Oxford Economic Papers Advance Access published October 16, 2007
! Oxford University Press 2007
All rights reserved
Oxford Economic Papers (2007), 1 of 14
doi:10.1093/oep/gpm037
Intrinsic comparative statics of a general
class of profit-maximizing rate-of-return
regulated firms
By Michael R. Caputo* and M. Hossein Partoviy
*Department of Economics, University of Central Florida, Orlando, FL
32816–1400, USA; e-mail: [email protected] (corresponding author)
yDepartment of Physics and Astronomy, California State University,
Sacramento CA 95819–6041, USA; e-mail: [email protected]
An exhaustive comparative statics analysis of a general rate-of-return regulated,
profit-maximizing model of the firm is carried out under a minimal set of assumptions. The resulting intrinsic comparative statics are contained in a positive semidefinite matrix. Each element of this matrix consists of a product of the A-J effect
term and a Slutsky-like expression, thereby permitting the familiar interpretation of a
compensated price change. The minimal set of assumptions allows a range of anomalous behavior that includes, inter alia, a reversal of the sign of the A-J effect and an
increase in the use of an unregulated factor as a result of a compensated own-price
increase. The implications of additional assumptions for the mathematical structure of
the model and its economic consequences are discussed, and the equivalency relations
among those assumptions are delineated. Throughout, mathematical results of the
analysis are interpreted with a view to elucidating their intuitive economic significance.
JEL classifications: C60, D21, L51.
1. Introduction
The seminal paper of Averch and Johnson (1962), which models the behavior of
a profit-maximizing firm operating under the rate-of-return regulation, has been
discussed and interpreted numerous times (see, e.g., Takayama 1969; Baumol and
Klevorick 1970; El-Hodiri and Takayama 1973; Takayama 1993; and references
cited therein) as well as extended to a dynamic setting (see, e.g., El-Hodiri and
Takayama 1981; Seiichi and Abe 1989). What is surprising, however, is that none of
this long line of research has sought to determine the intrinsic comparative statics
properties of the model, i.e., the complete set of observable and empirically verifiable comparative statics implications of the model resulting from the constrained
profit-maximizing assertion under the generic assumption of a locally differentiable
solution. Even the ostensibly comprehensive comparative statics analysis of
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intrinsic comparative statics
McNicol (1973) is limited to the empirically unrealistic case of two factors of
production and is conducted under ad hoc assumptions that are stronger than
those implied by the constrained optimization assertion. Recognizing that the
rate-of-return regulation wrecks havoc on the prototypical comparative statics
properties of the profit-maximizing firm, Hughes (1990) made use of conditional
profit and factor demand functions to uncover the nature of the pathology. It
is important to note, however, that these two extensive investigations into the
qualitative characteristics of the model failed to uncover its intrinsic comparative
statics properties.
In contrast, a complementary literature that contemplates a cost-minimizing
version of the Averch and Johnson (1962) model has been partially successful in
developing a complete qualitative characterization of it. In particular, Diewert
(1981), Fuss and Waverman (1982), and Lasserre and Ouellette (1994) have derived
the properties of the cost function and associated conditional factor demand functions for cost-minimizing firms facing the rate-of-return regulation. Unlike our
approach, however, these authors employ assumptions that are stronger than those
implied by the constrained minimization hypothesis. For example, they assume that
the marginal products of all the inputs are positive and that the production function is strongly quasi-concave in all the inputs. While neither of these assumptions
is unreasonable or objectionable from an economic point of view, both are extra
suppositions beyond what is implied by cost-minimizing behavior when firms face
the rate-of-return regulation. Consequently, certain properties of the cost function
would not unequivocally hold in the absence of the extra assumptions, e.g., the
property that the cost function is increasing in the unregulated input prices.
The comparative statics results derived in this paper, on the other hand, apply to
a general profit-maximizing firm employing multiple factors of production and
operating under a rate-of-return constraint. Moreover, no extraneous assumptions
beyond the customary smoothness conditions are imposed on the problem, thereby
guaranteeing that the resulting comparative statics properties of the generalized
model are intrinsic to it. Furthermore, the resulting comparative statics have the
preferred form of a constraint-free semi-definite matrix reminiscent of the Slutsky
matrix of neoclassical consumer theory. All comparative statics results obtained
here are derived using the new formalism of Partovi and Caputo (2006, 2007).
2. A generalized profit-maximizing, rate-of-return regulated
model of the firm
The Averch and Johnson (1962) model of the rate-of-return regulated firm is so
well known that we can be relatively succinct in defining our generalization of it.
Consider a profit-maximizing monopolist producing one homogeneous good, say
y 2 Rþ , employing M factors of production, say x 2 RM
þ , that are purchased
.
The
production function
in competitive factor markets at prices w 2 RM
þþ
M
ð2Þ
!
R
is
assumed
to
be
a
C
function
on
R
.
The
monopolist faces
f ðÞ : RM
þ
þ
þþ
the inverse demand function PðÞ : Rþ ! Rþ , likewise assumed to be a C ð2Þ
m. r. caputo and m. h. partovi
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M
function on
Rþþ . The revenue function RðÞ : Rþ ! Rþ is thus defined by
def
RðxÞ ¼ P f ðxÞ f ðxÞ which, given our assumptions of differentiability, is also
a C ð2Þ function on RM
þþ . The regulatory constraint is the archetypical condition
requiring that the rate of return on capital not exceed some fair value determined
by the regulatory agency. Defining xM as the capital of the firm, or equivalently,
as the regulated input, we may write the generalized rate-of-return regulated profitmaximizing model of the firm as
(
)
M
M
1
X
X
def
ð1Þ
wm xm s:t: RðxÞ wj xj 4 sxM ;
ðw, sÞ ¼ max RðxÞ x
m¼1
j¼1
where s 2 Rþþ is the fair rate of return on capital. Hereafter we will often refer to
the constrained optimization problem (1) as the generalized A-J model.
We now impose and maintain the following assumptions on problem (1)
throughout:
Assumption 1
s > wM .
def
ð1Þ
Assumption 2 For each a ¼ ðw, sÞ 2 Aopen RMþ1
þþ there exists an interior C
solution to problem (1), which we denote by x ¼ x ðaÞ, with l ¼ l ðaÞ being the
optimal value of the Lagrange multiplier corresponding to the rate-of-return
constraint.
Assumption 3
at x ¼ x ðaÞ.
The rate-of-return constraint is binding, and not just binding,
The assumption s > wM (A.1) is standard in the literature and fundamental to
problem (1), and amounts to asserting that the fair rate of return on capital is larger
than the rental price of one unit of capital, thereby permitting the monopolist to earn
a positive profit. Assumption 2 (A.2) describes the local differentiability of the
optimal solution, and is essential to any differentiable comparative statics analysis.
An alternative to (A.2) is to impose conditions on the primitives that imply it.
However, such an approach has the distinct disadvantage of injecting extraneous
elements into the model by way of sufficient conditions that transcend the basic
assumption of local differentiability. Assumption 3 (A.3), on the other hand, ensures
that the rate-of-return constraint is binding (and not just binding) at the optimum,
thereby implying that the solution to problem (1), namely x ¼ x ðaÞ, does not reduce
to the unconstrained solution of problem (1). Consequently, (A.3) is essential to
problem (1), since otherwise one would be dealing with the basic profit-maximizing
monopoly model, the qualitative properties of which are well-known. It is worth
noting here, however, that (A.3) would cease to hold for sufficiently large values of s,
as noted by Averch and Johnson (1962, p. 1056). In other words, (A.3) implies that s
is strictly less than the rate-of-return the firm would earn if it maximized profit
absent the regulatory constraint. We will see in Section 4 that this observation has
nontrivial implications for the economic consequences of the model.
It is important to point out here that assumptions (A.1)–(A.3) are weaker than
those commonly employed in the A-J model literature. In particular note that,
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intrinsic comparative statics
in contrast to the existing literature, we have not made any assumptions about the
slope of the inverse demand function, the slope or curvature of the revenue function
with respect to the factors of production or output, the marginal products of the
inputs, or the curvature of the production function, nor are we limiting the model to
two factors of production. Finally, it is worth emphasizing that no further assumption beyond (A.1)–(A.3) will be made in the following, although the consequences of
the observation made above in connection with (A.3) will be explored in Section 4.
As a preliminary step to our comparative statics analysis, we first establish a basic
property of problem (1) that will be important subsequently. The Lagrangian for
problem (1) is defined as
"
#
M
M
1
X
X
def
wm xm þ l
wj xj þ sxM RðxÞ :
ð2Þ
Lðx, l; w, sÞ ¼ RðxÞ m¼1
j¼1
Caputo and Partovi (2002, p.3) prove that the nondegenerate rank condition on
the constraint function of problem (1) holds at the solution. Hence, by
Theorem 2.3 of Takayama (1993), the first-order necessary conditions of problem
(1) are given by
@L
@R
¼
ðxÞ wi ½1 l ¼ 0, i ¼ 1, 2, . . . , M 1,
ð3Þ
@xi
@xi
@L
@R
@R
¼
ðxÞ wM l
ðxÞ s ¼ 0;
ð4Þ
@xM @xM
@xM
X
@L M1
@L
¼
l ¼ 0:
ð5Þ
wj xj þ sxM RðxÞ > 0, l > 0,
@l
@l
j¼1
Recall that with eqs (3)–(5) as necessary conditions, we have x ¼ x ðaÞ and
l ¼ l ðaÞ as their simultaneous solutions for each a 2 Aopen by assumption (A.2).
Next we establish the property that the factor demand functions x ¼ x ðaÞ
ðaÞ=@wM 0,
are independent of the rental rate on capital wM , i.e., @xm
m ¼ 1, 2, . . . , M. First, note that by Lemma 1 of Caputo and Partovi (2002),
the Lagrange multiplier is restricted according to l ðaÞ > 0 and l ðaÞ 6¼ 1. Using
these facts, we may write the first-order necessary conditions (3) and (5) as
@R
ðxÞ wi ¼ 0, i ¼ 1, 2, . . . , M 1,
@xi
M
1
X
wj xj þ sxM RðxÞ ¼ 0:
ð6Þ
ð7Þ
j¼1
These equations form a simultaneous set of M equations in the M unknowns x.
By (A.2), the solution of these necessary conditions is x ¼ x ðaÞ. Inasmuch
as eqs (6) and (7) are independent of wM , so too are the factor demand functions
ðaÞ=@wM 0, m ¼ 1, 2, . . . , M. McNicol
x ¼ x ðaÞ, thereby implying that @xm
(1973, Table 2B) established an analogous result in a two-input version of the
A-J model by deriving its comparative statics properties and then showing that
m. r. caputo and m. h. partovi
5 of 14
@xm
ðaÞ=@wM 0 under the assumptions that the marginal products are positive and
diminishing and the inverse demand and marginal revenue functions are downward
sloping in output. Our proof is superior to McNicol’s in that (i) it applies to the
generalized A-J model with M factors of production, (ii) it is realized directly by way
of a simple rearrangement of the first-order necessary conditions, and (iii) it is
derived under assumptions (A.1)–(A.3) only. Essentially the same remark applies
ðaÞ=@wM 0, although it should be noted
to Hughes’ (1990, p. 88) proof of @xm
that the latter is more general than McNicol’s in view of the facts that Hughes
studied the A-J model with many factors of production and made the more general
assumption that the production function is strictly quasi-concave.
Two more features of the generalized A-J model are important to underline
before we proceed to the analysis of its comparative statics. First, as stated above,
Lemma 1 of Caputo and Partovi (2002) establishes that l ðaÞ > 0 and l ðaÞ 6¼ 1
under assumptions (A.1)–(A.3). Furthermore, this is all that can be established
under assumptions (A.1)–(A.3). In particular, the possibility that l ðaÞ > 1,
which is excluded under the stronger assumptions adopted in the existing literature, may very well occur in the generalized A-J model (1). Indeed, Caputo
and Partovi (2002, p. 4) present a revenue function which satisfies assumptions
(A.1)–(A.3) and for which l ðaÞ > 1 holds. These authors (2002, Lemma 2 and
p. 6) also show that the assumptions commonly adopted in the existing literature
are in fact either equivalent to l ðaÞ 2 ð0, 1Þ or imply it, in agreement with
the above assertions. Second, Caputo and Partovi (2002, p.4) show that the
ðaÞ=@s < 0, does not necessarily hold in the generalso-called A-J effect, to wit, @xM
ized A-J model under assumptions (A.1)–(A.3). Indeed, by Lemma 2 of Caputo and
ðaÞ=@s < 0, thus implying that
Partovi (2002), l ðaÞ 2 ð0, 1Þ is equivalent to @xM
@xM ðaÞ=@s > 0 is equivalent to l ðaÞ > 1, since l ðaÞ is already restricted
by l ðaÞ > 0 and l ðaÞ 6¼ 1. In other words, the equivalent conditions
ðaÞ=@s > 0 and l ðaÞ > 1 can and do occur in the generalized A-J model,
@xM
confirming, inter alia, that assumptions (A.1)–(A.3) are weaker than those
ðaÞ=@s < 0 is not
employed in the existing literature. Clearly, the A-J effect @xM
intrinsic to the generalized A-J model inasmuch as it is not implied by the maximization assertion, the model’s mathematical structure, and the basic assumptions
(A.1)–(A.3). In the following, we refer to the conditions l ðaÞ 2 ð0, 1Þ and
ðaÞ=@s < 0 and @xM
ðaÞ=@s > 0, as the ‘normal’ and
l ðaÞ > 1, equivalently @xM
‘anomalous’ cases respectively.
3. Comparative statics
Our objective in this section is to derive a semi-definite matrix, with the rateof-return constraint already embodied, that involves the partial derivatives of the
decision functions with respect to the parameters for the generalized A-J model
under assumptions (A.1)–(A.3). Before doing so, we provide a brief exposition of
Theorem 1 of Partovi and Caputo (2006, 2007) which is the main result used in
our comparative statics calculations.
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intrinsic comparative statics
The key idea of the method of Partovi and Caputo (2006) originates in the observation that the partial derivatives of decision functions with respect to parameters, considered individually, are not necessarily susceptible to a comparative
statics characterization in a given optimization problem. This observation is manifest in the archetype utility maximization problem, where a linear combination of
such partial derivatives, to wit, those with respect to price and income, is required
for a complete comparative statics characterization of the problem as embodied in
the Slutsky matrix. On the other hand, aside from an inessential scale factor,
a general linear combination of partial derivatives with respect to the parameters
is simply a directional derivative pointing in some direction in parameter space.
Of course, not every arbitrary direction in parameter space will suffice, and it turns
out that the constraint structure of the optimization problem plays an important
role in determining the desired directions in parameter space if the comparative
statics properties are to emerge with the constraints already implemented. As shown
in Partovi and Caputo (2006, Lemma 1), the key to such constraint-free comparative
statics results is to find such directional derivatives with respect to the parameters
that return zero when applied to the constraint functions of the optimization
problem. Directional derivatives with this null property are defined as ‘generalized
compensated derivatives’ (GCD’s) by Partovi and Caputo (2006).
Using the null condition just described, one can characterize the desired directions
in parameter space in simple geometrical terms: all tangential directions with respect
to the level set of the constraint function in parameter space will yield directional
derivatives with the required null property. Equivalently, all directions on the tangent
hyperplane to the level surface of the constraint function in parameter space have the
requisite property. Since the gradient vector of a differentiable function is orthogonal
to its level set, any vector that is orthogonal to the gradient of the function with
respect to the parameters lies in the tangent hyperplane to the level set of that
function in parameter space. Thus the desired directions in parameter space are
those that are orthogonal to the gradient of the constraint function with respect to
the parameters. How many of these directions and corresponding GCD’s are necessary or desirable? In general, this number equals the dimension of the tangent hyperplane. Accordingly, a set of GCD’s constructed from the basis vectors of the tangent
hyperplane provides a ‘complete’ set of GCD’s (Partovi and Caputo, 2006). Using
this simple prescription, we will next construct a full set of GCD’s and use it to derive
a complete comparative statics characterization of the generalized A-J model.
To begin the construction, we recall the definition of the parameter vector a
def
for problem (1), videlicet, a ¼ ðw1 , w2 , . . . , wM , sÞ 2 RMþ1
þþ . The gradient operator
with respect to the parameter vector a is therefore given by the M þ 1 component
def
row vector of partial derivative operators ra ¼ ð@w@ 1 , @w@ 2 , . . . , @w@M , @s@ Þ.
Next, we define the constraint function gðÞ of problem (1) by
def
gðx; aÞ ¼
M
1
X
j¼1
wj xj þ sxM RðxÞ:
ð8Þ
m. r. caputo and m. h. partovi
7 of 14
As a result, the gradient vector of gðÞ with respect to a is given by
ra gðx; aÞ ¼ ðx1 ; x2 ; . . . ; xM1 ; 0; xM Þ: Because the solution to problem (1) is interior by assumption (A.2), it follows from continuity that ra gðx; aÞ 6¼ 0Mþ1 in
some neighborhood of the optimal solution, thereby implying, via the implicit
function theorem, that the constraint function defines a local M-dimensional manifold in RMþ1
þþ . This means that the tangent hyperplane to the level set of the constraint function in parameter space is of dimension M, and therefore that M basis
vectors are required for its description. But this is just what one expects under
typical circumstances, for the parameter space is of dimension M+1, the gradient
vector spans a space of one dimension since there is but one constraint function,
hence the tangent hyperplane must span the remaining M dimensions in parameter
space.
It is now a straightforward matter to confirm that the ensuing M
def
vectors in RMþ1 , to wit, t ¼ ð01 , 02 , . . . , 01 , 1 , 0þ1 , . . . , 0M1 , 0M , x
xM Þ,
M def
¼ 1, 2, . . . , M 1, and t ¼ ð01 , 02 , . . . , 0M1 , 1M , 0Mþ1 Þ, which Partovi and
Caputo (2006) define as ‘isovectors,’ are orthogonal to the gradient vector of
the constraint function in parameter space by verifying that ra gðx; aÞ t ¼ 0,
¼ 1, 2, . . . , M. Here the index attached to a numerical element of the isovector
P
signifies its position in the vector. Since the linear system M
¼1 c t ¼ 0Mþ1 has
only the trivial solution c ¼ 0M , the isovectors t , ¼ 1, 2, . . . , M, form a basis for
the tangent hyperplane to the level set of the constraint function in parameter
space. These isovectors define our choice of the desired directions in parameter
space for constructing the GCD’s.
The GCD’s are now constructed by taking the inner product of the gradient
operator ra with the M basis vectors t ; that is
@
x @
def D ðx, aÞ ¼ t ra ¼
, ¼ 1, 2, . . . , M 1;
ð9Þ
xM @s
@w
@
def
:
ð10Þ
DM ðx, aÞ ¼ tM ra ¼
@wM
Applying each of these GCD’s to the constraint function gðÞ defined in eq. (8)
yields zero identically, i.e., D ðx, aÞ gðx; aÞ 0, ¼ 1, 2 , . . . , M, thereby verifying that the directional derivatives D ðx, aÞ, ¼ 1, 2 , . . . , M, satisfy the required
null property and are in fact a complete set of GCD’s for problem (1).
Equipped with these M GCD’s, we are now in a position to apply Theorem 1 of
Partovi and Caputo (2006, 2007) to the derivation of a constraint-free negative
semi-definite comparative statics matrix for problem (1). Using the notation established above, Theorem 1 of Partovi and Caputo (2006, 2007) asserts that the typical
element of the said matrix takes the form
M X
@L D ðx ðaÞ, aÞ ðx ðaÞ, l ðaÞ; aÞ
@xm
m¼1
D 0 ðx ðaÞ, aÞ xm
ðaÞ , , 0 ¼ 1, 2, . . . , M:
ð11Þ
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intrinsic comparative statics
M
¼1;2;...;M1
...................................
Carrying out the computation implied in eq. (11), we find the following M M
negative semi-definite comparative statics matrix:
h h x ðaÞ i i
2
3
@x ðaÞ
@x ðaÞ
@x ðaÞ
½1 l ðaÞ @w x ðaÞ @s
½1 l ðaÞ @w M
h i 7
6
h ih M h x ðaÞ i i
x ðaÞ @x ðaÞ
6
@xM ðaÞ
x ðaÞ @xM ðaÞ
þ l ðaÞ x ðaÞ @wMM 7
6 þ l ðaÞ x ðaÞ @w
7
xM ðaÞ
@s
M
M
6
7
def
¼1;2;...;M1
7
;¼1;2;...;M1
ðaÞ ¼ 6
6 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 7: ð12Þ
6
7
h x ðaÞ i 6
7
@xM
ðaÞ
@xM ðaÞ
@xM
ðaÞ
4
5
@w x ðaÞ
@s
@wM
...................
By Theorem 4 of Partovi and Caputo (2006), rank ððaÞÞ 4 minðM 1, MÞ
¼ M 1, so that ðaÞ is singular and has at most M 1 negative eigenvalues in
its spectrum.
The form of ðaÞ given in eq. (12) is cumbersome and obscures its economic
implications. Fortunately, it is possible to derive a reduced form of ðaÞ which will
transparently reveal its economic significance.
In order to reduce ðaÞ, first recall that in Section 2 we established the result that
ðaÞ=@wM 0, m ¼ 1, 2, . . . , M. This implies that the last column of the matrix
@xm
ðaÞ is null. But given that ðaÞ is symmetric, the same is implied for the last row
of ðaÞ, i.e., that @xM
ðaÞ=@w ½x ðaÞ=xM
ðaÞ@xM
ðaÞ=@s 0, ¼ 1, 2, . . . , M 1.
Using these results we may rewrite ðaÞ as
h h x ðaÞ i i
2
3
@x ðaÞ
@x ðaÞ
½1 l ðaÞ @w x ðaÞ @s
0ðM1Þ1
M
7
def 6
, ¼1, 2, ..., M1
7
ðaÞ ¼ 6
ð13Þ
4 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5:
01ðM1Þ
0
Second, evaluate eq. (4) at x ¼ x ðaÞ, then rearrange it to arrive at
wM s
1 l ðaÞ ¼
:
@Rðx ðaÞÞ=@xM s
ð14Þ
Third, differentiate the identity form of the rate-of-return constraint with respect to
s and use eq. (6) to simplify the result. This sequence of steps yields the result
@xM
ðaÞ
xM
ðaÞ
¼
:
@s
@Rðx ðaÞÞ=@xM s
ð15Þ
Next, multiply eq. (14) by xM
ðaÞ=xM
ðaÞ, which is permissible because xM
ðaÞ > 0 by
(A.2). A combination of eqs (14) and (15) then gives
wM s @xM
ðaÞ
1 l ðaÞ ¼
:
ð16Þ
xM ðaÞ
@s
In view of the fact that s > wM by (A.1) and xM
ðaÞ > 0 by (A.2), an inspection of
eq. (16) reveals that
@x ðaÞ
sign ½1 l ðaÞ ¼ sign M
:
ð17Þ
@s
m. r. caputo and m. h. partovi
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Finally, using eq. (17) in eq. (13), we can conclude that the ðM 1Þ ðM 1Þ
comparative statics matrix ðaÞ, with typical element defined by
x ðaÞ @x ðaÞ
@x ðaÞ
def @xM ðaÞ
ðaÞ ¼
, , ¼ 1, 2, . . . , M 1,
ð18Þ
@s
@w
@s
xM ðaÞ
is positive semi-definite and constraint-free, with rank ð ðaÞÞ M 1. This
concludes the proof of the central result of this paper, which we summarize in
the following theorem.
Theorem 1 Under assumptions (A.1)–(A.3), the comparative statics of the rateof-return regulated model defined by eq. (1) et. seq. are summarized by the statement that the ðM 1Þ ðM 1Þ matrix ðaÞ, the typical element of which is
given by
x ðaÞ @x ðaÞ
@x ðaÞ
def @xM ðaÞ
ðaÞ ¼
, , ¼ 1, 2, . . . , M 1,
ð19Þ
@s
@w
@s
xM ðaÞ
is positive semi-definite, free of constraint. Furthermore, rank ð ðaÞÞ 4 M 1.
At this juncture it is useful to complement the above theorem with the associated
set of envelope equations for the model. These can be derived straightforwardly
using either the standard envelope theorem or Theorem 2 of Partovi and Caputo
(2006). They are
@ ðaÞ
¼ ½l ðaÞ 1xi ðaÞ, i ¼ 1, 2, . . . , M 1,
ð20Þ
@wi
@ ðaÞ
¼ xM
ðaÞ,
ð21Þ
@wM
@ ðaÞ
ðaÞ:
ð22Þ
¼ l ðaÞxM
@s
We leave the discussion of the economic implications of Theorem 1
and eqs (20)–(22) to Section 4, and conclude this section with a few general
remarks.
The reduced comparative statics matrix in eq. (19) has a rather complex
appearance. It is important to understand that this complexity is not an artifact
of the method used to drive it. Compared to the archetype models of profit maximization and cost minimization, the generalized A-J model is more complex structurally, so one should not expect its comparative statics to be as simple and
straightforward as those of the standard textbook models. On the other hand,
the complex structure of ðaÞ shows the power of the method of generalized
compensation of Partovi and Caputo (2006), since a traditional comparative
statics analysis (see, e.g., Silberberg 1990, ch.6) would not lead one to believe
that it is possible to construct a positive semi-definite matrix from the
partial derivative terms, let alone furnish a method for such an elaborate construction. It is also important to realize that computing ðaÞ from an empirical
estimate of x ðaÞ and l ðaÞ, although likely tedious, is a straightforward matter.
10 of 14
intrinsic comparative statics
Hence testing for the positive semi-definiteness of ðaÞ is likewise conceptually
straightforward.
ðaÞ=@s and
The existing literature, having focused on determining the sign of @xM
other such partial derivatives of the decision functions, has ignored the major
portion of the empirical content in the model as reflected in the positive semidefiniteness of ðaÞ. Only by probing the positive semi-definiteness of ðaÞ can
one conclude that a given set of data generated by profit maximizing rate-of-return
regulated firms is consistent with the behavioral implications of problem (1).
Finally, note that the work of Partovi and Caputo (2006) shows that since wM is
the only parameter that does not enter the constraint, one can only expect to sign
the GCD’s of the decision functions with respect to the other parameters. This
feature is borne out by the structure of ðaÞ.
4. Discussion
We are now in a position to consider the economic interpretation of Theorem 1
and the envelope equations of the preceding section. It is clear from eq. (19) that
ðaÞ=@s, whose sign is indeterminate under
the A-J comparative statics effect @xM
assumptions (A.1)–(A.3), plays a pivotal role in determining the economic con
ðaÞ@x ðaÞ=@s. For this
sequences of the Slutsky-like form @x ðaÞ=@w ½x ðaÞ=xM
reason, as well as the fact that the above-mentioned sign turns out to have
a profound effect on the economic behavior of the model, it is natural to
ðaÞ=@s < 0 and
discuss the implications of Theorem 1 for the cases of @xM
@xM ðaÞ=@s > 0, corresponding to the normal and anomalous cases defined in
Section 2, separately.
In the normal case l ðaÞ 2 ð0, 1Þ, @Rðx ðaÞÞ=@xM < wM (Caputo and Partovi, 2002,
ðaÞ=@s < 0 holds. To understand
eq. (8) and Lemma 2), and the A-J effect @xM
how the A-J effect arises in this case, note that the marginal condition
@Rðx ðaÞÞ=@xM < wM is a statement that the marginal revenue of capital is less
than its marginal cost at the regulated optimum, implying that the rate-of-return
regulation has caused the firm to employ capital at a higher level than would be
considered optimal for an unregulated firm. In this sense, the regulation has spurred
the firm to ‘over-employ’ capital. Consequently, any change that lessens the impact
of regulation on the firm should also serve to reduce the level of its capital utilization. Inasmuch as an increase in the value of the rate-of-return parameter s eases the
ðaÞ to result from
effect of regulation on the firm, one would expect a decrease in xM
such an increase, thus implying the standard A-J effect @xM ðaÞ=@s < 0. On the other
hand, in the anomalous case where l ðaÞ > 1, the firm would be expected to ‘under
ðaÞ=@s > 0. We
employ’ capital, thereby causing a reversal of the A-J effect, i.e., @xM
will find the foregoing connection between the A-J effect and the capital utilization
behavior of the firm useful in the following discussion.
According to Theorem 1, the ðM 1Þ ðM 1Þ matrix of Slutsky-like
ðaÞ@x ðaÞ=@s is negative semi-definite in the
terms @x ðaÞ=@w ½x ðaÞ=xM
m. r. caputo and m. h. partovi
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normal case. Consequently, the diagonal elements of this matrix must be nonpositive, i.e.,
@x ðaÞ
x ðaÞ @x ðaÞ
4 0,
ð23Þ
@w
xM ðaÞ
@s
where ¼ 1, 2, . . . , M 1. This inequality shows that a rate-of-return compensated increase in the price of an unregulated factor of production does not increase
the firm’s use of that factor. In other words, once the effect of the rate-of return
constraint is compensated for, the M 1 unregulated factors of production obey
the law of demand. Clearly, this conclusion is fully analogous to the incomecompensated law of demand from the archetypal utility maximization model, the
only nontrivial difference between the two being the opposite signs of the compensation terms. To understand why the compensation terms differ, write the
PM
budget and rate-of-return constraints as, respectively,
m¼1 pm xm I and
PM1
RðxÞ=xM j¼1 wj xj =xM 4 s, where pm 2 Rþþ , m ¼ 1, 2, . . . , M, is the price
the consumer pays for the mth good and I 2 Rþþ is the consumer’s income.
Inspection of the two constraints reveals that income and the rate of return play
equivalent roles in their respective models, namely, as parameters that limit the size
of the feasible set. For example, increases in I and s result in equivalent increases in
the size of the respective feasible sets. On the other hand, a unit increase in the price
of good increases a consumer’s expenditures by x , whereas a unit increase in the
price of unregulated input reduces a firm’s profit by x =xM . It is this difference
that accounts for the fact that the compensation terms appear in the Slutsky forms
with opposite signs and different weights in the two models.
Inasmuch as l ðaÞ 2 ð0, 1Þ in the normal case, the envelope result in eq. (20)
expresses the expected negative impact of a price increase in an unregulated input
on the firm’s profit. Similarly, eqs (21) and (22) express the negative and positive
effect, respectively, of an increase in the rental rate of capital and the rate-of-return
parameter on the firm’s profit. Equations (21) and (22) also show that the magnitude
of the impact on the firm’s profit of an increase in the rental rate of capital exceeds
that of a decrease in the rate of return. The intuition for this result lies in the fact that
the factor demand functions are independent of the rental rate of capital, as demonstrated in Section 2. This implies that the firm does not change its inputs when the
rental rate of capital changes. In contrast, a change in the rate-of-return parameter
generally results in changes in all inputs. As a result, the full impact of an increase in
the rental rate of capital on the firm’s profit is realized due to the lack of input
adjustment when wM changes, whereas only a partial impact on its profits is felt
when the rate-of-return parameter decreases because of the resulting change in the
firm’s inputs when s changes. Clearly, the comparative statics and envelope properties of the A-J model in the normal case are consistent with what might be intuitively
expected from a profit maximizing firm under the rate-of-return regulation.
By contrast, the behavior of the A-J model in the anomalous case for which
l ðaÞ > 1, @Rðx ðaÞÞ=@xM > wM , and @xM
ðaÞ=@s > 0 is rather bizarre. To highlight
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intrinsic comparative statics
this anomalous behavior in a general and intuitive manner, let us recall the
observation made in Section 2 that (A.3), which stipulates that the regulation constraint is binding on the firm, is expected to fail for sufficiently large values of s. Since
the optimal value of the Lagrange multiplier l ðaÞ vanishes as the binding threshold
of s is approached, and because the anomalous regime is characterized by the conditions l ðaÞ > 1 and l ðaÞ 6¼ 1, we see that the model cannot reach the binding
threshold from the anomalous regime no matter how large the rate-of-return parameter grows, assuming that l ðÞ 2 C ð0Þ in s. In short, in the anomalous case the
model fails to meet the basic, intuitive requirement that the constraint should unbind
if s is driven to sufficiently large magnitudes. This behavior is typical of the counterintuitive implications of the anomalous case in several respects, as shown below.
Recall that by Theorem 1 the ðM 1Þ ðM 1Þ matrix of Slutsky-like terms
ðaÞ@x ðaÞ=@s is positive semi-definite in the anomalous
@x ðaÞ=@w ½x ðaÞ=xM
case. This in turn implies that for unregulated inputs, a rate-of-return compensated
own-price increase does not decrease the firm’s use of that factor, i.e., once the
constraint is compensated for, the demand curves for the unregulated factors of
production do not slope downward in their own prices. This bizarre situation
cannot occur in the prototypical utility maximization model inasmuch as the
Slutsky matrix is negative semi-definite. In the generalized A-J model, however,
it is the product of the A-J comparative statics effect and the Slutsky-like matrix
which is positive semi-definite. Consequently, a determination of whether the rateof-return compensated demand functions are upward or downward sloping in their
own price will decide whether the normal or anomalous case prevails.
Other instances of counter-intuitive behavior are not hard to find in the anomalous regime. For example, consider the envelope result of eq. (20) which, for the
present case, implies an increase in the firm’s profit as a result of an increase in the
price of any of the unregulated inputs, echoing the above failure of the model to
produce downward sloping rate-of-return compensated demand functions. The
intuition for this result hinges on the facts that the revenue function of the firm
is locally convex in the unregulated inputs (Caputo and Partovi 2002, Lemma 2)
and that capital is ‘under-employed’ in this case. The increasing marginal revenue
of every unregulated factor along with the ‘under-employed’ capital suggest that the
firm will find it advantageous to expand its use of all inputs, thereby capitalizing on
the increasing marginal returns it enjoys with respect to the unregulated inputs and
increasing its profit as a result.
The envelope relations (21) and (22) do not change sign in the anomalous case,
so that no counter-intuitive qualitative results emerge in this case. On the other
hand, these equations show that the magnitude of the impact on the firm’s profit of
a decrease in the rate of return exceeds that of an increase in the rental rate of
capital, just the opposite of the normal case. The intuition here parallels that in the
normal case, except that now with capital ‘under-employed’ and marginal revenue
of the unregulated factors increasing, decreasing the rate-of-return parameter is
especially costly inasmuch as the resulting changes in the inputs must push the firm
even further from the unregulated profit maximum.
m. r. caputo and m. h. partovi
13 of 14
We close this section with a discussion of the relationship between the
regulated input xM and the unregulated inputs x , ¼ 1, 2, . . . , M 1. To
ðaÞ=@wM 0, m ¼ 1, 2, . . . , M, and the
this end, recall that the conditions @xm
symmetry of the matrix ðaÞ were shown to imply the null property
ðaÞ=@w ½x ðaÞ=xM
ðaÞ@xM
ðaÞ=@s 0,
¼ 1, 2, . . . , M 1.
Because
@xM
M
x ðaÞ 2 Rþþ by (A.2), the above identity implies that
@xM ðaÞ
@x ðaÞ
¼ sign M
sign
¼ 1, 2, . . . , M 1:
ð24Þ
@w
@s
Therefore, in the normal case, the use of the regulated input falls when the price of an
unregulated input increases, in which case the regulated input would typically be
deemed a complement to the set of unregulated inputs. On the other hand, in
the anomalous case, the use of the regulated input increases when the price of an
unregulated input increases, in which case the regulated input would be classified as a
substitute for the set of unregulated inputs. These classifications, however, may not be
appropriate in the generalized A-J model since the law of demand does not necessarily
hold for the factors of production. In particular, because @x ðaÞ=@w > 0 may occur
ðaÞ=@w < 0 holds (for ¼ 1, 2, . . . , M 1), one would typically regard
while @xM
ðaÞ as substitutes. By contrast, x ðaÞ and xM
ðaÞ in the aforementioned
x ðaÞ and xM
archetypal models would be deemed complements when @xM
ðaÞ=@w < 0, as the law
of demand would then lead to a reduction in the use of x ðaÞ as well.
5. Summary and conclusions
The Averch and Johnson (1962) model of the rate-of-return regulated, profitmaximizing firm has been in existence for over four decades, and a good deal of
effort has been expended in order to fully understand its economic content and
extend its reach. Remarkably, however, its intrinsic comparative statics properties
have heretofore gone undiscovered. We have addressed this deficiency by formulating a multivariate version of the model free of extraneous assumptions and developing a full comparative statics characterization of the resulting model using the
formalism of Partovi and Caputo (2006). The comparative statics results so obtained
are summarized in the semidefinite nature of a Slutsky-like matrix whose sign is
conditioned on the rate of change of the regulated input with respect to the fair rate
of return. The economic implications of these results are thus readily interpreted by
comparison and contrast to the archetypal utility maximization model.
The discussion in Section 4 clearly shows that while the three basic assumptions
defining the generalized A-J model allow the occurrence of the anomalous case, the
economic behavior of the model in that case is highly counterintuitive and fails to
meet a basic desideratum, namely the requirement that the model tend toward the
binding threshold as the rate-or-return parameter is increased to sufficiently large
magnitudes. Indeed, the exclusion of this behavior was likely part of the motivation
for the introduction of extra assumptions into the model in the early literature.
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intrinsic comparative statics
Acknowledgements
We thank two referees for their thoughtful comments and productive suggestions.
M. Hossein Partovi acknowledges partial support by a Research Award from California
State University, Sacramento.
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