# Considering Random Factors in Modeling Complex Microeconomic Systems

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## Abstract

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## 1. Introduction

- studying the existence of relations between functions and factors, and their closeness;
- classification of the economic phenomena factors;
- defining analytical forms of interrelation between the studied phenomena;
- determining the parameters of indexes’ periodic fluctuations conformity/law;
- study of quantitative changes of indexes/functions due to the influence of factors;
- economic interpretation of the obtained analytical dependences.

## 2. Model Building and Testing

- For enterprises 1 and 2: economy on scale due to vertical integration; benefits from access to resources (financial, material, technical, and also non-material), with the aim of strengthening financial status and technological upgrade of the enterprise; load leveling of production capacities during a year; transfer of organizational, production, and sales knowledge; use of existent practice of research and development (RnD) and production; and use of partner’s distribution trade mark in sales.
- For enterprise 3: possibility to exploit comparative advantages; risks distribution; potential advantages from entering new market with lower competition; using growth potential of the less mature market; economy on scale due to vertical integration; balancing production costs; common use of non-material resources, such as transfer of knowledge and experience; access to comparatively cheaper resources; and decrease of production costs.

- ${P}_{t}^{\left(i\right)}$: retained earnings (accrued profits) over t months for each of the enterprises $i,\left(i=1,2,3\right)$, where i is the index of the enterprises;
- ${K}_{t}^{\left(i\right)}$: own accrued capital of the enterprise, $i;$
- $K{A}_{t}^{\left(i\right)}$: accrued invested (accepted) capital over t months from enterprise $3\mathrm{to}\text{}\mathrm{enterprise}i,\left(i=1,2\right);$
- ${Q}_{t}^{\left(i\right)}$: level of output by production enterprises in month t (total output from the beginning of activity to the end of month t),$i=1,2$.

- $N{P}_{t}^{\left(i\right)}$: net profit of the enterprise $i,\left(i=1,2,3\right)$ in month t, not considering depreciation and amortization;
- $KPar{t}_{t}^{\left(i\right)}$: part of the enterprise’s 3 net profit, invested into the capital of enterprise $i,\left(i=1,2\right)$;
- $Npar{t}_{t}^{\left(3\right)}$: part of the enterprise’s 3 net profit remaining, after investing to the capital of other enterprises.

- ${v}_{t}^{(i)}$: transfer price (the price for which the production enterprise is selling its products to the sales enterprise within the group);
- ${p}_{t}{}^{(i)}$: the product sales price;
- ${f}_{t}^{(i)}$: monthly fixed general administrative expenses;
- $f{s}_{t}^{(i)}$: monthly fixed marketing and sales expenses;
- $s{c}_{t}^{(i)}$: variable unit sales costs;
- ${l}_{t}^{(i)}$: labor unit costs;
- ${L}_{t}^{(i)}$: number of direct production personnel at the enterprise;
- ${\eta}_{t}^{(i)}$: coefficient characterizing capital and labor productivity;
- ${\alpha}_{t}^{(i)}$: rate of profit reinvestment into the enterprise’s capital;
- ${\beta}_{t}^{\left(i\right)}$: rate of depreciation.

## 3. Results

#### 3.1. Level of Production Orders, Income, and Investments

#### 3.2. Considering Demand Fluctuations

#### 3.2.1. Seasonal Fluctuations

- Initial monthly volume of the market demand equals the average actual level, defined as the result of preliminary enterprises analysis.
- There is a general upward trend in the market volume of 15% annually; we make such an assumption based on analysis of the previous years’ data, as well as active marketing policy of the group.
- The fluctuations of market demand have a seasonal character, such as periodic increase and periodic decrease in certain months during the year.
- The level of market demand does not depend on the price level. We assume that the group is functioning at a market of perfect competition and is a price taker (does not essentially influence the price).
- Besides the (rather predictable) seasonal fluctuations, the level of market demand is also influenced by random external factors.
- The level of market demand for a certain period is the indicator of the maximum production output of the goods for this period, there is no work-in-progress considered.

#### 3.2.2. Random Demand Fluctuations

#### 3.3. Numeric Calculations

#### 3.3.1. Influence of Seasonal Demand Fluctuations on the System Dynamics

#### 3.3.2. Total Impact of the Seasonal and Random Fluctuations on the System Dynamics

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Dm (t,0.2,1)—market demand for the products of investigated enterprises considering seasonal and random fluctuations; kr = 0.3, ks = 1; D(t) is the guaranteed demand in month t.

**Figure 4.**Size of the guaranteed demand D(t) and its seasonal fluctuations at the level of coefficient ks = 0.9; 0.5.

**Figure 5.**Seasonal fluctuations of the invested capital level for the companies Mriya (k1a) and Emitex (k2a).

**Figure 6.**Seasonal fluctuations of the total capital level for the companies Mriya (k1) and Emitex (k2).

**Figure 7.**Impact of the seasonal demand fluctuations on the order size in month k. D(k): total guaranteed order size; Dem(k,0, ks): total order size considering seasonal demand fluctuations; x1, x2: order size for the companies Mriya and Emitex, respectively.

**Figure 8.**Dem(t,0.5,0)—consumer demand for products considering solely the random fluctuations; and the guaranteed monthly demand D(t).

**Figure 9.**Dm(t,0,0.8)—consumer demand for products not considering the random fluctuations; Dm(t,0.5,0.8)—consumer demand for products considering both the seasonal and the random fluctuations.

**Figure 10.**Level of investments from the company-client into the capital of production company Mriya (k1a) and the production company Emitex (k2a), considering both seasonal and random demand fluctuations.

**Figure 11.**Profit of the companies Mriya (p1), Emitex (p2), and the company-client Miltex (p3), considering both seasonal and random demand fluctuations.

**Figure 12.**Increase of the capital of the companies Mriya (k1) and Emitex (k2) considering both seasonal and random demand fluctuations.

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**MDPI and ACS Style**

Hoshovska, O.; Poplavska, Z.; Kryvinska, N.; Horbal, N.
Considering Random Factors in Modeling Complex Microeconomic Systems. *Mathematics* **2020**, *8*, 1206.
https://doi.org/10.3390/math8081206

**AMA Style**

Hoshovska O, Poplavska Z, Kryvinska N, Horbal N.
Considering Random Factors in Modeling Complex Microeconomic Systems. *Mathematics*. 2020; 8(8):1206.
https://doi.org/10.3390/math8081206

**Chicago/Turabian Style**

Hoshovska, Oksana, Zhanna Poplavska, Natalia Kryvinska, and Natalia Horbal.
2020. "Considering Random Factors in Modeling Complex Microeconomic Systems" *Mathematics* 8, no. 8: 1206.
https://doi.org/10.3390/math8081206