Abstract
In economics, \(X:= {\mathbb{R}}^{\ell}\) can be regarded as a commodity space of commodities \(x:= (x_{h})_{1\leq h\leq \ell}\) of amounts \(x_{h} \in {\mathbb{R}}^{}\) of units e h of goods or services labeled \(h = 1,\ldots,\ell\).
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Notes
- 1.
As well as in other domains, such that classical mechanics from which these ideas emerged: \(X:= {\mathbb{R}}^{\ell}\) can be regarded as a position or configuration space, its dual X ⋆ as the space of forces \(p: x\mapsto \left \langle p,x\right \rangle \in {\mathbb{R}}^{}\) associating with any position x ∈ X its duality product \(\left \langle p,x\right \rangle\) which is interpreted as a work. The bidual X ⋆ ⋆ is regarded as the space of velocities v and the duality product \(\left \langle p,v\right \rangle\) is interpreted as a power.
In his definition of happiness,31–74 Maupertuis actually chose \(X:= {\mathbb{R}}^{}\) as the space of durations d, its dual as the space of urges p, its bidual as the space of fluidities \(\varphi\) and its tridual as the space of hastes (see Sect. 5.3, p. 74).
- 2.
In physics, the gradient of a potential function \(U: x\mapsto U(x)\) is interpreted as a force: along an evolution \(t\mapsto x(t)\), \(\frac{d} {\mathit{dt}}U(x(t)) = \left \langle \frac{\partial U(x(t))} {\partial x},x^{\prime}(t)\right \rangle\) is a power. In economics, the variable x is replaced by the duration–allocation–price triple and the impetus plays the role of the mechanical power. For durations, the force is the urge and the impetuosity plays the role of power.
- 3.
We could add constraints of the form
$$\displaystyle{ \left \{\begin{array}{l} C(d(\cdot ),x(\cdot ),p(\cdot ))\;:=\;\int _{ T-\Omega (d(\cdot ))}^{T} \\ \left (\sum _{i=1}^{n}(d^{\prime}_{ i}(t)\left \langle p(t),x_{i}(t)\right \rangle + \left \langle d_{i}(t)p(t),x^{\prime}_{i}(t)\right \rangle + \left \langle p^{\prime}(t),d_{i}(t)x_{i}(t)\right \rangle )\right )\mathit{dt}\; \leq \; 0 \end{array} \right. }$$(7.7)on cumulated transactions and fluctuations on a temporal window, which are more classical since they involve integrals instead of suprema, which are difficult to deal with standard methods of calculus of variations. We observe that \(C(d(\cdot ),x(\cdot ),p(\cdot )) \leq \Omega (d(\cdot ))\sup _{t\in [T-\Omega (d(\cdot )),T]}E(d(\cdot ),x(\cdot ),p(\cdot ))\). It is also possible to add this cumulated cost to the stimulus (see Theorem 6.4.26, p. 232, of [15, Viability Theory. New Directions].).
- 4.
Which can be regarded a kind of fuzzy economy in the sense of fuzzy or toll-sets. See [22, Aubin & Dordan].
References
Aubin J-P (2011) Regulation of births for viability of populations governed by age-structured problems. J Evol Equat 99–117, DOI:http://dx.doi.org/10.1007/s00028-011-0125-z
Aubin J-P, Bayen A, Saint-Pierre P (2011) Viability theory. New directions. Springer
Aubin J-P, Chen LX, Dordan O (in preparation) Tychastic measure of viability risk. A viabilist portfolio performance and insurance
Aubin J-P, Chen LX, Dordan O (2012) Asset liability insurance management (ALIM) for risk eradication. In Bernhard P, Engwerda J, Roorda B, Schumacher H, Kolokoltsov V, Saint-Pierre P, Aubin J-P (eds) The interval market model in mathematical finance. Game-theoretic methods. Birkhäuser
Aubin J-P, Chen LX, Dordan O, Saint-Pierre P (2011) Viabilist and tychastic approaches to guaranteed ALM problem. Risk Decis Anal 3:89–113, DOI:10.3233/RDA-2011-0033
Aubin J-P, Chen LX, Dordan O, Faleh A, Lezan G, Planchet F (2012) Stochastic and tychastic approaches to guaranteed ALM problem. Bull Français d’Actuariat 12:59–95
Aubin J-P, Dordan O (1996) Fuzzy systems, viability theory and toll sets. In Nguyen H(ed) Handbook of fuzzy systems, modeling and control. Kluwer, pp 461–488
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Aubin, JP. (2014). Endowing Exchange Values: Adam Smith’s Invisible Man. In: Time and Money. Lecture Notes in Economics and Mathematical Systems, vol 670. Springer, Cham. https://doi.org/10.1007/978-3-319-00005-3_7
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