Essays in Index Number Theory, Volume I
W.E. Diewert and A.O. Nakamura (Editors)
c 1993 Elsevier Science Publishers B.V. All rights reserved.
Chapter 6
DUALITY APPROACHES TO MICROECONOMIC THEORY*
W.E. Diewert
1. Introduction
What do we mean when we say that there is a “duality” between cost and
production functions? Suppose that a production function F is given and that
u = F(x), where u is the maximum amount of output that can be produced
by the technology during a certain period if the vector of input quantities x ≡
(x1, x2, . . . , xN ) is utilized during the period. Thus, the production function
F describes the technology of the given firm. On the other hand, the firm’s
minimum total cost of producing at least the output level u given the input
prices (p1, p2, . . . , pN ) ≡ p is defined as C(u, p) and it is obviously a function
of u, p and the given production function F. What is not so obvious is that
(under certain regularity conditions) the cost function C(u, p) also completely
describes the technology of the given firm; i.e., given the firm’s cost function C,
it can be used in order to define the firm’s production function F. Thus, there
is a duality between cost and production functions in the sense that either of
these functions can describe the technology of the firm equally well in certain
circumstances.
In the first part of this chapter, we develop this duality between cost
and production functions in more detail. In Section 2, we derive the regularity
conditions that a cost function C must have (irrespective of the functional form
or specific regularity properties for the production function F), and we show
how a production function can be constructed from a given cost function. In
Section 3, we develop this duality between cost and production functions in a
more formal manner.
*This paper is an abridged version of IMSSS Technical Report No. 281, Stanford
University, 1978. An even more abridged version was published in K.J.
Arrow and M.D. Intriligator (eds.), Handbook of Mathematical Economics, Amsterdam:
North-Holland Publishing Co., 1982, pp. 535–599. The author would
like to thank C. Blackorby and R.C. Allen for helpful comments and the Canada
Council for financial support. My thanks also to May McKee and Shelley Hey
for typing a difficult manuscript. A preliminary version of this paper was presented
at the New York meetings of the Econometric Society, December 1977.
106 Essays in Index Number Theory 6. Duality Approaches 107
In Section 4, we consider the duality between a (direct) production function
F and the corresponding indirect production function G. Given a production
function F, input prices p ≡ (p1, p2, . . . , pN ) and an input budget of
y dollars, the indirect production function G(y, p) is defined as the maximum
output u = F(x) that can be produced, given the budget constraint on input
expenditures pT
x ≡
N
i=1 pixi ≤ y. Thus, the indirect production function
G(y, p) is a function of the maximum allowable budget y, the input prices
which the producer faces p, and the producer’s production function F. Under
certain regularity conditions, it turns out that G can also completely describe
the technology and thus there is a duality between direct and indirect production
functions.
The above dualities between cost, production and indirect production
functions also have an interpretation in the context of consumer theory: simply
let F be a consumer’s utility function, x be a vector of commodity purchases
(or rentals), u be the consumer’s utility level, and y be the consumer’s “income”
or expenditure on the N commodities. Then C(u, p) is the minimum cost of
achieving utility level u given that the consumer faces the commodity prices p
and there is a duality between the consumer’s utility function F and the function
C, which is often called the expenditure function in the context of consumer
theory. Similarly, G(y, p) can now be defined as the maximum utility that the
consumer can attain given that he faces prices p and has income y to spend
on the N commodities. In the consumer context, G is called the consumer’s
indirect utility function.
Thus, each of our duality theorems has two interpretations: one in the
producer context and one in the consumer context. In Section 2, we will use
the producer theory terminology for the sake of concreteness. However, in
subsequent sections, we use a more neutral terminology which will cover both
the producer and the consumer interpretations: we call a production or utility
function F an aggregator function, a cost or expenditure function C a cost
function, and an indirect production or utility function G an indirect aggregator
function.
In Section 5, the distance function D(u, x) is introduced. The distance
function provides yet another way in which tastes or technology can be characterized.
The main use of the distance function is in constructing the Malmquist
[1953] quantity index.
In Sections 2–5, we will provide proofs of theorems so that the reader will
be able to appreciate the techniques involved. In the remainder of the paper,
results will often only be stated (with some exceptions where new results are
presented).
In Section 6, we discuss a variety of other duality theorems: i.e., we
discuss other methods for equivalently describing tastes or technology, either
locally or globally, in the one output, N inputs context. The reader who is
primarily interested in applications can skip Sections 3–6.
The mathematical theorems presented in Sections 2–6 may appear to
be only theoretical results (of modest mathematical interest perhaps) devoid
of practical applications. However, this is not the case. In Sections 7–10, we
survey some of the applications of the duality theorems developed earlier. These
applications fall into two main categories: (i) the measurement of technology or
preferences (Sections 10 and 11), and (ii) the derivation of comparative statics
results (Sections 7–9 inclusive).
In Section 11, we consider firms that can produce many outputs while
utilizing many inputs (whereas earlier, we dealt only with the one output case).
We state some useful duality theorems and then note some applications of these
theorems.
Finally, in Section 12, we show how duality theory can be modified to
deal with noncompetitive situations and, in Section 13, we briefly note some of
the other areas of economics where duality theory has been applied.
2. Duality between Cost (Expenditure) and Production (Utility)
Functions: A Simplified Approach
Suppose we are given an N input production function F : u = F(x), where u is
the amount of output produced during a period and x ≡ (x1, x2, . . . , xN ) ≥ 0N
1
is a nonnegative vector of input quantities utilized during the period. Suppose,
further, that the producer can purchase amounts of the inputs at the fixed
positive prices (p1, p2, . . . , pN ) ≡ p 0N and that the producer does not
attempt to exert any monopsony power on input markets.2
The producer’s cost function C is defined as the solution to the problem
of minimizing the cost of producing at least output level u, given that the
producer faces the input price vector p:
(2.1) C(u, p) ≡ min
x
{pT
x : F(x) ≥ u}
In this section, it is shown that the cost function C satisfies a surprising
number of regularity conditions, irrespective of the functional form for the
production function F, provided only that solutions to the cost minimization
problem (2.1) exist. In a subsequent section, it is shown how these regularity
1
Notation: x ≥ 0N means each component of the N dimensional vector x is
nonnegative, x 0N means that each component is positive, and x > 0N
means that x ≥ 0N but x = 0N .
2
In Section 12 of this chapter, the assumption of competitive behavior is
relaxed.
108 Essays in Index Number Theory 6. Duality Approaches 109
conditions on the cost function may be used in order to prove comparative
statics theorems about derived demand functions for inputs (cf. Samuelson
[1947; ch. 4]).
Before establishing the properties of the cost function C, it is convenient
to place the following minimal regularity condition on the production function
F:
Assumption 1 on F: F is continuous from above; i.e., for every u ∈
textRange F,3
L(u) ≡ {x : x ≥ 0N , F(x) ≥ u} is a closed set.
If F is a continuous function, then of course F will also be continuous
from above. Assumption 1 is sufficient to imply that solutions to the cost
minimization problem (2.1) exist, as the following lemma indicates.
Lemma 1. If F satisfies assumption 1 above and p 0N , then for every
u ∈ range F, minx{pT
x : x ≥ 0N , F(x) ≥ u} exists.
Proof: Let u ∈ range F. Then there exists x∗
≥ 0N such that F(x∗
) ≥ u.
Define the set S∗
≡ {x : pT
x ≤ pT
x∗
, x ≥ 0N }. Since p 0N , S∗
is a closed
and bounded set. Thus
C(u, p) ≡ min
x
{pT
x : x ≥ 0N , F(x) ≥ u}
= min
x
{pT
x : x ∈ L(u)}
= min
x
{pT
x : x ∈ L(u) ∩ S∗
}
since if x ≥ 0N and x /∈ S∗
, then pT
x∗
< pT
x and x could not be a solution
to the cost minimization problem. Thus we can restrict attention to the closed
and bounded set of feasible x’s, L(u) ∩ S∗
, where the minimum of pT
x will be
attained.qed
The following seven properties for the cost function C can now be derived,
assuming only that the production function F satisfies assumption 1.
Property 1 for C: For every u ∈ range F and p 0N , C(u, p) ≥ 0;
i.e., C is a nonnegative function.
Proof:
C(u, p) ≡ min
x
{pT
x : x ≥ 0N , F(x) ≥ u}
= pT
x∗
say, where x∗
≥ 0N and F(x∗
) ≥ u
≥ 0 since p 0N and x∗
≥ 0N . qed
3
This simply means that the output u can be produced by the technology.
Throughout this section, range F can be replaced with the smallest convex set
containing range F.
Property 2 for C: For every u ∈ range F, if p 0N and k > 0, then
C(u, kp) = kC(u, p); i.e., the cost function is (positively) linearly homogeneous
in input prices for any fixed output level.
Proof: Let p 0N , k > 0 and u ∈ textRange F. Then
C(u, kp) ≡ min
x
{(kp)T
x : F(x) ≥ u}
= k min
x
{pT
x : F(x) ≥ u}
≡ kC(u, p). qed
Property 3 for C: If any combination of input prices increases, then
the minimum cost of producing any feasible output level u will not decrease;
i.e. if u ∈ textRange F and p1
> p0
, then C(u, p1
) ≥ C(u, p0
).
Proof:
C(u, p1
) ≡ min
x
{p1T
x : F(x) ≥ u}
= p1T
x1
say, where x1
≥ 0N and F(x1
) ≥ u
≥ p0T
x1
since p1
> p0
and x1
≥ 0N
≥ min
x
{p0T
x : F(x) ≥ u} since x1
is feasible for the cost
minimization problem but not necessarily optimal
≡ C(u, p0
). qed
Thus far, the properties of the cost function have been intuitively obvious
from an economic point of view. However, the following important property is
not an intuitively obvious one.
Property 4 for C: For every u ∈ textRange F, C(u, p) is a concave
function4
of p.
Proof: Let u ∈ textRange F, if p0
0N , p1
0N and 0 ≤ λ ≤ 1. Then
C(u, p0
) ≡ min
x
{p0T
x : F(x) ≥ u} = p0T
x0
say, and
C(u, p1
) ≡ min
x
{p1T
x : F(x) ≥ u} = p1T
x1
4
A function f(z) of n variables defined over a convex set S is concave iff z1
, z2
∈
S, 0 ≤ λ ≤ 1 implies f[λz1
+ (1 − λ)z2
] ≥ [λf(z1
) + (1 − λ)f(z2
)]. A set S is a
convex set iff z1
, z2
∈ S, 0 ≤ λ ≤ 1 implies [λz1
+ (1 − λ)z2
] ∈ S.
110 Essays in Index Number Theory 6. Duality Approaches 111
say. Now
C[u, λp0
+ (1 − λ)p1
] ≡ min
x
{[λp0
+ (1 − λ)p1
]T
x : F(x) ≥ u}
= [λp0
+ (1 − λ)p1
]T
xλ
, say,
= λp0T
xλ
+ (1 − λ)p1T
xλ
≥ λp0T
x0
+ (1 − λ)p1T
x1
,
since xλ
is feasible for the cost minimization problems associated with the input
price vectors p0
and p1
but is not necessarily optimal for those problems
= λC(u, p0
) + (1 − λ)C(u, p1
). qed
The basic idea in the above proof is used repeatedly in duality theory.
Owing to the nonintuitive nature of property 4, it is perhaps useful to provide
a geometric interpretation of it in the two input case (i.e., N = 2).
Suppose that the producer must produce the output level u. The u isoquant
is drawn in Figure 2.1. Define the set S0
as the set of nonnegative input
combinations which are either on or below the optimal isocost line when the
producer faces prices p0
; i.e., S0
≡ {x : p0T
x ≤ C(u, p0
), x ≥ 0N }, where
C0
≡ C(u, p0
) = p0T
x0
is the minimum cost of producing output u given that
the producer faces input prices p0
0N . Note that the vector of inputs x0
solves the cost minimization problem in this case. Now suppose that the producer
faces the input prices p1
0N and define S1
, C1
and x1
analogously;
i.e., S1
≡ {x : p1T
x ≤ C(u, p1
), x ≥ 0N }, C1
≡ C(u, p1
) = p1T
x1
, where the
vector of inputs x1
solves the cost minimization problem when the producer
faces prices p1
.
Let 0 < λ < 1 and suppose now that the producer faces the “average”
input prices λp0
+ (1 − λ)p1
. Define Sλ
, Cλ
and xλ
as before:
Sλ
≡ {x : [λp0
+ (1 − λ)p1
]T
x ≤ C[u, λp0
+ (1 − λ)p1
], x ≥ 0N },
Cλ
≡ C[u, λp0
+ (1 − λ)p1
] = [λp0
+ (1 − λ)p1
]T
xλ
where xλ
solves the cost minimization problem when the producer faces the
average prices λp0
+ (1 − λ)p1
. Finally, consider the isocost line which would
result if the producer spends an “average” of the two initial costs, λC0
+ (1 −
λ)C1
, facing the “average” input prices, λp0
+(1−λ)p1
. The set of nonnegative
input combinations which are either on or below this isocost line is defined as
the set
S∗
≡ {x : [λp0
+ (1 − λ)p1
]T
x ≤ [λC0
+ (1 − λ)C1
], x ≥ 0N }.
Figure 2.1
In order to show the concavity property for C, we need to show that Cλ
≥
λC0
+ (1 − λ)C1
or, equivalently, we need to show that Sλ
contains the set S∗
.
It can be shown that the isocost line associated with the set S∗
,
L∗
≡ {x : [λp0
+ (1 − λ)p1
]T
x = [λC0
+ (1 − λ)C1
]},
passes through the intersection of the isocost line associated with the sets S0
and S1
.5
On the other hand, the isocost line associated with the set Sλ
,
Lλ
≡ {x : [λp0
+ (1 − λ)p1
]T
x = Cλ
}
is obviously parallel to L∗
. Finally, Lλ
must be either coincident with or
lie above L∗
, since if Lλ
were below L∗
, then there would exist a point on
5
Let x∗
∈ L0
∩ L1
. Then p0T
x∗
= C0
and p1T
x∗
= C1
. Thus [λp0
+ (1 −
λ)p1
]T
x∗
= λC0
+ (1 − λ)C1
and x∗
∈ S∗
. This also follows from the readily
proven proposition that S0
∩S1
⊂ S∗
⊂ S0
∪S1
(cf. Diewert [1974a; 157–158]).
112 Essays in Index Number Theory 6. Duality Approaches 113
the u isoquant which would lie below at least one of the isocost lines L0
≡
{x : p0T
x = C0
} or L1
≡ {x : p1T
x = C1
} which would contradict the cost
minimizing nature of x0
or x1
.6
Property 5 for C: For every u ∈ textRange F, C(u, p) is continuous
in p for p 0N .
Proof: This property is a mathematical consequence of property 4 for C,
concavity in p for fixed u. For proofs, see Fenchel [1953; 75] or Rockafellar
[1970; 82].qed
Property 6 for C: C(u, p) is nondecreasing in u for fixed p; i.e., if
p 0N , u0
, u1
∈ textRange F and u0
≤ u1
, then C(u0
, p) ≤ C(u1
, p).
Proof: Let p 0N , u0
, u1
∈ textRange F and u0
≤ u1
. Thus
C(u1
, p) ≡ min
x
{pT
x : F(x) ≥ u1
}
≥ min
x
{pT
x : F(x) ≥ u0
}, since if u0
≤ u1
then
{x : F(x) ≥ u1
} ⊂ {x : F(x) ≥ u0
},
and the minimum of pT
x over a larger set cannot increase
≡ C(u0
, p). qed
In contrast to the previous properties for the cost function, the following
property requires some heavy mathematical artillery. Since these mathematical
results are useful not only in the present section, but also in subsequent sections,
we momentarily digress and state these results.
In the following definitions, let S denote a subset of RM
, T a subset of
RK
, {xn
} a sequence of points of S and {yn
} a sequence of points of T. For a
more complete discussion of the following definitions and theorems, see Green
and Heller [1981].
Definition: φ is a correspondence from S into T if, for every x ∈ S,
there exists a nonempty image set φ(x) which is a subset of T.
Definitions: A correspondence φ is upper semicontinuous (or alternatively,
upper hemicontinuous) at the point x0
∈ S if limn xn
= x0
, yn
∈ φ(xn
),
limn yn
= y0
implies y0
∈ φ(x0
). A correspondence φ is lower semicontinuous
at x0
∈ S if limn xn
= x0
, y0
∈ φ(x0
) implies that there exists a sequence {yn
}
such that yn
∈ φ(xn
) and limn yn
= y0
. A correspondence φ is continuous at
x0
∈ S if it is both upper and lower semicontinuous at x0
.
Lemma 2. (Berge [1963; 111-112]): φ is an upper semicontinuous correspondence
over S iff graph φ ≡ {(x, y) : x ∈ S, y ∈ φ(x)} is a closed set in S × T.7
6
It can be seen that the approximating set Sa
≡ {x : p0T
x ≥ C0
, x ≥ 0N }∩{x :
p1T
x ≥ C1
, x ≥ 0N } contains the true technological set L(u) ≡ {x : F(x) ≥ u}
and thus the minimum cost associated with Sa
will generally be lower than the
cost associated with L(u).
7
S × T is the set of (x, y) such that x ∈ S and y ∈ T.
Upper Semicontinuity Maximum Theorem. (Berge [1963; 116]): Let f
be a continuous from above function8
defined over S × T where T is a compact
(i.e., closed and bounded) subset of RK
. Suppose that φ is a correspondence
from S into T and that φ is upper semicontinuous over S. Then the function g
defined by g(x) ≡ maxy{f(x, y) : y ∈ φ(x)} is well defined and is continuous
from above over S.
Maximum Theorem. (Debreu [1952; 889–890], [1959, 19], Berge [1963, 116]):
Let f be a continuous real valued function defined over S × T, where T is
a compact subset of RK
. Let φ be a correspondence from S into T and let
φ be continuous at x0
∈ S. Define the (maximum) function g by g(x) ≡
maxy{f(x, y) : y ∈ φ(x)} and the (set of maximizers) correspondence ξ by
ξ(x) ≡ {y : y ∈ φ(x) and f(x, y) = g(x)}. Then the function g is continuous at
x0
and the correspondence ξ is upper semicontinuous at x0
.
Property 7 for C: For every p 0N , C(u, p) is continuous from
below9
in u; i.e., if p∗
0N , u∗
∈ textRange F, un
∈ textRange F for all n,
u1
≤ u2
≤ . . . and lim un
= u∗
, then limn C(un
, p∗
) = C(u∗
, p∗
).
Proof: Define the correspondence L for u ∈ textRange F by L(u) ≡ {x :
F(x) ≥ u, x ≥ 0N }. Since F is continuous from above (recall assumption 1),
it can be shown that (see Rockafellar [1970; 51] that the graph of L, graph
L ≡ {(u, x) : x ≥ 0N , u ∈ textRange F, u ≤ F(x)} is a closed set and
hence by Lemma 2 above, L is an upper semicontinuous correspondence over
range F. Let p∗
0N , u∗
∈ textRange F and let x∗
be a solution to the cost
minimization problem
C(u∗
, p∗
) ≡ min
x
{p∗T
x : x ≥ 0N , F(x) ≥ u∗
} = p∗T
x∗
.
Define S∗
≡ {x : p∗T
x ≤ p∗T
x∗
, x ≥ 0N }. For u ∈ textRange F and u ≤ u∗
,
it can be seen that
C(u, p∗
) ≡ min
x
{p∗T
x : x ≥ 0N , F(x) ≥ u}
= min
x
{p∗T
x : x ∈ L(u) ∩ S∗
}
= − max
x
{−p∗T
x : x ∈ φ(u)},
8
A real valued function f defined over S × T is continuous from above (or
alternatively, is upper semicontinuous) at z0
∈ S × T iff either of the following
conditions is satisfied: (i) for every ε > 0, there exists a neighborhood of z0
,
N(z0
), such that z ∈ N(z0
) implies f(z) < f(z0
) + ε, or (ii) if zn
∈ S × T,
limn zn
= z0
, f(zn
) ≥ f(z0
), then limn f(zn
) = f(z0
). f is continuous from
above over S × T if it is continuous from above at each point of S × T. See
Green and Heller [1981].
9
A function f is continuous from below iff −f is continuous from above.
114 Essays in Index Number Theory 6. Duality Approaches 115
where φ(u) ≡ L(u) ∩ S∗
, a compact set for u ∈ textRange F and u ≤ u∗
. It
can be verified that φ is an upper semicontinuous correspondence at u∗
and
that −p∗T
x is continuous in x and u and hence continuous from above. Thus
by the Upper Semicontinuity Maximum Theorem −C(u, p∗
) = maxx{−p∗T
x :
x ∈ φ(u)} is continuous from above in u at u∗
so that C(u, p∗
) is continuous
from below at u∗
.qed
In order to illustrate the last property of C, the reader may find it useful
to let N = 1 and let the production function F(x) be the following (continuous
from above) step function (cf. Shephard [1970; 89]):
F(x) ≡ {0, if 0 ≤ x < 1; 1, if 1 ≤ x < 2; 2, if 2 ≤ x < 3; . . . }.
For p > 0, the corresponding cost function C(u, p) is the following (continuous
from below) step function:
C(u, p) ≡ {0, if 0 = u; p, if 0 < u ≤ 1; 2p, if 1 < u ≤ 2; . . . }.
The above properties of the cost function have some empirical implications,
as we shall see later. However, one application can be mentioned at
this point. Suppose that we can observe cost, input prices and output for a
firm and suppose further that we have econometrically estimated the following
linear cost function:10
(2.2) C(u, p) = α + βT
p + γu
where α and γ are constants and β is a vector of constants. Could (2.2) be
the firm’s true cost function? The answer is no if the firm is competitively
minimizing costs and if either one of the constants α and γ is nonzero, for in
this case, C does not satisfy property 2 (linear homogeneity in input prices).
Suppose now that we have somehow determined the firm’s true cost function
C, but that we do not know the firm’s production function F (except
that it satisfies assumption 1). How can we use the given cost function C(u, p)
(satisfying properties 1–7 above) in order to construct the firm’s underlying
production function F(x)?
Equivalent to the production function u = F(x) are the family of isoproduct
surfaces {x : F(x) = u} or the family of level sets L(u) ≡ {x : F(x) ≥
u}. For any u ∈ textRange F, the cost function can be used in order to
construct an outer approximation to the set L(u) in the following manner. Pick
input prices p1
0N and graph the isocost surface {x : p1T
x = C(u, p1
)}. The
set L(u) must lie above (and intersect) this set, because C(u, p1
) ≡ minx{p1T
x :
10
This type of cost function is often estimated by economists; e.g., see Walters
[1961] survey article on cost and production functions.
x ∈ L(u)}; i.e., L(u) ⊂ {x : p1T
x ≥ C(u, p1
)}. Pick additional input price
vectors p2
0N , p3
0N , . . . and graph the isocost surfaces {x : piT
x =
C(u, pi
)}. It is easy to see that L(u) must be a subset of each of the sets
{x : piT
x ≥ C(u, pi
)}. Thus
(2.3) L(u) ⊂
p 0N
{x : pT
x ≥ C(u, p)} ≡ L∗
(u);
i.e., the true production possibilities set L(u) must be contained in the outer
approximation production possibilities set L∗
(u) which is obtained as the intersection
of all of the supporting total cost half spaces to the true technology set
L(u). In Figure 2.1, L∗
(u) is indicated by dashed lines. Note that the boundary
of this set forms an approximation to the true u isoquant and that this
approximating isoquant coincides with the true isoquant in part, but it does
not have the backward bending and nonconvex portions of the true isoquant.
Once the family of approximating production possibilities sets L∗
(u) has
been constructed, the approximating production function F∗
can be defined as
F∗
(x) ≡ max
u
{u : x ∈ L∗
(u)}
= max
u
{u : pT
x ≥ C(u, p) for every p 0N }(2.4)
for x ≥ 0N . Note that the maximization problem defined by (2.4) has an
infinite number of constraints (one constraint for each p 0N ). However,
(2.4) can be used in order to define the approximating production function F∗
given only the cost function C.
It is clear (recall Figure 2.1) that the approximating production function
F∗
will not in general coincide with the true function F. However, it is also
clear that from the viewpoint of observed market behavior, if the producer is
competitively cost minimizing, then it does not matter whether the producer
is minimizing cost subject to the production function constraint given by F
or F∗
: observable market data will never allow us to determine whether the
producer has the production function F or the approximating function F∗
.
It is also clear that if we want the approximating production function F∗
to coincide with the true production function F, then it is necessary that F
satisfy the following two assumptions:
Assumption 2 on F: F is nondecreasing; i.e., if x2
≥ x1
≥ 0N , then
F(x2
) ≥ F(x1
).
Assumption 3 on F: F is a quasiconcave function; i.e., for every u ∈
textRange F, L(u) ≡ {x : F(x) ≥ u} is a convex set.
If F satisfies assumption 2, then backward bending isoquants cannot occur,
while if F satisfies assumption 3, then nonconvex isoquants of the type
drawn in Figure 2.1 cannot occur.
116 Essays in Index Number Theory 6. Duality Approaches 117
It is not too difficult to show that if F satisfies assumptions 1–3 and
the cost function C is computed via (2.1), then the approximating production
function F∗
computed via (2.4) will coincide with the original production function
F; i.e., there is a duality between cost functions satisfying properties 1–7
and production functions satisfying assumptions 1–3. The first person to prove
a formal duality theorem of this type was Shephard [1953].
In the following section, we will prove a similar duality theorem after
placing somewhat stronger conditions on the underlying production function F.
The following result is the basis for most theoretical and empirical applications
of duality theory.
Lemma 3. (Hicks [1946; 331], Samuelson [1947; 68], Karlin [1959; 272] and
Gorman [1976]): Suppose that the production function F satisfies assumption 1
and that the cost function C is defined by (2.1). Let u∗
∈ textRange F,
p∗
0N and suppose that x∗
is a solution to the problem of minimizing the
cost of producing output level u∗
when input prices p∗
prevail; i.e.,
(2.5) C(u∗
, p∗
) ≡ min
x
{p∗T
x : F(x) ≥ u∗
} = p∗T
x∗
.
If in addition, C is differentiable with respect to input prices at the point
(u∗
, p∗
), then
(2.6) x∗
= pC(u∗
, p∗
)
where
pC(u∗
, p∗
) ≡ [∂C(u∗
, p∗
1, . . . , p∗
N )/∂p1, . . . , ∂C(u∗
, p∗
1, . . . , p∗
N )/∂pN ]T
is the vector of first order partial derivatives of C with respect to the components
of the input price vector p.
Proof: Given any vector of positive input prices p 0N , x∗
is feasible
for the cost minimization problem defined by C(u∗
, p) but it is not necessarily
optimal; i.e., for every p 0N , we have the following inequality:
(2.7) pT
x∗
≥ C(u∗
, p).
For p 0N , define the function g(p) ≡ pT
x∗
− C(u∗
, p). From (2.7), g(p) ≥ 0
for p 0N and from (2.5), g(p∗
) = 0. Thus, g(p) attains a global minimum at
p = p∗
. Since g is differentiable at p∗
, the first order necessary conditions for a
local minimum must be satisfied:
pg(p∗
) = x∗
− pC(u∗
, p∗
) = 0N
which implies (2.6).qed
Thus differentiation of the producer’s cost function C(u, p) with respect
to input prices p yields the producer’s system of cost minimizing input demand
functions, x(u, p) = pC(u, p).
The above lemma should be carefully compared with the following result.
Lemma 4. (Shephard [1953; 11]): If the cost function C(u, p) satisfies properties
1–7 and, in addition, is differentiable with respect to input prices at the
point (u∗
, p∗
), then
(2.8) x(u∗
, p∗
) = pC(u∗
, p∗
)
where x(u∗
, p∗
) ≡ [x1(u∗
, p∗
), . . . , xN (u∗
, p∗
)]T
is the vector of cost minimizing
input quantities needed to produce u∗
units of output given input prices
p∗
, where the underlying production function F∗
is defined by (2.4), u∗
∈
textRange F∗
and p∗
0N .
The difference between Lemma 3 and Lemma 4 is that Lemma 3 assumes
the existence of the production function F and does not specify the properties
of the cost function other than differentiability, while Lemma 4 assumes only
the existence of a cost function satisfying the appropriate regularity conditions
and the corresponding production function F∗
is defined using the given cost
function. Thus, from an econometric point of view, Lemma 4 is more useful
than Lemma 3: in order to obtain a valid system of input demand functions, all
we have to do is postulate a functional form for C which satisfies the appropriate
regularity conditions and differentiate C with respect to the components of
the input price vector p. It is not necessary to compute the corresponding
production function F∗
nor is it necessary to endure the sometimes painful
algebra involved in deriving the input demand functions from the production
function via Lagrangian techniques.11
For formal proofs of Lemma 4, see the following section and the accompanying
references.
Historical Notes:
The proposition that there are two or more equivalent ways of representing
preferences or technology forms the core of duality theory. The mathematical
basis for the economic theory of duality is Minkowski’s [1911] The-
orem:12
every closed convex set can be represented as the intersection of its
supporting halfspaces. Thus, under certain conditions, the closed convex set
L(u) ≡ {x : F(x) ≥ u, x ≥ 0N } can be represented as the intersection of the
halfspaces generated by the isocost surfaces tangent to the production possibilities
set L(u), p{x : pT
x ≥ C(u, p)}.
If the consumer (or producer) has a budget of y > 0 to spend on the N
commodities (or inputs), then the maximum utility (or output) that he can
11
For an exposition of the Lagrangian method for deriving demand functions
and comparative statics theorems, see Intriligator [1981].
12
See Fenchel [1953; 48–50] or Rockafellar [1970; 95–99].
118 Essays in Index Number Theory 6. Duality Approaches 119
obtain given that he faces prices p 0N can generally be obtained by solving
the equation y = C(u, p) or by solving
(2.9) 1 = C(u, p/y)
(where we have used the linear homogeneity of C in p) for u as a function of
the normalized prices, p/y. Call the resulting function G so that u = G(p/y).
Alternatively, G can be defined directly from the utility (or production) function
F in the following manner for p 0N , y > 0:
G∗
(p, y) ≡ max
x
{F(x) : pT
x ≤ y, x ≥ 0N }(2.10)
or
G(p/y) ≡ max
x
{F(x) : (p/y)T
x ≤ 1, x ≥ 0N }.
Houthakker [1951–52; 157] called the function G the indirect utility function,
and like the cost function C, it also can characterize preferences or technology
uniquely under certain conditions (cf. Section 4 below). Our reason for introducing
it at this point is that historically, it was introduced into the economics
literature before the cost function by Antonelli [1971; 349] in 1886 and then by
Kon¨us [1924]. However, the first paper which recognized that preferences could
be equivalently described by a direct or indirect utility function appears to be
by Kon¨us and Byushgens [1926; 157] who note that the equations u = F(x)
and u = G(p/y) are equations for the same surface, but in different coordinate
systems: the first equation is in pointwise coordinates while the second is in
planar or tangential coordinates. Kon¨us and Byushgens [1926; 159] also set up
the minimization problem that allows one to derive the direct utility function
from the indirect function and, finally, they graphed various preferences in price
space for the case of two goods.
The English language literature on duality theory seems to have started
with two papers by Hotelling [1932][1935], who was perhaps the first economist
to use the word “duality”:
Just as we have a utility (or profit) function u of the quantities consumed
whose derivatives are the prices, there is, dually, a function
of the prices whose derivatives are the quantities consumed.
Hotelling [1932; 594]
Hotelling [1932; 597] also recognized that “the cost function may be represented
by surfaces which will be concave upward”; i.e., he recognized that
the cost function C(u, p) would satisfy a curvature condition in p.
Hotelling [1932; 590] [1935; 68] also introduced the profit function Π which
provides yet another way by which a decreasing returns to scale technology can
be described. Using our notation, Π is defined as
(2.11) Π(p) ≡ max
x
{F(x) − pT
x}.
Hotelling indicated that the profit maximizing demand functions, x(p) ≡ [x1(p),
. . . , xN (p)]T
, could be obtained by differentiating the profit function Π; i.e.,
x(p) = − pΠ(p). Thus, if Π is twice continuously differentiable, one can readily
deduce Hotelling’s [1935; 69] symmetry conditions:
(2.12) −
∂xi
∂pj
(p) =
∂2
Π
∂pi∂pj
(p) =
∂2
Π
∂pj∂pi
(p) = −
∂xj
∂pi
(p).
The next important contribution to duality theory was made by Roy who
independently recognized that preferences could be represented by pointwise
or tangential coordinates:
Il vient alors tout naturellement `a l’esprit d’invoquer le principle
de dualit´e qui permet d’utiliser les ´equations tangentielles au lieu
des ´equations ponctuelles; ainsi apparaˆıt-il possible de pr´esenter les
´equations d’´equilibre sous une forme nouvelle et susceptible d’interpr´etation
f´econdes.
Roy [1942; 18–19]
Roy [1942; 20] defined the indirect utility function G∗
as in equation (2.10)
above and then he derived the counterpart to Lemma 3 above, which is called
Roy’s Identity [1942; 18–19]).
(2.13) x(p/y) = −
pG∗
(p, y)
yG∗(p, y)
,
where x(p/y) ≡ [x1(p/y), . . . , xN (p/y)]T
is the vector of utility (or output)
maximizing demand functions given that the consumer (or producer) faces
input prices p 0N and has a budget y > 0 to spend. Roy [1942; 24–
27] showed that G∗
was decreasing in the prices p, increasing in income y
and homogeneous of degree 0 in (p, y); i.e., G∗
(λp, λy) = G∗
(p, y) for λ > 0.
Thus, G∗
(p, y) = G∗
(p/y, 1) ≡ G(p/y) = G(v), where v ≡ p/y is a vector of
normalized prices. In his 1947 paper, Roy derived the following version of Roy’s
Identity [1947; 219] where the indirect utility function G is used in place of G∗
:
(2.14) xi(v) =
∂G(v)
∂vi
N
j=1
vj
∂G(v)
∂vj
; i = 1, 2, . . ., N.
120 Essays in Index Number Theory 6. Duality Approaches 121
The French mathematician Ville [1951–52; 125] also derived the useful
relations (2.14) in 1946, so perhaps (2.14) should be called Ville’s Identity.
Ville [1951–52; 126] also noted that if the direct utility function F(x) is linearly
homogeneous, then the indirect function G(v) ≡ maxx{F(x) : vT
x ≤ 1, x ≥
0N } is homogeneous of degree −1; i.e., G(λv) = λ−1
G(v) for λ > 0, v
0N , and thus −G(v) =
N
j=1vj(∂G(v)/∂vj). Substitution of the last identity
into (2.14) yields the simpler equations (see also Samuelson [1972]) if G(v) is
positive:
(2.15) xi(v) = −∂ ln G(v)/∂vi, i = 1, 2, . . ., N.
At this point, it should be mentioned that Antonelli [1971; 349] obtained a
version of Roy’s Identity in 1886 and Kon¨us and Byushgens [1926; 159] almost
derived it in 1926 in the following manner: they considered the problem of
minimizing indirect utility G(v) with respect to the normalized prices v subject
to the constraint vT
x = 1. As Houthakker [1951–52; 157–158] later observed,
it turns out that this constrained minimization problem generates the direct
utility function; i.e., we have for x 0N :
(2.16) F(x) = min
v
{G(v) : vT
x ≤ 1, v ≥ 0N }.
Kon¨us and Byushgens obtained the first order conditions for the problem (2.16):
vG(v) = µx. If the Lagrange multiplier µ is eliminated from this last system
of equations using the constraint vT
x = 1, we obtain x = vG(v) / vT
vG(v),
which is (2.14) written in vector notation. However, Kon¨us and Byushgens did
not explicitly carry out this last step.
Another notable early paper was written by Wold [1943, 1944] who defined
the indirect utility function G(v) (he called it a “price preference function”) and
showed that the indifference surfaces of price space were either convex to the
origin or possibly linear; i.e., he showed that G(v) was a quasiconvex function13
in the normalized prices v. Wold’s early work is summarized in Wold [1953;
145–148].
Malmquist [1953; 212] also defined the indirect utility function G(v) and
indicated that it was a quasiconvex function in v.
If the production function F is subject to constant returns to scale (i.e.,
F(λx) = λF(x) for every λ ≥ 0, x ≥ 0N ) in addition to being continuous
from above, then the corresponding cost function decomposes in the following
13
A function G is quasiconvex if and only if −G is quasiconcave.
manner: let u > 0, p 0N ; then
C(u, p) ≡ min
x
{pT
x : F(x) ≥ u}
= min
x
{upT
(x/u) : F(x/u) ≥ 1}
= u min
z
{pT
z : F(z) ≥ 1}
≡ uC(1, p).(2.17)
(The above proof assumes that there exists at least one x∗
> 0N such that
F(x∗
) > 0 so that the set {z : F(z) ≥ 1} is not empty). Samuelson [1953–54]
assumed that the production function F was linearly homogeneous and subject
to a “generalized law of diminishing returns,” F(x +x ) ≥ F(x )+F(x ), which
is equivalent to concavity of F when F is linearly homogeneous. Samuelson
[1953–54; 15] then defined the unit cost function C(1, p) and indicated that
C(1, p) satisfied the same properties in p that F satisfied in x. Samuelson
[1953–54; 15] also noted that a flat on the unit output production surface (a
region of infinite substitutability) would correspond to a corner on the unit cost
surface, a point which was also made by Shephard [1953; 27–28].
Shephard’s 1953 monograph appears to be the first modern, rigorous
treatment of duality theory. Shephard [1953; 13–14] notes that the cost function
C(u, p) can be interpreted as the support function for the convex set
{x : F(x) ≥ u}, and he uses this fact to establish the properties of C(u, p)
with respect to p. Shephard [1953; 13] also explicitly mentions Minkowski’s
[1911] Theorem on convex sets and Bonnesen and Fenchel’s [1934] monograph
on convex sets. It should be mentioned that Shephard did not develop a direct
duality between production and cost functions; he developed a duality between
production and distance functions (which we will define in a later section) and
then between distance and cost functions.
Shephard [1953; 41] defined a production function F to be homothetic if
it could be written as
F(x) = φ[f(x)]
where f is a homogeneous function of degree one and φ is a continuous, positive
monotone increasing function of f. Let us formally introduce the following
additional conditions on F (or f):
Assumption 4 on F: F is (nonnegatively) linearly homogeneous; i.e., if
x ≥ 0N , λ ≥ 0, then F(λx) = λF(x).
Assumption 5 on F: F is weakly positive; i.e., for every x ≥ 0N , F(x) ≥
0 but F(x∗
) > 0 for at least one x∗
> 0N .
Now let us assume that φ(f) is a continuous, monotonically increasing
function of one variable for f ≥ 0 with φ(0) = 0. Under these conditions the
inverse function φ−1
exists and has the same properties as φ, with φ−1
[φ(f)] =
122 Essays in Index Number Theory 6. Duality Approaches 123
f for all f ≥ 0. If f(x) satisfies assumptions 1, 4 and 5 above, then the cost
function which corresponds to F(x) ≡ φ[f(x)] decomposes as follows: let u > 0,
p 0N ; then
C(u, p) ≡ min
x
{pT
x : φ[f(x)] ≥ u}
= min
x
{pT
x : f(x) ≥ φ−1
(u)}
= φ−1
(u) min
x
{pT
(x/φ−1
(u)) : f(x/φ−1
(u)) ≥ 1},
where φ−1
(u) > 0 since u > 0,
= φ−1
(u)c(p),(2.18)
where c(p) ≡ minz{pT
z : f(z) ≥ 1} is the unit cost function which corresponds
to the linearly homogenous function f, a nonnegative, (positively) linearly homogenous,
nondecreasing, concave and continuous function of p (recall properties
1–5 above). As usual, we will not be able to derive the original production
function φ[f(x)] from the cost function (2.18) unless f also satisfies assumptions
2 and 3 above. Shephard [1953; 43] obtained the factorization (2.18) for
the cost function corresponding to a homothetic production function.
Finally, Shephard [1953; 28–29] noted several practical uses for duality
theory: (i) as an aid in aggregating variables, (ii) in econometric studies of
production when input data are not available but cost, input price and output
data are available, and (iii) as an aid in deriving certain comparative statics
results. Thus, Shephard either derived or anticipated many of the theoretical
results and practical applications of duality theory.
Turning now to the specific results obtained in this section, McFadden
[1966] showed that the minimum in definition (2.1) exists if F satisfies assumption
1. Property 1 was obtained by Shephard [1953; 14], property 2 by
Shephard [1953; 14] and Samuelson [1953–54; 15], property 3 by Shephard
[1953; 14], property 4 by Shephard [1953; 15] (our method of proof is due to
McKenzie [1956–57; 185]), properties 5 and 6 by Uzawa [1964; 217], and finally
property 7 was obtained by Shephard [1970; 83].
The method for constructing the approximating production possibilities
sets L∗
(u) in terms of the cost function is due to Uzawa [1964].
The very important point that the approximating isoquants do not have
any of the backward bending or nonconvex parts of the true isoquants was
made in the context of consumer theory by Hotelling [1935; 74], Wold [1943;
231] [1953; 164] and Samuelson [1950b; 359–360] and in the context of producer
theory by McFadden [1966] [1978a]. It is worth quoting Hotelling and
Samuelson at some length in order to emphasize this point:
If indifference curves for purchases be thought of as possessing a
wavy character, convex to the origin in some regions and concave in
others, we are forced to the conclusion that it is only the portions
convex to the origin that can be regarded as possessing any importance,
since the others are essentially unobservable. They can be
detected only by the discontinuities that may occur in demand with
variation in price-ratios, leading to an abrupt jumping of a point
of tangency across a chasm when the straight line is rotated. But,
while such discontinuities may reveal the existence of chasms, they
can never measure their depth. The concave portions of the indifference
curves and their many-dimensional generalizations, if they
exist, must forever remain in unmeasurable obscurity.
Hotelling [1935; 74]
It will be noted that any point where the indifference curves are convex
rather than concave cannot be observed in a competitive market.
Such points are shrouded in eternal darkness — unless we make our
consumer a monopsonist and let him choose between goods lying on
a very convex ‘budget curve’ (along which he is affecting the prices
of what he buys). In this monopsony case, we could still deduce the
slope of the man’s indifference curve from the slope of the observed
constraint at the equilibrium point.
Samuelson [1950b; 359–360]
Our proof of Lemma 3 follows a proof attributed by Diamond and McFadden
[1974; 4] to W.M. Gorman; however the same method of proof was
also used by Karlin [1959; 272]. Hicks’ and Samuelson’s proof of Lemma 3
assumed differentiability of the production function and utilized the first order
conditions for the cost minimization problem along with the properties of
determinants. Our earlier quotation by Hotelling [1932; 594] indicates that
he also obtained the Hicks [1946; 331], Samuelson [1947; 68] [1953–54; 15–16]
results in a slightly different context.
References to some of the more recent literature on duality will be given
in subsequent sections.
3. Duality between Cost and Aggregator (Production or Utility)
Functions
In this section, we assume that the aggregator function F satisfies the following
properties:
Conditions I on F:
(i) F is a real valued function of N variables defined over the nonnegative
orthant Ω ≡ {x : x ≥ 0N } and is continuous on this domain.
(ii) F is increasing; i.e., x x ≥ 0N implies F(x ) > F(x ).
124 Essays in Index Number Theory 6. Duality Approaches 125
(iii) F is a quasiconcave function.
Note that properties (i) and (ii) above are stronger than assumptions 1
and 2 on F made in the previous section, so that we should be able to deduce
somewhat stronger conditions on the cost function C(u, p) which corresponds
to an F(x) satisfying conditions I above.
Let U be the range of F. From I(i) and (ii), it can be seen that U ≡ {u :
u ≤ u < ou}, where u ≡ F(0N ) < ou. Note that the least upper bound ou
could be a finite number or +∞. In the context of production theory, typically
u = 0 and ou = +∞, but for consumer theory applications, there is no reason
to restrict the range of the utility function F in this manner.
Define the set of positive prices P ≡ {p : p 0N }.
Theorem 1. If F satisfies conditions I, then C(u, p) ≡ minx{pT
x : F(x) ≥ u}
defined for all u ∈ U and p ∈ P satisfies conditions II below.
Conditions II on C:
(i) C(u, p) is a real valued function of N + 1 variables defined over U × P
and is jointly continuous in (u, p) over this domain.
(ii) C(u, p) = 0 for every p ∈ P.
(iii) C(u, p) is increasing in u for every p ∈ P; i.e., if p ∈ P, u , u ∈ U,
with u < u , then C(u , p) < C(u , p).
(iv) C(ou, p) = +∞ for every p ∈ P; i.e., if p ∈ P, un
∈ U, limn un
= ou,
then limn C(un
, p) = +∞.
(v) C(u, p) is (positively) linearly homogeneous in p for every u ∈ U; i.e.,
u ∈ U, λ > 0, p ∈ P implies C(u, λp) = λC(u, p).
(vi) C(u, p) is concave in p for every u ∈ U.
(vii) C(u, p) is increasing in p for u > u and u ∈ U.
(viii) C is such that the function F∗
(x) ≡ maxu{u : pT
x ≥ C(u, p) for
every p ∈ P, u ∈ U} is continuous for x ≥ 0N .
Proof: (i) By I(i), F is continuous and hence continuous from above.
Thus, by Lemma 1 in the previous section, C(u, p) is well defined as a minimum
for (u, p) ∈ U × P. In order to prove the continuity of C, we will use the
Maximum Theorem, so it is first necessary to show that the correspondence
(3.1) L(u) ≡ {x : x ≥ 0N , F(x) ≥ u}
is continuous for u ∈ U. Since F is continuous from above, it can be seen that
graph L ≡ {(x, u) : x ≥ 0N , F(x) ≥ u} is a closed set in RN+1
, and thus by
Lemma 2, L is an upper semicontinuous correspondence over U. To show that
L is lower semicontinuous over U, let
(3.2) u0
∈ U, x0
∈ L(x0
), un
∈ U, lim
n
un
= u0
.
Since x0
∈ L(u0
), by (19), F(x0
) ≥ u0
. We must consider two cases.
Case 1: F(x0
) = u0
+λ where λ > 0. By (3.2), there exists n∗
such that
for n ≥ n∗
, un
≤ u0
+ λ. For n < n∗
, let xn
be any point such that xn
∈ L(un
)
while for n ≥ n∗
, define xn
≡ x0
so that F(xn
) = F(x0
) = u0
+ λ ≥ un
and
thus xn
∈ L(un
) and limn xn
= x0
.
Case 2: F(x0
) = u0
. If un
≤ u0
, then define xn
= x0
so that F(xn
) =
F(x0
) = u0
≥ un
and xn
∈ L(un
). If un
> u0
, then define the scalar kn
by
f(kn
) ≡ F(x0
+kn
1N ) = un
where 1N is an N dimensional vector of ones. Since
f(0) = u0
< un
and f(k) is a continuous monotonically increasing function of
k by I(i) and (ii), it can be seen that kn
is well defined. Note that as n becomes
large kn
tends to 0 since un
tends to u0
. Now define xn
= x0
+ kn
1N . Thus
xn
∈ L(un
) and limn xn
= x0
in this case also. Thus L(u) is both lower and
upper semicontinuous over U.
We cannot immediately apply the Maximum Theorem at this point since
L(u) is not a compact set.
Let u0
∈ U, p0
∈ P. Define the following sets:
(3.3)
Uδ(u0
) ≡ {u : u ≤ u ≤ u0
+ δ},
Pδ(p0
) ≡ {p : (p − p0
)T
(p − p0
) ≤ δ2
}.
Choose δ > 0 small enough so that Pδ(p0
) ⊂ P and Uδ(u0
) ⊂ U. Now let
x∗
> 0N be any point such that
(3.4) F(x∗
) ≥ u0
+ δ.
Now for every p ∈ Pδ(p0
), define the compact set Bp ≡ {x : pT
x ≤ pT
x∗
, x ≥
0N }. For i = 1, 2, . . ., N, define mi ≡ maxp{pT
x∗
/pi : p ≡ (p1, p2, . . . , pN )T
,
p ∈ Pδ(p0
)}. Since Pδ(p0
) is compact and each component of the vector p is
positive if p ∈ Pδ(p0
), mi is well defined as a maximum. Define m = max{mi :
i = 1, 2, . . ., N}. Define the compact set B as B ≡ {x : x ≥ 0N , x ≤ m1N }
where 1N is a vector of ones. It is obvious that
(3.5) Bp ≡ {x : pT
x ≤ pT
x∗
, x ≥ 0N } ⊂ B for p ∈ Pδ(p0
).
For (u, p) ∈ Uδ(u0
) × Pδ(p0
), we have
C(u, p) ≡ min
x
{pT
x : x ∈ L(u), x ≥ 0N }
= min
x
{pT
x : x ∈ L(u), x ≥ 0N , pT
x ≤ pT
x∗
}
since by (3.3) and (3.4), x∗
is feasible when u ∈ Uδ(u0
)
= min
x
{pT
x : x ∈ L(u) ∩ B}
for p ∈ Pδ(p0
) using (3.5).
126 Essays in Index Number Theory 6. Duality Approaches 127
Since L(u) is a continuous correspondence and since B is a (constant)
compact set, the correspondence φ(u, p) ≡ L(u)∩B for (u, p) ∈ Uδ(u0
)×Pδ(p0
)
is continuous with compact image sets and thus continuity of C follows via the
Maximum Theorem.
(ii) Let p ∈ P. From property 1 in the previous section, C(u, p) ≥ 0. Since
F(0N ) = u, C(u, p) ≡ minx{pT
x : F(x) ≥ u} ≤ pT
0N = 0. Thus C(u, p) = 0.
(iii) Let p ∈ P and u ≤ u < u < ou. Then
C(u , p) ≡ min
x
{pT
x : F(x) ≥ u }
= pT
x where F(x ) = u
> pT
k x where F(k x ) = u < u and 0 ≤ k < 1,
using I(i) and (ii)
≥ min
x
{pT
x : F(x) ≥ u } since k x is feasible but not
necessarily optimal for the cost minimization problem
≡ C(u p).
(iv) Let un
∈ U, limn un
= ou and p 0N . Then C(un
, p) = pT
xn
where xn
≥ 0N and F(xn
) = un
. Suppose the components of xn
remain
bounded from above for all n; i.e., xn
≤ k∗
1N for all n. Then each xn
∈ S ≡
{x : 0N ≤ x ≤ k∗
1N }, a compact set, and thus {xn
} contains at least one
convergent subsequence, {xnk
} say, with lim xnk
= x∗
. Thus ou = lim unk
=
lim F(xnk
) = F(lim xnk
) = F(x∗
) using the continuity of F. But then using
I(ii), F(x∗
+ 1N ) > F(x∗
) = ou, which is impossible since ou is the least upper
bound for the range of F. Thus our supposition is false, and at least one
component of xn
tends to +∞. Since p 0N , pT
xn
also tends to +∞.
(v) Since F is continuous, it is continuous from above and thus linear
homogeneity of C in p follows from property 2 of the previous section.
(vi) Concavity of C in p follows from property 4 of the previous section.
(vii) Let u ∈ U, u > u, p , p ∈ P. Then
C(u, p + p ) ≡ 2C(u,
1
2
p +
1
2
p ) using II(v)
≥ 2[
1
2
C(u, p ) +
1
2
C(u, p )] using II(vi)
= C(u, p ) + C(u, p )
> C(u, p ),
since for u > u, II(ii) and (iii) imply that C(u, p ) > 0.
(viii) It is first necessary to show that F∗
(x) is well defined as a maximum.
Let x ≥ 0N and p 0N . Then the set Ip(x) ≡ {u : u ∈ U, C(u, p) ≤ pT
x} is a
compact interval containing the point u, using II(i), (ii) and (iii). Thus
F∗
(x) ≡ max
u
{u : C(u, p) ≤ pT
x for every p ∈ P, u ∈ U}
= max
u
{u : u ∈ Ip(x) for every p ∈ P}
= max
u
{u : u ∈ I(x)},(3.6)
where I(x) ≡ p∈P {Ip(x)} is a compact interval containing u. Thus F∗
(x) is
well defined as a maximum.
At this point, it is useful to extend the domain of definition of C from
p 0N to p ≥ 0N . This can be done by utilizing the Fenchel closure operation:
for each u ∈ U, define the hypograph of C(u, p) as the (convex) set G(u) ≡
{(k, p) : p 0N , k ≤ C(u, p)}, let G(u) denote the closure of G(u) in RN+1
,
and now define C(u, p) for p ≥ 0N as C(u, p) ≡ maxk{k : (k, p) ∈ G(u)}. It can
be seen (cf. Fenchel [1953; 78] or Rockafellar [1970, 85]) that for each u ∈ U,
the extended C is continuous in p for p ∈ Ω ≡ {p : p ≥ 0N }.14
Once the domain of definition of C has been extended in the above continuous
manner, F∗
can now be defined as
(3.7) F∗
(x) ≡ max
u
{u : C(u, p) ≤ pT
x for every p ∈ Ω, u ∈ U}.
We now show that F∗
is continuous over Ω by showing that F∗
= F. Let
x ≥ 0N and u ≡ F(x ). Then for any p ∈ P,
(3.8) C(u , p) ≡ min
x
{pT
x : x ∈ L(u )} ≤ pT
x
since x is feasible but not necessarily optimal for the minimization problem.
By continuity, (3.8) is also valid for all p ∈ Ω. Thus F∗
(x ) ≡ maxu{u : u ∈
U, C(u, p) ≤ pT
x for every p ∈ Ω} ≥ u since by (3.8), u is feasible for all of
the constraints in the maximization problem.
Suppose F∗
(x ) = u > u . Then u satisfies the inequalities
(3.9) C(u , p) ≤ pT
x for every p ∈ Ω.
Since L(u ) ≡ {x : F(x) ≥ u } is a closed, convex set by I(i) and (iii), it is equal
to the intersection of its supporting halfspaces by Minkowski’s [1911] Theorem.
14
It can also be shown that the extended function C is jointly continuous
over U × Ω (see Rockafellar [1970, 89]. However, C(u, p) need not be strictly
increasing in u when p is on the boundary of Ω; e.g., consider the function
F(x1, x2) ≡ x1 which has the dual cost function C(u, p1, p2) ≡ p1u which is
not increasing in u when p1 = 0.
128 Essays in Index Number Theory 6. Duality Approaches 129
By I(ii), the surface {x : F(x) = u } never bends backwards. Hence L(u ) is
unbounded from above, and it can be seen that15
L(u ) = {x : C(u , p) ≤ pT
x
for every p ∈ Ω}. Thus by (3.9), x ∈ L(u ) which implies that F(x ) ≥ u > u ,
which is a contradiction since F(x ) = u . Thus our supposition is false and
F∗
(x ) = u = F(x ).qed
Note that we have proven the following corollaries to Theorem 1.
Corollary 1.1. If C(u, p) satisfies conditions II above, then the domain of
definition of C can be extended from U × P to U × Ω. The extended function
C is continuous in p for p ∈ Ω ≡ {p : p ≥ 0N } for each u ∈ U.
Corollary 1.2. For every x ≥ 0N , F∗
(x) = F(x), where F∗
is the function
defined by the cost function C in part (viii) of conditions II.
Corollary 1.2 shows that the cost function can completely describe a production
function which satisfies conditions I; i.e., to use McFadden’s [1966]
terminology, the cost function is a sufficient statistic for the production func-
tion.
The proof of Theorem 1 is straightforward, with the exception of parts (i)
and (viii), the parts that involve the continuity properties of the cost or production
function. These continuity complexities appear to be the only difficult
concepts associated with duality theory: this is why we tried to avoid them in
the previous section as much as possible. For further discussion on continuity
problems, see Shephard [1970], Friedman [1972], Diewert [1974a], Blackorby,
Primont and Russell [1978] and Blackorby and Diewert [1979].
Property I(ii), increasingness of F, is required in order to prove the correspondence
L(u) continuous and thus that C(u, p) is continuous over U ×P.16
If
property I(ii) is replaced by a weak monotonicity assumption (such as our old
assumption 2 on F of the previous section), then plateaus on the graph of F
(“thick” indifference surfaces to use the language of utility theory) will imply
discontinuities in C with respect to u (cf. Friedman [1972; 169]).
Note that II(ii) and (iii) imply that C(u, p) > 0 for u > u and p 0N
and that II(vii) is not an independent property of C since it follows from II(ii),
(iii), (v) and (vi). Note also that we have not assumed that F be strictly
quasiconcave; i.e., that the production possibility sets L(u) ≡ {x : F(x) ≥ u}
be strictly convex.
15
Recall our discussion of equation (2.3) in the previous section.
16
Friedman [1972] shows that I(ii) plus continuity from above (assumption 1
on F in the previous section) is sufficient to imply the joint continuity of C
over U × P (and indeed over U × Ω if we make use of Rockafellar’s [1970; 89]
result). However, unless we assume the additional property on F of continuity
from below, we cannot conclude that C(u, p) is increasing in u for p ∈ P, a
property which follows from I(i) and I(ii).
Finally, it is evident that given only a firm’s total cost function C, we
can use the function F∗
defined in terms of the cost function by (3.7) in order
to generate the firm’s production function. This is formalized in the following
theorem.
Theorem 2. If C satisfies conditions II above, then F∗
defined by (3.7) satisfies
conditions I. Moreover, if C∗
(u, p) ≡ minx{pT
x : F∗
(x) ≥ u} is the cost
function which is defined by F∗
, then C∗
= C.
Proof: (i) Extend the domain of definition of C from U × P to U × Ω via
the Fenchel closure operation. The extended C is then continuous over U × Ω
by Corollary 1.1 above. In the proof of Theorem 1 above, we have seen that
F∗
(x) defined by (3.7) is well defined for x ≥ 0N . Property II(viii) implies
that F∗
is continuous over Ω.
(ii) It is first necessary to define F∗
in yet another way: for x ≥ 0N ,
F∗
(x) ≡ max
u
{u : C(u, p) ≤ pT
x for every p ≥ 0N , u ∈ U}
= max
u
{u : C(u, p) − pT
x ≤ 0
for p ≥ 0N and 1T
N p = 1, u ∈ U}
using II(i) and (v)
= max
u
{u : H(u, x) ≤ 0, u ∈ U}(3.10)
where
(3.11) H(u, x) ≡ max
p
{C(u, p) − pT
x : p ≥ 0N , 1T
N p = 1}.
Since C(u, p) − pT
x is continuous in p over the compact set S ≡ {p :
p ≥ 0N , 1T
N p = 1}, H(u, x) is well defined as a maximum.17
Moreover, since
C(u, p) − pT
x is continuous in u, x and p, the Maximum Theorem implies that
H(u, x) will be continuous over U × Ω. We can also show that H(u, x) is
17
The function H(u, x) is called the difference function by Blackorby and Diewert
[1979]. It is equal to the negative of the conjugate function to the concave
function of p, C(u, p), for each u. For material on conjugate concave (or convex)
functions, see Fenchel [1953; 88–92], Karlin [1959; 226], Rockafellar [1970;
104] or Jorgenson and Lau [1974b].
130 Essays in Index Number Theory 6. Duality Approaches 131
nondecreasing in u. Let x ≥ 0N , u , u ∈ U with u < u . Then
H(u , x) ≡ max
p
{C(u , p) − pT
x : p ∈ S}
= C(u , p ) − p T
x where p ∈ S
≤ C(u , p ) − p T
x using property II(iii)18
≤ max
p
{C(u , p) − pT
x : p ∈ S} since p is feasible but
not necessarily optimal for the maximization problem
≡ H(u , x).
Also properties II(ii) and II(iv) imply that H(u, x) ≤ 0 and H(u, x) tends to
+∞ as u tends to ou. Thus if u∗
solves the maximization problem (3.10), then
H(u∗
, x) = 0.
Now let 0N ≤ x x . Then
F∗
(x ) ≡ max
u
{u : H(u, x ) ≤ 0, u ∈ U} = u say,
where
0 = H(u , x )
= C(u , p ) − p T
x for some p ≥ 0N such that 1T
N p = 1
< C(u , p ) − p T
x since x x and p > 0N
≤ H(u , x ) using definition (3.11).
Thus u is not a feasible solution for the maximization problem
max
u
{u : H(u, x ) ≤ 0, u ∈ U} = F∗
(x )
since H(u , x ) > 0. Since H is nondecreasing in u, if u ≥ u , H(u, x ) > 0
also. Thus F∗
(x ) < u = F∗
(x ).
(iii) Let x ≥ 0N , x ≥ 0N , 0 ≤ λ ≤ 1, F∗
(x ) ≥ u∗
and F∗
(x ) ≥ u∗
.
Then by the definition of F∗
, (3.7), and property II(iii) of C:
C(u∗
, p) ≤ C[F∗
(x ), p] ≤ pT
x for every p ∈ P and
C(u∗
, p) ≤ C[F∗
(x ), p] ≤ pT
x for every p ∈ P.
18
The continuity of C and property II(iii), C(u , p) < C(u , p) if u < u and
p ∈ P imply only that C(u , p) ≤ C(u , p) when p belongs to the boundary
of P.
Thus C(u∗
, p) ≤ λpT
x + (1 − λ)pT
x = pT
[λx + (1 − λ)x ] for every p ∈ P.
Hence
F∗
[λx + (1 − λ)x ] ≡ max
u
{u : C(u, p) ≤ pT
[λx + (1 − λ)x ] for every p ∈ P}
≥ u∗
since u∗
is feasible for the maximization problem.
Thus F∗
is a quasiconcave function.
It remains to show that C∗
, the cost function of F∗
, equals C. Let u∗
∈ U
and p∗
∈ P. Then
C∗
(u∗
, p∗
) ≡ min
x
{p∗T
x : F∗
(x) ≥ u∗
}
= min
x
{p∗T
x : F∗
(x) = u∗
} using properties I(i) and (ii)
= min
x
{p∗T
x : max
u
{u : C(u, p) ≤ pT
x for every p ∈ S} = u∗
,
x ≥ 0N } using definition (3.10) for F∗
= min
x
{p∗T
x : C(u∗
, p) ≤ pT
x for every p ∈ S with equality
holding for at least one p ∈ S, x ≥ 0N }
= p∗T
x∗
(3.12)
where x∗
is any supergradient19
of the concave function of p, g(p) ≡ C(u∗
, p),
at the point p∗
. The last equality in (3.12) follows since by the definition of x∗
being a supergradient,20
we have C(u∗
, p) ≤ pT
x∗
for every p ∈ P (and hence
also for every p ∈ S using the continuity of C) and C(u∗
, p∗
) = p∗T
x∗
. This last
equality in conjunction with (3.12) establishes that C∗
(u∗
, p∗
) = C(u∗
, p∗
).qed
Corollary 2.1. The set of supergradients to C with respect to p at the
point (u∗
, p∗
) ∈ U × P, ∂C(u∗
, p∗
), is the solution set to the cost minimization
problem minx{p∗T
x : F∗
(x) ≥ u∗
} where F∗
is the aggregator function
which corresponds to the given cost function satisfying conditions II via definitions
(3.6), (3.7), or (3.10). (The supergradients satisfy x∗
∈ ∂C(u∗
, p∗
) iff
C(u∗
, p) ≤ C(u∗
, p∗
) + x∗T
(p − p∗
) for every p 0N .)
Proof: If x∗
∈ ∂C(u∗
, p∗
), we have already shown that x∗
is a solution
to the cost minimization problem (3.12). On the other hand, if x ≥ 0N is not
19
In general, x∗
is a supergradient to a function g at the point p∗
iff g(p) ≤
g(p∗
)+x∗T
(p−p∗
) for all p ∈ P. If g is a concave function over the set P ≡ {p :
p 0N }, then Rockafellar [1970; 214–215] shows that for every p∗
∈ P, the
set of supergradients to g at the point p∗
, ∂g(p∗
), is a nonempty, closed convex
set. If g is differentiable at p∗
, then g(p∗
) reduces to the single point g(p∗
),
the gradient vector of g. Finally, if g is positively linearly homogeneous over P,
then it can be seen that x∗
is a supergradient to g at p∗
iff g(p) ≤ x∗T
p for
every p ∈ P and g(p∗
) = x∗T
p∗
.
20
Since C(u, p) is increasing in p for p ∈ P, x∗
≥ 0N also.
132 Essays in Index Number Theory 6. Duality Approaches 133
a supergradient to the linearly homogeneous function g(p) ≡ C(u∗
, p) at the
point p∗
0N , then we must have either C(u∗
, p ) > p T
x for some p ∈ P
in which case x is not feasible for the minimization problem above (3.12), or
C(u∗
, p∗
) < p∗T
x but, in this case, x cannot be optimal for the minimization
problem above (3.12), since at least one supergradient x∗
≥ 0N exists which
satisfies the constraints of (3.12) and C(u∗
, p∗
) = p∗T
x∗
< p∗T
x .qed
Corollary 2.2. (Shephard’s [1953; 11] Lemma): If C satisfies conditions II
and, in addition, is differentiable with respect to input prices at the point
(u∗
, p∗
) ∈ U × P, then the solution x∗
to the cost minimization problem
minx{p∗T
x : F∗
(x) ≥ u∗
} is unique and is equal to the vector of partial derivatives
of C(u∗
, p∗
) with respect to the components of the input price vector p;
i.e.,
(3.13) x∗
= pC(u∗
, p∗
).
Proof: Apply the above corollary, noting that ∂C(u∗
, p∗
) reduces to the
single point pC(u∗
, p∗
) when C is differentiable with respect to p at the point
(u∗
, p∗
).qed
The above two theorems provide a version of the Shephard [1953] [1970]
Duality Theorem between cost and aggregator functions. The conditions on
C which correspond to our conditions I on F seem to be straightforward with
the exception of II(viii), which essentially guarantees the continuity of the
aggregator function F∗
corresponding to the given cost function C. Condition
II(viii) can be dropped if we strengthen II(iii) to C(u, p) increasing in u for
every p in S ≡ {p : p ≥ 0N , 1T
N p = 1}. The resulting F∗
can be shown
to be continuous (cf. Blackorby, Primont and Russell [1978]); however, many
useful functional forms do not satisfy the strengthened condition II(iii).21
An
alternative method of dropping II(viii), which preserves continuity of the direct
aggregator function F∗
corresponding to a given cost function C, is to develop
local duality theorems; i.e., assume that C satisfies conditions II(i)–(vii) for
(u, p) ∈ U ×P, where P is now restricted to be a compact, convex subset of the
positive orthant. A (locally) valid continuous F∗
can then be defined from C
which in turn has C as its cost function over U × P. This approach is pursued
in Blackorby and Diewert [1979].
Historical Notes
Duality theorems between F and C have been proven under various regularity
conditions by Shephard [1953] [1970], McFadden [1962], Chipman [1970],
21
E.g., consider C(u, p) ≡ bT
pu where b > 0N but b is not 0N . This
corresponds to a Leontief or fixed coefficients aggregator function.
Hanoch [1978b], Diewert [1971a] [1974a], Afriat [1973a] and Blackorby, Primont
and Russell [1978].
Duality theorems between C and the level sets of F, L(u) ≡ {x : F(x) ≥
u}, have been proven by Uzawa [1964], McFadden [1966] [1978a], Shephard
[1970], Jacobsen [1970] [1972], Diewert [1971a], Friedman [1972], and Sakai
[1973].
4. Duality Between Direct and Indirect Aggregator Functions
We assume that the direct aggregator (utility or production) function F satisfies
conditions I listed in the previous section. The basic optimization problem
that we wish to consider in this section is the problem of maximizing utility
(or output) F(x) subject to the budget constraint pT
x ≤ y where p 0N
is a vector of given commodity (or input) prices and y > 0 is the amount
of money the consumer (or producer) is allowed to spend. Since y > 0, the
constraint pT
x ≤ y can be replaced with vT
x ≤ 1 where v ≡ p/y is the vector of
normalized prices. The indirect aggregator function G(v) is defined for v 0N
as
(4.1) G(v) ≡ max
x
{F(x) : vT
x ≤ 1, x ≥ 0N }.
Theorem 3. If the direct aggregator function F satisfies conditions I, then the
indirect aggregator function G defined by (4.1) satisfies the following conditions:
Conditions III on G: (i) G(v) is a real valued function of N variables
defined over the set of positive normalized prices V ≡ {v : v 0N } and is a
continuous function over this domain.
(ii) G is decreasing; i.e., if v v 0N , then G(v ) < G(v ).
(iii) G is quasiconvex over V .
(iv) G22
is such that the function F(x) ≡ minv{G(v) : vT
x ≤ 1, v ≥
0N } defined for x 0N is continuous over this domain and has a continuous
22
G here is the extension of G to the nonnegative orthant that is defined by
the Fenchel closure operation; i.e., define the epigraph of the original G as
Γ ≡ {(u, v) : v 0N , u ≥ G(v)}, define the closure of Γ as Γ and define the
extended G as G(v) ≡ infu{u : (u, v) ∈ Γ} for v ≥ 0N . The resulting extended
G is continuous from below (the sets {v : G(v) ≤ u, v ≥ 0N } are closed for all
u). If the range of F is U ≡ {u : u ≤ u < ou} where u < ou, then the range
of the unextended G is {u : u < u < ou} and the range of the extended G
is {u : u < u ≤ ou} so that if ou = +∞, then G(v) = +∞ for v = 0N and
possibly for other points v on the boundary of the nonnegative orthant.
134 Essays in Index Number Theory 6. Duality Approaches 135
extension23
to the nonnegative orthant Ω ≡ {x : x ≥ 0N }.
Proof: (i) For v 0N , the constraint set ρ(v) ≡ {v : x ≥ 0N , vT
x ≤ 1}
for the maximization problem (4.1) is compact so that G(v) is well defined as
a maximum. It can be verified that ρ(v) is a continuous correspondence for
v 0N and that there exists a compact set B such that ρ(v) ⊂ B if v0
0N ,
v ∈ Nδ(v0
) where Nδ(v0
) ≡ {v : (v − v0
)T
(v − v0
) ≤ δ2
} and δ > 0 is chosen
small enough so that Nδ(v0
) ⊂ V ≡ {v : v 0N }. Thus for v ∈ Nδ(v0
),
G(v) ≡ maxv{F(x) : x ∈ ρ(v)} = maxv{F(x) : x ∈ ρ(v) ∩ B} and continuity of
G follows from the continuity of F and the Maximum Theorem.
(ii) Let 0N v v . Then
G(v ) ≡ max
x
{F(x) : v T
x ≤ 1, x ≥ 0N }
= max
x
{F(x) : v T
x = 1, x ≥ 0N } using I(ii)
= F(x ) say where v T
x = 1 and x ≥ 0N .
Since v v , v T
x < 1 and thus ε∗
≡ (1 − v T
x )/v T
1N > 0. Thus
G(v ) ≡ max
x
{F(x) : v T
x ≤ 1, x ≥ 0N }
≥ F(x + ε∗
1N ) since x + ε∗
1N ≥ 0N is feasible for the
maximization problem as v T
(x + ε∗
1N ) = 1
> F(x ) using condition I(ii) on F
= G(v ).
(iii) Let v 0N , v 0N , 0 ≤ λ ≤ 1, G(v ) ≤ u∗
and G(v ) ≤ u∗
.
Define the sets H ≡ {x : v T
x ≤ 1, x ≥ 0N }, H ≡ {x : v T
x ≤ 1, x ≥ 0N }
and Hλ
≡ {x : [λv + (1 − λ)v ]T
x ≤ 1, x ≥ 0N }. Then, as in Section 1, it can
be seen that Hλ
⊂ H ∪ H . Thus
G[λv + (1 − λ)v ] ≡ max
x
{F(x) : x ∈ Hλ
}
≤ max
x
{F(x) : x ∈ H ∪ H } since Hλ
⊂ H ∪ H
≤ u∗
since F(x) ≤ u∗
if x ∈ H or if x ∈ H .
23
Again F is extended to the nonnegative orthant by the Fenchel closure
operation: define the hypograph of the original F as ∆ ≡ {(u, x) : x
0N , u ≤ F(x)}, define the closure of ∆ as ∆ and define the extended F as
F(x) ≡ supu{u : (u, x) ∈ ∆} for x ≥ 0N . Since the unextended F is continuous
for x 0N , the extended F can easily be shown to be continuous from
above for x ≥ 0N . Condition III(iv) implies that the extended F is continuous
from below for x ≥ 0N as well.
(iv) Extend G to v ≥ 0N using the Fenchel closure operation. The resulting
extended G is continuous from below and thus minv{G(v) : vT
x ≤ 1, v ≥
0N } will exist and be finite for x 0N using a result due to Berge [1963;
76]. Thus F(x) is well defined for x 0N . We show that F has a continuous
extension to x ≥ 0N by showing that F(x) = F(x) for x 0N .
Let x∗
0N and u∗
≡ F(x∗
). Since x∗
is on the boundary of the closed
convex set L(u∗
) ≡ {x : F(x) ≥ u∗
, x ≥ 0N } (where we have used I(i), (ii) and
(iii)), there exists at least one supporting hyperplane v∗
= 0N to L(u∗
) at the
point x∗
; i.e., v∗
is such that x ∈ L(u∗
) implies v∗T
x ≥ v∗T
x∗
. By property
I(ii) on F, v∗
> 0N and we can normalize v∗
so that
(4.2) v∗T
x∗
= 1.
By property I(ii) on F, v∗
also has the property that
(4.3) x ∈ interior L(u∗
) implies v∗T
x > v∗T
x∗
= 1.
Now
G(v∗
) ≡ sup
x
{F(x) : v∗T
x ≤ 1, x ≥ 0N }(4.4)
≥ F(x∗
) ≡ u∗
since by (4.2), x∗
is feasible.
If F(x) > u∗
, then x ∈ interior L(u∗
) and (4.3) implies that v∗T
x > 1 so that
x is not feasible for the maximization problem in (4.4). Thus
(4.5) G(v∗
) = F(x∗
) = u∗
.
Now
F(x∗
) ≡ min
v
{G(v) : vT
x∗
≤ 1, v ≥ 0N }
≤ G(v∗
) since by (4.2), v∗
is feasible.(4.6)
Also
F(x∗
) ≡ min
v
{G(v) : vT
x∗
≤ 1, v ≥ 0N }
= G(v ) say where v T
x∗
= 1, v ≥ 0N(4.7)
≡ sup
x
{F(x) : v T
x ≤ 1, x ≥ 0N }
≥ F(x∗
) since by (4.7), x∗
is feasible
= G(v∗
) by (4.5).(4.8)
(4.5), (4.6) and (4.8) imply that F(x∗
) = G(v∗
) = F(x∗
).qed
136 Essays in Index Number Theory 6. Duality Approaches 137
Corollary 3.1. The direct aggregator function F can be recovered from the
indirect aggregator function G; i.e., for x 0, F(x) = minv{G(v) : vT
x ≤ 1,
v ≥ 0N }.
Corollary 3.2. Let F satisfy conditions I and let x∗
0N . Define the
closed convex set of normalized supporting hyperplanes at the point x∗
to
the closed convex set {x : F(x) ≥ F(x∗
), x ≥ 0N } by H(x∗
).24
Then: (i)
H(x∗
) is the solution set to the indirect utility (or production) minimization
problem minv{G(v) : vT
x∗
≤ 1, v ≥ 0N }, where G is the indirect function
which corresponds to F via definition (4.1), and (ii) if v∗
∈ H(x∗
), then
x∗
is the solution to the direct utility (or production) maximization problem
maxx{F(x) : v∗T
x ≤ 1, x ≥ 0N }.
Proof: (i) If v∗
∈ H(x∗
), we have shown in the proof of Theorem 3(iv)
that v∗
is a solution to
(4.9) min
v
{G(v) : vT
x∗
≤ 1, v ≥ 0N } = min
v
{G(v) : vT
x∗
= 1, v ≥ 0N }
where the equality in (4.9) follows from III(ii) on G. Now assume that v
is feasible for the second minimization problem in (4.9); i.e., v ≥ 0N and
v T
x∗
= 1. Then
G(v ) ≡ max
x
{F(x) : v T
x ≤ 1, x ≥ 0N } > F(x∗
)
where the above inequality follows if v is not a supporting hyperplane to the
set {x : F(x) ≥ F(x∗
)}. Thus v will not be a solution to (4.9) since G(v∗
) =
F(x∗
) < G(v ) where v∗
∈ H(x∗
). Thus the solution set to (4.9) is precisely
H(x∗
).
(ii) This part follows directly from (4.4) and (4.5).qed
Corollary 3.3. (Hotelling [1935; 71], Wold [1944; 69–71] [1953; 145] Identity):
If F satisfies conditions I and in addition is differentiable at x∗
0N
with a nonzero gradient vector F(x∗
) > 0N , then x∗
is a solution to the direct
(utility or production) maximization problem maxx{F(x) : v∗T
x ≤ 1, x ≥ 0N }
where
(4.10) v∗
≡
F(x∗
)
x∗T F(x∗)
.
Proof: Under the stated conditions, the set of normalized supporting
hyperplanes H(x∗
) reduces to the single point v∗
defined by (4.10) (note that
24
If v∗
∈ H(x∗
), then v∗T
x∗
= 1, v∗
≥ 0N and F(x) ≥ F(x∗
) implies v∗T
x ≥
v∗T
x∗
= 1. The closedness and convexity of H(x∗
) is shown in Rockafellar
[1970; 215].
x∗T
v∗
= x∗T
F(x∗
)/x∗T
F(x∗
) = 1). The present corollary now follows from
part (ii) of the previous corollary.qed
The system of equations (4.10) is known as the system of inverse demand
functions; the ith equation
pi/y ≡ v∗
i = [∂F(x∗
)/∂xi]
N
j=1
x∗
j ∂F(x∗
)/∂xj
gives the ith commodity price pi divided by expenditure y as a function of the
quantity vector x∗
which the producer or consumer would choose if he were
maximizing F(x) subject to the budget constraint v∗T
x = 1.
We now assume that a well behaved indirect aggregator function G is
given and we show that it can be used in order to define a well behaved direct
aggregator function F which has G as its indirect function.
Theorem 4. Suppose G satisfies conditions III. Then F(x) defined for x 0N
by
(4.11) F(x) ≡ min
v
{G(v) : vT
x ≤ 1, v ≥ 0N }
has an extension to x ≥ 0N which satisfies conditions I. Moreover, if we define
G∗
(v) ≡ maxx{F(x) : vT
x ≤ 1, x ≥ 0N } for v 0N , then G∗
(v) = G(v) for
all v 0N .
Proof: For x 0N , F(x) is well defined as a minimum (see the proof
of Theorem 3). Now extend F to Ω ≡ {x : x ≥ 0N } via the Fenchel closure
operation. Continuity of the extended F follows directly from III(iv). To
show that F is increasing and quasiconcave over x 0N , repeat the proofs
of parts (ii) and (iii) of Theorem 3 with the obvious changes due to the fact
that we are now dealing with the minimization problem (4.11) instead of the
maximization problem (4.1). The extended F will also have the properties of
increasingness and quasiconcavity over Ω. Finally, the proof that G∗
(v) = G(v)
for v 0N proceeds analogously to the proof in Theorem 3 that F(x) = F(x)
for x 0N .qed
Corollary 4.1. Let G satisfy conditions III and let v∗
0N . Define the
closed convex set of normalized supporting hyperplanes at the point v∗
to the
closed convex set {v : G(v) ≤ G(v∗
), v ≥ 0N } by H∗
(v∗
). Then: (i) H∗
(v∗
)
is the solution set to the direct maximization problem maxx{F(x) : v∗T
x ≤
1, x ≥ 0N }, where F is the direct function which corresponds to the given
indirect function G via definition (4.11), and (ii) if x∗
∈ H(v∗
), then v∗
is a
solution to the indirect minimization problem minv{G(v) : vT
x∗
≤ 1, v ≥ 0N }.
The proof of Corollary 4.1 follows in an analogous manner to the proof
of Corollary 3.2.
138 Essays in Index Number Theory 6. Duality Approaches 139
Corollary 4.2. (Ville [1946; 35], Roy [1947; 222] Identity): If G satisfies
conditions III and, in addition, is differentiable at v∗
0N with a nonzero
gradient vector G(v∗
) < 0N , then x∗
is the unique solution to the direct
maximization problem maxx{F(x) : v∗T
x ≤ 1, x ≥ 0N }, where
(4.12) x∗
≡ G(v∗
)/v∗T
G(v∗
).
Proof: Under the stated conditions, the set of normalized supporting
hyperplanes H∗
(v∗
) reduces to the single point x∗
> 0N defined by (4.12).
Thus from part (i) of the previous corollary, x∗
is the unique solution to the
direct maximization problem.qed
It can be seen that (4.12) provides the counterpart to Shephard’s Lemma
in the previous section. Shephard’s Lemma and Roy’s Identity are the basis for
a great number of theoretical and empirical applications as we shall see later.
Finally, we note that although condition III(iv) appears to be a bit odd,
it enables us to derive a continuous direct aggregator function from a given
indirect function satisfying conditions III.25
Historical Notes
Duality theorems between direct and indirect aggregator functions have been
proven by Samuelson [1965] [1969b] [1972], Newman [1965; 138–165], Lau
[1969], Shephard [1970; 105–113], Hanoch [1978b], Weddepohl [1970; ch. 5],
Katzner [1970; 59–62], Afriat [1972c] [1973c] and Diewert [1974a].
For closely related work relating assumptions on systems of consumer demand
functions to the direct aggregator function F (the integrability problem),
see Samuelson [1950b], Hurwicz and Uzawa [1971], Hurwicz [1971] and Afriat
[1973a] [1973b].
For a geometric interpretation of Roy’s Identity, see Darrough and Southey
[1977], and for some extensions, see Weymark [1980].
25
Without condition III(iv), we could still deduce continuity of F(x) over x
0N but the resulting F would not necessarily have a continuous extension to
x ≥ 0N (since F is not necessarily concave, but is only quasiconcave over x
0N , its extension is not necessarily continuous). For discussion and examples
of these continuity problems, see Diewert [1974a; 121–123].
5. Duality between Direct Aggregator Functions and Distance or
Deflation Functions
In this section, we consider a fourth alternative method of characterizing tastes
or technology, a method which proves to be extremely useful for defining a
certain class of index number formulae due to Malmquist [1953; 232].
As usual, let F(x) be an aggregator function satisfying conditions I listed
in Section 3 above. For u belonging to the interior of the range of F (i.e.,
u ∈ Int U, where U ≡ {u : u ≤ u < ou}) and x 0N , define the distance or
deflation function26
D as
(5.1) D(u, x) ≡ max
k
{k : F(x/k) ≥ u, k > 0}.
Thus D(u∗
, x∗
) is the biggest number which will just deflate (inflate if F(x∗
) <
u∗
) the given point x∗
0N onto the boundary of the utility (or production)
possibility set L(u∗
) ≡ {x : F(x) ≥ u∗
}. If D(u∗
, x∗
) > 1, then x∗
0N
produces a higher level of utility or output than the level indexed by u∗
.
In turns out that the mathematical properties of D(u, x) with respect to
x are the same as the properties of C(u, p) with respect to p, but the properties
of D with respect to u are reciprocal to the properties of C with respect to u,
as the following theorem shows.
Theorem 5. If F satisfies conditions I, then D defined by (5.1) satisfies conditions
IV below.
Conditions IV on D: (i) D(u, x) is a real valued function of N + 1
variables defined over Int U × Int Ω = {u : u < u < ou} × {x : x 0N } and is
continuous over this domain.
(ii) D(u, x) = +∞ for every x ∈ Int Ω; i.e., un
∈ Int U, lim un
= u,
x ∈ Int Ω implies limn D(un
, x) = +∞.
(iii) D(u, x) is decreasing in u for every x ∈ Int Ω; i.e., if x ∈ Int Ω,
u , u ∈ Int U with u < u , then D(u , x) > D(u , x).
(iv) D(ou, x) = 0 for every x ∈ Int Ω; i.e., un
∈ Int U, lim un
= ou,
x ∈ Int Ω implies limn D(un
, x) = 0.
(v) D(u, x) is (positively) linearly homogeneous in x for every u ∈ Int U;
i.e., u ∈ Int U, λ > 0, x ∈ Int Ω implies D(u, λx) = λD(u, x).
26
Shephard [1953; 6][1970; 65] introduced the distance function into the economics
literature, using the slightly different but equivalent definition: D(u, x)
≡ 1/ minλ{λ : F(λx) ≥ u, λ > 0}. McFadden [1978a] and Blackorby, Primont
and Russell [1978] call D the transformation function, while in the mathematics
literature (e.g., Rockafellar [1970; 28]), D is termed a guage function. The term
deflation function for D would seem to be more descriptive from an economic
point of view.
140 Essays in Index Number Theory 6. Duality Approaches 141
(vi) D(u, x) is concave in x for every u ∈ Int U.
(vii) D(u, x) is increasing in x for every u ∈ Int U; i.e., u ∈ Int U,
x , x ∈ Int Ω implies D(u, x + x ) > D(u, x ).
(viii) D is such that the function
(5.2) F(x) ≡ {u : u ∈ Int U, D(u, x) = 1}
defined for x 0N has a continuous extension to x ≥ 0N .
Proof: (i) We first show that D defined by (5.1) is well defined. Let
x∗
0N and define the function gx∗ (k) ≡ F(x∗
/k) for k > 0. Note that
limk→∞ gx∗ (k) = limk→∞ F(x∗
/k) = F(0N ) = u using I(i) and limk→0 gx∗ (k) =
limk→0 F(x∗
/k) = ou using x∗
0N , I(i), I(ii) and the definition of ou. Using
I(ii), it is easy to show that gx∗ (k) is a monotonically decreasing function
of k. Finally, from I(i), gx∗ (k) is a continuous function of k. Thus range
gx∗ = Int U and for every u∗
∈ Int U, there exists a unique k∗
> 0 such that
gx∗ (k∗
) ≡ F(x∗
/k∗
) = u∗
. By I(ii), if k > k∗
, then F(x∗
/k) < F(x∗
/k∗
) = u∗
.
Thus
D(u∗
, x∗
) ≡ max
k
{k : F(x∗
/k) ≥ u∗
, k > 0}
= {k∗
: F(x∗
/k∗
) = u∗
, k∗
> 0}.(5.3)
In what follows, we will use (5.3) in order to define D instead of (5.1).27
We
show that D(u, x) is continuous in (u, x) over Int U × Int Ω by showing that
the upper and lower level sets are closed in Int U × Int Ω.
Let (un
, xn
) ∈ Int U × Int Ω with limn(un
, xn
) ≡ (u0
, x0
) ∈ Int U × Int Ω
and D(un
, xn
) ≤ k∗
> 0 for every n. Define k0
> 0 by F(x0
/k0
) = u0
, and
kn
> 0 by F(xn
/kn
) = un
. Thus D(un
, xn
) ≡ kn
≤ k∗
implies, using I(ii), that
F(xn
/k∗
) ≤ F(xn
/kn
) = un
. Thus u0
≡ limn un
≥ limn F(xn
/k∗
) = F(x0
/k∗
)
using I(i). But u0
= F(x0
/k0
) ≥ F(x0
/k∗
) implies k∗
≥ k0
≡ D(u0
, x0
) using
I(ii).
Now let (un
, xn
) ∈ Int U × Int Ω with limn(un
, xn
) ≡ (u0
, x0
) ∈ Int U ×
Int Ω and D(un
, xn
) ≥ k∗
> 0 for every n. Define k0
> 0 by F(x0
/k0
) = u0
and kn
> 0 by F(xn
/kn
) = un
. Thus D(un
, xn
) ≡ kn
≥ k∗
implies that
F(xn
/k∗
) ≥ F(xn
/kn
) = un
. Thus u0
≡ limn un
≤ limn F(xn
/k∗
) = F(x0
/k∗
)
using I(i) again. But u0
= F(x0
/k∗
) ≤ F(x0
/k∗
) implies k∗
≤ k0
≡ D(u0
, x0
).
(ii) Let x∗
0N , u < un
< ou and limn un
= u. Define kn
≡ D(un
, x∗
)
so that F(x∗
/kn
) = un
. Since limn un
= u and since x = 0N is the unique
solution for the equation F(x) = u, continuity of F implies that limn x∗
/kn
= 0
so that limn kn
= +∞.
27
The reason why we did not define D directly by (5.3) is that definition (5.1)
provides a valid definition for D when F satisfies weaker regularity conditions
(such as our assumptions 1, 2 and 3 in Section 1).
(iii) This follows directly from (5.3) and property I(ii) on F.
(iv) Let x∗
0N , u < un
< ou and limn un
= ou. Define kn
≡ D(un
, x∗
)
so that F(x∗
/kn
) = un
. If x ≥ 0N satisfies the equation F(x) = ou, then at
least one component of x must be +∞. Since ou = limn = limn F(x∗
/kn
) =
F[limn(x∗
/kn
)] using I(i), it follows that we must have limn kn
= 0.
(v) Let u ∈ Int U, x 0N , λ > 0. Then using (5.3), D(u, λx) ≡ {k :
k > 0, F(λx/k) = u} = λ{λ−1
k : λ−1
k > 0, F(x/λ−1
k) = u} = λ{k : k > 0,
F(x/k ) = u} ≡ λD(u, x).
(vi) Let u ∈ Int U, x 0N , x 0N and 0 ≤ λ ≤ 1. Define k ≡
D(u, x ), k ≡ D(u, x ) and kλ
≡ D[u, λx + (1 − λ)x ]. Then F(x , k ) = u,
F(x /k ) = u and F [λx + (1 − λ)x ]/kλ
= u. If we define λ∗
≡ k λ/[(1 −
λ)k + λk ] and k∗
≡ (1 − λ)k + λk , it can be verified that 0 ≤ λ∗
≤ 1,
k∗
> 0, and [λx + (1 − λ)x ]/k∗
= λ∗
(x /k ) + (1 − λ∗
)(x /k ). Thus by
I(iii), F [λx + (1 − λ)x ]/k∗
≥ u. Using I(ii), the last inequality implies
D[u, λx +(1−λ)x ] ≡ kλ
≥ k∗
≡ λk +(1−λ)k ≡ λD(u, x )+(1−λ)D(u, x ).
(vii) Let u ∈ Int U, x 0N , x 0N . Properties IV(ii) to (iv) imply
that D(u, x ) > 0. Thus
D(u, x + x ) = 2D[u, (1/2)x + (1/2)x ] using (v) above
≥ 2[(1/2)D(u, x ) + (1/2)D(u, x )] using (vi) above
> D(u, x ) using D(u, x ) > 0.
(viii) Let x∗
0N and define u∗
≡ F(x∗
). Thus using I(ii) and definition
(5.3) it can be seen that D(u∗
, x∗
) ≡ {k : F(x∗
/k) = u∗
, k > 0} = 1. Using
IV(iii), F(x∗
) ≡ {u : D(u, x∗
) = 1, u ∈ Int U} = u∗
. Thus F(x∗
) = F(x∗
) for
every x∗
0N and since F is continuous over Ω, F has F as its continuous
extension.qed
Corollary 5.1. F(x) ≡ {u : u ∈ Int U, D(u, x) = 1} = F(x) for every
x 0N and thus F = F; i.e., the original aggregator function F is recovered
from the distance function D via definition (5.2) if F satisfies conditions I.
As was the case with the cost function C(u, p) studied in Section 3 above,
D satisfying conditions IV over Int U × Int Ω can be uniquely extended to
Int U × Ω using the Fenchel closure operation. It can be verified that the
extended D satisfies conditions (v), (vi) and (vii) over Int U × Ω, but the
joint continuity condition IV(i) and the monotonicity conditions in u are no
longer necessarily satisfied.28
It should also be noted that if condition I(iii)
28
If conditions IV(i) to (vii) were satisfied by D over Int U × Ω, then we could
derive the corresponding continuous F from D without using the somewhat
unusual condition IV(viii).
142 Essays in Index Number Theory 6. Duality Approaches 143
(quasiconcavity of F) were dropped, then Theorem 5 would still be valid except
that condition IV(vi) (concavity of D in x) would have to be dropped.
The following theorem shows that the deflation function D can also be
used in order to define a continuous aggregator function F.
Theorem 6. If D satisfies conditions IV above, then F defined by (5.2) for
x ∈ Int Ω has an extension to Ω which satisfies conditions I. Moreover, if we
define the deflation function D∗
which corresponds to F by
(5.4) D∗
(u, x) ≡ {k : F(x/k) = u, k > 0},
then D∗
(u, x) = D(u, x) for (u, x) ∈ Int U × Int Ω.
Proof: (i) Since for every x ∈ Int Ω, range D(u, x) as a function of
u ∈ Int U is (0, +∞) and since D is a continuous, monotonically decreasing
function of u over this domain (we have used IV(i)–(iv) here), we see that F(x)
is well defined by (5.2) for x 0N . Property IV(viii) implies that F has a
continuous extension to x ≥ 0N . Since F is continuous over Ω, we need only
prove properties I(ii) and (iii) for x ∈ Int Ω.
(ii) Let 0N x x and define u , u ∈ Int U by the equations
D(u , x ) = 1 and D(u , x ) = 1. Thus by (5.2), F(x ) = u and F(x ) = u .
Now
1 = D(u , x )
= D(u , x )
< D(u , x ) since x x using IV(vii).
But D(u , x ) < D(u , x ) implies F(x ) = u > u = F(x ) using IV(iii).
(iii) Let x , x ∈ Int Ω, 0 ≤ λ ≤ 1, u∗ ∈ Int U with F(x ) ≥ u∗
and
F(x ) ≥ u∗
. Then D(u∗
, x ) ≥ 1 and D(u∗
, x ) ≥ 1, since D[F(x ), x ] = 1
and D[F(x ), x ] = 1 using (5.2) and IV(iii). Note that D F[λx + (1 − λ)x ],
λx + (1 − λ)x = 1 also using definition (5.2). Thus D[u∗
, λx + (1 − λ)x ] ≥
λD(u∗
, x ) + (1 − λ)D(u∗
, x ) using IV(vi) ≥ λ1 + (1 − λ)1 = 1. Again using
IV(iii), we conclude that F[λx + (1 − λ)x ] ≥ u∗
.
To prove the moreover part of the theorem, let u ∈ Int U, x ∈ Int Ω and
define k ≡ D∗
(u, x) > 0. Then by definition (5.4), F(x/k) = u. By definition
(5.2), the last equality implies
1 = D(u, x/k) = (1/k)D(u, x) using IV(v)
or
k ≡ D∗
(u, x) = D(u, x). qed
Corollary 6.1. (Shephard29
[1953; 10–13], Hanoch [1978b; 7]: If D satisfies
conditions IV and, in addition, is continuously differentiable at (u∗
, x∗
) ∈
Int U × Int Ω with D(u∗
, x∗
) = 1 and ∂D(u∗
, x∗
)/∂u < 0, then x∗
is a solution
to the direct maximization problem maxx{F(x) : v∗T
x ≤ 1, x ≥ 0N }, where F
is defined by (5.2) and v∗
> 0N is defined by
(5.5) v∗
≡ xD(u∗
, x∗
).
Moreover, F is continuously differentiable at x∗
with
(5.6) xF(x∗
) = −
xD(u∗
, x∗
)
∂D(u∗, x∗)/∂u
.
Proof: Since F(x) is implicitly defined by the equation D[F(x), x] = 1
for x in a neighborhood of x∗
0N , the Implicit Function Theorem (see
Rudin [1953; 177–182]) implies that F is continuously differentiable at x∗
with
partial derivatives given by (5.6), since the Jacobian of the transformation,
∂D(u∗
, x∗
)/∂u, is nonzero. Since D(u∗
, x) is linearly homogenous in x, multiplying
both sides of (5.6) by x∗T
yields x∗T
F(x∗
) = −x∗T
xD(u∗
, x∗
) /
uD(u∗
, x∗
) = −D(u∗
, x∗
)/ uD(u∗
, x∗
) = −1/ uD(u∗
, x∗
) > 0 using Euler’s
Theorem on homogeneous functions. Therefore,
(5.7) xD(u∗
, x∗
) = − xF(x∗
) uD(u∗
, x∗
) = xF(x∗
)/x∗T
xF(x∗
).
Since xD(u∗
, x∗
) > 0N , xF(x∗
) > 0N also. Now apply Corollary 3.3.
Equations (4.10) and (5.7) imply (5.5).qed
Thus, the consumer’s system of inverse demand functions can be obtained
by differentiating the deflation function D satisfying conditions IV (plus differentiability)
with respect to the components of the vector x.
Historical Notes
Duality theorems between distance or deflation functions D and aggregator
functions F have been proven by Shephard [1953] [1970], Hanoch [1978b], McFadden
[1978a] and Blackorby, Primont and Russell [1978].
There are a number of interesting relationships (and further duality theorems)
between direct and indirect aggregator, cost and deflation functions. For
example, Malmquist [1953; 214] and Shephard [1953; 18] showed that the deflation
function for the indirect aggregator function, maxk{k : G(v/k) ≤ u, k >
29
The result can readily be deduced from several separate equations in Shephard
but it is explicit in Hanoch’s paper.
144 Essays in Index Number Theory 6. Duality Approaches 145
0}, equals the cost function, C(u, v). A complete description of these interrelationships
and further duality theorems under various regularity conditions
may be found in Hanoch [1978b] and Blackorby, Primont and Russell [1978].
For some applications see Deaton [1979].
For a local duality theorem between deflation and aggregator functions,
see Blackorby and Diewert [1979].
6. Other Duality Theorems
Concave functions can be characterized by their conjugate functions. Furthermore,
it turns out that closed convex sets can also be described by a conjugate
function under certain conditions.30
Thus, a direct aggregator function F,
having convex level sets L(u) ≡ {x : F(x) ≥ u}, can also be characterized by
its conjugate function as well as by its cost, deflation or indirect aggregator
function. This conjugacy approach was initiated by Hotelling [1932; 590–592]
[1935; 68–70] and extended by Samuelson [1947; 36–39] [1960] [1972], Lau [1969]
[1976] [1978a], Jorgenson and Lau [1974a] [1974b], and Blackorby, Primont and
Russell [1978]. We will not review this approach in detail, although in a later
section we will review the closely related duality theorems between profit and
transformation functions.
Another class of duality theorems (which also has its origins in the work
of Hotelling [1935; 75] and Samuelson [1960]) is obtained by partitioning the
commodity vector x ≥ 0N into two vectors, x1
and x2
say, and then defining
the consumer’s variable indirect aggregator31
function g as
(6.1) g(x1
, p2
, y2
) ≡ max
x2
{F(x1
, x2
) : p2T
x2
≤ y2
, x2
≥ 0N2 }
where p2
0N2 is a positive vector of prices the consumer faces for the goods
x2
and y2
> 0 is consumer’s budget which he has allocated to spend on the x2
goods. The solution set to (6.1), x2
(x1
, p2
, y2
), is the consumer’s conditional
(on x1
) demand correspondence. If g satisfies appropriate regularity conditions,
conditional demand functions can be generated by applying Roy’s Identity
(4.12) to the function G(v2
) ≡ g(x1
, v2
, 1), where v2
≡ p2
/y2
. For formal
duality theorems between direct and variable indirect aggregator functions, see
Epstein [1975], Diewert [1978a] and Blackorby, Primont and Russell [1977a].
For various applications of this duality, see Epstein [1975] (for applications
to consumer choice under uncertainty) and Pollak [1969] and Diewert [1978a]
30
See Rockafellar [1970; 102–105] and Karlin [1959; 226–227].
31
Pollak [1969] uses the alternative terminology, “conditional indirect utility
function.”
(estimation of preferences for public goods using market demand functions).
Finally, the variable indirect utility function can be used to prove versions
of Hicks’ [1946; 312–313] composite good theorem — that a group of goods
behaves just as if it were a single commodity if the prices of the group of goods
change in the same proportion — under less restrictive conditions than were
employed by Hicks; see Pollak [1969], Diewert [1978a] and Blackorby, Primont
and Russell [1977a].
We turn now to a brief discussion of a vast literature; i.e., the implications
of various special structures on one of our many equivalent representations
of tastes or technology (such as the direct or indirect aggregator function or
the cost function) on the other representations. For example, Shephard [1953]
showed that homotheticity of the direct function implied that the cost function
factored into φ−1
(u)c(p) (recall equation (2.18) above). Another example of a
special structure is separability.32
References which deal with the implications
of separability and/or homotheticity include Shephard [1953] [1970], Samuelson
[1953–54], [1965] [1969b] [1972], Gorman [1959] [1976], Lau [1969] [1978a],
McFadden [1978a], Hanoch [1975] [1978b], Pollak [1972], Diewert [1974a], Jorgenson
and Lau [1975], and Blackorby, Primont and Russell [1975a] [1975b]
[1977a] [1977b] [1977d] [1978]. For the implications of separability and/or homotheticity
on Slutsky coefficients or on partial elasticities of substitution,33
see Sono [1945], Pearce [1961], Goldman and Uzawa [1964], Geary and Morishma
[1973], Berndt and Christensen [1973a], Russell [1975], Diewert [1974a;
150–153] and Blackorby and Russell [1976]. For the implications of Hicks’
[1946; 312–313] Aggregation Theorem on aggregate elasticities of substitution,
see Diewert [1974c].
32
Loosely speaking, F(x) = F(x1
, x2
, . . . , xM
) is separable in the partition
(x1
, x2
, . . . , xM
) if there exist functions F, F1
, . . . , FM
such that F(x) =
F[F1
(x1
), F2
(x2
), . . . , FM
(xM
)] and F is additively separable if there exist
functions F∗
, F1
, F2
, . . . , FM
such that F(x) = F∗
[F1
(x1
) + F2
(x2
) + · · · +
FM
(xM
)]. For historical references and more precise definitions, see Blackorby,
Primont and Russell [1977d][1978].
33
Uzawa [1962] observed that the Allen [1938; 504] partial elasticity of substitution
between inputs i and j, σij (u, p) = C(u, p)Cij(u, p)/Ci(u, p)Cj(u, p) where
C(u, p) ≡ minx{pT
x : F(x) ≥ u} is the cost function dual to the aggregator
function F and Ci denotes the partial derivative with respect to the ith price
in p, pi, and Cij denotes the second order partial derivative of C with respect
to pi and pj. Shephard’s Lemma implies that Cij(u, p) = ∂xi(u, p)/∂pj =
∂xj(u, p)/∂pi = Cji(u, p) assuming continuous differentiability of C, where
xi(u, p) and xj(u, p) are the cost minimizing demand functions. Thus σij can
be regarded as a normalization of the response of the cost minimizing xi to a
change in pj.
146 Essays in Index Number Theory 6. Duality Approaches 147
For empirical tests of the separability assumption, see Berndt and Christensen
[1973b] [1974], Burgess [1974] and Jorgenson and Lau [1975]; for theoretical
discussions of these testing procedures, see Blackorby, Primont and
Russell [1977c], Jorgenson and Lau [1975], Lau [1977c], Woodland [1978] and
Denny and Fuss [1977].
For the implications of assuming concavity of the direct aggregator function
or of assuming convexity of the indirect aggregator function, see Diewert
[1978a].
The duality theorems referred to above have been “global” in nature. A
“local” approach has been initiated by Blackorby and Diewert [1979], where it
is assumed that a given cost function C(u, p) satisfies conditions II above over
U ×P, where U is a finite interval and P is a closed, convex, and bounded subset
of positive prices. They then construct the corresponding direct aggregator,
indirect aggregator and deflation functions which are dual to the given “locally”
valid cost function C. The proofs of these “local” duality theorems turn out to
be much simpler than the corresponding “global” duality theorems presented
in this paper (and elsewhere), since troublesome continuity problems do not
arise due to the assumption that U × P is compact. These “local” duality
theorems are useful in empirical applications, since econometrically estimated
cost functions frequently do not satisfy the appropriate regularity conditions
for all prices, but the conditions may be satisfied over a smaller subset of prices
which is the empirically relevant set of prices.
Epstein has extended duality theory to cover more general maximization
problems. In an unpublished working paper of his, the following utility maximization
problem which arises in the context of choice under uncertainty is
considered:
(6.2) max
x,x1,x2
F(x, x1
, x2
) : x ≥ 0N , x1
≥ 0N1 , x2
≥ 0N2 ,
pT
x + p1T
x1
≤ y1
, pT
x + p2T
x2
≤ y2
where x represents current consumption, there are two future uncertain states
of nature, xi
represents consumption in state i (i = 1, 2), p is the current
price vector, pi
is the discounted future price vector which will prevail if state i
occurs, and yi
> 0 is the consumer’s discounted income if state i occurs. In
Epstein [1981a], the following maximization problem is considered:
(6.3) max
x
{F(x) : x ≥ 0N , c(x, α) ≤ 0}
where c is a given constraint function which depends on a vector of parameters
α.
We will not attempt to provide a detailed analysis of Epstein’s results but
rather we will present a more abstract version of his basic technique which will
hopefully capture the essence of duality theory.
The basic maximization problem we study is maxx{F(x) : x ∈ B(v)}
where F is a function of N real variables x defined over some set S and B(v) is
a constraint set which depends on a vector of M parameters v, which in turn
can vary over a set V . Our assumptions on the sets S and V and the constraint
set correspondence B are:
(i) S and V are nonempty compact sets in RN
and RM
.
(6.4)
(ii) For every v ∈ V , B(v) is nonempty and B(v) ⊂ S.
(iii) For every x ∈ S, the inverse correspondence34
B−1
(x) is
nonempty and B−1
(x) ⊂ V .
(iv) The correspondence B is continuous over V .
(v) The correspondence B−1
is continuous over S.
Our assumptions on the primal function F are:
(i) F is a real valued function of N variables defined over S
and is continuous over S.
(6.5)
(ii) For every x∗
∈ S, there exists v∗
∈ V such that
F(x∗
) = max
x
{F(x) : x ∈ B(v∗
)}.
The function G dual to F is defined for v ∈ V by
(6.6) G(v) ≡ max
x
{F(x) : x ∈ B(v)}.
Theorem 7. If S, V and B satisfy (6.4) and F satisfies (6.5), then G defined
by (6.6) satisfies the following conditions:
(i) G is a real valued function of M variables defined over V
and is continuous over V .
(6.7)
(ii) For every v∗
∈ V , there exists x∗
∈ S such that
G(v∗
) = min
v
{G(v) : v ∈ B−1
(x∗
)}.
Moreover, if we define the function F∗
dual to G for x ∈ S by
(6.8) F∗
(x) ≡ min
v
{G(v) : v ∈ B−1
(x)},
34
B−1
(x) ≡ {v : v ∈ V and x ∈ B(v)}.
148 Essays in Index Number Theory 6. Duality Approaches 149
then F∗
(x) = F(x) for every x ∈ S.
Proof: (6.7)(i) follows from (6.4)(i), (ii), (iv); (6.5)(i) and the Maximum
Theorem.
(ii) Let v∗
∈ V and let x∗
∈ S be any solution to the following maximization
problem:
(6.9) max
x
{F(x) : x ∈ B(v∗
)} = F(x∗
) = G(v∗
)
where (6.9) follows using definition (6.6). By definition (6.8)
(6.10) F∗
(x∗
) ≡ min
v
{G(v) : v ∈ B−1
(x∗
)} ≤ G(v∗
)
since by (6.9), x∗
∈ B(v∗
) and thus v∗
∈ B−1
(x∗
) and thus v∗
is feasible for
the minimization problem in (6.10). For every v ∈ B−1
(x∗
), x∗
∈ B(v) and
thus using (6.9), we have
(6.11) G(v) ≡ max
x
{F(x) : x ∈ B(v)} ≥ F(x∗
) = G(v∗
)
since x∗
is feasible for the maximization problem in (6.11) and the last equality
in (6.11) follows from (6.9). Since (6.11) is satisfied for every v ∈ B−1
(x∗
),
(6.12) F∗
(x∗
) ≡ min
v
{G(v) : v ∈ B−1
(x∗
)} ≥ G(v∗
).
Thus (6.9), (6.10) and (6.12) imply that F(x∗
) = G(v∗
) = F∗
(x∗
) and thus
(6.10) becomes an equality, which establishes property (6.7)(ii) for G.
Notice that we have not yet used property (6.5)(ii) for F. However, it
will be used in order to prove the moreover part of the theorem. Let x∗
∈ S.
Then using (6.5)(ii), there exists v∗
∈ V such that F(x∗
) = max{F(x) : x ∈
B(v∗
)} ≡ G(v∗
). Thus we have an x∗
∈ S and v∗
∈ V such that (6.9) is
satisfied and now we can repeat the proof of part (ii) above, showing that
F(x∗
) = F∗
(x∗
).qed
Corollary 7.1. Let x∗
∈ S and define H(x∗
) to be the set of v∗
∈ V such
that F(x∗
) = maxx{F(x) : x ∈ B(v∗
)}. If v∗
∈ H(x∗
), then x∗
is a solution to
maxx{F(x) : x ∈ B(v∗
)}, and v∗
is a solution to minv{G(v) : v ∈ B−1
(x∗
)}.
Notice that property (6.5)(ii) of F is the replacement for our old quasiconcavity
assumption in Section 4, and the set H(x∗
) defined in Corollary 7.1
replaces the set of normalized supporting hyperplanes which occurred in Corollary
3.2.
Owing to the symmetric nature of our assumptions, it can be seen that
the proof of the following theorem is the same as the proof of Theorem 7, except
that the inequalities are reversed.
Theorem 8. If S, V and B satisfy (6.4) and G satisfies (6.7), then F∗
defined
by (6.8) satisfies (6.5). Moreover, if we define the function G∗
dual to F∗
for
v ∈ V by
(6.13) G∗
(v) ≡ max
x
{F∗
(x) : x ∈ B(v)},
then G∗
(v) = G(v) for every v ∈ V .
Corollary 8.1. Let v∗
∈ V and define H∗
(v∗
) to be the set of x∗
∈ S such
that G(v∗
) = minv{G(v) : v ∈ B−1
(x∗
)}. If x∗
∈ H∗
(v∗
), then v∗
is a solution
to minv{G(v) : v ∈ B−1
(x∗
)}, and x∗
is a solution to maxx{F∗
(x) : x ∈ B(v∗
)}.
Note that condition (6.7)(ii) on G replaces our old quasiconvexity condition
on G in Section 4, and the set H∗
(v∗
) defined in Corollary 8.1 replaces the
set of normalized supporting hyperplanes which occurred in Corollary 4.1.
We cannot establish counterparts to Corollary 3.3 (Hotelling–Wold Identity)
and Corollary 4.2 (Ville–Roy Identity) since these corollaries made use
of the differentiable nature of F or G and the relevant constraint function.
Thus in order to derive counterparts to Corollaries 3.3 and 4.2 in the present
context, we need to make additional assumptions on F (or G) and the constraint
correspondence B.35
However, the above theorems (due essentially to
Epstein) do illustrate the underlying structure of duality theory. They can also
be interpreted as examples of local duality theorems.
7. Cost Minimization and the Derived Demand for Inputs
Assume that the technology of a firm can be described by the production
function F where u = F(x) is the maximum output that can be produced using
the nonnegative vector of inputs x ≥ 0N . Assume that F satisfies assumption 1
of Section 2 (i.e., the production function is continuous from above). If the
firm takes the prices of inputs p 0N as given (i.e., the firm does not behave
monopsonistically with respect to inputs), then we saw in Section 2 that the
firm’s total cost function C(u; p) ≡ minx{pT
x : F(x) ≥ u} was well defined for
all p 0N and u ∈ textRange F. Moreover, C(u, p) was linearly homogeneous
and concave in prices p for every u and was nondecreasing in u for each fixed p.
Now suppose that C is twice continuously differentiable36
with respect
to its arguments at a point (u∗
, p∗
) where u∗
∈ textRange F and p∗
≡
35
Epstein [1981a] derives counterparts to 4.2 in the context of his specific
models.
36
By this assumption, we mean that the second order partial derivatives of C
exist and are continuous functions for a neighborhood around (u∗
, p∗
).
150 Essays in Index Number Theory 6. Duality Approaches 151
(p∗
1, . . . , p∗
N ) 0N . From Lemma 3 in Section 2, the cost minimizing input
demand functions x1(u, p), . . . , xN (u, p) exist at (u∗
, p∗
) and they are in fact
equal to the partial derivatives of the cost function with respect to the N input
prices:
(7.1) xi(u∗
, p∗
) = ∂C(u∗
, p∗
)/∂pi; i = 1, . . . , N.
Thus, the assumption that C be twice continuously differentiable at (u∗
, p∗
)
ensures that the cost minimizing input demand functions xi(u, p) exist and are
once continuously differentiable at (u∗
, p∗
).
Define (∂xi/∂pj) ≡ [∂xi(u∗
, p∗
)/∂pj] to be the N×N matrix of derivatives
of the N input demand functions xi(u∗
, p∗
) with respect to the N prices p∗
j ,
i, j = 1, 2, . . . , N. From (7.1), it follows that
(7.2) (∂xi/∂pj) = 2
ppC(u∗
, p∗
)
where 2
ppC(u∗
, p∗
) ≡ [∂2
C(u∗
, p∗
)/∂pi∂pj] is the Hessian matrix of the cost
function with respect to the input prices evaluated at (u∗
, p∗
). Twice continuous
differentiability of C with respect to p at (u∗
, p∗
) implies (via Young’s
Theorem) that 2
ppC(u∗
, p∗
) is a symmetric matrix, so that using (7.2),
(7.3) (∂xi/∂pj) = (∂xi/∂pj)T
= (∂xj /∂pi),
i.e., ∂xi(u∗
, p∗
)/∂pj = ∂xj(u∗
, p∗
)/∂pi for all i and j.
Since C is concave in p and is twice continuously differentiable with respect
to p around the point (u∗
, p∗
), it follows37
that 2
C(u∗
, p∗
) is a negative
semidefinite matrix. Thus by (7.2),
(7.4) zT
(∂xi/∂pj)z ≤ 0 for all vectors z.
Thus, in particular, letting z = ei (the ith unit vector), (7.4) implies
(7.5) ∂xi(u∗
, p∗
)/∂pi ≤ 0, i = 1, 2, . . ., N;
i.e., the ith cost minimizing input demand function cannot slope upwards with
respect to the ith input price for i = 1, 2, . . . , N.
Since C is linearly homogeneous in p, we have C(u∗
, λp∗
) = λC(u∗
, p∗
)
for all λ > 0. Partially differentiating this last equation with respect to pi for
λ close to 1 yields the equation Ci(u∗
, λp∗
)λ = λCi(u∗
, p∗
), where Ci(u∗
, p∗
) ≡
∂C(u∗
, p∗
)/∂pi. Thus, Ci(u∗
, λp∗
) = Ci(u∗
, p∗
) and differentiation of this last
equation with respect to λ yields (when λ = 1)
N
j=1
p∗
j ∂2
C(u∗
, p∗
)/∂pi∂pj = 0 for i = 1, 2, . . ., N.
37
See Fenchel [1953; 87–88] or Rockafellar [1970; 27].
Thus, using (7.2), we find that the input demand functions xi(u∗
, p∗
) satisfy
the following N restrictions:
(7.6) (∂xi/∂pj)p∗
= 2
ppC(u∗
, p∗
)p∗
= 0N
where p∗
≡ (p∗
1, p∗
2, . . . , p∗
N )T
.
The final general restriction that we can obtain on the derivatives of the
input demand functions is obtained as follows: for λ near 1, differentiate both
sides of C(u∗
, λp∗
) = λC(u∗
, p∗
) with respect to u and then differentiate the
resulting equation with respect to λ. When λ = 1, the last equation becomes
N
j=1
p∗
j ∂2
C(u∗
, p∗
)/∂u∂pj = ∂C(u∗
, p∗
)/∂u.
Note that the twice continuous differentiability of C and (7.1) implies that
∂2
C(u∗
, p∗
)/∂u∂pj = ∂2
C(u∗
, p∗
)/∂pj∂u
= ∂[∂C(u∗
, p∗
)/∂pj]/∂u = ∂xj (u∗
, p∗
)/∂u.
Thus
N
j=1
p∗
j
∂2
C(u∗
, p∗
)
∂u∂pj
=
N
j=1
p∗
j
∂xj (u∗
, p∗
)
∂u
=
∂C(u∗
, p∗
)
∂u
≥ 0.(7.7)
The inequality ∂C(u∗
, p∗
)/∂u ≥ 0 follows from the nondecreasing in u property
of C. The inequality (7.7) tells us that the changes in cost minimizing input
demands induced by an increase in output cannot all be negative; i.e., not all
inputs can be inferior.
With the additional assumption that F be linearly homogeneous (and
there exists x > 0N such that F(x) > 0), we can deduce (cf. Section 2) that
C(u, p) = uc(p), where c(p) ≡ C(1, p). Thus, when F is linearly homogeneous,
(7.8) xi(u∗
, p∗
) = u∗
∂c(p∗
)/∂pi, i = 1, . . . , N,
and ∂xi(u∗
, p∗
)/∂u = ∂c(p∗
)/∂pi. Thus if x∗
i ≡ xi(u∗
, p∗
) > 0 for i =
1, 2, . . ., N, using (69) we can deduce the additional restrictions
(7.9)
∂xi(u∗
, p∗
)
∂u
u∗
x∗
i
=
u∗
∂c(p∗
)/∂pi
x∗
i
= 1
if F is linearly homogeneous; i.e., all of the input elasticities with respect to
output are unity.
152 Essays in Index Number Theory 6. Duality Approaches 153
For the general two input case, the general restrictions (7.3)–(7.7) enable
us to deduce the following restrictions on the partial derivatives of the
two input demand functions, x1(u∗
, p∗
1, p∗
2) and x2(u∗
, p∗
1, p∗
2) : ∂x1/∂p1 ≤ 0,
∂x2/∂p2 ≤ 0, ∂x1/∂p2 ≥ 0, ∂x2/∂p1 ≥ 0 (and if any one of the above inequalities
holds strictly, then they all do, since p∗
1∂x1/∂p1 = −p∗
2∂x1/∂p2 =
−p∗
2∂x2/∂p1 = (p∗
2)2
(p∗
1)−1
∂x2/∂p2) and p∗
1∂x1/∂u+p∗
2∂x2/∂u ≥ 0. Thus, the
signs of ∂x1/∂u and ∂x2/∂u are ambiguous, but if one is negative, then the
other must be positive. For the constant returns to scale two input case, the
ambiguity disappears: we have ∂x1(u∗
, p∗
)/∂u ≥ 0, ∂x2(u∗
, p∗
)/∂u ≥ 0 and at
least one of the inequalities must hold strictly if u∗
> F(02).
An advantage in deriving these well known comparative statics results
using duality theory is that the restrictions (7.2)–(7.7) are valid in cases where
the direct production function F is not even differentiable. For example, a
Leontief production function has a linear cost function C(u, p) = uaT
p, where
aT
≡ (a1, a2, . . . , aN ) > 0T
N is a vector of constants. It can be verified that the
restrictions (7.2)–(7.7) are valid for this nondifferentiable production function.
Historical Notes
Analogues to (7.3) and (7.4) in the context of profit functions were obtained by
Hotelling [1932; 594] [1935; 69–70]. Hicks [1946; 311 and 331] and Samuelson
[1947; 69] obtained all of the relations (7.2)–(7.6) and Samuelson [1947; 66] also
obtained (7.7). All of these authors assumed that the primal function F was
differentiable and their proofs used the first order conditions for the cost minimization
(or utility maximization) problem plus the properties of determinants
in order to prove their results.
Our proofs of (7.3)–(7.6), using only differentiability of the cost function
plus Lemma 3 in Section 1, are due to McKenzie [1956–57; 188–189] and Karlin
[1959; 273]. McFadden [1978a] also provides alternative proofs.
If F is only homothetic rather than being linearly homogeneous, then the
relations (7.9) are no longer true. If F is homothetic, then by (2.18), C(u, p) =
φ−1
(u)c(p) where φ−1
is a monotonically increasing function of one variable.
Thus, under our differentiability assumptions, xi(u∗
, p∗
) = φ−1
(u∗
)∂c(p∗
)/∂pi
and ∂xi(u∗
, p∗
)/∂u = [dφ−1
(u∗
)/du][∂c(p∗
)/∂pi], so that if x∗
i ≡ xi(u∗
, p∗
) > 0,
(7.10)
∂xi(u∗
, p∗
)
∂u
u∗
x∗
i
=
u∗
[dφ−1
(u∗
)]/du
φ−1(u∗)
≡ η(u∗
) ≥ 0 for i = 1, 2, . . . , N.
Thus, in the case of a homothetic production function, the input elasticities
with respect to output are all equal to the same nonnegative number independent
of the input prices, but dependent in general on the output level
u∗
. Furthermore, assuming homotheticity of F, we can solve the equation
C(u, p) = φ−1
(u)c(p) = y for u = φ[y/c(p)] = φ[1/c(p/y)] ≡ G(p/y), where
y > 0 is the producer’s allowable expenditure on inputs. If we replace u∗
by
φ(y∗
/c(p∗
)) in the system of input demand functions xi(u∗
, p∗
), we obtain the
system of “market” demand functions
xi φ[y∗
/c(p∗
)], p∗
= φ−1
[φ(y∗
/c(p∗
)]∂c(p∗
)/∂pi
= [y∗
/c(p∗
)]∂c(p∗
)/∂pi for i = 1, 2, . . ., N.
Thus if x∗
i ≡ xi(u∗
, p∗
) > 0,
(7.11)
∂xi
∂y
φ
y∗
c(p∗)
, p∗ y∗
x∗
i
= 1, i = 1, 2, . . ., N;
i.e., all inputs have unitary “income” (or expenditure) elasticity of demand if
the underlying aggregator function F is homothetic. Note the close resemblance
of (7.11) to (7.9). That homotheticity of F implies the relations (7.11) dates
back to Frisch [1936; 25] at least. For further references, see Chipman [1974a;
27].
8. The Slutsky Conditions for Consumer Demand Functions
Assume that a consumer has a utility function F(x) defined over x ≥ 0N which
is continuous from above. Then we have seen in Section 2 that C(u, p) ≡
minx{pT
x : F(x) ≥ u} is well defined for u ∈ textRange F and p 0N .
Moreover, the cost function C has a number of properties including nondecreasingness
in u for each p 0N and linear homogeneity and concavity in p
for each u ∈ textRange F.
Assume that the consumer faces prices p∗
0N and has income y∗
> 0
to spend on commodities. Then the consumer will wish to choose the largest
u such that his cost minimizing expenditure on the goods is less than or equal
to his available income. Thus, the consumer’s equilibrium utility level will be
u∗
defined by
u∗
≡ max
u
{u : C(u, p∗
) ≤ y∗
, u ∈ textRange F}.
Now assume that C is twice continuously differentiable with respect to
its arguments at the point (u∗
, p∗
) with
(8.1) ∂C(u∗
, p∗
)/∂u > 0.
The fact that C is nondecreasing in u implies that ∂C(u∗
, p∗
)/∂u ≥ 0; however,
the slightly stronger assumption (8.1) enables us to deduce that the consumer
154 Essays in Index Number Theory 6. Duality Approaches 155
will actually spend all of his income on purchasing (or renting) commodities;
i.e., (8.1) implies that
(8.2) C(u∗
, p∗
) = y∗
.
Furthermore, since C is linearly homogeneous in p, (8.2) implies
(8.3) C(u∗
, p∗
/y∗
) = 1.
Our differentiability assumptions plus (8.1) and (8.3) imply (using the
Implicit Function Theorem) that (8.3) can be solved for u as a function of p/y
in a neighborhood of p∗
/y∗
. The resulting function G(p/y) is the consumer’s
indirect utility function, which gives the maximum utility level the consumer
can attain, given that he faces commodity prices p and has income y to spend
on commodities. The Implicit Function Theorem also implies that G will be
twice continuously differentiable with respect to its arguments at p∗
/y∗
. Note
that
(8.4) u∗
= G(p∗
/y∗
).
The consumer’s system of Hicksian [1946; 331] or constant real income
demand functions38
f1(u, p), . . . , fN (u, p) is defined as the solution to the
expenditure minimization problem minx{pT
x : F(x) ≥ u}. Since we have
assumed that C is differentiable with respect to p at (u∗
, p∗
), by Lemma 3 in
Section 2,
(8.5) fi(u∗
, p∗
) = ∂C(u∗
, p∗
)/∂pi, i = 1, . . . , N;
i.e., the Hicksian demand functions can be obtained by differentiating the cost
function with respect to the commodity prices. On the other hand, the consumer’s
system of ordinary market demand functions, x1(y, p), . . . , xN (y, p),
can be obtained from the Hicksian system (8.5), if we replace u by G(p/y),
the maximum utility the consumer can obtain when he has income y and faces
prices p. Thus,
(8.6) xi(y∗
, p∗
) ≡ fi[G(p∗
/y∗
), p∗
], i = 1, . . . , N.
Thus, the consumer’s system of market demand functions can be obtained
from the cost function as well as by using the Ville–Roy Identity (4.12). Finally,
it can be seen that if we replace y in the consumer’s system of market demand
38
In the previous section, these functions are denoted as x1(u, p), . . . , xN (u, p).
functions by C(u, p), then we should obtain precisely the system of Hicksian
demand functions (8.5); i.e., we have
(8.7) xi[C(u∗
, p∗
), p∗
] = ∂C(u∗
, p∗
)/∂pi, i = 1, . . . , N.
Differentiating both sides of (8.7) yields:
∂2
C(u∗
, p∗
)
∂pi∂pj
=
∂xi(y∗
, p∗
)
∂pj
+
∂xi(y∗
, p∗
)
∂y
∂C(u∗
, p∗
)
∂pj
using (8.2)
=
∂xi(y∗
, p∗
)
∂pj
+ fj(u∗
, p∗
)
∂xi(y∗
, p∗
)
∂y
using (8.5)
=
∂xi(y∗
, p∗
)
∂pj
+ xj (y∗
, p∗
)
∂xi(y∗
, p∗
)
∂y
using (8.4) and (8.6)
≡ k∗
ij, i, j = 1, 2, . . ., N,(8.8)
where k∗
ij is known as the ijth Slutsky coefficient. Note that the N × N matrix
of these Slutsky coefficients, K∗
≡ [k∗
ij ], can be calculated from a knowledge of
the market demand functions, xi(y, p), and their first order derivatives at the
point (y∗
, p∗
). (8.8) shows that K∗
≡ 2
ppC(u∗
, p∗
) and thus (recall equations
(7.3), (7.4) and (7.6) of the previous section) K∗
satisfies the following Slutsky–
Samuelson–Hicks Conditions:
(i) K∗
= K∗T
.(8.9)
(ii) zT
K∗
z ≤ 0 for every z.
(iii) K∗
p∗
= 0N .
Historical Notes
Slutsky [1915] deduced (8.9)(i) and part of (8.9)(ii); i.e., that k∗
ii ≤ 0. Samuelson
[1938; 348] and Hicks [1946; 311] deduced the entire set of restrictions
(8.9) under the assumption that F was twice continuously differentiable at
an equilibrium point x∗
> 0N and F satisfied the additional property that
vT 2
xxF(x∗
)v < 0 for all v = 0N such that v = k F(x∗
) for any scalar k.
In fact, under these hypotheses, Samuelson and Hicks were able to deduce the
following strengthened version of (8.9)(ii): zT
K∗
z < 0 for every z = 0N such
that z = kp∗
for any scalar k.
Our proof of conditions (8.9) is due to McKenzie [1956–57] and Karlin
[1959; 267–273]. See also Arrow and Hahn [1971; 105]. This method of
proof again has the advantage that differentiability of F does not have to be
assumed; essentially, all that is required is differentiability of the demand functions.
Afriat [1972c] makes this point.
156 Essays in Index Number Theory 6. Duality Approaches 157
For a derivation of conditions (8.9) which utilizes only the properties of
the indirect utility function G, see Diewert [1977; 356].
For a “traditional” derivation of (8.9), see Intriligator [1981].
9. Consumption Theorems in Terms of Over and Under
Compensation Revisited
The task of this section is to cast some light on the following somewhat enigmatic
footnote in a paper by Samuelson:
This can be seen by writing utility as a function of the overcompensating
changes in prices or U = U(q1, . . . ) = U[F1
(p1, . . . , pn), . . . ]
= V (p1, . . . , pn) with [∂V (p1, . . . , pn)/∂pi] proportional to (qi − q0
i )
and vanishing at pi = p0
i . Hence, at p0
i , [∂(qi − q0
i )/∂pj] is proportional
to [∂2
V (p0
1, . . . , p0
n) / ∂pi∂pj], which is symmetric; this last
matrix is also negative semi-definite39
because the price ratios at
(p0
) give the lowest utility possible . . . .
To handle the case of undercompensation, note that around any
initial point, q0
i = Di
(p0
i , . . . , p0
n, I0
), we can solve the implicit set
of equations
qi = Di
(p1, . . . , pn, X)Σp0
i Di
(p1, . . . , pn, X) = Σp0
i q0
i
for qi = fi
(p1, . . . , pn) and X = X(p1, . . . , pn); then it can be shown
that for U = U(q1, . . . ) = U(f1
, . . . ) = W(p1, . . . , pn), [∂fi
(p0
1, . . . ,
p0
n) / ∂pj] = (∂2
W/∂pi∂pj) is symmetric, and negative semi-definite
by virtue of the fact that the price ratios at (p0
) maximize U or W.
It can be shown that taking a mean of overcompensated and undercompensated
changes — as e.g. 1
2 [f(p1, . . . , pn) + F(p1, . . . , pn)] —
gives a change that agrees locally around (p0
) with an indifference
change up to derivatives of still higher order: such a locus osculates
the indifference surface so as to have not only the same slope but
also the same curvature.40
Samuelson [1953; 8]
39
This is an obvious slip; Samuelson means positive rather than negative semi-
definiteness.
40
Samuelson and Swamy [1974; 582] add the following explanatory note on
the above footnote: “The truth of this finding, that the Ideal index gives a
second-order or osculating approximation to the true homothetic index, could
have been vaguely suspected from the finding in Samuelson [1953; p. 8, n. 1]
Suppose that the consumer’s utility function F is defined and continuous
from above for nonnegative commodity vectors x ≥ 0N . Then, as we have seen
in Section 2, the consumer’s cost or expenditure function C(u, p) ≡ minx{pT
x :
F(x) ≥ u} is well defined for all positive commodity price vectors p 0N and
all utility levels u ∈ U, where U is the smallest convex set containing the
range of F. Moreover, C satisfies properties 1–7 of Section 2. Suppose that
C is twice continuously differentiable in some neighborhood around the point
(u0
, p0
) where u0
∈ U and p0
0N , and in addition:
∂C(u0
, p0
)/∂u ≡ uC(u0
, p0
) > 0 and(9.1)
pC(u0
, p0
) ≡ x0
> 0N .(9.2)
By Lemma 3, the consumer’s system of constant utility (or Hicksian)
demand functions, x(u, p) ≡ [x1(u, p), . . . , xN (u, p)]T
, can be obtained by differentiating
C with respect to the commodity prices; i.e., for (u, p) close to
(u0
, p0
), we have x(u, p) = pC(u, p). Thus x0
≡ (x0
1, . . . , x0
N )T
in (9.2) can
be interpreted as the consumer’s initial demand vector.
Samuelson’s overcompensated indirect utility function, u = V (p), can be
defined as the solution to the following equation involving u and p:
(9.3) C(u, p) = pT
x0
.
Thus the consumer is given a new budget constraint indexed by the commodity
price vector p and given just enough income, y ≡ C(u, p), so that he can
purchase his initial consumption vector x0
at the new prices: this is the economic
interpretation of equation (9.3) which implicitly defines u = V (p). Our
differentiability assumptions on C plus assumption (9.1) are sufficient to imply
the existence of V (p) for p ∈ Bδ(p0
) where Bδ(p0
) ≡ {p : (p−p0
)T
(p−p0
) < δ2
}
is the open ball of radius δ > 0 around the point p0
. We choose δ > 0 small
enough so that Bδ(p0
) is a subset of the positive orthant and so that C(V (p), p)
is twice continuously differentiable with respect to the components of p with
uC(V (p), p) > 0 for p ∈ Bδ(p0
). This will imply that for p ∈ Bδ(p0
),41
(9.4) V (p) = max
u
{u : C(u, p) ≤ pT
x0
, u ∈ U}.
that the symmetric mean of overcompensated and undercompensated demand
functions provides a high-order osculating approximation to the Slutsky-Hicks
just-compensated demand along the indifference contours.” A symmetric mean
m(x, y) of two nonnegative numbers x and y is usually defined to be any
function which satisfies (i) m(x, y) = m(y, x), (ii) m(x, x) = x, and (iii)
Min{x, y} ≤ m(x, y) ≤ Max{x, y}.
41
In fact, (9.4) can be used to define V (p) for all p 0N .
158 Essays in Index Number Theory 6. Duality Approaches 159
The following inequality is valid for every p 0N :
(9.5) pT
x0
≥ min
x
{pT
x : F(x) ≥ u0
≡ F(x0
)} = C(u0
, p),
since x0
is feasible for the minimization problem in (9.5), but is not necessarily
optimal. Thus, since C is nondecreasing in u, u0
is feasible for the maximization
problem in (9.4), and thus, for every p 0N ,
V (p) ≥ u0
with V (P0
) = u0
. Thus V does in fact attain a global minimum with respect
to prices p when p = p0
.
The partial derivatives of V (p) for p ∈ Bδ(p0
) can be obtained by replacing
u in (9.3) by V (p) and differentiating the resulting equation. For p ∈ Bδ(p0
),
we find that
(9.6) pV (p) = x0
− pC[V (p), p] / uC[V (p), p]
so that when p = p0
, (using (9.2) as well):
(9.7) pV (p0
) = 0N .
Now differentiate the system of equations (9.6) with respect to p. When
p = p0
, using (9.7) we find that:
(9.8) 2
ppV (p0
) = − 2
ppC(u0
, p0
)/ uC(u0
, p0
).
Note that 2
ppV (p0
) is a positive semidefinite symmetric matrix, since
C(u0
, p) is concave and twice continuously differentiable with respect to p at
p = p0
, and thus 2
ppC(u0
, p0
) is a negative semidefinite symmetric matrix.
Now define the consumer’s system of overcompensated demand functions,
d(p) ≡ [d1(p), . . . , dN (p)]T
, for p ∈ Bδ(p0
), by replacing u in the consumer’s
system of Hicksian demand functions, x(u, p) ≡ pC(u, p), by u = V (p); i.e.,
d(p) ≡ pC(V (p), p). Now differentiate this last system of equations with
respect to p in order to form the N × N matrix of overcompensated demand
derivatives [∂di(p)/∂pj] ≡ pd(p). Using (9.7) when evaluating the derivatives
at p = p0
, we find that
(9.9) pd(p0
) = 2
ppC(u0
, p0
) = k0 2
ppV (p0
)
where the last equality follows from (9.8) with k0
≡ −1/ uC(u0
, p0
) < 0.
Thus, the matrix of derivatives of the overcompensated demand functions is
precisely equal to the matrix of derivatives of the Hicksian demand functions
(which in turn is equal to the matrix of Slutsky coefficients42
), when both
matrices are evaluated at p = p0
.
We turn now to the system of undercompensated demand functions. The
undercompensated indirect utility function u = W(p) is defined for p ∈ Bδ(p0
)
to be the solution to the following equation involving u and p (if the solution
exists):
(9.10) p0T
pC(u, p) = p0T
x0
.
An economic interpretation of W(p) can be obtained as follows: given a
price vector p ∈ Bδ(p0
) and a utility level u near u0
, calculate the consumer’s
Hicksian demand vector x(u, p) ≡ pC(u, p). Then choose u ≡ W(p) so that
the resulting demand vector x[W(p), p] will just be on the consumer’s original
budget constraint p0T
pC(u, p0
) = p0T
x0
.
When p = p0
, (9.10) becomes p0T
pC(u, p0
) = p0T
x0
, and this last
equation has the unique solution u = u0
. (Using p0T
pC(u, p0
) = C(u, p0
),
by Euler’s Theorem on homogeneous functions, p0T
x0
= p0T
pC(u0
, p0
) =
C(u0
, p0
), uC(u0
, p0
) > 0, and C(u, p0
) is nondecreasing in u). Thus since
∂[p0T
pC(u0
, p0
)] / ∂u = uC(u0
, p0
) > 0 by (9.1), our differentiability assumptions
on C plus the Implicit Function Theorem imply the existence of
u = W(p) satisfying (9.10) for p ∈ Bδ(p0
) for some δ > 0.
Since the maximum utility the consumer could attain in the original budget
constraint was u0
≡ F(x0
) = maxx{F(x) : p0T
x ≤ p0T
x0
, x ≥ 0N }, it is
easy to see that W(p) ≤ u0
= W(p0
) for all p ∈ Bδ(p0
). Thus maxp{W(p) : p ∈
Bδ(p0
)} = W(p0
) and thus W attains at least a local maximum at p = p0
.43
The partial derivatives of W(p) can be obtained by replacing u in (9.10) by
W(p) and differentiating the resulting equation with respect to p for p ∈ Bδ(p0
):
(9.11) p0T 2
ppC[W(p), p] + p0T 2
puC[W(p), p] T
p W(p) = 0T
N .
When p = p0
, W(p0
) = u0
, p0T 2
puC(u0
, p0
) = uC(u0
, p0
) > 0 and
p0T 2
ppC(u0
, p0
) = 0T
N , so that (9.11) yields
(9.12) pW(p0
) = 0N .
Now differentiate (9.11) with respect to the components of p and evaluate
the resulting system of equations when p = p0
. Using (9.12) and the identities
p0T 2
pu C(u0
, p0
) = uC(u0
, p0
) and p p0T 2
ppC[W(p), p] = − 2
pp
C(u0
, p0
), when p = p0
,44
we find that
(9.13) 2
ppW(p0
) = 2
ppC(u0
, p0
)/ uC(u0
, p0
) = − 2
ppV (p)
42
Recall equation (8.8) in Section 8.
43
W will not in general be defined for all p 0N whereas V will be.
44
Since for all p ∈ Bδ(p0
), pT 2
ppC[W(p), p] = 0T
N , then p pT 2
ppC[W(p), p]
= 0N×N = 2
ppC[W(p), p] + p p0T 2
ppC[W(p), p] where p0
≡ p is treated as
a constant vector when differentiating the last term.
160 Essays in Index Number Theory 6. Duality Approaches 161
where the last equality follows from (9.8). Thus the Hessian matrix of W evaluated
at p0
is a negative semidefinite symmetric matrix which is proportional
to the matrix of Slutsky coefficients 2
ppC(u0
, p0
) and is equal to minus the
Hessian matrix of V evaluated at p0
.
Define the consumer’s system of undercompensated demand functions,
D(p) ≡ [D1(p), . . . , DN (p)]T
for p ∈ Bδ(p0
), by replacing u in the consumer’s
system of Hicksian demand functions, x(u, p) ≡ pC(u, p), by u = W(p); i.e.
D(p) ≡ pC[W(p), p]. Now differentiate this last system of equations with
respect to p in order to form the N × N matrix of undercompensated demand
derivatives [∂Di(p)/∂pj] ≡ pD(p). Using (9.12), at p = p0
(9.14) pD(p0
) = 2
ppC(u0
, p0
).
The above results establish counterparts to the Samuelson results using
duality theory. We have obtained a strengthening of Samuelson’s results in the
sense that it is not necessary to take a symmetric mean of the over and under
compensated demand systems: both systems when differentiated with respect
to p yield precisely the consumer’s matrix of Hicksian demand derivatives when
p = p0
.
10. Empirical Applications using Cost or Indirect Utility Functions
Suppose that the technology of an industry can be characterized by a constant
returns to scale production function f which has the following properties:45
(10.1)
f is a (i) positive, (ii) linearly homogeneous, and (iii) concave
function defined over the positive orthant in RN
.
It can be shown46
that the cost function which corresponds to f has the
following form: for u ≥ 0, p 0N ,
C(u, p) ≡ min
x
{pT
x : f(x) ≥ u, x ≥ 0N }(10.2)
= uc(p)
where c(p) ≡ C(1, p) is the unit cost function and it also satisfies the three
properties listed in (10.1).
45
f can be uniquely extended to the nonnegative orthant by using the Fenchel
closure operation.
46
See Samuelson [1953–54] and Diewert [1974a; 110–112].
The producer’s system of input demand functions, x(u, p) ≡ [x1(u, p),
. . . , xN (u, p)]T
, can be obtained as the set of solutions to the programming
problem (10.2) if we are given a functional form for the production function f.
Thus, one method for obtaining a system of derived input demand functions
that are consistent with the hypothesis of cost minimization is to postulate
a (differentiable) functional form for f and then use the usual Lagrangian
techniques in order to solve (10.2).
The problem with this first method for obtaining the system of input demand
functions x(u, p) is that it is usually very difficult to obtain an algebraic
expression for x(u, p) in terms of the (unknown) parameters which characterize
the production function f, particularly if we assume that f is a flexible47
linearly homogeneous functional form.
A second method for obtaining a system of input demand functions x(u, p)
makes use of Lemma 4 (Shephard’s Lemma): simply postulate a functional form
for the cost function C(u, p) which satisfies the appropriate regularity conditions
and, in addition, is differentiable with respect to input prices. Then,
x(u, p) = pC(u, p) and the system of derived demand functions can be obtained
by differentiating the cost function with respect to input prices.
For example, suppose that the unit cost function is defined by
c(p) ≡
N
i=1
N
j=1
bijp
1
2
i p
1
2
j with bij = bji ≥ 0 for all i, j.
Then, if at least one bij > 0, the resulting function c satisfies (10.1), and the
input demand functions are
(10.3) xi(u, p) =
N
j=1
bij(pj/pi)
1
2 u; i = 1, 2, . . ., N.
Note that the system of input demand equations (10.3) is linear in the
unknown parameters, and thus linear regression techniques can be used in order
47
f is a flexible functional form if it can provide a second order (differential)
approximation to an arbitrary twice continuously differentiable function f∗
at
a point x∗
. f differentially approximates f∗
at x∗
iff (i) f(x∗
) = f∗
(x∗
), (ii)
f(x∗
) = f∗
(x∗
), and (iii) 2
f(x∗
) = 2
f∗
(x∗
), where both f and f∗
are
assumed to be twice continuously differentiable at x∗
(and thus the two Hessian
matrices in (iii) will be symmetric). Thus a general flexible functional form f
must have at least 1 + N + N(N + 1)/2 free parameters. If f and f∗
are
both linearly homogeneous, then f∗
(x∗
) = x∗T
f∗
(x∗
) and 2
f∗
(x∗
)x∗
= 0N ,
and thus a flexible linearly homogeneous functional form f need have only
N + N(N − 1)/2 = N(N + 1)/2 free parameters. The term “flexible” is due to
Diewert [1974a; 113] while the term “differential approximation” is due to Lau
[1974; 183].
162 Essays in Index Number Theory 6. Duality Approaches 163
to estimate the bij, if we are given data on output, inputs and input prices.
Note also that bij in the ith input demand equation should equal bji in the jth
equation for j = i. These are the Hotelling [1932; 594], Hicks [1946; 311 and
331], Samuelson [1947; 64] symmetry restrictions (7.3) and we can statistically
test for their validity. If some of the bij are negative, then the system of input
demand equations can still be locally valid.48
Finally, note that if bij = 0 for
i = j, then (10.3) becomes xi(u, p) = biiu, i = 1, 2, . . . , N, which is the system
of input demand functions that corresponds to the Leontief [1941] production
function, f(x1, x2, . . . , xN ) ≡ min{xi/bii : i = 1, 2, . . ., N}. In the general case,
the production function which corresponds to (10.3) is called the generalized
Leontief production function.49
It can also be shown that the corresponding
unit cost function, i j bijp
1
2
i p
1
2
j , is a flexible linearly homogeneous functional
form.50
As another example of the second method for obtaining input demand
functions, consider the following translog cost function:
ln C(u, p) ≡ α0 +
N
i=1
αi ln pi +
1
2
N
i=1
N
j=1
γij ln pi ln pj
(10.4)
+ δ0 ln u +
N
i=1
δi ln pi ln u +
1
2
ε0(ln u)2
where the parameters satisfy the following restrictions:
N
i=1
αi = 1; γij = γji for all i, j;
N
j=1
γij = 0 for i = 1, 2, . . . , N; and
N
i=1
δi = 0.(10.5)
The restrictions (10.5) ensure that C defined by (10.4) is linearly homogeneous
in p. The additional restrictions
(10.6) δ0 = 1; δi = 0, for i = 1, 2, . . ., N; and ε0 = 0
ensure that C(u, p) = uC(1, p) so that the corresponding production function
is linearly homogeneous. Finally, with the additional restrictions γij = 0 for
all i, j and αi ≥ 0 for i = 1, 2, . . ., N, C defined by (10.4) reduces to a CobbDouglas
cost function.
The “translog” functional form defined by (10.4) is due to Christensen,
Jorgenson and Lau [1971], Griliches and Ringstad [1971] (for two inputs) and
48
See Blackorby and Diewert [1979] and Diewert [1974a; 113-114].
49
See Diewert [1971a].
50
See Diewert [1974a; 115].
Sargan [1971; 154–146] (who calls it the log quadratic production function). In
general, C defined by (10.4) will not satisfy the appropriate regularity conditions
(e.g., conditions II in Section 3) globally, but it can provide a good local
approximation to an arbitrary twice differentiable, linearly homogeneous in p,
cost function;51
i.e., the translog function form (10.4) is flexible.
The cost minimizing input demand functions xi(u, p) which (10.4) generates
via Shephard’s Lemma are not linear in the unknown parameters. However,
it is easy to verify that the factor share functions
si(u, p) ≡ pixi(u, p)/
N
k=1
pkxk(u, p) = pixi(u, p)/C(u, p) = ∂ ln C(u, p)/∂ ln pi
are linear in the unknown parameters:
(10.7) si(u, p) = αi +
N
j=1
γij ln pj + δi ln u, i = 1, . . . , N.
However, since the shares sum to unity, only N − 1 of the N equations
defined by (10.7) can be statistically independent. Moreover, notice that the
parameters α0, δ0 and ε0 do not appear in (10.7). However, all of the parameters
can be statistically determined given data on output, inputs and input
prices if we append equation (10.4) (which is also linear in the unknown parameters)
to N − 1 of the N equations in (10.7).
The above two examples illustrate how simple it is to use the second
method for generating systems of input demand functions which are consistent
with the hypothesis of cost minimization.
Just as Shephard’s Lemma (3.13) can be used to derive systems of cost
minimizing input demand functions, Roy’s Identity (4.12) can be used to derive
systems of utility maximizing commodity demand functions in the context of
consumer theory. For example, consider the following translog indirect utility
function:52
for v ≡ p/y 0N , define
(10.8) G(v) ≡ α0 +
N
i=1
αi ln vi +
1
2
N
i=1
N
j=1
γij ln vi ln vj ; γij = γji.
Roy’s Identity (4.12) applied to G defined by (10.8) yields the following
system of consumer demand functions where v ≡ (v1, . . . , vN )T
= pT
/y,
51
See Lau [1974; 186].
52
See Jorgenson and Lau [1970] and Christensen, Jorgenson and Lau [1975].
This translog function can locally approximate any twice continuously differentiable
indirect utility function. However, G(v) defined by (10.8) will, in general,
not satisfy conditions III globally.
164 Essays in Index Number Theory 6. Duality Approaches 165
pT
≡ (p1, . . . , pN ) is a vector of positive commodity prices, and y > 0 is the
consumer’s expenditure on the N goods:
(10.9) xi(p/y) =
p−1
i y αi +
N
j=1γij ln pj −
N
j=1γij ln y
N
k=1αk +
N
k=1
N
m=1 γkm ln pm −
N
k=1
N
m=1 γkm ln y
,
i = 1, 2, . . . , N.
Note that the demand functions are homogeneous of degree 0 in all of
the parameters taken together. Thus, in order to identify the parameters, a
normalization such as
(10.10)
N
i=1
αi = −1
must be appended to equations (10.9). Note also that the parameter α0 which
occurs in (10.8) cannot be identified if we have data only on consumer purchases
(rentals in the case of durable goods) x, prices p, and total expenditure
y. Moreover, only N − 1 of the N equations in (10.9) are independent
and equation (10.8) cannot be added to the independent equations in (10.9) to
give N independent estimating equations because the left hand side of (10.8)
is the unobservable variable, utility u. Thus the econometric procedures used
to estimate consumer preferences are not entirely analogous to the procedures
used to estimate production functions, even though from a theoretical point of
view, the duality between cost and production functions is entirely isomorphic
to the duality between expenditure and utility functions.
The system of commodity demand functions defined by (10.9) is not linear
in the unknown parameters and thus nonlinear regression techniques will have
to be used in order to estimate econometrically the unknown parameters. We
generally obtain nonlinear demand equations using Roy’s Identity if we assume
that G is defined by a flexible functional form.53
The system of demand equations defined by (10.9) could be utilized given
microeconomic data on a single utility maximizing consumer (with constant
53
However, if we assume that the direct utility function F is linearly homogeneous,
then the corresponding indirect utility function G will be homogeneous
of degree −1. An indirect utility function G which is flexible homogeneous
of degree −1 can be obtained by using the translog functional form (10.8)
with the additional restrictions α0 = 0,
N
i=1 αi = −1 and
N
j=1γij = 0 for
i = 1, . . . , N. In this case, the consumer’s system of commodity share equations
becomes si ≡ pixi/y = −αi −
N
j=1γij ln pj, i = 1, . . . , N, which is linear in
the unknown parameters. However, the assumption of linear homogeneity (or
even homotheticity) for F is highly implausible in the consumer context, since
it leads to unitary income elasticities for all goods (cf. Frisch [1936; 25]).
preferences) or given cross section data on a number of consumers, assuming
that each utility maximizing consumer in the sample had the same preferences.
However, could we legitimately apply the system (10.9) to market data; i.e.,
assume that xi represented total market demand for commodity i divided by
the number of independent consuming units, pi is the price of commodity i, and
y is total market expenditure on all goods divided by the number of consumer
units? The answer is no in general.54
However, if we have information on the
distribution φ(y) of expenditure y by the different households in the market
and we are willing to assume that each household has the same tastes, then the
market demand functions Xi can be obtained by integrating over the individual
demand functions xi(p/y):
(10.11) Xi(p) = N
∞
0
xi(p/y)φ(y)dy, i = 1, . . . , N,
where N is the number of households in the market and
∞
0
φ(y)dy = 1.
The integrations in (10.11) can readily be performed using the xi(p/y) defined
by (10.9) if we impose the following normalizations on the parameters
of the translog indirect utility function defined by (10.8): (i) α0 = 0; (ii)
N
i=1 αi = −1; and (iii) N
i=1
N
j=1γij = 0. The effect of these three normalizations
is to make G homogeneous of degree −1 along the ray of equal prices;
i.e., G(λ1N ) = λ−1
G(1N ) for all λ > 0, and this in turn is simply a harmless
(from a theoretical point of view but not necessarily from an econometric point
of view) cardinalization of utility so that utility is proportional to income when
the prices the consumer faces are all equal. This approach for obtaining systems
of market demand functions consistent with microeconomic theory has
been pursued by Diewert [1974a; 127–130] and Berndt, Darrough and Diewert
[1977].
There is a simpler method for obtaining systems of market demand functions
consistent with individual utility maximizing behavior which is due to
Gorman [1953]: assume that each household’s preferences can be represented
by a cost function of the form
(10.12) C(u, p) = b(p) + uc(p)
where b and c are unit cost functions which satisfy conditions (10.1), p
0N and c(p)u ≥ y − b(p) ≥ 0 where y is household expenditure. Blackorby,
Boyce and Russell [1978] call a functional form for C which has the structure
(10.12) a Gorman polar form. If y − b(p) ≥ 0, the indirect utility function
54
The reader is referred to the considerable body of literature on the implications
of microeconomic theory for systems of market (excess) demand functions,
which is reviewed by Shafer and Sonnenschein [1980] and Diewert [1976c].
166 Essays in Index Number Theory 6. Duality Approaches 167
which corresponds to (10.12), is G(v) ≡ [1/c(v)] − [b(v)/c(v)] = [y/c(p)] −
[b(p)/c(p)] where v ≡ p/y, then Roy’s Identity (4.12) yields the following system
of individual household demand functions if the unit cost functions b and c are
differentiable:
(10.13) x(p/y) = pb(p) + [c(p)]−1
[y − b(p)] pc(p); y ≥ b(p).
The interesting thing about the system of consumer demand functions
defined by (10.13) is that they are linear in the household’s income or expenditure
y. Thus, if every household in the market under consideration has the
same preferences which are dual to C defined by (10.12) and each household
has income y ≥ b(p), then the system of market demand functions X(p) defined
by (10.11) is independent of the distribution of income; in fact
(10.14) X(p)/N∗
= pb(p) + [c(p)]−1
[y∗
− b(p)] 0c(p)
where X(p)/N∗
is the per capita market demand vector and y∗
≡ yφ(y)dy
is average or per capital expenditure. Comparing (10.14) with (10.13), we see
that the per capita market demand system has the same functional form as the
individual demand vector for a single decision making unit. The advantage of
this approach over the previous approach is that it does not require information
on the distribution of expenditure: all that is required is information on
market expenditure by commodity, commodity (rental) prices, and the number
of consumers or households.55
Several flexible functional forms for cost functions have been estimated
empirically, using Shephard’s Lemma in order to derive systems of input demand
functions: see Parks [1971], Denny [1972][1974], Binswanger [1974], Hudson
and Jorgenson [1974], Woodland [1975], Berndt and Wood [1975], Burgess
[1974] [1975], and Khaled [1978]. Khaled also develops a very general class of
functional forms which contains most of the other commonly used functional
forms as special cases.
There are also many applications of the above theory to the problem of
estimating consumer preferences. For empirical examples, see Lau and Mitchell
[1970], Diewert [1974d], Christensen, Jorgenson and Lau [1975], Jorgenson and
Lau [1975], Boyce [1975], Boyce and Primont [1976], Christensen and Manser
[1977], Darrough [1977], Blackorby, Boyce and Russell [1978], Howe, Pollak and
Wales [1979], Donovan [1977] and Berndt, Darrough and Diewert [1977].56
55
For other generalizations of the Gorman polar form (10.12) which have useful
aggregation properties, see Gorman [1959; 476], Muellbauer [1975][1976] and
Lau [1977a][1977b]. Diewert [1978a] shows that functional forms of the type
(10.12) are flexible. See also Lau [1977c].
56
Lau [1978b] considers the problems of testing for or imposing the various
11. Profit Functions
Up to now, we have considered the case of a firm which produces only a single
output, using many inputs. However, in the real world most firms produce
a variety of outputs, so that it is now necessary to consider the problems of
modelling a multiple output, multiple input firm.
For econometric applications, it is convenient to introduce the concept of a
firm’s variable profit function Π(p, x): it simply denotes the maximum revenue
minus variable input expenditures that the firm can obtain given that it faces
prices p 0I for variable inputs and outputs and given that another vector of
inputs x ≥ 0J is held fixed. We denote the variable inputs and outputs by the I
dimensional vector u ≡ (u1, u2, . . . , uI ), the fixed inputs by the J dimensional
vector −x ≡ (−x1, . . . , −xJ ), and the set of all feasible combinations of inputs
and outputs is denoted by T, the firm’s production possibilities set. Outputs
are denoted by positive numbers and inputs are denoted by negative numbers,
so if ui > 0, then the ith variable good is an output produced by the firm.
Formally, we define Π for p 0I and −x ≤ 0J by
(11.1) Π(p, x) ≡ max
u
{pT
u : (u, −x) ∈ T}.
If T is a closed nonempty, convex cone in Euclidean I + J dimensional
space with the additional properties: (i) if (u, −x) ∈ T, then x ≥ 0J (the last
J goods are always inputs), (ii) if (u , −x ) ∈ T, u ≤ u and −x ≤ −x , then
(u , −x ) ∈ T (free disposal) and (iii) if (u, −x) ∈ T, then the components of u
are bounded from above (bounded outputs for bounded fixed inputs), then Π
has the following properties: (i) Π(p, x) is a nonnegative real valued function
defined for every p 0I and x ≥ 0J such that Π(p, x) ≤ pT
b(x) for every
p 0J ; (ii) for every x ≥ 0J , Π(p, x) is (positively) linearly homogeneous,
convex and continuous in p; and (iii) for every p 0I, Π(p, x) is (positively)
linearly homogeneous, concave, continuous and nondecreasing in x. Moreover,
it can be shown57
that T can be constructed using Π as follows:
(11.2) T = {(u, −x) : pT
u ≤ Π(p, x), for every p 0I; x ≥ 0J }.
Thus, there is a duality between production possibilities sets T and variable
profit functions Π satisfying the above regularity conditions. Moreover,
in a manner which is analogous to the proof of Shephard’s Lemma (3.13) and
Roy’s Identity (4.12), the following result can be proven:
monotonicity and curvature conditions on the cost of indirect utility functions.
On the issue of how flexible are flexible functional forms, see Wales [1977] and
Byron [1977].
57
See Gorman [1968b], McFadden [1966], or Diewert [1973a].
168 Essays in Index Number Theory 6. Duality Approaches 169
Hotelling’s Lemma. [1932; 594]: If a variable profit function Π satisfies the
regularity conditions below (11.1) and is in addition differentiable with respect
to the variable quantity prices at p∗
0I and x∗
≥ 0J , then ∂Π(p∗
, x∗
)/∂pi =
ui(p∗
, x∗
) for i = 1, 2, . . ., I, where ui(p∗
, x∗
) is the profit maximizing amount
of net output i (of input i if ∂Π(p∗
, x∗
)/∂pi < 0) given that the firm faces the
vector of variable prices p∗
and has the vector x∗
of fixed inputs at its disposal.
Hotelling’s Lemma can be used in order to derive systems of variable output
supply and input demand functions. We need only postulate a functional
form for Π(p, x) which is consistent with the appropriate regularity conditions
for Π and is differentiable with respect to the components of p. For example,
consider the translog variable profit function Π defined as:
ln Π(p, x) ≡ α0 +
I
i=1
αi ln pi +
1
2
I
i=1
I
h=1
γih ln pi ln ph
+
I
i=1
J
j=1
δij ln pi ln xj +
J
j=1
βj ln xj
+
1
2
J
j=1
J
k=1
φjk ln xj ln xk(11.3)
where γih = γhi and φjk = φkj . It is easy to see that Π defined by (11.3) is
homogeneous of degree one in p if and only if
(11.4)
I
i=1
αi = 1;
I
i=1
δij = 0 for j = 1, . . . , J;
I
h=1
γih = 0 for i = 1, . . . , I.
Similarly, Π(p, x) is homogeneous of degree one in x58
if and only if
(11.5)
J
j=1
βj = 1;
J
j=1
δij = 0 for i = 1, . . . , I;
J
k=1
φjk = 0 for j = 1, . . . , J.
If Π(p, x) > 0, define the ith variable net supply share by si(p, x) ≡
piui(p, x)/Π(p, x). Hotelling’s Lemma applied to the translog variable profit
58
If we drop the assumption that the production possibilities set T be a cone (so
that we no longer assume constant returns to scale in all inputs and outputs),
then Π(p, x) does not have to be homogeneous of degree one in x. Thus the
restrictions (11.5) can be used to test whether T is a cone or not. If we drop
the assumption that T be convex, then Π(p, x) need not be concave (or even
quasiconcave) in x.
function defined by (11.3) yields the following system of net supply share func-
tions:
(11.6) si(p, x) = αi +
I
h=1
γih ln ph +
J
j=1
δij ln xj; i = 1, . . . , I.
Since the shares sum to unity, only I − 1 of the equations (11.6) are
independent. I −1 of equations (11.6) plus equation (11.3) can be used in order
to estimate the parameters of the translog variable profit function. Note that
these equations are linear in the unknown parameters as are the restrictions
(11.4) and (11.5) so that modifications of linear regression techniques can be
used.
Alternative functional forms for variable profit functions have been suggested
by McFadden [1978b], Diewert [1973a] and Lau [1974]. Empirical applications
have been made by Kohli [1978], Woodland [1977c], Harris and Appelbaum
[1977], and Epstein [1977].
A concept which is closely related to the variable profit function notion,
is the concept of a joint cost function, C(u, w) ≡ minx{wT
x : (u, −x) ∈ T},
where T is the firm’s production possibilities set as before, and w 0J is
a vector of positive input prices. As usual, if C(u, w) is differentiable with
respect to input prices w (and satisfies the appropriate regularity conditions),
then Shephard’s Lemma can be used in order to derive the producer’s system
of cost minimizing input demand functions x(u, w); i.e., we have
(11.7) x(u, w) = wC(u, w).
Joint cost functions have been empirically estimated by Burgess [1976a]
(who utilized a functional form suggested by Hall [1973]), Brown, Caves and
Christensen [1979], and Christensen and Greene [1976] (who utilized a translog
functional form for C(u,w) analogous to the translog variable profit function
defined by (11.3)).
Historical Notes
Samuelson [1953–54; 20] introduced the concept of the variable profit func-
tion59
and stated some of its properties. Gorman [1968b] and McFadden [1966]
[1978a] established duality theorems between a production possibilities set
satisfying various regularity conditions and the corresponding variable profit
function.60
Alternative duality theorems are due to Shephard [1970], Diewert
59
Samuelson called it the national product function.
60
Gorman uses the term “gross profit function” and McFadden uses the term
“restricted profit function” to describe Π(p, x).
170 Essays in Index Number Theory 6. Duality Approaches 171
[1973a] [1974b], Sakai [1974] and Lau [1976]. For the special case of a single
fixed input, see Shephard [1970; 248–250] or Diewert [1974b].
McFadden [1966] [1978a] introduced the joint cost function, stated its
properties, and proved formal duality theorems between it and the firm’s production
possibilities set T, as did Shephard [1970] and Sakai [1974].
There are also very simple duality theorems between production possibilities
sets and transformation functions, which give the maximum amount of one
output that the firm can produce (or the minimum amount of input required)
given fixed amounts of the remaining inputs and outputs. For examples of
these theorems, see Diewert [1973a], Jorgenson and Lau [1974a] [1974b], and
Lau [1976].
As usual, Hotelling’s Lemma can be generalized to cover the case of a
nondifferentiable variable profit function: the gradient of Π with respect to p
is replaced by the set of subgradients. This generalization was first noticed by
Gorman [1968b; 150–151] and McFadden [1966; 11] and repeated by Diewert
[1973a; 313] and Lau [1976; 142].
If Π(p, x) is differentiable with respect to the components of the vector
of fixed inputs, then wj ≡ ∂Π(p, x)/∂xj can be interpreted as the worth to the
firm of a marginal unit of the jth fixed input; i.e., it is the “shadow price” for
the jth input (cf. Lau [1976; 142]). Moreover, if the firm faces the vector of
rental prices w 0J for the “fixed” inputs, and during some period the “fixed”
inputs can be varied, then if the firm minimizes the cost of producing a given
amount of variable profits we will have (cf. Diewert [1974a; 140])
(11.8) w = xΠ(p, x)
and these relations can also be used in econometric applications.
The translog variable profit was independently suggested by Russell and
Boyce [1974] and Diewert [1974a; 139]. Of course, it is a straightforward modification
of the translog functional form due to Christensen, Jorgenson and Lau
[1971], and Sargan [1971].
The comparative statics properties of Π(p, x) or C(u, w) have been developed
by Samuelson [1953–54], McFadden [1966] [1978a], Diewert [1974a;
142–146], and Sakai [1974].
In international trade theory, it is common to assume the existence of sectoral
production functions, fixed domestic resources x, and fixed prices of internationally
traded goods p. If we now attempt to maximize the net value of internationally
traded goods produced by the economy, we obtain the economy’s
variable profit function, Π(p, x), or Samuelson’s [1953–54] national product
function. If the sectoral production functions are subject to constant returns
to scale, Π(p, x) will have all the usual properties mentioned above. However,
the existence of sectoral technologies will imply additional comparative statics
restrictions on the national product function π: see Chipman [1966], [1972],
[1974b], Samuelson [1966], Ethier [1974], Woodland [1977a][1977b], Diewert
and Woodland [1977], and Jones and Scheinkman [1977] and the many references
included in these papers.
Finally, note that the properties of Π(p, x) with respect to x are precisely
the properties that a neoclassical production function possesses. If x is a vector
of primary inputs, then Π(p, x) can be interpreted as a value added function. If
the prices p vary (approximately) in proportion over time, then Π(p, x) can be
deflated by the common price trend and the resulting real value added function
has all of the properties of a neoclassical production function; see Khang [1971],
Bruno [1978] and Diewert [1978a][1980].
12. Duality and Noncompetitive Approaches to Microeconomic
Theory
Up to now, we have assumed that producers and consumers take prices as given
and optimize with respect to the quantity variables they control. We indicate
in this section how duality theory can be utilized even if there is monopsonistic
or monopolistic behavior on the part of consumers or producers. We will not
attempt to be comprehensive but will illustrate the techniques involved by
means of our four approaches to modeling nonprice taking behavior.
Approach 1: The Monopoly Problem
Suppose that a monopolist produces output x0 by means of the production
function F(x), where x ≥ 0N is a vector of variable inputs. Suppose, further,
that he faces the (inverse) demand function p0 = wD(x0); i.e., p0 ≥ 0 is
the price at which he can sell x0 > 0 units of output, D is a continuous
positive function of x0, and the variable w > 0 represents the influence on
demand of “other variables”. That is to say, if the monopolist is selling to
consumers, w might equal disposal income for the period under consideration;
if the monopolist is selling to producers, w might be a linearly homogeneous
function of the prices that those other producers face.61
Finally, suppose that
the monopolist behaves competitively on input markets, taking as given the
vector p 0N of input prices. The monopolist’s profit maximization problem
61
If nonquantity variables do not influence the inverse demand function that
the monopolist faces for the periods under consideration, then w can be set
equal to 1 in each period.
172 Essays in Index Number Theory 6. Duality Approaches 173
may be written as
max
p0,x0,x
{p0x0 − pT
x : x0 = F(x), p0 = wD(x0), x ≥ 0N }
(12.1)
= max
x
{wD[F(x)]F(x) − pT
x : x ≥ 0N }
= max
x
{wF∗
(x) − pT
x : x ≥ 0N }
≡ Π∗
(w, p)
where F∗
(x) ≡ D[F(x)]F(x) = p0x0/w is the deflated (by w) revenue function
or pseudo production function and Π∗
is the corresponding pseudo profit function
(recall Section 11) which corresponds to F∗
.62
Notice that w plays the
role of a price of F∗
(x). If F∗
is a concave function, then Π∗
(1, p/w) will be the
conjugate function to F∗
(recall the Samuelson [1960], Lau [1969][1978a], and
Jorgenson and Lau [1974a][1974b] conjugacy approach to duality theory) and
Π∗
will be dual to F∗
(i.e., F∗
can be recovered from Π∗
). Even if F∗
is not
concave, if the maximum in (12.1) exists over the relevant range of (w, p) prices,
then Π∗
can be used to represent the relevant part of F∗
(i.e., the free disposal
convex hull of F∗
can be recovered from Π∗
). Moreover if Π∗
is differentiable
at (w∗
, p∗
) and w∗
0, p∗
0, x∗
solve (12.1), then Hotelling’s Lemma implies
(12.2) u∗
0 ≡ p∗
0x∗
0/w∗
= wΠ∗
(w∗
, p∗
) and − x∗
= pΠ∗
(w∗
, p∗
).
Moreover, if Π∗
is twice continuously differentiable at (w∗
, p∗
), then we can
deduce the usual comparative statics results on the derivatives of the deflated
sales function u0(w∗
, p∗
) ≡ wΠ∗
(w∗
, p∗
) and the input demand functions
−x(w∗
, p∗
) ≡ pΠ∗
(w∗
, p∗
): namely 2
Π∗
(w∗
, p∗
) is a positive semidefinite
symmetric matrix and (w∗
, p∗T
) 2
Π∗
(w∗
, p∗
) = 0T
N+1.
Equations (12.2) can be used in order to estimate econometrically the
parameters of Π∗
and hence indirectly of F∗
: simply postulate a functional form
for Π∗
, differentiate Π∗
, and then fit (12.2), given a time series of observations
on p0, p, w, x0 and x. The drawbacks to this method are: (i) we cannot
disentangle D from F; (ii) we cannot test whether the producer is in fact
behaving competitively on the output market; and (iii) we cannot use our
estimated equations to predict output x0 or selling price p0 separately.
Approach 2: The Monopsony Problem
Consider the problem of a consumer maximizing a utility function F(x)
satisfying conditions I but now we no longer assume that the consumer faces
62
Note that we have suppressed mention of any fixed inputs. We assume sufficient
regularity on F and D so that the maximum in (12.1) exists.
fixed prices for the commodities he purchases, but rather he is able to monopsonistically
exploit one or more of the suppliers that he faces. Then in period r,
he faces a nonlinear budget constraint of the form hr(x) = 0 where x ≥ 0N is
his vector of purchases (or rentals). Let xr
> 0N be a solution to the period r
constrained utility maximization problem, so that
(12.3) max
x
{F(x) : hr(x) = 0, x ≥ 0N } = F(xr
); r = 1, . . . , T.
Suppose, further, that the rth budget constraint function hr is differentiable
at xr
with xh(xr
) 0N for each r. Then we may linearize the rth budget
constraint around x = xr
by taking a first order Taylor series expansion.
The linearized rth budget constraint is hr(xr
) + [ xhr(xr
)]T
(x − xr
) = 0 or
[ hr(xr
)]T
(x − xr
) = 0 since hr(xr
) = 0 using (12.3). It is easy to see that the
utility surface {x : F(x) = F(xr
), x ≥ 0N } is tangent not only to the original
nonlinear budget surface {x : hr(x) = 0, x ≥ 0N } at x = xr
, but also to the
linearized budget constraint surface {x : [ hr(xr
)]T
(x − xr
) = 0, x ≥ 0N } at
x = xr
. Since we assume F is quasiconcave, the set {x : F(x) ≥ F(xr
), x ≥ 0N }
is convex and the linearized budget constraint is a supporting hyperplane to
this set; i.e.,
(12.4) max
x
{F(x) : prT
x ≤ prT
xr
, x ≥ 0N } = F(xr
), r = 1, . . . , T
where pr
≡ hr(xr
) for r = 1, 2, . . ., T. But now (12.4) is just a series of
aggregator maximization problems of the type we have studied in Section 4
(the rth vector of normalized prices is defined as vr
≡ pr
/prT
xr
) and the
estimation techniques outlined in Section 10 above (recall equations (10.9) for
example) can be used in order to estimate the parameters of the indirect utility
function dual to F.
When we were dealing with linear budget constraints in Section 4, it
was irrelevant whether F was quasiconcave or not (recall our discussion and
diagram in Section 2). However, now we require the additional assumption that
F be quasiconcave in order to rigorously justify the replacement of (12.3) by
(12.4). Note also that in order to implement the above procedure, it is necessary
to know the vector of derivatives xhr(xr
) for each r; i.e., we have to know
the derivatives of the supply functions that the consumer is “exploiting” each
period — information which was not required in approach 1.
The monopsony model presented here is actually much broader than the
classical model of monopsonistic exploitation: prices that the consumer faces
can vary with the quantity purchased for a large number of reasons, including
search and transactions costs, quantity discounts, and the existence of progressive
taxes on labor earnings. Most tax systems lead to budget constraints
with “kinks” or nondifferentiable points. This does not cause any problems
174 Essays in Index Number Theory 6. Duality Approaches 175
with the above procedure unless the consumer’s observed consumption-leisure
choice falls precisely on a kink in his budget constraint.63
Approach 3: The Monopoly Problem Revisited
Consider again the monopoly problem outlined above. Suppose xr
0 > 0,
xr
> 0N is a solution to the period r monopoly profit maximization problem
which can be rewritten as
(12.5) max
x0,x
{wr
D(x0)x0 − prT
x : x0 = F(x), x ≥ 0N } = wr
D(xr
0)xr
0 − prT
xr
;
r = 1, 2, . . ., T,
where pr
0 ≡ wr
D(xr
0) > 0 is the observed selling price of the output during
period r, wr
D(x0) is the period r inverse demand function, and pr
0N is
the period r input price vector. If the production function F is continuous and
concave (so that the production possibility set {(x0, x) : x0 ≤ F(x), x ≥ 0N } is
closed and convex) and if the inverse demand function D is differentiable at xr
0
for r = 1, . . . , T, then the objective function for the rth maximization problem
in (12.5) can be linearized around (xr
0, xr
) and this linearized objective function
will be tangent to the production surface x0 = F(x) at (xr
0, xr
). Thus,
(12.6) max
x0,x
{˜pr
0x0 − prT
x : x0 = F(x), x ≥ 0N } ≡ Π(˜pr
0, pr
) = ˜pr
0xr
0 − prT
xr
,
r = 1, . . . , T,
where ˜pr
0 ≡ wr
D(xr
0) + wr
D (xr
0)xr
0 = pr
0 + wr
D (xr
0)xr
0 > 0 is the period r
shadow or marginal price of output (˜pr
0 < pr
0 if wr
> 0 and D (xr
0) < 0) and
Π is the firm’s true profit function which is dual to the production function F
(recall Π∗
defined in approach 1 was dual to the convex hull of D[F(x)]F(x) ≡
F∗
(x)). Thus, the true nonlinear monopolistic profit maximization problems
(12.6) which have the usual structure once the appropriate marginal output
prices ˜pr
0 have been calculated so that the usual econometric techniques can be
applied (recall equations (11.6) in Section 11).64
Comparing approach 3 with approach 1, it can be seen that approach 3
requires the extra assumption that the production function be concave (convex
technology) and requires additional information; i.e., a knowledge of the slope
of the demand curve the monopolist is exploiting is required for each period.
It is easy to see how this approach can be generalized to a multiproduct
firm which simultaneously exploits several output and input markets: all that
63
See Wales [1973] and Wales and Woodland [1976][1977][1979] for econometric
treatments of this last problem.
64
The notation has been changed and we are now holding fixed inputs fixed for
all r, so that we can suppress mention of these fixed inputs in (12.5).
is required is the assumption of a convex technology and a (local) knowledge of
the demand and supply curves that the firm is exploiting so that the appropriate
shadow prices can be calculated.
Of course, the above techniques can also be used in situations where
the firm is not behaving monopolistically or monopsonistically in an exploitive
sense, but merely faces prices for its outputs or inputs that depend on the
quantity sold or purchased for any number of reasons, including transactions
costs or quantity discounts.
Approach 4: The Monopoly Problem Once Again
Suppose now that the production function satisfies conditions I and, as
usual, we suppose that xr
0 > 0, xr
> 0N is the solution to the period r monopolistic
profit maximization problem (12.5), which we rewrite as
(12.7) max
x0
{wr
D(x0)x0 − C(x0, pr
) : x0 ≥ 0} = wr
D(xr
0)xr
0 − prT
xr
,
r = 1, . . . , T
where C is the cost function dual to F. If the inverse demand function D
is differentiable at xr
0 > 0 and ∂C(xr
0, pr
)/∂x0 exists, then the first order
conditions for the rth maximization problem in (12.7) yield the condition
wr
D(xr
0) + wr
D(xr
0)xr
0 − ∂C(xr
0, pr
)/∂x0 = 0 or, recalling that pr
0 ≡ wr
D(xr
0)
is the observed selling price of output in period r,
(12.8) pr
0 = −wr
D(xr
0)xr
0 + ∂C(xr
0, pr
)/∂x0, r = 1, . . . , T.
If the cost function C is differentiable with respect to input prices at
(xr
0, pr
) for each r, then Shephard’s Lemma implies the additional equations
(12.9) xr
= pC(xr
0, pr
), r = 1, . . . , T.
Suppose that the part of the inverse demand function that depends on
x0, D(x0), can be adequately approximated over the relevant x0 range by the
following function:
(12.10) D(x0) ≡ α − β ln x0
where α > 0, β ≥ 0 are constants. Substitution of (12.10) into (12.8) yields
the equations
(12.11) pr
0 = wr
β + ∂C(xr
0, pr
)/∂x0, r = 1, . . . , T.
Given the observable price and quantity decisions of the firm, pr
0, pr
, xr
0,
xr
and data on wr
(we can assume wr
≡ 1 if this is appropriate), the system
176 Essays in Index Number Theory 6. Duality Approaches 177
of equations (12.9) and (12.11) can be jointly econometrically estimated once
we assume a differentiable functional form for the cost function C(x0, p). Note
that if β = 0 in equations (12.11), then the producer is behaving competitively,
selling output at a price pr
0 equal to marginal cost, ∂C(xr
0, pr
)/∂x0. Equations
(12.11) are also consistent with the producer behaving like a “naive” markup
monopolist (depending on what wr
is). Thus, we now have the basis for a
statistical test of market structure: (i) if β = 0, then the producer’s behavior is
consistent with competitive price taking behavior, (ii) if β > 0 and βwr
/pr
0 < 1
for r = 1, 2, . . ., T, then we have consistency with classical monopolistic behav-
ior,65
(iii) if β > 0 but βwr
/pr
0 ≥ 1 for some r, then we have consistency with
markup monopolistic behavior, and (iv) if β < 0, then we have inconsistency
with all three of the above types of behavior.66
This approach offers several advantages over the previous approaches:
(i) we can now statistically test for competitive behavior, (ii) informational
requirements are low — we do not require exogenous information on the elasticity
of demand (this information is endogenously generated), (iii) we do not
have to assume that the production function F is concave so that the model is
consistent with an increasing returns to scale production function, and finally,
(iv) the procedure is particularly simple — just insert the term βwr
into the
competitive equation, price equals marginal cost.
Historical Notes
Approach 1 is essentially due to Lau [1974; 193–194] [1978]67
but it has its
roots in Hotelling [1932; 609]. Approach 2 is in Diewert [1971b] but it has its
roots in the work of Frisch [1936; 14–15]. Approach 3 (which is isomorphic
to approach 2) is outlined in Diewert [1974a; 155]. Approach 4 is due to Appelbaum
[1975], who makes somewhat different assumptions on the functional
65
Using (12.10) and pr
0 ≡ wr
D(xr
0), we find that the first order conditions
(12.8) translate into wr
D(xr
0) 1 + [D(xr
0)xr
0/D(xr
0)] = pr
0[1 − (βwr
/pr
0)] =
∂C(xr
0, pr
)/∂x0 > 0 or [1 − (βwr
/pr
0)] = (pr
0)−1
∂C/∂x0 > 0 which implies
βwr
/pr
0 < 1. The second order necessary conditions for (12.7) require −β ≤
(xr
0/wr
)∂2
C(xr
0, pr
)/∂x2
0 which will be satisfied if β ≥ 0 and ∂2
C(xr
0, pr
)/∂x2
0 ≥
0 (nondecreasing marginal costs or nonincreasing returns to scale).
66
Of course, these tests are conditional on the assumed functional form for C,
the assumed functional form for the inverse demand function wf D(x0) where
D is defined by (12.10), and the assumption of price taking behavior on input
markets.
67
Lau uses a normalized profit function and does not assume that p0 = wD(x0),
but simply assumes that p0 = D(x0).
form of the inverse demand function.68
Appelbaum [1975][1979] also indicates
how his approach can be extended to several monopolistically supplied outputs
or monopsonistically demanded inputs and he presents an empirical example
based on the U.S. crude petroleum and natural gas industry. Another empirical
example of his technique based on Canada–U.S. trade is in Appelbaum
and Kohli [1979]. Approach 4 has also been applied by Schworm [1980] in the
context of investment theory where the price of investment goods purchased
by a firm depends on the quantity purchased.
13. Conclusion
We have attempted to give a fairly comprehensive treatment of the foundations
of the duality approach to microeconomic theory in Sections 2–6 of this chapter.
In Sections 7 and 8 we showed how duality theory could be used in order to
derive the usual comparative statics theorems for producer and consumer theory,
while in Section 9 some additional partial equilibrium comparative statics
theorems were derived. In Sections 10 and 11, we showed how duality theory
has been used as an aid in the econometric estimation of preferences and
technology. Finally, in Section 12, we indicated how duality theory could be
applied in certain noncompetitive situations.
The number of papers using duality theory during the last decade is so
large that, unfortunately, we are unable to review (or even reference) them. Additional
topics and references can be found in my earlier survey paper (Diewert
[1974a]) and the comments on it (Jacobsen [1974], Lau [1974] and Shephard
[1974]) as well as in Fuss and McFadden [1978] which provides a comprehensive
treatment of the duality approach to production theory.
We have mentioned the aggregation over consumers problem in Section 10
above but we have not mentioned the corresponding aggregation over producers
problem: for results and references to this literature, see Hotelling [1935; 67–
70], Gorman [1968b], Sato [1975] and Diewert [1980; Part III].
Although we have used duality theory to derive several partial equilibrium
comparative statics theorems, we have not mentioned the corresponding general
equilibrium literature: see Jones [1965][1972], Diewert [1974e][1974f][1978d],
Epstein [1974], Woodland [1974] and Burgess [1976b] for various applications.
The related literature on optimal taxation often makes use of duality theory:
for references to this literature, see Mirrlees [1981], and Deaton [1979].
Finally, some recent references that utilize duality theory in the context
of continuous time optimization problems are Lau [1974; 190–193], Appelbaum
[1975], Cooper and McLaren [1977], Epstein [1978] [1981b], McLaren
68
He models more explicitly the demand function that the producer is exploiting.
178 Essays in Index Number Theory 6. Duality Approaches 179
and Cooper [1980], and Schworm [1980].
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