UDC 004.942

Dynamic model of competitive interaction of firms with taxation in the market

Малафеев Олег Алекссевич – доктор физико-математических наук, профессор кафедры Моделирования социально-экономических систем Санкт-Петербургского государственного университета.

Акрамова Гулёра Абдихаликовна – аспирант кафедры Моделирования социально-экономических систем факультета Прикладной математики – процессов управления Санкт-Петербургского государственного университета.

Abstract: The article considers a dynamic model of competitive interaction of firms with taxation in the market - this is a model that describes the interaction between firms in the market, taking into account the tax impact. This model takes into account the dynamic aspects of firms behavior and their response to changes in tax policy.

Keywords: model, describes the market, taxation, dynamic preferences of firms, interaction between firms, tax policy, firms react to changes in tax policy, market share over time.

Introduction

The article examines the problem of competitive interactions in the market through mutual taxation between firms and the state. And any business and state actually has one goal - to make a profit. The enterprise wants to get as much profit as possible in the form of profit, and the goal of the state is to get as much profit as possible through taxation of enterprises, institutions and organizations. The ways to achieve these goals seem to be mutually exclusive. It is unprofitable for the enterprise to give part of the profit to the state, therefore it reduces its income. On the other hand, it is beneficial for the state to transfer all proceeds to the enterprise. But both business and the state need a compromise for a normal life. If the state receives all the profit from the enterprise, then the enterprise will soon cease to exist, because it will not have the means for further development and existence. In turn, the funds that an enterprise provides to the state in the form of taxes are spent by the state in favor of this enterprise. The state sets the rules in force in a particular country, conducts an anti-monopoly policy to maintain healthy competition in the market, protects the interests of local producers and, of course, guarantees that the state protects the enterprise in the event of a violation of the law. interests of other market participants. Maintaining competition in the market.

This work is devoted to finding a compromise between the interests of the state and the enterprise.

Informal problem statement.

The article considers three models; in the simplest first model, two firms produce the same product and do not compete with each other. The second model is a generalization of the first, but in it, firms are already competing with each other in the market. The third model is a competitive interaction model in which firms interact through mutual taxation in the market and between the state. Nowadays, any product, especially consumer goods, has a spare product. Products may vary in design, price, and other parameters; how the manufacturer (or seller) puts the goods up for sale (positions); but the main purpose of purchasing the product remains unchanged. Allow firms to produce homogeneous (interchangeable) goods. Then the rate of growth of a firm's capital will depend on the volume of demand for goods produced by that firm, the demand for goods - stocks produced by a competing firm, and the total volume of demand. Then the existence and specificity of the solution of this differential equation is shown. Consider a solution similar to the Lotka-Volterra model.

is the capital of the first firm, is the capital of the second firm.

- aggregate demand for goods, i.e. these coefficients determine the capital growth rates of firms 1 and 2, respectively.

- coefficients for modeling the process of demand saturation, i.e. satisfaction of demand with goods produced by firms 1 and 2, respectively.

We obtain equations showing the growth rate of the capital of firms 1 and 2, respectively.

(1)

Let us consider in more detail: (1) The first equation in the system is a factor that models demand saturation due to products produced by firm 1 itself and is a term that models demand saturation opposition from a rival firm. Accordingly, in the second equation above the system shown, is a factor that simulates demand saturation due to goods produced by firm 2 itself and is a term for stimulating demand saturation due to manufactured goods competing firm,

- are coefficients that model the demand for substituted goods due to competition between two firms producing substituted goods.

Equation (1) - Voltter's equation describes a model of competition between two firms producing interchangeable goods.

Investigation of the solution of a differential equation describing a model of interaction between two firms.

This system can be rewritten as ( ):

Let us investigate the solution of this system for the initial data positive and .

It can be shown that for any limited time interval ( ) there is a unique solution of two continuous functions between two positive numbers, the value of which depends on the end of the interval (i.e., are limited)).

Consider what happens with an infinite increase in time in the form of an overwrite ( ):

deduce

further, we get

(2)

We neglect the almost impossible case when

and suppose that

or

then according to (2)

(3)

Since is limited, so tends to zero. Thus, the capital value of firm 2, which has a lower value , i.e., the demand for the goods of this company is fully satisfied, decreases, and over time, firm 2 loses its capital, but the presence of the 1st firm will continue.

Indeed, when a firm loses all of its capital, the opportunity is rare. In many cases, the owner of the company (or its management) quickly begins to understand that the costs of producing and promoting goods exceed the proceeds from its sales, and takes any action to resolve the situation.

The work of non-competitive firms.

If firms 2 and 1 exist on the market independently of each other, that is, they do not compete with each other, then after a sufficiently long period of time, other people's capitals, i.e., do not compete with each other, then after a sufficiently long time, capital of firm 1 is regulated by law

from the moment , which received the value .

Let be a root of the equation

If , then the expression from this moment becomes positive, and increases to the value according to the law.

Since it is assumed that for around

at

where is positive, and therefore

It can be concluded that never reaches it cannot reach any small value in a limited time. The amount of that has a boundary tends to because it is increasing and limited.

Similarly, if then tilts, decreases and becomes If then the differential equation satisfies Since the solution for these initial conditions is unique, v_1 remains constant.

Thus, for firm 1, capital tends to a limit that is not equal to finite zero, for example,

The work of competing firms.

Consider a situation where two firms producing similar goods are competing in the market.

If we take the first function as an -function

,

where ,

then from the moment of time changes according to the law

If the value of , equal to differs from which is the root of the expression at this initial moment, then

As

then let's say

From here

Since at time the expression

it is obvious

Solving this equation for . We get

where

and tends as to

Thus, if the capital of firm 1 has a non-zero final distribution, then firm 2 ceases to exist. This firm disappears with a low , i.e. the demand for the goods of this firm is fully satisfied.

As mentioned earlier, this does not always mean bankruptcy. As a rule, companies now have not one, but several areas of activity.

The management of a company with only one direction of activity cannot lead the business the bankruptcy, but can simply decide to liquidate the company, change the scope of activity or change the manufactured product in a positive direction.

A company with several fields of activity has the same choice - to close harmful production or change the product, or organize a more successful promotion of this product on the market; however, it is much easier for a large company to implement any of these measures since it is financially stable and has the capital to invest in the above activities.

Based on the above, we calculate the total revenues of firms competing with each other in the market for a limited period of time without tax:

where

is the cost of the initial capital of firm 1, is the cost of the initial capital of firm 2.

Literature list

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