文档介绍：Chapter 1: Introduction
z How to build an economic model? (Hal )
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
1
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. the union of A and B is the set A∪ B = {x : x∈ A or x∈ B}
2. the intersection of A and B is A∩ B = {x : x∈ A and x∈ B}
3. the difference of A and B is A \ B = {x : x∈ A and x∉ B}
4. the symmetric difference of A and B is A∆B = (A∪ B)\ (A∩ B)
plement of A is Ac = U \ A
Theorem 1 Let A, B and C be sets,
1. A \ ()()()B ∪C = A \ B ∩ A \ C
2. A \ ()()()B ∩C = A \ B ∪ A \ C
A
C B
Corollary 2 (an’s Law)
()A∪ B c = Ac ∩ Bc and ()A∩ B c = Ac ∪ Bc
2
Generalizing theorem 1 to theorem 3:
⎛⎞⎛⎞
A \ ⎜∪ Si ⎟= ∩()A \ Si and A \ ⎜∩ Si ⎟= ∪()A \ Si
⎝ i∈I ={}1,2,3... ⎠ i∈I ⎝ i∈I ={}1,2,3... ⎠ i∈I
Given any set A, the power set of A, written by Ρ(A) is the set consisting of all subsets of A,
., 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: R 2 ≡ R× R
n
R ≡ R× R× R×...R = {}()x1, x2 ,..., xn | xi ∈ R,i =1,2,...n , where the element
n
()x1, x2 ,...xn of R 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