A COMPLETE THEORY OF COMPARATIVE STATICS FOR

Metroeconomica 57:1 (2006)
31–67
A COMPLETE THEORY OF COMPARATIVE STATICS FOR
DIFFERENTIABLE OPTIMIZATION PROBLEMS
M. Hossein Partovi and Michael R. Caputo*
California State University and University of Central Florida
(February 2003; revised September 2004)
ABSTRACT
A new comparative statics formalism using generalized compensated derivatives is presented that, in
contrast to existing methodologies, directly yields constraint-free semidefiniteness results for any differentiable, constrained optimization problem. The formalism provides a natural and powerful method
of constructing comparative statics results, free of constraints and unrestricted in scope. New results
on envelope relations, invariance conditions, rank inequalities and non-uniqueness are derived that
greatly extend their utility and reach. The methodology is illustrated by deriving the comparative statics
of multiple linear constraint utility maximization models and the principal-agent problem with hidden
actions, both highly nontrivial and hitherto unsolved problems.
1. INTRODUCTION
With the publication of Samuelson’s (1947) dissertation, the method of
comparative statics became a standard part of the economist’s tool kit.
Samuelson (1947) essentially formulated the following strategy for
deriving comparative statics results in differentiable optimization problems:
* Earlier versions of this work (Partovi and Caputo, 1998) have been presented at North
Carolina State University, University of California, Berkeley, University of California, Davis,
University of Tennessee, Knoxville, University of Maryland, East Carolina University and
University of Central Florida, as well as at the North American Meeting of the Econometric
Society held at the California Institute of Technology, 25–29 June 1997, the 72nd Annual
Western Economic Association International Conference held in Seattle, Washington, 9–13 July
1997, and the Southeast Economic Theory Conference held at Georgetown University, 12–14
November 1999. We are grateful to several colleagues and an anonymous referee for helpful
comments on the organization and presentation of the contents of this paper. In particular,
we would like to mention Susan Athey, Martine Quinzii, Walter N. Thurman and Juaquim
Silvestre. M.H.P.’s work was in part supported by research awards from California State
University, Sacramento.
© Blackwell Publishing Ltd 2006, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main
Street, Malden, MA 02148, USA.
32
M. Hossein Partovi and Michael R. Caputo
(1) assume the second-order sufficient conditions hold at the optimal solution and apply the implicit function theorem to the first-order necessary conditions to characterize the optimal choice functions, (2) differentiate the
identity form of the first-order necessary conditions with respect to the
parameter of interest, and (3) solve the resulting linear system of comparative static equations. An important theorem of Samuelson’s analysis (1947,
p. 32, his equation (21)) states that every unconstrained differentiable optimization problem possesses a semidefinite matrix that embodies its comparative statics relations. The elements of this matrix are in general linear
combinations of partial derivatives of the decision variables with respect to
a parameter and obey the inequality and symmetry conditions that follow
from the semidefiniteness property. In special cases these conditions may
reduce to the slope of a decision variable with respect to a parameter, e.g.
the slope of the factor demand functions with respect to their prices in the
profit maximization model. However, the general case involves a linear combination of such single terms, a situation that is familiar from the utility maximization model where the said linear combinations are referred to as
compensated derivatives.
This primal approach pioneered by Samuelson is indicative of most comparative statics analyses in economics to this day, and provides the general
framework within which subsequent generalizations and refinements have
been carried out. As explained later, this is largely due to the fact that none
of these generalizations succeeded in deriving a result analogous to the
above-mentioned theorem of Samuelson for the constrained problem. In fact,
the existing literature does not really contemplate the general existence of
such an extension (Silberberg, 1990, pp. 213, 216), and in certain instances
flatly denies its existence (Blomquist, 1989).
The central result of our paper is the generalization of Samuelson’s (1947)
theorem to the class of constrained differentiable optimization problems. The
formalism for the generalization is natural and grounded in intuition, and
the comparative statics properties derived from it are sufficiently broad and
powerful as to render it an effective tool in the analysis of economic models
and in confronting them with empirical data. Indeed, only one conceptual
ingredient beyond Samuelson’s (1947) basic framework is needed in its
construction, namely the geometrical significance of generalized compensated
derivatives (GCDs) in formulating a constraint-free comparative statics
matrix (CSM) for constrained problems. This idea in turn leads to a method
for identifying a suitable class of GCDs and constructing the desired semidefinite matrix in a constraint-free form. Specifically, our principal theorem
demonstrates that linear combinations of partial derivatives of the decision
variables with respect to a parameter constitute the fundamental compara-
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
33
tive statics objects of the model, precisely as in Samuelson’s theorem for
unconstrained problems. In our case, these linear combinations involve
GCDs, and as such point to a rather remarkable fact: the compensated comparative statics structure already familiar from the basic utility maximization
model is typical of the most general case. For example, we find for the
principal-agent problem, treated in section 6, that a rise in the probability of
a given profit level resulting from a low-level effort by the agent causes a drop
in the compensated wages offered by the principal. Generally speaking,
uncompensated derivatives of the decision variables do not possess definite
signs in constrained optimization models, a fact that is already familiar from
the prototype utility maximization problem where only income-compensated
derivatives possess definite signs. Thus, it is entirely natural that appropriately defined GCDs should play a key role in a general theory of comparative statics. It is worth adding here that our method provides not just an
existence result, but also an explicit method for constructing suitable compensated derivatives, as well as the resulting semidefinite matrix in constraintfree form, for any differentiable optimization problem.
At this point it is appropriate to review, in some detail, the two main
contributions to the differentiable comparative statics literature since
Samuelson’s (1947) original work, partly with a view to the concepts and
methods presented in this paper. Some 27 years after its inception, Samuelson’s primal method was significantly advanced by Silberberg (1974). By
simultaneously considering the set of parameters and choice variables of the
original, or primal, problem as decision variables, Silberberg (1974) set up a
primal-dual problem that, upon optimization with respect to the choice variables, implies the primal optimality conditions. Optimization with respect
to the parameters, on the other hand, implies the envelope properties of
the original problem as first-order necessary conditions and the comparative statics of the primal problem as second-order necessary conditions.
Undoubtedly, the most significant result of this method was the construction
of a semidefinite matrix that contains the comparative statics relations of a
general constrained optimization problem. Its main shortcoming, on the
other hand, is the fact that the resulting semidefinite CSM is subject to constraints if the constraint functions depend on the parameters of interest in
the problem, the prototypical case in economics. That this is a severe limitation may be seen by applying the primal-dual method to the basic utility maximization problem and observing that the resulting semidefinite matrix bears
little resemblance to the familiar Slutsky matrix. Indeed, it is precisely the
imposition of the constraints—left unimplemented in Silberberg’s (1974)
method—that gives the Slutsky matrix its familiar form and economic
interpretation.
© Blackwell Publishing Ltd 2006
34
M. Hossein Partovi and Michael R. Caputo
A few years after Silberberg’s (1974) innovation, Hatta (1980) introduced
the gain method to deal with a class of optimization problems that are essentially a non-linear, multiple constraint generalization of the archetype utility
maximization problem. For unconstrained problems, Hatta’s (1980) method
is identical to Silberberg’s (1974), while for constrained problems, the gain
method succeeds in deriving constraint-free comparative statics results for
the above-mentioned class of problems. The procedure used by Hatta (1980)
amounts to applying a modified form of the compensation scheme used in
the standard utility maximization problem. While the flavor of Hatta’s (1980)
analysis is quite similar to that of Silberberg’s (1974), it does represents a significant advance over the latter in that it succeeds in overcoming the important shortcoming in the primal-dual method, at least for a restricted class of
constrained optimization problems. However, Hatta’s (1980) method has not
spurred further progress in the subject, nor has it gained wide acceptance
by workers in the field, primarily due to its restricted scope and lack of a
compelling conceptual basis.
A derivation of the comparative statics results of Silberberg (1974) and
Hatta (1980) vis-à-vis the main theorem of this paper is given in section 5.
This derivation clarifies the scope and limitations of those methods. Moreover, it demonstrates that the results of our paper subsume all previous
comparative statics schemes for dealing with differentiable optimization
problems.
Before closing this section, it is appropriate to review the use of compensated derivatives in differentiable comparative statics analysis. The interest in
compensated comparative statics properties of economic models has its
genesis in the Slutsky matrix of compensated derivatives of the Marshallian
(or ordinary) demand functions. Research on compensated comparative
statics properties of general optimization problems, however, is of a more
recent origin. The best-known contribution of this ilk is a set of three papers
by Kalman and Intriligator (1973) and Chichilnisky and Kalman (1977,
1978), which introduced generalizations that actually predate the contributions described above, although within a restricted framework. In particular,
these authors emphasized the significance of compensated derivatives in the
context of a general class of constrained optimization problems and established the existence of a generalized Slutsky matrix for such problems.
However, they did not succeed in establishing the crucial semidefiniteness
properties of this matrix in general. Perhaps because their analysis was primarily concerned with establishing the existence of solutions using primal
methods, and because their comparative statics results were restricted to
special forms, their work was largely superseded by the aforementioned subsequent developments. Similarly, their construction and use of compensated
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
35
derivatives, although a significant advance toward a general method of
dealing with constraints, was rendered in the same limited context and was
not much pursued by others.
The last contribution to be mentioned is that of Houthakker (1951–52).
In an attempt to quantify the role of quality in consumer demand,
Houthakker (1951–52) clearly recognized the important role played by compensated derivatives, the large number of ways in which they can be constructed and how they are related to a differential characterization of the
constraints present in the problem, although in the context of specific examples. However, he only succeeded in deriving the desired semidefiniteness
condition for a restricted class of problems, and his contribution did not lead
to any significant development in the subject.
2. GENERALIZED COMPENSATED DERIVATIVES
The method presented here is based on a geometric generalization of the
concept of compensated derivatives. As is often the case, this generalization
leads to a conceptually simpler structure, while yielding a powerful method
of deriving constraint-free comparative statics results for any differentiable
optimization problem. The basic idea originates in the observation that
the given parameters of the optimization problem are in general not the
natural ones for formulating comparative statics results. This is plainly
obvious in the prototype utility maximization problem, where a linear combination of partial derivatives in the form of a compensated derivative must
be used in order to obtain the desired semidefiniteness property. On the other
hand, a linear combination of partial derivatives is, aside from an inessential
scale factor, simply a directional derivative pointed in some direction in
parameter space. Since an uncompensated, i.e. a partial, derivative is also a
directional derivative, it follows that the distinction between the two is merely
a matter of the choice of coordinates in parameter space and has no intrinsic standing. Indeed, a rotation in parameter space whose magnitude and
direction may vary from point to point can interchange the role of the two.
Such a rotation is equivalent to adopting a new set of parameters for the
optimization problem. Clearly then, any general formulation of differential
comparative statics must consider the possibility of choosing from this vastly
enlarged class of directional derivatives in parameter space.
How, then, does one choose the compensated derivatives so as to guarantee the desired semidefiniteness property free of constraints, and without
requiring any restriction on the structure of the optimization problem?
Remarkably, there is a simple and natural answer to this question. One simply
© Blackwell Publishing Ltd 2006
36
M. Hossein Partovi and Michael R. Caputo
chooses the compensated derivatives in conformity with the constraints, i.e.
along directions that are tangent to the level surfaces of all the constraint
functions at each point of the parameter space. Equivalently, acting on
the constraint functions, the compensated derivatives are required to
return zero at all points of parameter space. We will see in section 3 that
this requirement is in effect the ab initio differential implementation of the
constraints.
3. DEVELOPMENT OF GENERALIZED COMPENSATED DERIVATIVES
To convey a clear picture of how the ideas described in section 2 are implemented, we begin by briefly describing the geometrical aspects of GCDs.
Consider a finite number of real-valued C (1) functions (x, a) |→ fk(x, a), k =
1, 2, . . . , K, K < N, defined for a ∈ Popen ⊂ ℜN and x ∈ D ⊂ ℜM. The functions fk(·), k = 1, 2, . . . , K, will later be identified with objective or constraint
functions, with x representing the decision vector in decision space and a representing the parameter vector in parameter space. For most of this section
def
the dependence of f (⋅) = (f 1 (⋅), f 2 (⋅), f 2 (⋅), . . . , f K (⋅)) on x plays a secondary
role, so that it is useful to consider x as fixed and each fk(·) as a C (1)
function defined for a ∈ Popen ⊂ ℜN.
Given a fixed value of x ∈ D, let a ∈ Popen and assume that the
gradient vectors ∇afk(x, a ), k = 1, 2, . . . , K, are independent, i.e. the K ×
N Jacobian matrix ∂f(x, a )/∂a has full rank. Then the implicit function
theorem implies that the level surface of f(·) passing through a , namely
def
S (a ) = {a ∈ P open : f (x, a ) = f (x, a )}, is (N − K)-dimensional. Hence the normal
hyperplane N(a), which is defined to be the vector space generated by the set
of normal vectors ∇afk(x, a ), k = 1, 2, . . . , K, is K-dimensional. The tangent
hyperplane to the level surface at a , denoted by I( a ), is generated by the set
of vectors that are tangent to S( a ) at a , and represents the directions of no
change, or the null directions, for f(·) at a . We shall refer to a vector in the
tangent hyperplane as an isovector. Thus, an isovector is any vector that
points in a null direction. Together, the isovectors and normal vectors span
all possible directions at the point a of the parameter space. Thus, we have
associated with each point a of the parameter space a pair of orthogonal
vector spaces N( a ) and I( a ). Figure 1 is an illustration of the structure just
described in a three-dimensional parameter space, with t1 and t2 depicting
two isovectors in the tangent hyperplane I( a ).
If we now consider all points of Popen as endowed with the structure just
described, there emerges a configuration of orthogonal vector spaces N(a)
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
37
Figure 1. An illustration of the tangent plane, I(ā), and normal direction, ∇ aφ(x, ā), in a threedimensional parameter space; isovectors t1 and t2 generate I(ā) and ∇ aφ(x, ā) generates
(ā).
and I(a) covering all of Popen. Hence at every point a ∈ Popen and for any direction specified by the unit vector u ∈ ℜN, the directional derivative of fk(·) at
def
the point a ∈ Popen in the direction u is given by Duf k (x, a ) = u ⋅ ∇ af k (x, a ) ,
thereby implying that if u ∈ I(a), then Dufk(x, a) = 0, k = 1, 2, . . . , K. We
refer to this condition as the null property of a directional derivative. Note
that we have suppressed the dependence on a in Du to avoid cluttered notation. Furthermore, note that the length of u plays no role with respect to the
null property, only the fact that it points in a null direction, or, equivalently,
that it is an isovector, matters. Therefore, if t is an isovector of any length,
then t · ∇afk(x, a) = 0, k = 1, 2, . . . , K, so that t · ∇a possesses the null property as well. We have thus shown that any directional derivative of a function f(·) in the direction of one of its isovectors has the null property with
respect to that function.
© Blackwell Publishing Ltd 2006
38
M. Hossein Partovi and Michael R. Caputo
Recall that when N(a) has dimension K, that is, when dim[N(a)] = K,
then dim[I(a)] = N − K with K < N, which is the case of interest for economic
problems. In any case, one requirement for the construction of directional
derivatives possessing the null property is that the set of isovectors used in
their construction span I(a). Such isovectors can be characterized as a set
of N − dim[N(a)] N-dimensional vectors, each of which is orthogonal to
∇afk(x, a) for k = 1, 2, . . . , K. The spanning property of the isovectors
permits the comparative statics information contained in the primal secondorder necessary conditions to be recovered using directional derivatives constructed with such isovectors. Another desirable requirement is that the
spanning isovectors be the smallest such set, as this permits the most efficient
recovery of the comparative statics information. This implies that the spanning isovectors must also be linearly independent. As such, the resulting
isovectors represent a complete and efficient scaffolding of I(a) in the sense
that they form a basis for I(a). Therefore, by using a basis of I(a) in constructing directional derivatives possessing the null property, one deals with
the most efficient set that spans I(a). With these definitions and results in
mind, we now introduce the definition of a complete set of GCDs for a
vector-valued function.
Definition: If ta ∈ I(a), a = 1, 2, . . . , dim[I(a)], form a basis for
I(a) corresponding to the vector-valued function f(·), then
def
N
Da (x, a ) = ta ⋅ ∇ a = ∑m =1tma ∂ ∂ am , a = 1, 2, . . . , dim[I (a )] , are a complete set
of GCDs with respect to the function f(·) at the point a, holding x fixed.
Thus, any directional derivative in a null direction in parameter space is a
GCD. It is worth emphasizing here that, mathematically, GCDs are simply
standard directional derivatives in parameter space, distinguished only by the
set of null directions for which they are defined. Observe that for N ≥ 3 any
linear combination of Da(x, a) will yield a GCD. This fact corresponds to
the infinite number of ways one can choose N − K basis vectors in I(a) when
N ≥ 3. Moreover, this observation hints at the generality of the present
method as well as the diversity of comparative statics results it can generate.
Also note that we have made the dependence of the GCDs on the points
x of the decision space and a of the parameter space explicit in our notation, a practice that is appropriate for our particular application and one that
we shall henceforth follow.
As an example, consider the case illustrated in figure 1 again where t1 and
2
t depict a pair of isovectors at point a , which is located on the level surface
S( a ) of f(x, a). Since t1 and t2 form a basis for I(a), the directional derivatives t1 · ∇a and t2 · ∇a are a complete set of GCDs with respect to f(·). Their
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
39
null property simply reflects the fact that the rate of change of a function is
zero in directions tangent to its level surface. Moreover, the reason for making
the null property a defining characteristic of GCDs is the crucial fact that
when the GCDs possess this property with respect to the constraint functions, the resulting semidefiniteness comparative statics results emerge free of
constraints. In effect, GCDs automatically and differentially account for the
presence of the constraints when they are used to conduct the comparative
statics analysis. These basic results will be established in this and the following sections. At this point, we turn to an explicit construction of isovectors
and GCDs for three economic problems.
For the first problem the function f(·) is given by the generic budget condef
N−1
N−1
straint, namely f (x, a ) = m − p ⋅ x, where p ∈ ℜ++
and x ∈ ℜ++
. The variables
def
of interest are the N parameters a = ( p1, p2 , . . . , pN −1, m) = (p, m) . The normal
direction in parameter space is therefore given by ∇af(x, a) = (-x,1) ∈ ℜN.
Since ∇af(x, a) ≠ 0N, the implicit function theorem implies that the tangent
hyperplane is of dimension N − 1 and the normal hyperplane is of dimension 1. Thus, we must choose a set of N − 1 vectors, all orthogonal to the
normal direction ∇af(x, a) = (-x,1), as a basis for the tangent hyperplane. A
def
convenient choice is ta = (01, 0 2 , . . . , 0a −1, 1a , 0a +1, . . . , 0 N −1, xa ), a = 1, 2, . . . ,
N − 1, where the subscript on the element of the isovector indicates its position within the vector. These vectors are in fact isovectors, as can be seen
by verifying that ta · ∇af(x, a) = 0 for a = 1, 2, . . . , N − 1. That these isovectors are linearly independent and thus form a basis for the tangent hyperplane follows from the fact that the only solution to the linear system of
a
N−1
equations ΣN−1
. Recalling that
a=1cat = 0N is the null vector c = 0N−1 in ℜ
∂
∂
∂ 
a def  ∂
∇ =
,
,...,
,
 , we therefore see that a complete set of
 ∂ p1 ∂ p2
∂ pN −1 ∂ m 
∂
∂
+ xa
, a = 1, 2, . . . ,
∂ pa
∂m
N − 1, which are precisely the compensated derivatives that appear in the
prototype utility maximization problem. That these GCDs possess the null
property with respect to f(·) can be seen by verifying that Da(x, a) ° f(x, a) =
0 for a = 1, 2, . . . , N − 1.
The second problem is a natural generalization of the first to a pair of
def
linear budget constraints. In this case let f k (x, a ) = m k − p k ⋅ x, where pk
def
M
M
∈ ℜ++
, x ∈ ℜ++
, m ∈ ℜ++ and a k = (p k , m k ), k = 1, 2. We assume that the two
price vectors are linearly independent. The construction of the GCDs for
this problem parallels the treatment of the prototype model except
def
GCDs is given by Da (x, a ) = ta ⋅ ∇ a =
2 ( M +1)
for the doubling of the parameter set here to a = (p1, m1, p 2 , m 2 ) ∈ℜ ++
.
def
© Blackwell Publishing Ltd 2006
40
M. Hossein Partovi and Michael R. Caputo
def  ∂
∂
Because the gradient operator is now of the form ∇ a =  1 , 1 , . . . ,
 ∂ p1 ∂ p2
∂
∂
∂
∂
∂
∂ 
,
,
,
,..., 2 ,
 , the normal vectors in parameter space
1
∂ pM
∂ m1 ∂ p12 ∂ p22
∂ pM ∂ m 2 
are ∇af1(x, a1) = (-x, 1M+1, 0M, 0M+1) and ∇af2(x, a2) = (0M, 0M+1, -x, 12(M+1)). In
view of the fact that ∇af1(x, a) · ∇af2(x, a) = 0 the gradient vectors are linearly
independent, thereby implying via the implicit function theorem that the two
constraints define a 2M-dimensional manifold in the 2(M + 1)-dimensional
parameter space, hence dim[I(a)] = 2M. Consequently, we must now choose a
set of 2M vectors, all orthogonal to the two normal directions, as a basis
def
for I(a). Letting e aM = (01, 0 2 , . . . , 0a −1, 1a , 0a +1, . . . , 0M ) be a standard basis
vector for ℜM and then following the pattern used for the prototype model,
def
we find that the isovectors are given by ta = (e aM , xa , 0 M , 0 2( M +1) ), a = 1, 2,
def
. . . , M, and ta = (0 M , 0M +1, e aM−M , xa −M ), a = M + 1, M + 2, . . . , 2M, since
2M
∇afk(x, ak) · ta = 0, k = 1, 2 and a = 1, 2, . . . , 2M. Moreover, since Σa=1
cata =
a
02(M+1) has only the trivial solution the 2M isovectors t are linearly independent and thus form a basis for I(a). It therefore follows that a complete
def
∂
∂
set of GCDs is given by Da (x, a ) = ta ⋅ ∇ a = 1 + xa
, a = 1, 2, . . . ,
∂ pa
∂ m1
def
∂
∂
+ xa −M
M, and Da (x, a ) = ta ⋅ ∇ a =
, a = M + 1, M + 2, . . . , 2M.
2
∂ pa −M
∂m2
Noting that Da(x, a) ° fk(x, ak) = 0, k = 1, 2 and a = 1, 2, . . . , 2M, we conclude that the GCDs Da(x, a), a = 1, 2, . . . , 2M, posses the null property
with respect to f1(·) and f2(·). The above GCDs are essentially identical to
the neoclassical Slutsky compensated derivatives. This is not unexpected since
the additional constraint is just another linear budget constraint that is independent of the first, and whose parameter set (p2, m2) does not overlap with
that in the first constraint.
The third problem deals with an unconstrained model, to wit the profit
maximizing model of the firm, and gives an indication of the calculational
def
novelty of the present method. Consider f̃ (x, a ) = s[ pF (x ) − w ⋅ x ], where x ∈
M
M
ℜ++
, w ∈ ℜ++
and s > 0. Clearly, the magnitude of s does not affect the optimal
values of the decision variables, nor does it have any effect on the comparative statics of the problem. In any case, s will be treated as a parameter whose
value will eventually be set equal to unity. The scale factor s thus serves an
auxiliary purpose in this calculation, although there are economically meaningful interpretations of its role as will be discussed in section 4. Define the
def
+2
parameter vector as a = (w1, w2 , . . . , wM , p, s ) = (w, p, s ) ∈ℜ M
++ . Consequently,
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
41
def  ∂
∂
∂
∂ ∂
,
,...,
,
,  , therefore
the gradient operator is given by ∇ a = 
 ∂ w1 ∂ w2
∂ wM ∂ p ∂ s 
def
implying that ∇ af˜ (x, a ) = ( − sx, sF (x ), f (x, a )), where f (x, a ) = pF (x ) − w ⋅ x.
Note that there are M + 2 parameters under consideration, so that N =
M + 2. Given that ∇a f̃ (x, a) ≠ 0M+2, the implicit function theorem implies
that N(a) is of dimension unity and I(a) is of dimension M + 1. The M + 1
def 
sxa 
vectors defined by ta =  01, 0 2 ,. . . , 0a −1, 1a , 0a +1, . . . , 0M , 0M +1,
 , a = 1,

f ( x, a ) 
def 
sF (x ) 
2, . . . , M, and t M +1 =  01, 0 2 ,. . . , 0M , 1M +1, −
 , are isovectors since

f ( x, a ) 
ta · ∇a f̃ (x, a) = 0 for a = 1, 2, . . . , M + 1. These M + 1 isovectors
a
also form a basis for I(a) because the only solution of ΣM+1
a=1 cαt = 0M+2
is c = 0M+1. Accordingly, the associated complete set of GCDs is
def
∂
 sxa  ∂
+
found to be Da (x, a ) = ta ⋅ ∇ a =
, a = 1, 2, . . . , M, and
∂wa  f (x, a )  ∂ s
∂  sF (x )  ∂
−
. In view of the fact that Dα(x, a) °
∂ p  f (x, a )  ∂ s
f̃ (x,a) = 0, a = 1, 2, . . . , M + 1, the GCDs posses the null property with respect to f̃(·).
It is appropriate at this juncture to emphasize that while the customary
meaning of compensation refers to a correction term that accounts for the
effect of a constraint in the problem, the archetypal example being the correction for the income effect in the utility maximization problem, no necessary
connection with constraints is implied in the case of generalized compensation,
as is clearly illustrated in the profit maximization problem treated above.
Indeed any problem, constrained or not, will admit the use of generalized compensation if its parameter space is larger than one-dimensional. Furthermore,
as the introduction of the scale parameter s for the profit maximization
problem shows, the parameter space can always be enlarged, so that the restriction to more than one dimension is really no restriction at all.
We conclude this section by establishing the most important property of a
GCD, namely its constraint conformance property mentioned earlier. To that
end, consider a restriction of the above construction in parameter space
to the case where the two vector arguments x and a of the functions fk(·)
are functionally related. Specifically, let a |→ x(a) be C (1) ∀ a ∀ ∈ Popen, and
consider the resulting restricted set of functions (x(a), a) |→ fk(x(a), a). In
def
DM +1 (x, a ) = t M +1 ⋅ ∇ a =
© Blackwell Publishing Ltd 2006
42
M. Hossein Partovi and Michael R. Caputo
applications, the vector-valued function x(·), which we shall refer to as a decision function, is derived from an optimality condition. Next, let a subset of
the restricted set of functions (x(a), a) |→ fk(x(a), a) serve as constraint functions by virtue of vanishing identically, i.e. fk(x(a), a) = 0 ∀ a ∈ Popen and
k = 1, 2, . . . , C, with C ≤ K.
We now exploit the constraint identities fk(x(a), a) = 0 ∀ a ∈ Popen
and k = 1, 2, . . . , C, by applying the restricted parameter space
def
GCDs Da (a ) = Da (x(a ), a ) to them. We begin by introducing the compact
def
def ∂
f k (x, a ) , xi ;a (a ) = Da (a ) o xi (a ) = ta ⋅ ∇ a xi (a ) and
notation f ,ki (x, a ) =
∂ xi
def
f ;ka (x, a ) = Da (a ) o f k (x, a ). This notational convention implies that a subscript occurring to the right of a comma signifies partial differentiation,
whereas a subscript occurring to the right of a semicolon signifies directional
differentiation corresponding to a GCD. Moreover, Latin subscripts are used
to denote differentiation with respect to decision variables, while Greek
indices are used to denote differentiation with respect to parameters. The
above definitions yield a considerable simplification of the ensuing fundamental result.
Lemma 1: Every GCD of a decision function conforms to the constraints in
decision space, i.e. x;a(a) · ∇xfk(x(a), a) = 0 ∀ a ∈ Popen and for k = 1, 2, . . . ,
C and a = 1, 2, . . . , N − dim[N(a)].
Proof: First, differentiate the kth constraint identity fk(x(a), a) = 0 with
respect to am to get
M
∑f
i =1
k
,i
(x(a ), a )xi ,m (a ) + f ,km (x(a ), a ) = 0
Next, multiply this identity by tam and sum over m to arrive at
M
N
N
i =1
m =1
m =1
∑ f,ki (x(a ), a )∑ xi ,m (a )tma + ∑ f,km (x(a ), a )tma = 0
Using the definitions xi ;a (a ) = ∑m =1 xi ,m (a )tma and f ;ka (x(a ), a ) = ∑m =1f ,km
def
N
(x(a ), a )tma, the preceding identity can be rewritten as
M
Da (a ) o f k (x(a ), a ) = ∑ f ,ki (x(a ), a )xi ;a (a ) + f ;ka (x(a ), a ) = 0
i =1
© Blackwell Publishing Ltd 2006
def
N
Theory of Comparative Statics
43
which holds ∀ a ∈ Popen, k = 1, 2, . . . , C and a = 1, 2, . . . , N − dim[N(a)].
k
Due to the null property of the GCDs, it follows that f ;a
(x(a), a) = 0 identically. The above equation therefore reduces to
M
∑f
i =1
k
,i
(x(a ), a )xi ;a (a ) = x ;a (a ) ⋅ ∇ xf k (x(a ), a ) = 0 ∀ a ∈ P open
k = 1, 2, . . . , C and a = 1, 2, . . . , N − dim[N(a)].
Since the last equation is the inner product of x;a(a) with the decision space
normal vectors ∇xfk(x(a), a), its vanishing for every value of k and a implies
the orthogonality of every GCD of x(a) to every normal vector associated
with the constraint surfaces in decision space, implying in turn that x;a(a) lies
in the null space of ∇xfk(x(a), a). We have thus established that the application of parameter space GCDs to the decision functions produces isovectors
in decision space, i.e. vectors that conform to the constraints. This crucial
property of a GCD underlies the central result of this paper.
We pause here to emphasize the profoundly dual nature of this result: a
directional derivative that annihilates the constraint functions in parameter
space, when applied to decision functions, will produce a vector that conforms
to the constraints in decision space.
4. THE MAIN THEOREMS
Having described the construction and properties of GCDs in some detail,
we are now in a position to establish the main theorem of this paper. Consider the optimization problem
V (a ) = max{ f (x, a ) s.t. g k (x, a ) = 0, k = 1, 2, . . . , K }
def
x
(1)
where f(·) ∈ C (2), gk(·) ∈ C (2) for k = 1, 2, . . . , K and M, N > K. To avoid trivialities, we also assume that the standard constraint qualification condition
holds at the optimum point, i.e. the rank of the K × M matrix ∂g(x, a)/∂x
is equal to K at the critical points of problem (1). Furthermore, we assume
that their exists unique C(1) decision functions x(·), with values defined by
x(a ) = arg max{ f (x, a ) s.t. g k (x, a ) = 0, k = 1, 2, . . . , K }
def
(2)
x
and that x(a) ∈int[D] for each value of a ∈ Popen. Similarly, we assume that
the set of parameter space normal vectors ∇agk(x(a), a) is linearly independ-
© Blackwell Publishing Ltd 2006
44
M. Hossein Partovi and Michael R. Caputo
ent, i.e. the rank of the K × N matrix ∂g(x(a), a)/∂a is equal to K, for a ∈
Popen. By the implicit function theorem, the latter assumption implies that
dim[N(a)] = K and dim[I(a)] = N − K at the optimum point. We also define
the value of the Lagrangian function L(·) as
K
L ( x, l , a ) = f ( x, a ) + ∑ l k g k ( x, a )
def
(3)
k =1
where l ∈ ℜK is the vector of Lagrange multipliers for problem (1). We note
in passing that the above set of assumptions is redundant. For example, the
linear independence of the set of vectors ∇agk(x(a), a) implies the same for
the set of vectors ∇xgk(x(a), a), a fact that can be established by a contrapositive argument.
In order to characterize the comparative statics implied by the constrained
maximum property of x(a), we construct a set of GCDs with respect to the
constraint functions given in equation (1), possibly including the objective
function as well, denoted by Da(x, a), a = 1, 2, . . . , A, according to the
procedure explained in section 3. Here, A is the dimension of the tangent hyperdef
plane, i.e. A = dim[I (a )] = N − K . It is quite useful at this juncture to introdef
A
duce the definition h(a ) = ∑a =1ha x ;a (a ), where h ∈ ℜA is an arbitrary vector.
By Lemma 1, x;a(a) conforms to the constraints in decision space ∀ a ∈ Popen
and for a = 1, 2, . . . , A. But then the same is implied for h(a) since the latter
is a linear combination of conforming vectors. By construction, then, the
vector h(a) conforms to the constraints in decision space ∀ a ∈ Popen. Finally,
recall that a real matrix L, with typical element Lij, is by definition positive
definite or semidefinite if (1) it is symmetric and (2) for every real vector
v ≠ 0, the quadratic form v†Lv is positive definite or semidefinite, respectively,
where ‘†’ signifies transposition. We are now in a position to state the central
result of our paper, the proof of which, as well as of all the succeeding
theorems, can be found in the Appendix.
Theorem 1: The constrained optimization problem defined by equation (1) et
seq. admits of an A × A constraint-free positive semidefinite CSM W(a), the
typical element of which is given by
M
Ωab (a ) = ∑ xi ,b (a )L,i ;a (x(a ), l (a ), a )
i =1


= ∑ xi ,b (a ) f,i ;a (x(a ), a ) + ∑ l k (a )g,ki ;a (x(a ), a ), a , b = 1, 2, . . . , A.


i =1
k =1
M
© Blackwell Publishing Ltd 2006
K
Theory of Comparative Statics
45
Theorem 1 asserts that the matrix W(a), which is a linear combination
of GCDs of the decision functions with respect to the parameters, is
positive semidefinite, free of constraints. We refer to a matrix possessing
these properties as a CSM for the optimization problem. The unrestricted
existence of a CSM for a general constrained optimization problem is the
main result of our analysis. It is worth re-emphasizing here that there is a
large freedom of choice in the construction of CSMs, a feature that will be
explored in the following. While this freedom may be exploited to generate
different forms of comparative statics for a given optimization problem, it is
well to remember that all such matrices convey no more information than is
contained in the second-order necessary conditions expressed in equation
(A2). These conditions, in turn, originate in the local concavity of the underlying constrained maximization problem defined in equation (1). In sum,
Theorem 1 shows that comparative statics results in the form of linear combinations of GCDs are basic to all differentiable optimization problems, and
thus form fundamental, testable implications of models posited in economic
theory.
As one might surmise, the null property of the GCDs ensures that the envelope theorem holds for the constrained optimization problem defined by
equation (1) without the intrusion of the constraint functions. This result is
summarized in
Theorem 2: For the constrained optimization problem defined by equation (1)
et seq., the indirect objective function V(·) satisfies the envelope property
V;a (a ) = f;a (x(a ), a )
(i)
Furthermore, if the GCDs possess the (optional) null property with respect to
the objective function f (·) as well, then the indirect objective function V(·) satisfies the envelope property
V;a (a ) = 0
(ii)
Next we explore the consequences of any symmetry or invariance property that the objective and constraint functions might possess. We focus on
those invariances that are likely to play a significant role in economic applications. Our result is summarized by
Theorem 3: Suppose there exist a pair of C (1) vector-valued functions
X(·):D → ℜM and A(·):Popen Æ ℜN, and a differential operator J(·) defined
by
© Blackwell Publishing Ltd 2006
46
M. Hossein Partovi and Michael R. Caputo
M
J ( x, a ) = ∑ X i ( x )
def
i =1
N
∂
∂
+ ∑ Am (a )
∂ xi m =1
∂ am
(4)
Suppose further that the action of J(·) on f(·) and gk(·), k = 1, 2, . . . , K, can
be described by
M
N
i =1
m =1
J (x, a ) o f (x, a ) = ∑ X i (x ) f,i (x, a ) + ∑ Am (a ) f,m (x, a ) = F ( f (x, a ))
M
N
i =1
m =1
J (x, a ) o g k (x, a ) = ∑ X i (x )g,ki (x, a ) + ∑ Am (a )g,km (x, a )
= G ( g (x, a )), k = 1, 2, . . . , K
k
(5)
(6)
k
for every x ∈ D ⊂ ℜM and a ∈ Popen ⊂ ℜN, where F(·):ℜ → ℜ and Gk(·):ℜ →
ℜ are C (1) functions, with Gk(0) = 0, k = 1, 2, . . . , K. Then if the second-order
sufficient condition of problem (1) holds, the decision functions x(·) possess the
invariance property
N
X i (x(a )) − ∑ Am (a )xi ,m (a ) = 0, i = 1, 2, . . . , M
m =1
(7)
These conditions have a straightforward interpretation as invariance
conditions. They essentially state that if the objective and constraint
functions are evaluated at the ‘slightly’ displaced values of their arguments
x + eJ(x, a)x = x + eX(x) and a + eJ(x, a)a = a + eA(a), instead of x and a,
respectively, where e is a ‘small’ real number, then the underlying optimization problem remains unchanged to first order in e. Given that the secondorder sufficient condition holds, this first-order invariance condition
implies that the modified objective and constraint functions define a solution
that differs from the solution of the original problem only by quantities
of second order in e. In other words, they imply that the decision vector
x(a) + eX(x) differs from the solution x(a + eA(a)) by second-order quantities only.
To demonstrate the power of Theorem 3, consider how it applies to the
case of homogeneity. Suppose the objective and constraint functions of a
given optimization problem satisfy the invariance conditions given in equations (5) and (6) with X(x) = hx and A(a) = a. Then according to equation
(7), the solution to this problem will satisfy the invariance condition
ΣNm=1amxi,m(a) = hxi(a). This last condition characterizes x(a) as a homogeneous
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
47
function of degree h by Euler’s Theorem. As an example of this, consider
the prototype utility maximization problem. Formally, the underlying symmetry of this problem corresponds to the fact that the invariance conditions
given in equations (5) and (6) are satisfied by the utility function and budget
constraint with X(x) = 0M and A(a) = a, hence h = 0 in this case. The resulting invariance condition then characterizes the demand functions as homogeneous of degree zero in prices and income.
An even simpler case of scale invariance is one that we have already
exploited, namely the use of the scale parameter s in the profit maximization
model in section 3. This invariance can be interpreted in several economically meaningful ways. For example, one can consider the fact that the
amount of revenues minus expenditures being maximized in that model may
be expressed in various multiples of a given currency, or even in different currencies, and this should have no effect on the decision functions. Alternatively, one might interpret 1 − s > 0 as a flat tax rate on profits and the
objective function as the net profit of the firm, which again has no effect on
the decision functions. Mathematically, on the other hand, one chooses
∂
and h = 0. Then
X(x) = 0M and A(a) = (0, 0, . . . , 0, s), so that J (x, a ) = s
∂s
the invariance conditions in equations (5) and (6) are satisfied, and for s ≠ 0
we have the result that ∂x(a)/∂s = 0M. Another example of the use of Theorem
3 will be considered in section 7.
We now proceed to study several structural features of Ω(a). The first
question concerns the definiteness of Ω(a), or more specifically, whether
its rank is lower than its order, and if so, whether there exists an upper
bound on this rank. Recall that we are dealing with M decision variables,
N parameters, K independent constraints, and a CSM of order A ≤ N
defined by
M
M
Ωab (a ) = − ∑ ∑ xi ;a (a )L,ij (x(a ), l (a ), a )x j ;b (a )
def
i =1 j =1


= − ∑ ∑ xi ;a (a ) f,ij (x(a ), a ) + ∑ l k (a )g,kij (x(a ), a )x j ;b (a )


k =1
i =1 j =1
M
M
K
as given in equation (A3). The task before us is to establish an upper bound
for the rank of this matrix. Let us emphasize that an upper bound is all that
can in general be established, since the M × A matrix xi;a(a) that appears
in equation (A3) can have an arbitrarily small rank, including zero, implying the same for Ω(a) For example, for the optimization problem
maxx[G(x) + H(a)], the decision functions do not depend on the parameters,
© Blackwell Publishing Ltd 2006
48
M. Hossein Partovi and Michael R. Caputo
thus causing xi;a(a) to vanish identically. This implies the vanishing of Ω(a)
and its rank.
Let the rank of Ω(a) be denoted by r(Ω(a)). Then an upper bound to
r(Ω(a)) can be derived from the standard theorems that (1) the rank of an
NR × NC matrix cannot exceed min(NR, NC ), and (2) the rank of a product
cannot exceed that of any of its factors.
Theorem 4: The CSM Ω(a) of the constrained optimization problem defined
by equation (1) et seq. has the property r(Ω(a)) ≤ min(M − K, A).
As a first example of Theorem 4, consider the profit maximization problem
of section 3, where K = 0 and A = M + 1. For this problem Theorem 4 yields
r(Ω(a)) ≤ min(M, M + 1) = M, implying that the full, (M + 1) × (M + 1) CSM
will be singular. If, however, one uses the M × M submatrix of the full CSM
corresponding to the partial derivatives of the input factors with respect to
the input prices, then there will no longer be a necessary rank reduction. As
a second example, consider the prototype utility maximization problem, also
considered in section 3, where K = 1. For this problem Theorem 4 yields
r(Ω(a)) ≤ min(M − 1, M) = M − 1, implying that the M × M Slutsky matrix
is necessarily singular since its order exceeds its rank by at least one.
More generally, when N >> M, the resulting CSM will be highly redundant
with r(Ω(a)) << A. It is important to emphasize that Theorem 3 represents
the rank reduction that is imposed by the underlying geometry of the GCDs,
independently of the specific properties of the objective function. Thus, there
may very well be further rank reductions of Ω(a) in specific cases resulting
from the special properties of the objective function.
Our final task in this section is a characterization of the non-uniqueness
in Theorem 1. Specifically, we intend to classify and characterize all possible
CSMs associated with problem (1). In doing so, we take the set of decision
variables and parameters as given and fixed, thereby excluding from the
present discussion the non-uniqueness associated with these choices. It must
be emphasized that although ordinarily there is a ‘natural’, or ‘sensible’,
choice of decision and parameter sets associated with a given problem, there
does exist in principle the possibility of considering other sets constructed
from the given ones, or even considering smaller or larger sets. For example,
one could ignore certain parameters as uninteresting or irrelevant, or conversely, one could augment the parameter set by introducing auxiliary parameters. For the decision variables in a constrained problem, one could discard
some of the constraint equations by solving for a subset of the decision variables and conversely. Moreover, these alternative choices are not always mere
mathematical curiosities devoid of meaning or use. Indeed, we will exploit
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
49
these extra degrees of freedom in our treatment of the principal-agent
problem in section 7, where the usefulness of an alternative choice for the
parameter set will be evident.
In preparation for the ensuing theorem, recall that we have defined and
def
N
used a complete set of GCDs according to Da (a ) = ∑m =1tma ∂ ∂ am. Also recall
that the set of isovectors ta, a = 1, 2, . . . , A, is a basis for the tangent hyperplane in parameter space. Consider a different choice for the set of isovectors, say t̃a, a = 1, 2, . . . , A, with the same properties as ta. Since each set of
isovectors ta and t̃ a, a = 1, 2, . . . , A, form a basis for the tangent hyperplane in parameter space, there exists a non-singular matrix C of order A
that expresses the new isovectors as a linear combination of the old, and condef
A
versely under the inverse matrix C −1, i.e. t˜ma = ∑g =1Cag tmg . With these results
in mind, we have the following theorem:
def
def
N
N
Theorem 5: The GCDs Da (a ) = ∑m =1tma ∂ ∂ am and D˜ a (a ) = ∑m =1t˜ma ∂ ∂ am are
related by
A
D̃a (a ) = ∑ Cag Dg (a )
g =1
while the CSMs Ω(a) and Ω̃ (a) are related by the formula
A
A
Ω̃ab (a ) = ∑ ∑ Cag Ωgd (a )C bd
d =1 g =1
According to Theorem 5 the two matrices Ω(a) and Ω̃ (a) are congruent.
As a pair of CSMs, on the other hand, Ω(a) and Ω̃ (a) are essentially
equivalent in the sense that they are of equal rank and the semidefiniteness
of one implies that of the other. These properties follow from the observation that each of the two sets of isovectors from which Ω(a) and W̃ (a) are
constructed forms a basis for the tangent hyperplane in the parameter space,
and as such must provide a description fully equivalent to the other. It should
be noted, however, that congruency does not imply similarity of economic
implications, as the two matrices can be quite different with respect to such
matters as observability and empirical verification.
Thus far we have only considered complete sets of GCDs, i.e. those constructed from a set of isovectors that constitute a basis for the tangent hyperplane. It is of interest to know the implications of an incomplete or
© Blackwell Publishing Ltd 2006
50
M. Hossein Partovi and Michael R. Caputo
dependent set of isovectors for the information content of the resulting
CSMs. Similarly, one could contemplate the consequences of generalized
transformations of the decision variables or parameters on the CSMs.
Readers interested in such matters are referred to the unabridged version
this paper (Partovi and Caputo, 1998). This paper also establishes the
existence of a ‘universal CSM’, the details of which can also be found in
Partovi and Caputo (2006), that embodies the comparative statics information of a given model exhaustively, i.e. from which any other CSM can be
deduced.
5. FORMAL RELATIONSHIP WITH THE COMPARATIVE STATICS
METHODS EXTANT
As mentioned in section 1, Samuelson (1947) established the foundations
of differentiable comparative statics methodologies, while Silberberg
(1974) generalized and advanced that work to the point of constructing
a semidefinite matrix conveying the comparative statics properties of a
general constrained optimization problem. However, Silberberg’s (1974)
construction has a serious shortcoming in dealing with constrained optimization problems, namely the subjection of the said matrix to the constraints. The method of generalized compensation summarized by Theorem
1 removes this limitation in a general way. Naturally, this raises the question
of just how, if at all, Theorem 1 is related to the central result of Silberberg
(1974). We will answer this question by deriving the relationship between
Theorem 1 and Silberberg’s (1974) main theorem.
To that end, recall Silberberg’s (1974, equation (10)) result as applied to
the constrained optimization problem defined in equation (1) et seq. Stated
N
in our notation, Silberberg’s (1974) result is that ΣNm=1Σn=1
qmSmn(a)qn ≥ 0∀q ∈
N
N
k
ℜ
Σ m=1qmg,m(x(a), a) = 0, k = 1, 2, . . . , K, where
∈
def


Smv (a ) = ∑ xi ,v (a ) f,mi (x(a ), a ) + ∑ l k (a )g,kmi (x(a ), a )


i =1
k =1
M
K
K
+ ∑ l k ,v (a )g,km (x(a ), a ), m, v = 1, 2, . . . , N
(8)
k =1
Restated in geometrical terms, his result is that the quadratic form q†S(a)q is
non-negative provided that the vector q lies in the parameter space tangent
hyperplane defined by the constraint functions, i.e. provided q ∈ I(a). Now
recall that by construction the isovectors ta ∈ I(a), a = 1, 2, . . . , A, and like-
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
51
wise for any linear combination of them. Hence, for any arbitrary vector
h ∈ ℜA, if we let q = ΣAa=1hata, then q†S(a)q ≥ 0 can be rewritten as
ΣAa=1ΣAb=1hata†S(a)tbhb ≥ 0. Because h ∈ ℜA is arbitrary, we may conclude that
the A × A symmetric matrix T †S(a)T, with typical element ta†S(a)tb, a, b = 1,
2, . . . , A, is positive semidefinite. Using equation (8) and the C (2) nature of
f(·) and gk(·), k = 1, 2, . . . , K, we find that ta†S(a)tb can be written as
N
N


Smv (a )tvb = ∑ ∑ ∑ xi ,v (a )tvb  f,im (x(a ), a )tma


m =1 v =1 i =1
N
∑ ∑t
a
m
m =1 v =1
N
N
(9)



+ ∑ l k (a )g,kim (x(a ), a )tma   + ∑ ∑ ∑ l k ,v (a )tvb g,km (x(a ), a )tma 
  m =1 v =1  k =1

k =1
N
K
N
K
Using the definition of a GCD, that is to say, Da (x, a ) = ta ⋅ ∇ a = ∑m =1tma ∂ ∂ am ,
def
N
def
along with our notational convention that, say, xi ;a (a ) = Da (a ) o xi (a ) = ta ⋅
∇ a xi (a ) = ∑m =1tma xi ,m (a ), we find that equation (9) may be expressed as
N
N
N
∑ ∑t


Smv (a )tvb = ∑ xi ; b (a ) f,i ;a ( x(a ), a ) + ∑ l k (a )g,ki ;a ( x(a ), a )


i =1
k =1
M
a
m
m =1 v =1
K
K
(10)
+ ∑ l k; b (a )g;ka ( x(a ), a )
k =1
k
By the null property of a GCD it then follows that g;a
(x(a), a) = 0 ∀ a ∈ Popen,
a = 1, 2, . . . , A, and k = 1, 2, . . . , K, thus implying that the last term in
equation (10) vanishes identically. Consequently, inspection of Theorem 1
shows that equation (10) reduces to ΣNm=1ΣNn=1tmaSmn(a)tnb = Ωab(a), thereby completing the demonstration of the relationship between Theorem 1 and the
main comparative statics result of Silberberg (1974).
Next we consider the work of Hatta (1980), and show that its main comparative statics result is in fact a special case of Theorem 1. The optimization problem treated by Hatta (1980), given in his equation (10), is a special
case of our equation (1), and appears as
def
max  f (x, p) s.t. g l (x, p, k l ) = k l − k l (x, p) = 0, l = 1, 2, . . . , K 
x 

def
in our notation, where x ∈ ℜM, p ∈ ℜM, k ∈ ℜK and a = (p, k ) ∈ℜ M + K.
The crucial property of this problem is the occurrence of the parameters
© Blackwell Publishing Ltd 2006
52
M. Hossein Partovi and Michael R. Caputo
k in a separable, linear manner in the constraint equations, and their
absence from the objective function. This special structure makes it
possible to construct a complete set of GCDs patterned after those
customarily used for the prototype utility maximization model, namely
Da (x, a ) = ∂ ∂ pa + ∑l=1[∂ k l (x, p) ∂ pa ] ∂ ∂ k l , a = 1, 2, . . . , M. Using these
def
K
GCDs in Theorem 1 we find that Ωab(a) takes the special form


Ωab (a ) = ∑  f,ia (x(a ), p) − ∑ l l (a )k,lia (x(a ), p)

i =1 
l =1
l
K
 ∂ x (a )
∂ k (x(a ), p) ∂ xi (a ) 
× i
+∑
∂
p
∂ pb
∂k l 

l =1
b
M
K
a, b = 1, 2, . . . , M. This form of Ωab(a) is identical to Hatta’s (1980) Theorems 6 and 7, his chief comparative statics results. Note that because of
the special structure of the problem, compensation terms appear only in
the partial derivatives of the decision functions. These compensated
derivatives are denoted by sp(p, x*(p, k)) and termed ‘the Slutskian substitution matrix’ by Hatta (1980), while the Lagrange multipliers l are represented by fk in his notation. An examination of the manner in which the
matrix sp(p, x*(p, k)) is derived by Hatta (1980), on the other hand, reveals
that its elements are constructed in conformity to the constraints, i.e. precisely according to the definition of our GCDs, although this property is
obscured by the presentation. Moreover, the method of their construction
specifically relies on the special role played by the parameters k and is therefore limited to the assumed form of the problem.
This completes our demonstration of how Theorem 1 is related to the
two main comparative statics methods. In passing, note that Caputo (1999)
provides a complementary exposition to that above, in that he shows that the
central comparative statics results of Hatta (1980) are a special case of those
of Silberberg (1974).
6. COMPARATIVE STATICS OF MULTIPLE LINEAR CONSTRAINT
UTILITY MAXIMIZATION PROBLEMS
The primary purpose of this and the ensuing section is to illustrate, in some
detail, the workings of Theorems 1–5. A second, parallel objective is to
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
53
present and discuss certain novel results that naturally emerge from the application of these theorems to well-known economic models. In this section we
examine the comparative statics of a general class of multiple linear constraint utility maximization problems and in the next we do the same for the
principal-agent problem with hidden information, both hitherto unsolved
problems. For examples dealing with other areas, we refer the reader to the
unabridged version of this paper (Partovi and Caputo, 1998), Caputo and
Paris (2000), which studies the generalized maximum entropy formalism, and
Caputo and Partovi (2002), which deals with models of profit-maximizing
rate-of-return-regulated firms. We assume throughout that the regularity
conditions stipulated in section 4 are in effect and remind the reader that the
GCDs for the class of utility maximization problems presently under consideration have already been constructed in section 3.
The general utility maximization problem under consideration here can be
stated as
V (a ) = max
{U (x ) s.t. m k − p k ⋅ x = 0, k = 1, 2, . . . , K ≤ M }
M
def
x ∈ℜ ++
(11)
1
1
2
2
K
K
k
where pk ∈ ℜM
++, m ∈ ℜ++, k = 1, 2, . . . , K, and a = ( p , m , p , m , . . . , p , m )
def
K ( M +1)
is the parameter vector. For K = 2, the first-order necessary condi∈ℜ ++
tions include U,i(x) − l1p1i − l2p2i = 0, i = 1, 2, . . . , M, where l1 and l2 are the
Lagrange multipliers. Equipped with the 2M GCDs from section 3, we may
apply them to the above first-order necessary conditions and then use
Theorem 1 to derive the ensuing 2M × 2M negative semidefinite and constraint-free CSM:
1
l1 (a )S 2 (a ) 
def  l (a )S (a )
S(a ) =  1

1
2
l 2 (a )S (a ) l 2 (a )S (a )
def
(12)
where S mkv (a ) = ∂ xm (a ) ∂ pvk + xv (a )∂ xm (a ) ∂ m k , k = 1, 2, and m, n = 1,
2, . . . , M.
Since the full CSM S(a) is negative semidefinite, the same must be true
of its diagonal blocks, l1(a)S1(a) and l2(a)S2(a). Moreover, the symmetry
of S(a) implies that one off-diagonal matrix is the transpose of the other,
i.e. l1(a)S2(a) = l2(a)S1(a)† = l2(a)S1(a), where use has been made of the
symmetry of S1(a). Using this symmetry property and factoring out
l1(a)−1 in equation (12), we arrive at the following structure for the CSM when
K = 2:
© Blackwell Publishing Ltd 2006
54
M. Hossein Partovi and Michael R. Caputo
1
def
l1 (a )l 2 (a )S1 (a )
−1  l (a ) S (a )
S(a ) = l1 (a )  1

2
1
l 2 (a ) S1 (a ) 
l 2 (a )l1 (a )S (a )
2
(13)
The redundancy of S(a) is now fully manifest in equation (13), since its rank
is seen to be at most equal to that of its building block matrix S1(a). Moreover, S1(a) must obey the two independent constraint conditions, scilicet,
1 1
M
2 1
ΣM
m=1pmS mn(a) = 0 and Σ m=1pmSmn(a) = 0. The first of these can be verified by dif1
1
ferentiating the constraint identity m1 − ΣM
m=1pmxm(a) = 0 with respect to pn and
m1, or by using the symmetry of S1(a) and the fact that x(a) is positively
homogeneous of degree zero in (p1, m1) by Theorem 3. The second can be
2
verified by differentiating the constraint identity m2 − ΣM
m=1pmxm(a) = 0 with
1
1
1
respect to pn and m . Consequently, the rank of S (a) cannot exceed M − 2,
implying the same for S(a). This latter conclusion can also be confirmed by
appealing to Theorem 4, which asserts that r(S(a)) ≤ min(M − 2, 2M) =
M − 2. Thus, even though S(a) is of order 2M, it has at least M + 2 zero
eigenvalues in its spectrum, thereby implying that at least M + 2 rows and
their corresponding columns can be deleted from S(a) without loss of information. Equivalently, the full 2M × 2M CSM S(a) has at most M − 2 linearly
independent rows and columns.
The generalization of the above results to the case K ≤ M is straightforward and immediate. In this case the GCDs take the form
 ∂
∂
,
a = 1, 2, . . . , M ,
 1 + xa
∂ m1
 ∂ pa
∂
 ∂ +x
,
a = M + 1, M + 2,
def a
 2
a −M
a
Da (x, a ) = t ⋅ ∇ =  ∂ pa −M
∂m2
. . . , 2M ,
 M

a = ( K − 1)M + 1, ( K − 1)M
∂
∂

+ xa −( K −1)M
,
K
K
∂ p
+ 2, . . . , KM .
∂m
 a −( K −1)M
The corresponding first-order necessary conditions include U,i(x) − ΣKk=1lkpik
= 0, i = 1, 2, . . . , M, where lk, k = 1, 2, . . . , K, are the Lagrange multipliers.
Equipped with the aforementioned KM GCDs, we apply them to the
above first-order necessary conditions and then use Theorem 1 to derive the
ensuing KM × KM negative semidefinite and constraint-free CSM for
problem (11):
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
1
2
3
 l1 (a )S (a ) l1 (a )S (a ) l1 (a )S (a )
 l a S1 a
2
3
 2 ( ) ( ) l 2 (a )S (a ) l 2 (a )S (a )
def
S(a ) =  l3 (a )S1 (a ) l3 (a )S 2 (a ) l3 (a )S3 (a )

M
M
M

l K (a )S1 (a ) l K (a )S 2 (a ) l K (a )S3 (a )
55
. . . l1 (a )S K (a ) 
. . . l 2 (a )S K (a ) 
. . . l3 (a )S K (a ) 

O
M

. . . l K (a )S K (a )
(14)
def
where now S mkv (a ) = ∂ xm (a ) ∂ pvk + xv (a )∂ xm (a ) ∂ m k, k = 1, 2, . . . , K, and
m, n = 1, 2, . . . , M. Following the reasoning that preceded equation (13), we
arrive at the ensuing structure for the CSM of problem (11):
 l1 (a ) S1

1
 l 2 (a )l1 (a )S
def
−1
S(a ) = l1 (a )  l3 (a )l1 (a )S1

M

l (a )l (a )S1
K
1
2
l1 (a )l 2 (a )S1
2
l 2 (a ) S1
l1 (a )l3 (a )S1
l 2 (a )l3 (a )S1
l3 (a )l 2 (a )S1
l3 (a ) S1
M
l K (a )l 2 (a )S1
M
l K (a )l3 (a )S1
2
. . . l1 (a )l K (a )S1 

. . . l 2 (a )l K (a )S1 
. . . l3 (a )l K (a )S1 

O
M

2
...
l K (a ) S1 
(15)
The structure of the full CSM S(a) as exhibited in equation (15) has an
important consequence for empirical work based on the K-linear constraint
model. In particular, equation (15) shows that if one is willing to accept (1)
the symmetry conditions lk(a)S(a) = l(a)Sk(a) as applying to the empirical
demand functions x(a), and (2) l1(a) > 0, which is the archetypal condition
under the usual assumptions, then testing for the negative semidefiniteness
of the building block matrix S1(a) would be equivalent to testing for the
negative semidefiniteness of the full CSM S(a). In other words, once
the symmetry condition of the off-diagonal blocks of S(a) is imposed on the
empirical demand functions, testing for the negative semidefiniteness of S(a)
reduces to testing for the negative semidefiniteness of S1(a). Clearly, this
affords a rather drastic simplification of the hypothesis testing process for the
multiple linear constraint utility maximization model.
According to Theorem 3, the building block matrix S1(a), and consequently the full CSM S(a) as well, must obey the K independent constraint
k 1
equations ΣM
m=1pmSmn(a) = 0, k = 1, 2, . . . , K. This conclusion can be verified
by differentiating the K budget constraints in identity form with respect to
p1n and m1. Accordingly, the rank of the building block matrix S1(a) cannot
exceed M − K, implying the same for the CSM matrix S(a). This conclusion
can also be confirmed by appealing to Theorem 4, which asserts that r(S(a))
© Blackwell Publishing Ltd 2006
56
M. Hossein Partovi and Michael R. Caputo
≤ min(M − K, KM) = M − K. Clearly, each added constraint lowers the rank
of the full CSM S(a) by adding a new zero to its spectrum, while in general
reducing the maximized utility level. As the number of constraints
approaches the dimension of the consumption bundle, i.e. as K → M, the
optimization process becomes progressively less relevant in determining the
chosen bundle, while, correspondingly, the full CSM S(a) loses rank and
information, until at last it vanishes altogether when K = M.
The above results for the K-linear constraint utility maximization problem
(11) are new. We summarize them here as follows:
Proposition 1: The comparative statics of the K-linear constraint utility maximization model defined by equation (11) are summarized by the statement that
the KM × KM matrix S(a) defined in equation (14) is negative semidefinite
and free of constraints. Moreover, r(S(a)) ≤ M − K.
We conclude this section by applying Theorem 2 to problem (11).
Because the direct utility function U(·) is not a function of the parameter
vector a, it follows from Theorem 2 that V;a(a) = 0 for a = 1, 2, . . . ,
KM. Thus, an income-compensated change in any price leaves the
utility-maximizing position unchanged, exactly as it does in the archetypal
model.
7. COMPARATIVE STATICS OF THE PRINCIPAL-AGENT PROBLEM
WITH HIDDEN ACTIONS
The principal-agent problem is an example of a class of models that involve
asymmetric information and uncertainty, as well as parameter overlap between the objective and constraint functions. Models involving uncertainty
typically entail a constraint that originates in the fact that the probability set
has unit measure, i.e. that the sum of all the probabilities equals unity.
Such constraints restrict the parameters but not the decision variables, and
therefore do not qualify as constraints in the usual sense. Indeed, they play
no role in determining the solution to the optimization problem. If, however,
one is interested in comparative statics involving the probabilities, then there
arises the question of how the constraint is to be implemented in parameter
space. We will show in what follows that there is a natural method of implementing such constraints which maintains the intrinsic symmetries of the
problem.
The model in question is the principal-agent problem with hidden actions,
where a firm, the principal, intends to hire an individual, the agent, to work
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
57
on a specific venture on a contractual basis. Since the agent’s effort level is a
choice variable in the principal’s profit maximization problem, the latter can
be formulated as a pair of maximization problems, one for each effort level,
and the optimum decided by comparing the results.
The basic problem is to maximize the principal’s expected profits, assuming a given effort level for the agent. Accordingly, it can be formulated as
follows:
M
M
max ∑ [p i − xi ] piI s.t. ∑ u(xi ) piI − c I ≥ u ,
x
i =1
i =1
M
M
M
i =1
i =1
i =1
∑ u(xi ) piI − c I ≥ ∑ u(xi ) piII − c II , ∑ pik = 1, k = I , II
Here pik ∈ (0, 1), i = 1, 2, . . . , M, k = I, II, is the probability that the ith profit
level pi is realized for the firm given that the agent performs at effort level k,
with I and II corresponding to high and low effort, respectively. The decision
variable xi is the agent’s compensation when the ith profit level is realized, ck
is the agent’s disutility of working at effort level k, ū is the market price of
the agent’s services and u(·) − ck is the agent’s utility function. Observe that
we have assumed that cI > cII, corresponding to the fact that the agent prefers
low effort to high effort, ceteris paribus. We have also assumed a high effort
level (I) for the agent, as is apparent from the objective function and the
second constraint. Once the optimum contract for the above problem and its
conjugate, which is arrived at by interchanging I and II in the above problem,
are determined, the principal chooses the more profitable compensation
schedule and offers the corresponding contract to the agent. Furthermore,
we assume, without loss of generality, that the principal finds k = I to be
the more profitable choice. We also assume that both inequality constraints
bind, since an inequality constraint that does not bind has no bearing on
the comparative statics of the problem. Finally, we assume that u′(xi) > 0,
i = 1, 2, . . . , M.
Under the foregoing assumptions, the principal’s problem can be
rewritten as
def


C (a ) = min ∑ piI xi s.t. B k − ∑ pik u(xi ) = 0, s k − ∑ pik = 0, k = I , II 
x 

i =1
i =1
i =1
M
def
M
M
(16)
where B k = c k + u , k = I, II, and the parameters sk > 0, k = I, II, are a
pair of auxiliary variables that will be set equal to unity at a convenient
© Blackwell Publishing Ltd 2006
58
M. Hossein Partovi and Michael R. Caputo
point in the course of the analysis. The parameter a is defined as
def
a = (p I , B I , s I , p II , B II , s II ) ∈ℜ 2( M + 2 ). We thus have a total of 2(M + 2)
parameters, to be eventually reduced to 2(M + 1) upon setting sk = 1,
k = I, II.
Our strategy here is to find GCDs for problem (16) by developing an intuitive generalization of the GCDs already constructed for the prototype utility
maximization problem. We accomplish this in two steps. In the first step, note
M k
that for each value of k the constraint Bk − Σi=1
pi u(xi) = 0 is analogous to the
prototype budget constraint. This analogy suggests the corresponding
∂
∂
+ u(xi ) k , k = I, II, which possess the null property
derivative
k
∂ pi
∂B
k
with respect to the constraints Bk − ΣM
i=1p i u(xi) = 0, k = I, II. The second
step is to amend this derivative so as to extend the null property to the
M k
constraints sk − Σi=1
pi = 0, k = I, II. The addition of the partial derivative
∂
∂
∂
+ u(xi ) k provides the desired extension of the null property
to
k
k
∂s
∂ pi
∂B
M k
to the constraints sk − Σi=1
pi = 0, k = I, II. Therefore, the directional
derivatives
d ik (x, a ) =
def
∂
∂
∂
+ u(xi ) k + k , i = 1, 2, . . . , M , k = I , II
k
∂ pi
∂B
∂s
have the required null property with respect to all the constraints and represent a complete set of GCDs for the principal-agent problem.
To proceed with the construction of the CSM, we note that the pertinent
first-order necessary conditions are piI − lIpiIu′(xi) − lIIpiIIu′(xi) = 0, i = 1, 2,
. . . , M, where lk, k = I, II, are the Lagrange multipliers. The CSM for this
problem is found by using the above GCDs and the first-order necessary
conditions in conjunction with Theorem 1. The result is a negative semidefinite matrix that may be written in the 2M × 2M block matrix form
11
Φ12 (a )
def  Φ (a )
Φ(a ) =  21

 Φ (a ) Φ 22 (a )
(17)
where Φ ijkk ′ (a ) = [2 − k − l k (a )u ′(xi (a ))]d kj ′ (a ) o xi (a ), i, j = 1, 2, . . . , M and k,
def
def
def
k′ = I, II, and where I = 1 and II = 2. By Theorem 4, r(Φ(a)) ≤ min(M − 2,
2M) = M − 2, so that Φ(a) is a highly redundant CSM since its order exceeds
its rank by at least M + 2. In other words, there are at least M + 2 zeros in
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
59
the spectrum of Φ(a), thereby implying that the properties of the full CSM
Φ(a) are fully conveyed by the statement that Φ11(a) is a negative semidefinite matrix whose rank does not exceed M − 2.
It is advantageous at this point to eliminate the auxiliary parameters sk,
k = I, II, from our results. Since partial derivatives with respect to these
parameters occur only in terms of the form djk(a) ° xi(a), they can be eliminated by recourse to a scale symmetry of the underlying problem. Inspection
of equation (16) reveals that rescaling the augmented parameter set
def
def
a = (a I , a II ) = (p I , B I , s I , p II , B II , s II ) ∈ℜ 2( M + 2 ) according to ak → mkak, where
k
m > 0, k = I, II, leaves the problem unchanged. By Theorem 3 this implies
the homogeneity condition
M k ∂
∂
∂ 
+ Bk
+ s k k x(a ) = 0M , k = I , II
∑ p j
k
k
∂B
∂s 
 j =1 ∂ p j
Using this equation for eliminating all the derivative terms with respect to
the auxiliary parameters sk and then setting the latter equal to unity results
in the replacement of the GCD set djk(x, a) with
Dik (x, a ) =
def
M
∂
∂
∂
− ∑ p kj
+ [u(xi ) − B k ] k , i = 1, 2, . . . , M , k = I , II (18)
k
k
p
B
∂ pi
∂
∂
j =1
j
We pause at this point to underline the role played by the auxiliary paramM k
eters in dealing with the pair of constraints Σi=1
pi = 1, k = I, II, which constrain the parameter space but not the decision space. Because these
constraints do not allow a change in one of the probabilities while the others
are kept fixed, a partial derivative of the sort ∂x(a)/∂pik does not correspond
to a realizable scenario in the real world. On the other hand, one can envision a change in a given probability with the compensating change required
by the constraint symmetrically allocated to all the probabilities. It is precisely this objective of enforcing the constraint in a symmetrical manner that
is accomplished by the introduction of the auxiliary parameters sk, k = I, II.
The end result, which is seen in equation (18), is the replacement of the
uncompensated derivative ∂/∂pik by the constraint-conforming combination
k
k
∂/∂pik − ΣM
j=1 pj ∂/∂pj . Observe that the role played by the auxiliary variables
here is entirely analogous to that of Lagrange multipliers in a constrained
optimization problem, the major distinction being that the latter are used to
enforce decision space constraints. Needless to say, the parameter constraints
present here are typical of any optimization problem that involves uncer-
© Blackwell Publishing Ltd 2006
60
M. Hossein Partovi and Michael R. Caputo
tainty. The method of auxiliary variables introduced above is thus a natural
and effective way of dealing with problems involving uncertainty.
def
To interpret the CSM Φ(a), observe that since B k = c k + u , it follows from
the envelope theorem that ∂C(a)/∂ck = lk(a), where ck is the agent’s
disutility for performing at effort level k. Thus, lk(a) represents the expected
marginal cost to the principal of the agent’s disutility for working at effort
level k. Recalling that cI > cII and that the contract offered to the agent
induces level I performance, we conclude that lI(a) ≥ 0 and lII(a) ≤ 0. A
detailed analysis using the first-order necessary conditions and the constraints, along with the properties of C(a), confirms these intuitive conclusions. In addition, since lII(a) ≤ 0, pik > 0 and u′(xi) > 0, k = I, II and i = 1,
2, . . . , M, the first-order necessary conditions imply 1 − lI(a)u′(xi(a)) ≤ 0,
i = 1, 2, . . . , M.
We are now in a position to state the comparative statics results for the principal-agent problem in a more useful form. First, let us consider the negative
def
semidefinite matrix Φ22(a), where Φ ij22 (a ) = − l II (a )u ′(xi (a ))D IIj (a ) o xi (a ). Since
lII(a) ≤ 0 and u′(xi) > 0, the M × M matrix H(a), with typical element
H ij (a ) =
def
∂ xi (a ) M II ∂ xi (a )
− ∑ pl
∂ p IIj
∂ plII
l =1
∂ x (a )
+ [u(x j (a )) − c − u ] i II , i , j = 1, 2, . . . , M
∂c
(19)
II
is also negative semidefinite and has a rank no larger than M − 2. Since
def
B k = c k + u , k = I, II, it follows that ∂xj(a)/∂BII = ∂xj(a)/∂cII, a fact that was
used in equation (19). The structural similarity of Hij(a) to the prototype
Slutsky equation is now evident, thereby implying a similar economic
interpretation. The extra terms here result from the constraint on the probabilities as well as the generally non-linear function u(·). For example, Hii(a) ≤ 0
implies that the appropriately compensated change in xi(a), the wage offered
by the principal in case profit level I is realized, as a result of an increase in
pjII, the probability of the jth profit level conditional on effort level II, is nonpositive. This is of course the expected response inasmuch as the principal’s
objective is to induce the agent to effort level I, and hence away from effort
level II.
To complete the analysis, consider the negative semidefinite matrix Φ11(a),
def
I
where Φ11
ij (a ) = [1 − l I (a )u ′( xi (a ))]D j (a ) o xi (a ). Given that 1 − lI(a)u′(xi(a)) ≤
0, i = 1, 2, . . . , M, the M × M matrix G(a), with typical element
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
Gij (a ) =
def
∂ xi (a ) M I ∂ xi (a )
− ∑ pl
∂ p Ij
∂ plI
l =1
+ [u(x j (a )) − c I − u ]
∂ xi (a )
, i , j = 1, 2, . . . , M ,
∂c I
61
(20)
is positive semidefinite and has a rank no larger than M − 2. Note that for
the elements on the main diagonal of the matrices G(a) and H(a), there is a
sign reversal in going from level I derivatives to those of level II. We summarize our comparative statics results for the principal-agent problem in the
ensuing proposition.
Proposition 2: The comparative statics of the principal-agent problem with
hidden action are conveyed by the M × M positive semidefinite matrix G(a) and
M × M negative semidefinite matrix H(a), neither of which has a rank larger
than M - 2.
To impart an economic interpretation to the terms Gii(a) ≥ 0, it is useful
to recall that the wage contract is designed to induce the agent to level I
effort. Then Gii(a) ≥ 0 asserts that given level I effort, an increase in the
probability that profit level i is realized, once compensated for the unit probability measure and the cost of level I effort, results in the principal offering
the agent a corresponding wage contract as least as lucrative as before. That
is, a probability- and cost-compensated level I wage contract is upward
sloping in its level I probability of profit. This conclusion therefore is entirely
analogous to the conclusion that income-compensated demand curves are
downward sloping in their own prices in the basic utility maximization
model.
As already emphasized, the above analysis and its results pertain to
the case where the contract is designed to induce effort level I. While
the analysis for the complementary case is parallel to the foregoing, the
results are not expected to be symmetrical with respect to an interchange
of I and II since the condition cI > cII breaks the symmetry between the two
cases.
8. CONCLUDING REMARKS
The main objective of this work, namely the derivation of unconstrained
comparative statics matrices for a general, differentiable, constrained optimization problem, has been fully realized. The result, stated in Theorem 1,
© Blackwell Publishing Ltd 2006
62
M. Hossein Partovi and Michael R. Caputo
is not merely an existence result, for it provides a natural and powerful
method for constructing CSMs with constraints already implemented. In
effect, Theorem 1 completes the program initiated and developed by Samuelson (1947), generalized and streamlined by Silberberg (1974), and further
advanced by Hatta (1980). Moreover, we have established in detail how the
comparative statics results of these authors are subsumed in Theorem 1. We
have also developed a number of other new results and extensions summarized in Theorems 2–5 that further characterize the properties of the CSMs
and thereby serve to broaden the power and reach of the analysis. In particular, the realization that comparative statics results for a given problem can
assume a wide range of forms and textures significantly strengthens their role
in hypothesis testing, the primary raison d’être for all comparative statics
analyses.
Throughout, we have dealt with the comparative statics of a given, interior solution to an optimization problem. As such, there is no need to deal
with inequality constraints, since those that bind can be treated as equality
constraints, and those that do not can be ignored altogether. Nor have we
concerned ourselves with issues of integrability, since these are primarily relevant to utility maximization problems of a particular structure. Similarly,
although we have not dealt with problems involving discrete-time, finitehorizon, intertemporal optimization, these and other categories can be
treated straightforwardly by our method.
APPENDIX
Proof of Theorem 1: When restricted to the solution x(a) of problem (1), the
first-order and second-order necessary conditions are given by
K
L,i (x(a ), l (a ), a ) = f,i (x(a ), a ) + ∑ l k (a )g,ki (x(a ), a ) = 0,
k =1
(A1)
i = 1, 2, . . . , M
M
M
∑ ∑l L
i
i =1 j =1
,ij
M
(x(a ), l (a ), a )l j ≤ 0 ∀ l ∈ℜ M ∋ ∑ l i g,ki (x(a ), a ) = 0,
i =1
k = 1, 2, . . . , K
(A2)
Recall that since h(a ) = ∑a =1ha x ;a (a ) conforms to the constraints by condef
A
struction, equation (A2) holds free of constraints if h(a) is substituted for l.
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
63
Since h ∈ ℜA is arbitrary, equation (A2) implies that the matrix Ω(a), the
typical element of which is given by
M
M
Ωab (a ) = − ∑ ∑ xi ;a (a )L,ij (x(a ), l (a ), a )x j ;b (a ), a , b = 1, 2, . . . , A
def
i =1 j =1
(A3)
is positive semidefinite free of constraints. The symmetry of Ω(a) is a consequence of the assumptions f(·) ∈ C (2) and gk(·) ∈ C (2) for k = 1, 2, . . . , K.
Equation (A3) can be rewritten by first applying the GCD Da(a) to the firstorder necessary condition (A1), obtaining
M
∑L
,ij
j =1
(x(a ), l (a ), a )x j ;a (a ) = − L,i ;a (x(a ), l (a ), a )
(A4)
K
− ∑ g,ki (x(a ), a )l k ;a (a )
k =1
If equation (A4) is now multiplied by xi;b(a) and summed over i, the second
term on the right-hand side vanishes by Lemma 1. Upon using the definition
of Ω(a) from equation (A3), this operation leaves a simplified expression for
Ω(a), i.e.
M
Ωab (a ) = ∑ xi ;b (a )L,i ;a (x(a ), l (a ), a )
i =1


= ∑ xi ;b (a ) f,i ;a (x(a ), a ) + ∑ l k (a )g,ki ;a (x(a ), a ), a , b = 1, 2, . . . , A.


i =1
k =1
M
K
It is appropriate to recall here that Ω(a) is a symmetric matrix, that is to say,
Ωab(a) = Ωba(a), a fact that we have already used in writing Theorem 1 and
will continue to use throughout this work.
Proof of Theorem 2: By equations (1) and (2) we have V (a ) = f (x(a ), a ).
def
Apply the GCD Da(a) to the definition V (a ) = f (x(a ), a ) and the constraint
k
g (x(a), a) = 0, k = 1, 2, . . . , K, to get
def
M
V;a (a ) = ∑ f,i (x(a ), a )xi ;a + f;a (x(a ), a )
i =1
© Blackwell Publishing Ltd 2006
64
M. Hossein Partovi and Michael R. Caputo
M
∑g
k
,i
i =1
(x(a ), a )xi ;a (a ) + g;ka (x(a ), a ) = 0, k = 1, 2, . . . , K
Next, multiply the second equation by lk(a), sum over k and recall that
k
g;a
(x(a), a) = 0 by the definition of a GCD. This sequence of operations yields
K
k
ΣM
i=1Σ k=1lk(a)g,i (x(a), a)xi;a(a) = 0. Now add this latter result to the first equation above to get


V;a (a ) = ∑  f,i (x(a ), a ) + ∑ l k (a )g,ki (x(a ), a )xi ;a (a ) + f;a (x(a ), a )

k =1
i =1 
K
M
But each term of the summand is identically zero by the first-order necessary conditions (A1), hence V;a(a) = f;a(x(a), a). If the GCDs have the null
property with respect to the objective function, i.e. if f;a(x(a), a) = 0, then the
envelope property reduces to V;a(a) = 0.
Proof of Theorem 3: Differentiate equations (5) and (6) with respect to xj:
M
∑ [X ( x) f
,ij
i
i =1
( x, a ) + X i , j f,i ( x, a )]
(A5)
N
+ ∑ Am (a ) f, mj ( x, a ) = F ′( f ( x, a )) f, j ( x, a )
m =1
M
∑ [X (x )g
i
i =1
k
,ij
(x, a ) + X i , j (x )g,ki (x, a )]
(A6)
N
+ ∑ Am (a )g (x, a ) = G ′ ( g k (x, a ))g,kj (x, a ), k = 1, 2, . . . , K
k
,mj
k
m =1
Next multiply equation (A6) by lk and sum over k, add the result to equation
(A5), and simplify using the definition L(x, l , a ) = f (x, a ) + ∑k =1 l k g k (x, a )
def
K
from equation (3). Then use the first-order necessary conditions L,i(x(a), l(a),
a) = f,i(x(a), a) + ΣKk=1lk(a)gk,i(x(a), a) = 0, i = 1, 2, . . . , M, to simplify the resulting equation, and evaluate it at x(a) to get the formula
M
∑ X (x(a ))L
,ij
i
i =1
N
(x(a ), l (a ), a )) + ∑ Am (a )L,mj (x(a ), l (a ), a ))
m =1
K
= − ∑ l k (a )g (x(a ), a ) F ′( f (x(a ), a )) − G k ′ ( g k (x(a ), a )).


k =1
k
,j
© Blackwell Publishing Ltd 2006
(A7)
Theory of Comparative Statics
65
Next, evaluate equation (6) at x(a) and use Gk(0) = 0, k = 1, 2, . . . , K,
to get
M
∑ X (x(a ))g
i
i =1
N
k
,i
(x(a ), a ) + ∑ Am (a )g,km (x(a ), a ) = 0, k = 1, 2, . . . , K
m =1
Then differentiate the kth constraint identity gk(x(a), a) = 0, k = 1, 2, . . . , K,
with respect to am:
M
∑g
k
,i
i =1
(x(a ), a )xi ,m (a ) + g,km (x(a ), a ) = 0, k = 1, 2, . . . , K
Now eliminate g ,mk (x(a), a) from the last two equations to get the expression
M
∑ Z (a )g
i
k
,i
i =1
(x(a ), a ) = 0, k = 1, 2, . . . , K
(A8)
where Z(a ) = X(x(a )) − ∑m =1 Am (a )x ,m (a ). Equation (A8) implies that the
N
def
vector Z(a) lies in the tangent hyperplane to the constraint functions in decision space. Next multiply equation (A7) by Zj(a), sum over j and take account
of equation (A8) to arrive at the result
M
M
∑ ∑ X (x(a ))L
,ij
i
i =1 j =1
M
(x(a ), l (a ), a ))Z j (a )
(A9)
N
+ ∑ ∑ Am (a )L,mj (x(a ), l (a ), a ))Z j (a ) = 0
j =1 m =1
Now differentiate the first-order necessary condition L,j (x(a), l(a), a) = 0,
j = 1, 2, . . . , M, with respect to am, multiply the result by Zj(a), sum over j
and apply equation (A8) to eliminate one set of terms. Next, multiply the
latter result by Am(a), sum over m and use the symmetry of the second-order
partial derivatives of the Lagrangian function to arrive at the expression
M
M
N
∑ ∑ ∑ A (a )L
m
i =1 j =1 m =1
M
,ij
(x(a ), l (a ), a ))xi ,m (a )Z j (a )
N
+ ∑ ∑ Am (a )L,mj (x(a ), l (a ), a ))Z j (a ) = 0
j =1 m =1
© Blackwell Publishing Ltd 2006
(A10)
66
M. Hossein Partovi and Michael R. Caputo
Finally, substitute equation (A10) into equation (A9) to get
M
M
∑ ∑ Z (a )L
,ij
i
i =1 j =1
(x(a ), l (a ), a ))Z j (a ) = 0
(A11)
If the second-order sufficient condition holds, then since Z(a) lies in the
tangent hyperplane to the constraint functions in decision space, equation
(A11) implies Zi(a) = 0, i = 1, 2, . . . , M.
Proof of Theorem 4: By Lemma 1 the A vectors x;a(a) lie in the tangent hyperplane to the constraints in decision space. Since the K vectors xgk(x(a),
a) are linearly independent, the implicit function theorem implies that
the tangent hyperplane to the constraints in decision space is of dimension
M − K, and thus at most M − K of the A vectors x;a(a) can be linearly
independent, i.e. r(xi;a(a)) ≤ M − K. Recalling that Ω(a) is an A × A matrix,
the result follows.
def
Proof of Theorem 5: First, substitute the isovectors t˜ma = ∑g =1Cag tmg into
A
N
N
A
def
def
the GCDs D˜ a (a ) = ∑m =1t˜ma ∂ ∂ am to get the formula D̃a (a ) = ∑m =1 ∑g =1
Cag tmg ∂ ∂ am = ∑g =1Cag Dg (a ).
A
Next,
substitute
the
resulting
GCDs
A
def
˜ ab (a ) = ∑M [D˜ b (a ) o xi (a )]
D̃a (a ) = ∑g =1Cag Dg (a ) into the matrix element Ω
i =1
[D˜ a (a ) o L,i (x(a ), l (a ), a )] to arrive at
M
A
A
˜ ab (a ) = ∑ ∑ C bd Dd (a ) o xi (a ) ∑ Cag Dg (a ) o L,i (x(a ), l (a ), a )
Ω


 
 g =1

i =1  d =1
A
A
M
A
A
= ∑ ∑ ∑ C bd xi ;d (a )L,i ;g (x(a ), l (a ), a )Cag = ∑ ∑ Cag Ωgd (a )C bd ,
d =1 g =1 i =1
which is what we wished to show.
d =1 g =1
REFERENCES
Blomquist, N. S. (1989): ‘Comparative statics for utility maximization models with nonlinear
budget constraints’, International Economic Review, 30, pp. 275–96.
© Blackwell Publishing Ltd 2006
Theory of Comparative Statics
67
Caputo, M. R. (1999): ‘The relationship between two dual methods of comparative statics’,
Journal of Economic Theory, 84, pp. 243–50.
Caputo, M. R., Paris, Q. (2000): Comparative Statics of the Generalized Maximum Entropy Estimator of the General Linear Model, University of Central Florida, Orlando, FL.
Caputo, M. R., Partovi, M. H. (2002): Fundamental Comparative Statics of a General Class of
Profit-maximizing Rate-of-return Regulated Firms, University of Central Florida, Orlando,
FL.
Chichilnisky, G., Kalman, P. J. (1977): ‘Properties of critical points and operators in economics’, Journal of Mathematical Analysis and Applications, 57, pp. 340–9.
Chichilnisky, G., Kalman, P. J. (1978): ‘Comparative statics of less neoclassical agents’, International Economic Review, 19, pp. 141–8.
Hatta, T. (1980): ‘Structure of the correspondence principle at an extremum point’, Review of
Economic Studies, 47, pp. 987–97.
Houthakker, H. S. (1951–52): ‘Compensated changes in quantities and qualities consumed’,
Review of Economic Studies, 19, pp. 155–64.
Kalman, P., Intriligator, M. D. (1973): ‘Generalized comparative statics with applications to consumer and producer theory’, International Economic Review, 14, pp. 473–86.
Partovi, M. H., Caputo, M. R. (1998): A Complete Method of Comparative Statics for Optimization Problems, University of Central Florida, Orlando, FL.
Partovi, M. H., Caputo, M. R. (2006): ‘Existence of a universal comparative statics matrix for
differential optimization problems’, Economic Theory, forthcoming.
Samuelson, P. A. (1947): Foundations of Economic Analysis, Harvard University Press,
Cambridge, MA.
Silberberg, E. (1974): ‘A revision of comparative statics methodology in economics, or, how
to do comparative statics on the back of an envelope’, Journal of Economic Theory, 7,
pp. 159–72.
Silberberg, E. (1990): The Structure of Economics: A Mathematical Analysis, 2nd edn, McgrawHill Publishing Company, New York.
M. Hossein Partovi
Department of Physics and Astronomy
California State University
Sacramento
CA 95819-6041
USA
E-mail: [email protected]
© Blackwell Publishing Ltd 2006
Michael R. Caputo
Department of Economics
University of Central Florida
PO Box 161400
Orlando
FL 32816-1400
USA
E-mail: [email protected]
Metroeconomica 58:2 (2007)
360
ERRATUM
An error was introduced in the article by M. Hossein Partovi and Michael R.
Caputo entitled ‘A Complete Theory of Comparative Statics for Differentiable Optimization Problems’, published in Volume 57, Number 1 of Metroeconomica. The error in question appeared on the right-hand side of the
(unnumbered) equation in Theorem 1 on p. 44 of this paper.
In this equation the term xi;b(a) erroneously appeared as xi,b(a). It is important to note that the change from the semicolon (correct) to a comma (incorrect) in this term corresponds to replacing a generalized compensated
derivative (correct) with an ordinary partial derivative (incorrect), thereby
rendering the equation false. The correct form, which also appeared on p. 63
of the paper, reads:
M
Ω αβ (a ) = ∑ xi ;β (a )L,i ;α ( x (a ) , l (a ) , a )
i =1
M
K
⎡
⎤
= ∑ xi ;β (a ) ⎢ f,i ;α ( x (a ) , a ) + ∑ λ k (a ) g,ki ;α ( x (a ) , a )⎥ , α , β = 1, 2, . . . , A
⎣
⎦
i =1
k =1
REFERENCE
Partovi, M. H., Caputo, M. R. (2006): ‘A Complete Theory of Comparative Statics for Differentiable Optimization Problems’, Metroeconomica, 57 (1), pp. 31–67.
Journal compilation © 2007 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford, OX4
2DQ, UK and 350 Main St, Malden, MA 02148, USA