ch. 5b linear models & matrix algebra
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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
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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)
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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)
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 4
5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (3x3)(3x3)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 5
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5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (3x3)(3x3)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 6
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5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (3x3)(3x3)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 7
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5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (nxn)(nxn)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 8
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5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (nxn)(nxn)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 9
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5.5 Deriving Cramer’s 5.5 Deriving Cramer’s Rule Rule (nxn)(nxn)
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 10
5.65.6 Applications to Market and National-Applications to Market and National-income Models: Matrix Inversionsincome Models: Matrix Inversions
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 11
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
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 12
5.6 The macro model 5.6 The macro model (3.5-2, (3.5-2, p. 47)p. 47)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 13
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 14
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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)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 15
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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)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 16
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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)
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 17
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 18
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
Intermediate inputs: Sector 2
200 100 1700 2000
Primary inputs (wi)
650 1400 1100 3150
Total outlays(xi)
1000 2000 3150 6150
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 19
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)
Intermediate inputs: Sector 1
0.15 0.25 350 1000
Intermediate inputs: Sector 2
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
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 20
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 21
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 22
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 23
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 24
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 25
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 26
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 27
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 28
5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 29
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 30
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5.7 Leontief Input-Output Models5.7 Leontief Input-Output ModelsStructure of an input-output Structure of an input-output
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Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 31
multiplieroutput s1'sector 52.100.1$
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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
Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 32
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 and Matrix Algebra 5.5 - 5.8 33
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