Approximate algorithm for modeling optimal renewal times in economic systems.

*(English. Russian original)*Zbl 0875.90269
Cybern. Syst. Anal. 28, No. 6, 945-949 (1992); translation from Kibern. Sist. Anal. 1992, No. 6, 167-172 (1992).

Summary: Systematic analysis of various optimization problems in one-, two-, and multi-sectoral integral dynamic models proposed by Glushkov suggests that in many important cases in practice the optimal (turnpike) renewal times of the elements of an economic systems under conditions of technological progress are approximately described by an integro-functional equation with memory for the unknown \(x(t)\):
\[
\Phi (t) \overset \text{df} = \int_t^{x^{-1} (t)} [F(t)- F(x (\tau)) ]d\tau - f(t) =0,
\]
where \(x^{-1} (\cdot)\) is the inverse of the function \(x(\cdot)\) and \(t- x(t)\) is the useful service life of the elements. Equation (1) was first derived for the optimization problem in a two-commodity integral dynamic macroeconomic model. Further studies have shown that Eq. (1) arises also in many other important applications in integral models. This equation relates the cost of development and introduction of new production capacity with a rollover estimate of efficiency – the incremental output produced by new production capacity compared with the output from existing capacity during its future operation. A qualitative analysis of Eq. (1) has shown that it has highly nontrivial properties that complicate its numerical solution.

##### MSC:

91B74 | Economic models of real-world systems (e.g., electricity markets, etc.) |

90C40 | Markov and semi-Markov decision processes |

##### Keywords:

optimization; multi-sectoral integral dynamic models; renewal times; economic systems; integro-functional equation with memory; macroeconomic model
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\textit{D. D. Ajstrakhanov} and \textit{Yu. P. Yatsenko}, Cybern. Syst. Anal. 28, No. 6, 945--949 (1992; Zbl 0875.90269); translation from Kibern. Sist. Anal. 1992, No. 6, 167--172 (1992)

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##### References:

[1] | V. M. Glushkov, ?A class of dynamic macroeconomic models,? Uprav. Sist. Mash., No. 2, 3-6 (1977). |

[2] | V. M. Glushkov, V. V. Ivanov, and Yu. P. Yatsenko, ?Analytical study of a class of dynamic models, I, II,? Kibernetika, No. 2, 1-12 (1980); No. 3, 104-112 (1982). · Zbl 0461.90014 |

[3] | Yu. P. Yatsenko, Integral Models of Systems with Controlled Memory [in Russian], Naukova Dumka, Kiev (1991). · Zbl 0852.93002 |

[4] | L. V. Kantorovich, V. I. Zhiyanov, and A. G. Khovanskii, ?Differential optimization principle in application to a onecommodity dynamic model,? Sib. Mat. Zh.,19, No. 5, 1053-1064 (1978). · Zbl 0402.90025 |

[5] | Yu. P. Yatsenko, ?Analysis of an integro-functional equation arising in optimization problems,? Kibernetika, No. 6, 117-118 (1989). |

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