APPLICATION OF DIFFERENTIAL EQUATIONS IN SOLVING ECONOMIC PROBLEMS

ABSTRACT

This article discusses the use of differential equations in the study of indicators in the economy that change over time, and the issues of approximation of forecasting results to reality.

АННОТАЦИЯ

В данной статье рассматривается использование дифференциальных уравнений при изучении показателей в экономике, которые изменяются с течением времени, и вопросы приближения результатов прогнозирования к реальности.

Keywords: continuous functions, differential equations, model, derivative value, product index change, inflation rate.

Ключевые слова: непрерывные функции, дифференциальные уравнения, модель, значение производной, изменение индекса продукта, уровень инфляции.

In later years, foreign literature has been widely used to construct and infer time-dependent continuous functions rather than study the dynamics of traditional tables and numbers in solving economic issues. Until this period, all processes that change in time were expressed through differential and integral Equations, and all processes that change in time in the fields of mechanics, chemistry, biology were transferred to the language of equations. While economics also has the practice of studying indicators that change in relation to time, it is with differential equations that these processes are little studied. Calculations show that the difference between the values of the continuous function to be constructed and the real numbers of the real processes is 2-5%. Continuous functions are widely used in the study of time-varying processes in foreign literature. [7-9]

This article will focus on the problem of expressing the relationship between market price and production volume using continuous functions. As the main indicator, we received a change in the index of the amount of products per person. That is:

 =   received in the form of

In the next steps, it was proved that such an assumption is justified and the use of continuous functions is very useful in drawing important conclusions, that is, it is possible to determine the value of prices in different periods, the dynamics of changes in the volume of output per person . In addition, the necessary information can be obtained by taking the derivative from the defined function.

The task of economics is not only to determine positive or negative situations, but also to find solutions to problems in economic processes. In this sense, the article presents proposals and results of scientific calculations on measures to stabilize price growth, taking into account the influence of factors.  Taking into account that the market price of agricultural products depends on the volume of production, import and inflation, we express the price of product (i) in the year (t+1) using the following formula:

                                                                (1)

Here:

  - (i) the expected price for the product in the next year

  - (i) market price of the product in the base year

  - index of change in the amount of (i) output produced per person

 – (t) inflation rate in the year.

Above, the  index is the coefficient proposed by the authors, which is determined from the following formula:

 =                                                                  (2)

here:

(i) the values of the function representing the production of goods calculated in the corresponding base and reporting years;

 – the values of the population calculation function in the studied area in years t and t+1.

Now let's consider the algorithm for calculating prospective prices for a specific product and region using the above formulas.

We take into account that the price of wheat in the markets of Bukhara region is = 3.5 thousand soums and = 4.5 thousand soums and the average inflation rate is 12%. In addition, we rely on statistics on population and wheat production for 2016-2020..

Table 1 presents information for finding the function in the form of   = ao+bt based on the differential method for population and Y for the years 2016-2020 at the scale of Bukhara region. [1-3] This table compares theoretical and model-calculated numbers.

Table 1

Table of necessary data for the compilation of the population forecasting function of the Bukhara region

 we accept that and = 27.175 the equation (1)= 1815.2 solve on condition  we will have a function.

Table 2 shows wheat production statistics in Bukhara region for the period 2016-2020 and initial data for constructing a linear production function based on these numbers.

Table 2

The necessary calculations for the compilation of the forecasting function of wheat production in the Bukhara region

Based on the methodology used above, we create a production function in the direction of the differential method and perform the relevant calculations.[1,2,3]

 as it is being seen = -33,8 equation

y(0) = 559,1 we create the corresponding function by solving based on the condition.

 = - 33,8t+592,9

Now we will make calculations based on the model and fill in the table.

The coefficient   included in our proposal generally indicates the volume of output per person.

In our case now:

=  (3) has a view function is created.

 To show that the calculation methodology can be applied in practice, we present the numbers that represent the actual situation in the region under consideration.

Taking into account that the price of wheat in the regional market in 2020 is = 3.5 thousand soums, and in 2021 it is = 4.8 thousand soums, we will calculate the prices calculated based on the above methodology:

 =  = 0,235

 =  = 0,195

So,  =  = 0,83

(1)                                             according to the formula:

 =  + S*

Wheat prices in Bukhara markets in 2022:

 =  + 3,5*0,1= 4,5668 thousand soums:

This is very close to the current prices.

We calculate the coefficient  in another way.  is a function that in our case depends on t from which we take derivative.

According to the definition of the derivative:

Or

 =

In our case, since ∆x=∆t=1 is a condition, the value of the derivative only represents the difference between the values of this function, that is, the decrease or increase of   in the time interval. Using this method also gives results and can be used.

It can be noted separately that the change in the index of the product per person over a certain period of time can also be calculated using this formula. In 2010, the price of meat offered to the market by entrepreneurs in the Bukhara region was 25 thousand soums, currently this value is 75 thousand soums. If the annual inflation rate is estimated at 14%, then one can calculate the meat per capita change index between 2010 and 2022. (1) from the formula we find  and perform the calculation:

 =

That is, meat production per person is in Bukhara region:

= 1,18 

Decreased by 18 percent. In the calculations, the price is taken as S, where the initial price (2010) is 25,000 soums, and if the annual inflation is 14% on average until the present period, then the inflation sur It is determined that the price of meat will increase to 25*0.14*13=45.5 soums under the influence of 'ats. We can say that the increase in the price of meat to 45.5 soums depends on the increase in wages, production costs and other reasons.

Then the increase in the amount of  ∆=75-45.5=29.5 soums can be interpreted as a result of the imbalance between supply and demand.

We will focus on the mechanisms of keeping product prices as stable as possible in the markets. One of the necessary conditions for stabilizing prices according to the law of supply and demand is to maintain the same amount of products per person. As a result of long-term observations, the following interesting information can be given as an example. In July-August 2011, the price of tomatoes in Bukhara markets was 3,000 soums, and meat was 25,000 soums. Despite the fact that the price of meat has increased 3 times over the last 12 years, the annual increase in the number of greenhouses has made it possible to keep the price of tomatoes stable. Thus, the stability of the production volume per person can be considered as one of the fundamental reasons for price stability. Therefore, we accept the condition  ≥1 as a necessary condition for price stability.

Let's consider the mechanism of using the above method. The function calculated by us is   shows that it represents a change. According to the formula for calculating the derivative if y=u/  was.

that is why

So:

or:

If we take into account , it means that K= decreases by -0.018 every year. So K ̅=1- 0.018≈0.98. In previous simple calculations, K ̅=0.83

If the price is predictable then:

 =  + S*  from

 =

It appears that the prediction values took very close values in both cases.

The solution to the second problem follows from this. That is, since the value of () is in the computations or ()∆x = (+∆x) -(), it can be taken as approximately annual increase or decrease. [4-6]

We believe that the analysis of economic processes using mathematical methods will give a positive result for use in the practice of forecasting. Especially the analysis of dynamic rows is very important for the implementation of these methods. Difficulties in the expression of any time-dependent economic processes in differential equations are due to the accounting of many factors affecting them.  The problem of not accurately reflecting real life in modeling economic processes using the previous method and is considered to be a special effect of various factors.

The processes studied in the proposed [10-12] methods are expressed in differential equations. In this case, all the studied processes were first put into the form of a differential equation, and then an equation reflecting the real process was created. That is,  y(0)=1815 for population and y'=-33.8 and y(0)=559.1 for wheat production were expressed in equations. However, it should be noted that only production costs are considered as the main factor when creating a model of economic processes, but there are other factors that can influence the process. Among these, we can see the effect of weather on productivity.

In general, if we express the productivity function as , then represents the external influence function. However,  has not been fully studied or its expression based on certain laws has not been fully studied, including the function of the effect of weather on productivity has not been determined.

In conclusion, it can be noted that the use of differential equations in the study of time-varying indicators in the economy makes it possible to bring forecast results closer to reality and reduce errors in calculations.         

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