Introduction to Mathematical Economics Lecture 6

Introduction to Mathematical Economics

Lecture 6

Undergraduate Program Faculty of Economics & Business Universitas Padjadjaran

Ekki Syamsulhakim, SE., MApplEc.

Previously

• Cubic Function: Application • _{Rational Function: Application}

• _{Other polynomial function  bivariate} • _{Exponential function}

– _Univariate – _Bivariate

Today

•

_{Totally review topics:}

– application of a 3x3 linear equation system in

economics

– _{Basics of matrix algebra}

SYSTEM OF LINEAR

EQUATIONS

System of Linear Equations

•

_{If each constraint is expressed as a linear}

equation the constraints form a system of

linear equations

•

Example:

– _{Linear Demand Function and Linear Supply}

Function  market system

– _{Linear Revenue Function and Linear Cost}

System and Solution

•

The solution to a system of linear

equations is a set of values which

simultaneously satisfy all the equations (or

conditions) of the system

•

For any system of linear equation, 3

possibilities exist:

– There may be no solutions – A single solution

System and Solutions Graphically

• The solution of a system of two linear equations

in two variables is represented graphically by the point of intersection of the two lines (equations)

• _{The no-solution case for a system of two}

equations and two variables is represented by two lines which are parallel (they do not

intersect)

• _{For the unlimited number of solutions, the lines}

Solutions graphically

Equation 1 Equation 2

Solutions graphically

Equation 1 Equation 2

Solutions graphically

System and Solution

•

_{The graphical description apply to systems}

of equations with 2 variables, regardless

the number of equations

•

For 3 variables, graphical description of

the three solution possibilities requires the

use of planes

Solution possibilities

• A system of linear equations having a unique

solution is a consistent system of equation

• A system of linear equations not having a unique

solution is an inconsistent system of equation

• A system of linear equations having an unlimited

number of solutions is a dependent system of

Exactly, Overconstrained, and

Underconstrained Systems of Eq

•

Exactly constrained systems of equation

are systems which have an equal number

of equations and unknowns (variables)

•

Overconstrained systems of equation are

systems which have number of equations

exceeds unknowns (variables)

•

Underconstrained systems of equation are

Exactly, Overconstrained, and

Underconstrained Systems of Eq

• _{Abbreviated way for describing the number of}

equations and variables in the system: A system with three equations and three variables is

referred to as being 3 X 3

• A system with four equations and two variables

is referred to as being 4 X 2

• _{The abbreviation condenses the expression}

3 X 3 System

• Solve the following system of three equations in

the three variables X, Y, Z!

Eq 1 : 2X + 3Y + Z = 6 Eq 2 : X + 4Y + 3Z = 12 Eq 3 : 3X + Y + 2Z = 10

• For a 3X3 system, the elimination method

requires selecting two different pairs of

equations and eliminating the same variables for each pairs

• _{This process results in a 2X2 system which is}

3 X 3 System

Eliminate Z from the equation

3 X 3 System

From equation 4 and 5:

Eq 4 : 5X + 5Y = 6

_{Finally, substitution of X=1 and Y=0.2 into}

Applications

• An aircraft manufacturer uses 3 machines (X,Y,

and Z) to manufacture three different parts, referred to as A, B, and C. The machines are

used to make other parts, but no other machines are required for the production of A, B, and C

• _{The manufacturer wants to determine the}

number of hours per month to use each machine for A, B, and C production under the following

Applications

•

_{To produce one part A requires the use of}

3 hours of X, 4 hours of Y, and 1 hour of Z

– _{Each month, 380 A must be produced}

– _{If X, Y, Z are used to represent hours of use}

Applications

3X + 4Y + Z = 380

– In this form, X, Y, and Z represent the hours

that each machine is operated per month to make the 380 A parts

•

_{One part B requires 2 hours of use by}

machine X, 4 hours of Y use, and 2 hours

use of machine Z.

Applications

2X + 4Y + 2Z = 400

•

Finally, each part C requires 6 hours of X,

2 hours of Y, and 2 hours of Z.

– _{There must be 520 Cs produced each month} – _{This constraint is the following:}

Applications

•

_{To determine how many hours per month}

each machine should be operated for the

three products, the system of equations

Applications

• Thus, the system of 2 equations and 2

unknowns with Y eliminated is as follows:

• _{Eq4 : X – Z = –20}

• Eq5 : 10X + 2Z = 640

• _{2Eq4 : 2X – 2Z = –40}

• _{Eq5 : 10X + 2Z = 640+}

Matrix Application

•

_Econometrics

•

Input – Output table

•

Social Accounting Matrix

Topic Outline

•

_{Basic Elements of Matrix Algebra}

•

Elementary Matrix Arithmetic

Definition of a Matrix

• _{A matrix is a rectangular array of numbers}

enclosed in parentheses

• _{It is conventionally denoted by a capital letter}

Size / dimension / order

of a matrix

•

We describe a matrix size by specifying the

Elements of a matrix

• _{We denote the element in row i and column j of}

Elements of a matrix

• _{We denote the element in row I and column j of}

Elements of a matrix

• _{We denote the element in row i and column j of}

Further definitions

• _{Square Matrix: matrix A}

p,q is said to be square if

if has the same number of rows as columns. i.e.if p=q

• _{Zero Matrix: matrix A}

p,q is a zero matrix, written

0 p, q if a_i,j= 0 for all i and j

• _{Diagonal Matrix: matrix A}

Matrix arithmethic

• _{Matrix Equality}

– We say that matrix A_p,q is equal to matrix B_r,s if A is the

same size as B, i.e. p=r and q=s

– _{The elements of A are equal to the elements of B, i.e.}

Matrix arithmethic

• Matrix Addition

– _{We may add matrix A to matrix B if A is the same size}

as B

– The sum is written A_p,q+ B_p,q= C_p,q where c_i,j= a_i,j+ b_i,j for

Matrix arithmethic

Matrix arithmethic

•

_{Matrix Addition Laws}

•

Consider three p × q matrices, A, B and C

– _{Commutative Law}

A+ B = B+ A

– _{Associative Law}

Matrix arithmethic

•

_{Scalar Multiplication of Matrices}

– A scalar is a single number

– Given a scalar kand a matrix A_p,q, the product

k × A= kA= C

– _{where C is a p×q matrix}

Matrix arithmethic

• Scalar Multiplication Laws

– _{Consider scalars, k and m and matrices A}

Matrix multiplication

•

Inner Product

– _{Let a be a row vector and b be a column vector}

of the same order.

– _{If both vectors are of order n, the inner product}

(sometimes also referred to as the scalar

Matrix multiplication

•

For example, suppose that an individual

consumes

n

goods.

•

The quantity of the

i-

th good consumed is

x

_i

and its price is

p

_i

.

•

Then

p

= (

p

₁

…

p

_n

)

is the price vector and

x

=

(x

₁

…

x

_n

) is the vector of quantities

Matrix multiplication

– _{Inner product = scalar}

Matrix multiplication

•

_{In order to form the matrix product}

_AB

_the

two matrices must be

conformable

Matrix multiplication

– _{outer product = matrix}

Matrix multiplication

• Matrix Multiplication Laws

– _{Consider matrices A}

p,q , Bq, r , Cq, r and Dr, s • Commutative Law

– _{Matrix multiplication is not commutative} – _{AB  BA}

• Associative Law

A( BD) = AB( D)

• Distributive Law

Matrix Arithmethic, in general