# Mathematical Functions of Economics

## Introduction

The math functions, which are generally used for expressing the main economic tendencies may be either linear or inverse, nevertheless, some of the functions, which are used for describing non-linear processes may be curvilinear and functions of two or more independent variables. Originally, all the functions are used for analyzing the economic output of a company, or the entire branch of the industry. Thus, the meta-production functions are used for comparing the performance practice of the existing entities and for the practices of determining the most efficient production function. In general, the functions are used for defining the maximum of a technologically defined process, and the production output factors, which require exclusive precision, may be calculated only by the means of math functions.

This paper aims to concentrate on the four types of mathematical functions, such as:

1. Inverse function,
2. Curvilinear function,
3. Functions of two or more independent variables
4. Linear functions

The linkage between functions and one economic concept will be explained based on the interpretation and analysis of why and how each function promotes a better understanding of managerial economics.

## Math Functions

The most universal aim of math functions in economics is the capability of optimization. While optimization in any sphere is regarded to be the selection of the best element from the available alternatives, math functions are intended for searching this alternative and applying the gained results for the economic principles. In other words, this means the solving of a problem, where minimization or maximization of a real function is required.

The original value of the math functions in the economic sphere is covered by the fact that functions may be defined by the specification of the input requirements, which are required to outline and precisely formulate the properly defined quantities of the economic output based on the available technology. As it is stated by Bowley (2003, p. 561):

It is usually presumed that unique production functions can be constructed for every production technology. Thus, there are several ways of specifying the production function. In a general mathematical form, a production function can be expressed as:

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Q = f(X1,X2,X3,…,Xn)

where:

• Q = quantity of output
• X1,X2,X3,…,Xn = factor inputs (These are the values of the factors, required for the economic functioning). Nevertheless, this general form does not encompass joint production that is a production process, which has multiple co-products or outputs.

In the light of this consideration, there is a strong necessity to emphasize that specifying a production function from the position of a discrete outputs and inputs combination does not require the formula equation. Nevertheless, using an equation generally requires the continual variation of output, considering the possible valuations and changes in the range of inputs. Considering the necessity to use linear or curvilinear functions, it should be emphasized that the fixed ratios of factors, which express the input of labor force and working tools, may be used for the discrete input combinations, consequently, the maximum outputs of the functions are of practical interest for the economic analysis from the position of math functions.

## Inverse Functions

The inverse functions are generally used for defining the inputs and outputs of demand values. Originally, the inverse demand function is applied for mapping from the values of quantity of output in association with the market prices of the demanded and consumed goods. The objective of the function is to model the demand which the economic issues face. The imaging, which is defined by the inverse functioning is further analyzed from the position of the output quantity, consequently, it should be emphasized that the inverse demand function is generally applied from the position of the input and output analysis. Thus, it is evident that the inverse function may be used for deriving the marginal revenue factors from the aspects of demand and consumption. By Psalidopoulos (2004, p. 289):

Total revenue equals price, P, times quantity, Q, or TR = P×Q. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 -.5Q) × Q = 120Q – 0.5Q². The marginal revenue function is the first derivative of the total revenue function or MR = 120 – Q. Note that the MR function has the same y-intercept as the inverse demand function, the x-intercept of the MR function is one-half the value of the demand function, and the slope of the MR function is twice that of the inverse demand function.

In the light of this fact, it should be emphasized that the relations hold the necessity for linear demand function, nevertheless, the importance of the quick calculation of the profit-maximizing conditions requires the inverse approach regardless of the market structure and the realities of the demand tendencies. Thus, to derive the values of marketing consumption, the total cost functions are taken from the inverse function calculating position. In the light of this perspective, the considerations by Guillebaud and Marshall (2006, p 184) should be emphasized:

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The inverse demand function appears in this form because economists place the independent variable on the y axis and the dependent variable on the x-axis. The slope of the inverse function is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for calculating elasticity is PED = (∆Q/∆P) × (P/Q).

Consequently, the reverse function and the calculation of the demand, supply, and consumption issued should be represented from the perspective of the elasticity, otherwise, the demand is calculated by the means of linear functions.

## Linear

The linear demand and supply functions are simpler than the inverse calculations, and these are simpler for defining the elasticity of these economic perspectives. As it is stated in Bowley (2003, p. 612):

If there is a function f, where A R from some set A to the real numbers, then, An element x0 in A such that f(x0) ≤ f(x) which is represented as x in A or such that f(x0) ≥ f(x) for all x in A requires to be found.

Such formulation is regarded as an optimization tool, which is the aspect of the linear calculation of the economic functions. Moreover, by the Review of Fundamental Mathematics any linear relation between y and x can be expressed in the algebraic form:

y = a+bx

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where:

•  y is the dependent variable,
• x is an independent variable,
• a is the intercept parameter,
•  b is the slope parameter.

The linear functions may be of three various types.

The most commonly applied function is the downward sloping, which is aimed to represent the demand level when the price of the goods rises. Originally, this slope is used in the case when the level of the sales is decreasing, consequently, this slope clearly demonstrates the tendencies of the monopolistic market, when a trader has an opportunity to increase the price.

The horizontal demand linear function defines the image when the demand drops to zero, and, the price stays at the same level. Thus, the elasticity is hard to define in accordance with these conditions, nevertheless, a situation when a competitor’s product of high quality is present on the market, may be featured.

A special case of vertical linear function is aimed at defining the situation, when independently on the price, the demanded quality is the same. This function represents the economic aspects of the goods of the first need, when the consumers do not mention the price.

## Curvilinear Functions

These are the functions of the unit related cost behavior. The curvilinear functions are generally used jointly with the linear functions, and define the relations of the cost aspects jointly with the quantity of the offered goods. Thus, Psalidopoulos (2004, p. 271) gives the following considerations:

It should be noted that Marginal cost is not equal to Average Variable Cost. Moreover, it should be noted that when TC is nonlinear, the slope of the curve and, therefore MC, is continuously changing. Because MC is changing, AVC also is changing and MC ≠ AVC. When TC is linear, the slope is constant, therefore MC is constant, and AVC must equal MC

The graphic representation of this function is the following:

Where:

• AVC = average variable costs (per unit)
• AFC = average fixed costs (per unit)
• ATC = average total costs (ATC = AVC + AFC)

The economic value of these functions is generally associated with the matters of the Total Cost calculation, while the accountants prefer applying them for the calculation of the approximation of the total costs, based on the output of the relevant range.

The curvilinear functions are also used for calculating the linearity assumptions of the relevant price range. In accordance with the graphic representation of the linearity assumption is similar to the inverse linear function of the price level, associated with the demand levels.

Considering the necessity to assess the values of the curvilinear function in the context of the total cost levels, these functions represent the value of the relevant range of the prices and the level of the demand, associated with this level, as well as the activity of the consumers and the target audience, which is oriented. In the light of this consideration, it should be stated that the curvilinear analysis of the demand levels and the ranges of the relevant prices are associated with the activity of the consumers. Thus, due to the fixed cost per unit, the range of the prices changes by the changes in the production volume, so, the total cost per unit will be also varying, depending on the production volume.

## Functions of Two or More Independent Variables

These functions are used for the calculations of the expected values and the principles of the cost changes, based on the valuations of the costs, supply and demand levels. Thus, the discrete random variables in these equivalents to the probability calculated sum of the probable range of variables. By McGuigan, Moyer, and Harris (2001, p. 234) should be emphasized the following statement:

For continuous random variables with a density function, it is the probability density-weighted integral of the possible values. The term “expected value” can be misleading. It must not be confused with the “most probable value.” The expected value is in general, not a typical value that the random variable can take on. It is often helpful to interpret the expected value of a random variable as the long-run average value of the variable over many independent repetitions of an experiment.

In the light of this perspective, it should be stated that the empirical estimation of the random variable of the cost, supply, and demand values. The repeated observations of the economic values are considered from the position of the arithmetic mean calculations, and, it should be processed of the optimization of the value, and the necessities of minimizing the sum of the squares and the residuals of the unbiased calculation manner. Nevertheless, as McGuigan, Moyer, and Harris (2001, p. 261) emphasize:

If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller.

Moreover, the functions with independent variables may be used for the calculation of the regression, and analysis of the regression rates, and the necessity to calculate the regression of the price variables. Thus, the regression analysis based on the functional analysis is generally performed with the estimated coefficients of an Operated Limited Static regression. To go on with the functions of the independent variables, McGuigan, Moyer, and Harris (2001, p. 274) made the following assumption:

The amount, which is selected for the analysis, depends not only on the supply and demand factors but also on the income and the flexibility factors, which are defined by the investments in the stock market and the activities of the consumers. Originally, three variables are used: the average income of the consumers, the investment income (which is an independent variable), and the sums, which consumers are ready to spend.

In the light of this perspective the functions of the independent variables, which are regarded from the perspective of graphic representation of the calculations, should be joined with the linear functions, similarly with the curvilinear functions.

## Conclusion

Finally, it should be stated that all the analyzed functions, which are used in economic studies, are generally aimed at the representation of the economic tendencies and principles, in a graphic manner. Consequently, the visual information may be used for deeper analysis. The fact is that any separate function can not be applied for the deep and thorough analysis, while the complex and multi-angled approach will help to achieve the required results. Thus, the functions may be used for representing the valuable image of the economic perspective and math tools for calculating this perspective.

## References

Bowley, A. L. (2003). The Mathematical Groundwork of Economics: An Introductory Treatise. Oxford: Clarendon Press.

Guillebaud, C. W., & Marshall, A. (2006). Principles of Economics (Vol. 6). London: Macmillan for the Royal Economic Society.

McGuigan, J.R., Moyer, R.C., Harris, F. H., (2001)”Managerial Economics: Applications, Strategy and Tactics”

Psalidopoulos, M. (Ed.). (2004). The Canon in the History of Economics: Critical Essays. London: Routledge.

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