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```							 Chiang & Wainwright
Mathematical Economics
Chapter 4
Linear Models and Matrix Algebra

Chiang_Ch4.ppt Stephen Cooke U. Idaho   1
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   2
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   3
One Commodity Market Model
(2x2 matrix)
   Economic Model                         ac
(p. 32)
P  *

1) Qd=Qs                                   bd
2) Qd = a – bP (a,b >0)                    ad  bc
3) Qs = -c + dP (c,d >0)               Q 
*

bd
 Find P* and Q*
Scalar Algebra                        Matrix Algebra
Endog. ::      Constants
1 b  Q   a 
4) 1Q + bP = a                      1  d   P    c 
5) 1Q – dP = -c                              
Ax  d
x*  A1d
Chiang_Ch4.ppt Stephen Cooke U. Idaho   4
One Commodity Market Model
(2x2 matrix)
Matrix algebra

1 b  Q   a 
1  d   P    c 
         
Ax  d
1
Q  1 b   a 
*

 *         c 
 P  1  d   
1
x A d
*

Chiang_Ch4.ppt Stephen Cooke U. Idaho   5
General form of 3x3 linear matrix
Scalar algebra form
parameters & endogenous variables                     exog. vars
& const.
a11x        + a12y               + a13z                 = d1
a21x        + a22y               + a23z                 = d2
a31x        + a32y               + a33z                 = d3
Matrix algebra form
parameters               endog.          exog. vars.
vars           & constants
 a11      a12         a13   x   d1 
a         a22               y   d 
a23     2 
 21
 a31
          a32         a33   z   d 3 
   
Chiang_Ch4.ppt Stephen Cooke U. Idaho                6
1. Three Equation National Income Model
(3x3 matrix)

   Let (Exercise 3.5-1, p. 47)
Y = C + I0 + G0
C = a + b(Y-T)     (a > 0, 0<b<1)
T = d + tY         (d > 0, 0<t<1)
 Endogenous variables?
 Exogenous variables?
 Constants?
 Parameters?
 Why restrictions on the parameters?

Chiang_Ch4.ppt Stephen Cooke U. Idaho   7
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   8
6. Three Equation National Income Model
Exercise 3.5-1 p. 47
Parameters &                               Exog.
Endogenous vars.                            vars.
   Given
Y              C           T                &cons.
Y = C + I0 + G0
1Y          -1C +0T                  =      I0+G0
C = a + b(Y-T)        -bY +1C +bT                           =        a
T = d + tY             -tY +0C +1T                          =        d
   Find Y*, C*, T*     1  1 0 Y   I 0  G0 
 b 1 b  C    a 
Ax  d                                
  t 0 1 T   d 
                      
x*  A1d
Chiang_Ch4.ppt Stephen Cooke U. Idaho         10
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 
*

 *             a 
C    b 1 b          
T *    t 0 1  d 
                      
1
x A d
*

Chiang_Ch4.ppt Stephen Cooke U. Idaho   11
3. Two Commodity Market Equilibrium
Section 3.4, p. 42
 Section 3.4, p. 42            Scalar algebra
 Given                         1Q1 +0Q2 +2P1 - 1P2 = 10
Qdi = Qsi,     i=1, 2
1Q1 +0Q2 - 3P1 +0P2= -2
Qd1 = 10 - 2P1 + P2
0Q1+ 1Q2 - 1P1 + 1P2= 15
Qs1 = -2 + 3P1
Qd2 = 15 + P1 - P2              0Q1+ 1Q2 +0P1 - 2P2= -1
Qs2 = -1 + 2P2                  1      0 2         1   Q1   10 
 Find Q1*, Q2*, P1*, P2*       1
       0 3       0  Q2   2
    
Ax  d                      0      1       1 1   P   15 
1
                         
x*  A1d                   0      1      0  2  P2    1 

Chiang_Ch4.ppt Stephen Cooke U. Idaho   12
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   P   15 
1
                    
0 1 0  2  P2    1
Ax  d
1
Q1*  1     0 2  1               10 
 * 
Q 2   1   0 3 0 

  2
 
 P *  0    1 1 1                15 
 1*                              
 P2  0
            1 0  2                1
x*  A1d        Chiang_Ch4.ppt Stephen Cooke U. Idaho   13
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
                                                   
an1     x1        a n 2 x2              anm xn  d n

Chiang_Ch4.ppt Stephen Cooke U. Idaho      14
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    a22  a2 m   x2  d 2 
 21                 
               
                   
an1 an 2  anm   xn  d n 
Ax  d
x*  A1d
Chiang_Ch4.ppt Stephen Cooke U. Idaho     15
3.4 Solution of a General-equation
System
  Why?
 Given (p. 44)
2x + y = 12            4x + 2y =24
2(2x + y) = 2(12)
4x + 2y = 24
 one equation with two
Find x*, y*               unknowns
y = 12 – 2x            2x + y = 12
4x + 2(12 – 2x) =      x, y
24
Conclusion:
4x +24 – 4x = 24          not all simultaneous equation
0=0?                      models have solutions
indeterminant!

Chiang_Ch4.ppt Stephen Cooke U. Idaho   16
4.3 Linear dependence
v1  5 12 
'

v2  10 24 
'
   A set of vectors is
linearly dependent if any                      5 10   v1 
'
12 24    ' 
one of them can be                                   v 2 
 
expressed as a linear                         2v1  v2  0 /
/    /

combination of the
remaining vectors;                            2      1     4
v1    v2    3   
otherwise it is linearly                      7      8     5 
independent.
3v1  2v2
    Dependence prevents
solving the system of                     6 21  2 16 
equations. More                           4 5  v3
unknowns than                            3v1  2v2  v3  0
independent equations.
Chiang_Ch4.ppt Stephen Cooke U. Idaho                 17
4.2 Scalar multiplication
2         4 16 32
8             48 8 
6         1         
1 2         4 1 4 1 2
8 6            3 4 1 8 
          1           

 a11      a12    a11              a12 
 1                  a
a 21      a 22   21                a 22 


Chiang_Ch4.ppt Stephen Cooke U. Idaho   18
4.3 Geometric interpretation (2)
x2

 Scalar                                    6

multiplication                             5

 Source of linear
4
6 4  2U
dependence
3

2
3 2  U
1
x1

-4       -3   -2   -1        1   2   3     4    5     6


 1 U   3  2              -2

Chiang_Ch4.ppt Stephen Cooke U. Idaho                                 19
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 
   Matrix addition                      7 9  0 2  7 11
                      
A2 x 2  B2 x 2 C 2 x 2

   Matrix subtraction 2 1  1 0  1 1
7 9 2 3 5 6
               

Chiang_Ch4.ppt Stephen Cooke U. Idaho           20
4.3 Geometric interpretation
x2

 v' = [2 3]                     5

 u' = [3 2]                     4

 v'+u' = [5 5]                  3

2

1
x1

1    2    3   4   5

Chiang_Ch4.ppt Stephen Cooke U. Idaho               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   22
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 
aB  a11a12 
c11   c12 c13                           b21
          22   b23 


c

 a11b11  a12b21 a11b12  a12b22 a11b13  a12b23 
• Dimensions: a(1x2), B(2x3), c(1x3)
Chiang_Ch4.ppt Stephen Cooke U. Idaho            23
4.3 Notes on Vector Operations
Multiplication of Vectors
Geometric Interpretation of Vector Operations
Linear Dependence
Vector Space

An [m x 1] column vector u
3                           and a [1 x n] row vector v,
u  
2 x1
2                           yield a product matrix uv of
dimension [m x n].
v  1 4 5
1x 3

 3                              31215 
uv                             5  
2 8 10 
1             4
2 x3    2                                      

Chiang_Ch4.ppt Stephen Cooke U. Idaho      24
4.4 Laws of Matrix Addition &
Multiplication
Matrix Multiplication

   Commutative law: A + B = B + A

 a11 a12  b11 b12   a11  b11 b12  a12 
A B               b       a  a
 a 21 a 22   21 b22   21 21 b22  a 22 

b11 b12  a11 a12   b11  a11 b12  a12 
B A          b       b  a
b21 b22   21 b22   21 21 b22  a 22   
Chiang_Ch4.ppt Stephen Cooke U. Idaho   25
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  6 7 
3 4            
10  26 1 1  27  12 13 
AB                              24 25
30  46 3 1  47         
01   13 02   14  3  4
BA                                   27 40 
 61  73     62  7 4           
Chiang_Ch4.ppt Stephen Cooke U. Idaho   26
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
P     PAB  .7 .3
M   AA         
 PBA PBB  .4 .6
        
At the end of one year, how many employees will be at each plant?
PAA     PAB 
A1    B1   x M
/
 A0    B0                A0 PAA  A0 PBA    B0 PAB  B0 PBB 
PBB 
0
 PBA        
.7 .3
 100 100                     .7 *100  .4 *100,         .3 *100  .6 *100
.4 .6
 110 90

Chiang_Ch4.ppt Stephen Cooke U. Idaho                         27
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
P     PAB  .7 .3
M   AA         
 PBA PBB  .4 .6
        
At the end of two years, how many employees will be at each plant?
PAA     PAB 
A1   B1   x M
/
 A0   B0                110 90
PBB 
0
 PBA        
PAA     PAB  PAA PAB 
A2   B2   x 0 M 2
/
 A0   B0 
 PBA    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               28
4.5 Identity and Null Matrices
Identity Matrices
Null Matrices
Idiosyncrasies of Matrix Algebra

 Identity Matrix is a
1           0
0               or
square matrix and also                                    1
it is a diagonal matrix
with 1 along the                              1           0    0
diagonals                                     0           1    0 etc.
similar to scalar “1”                                           
 Null matrix is one in                         0
            0    1

which all elements are
zero                                          0           0    0
similar to scalar “0”                       0                 
Both are “idempotent”                                       0    0
matrices                                      0
            0    0

A = AT        and
A = A2 = A3 = …
Chiang_Ch4.ppt Stephen Cooke U. Idaho             29
4.6 Transposes & Inverses
Properties of Transposes Inverses and Their Properties
Inverse Matrix and Solution of Linear-equation Systems

Transposed matrices                          3 8  9

A       
 (A')' = A                                    1 0   4
 Matrix rotated along its
principle major axis
 3 1
(running nw to se)
A   8 0
 Conformability changes                                     
unless it is square                                    9 4
      

Chiang_Ch4.ppt Stephen Cooke U. Idaho       30
4.6 Inverse matrix
   AA-1 = I                         • Ax=d
   A-1A=I                           • A-1A x = A-1 d
   Necessary for                    • Ix = A-1 d
matrix to be                     • x = A-1 d
square to have
• Solution depends on
inverse
A-1
   If an inverse exists
• Linear independence
it is unique
• Determinant test!
   (A')-1=(A-1)'
Chiang_Ch4.ppt Stephen Cooke U. Idaho   31
4.2 Matrix inversion
   It is not possible to • In matrix algebra
divide one matrix by      AB-1  B-1 A. Thus
another. That is, we      writing does not
can not write A/B.        clearly identify
This is because for       whether it
represents
two matrices A and
AB-1 or B-1A
B, the quotient can
be written as AB  -1 or • Matrix division is
-1A.                    matrix inversion
B
• (topic of ch. 5)
Chiang_Ch4.ppt Stephen Cooke U. Idaho   32
Ch. 4 Linear Models & Matrix Algebra
 Matrix algebra can be
used:
Ax  d
1
a. to express the system
of equations in a
x A d
*

compact notation;
b. to find out whether                      A 
solution to a system of                         det A
equations exist; and
c. to obtain the solution if it                  adjA
exists. Need to invert the                x 
*
d
A matrix to find the                            A
solution for x*

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                      z1 
z 
y  c 1 c2        c3     c4  2 
 z3 
 
 z4 
y = c'z

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
a1b1 +a2b2 +a3b3 =         a b
i 1
i i

c11   c12   c13   a11b11  a12b21 a11b12  a12b22                 a11b13  a12b23 
2                         2

a                        a
2

a
k 1
1k   bk1
k 1
1k   bk 2
k 1
1k   bk 3

Chiang_Ch4.ppt Stephen Cooke U. Idaho                      35

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