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

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

3. 5.5 Deriving Cramer’s Rule (nxn)
Ch. 5b Linear Models and Matrix Algebra

4. 5.5 Deriving Cramer’s Rule (3x3)
Ch. 5b Linear Models and Matrix Algebra

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

6. 5.5 Deriving Cramer’s Rule (3x3)
Ch. 5b Linear Models and Matrix Algebra

7. 5.5 Deriving Cramer’s Rule (3x3)
Ch. 5b Linear Models and Matrix Algebra

8. 5.5 Deriving Cramer’s Rule (nxn)
Ch. 5b Linear Models and Matrix Algebra

9. 5.5 Deriving Cramer’s Rule (nxn)
Ch. 5b Linear Models and Matrix Algebra

10. 5.5 Deriving Cramer’s Rule (nxn)
Ch. 5b Linear Models and Matrix Algebra

11. 5.6 Applications to Market and National-income Models: Matrix Inversions
Ch. 5b Linear Models and Matrix Algebra

12. 5.6 Macro model Section 3.5, Exercise 3.5-2 (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) Cramer’s rule
Ch. 5b Linear Models and Matrix Algebra

13. 5.6 The macro model (3.5-2, p. 47)
Ch. 5b Linear Models and Matrix Algebra

14. 5.6 The macro model (3.5-2, p. 47)
Ch. 5b Linear Models and Matrix Algebra

15. 5. 6 Application to Market & National Income Models: Cramer’s rule (3
5.6 Application to Market & National Income Models: Cramer’s rule (3.5-2, p. 47)
Ch. 5b Linear Models and Matrix Algebra

16. 5.6 Application to Market & National Income Models: Matrix Inversion (3.5-2, p. 47)
Ch. 5b Linear Models and Matrix Algebra

17. 5.6 Application to Market & National Income Models: Matrix Inversion (3.5-2, p. 47)
Ch. 5b Linear Models and Matrix Algebra

18. Total gross output (xi)
5.7 Leontief Input-Output Models
Miller 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)
6150
Ch. 5b Linear Models and Matrix Algebra

19. Primary inputs (wi/xi)
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)
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
6150
Ch. 5b Linear Models and Matrix Algebra

20. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

21. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

22. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

23. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

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

25. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

26. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

27. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

28. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

29. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

30. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

31. 5.7 Leontief Input-Output Models Structure of an input-output model
Ch. 5b Linear Models and Matrix Algebra

32. 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. 5b Linear Models and Matrix Algebra

33. Why use matrix method at all? Compact notation
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