**Mathematical Economics ** Case Solution

### Abstract

Taking under consideration the growing importance of mathematics in the field of economics; it is analyzed that mathematics permits the economists to create models and conduct quantifiable to test the future economic activity. In economics, the types of mathematics in economics include: statistics, calculus and algebra. The significant advancements in the big data techniques, computing power and other advanced mathematics applications have played significant roles in making quantifiable methods a standard economics element.

The differential calculus is based on some fundamental concepts of the real numbers; limits, functions and continuity, which are highly significant concepts of mathematics, which have been formulated and had their modern content assigned during mathematical analysis development and during the studies of its foundation. The linear model is the form of the mathematical models, which helps in understanding as well as exploring the meaning of the functional relationships and equations.

In addition, the mathematical optimization tends to provide the self-constrained introduction to as well as the control techniques and survey of mathematical programming and their application to dynamic and static problems in economics. The wider application of the theory of game in economics ranges from analysis of economic competition to the economic phenomenon, such as: voting theory, bargaining mechanism design, political economy, behavioral economics and experimental economics. The econometrics modeling could drive the business ahead, helping in optimization of cost, better or smarter decision making and better understanding of the customer demand in the market. The matrix algebra involves various problems that tend to demonstrate the foundation expressions of matrix algebra as well as the illustration of using algebra for a number of economic problems and issues.

### Introduction

Economics is the social science that does not only explain what goes on in economy of cone country but also describes how the economy operates and makes predictions related to what might happen to the specific variables of economy if certain changes take place. Thus, the use and application of mathematics are fundamental to any serious application of economies. The economics file is riddled with mathematics applications and equations to a greater extent. In economics, the types of mathematics include: statistics, calculus and algebra.

Algebra is widely used as a mean of making computations, such as: the total revenue and total cost; whereas, the calculus is used in order to find the growth models, profit maximization curves as well as utility curves. The Mathematical economics is the economics model that utilizes the principles and methods of Math, to create economic theories as well as investigates the economic quandaries. It is pertinent to note that the mathematics permits the economists to create model and conduct quantifiable test in order to predict the economic activity in near future. Following are some mathematical representation of the economic problems:

### Differential calculus

In mathematics, differential calculus is used to find the rate of change of quantity with respect to other. The differential calculus makes it possible to easily compute the function’s limit in many cases. Differentiation is one of the methods that is critically used to find the slope of the function at any point. The basis differentiation techniques are easy to apply, and are quite straightforward. In order to drive the expression for slope of y=6x^2 function for any value of x the basis differentiation rules require you to:

- Multiply the whole term by the value of x power, and
- Subtract 1 from the x power.

With the use of the aforementioned rule, the slope of the function becomes;

2 * 6x^{2-1 }= 12x

This is called y derivative with respect to x and is written dy/dx.

y = 6x^{2}

216

54

24

6

0 1 2 3 4 5 6 x

The graph of the y=6x^2 function shows that any term in x^{2 }would be increasing at an ever increasing rate as x is increased. In other words, the slope of the function must rise as x increases. The actual value of the slope could be estimated for any given value of x by entering the given value of x into the formula of Slope = 12x.

Consequently, the differential calculus is based on some fundamental concepts of the real numbers; limits, functions and continuity, which are highly significant concepts of mathematics that have been formulated and assigned with their modern content during the mathematical analysis development as well as during the studies of its foundation.(Rosser, 2018).

### Linear models

The linear-equations are of paramount importance in the file of mathematics because they could be easily used to model any real world phenomenon,involving variable. The linear model is one form of the mathematical models that helps in understanding as well as exploring the meaning of the functional-relationships and equations. The form of the linear function is as follows:

**y = f(x) = a + bx**

The graph shows negative, positive and zero relationship between dependent and independent-variables, which are illustrated below:

Negative Zero Positive

The equation of the straight also referred to mathematical model of deterministic nature. In other words, the linear regression analysis is considered as the predictive tool for modeling the relationship between the variable with the set of the independent variable. The applications of the linear mathematical models are as follows:

- To model the casual relationship between parameters; the least square regression is used.
- The linear regression is used in customer survey results analysis and market research studies.(Chakure, 2019).

Thus, the application of the linear mathematical models in many-areas, helps in understanding& predicting the behavior of complicated and complex systems or analyzing the financial, experimental and biological data.(Krukrubo, 2019). Additionally, the linear algebra is the mathematics of modern technological world of complicated multi-variable systems. The linear-trans formation and the vector spaces are central themes of much of mathematics.(Tucker, 1993).

### Mathematical-optimization

In economics, the mathematical optimization tends to provide the self-constrained introduction and it controls the techniques and surveys of mathematical programming and their application to dynamic and static problems in economics.(Intriligator, 2002 ).In addition to this, the mathematical optimization provides statistical foundation of an efficient usage as well as the utilization of the resources, and the applicability of the optimization theory to economics stems from its effectiveness in solving the problems of economic allocations. Not only this, it also helps in solving calculus optimization-problems, which include:optimizing to both reduce the cost and maximize the revenue or to maximize the profit returns…………………

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