CHAPTER ONE

1. INTRODUCTION
1.1. Mathematics as a Language for Economists
Economics is a social science. It does not just describe what goes on in the economy. It attempts
to explain how the economy operates and to make predictions about what may happen to
specified economic variables if certain changes take place; for example, what effect a crop
failure will have on crop prices, what effect a given increase in sales tax will have on the price of
finished goods, what will happen to unemployment if government expenditure is increased. It
also suggests some guidelines that firms, governments or other economic agents might follow if
they wished to allocate resources efficiently.
In introductory economic analysis predictions are often explained with the aid of sketch
diagrams. For example, supply and demand analysis predicts that in a competitive market if
supply is restricted then the price of a good will rise. However, this is really only common sense
as any market trader will tell you. An economist, further to this, also needs to be able to say by
how much price is expected to rise if supply contracts by a specified amount. This quantification
of economic predictions requires the use of mathematics. An economist without the tools of
mathematics is like a blind person swimming in the middle of ocean. Although non-
mathematical economic analysis may sometimes be useful for making qualitative predictions
(i.e. predicting the direction of any expected changes), it cannot by itself provide the
quantification that users of economic predictions require. A firm needs to know how much
quantity sold is expected to change in response to a price increase. The government wants to
know how much consumer demand will change if it increases a sales tax. Therefore,
mathematics is fundamental to any serious application of economics to these areas though it,
sometimes, believed that mathematics makes economics more complicated. Algebraic notation,
which is essentially a form of shorthand, can, however, make certain concepts much clearer to
understand than if they were set out in words. It can also save a great deal of time and effort in
writing out tedious verbal explanations. For example, the relationship between the quantity of
apples consumers wish to buy and the price of apples might be expressed as: ‘the quantity of
apples demanded in a given time period is 1,200kg when price is zero and then decreases by
10kg for every 1 Birr rise in the price of a kilo of apples. It is much easier, however, to express
this mathematically as: Qd =1200−10 p where Qd is the quantity of apples demanded in
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kilograms and p is the price in birr per kilogram of apples. This is a very simple example. The
relationships between economic variables can be much more complex and mathematical
formulation then becomes the only feasible method for dealing with the analysis. Because of
these, it is a language for economists.

In the other direction, as to the nature of mathematical economics, it should note that economics
is unique among the social sciences to deal more or less exclusively with metric concepts. Prices,
supply and demand quantities, incomes, employment rates, interest rates, whatever studied in
economics, are naturally quantitative metric concepts, where other social sciences need contrived
concepts in order to apply any quantitative analysis. So, if one believes in systematic relations
between metric concepts in economic theory, mathematics is a natural language in which to
express them.

1.2. Mathematical Economics: Basics and Purposes

The development of mathematical economics is not revolutionary step. It took several centuries
to develop the present stage of mathematical economics. Sir William Petty (1623-1687) is
believed to be the first participant in this field. He used the terms of symbols in his studies, but
he was not successful. The first successful attempt was made by an Italian, named Giovanni
Ceva (1647—1734). After these earlier developments, Antoine Augustin Cournot (1801-1877)
made use of symbols in his theory of wealth. After his work, Alfred Marshall in his “Principles
of Economics” (1890), and Irving Fisher in his Ph.D. thesis “Mathematical Investigations in The
Theory of Value and Prices” showed a great interest in mathematical formulation of the
economic theory. After their work, a race have been began in this field that everyone specialized
in mathematics and with less knowledge in the economic theory jumped in this new field and
more and more articles started publishing with excessive use of mathematics and lacking theory.
As a result, an economist of 19th century cannot even understand the economic journals of
present times.

It is almost as hard to define mathematics as it is to define economics. An easy definition of

economics is given by Jacob Viner, “Economics is what economists do”, so we can say that
mathematics is what mathematicians do. Mathematical economics is not an individual branch of

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economics in the sense that international trade, public finance, or urban economics, but it is an
approach to economic theory.

In mathematical economics, mathematical symbols and equation are used in the statement of the
problem. Since mathematical economics is just an approach to economic analysis, it must not
differ from the non-mathematical approach in the conclusion but we observe entirely opposite
situation and here the problems starts. The major difference between mathematical economics
and literary economics is that in the former, the assumptions and conclusions are described in
mathematical symbols and equations whereas, in the later, words and sentences are used to
achieve the desired goal. Because of these differences, however, it can only define mathematical
economics saying that it is the application of mathematical methods in economic theory.

The term “mathematical economics” is sometimes confused with a related term, “econometrics”.
As the “metric” part of the latter term implies, econometrics is concerned mainly with the
measurement of economic data. Hence it deals with the study of empirical observations using
statistical methods of estimation and hypothesis testing. Mathematical economics on the other
hand refers to the analysis, with little or no concern about such statistical problems as the errors
of measurement of the variable under study.
1.3. Economic Models
The term “Model” is very common in economics. In-fact “model” and “theory” are two different
names for the same thing: the former is simply less ostentatious. It can be defined as a set of
assumptions from which the conclusions can be drawn. In simple words we can say that model is
simply a representation of some aspects of the real world. Economic theory is descriptive as well
as analytical. It does not give us complete descriptions of economic phenomenon, but by making
certain assumptions, we can construct models. The models then help in representing reality and
help in understanding the characteristics of economic behavior. In-fact, a model cannot explain
everything, only a subset of everything; usually a model explains only a small subset of
everything. As a result, in building a model, one must begin by identifying the system of interest
(the system you what to explain).

In economic analysis, generally, four types of models are used; visual models, mathematical
models, empirical models, and simulation models. Visual models are simply pictures of an

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abstract economy; graphs with lines and curves that tell an economic story. Mathematical
Models are the most formal and abstract of the economic models are the purely mathematical.
These models are systems of simultaneous equations with an equal or greater number of
economic variables. Empirical models are mathematical models designed to be used with data.
The fundamental model is mathematical, exactly as described above. With an empirical model,
however, data is gathered for the variables, and using accepted statistical techniques, the data are
used to provide estimates of the model's values. For example, suppose in an economic study the
following question is asked: "What will happen to investment if income rises one percent?" The
purely mathematical model might only allow the analyst to say, "Logically, it should rise." The
user of the empirical model, on the other hand, using actual historical data for investment,
income, and the other variables in the model, might be able to say, "By my best estimate,
investment should rise by about two percent. Simulation models, where, are used with
computers, embody the very best features of mathematical models without requiring that the user
be proficient in mathematics.
Generally, mathematical economic model has three basic components: variable, parameters and
equation.

1. Variable: is simply a symbol which take any value (informally) and the value varies (not
fixed). In economics, it is something which has any magnitude that determined after goes
through and hence its magnitude changed. Example: Price (P), Demand (D/Qd), Revenue
(R), Cost (C), Consumption (C), National Income (Y), Investment (I), etc. Variable can be
divided based on different criteria.
Based on the nature of the value, variable can be divided as Quantitative and Qualitative
variables. Quantitative variable are a variable which expressed in terms of number or
numerically; for example Height, Price etc. Qualitative variables are a variable which are not
expressed in terms of number or numerically; for example Sex, Religion etc. Based on the
behavior, variable can be divided as Dependent and Independent variables. Dependent variable
is the effect in the model and independent variable is cause in the model. Based on the source of
the value, variable can be classified as Endogenous and exogenous variables. In models make a
distinction between those variables whose levels you want to explain in your model, and those
variables you want to include in your model, but not explain. The former are called endogenous
variables, the latter exogenous variables. In more detail: you, the model builder, choose what
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variables you want your model to explain. The variables whose levels you want to explain are
call endogenous variables. The adjective endogenous means inside – what is determined inside
your model. The prefix “ex” mean “out, from, or away” The prefix “end” means “in, into, into”
hence, the adjective “exogenous” means the value of the variable is determined outside of the
system The adjective “endogenous” means the value of the variable is determined within the
system. The economic model captures the link between the exogenous and endogenous variables.

A simple economic model illustrates the distinction between endogenous and exogenous
variables. Consider a simple demand and supply analysis of the market for the familiar mythical
good the “widget” (an unnamed, unspecified, or hypothetical manufactured good or product) the
endogenous variables in this model are the price of a widget and the quantity of widgets sold.
The exogenous variables in this example include the price of the input to widget production and
the price of the good that the consumers consider as a possible substitute for widgets. The
purpose of the model determines which variables are endogenous and which are exogenous. You
include exogenous variables in a model because you assume they, together, will determine /
explain the level of endogenous variables in your model.

2. Parameter or Constant: is a symbol which is not variable. Informally, it is a number that is

in a model and not changed (it is fixed). For example, in the demand model Qd = a – bP;
“Qd” and “P” are variables and “a” and “b” are parameters.
3. Equation: is a mathematical statement that shows an economic expression. Generally, in
economics, equation can be classified in to three as identity, behavioral and equilibrium
equation.
a. Identity equation: is a definition or a statement which shows that two or more variables
or expression are always equivalent. It sets an identity between two or alternative
expression that have exactly the same meaning. For example: π=R−C ; Y = C + I + G.
b. Behavioral equation: is a statement which shows how a variable can change in
response to the change in another variable. For example: Qs = a + bP; Qd = a – bP.
c. Equilibrium equation: is a mathematical statement which shows the condition that two
or more variables are equal to one another. Equilibrium equations appear in a model to
show the existence of equilibrium or state of rest. For example Qd = Qs; S = I.