Ch' 5 Linear Models

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Ch. 5 Linear Models Matrix Algebra

• 5.1 Conditions for Nonsingularity of a
  Matrix5.2 Test of Nonsingularity by Use of
  Determinant
• 5.3 Basic Properties of Determinants
• 5.4 Finding the Inverse Matrix
• 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

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5.1 Conditions for Nonsingularity of a Matrix3.4
Solution of a General-equilibrium System (p. 44)

• x y 8x y 9(inconsistent dependent)
• 2x y 124x 2y 24(dependent)
• 2x 3y 58y 18x y 20(over identified
  dependent)

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5.1 Conditions for Non-singularity of a
Matrix3.4 Solution of a General-equilibrium
System (p. 44)

• Sometimes equations are not consistent, and they
  produce two parallel lines. (contradict)
• Sometimes one equation is a multiple of the
  other. (redundant)

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5.1 Conditions for Non-singularity of a
MatrixNecessary versus sufficient
conditionsConditions for non-singularityRank of
a matrix

• A) Square matrix , i.e., n. equations n.
  unknowns. Then we may have unique solution. (nxn
  , necessary)
• B) Rows (cols.) linearly independent (rankn,
  sufficient)
• A B (nxn, rankn) (necessary sufficient),
  then nonsingular

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5.1 Elementary Row Operations (p. 86)

• Interchange any two rows in a matrix
• Multiply or divide any row by a scalar k (k ? 0)
• Addition of k times any row to another row
• These operations will
• transform a matrix into a reduced echelon matrix
  (or identity matrix if possible)
• not alter the rank of the matrix
• place all non-zero rows before the zero rows in
  which non-zero rows reveal the rank

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5.1 Conditions for Nonsingularity of a
MatrixConditions for non-singularity, Rank of a
matrix (p. 86)
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5.1 Conditions for Non-singularity of a
MatrixConditions for non-singularity, Rank of a
matrix (p. 96)
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5.1 Conditions for Nonsingularity of a
MatrixConditions for non-singularity, Rank of a
matrix (p. 96)
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5.1 Conditions for Non-singularity of a
MatrixConditions for non-singularity, Rank of a
matrix (p. 96)
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5.2 Test of Non-singularity by Use of
DeterminantDeterminants and non-singularityEvalu
ating a third-order determinantEvaluating an nth
order determent by Laplace expansion

• Determinant A is a uniquely defined scalar
  associated w/ a square matrix A(Chiang
  Wainwright, p. 88)
• A defined as the sum of all possible products
  ?t(-1)t a1j a2kang, where the series of second
  subscripts is a permutation of (1,.., n)
  including the natural order (1, , n), and t is
  the number of transpositions required to change a
  permutation back into the original order
  (Roberts Schultz, p. 93-94)
• t equals P(n,r)n!/(n-r)!, i.e., the permutation
  of n objects taken r at a time

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5.2 Test of Non-singularity by Use of Determinant

• P(n,r) n!/(n-r)! P(2,2) 2!/(2-2)! 2
• There are only two ways of arranging subscripts
  (i,k) of product (-1)ta1ja2k either (1,2) or
  (2,1)
• The first permutation is even positive (-1)2
  and second is odd and negative (-1)1
• 0!(1) 11!(1) 12!(2)(1)
  23!(3)(2)(1) 64!(4)(3)(2)(1)
  245!(5)(4)(3)(2)(1) 120 6!(6)(4)(3)(2)(1)
  720 10! 3,628,800

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5.2 Test of Non-singularity by Use of Determinant
and permutations 2x2 and 3x3
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5.2 Test of Non-singularity by Use of Determinant
4 x 4 permutations 24
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5.2 Evaluating a third-order determinantEvaluati
ng an 3 order determent by Laplace expansion

• Laplace Expansion by cofactors if /A/ 0, then
  /A/ is singular, i.e., under identified

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5.2 Determinants

• Pattern of the signs for cofactor minors

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5.1 Conditions for Nonsingularity of a
MatrixConditions for non-singularity, Rank of a
matrix (p. 96)
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5.1 Conditions for Non-singularity of a
MatrixConditions for non-singularity, Rank of a
matrix (p. 96)
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5.2 Evaluating a determinant

• Laplace expansion of a 3rd order determinant by
  cofactors. If /A/ 0, then singular

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5.2 Test of Non-singularity by Use of Determinant

• P(3,3) 3!/(3-3)! 6
• A 1(5)9 2(6)7 3(8)4 -3(5)7 6(8)1
  9(4)2
• Expansion by cofactorsA (1)c11 (2)c12
  (3)c13
• C11 5(9) 6(8)
• C12 -4(9) 6(7)
• C13 4(8) 7(5)
• Expansion across any row or column will give the
  same for the determinant

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5.3 Basic Properties of DeterminantsProperties
I to III (related to elementary row operations)

• The interchange of any two rows will alter the
  sign but not its numerical value
• The multiplication of any one row by a scalar k
  will change its value k-fold
• The addition of a multiple of any row to another
  row will leave it unaltered.

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5.3 Basic Properties of DeterminantsProperties
IV to VI

• The interchange of rows and columns does not
  affect its value
• If one row is a multiple of another row, the
  determinant is zero
• The expansion of a determinant by alien cofactors
  produces a result of zero

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5.3 Basic Properties of DeterminantsProperties
I to V

• If /A/ ? 0
• Then
• A is nonsingular
• A-1 exists
• A unique solution to
• XA-1d exists

• /A/ /A'/
• Changing rows or col. does not change but
  changes the sign of /A/
• k(row) k/A/
• ka row or col.b /A/
• If row or col akb, then /A/ 0

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5.4 Finding the Inverse aka the hard way

• Steps in computing the Inverse Matrix and solving
  for x
• 1.  Find the determinant A using expansion by
  cofactors.
• If A 0, the inverse does not exist.
• 2.  Use cofactors from step 1 and complete the
  cofactor matrix.
• 3.   Transpose the cofactor matrix gt adjA 
• 4. Divide adj.A by A gt A-1
• 5. Post multiply matrix A-1 by column vector of
  constants d to solve for the vector of variables x

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5.4 Finding the Inverse MatrixExpansion of a
determinant by alien cofactors, Property VI,
Matrix inversion

• Expansion by alien cofactors yields /A/0
• This property of determinants is important when
  defining the inverse (A-1)

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5.4 A Inverse (A-1)

• Inverse of A is A-1
• if and only if A is square (nxn) and rank n
• AA-1 A-1A I
• We are interested in A-1 because xA-1d

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5.4 matrix A matrix of parameters from the
equation Axd
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C Matrix of cofactors of A
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C' or adjoint A Transpose matrix of the
cofactors of A
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AC'
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Matrix AC'
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Inverse of A
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Solving for X using Matrix Inversion
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5.4 A Inverse, solving for P
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Cramers rule
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Cramer's rule
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Cramer's rule
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Deriving Cramers Rule
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5.4 Finding the Inverse aka the hard way

• Steps in computing the Inverse Matrix and solving
  for x
• 1.  Find the determinant A using expansion by
  cofactors.
• If A 0, the inverse does not exist.
• 2.  Use cofactors from step 1 and complete the
  cofactor matrix.
• 3.   Transpose the cofactor matrix gt adjA 
• 4. Divide adj.A by A gt A-1
• 5. Post multiply matrix A-1 by column vector of
  constants d to solve for the vector of variables x

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Derivation of matrix inverse formula

• A ai1ci1 . aincin (scalar)
• Adj. A transposed cofactor matrix of A
• A(adj.A)AI (expansion by alien cofactors 0
  for off diagonal elements)
• A(adj.A)/A I
• A-1 (adj.A)/A QED Roberts Schultz, p.
  97-8)

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5.4 Finding the Inverse aka the hard way

• Steps in computing the Inverse Matrix and solving
  for x
• 1.  Find the determinant A using expansion by
  cofactors.
• If A 0, the inverse does not exist.
• 2.  Use cofactors from step 1 and complete the
  cofactor matrix.
• 3.   Transpose the cofactor matrix gt adjA 
• 4. Divide adj.A by A gt A-1
• 5. Post multiply matrix A-1 by column vector of
  constants d to solve for the vector of variables x

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5.4 A Inverse

• A(adjA) AI
• A(adjA)/A I ( A is a scalar)
• A-1A(adjA)/A A-1I
• adjA/A A-1

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Finding the Determinant

• 1Y 1C1G I0
• -bY1C 0G a-bT0
• -gY0C 1G 0

• Y CI0G
• C a b(Y-T0)
• G gY

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Derivation of matrix inverse formula

• A ai1ci1 . aincin (scalar)
• Adj. A transposed cofactor matrix of A
• A(adj.A)AI (expansion by alien cofactors 0
  for off diagonal elements)
• A(adj.A)/A I
• A-1 (adj.A)/A QED Roberts Schultz, p.
  97-8)

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5.4 Finding the Inverse aka the hard way

• Steps in computing the Inverse Matrix and solving
  for x
• 1.  Find the determinant A using expansion by
  cofactors.
• If A 0, the inverse does not exist.
• 2.  Use cofactors from step 1 and complete the
  cofactor matrix.
• 3.   Transpose the cofactor matrix gt adjA 
• 4. Divide adj.A by A gt A-1
• 5. Post multiply matrix A-1 by column vector of
  constants d to solve for the vector of variables x

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5.4 A Inverse

• A(adjA) AI
• A(adjA)/A I ( A is a scalar)
• A-1A(adjA)/A A-1I
• adjA/A A-1

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5.4 Inverse, an example
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Finding the Determinant

• 1Y 1C1G I0
• -bY1C 0G a-bT0
• -gY0C 1G 0

• Y CI0G
• C a b(Y-T0)
• G gY

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The macro model

• YCI0G 1Y - 1C 1G I0
• Cab(Y-T0) -bY 1C 0G a-bT0
• GgY -gY 0C 1G 0

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Macro model

• Section 3.5, Exercise 3.5-2 (a-d), p. 47
• 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) Cramers
  rule

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The macro model

• YCI0G 1Y - 1C 1G I0
• Cab(Y-T0) -bY 1C 0G a-bT0
• GgY -gY 0C 1G 0

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3.4 Solution of General Eq. System

• (1)(1)-(1)(1) 0(inconsistent dependent)
• (2)(2)-(1)(4) 0(dependent)
• (2)(1)-(1)(3) -1(independent as rewritten)

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5.7 Leontief Input-Output Models Structure of an
input-output model The open model, A numerical
example Finding the inverse by approximation,
The closed model

• (I -A)x d x (I -A)-1 d

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Miller and Blair 2-3, Table 2-3, p 15 Economic
Flows ( millions)
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Leontief Input-output Analysis
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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

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3.4 Solution of General Eq. System

• (1)(1)-(1)(1) 0(inconsistent dependent)
• (2)(2)-(1)(4) 0(dependent)
• (2)(1)-(1)(3) -1(independent as rewritten)

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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