Handout 12th Lecture, ECON 5200
Positive theory of general equilibrium
Kjell Arne Brekke
November 12, 2010
1
Positive theory of equilibrium
1.1
The excess demand function
The excess demand function of consumer i is
zi (p) = xi (p; p ! i )
!i
and the (aggregate) excess demand for the economy is
X
z(p) =
zi (p)
i
Proposition 1 If preferences are continuous, strictly convex and strongly monotone, and
if also ! i
0 for all i, Then the for all p
0 the excess demand satis…es
1. z( ) is continuous
2. z( ) is homogenous of degree zero
3. pz(p) = 0 for all p. (Walras law)
4. There is an s > 0 such that zl (p) >
s for every l and all p:
5. If pn ! p where p 6= 0 and pl = 0 then
maxfz1 (pn ); :::; zL (pn )g ! 1
1.2
Equilibriuim in the exchange economy
An exchange economy is one where
L
R+
Y =
Now
Proposition 2 If preferences are continuous, convex and locally nonsatiated, p
stitute a walrasian equilibrium if and only if
z(p)
1
0:
0 con-
1.3
Excistence of equilibrium
Proposition 3 Suppose that z(p) is de…ned for p
excess demand function, then
z(p) = 0
0, and satis…es the properties of an
has a solution
1.4
The Sonnenshein - Debreu - MasCollell, Anything goes
While an equilibrium exist, what about stability and uniqueness. Consider the two dimensional case, then since
p1 z1 (p) + p2 z2 (p) = 0
everything is de…ned by z1 (p) and we may simplify by normalizing such that p2 = 1.
Problem 4 Can we say anything about z1 (p1 ; 1) as p1 ! 0 and p1 ! 1? (Guess, based
on economic intuition, and then try to use the properties of excess demand derived above
to prove your guess.)
Equilibrium is thus the prices such that z1 (p) = 0. It follows from the solution to the
problem that equilibrium exist if z(p) is continuous. But is it unique, even locally? We
can easily plots potential functions where it is not. (E.g. Fig 17.D.3)
What about stability? We then need to model how the economy approaches an equilibrium. One approach is that prices increases when there is excess demand and decreases
when there is excess supply
dpl
= cl zl (p)
dt
again this is all de…ned by the excess demand function. Within the class of functions that
satis…es the properties 1-5 above, there is no guarrantee that this "Tatonnement" process
will converge.
An even larger concern: Can the theory …t any dataset (hence being without real
content)?
Theorem 5 (Debreu Sonnenschein Mas-Colell) Suppose that z(p) is a continuous
function de…ned on
pl
" for every l and l0 g
P" = fp
0:
pl 0
and z(p) has values in RL . Assume further that z( ) is homogenous of degree 1 and
satsi…es Walras law (z(p)p = 0). Then there is an economy of at least L consumers
whose aggregate excess demand function coincides with z(p) on P" .
The proof will not be given.
So what do we know about stability and uniqueness? Nothing whatsoever! In the
aggregate anything goes.
Essentially, as we can choose perferences individually, the individual excess demand
function thus can be chosen to form a basis for the entire function space. As long as
2
there is no restriction on how we specify preferences for individual i given the preferences
of others, we are free to span the entire space of excess demand functions. Kirman’s
argument is that we have to leave the methodological individualism. I think that this is
misplaced. I’d rather have more substantive rationality in our theories. There are some
rather common objectives we all try to maximize, one way or another, the individual
preferences thus cannot span such a large space. But Kirman is right that we lack that
theory.
3