Chapter 1: Introduction
z How to build an economic model? (Hal R.Varian)
1. An economic model: an idealization of the reality, but not the reality.
2. Why do we need an economic model?
3. How to build an economic model? z Getting ideas from reality: An interesting one? Is the idea worth pursuing? z Don’t look at the literature too soon z Simplifying and Generalizing your model z Making mistakes: team work z Searching the literature z Giving a seminar
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z Mathematics 1. Set theory
A Set (A) is a collection of objects called elements (a): a ∈ A The empty set is Φ , and the universal set is U .
Binary operations on set: 1. 2. 3. 4. the union of A and B is the set A ∪ B = {x : x ∈ A or x ∈ B} the intersection of A and B is A ∩ B = {x : x ∈ A and x ∈ B} the difference of A and B is
A \ B = {x : x ∈ A and x ∉ B}
the symmetric difference of A and B is A∆B = ( A ∪ B ) \ ( A ∩ B )
The complement of A is Ac = U \ A
Theorem 1
Let A, B and C be sets,
1. 2.
A \ (B ∪ C ) = ( A \ B ) ∩ ( A \ C ) A \ (B ∩ C ) = ( A \ B ) ∪ ( A \ C )
A
C
B
Corollary 2 (DeMorgan’s Law)
( A ∪ B )c = Ac ∩ B c
and ( A ∩ B ) = Ac ∪ B c
c
2
Generalizing theorem 1 to theorem 3: A\⎛ ⎜ ∪ Si ⎞ ⎟ = ∩( A \ S i ) and A \ ⎛ ⎜ ∩ Si ⎞ ⎟ = ∪( A \ S i ) ⎝ i∈I ={1, } ⎠ i∈I ⎝ i∈I ={1, } ⎠ i∈I
Given any set A, the power set of A, written by Ρ( A) is the set consisting of all subsets of A, i.e., Ρ( A) = {B | B ⊂ A}
Question : If a set A has n elements, how many elements are there in Ρ( A) ?
The Cartesian Product of two sets A and B (also called the product set or cross product) is defined to be the set of all points (a, b ) where a ∈ A and b ∈ B . It is denoted A × B .
Example:
R2 ≡ R × R
R n ≡ R × R × R × ...R = {( x1 , x2 ,..., xn ) | xi ∈ R, i = 1,2,...n} ,
where
the
element
(x1 , x2 ,...xn ) of
R n is an n-dimensional ordered vector. We denote: x
S ⊂ R n is a convex set if ∀x, y ∈ S , we have tx + (1 − t ) y ∈ S for all t ∈ [0,1]
The intersection of convex sets is convex, but the union of them is not.
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2. Topology
A metric space is a set S with a global distance function (the metric d ) that, for every points
x and y in S , gives the distance between them as a nonnegative real number d (x, y ) . A
metric space must satisfy: 1. d (x, y ) = 0 iff x = y 2. d (x, y ) = d ( y, x ) 3. d ( x, y ) + d ( x, z ) ≥ d (x, z ) Example: Euclidean metric in R 2 : d (x, y ) =
(x1 − y1 )2 + (x2 − y2 )2
Open and Closed ε − Balls: let ε be a real positive number, then 1. The open ε − ball with center x 0 and radius ε > 0 is
Bε x 0 = x ∈ R n | d x 0 , x < ε
( ) {
(
) }
) }
2. The closed ε − ball with center x 0 and radius ε > 0 is
Bε x 0 = x ∈ R n | d x 0 , x ≤ ε
( ) {
(
Open and Closed sets in R n : A set S ⊂ R n is open if ∀x ∈ S , ∃ε > 0,
Bε ( x ) ⊂ S .
A set S ⊂ R n is closed if its complement, S c , is open.
Some important properties of open and closed sets: 1. The union of open sets is open. 2. The intersection of any finite number of open sets is open. 3. The union of any finite number of closed sets is closed.
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4. The intersection of closed sets is closed.
Question: what if the collection is infinite for 2. and 3.?
Theorem 4: Every open set is a collection of open balls.
Bounded sets in R n : A set S ⊂ R n is bounded if ∃ε > 0 and x ∈ R n , S ⊂ Bε ( x ) .
Let S ⊂ R be a nonempty set of real numbers: 1. Any real number l is a lower bound if ∀ x ∈ S , 2. Any real number u is an upper bound if ∀ x ∈ S ,
x ≥ l . The set is bounded from below. x ≤ u . The set is bounded from above.
3. The largest number among lower bounds is called the greatest lower bound of S. 4. The smallest number among upper bounds is called the least upper bound of S.
A bounded set is bounded both from below and above. We can show that for any bounded subsets of the real line, there always exists a g.l.b. and l.u.b.
Let S ⊂ R be a bounded set and let a be the g.l.b of S and b be the l.u.b. of S, then we have: 1. If S is open, then a ∉ S and b ∉ S 2. if S is closed, then a ∈ S and b ∈ S
Compact sets : A set is compact if it is closed and bounded.
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3. Relations and functions
Consider an ordered pair (s, t ) that associated an element s ∈ S to another element t ∈ T . Any collection of such ordered pairs is said to constitute a binary relation between the sets S and
T . Note that a binary relation R is a subset of the cross product S × T .
Some properties of relations: 1. The relation is complete if either xRy or yRx. 2. 3. 4. transitive if xRy and yRz implies xRz. reflexive if xRx. symmetric if xRy ⇔ yRx
Examples: The preference relation ( ≿ ) is complete, transitive and reflexive.
The function is a mapping from one set D (domain) to another set R (range) denoted as:
f :D→ R
The image of f : I ≡ {y | y = f ( x)} ⊂ R The inverse image of a set of points S ⊂ R is: f −1 (S ) ≡ {x | x ∈ D, f ( x ) ∈ S } The graph of f : G ≡ {(x, y ) | x ∈ D, y = f ( x ) ∈ S } A function is a surjective function if
the range ran( f ) = R A function is an injective function (or one to one) if f (a) = f (b) implies a = b A function is bijective if it is both surjective and injective. In a sense, the domain and the range must have the same number of elements.
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