Ch. 5b Linear Models & Matrix Algebra

Ch. 5b Linear Models & Matrix Ch. 5b Linear Models & Matrix Algebra Algebra

5.5 Cramer's Rule5.6 Application to Market and National-Income Models5.7 Leontief Input-Output Models5.8 Limitations of Static Analysis

1

5.2 Evaluating a third-order determinant5.2 Evaluating a third-order determinantEvaluating a 3rd order determinant by Laplace Evaluating a 3rd order determinant by Laplace expansionexpansion

2

ijji

iji

ii MCCaA

CaCaCaA

aa

aaa

aa

aaa

aa

aaaA

aaa

aaa

aaa

A

1 where

1column down Expanding

A findGiven

3

111

313121211111

2322

131231

3332

131221

3332

232211

333231

232221

131211

Ch. 5a Linear Models and Matrix Algebra 5.1 - 5.4

5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (nxn)(nxn)

dAA

x

d

d

d

CCC

CCC

CCC

A

x

x

x

dAA

x

nxn

nxn

nnnn

n

n

x

nxn

*adjoint 1

find

tion,multiplica of law eassociativ theApplying

1

adjoint 1

*

1

2

1

21

22212

12111

11

1

*

*

*

*

2

1

Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 3

5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (3x3)(3x3)

AAx

aad

aad

aad

A

Aaa

aad

aa

aad

aa

aadCd

MCCdCdCdCd

Cd

Cd

Cd

ACdCdCd

CdCdCd

CdCdCd

Ax

x

x

*

iii

ijji

ijii

ii

iii

iii

iii

111

33323

23222

13121

1

12322

13123

3332

13122

3332

23221

3

11

31212111

3

11

3

13

3

12

3

11

333232131

323222121

313212111

*3

*2

*1

such that A Find.4)

)3

1 where)2

11)1




A

A

aaa

aaa

aaa

aad

aad

aad

Ca

Cdx

iii

iii

1

333231

232221

131211

33323

23222

13121

3

111

3

11

*1



A

A

aaa

aaa

aaa

ada

ada

ada

Ca

Cdx

iii

iii

2

333231

232221

131211

33331

23221

13111

3

122

3

12

*2



A

A

aaa

aaa

aaa

daa

daa

daa

Ca

Cdx

iii

iii

3

333231

232221

131211

33231

22221

11211

3

133

3

13

*3



A

A

Ca

Cdx

A

A

Ca

Cdx

A

A

Ca

Cdx

nn

iinin

n

iini

n

n

iii

n

iii

n

iii

n

iii

1

1*

2

122

12

*2

1

111

11

*1



n

iinin

n

iini

n

n

iinin

n

iinin

n

iii

n

iiin

iii

n

iii

n

iii

n

iiin

iii

n

iii

n

iijij

Ca

CdxCaACd

Ax

Ca

CdxCaACd

Ax

Ca

CdxCaACd

Ax

scalarCaA

1

1*th

11

*

122

12

*2

nd

122

12

*2

111

11

*1

st

111

11

*1

1

exp. col. n for the 1

exp. col. 2 for the 1

exp. col. 1 for the 1

expansioncolumn afor j) (j where:pattern a

Given thatfor Looking


AA

AA

AA

AA

Cd

Cd

Cd

Cd

A

x

x

x

x

nn

iini

n

iii

n

iii

n

iii

n

3

2

1

1

13

12

11

*

*

*

*

13

2

1


5.65.6 Applications to Market and National-Applications to Market and National-income Models: Matrix Inversionsincome Models: Matrix Inversions

65z 3;1y ;21

3

4

4

312-

61014-

3-3-6

6

1

z

y

x

312-

61014-

3-3-6

6

1A

3

4

4

d ;

z

y

x

x;

301

125

214

InversionMatrix

1-

1

x

dAx

A

dAx

65

31

21

;6

;5 ;2 ;3

3

4

4

d ;

z

y

x

x;

301

125

214

Rule sCramer'

3

2

1

321

AAz

AAy

AAx

A

AAA

A

dAx


5.6 Macro model5.6 Macro modelSection 3.5, Exercise 3.5-2 (a-d), p. 47 andSection 5.6, Exercise 5.6-2 (a-b), p. 111Given the following model

(a) Identify the endogenous variables(b) Give the economic meaning of the parameter g(c) Find the equilibrium national income (substitution)(d) What restriction on the parameters is needed for a

solution to exist?Find Y, C, G by (a) matrix inversion (b) Cramer’s rule


5.6 The macro model 5.6 The macro model (3.5-2, (3.5-2, p. 47)p. 47)


010

01

111

form Standard

)10(*

)10 ,0()(*

model The

0

0

0

0

GCgY

bTaGCbY

IGCY

gYgG

baTYbaC

GICY

5.6 The macro model 5.6 The macro model (3.5-2, (3.5-2, p. 47)p. 47)


gb

g

bA

1

10

01

111

tdeterminan The

010

01

111

formMatrix

0

0

bTa

I

G

C

Y

g

b

5.6 Application to Market & National 5.6 Application to Market & National Income Models: Cramer’s rule (3.5-2, Income Models: Cramer’s rule (3.5-2, p. 47)p. 47)


)(1

00

1

11

)(1

11

10

0

11

)(1100

01

11

constants of tor column vec with the

t determinan theofcolumn jth thereplacingby xofsolution theFinding

00*000

0

00*000

0

00*000

0

*j

gb

IbTag

A

AGIbTag

g

bTab

I

A

gb

bTagbI

A

ACbTagbI

g

bTab

I

A

gb

bTaI

A

AYbTaIbTa

I

A

GG

CC

YY

5.6 Application to Market & National 5.6 Application to Market & National Income Models: Matrix Inversion (3.5-2, Income Models: Matrix Inversion (3.5-2, p. 47)p. 47)


gb

g

bA

1

10

01

111

10

01

111

g

bA

bb

gg

gb

C

11

11

1

bgg

bgbCT

1

1

111

5.6 Application to Market & National 5.6 Application to Market & National Income Models: Matrix Inversion (3.5-2, Income Models: Matrix Inversion (3.5-2, p. 47)p. 47)


gb

bTaIgG

gb

bTagbIC

gb

bTaIY

1

1

1

1

00*

00*

00*

01

1

111

1

10

0

*

*

*

bTa

I

bgg

bgbgb

G

C

Y


5.7 Leontief Input-Output ModelsMiller and Blair 2-3, Table 2-3, p 15 Economic Flows ($ millions)

Inputs (col’s)

Outputs (rows)

Sector 1(zi1)

Sector 2(zi2)

Final demand

(di)

Total gross output

(xi)

Intermediate inputs: Sector 1

150 500 350 1000


200 100 1700 2000

Primary inputs (wi)

650 1400 1100 3150

Total outlays(xi)

1000 2000 3150 6150


5.7 Leontief Input-Output ModelsMiller and Blair 2-3, Table 2-3, p 15 Inter-industry flows as factor shares

Inputs (col’s)

Outputs (rows)

Sector 1(zi1/x1

=ai1)

Sector 2(zi2/x2

=ai2)

Final demand

(di)

Total output

(xi)


0.15 0.25 350 1000


0.20 0.05 1700 2000

Primary inputs (wi/xi)

0.65 0.70 1100 3150

Total outlays(xi/xi)

1.00 1.00 3150 6150

5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output

modelmodel

nnnnnnn

nn

nn

dxaxaxax

dxaxaxax

dxaxaxax

2211

222221212

112121111

model The


17002000*05.01000*20.020002sector

3502000*25.01000*15.01000 1sector

d a a x 2i21i1i

xx


modelmodel

nnnnnnn

nn

nn

dxaxaxax

dxaxaxax

dxaxaxax

2211

222221212

112121111

form standard The


17002000*05.01000*20.020002sector

3502000*25.01000*15.01000 1sector

d a a x 2i21i1i

xx


modelmodel


nnnnnn

n

n

d

d

d

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

1

1

1

formMatrix

1700

350

2000

1000

05.0120.0

25.015.01

2sector

1sector


modelmodel


nnnnnn

n

n

d

d

d

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

100

010

001

decomposed formmatrix The

1700

350

2000

1000

05.020.0

25.015.0

10

01

dxAI


1221126400

3300025411

050200

250150

050200

250150

10

01

analogy by

algebrascalar 1x if 11

2(a))-9.5 250, p. t, Wainwrigh& (Chiangion approximat seriesTaylor

2

12

0

1

12

0

1

00111

..

.....

..

..

..

..

A...AAIAAI

x...xxxx

IAAAAIAA

n

n

n

n

n

n

-


modelmodelMiller & Blair, p. 102Miller & Blair, p. 102


1700

350

2000

1000

95.020.0

25.085.0

1700

350

2000

1000

05.0120.0

25.015.01

1700

350

2000

1000

05.020.0

25.015.0

10

01

simplified formMatrix

dxAI


modelmodel


1700

350

1221.12640.0

3300.02541.1

2000

1000

1700

350

85.020.0

25.095.0

7575.0

1

2000

1000

7575.0)25.0*20.0()095.*85.0(

1700

350

95.020.0

25.085.0

2000

1000

matrix theInverting

1

1*

AI

dAIx


modelmodel


2000208170092

output base100056189350

20001700*1221.1350*2640.0

10001700*3300.0350*2541.1

1700

350

1221.12640.0

3300.02541.1

inversionmatrix with results theChecking

*2

*1

*2

*1

*

*

2

1

x

x

x

x

x

x


modelmodel



modelmodel

20007575.0

1515

7575.0

170020.0

35085.0

10007575.0

5.757

7575.0

95.01700

25.0350

7575.0 ;1700

350

95.020.0

25.085.0

rule sCramer' with results theChecking

2*2

1*1

1

*2

*1

AI

AIx

AI

AIx

AIx

x


400,2$2092*70.01439*65.0

20921700*1221.1700*2640.0

14391700*3300.0700*2541.1

1700

700

1221.12640.0

3300.02541.1

in total? need wedoinput primary much how

1,sector fromn consumptio double weIf

*0

*202

*101

*

*

*

*

2

1

2

1

wxaxa

x

x

x

x


modelmodel


20922081700184

1439561178700

20921700*1221.1700*2640.0

14391700*3300.0700*2541.1

1700

700

1221.12640.0

3300.02541.1

in total? need wedoinput primary much how

1,sector fromn consumptio double weIf

*

*

*

*

*

*

2

1

2

1

2

1

x

x

x

x

x

x


modelmodel


multiplieroutput s1'sector 52.100.1$

52.1$

52.1$264.0$2541.1$

2sector in 264.0$0*1221.11*2640.0

1sector in 2541.1$0*3300.01*2541.1

0

1

1221.12640.0

3300.02541.1

economy? in the production increase wedomuch how

$1,by 1sector fromn consumptio increase weIf

*

*

*

*

2

1

2

1

x

x

x

x


modelmodel

5.85.8 Limitations of Static Limitations of Static AnalysisAnalysis Static analysis solves for the

endogenous variables for one equilibrium

Comparative statics show the shifts between equilibriums

Dynamics analysis looks at the attainability and stability of the equilibrium


5.6 Application to Market and National-Income 5.6 Application to Market and National-Income ModelsModels

Market modelMarket modelNational-income modelNational-income modelMatrix algebra vs. elimination of variablesMatrix algebra vs. elimination of variables

Why use matrix method at all?Compact notationTest existence of a unique

solutionHandy solution expressions

subject to manipulation


Ch. 5b Linear Models & Matrix Algebra

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