Unformatted text preview: uilibrium
By
conditions) we can reduce the system of six equations to that of only two:
conditions) ( a0 −b0 ) + ( a1 −b1 ) P + ( a2 −b2 ) P2 = 0
1 1
(α0 − β0 ) + (α1 − β1 ) P + (α2 − β2 ) P2 = 0
We have a system of two equations with six parameters: is there a problem?
18 Reducing the Number of Parameters
We can reduce the number of parameters by introducing new parameters: ci ≡ ai − bi , i = 0,1,2 γ i ≡ αi − βi , i = 0,1,2
Our system then becomes c1 P + c2 P2 = −c0
1 γ1 P + γ 2 P2 = −γ 0
1
We solve this system again by eliminating the variables:
We
Find
Find P from the first equation: P * = c2γ 0 −c0γ 2
1
1
c1γ 2 − c2γ1 Substitute P
Substitute 2 *
into the second equation to find P2 = c0γ 1 − c1γ 0
c1γ 2 − c2γ 1 What should be the restrictions on the parameters for
What
these prices to make sense?
these
19 General Market Equilibrium
In case we have n commodities, we are talking about a model of
In
general market equilibrium. We will then have n equilibrium conditions (one
for each good):
for Qdi − Qsi = Qdi ( P , P2 ,..., Pn ) − Qsi ( P , P2 ,..., Pn ) = 0, i = 1..n
1
1 Pi = Pi ( a1 , a2 ,..., am ) , i = 1..n, m ≠ n
* * *
Equilibrium prices P will in the end be functions of the model’s
Equilibrium
i
parameters ai The number of the model’s parameters may not be equal to the number
The
may
of the model’s equations representing equilibrium conditions.
of 20 Consistency and Independence
It may happen that a particular equation system (1) may not even have a
It
solution—consistency problem (2) may have more than one solution—
solution—consistency
independence problem
Consistency problem: the two equations below
the
are contradicting each other
are
x
y
8
+ =
y x
y
9
+ = 9 y =9−x y = 8− x
8 The two lines are parallel to each
The
other, so no intersection point
exists:
exists:
our system is inconsistent
our
inconsistent
x 21 Consistency and Independence
Independence problem: there may be fewer (independent) equations
there
compared to the number of parameters.
compared
There are an infinite number
There
2
12 x +y =
of (x,y) that satisfy these two equations.
4 x +2 y =24
The intersection of a straight
y
line with itself makes an
infinite number of points.
infinite
12
y = 12 − 2 x
2 y = 24 − 4 x 6 x These two equations are functionally
These
dependent, so the system of these two
dependent so
equations has an infinite number of
solutions.
solutions.
22...
View
Full
Document
This note was uploaded on 09/14/2013 for the course STAT 1001 taught by Professor Kim during the Fall '11 term at Yonsei University.
 Fall '11
 Kim

Click to edit the document details