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.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
In order to provide a simple and direct proof of the fundamental identities, the following assumptions are imposed on problems P and T, and are discussed subsequently. They are sufficient
for the ensuing analysis to hold, but are not the weakest possible:
M .R. Caputo Economics Letters 66 2000 265 –273
267
m n
Assumption 1. f : X3A
→ R and g: X3A
→ R, where A,R and X,R are open sets.
+ +
+ +
+
Assumption 2. a solution to P when a, b 5 a ,b [ A 3 R, namely x 5 xa ,b defined in
P
1.
+ +
+ +
+
ˆ Assumption 3. a solution to T when a,
g 5 a ,g [ A 3 R, namely x 5 xa ,g defined in
T
2.
2 2
Assumption 4. f [C ;x,a[X3A and g[C
;x,a[X3A.
+ +
+ +
+
Assumption 5. l ±0 and m ±0, where l 5 la ,b is the optimal value of the Lagrange
+ +
+ +
+
ˆ multiplier in P when a,
b 5 a ,b and m 5 ma ,g is the optimal value of the Lagrange
+ +
multiplier in T when a, g 5 a ,g .
Assumption 6. The n 1 1 3 n 1 1 bordered Hessian determinants
+ +
+ +
+ +
+
f x ;a 2 l g x ;a
2 g x ;a 9
xx P
xx P
x P
n 3n n 31
±
+ +
2 g x ;a
x P
131 13n
+ +
+ +
+ +
+
g x ;a 2 m f x ;a
2 f x ;a 9
xx T
xx T
x T
n 3n n 31
and ±
0,
+ +
2 f x ;a
x T
131 13n
where the symbol 9 denotes transposition.
Assumption 7. For all z [ X and ; e . 0, there exists some x [ Bz;e X such that fx;a . fz;a
and gx;a , gz;a for each a [ A, where Bz; e is the open n-ball centered at z [ X of radius e . 0.
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 and a. 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 +
+ 1
+ +
ˆ theorem, imply that x [ C
;a, b [ Ba ,b ;d and x [ C
;a, g [ Ba ,g ;d . Moreover,
P T
Assumptions 2–6 imply that the second-order sufficient condition holds at the solution to P and T
+ +
ˆ and, therefore, that xa,
b is the unique solution to P ;a,b [ Ba ,b ;d and that xa,g is
P +
+
the unique solution to T ;a, g [ Ba ,g ;d . Finally, Assumption 7 essentially requires that the
T
functions f and g exhibit local nonsatiation. That is, it asserts that at each point z in the domain of the functions f and g, one can always find another point x in a neighborhood of z, no matter how large or
small that neighborhood is, such that the value of the function f will increase and the value of the function g will decrease. In passing, note that under slightly altered assumptions, one could invoke the
Theorem of the Maximum Berge, 1963, p. 116 to deduce certain continuity properties of the indirect objective functions and solutions functions of problems P and T.
268 M
.R. Caputo Economics Letters 66 2000 265 –273
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 :
+ +
ˆ xa,
b
; xa,Fa,b ;a,b [ Ba ,b ;d
a
P +
+
Ga,Fa, b
; b;a,b [ Ba ,b ;d b
P +
+
ˆ xa,
g
; xa,Ga,g ;a,g [ Ba ,g ;d
c
T +
+
Fa,Ga, g
; g ;a,g [ Ba ,g ;d d
T
Proof 1. Letting
g 5 Fa,b it follows from 2 and Assumption 3 that
ˆ xa,Fa,
b : 5arg min gx;a
s.t. fx;a Fa,
b , h
j
x
+ +
ˆ so that from the constraint and Assumption 5 fxa,Fa,
b ;a ; Fa,b ;a,b [ Ba ,b ;d . But
P +
+
by P, 1, and Assumption 2, Fa, b
; fxa,b ;a;a,b [ Ba ,b ;d . The last two identities
P +
+
ˆ ˆ
imply that fxa,Fa,
b ;a
; fxa,b ;a;a,b [ Ba ,b ;d . If xa,Fa,b is feasible in P,
P
ˆ i.e., if g xa,Fa,
b ;a b, then it follows from the last identity and uniqueness that xa,b
; s
d
+ +
ˆ ˆ
xa,Fa, b ;a,b [ Ba ,b ;d . To prove that xa,Fa,b is feasible in P and thus complete
P
ˆ the proof, assume to the contrary that g xa,Fa,
b ;a . b. By the continuity of g from Assumption s
d
¯ ˆ
¯ 4, there exists a point x such that g xa,Fa,
b ;a . gx;a . b, and furthermore by Assumption 7 it
s d
¯ ˆ
also follows that fx;a . f xa,Fa,
b ;a ; Fa,b . But this contradicts the fact noted at the
s d
ˆ beginning of this part of the proof that xa,Fa,
b solves problem T when g 5 Fa,b , thereby
ˆ implying that xa,Fa,
b is feasible in P.
Proof 2. Letting
g 5 Fa,b it follows from T, 2, and Assumption 3 that
+ +
ˆ
Ga,Fa, b
; gxa,Fa,b ;a;a,b [ Ba ,b ;d .
P
From 1, Assumption 2, and the constraint of P it follows that
+ +
gxa, b ;a
; b;a,b [ Ba ,b ;d .
P +
+
Using part a and combining the two identities gives Ga,Fa, b
; b;a,b [ Ba ,b ;d . The
P
proof of parts c and d is reciprocal to that of parts a and b, respectively. Q.E.D.
M .R. Caputo Economics Letters 66 2000 265 –273
269
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, Fa, 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, Ga,g .
Part b asserts that the functions G and F are inverse functions of one another with respect to the constraint parameter
b of P, holding a fixed. Similarly, part d asserts that the functions F and G are inverse functions of one another with respect to the constraint parameter
g of T, holding a fixed. 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 :
ˆ ˆ
≠x ≠x
≠x ≠F
≠x
+ +
]] ]
] ]
]] a,
b ;
a,Fa, b 1
a,Fa, b
a, b ;a,b [ Ba ,b ;d
a, b
P
≠a ≠a
≠ g
≠a ≠
b
ˆ ≠x
≠F
+ +
] ]
; a,Fa,
b a,
b ;a,b [ Ba ,b ;d a
P
≠ g
≠ b
≠G ≠G
≠F ≠G
≠F
+ +
] ]
] ]
] a,Fa,
b 1 a,Fa,
b a,
b
; 0;a,b [ Ba ,b ;d
a,Fa, b
a, b
P
≠a ≠
g ≠a
≠ g
≠ b
+ +
; 1;a,b [ Ba ,b ;d b
P
ˆ ˆ
≠x ≠x
≠x ≠G
≠x
+ +
] ]]
]] ]
] a,
g ;
a,Ga, g 1
a,Ga, g
a, g ;a,g [ Ba ,g ;d
a, g
T
≠a ≠a
≠ b
≠a ≠
g
≠x ≠G
+ +
]] ]
; a,Ga,
g a,
g ;a,g [ Ba ,g ;d c
T
≠ b
≠ g
≠F ≠F
≠G ≠F
≠G
+ +
] ]
] ]
] a,Ga,
g 1 a,Ga,
g a,
g
; 0;a,g [ Ba ,g ;d
a,Ga, g
a, g
T
≠a ≠
b ≠a
≠ b
≠ g
+ +
; 1;a,g [ Ba ,g ;d . d
T
270 M
.R. Caputo Economics Letters 66 2000 265 –273
The first identity in part c of Theorem 2 evaluated at g 5 Fa,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 at
g 5 Fa,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 ≠Fa,
b ≠b ; la,b ± 0 is the optimal value of the Lagrange multiplier from P, and
ˆ similarly ≠Ga,
g ≠g ; ma,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,
la,b ; [ma,g ]
when g 5 Fa,b or b 5 Ga,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
