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CHAPTER 9
General Equilibrium Theory
The previous chapters deal with theories of individual agents’ behavior. In the
rest of the book, we turn to economic theories that predict group or societal
outcomes. We ﬁrst turn our attention to general equilibrium theory.
General equilibrium theory can often be studied through a reduced-form
model, the excess demand function of an economy. The equilibrium outcomes
of the economy are given as zeroes of the excess demand function. There are
two immediate questions about the scope of the model: What is the class of
excess demand functions that can arise from a well-behaved economy? And
which sets of prices can be equilibrium prices?
The answers to these questions carry a largely negative message about
general equilibrium theory. The Sonnenschein–Mantel–Debreu Theorem (as
we shall refer to it) shows that, roughly speaking, any continuous function
that satisﬁes Walras’ law can be the aggregate excess demand function of
a very well-behaved economy. The result implies that any compact set of
strictly positive prices can be the set of Walrasian equilibrium prices of a
well-behaved economy. No additional constraints are obtained by insisting on
basic regularity properties of the equilibria.
Considered as data on an economy, an excess demand function, or a set of
putative equilibrium prices, may seem odd. The next set of questions under
study is much more similar to the approach in Chapter 3. If we assume that
we can observe equilibria for different vectors of endowments (in a sense,
we can sample from the “equilibrium manifold”), then the theory of general
equilibrium can be refuted: There are nonrationalizable datasets. The theory
is testable if we can observe prices from different endowment vectors. The
nature of the testable implications follow from a very general principle, the
Tarski–Seidenberg Theorem, which we shall also review here.
We focus on a model of an economy where all economic activity takes the
form of exchange. There are I consumers; each consumer i is described by
a pair (⪰i,ωi), where ⪰i is a preference relation on Rn
+, and ωi ∈Rn
+ is an
endowment vector. An exchange economy is a tuple E = (⪰i,ωi)I
i=1.


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General Equilibrium Theory
When ⪰i is continuous, strictly convex, and locally nonsatiated, the demand
function of agent i is well deﬁned as di(p,M) = argmax⪰i{x ∈Rn
+ : p · x ≤M},
for p ∈Rn
++ and M > 0. The excess demand function of agent i is Zi(p) =
di(p,p·ωi)−ωi. Finally, the aggregate excess demand function of the economy
E is Z = I
i=1 Zi. The function Z is continuous and satisﬁes p·Z(p) = 0 for all p
in its domain, a property called Walras’ Law. A vector p ∈Rn
++ is a Walrasian
equilibrium price if Z(p) = 0.
One can deﬁne the weak axiom of revealed preference for excess demand
functions. Indeed, note that WARP for individual demand functions says that
there cannot exist prices p and q such that q·di(p,p·ωi) < q·ωi and p·di(q,q·
ωi) ≤p · ωi. So we say that an individual excess demand function Zi satisﬁes
WARP if there are no p and q such that q·Zi(p) < 0 and p·Zi(q) ≤0. Moreover,
Zi satisﬁes a version of the strong axiom of revealed preference, which states
that there is no sequence of prices p1,...pK with pk · Zi(pk+1) ≤0, k = 1,...,
K −1 and pK · Zi(p1) < 0.
9.1
THE SONNENSCHEIN–MANTEL–DEBREU THEOREM
We know that the aggregate excess demand function of an exchange economy
satisﬁes Walras’ Law and has homogeneity of degree zero. We ask here
whether there are any other properties which are systematically satisﬁed by
excess demand. Let S = {p ∈Rn
+ : ∥p∥= 1} be the intersection of the unit
sphere with the non-negative orthant, and let  = {p ∈Rn
+ : 
i pi = 1}. By
homogeneity, one can, without loss of generality, restrict attention to prices
either in S or , as demand functions and excess demand functions are
homogeneous of degree zero. Depending on the context, it is easier to work
with S or . The relative interior of S is denoted by int S. Likewise the relative
interior of  is denoted by int .
Sonnenschein–Mantel–Debreu Theorem
Suppose Z : S →X is a continu-
ous function satisfying Walras’ Law and let K ⊆int S be compact. Then there
exists an exchange economy E = (⪰i,ωi)n
i=1, in which each ⪰i is continuous,
strictly convex, and monotonic, such that the sum of individuals agents’ excess
demand functions in E coincides with Z on K.
The Sonnenschein–Mantel–Debreu (SMD) Theorem has a complicated proof,
but we present the gist of it in Section 9.1.1. Note that the economy E in the
SMD Theorem has a number of agents that is equal to the number of goods.
The next result shows that the conclusion of the theorem does not hold with
fewer consumers.
Proposition 9.1
There is a function Z satisfying the hypotheses of the SMD
Theorem, and a nonempty compact set K ⊆int S such that Z cannot be written
as the sum of fewer than n individual agents’ excess demand functions on K.


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9.1 The Sonnenschein–Mantel–Debreu Theorem
131
Proof. For any p ∈S, let T(p) denote the subspace that is orthogonal to p in
Rn
+. Choose p0 ∈int S arbitrarily, and let ε > 0 be such that p0
i > ε for all i. Let
K be the set of p ∈int S with pi ≥ε for all i. The set K is compact.
Deﬁne the function Z : S →Rn
+ by letting Z(p) be the projection of the vector
(p −p0) on to T(p).1 Note that Z is continuous and satisﬁes Walras’ Law, as
p · Z(p) = 0 because Z(p) is orthogonal to p. Observe that, for any p ∈S\{p0},
(p −p0) · Z(p) > 0,
(9.1)
because p −p0 can never be orthogonal to p, as both p and p0 are in the unit
sphere. This is a basic property of projections.
Suppose, toward a contradiction, that there are k < n individual excess
demand functions such that Z(p) = k
i=1 Zi(p) for all p ∈K. Walras’ Law
implies that Zi(p0) ∈T(p0). The vectors Zi(p0), i = 1,...,k, form a linearly
dependent set, as 0 = Z(p0) = k
i=1 Zi(p0). Let  be the linear subspace of
T(p0) spanned by the vectors Zi(p0), i = 1,...,k.
The dimension of  is strictly smaller than the dimension of T(p0), because
dim < k ≤n −1 = dimT(p0). Then the orthogonal complement ⊥of 
in T(p0) is nontrivial. Since p0 projects to 0 in T(p0), one can choose η ∈⊥
small enough so that there is ¯p ∈K such that ¯p ̸= p0, and ¯p −p0 projects to η
in T(p0).
Choose an arbitrary i. Then
(¯p −p0) · Zi(p0) = (¯p −p0 −η) · Zi(p0) + η · Zi(p0) = 0.
This follows because we know (¯p −p0 −η) · Zi(p0) = 0 as Zi(p0) ∈T(p0) and
(¯p−p0 −η) is the orthogonal projection of (¯p−p0) onto p0. Further, we know
η · Zi(p0) = 0 as Zi(p0) ∈ and η ∈⊥. The function Zi satisﬁes the weak
axiom of revealed preference, because it is an individual agent’s excess demand
function. Then by Walras’ Law, (¯p−p0)·Zi(p0) = 0 implies that ¯p·Zi(p0) ≤0.
By the weak axiom then, p0 · Zi(¯p) ≥0 = ¯p · Zi(¯p). Thus (¯p −p0) · Zi(¯p) ≤0.
Since this holds for all i, we obtain that (¯p −p0) · Z(¯p) ≤0, in contradiction
of (9.1).
The SMD Theorem talks about the behavior of Z on a compact subset of the
sphere; but students of general equilibrium theory know that many results rely
on the behavior of Z close to the boundary of its domain, when some prices
are close to zero. The next theorem says that one can decompose Z on all of its
domain as the sum of individual excess demand functions. The decomposition
is of a weaker nature, though.
Theorem 9.2
Suppose Z : S →X is a function satisfying Walras’ Law that
is bounded below. Then there are functions Zi : S →X, i = 1,...,n, satisfying
Walras’ Law and the Strong Axiom of Revealed Preference, for which Z =
n
i=1 Zi.
1 Formally, Z(p) = p −p0 −pp′(p −p0) = pp′p0 −p0, since ||p|| = 1;p′ is the transpose of p.


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General Equilibrium Theory
A proof of Theorem 9.2 can be obtained with arguments similar to those in
Section 9.1.1.
The SMD Theorem says that general equilibrium theory does not restrict
aggregate excess demand functions. It may not seem natural to assume that
someone could “observe” an excess demand function. It makes more sense that
one could observe a set of putative equilibrium prices. The theorem implies,
however, that any compact set of prices can be Walrasian equilibrium prices:
Corollary 9.3
Let K ⊆int S be compact. For any compact set K′ ⊆int S with
K ⊆K′, there exists an exchange economy E = (⪰i,ωi)n
i=1, in which each ⪰i is
continuous, strictly convex, and monotonic, for which K is the set of Walrasian
equilibrium prices of E in K′.
Proof. Let g(p) = (11 −pp′11) (one of the projections in the proof of the SMD
Theorem in Section 9.1.1, below),
f(p) = inf
q∈K ∥p −q∥,
and set Z(p) = f(p)g(p). Note that Z is continuous, p·Z(p) = f(p)(p·g(p)) = 0,
and Z(p) = 0 iff p ∈K ∪{11}. So for p ∈int S, Z(p) = 0 iff p ∈K. The result
follows from the SMD Theorem applied to Z and K′.
Some classic results of Mas-Colell strengthen Corollary 9.3. First, he
shows that any compact set in the interior of the price sphere can be the
equilibrium price set of a well-behaved economy (even taking the boundary
into consideration). Further, if there are at least three commodities, then for
any ﬁnite set A ⊆int S with an odd number of elements, and any function
d : A →{−1,1} such that 
p∈A d(p) = 1, there is a well-behaved exchange
economy with a smooth excess demand function such that A is its set of
Walrasian equilibrium prices, and d(p) is the index (see Mas-Colell, Whinston,
and Green, 1995) of equilibrium p.
9.1.1
Sketch of the proof of the Sonnenschein–Mantel–Debreu Theorem
Consider ﬁrst the excess demand function of a single consumer, Z(p) =
d(p,p·ω)−ω. As discussed at the beginning of the chapter, the excess demand
function Z satisﬁes WARP if there are no p and q such that q · Z(p) < 0 and
p·Z(q) ≤0. Observe that if Z satisﬁes the weak axiom of revealed preference,
and g : S →R+ is a strictly positive function, then p →g(p)Z(p) also satisﬁes
the weak axiom of revealed preference.
In preparation for the proof, we need to consider the function ˜Zi(p) =
1i −pip. We intend for ˜Zi to be the excess demand function of an individual
consumer. Recall that 1i is the unit vector with a one in the ith coordinate.
Recall that ∥p∥= 1 for p ∈S; thus for any vector x ∈X, pp′x is the orthogonal
projection of x onto the subspace deﬁned by the vector p. To see this, note that


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9.1 The Sonnenschein–Mantel–Debreu Theorem
133
pp′ satisﬁes (pp′)(pp′) = pp′, so that for any x, if y = pp′x, then y = pp′y (a
property called idempotence). Moreover, for any x, pp′(x −pp′x) = 0. So
˜Zi(p) = 1i −pp′1i = 1i −pip
is the “residual” of projecting 1i on the linear subspace spanned by p. In other
words, it is the projection of 1i on the subspace of vectors orthogonal to p.
Observe that ˜Zi is continuous and satisﬁes Walras’ Law, as p· ˜Zi(p) = pi −pip·
p = 0.
Note that ˜Zi is obtained as the result of a maximization program because
˜Zi(p) is the projection of 1i on to a subspace. (The projection results from
maximizing the negative of the distance of 1i to the subspace of vectors
orthogonal to p.) As a consequence of being the solution to a maximization
program, ˜Zi will satisfy the weak axiom of revealed preference.
So we see that ˜Zi is continuous, and satisﬁes Walras’ law and WARP. The
idea in the rest of the proof is to use the functions ˜Zi as a “basis” on which one
can decompose the function Z.
The set K from the hypothesis of the theorem is a compact set in the interior
of the sphere. Let f : S →R be a continuous function for which
Zi(p) + f(p)pi > 0
for all p ∈K.2 There is such a function because K is compact and pi > 0. Note
that the function
p →(Zi(p) + f(p)pi) ˜Zi(p) = (Zi(p) + f(p)pi)(1i −pip)
satisﬁes the weak axiom of revealed preference on K by the observation made
above. Deﬁne then Zi(p) = (Zi(p) + f(p)pi)(1i −pip).
We are not going to prove that Zi(p) on K can be generated by preferences
⪰i which are continuous, strictly convex, and monotonic. This proof is quite
involved and depends on the fact that K is a compact subset of the relative
interior of the unit sphere. One can see that it is “almost” generated by
preferences satisfying these properties (since it is based on a maximization
problem). We hope that having established that Zi(p) satisﬁes the weak axiom
is instructive enough.
Finally, we must verify that Z = n
i=1 Zi. Observe that
n

i=1
(Zi(p) + f(p)pi)(pip)
=
n

i=1
(piZi(p) + p2
i f(p))p
= (p · Z(p) + p · pf(p))p = f(p)p.
2 Note that Zi is the ith component of the function Z, not to be confused with Zi.


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General Equilibrium Theory
Likewise,
n

i=1
(Zi(p) + f(p)pi)1i
= Z(p) + f(p)p.
Thus,
n

i=1
(Zi(p) + f(p)pi)(1i −pip) = Z(p).
9.2
HOMOTHETIC PREFERENCES
We now turn to a different (and simpler) construction than the one in SMD. It
requires that Z be smooth, but it delivers a rationalizing economy in which all
agents’ preferences are homothetic.
Homotheticity has, generally speaking, strong implications. For example,
it is a crucial ingredient in aggregation theorems, from which the existence
of a representative consumer follows. It is therefore striking that there is a
version of the SMD Theorem even when we ask that agents’ preferences be
homothetic.
For convenience, we now take the domain of Z to be . The next result is
due to Rolf Mantel.
Theorem 9.4
Suppose that Z :  →X is a C2 function satisfying Walras’
Law, and let K ⊆int  be compact. Then there exists an exchange economy
E = (⪰i,ωi)n
i=1, in which each ⪰i is continuous, convex, homothetic, and
monotonic, such that the sum of individuals agents’ excess demand functions
in E coincides with Z on K.
Proof. Let Z be as in the hypothesis of the theorem. We construct an exchange
economy with n agents, each one endowed with m units of one good: the
endowment of agent i is m1i. As we shall see, the parameter m plays an
important role in this construction because it scales up an economy in which
all prices are equilibrium prices.
Let A be an n × n matrix and denote by ai the vector formed from its ith
column. We choose A such that the ai vectors are linearly independent and
ai · 1 = 1 (we use the notation 1 for a vector of ones). For an n-vector x, log(x)
denotes the vector whose entries are the logarithms of the entries of x.
Deﬁne the following functions:
gi(p) = 1
mZi
1
1

j pj
p
2
−ai · log(Ap),
on the domain consisting of p ∈Rn
++ with
1

j pj p ∈K. Denote by K′ this domain
of prices. We intend gi to be the indirect utility function of agent i in our
construction, when her income is 1.


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9.2 Homothetic preferences
135
Observe now that ai log(Ap) is a concave function of p, and that the second
derivatives of Z are bounded on K. Then by choosing m large enough we know
that gi(p) is a convex function on K.
Deﬁne the utility function of agent i to be
ui(x) = inf{gi(p) : p · x ≤1,p ∈K′}.
It is routine to verify that ui is monotonic, continuous, and quasiconcave.
To see that the preferences represented by ui are homothetic, note that ui is
log-homogeneous:
ui(λx) = inf{gi(p) : λp · x ≤1,p ∈K′}
= inf{gi((1/λ)q) : q · x ≤1,q ∈K′}
= inf{ 1
mZi
1
1

j qj
q
2
−ai · log(Aq(1/λ)) : q · x ≤1,q ∈K′},
but
ai · log(Aq(1/λ)) = ai · log(Aq) −ai · 1log(λ) = ai · log(Aq) −log(λ),
so that ui(λx) = ui(x) + log(λ). This means that a monotonic transformation
of ui is homogeneous, and therefore that the preferences represented by ui are
homothetic.
As a result, if we let vi(p,M) be the indirect utility function derived from ui,
then
vi(p,M) = gi(p) + log(M),
for M > 0.
Using Roy’s identity we obtain the demand of agent i as
di(p,p · m1i) = −∇pvi(p,p · m1i)
∇Mvi(p,p · m1i) = −p · m1i∇pgi(p).
Now,
∇pgi(p) = 1
m∇Zi(p) −A′L(p)−1ai,
where L(p) is the n×n diagonal matrix which has n
j=1 ajipj in its ith row and
column. So
di(p,p · m1i) = −p · m1i∇gi(p) = −pi∇Zi(p) + (pim)A′L(p)−1ai.
Then aggregate excess demand for this economy is
n

i=1
di(p,p · m1i) −m1 = −
n

i=1
pi∇Zi(p) +
n

i=1
(pim)A′L(p)−1ai −m1
= Z(p) + mA′L(p)−1Ap −m1
= Z(p) + mA′1 −m1 = Z(p),
where we have used the facts that Z(p) + n
i=1 pi∇Zi(p) = 0 (which follows
from Walras’ Law) and that A′1 = 1.


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General Equilibrium Theory
Remark 9.5
The proof of Theorem 9.4 uses an interesting construction. It
uses an economy in which agents’ preferences are homothetic and all prices
are equilibrium prices. One can verify that if gi in the proof is deﬁned to be
ai · log(Ap), and agents’ endowments are as above, then the construction used
in the proof gives an economy in which all prices are zero.
The proof works by adding a scaled-down version of Z(p) to the excess
demand function, and rationalizes such a “perturbed” excess demand by a
perturbation of the original homothetic preferences. The new economy has
the zeroes of Z as equilibrium prices, but its excess demand function is a
scaled-down version of Z. Now homotheticity guarantees that by scaling up
endowments we obtain an economy in which Z is the excess demand function.
9.3
PRICES AND ENDOWMENTS
We have looked at the implications of general equilibrium theory when one is
given either a set of prices or an aggregate excess demand function. We now
turn to a different set of givens.
We assume that we observe a ﬁnite collection of prices and endowments (or,
equivalently, of aggregate endowment and individual agents’ incomes). Under
the assumption that one can observe prices and endowments, we are going to
show that general equilibrium theory is testable. This observation is due to
Brown and Matzkin, as is most of the discussion in Section 9.3.
Consider an exchange economy E = (⪰i,ωi)I
i=1, where each pair (⪰i,ωi)
describes one consumer; as before, each agent i is described by a preference
relation ⪰i and a vector of endowments ωi ∈Rn
+. I is a positive integer
specifying the number of consumers. An allocation of 
i ωi is a vector
(xi)i∈I ∈RnI
+ such that 
i xi = 
i ωi. A Walrasian equilibrium is a pair ((xi),p)
such that (xi) is an allocation and p ∈Rl
++ satisﬁes that xi is maximal for ⪰i in
the set
{z ∈Rn
+ : p · z ≤p · ωi}.
We assume that we have data on prices, incomes, and resources. Speciﬁcally,
an economy-wide dataset is a collection DW = (pk,(ωk
i )I
i=1)K
k=0. If it seems
unreasonable to assume that individual endowments are observable, note that
one can instead work with individual incomes and aggregate endowment. The
results will be the same.
A dataset DW is Walras rationalizable if there are locally nonsatiated
preference relations ⪰i and, for each k an allocation (xk
i ) of 
i ωk
i such that
((xk
i ),pk) is a Walrasian equilibrium of the exchange economy (⪰i,ωk
i )I
i=1.
There are economy-wide datasets that are not Walras rationalizable.
Consider for example the dataset represented in Figure 9.1. In the ﬁgure, there
are two observations (pk,(ωk
1,ωk
2)), k = 1,2. For each k, ¯ωk = ωk
1 + ωk
2 deﬁnes
an Edgeworth box. Suppose that k = 1 gives the taller box, while k = 2 deﬁnes
the wider of the two boxes. The boxes are represented so that the consumption
space of agent 1 is the same in the two boxes (the (0,0) consumption vector


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9.3 Prices and endowments
137
p1
p2
B
A
ω1
ω2
Fig. 9.1 Two Edgeworth boxes representing a non-rationalizable dataset.
for agent 1 is the same in the two boxes). Note that pk and ωk
i deﬁne a budget
set for each consumer.
If the data in Figure 9.1 were Walras rationalizable, there would need to
exist some (unobserved) allocation of ¯ωk, for k = 1,2. These allocations must
lie inside each Edgeworth box. Then, if the dataset in Figure 9.1 were Walras
rationalizable, the allocation of ¯ω1 would have to lie on the segment A of the
budget line for consumer 1, while the allocation of ω2 would have to lie on the
segment B of the budget line for consumer 1 at prices p2. Any such allocation
would imply that consumer 1 violates WARP. So the allocation could not be
part of a Walrasian equilibrium, as the resulting choices by consumer 1 would
be incompatible with any preference relation for consumer 1.
The observation that the theory of Walrasian equilibrium is testable is subtle,
and in sharp contrast with the message of the SMD Theorem, but it is not
different in nature than the idea that there are individual observations that
violate WARP. One would ideally want to characterize all datasets that are
Walras rationalizable: such a test exists, but a “closed form” test is not known.
The following theorem (due to Brown and Matzkin) is a direct conse-
quence of Afriat’s Theorem. It is useful because it sets up the problem of
Walras-rationalizing data as a system of polynomial inequalities.
Theorem 9.6
A dataset DW = (pk,(ωk
i )I
i=1)K
k=0 is rationalizable iff there are
numbers (Uk
i ,λk
i )I
i=1 for k = 1,...,K, and vectors (xk
i )I
i=1 ∈XI for k = 1,...,K,


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General Equilibrium Theory
such that
Uk
i ≤Ul
i + λl
ipl · (xk
i −xl
i)
λk
i > 0
pk · xk
i = pk · ωk
i

i
xk
i =

i
ωk
i .
The characterization in Theorem 9.6 is not practical as a test because to
determine if a dataset is rationalizable requires solving a large system of
polynomial inequalities. This is hard to do. The point made by Brown and
Matzkin is that there exists a different test: one that is similar in nature
to checking GARP or SARP. We proceed to introduce the mathematical
framework in which we can express such a test. We need to introduce the
mathematical theory that deals with systems of polynomial inequalities.
Consider a system of polynomial inequalities:
I :
⎧
⎪⎪⎨
⎪⎪⎩
p1(a,x) S1 0
...
pk(a,x) Sk 0,
where a ∈Rm and x ∈Rn, so that pi is a polynomial of m + n variables and
Si ∈{≥,=,>,̸=}. Here we interpret a ∈Rm as a parameter, and the variables
x ∈Rn as unknowns: We want to know whether, for given a, there is x such
that (a,x) solves system I.
The following is a celebrated theorem due to Tarski and Seidenberg.
Theorem 9.7
There are systems of polynomial inequalities J1,...,JL in m
variables with the following property: For all a ∈Rm, there is x ∈Rn for which
(a,x) solves system I iff there is l such that a solves system Jl.
Remark 9.8
The theorem actually says more. It says that if we ﬁx a real
closed ﬁeld F, then there is a solution to I in F iff the coefﬁcients of I are a
solution to some Jl in F. Perhaps more importantly, the theorem is not just an
existence result. It is also associated with an algorithm which can be used, in
principle, to derive the systems J1,...,JL from the system I. The algorithm
is, however, not computationally efﬁcient. It is known that the problem of
eliminating quantiﬁers for real closed ﬁelds in general is computationally
complex. We discuss related ideas in Chapter 12.
We present two examples to illustrate the theorem. The ﬁrst is an example in
elementary algebra. Consider the second-degree polynomial of one variable:
p(x) = ax2 + bx + c. Suppose that a ̸= 0. The equation p(x) = 0 has a
solution (in R) iff the inequality b2 −4ac ≥0 is satisﬁed. The example is
an instance of quantiﬁer elimination since the statement ∃x(p(x) = 0), which
involves an existential quantiﬁer over x, is seen to be equivalent (when a ̸= 0)


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9.3 Prices and endowments
139
to the statement b2 −4ac ≥0.3 We can call the statement ∃x(p(x) = 0)
an existential statement because it is preceded by an existential quantiﬁer;
while b2 −4ac ≥0 is non-existential. As a statement that describes the
empirical content of a theory, ∃x(p(x) = 0), or any other existential statement,
is very problematic because its veriﬁcation requires checking all possible real
numbers. By eliminating quantiﬁers we can go from an existential statement to
a statement that can be directly veriﬁed on the data. Chapter 13 discusses this
issue in more depth.
The second example is familiar from our discussion of demand theory in
Chapter 3. Let D = {(xk,pk)}K
k=1 be a consumption dataset. Afriat’s Theorem
says that there is a solution (U1,...,UK,λ1,...,λK) to the (linear) system of
Afriat inequalities
Uk ≤Ul + λlpl · (xk −xl),k ̸= l
and λk > 0 iff the data satisfy GARP. Afriat’s Theorem is an example of
elimination of quantiﬁers: The statement that there is a solution to the system
of Afriat inequalities is existential. And Afriat’s Theorem says that such
an existential system is equivalent to GARP – observe that GARP is the
statement that for any sequence xh1,...,xhH of distinct observations ph1 · xh1 ≥
ph1 · xh2,...,phH−1 · xhH−1 ≥phH−1 · xhH implies that phH · xhH ≤phH · xh1. Saying
that a dataset satisﬁes GARP is not an existential statement.
We can formulate the satisfaction of GARP as the satisfaction of a system of
polynomial inequalities. To write down this system, let s denote a sequence
xhs
1,...,xhs
Hs of distinct observations in D. Let  be the set of all such
sequences. For each s ∈ we need to check that there is not a cycle xhs
1 ⪰R ...
⪰R xhs
Hs ≻R xhs
1. Satisfaction of GARP is the same as saying that no sequence
s ∈ gives rise to a cycle.
Let ri
s be the inequality phs
i · xhs
i < phs
i · xhs
i+1, i = 1,...Hs −1, and qs the
inequality phs
Hs ·xhs
Hs ≤phs
Hs ·xhs
1. Note that, for a sequence s, the satisfaction of
any one of these inequalities rules out the potential cycle xhs
1 ⪰R ... ⪰R xhs
Hs ≻R
xhs
1.
Deﬁne L = 
s∈{r1
s ,...,rHs
s ,qs}. Note that each l ∈L selects, for every
sequence s, one inequality: either one of the ri
s, or qs. If we satisfy the inequality
selected by l for s then we rule out the cycle indicated by sequence s. Let Jl be
the system of inequalities obtained by collecting the inequalities selected by l.
Satisfaction of just one system Jl rules out all potential cycles.
We know from Afriat’s Theorem that there is a solution to the system of
Afriat inequalities iff there is l such that the observed data (which are the
coefﬁcients of the system of Afriat inequalities) satisfy the corresponding
Jl. Thus Afriat’s Theorem illustrates an instance of quantiﬁer elimination, as
formalized in Theorem 9.7.
3 Formally, to apply the theorem, we need to treat a ̸= 0 as an inequality in the original system,
although the existentially quantiﬁed x does not appear there. The inequality a ̸= 0 then also
appears in the system in which x has been eliminated.


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140
General Equilibrium Theory
9.4
THE CORE OF EXCHANGE ECONOMIES
We ﬁnalize the discussion of exchange economies by brieﬂy looking instead
into the core of Walrasian equilibrium. Much less is known about the core than
about equilibrium, and we need to restrict attention to the case of two agents.
The material here is due to Bossert and Sprumont.
Consider an exchange economy with two consumers. Suppose that there is
an aggregate endowment ¯ω ∈Rn
++. In any allocation, consumption of agent 2
is determined by the consumption of agent 1. The set of possible endowments
and consumptions of agent 1 is E = {x ∈Rn
+ : x ≤¯ω} (either agent is permitted
to have zero endowment). For a set of preferences ⪰1 and ⪰2, we deﬁne the
core
C(⪰1,⪰2,ω) =
{x ∈E : x ⪰1 ω1 and ( ¯ω −x) ⪰2 ( ¯ω −ω1)}
∩{x ∈Rn
+ :̸ ∃y ∈E such that y ≻1 x and ( ¯ω −y) ≻2 ( ¯ω −x)}.
The ﬁrst part of this equation speciﬁes the usual individual rationality
constraint. The second expresses Pareto optimality. In fact, it expresses weak
Pareto optimality, but note that for strictly convex preferences, the notions
coincide.
The question is, given a correspondence c : E ⇒E, when do there exist
preferences ⪰1 and ⪰2, satisfying natural properties, such that for all e ∈E,
c(e) = C(⪰1,⪰2,e)? The properties under consideration are continuity, strict
monotonicity, and strict convexity.
We say that a set S ⊆E is connected if it is connected in the usual topological
sense; that is, the set cannot be partitioned into nontrivial disjoint open sets.
Viewing c as a correspondence, we can also deﬁne continuity in the usual
sense (of the joint hypotheses of lower and upper hemicontinuity). We say
that the correspondence c is regular if it is continuous in e, and for all e, c(e)
is connected.
For the core with strictly convex and monotone preferences, if some vector
e is ever selected for an endowment e′, then e must be the uniquely chosen
element when e′ is the endowment. We say that c satisﬁes persistence if for all
e,e′ ∈E, if e ∈c(e′), then c(e) = {e}.
Two more technical conditions are speciﬁed. The conditions are a method
of “identifying” the best and worst elements in the correspondence, and
postulating that they behave naturally with respect to the usual order on Rn.
Deﬁne, for e ∈E,
S0(e) = {S ⊆E : c(e) ∪{0} ⊆S,S path connected }
and
Sω(e) = {S ⊆E : c(e) ∪{ω} ⊆S,S path connected }.


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9.5 Chapter references
141
We say that c satisﬁes strict path monotonicity if for all e,e′ ∈E for which
e ≥e′ and e ̸= e′, we have

S∈S0(e)
S ⊊

S∈S0(e′)
S
and

S∈Sω(e′)
S ⊊

S∈Sω(e)
S.
Finally, we deﬁne S(e,e′) = {S ⊆E : c(e) ∪c(e′) ⊆S,S path connected }.
Say that c satisﬁes strict averaging reduction if for all e,e′ ∈E for which e ̸= e′
and all λ ∈(0,1), c(λe + (1 −λ)e′) is contained in the interior of %
S∈S(e,e′) S
relative to 
e∈E c(e).
Theorem 9.9
There exist strictly convex, strictly monotonic, and continuous
⪰1 and ⪰2 for which c(e) = C(⪰1,⪰2,e) for all e ∈E iff c : E ⇒E
satisﬁes regularity, persistence, strict path monotonicity, and strict averaging
reduction.
9.5
CHAPTER REFERENCES
The Sonnenschein–Mantel–Debreu Theorem was ﬁrst established for the case
when Z is a polynomial in Sonnenschein (1972). Mantel (1974) proved a
decomposition of Z as in the statement of the theorem, but using 2n agents.
The ﬁnal statement was obtained by Debreu (1974). Proposition 9.1 is due to
Debreu (1974). Corollary 9.3 is proved in Mas-Colell (1977); a much more
general statement appears there: that any compact subset of the interior of
the price sphere can be the equilibrium price set of a well-behaved economy.
The strengthening follows from a strengthening of the SMD Theorem. The
result pertaining to indices can also be found in this work; see also Mas-Colell,
Whinston, and Green (1995). Geanakoplos (1984) constructs an explicit utility
function to rationalize the individual excess demand functions constructed in
Debreu’s proof. Theorem 9.2 is from McFadden, Mas-Colell, Mantel, and
Richter (1974).
Theorem 9.4 is due to Mantel (1976). Mantel shows more. The endowments
in the construction in the theorem can be chosen arbitrarily, as long as they
are linearly independent. In particular, they can be close to proportional: when
they are exactly proportional (and preferences homothetic) we know that the
economy admits a representative consumer, which of course strongly limits the
excess demand function. A graphical illustration of the proof of Theorem 9.4
can be found in Shafer and Sonnenschein (1982) (attributed to Mas-Colell).
The SMD Theorem has a host of important implications. Uzawa (1960b)
has shown that Brouwer’s Fixed-Point Theorem is equivalent to the existence
theorem of the zeroes of a continuous function satisfying the properties
usually obtained from exchange economies. The SMD Theorem implies that


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142
General Equilibrium Theory
the Fixed-Point Theorem is equivalent to a standard equilibrium existence
theorem. In recent years, this has been exploited by computer scientists
studying the computational complexity of ﬁnding competitive equilibria (see
Papadimitriou and Yannakakis (2010); this possibility was anticipated by
Mantel (1977)).
Theorem
9.6,
the
example
in
9.3,
and
the
idea
of
using
the
Tarski–Seidenberg result appear in Brown and Matzkin (1996). The
Tarski–Seidenberg result is due to Tarski (1951), and was popularized by
Seidenberg (1954). The computational complexity of quantiﬁer elimination in
this environment was described in Davenport and Heintz (1988). Brown and
Shannon (2000) show that, in the setting assumed by Brown and Matzkin,
certain regularity conditions of equilibria (that all equilibria are locally
stable, and that the equilibrium correspondence is monotone) have no testable
implications in addition to Walrasian rationalizability.
The discussion in 9.4, and in particular, Theorem 9.9, is due to Bossert and
Sprumont (2002).
We have not discussed the literature on public goods. Snyder (1999)
provides a discussion of the relevant ideas.
The survey paper by Carvajal, Ray, and Snyder (2004) covers in depth some
of the same material as this chapter.
