The Simplest Models of the Macroeconomics Dynamics in a Composition with the Principle of the Subjective Entropy Maximum

Kasianov, Vladimir Aleksandrovich; Goncharenko, Andriy Viktorovich

INTRODUCTION

Aviation industry and airlines are undergoing hard times due to the SArS COVID-19 pandemic period as well as the tragedy of the heroic rebuff of Ukraine to the fascist–russist full-scale warfare invasion.

The current circumstances require indispensable measures to be taken in the major macroeconomic airline industry components. The presented paper is dedicated to the simplest macroeconomic problem setting in the framework of the Solow [1–3] and Cobb–Douglas [4,5] models, similar to the approach adopted for the estimation of economic growth [6], together with consideration of the individuals’ subjective preferences functions of the available alternatives obtained based on the subjective entropy maximum principle, which has been previously proposed in the literature [7–10]. This principle was applied to the simplest problems concerning the macroeconomics dynamics, and those were the continuous models; they were in the type of the Walras–Leontief ones [11].

Nevertheless, some important problems are neither included within nor converge into those classes of the simplest macroeconomics models: Solow [1–3], Mankiw–Romer–Weil [6] and others.

A combination of macroeconomics models incorporating the principle of the subjective entropy maximum is tried in the present work. The principle was developed during 1990–2010. Although this principle formally hardly deviates from the Jaynes’ principle [12–14], the combination utilised in the present study widens the horizon of the practical applications of the obtained results, especially in psychology [7–10], economics [7–10,15], theory of conflicts, etc. [16–23]. Further, the approach could be recommended for implementation in other spheres of study that share similar characteristics [24–32].

PROBLEM STATEMENT

Let us consider a macroeconomics problem by Solow.

The production function is given by the expression:

1 $Y_{t + 1} = A {(K_{t})}^{α} {(L_{t})}^{β},$

where Y_t +1 is the production function Y at the time (step of iteration, recursion stage) t + 1, the time following the previous timet; A is the coefficient; K_t is the capital gained at the preceding point of time t; α is the power index for the capital K_t; L_t is the labour component L of the production process Y at the earlier point of time t; and β is the power index for the labour component L, analogously to α coefficient.

For some reasons, the power indices of α and β can be bound by the relation of

2 $β = 1 - α .$

For the components of K_t and L_t there should be satisfied the balance equation:

3 $Y_{t} = K_{t} + L_{t} .$

The division envisaged by eq. (3) is provided with the use of the individuals’ subjective preferences π:

4 $π_{t}^{K} a n d π_{t}^{L} .$

PROPOSED SOLUTION

3.1.

Entropy conditional optimisation

The objective functional for the individuals’ subjective preferences π, as ascertained in eq. (4), is determined using the expression:

5 $Φ_{π} = - \sum_{i = 1}^{N = 2} π_{t}^{i} \ln π_{t}^{i} + β_{π} [π_{t}^{K} K_{t} + π_{t}^{L} L_{t}] + γ [\sum_{i = 1}^{N = 2} π_{t}^{i} - 1],$

where

6 $- \sum_{i = 1}^{N = 2} π_{t}^{i} \ln π_{t}^{i}$

is the subjective entropy of the individuals’ subjective preferences π, representing the measure of the available alternatives’ effectiveness preferences functions uncertainty; and β_π and γ are the corresponding cognitive coefficients.

Then, on condition of

7 $\frac{\partial Φ_{π}}{\partial π_{t}^{i}} = 0$

it is possible to find

8 $- \ln π_{t}^{K} - 1 + β_{π} K_{t} + γ = 0.$

This yields

9 $\ln π_{t}^{K} = γ - 1 + β_{π} K_{t} .$

Thus,

10 $π_{t}^{K} = e^{γ - 1 + β_{π} K_{t}} .$

On the other hand,

11 $- \ln π_{t}^{L} - 1 + β_{π} L_{t} + γ = 0.$

This yields

12 $\ln π_{t}^{L} = γ - 1 + β_{π} L_{t} .$

Thus,

13 $π_{t}^{L} = e^{γ - 1 + β_{π} L_{t}} .$

The procedure outlined in eqs (7)–(13) leads to the expression:

14 $π_{t}^{K} + π_{t}^{L} = e^{γ - 1 + β_{π} K_{t}} + e^{γ - 1 + β_{π} L_{t}} = 1.$

The normalising condition, i.e. eq. (14), means that

15 $e^{γ - 1} = \frac{1}{e^{β_{π} K_{t}} + e^{β_{π} L_{t}}} .$

Because of eq. (15),

16 $π_{t}^{K} = \frac{e^{β_{π} K_{t}}}{e^{β_{π} K_{t}} + e^{β_{π} L_{t}}} .$

In turn

17 $π_{t}^{L} = \frac{e^{β_{π} L_{t}}}{e^{β_{π} K_{t}} + e^{β_{π} L_{t}}} .$

3.2.

Simulation

In the simplest case, the recursive system could be used:

18 $(\begin{array}{l} Y_{t + 1} \\ K_{t + 1} \\ L_{t + 1} \end{array}) = (\begin{array}{l} A {(K_{t})}^{α} {(L_{t})}^{β} \\ π^{K} Y_{t + 1} \\ Y_{t + 1} - K_{t + 1} \end{array}) .$

For the accepted data:

19 $A = 1.872, α = 0.3, β = 0.7.$

The initial conditions would then be represented by:

20 $(\begin{array}{l} Y_{0} \\ K_{0} \\ L_{0} \end{array}) = (\begin{array}{l} 10 \\ 6 \\ 4 \end{array});$

and

21 $π^{K} = 0.2;$

The results of the computer simulation are shown in Figs 1–3.

In the case when

22 $π^{K} = 0.34,$

the picture drastically changes.

The results of modelling for eqs (1)–(20) with eq. (22) instead of eq. (21) are illustrated in Figs 4–6.

The results of calculations obtained in pursuance of eqs (1)–(22) are portrayed in Figs 1–6, and they prompt a conclusion that there must be some preferences function value that ensures neither decrease nor increase of the characteristics, but stable magnitudes.

Indeed, at the value of

23 $π^{K} = 0.22255$

there is a situation of stability.

The results indicated by eq. (23) prove the supposition described above. It is represented in Figs 7–9.

Moreover, at the value of

24 $π^{K} = 0.386,$

there is one more situation of stability.

The results arising pursuant to eq. (24) are shown in Figs 10–12.

Thus, there are at least two situations of the stability; and moreover, at practically the same production (compare Figs 7 and 10), the distribution between the components of Y_t, namely K_t and L_t, is different (compare Figs 8 and 11 with 9 and 12, respectively).

The other decrease is for the values of

25 $π^{K} > 0.386.$

For instance,

26 $π^{K} = 0.3986.$ $${\pi ^K} = 0.3986.$$

The results are represented in Figs 13–15.

Therefore, we have arrived at a means to formulate an optimisation problem for the production maximum.

3.3.

Further developments

One way in which the development of the model can be linked with the results following from eqs (1)–(26) arises when the preferences are considered not constant.

For example,

27 $(\begin{array}{l} Y_{t + 1} \\ K_{t + 1} \\ L_{t + 1} \\ π_{t + 1}^{K} \end{array}) = (\begin{array}{l} A {(K_{t})}^{α} {(L_{t})}^{β} \\ π_{t}^{K} Y_{t + 1} \\ Y_{t + 1} - K_{t + 1} \\ \frac{e^{β_{π} K_{t}}}{e^{β_{π} K_{t}} + e^{β_{π} L_{t}}} \end{array}) .$

The initial value is given as:

28 $π_{0}^{K} = 0.3.$

For the parameter,

29 $β_{π} = 0.3.$

The computer simulation results arising pursuant to eqs (27)–(29) are shown in Figs 16–20.

ANALYSIS

It is discernible from the results of the simulation that, when the preferences functions obtained the values belonging to the diapason from eqs (23)–(24), the production and its characteristic components increase (see Figs 1–20).

It is supposed that the parameters of the income (the result of production) Y, physical capital K and labour L are known and controlled in the moment of timet.

Moreover, consumption C is also assumed to be known; and labour L is some function of C, being such that if C = 0, then L = L_min or L = 0. In addition, L has a zone of saturation.

The logistic function is a good one for a description of similar processes.

In such a problem, setting consumption C would be rendered possible after the procurement of production Y and capital K.

The problem is that consumption C can be determined from the balance equation or the formula by Cobb–Douglas if the latter is accepted as a part of the model. These two values do not coincide in the general case.

The Solow model contains a detailed description of all characteristics depending on the following two varying factors: physical capital K and labour L. Consumption C could be present in such types of models. However, it may ideally be considered via the labour L model. There are two conditions that are accepted:

When consumption C tends to 0 then labour L tends to L_min. There is the so-called slavery labour that enables the sustenance of this relation.
If consumption C tends to ∞ then labour L tends to L_max, and this could be obtained through the constraint for physical capital K.

In the framework of the above constraints, the dynamical model has been constructed, which realises some properties of the process:

There is some value of the labour L tension when the economic is characterised with the decrease of the system through time.
On the other hand, there are some values ensuring the increase of the economical characteristics.
Also, it is a very important moment when the labour L tension belongs to a certain diapason, for then, the economical characteristics are stable in time.

Qualitative investigations based on the Mankiw–Romer–Weil economics model give a larger number of such diapasons and boundaries. This issue is a problem for further study. Moreover, it is possible to construct models where the areas of physical capital K and human capital H are separated into the spheres of the special economic problems. For example, this approach could be applied to the agricultural problems or the transportation ones (including aviation), civil engineering and construction business systems, computer technologies and finally to the area of various scientific technologies.

It is important to bear in mind that the procedures outlined in the present study need not be confined in their scope to the choosing of the values of the preference function occurring in the examples discussed herein; rather, the preference functions are obtained in the explicit view following from the optimisation. In the examples discussed here, it is possible to show the change in the subjective entropy of the processes. This would explain the behaviour of the processes under study in the article but it could be further elaborated in further research.

CONCLUSIONS

At the current stage, studies into the creation of the macroeconomics classes of models, as well as their qualitative and quantitative investigations, are still underway. In addition, the applicability of these ideas extends predominantly to the higher educational issues.

The seminal feature is the production of the scientific product, which relates mostly to the intellectual product rather than equipment. Production of ‘discovery’ that can be evaluated though the years ahead is the peculiarity of such kind of human activity.

The Simplest Models of the Macroeconomics Dynamics in a Composition with the Principle of the Subjective Entropy Maximum

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