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Ch. 5b Linear Models & Matrix Algebra 5.5Cramer's Rule 5.6Application to Market and National-Income Models 5.7Leontief Input-Output Models 5.8Limitations of Static Analysis 1

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5.2 Evaluating a third-order determinant Evaluating a 3rd order determinant by Laplace expansion 2 Ch. 5a Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (nxn) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (3x3) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (3x3) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (3x3) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (3x3) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (nxn) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (nxn) Ch. 5b Linear Models and Matrix Algebra

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5.5 Deriving Cramers Rule (nxn) Ch. 5b Linear Models and Matrix Algebra

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5.6Applications to Market and National-income Models: Matrix Inversions Ch. 5b Linear Models and Matrix Algebra

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5.6 Macro model Section 3.5, Exercise (a-d), p. 47 and Section 5.6, Exercise (a-b), p. 111 Given 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) Cramers rule Ch. 5b Linear Models and Matrix Algebra

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5.6 The macro model (3.5-2, p. 47) Ch. 5b Linear Models and Matrix Algebra

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5.6 The macro model (3.5-2, p. 47) Ch. 5b Linear Models and Matrix Algebra

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5.6 Application to Market & National Income Models: Cramers rule (3.5-2, p. 47) Ch. 5b Linear Models and Matrix Algebra

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5.6 Application to Market & National Income Models: Matrix Inversion (3.5-2, p. 47) Ch. 5b Linear Models and Matrix Algebra

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5.6 Application to Market & National Income Models: Matrix Inversion (3.5-2, p. 47) Ch. 5b Linear Models and Matrix Algebra

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Miller and Blair 2-3, Table 2-3, p 15 Economic Flows ($ millions) Inputs (cols) Outputs (rows) Sector 1 (z i1 ) Sector 2 (z i2 ) Final demand (d i ) Total gross output (x i ) Intermediate inputs: Sector Intermediate inputs: Sector Primary inputs (w i ) Total outlays (x i )

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Miller and Blair 2-3, Table 2-3, p 15 Inter-industry flows as factor shares Inputs (cols) Outputs (rows) Sector 1 (z i1 /x 1 =a i1 ) Sector 2 (z i2 /x 2 =a i2 ) Final demand (d i ) Total output (x i ) Intermediate inputs: Sector Intermediate inputs: Sector Primary inputs (w i /x i ) Total outlays (x i /x i )

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5.7 Leontief Input-Output Models Structure of an input-output model Ch. 5b Linear Models and Matrix Algebra

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5.7 Leontief Input-Output Models Structure of an input-output model Ch. 5b Linear Models and Matrix Algebra

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5.7 Leontief Input-Output Models Structure of an input-output model Ch. 5b Linear Models and Matrix Algebra

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5.7 Leontief Input-Output Models Structure of an input-output model Ch. 5b Linear Models and Matrix Algebra

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model Miller & Blair, p. 102

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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Ch. 5b Linear Models and Matrix Algebra Leontief Input-Output Models Structure of an input-output model

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5.8Limitations of Static Analysis 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

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5.6 Application to Market and National-Income Models Market model National-income model Matrix algebra vs. elimination of variables Why use matrix method at all? Compact notation Test existence of a unique solution Handy solution expressions subject to manipulation Ch. 5b Linear Models and Matrix Algebra

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