Abstract
Real economic problems are complex and involve too many variables and constraints to be both computable and reasonably relevant. Mathematics cannot be “applied” to “economics,” at best, economics can motivate mathematics, in other ways than physical sciences did, by offering at least “qualitative” mathematical metaphors of economic evolutions, not quantitative ones4 − 11.
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- 1.
In which case k ≥ 0.
- 2.
According to a suggestion of Efim Galperin, derivatives from the right \(x^{\prime}(t) =\lim_{h\mapsto 0+}\frac{x(t + h) - x(t)} {h}\) are “physically non-existent” since time t + h is not yet known, following the Czech Count Jiri Buquoy, who in 1812, formulated the equation of motion of a body with variable mass, which retained only the attention of Poisson before being forgotten.
- 3.
If we regard the number x(t) of units of numéraire as a value, then x′(t) is regarded as an impetus, which is the derivative of a value, in the same way than in mechanics, the force is the derivative of a potential, the analog of value.
- 4.
Stating that the acceleration x′(t) = ρ(x′(t)).
- 5.
This issue, at the origin of the gap between specialists of dynamical systems and those on control theory (where the parameters evolve), is rarely explicitly dealt with. This is the reason for mentioning the following quotation of the book [127, Introduction to Applied Nonlinear Systems and Chaos] by Stephen Wiggins: On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution: “[…] In all of our analysis thus far the parameters have been constant. The point is that we cannot think of the parameter as varying in time, even though this is what happens in practice. Dynamical systems having parameters that change in time (no matter how slowly!) and that pass through bifurcation values often exhibit behavior that is very different from the analogous situation where the parameters are constant.”
- 6.
The sign of the growth rate is negative instead of positive, since we want to bound below the number of commodity units instead of bounding above the members of the population: “Population, when unchecked, increases in a geometrical ratio” and his solution: “By moral restraint, I mean a restraint from marriage […]”. In economics, this would be a restraint from consumption. See Chaps. 6 and 7 of [15, Aubin, Bayen & Saint-Pierre] for more details.
References
Aubin J-P, Bayen A, Saint-Pierre P (2011) Viability theory. New directions. Springer
Evans LC (1998) Partial differential equations. American Mathematical Society
Goebel R, Sanfelice RG, Teel AR (2012) Hybrid dynamical systems. modeling, stability and robustness. Princeton University Press, Princeton
Krehl POK (2008) History of shock waves, explosions and impact. Springer
Wiggins S (1990) Introduction to applied nonlinear systems and chaos. Springer
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Aubin, JP. (2014). How Long and How Much Endowing One Commodity. 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_2
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