Ch 5 b Linear Models Matrix Algebra 5

Ch. 5 b Linear Models & Matrix Algebra 5. 5 5. 6 5. 7 5. 8 Cramer's Rule Application to Market and National-Income Models Leontief Input-Output Models Limitations of Static Analysis 1

5. 2 Evaluating a third-order determinant Evaluating a 3 rd order determinant by Laplace expansion Ch. 5 a Linear Models and Matrix Algebra 5. 1 - 5. 4 2

5. 5 Deriving Cramer’s Rule (nxn) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 3

5. 5 Deriving Cramer’s Rule (3 x 3) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 4

5. 5 Deriving Cramer’s Rule (3 x 3) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 5

5. 5 Deriving Cramer’s Rule (3 x 3) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 6

5. 5 Deriving Cramer’s Rule (3 x 3) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 7

5. 5 Deriving Cramer’s Rule (nxn) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 8

5. 5 Deriving Cramer’s Rule (nxn) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 9

5. 5 Deriving Cramer’s Rule (nxn) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 10

5. 6 Applications to Market and National-income Models: Matrix Inversions Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 11

5. 6 Macro model Section 3. 5, Exercise 3. 5 -2 (a-d), p. 47 and Section 5. 6, Exercise 5. 6 -2 (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) Cramer’s rule Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 12

5. 6 The macro model (3. 5 -2, p. 47) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 13

5. 6 The macro model (3. 5 -2, p. 47) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 14

5. 6 Application to Market & National Income Models: Cramer’s rule (3. 5 -2, p. 47) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 15

5. 6 Application to Market & National Income Models: Matrix Inversion (3. 5 -2, p. 47) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 16

5. 6 Application to Market & National Income Models: Matrix Inversion (3. 5 -2, p. 47) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 17

5. 7 Leontief Input-Output Models Miller and Blair 2 -3, Table 2 -3, p 15 Economic Flows ($ millions) Inputs (col’s) Sector 1 (zi 1) Sector 2 (zi 2) Final demand (di) Total gross output (xi) Intermediate inputs: Sector 1 150 500 350 1000 Intermediate inputs: Sector 2 200 1700 2000 Primary inputs (wi) 650 1400 1100 3150 Total outlays (xi) 1000 2000 3150 6150 Outputs (rows) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 18

5. 7 Leontief Input-Output Models Miller and Blair 2 -3, Table 2 -3, p 15 Inter-industry flows as factor shares Inputs (col’s) Sector 1 (zi 1/x 1 =ai 1) Sector 2 (zi 2/x 2 =ai 2) Final demand (di) Total output 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 3150 6150 Outputs (rows) (xi) Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 19

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 20

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 21

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 22

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 23

5. 7 Leontief Input-Output Models Structure of an input-output model Miller & Blair, p. 102 Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 24

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 25

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 26

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 27

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 28

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 29

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 30

5. 7 Leontief Input-Output Models Structure of an input-output model Ch. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 31

5. 8 Limitations 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. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 32

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. 5 b Linear Models and Matrix Algebra 5. 5 - 5. 8 33