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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 Addition
   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;
                                              1  adjA
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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