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Lagrangian transposition identities and reciprocal pairs of
constrained optimization problems
*
Michael R. Caputo
Department of Agricultural and Resource Economics, University of California, One Shields Avenue, Davis, CA 95616, USA
Received 9 September 1998; accepted 9 September 1999
Abstract
A simple but rigorous proof of the Lagrangian Transposition Identities is given, and symmetric and semidefinite comparative statics matrices are derived, thereby providing a comprehensive qualitative characteri-zation of any sufficiently smooth reciprocal pair of constrained optimicharacteri-zation problems. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Comparative statics; Optimization theory
JEL classification: C60; C61
1. Introduction
Loosely speaking, the Lagrangian Transposition Principle (Panik, 1976, p. 209) asserts that the solution of the primal constrained maximization problem
F(a,b):5max f(x;h a) s.t. g(x;a)#bj (P)
x
where
x*(a,b):5arg max f(x;h a) s.t. g(x;a)#bj, (1)
x
is identical to the solution of the transposed (or reciprocal, or mirrored) constrained minimization problem
*Tel.:11-530-752-1519; fax: 11-530-752-5614.
E-mail address: [email protected] (M.R. Caputo)
G(a,g):5min g(x;h a) s.t. f(x;a)$gj (T)
x
where
ˆ
x(a,g):5arg min g(x;h a) s.t. f(x;a)$gj. (2)
x
The corresponding Lagrangian functions for (P) and (T) are given by, respectively,
L(x,l;a,b):5f(x;a)1l[b2g(x;a)], (3)
M(x,m;a,g):5g(x;a)1m[g2f(x;a)]. (4)
It appears that it was not until Henderson and Quandt’s (1958, p. 52) book that economists noticed the reciprocity or symmetry inherent between pairs of constrained optimization problems such as (P) and (T). Along the same vein, Silberberg (1978, pp. 234–238) proves that the local necessary and sufficient conditions are identical for the two-variable reciprocal pair of consumer problems, utility maximization and expenditure minimization. Taking a different approach, Varian (1978, p. 112) gives conditions for which the solution to the utility maximization problem is a solution to the expenditure minimization problem, and vice versa. In a general setting, Newman (1982) lays out conditions under which a solution to a primal constrained maximization problem is a solution to the reciprocal constrained minimization problem, and vice versa. The contribution of this note is in establishing a complete qualitative characterization of, and the relationship between, the solution functions and indirect objective functions of the reciprocal pair of constrained optimization problems (P) and (T). This paper thus represents the logical subsequent step to Newman (1982) in analyzing reciprocal pairs of constrained optimization problems.
In a simple but rigorous manner, this paper establishes four fundamental identities linking the values of the indirect objective functions and values of the solution functions for (P) and (T). By presenting a proof of the identities in a general setting, the necessity of proof for each separate application is obviated. With differentiability, the reciprocal nature of the Lagrange multipliers for (P) and (T), as well as a vastly simplified proof of the existence of the generalized Slutsky matrix of Kalman and Intriligator (1973, Proposition 2) follow from the identities. As a result, the compensation operation of Kalman and Intriligator (1973) is shown to have an intuitive and natural interpretation. Moreover, Kalman and Intriligator’s (1973, Theorem 3) proof of the negative semidefiniteness of their generalized matrix of substitution effects is shown to be incorrect, and a new proof is offered. The value of their theorem is of limited use, however, for it requires strong sufficient conditions on the structure of the optimization problem which are often times not satisfied. In order to rectify this situation, a general constraint-free symmetric and semidefinite comparative statics matrix is derived for problems (P) and (T) — even though problems (P) and (T) are constrained optimization problems — thereby facilitating empirical testing of the underlying economic theory.
2. Assumptions and fundamental identities
m n
Assumption 6. The (n11)3(n11) bordered Hessian determinants
+ + + + + + +
Assumption 1 defines the domain and target space of the functions f and g, and also serves to define the dimensionality of the vectors x anda. Assumptions 2 and 3 assert the existence of a solution to
(P) and (T) at a given point in the appropriate space. Assumption 5 asserts that the optimal values of the Lagrange multipliers from (P) and (T) are nonzero, which in turn implies that the constraints are binding (or tight) at the optimum, rather than just binding, as would be the case for prototype
ˆ
investigations of the Le Chatelier principle (see, e.g., Silberberg, 1978, pp. 293–298). Assumptions 2–6, in conjunction with the first-order necessary conditions for (P) and (T) and the implicit function
( 1 ) + + ˆ ( 1 ) + +
theorem, imply that x*[C ;(a,b)[B((a,b );d ) and x[C ;(a,g)[B((a,g );d ). Moreover,
P T
Assumptions 2–6 imply that the second-order sufficient condition holds at the solution to (P) and (T)
+ + ˆ
Assumptions 1–7 are not the most general sufficient conditions under which the subsequent results will hold. That being said, it is still true that when the focus is on the qualitative properties of a model, as it is here, these assumptions are often (but not always) maintained either implicitly or explicitly in optimization problems in economics, or are implied by other stronger sufficient conditions imposed on the primitives. As a result, remarks will be offered after the proof of Theorem 1 as to which assumptions can be relaxed and which are crucial for the conclusions of the theorem. The convention that the derivative of a scalar-valued function with respect to a column vector is a row vector is adopted.
Theorem 1. (Lagrangian Transposition Identities). Under Assumptions 1–7, the following identities link the values of the indirect objective functions and the values of the solution functions for the reciprocal pair of constrained optimization problems (P) and (T ):
+ +
From (1), Assumption 2, and the constraint of (P) it follows that
The exact interpretation of Theorem 1 is important. For example, part (a) asserts that the value of the choice vector that solves (P) is identically equal to the value of the choice vector that solves (T) when the value of the constraint function in (T), namely g, is set equal to the value of the indirect objective function in (P), F(a,b). Notice that the values of the choice functions are asserted to be
ˆ
equal, not the functions themselves, as the notation (x* vs. x ) and the parameters which the functions depend on make clear. Likewise, part (c) asserts that the value of the choice vector that solves (T) is identically equal to the value of the choice vector that solves (P) when the value of the constraint function in (P), namely b, is set equal to the value of the indirect objective function in (T), G(a,g).
Part (b) asserts that the functions G and F are inverse functions of one another with respect to the constraint parameterb of (P), holdinga fixed. Similarly, part (d) asserts that the functions F and G
are inverse functions of one another with respect to the constraint parameterg of (T), holdingafixed.
As noted earlier, Theorem 1 holds under more general conditions than those stated in Assumptions ( 2 )
1–7. For example, the C nature of f and g can be dispensed with, without necessarily invalidating Theorem 1. Continuity of f and g, however, cannot be dropped since it is crucial to the proof of parts (a) and (c). Similarly, uniqueness of the solutions is needed in the proof of parts (a) and (c). Parts (b) and (d) of Theorem 1, on the other hand, will hold under more general conditions than parts (a) and (c). For instance, the presence of multiple solutions to (P) and (T) would not affect parts (b) and (d), but it would invalidate parts (a) and (c) as stated.
In economic theory, much of the interest in optimization models centers on the derivatives of the
ˆ
functions (x*,F,x,G) and the corresponding relationships between the derivatives. Such derivative relationships are the content of the following theorem, whose proof follows by differentiating the identities in Theorem 1 using the chain rule.
Theorem 2. (Generalized Comparative Statics). Under Assumptions 1–7, the following derivative decompositions hold for the reciprocal pair of constrained optimization problems (P) and (T ):
The first identity in part (c) of Theorem 2 evaluated at g5F(a,b) is a generalization of the
derivation of the Slutsky matrix a la Cook (1972), which has been subsequently repeated by Jehle (1991, p. 175), Silberberg (1978, pp. 248–250), Takayama (1985, p. 143), and Varian (1978, pp. 130–134), among others. An alternative and also more general proof of the Slutsky matrix follows from using both parts of (a), the first part of (d) evaluated atg5F(a,b), and part (b) of Theorem 1, a
route of proof noted only by Silberberg (1978, p. 261). Moreover, this last derivation shows that not all of the derivative decompositions in Theorem 2 are independent of one another. The second identities in (a) and (c) are the generalized Slutsky-like decompositions for ‘income’ and the ‘utility level.’
The first identities in parts (b) and (d) are the generalized Roy-like identities. By the Envelope Theorem ≠F(a,b) /≠b;l*(a,b)±0 is the optimal value of the Lagrange multiplier from (P), and
ˆ
similarly ≠G(a,g) /≠g;m(a,g)±0 is the optimal value of the Lagrange multiplier from (T). Thus,
the second identities of (b) and (d) establish the reciprocal nature of the Lagrange multipliers from (P)
21
ˆ
and (T), namely, l*(a,b);[m(a,g)] wheng5F(a,b) or b5G(a,g). The reciprocal nature of
the Lagrange multipliers in (P) and (T) is now no longer surprising, since it was pointed out in the discussion of Theorem 1 that the functions F and G are inverses of one another with respect to the constraint parameters (b,g), holding a fixed.
3. Comparative statics and the generalized Slutsky matrix
One important aspect achieved by Theorem 2 is the vast simplification of the proof of the existence of the generalized Slutsky matrix of Kalman and Intriligator (1973, Proposition 2). Their proof requires slugging through three journal length pages of matrix algebra and partitioned inversion of an (n11)3(n11) matrix. Moreover, they defined the operation of a compensated change in a parameter vector in such a way so as to hide the fact that the compensated comparative statics matrix is more naturally found by solving for a particular comparative statics matrix of the reciprocal constrained optimization problem, thereby concealing the fundamental nature of their mathematical operation of compensation.
In order to see the simplicity afforded by the Lagrangian Transposition Identities in proving the existence of the generalized Slutsky matrix, use the second part of (a) of Theorem 2 to get
21
which is identical to Proposition 2 of Kalman and Intriligator (1973) due to the envelope result
≠F(a,b) /≠b;l*(a,b)±0 noted earlier. The fundamental nature of the mathematical operation
ˆ
implicit in the compensated comparative statics matrix ≠x(a,F(a,b)) /≠a of Kalman and Intriligator
(1973) is, therefore, made explicit and intuitive via the Lagrangian Transposition Identities: it is the comparative statics matrix of the decision vector with respect to a associated with the transposed
problem (T), in which the compensation operation holdsg constant, as is clear given the definition of
ˆ
Symmetrically, since the transpose of (T) is (P), the comparative statics matrix ≠x* /≠a from
ˆ
problem (P) is the compensated comparative statics matrix of≠x /≠a from problem (T), only now the
compensation operation holdsb constant, as is obvious given the definition of x*(a,b) in (1) and the
definition of the partial derivative. Thus, the term compensation, as used by Kalman and Intriligator (1973), means that one has solved for the comparative statics matrix with respect toaof the transpose
of the constrained optimization problem under consideration, whether that be (P) or (T).
The error in Kalman and Intriligator’s (1973) proof of the negative semidefiniteness of the generalized Slutsky matrix is their assertion (p. 481) ‘‘ . . . that the second-order conditions for a local maximum of L are the conditions that the quadratic forms of the matrix
≠g 9
be negative semidefinite,’’ where L(x, y):5F(x,a)1y[b2g(x,a)] is the Lagrangian function for their
problem and their notation is adopted for this demonstration. To see the error, note that the
2 2 n
second-order necessary condition for their problem h9[≠ L /≠x ]h#0;h[R ][≠g /≠x]h50 (see,
e.g., Takayama, 1985, Theorem 1.E.16) is not equivalent to, nor does it imply, the incorrect second-order necessary condition asserted by Kalman and Intriligator (1973) in the above quote. This error invalidates the remaining steps in their proof since they are based on the incorrectly postulated negative semidefiniteness of (6).
The key to obtaining a proof of the negative semidefiniteness of the generalized Slutsky matrix is to show that the matrix
First recall that there exists an optimal solution to their constrained optimization problem under their assumptions, so that by Theorem 1.E.16 of Takayama (1985) noted above, the second-order
2 2 n
necessary condition h9[≠ L /≠x ]h#0;h[R ][≠g /≠x]h50 holds at the solution. By their
Assumption 4, the matrix in (6) is nonsingular, so it follows that the second-order sufficient condition
2 2 n
Takayama (1985, footnote 16, p. 167) that D is symmetric negative semidefinite. It immediately follows that the generalized Slutsky matrix of Theorem 3 in Kalman and Intriligator (1973) is negative semidefinite, given that the sufficient conditions stated in their Theorem 3 hold.
Even though Theorem 3 in Kalman and Intriligator (1973) has now been shown to be correct, this does not assuage any concerns about its usefulness. Such concerns center around the strong sufficient conditions that must hold for the theorem to be valid. In order to mollify such legitimate concerns, the following theorem establishes a general comparative statics symmetry and semidefiniteness result for the reciprocal pair of constrained optimization problems (P) and (T), the proof of which follows from Theorem 2 of Caputo (1999).
Theorem 3. (Symmetry and Semidefiniteness). Under Assumptions 1–7, the following comparative statics matrices are symmetric and obey the semidefiniteness properties below in the reciprocal pair of constrained optimization problems (P) and (T ), respectively:
≠x*(a,b) ≠x*(a,b)
m
]]] ]]]
h9
H
La sx*(a,b),l*(a,b);a,bdF
1 g x*(s a,b);adGJ
h$0;h[Rx ≠a ≠ a
b
+ +
and ;(a,b)[B (
s
a,b );dd
, (a)P
ˆ ˆ
≠x(a,g) ≠x(a,g)
m
ˆ ˆ ]]] ]]] ˆ
h9
H
Ma sx(a,g),m(a,g);a,gdF
1 f x(s a,g);adGJ
h#0;h[Rx a
≠a ≠g
+ +
and ;(a,g)[B (
s
a,g );dd
. (b)T
This is the kind of general result that Kalman and Intriligator (1973) were probably hoping for in their analysis, but were unfortunately not able to achieve. This was due to (i) their insistence in deriving a generalized Slutsky equation (given in Eq. (5)) that was basically of the same structural form as the prototype Slutsky equation and (ii) their primal vista of the constrained optimization problem (P). The dual view adopted in Theorem 3 leads naturally to the basal form of the comparative statics matrix that possesses the requisite symmetry and semidefiniteness property. Moreover, Theorem 3 also achieves the noteworthy feat of expressing the comparative statics properties of the reciprocal pair of constrained optimization problems (P) and (T) in the form of unconstrained symmetric and semidefinite matrices, thereby facilitating empirical testing of the underlying economic theory. Furthermore, Theorem 3 does not restrict the parameter vectorato be of the same dimension
as the decision vector x, whereas Kalman and Intriligator (1973, Theorem 3) had to impose such a restriction. Theorem 3, therefore, completes the mission of this paper in that a complete qualitative characterization of the reciprocal pair of constrained optimization problems (P) and (T) has now been achieved.
Acknowledgements
References
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