Notes V
General Equilibrium: Positive Theory
In this lecture we go on considering a general equilibrium model of a private
ownership economy. In contrast to the Notes IV, we focus on positive issues
such as existence, uniqueness, or stability of equilibria. The notes are based
on MWG, chapter 17.
1
Walrasian Equilibrium and Excess Demand
I > 0, J > 0, L > 0, all of them are finite. An economy is defined by
({Xi , %i }Ii=1 , {Yj }Jj=1 , {(ωi , θi1 , ..., θiJ )}Ii=1 ).
Definition 1 A Walrasian (or price-taking) equilibrium is an allocation (x∗ , y ∗ )
and a price vector p = (p1 , ..., pL ) if
(i) For every j, yj∗ ∈ Yj maximizes profits in Yj : p · yj∗ ≥ p · yj for all yj ∈ Yj .
(ii) For every i, x∗i ∈ XP
i is maximal for %i in the respective budget set
∗
{xi ∈P
Xi : p · xP
≤
p
·
ω
+
i
iP
j θij p · yj }.
(iii) i x∗i = i ωi + j yj∗ .
For a while, let us consider exchange economies. Most results obtained from
this exercise easily carry over to economies with production.
Definition 2 An exchange economy is defined by
E ≡ ({Xi , %i }Ii=1 , Y1 = −RL+ , {ωi }Ii=1 ).
We assume that preferences are continuous, strictly convex, and locally
nonP
satiated (before long: strictly monotone). Moreover, we assume: i ωi À 0.
Notice also that we assume free disposal, which is taken into account by allowing for one firm whose only available technology is that of free disposal:
Y1 = −RL+ ⇔ yl1 ≤ 0 for all l = 1, ...L.
Query V.1 What is the relationship between the assumption of free disposal
and prices?
In the setting of an exchange economy, an allocation (x∗ , y ∗ ) and a price
vector p constitute a Walrasian equilibrium if and
only if P
(i) y1∗ ≤ 0, p · y1∗ =
P
0, p ≥ 0, (ii) x∗i = xi (p, p · ωi ) for all i, and (iii) i x∗i = i ωi + y1∗ .
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Notice that (i) we will prove in class. However, by (i) and (iii) aggregate
demand cannot exceed aggregate supply of a commodity (as y1 ≤ 0). Thus,
by (i), a price pl is zero if and only if aggregate demand is smaller than
aggregate supply (i.e., only free goods can be in excess supply).
Proposition 1 Suppose, preferences in E are strictly convex and locally
nonsatiated.
Then, p is a Walrasian equilibrium price vector if and only
P
if:
(x
(p,
p
· ωi ) − ωi ) ≤ 0.
i
i
Query V.2 Prove Proposition 1.
Definition 3 Consumer i’s excess demand function is zi (p) = xi (p, p · ωi ) −
ωi , where xi (p, p·ωi ) is her Walrasian
demand function. The aggregate excess
P
demand function is z(p) = i zi (p).
From here on, we’ll state most results in terms of the excess demand (rather
than Walrasian demand).
Definition 4 E + defines an exchange economy where %i are strictly monotone, continuous, and strictly convex.
In E + , p constitutes a Walrasian price vector if and only if z(p) ≤ 0. If,
moreover, preferences are strictly monotone — this we’ll assume from here
on — a Walrasian price vector has the property that p À 0.
Query V.3 If %i are strongly monotone for all i, why must a Walrasian price
vector be strictly positive: p À 0?
Under strong monotonicity of preferences, p is a Walrasian equilibrium price
vector if and only if zl (p) = 0 for every l = 1, ..., L. I.e., z(p) = 0.
P
Proposition 2 Consider an economy E + where i ωi À 0. Then z(p) is
defined on p À 0 and satisfies:
(i) z(p) is continuous.
(ii) z(p) is HD0 .
(iii) Walras law: p · z(p) = 0 .
(iv) z(p) is bounded below. I.e., there is some number s : zl (p) > −s for
every l = 1, ..., L and all p.
(v) If pn → p, where p 6= 0 and pl = 0 for some l, then:
{max{z1 (pn ), ..., zL (pn )}}∞
n=1 → ∞.
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Property (i) follows from the fact that xi (p, p · ωi ) is continuous.
Query V.4 From which property about %i does it follow that xi (p, p · ωi ) is
continuous?
Property (ii) follows from the fact that xi (p, p · ωi ) is HD0 , (iii) comes from
strong monotonicity of %i , (iv) stems from the fact that demand cannot be
negative. Finally, (v) we’ll show in class.
2
Some Mathematical Prerequisites
for Existence Proofs
Correspondence. A correspondence is a “multi-valued function”. Suppose
our domain is A ⊆ RN . A (real valued) function f : A → R is a rule that
assigns to every x ∈ A a single value f (x) ∈ R (a singleton). In contrast, a
(real valued) correspondence ϕ(x) : A → RK is a rule that assigns to every
x ∈ A a set ϕ(x) ∈ RK (which is not necessarily a singleton). Obviously,
every function is a correspondence. But a correspondence is a function if and
only if for every x ∈ A we have that ϕ(x) is a singleton.
Convex-Valuedness of a Correspondence. Suppose, a correspondence ϕ(x) :
A → RK assigns to every x ∈ A a set ϕ(x) ∈ RK . This correspondence is
convex valued at x if ϕ(x) is a convex set. This correspondence is convex
valued if ϕ(x) is a convex set for all x ∈ A.
Upper Hemicontinuity (uhc) of a Correspondence. Let ϕ : A → Y be a
correspondence, where A ⊂ RN , Y ⊂ RK , both A and Y are closed, and
Y is bounded. Consider any two converging sequences {xn } and {y n } such
that for all n, yn ∈ ϕ(xn ), where xn → x and x, xn ∈ A, and y n → y and
y, y n ∈ Y for n = 1, 2, .... The correspondence ϕ : A → Y is said to be uhc
at x if y ∈ ϕ(x). The correspondence ϕ : A → Y is said to be uhc if it is uhc
at all x ∈ A.
Brouwer’s Fixed-Point Theorem. Suppose that A ⊂ RN is nonempty, compact, and convex. If f : A → A is a continuous function from A to itself,
then f (.) has a fixed point; i.e., there is an x ∈ A such that: x = f (x).
Kakutani’s Fixed-Point Theorem. Suppose that A ⊂ RN is nonempty, compact, and convex. If ϕ : A → A is an upper hemicontinuous correspondence
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from A to itself, with ϕ(x) ⊂ A being nonempty and convex for every x ∈ A
(i.e., “convex-valued”), then ϕ(.) has a fixed point; i.e., there is an x ∈ A
such that: x ∈ ϕ(x).
3
Existence of Equilibrium
This is the first (positive) question. We cannot use our GE-framework unless
there is an equilibrium. The conditions for which an equilibrium exists are
clarified in this section.
By HD0 of an excess function, we are allowed to normalize the price vector
(e.g., set one price equal to unity, or normalize pricesP
to the unit simplex in
L
L
R+ ). The unit simplex is defined by ∆ ≡ {p ∈ R+ : l pl = 1}. Moreover,
denote the interior of ∆ by ∆i , and the boundary of the simplex by ∂∆.
Before going to the propositions, please be sure you understand the following concepts: convex-valuedness of a correspondence, (upper) hemicontinuity of a correspondence, Brouwer’s Fixed-Point Theorem, and Kakutani’s
Fixed-Point Theorem.
I start with the general result first, and give a simplified (more special,
but probably more instructive) version thereafter. However, for the “real”
existence proof — which is also applicable for production economies — I ask
you to read my Notes VI.
Proposition 3 Consider an exchange economy E + with ω À 0. There exists
a Walrasian equilibrium, i.e., there exists an allocation (x∗ , y ∗ ) and a price
vector p that constitute a solution to the system of equations z(p) = 0.
Proof. First, construct a correspondence f (p) from all p ∈ ∆ into ∆. Step
(i) considers f (p) : ∆i → ∆, step (ii) considers f (p) : ∂∆ → ∆.
(i) Construct a correspondence for all p ∈ ∆i :
f (p) = {q ∈ ∆ : z(p) · q ≥ z(p) · q 0 for all q 0 ∈ ∆}, which assigns an element
(a set) of ∂∆ to every p ∈ ∆i . Observe that if z(p) = 0 (i.e., we are having
a Walrasian equilibrium), f (p) = ∆. However, if z(p) 6= 0, then f (p) ⊂ ∂∆.
In particular, ql = 0 if zl (p) < max{z1 (p), ..., zL (p)}.
(ii) Construct a correspondence for all p ∈ ∂∆:
f (p) = {q ∈ ∆ : p · q = 0} = {q ∈ ∆ : ql = 0 if pl > 0}. As for any
p ∈ ∂∆ : p · p > 0, no fixed point can be represented by a price vector
p ∈ ∂∆.
(iii) Certainly, a fixed point of f (p) is a Walrasian equilibrium. Notice that
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a fixed point means p∗ ∈ f (p∗ ). In this case, p∗ 6∈ ∂∆. Thus, p∗ À 0. But
if z(p∗ ) 6= 0, then p∗ ∈ ∂∆. Hence, a fixed point represents a Walrasian
equilibrium.
(iv) The fixed point correspondence is convex-valued and upper hemicontinuous (as will be shown in class).
(v) Now we can apply Kakutani’s Fixed-Point Theorem to establish that
there is a fixed point. By (iii), then, there is a Walrasian equilibrium.
.
W.H.O.W.
All right, this was pretty tough. The difficulty in the preceding proof arose
from boundary complications, i.e., excess demand is not well defined when
p ∈ ∂∆, as the maximum zl (p) is going to infinity. For purely instructive
reasons, we proceed as follows. Assume properties (i) to (iii) from Proposition 2, and z(p) is well defined for all nonzero p ∈ RL+ .1 Remember that in
equilibrium we have z(p) ≤ 0.
Corollary 1 Consider an exchange economy E with ω À 0 and z(p) being
well defined for all p ∈ RL+ . Then there exists a Walrasian equilibrium, i.e.,
there exists an allocation (x∗ , y ∗ ) and a price vector p ≥ 0 that constitute a
solution to the system of equations z(p) ≤ 0.
(i) As z(p) are HD0 inP
prices, we can restrict our attention to the price
simplex: ∆ = {p ∈ RL+ | l pl = 1}.
(ii) Define the function zl+ (p) = max {zl (p), 0}. The function z + (p) is continuous, and z + (p) P
· z(p) = 0 implies z(p) ≤ 0.
(iii) Define α(p) = l (pl + zl+ (p)) ≥ 1.
(iv) f (p) = (p + z + (p))/α(p) is a continuous function from the price simplex
to itself.
(v) By Brouwer’s Fixed-Point Theorem there exists a p∗ ∈ ∆ such that
p∗ ∈ ∆ = f (p∗ ).
(vi) By Walras law: 0 = p∗ · z(p∗ ) = f (p∗ ) · z(p∗ ) = (1/α(p∗ )) (p∗ + z + (p∗ )) ·
z(p∗ ) = (1/α(p∗ )) z + (p∗ ) · z(p∗ ). But then, z + (p∗ ) · z(p∗ ) = 0, which implies,
by (i), that z(p∗ ) ≤ 0.
W.H.O.W.
Query V.5 Show that f (p) : ∆ → ∆, as claimed in step (iv).
1
Such excess demand functions are not possible with monotone preferences, yet they
exist with locally nonsatiated preferences.
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4
Bonus Stuff
Uniqueness – A few Results
Suppose there exist Walrasian equilibria. The question then is: “How many
equilibria are there?” If there is a (globally) unique equilibrium, we can
perform meaningful comparative static analysis. However, if there is more
than one equilibrium (i.e., multiplicity) – the next best thing is to have a
finite number of equilibria. In this case, we have local uniqueness, i.e., at
every Walrasian equilibrium (x∗ , y ∗ ), there exists an ² > 0 and an ²−ball
about (x∗ , y ∗ ), B² (x∗ , y ∗ ), such that there is no other Walrasian equilibrium
within B² (x∗ , y ∗ ). More precisely, a Walrasian equilibrium price vector p 6= 0
is locally unique, if there is an ² > 0 such that if p0 6= p, and ||p0 − p|| < ²
then z(p0 ) 6= 0.
In contrast to local uniqueness, we might encounter indeterminate equilibria, in which case for every ² > 0 – however small – there is an infinite
number of Walrasian equilibrium price vectors in ||p0 − p|| < ².
Indeterminateness is not a desirable property. If the economy is regular, all equilibria are locally unique (determinate). Moreover, an economy
is regular, if the Jacobian matrix of price effects Dẑ(p) has rank L − 1 (is
nonsingular).2
Query. Suppose, L = 2. Under which condition is E + regular? Under which
condition does E + face indeterminate equilibria?
We now consider a condition that guarantees global uniqueness of equilibrium.
Definition 5 (Gross Substitution) The excess demand function has the
gross substitution (GS) property if whenever p and p0 are such that, for some
l, p0l > pl and p0k = pk for all k 6= l, we have zk (p0 ) > zk (p) for all k 6= l.
Notice that the gross substitution property (as defined above) implies: zl (p0 ) <
zl (p)! In a differential version, GS implies: δ zk (p)/δ pl > 0, i.e., all the offdiagonal entries of Dz(p) are positive.
Proposition 4 In E + , there is a globally unique equilibrium, if z(p) satisfies
the gross substitution property.
2
Normalize the price vector such that the price of good L = 1: p = (p1 , p2 , ..., pL−1 , 1).
The normalized excess demand function is then: ẑ(p) = (z1 (p), z2 (p), ..., zL−1 (p)). Then,
p 6= 0 is a Walrasian equilibrium price vector if ẑ(p) = 0.
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Observe that the GS property is sufficient, not necessary!
Excess Demand in Economies with Production
Definition 6 An economy with production is defined by
P ≡ ({Xi , %i }Ii=1 , {Yj }Jj=1 , {(ωi , θi1 , ..., θiJ )}Ii=1 ). Let P + be an economy with
production, where all production sets are closed, strictly convex and bounded.
Consider anPeconomy P. The
excess demand is given
P production-inclusive
P
P
by: z̄(p) = i xi (p, p · ωi + j θij πj (p)) − i ωi − j yj (p) .
Proposition 5 Consider an economy P + . Then, z̄(p) satisfies properties
(i) to (v), as given by Proposition 2.
See Exercise 17.B.4 (MWG, p.642).
Local Nonsatiation and Positivity of Prices
Notice that local nonsatiation implies that there is at least one desirable
good, otherwise 0 would be a global satiation point. Thus, p · x∗i = p · ωi .
Proposition 6 Suppose, preferences in E are strictly convex and locally
nonsatiated.
P Then, p is a Walrasian equilibrium price vector if and only
if: z(p) ≡ i (xi (p, p · ωi ) − ωi ) ≤ 0.
Proof (Sketch). It can easily be shown that
≤P
0 ⇔ [(y1∗ ≤ 0, p · y1∗ =
P z(p)
∗
∗
0, p ≥ 0) & (xi = xi (p, p · ωi ) for all i) & ( i xi = i ωi + y1∗ )].||
Proposition 7 Let p be a Walrasian price vector in E. Then, no commodity
has a negative price: pl ≮ 0 for all l = 1, ..., L.
Proof (direct). Because of the possibility of free disposal, there are no transactions with a negatively priced commodity (nobody is willing to sell). So
there are is no trade with such commodities – hence, no good has a negative
price.||
Proposition 8 Let p be a Walrasian price vector in E. Then, p 6= 0.
Proof (direct). Suppose p = 0. By local nonsatiation there is a desirable
commodity, say l. But then, as the budget set is unbounded, there exists no
maximal element, x∗i , in the budget set.||
From HD0 of the excess demand functions, P
and from Proposition 3, we
can normalize prices without loss of generality: l pl = 1.
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Corollary 2 Let p be a Walrasian price vector in E. Then, the price of
every desirable commodity l is strictly positive: pl > 0.
Corollary 3 Let p be a Walrasian price vector in E. If all commodities are
desirable (strong monotonicity), p À 0.
Proposition 9 Let p be a Walrasian price vector in E. If some commodities
are not desirable, i.e., zl (p) < 0, then, pl = 0 and the price vector is not
strictly positive.
Proof (direct). Suppose first, all goods are desirable. Then, p À 0. As
p · z(p) = 0 we have z(p) = 0, i.e., zl (p) = 0 for
P all l = 1, ..., L.
0
Next, suppose that l0 is not desirable, i.e.:
i zl0 i (p) < 0. Define z (p) =
(z1 , z2 , ...zl0 −1 , zl0 +1 , ..., zL ), and p0 = (p1 , ...pl0 −1 , pl0 +1 , ..., pL ). Then, pl0 zl0 (p)+
p0 · z 0 (p) = 0. By Corollary 1, p0 À 0. Moreover, z 0 (p) = 0, as all those goods
are desirable (and z 0 (p) ≥ 0). Thus, p0 · z 0 (p) = 0. As zl0 (p) < 0, we must
have pl0 = 0.
The argument can easily be extended to the case with several bads.||
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