 # Mathematics for Economics

Michael Hoy
John Livernois
Chris McKenna
Ray Rees
Thanasis Stengos
Pages: 976
https://www.jstor.org/stable/j.ctt5hhc2f

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xiv)
4. ### Part I Introduction and Fundamentals

• Chapter 1 Introduction
(pp. 3-10)

Almost for as long as economics has existed as a subject of study, mathematics has played a part in both the exploration and the exposition of economic ideas.¹ It is not simply that many economic concepts arequantifiable(examples include prices, quantities of goods, volume of money) but also that mathematics enables us to explore relationships among these quantities. These relationships are explored in the context ofeconomic models, and how such models are developed is one of the key themes of this book. Mathematics possesses the accuracy, the rigor, and the capacity to deal clearly with complex systems, which...

• Chapter 2 Review of Fundamentals
(pp. 11-60)

In this chapter we give a concise overview of some fundamental concepts that underlie everything we do in the rest of the book.

In section 2.1 we present the basic elements of set theory. We then go on to discuss the various kinds of numbers, ending with a concise treatment of the properties of real numbers, and the dimensions of economic variables. We then introduce the idea of point sets, beginning with the simplest case of intervals of the real line, and define their most important properties from the point of view of economics: closedness, boundedness, and convexity. Next we...

• Chapter 3 Sequences, Series, and Limits
(pp. 61-100)

Studying sequences and series is the best way to gain intuition about the rather perplexing notions of arbitrarily large numbers (infinity) and infinitesimally small (but nonzero) numbers. We gain such understanding by using the idea of the limit of a sequence of numbers. Thus, from a mathematical perspective, this chapter provides very useful background to the important property of continuity of a function, which we will explore fully in chapter 4. There are also some interesting economic applications of series and sequences, in particular the notion of discounting a future stream of payments or receipts, which is a critical aspect...

5. ### Part II Univariate Calculus and Optimization

• Chapter 4 Continuity of Functions
(pp. 103-126)

The idea of continuity of a function is extremely important in mathematics. Many convenient techniques of analysis can be used if a function is continuous but not if it is discontinuous. In modeling economic problems, we often assume that we can represent various economic concepts by continuous functions (e.g., the relationship between the quantity of some commodity produced by the firm and its profit level). Thus it is important to know precisely what is the content of this assumption, especially since in many instances there is a natural reason to believe that the function willnotbe continuous everywhere, and...

• Chapter 5 The Derivative and Differential for Functions of One Variable
(pp. 127-194)

The purpose of the derivative is to express in a convenient way how a change in the level of one variable (e.g.,x) determines a change in the level of another variable (e.g.,y). Much of economics is in fact concerned with just this sort of analysis. For example, we study how a change in a firm’s output level affects its costs and how a change in a country’s money supply affects the rate of inflation. Although expressing the relationship betweenxandyas a functiony=f(x) does capture this idea implicitly, it is much more convenient...

• Chapter 6 Optimization of Functions of One Variable
(pp. 195-232)

Many economic models are based on the idea that an individual decision maker makes anoptimal choicefrom some given set of alternatives. To formalize this idea, we interpret optimal choice as maximizing or minimizing the value of some function. For example, a firm is assumed to minimize costs of producing each level of output and to maximize profit; a consumer to maximize utility; a policy maker to maximize welfare or the value of national output; and so on. It follows that the mathematics of optimization is of central importance in economics, and in this and chapters 12 and 13...

6. ### Part III Linear Algebra

• Chapter 7 Systems of Linear Equations
(pp. 235-266)

In chapter 2 we defined alinear functionas one that takes the form$y=a+bx \caption {(7.1)}$for known constantsaandb, and wherexis theindependent variablethat takes on values over some specified domain, andyis the resulting value of the function at eachx-value. We also know that by taking specific values ofx, we can draw the graph ofxandyin a two-dimensional picture. The graph is a straight line: hence the phraselinear function. There are many examples of functions in economics that can be represented in a linear form. The market...

• Chapter 8 Matrices
(pp. 267-300)

A matrix provides a very powerful way of organizing and manipulating data. In chapter 7 matrices were used to focus attention on the parameters and constants of a simultaneous-equation system. The rows of a matrix could be manipulated to find solutions to the unknown variables in the original equations using, for example, the Gauss-Jordan elimination approach. There are clearly many instances where a large amount of information can be summarized in matrix form. Moreover there are procedures or operations on matrices that allow us to discover important properties of systems of equations.

As with any area of mathematics, there are...

• Chapter 9 Determinants and the Inverse Matrix
(pp. 301-346)

In chapter 8 we defined the operations of addition, subtraction, and multiplication of matrices. What about division? Can we define rules for dividing one matrix by another? The answer is yes, but only under certain restrictions. Division is restricted only tosquarematrices, and then only to those square matrices that satisfy a condition known asnonsingularity. The reason for all this can again be traced to the relation between matrix algebra and the problem of solving a system of simultaneous linear equations.

Consider, first, the division of two numbers. If we dividebintoa, we can write this...

• Chapter 10 Some Advanced Topics in Linear Algebra
(pp. 347-390)

In this chapter we consider three important advanced topics in matrix algebra: vector spaces, eigenvalues, and quadratic forms. All play important roles in a variety of contexts in economic theory and in econometrics. Vector spaces enable us to talk about distance between points, and linear dependence between vectors. They are therefore closely linked to the study of systems of linear equations of chapter 7. Eigenvalues play an important role in determining the stability properties of dynamic, linear systems and so this topic is of use in chapters 18, 20, 21, 23, and 24. Quadratic forms have applications in econometrics, and...

7. ### Part IV Multivariate Calculus

• Chapter 11 Calculus for Functions of n-Variables
(pp. 393-472)

We have already discussed at length the basic principles of calculus for functions of one variable,y=f(x) with$x\in \mathbb{R}$. Continuity was presented in chapter 4, and the derivative was presented in chapter 5. Economic analysis, however, often demands consideration of functions of more than one variable. For example, it is often important to model how the level of output produced by a firm depends on several inputs rather than just one. In this chapter we consider the fundamental relationships of differential calculus for functions of more than one variable. Fortunately, much of what was learned in chapters...

• Chapter 12 Optimization of Functions of n-Variables
(pp. 473-502)

The idea of optimization is fundamental in economics, and the mathematical methods of optimization underlie most economic models. For example, the theory of demand is based on the model of a consumer who chooses the best (“most preferred”) bundle of goods from the set of affordable bundles. The theory of supply is based on the model of a firm choosing inputs in such a way as to minimize the cost of producing any given level of output, and then choosing output to maximize profit. Rationality and optimization are virtually synonymous in economics.

In a formal sense, by optimization we mean...

• Chapter 13 Constrained Optimization
(pp. 503-528)

If, when maximizing or minimizing a function, we are free to consider any value of anx-variable on the real line as a possible solution, then the problem is said to be unconstrained. Most of the techniques developed in chapters 6 and 12 related to this case. In many, probably most, economic problems, however, there exist one or moreconstraintswhich restrict the set ofx-values we are allowed to consider as possible solutions. We already examined one type of constraint in chapters 6 and 12, namely that wherex-values are restricted to lie in some interval. The examples we...

• Chapter 14 Comparative Statics
(pp. 529-566)

As we discussed in chapter 1, economic models have two types of variables: endogenous variables, whose values the model is designed to explain, and exogenous variables, whose values are taken as given from outside the model. The solution values we obtain for the endogenous variables will typically depend on the values of the exogenous variables, and a central part of the analysis will often be to show how the solution values of the endogenous variables change with changes in the exogenous variables. This is the problem of comparative-static equilibrium analysis or comparative statics.

In the first section of this chapter...

• Chapter 15 Concave Programming and the Kuhn-Tucker Conditions
(pp. 567-582)

In the constrained optimization problems of chapter 13, we used the case where the function constraints are alwaysequalities. This is usually referred to as the “classical optimization problem.” However, sometimes this is not the most sensible formulation of a problem from the point of view of economics, and problems can arise that require us to set the constraints asweak inequalities. In this chapter we develop the necessary conditions for solutions of this type of problem. Because it is assumed that the objective and constraint functions are all concave, it is generally referred to as theconcave-programming problem.

We...

8. ### Part V Integration and Dynamic Methods

• Chapter 16 Integration
(pp. 585-632)

In this chapter, we address the question of whether knowing the derivative of a function,${f}'(x)$, allows one to determine, or recover, the original functionf(x). Since this process is the reverse of differentiation it is referred to as antidifferentiation, although it is also referred to as finding the indefinite integral. Related to this concept is the definite integral of a function, which is the area beneath a curve between two points. The process of integration is very useful in economics as it reflects the relationship between stocks and flows (e.g., investment and capital stock) and marginal and total...

• Chapter 17 An Introduction to Mathematics for Economic Dynamics
(pp. 633-642)

Economic dynamics is a study of how economic variables evolve over time. Unlike economic statics, which is a study of economic systems at rest, the focus of attention in economic dynamics is on how economic systems change as they move from one position of rest (i.e., equilibrium) to another. In this sense, economic dynamics, in adding the dimension of time to economic models, goes a step beyond economic statics. Often, however, this added realism and complexity can be managed only by reducing the complexity of the economic model in some other direction.

Once we introduce time to economic models, we...

• Chapter 18 Linear, First-Order Difference Equations
(pp. 643-664)

In the next three chapters we introduce some elementary techniques for solving and analyzing the kinds of difference equations that are common in economics. We begin in this chapter with linear, first-order difference equations. In the next chapter we introduce nonlinear, first-order difference equations, including the famous logistic equation used extensively in the study ofchaos. In chapter 20 we examine linear, second-order difference equations.

In this section we explain how to solve linear, first-order difference equations that are autonomous.

Definition 18.1 The general form of the linear, first-order, autonomous difference equation is given by${{y}_{t+1}}=a{{y}_{t}}+b,\quad \ t=0,1,2,\ldots \caption {(18.1)}$whereaandb...

• Chapter 19 Nonlinear, First-Order Difference Equations
(pp. 665-680)

In the previous chapter we saw that linear, first-order difference equations can be solved explicitly. We will see in the next chapter that this is also true for linear, second-order difference equations.Nonlineardifference equations, on the other hand, cannot be solved explicitly in general. However, it is still possible to obtain qualitative information about the solution by analyzing the nonlinear difference equation with the aid of a phase diagram. This technique can be very useful in economics because we are often mainly concerned with the qualitative properties of dynamic models. In this chapter we do this analysis for first-order...

• Chapter 20 Linear, Second-Order Difference Equations
(pp. 681-714)

In this chapter we turn to linear difference equations of the second order. We focus our attention on theautonomouscase in section 20.1 and consider a special nonautonomous case in section 20.2. In addition we introduce a new solution technique in this chapter. The technique involves breaking up the relatively difficult problem of finding the general solution to the difference equation into two parts, each of which is easier to solve than the whole. Not only does this simplify matters in this chapter, but it proves to be indispensable in later chapters in solving differential equations, and systems of...

• Chapter 21 Linear, First-Order Differential Equations
(pp. 715-738)

In the next three chapters we explain techniques for solving and analyzing ordinary differential equations. We do not attempt to provide exhaustive coverage of the topic but instead focus on the types of differential equations and techniques of analysis that are most common in economics. We begin in this chapter with linear, first-order differential equations. In the next chapter we turn to an examination of nonlinear, first-order differential equations, and in the chapter after that we examine linear, second-order differential equations. In this chapter and throughout, we will solve a large number of examples and economic applications to illustrate the...

• Chapter 22 Nonlinear, First-Order Differential Equations
(pp. 739-752)

In chapter 21 we saw that we could apply a single solution technique to solve any first-order differential equation that islinear. When the differential equation isnonlinear, however, no single solution technique will work in all cases. In fact only a few special classes of nonlinear, first-order differential equations can be solved at all. We will examine two of the more common classes in section 22.2. Even though solutions are known to exist for any nonlinear differential equation of the first order that satisfies certain continuity restrictions, it is simply not possible to find that solution in many cases....

• Chapter 23 Linear, Second-Order Differential Equations
(pp. 753-780)

Until now we have confined our analysis of differential equations to those of the first order. In this chapter we will examine linear, second-order differential equations with constant coefficients. We focus our attention on theautonomouscase in section 23.1 and consider a specialnonautonomouscase in section 23.2.

We begin by explaining how to solve a linear, autonomous, second-order differential equation.

Definition 23.1 The linear, autonomous, second-order differential equation (constant coefficients and a constant term) is expressed as$\ddot{y}+{{a}_{1}}\dot{y}+{{a}_{2}}y=b \caption {(23.1)}$

Equation (23.1) is linear becausey,, andÿare not raised to any power other than one. It is...

• Chapter 24 Simultaneous Systems of Differential and Difference Equations
(pp. 781-844)

It is common in economic models for two or more variables to be determined simultaneously. When the model is dynamic and involves two or more variables, asystemof differential or difference equations arises. The purpose of this chapter is to extend our single equation techniques to solve systems of autonomous differential and difference equations.

We begin with the simplest case—a system of two linear differential equations—and solve it using the substitution method. We then proceed to a more general method, known as the direct method, that can be used to solve a system of linear differential equations...

• Chapter 25 Optimal Control Theory
(pp. 845-920)

In this chapter we take up the problem of optimization over time. Such problems are common in economics. For example, in the theory of investment, firms are assumed to choose the time path of investment expenditures to maximize the (discounted) sum of profits over time. In the theory of savings, individuals are assumed to choose the time path of consumption and saving that maximizes the (discounted) sum of lifetime utility. These are examples of dynamic optimization problems. In this chapter, we study a new technique, optimal control theory, which is used to solve dynamic optimization problems.

It is fundamental in...