MATH ASSIGNMENTAPPLICATIONS OF LINEAR ALGEBRA IN ECONOMICS

BY:- AMIT GARG

Economics is the branch of knowledge concerned with the production, consumption, and transfer of wealth .

Many economic relationships can be approximated by linear equations and others can be converted to linear relationships. So the analysis of many economic models reduces to the study of systems of linear equations.

Relationship between Linear algebra and Economics ….

What is Economics ????

Applications of linear algebra in economics...• Leontiff Input-Output Model It is a model that show interdependencies between different branches of economy• Developed by :- Wassily Leontief He devided the economy ito different sectors (like :- coal industry , agricultural industry , manufacturing industry etc).• Use of Linear Equation For each sector , he wrote linear equation describing how sector distributes Output to the other sectors

Leontiff Input-Output Model…….

Consumption Matrix

A consumption matrix shows the quantity of inputs neededto produce one unit of a good.

From\To Agriculture Manufacturing Labour

Agriculture 0.25 0.083 0.2

Manufacturing 0.25 0.167 0.4

Labour 1.25 0.4167 0.2

Represents Producing Sector of the economy

Represents Consuming Sector of the economy

TOTAL PRODUCTION , INTERNAL DEMAND , FINAL DEMAND...

Acc to Leontiff model

= +Total productproduced

Internal demand .

Final Demand

(F)(X)

• f is demand from the non-producing sector of the economy.• x is the total amount of the product produced.• They both can be represented as vectors (combining demands

from different industries) .

MATHEMATICS INVOLVED……

The internal demand is equal to the consumption matrixmultiplied by the total production vector .

X (Amount produced)]= [Cx] + final demand [F]

Therefore :- x = cx +f _ _ _ _ _ (1) By using the algebric properties of Rn. Ix = Cx + f _ _ _ _ _ (2) (I-C)x = f _ _ _ _ _ (3)

Let C be the consumption matrix for an economy, and let fthe final demand. If C and f have nonnegative entries, andif C is economically feasible, then the inverse of the matrix(I-C) exists and the production vector:

X = (I - C)^-1*f .......(4)

Therefore there exist a nonnegative entries and is the unique solution of the equation :-

X = Cx +f

USES OF INPUT – OUTPUT MODEL…

DEMAND SUPPLY PRICE

PREDICTION OF THESE THREE THINGS …..

REFERENCES :- (1) Research Paper of Sir Lucas Davidson . (2) WIKIPEDIA .