--- Page 1 ---
CHAPTER 4
Topics in Rational Demand
Chapter 3 established the testable implications of the hypothesis that con-
sumers are rational. We are often interested in situations where the rational-
izing preference, or demand function, satisﬁes additional properties. We want
to know what additional structure is imposed on a dataset from demanding
that the rationalization has to use a utility function with some given property.
In particular, we focus on the properties of supermodularity, submodularity,
homotheticity, separability, complements, and substitutes.
4.1
DISCRETE GOODS: SUPERMODULAR AND SUBMODULAR
RATIONALIZATIONS
In Chapter 3 we discussed a collection of results in the spirit of Afriat’s
Theorem. In these results, one obtains a concave rationalization from the
rationalizability of the data. Arguably, though, most consumption goods come
in discrete units. Some goods seem particularly “lumpy,” such as cars and
houses. For such goods, the notion of concavity is not well deﬁned. One can
instead investigate super- and submodularity. Supermodularity corresponds
to the notion that goods are complements: speciﬁcally, that increases in the
consumption of one good become more valuable when one consumes more
of the other goods. Submodularity corresponds to the property of substitute
goods.
The meaning of super- and submodularity can be understood from
Figure 4.1. In the ﬁgure, there are two goods. For any two bundles x and y,
x ∧y is the component-wise minimum of the bundles x and y, and x ∨y is the
component-wise maximum (see the deﬁnitions in Chapter 1). The function
u : X →R is supermodular if for all x,y ∈X,
u(x) + u(y) ≤u(x ∨y) + u(x ∧y),
and submodular if −u is supermodular.
If u is supermodular then the change in utility u(x)−u(x∧y) cannot exceed
the change u(x∨y)−u(y). Note that in Figure 4.1, the increase in good 2 when
we go from x∧y to x is the same as when we go from y to x∨y. This means that


--- Page 2 ---
58
Topics in Rational Demand
x
y
u(x) – u(x    y)
x    y
>
x    y
>
>
>
u(x    y) – u(y) 
Fig. 4.1 Supermodularity.
the change in utility resulting from adding the amount of good 2 in the change
from x ∧y to x, can only be larger when the quantity of good 1 is larger. This
is a notion of complementary goods: the increases in utility due to increases in
consumption of good 2 are larger as we consume more of good 1.
Submodularity has the opposite interpretation. If u is submodular in the
ﬁgure, then the increase in utility due to the increase in consumption of good
2 is diminished by a higher consumption of good 1. Thus the two goods are
substitutes.
It turns out that, when goods are discrete, supermodularity and submodu-
larity of utility are empirically indistinguishable from rationalizability of the
data. This result is in the spirit of Afriat’s, but it comes from a rather different
set of ideas.
Let X ⊆Zn
+. A (discrete) dataset is a ﬁnite list of observations (xk,Bk),
k = 1,...,K, where for each k, xk ∈Bk ⊆X. The interpretation is that xk is
the chosen element from the budget Bk. Assume:
• xk is maximal in Bk, in the sense that if y > xk then y /∈Bk.
• Bk satisﬁes the property that z ∈Bk whenever there is y ∈Bk and z ≤y.
• |Bk| < +∞.
Given a dataset D, we can deﬁne the strong revealed preference pair ⟨⪰S,≻S⟩
as in Chapter 3 by x ⪰S y iff there is k such that x = xk and y ∈Bk, and x ≻S y
iff x ⪰S y and x ̸= y. Then the strong axiom of revealed preference (SARP) is
the requirement that ⟨⪰S,≻S⟩is acyclic.
Note that Theorem 3.9 is valid for the model discussed here. The notion of
data is different, but the assumptions we have made on the data guarantee that
the proof of the theorem applies as written.
Theorem 4.1
Let X ⊆Zn
+ be a lattice. The following statements are
equivalent:


--- Page 3 ---
4.1 Discrete goods: Supermodular and submodular rationalizations 59
I) D is strongly rationalizable.
II) D satisﬁes SARP.
III) D has a strictly monotonic and supermodular strong rationalization
u : X →R.
IV) D has a strictly monotonic and submodular strong rationalization v :
X →R.
Proof. The equivalence of (I) and (II) is clear from previous results (Theo-
rem 3.9). We need to prove that (II) implies (III) and (IV).
Because K is ﬁnite, and each Bk is ﬁnite, there is M such that for all x ∈
k Bk
and all i, xi < M. Let XM = {x ∈Zn
+ : xi < M for all i}.
Let D satisfy SARP. Imagine ﬁrst that consumption space is XM. By
Theorem 3.9, D has a strictly monotonic rationalization ⪰on XM. Since XM
is ﬁnite, there is an integer-valued utility function v : XM →Z+ deﬁned by
v(x) = |{y ∈XM : x ⪰y}|. Let V = supx∈XM v(x). Deﬁne g : X →R as follows.
For x ∈XM, g(x) = v(x). Otherwise, g(x) = V + 
i xi. Note that g is a strictly
monotonic, integer-valued function which strongly rationalizes the data.
Let u(x) = 2g(x); we claim that u is a supermodular strong rationalization of
D. It is clearly a strong rationalization because it is a monotonic transformation
of g. To see that it is supermodular, let x and y in X and suppose without loss
of generality that g(x) ≥g(y). If x = x ∨y then y = x ∧y so there is nothing to
prove. Suppose then that x ̸= x∨y, which implies that x∨y ≻x, as ⪰is strictly
monotonic. Thus g(x ∨y) ≥g(x) + 1. Then,
u(x) + u(y) = 2g(x) + 2g(y) ≤2g(x) + 2g(x) ≤2g(x)+1
≤2g(x∨y) = u(x ∨y) ≤u(x ∨y) + u(x ∧y);
so u is supermodular.
To exhibit a submodular strong rationalization, deﬁne v : X →R by v(x) =
g(x)
i=0 2−i. The function v is a strong rationalization of D because v is a
monotonic transformation of g. To see that v is submodular, suppose that
x ̸= x ∨y and x ̸= x ∧y (otherwise we have nothing to prove). Then,
v(x) −v(x ∧y) =
g(x)

j=g(x∧y)+1
2−j ≥
 1
2
!g(x∧y)+1
.
On the other hand,
v(x ∨y) −v(y) =
g(x∨y)

j=g(y)+1
2−j ≤
∞

j=g(y)+1
2−j
=
 1
2
!g(y)
≤
 1
2
!g(x∧y)+1
where the last inequality follows because x ̸= x ∨y implies that y ̸= x ∧y, so
y ≻x∧y by the strict monotonicity of ⪰. Thus v(x∨y)−v(y) ≤v(x)−v(x∧y),
which shows that v is submodular.


--- Page 4 ---
60
Topics in Rational Demand
x
Data {(p, x), (p′, x ′)} satisfying GARP.
Projected data violate GARP.
x
x ′
x'
(b) 
(a) 
Fig. 4.2 Testable implications of homothetic preferences.
4.2
DIVISIBLE GOODS
Super- and submodularity are properties of particular interest when studying
discrete goods. Other properties are commonly studied in environments with
divisible goods. We now turn to homotheticity, separability, complements, and
substitutes.
Assume throughout that consumption space X ⊆Rn
+ is a convex set. The
notion of dataset is the same as in 3.1: A consumption dataset D is a collection
(xk,pk), k = 1,...K, with K ≥1 an integer, xk ∈X and pk ∈Rn
++.
4.2.1
Homotheticity
Here we assume X is a cone, so that whenever x ∈X and α ≥0, we have αx ∈X.
A preference ⪰is homothetic if, for all α > 0 and all x,y, x ⪰y implies that
αx ⪰αy. A function u : X →R is homothetic if the preference that it represents
is homothetic.
We can begin to understand the testable implications of homotheticity from
Figure 4.2. Suppose that we are interested in whether the data {(p,x),(p′,x′)}
have a homothetic rationalization. Clearly, the data satisfy GARP so they
have a rationalization. It is easy to see, however, that no homothetic
rationalization exists. The reason can be gleaned from Figure 4.2(b): a
homothetic rationalization would imply that demand would have to lie on the
ray joining x and 0, for any budget line that is parallel to the budget line for
(p,x). So if we “deﬂate” the budget line for (p,x) until it crosses the point
where the ray crosses the budget line for (p′,x′) the demand would have to lie
at that intersection. We would then have a violation of WARP.
Somewhat more formally, Figure 4.2 presents the following situation. If
there were a homothetic rationalization, then any supporting hyperplane of the
upper contour set {y ∈X : y ⪰x} would also support {y ∈X : y ⪰θx}, for θ > 0.
Choose θ such that θx lies on the budget line for (p′,x′), i.e. p′ · (θx) = p′ · x′;
thus x′ is revealed preferred to θx. By homotheticity, p supports the upper


--- Page 5 ---
4.2 Divisible goods
61
contour set of ⪰at θx, so θx should be chosen at prices p and income p · θx.
Then WARP would require that p · (θx) ≤p · x′.
The reasoning above leads to the following necessary condition for
rationalization by a homothetic preference:
θ = p′ · x′
p′ · x ⇒θp · x ≤p · x′;
or,
1 ≤p · x′
p · x
p′ · x
p′ · x′ .
A strengthening of this condition turns out to be necessary and sufﬁcient for
rationalization by a homothetic preference. To simplify the notation, we shall
normalize the prices in the data so that pk ·xk = 1 for all k. If we have prices so
that pk · xk ̸= 1, it is simple to redeﬁne prices qk =
pk
pk·xk provided each xk ̸= 0.
Theorem 4.2 will then apply to these normalized prices.
A dataset D satisﬁes the homothetic axiom of revealed preference (HARP)
if, for any sequence k1,...,kL of numbers in {1,...,K}, we have
pkL · xk1
L−1
"
i=1
pki · xki+1 ≥1.
Observe that HARP is stronger than GARP: Fix a sequence k1,...,kL such
that GARP is violated; that is pki ·xki+1 ≤pki ·xki, i = 1,...L−1, and pkL ·xk1 <
pkL ·xkL. Then pki ·xki = 1 implies that the product of the left-hand side of these
inequalities is strictly less than one. Thus HARP is violated.
Finally, let us deﬁne the homothetic revealed preference order pair ⟨⪰H,≻H⟩
as x ⪰H y when there are k and α > 0 for which αx = xk and pk · xk ≥pk · (αy)
and x ≻H y when there are k and α > 0 for which αx = xk and pk ·xk > pk ·(αy).
The meaning of ⪰H should be intuitive: x ⪰H y when the data, together with
the hypothesis of homothetic preferences, imply that x must be preferred to y.
Speciﬁcally, if x and xk are on the same ray, then homotheticity requires that
x be demanded at the same prices as xk. This is like adding the observation
(x,pk) to the data. Then pk · x > pk · y implies that x is “revealed preferred” to
y, which is precisely the meaning of x ⪰H y.
The following result is due to Hal Varian.
Theorem 4.2
Let X be a convex cone such that for all x ∈X and all ε > 0,
there is ε′ ∈(0,ε) with x + ε′1 ∈X. Let D = {(xk,pk)}K
k=1 be a consumption
dataset, where for all k, pk · xk = 1. The following statements are equivalent:
I) D has a locally nonsatiated and homothetic weak rationalization.
II) D satisﬁes the homothetic axiom of revealed preference.
III) There is a strictly positive real number Uk for each k, such that
Uk ≤Ulpl · xk
for each pair of observations (xk,pk) and (xl,pl) in D.


--- Page 6 ---
62
Topics in Rational Demand
IV) D has a concave, homothetic, continuous, and monotonic rationaliza-
tion u : X →R.
V) ⟨⪰H,≻H⟩is acyclic.
Proof. We ﬁrst prove that I implies II. Let ⪰be a locally nonsatiated,
homothetic rationalization. Fix any sequence k1,...,kL in {1,...,K}.
Let the sequence s1,...,sL−1 be deﬁned by
s1 = pk1 · xk2
s2 = (pk1 · xk2)pk2 · xk3 = s1pk2 · xk3
...
sL−1 = sL−2pkL−1 · xkL.
We ﬁrst argue that xk1 ⪰xk2/s1. Notice that the bundle xk2/s1 is affordable in
the budget at which xk1 was purchased: pk1 · (xk2/s1) = 1. Then xk1 ⪰(xk2/s1).
Next, we argue that homotheticity implies (xk2/s1) ⪰(xk3/s2). The reason is
that
pk2 · xk2 = pk2 ·
 s1
s2 xk3
!
= 1,
so that xk2 ⪰(s1/s2)xk3, and hence by homotheticity, (xk2/s1) ⪰(xk3/s2).
By repeating this argument we obtain that
xk1 ⪰(xk2/s1) ⪰(xk3/s2) ⪰··· ⪰(xkL/sL−1).
By assumption, xkL is ⪰maximal in the set {x : pkL ·x ≤pkL ·xkL}, so xkL/sL−1
is ⪰maximal in the set {x : sL−1pkL ·x ≤sL−1pkL ·xkL/sL−1} by the homotheticity
of ⪰. Then xk1 ⪰(xkL/sL−1) and local nonsatiation of ⪰implies sL−1pkL ·xk1 ≥
sL−1pkL · (xkL/sL−1). Since sL−1pkL · (xkL/sL−1) = 1 we have established that
HARP is satisﬁed.
Second, we prove that II implies III. Let the number Ul be deﬁned as the
inﬁmum of
L−1
"
i=1
pki · xki+1
over all sequences k1,...,kL in {1,...,K} with kL = l. The inﬁmum is achieved
for some sequence because HARP guarantees that removing a cycle only
makes the product L−1
i=1 pki ·xki+1 smaller, so we can without loss of generality
consider the inﬁmum over sequences with no cycles, and there are ﬁnitely
many such sequences. Let Ul = L−1
i=1 pki ·xki+1 and Um = L′−1
i=1 pk′
i ·xk′
i+1; then
Ul =
L−1
"
i=1
pki · xki+1 ≤(
L′−1
"
i=1
pk′
i · xk′
i+1)(pm · xl) = Um(pm · xl).
Note that, following this construction, Uk > 0 for all k.


--- Page 7 ---
4.2 Divisible goods
63
Third, we prove that III implies IV. Since we assumed that pl · xl = 1 for all
l, we have Ulpl · xk = Ul + Ulpl · (xk −xl) for all k,l, so that we have a solution
to Afriat’s inequalities. Deﬁne a utility function u : X →R by
u(x) = min{Uk + Ukpk · (x −xk) : k = 1,...,K},
a construction analogous to the one in the proof of Afriat’s Theorem. Then
pk · xk ≥pk · y implies that u(xk) ≥u(y), so u rationalizes the data.
Clearly, u is continuous, monotonic, and homothetic. The proof that it is
concave is the same as in Afriat’s Theorem.
Finally, we demonstrate that (II) and (V) are equivalent. Suppose by means
of contradiction that there is a cycle z1 ⪰H ... ⪰H zL ≻H z1. For each i = 1,...,L,
suppose αizi = xki. Now, zi ⪰H zi+1 means that pki · xki ≥pki ·

αizi+1
=
pki ·
#
αkixki+1
αki+1
$
. Then for all i = 1,...,L −1, 1 = pki · xki ≥pki ·
#
αkixki+1
αki+1
$
, and
1 = pkL · xkL > pkL ·
#
αkLxk1
αk1
$
. Multiplying the inequalities and canceling the α
terms obtains pkL · xk1 L−1
i=1 pki · xki+1 < 1, violating HARP.
On the other hand, suppose (II) is violated, and let k1,...,kL be a sequence
for which pkL · xk1 L−1
i=1 pki · xki+1 < 1. Deﬁne z1 = xk1, α1 = 1, and for each
i = 2,...,L, deﬁne αi = pki−1 ·xki, and zi =
xki
i
j=1 αj . Observe that for all i = 1,...,
L −1, zi ⪰H zi+1, since pki · (αizi) = pki · (αizi+1). Observe also that
#L
i=1 αL$
pkL ·zL = pkL ·xkL = 1, and that
#L
i=1 αL$
pkL ·z1 = pkL ·xk1 L−1
i=1 pki ·
xki+1 < 1 by assumption, so that zL ≻H z1, constituting a ⟨⪰H,≻H⟩cycle.
Remark 4.3
The preceding does not allow observations of xk = 0. If we
did, the “non-normalized” version of HARP and condition (III) are necessary
and sufﬁcient here. The non-normalized version of HARP would read that for
all sequences {k1,...,kn}, we have pkn · xk1 n−1
i=1 pk1 · xki+1 ≥n
i=1 pki · xki. The
non-normalized version of condition (III) is that Ukpl · xl ≤Ulpl · xk for all k
and l.
4.2.2
Separability
In practical analysis of consumer demand, separability is a very important
property. In principle, a consumer chooses among many different goods,
solving inter- as well as intratemporal optimization problems. Practical
researchers abstract away from this complexity by considering some subset of
goods in isolation. For example, classical studies of applied demand, such as
Deaton (1974), work with only 9 goods. Modern applied papers often consider
more goods, but still greatly simplify the universe of possible goods.
The simpliﬁcation of focusing on a small subset of goods and assuming
separability avoids the issues raised in Section 3.2.3. In fact, assuming
separability of some kind seems unavoidable for any tractable empirical study
of consumption.


--- Page 8 ---
64
Topics in Rational Demand
Let X ⊆X1 ×X2, with Xi ⊆Rni
+; and write vectors in X as (x1,x2), where xi ∈
Xi. A preference relation ⪰is separable in X1 if for all x1,x′
1 ∈X1 and all x2, ˆx2 ∈
X2 (x1,x2) ⪰(x′
1,x2) if and only if (x1, ˆx2) ⪰(x′
1, ˆx2). A utility function u: X →R
is separable in X1 if there are functions f : X1 →R and g : f(X1) × X2 →
R, where for all x2, y →g(y,x2) is strictly monotonic, such that u(x1,x2) =
g(f(x1),x2).
When preferences are separable, the consumer’s maximization problem
reduces into a “subproblem” for the goods in X1. Given a budget to spend on
goods x1, the consumer can solve the problem of choosing x1 independently
of the speciﬁc bundle x2 chosen: only the budget left over to spend on x1
matters, not the actual quantities of the goods in x2. Thus separability allows
the separate analysis of demand for x1. Of course, here we are concerned with
testing for separability, so we cannot ignore the relation between choosing x1
and x2.
Theorem 4.4
The following statements are equivalent:
I) There are strictly positive real numbers Uk, Vk, λk, μk, k = 1,...,K
such that
Uk ≤Ul + λlpl
1 · (xk
1 −xl
1) + λl
μl (Vk −Vl),
(4.1)
Vk ≤Vl + μlpl
2 · (xk
2 −xl
2).
(4.2)
II) The dataset {(pk
2,xk
2)} satisﬁes GARP; and there is a solution (Vk,μk)
to 4.2 above such that the dataset {((pk
1,1/μk),(xk
1,Vk)} satisﬁes
GARP.
III) D has a concave, continuous, and monotonic rationalization
u : X →R that is separable in X1.
We present Theorem 4.4 without proof. It is important to note that Theorem 4.4
focuses on concave rationalizations (note the contrast with Afriat’s Theorem,
where concavity comes for free). The Afriat inequalities for this problem are
described by Equations (4.1) and (4.2). These inequalities are not linear, in
contrast with the original Afriat inequalities.
A special kind of separability is additive separability. A test for additive
separability is particularly interesting for time-series data (see Section 5.3.3),
in which there is a single observation (a dataset of size 1).
Let X ⊆X1 × X2 × ··· × XT, with Xt = Rn
+, t = 1,...,T. Write vectors
in X as (x1,...,xT), where xt ∈Rn
+. A preference relation ⪰is additively
separable if there is a function u : Rn
+ →R such that (x1,...,xT) ⪰(x′
1,...,x′
T)
iff T
t=0 u(xt) ≥T
t=0 u(x′
t).
Say that a dataset {(xk,pk)}K
k=1 is additively separably rationalizable by the
function u : Rn
+ →R if the additively separable preferences deﬁned from u
weakly rationalize the data {(xk,pk)}K
k=1. We provide a test for the case when
K = 1, so the dataset is (x,p). Note that p = (p1,...,pT), with pt ∈Rn
++.


--- Page 9 ---
4.2 Divisible goods
65
Proposition 4.5
Data (x,p) (a dataset with K = 1) with x ≫0 is additively
separably rationalizable by a concave and strictly increasing function iff the
correspondence ρ(x) = 
t:x=xt{pt} satisﬁes cyclic monotonicity.
Proposition 4.5 follows from using the ﬁrst-order conditions for maximiza-
tion of 
t u(xt), and Corollary 1.10.
Note that general utility maximization is not testable with a single
observation, but the theory of additive separability has testable implications
even when K = 1.
4.2.3
Quasilinear utility
Many economic models assume that utility takes a quasilinear form: u(x) + y,
for x ∈Rn
+ and y ∈R. There are n+1 goods, and the (n+1)-st is a numeraire.
Wealth is measured in the same units as the numeraire good, which therefore
always has a price of one. We shall allow consumption of the numeraire to be
negative, which is a common assumption in applications of quasilinear utility.
Given the assumption that y can be negative, the maximization of u(x) + y
subject to a budget constraint with prices p (the problem max(x,y) u(x) + y
subject to the constraint p · x + y ≤I) is, when u is locally nonsatiated,
equivalent to the maximization of u(x) −p · x.
As in Proposition 4.5, the property of cyclic monotonicity described in 1.4
can be used to characterize datasets that could be rationalized by a quasilinear
utility.
We take as primitive a dataset D: a collection (xk,pk), k = 1,...K, with
K ≥1 an integer, xk ∈X and pk ∈Rn
++. There are now, however, n + 1 goods
and we seek a rationalization by a quasilinear utility function. A dataset D is
quasilinear rationalizable if there exists a locally nonsatiated utility function
u : X →R such that for all k and x ∈X,
u(x) −pk · x ≤u(xk) −pk · xk.
Theorem 4.6
Let X be a convex consumption space such that for all x ∈X
and all ε > 0, there is ε′ ∈(0,ε) with x+ε′1 ∈X. The following statements are
equivalent:
I) D is quasilinear rationalizable.
II) For each k, there is Uk such that for all k,l,
Uk ≤Ul + pl · (xk −xl).
III) The correspondence ρ(x) = 
k:x=xk{pk} satisﬁes cyclic monotonicity.
IV) The data are quasilinear rationalizable by a continuous, strictly
increasing, concave utility function.
Proof. That (I) implies (II) is as follows. Let u rationalize the data, and
let Uk = u(xk). In particular, we have Ul −pk · xl ≤Uk −pk · xk, which
establishes II.


--- Page 10 ---
66
Topics in Rational Demand
That (II) implies (III) follows from adding up Afriat inequalities correspond-
ing to cycles, as we discussed after stating Afriat’s Theorem (and by the same
argument as in the proof of Theorem 1.9). The proof that (III) implies (IV) is
exactly as in Theorem (1.9): let
u(x) = inf{pk1 · (x −xk1) + pk2 · (xk1 −xk2) + ... + pkM−1 · (xkM −xkM−1)},
where the inﬁmum is taken over all sequences k1,...,kM with xk1 ﬁxed at some
arbitrary x0 ∈X. (See Corollary 1.10).
That (IV) implies (I) is trivial.
Remark 4.7
If one assumes that income and quasilinear good consumption
must be positive, then the equivalence of (II)–(IV) in the preceding still
hold. That (IV) implies (II) holds from the ﬁrst-order conditions of concave
optimization. Condition (IV) could be proved by choosing I large enough so
that I −pi · xi ≥0 for all i, and deﬁning yi = I −pi · xi.
The idea behind Theorem 4.6 is similar to the argument in Theorem 1.9, and it
is instructive to see why. If we reason as in the discussion of Afriat’s Theorem,
and assume that there is a differentiable rationalization u, then the relevant
ﬁrst-order condition for maximizing u(x) −p · x is
∇u(x) = p.
As was the case for Afriat’s Theorem, we seek to infer marginal utilities from
data, but the difference is that now the Lagrange multiplier (λk in Afriat’s
Theorem) is known and equal to one. The dataset therefore already tells us
what the marginal utilities must be, if the data are to be rationalized by a
quasilinear utility. Theorem 4.6 therefore asks for prices that could be marginal
utilities for a concave utility function – namely cyclic monotonicity. Most of
the work in proving Afriat’s Theorem went into establishing the existence of
multipliers λk such that ρ(x) = 
k:x=xk{λkpk} is cyclically monotone. In the
case of quasilinear utility, this is already taken care of by the quasilinearity
assumption.
4.2.4
Gross complements and substitutes
We now turn to two basic properties of demand: the complementarity or
substitutability between a pair of goods. Roughly speaking, two goods are
“gross” complements if a price change that favors the consumption of one
good also induces higher consumption of the second good. Common examples
of complementary goods include coffee and sugar, or gin and tonic. Instead
of being properties of preference, gross complements and gross substitutes are
properties of demand functions.
Formally, we say that a demand function on R2
+ satisﬁes gross complements
if for all m, d(p,m) is a weakly decreasing function of p. We say that the
demand function is rational if there exists a locally nonsatiated preference


--- Page 11 ---
4.2 Divisible goods
67
relation ⪰for which there is a unique ⪰-maximal element of {x ∈Rn
+ : p · x ≤
m}, and d(p,m) is equal to this element. A demand function satisfying gross
complements is one for which both goods respond in the same direction to
changes in price. On the other hand, we will say a demand function on R2
+
satisﬁes gross substitutes if d1(p,m) increases in p2, and d2(p,m) increases
in p1. Note that for these deﬁnitions to be at all meaningful, demand must be
single-valued. We say a demand function is a rational demand function if there
is a monotonic preference which generates d as its demand correspondence.
It is convenient here to suppose that m is normalized; this is without loss of
generality as any rational demand function is homogeneous. For the remainder
of this subsection, we always assume m = 1, and drop the dependence of d
on m.
Data come in the form of observed price/demand pairs, D = {(pk,xk)}K
k=1. We
turn to the question of when data are consistent with a rational demand function
exhibiting the property of gross complements. In line with the normalization of
income, assume that data has been normalized such that, for all k, pk · xk = 1.
Consider Figure 4.3(a), which depicts a hypothetical observation of demand
x = (x1,x2) at prices p = (p1,p2). In principle, the two budgets in Figure 4.3(a)
are not comparable, and the observations might be consistent with gross
complements. However, the dotted budget line in the ﬁgure can be obtained
by either starting from (x,p) and making one good cheaper, or by starting from
(x′,p′) and making the other good cheaper. Either way, demand at the dotted
budget line should be larger than both x and x′. As Figure 4.3(b) illustrates,
this is not possible. In this fashion we obtain a condition on the data that is
necessary for consistency with gross complements: the pointwise maximum
of demands, x ∨x′, must be affordable for any budget larger than the p and p′
budgets.
There is a second necessary condition. Consider the observed demands in
Figure 4.3(c). This a situation where, when we go from p to p′, demand for
the good that gets cheaper decreases while demand for the good that gets
more expensive increases. This is not in itself a violation of complementarity.
However, consider Figure 4.3(d): were we to increase the budget from p to the
dotted prices, complementarity would imply a demand at the dotted prices that
is larger than x. But no point in the dotted budget line is both larger than x and
satisﬁes the weak axiom of revealed preference (WARP) with respect to the
choice of x′.
So a simultaneous increase in one price and decrease in another cannot yield
opposite changes in demand. This property is a strengthening of WARP: Fix
p, p′ and x as in Figure 4.3(c). Then WARP requires that x′ not lie below the
point where the p and p′ budget lines cross. Our property requires that x′ not
lie below the point on the p′-budget line with the same quantity of good 2 as x.
The following result states that the two necessary conditions illustrated
in Figure 4.3 are in fact both necessary and sufﬁcient. These constitute a
nonparametric test of gross complements in the demand for a pair of goods.


--- Page 12 ---
68
Topics in Rational Demand
x
A larger budget.
(a)
(b)
(c)
(d)
Inconsistency with gross complements.
Two observations: (x, p) and (x′, p′).
Gross complements would induce a
WARP violation.
x
x
x
x′
x′
x′
x′
Fig. 4.3 Implications of gross complements.
Theorem 4.8
Let {(pk,xk)}K
k=1 be a dataset with n = 2 and, for all k, pk ·
xk = 1 (a normalization). There exists a rational demand d, satisfying gross
complements, such that for all k, d(pk) = xk iff for all k,l, the following are
satisﬁed:
• (pk ∧pl) · (xk ∨xl) ≤1
• For all i ̸= j, if pk · xl ≤1 and pk
i > pl
i, then xk
j ≥xl
j.
We now turn to gross substitutes. To illustrate the implications of gross
substitutes for observed demand, consider the example in Figure 4.4. We have
two observations: x is the bundle purchased at prices p, and x′ is purchased at
prices p′. These purchases do not appear to directly violate gross substitutes.
The observed choices are also consistent with the weak axiom of revealed
preference, so there is an extension of these purchases to a rational demand
function that is deﬁned for all prices. There is, however, no demand function
compatible with these observations which satisﬁes gross substitutes: Consider
the prices p′′ given by the dotted budget line. Gross substitutes and the choice
of x at p require a decrease in the consumption of the good whose price is the
same in p and in p′′, so demand at p′′ should lie in the northwest segment of the


--- Page 13 ---
4.3 Chapter references
69
x
x′
x
x′
Two observations: (x, p) and (x′, p′).
Implications of gross substitutes.
(b)
(a)
Fig. 4.4 Implications of gross substitutes.
budget line. On the other hand, gross substitutes and the choice of x′ require
that demand at p′′ lies in the southeast segment of the budget line. Since these
two segments are disjoint, there is no demand function that extends the data
and satisﬁes gross substitutes.
Figure 4.4 then suggests a necessary condition for observations to be
consistent with a rational demand exhibiting gross substitutes. The next
theorem states that the condition is both necessary and sufﬁcient. Smoothness
of demand comes for free in this case.
Theorem 4.9
Let {(pk,xk)}K
k=1 be a dataset with n = 2 and, for all k, pk·xk = 1
(a normalization). There exists a smooth and rational demand function d
satisfying gross substitutes such that for all k, d(pk) = xk iff for all k,l such
that pk
1 ≤pl
1 and pl
2 ≤pk
2, we have pl
1xl
1 ≤pk
1xk
1.
4.3
CHAPTER REFERENCES
A version of Theorem 4.1 appeared ﬁrst in Chambers and Echenique (2009b),
and was extended by Shirai (2010). Our proof of the supermodular result
follows suggestions by Eran Shmaya and John Quah. The submodular utility
used in the proof is a construction due to Rader (1963) (Shirai noted that this
construction yields a submodular utility).
The results in Sections 4.2.1 and 4.2.2 are from Varian (1983a). The
equivalence between conditions (III) and (I) in Section 4.2.1 was shown by
Afriat (1972) and Diewert (1973); HARP is due to Varian, though Diewert
mentions an equivalent test in a footnote. Knoblauch (1993) describes a
method for predicting responses to price changes for data consistent with
homotheticity.
Theorem 4.4 is due to Hal Varian. The recent paper of Quah (2013) is
the ﬁrst paper to treat the case of a general, possibly nonconcave, separable
rationalization. In fact Quah presents an example of a dataset that has a


--- Page 14 ---
70
Topics in Rational Demand
separable rationalization, but not a concave and separable one. Quah shows
that to test for separability one needs to verify that data satisfy some ﬁnite
set of conﬁgurations. The paper by Cherchye, Demuynck, Hjertstrand, and
De Rock (2014) shows that testing for separability in Varian’s concave setting
is computationally hard, and provides a computational approach to dealing
with the system of inequalities in Theorem 4.4. Echenique (2013) proves that
the nonconcave case is also computationally hard, even when the number of
goods is as small as in Deaton (1974) (i.e. 9 goods).
Proposition 4.5 appears in Browning (1989), who uses it to test for
additive separability in time series data. He focuses on the case of a single
observation because households in consumption surveys only make a single
choice. The paper of Echenique, Imai, and Saito (2013) presents a result on
time separability for multiple observations (as well as for other models of
intertemporal choice).
The result on quasilinear utility is from Brown and Calsamiglia (2007).
Theorem 4.8 is due to Chambers, Echenique, and Shmaya (2010), while
Theorem 4.9 appears in Chambers, Echenique, and Shmaya (2011). Related
are the papers of Kehoe and Mas-Colell (1984) and Kehoe (1992), which show
that gross substitutability of demand implies a version of the weak axiom.
