44, Theorem 7], which here must be applied to the function \(\varDelta k\mapsto \Pi _{\mathrm{SR}}\left (k +\varDelta k\right )\) as Rockafellar’s primal value (his is a function of the parameter increment, rather than of the parameter point as here, and this is what shifts the argument by k and adds the term \(\left \langle r\,\vert \,k\right \rangle\) to minus the conjugate).
- 16.
- 17.
- 18.
- 19.
The PIR would give the same result, but it would require establishing first that \(C_{\mathrm{SR}}\left (\cdot,k,w\right )\) is l.s.c. to invert the conjugacy relationship ( 3.1.13), i.e., to show that the saddle function \(C_{\mathrm{SR}}\left (\cdot,k,\cdot \right )\) is indeed a partial conjugate of the bivariate convex function \(\Pi _{\mathrm{SR}}\left (\cdot,k,\cdot \right )\). This can be problematic (as is noted in the Comment after Corollary B.7.5).
- 20.
The PIR would give the same result, but it would require establishing first that \(\overline{\Pi }_{\mathrm{SR}}\left (\cdot,k,w\right )\) is l.s.c. to invert the conjugacy relationship ( 3.3.8), i.e., to show that the saddle function \(\overline{\Pi }_{\mathrm{SR}}\left (\cdot,\cdot,w\right )\) is indeed a partial conjugate of the bivariate convex function \(\underline{C}_{\mathrm{SR}}\left (\cdot,\cdot,w\right )\). This can be problematic (as is noted in the Comment after Corollary B.7.5).
- 21.
The three systems on the left in Table 3.2 do not yield new ones (when \(\Pi _{\mathrm{SR}}\) is replaced by C LR) simply because they do not involve \(\Pi _{\mathrm{SR}}\) at all. So there are not ten but seven of the “mirror images”.
- 22.
The subsystem’s condition that C SR = C SR at \(\left (y,k,w\right )\) rules out a different duality gap, and on its own it does not imply that \(\overline{\Pi }_{\mathrm{SR}} = \Pi _{\mathrm{SR}}\) at \(\left (\,p,k,w\right )\) when y maximizes short-run profit at \(\left (\,p,k,w\right )\): see Appendix A for an example (in which C SR = C SR trivally because the technology has no variable input).
- 23.
- 24.
- 25.
- 26.
Here, two-stage solving means first minimizing \(\left \langle w\,\vert \,v\right \rangle\) over v (subject to \(\left (y,-k,-v\right ) \in \mathbb{Y}\)) to find the solution \(\check{v}\) and the minimum value \(C_{\mathrm{SR}} = \left \langle w\,\vert \,\check{v}\right \rangle\) as functions of \(\left (y,k,w\right )\), and then minimizing \(\left \langle r\,\vert \,k\right \rangle + C_{\mathrm{SR}}\left (y,k,w\right )\) over k to find the solution \(\check{k}\left (y,r,w\right )\). This gives the complete solution (in terms of y, r and w) as the pair \(\check{k}\left (y,r,w\right )\) and \(\check{v}\left (y,\check{k}\left (y,r,w\right ),w\right )\).
- 27.
See the Comment on proper and improper solutions in Sect. B.6 of Appendix B.
- 28.
For another proof of this, note that: (i) by ( 3.3.8), \(\underline{C}_{\mathrm{SR}} = \overline{\Pi }_{\mathrm{SR}}^{\#_{1}} \leq \Pi _{\mathrm{SR}}^{\#_{1}} = C_{\mathrm{SR}}^{\#_{1}\#_{1}}\) by ( 3.1.13), with the inequality holding because \(\overline{\Pi }_{\mathrm{SR}} \geq \Pi _{\mathrm{SR}}\), and (ii) \(C_{\mathrm{SR}}^{\#_{1}\#_{1}} \leq C_{\mathrm{SR}}\) by ( B.2.4) without the middle term. So \(\underline{C}_{\mathrm{SR}} \leq C_{\mathrm{SR}}^{\#_{1}\#_{1}} \leq C_{\mathrm{SR}}\) everywhere (and it follows that all three are equal if the outer two are).
- 29.
Although it follows that \(\mathbb{Y}^{\circ }\) is the convex cone generated by \(\mathcal{G}^{{\prime}}\cup \mathcal{G}^{{\prime\prime}}\cup \left (-\mathcal{G}^{{\prime\prime}}\right )\), it is better to keep \(\mathcal{G}^{{\prime}}\) and \(\mathcal{G}^{{\prime\prime}}\) separate when it comes to parameterizing the programme ( 3.12.2)–( 3.12.4) in the standard way: for this purpose, an equality constraint should not be converted to a pair of opposite inequalities. To do so would complicate the dual solution by making it nonunique and unbounded: a primal equality constraint (say a ⋅ y = 0) may have a unique multiplier \(\hat{\sigma }^{{\prime\prime}}\), but if it were replaced by a pair of inequalities (a ⋅ y ≤ 0 and − a ⋅ y ≤ 0), then a corresponding multiplier pair would be any \(\left (\hat{\sigma }_{1}^{{\prime}},\hat{\sigma }_{2}^{{\prime}}\right ) \geq 0\) with \(\hat{\sigma }_{1}^{{\prime}}-\hat{\sigma }_{2}^{{\prime}} =\hat{\sigma } ^{{\prime\prime}}\), i.e., it would be any point of a half-line. Its unboundedness expresses the fact that the programme would become infeasible if one inequality constraint of the pair were tightened without relaxing the other by the same amount (i.e., if the constraints were perturbed to a ⋅ y ≤ ε 1 and − a ⋅ y ≤ ε 2 for ε 1 < −ε 2).
- 30.
Formally, A ′ and B ′ are the \(\mathcal{G}^{{\prime}}\times T\) and \(\mathcal{G}^{{\prime}}\times \Phi \) matrices with entries A gt ′ = g t and B gϕ ′ = g ϕ for t ∈ T, \(\phi \in \Phi \) and \(g \in \mathcal{G}^{{\prime}}\) (and the same goes for′ ′ instead of′). For Farkas’s Lemma, see, e.g., [12, 2.2.6], [42, 22.3.1], [45, 6.45] or [48, 4.19].
- 31.
The identity ( 3.12.9) reduces to ( 3.12.5) when the primal and dual values are equal, i.e., when \(\tilde{\Pi } = \overline{\tilde{\Pi }}\) and \(\Pi = \overline{\Pi }\) at \(\left (\,p,k\right )\). This always applies to (feasible) finite LPs, but not always to infinite LPs. To prove ( 3.12.9) without relying on absence of a duality gap, note that the change of variables from r to σ by r = B T σ transforms ( 3.3.13)–( 3.3.14) into ( 3.12.6)–( 3.12.8). This is detailed in the first Comment after ( 3.12.14). The argument extends to infinite LPs (and it applies also when there is a duality gap).
- 32.
- 33.
- 34.
See, e.g., [11, 5.1 and 9.1] or [44, Example 1’, p. 24] for proofs based on the simplex algorithm or on polyhedral convexity, respectively. This is not so with a pair of infinite LPs: both can be feasible without having the same value (i.e., the primal and dual values can both be finite but unequal). See Appendix A for an example.
- 35.
Also, the nonnegativity constraint on k ϕ will make it appear a second time even if k ϕ imposes just one constraint on y (i.e., 0 ≤ k ϕ in addition to a ⋅ y ≤ k ϕ for some a ≠ 0).
- 36.
First, the intrinsic dual’s constraint \(\left (\,p,r\right ) \in \mathbb{Y}^{\circ }\) is rewritten as: p = A T σ and r = B T σ for some \(\sigma = \left (\sigma ^{{\prime}},\sigma ^{{\prime\prime}}\right )\) with σ ′ ≥ 0. Then σ is made an explicit decision variable alongside r (and so the existential quantifier on σ is dropped); this produces the inclusive standard dual LP ( 3.12.13)–( 3.12.14). Finally, r is replaced by B T σ; this produces the standard dual LP ( 3.12.6)–( 3.12.8).
- 37.
In this case, the matrix \(M^{\mathrm{R}}:= M^{\mathrm{T}}\left (MM^{\mathrm{T}}\right )^{-1}\) is a right inverse of M. The parametric equations of \(\mathbb{Y}^{\circ }\) imply that \(\sigma ^{\mathrm{T}} = \left [p^{\mathrm{T}}\,r^{\mathrm{T}}\right ]M^{\mathrm{R}}\), and, after partitioning M R into \(\left [R^{{\prime}}\,R^{{\prime\prime}}\right ]\) to match the partition \(\sigma = \left (\sigma ^{{\prime}},\sigma ^{{\prime\prime}}\right )\), the required system of equations and inequalities is: \(\left [p^{\mathrm{T}}\,r^{\mathrm{T}}\right ]\left (M^{\mathrm{R}}M -\mathrm{ I}\right ) = 0\) and \(\left [p^{\mathrm{T}}\,r^{\mathrm{T}}\right ]R^{{\prime}}\geq 0\).
- 38.
To see this, let the original primal LP be to maximize p ⋅ y over \(y \in \mathbb{R}^{n}\) subject to Ay ≤ k, given arbitrary vectors \(p \in \mathbb{R}^{n}\) and \(k \in \mathbb{R}^{m}\), and given an m × n matrix A (i.e., assume for simplicity that A = A ′, \(B = B^{{\prime}} =\mathrm{ I}\), and so \(\sigma =\sigma ^{{\prime}} = r\) and the standard and the intrinsic duals are the same). The dual LP is to minimize r ⋅ k over r ≥ 0 subject to r T A = p T. The FFE (Complementarity) Conditions on \(\left (y,r\right )\) are: Ay ≤ k, r ≥ 0, r T A = p T and p ⋅ y ≥ r ⋅ k (or, equivalently, p ⋅ y = r ⋅ k). This is a system with n + m variables and \(2m + 2n + 1\) inequalities (counting an equality as two inequalities). Its auxiliary LP has \(n + m + 1\) decision variables (viz., y, r and an artificial variable, say z ≥ 0, as the minimand, whose minimum value is zero if and only if the FFE system is soluble) and \(2\left (m + n + 1\right )\) inequality constraints (viz., z ≥ 0 and all the complementarity inequalities but with z subtracted from the lesser side, i.e., p ⋅ y ≥ r ⋅ k − z, etc.): see [11, (16.2), p. 240]. So the auxiliary LP has one more variable and one more constraint than the original primal and dual LPs together. Solving the auxiliary LP by a primal-dual algorithm (such as the simplex method) gives a solution to the original LP “in duplicate”.
- 39.
The weak topologies do not enter the analysis explicitly, but they make the adjoint operators continuous: see, e.g., [18, 16C].
- 40.
The output space is \(Y = L^{\infty }\left [0,T\right ]\), which has a Banach predual \(Y ^{{\prime}} = L^{1}\left [0,T\right ]\). The fixed-input space K depends on the technique: it is either \(\mathbb{R}\) for a thermal technique, or \(\mathbb{R}^{2}\) for pumped storage, or \(\mathbb{R}^{2} \times L^{\infty }\left [0,T\right ]\) for hydro. As for L (the space of standard perturbations of the inequality constraints), it is either \(L^{\infty }\left [0,T\right ]\) or its Cartesian product with \(\mathcal{C}\left [0,T\right ]\) when, in the case of an energy storage technique, there are reservoir constraints in addition to the generation constraints. And the balance constraint of a storage technique has \(\mathbb{R}\) as X (the space of standard perturbations of the equality constraint).
- 41.
Formula ( 3.12.16) adapts [12, p. 154, line 11 f.b.], where the construction is mistakenly proposed as a possible way of dealing with a non-solid cone (in such a case the polar cannot have a compact base, so the analysis does not apply). The construction can, however, be extended to the case that \(\mathbb{Y}\) is only relatively solid, i.e., has a nonempty interior in the linear subspace \(\mathbb{Y} - \mathbb{Y}\) (assumed to be closed in Y × K); the polar \(\mathbb{Y}^{\circ }\) is then the sum of the annihilator \(\left (\mathbb{Y} - \mathbb{Y}\right )^{\perp }\) and a cone with a compact base.
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Horsley, A., Wrobel, A.J. (2016). Characterizations of Long-Run Producer Optimum. 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_3
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