Economics Letters 66 2000 265–273
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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 optimization 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
Fa,
b : 5max fx;a
s.t. gx;a
b P
h j
x
where
xa, b : 5arg max fx;a
s.t. gx;a
b , 1
h j
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
: caputoprimal.ucdavis.edu M.R. Caputo 0165-1765 00 – see front matter
2000 Elsevier Science S.A. All rights reserved.
P I I : S 0 1 6 5 - 1 7 6 5 9 9 0 0 2 2 9 - 3
266 M
.R. Caputo Economics Letters 66 2000 265 –273
Ga, g : 5min gx;a
s.t. fx;a
g T
h j
x
where
ˆ xa,
g : 5arg min gx;a s.t.
fx;a g .
2 h
j
x
The corresponding Lagrangian functions for P and T are given by, respectively,
Lx, l;a,b : 5 fx;a 1 l[b 2 gx;a],
3
Mx, m;a,g : 5 gx;a 1 m[g 2 fx;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
