"Preface...

To the First Ed

Economic theory has changed significantly its character in the present century. There have been four major revolutions in Economics : (i) Mathematisation of Economics, (ii) Dynamisation of Economics, (iii) Estimation, and (iv) Programming.

Mathematisation of Economics : Economics is basically quantitative and involves relationships among variables. This necessitates as well as facilitates the use of mathematics in Economics. However disdainful and painful it may be to learn mathematical techniques, one cannot dispense with the use of such techniques in this science. Increased mathematisation of Economics has led to completely new branches of Economics such as Mathematical Economics and Econometrics. Strictly speaking, there is no fundamental difference between Mathematical Economics and Economic Theory. Mathematical Economics puts the literary form of economic theory in mathematical language—the short-hand of logic. In doing so it provides concrete form to economic laws and relationships, and makes it more precise and practical. More than this, use of mathematics helps in systematic understanding of the relationships and in derivation of certain results which would either be impossible through verbal logic or would involve clumsy, complex and circular process.

Illustrations of the use of mathematics in Economics are not far to seek.

Economics is the science of choice-making. The consumer makes his choice given his tastes, income, and prices, other things remaining the same. Preference scales and indifference curves can then be constructed. But all these involve use of mathematics, ranging from simple Geometry to Calculus. Consumer’s equilibrium is studied in the context of iven income and prices, but ‘Equilibrium’ itself is a mathematical concept derived from Statics and Dynamics.

Maximisation of profits or minimisation of costs by a firm under some constraints is the basic problem discussed in the theory of firm. This problem can also be solved with the help of Differential Calculus, Linear Programming or Theory of Games.
Consumer’s equilibrium and equilibrium of the firm involve decisions at the margin and marginal analysis is nothing but extension of differential calculus. We also know that price change on account of output change depends upon elasticity of demand and supply and ‘Elasticity’ is, in fact, a mathematical concept.

Basic relationships in Micro and Macro-Economics can usually be put in the form of functions and equations : Quadratic, Simultaneous, Differential and Difference. For example, in Growth Economics where the time-paths of national income are to be found out (depending on some crucial factors) difference and differential equations are indispensable. In planning models, sectoral targets are fixed only with the help of Input-Output Analysis and Linear Programming. Determinants and Matrix Algebra are of immense use in such techniques.

Thus in almost all fields of economics, mathematics is useful. Economic phenomena are so complex and interdependent that some abstractions— ceteris paribus assumptions — have to be made. It is often not possible for a student of Economics to understand the effect of many factors simultaneously on particular economic phenomenon. Mathematics is especially helpful in these situations. A simplified model is first built wherein relationship between the different variables is established by making assumptions and keeping many factors constant. Thereafter, the ceteris paribus assumptions are relaxed one by one and the effects analysed.

The proof of pudding lies in eating. After going through this book the reader will find out that how easy, smooth and systematic it becomes understand the economic processes and derivation of various results by learning a little of mathematics. For example, the cobweb theorem and the stability conditions can be derived and understood very neatly and briefly in not more than two pages. This cannot be possible without the use of the tools which have been used in this volume.

Dynamisation of Economics : Traditional economics explains the situation at a point of time, whereas actual economy witnesses process of change over time. Hence ‘time’ must be accounted for and, therefore, dynamisation a step towards reality. Growth economics is an outgrowth of dynamisation of Economics.

Estimation : Since economics does not deal with pure theorizing alone but also with the relevance of theory in practice, there must be some methods of testing the correspondence between theory and reality. And here comes new branch of knowledge, i.e., Econometrics, to our rescue. In Econometrics we use mathematics to specify the relationships as given by economic theory and apply tools of statistics and theory of probability to test theorems of economic theory, estimate parameters and then predict the results. In the words of J. Tinbergen, Econometrics is mathematical economics working with measured data to give empirical content to economic theory.

Programming Research : Problems of maximisation and minimisation are basic to Economics. For example, we have to minimise cost of transportation of goods from different godowns to different markets. But against this, maximum capacity of each godown is fixed so also the maximum demand in each of the markets. These are termed as constraints or side conditions. Such problems are solved by programming techniques making economics a complete practical science.

All these revolutions thus involve increasing use of mathematics.

A. Chiang in his book, “Fundamental Methods of Mathematical Economics”, rightly considers the mathematical approach as a quick mode of transportation from a set of postulates to a set of conclusions. For example, if a person intends to go to a place five miles away, he will definitely prefer driving to walking if he possesses a car unless he has time to kill or wants to exercise his legs. Similarly, a theorist who wishes to get to his conclusions more quickly will find it convenient to drive the vehicle of mathematical techniques appropriate for his particular purpose. But all the same he must know the driving and for this driving lessons are essential. In other words, the theorist must accustom himself with mathematical techniqes and their applications which are driving lessons to elucidate the problems of economic theory.

One must also be on guard lest the use of mathematics should turn into a disadvantage. Mathematics, like fire, is a bad master although a good servant. It is also an art and naturally one gets fascinated by its aesthetic values after perfecting its techniques. At times, therefore, addiction to mathematics tempts us to make inappropriate economic assumptions and limit ourselves only to the problems that can be solved mathematically. In this connection we would like to reproduce here the conversation between Prof. Samuelson and his student.

Student: “I am interested in economic theory. I know little mathematics. And when I look at the journals I am greatly troubled. Must I give up hope of being a theorist? Must I learn mathematics?”

Samuelson : “Some of the most distinguished economic theorists, past I and present, havebeen innocent of mathematics. Some of the most distinguished theorists have known some mathematics. Obviously you can become a great theorist without knowing mathematics. Yet it is fair to say that you will have to be that much more clever and brilliant ... Mathematics is neither necessary nor a sufficient condition for a fruitful career in economic theory. It can help .... ”

Thus use of mathematics can only better good economics; it can never compensate for bad economics.

The book is specifically meant for use by students and teachers in India who have got a rudimentary knowledge of mathematics and as such it is presumed that the readers are acquainted only with elementary Algebra and Geometry.

In the first part we have introduced the highly useful Algebra of Matrices and extended the discussion to cover Input-Output Analysis and Linear Programming. In the second part, Differential Calculus — the Calculus of related changes — is given in great detail with separate chapters dealing with applications in the various branches of micro-economic theory. In the third part, Integral Calculus, the technique of deriving the macro from the micro analysis, has been dealt with. Differential and Difference equations are covered in the last four chapters with illustrations from macro-economic theory and the theory of economic growth.

Though we have tried to make the book self-contained, we do not claim that we have achieved perfection. It is almost impossible to compress the entire tool-box provided by mathematics in a single volume intended to be studied in a single academic session. The purpose of this work, then, is just to give an introduction to basic mathematics capable of providing tools for use in economic theory and Econometrics.

This is not a book on economic theory. Hence understanding of economic theory is presumed. Only those who have cared to study their economics well will be benefited by this book;it is in no way a substitute for books on economic theory.

We will be glad to receive suggestions for improvement of the volume from those who may use the book as students or as teachers.

JAIPUR
July 25, 1973

MEHTA MADNANI"

--- Mathematics for Economists, Mehta and Madnani (8th Ed., 1997)


Also, these 2 notes opposite the first page of the preface are amusing:

"There is no substitute for hard work. You must supplement reading by practising questions.

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