University of Nottingham

1
School of Economics
University of Nottingham
Mathematical Economics 1st Year
Undergraduate 2007-2008
Dr Alex Possajennikov
Week 4 Linear Algebra
2
Linear Algebra
Examples
Demand function
Q ad - bdP
linear
Supply function
Q as bsP
linear
Equilibrium price and quantity both equations
are satisfied
Two interrelated markets
Demand for good 1
Q1D 10 - 2P1 3P2
Supply of good 1
Q1S -1 4P1 P2
Demand for good 2
Q2D 12 - 3P1 2P2
Supply of good 2
Q2S -3 2P1 3P2
Equilibrium Q1D Q1S, Q2D Q2S all equations
are satisfied
3
Linear Algebra
A macroeconomic model of an economy
Total output
Y C I G0
Consumption
C 50 0.6Y
Investment
I 20 0.2Y
Government
G0 90
Equilibrium all equations are satisfied
What are the equilibrium values of output,
consumption, and investment?
If government spending G0 increases, how does it
affect equilibrium output?
4
Linear Algebra
Systems of linear equations
a11x1 a12x2 a1nxn b1 a21x1 a22x2
a2nxn b2 am1x1 am2x2 amnxn bm
m equations
n unknowns
aij, bi are parameters
Solution such values for x1,,xn that satisfy
all m equations
A system can have zero, one, or many solutions
The macroeconomic model
Y C I 90 C 50 0.6Y I 20 0.2Y
Y - C - I 90 - 0.6Y C
50 - 0.2Y I 20
5
Linear Algebra
Solving systems of linear equations two basic
methods
Substitution method
Y - C - I 90 G0 - 0.6Y C
50 - 0.2Y I 20
Y C I 90 C 50 0.6Y I 20 0.2Y
Gaussian elimination method
Y - C - I 90 - 0.6Y C
50 - 0.2Y I 20
Solution Y 800, C 530, I 180
Does a system have a (unique) solution?
How does the solution depend on the parameters
(G0)?
6
Linear Algebra
Vectors
An n-vector is a column of n elements
Matrices
An n ? m matrix is an rectangular array of
elements with n rows and m columns
n, m are dimension(s), order, size(s) of matrix
A vector can be seen as matrix with n rows and 1
column (n ? 1)
7
Linear Algebra
Economic examples
Price (row) vector
p (1 3 4)
Consumption vector
Production matrix 3 products requiring 4 inputs
Transpose
aij aji
Transpose of n ? m matrix is a m ? n matrix
If x is a vector (n ? 1 matrix), x is a row
vector (1 ? n matrix)
(A) A
Symmetric matrix A A
8
Linear Algebra
Operations
Sum of matrices of the same dimensions
Multiplication by scalar
(A B) C A (B C) A B B A
?(A B) ?A ?B (? ?)A ?A ?A
(A B) A B
Example
9
Linear Algebra
Product of matrices
The product of n ? m matrix A and m ? l matrix B
is n ? l matrix C with elements cij ai1b1j
aimbmj
C AB
Conformability only n ? m and m ? l matrices can
be multiplied
n ? m matrix and m ? l matrix give n ? l matrix
The product of a row vector (1 ? n matrix) and a
column vector of the same size (n ? 1 matrix) is
a scalar (1 ? 1 matrix)
c11 a? b a1b1 anbn
Scalar (inner) product of vectors
10
Linear Algebra
The product of n ? m matrix A and m ? l matrix B
is n ? l matrix C with elements cij ai1b1j
aimbmj
Take row a(i) of A and column b(j) of B and find
a(i)? b(j)
Example
11
Linear Algebra
Properties of the product of matrices
AB ? BA in general
AB 0 does not imply that A 0 or B 0
Example
(AB)C A(BC)
A(B C) AB AC (A B)C AC BC
(AB) BA
AIn A InA A
Identity matrix
12
Linear Algebra
Economic examples
Consumption vector 3 goods
Price (row) vector 3 goods
p (p1 p2 p3)
Total expenditure
p1x1 p2x2 p3x3 p? x
Production matrix 3 products, 2 inputs
Production plan of 3 products
Input requirements
Input 1 a11x1 a12x2 a13x3
Ax
Total expenditure
p? Ax
13
Linear Algebra
Matrices and systems of linear equations
a11x1 a12x2 a1nxn b1 a21x1 a22x2
a2nxn b2 am1x1 am2x2 amnxn bm
m n
n 1
m 1
Ax b
14
Linear Algebra
Example
Y - C - I 90 - 0.6Y C
50 - 0.2Y I 20
Does a system have a (unique) solution?
How does the solution depend on the parameters?
Y - C - I G0 - ? Y C
50 - 0.2Y I 20
For which values of ? and G0 does the system have
a (unique) solution?
How does it depend on ?? How does it depend on G0?