Prof. Dr. Olivier Bochet
Room: A.314
Phone: 031 631 4176
E-mail: [email protected]
Webpage: http://sta¤.unibe.ch/bochet
Advanced Microeconomics
Fall 2010
Lecture Note 4
General Equilibrium III
1
Introduction
In Lecture Note 2 and 3, we presented a separate introduction to economies
with pure exchange on the one hand, and economies with pure production
on the other hand. We will now merge these two sides together and study
what we will refer to as private ownership economies. Our goal here is, with
the help of a full blown general equilibrium model, to cover the two welfare
theorems. While the proof of the …rst welfare theorem will be a simple extension of the proof we saw for exchange economies, we will spend a good
deal of time covering the proof of the second welfare theorem. Based on the
two leading examples of failure of the second welfare theorem that we covered for exchange economies –failure coming from non-convexities and failure
coming from the lack of strict monotonicity/zero wealth– we will construct
a two-part proof. The …rst part will be based mainly on the assumption
of convexity and local non-satiation and will allow for failure of the second
kind, while ruling out failure of the …rst kind. To this end, this …rst part
of the proof can be seen as a version of the second welfare theorem that is
1
based on a weaker price equilibrium notion which does not require agents
to be fully optimizing –a price quasi-equilibrium. Next, the second part of
the proof will establish conditions under which a price quasi-equilibrium is a
price equilibrium with transfers/a Walrasian equilibrium.
As usual, we will …rst introduce the notations and de…nitions that are
useful for the lecture note.
2
Private ownership economies
There is a set of agents N = f1; ::; ng, a set M = f1; :::; mg of …rms, and L
in…nitely divisible goods. The consumption set of each agent i 2 N is RL+ .
Each agent i 2 N has a preference relation i over RL+ that is rational –i.e.
complete and transitive. Each i 2 N has some initial endowment –a stock
of resources–P
! i = (! 1i ; :::; ! Li ) 2 RL+ n f0g: The aggregate endowment is
denoted ! = i2N ! i . We assume that ! 2 RL++ , i.e. each good is available
in some quantities. The endowment point ! = (! 1 ; :::; ! n ) 2 RLn
+ is the list of
agents’initial endowments. Each …rm j 2 M is endowed with a production
technology described by a production set Yj that is non-empty and closed.
Each i 2 N has aPclaim to a share ij 2 [0; 1] to the pro…ts of …rm j 2 M .
For each j 2 M , i2N ij = 1.
A private ownership economy is E = h( i ; ! i ; ( ij )j2M )i2N ; (Yj )j2M i. An
allocation is a vector (x; y) = (x1 ; :::; xn ; y1 ; :::; ym ) 2 RLn
RLm composed
+
of a bundle xi for each agent i 2 N , and a production
plan yjPfor each …rm
P
= ! + j2M yj . Let
j 2 M . An allocation (x; y) is feasible if
i2N xi P
P
Ln
Y1 ::: Ym : i2N xi = ! + j2M yj g be the set of
FE f(x; y) 2 R+
feasible allocations for E.
Notice that an exchange economy with free disposal E = h( i ; ! i ; i1 )i2N ; Y1 i
is a special case of a private ownership economy in which J = f1g and the
only available technology is Y1 = RL+ , i.e. the free disposal technology –no
production can take place, only destruction.
We now want to investigate among all allocations that are feasible, the
ones that are economically meaningful. For this, we introduce the central
notion of economic e¢ ciency, namely Pareto e¢ ciency.
2
Pareto e¢ ciency: Allocation (x; y) 2 FE is Pareto e¢ cient if there
does not exist another allocation (x0 ; y 0 ) 2 FE such that
x0i
x0j
xi for each i 2 N
j xj for at least one j 2 N
(1)
(2)
i
We now turn to several de…nitions of a market equilibrium.
P
p ! i + j2M ij p yj g
For each i 2 N , let Bi (p) = fxi 2 RL+ : p xi
be agent i’s budget set at prices p. Notice how the de…nition of agent i’s
budget set has changed. We need now to incorporate the pro…ts generated
from production activities and on which agent i has a claim.
Walrasian equilibrium: Given an economy E, A triple (p ; x ; y ) 2
RLm is a Walrasian equilibrium if the following three conR n f0g RLn
+
ditions hold
X
X
1)
xi = ! +
yj
L
i2N
|
{z
j2M
}
Supply=Demand
yj p yj0 for
2) For each j 2 M , p
|
{z
all yj0 2 Yj
}
yj is pro…t maximizing at p
3) For each i 2 N , xi
|
i
x0i for all x0i 2 Bi (p )
{z
}
xi is maximal for
i
over Bi (p)
We now state a more general notion of equilibrium.
Price equilibrium with transfers: Given an economy E, (p ; x ; y ) 2
RLm is supportable as a price equilibrium
R n f0g RLn
+
P with transfers if
there
is
an
assignment
of
wealth
levels
(w
;
:::;
w
)
with
!+
1
n
i2N wi = p
P
yj such that
j2M p
X
X
1)
xi = ! +
yj
L
i2N
j2M
2) For each j 2 M; p
3) For each i 2 N , xi
yj
i
p
yj0 for all yj0 2 Yj
x0i for all x0i 2 fx0i 2 RL+ : p
x0i
wi g
As before, a Walrasian equilibrium is a special case of a price equilibrium with transfers. A price equilibrium with transfers only stipulate that
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there exists some wealth distribution such that (p ; x ; y ) is a Walrasian
equilibrium.
Theorem 1 (First welfare theorem) Let E be a private ownership economy.
Suppose preferences are locally non-satiated. Then any price equilibrium with
transfers is Pareto e¢ cient. In particular, any Walrasian equilibrium is
Pareto e¢ cient.
Proof. Pick a private ownership economy E and suppose that (p ; x ; y )
is price equilibrium
with transfers
P with associate wealth levels (w1 ; :::; wn )
P
such that i2N wi = p ! + j2M p yj . Assume by contradiction that
(x ; y ) is not e¢ cient. Then there exists (x0 ; y 0 ) 2 FE such that x0i i xi
for each i 2 N , and x0j j xj for at least one j 2 N . The preference
maximization part in the de…nition of a Walrasian equilibrium implies that
x0j j xj =) p x0j > p ! j . Local non-satiation implies also an additional
p ! i . Suppose this is not true. That
property: if x0i i xi , then p x0i
0
0
is there exists xi i xi and p xi < p ! i . By local non-satiation, there
exists x00i and > 0, arbitrarily small, such that kx00i x0i k
, x00i i x0i , and
p x00i
p ! i . By transitivity x00i i xi . But this is in contradiction with xi
being a maximal element in agent i’s budget set. Hence the claim is true.
This gives us that
X
X
X
p x0i >
wi = p ! +
p yj
i2N
i2N
j2M
Now, because yj is pro…t maximizing at price p for each j 2 M , we have
p
!+
X
p
yj
p
!+
j2M
Therefore,
X
p
j2M
X
p
x0i > p
!+
i2N
X
j2M
This inequality can be true if and only if
X
X
yj0
x0i > ! +
i2N
j2M
4
yj0 for all yj0 2 Yj
p
yj0
But then allocation (x0 ; y 0 ) 2
= FE , unlike we assumed. We conclude that
(x ; y ) is an e¢ cient allocation.
Q.E.D.
Although assumptions on the primitives of the economy are very weak
–i.e. only local non-satiation is required– we must keep in mind all the
exogenous assumptions underlying the model:
1) Markets are complete
2) No externalities
3) No uncertainty
4) Price-taking behavior (very strong assumption when economy is small)
Moreover, the …rst welfare theorem is silent regarding the desirability of
the equilibrium allocation –aside from Pareto e¢ ciency. We next investigate whether markets are possibly biased towards some …nal allocation of
resources –thereby favoring some agent (or group of agents) over others. As
it turns out, markets are in fact unbiased. This is the content of the second
welfare theorem whose message is that any distributional objectives can be
achieved through the use of competitive markets –under some assumptions
on the economy.1
In Lecture note 2, we saw two di¤erent kind of failures of the second
welfare theorem. One came from the lack of convexity of the economy, while
the second one was linked to the lack of strong monotonicity/zero wealth in
equilibrium. These two problems are recalled in Figure 1. We would like to
have versions of the second welfare theorem that underline how these two
failures can arise. In particular, the second type of failure seems to have
di¤erent sources. As such it is not easy to disentangle the driving forces
behind the second kind of failure. To do so, we will split the proof of the
second welfare theorem in two di¤erent parts. The …rst part will make use
of convexity –and thus rule out failures of the …rst kind–and will allow for
failures of the second kind.
To allow for failure of the second kind while ruling out failures of the
…rst kind, we will look at a weaker notion of market equilibrium called quasiequilibrium with transfers. The …rst part of the proof will then aim at proving
at weaker version of the second welfare theorem. The second part of the proof
1
Competitive markets refer to the assumption of price-taking behavior.
5
will be concerned with establishing under which conditions a quasiequilibrium is actually an equilibrium. Doing so will provide a full
understanding of the driving forces behind failures linked to lack of strict
monotonicity/zero wealth at equilibrium. As it turns out, having a consumer
with zero wealth is often problematic.
Price quasi-equilibrium with transfers: Given an economy E, (p ; x ; y ) 2
R n f0g RLn
RLm is a price quasi-equilibrium
+
P with transfers if there
P is an
assignment of wealth levels (w1 ; :::; wn ) with i2N wi = p ! + p
j2M yj
and such that
X
X
xi = ! +
yj
1)
L
i2N
j2M
2) For each j 2 M; p
3) For each i 2 N , x0i
yj p yj0 for all yj0 2 Yj
x0i wi
i xi implies that p
The only di¤erence with the de…nition of a price equilibrium with transfers is item 3). In the de…nition of a price quasi-equilibrium, item 3) requires
that anything that is preferred to xi cannot cost less to agent i than xi .
Notice then that agent i does not necessarily fully optimize at xi since item
3) does not rule out that there may be bundles that are strictly preferred
and which cost the same as xi does. Let us call this “quasi-preference maximization”.
Based on this de…nition, we can now state a weaker version of the second
welfare theorem that uses convexity but allows for failures of the second kind.
This is the …rst part of the proof of the standard version of the second welfare
theorem. Before going to the proofs, let us recall below what the two-part
Theorem will be.
First part: If all preferences and technology are convex, then any e¢ cient
allocations can be supported as a price quasi-equilibrium with transfers.
Second part: Giving su¢ cient conditions for a price quasi-equilibrium
to actually be an equilibrium
Theorem 2 (A version of the second welfare theorem) Let E be an economy.
Suppose that for each j 2 M , Yj is convex, and for each i 2 N . i is both
convex and locally non-satiated. Then each Pareto e¢ cient allocation (x ; y )
can be supported as a price quasi-equilibrium with transfers for some p 6= 0
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Proof: The proof is divided into several steps. Before going to the steps,
let us de…ne theP
following items.
each i 2 N , let Vi fx0i 2 RL+ : x0i i
P For
0
xi g: Then
Vi = f i2 x0i 2 RLn
+ : xi 2 Vi for each i 2 N g. Next,
P V = i2NP
let Y = j2M Yj = f j2M yj0 : yj0 2 Yj for each j 2 M g. Note that V is the
set of aggregate consumption bundles that can be split into agents so that
x0i i xi for each i 2 N . The set Y is simply the aggregate production set.
Finally, the set Y + f!g is the aggregate production set with origin shifted at
!. This is the set of aggregate bundles producible with the given technology
and endowment, and usable for consumption.
Step 1: Every Vi is a convex set
Take x0i ; x00i 2 Vi : By construction, we have that x0i i xi and x00i i xi .
Pick
2 [0; 1] and assume that x0i i x00i . By convexity of preferences,
x0i + (1
)x00i i x00i . By transitivity, x0i + (1
)x00i i xi . Hence Vi is a
convex set.
Step 2: V and Y + f!g are convex sets
The sum of any convex sets is a convex set
Step 3: V \ (Y + f!g) = ;
Since (x ; y ) is e¢ cient, V \ (Y + f!g) 6= ; would be a contradiction.
Step 4: There exists p 6= 0 and a number r such that p z r for each
z 2 V , and p z r for each z 2 Y + f!g
This is a simple consequence of the separating hyperplane theorem which
establish the implication above for any two disjoint convex sets.
The separation argument is illustrated in Figure 2. The remaining steps
are meant at establishing that (p ; x ; y ) is a price quasi-equilibrium with
transfers.
P
Step 5: If x0i i xi for each i 2 N , then p ( i2N x0i ) r
Suppose that x0i
xi for each i 2 N . By local non-satiation, for each
i 2 N , there is a x^i arbitrarilyP
close to x0i and such thatP
x^i i x0i . Therefore
x^i 2 Vi for each i 2 N . Hence i2N x^i 2 V and so p ( i2N x^i ) r.
P
That is, the set i2N fx0i : x0i i xi g is contained in the closure of V
–note that V is an open set–itself contained in the half-space fz : p z rg.
Therefore, the set of bundles that are at least as good as xi for each i is
“above”the price hyperplane.
7
P
P
P
Step 6: p ( i2N xi ) = p P
( i2N ! + j2M yj ) = r
ByPStep 5, we haveP
that p ( i2N xi ) r: On the other hand, we know
that P i2N xi = ! + j2M yjP2 Y + f!g. Therefore, we also have that
p ( i2N xi ) r. Thus p ( i2N xi ) = r. And,
p
X
(
xi ) = p
!+p
i2N
X
yj = r
j2M
We now proceed to establish pro…t maximization and preferences “quasi”maximization.
Step 7: For each j 2 M , p yj0 p yj for all yj0 2 Yj
For each j 2 M , and each yj0 2 Yj , we have that
yj0 +
X
k6=j
Hence
p
(! + yj0 +
X
yk )
X
yk )
p
yj0
yk 2 Y
r=p
(! +
p
(yj0 +
yj )
j2M
k6=j
Thus
X
p
(yj +
k6=j
X
yk )
k6=j
We therefore conclude that
p
yj
This holds for each yj0 2 Yj and each j 2 M .
Step 8: For each i 2 N , if x0i
Pick i 2 N and x0i such that x0i
5 and 6, we have
X
p (x0i +
xk )
i
i
xi , then p x0i p xi
xi . By construction, x0i 2 Vi . By Steps
r=p
(xi +
k6=i
k6=i
We therefore conclude that
p
X
x0i
p
8
xi
xk )
This holds for each x0i 2 Vi and each i 2 N .
Step 9: Wealth levels wi = p xi for each i 2 N support (p ; x ; y ) as
a price quasi-equilibrium with transfers
By Steps 7 and 8, we have pro…t maximization and quasi-preference
maximization of the de…nition of a price quasi-equilibrium. Next, because
(x ; y ) 2 FE by de…nition, we have
X
X
X
wi
yj =
xi = ! +
i2N
j2M
i2N
Q.E.D.
Convexity and local non-satiation allow us to only establish decentralization as a quasi-equilibrium with transfers. The assumptions made on the
economy do not allow us to rule out failures of the second kind as illustrated
again in Figure 3. There, notice that price p2 = (0; 1) cannot be a candidate
for a price quasi-equilibrium. Notice that w1 = p2 x1 = 1 and w2 = 0.
However, for agent 1 there are bundles that cost less than x1 and that are
strictly preferred to x1 . The price p1
0 cannot be a candidate for the same
reason. In fact the only candidate is p3 = (1; 0), i.e. good 2 is free. Notice
that w1 = p3 x1 = 0 while w2 = p3 x2 = 1. Hence, agent 1 has zero wealth.
0 = w1 for any x01 while x2 is already agent 2’s best
Observe that p3 x0i
bundle. Hence the quasi-preference maximization part is satis…ed. (p3 ; x ) is
a price quasi-equilibrium. But as emphasized before, this is not agent 1’s best
a¤ordable bundle. At p3 his demand for good 2 is in…nite because good 2 is
free. For this very reason (p3 ; x ) cannot be a price equilibrium. Therefore,
the second welfare theorem fails.
An important feature of the example is that at equilibrium, agent 1 has
zero wealth. This is key to the failure of the second welfare theorem. We
now gives conditions under which a price quasi-equilibrium is actually a price
equilibrium.
Theorem 3 (Existence of a cheaper consumption bundle) Let E be an economy. Suppose that i is continuous for each i 2 N , and that (p ; x ; y ) with
wealth levels (wi )i2N form a price quasi-equilibrium with transfers. Then if
there exists x0i such that p x0i < p xi (a cheaper consumption bundle), then
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xi i xi implies p xi > wi . In particular, if there exists a cheaper consumption bundle for each i 2 N , then (p ; x ; y ) with wealth levels (wi )i2N form
a price equilibrium with transfers.
Proof. Suppose by contradiction that there exists xi such that xi i xi
and p xi = wi . By the cheaper consumption bundle assumption, there exists
x0i such that p x0i < wi . For all 2 [0; 1), we have
p
If
( xi + (1
)x0i ) < wi
is close enough to 1, continuity of preferences implies that
xi + (1
)x0i
i
xi and p
( xi + (1
)x0i ) < wi
A contradiction.
Q.E.D.
The proof is illustrated in Figure 4. We have now a direct corollary to
the previous result.
Theorem 4 (Corollary to the previous result) Suppose that i is continuous
for each i 2 N . Then any price quasi-equilibrium with transfers that has
(w1 ; :::; wn )
0 is a price equilibrium with transfers.
Let us go back one last time to the example of Figure 3. In Figure 5,
we ask ourselves whether there are some e¢ cient allocations which can be
supported as price equilibria.2 Allocation x is such that x1
0 while
2
x22 = 0. At p = (0; 1), there are some bundles which cost less than x1 does
and which are preferred to x1 –these are above the indi¤erence curve of agent
1 passing through x1 and below the budget line. Hence (p2 ; x ) cannot be a
price quasi-equilibrium. For the same reason, (p3 ; x ) cannot be a price quasiequilibrium either. In fact the only candidate for a price quasi-equilibrium is
(p1 ; x ) where the budget line is tangent to agent 1’s indi¤erence curve passing
2
Notice that the discussion which followed is di¤erent from what I said in class. Indeed,
in class I made the mistake of saying that no e¢ cient allocations could be supported as
price equilibrium. This is not true as the following discussion will establish. All apology
for the misleading discussion provided in class!
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through x1 . One quickly notices that for agent 1 (and respectively for agent
2), anything that is preferred to x1 (respectively, x2 ) cost at least as much as
x1 (respectively, x2 ). Hence (p1 ; x ) is a price quasi-equilibrium. Also notice
that a cheaper consumption bundle exists for both agents –because prices are
all positive, both agents have positive wealth at equilibrium. Given the above
two theorems, we readily conclude that (p1 ; x ) is in fact a price equilibrium.
The same reasoning applies to any allocation x with x1
0 and x22 = 0.
Therefore, while the second welfare theorem fails in this economy, it fails at
only one allocation x = ((0; ! 2 ); (! 1 ; 0)).
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