ECON 117 Mathematical Economics
Spring 2008
Week 1
I. The Basics on Sets and Real Functions
1. Vectors
(a) Denote a = (1,1) in the following X-Y space. y
x
(b) Denote 2 a in the above diagram.
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ECON 117 Mathematical Economics (c) a
ECON 117 Mathematical Economics
Spring 2008
Week 2
3.2
Differentiation of a Real Function
Y
Def) Real function: f : X Y is a real function if
Q) Draw the following real function:
f = 2 x2 8x + 2 y = 2x + 2
(1)
(2)
Def) Differentiation of a
ECON 117 Mathematical Economics
Spring 2008
Week 3
3.5
Total Derivatives and Partial Derivatives
, i.e., f :
2
Def) For z = f ( x, y ) , x, y , z
,
z (partial derivative of z with respect to x ) is the marginal rate of change of z with resp
ECON 117 Mathematical Economics
Spring 2008
Week 4
Def) Linear combination (broad meaning) Let {xi } = {x1 , , x m } where If y =
xi
n
.
x ,
i =1 i i i
m
, then we call y is a linear combination of {x i } or y is linearly
dependent o
ECON 117 Mathematical Economics
Spring 2008
Week 6
3.1
The Rank of a Matrix
A , ( A ) , is the maximum number of rows (or columns) in A that are
Def) The rank of matrix
linearly independent.
Theorem) A , the maximum number of linearly indepe
ECON 117 Mathematical Economics
Spring 2008
Week 5
Def) Space We call
A that is spanned by k (< n) independent n-dimensional vectors has dimension k . A is a subspace of
n
.
2 Q) Find the space that is spanned by x = 1
2
.
Q) Find the sp
ECON 117 Mathematical Economics
Spring 2008
Week 7
3.4 The Properties of Determinant
1. A = A
2. If B is obtained from A by interchanging any two rows (or columns) of A , then
B = A
1 2 1 Ex) A = 2 1 2 0 1 1
2 1 2 B = 1 2 1 0 1 1
ECON 117 Mathematical Economics
Spring 2008
Week 8 (Half)
3.6 Cramers Rule
Q) x + y + z = 4 x z =1
x y
=1
Find x, y, z by using the Cramers Rule.
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ECON 117 Mathematical Economics
Spring 2008
4.
Eigenvalue Problems
4.1 Eigenva
ECON 117 Mathematical Economics
Spring 2008
Week 9
4.2 Properties of eigenvalues and eigenvectors
Let A be symmetric
( nn )
(1)
Eigenvalues are real
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ECON 117 Mathematical Economics
Spring 2008
(2)
The two eigenvectors x i , x j
ECON 117 Mathematical Economics
Spring 2008
III. Mathematical Programming
1. Maximization and Minimization without Constraint 1.1 One variable case: f :
( f is continuous and continuously differentiable)
Q) Find all local max points and a global
ECON 117 Mathematical Economics
Spring 2008
Week 10
1.2
Multiple variable case: y = f (x), f :
n
Def) f ( x) has a local max at x 0 if
f (x0 ) > f (x0 + h x) x 0
n
, small h 0
Theorem) f ( x) has a local max at x 0 if
f f f Df (x
ECON 117 Mathematical Economics
Spring 2008
Week 12
Q) Solve the following program
max x1 x2
x1, x2
s.t.
x1 + x2 = 1
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ECON 117 Mathematical Economics
Spring 2008
2.2
Ex)
Utility Maximization
( x1 , x2 ) .
There are two goods
ECON 117 Mathematical Economics
Spring 2008
Week 11
2.
2.1
f:
Ex)
Optimization with equality constraints
One constraint case
n n
f , g : continuous & differentiable
g: b
[OP: Original Program]
x1 , x2 , xn
max f ( x1 , x2 , , xn ) s.t. g (
ECON 117 Mathematical Economics
Spring 2008
Week 13
3.
Optimization with inequality constraints
3.1 The basic idea
Q)
y = f ( x ), f :
x
is convex in x
Let x solve min f ( x )
Find the F.O.C. for x
x1
Q) Let x solve:
min f ( x )
x
s.t.