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. 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(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
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