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CHAPTER 6
Production
Production theory is another classical environment in which revealed pref-
erence theory is applied. The case of production is simpler than the case
of demand treated in the previous chapters, mainly because ﬁrm output is
a cardinally measurable and observable concept, whereas utility is not. In
the case of production, we shall assume that ﬁrm output and prices are both
observed, while the set of all feasible production vectors, that is the ﬁrm’s
technology, is not.
We will consider two approaches to production theory: the cost minimiza-
tion model and the proﬁt maximization model. In the ﬁrst model, factor prices
and factor demands are observed, and (single-dimensional) output is observed
as well. This environment is very similar to the consumer case, but, as we have
noted, simpler. We want to know whether the model is consistent with the cost
minimization hypothesis, meaning that the cost of production is minimized for
a given level of output.
In the second model, the model of proﬁt maximization, we want to test the
hypothesis that producers maximize proﬁt. This model is in a sense “dual” to
the consumer case. In the consumer case, we needed to solve for the function
being maximized, but we know the budget set. In contrast, in the producer case,
we know the function being maximized: it is a linear proﬁt function; but we do
not necessarily know the available technology (the constraint set faced by the
ﬁrm).
6.1
COST MINIMIZATION
We take as primitive a dataset comprising the input–output decisions of a ﬁrm.
The ﬁrm uses n factors, and produces a single good. An input–output dataset
D consists of a collection (yk,xk,pk), k = 1,...,K, where yk ∈R, xk ∈Rn
+, and
pk ∈Rn
++. Each observation k consists of a quantity of output yk, a vector of
factor demands xk, and factor prices pk. Note that output can be negative, but
inputs are always positive.
A production function is a mapping f : Rn
+ →R. We say that a production
function f cost rationalizes input–output dataset D if f(xk) = yk for all k, and


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84
Production
f(x) ≥f(xk) implies that pk · x ≥pk · xk. In other words, given prices pk, xk is a
cost-minimizing bundle across all bundles which can, according to f, produce
at least yk.
Theorem 6.1
The following statements are equivalent:
I) There is a continuous production function that cost rationalizes D.
II) yj ≤yk implies pj · xj ≤pj · xk and yj < yk implies pj · xj < pj · xk.
III) There are real numbers Uk,λk > 0 for which yj ≤yk implies Uj ≤Uk
and yj < yk implies Uj < Uk, and for all j,k,
Uj ≤Uk + λkpk · (xj −xk).
IV) There is a continuous, monotonic, and quasiconcave production
function that cost rationalizes D.
Equation (II) plays the role of GARP in this context. If we were to deﬁne the
revealed preference by xk ⪰R xj if pk ·xj ≤pk ·xk, and xk ≻R xj if pk ·xj < pk ·xk,
then (II) guarantees that ⟨⪰R,≻R⟩is acyclic.
The equivalence between (III) and (IV) in Theorem 6.1 is reminiscent
of Afriat’s Theorem. The statement in (III) gives the “Afriat inequalities”
corresponding to the problem under consideration. Note, however, that (III)
says more than in Afriat’s Theorem. The reason is that we observe production
output, the analogue of utility in demand theory, while utility is not observable.
Proof. To see that (I) implies (II), we ﬁrst note that if yj ≤yk then since f cost
rationalizes D, we have pj·xj ≤pj·xk. If yj < yk, then we know that pj·xj ≤pj·xk,
and if in fact pj · xj = pj · xk, we cannot have xk = 0, as otherwise, xk = xj = 0,
which would imply yk = f(xk) = f(xj) = yj, a contradiction. Since f(xk) =
yk > yj = f(xj), and f is continuous, there is x < xk for which f(x) > f(xj),
yet pj · x < pj · xj, a contradiction to the fact that f cost rationalizes D.
To see that (II) implies (III), we refer to Afriat’s Theorem. We may deﬁne
the revealed preference relations ⪰R and ≻R in the same way they are deﬁned
in Chapter 3, so that xj ⪰R xk if pj · xk ≤pj · xj, and xj ≻R xk if pj · xk < pj · xj.
By (II), xj ⪰R xk implies yj ≥yk and xj ≻R xk implies yj > yk. It follows that
the preference relation ⪰on X0 = {xk : k = 1,...,K} deﬁned by xj ⪰xk iff
yj ≥yk is such that xj ⪰R xk implies xj ⪰xk and xj ≻R xk implies xj ≻xk. The
result then follows by the constructive proof of Afriat’s Theorem, so that the
desired numbers exist. It is easily veriﬁed that yj ≤yk implies Uj ≤Uk and
yj < yk implies Uj < Uk.
Finally, to see that (III) implies (IV), let u : Rn
+ →R be the utility function as
constructed in Afriat’s Theorem. Recall that u is strictly increasing and concave
and that u(xk) = Uk. Observe ﬁrst that if u(x) ≥u(xk), then it follows that
Uk + λkpk · (x −xk) ≥u(x) ≥Uk, so that pk · xk ≤pk · x.
We now let ϕ be any strictly increasing transformation of u for which
ϕ(u(xk)) = yk (that this is possible follows as yj ≤yk implies uj ≤uk and yj < yk
implies uj < uk). Then let f = ϕ ◦u, and note that f is strictly increasing and


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6.1 Cost minimization
85
quasiconcave. Finally, it cost rationalizes the data: f(x) ≥f(xk) implies that
u(x) ≥u(xk), which we have shown implies pk · xk ≤pk · x.
Theorem 6.1 describes the datasets that are rationalizable by a quasiconcave
production function. Unlike the demand context, concavity here will impose
additional testable restrictions. For example, consider an environment with
one input. Suppose three observations are given in D: (y1,x1,p1) = (0,0,1),
(y2,x2,p1) = (1,1,1), and (y3,x3,p3) = (3,2,1). Note that D is cost rationaliz-
able by a quasiconcave production function; for example, f(x) = max{x,2x−1}
cost rationalizes the data. However, any f which cost rationalizes D must
satisfy f(0) = 0, f(1) = 1, and f(2) = 3. No such function can be concave.
To this end, we can also test when an input–output dataset can be cost
rationalized by a concave production function.
Theorem 6.2
The following are equivalent:
I) There is a concave production function that cost rationalizes D.
II) For each k, and all αj ≥0 for which 
j̸=k αj = 1, if pk ·(
j̸=k αjxj) ≤
pk · xk, then 
j̸=k αjyj ≤yk, and if pk · (
j̸=k αjxj) < pk · xk, then

j̸=k αjyj < yk.
III) There is a concave, monotonic, and continuous production function
that cost rationalizes D.
Proof. We ﬁrst show that (I) implies (II). Suppose D can be cost rationalized
by a concave production function f, and suppose that pk ·xk ≥pk ·
#
j̸=k αjxj$
for some αj as in statement (II). Then suppose by way of contradiction
that 
j̸=k αjyj > yk. In particular this implies that 
j̸=k αjf(xj) > f(xk). By
concavity, we know that f
#
j̸=k αjxj$
≥
j̸=k αjf(xj) > f(xk). First, we show
that 
j̸=k αjxj ̸= 0. Suppose, toward a contradiction, that 
j̸=k αjxj = 0.
Choose some j for which αj > 0 and yj > yk. We must have xj = 0, as

j̸=k αjxj = 0, which allows us to conclude that yj = f(0) > f(xk). By cost
rationalization, we then have that 0 = pk ·0 ≥pk ·xk, which implies that xk = 0,
contradicting f(0) > f(xk). This shows that 
j̸=k αjxj > 0.
Next, on the interior of any one-dimensional subset of Rn, f is continuous
(as it is concave). In particular, except possibly at the origin, f is continuous
on the ray passing through the origin and 
j̸=k αjxj. So, consider β < 1 for
which f
#
β
#
j̸=k αjxj$$
> f(xk). Then since pk ∈Rn
++, we know that pk ·
#
β
#
j̸=k αjxj$$
< pk·xk, a contradiction to the cost rationalization hypothesis.
This establishes that 
j̸=k αjyj ≤yk.
To complete the proof of (II), suppose that that pk·xk > pk·
#
j̸=k αjxj$
. Sup-
pose, toward a contradiction, that 
j̸=k αjyj ≥yk. Then f(xk) ≤
j̸=k αjf(xj) ≤
f
#
j̸=k αjxj$
, where the inequality follows by concavity. This is a direct


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86
Production
contradiction to the cost rationalization hypothesis, as 
j̸=k αjxj can produce
yk at a lower cost than xk.
Second, we prove that (II) implies (III). To that end, we show the existence
of λk > 0 such that for all j,k,
yk ≤yj + λjpj · (xk −xj).
(6.1)
The existence of such λk is equivalent to the existence of λk > 0 and μ > 0
such that for all j,k,
μ(yj −yk) + λjpj · (xk −xj) ≥0.
If we can solve these inequalities, we can renormalize, setting μ = 1, and
obtain a solution to (6.1). Now, by Lemma 1.12, there is no solution exactly
when, for each (j,k) where j ̸= k, there is α(j,k) ≥0 for which 
(j,k) α(j,k)(yj −
yk) ≤0 and for all j, 
k̸=j α(j,k)pj · (xk −xj) ≤0, and at least one of these
inequalities is strict. Since for all j, 
k̸=j α(j,k)pj · (xk −xj) ≤0, we get by (II)
that for every j, 
k̸=j α(j,k)(yk −yj) ≤0, with a strict inequality if the original
inequality is strict. By summing across j, we have 
(j,k) α(j,k)(yk −yj) ≤0, with
a strict inequality if any of the inequalities corresponding to some j is strict, a
contradiction.
The construction of a production function is the same as in Afriat’s Theorem.
Let f(x) = mink yk + λkpk · (x −xk), a concave, continuous, and strictly
monotonic function. Cost rationalization is veriﬁed as in Theorem 6.1. Finally,
for all k, f(xk) = yk.
Often we want to ensure that f(x) ≥0 for all x. A test for this is provided by
weakening the equality 
j̸=k αj = 1 in Theorem 6.2.
Theorem 6.3
The following are equivalent:
I) There is a non-negative concave production function that cost
rationalizes D.
II) For each k, and all αj ≥0 for which 
j̸=k αj ≤1, if pk ·(
j̸=k αjxj) ≤
pk · xk, then 
j̸=k αjyj ≤yk, and if pk · (
j̸=k αjxj) < pk · xk, then

j̸=k αjyj < yk.
III) There is a non-negative, concave, monotonic, and continuous produc-
tion function that cost rationalizes D.
Proof. We ﬁrst establish that (I) implies (II). The only difference from the
proof of Theorem 6.2 is that the equation f
#
j̸=k αjxj$
> f(xk) is established
by observing that f
#
j̸=k αjxj$
= f
#
(1 −
j̸=k αjxj)0 + 
j̸=k αjxj$
≥(1 −

j̸=k αj)0 + 
j̸=k αjf(xj) follows from non-negativity and concavity instead
of concavity alone.


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6.2 Proﬁt maximization
87
To see that (II) implies (III), we add, for every j, an inequality of the
form: yj + λjpj · (−xj) ≥0 to the list of inequalities described in Theorem 6.2.
Equivalently, in terms of the expression involving μ, this adds, for every j, an
inequality of the form:
μyj + λjpj · (−xj) ≥0.
Again, by Lemma 1.12, there is no solution exactly when, for each j,
there is ηj and for each (j,k) where j ̸= k, there is α(j,k) ≥0 for which
#
j ηjyj$
+
#
(j,k) α(j,k)(yj −yk)
$
≤0 and for all j, ηjpj ·(−xj)+
k̸=j α(j,k)pj ·
(xk −xj) ≤0, and at least one of these inequalities is strict. Since for all j,
ηjpj·(−xj)+
k̸=j α(j,k)pj·(xk−xj) ≤0, we get by (II) that for every j, ηj(−yj)+

k̸=j α(j,k)(yk−yj) ≤0, with a strict inequality if the original inequality is strict.
By summing across j, we have
#
j ηj$
+
#
yj 
(j,k) α(j,k)(yk −yj)
$
≤0, with a
strict inequality if any of the inequalities corresponding to some j is strict, a
contradiction.
The construction of f
used in the proof of Theorem 6.2 ensures
non-negativity; indeed, observe that f(0) = mink yk + λkpk · (−xk) ≥0.
Non-negativity then follows from monotonicity.
6.2
PROFIT MAXIMIZATION
The previous section assumed a ﬁrm with a single output y using n factors
of production. We now turn to a more ﬂexible formulation in which a ﬁrm
operates in n goods, choosing a net production vector y ∈Rn. If yi > 0 then
good i is produced in quantity yi by the ﬁrm. If yi < 0 then the ﬁrm uses good
i as an input.
A production dataset D is a collection (yk,pk), k = 1,...K, with yk ∈Rn and
pk ∈Rn
++.
We are interested in when a production dataset D is consistent with the
hypothesis of proﬁt maximization. A production set Y is a subset of Rn.
Production sets consist of all potential combinations of inputs and outputs
which are feasible. We say that production set Y rationalizes production dataset
D if for all k, yk ∈Y and pk ·yk ≥pk ·y for all y ∈Y. We say a production dataset
D is rationalizable if it is rationalizable by a production set. We will say that Y
is comprehensive if whenever y ∈Y and y′ ≤y, then y′ ∈Y.
Given is a production dataset D.
Theorem 6.4
The following statements are equivalent:
I) For all j,k, pk · yk ≥pk · yj.
II) D is rationalizable.
III) D is rationalizable by a closed, convex, and comprehensive produc-
tion set.


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88
Production
Proof. That (III) implies (II) and (II) implies (I) are obvious. To see that (I)
implies (III), let Y be the convex and comprehensive hull of {yk}K
k=1.1 Y is
obviously closed, convex, and comprehensive. We claim that Y rationalizes D.
To see this, it is ﬁrst clear that yk ∈Y for all k, by deﬁnition. Second, suppose
that y ∈Y. We want to show that for any j, pj · yk ≥pk · y. By deﬁnition of the
convex and comprehensive hull, there exists y′ = K
k=1 λkyk, where λk ≥0 for
all k and K
k=1 λk = 1, where y ≤y′ (this is an easy set-theoretic argument).
Consequently,
pj · y ≤pj · y′ =
K

k=1
λk(pj · yk) ≤
K

k=1
λk(pj · yj) = pj · yj,
where the last inequality follows from (I).
It is usually assumed that 0 ∈Y, because a ﬁrm can always choose to do
nothing. This adds the additional implication that proﬁts must be non-negative.
Corollary 6.5
Given is a production dataset D. Then the following are
equivalent:
I) For all j,k, pk · yk ≥0 and pk · yk ≥pk · yj.
II) D is rationalizable by a production set containing 0.
III) D is rationalizable by a closed, convex, and comprehensive produc-
tion set containing 0.
Proof. Let Y be the convex hull of the elements of D and the origin, and
proceed as above.
Theorem 6.4 illustrates an important distinction between the cases of
production and demand. We can say that a dataset D satisﬁes the weak axiom of
production if for all j,k, pk ·yk ≥pk ·yj; that is, if condition (I) in Theorem 6.4 is
satisﬁed. The weak axiom of production is also called the weak axiom of proﬁt
maximization. Note that the weak axiom of production is a binary condition –
that is, we only need to check pairs of data points to verify its satisfaction. In
our analysis of rational demand, we found that WARP was too weak, and that
rationalizability instead required GARP. The reason behind the sufﬁciency of
the weak axiom of production lies in the simpler structure of the problem of
production. In the case of demand, utility and the “shadow price” on utility
are unknowns. The Afriat inequalities of Chapter 3 state that such unknowns
must be extracted from the data. In contrast, in the case of production, proﬁt is
directly observable from the data.
An interesting and simple result on identiﬁcation of production sets is
possible. Given a rationalizable dataset D, we can deﬁne the lower production
set Y(D) to be the set deﬁned in the proof of Theorem 6.4: Let Y(D) be the
1 This is the smallest convex and comprehensive set containing these points. See Figure 6.2(a)
for an example.


--- Page 7 ---
6.2 Proﬁt maximization
89
p1
y1
y2
p2
p3
y3
Fig. 6.1 A rationalizable production dataset.
p1
p1
y1
y1
y2
y2
p2
p2
p3
p3
The lower production set, Y(D).
y3
y3
The upper production set, Y(D).
(a)
(b)
Fig. 6.2 Range of rationalizing production sets.
convex and comprehensive hull of the elements yk of D. We can deﬁne the
upper production set Y(D) to be the polyhedron generated by data D; that is,
let
Y(D) = {y ∈Rn : pj · y ≤pj · yj for all j}.
These rationalizing production sets are illustrated in Figure 6.2, based on the
example from Figure 6.1.
Theorem 6.6
Let D be a rationalizable production dataset, and Y ⊆Rn be
convex and comprehensive. Then Y rationalizes D iff Y(D) ⊆Y ⊆Y(D).


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90
Production
Proof. It is clear that if Y rationalizes D, then the set inclusions must hold.
Conversely, suppose that Y is convex and comprehensive and that the set
inclusions hold. Let y ∈Y. Because Y(D) ⊆Y, we know that yk ∈Y for all
k. And because Y ⊆Y(D), we know that pk · y ≤pk · yk for all k.
As in the case of demand theory, if one wishes to place additional assumptions
on Y, these may add restrictions on the rationalizable datasets. One assumption
of interest is constant returns to scale: A production set Y exhibits constant
returns to scale if for all y ∈Y and all λ ∈R+, λy ∈Y. It turns out that the
only additional implication imposed by this restriction is that the observed
ﬁrm always earns zero proﬁts (contrast with the discussion of homotheticity in
4.2.1).
Theorem 6.7
The following statements are equivalent:
I) For all j,k, 0 = pk · yk ≥pk · yj.
II) D is rationalizable by a production set satisfying constant returns to
scale.
III) D is rationalizable by a closed, convex, and comprehensive produc-
tion set satisfying constant returns to scale.
Proof. That (III) implies (II) is obvious. To see that (II) implies (I), we must
have 0 = pj · yj for all j, otherwise, if pj · yj > 0, then yj could not maximize
proﬁts. And because a ﬁrm can always choose 0 ∈Y as production, proﬁts
must be non-negative.
To see that (I) implies (III), deﬁne Y to be the convex, comprehensive
cone generated by the vectors yk of D; that is, the smallest convex and
comprehensive cone containing {yk : k = 1,...,K}. Note that Y satisﬁes the
required properties. It is clear that yk ∈Y for all k. Now, let y ∈Y. We will
show that pj · yj ≥pj · y. There exists λ ∈Rn
+ for which y ≤K
k=1 λkyk. Now,
pj · y ≤K
k=1 λk(pj · yk) ≤K
k=1 λk(pj · yj) = 0 = pj · yj.
Remark 6.8
An interesting distinction from the consumer case is that each
of the results of this section could be proved with inﬁnite datasets.
6.2.0.1 Nonlinear pricing
Just as in demand theory, we can consider an environment of nonlinear pricing.
To this end, for each k, we suppose we have a proﬁt function gk : Rn →R which
is weakly increasing (so that x ≤y implies g(x) ≤g(y)) and continuous. In this
context, we say that D is rationalizable if there exists Y for which yk ∈Y for all
k, and for all y ∈Y, gk(yk) ≥gk(y).
Theorem 6.9
For a given production dataset D, the following are
equivalent:
I) For all j,k, gk(yk) ≥gk(yj).
II) D is rationalizable.
III) D is rationalizable by a closed and comprehensive production set.


--- Page 9 ---
6.2 Proﬁt maximization
91
Proof. That (III) implies (II) and (II) implies (I) are again obvious. To see
that (I) implies (III), deﬁne Y = {x ∈Rn : gk(x) ≤gk(yk) for all k}. Note
that Y is closed, as the intersection of a collection of closed sets (that is,
Y = %K
k=1{x ∈Rn : gk(x) ≤gk(yk)}, each of which is closed by continuity).
Further, it is comprehensive since each gk is weakly increasing. By (I), for all
k, yk ∈Y. And if y ∈Y, then by deﬁnition, gk(yk) ≥gk(y).
Theorem 6.9 has an interesting connection with the theory of fairness. If each
gk is a utility function, and yk is agent k’s consumption, then the inequality
gk(yk) ≥gk(yj) states that the allocation (y1,...,yK) is envy-free. The result
characterizes envy-free allocations as those for which there is some set Y from
which each agent is allowed to maximize preference.
6.2.1
Unobserved factors of production
In contrast with demand theory (see 3.2.3), we can obtain restrictions from
proﬁt maximization even when the choices of some goods are unobserved.
Assume that production takes place in Rm+n; prices and output of the ﬁrst
m goods are observed, while of the last n goods, only prices are observed.
We shall deﬁne a partial production dataset to be a collection D of vectors
(yk,(pk,πk)), k = 1,...K, with yk ∈Rm and (pk,πk) ∈Rm+n
++ . Say that partial
production dataset D is rationalizable if there exists Y ⊆Rm+n and, for each k,
xk ∈Rn such that for all k, (yk,xk) ∈Y and (pk,πk)·(yk,xk) ≥(pk,πk)·(y,x) for
all (y,x) ∈Y.
By Theorem 6.4, we know that a partial production dataset is rationalizable
iff it is rationalizable by a closed, convex, and comprehensive production set.
Theorem 6.10
A partial production dataset is rationalizable iff for all
 = (λj,k) ∈RK×K
+
such that for all k,

j∈K
λ(j,k)πk =

j∈K
λ(k,j)πj
and λ(k,k) = 0, we have

(j,k)∈K×K
λ(j,k)pk · (yj −yk) ≤0.
Proof. By Theorem 6.4, rationalizability is equivalent to the existence of xk
solving the following inequalities, one for each pair j,k where j ̸= k:
πk · xk −πk · xj ≥pk · (yj −yk).
By Lemma 1.14, this inequality has no solution iff there are non-negative
real numbers λ(j,k) such that for all k,

{j∈K:j̸=k}
λ(j,k)πk −

{j∈K:j̸=k}
λ(k,j)πj = 0


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92
Production
and

{(j,k)∈K×K:j̸=k}
λ(j,k)pk · (yj −yk) > 0.
This is exactly what the statement of the theorem precludes (note that λ(k,k) is
unconstrained so we can take λ(k,k) = 0).
Theorem 6.10 has two immediate important corollaries. The ﬁrst is the
following negative result, which concerns prices of unobserved commodities
that are conically independent across observations, in the sense that no such
price vector is in the convex cone spanned by the remaining price vectors. For
example, every linearly independent set of vectors is conically independent,
but not conversely. In the case of conic independence, there are no testable
implications to the proﬁt maximization hypothesis. This is problematic if we
do not know which factors may not be observed. As a practical matter, it seems
likely that the larger the number of unobserved commodities, the more likely
the price vectors are to be conically independent. In such an environment, the
hypothesis of proﬁt maximization is not falsiﬁable.
Corollary 6.11
Suppose that the vectors {πk}K
k=1 have the property that for
each j, πj is not in the convex cone spanned by {πk}k̸=j. Then (yk,(pk,πk)) is
rationalizable.
The second is the case in which for all j,k, we have πj = πk. The matrix 
in Theorem 6.10 then satisﬁes the condition that, for all k, 
{j:j̸=k} λ(j,k) =

{j:j̸=k} λ(k,j). The diagonal of  is identically zero. Consider a modiﬁed matrix
′ which differs only from  on the diagonals, and the diagonals are chosen to
be non-negative and so that for all j,k, 
{i∈K} λ(i,j) = 
{i∈K} λ(i,k). The matrix
′ now has the feature that all columns and rows sum to the same number.
In combinatorics, such a matrix is a non-negative multiple of a bistochastic
matrix.2 The reason this is of interest is that, by a theorem of Birkhoff and
von Neumann (which can be found in Berge 1963, for example), a bistochastic
matrix is known to be a convex combination of permutation matrices.3
It therefore follows that ′ is a non-negative linear combination of
permutation matrices. As a result of this, it is easily seen that the condition in
Theorem 6.10 reverts to a cyclic monotonicity condition as in Theorem 1.9.
However, the inequality here is of the opposite sign as the version in
Theorem 1.9. This should not be a surprise, as the following corollary relates to
proﬁt maximization, and the proﬁt function is well known to be convex, rather
than concave.
2 A matrix is bistochastic if it consists of only non-negative entries, and the row and column
sums are equal to one.
3 A permutation matrix is a matrix of ones and zeroes, with exactly one 1 in each row and each
column.


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6.3 Chapter references
93
Corollary 6.12
Suppose that for all j,k, πj = πk. Then D is rationalizable iff
for all sequences i1,...,ik,
k

j=1
pij · (yij+1 −yij) ≤0.
It is possible to give a “cyclic” version (one ruling out certain cycles, that
is) of Theorem 6.10 in the general case, but this condition does not seem to be
any more illuminating than the one stated in the theorem.
Similar exercises can be undertaken with partial production datasets, which
can derive the testable implications of constant returns to scale and other such
hypotheses. We shall not undertake such an exercise here.
6.2.2
Measuring violations from rationalizability
The most common approach to measuring deviations from rationalizability is
to ﬁnd an approximate technology which would have generated the observed
data. There are generally two approaches: a most conservative, and a least
conservative, corresponding to the ideas in Theorem 6.6. Given an approximate
production set, a measure of efﬁciency as in Debreu (1951) or Farrell (1957)
can be applied to the observed data given this production set.
Banker and Maindiratta (1988) suggest that one should construct a
technology so that as many data points as possible are consistent with
proﬁt maximization. So, they ask that all observed vectors are feasible for
the technology, and as many of possible maximize proﬁts. They establish
that Y(D) generates such a technology. The axiomatic approach of Banker,
Charnes, and Cooper (1984) justiﬁes Y(D).4
Varian (1990) describes a very similar idea. That is, if there is a pair (pk,yk)
(pj,yj) for which pk · yk < pk · yj, Varian recommends that the measure of
inefﬁciency for this violation should be pk·yj
pk·yk −1. This measure is economically
meaningful, in that it represents the percentage gain in proﬁt that the ﬁrm could
have obtained by producing yj instead of yk.
6.3
CHAPTER REFERENCES
Theorem 6.1 can be found in Hanoch and Rothschild (1972) and Varian (1984).
Hanoch and Rothschild (1972) considers the case when the rationalizing
production function is only required to be weakly monotonic.
Theorems 6.4 and 6.7 and Corollary 6.5 are also found in Hanoch and
Rothschild (1972). Again, Varian (1984) also discusses this result.
4 Their axioms are convexity, comprehensivity, yk ∈Y for all k, and minimality with respect
to set inclusion while satisfying these properties. That is, their axioms deﬁne the convex,
comprehensive hull of yk.


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94
Production
Theorems 6.2 and 6.3 are basically due to Afriat (1972) and Diewert and
Parkan (1983). The idea behind Theorem 6.6 is due to Diewert and Parkan
(1983) and Varian (1984).
The ideas of this section ﬁrst appear in Afriat (1972) and Hanoch and
Rothschild (1972). These papers also consider the case where no price data
are observed; and the goal is simply to test whether there is a production
function or production set consistent with the observed inputs and outputs
being produced efﬁciently. Tests where there are data only on certain of the
variables are considered in detail in Afriat (1972), Hanoch and Rothschild
(1972), and Diewert and Parkan (1983). Theorem 6.10 is based on the linear
programming ideas found in these works. Issues we have not discussed, which
are discussed in detail in these papers, are the implications of assumptions on
production when proﬁts may be observable, or when no prices are observable,
and so on. F¨are, Grosskopf, and Lovell (1987), for example, talks about tests
for non-monotonicity.
Many of the early results described in this chapter have evolved into the
ﬁeld of data envelopment analysis (DEA), which seeks to measure and test
efﬁciency of productive units via linear programming and convex analytic
techniques. See, for example, Cooper, Seiford, and Tone (2007). Stochastic
frontier analysis studies similar questions from a stochastic production
standpoint; see Kumbhakar and Lovell (2003) for an exposition.
The ideas on which production set to use for efﬁciency measurement are due
to Banker, Charnes, and Cooper (1984) and Banker and Maindiratta (1988).
These ideas are fundamental in DEA. The measure of inefﬁciency of observed
data brieﬂy discussed is due to Varian (1990).
One may be interested in studying particular properties of the unobserved
technologies: The papers by Chambers and Echenique (2009a) and Dziewulski
and Quah (2014) focus on supermodularity of the production function.
There is a large empirical literature testing the ideas in this chapter in a
nonparametric fashion. A few such works which test the ideas in agricultural
settings include Fawson and Shumway (1988), F¨are, Grosskopf, and Lee
(1990), Ray and Bhadra (1993), Featherstone, Moghnieh, and Goodwin
(1995), and Tauer (1995).
The Birkhoff–von Neumann Theorem is an important result in graph theory,
and is attributed to Birkhoff (1946) and Von Neumann (1953).
