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ECON 301 – LECTURE #7
WALRAS LAW
Is there a particular reason that we can solve the equilibrium price ratio from the market
equilibrium condition of either good X or good Y?
Remember, in our example we got the same value for
_ P
X
__
= 3/5 from the market
P
Y
equilibrium condition of both good X and good Y.
This is not a coincidence!
Indeed, it
comes from a general theoretical result, namely, Walras Law.
Walras Law can be stated in the context of a pure exchange economy as follows:
Walras Law states that if the market for good X is already in equilibrium, then so is the
market for good Y.
Therefore, if we have already solved for the equilibrium price ratio
from the market equilibrium equation of good X, then we do not need to solve the other
market equilibrium equation since it will be in equilibrium as well (and give us the same
answer in terms of the equilibrium price ratio…needless suffering).
The logic behind Walras Law lies in the close relationship between the two concepts of
consumer equilibrium and market equilibrium.
It is not difficult to show (prove) that
Walras Law works within the context of the pure exchange economy.
To do so, we start
with the individual budget constraints
P
X
X
A
+ P
Y
Y
A
=
P
X
!
X
A
+ P
Y
!
Y
A
P
X
X
B
+ P
Y
Y
B
=
P
X
!
X
B
+ P
Y
!
Y
B
If we sum these two budget constraints together, we get…
P
X
(X
A
+ X
B
) + P
Y
(Y
A
+
Y
B
)
=
P
X
(
!
X
A
+
!
X
B
) + P
Y
(
!
Y
A
+
!
Y
B
)
[market X]
[market Y]
[endowment X]
[endowment Y]
P
X
X + P
Y
Y
=
P
X
!
X
+ P
Y
!
Y
P
X
(X –
!
X
) + P
Y
(Y –
!
Y
) = 0
(10)
Now, let’s suppose the market for good X is in equilibrium already.
That is the aggregate
demand = the aggregate supply of X.
If the market for one good (say, good Y) is in equilibrium, then the market for the
other good (i.e. good X) is also in equilibrium.
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X
A
+ X
B
=
!
X
[market X]
X =
!
X
X –
!
X
= 0
“excess demand” = 0
Thus,
P
X
(X –
!
X
) = 0
“value of excess demand” = 0
Substituting this result into (10), we get:
P
X
(X –
!
X
) + P
Y
(Y –
!
Y
) = 0
(10)
P
Y
(Y –
!
Y
) = 0
“value of excess demand” = 0
Y –
!
Y
= 0
“excess demand” = 0
Y =
!
Y
“demand” = “supply”
And the market for Y is in equilibrium!
What we have done here is to show that if we
know the market for X is in equilibrium then we can derive that the market for Y is also
in equilibrium.
What do I mean by excess demands?
We can define excess demands for both goods by
saying that “market demand – market supply = excess demand”
1
.
Z
X
= X –
!
X
Z
Y
= Y –
!
Y
and Walras Law can be written as
P
X
Z
X
+ P
Y
Z
Y
= 0
(11)
In this context, we can state Walras Law as:
For those of you who are wondering, Walras Law works for any number of markets (not
just two).
The statement of Walras Law for the general case of multiple markets is:
1
Of course, if the expression is negative we have excess supply.
The value of all market excess demands must sum to zero.
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 Winter '09
 COREYVANDEWAAL
 Market Equilibrium

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