Unformatted text preview: uilibrium
conditions) we can reduce the system of six equations to that of only two:
conditions) ( a0 −b0 ) + ( a1 −b1 ) P + ( a2 −b2 ) P2 = 0
(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
We solve this system again by eliminating the variables:
Find P from the first equation: P * = c2γ 0 −c0γ 2
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
these prices to make sense?
19 General Market Equilibrium
In case we have n commodities, we are talking about a model of
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 Pi = Pi ( a1 , a2 ,..., am ) , i = 1..n, m ≠ n
* * *
Equilibrium prices P will in the end be functions of the model’s
parameters ai The number of the model’s parameters may not be equal to the number
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
solution—consistency problem (2) may have more than one solution—
Consistency problem: the two equations below
are contradicting each other
+ = 9 y =9−x y = 8− x
8 The two lines are parallel to each
other, so no intersection point
our system is inconsistent
x 21 Consistency and Independence
Independence problem: there may be fewer (independent) equations
compared to the number of parameters.
There are an infinite number
12 x +y =
of (x,y) that satisfy these two equations.
4 x +2 y =24
The intersection of a straight
line with itself makes an
infinite number of points.
y = 12 − 2 x
2 y = 24 − 4 x 6 x These two equations are functionally
dependent, so the system of these two
equations has an infinite number of
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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