Chiang & Wainwright Mathematical Economics Chapter 4 Linear Models and Matrix Algebra...

Chiang & WainwrightChiang & WainwrightMathematical Mathematical EconomicsEconomics

Chapter 4Linear Models and Matrix Algebra

1Chiang_Ch4.ppt Stephen Cooke U. Idaho

Ch 4 Linear Models and Ch 4 Linear Models and Matrix AlgebraMatrix Algebra4.1 Matrices and Vectors4.2 Matrix Operations4.3 Notes on Vector Operations4.4 Commutative, Associative,

and Distributive Laws4.5 Identity Matrices and Null

Matrices 4.6 Transposes and Inverses4.7 Finite Markov Chains


Objectives of math for Objectives of math for economists economists

To understand mathematical economics problems by stating the unknown, the data and the conditions

To plan solutions to these problems by finding a connection between the data and the unknown

To carry out your plans for solving mathematical economics problems

To examine the solutions to mathematical economics problems for general insights into current and future problems

(Polya, G. How to Solve It, 2nd ed, 1975) 3Chiang_Ch4.ppt Stephen Cooke U. Idaho

One Commodity Market One Commodity Market ModelModel (2x2 matrix)(2x2 matrix)

Economic Model (p. 32)

1) Qd=Qs

2) Qd = a – bP (a,b >0)

3) Qs = -c + dP (c,d >0)

Find P* and Q*Scalar AlgebraEndog. :: Constants4) 1Q + bP = a

5) 1Q – dP = -c 4

dAx

dAx

c

a

P

Q

d

b

1*

1

1

db

bcadQ

db

caP

*

*

Matrix Algebra

Chiang_Ch4.ppt Stephen Cooke U. Idaho

One Commodity Market One Commodity Market ModelModel (2x2 matrix)(2x2 matrix)

5

dAx

c

a

d

b

P

Q

dAx

c

a

P

Q

d

b

1*

1

*

*

1

1

1

1

Matrix algebra


General form of 3x3 linear General form of 3x3 linear matrixmatrix

parameters endog.vars

exog. vars. & constants

Scalar algebra form

parameters & endogenous variables exog. vars

& const.

a11x + a12y + a13z = d1

a21x + a22y + a23z = d2

a31x + a32y + a33z = d3

6

3

2

1

333231

232221

131211

d

d

d

z

y

x

aaa

aaa

aaa

Matrix algebra form


1. Three Equation National Income 1. Three Equation National Income Model Model (3x3 matrix) (3x3 matrix)Let (Exercise 3.5-1, p. 47)Y = C + I0 + G0

C = a + b(Y-T) (a > 0, 0<b<1)

T = d + tY (d > 0, 0<t<1)Endogenous variables?Exogenous variables?Constants?Parameters?Why restrictions on the

parameters?7Chiang_Ch4.ppt Stephen Cooke U. Idaho

2. Three Equation National Income 2. Three Equation National Income Model Model

Exercise 3.5-2, p.47Exercise 3.5-2, p.47 Endogenous: Y, C, T: Income (GNP), Consumption,

and Taxes Exogenous: I0 and G0: autonomous Investment &

Government spending Constants a & d: autonomous consumption and

taxes Parameter t is the marginal propensity to tax gross

income 0 < t < 1 Parameter b is the marginal propensity to consume

private goods and services from gross income 0 < b < 1

8

btb

GIbdaY

1

)8 00*


6. Three Equation National Income 6. Three Equation National Income Model Model Exercise 3.5-1 p. 47Exercise 3.5-1 p. 47Parameters &

Endogenous vars.Exog. vars.

Y C T &cons.

1Y -1C +0T = I0+G0

-bY +1C +bT = a

-tY +0C +1T = d

Given

Y = C + I0 + G0

C = a + b(Y-T)

T = d + tY

Find Y*, C*, T*

10

d

a

GI

T

C

Y

t

bb00

10

1

011

dAx

dAx1*


7. Three Equation National Income 7. Three Equation National Income Model Model Exercise 3.5-1 p. 47Exercise 3.5-1 p. 47

11

dAx

d

a

GI

t

bb

T

C

Y

dAx

d

a

GI

T

C

Y

t

bb

1*

00

1

*

*

*

00

10

1

011

10

1

011


3. Two Commodity Market 3. Two Commodity Market EquilibriumEquilibrium

Section 3.4, p. 42Section 3.4, p. 42Section 3.4, p. 42 GivenQdi = Qsi, i=1, 2

Qd1 = 10 - 2P1 + P2

Qs1 = -2 + 3P1

Qd2 = 15 + P1 - P2

Qs2 = -1 + 2P2

Find Q1*, Q2

*, P1*, P2

*

Scalar algebra1Q1 +0Q2 +2P1 - 1P2 = 10

1Q1 +0Q2 - 3P1 +0P2= -2

0Q1+ 1Q2 - 1P1 + 1P2= 15

0Q1+ 1Q2 +0P1 - 2P2= -1

12

1

15

2

10

2010

1110

0301

1201

2

1

2

1

P

P

Q

Q

dAx

dAx1*


4. Two Commodity Market 4. Two Commodity Market EquilibriumEquilibrium

Section 3.4, p. 42 (4x4 matrix)Section 3.4, p. 42 (4x4 matrix)

13dAx

P

P

Q

Q

dAx

P

P

Q

Q

1*

1

*

*

*

*

2

1

2

1

1

15

2

10

2010

1110

0301

1201

1

15

2

10

2010

1110

0301

1201

2

1

2

1


4.1 Matrices and Vectors4.1 Matrices and VectorsMatrices as ArraysMatrices as ArraysVectors as Special MatricesVectors as Special Matrices

Assume an economic model as system of linear equations in which aij parameters, where i = 1.. n rows, j = 1.. m columns, and n=mxi endogenous variables, di exogenous variables and constants

nn

n

n

nm

m

m

nn d

d

d

x

x

x

ax

ax

ax

axa

axa

axa

2

1

2

22

12

211

22121

12111


4.1 Matrices and Vectors4.1 Matrices and Vectors A is a matrix or a rectangular array of elements in which

the elements are parameters of the model in this case. A general form matrix of a system of linear equations

Ax = d where A = matrix of parameters (upper case letters => matrices)x = column vector of endogenous variables, (lower case => vectors)d = column vector of exogenous variables and constants

Solve for x*

dAx

dAx

d

d

d

x

x

x

aaa

aaa

aaa

nnnmnn

m

m

1*

2

1

2

1

21

22221

11211


3.4 Solution of a General-3.4 Solution of a General-equation Systemequation System

Given (p. 44)2x + y = 124x + 2y = 24Find x*, y*y = 12 – 2x4x + 2(12 – 2x)

= 244x +24 – 4x =

240 = 0 ?

indeterminant!

Why?4x + 2y =242(2x + y) = 2(12)one equation with two

unknowns2x + y = 12x, yConclusion:

not all simultaneous equation models have solutions


4.3 Linear dependence4.3 Linear dependence

A set of vectors is linearly dependent if any one of them can be expressed as a linear combination of the remaining vectors; otherwise it is linearly independent.

Dependence prevents solving the system of equations. More unknowns than independent equations.

//2

/1

'2

'1

'2

'1

02

2412

105

2410

125

vv

v

v

v

v

17

023

54

162216

23

5

4

8

1

7

2

321

3

21

321

vvv

v

vv

vv


4.2 Scalar multiplication4.2 Scalar multiplication

8143

2141

16

42

8

1

848

3216

16

428

18

2221

1211

2221

12111aa

aa

aa

aa


4.3 Geometric 4.3 Geometric interpretation (2)interpretation (2)

Scalar multiplication

Source of linear dependence

19

6 4 2 U

3 2 U

1 3 2U

x2

x1

-4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-2


4.2 Matrix Operations4.2 Matrix OperationsAddition and Subtraction of MatricesAddition and Subtraction of MatricesScalar MultiplicationScalar MultiplicationMultiplication of MatricesMultiplication of MatricesThe Question of DivisionThe Question of DivisionDigression on Σ NotationDigression on Σ Notation

222222

117

25

20

13

97

12

xxx CBA

Matrix addition

Matrix subtraction

20

65

11

32

01

97

12


4.3 Geometric 4.3 Geometric interpretationinterpretationv' = [2 3]u' = [3 2]v'+u' = [5 5]

21

x1

x2

5

4

3

2

1

1 2 3 4 5


4.4 Matrix multiplication4.4 Matrix multiplication

Exceptions AB=BA iff

B = a scalar,B = identity matrix I, orB = the inverse of A, i.e., A-1


4.2 Matrix multiplication4.2 Matrix multiplicationMultiplication of matrices require conformability

conditionThe conformability condition for multiplication is that the

column dimensions of the lead matrix A must be equal to the row dimension of the lag matrix B.

What are the dimensions of the vector, matrix, and result?

c

bb

bbbaaaB

232221

131211

1211

23

231213112212121121121111

131211

babababababa

ccc

• Dimensions: a(1x2), B(2x3), c(1x3)


4.3 Notes on Vector Operations4.3 Notes on Vector OperationsMultiplication of VectorsMultiplication of VectorsGeometric Interpretation of Vector OperationsGeometric Interpretation of Vector OperationsLinear DependenceLinear DependenceVector SpaceVector Space

2

312xu

An [m x 1] column vector u and a [1 x n] row vector v, yield a product matrix uv of dimension [m x n].

24

54131

xv

10

15

8

12

2

3541

2

3

32xvu


4.4 Laws of Matrix Addition & 4.4 Laws of Matrix Addition & MultiplicationMultiplicationMatrix AdditionMatrix AdditionMatrix MultiplicationMatrix Multiplication

22222121

12121111

2221

1211

2221

1211

abaa

abba

bb

bb

aa

aaBA

Commutative law: A + B = B + A

25

22222121

12121111

2221

1211

2221

1211

abab

abab

bb

aa

bb

bbAB


4.4 Matrix Multiplication4.4 Matrix MultiplicationMatrix multiplication is generally not commutative.

That is, AB BA even if BA is conformable (because diff. dot product of rows or col. of A&B)

76

10,

43

21BA

26

2524

1312

74136403

72116201AB

4027

43

47263716

41203110BA


4.7 Finite Markov Chains4.7 Finite Markov ChainsMarkov processes are used to measure

movements over time, e.g., Example 1, p. 80

Chiang_Ch4.ppt Stephen Cooke U. Idaho 27

90110

100*6.100*3.,100*4.100*7.6.4.

3.7.100100

PP

PPx

plant?each at be willemployeesmany how year, one of end At the

6.4.

3.7.

PP

PPM

yprobabilitknown a w/ plantseach between move andstay employees The

100100x

B &A plants over two ddistribute are 0 at time Employees

0000BBBA

ABAA00

/011

BBBA

ABAA

00/0

BBABBAAA PBPBPAPABAMBA

BA

4.7 Finite Markov Chains4.7 Finite Markov Chainsassociative law of multiplication

Chiang_Ch4.ppt Stephen Cooke U. Idaho 28

8711390*6.110*3.90*4.110*7.6.4.

3.7.90110

PP

PP

PP

PPx

90110PP

PPx

plant?each at be willemployeesmany how years, twoof end At the

6.4.

3.7.

PP

PPM

yprobabilitknown a w/ plantseach between move andstay employees The

100100x

B &A plants over two ddistribute are 0 at time Employees

BBBA

ABAA

BBBA

ABAA00

2/022

BBBA

ABAA00

/011

BBBA

ABAA

00/0

BAMBA

BAMBA

BA

4.5 Identity and Null Matrices4.5 Identity and Null MatricesIdentity MatricesIdentity MatricesNull MatricesNull MatricesIdiosyncrasies of Matrix AlgebraIdiosyncrasies of Matrix Algebra

000

000

000

.

100

010

001

10

01

etc

or Identity Matrix is a square matrix and also it is a diagonal matrix with 1 along the diagonals similar to scalar “1”

Null matrix is one in which all elements are zero

similar to scalar “0”Both are “idempotent”

matricesA = AT andA = A2 = A3 = …


4.6 Transposes & Inverses4.6 Transposes & InversesProperties of Transposes Inverses and Their Properties of Transposes Inverses and Their PropertiesProperties Inverse Matrix and Solution of Linear-equation Inverse Matrix and Solution of Linear-equation SystemsSystems

4 01

983ATransposed matrices

(A')' = AMatrix rotated along

its principle major axis (running nw to se)

Conformability changes unless it is square

30

49

08

13

A


4.6 Inverse matrix4.6 Inverse matrix

AA-1 = I A-1A=INecessary for

matrix to be square to have inverse

If an inverse exists it is unique

(A')-1=(A-1)'

31

• A x = d• A-1A x = A-1 d• Ix = A-1 d• x = A-1 d• Solution depends on

A-1

• Linear independence• Determinant test!


4.2 Matrix inversion 4.2 Matrix inversion

It is not possible to divide one matrix by another. That is, we can not write A/B. This is because for two matrices A and B, the quotient can be written as AB-1 or B-1A.

32

• In matrix algebra AB-1 B-1 A. Thus writing does not clearly identify whether it represents AB-1 or B-1A

• Matrix division is matrix inversion

• (topic of ch. 5)


Ch. 4 Linear Models & Matrix Ch. 4 Linear Models & Matrix AlgebraAlgebra

Matrix algebra can be used:

a. to express the system of equations in a compact notation;

b. to find out whether solution to a system of equations exist; and

c. to obtain the solution if it exists. Need to invert the A matrix to find the solution for x*

33

dA

adjAx

A

adjAA

dAx

dAx

*

1

1*

det


4.1Vector multiplication4.1Vector multiplication (inner or dot product) (inner or dot product)

y = c'z

34

44332211 zczczczcy

4

1iii zcy

4

3

2

1

4321

z

z

z

z

ccccy

1x1 = (1x4)( 4x1)


4.2 4.2 Σ notationΣ notation

Greek letter sigma (for sum) is another convenient way of handling several terms or variables

i is the index of the summationWhat is the notation for the dot product?

231213112212121121121111131211 babababababaccc

2

111

kkkba

35

3

1iiiba

j

a1b1 +a2b2 +a3b3 =

2

121

kkkba

2

131

kkkba


Chiang & Wainwright Mathematical Economics Chapter 4 Linear Models and Matrix Algebra...

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Transcript of Chiang & Wainwright Mathematical Economics Chapter 4 Linear Models and Matrix Algebra...