ECO606 Updated Handouts

Lesson 01 INTRODUCTION TO MATHEMATICAL ECONOMICS

TOPIC 001: DEMYSTIFYING MATHEMATICS AND MATHEMATICAL ECONOMICS

Who’s Afraid of Mathematics (The oldest friend)?

Unity of Allah and His Book in ‘Kalma’ teaches us Math. Inception of Life on Earth teaches us Math. Adam gave us 1st number. Eve gave us addition. Their children gave us multiplication. Total number of Prophets give us idea of large numbers.

Origin of Mathematics

From Latin ‘mathēmatica’ that stands for mathematics and ‘mathēmaticus’ stands for mathematical.

Age of Mathematics

“Mathematics is as old as man.” (Stefan Banach)

Universality of Mathematics

Film is one of the three universal languages, the other two: mathematics and music. (Frank Capra)

The study of mathematics, like the Nile, begins in minuteness but ends in magnificence. (Charles Caleb Colton)

“Pure mathematics is, in its way, the poetry of logical ideas.” (Albert Einstein)

Mathematics is the music of reason. (James Joseph Sylvester)

God used beautiful mathematics in creating the world. (Paul Dirac)

All science requires mathematics. The knowledge of mathematical things is almost innate in us. This is the easiest of sciences, a fact which is obvious in that no one's brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon. (Roger Bacon)

" Without mathematics, there's nothing you can do. Everything around you is mathematics. ‘Everything’ around you is numbers." (Shakuntala Devi)

Human beings have a certain number of eyes, ears, hands, arms, legs, feet. And there is also a certain number of skies i.e. seven skies.

Precision via Mathematics

My belief is that nothing that can be expressed by mathematics cannot be expressed by careful use of literary words. (Paul Samuelson)

Mathematics allows for no hypocrisy and no vagueness. (Stendhal) “A problem well stated is a problem half solved.” (Charles Franklin Kettering)

Applicability of Mathematics

The true mathematician is not a juggler of numbers, but of concepts. (I. Stewart (1975))

Mathematics as a Subject

Despite, its primitive origin, students are usually afraid of mathematics. Opinion of Shakuntala Devi (a.k.a. Human Computer), “Why do children dread mathematics? Because of the wrong approach, because it is looked at as a subject.” (Shakuntala Devi)

Good news for young people as they like adventures. “Mathematics is an adventurous and satisfying route to doing things.” Every mathematical problem starts with uncertainty of not having a definite solution but the solution, if obtained, gives a sense of fulfillment and the journey from the problem to the solution becomes an adventure.

A central figure in journey of scientific revolution is Galileo Galilei (a.k.a mathematical Platonist). He says, “If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.” “Don’t worry about your difficulties in mathematics. I can assure you mine are still greater”. (Albert Einstein)

I've always been interested in using mathematics to make the world work better. (Alvin E. Roth) This sums the learning and inspiration from the elders.

Mathematics for Economics vs. Mathematical Economics

No carving in the stone. However; Former is about ‘Comprehension’ of the mathematical tools applied to economic situations. [Algebra, matrices, differential calculus etc.] Whereas, latter deals with the ‘Application’ of mathematical tools to comprehend and solve of economic situations [e.g. application of algebra on market equilibrium, application of matrices on national income analysis, application of differentiation for marginal utility/cost/product etc].

Mathematical Economics: An Approach to Economics rather than its Branch

Can include problems from microeconomics, macroeconomics, public finance, international trade and other branches of economics. e.g. Equilibrium for demand and supply curves, calculation of taxation revenue, local price elasticity of foreign demand of exports, among others.

TOPIC 002: MATHEMATICS VS. NON-MATHEMATICAL ECONOMICS

Non-mathematical economics can also be termed as literary economics. Both remain approaches and should not fundamentally differ from each other. However, there are two noteworthy differences: Former uses symbols/equations to denote assumptions and conclusions instead of words/sentences.

An Assumption in Mathematical Economics

Rather stating an assumption that a consumer can consume whole of any increase in his income, or a part of it or none of it is mentioned in mathematical fashion as follows:

A Conclusion in Mathematical Economics

Rather stating a conclusion that an increase in private investment has brought about 3 times of increase in national income can be expressed in mathematical expression as follows:

∆𝒀= Change (increase) in national income. ∆𝑰 = Change (increase) in private investment.

Theorems in Mathematical Economics

Theorem: A rule in mathematics expressed in terms of symbols and formula. Envelope Theorem: Applied to producer theory and auction theory. Roy's Identity: Applied to Marshallian demand function. Shephard Lemma: Used to explain relationship between expenditure (or cost) functions and Hicksian demand functions.

Advantage(s) of Mathematical Economics

Mathematical economics utilizes the symbols that are more convenient to use in deductive reasoning. The use of symbols also increases conciseness and preciseness of statement. Use of mathematical functions allow to incorporate more than two economic variables in a situation without resorting to 3-dimensional or hyperspace graphs i.e. allows for ‘𝒏’ number of variables.

Practical Importance of Mathematical Economics

During current times, economics is said to be highly mathematized. Enables the comprehension of the professional articles one comes across in such periodicals as the American Economic Review, Quarterly Journal of Economics, Journal of Political Economy, Review of Economics and Statistics, and Economic Journal.

TOPIC 003: MATHEMATICAL ECONOMICS VERSUS ECONOMETRICS

Mathematical economics is often confused with econometrics. Etymology of Econometrics: A portmanteau of two words ‘Econo’ from economics and 'metrics’ which implies measurement. In Jargonized language: Econometrics is the study of empirical observations using statistical methods of estimation and hypothesis testing. Mathematical economics is, however, concerned with application of mathematical tools on economic theory without much concern to measurement of variables and errors in it. Thin line between a mathematical model and econometrics model: