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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.

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 — and , — and in an analysis of demand and supply, — and such as and . If one variable, say , changes in an entirely predictable way in terms of another variable, say , then, under certain conditions (to be defined precisely in Chapter 4), we say that is a function of . A function provides a rule for providing values of given values of . The simplest function that relates two or more variables is a linear function. In the case of two variables, the linear function takes the form of the linear equation = + for ≠ 0. For example, = 3 + 5 is an example of a linear function. Given a value of , one can determine the corresponding value of y using this functional relationship. For instance, when = 2, = 3 × 2 + 5 = 11 and when = −3, = 3 × (−3) + 5 = −4. We will say more about functions in Chapter 4. Linear equations or functions may be portrayed by a straight line on a graph. In this chapter, we introduce graphs and give a number of examples showing how linear equations can be used to model situations in economics and how to interpret properties of their graphs.

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. This is achieved through the introduction of nonlinear functions. The simplest nonlinear function is the quadratic function. This function takes the general form () = + + , (3.1) where ≠ = 0, b and c are constants. The condition ≠ = 0 is to prevent the occurrence of the degenerate case in which (3.1) reduces to a linear function.

The concept of a function is fundamental to many of the applications that we will encounter in economics. As we have already seen in Chapters 2 and 3, it is a convenient way of expressing a relationship between two variables in terms of a prescribed mathematical rule.

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 £3,000 in an investment bond for which there is a guaranteed annual rate of interest of 5% for two 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). We begin this chapter by sketching the graphs of some exponential functions and highlighting some of their important properties. Exponential functions are functions in which a constant base a is raised to a variable exponent . The general form of an exponential function is given by = , where > 0 and ≠ 1. (5.1) The parameter a is known as the base of the exponential function. The independent variable x occurs as the exponent of the base.

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 Chapter 7, differentiation is also an important tool in the determination of the maximum or minimum values of economic functions such as profit and cost.

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.

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, = 100−5, the demand equation (or demand function) for some good describes a model where the demand depends only on the price of the good. In practice, 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 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.

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.

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.