Dynamic model of firms competitive interaction on the market with taxation

In this article dynamic model of competitive interaction of firms on the market with taxation is explored. Any enterprise such as any state has the same goal -to generate income. The company wants to get as much income as possible in the form of profit, and the goal of a state is to get as much income as possible through taxation of enterprises, institutions and organizations. It is not profitable for an enterprise to give part of its profit to the state, therefore, it reduces its income. But it is beneficial for the state to receive all the profit from the enterprise. For a normal existence, both of enterprise and states must seek a compromise. If the state takes all the profits from the enterprise, the enterprise will cease to exist. The funds that the company gives to the state in the form of tax, the state spends in favor of the same enterprise. The state establishes the rules in force on the market in this country, conducts antimonopoly policy to maintain healthy competition on the market and protects the interests of domestic producers. All in all, the state guarantees the enterprise protection in case of violation of its legitimate interests by other market participants by the rules, which maintains competition on the market.

The first equation in the system (1) 𝑘 1 𝑉 1 is a factor that simulates the saturation of demand at the expense of the goods produced by the first firm. 𝑘 2 𝑉 2 is a term that simulates the saturation of demand by counteracting the competing firm. In the second equation the 𝑘 2 𝑉 2 system is a factor that simulates the saturation of demand at the expense of the goods produced by the second company and 𝑘 1 𝑉 1 is a term that simulates the saturation of demand at the expense of goods produced by a competing company.

𝑘 1 𝑘 2 -coefficients model how the demand is met with interchangeable goods at the expense of competitors. Equations (1) -Volterra equations, describe the competition model of two firms producing interchangeable goods.

3 Analysis of the differential equation solution of the model of interaction between two firms.

This system can be rewritten as (1 ‘):

We study the solution of this system with initial data 𝑉 1 0 , 𝑉 2 0 , positive for 𝑡 = 𝑡 0

It can be shown that for any finite time interval (𝑡 0 , T), there is a unique solution of two continuous functions between two positive numbers, which depends on the end of the interval T (i.e., 𝑉 1 and 𝑉 2 are bounded) .

Consider what happens with an unlimited increase in time.

We get then,

We can say that, if

We get So, the value of the capital of the second company, in which 𝑝 𝑘 is less important, the demand for the goods of this company is fully satisfied. It decreases and with time the second company loses all its capital, while the first company continues to exist. When a firm really loses all its capital, it is extremely rare situation. The owner of the company understands that the costs of the production and promotion of goods exceed the income from its sale, and take any steps to solve.

If the second and the first firms exists on the market independently of each other, after large period of time the capital of the first firm obeys to the law.

Starting from the moment 𝑡 1 , when 𝑉 1 takes the value 𝑉 1 ′ .

Let’s say 𝑉 𝑡 𝑙 is the root of the equation.

(

If 𝑉 1 ′ < 𝑉 2 ′ , the term (3), starting from the moment, will increase to the value 𝑉 𝑡 𝑙 following the law.

As was considered, that 𝐹 𝑉 1 ′ ≥ 0, in neighbourhood 𝑉 If the capital of the first firm has a finite redistribution, different from zero, then the second firm ceases to exist.

This does not always mean the bankruptcy of the company. In nowadays companies have not one, but companies have several areas of activity.

The management of the company, which has only one line of business, may not bring the case to bankruptcy, but simply makes a decision to liquidate the company, the line of business, or change the product being produced for the better.

A company with several fields of activity has the same choice -to close unprofitable production, or to change a product, or to organize a more successful promotion of this product on the market; but it is much easier for a large company to implement any of these measures, since it is financially more stable and has the money that it can invest in the above measures. Based on the above, we calculate the total income of firms competing with each other on the market for a finite period of time without taxation:

Where, / 𝑉 1 0 -first company capital value, 𝑉 2 0 -second company capital value.

4 Study based on Lotka Volterra model.

When we neglect the case in which the firm itself saturates the market with its goods, the demand for goods produced by the firm is reduced only by the products of a competing firm. Then equation ( 1) has the form:

(4) So, from system (4), we can write out three solutions to this system:

If the values of 𝑉 1 or 𝑉 2 are equal to zero at any given time, then they