Mathematical
Economics (A relation between Mathematics and Economics)
Renu Bala
Associate
Professor, Govt College, Solan, H.P
*Corresponding
Author E-mail:
ABSTRACT:
Mathematical
economics relies on defining all the relevant assumptions, conditions and
casual structures of economic theories in mathematical terms. There are two
main benefits from doing this. It is based on symbols and drawings of
mathematics. There is as much advantage of using them in industry labour for
us. The English economist William Stanley Jevons was one of the strongest and
most successful proponent of mathematical economics. It attempts to translate
this theory into useful tools for everyday economic policy making by combining
mathematical economics with statistical methods. It is essential to every
economics student to be acquainted with the introduction and the development of
the methods and techniques. The objective of econometrics as a whole is to
convert qualitative statements - such as the relationship between two and more
variables is positive. It is particularly useful in solving optimization
problems where a policymaker is looking for the best tweak out of a range of
affect a specific outcome. As we are flooded with ever more information,
econometric methods have become ubiquitous in economics. Econometric methods
are used in many branches of economics including finance and labour economics.
KEYWORDS: mathematics,
economics, models, equilibrium.
INTRODUCTION
Mathematical
economics is a branch of economics that engages mathematical tools and methods
to analyses economic theories. Mathematical economics is best defined as a sub
field of economics that examines the mathematical aspects of economies &
economic theories. It may be interesting to begin the study of mathematical
economics with an enquiry into the history of mathematical economics. It is
generally believed that the use of mathematics as a tool of economics dates
from the pioneering work of Cournot (1838). However there were many others who
used mathematics in the analysis of economic ideas before Cournot. We shall
make a quick survey of the most important contributors.
The French engineer,
A. J. E. Dupuit, used mathematical symbols to express his concepts of supply
and demand. Even though he had no systematic theory he did develop the concepts
of utility and diminishing utility, which were clearly stated and presented in
graphical form.
The
use of Mathematics in economics can be accounted by the three following reasons
1.
Economic Fundamentals, articles and
prepositions can be easily understood with the help of mathematics.
2.
The application of maths helps in making
the subject look a bit easier.
3.
The application of Maths helps to develop
logical thinking for understanding of the subject as the subject Maths itself
is based on logics thus helping in wide spread improvement of inductive,
rational and logical thinking of the individual.
Mathematical
Economics: Meaning and Importance:
Mathematical
economics is the application of mathematical meth economic theories and
analyses odds to represent problems posed in economics. It allows formulation
and derivation of key relationships in a theory with clarity, generality,
rigor, and simplicity. By convention, the methods refer to those beyond simple
geometry, such as differential and integral calculus, difference and
differential equations, matrix algebra, and mathematical programming and other
computational methods.
Advantages
of Mathematical economics:
(1)
The ‘language’ used is more concise and
precise (2) a number of mathematical theorems help us to prove or disprove
economic concepts (3) helps us in giving focus to the assumptions used in economics
(4) it make the analysis more rigors (5) it allows us to treat the general
variable case, otherwise the number of variables in economic analysis will be
very limited.
Mathematical
Representation of Economic Models:
Economic models
generally consist of a set of mathematical equations that describe a theory of
economic behavior. The aim of model builders is to include enough equations to
provide useful clues about how rational agents behave or how an economy works.
An economic model is a simplified description of reality, designed to yield
hypotheses about economic behavior that can be tested. An important feature of
an economic model is that it is necessarily subjective in design because there
are no objective measures of economic outcomes. Different economists will make
different judgments about what is needed to explain their interpretations of
reality. There are two broad classes of economic models theoretical and
empirical. Theoretic models seek to derive verifiable implications about
economic behavior al under the assumption that agents maximize specific
objectives subject to constraints that are well defined in the model. They
provide qualitative answers to specific questions such as the implications of
asymmetric information (when person on one side of a transaction knows more
than the other person) or how best to handle market failures.
ECONOMIC
FUNCTION:
A function is a
mathematical relationship in which the values of a single dependent variable
are determined by the values of one or more independent variables. Function
means the dependent variable is determined by the independent variable.
We say y is a
function of x. This means y depends on or is determined by x. mathematically we
write y = f(x)
Demand
function:
Demand function express
the relationship between the price of the commodity (independent variable) and
quantity of the commodity demanded (dependent variable).It indicate how much
quantity of a commodity will be purchased at its different prices. Hence,
represent the quantity demanded of a commodity and Demand function = f (x) the
basic determinants of demand function = f (Px, Pr, Y, T, W, E)
Supply
function:
The functional
relationship between the quantity of commodities supplied and various
determinants is known as supply function. It is the mathematical expression of
the relationship between supply and factors that affect the ability and
willingness of the producer to offer the product. Mathematically, a supply
function can be expressed as Supply S x = f (P x)
Utility
function:
Utility function
is a mathematical function which ranks alternatives according to their utility
to an individual. The utility function measures welfare or satisfaction of a
consumer as a function of consumption of real goods, such as food, clothing and
composite goods rather than nominal goods measured in nominal terms. Thus the
utility function shows the relation between utility derive d from the quantity
of different commodity consumed. A utility function for a consumer consuming
three different goods may be represented: U = f (X ₁, X ₂ , X ₃ ………) Example:
Given the utility unction of a consumer U = 2x 2 +5, find the marginal utility.
Marginal utility is given by the first order derivative of the total utility
function.
Consumption
Function:
The consumption
function refers to the relationship between income and consumption. It is a
functional relationship between consumption and income. Symbolically, the
relationship is represented as C= f(Y), where С is consumption, Y is income.
Thus the consumption function indicates a functional relationship between С and
Y, where С is the defendant variable and Y is the independent variable, i.e., С
is determined by Y. In fact, propensity to consume or consumption function is a
sketch of the va expenditure corresponding to different levels of income. ross
national income. various amounts of consumption In the Keynesian framework, the
consumption function or propensity to consume, refers to a functional
relationship between two aggregates, i.e., total consumption and g The
Keynesian Consumption function C = a +bY
Maxima
and Minima:
Let f(x) be a real
valued function defined on an interval I. Then, f(x) is said to have the
maximum value in the interval I, if there exists a point c in I such that f(x)
≤ f(c) for all x ∈I. The
number f(c) is called the maxima or the maximum value of f(x) in the interval I
and the point c is called a point of maxima off in the interval I. Similarly if
f(x) ≥ f(c) for all x ∈ I then
c is called the point of maxima and f(c) is called the maximum value of f for
all x in I. Similarly if f is a function of two variables say z = f(x,y) then
the function is said to achieve its maximum value at (a,b) if f(x,y) ≤ f(a,b)
for all (x,y) in the domain of function. Similarly if the function f(x,y) ≥
f(a,b) for all (x,y) in the domain of function. Maxima and Minima plays an
important role in studying the firm equilibrium and consumer equilibrium.
Maxima and minima often called as optimization techniques helps in calculating
maximum utility of consumer which is maximum when utility is maximum. Similarly
achievement of maximum profit through differentiation we are able to achieve
the equilibrium of the firm.
Differential
Equations:
The theory of
differential equations has become an essential tool of economic analysis
particularly since computer has become commonly available. It would be
difficult to comprehend the contemporary literature of economics if one does
not understand basic concepts (such as bifurcations and chaos) and results of
modem theory of differential equations. Difference and Differential equations
are very helpful to study the “Macro Economic Theories” and the ‘Theories of
Economic Growth.” Application of “Differential Equations’ to economic theories
are abundant. A few of them are Multiplier and Accelerator Interaction and
Cob-web Model and trade cycle Likewise, the application of differential
equations to economic analysis is also much. For instance, a differential
equation expresses the rate of change of the current state as a function of the
current state. A simple illustration of this type of dependence is change of in
(GDP) over time. Consider state x of the GDP of the economy. The rate of change
of the GDP is proportional to the current GDP x’ (t) = g (x (t)) where t stands
for time and x'(t) the derivative of the function x with respect to t. The
growth rate of the GDP is x’ (t)/x. If the growth rate g is given at any time
t, the GDP at time t is given by solving the differential equation. This is a
first order differential equation which can be solved by separation of
variables . The solution involves exponential function which states that t the
GDP decays (increases) exponentially in time when g is negative (positive)
Econometrics:
Econometrics
literally means economic measurement. It is a combination of mathematical
economics, statistics, economic statistics and economic theory. Econometrics
literally means economical measurement though the grammatically correct term
from Greek would be econometrics; the word has been shortened in English.
Econometricians are concerned with the tasks of developing and applying
quantitative or statistical methods to the study and elucidation of economics
principles. Econometrics is derived from mathematical economics, statistics,
statistical economics, and economics theory. Osker Lange defines econometrics
as “The science, which deals with the determination quantitative laws occurring
in economic life.” Econometric can measure the statistical importance of the
economic relation. The most important statistical method in econometrics is
regression and co-relation analysis, Regression methods are important in
econometrics because economists cannot perform experiments under control
environment and observational data is biased and may contain experimental
errors which can be solved using statistical techniques like regression and
other statistical methods. Econometric analysis is divided into time series
analysis and cross sectional analysis. Time series analysis examines variables
over time, such as the effects of population growth on a nation's GDP. Cross
sectional analysis examines the relationship between different variables at a
point in time, for instance, the relationship between individual’s income and
food expenditures. When time series analysis and cross sectional analysis are
conducted simultaneously on the same sample, it is called panel analysis. If
the sample is different each time, it is called repeated cross section data.
Multidimensional panel data analysis is conducted on data sets that have more
than two dimensions. Econometric analysis may also be classified on the basis
of the number of relationships modeled. Single equation methods model a single
variable (the dependent variable) as a function of one or more explanatory
variables. In many econometric contexts such single equation methods may not be
able to recover estimates of causal relationships because either the dependent
variable causes changes in one of the explanatory variables or because
variables not included in the model cause both the dependent and at least one
of the independent variables. Simultaneous equation methods have been developed
as one means of addressing these problems. Many of these methods use variants
of instrumental variables models to make estimates. Much larger econometric
models are used in an attempt to explain or predict the behavior of national
economies. In recent times econometrics, has emerged as an important tool in
economics.
CONCLUSION:
Mathematics, used
correctly in economy, is a tool of thought; it is a way of quickly reaching the
goal, without becoming mandatory to be itself a goal of economic science It is
quite clear that without having a good understanding of the mathematical
concepts it will be difficult to have good grasp on economics. This shows that
economics and Mathematics co-exist and sound knowledge of Maths is required for
developing interest in economics and helps in making the subject more lucid to
understand.
REFRENCES:
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S. “Mathematics in economics: A perspective on necessity and sufficiency
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for Economists. Book by Carl P. Simon and Lawrence E. Blume
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