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Chiang & Wainwright
Mathematical Economics
Chapter 4
Linear Models and Matrix Algebra
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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Ch 4 Linear Models and Matrix
Algebra
4.1 Matrices and Vectors
 4.2 Matrix Operations
 4.3 Notes on Vector Operations
 4.4 Commutative, Associative, and
Distributive Laws
 4.5 Identity Matrices and Null Matrices
 4.6 Transposes and Inverses
 4.7 Finite Markov Chains

Chiang_Ch4.ppt Stephen Cooke U. Idaho
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Objectives of math for economists
To understand mathematical economics problems
by stating the unknown, the data and the
conditions
 To plan solutions to these problems by finding a
connection between the data and the unknown
 To carry out your plans for solving mathematical
economics problems
 To examine the solutions to mathematical
economics problems for general insights into
current and future problems
(Polya, G. How to Solve It, 2nd ed, 1975)

Chiang_Ch4.ppt Stephen Cooke U. Idaho
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One Commodity Market Model
(2x2 matrix)

Economic Model
(p. 32)
1) Qd=Qs
2) Qd = a – bP (a,b >0)
3) Qs = -c + dP (c,d >0)
 Find P* and Q*
Scalar Algebra
Endog. ::
Constants
4) 1Q + bP = a
5) 1Q – dP = -c
ac
P 
bd
ad  bc
*
Q 
bd
*
Matrix Algebra
1 b  Q   a 
1  d   P    c 

   
Ax  d
x*  A1d
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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One Commodity Market Model
(2x2 matrix)
Matrix algebra
1 b  Q   a 
1  d   P    c 

   
Ax  d
1
Q  1 b   a 
 *  



 P  1  d   c 
*
1
x A d
*
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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General form of 3x3 linear matrix
Scalar algebra form
parameters & endogenous variables
a11x
a21x
a31x
+ a12y
+ a22y
+ a32y
Matrix algebra form
parameters
 a11
a
 21

 a31
a12
a22
a32
exog. vars
& const.
= d1
+ a13z
+ a23z
+ a33z
endog.
vars
= d2
= d3
exog. vars.
& constants
a13   x   d1 
 y   d 
a23 
   2 
a33 

z
 
d3 

Chiang_Ch4.ppt Stephen Cooke U. Idaho
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1. Three Equation National Income Model
(3x3 matrix)

Let (Exercise 3.5-1, p. 47)
Y = C + I0 + G0
C = a + b(Y-T)
T = d + tY
(a > 0, 0<b<1)
(d > 0, 0<t<1)
Endogenous variables?
 Exogenous variables?
 Constants?
 Parameters?
 Why restrictions on the parameters?

Chiang_Ch4.ppt Stephen Cooke U. Idaho
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2. Three Equation National Income Model
Exercise 3.5-2, p.47





Endogenous: Y, C, T: Income (GNP), Consumption, and
Taxes
Exogenous: I0 and G0: autonomous Investment &
Government spending
Constants a & d: autonomous consumption and taxes
Parameter t is the marginal propensity to tax gross
income 0 < t < 1
Parameter b is the marginal propensity to consume
private goods and services from gross income 0 < b < 1
a  bd  I 0  G0
8) Y 
1  b  bt
*
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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6. Three Equation National Income Model
Exercise 3.5-1 p. 47

Given
Y = C + I0 + G0
Parameters &
Endogenous vars.
Exog.
vars.
Y
&cons.
T
-1C +0T
=
I0+G0
C = a + b(Y-T)
-bY +1C +bT
=
a
T = d + tY
-tY +0C +1T
=
d

Find Y*, C*, T*
Ax  d
x*  A1d
1Y
C
 1  1 0 Y   I 0  G0 
 b 1 b C    a 

  

  t 0 1 T   d 
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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7. Three Equation National Income Model
Exercise 3.5-1 p. 47
 1  1 0 Y   I 0  G0 
 b 1 b C    a 

  

  t 0 1 T   d 
Ax  d
1
Y   1  1 0  I 0  G0 
 * 



C    b 1 b  a 
T *    t 0 1  d 
 
*
1
x A d
*
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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3. Two Commodity Market Equilibrium
Section 3.4, p. 42
Section 3.4, p. 42
 Given
Qdi = Qsi,
i=1, 2
Qd1 = 10 - 2P1 + P2
Qs1 = -2 + 3P1
Qd2 = 15 + P1 - P2
Qs2 = -1 + 2P2
 Find Q1*, Q2*, P1*, P2*

Ax  d
x*  A1d
Scalar algebra
1Q1 +0Q2 +2P1 - 1P2 = 10
1Q1 +0Q2 - 3P1 +0P2= -2
0Q1+ 1Q2 - 1P1 + 1P2= 15
0Q1+ 1Q2 +0P1 - 2P2= -1

1
1

0

0
 1   Q1   10 
0  3 0  Q2   2

1  1 1   P1   15 
   
1 0  2  P2    1 
0
2
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4. Two Commodity Market Equilibrium
Section 3.4, p. 42 (4x4 matrix)
1 0 2  1  Q1   10 
1 0  3 0  Q   2

 2    
0 1  1 1   P1   15 

   
0 1 0  2  P2    1
Ax  d
Q1*  1
 * 
Q 2   1
 P *  0
 1*  
 P2  0
x*  A1d
 1
0  3 0 
1 1 1 

1 0  2
0
2
1
 10 
  2
 
 15 
 
  1
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.1 Matrices and Vectors
Matrices as Arrays
Vectors as Special Matrices

Assume an economic model as system of
linear equations in which
aij parameters, where
i = 1.. n rows, j = 1.. m columns, and n=m
xi endogenous variables,
di exogenous variables and constants
a11
x1

a12 x2
  a1m xn  d1
a21
x1
 a22 x2
  a 2 m xn  d 2


 a n 2 x2
  anm xn  d n

an1
x1
Chiang_Ch4.ppt Stephen Cooke U. Idaho

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4.1 Matrices and Vectors
A is a matrix or a rectangular array of elements in which the
elements are parameters of the model in this case.
 A general form matrix of a system of linear equations
Ax = d
where
A = matrix of parameters (upper case letters => matrices)
x = column vector of endogenous variables, (lower case => vectors)
d = column vector of exogenous variables and constants
Solve for x*

 a11 a12  a1m   x1   d1 
a
  x  d 
a

a
22
2m   2 
 21
  2
 


      

   
an1 an 2  anm   xn  d n 
Ax  d
x*  A1d
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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3.4 Solution of a General-equation
System
Given (p. 44)
2x + y = 12
4x + 2y = 24
Find x*, y*
y = 12 – 2x
4x + 2(12 – 2x) =
24
4x +24 – 4x = 24
0=0?
indeterminant!

Why?
4x + 2y =24
2(2x + y) = 2(12)
 one equation with two
unknowns
2x + y = 12
x, y
Conclusion:
not all simultaneous equation
models have solutions

Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.3 Linear dependence
v1'  5 12 


A set of vectors is
linearly dependent if any
one of them can be
expressed as a linear
combination of the
remaining vectors;
otherwise it is linearly
independent.
Dependence prevents
solving the system of
equations. More
unknowns than
independent equations.
v2'  10 24 
 5 10   v1' 
12 24    ' 

 v2 
2v1/  v2/  0 /
2
1
4
v1    v 2    3   
7 
8
5 
3v1  2v 2
 6 21  2 16 
 4 5  v3
3v1  2v 2  v3  0
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.2 Scalar multiplication
2
8
6
1 2
8 6
4 16


1  48
4 1 4


1  3 4
 a11
 1
a21
a12    a11


a22   a21
32
8 
1 2
1 8 
 a12 
 a 22 
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.3 Geometric interpretation (2)
x2
Scalar
multiplication
 Source of linear
dependence

6
5
4
 6 4  2 U
3
3 2  U
2
1
x1
-4

 1 U   3  2
-3

-2
-1
1
2
3
4
5
6
-2
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.2 Matrix Operations
Addition and Subtraction of Matrices
Scalar Multiplication
Multiplication of Matrices
The Question of Division
Digression on Σ Notation
 2 1 3 1  5 2 
7 9  0 2  7 11

 
 

A2 x 2  B2 x 2 C 2 x 2

Matrix addition

Matrix subtraction 2 1  1 0  1 1
7 9 2 3 5 6

 
 

Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.3 Geometric interpretation
x2
v' = [2 3]
 u' = [3 2]
 v'+u' = [5 5]

5
4
3
2
1
x1
1
2
Chiang_Ch4.ppt Stephen Cooke U. Idaho
3
4
5
21
4.4 Matrix multiplication
Exceptions
 AB=BA iff

B = a scalar,
B = identity matrix I, or
B = the inverse of A, i.e., A-1
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.2 Matrix multiplication



Multiplication of matrices require conformability
condition
The conformability condition for multiplication is
that the column dimensions of the lead matrix A
must be equal to the row dimension of the lag
matrix B.
What are the dimensions of the vector, matrix,
and result?
b11 b12 b13 
c11
c12 c13 
aB  a11a12 
b21
22
c
b23 

 a11b11  a12b21 a11b12  a12b22 a11b13  a12b23 
• Dimensions: a(1x2), B(2x3), c(1x3)
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.3 Notes on Vector Operations
Multiplication of Vectors
Geometric Interpretation of Vector Operations
Linear Dependence
Vector Space
 3
u  
2 x1
 2
v  1 4 5
An [m x 1] column vector u
and a [1 x n] row vector v,
yield a product matrix uv of
dimension [m x n].
1x3
 3
uv   1
2 x3
 2
4
31215 
5  

2
8
10


Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.4 Laws of Matrix Addition &
Multiplication
Matrix Addition
Matrix Multiplication

Commutative law: A + B = B + A
 a11 a12  b11 b12   a11  b11 b12  a12 
A B  





a
a
b
b
a

a
b

a
22 
 21 22   21 22   21 21 22
b11 b12  a11 a12   b11  a11 b12  a12 
B A 





b
b
b
b
b

a
b

a
22 
 21 22   21 22   21 21 22
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.4 Matrix Multiplication

Matrix multiplication is generally not commutative. That is,
AB  BA even if BA is conformable
(because diff. dot product of rows or col. of A&B)
1 2
0  1
A
,B  


3
4
6
7




10  26 1 1  27  12 13 
AB  











3
0

4
6
3

1

4
7
24
25

 

01   13 02   14  3  4
BA  


62  7 4   27 40 
 61  73
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.7 Finite Markov Chains

Markov processes are used to measure
movements over time, e.g., Example 1, p. 80
Employees at time 0 are distribute d over two plants A & B
x 0/  A0
B0   100 100
The employees stay and move between each plants w/ a known probabilit y
PAB  .7 .3
P
M   AA



 PBA PBB  .4 .6
At the end of one year, how many employees will be at each plant?
A1
B1   x 0/ M
 A0
.7 .3
 100 100

.4 .6
P
B0  AA
 PBA
PAB 
 A0 PAA  A0 PBA
PBB 
 .7 *100  .4 *100,
B0 PAB  B0 PBB 
.3 *100  .6 *100
 110 90
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.7 Finite Markov Chains

associative law of multiplication
Employees at time 0 are distribute d over two plants A & B
x 0/  A0
B0   100 100
The employees stay and move between each plants w/ a known probabilit y
PAB  .7 .3
P
M   AA
  .4 .6
P
P


BB 
 BA
At the end of two years, how many employees will be at each plant?
A1
B1   x M
 A0
A2
B2   x 0/ M 2
 A0
/
0
PAA
B0 
 PBA
PAA

B0 
 PBA
PAB 
 110 90
PBB 
PAB  PAA PAB 
PBB   PBA PBB 
.7 .3
 110 90
 .7 *110  .4 * 90 .3 *110  .6 * 90  113 87

.4 .6
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.5 Identity and Null Matrices
Identity Matrices
Null Matrices
Idiosyncrasies of Matrix Algebra
Identity Matrix is a
square matrix and also
it is a diagonal matrix
with 1 along the
diagonals
similar to scalar “1”
 Null matrix is one in
which all elements are
zero
similar to scalar “0”
Both are “idempotent”
matrices
A = AT
and
A = A2 = A3 = …

1
0

1
0


0
0
0


0
Chiang_Ch4.ppt Stephen Cooke U. Idaho
0
or

1
0 0
1 0
etc
.

0 1

0
0
0
0

0
0

29
4.6 Transposes & Inverses
Properties of Transposes Inverses and Their Properties
Inverse Matrix and Solution of Linear-equation Systems
Transposed matrices
 (A')' = A
 Matrix rotated along its
principle major axis
(running nw to se)
 Conformability changes
unless it is square

3 8  9
A

4
1 0
 3 1



A  8 0


 9 4
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.6 Inverse matrix





AA-1 = I
A-1A=I
Necessary for
matrix to be
square to have
inverse
If an inverse exists
it is unique
(A')-1=(A-1)'
•
•
•
•
•
Ax=d
A-1A x = A-1 d
Ix = A-1 d
x = A-1 d
Solution depends on
A-1
• Linear independence
• Determinant test!
Chiang_Ch4.ppt Stephen Cooke U. Idaho
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4.2 Matrix inversion

It is not possible to • In matrix algebra
AB-1  B-1 A. Thus
divide one matrix by
writing does not
another. That is, we
clearly identify
can not write A/B.
whether it
This is because for
represents
two matrices A and
AB-1 or B-1A
B, the quotient can
• Matrix division is
-1
be written as AB or
matrix inversion
-1
B A.
• (topic of ch. 5)
Chiang_Ch4.ppt Stephen Cooke U. Idaho
32
Ch. 4 Linear Models & Matrix Algebra
Matrix algebra can be
used:
a. to express the system
of equations in a
compact notation;
b. to find out whether
solution to a system of
equations exist; and
c. to obtain the solution if it
exists. Need to invert the
A matrix to find the
solution for x*

Ax  d
1
x A d
*
adjA
A 
det A
adjA
*
x 
d
A
1
Chiang_Ch4.ppt Stephen Cooke U. Idaho
33
4.1Vector multiplication
(inner or dot product)
y  c1 z1  c2 z 2  c3 z 3  c4 z 4
4
y   ci z i
i 1
y  c 1 c2
c3
y = c'z
 z1 
z 
c4  2 
 z3 
 
 z4 
1x1 = (1x4)( 4x1)
Chiang_Ch4.ppt Stephen Cooke U. Idaho
34
4.2 Σ notation



Greek letter sigma (for sum) is another convenient way of
handling several terms or variables
i is the index of the summation
What is the notation for the dot product?
j
3
a b
a1b1 +a2b2 +a3b3 =
c11
i 1
c12
i i
c13   a11b11  a12b21
k 1
1k
bk1
a
k 1
1k
a11b13  a12b23 
2
2
2
a
a11b12  a12b22
bk 2
Chiang_Ch4.ppt Stephen Cooke U. Idaho
a
k 1
1k
35
bk 3