Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
At 0, the normal cone to Y 0 equals the polar cone Y 0 ∘. When Y 0 is a vector subspace of Y, as in (5.1.7), the normal cone is the same at every y: it is the annihilator space (a.k.a. orthogonal complement) Y 0 ⊥ .
- 2.
For example, the function \(\check{k}\left (y\right ):=\mathop{ \mathrm{EssSup}}\nolimits \left (y\right ):=\mathop{ \mathrm{ess}}\nolimits \sup _{t}y\left (t\right )\), for \(y \in Y:= L^{\infty }\left [0,T\right ]\), has no subgradient in \(P:= L^{1}\left [0,T\right ]\) at any y with \(\mathop{\mathrm{meas}}\nolimits \left \{t: y\left (t\right ) =\mathop{ \mathrm{EssSup}}\nolimits \left (y\right )\right \} = 0\). This is because \(\gamma \in L^{1} \cap \partial ^{\mathrm{a}}\mathop{ \mathrm{EssSup}}\nolimits \left (y\right )\) if and only if γ ≥ 0, \(\int _{0}^{T}\gamma \left (t\right )\mathrm{d}t = 1\) and γ = 0 on \(\left \{t: y\left (t\right ) <\mathop{ \mathrm{EssSup}}\nolimits \left (y\right )\right \}\): see, e.g., [32, 4.5.1: Example 3].
- 3.
To see that (7.1.18)–(7.1.21) are indeed the FFE Conditions, recall from (3.1.5) that primal and dual feasibilities mean that \(\left (y,-k,-v\right ) \in \mathbb{Y}\) and \(\left (\,p,r,w\right ) \in \mathbb{Y}^{\circ }\). In the c.f.c. case, the two feasibility conditions expand into (7.1.18) and (7.1.19)–(7.1.20).
- 4.
- 5.
- 6.
This is the borderline case between Hicks-Allen complements and substitutes: see, e.g., [47, 1.F.d].
- 7.
Formally, the multi-valued rate of substitution equals \(\mathbb{R}_{+} = \left [0, +\infty \right )\).
- 8.
- 9.
When y is a decision variable, as in the SRP programme, this is Slater’s Condition on the constraints (inequalities and equations) that define Y0.
- 10.
For a linear functional aj, its \(\mathrm{m}\left (Y,P\right )\)-continuity is equivalent to its \(\mathrm{w}\left (Y,P\right )\)-continuity (and it simply means that aj ∈ P). But a concave function (bl) or a convex function (\(\check{k}_{\phi }\) or \(\check{v}_{\xi }\)) can be \(\mathrm{m}\left (Y,P\right )\)-continuous (and hence \(\mathrm{w}\left (Y,P\right )\)-u.s.c. or l.s.c., respectively) without being \(\mathrm{w}\left (Y,P\right )\)-continuous. The weak and Mackey topologies are briefly discussed in Sect. 6.2
- 11.
- 12.
For ϱ = 0, this fails if and only if D ≠ ∅ but P ∩ D = ∅ (in which case \(P \cap 0D = \left \{0\right \}\) but \(0\left (P \cap D\right ) =\emptyset\)).
- 13.
- 14.
\(\Pi _{\mathrm{Exc}}\left (y\right )\) is the excess a.k.a. pure profit from an output y (i.e., revenue at prices p less minimum input cost at prices r and w).
- 15.
The partial derivatives \(\partial \Pi _{\mathrm{SR}}/\partial \zeta _{j}^{{\prime}}\) and \(\partial \Pi _{\mathrm{SR}}/\partial \zeta _{l}^{{\prime\prime}}\) exist if the α j and β l associated by (7.2.1) with the optimal y are unique. If not, the derivative property still holds for the superdifferential, i.e., \(\hat{\partial }_{\zeta ^{{\prime}},\zeta ^{{\prime\prime}}}\Pi \) contains each \(\left (\alpha,\beta \right )\) that satisfies (7.2.1) for some optimal y (and hence for every optimal y).
- 16.
In other words, each \(\check{k}_{\phi }\) or \(\check{v}_{\xi }\) is a p.l.h. convex finite function.
- 17.
When each \(\check{k}_{\phi }\) is weakly* l.s.c. and Y 0 is weakly* closed as in Part 2 , this is equivalent to weak* compactness of \(\left \{y \in Y _{0}:\check{ k}\left (y\right ) \leq k\right \}\).
- 18.
- 19.
- 20.
- 21.
Being also weakly* closed, the set \(\left \{y \in Y _{0}:\check{ k}\left (y\right ) \leq k\right \}\) is actually weakly* compact by the Banach-Alaoglu Theorem.
- 22.
Similarly, if a unit output requires a unit of a costlessly storable variable input, whose total amount available, v ξ , can be spread as an input flow \(\tilde{v}_{\xi }\left (\cdot \right )\) over the period, then the output rate is constrained to a nonnegative \(y\left (t\right ) \leq \tilde{v}_{\xi }\left (t\right )\) for some \(\tilde{v}_{\xi }\left (t\right ) \geq 0\) with \(\int _{0}^{T}\tilde{v}_{\xi }\left (t\right )\mathrm{d}t = v_{\xi }\). This can be summarized in the single constraint \(v_{\xi } \geq \check{ v}_{\xi }\left (y\right ):=\int _{ 0}^{T}y\left (t\right )\mathrm{d}t\).
References
Bair J, Fourneau R (1975) Etude géometrique des espaces vectoriels (Lecture notes in mathematics, vol 489). Springer, Berlin-Heidelberg-New York
Holmes RB (1975) Geometric functional analysis and its applications. Springer, Berlin-Heidelberg-New York
Horsley A, Wrobel AJ (1996) Efficiency rents of storage plants in peak-load pricing, I: pumped storage. STICERD Discussion Paper TE/96/301, LSE (This is a fuller version of Ref. [27])
Horsley A, Wrobel AJ (2002) Efficiency rents of pumped-storage plants and their uses for operation and investment decisions. J Econ Dyn Control 27:109–142. DOI: 10.1016/S0165-1889(01)00030-6
Ioffe AD, Tihomirov VM (1979) Theory of extremal problems. North-Holland, Amsterdam-New York-Oxford
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Rockafellar RT (1974) Conjugate duality and optimization. SIAM, Philadelphia
Takayama A (1985) Mathematical economics. Cambridge University Press, Cambridge-London-New York
Tiel J van (1984) Convex analysis. Wiley, Chichester-New York-Brisbane
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Horsley, A., Wrobel, A.J. (2016). Production Techniques with Conditionally Fixed Coefficients. 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_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-33398-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33397-7
Online ISBN: 978-3-319-33398-4
eBook Packages: Economics and FinanceEconomics and Finance (R0)