## APPLICATION OF MATHEMATICAL METHODS IN ECONOMIC GEOGRAPHY

After studying Chapter 11, students should:

* know *

• the essence and directions of mathematical-geographical modeling;

• models of regional and national economy;

• the main types of forecasts in economic geography;

* be able to *

• develop the main types of models used in economic and geographical analysis;

• analyze the problems of location and development of productive forces on the basis of mathematical and geographic models;

* own *

• methods of collecting, processing, analyzing and interpreting economic geographic information for the development of mathematical and geographical models;

• skills in the analysis and forecasting of economic and geographical processes.

## Essence and directions of mathematical-geographical modeling

Modeling is of special importance for the development of the methodology of economic and geographical studies. The active penetration into the geography of the system approach, the concretization of research methods, the need for a clear formulation of problems and their rigorous solutions led to the development of mathematical methods in geography.

First of all, we make one fundamental remark concerning the differences in the concepts method and model .

* The method *, as noted above, is a way of knowing, exploring the phenomena of nature and social life. * The model * (from Latin "modulus" - a measure, a sample) is a diagram, image or description of a phenomenon or process in nature and society. Taking into account both definitions, it can be argued that the model is an integral part of the method. In this regard, in the course of explaining the nature of a method, model terms may appear.

The term model is widely distributed both in scientific and in common language, and in different situations it makes a different sense. In the most general formulation, the model is a conditional image of the research object designed to simplify this study.

Modeling a geosystem or some of its elements (for example, a structure or process of functioning), we create an artificial analogue (copy) in a simplified form. It is almost impossible to imitate a complex system in all its diversity. It is important that the model describes the main properties of the object, reproduce the relationships between its elements, the types and nature of intrasystemic and external relationships, and in those or other quantitative indicators. In this connection, the simulation contains not only qualitative, but also quantitative criteria for analyzing the object under study. This makes it possible to improve the modeled object and on this basis to develop a strategy for regulating the development of the geosystem itself.

Mathematical-geographical modeling is a complex process, consisting of a number of consecutive stages.

1. The preparatory stage, during which the goal is set and the research tasks are determined. Based on preliminary data on the system, the need to apply the mathematical modeling method is revealed. Then, in accordance with the research objectives, formulate the modeling tasks for the given system.

2. The second stage consists of a series of sequential operations:

- identifying the nature of the interrelatedness of the tasks formulated at the preparatory stage;

- the establishment of a sequence of solving these problems.

3. The third stage is the construction of models for solving each of the tasks posed.

Examples of models in geography can serve as the following.

* 1. Models of a class of geographic problems to identify the links between the elements of the system: assessment of the balance of energy and biomass of the Earth, measurement of the displacement of individual points on the earth's surface, characteristics of the development of individual erosion forms in river basins, study of relationships in landscapes, assessment of natural regimes of the geographical environment, assessment of the relationships between the radiation balance in various geographical facies, assessing the intensity of development of gully erosion, identifying the leading factors of complex connections in the system, the city's population, the level of its development, the structure of the territorial production complex, etc.*

* 2. Models of the problem analysis class of the system elements *: statistical analysis of the boundaries of avalanche zones, finding the basics of designing forest shelterbelts, morphometric analysis for the design of various engineering structures, determining the density of the city population, etc.

* 3. Models of the class of zoning (and zoning) tasks: * identification of types of natural landscapes as a function of the radiation index of dryness, recognition of samples for the identification of areas and analysis of taxonomic distance in the n-dimensional space, identification of zones of influence of cities, zoning of territories,

One of the first macroeconomic models was the model of F. Ken's reproduction (1758). Modern development of modeling of economic processes is connected with a complex of economic-mathematical methods and econometrics. Econometric models are developed to predict economic indicators such as GNP, personal income, personal consumption expenditure, investment, government purchases of goods and services, net exports, etc.

Investigation of dependencies and relationships between objectively existing phenomena and processes plays an important role in the economy. The method of studying connections in statistics is called regression analysis, and connections are called regression analysis.

Mathematically, the problem is formulated as follows.

It is required to find an analytical expression of the dependence of the economic phenomenon (for example, labor productivity) on the factors determining it; those. We look for a function * y * = * f * (* x * 1 *, x * 2 *, ..., x * n *), * reflecting on the average the dependence on which, knowing the values of the independent factors * x * i, we can find an approximate value of the dependent indicator * y *

In order to find the regression equation, it is necessary to determine the general form of the functional dependence and to calculate the parameters of the equation (the method of least squares is often used). When choosing the type of dependence are guided by the following: it must be consistent with professional and logical considerations regarding the nature and nature of the links studied; if possible, use simple dependencies that do not require complex calculations, easily interpretable and practical. The technology of developing the regression model is described in almost every textbook on the general theory of statistics and econometrics.

Regression models of socio-economic objects and processes are used in many areas, depending on the type of functions used.

Pairwise linear regression is used in the study of the consumption function:

where * С - * consumption; * y - * income; * K * and * L - * parameters of the function.

Among the class of non-linear functions one should call an equilateral hyperbole

which is used to characterize many kinds of relationships. For example, to characterize the relationship between the specific costs of raw materials, materials, fuel, and the volume of output, the time of circulation of goods from the value of turnover at both the micro- and macro-levels.

A classic example is the Philips curve, which characterizes the non-linear relationship between the rate of unemployment * x * and the percentage of wage growth * y: *

where ε is a random deviation.

This equation, reflecting the inverse relationship of these indicators, was obtained in the late 1950's. English economist AV Philips on the basis of data generalized more than for a 100-year period.

An analogous example is also the relationship between the share of spending on durable goods and the total amount of expenses (or revenues). A mathematical description of this kind of relationship was called the Engel curves. In 1857, the German statistician E. Engel, on the basis of a study of family expenditure, formulated the following regularity: with increasing income, the share of expenditure on food decreases. Consequently, with increasing income, the share of their spending on non-food items will increase.

In studies of productivity, labor intensity of agricultural production, equations with square roots, combinations of trigonometric functions are used.

Multiple regression is used to solve problems of demand, stock returns, when studying the function of production costs, in macroeconomic calculations, etc.

In the 1930s. JM Keynes formulated the hypothesis of the consumer function. Subsequently, researchers repeatedly turned to the problem of its improvement. The modern consumer function is considered as a model of the following kind:

where * С * - consumption; * y * - income: * P * - the cost of living costs; * M * - cash; * Z * - liquid assets.

A special role in the modeling of socio-economic systems is played by production functions that connect the values of the product vector * y * with the values of the resource vector * x *:

where * a = a * 1, ..., * a * n is the vector of the parameters of the production function.

Instead of a general representation of a production function, two particular cases are often used.

1. Release functions in which the resource costs are taken as independent variables, and the function is output:

2. Production cost functions in which the independent variable is output and the function is the cost: .

A detailed description of the use of production functions in modeling is given in paragraph 11.3, where models of regional and national economies are considered, since it is at these levels of the economic hierarchy that they find the greatest application.

In some cases, the dependence of the phenomenon or object studied on time is investigated, i.e. its dynamics are modeled (see paragraph 11.2).

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