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2023 | Buch

Elements of Mathematics for Economics and Finance

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Über dieses Buch

Based on over 15 years’ experience in the design and delivery of successful first-year courses, this book equips undergraduates with the mathematical skills required for degree courses in economics, finance, management, and business studies. The book starts with a summary of basic skills and takes its readers as far as constrained optimisation helping them to become confident and competent in the use of mathematical tools and techniques that can be applied to a range of problems in economics and finance.

Designed as both a course text and a handbook, the book assumes little prior mathematical knowledge beyond elementary algebra and is therefore suitable for students returning to mathematics after a long break. The fundamental ideas are described in the simplest mathematical terms, highlighting threads of common mathematical theory in the various topics.

Features of the book include:

a systematic approach: ideas are touched upon, introduced gradually and then consolidated through the use of illustrative examples; several entry points to accommodate differing mathematical backgrounds; numerous problems, with full solutions, and exercises to illustrate the theory and applications; key learning objectives and self-assessment questions provided for each chapter; full solutions to exercises, available to lecturers via the web.

Vass Mavron is Emeritus Professor of Mathematics at Aberystwyth University. Tim Phillips is Professor of Applied Mathematics in the School of Mathematics at Cardiff University.

Inhaltsverzeichnis

Frontmatter
1. Essential Skills
Abstract
Many models and problems in modern economics and finance can be expressed using the language of mathematics and analysed using mathematical techniques. This book introduces, explains, and applies the basic quantitative methods that form an essential foundation for many undergraduate courses in economics and finance. The aim throughout this book is to show how a range of important mathematical techniques work and how they can be used to explore and understand the structure of economic models.
Vassilis C. Mavron, Timothy N. Phillips
2. Linear Equations
Abstract
In this book, we will be concerned primarily with the analysis of the relationship between two or more variables. For example, we will be interested in the relationship between economic entities or variables such as
  • total cost and output,
  • price and quantity in an analysis of demand and supply,
  • production and factors of production such as labour and capital.
Vassilis C. Mavron, Timothy N. Phillips
3. Quadratic Equations
Abstract
Linear equations and methods for their solution were introduced in the previous chapter. As we have seen, the graphs of linear functions are straight lines and therefore their slopes are constant. This means that the function changes by a constant amount whenever the dependent variable changes by the same fixed value. This type of behaviour is not always observed in real-life applications in economics. It is, therefore, necessary to introduce an added level of sophistication to the mathematical modelling.
Vassilis C. Mavron, Timothy N. Phillips
4. Functions of a Single Variable
Abstract
The concept of a function is fundamental to many of the applications that we will encounter in economics. As we have already seen in Chaps. 2 and 3, it is a convenient way of expressing a relationship between two variables in terms of a prescribed mathematical rule. More formally, we have the following definition.
Vassilis C. Mavron, Timothy N. Phillips
5. The Exponential and Logarithmic Functions
Abstract
An important class of nonlinear functions that is of particular interest in economics comprises the exponential and logarithmic functions. These functions are useful for investigating problems associated with economic growth and decay and mathematical problems in finance such as the compounding of interest on an investment or the depreciation of an asset. For example, if a person invests £3000 in an investment bond for which there is a guaranteed annual rate of interest of 5% for 2 years, the evaluation of an exponential function will provide the return at the end of that period. If a credit card company charges interest on an outstanding balance, the evaluation of an exponential function will provide information on the AER (annual equivalent rate).
Vassilis C. Mavron, Timothy N. Phillips
6. Differentiation
Abstract
Economists are interested in the effects of change. Therefore, the concept of the derivative of a function, which provides information about how a function changes in response to changes in the independent variable, is an important one in economic analysis. For example, the derivative of a production function provides information about the manner in which the output of a production process changes as the number of workers employed by the company changes. Differentiation is the mathematical tool that allows us to quantify such rates of change. As we will see in Chap. 7, differentiation is also an important tool in the determination of the maximum or minimum values of economic functions such as profit and cost.
Vassilis C. Mavron, Timothy N. Phillips
7. Maxima and Minima
Abstract
In this book, the concept of the derivative of a function has been introduced, and its application in economics has been described. However, the primary use of the derivative in economic analysis is related to the process of optimization. Optimization is defined to be the process of determining the local or relative maximum or minimum of a function.
Vassilis C. Mavron, Timothy N. Phillips
8. Partial Differentiation
Abstract
Economic models that we have encountered so far have assumed that a quantity under consideration depends only on the value of one variable; i.e., the quantity is a function of one variable. For example, \(Q=100-5P,\) the demand equation (or demand function) for some good describes a model where the demand Q depends only on the price P of the good. In practice, Q will depend on other variables such as consumer income or the price of a substitutable good. To take into account all variables affecting the value of Q would make an economic model too difficult to analyse or use. Useful models should lend themselves readily to analysis, perhaps with the aid of computers, while at the same time give a reasonably accurate model of the real situation.
Vassilis C. Mavron, Timothy N. Phillips
9. Optimization
Abstract
Optimization is a concept of prime importance in economic analysis. Companies endeavour to maximize profit and minimize costs. Governments hope to minimize unemployment and inflation while maximizing tax revenue. Consumers are assumed to want to obtain maximum utility (satisfaction or benefit) from their consumption of particular products.
Vassilis C. Mavron, Timothy N. Phillips
10. Matrices and Determinants
Abstract
Matrix theory is a powerful mathematical tool for dealing with data as a whole rather than the individual items of data. Matrices are especially useful in the theory of equations. They can be used to solve systems of simultaneous linear equations. Determinants are related to matrices and are useful for determining whether or not a unique solution exists. In some cases, using determinants, the solution for each unknown can be expressed explicitly in terms of the coefficients of the equations by applying what is known as Cramer’s rule. Systems of simultaneous linear equations occur, for example, when optimizing a function using Lagrange multipliers or when trying to find the equilibrium prices of interdependent commodities. As we shall see, matrices can be added and in some cases multiplied together. In economics, business, and finance, many basic theoretical models are linear in that they are described in some way by linear functions. Analyzing these models is made simpler by matrix algebra.
Vassilis C. Mavron, Timothy N. Phillips
11. Integration
Abstract
Differentiating a function \(f(x)\) gives its derivative \(f'(x)\), which is also a function of x. Geometrically, we can view \(f'(x)\) as giving the slope of the tangent at any point on the graph of \(y=f(x)\) or, equivalently, the rate of change of \(f(x)\) with respect to x at that point.
Vassilis C. Mavron, Timothy N. Phillips
12. Linear Difference Equations
Abstract
Problems encountered so far have mostly been static in that the quantities and equations involved are for a particular period of time. For instance, the current price of a good depends on the current demand of consumers.
Vassilis C. Mavron, Timothy N. Phillips
13. Differential Equations
Abstract
There are close similarities between the theories of linear difference equations and linear differential equations. Indeed, differential equations may be regarded as the continuous analogues of difference equations where the variable quantity, such as time, is assumed to flow continuously rather than occurring in discrete intervals.
Vassilis C. Mavron, Timothy N. Phillips
Backmatter
Metadaten
Titel
Elements of Mathematics for Economics and Finance
verfasst von
Vassilis C. Mavron
Timothy N. Phillips
Copyright-Jahr
2023
Electronic ISBN
978-3-031-43910-0
Print ISBN
978-3-031-43909-4
DOI
https://doi.org/10.1007/978-3-031-43910-0