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2020 | OriginalPaper | Buchkapitel

3. Pontryagin’s Principle for a Class of Discrete Time Infinite Horizon Optimal Growth Problems

verfasst von : Ayşegül Yıldız Ulus

Erschienen in: Mathematical Modelling and Optimization of Engineering Problems

Verlag: Springer International Publishing

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Abstract

In this chapter, we aim to apply the approach of weak Pontryagin’s principles to a class of discrete time infinite horizon optimal growth problems. The idea of this approach is to transform the optimal growth problem into a dynamical system which is governed by a difference equation or a difference inequation. We establish necessary and sufficient conditions of optimality in terms of weak Pontryagin’s principles.

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Vorheriges Kapitel Developing a Nationwide Energy Storage Policy by Optimal Size and Site Selection

Nächstes Kapitel A Medical Modelling Using Multiple Linear Regression

A discussion can be found in [11] which gives the mathematical background of the classical approaches for the optimal growth problem together with the recent developments.

These are also Theorem 3.3 and Theorem 3.8 of [8].

Essentially the Assumption (H1) in [9]. Note that Assumption (H4) is always satisfied in our case since $\frac {\partial g}{\partial c}(k_t, c_t) = -1\neq 0$ for all t = 0, 1…

(Co) holds if the utility function u is supposed to be concave since we have f(k _t) = c _t + k _t+1, we will have the following concave Pontryagin’s Hamiltonian function:

$$\displaystyle \begin{aligned}H_t(k,c,1, \lambda_{t+1}) = \beta^{t}u(c_t) + \lambda_{t+1} (f(k_t) -c_t) = \beta^{t}u(c_t) + k_{t+1}\end{aligned}$$

In general, the production function is supposed to be of type Cobb-Douglas f(k, r) = Ak ^αr ^β where the exponents α and β represent the elasticity of production with respect to capital and the extracted natural resource, respectively.

Sometimes, a stronger conservation constraint is taken into account in the sense that $s_t\geq \bar {s}$ where $\bar {s}>0$ for all t = 0, 1, …. This refers to a strong conservation concern implying sustainability. In this chapter, we suppose only the less strong one in order to be able to obtain the analytical results.

In [1], they propose a bounded case with a difference equation system; here, with a slight change we give the result for the bounded case with a difference inequation system.

The mapping (s _t, k _t, r _t, c _t)↦Dg _(r,c)(s _t, k _t, r _t, c _t) is continuous since g is continuously differentiable and the composition with the determinant is also continuous; therefore, there exists a neighborhood containing (s ^∗, k ^∗, r ^∗, c ^∗) such that the set $\{(s_t, k_t, r_t, c_t)| \det Dg_{(r,c)}(s_t, k_t, r_t, c_t)\neq 0 \}$ is open. Here,

$$\displaystyle \begin{gathered} \left |{\begin{array}{cc} \frac{\partial g_1}{\partial r_t}(s_t^*, k_t^*, r_t^*, c_t^*) & \frac{\partial g_1}{\partial c_t}(s_t^*, k_t^*, r_t^*, c_t^*) \\ \frac{\partial g_2}{\partial r_t}(s_t^*, k_t^*, r_t^*, c_t^*) & \frac{\partial g_2}{\partial c_t}(s_t^*, k_t^*, r_t^*, c_t^*) \end{array} } \right| = \left |{\begin{array}{cc} -1 & 0 \\ \frac{\partial f}{\partial r_t}(k_t^*, r_t^*) & 1 \end{array} } \right| \neq 0 \end{gathered} $$

Moreover, $\sigma :=\sup _{t\in \mathbb N} ||Dg_{(r,c)}(s^*, k^*, r^*, c^*)^{-1} ||\in (0, +\infty )$ since Dg _(r,c)(s ^∗, k ^∗, r ^∗, c ^∗)⁻¹ = Dg _(r,c)(s ^∗, k ^∗, r ^∗, c ^∗) with a convenient production function f(k _t, r _t).

This assumption on mortality and the recruitment parameters m _j, ν _j ∈ [0, 1] is essential for the result.

Since $\lambda ^*\in \ell ^{1}(\mathbb N, \mathbb R^{n*})$ the Transversality Condition (TC) holds, that is, $\lim _{t\to +\infty } \lambda _t^{*}= 0$ (3.37)

M. De Lara, L. Doyen, Sustainable Management of Natural Resources: Mathematical Models and Methods (Springer Science and Business Media, Berlin, 2008)CrossRef

P. Dasgupta, G. Heal, The optimal depletion of exhaustible resources. Rev. Econ. Stud. 41, 1–28 (1974)CrossRef

R.M. Solow, Intergenerational equity and exhaustible resources. Rev. Econ. Stud. 41, 29–45 (1974)CrossRef

C.W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation (Wiley, Hoboken, 2010)

N. Stokey, R.E. Lucas Jr., E.C. Prescott, Recursive Methods in Economic Dynamics (Harvard University Press, New York, 1989)CrossRef

R.A. Dana, C. Le Van, Dynamic Programming in Economics (Springer Science & Business Media, Dordrecht, 2003)MATH

C. Le Van, C. Saglam, Optimal growth models and the Lagrange multiplier. J. Math. Econ. 40(3), 393–410 (2004)MathSciNetCrossRef

J. Blot, N. Hayek, Infinite-horizon Optimal Control in the Discrete-time Framework. Springer Briefs in Optimization (Springer, New York, 2014)

J. Blot, N. Hayek, F. Pekergin, N. Pekergin, Pontryagin principles for bounded discrete-time processes. Optimization 64(3), 505–520 (2015)MathSciNetMATH

10.

J. Blot, N. Hayek, F. Pekergin, N. Pekergin, The competition between Internet service qualities from a difference game viewpoint. Int. Game Theory Rev. 14(01), 1250001 (2012)

11.

A.Y. Ulus, On discrete time infinite horizon optimal growth problem. Int. J. Optim. Control 8(1), 102–116 (2017)MathSciNet

12.

H. Hotelling, The economics of exhaustible resources. J. Polit. Econ. 39(2), 137–175 (1931)CrossRef

13.

H. Caswell, Matrix Population Models, 2nd edn. (Sinauer Associates, Sunderland, 2001), pp. 91–92

Titel: Pontryagin’s Principle for a Class of Discrete Time Infinite Horizon Optimal Growth Problems
verfasst von: Ayşegül Yıldız Ulus
Verlag: Springer International Publishing
Buch: Mathematical Modelling and Optimization of Engineering Problems
Print ISBN: 978-3-030-37061-9

Electronic ISBN: 978-3-030-37062-6

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-37062-6_3

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