Abstract

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Researchers should cite this work as follows:

• Chernyshov, S. I. (2023). 2008-Chernyshov-EtAl-The Problem of Modelling of Economic Dynamics in Differential Form. SIMIODE, QUBES Educational Resources. doi:10.25334/XABV-8685

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Chernyshov, S. I.. V. S. Ponomarenko, and  A. V. Voronin. 2008. The Problem of Modelling of Economic Dynamics in Differential Form. Notes. 35 pp.

See https://arxiv.org/ftp/arxiv/papers/0904/0904.0756.pdf . Accessed on 278 March 2023.

Introduction:

The authors have tried to analyze procedures of the construction of differential equations that are employed for modeling of macroeconomic processes. The results prove to be rather unexpected. Thus, the derivation of the differential equation of Harrod’s model is based on a linear relation between capital and income. As a result, there arises a contradiction in terms of dimension that is rooted in incorrect treatment of the fundamental notion of the infinitesimal quantity. One can overcome this contradiction by relating capital to the integral of income over a corresponding time interval. However, in this case, the solution is by no means an exponential growth but a much more realistic relation that reflects, in particular, objective finiteness of the prognostic period.

An analysis of the models of Harrod-Domar, Phillips, as well as of other models (see the well-known treatise by R. Allen), leads us to the conclusion that analogous deficiencies are, in principle, inherent in these models too. In general, the refraction in the sphere of economic dynamics of the methodology of the construction of mathematical models borrowed from the field of natural sciences, such as dynamics, electrodynamics, etc., proves to be absolutely unjustified. As a matter of fact, differential equations adequate to these models follow naturally from the consideration of an infinitesimal element. However, as regards the problems of economics, such an approach is objectively senseless. Nevertheless, economics, in its turn, has intrinsic advantages from the point of view of possibilities of mathematical modeling, which is embodied in the notion of balance. As we will show, there exist formal means to reduce Leontief’s model of ”expenses-output” in its canonical interpretation to a system of linear differential equation (of, generally speaking, arbitrary order with respect to the derivatives).

At the same time, the scantiness of the arsenal of the means of linear theory that are used in representative modeling of macroeconomic processes is almost universally recognized nowadays. In this regard, we will characterize briefly those areas of systems analysis that are devoted to the construction of nonlinear models that are adequate to a given ”input-output” mapping. In what follows, we nonetheless note that Leontief’s model in the differential form can be elementary reduced to a Fredholm integral equation of the second kind (with respect to a vector function), whose theory and algorithms of numerical realization are as constructive as possible. In the case, when the kernel of such an equation depends on a parameter, which is quite naturally interpreted in terms of the object sphere, the spectrum of its possible solutions becomes extremely wide. We think that the development of the theory of Fredholm integral equations of the second kind, whose kernels contain parameters, and its application to the modeling of the processes of economic dynamics is rather promising.