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Profit-maximizing input demand under rate-of-return regulation
Resources and Energy 12 (1990) 79-95. North-Holland
PROFIT-MAXIMIZING INPUT DEMAND UNDER RATE-OF-RETURN REGULATION Pathological Substitution and Output Effects Joseph P. HUGHES Rutgers
U~tversit~~,
New
~ru~s~~ck.
NJ
~89~3, USA
Received March 1988, final verston recetved May 1989 Cost-mmimrzing and profit-maximtzing input demand for the firm subJect to rate-of-return regulation are examined. Unregulated cost and profit functtons which are condittonal on the rate of employment of the rate-base input are shown to be Identically equal to the regulated cost and profit functions, evaluated at the regulated opttmum. Shephard’s lemma and Hotelling’s lemma applied to the conditional cost and profit functions, evaluated at the regulated opttmum, yield the regulated input demands. For both the profit-maxtmizing and cost-mimmizing demands, own-prtce effects are not necessartly negative nor are the cross-price effects in general equal. Moreover, the profit-maximizing output effect is not necessartly negative. Various decompositions of input demand are explored to explain these conclustons.
1. Introduction Rate-of-return regulation introduces a pervasive pathology to the profit function and to the profit-maximizing input demands. When the regulatory constraint is binding, the profit function is not necessarily convex. Hotelling’s lemma cannot be applied directly.’ The own-price effect on profit-maximizing input demand is not necessarily negative. Neither component of the ownprice effect, the substitution effect and the profit-maximizing output effect, is necessarily negative.2 Moreover, the cross-price effects between pairs of inputs are not necessarily equal. ‘See Cowmg (1978) and for the equivalent result for Shephar~s lemma, Cowmg ( 1979). Fuss and Waverman (1981). Smith (1981). and Nelson and Wohar f 1983). ‘For analyses of the comparative-static restrictions on the demand for inputs, see. for example, Baumol and Klevonck (1970) and McNicoi (1973). who assume that there are only two inputs, and Ofori-Mensa (1982) who treats the n-input case. Ofori-Mensa shows that the results of the two-input case do not generalize to the n-input case for cost minimrzation. He does not explore the comparative-static restricttons for profit maxtmization nor does he investigate the productive relationshtps whtch lead to pathologtcal results. None of these papers decomposes the profit-maximizing response of input demand to prtce changes into substitutton and output effects. Hughes (1984. 1986) examines the n-input case of cost minimization and, usmg the conditional (or restricted) cost function, investigates the productive relationships which lead to pathological substitutton effects. 0165-0572/90/%3.50 8 1990, Elsevier Science Publishers B.V. (North-Holland)
80
J.P. Hughes, Profit-maximizing
input demand
The workings of this pathology can be uncovered by formulating an unregulated profit function which is conditional on the rate of employment of the rate-base input. When this unregulated conditional profit function is maximized subject to the rate-of-return constraint and the resulting demand function for the rate-base input is substituted into the conditional profit function, the conditional profit function becomes identically equal to the unconditional regulated profit function3 Applying Hotelling’s lemma directly to the unconditional regulated profit function fails to yield these demands. However, Hotelling’s lemma can be directly applied to the conditional profit function, evaluated at the regulated optimum, to yield the conditional rates of employment of the nonrate-base inputs, which at the regulated optimum, are equal to the regulated values of input demand. When the regulated demand function for the rate-base input is substituted into these unregulated conditional demand functions, the conditional demand functions are identically equal to the unconditional regulated demand functions. Using this identity, the effect on the demand for a nonrate-base input of a change in an input’s price can be decomposed into two components, one the result of holding the rate of employment of the rate-base input constant and the other the result of adjusting the rate base to satisfy the rate-of-return constraint. The latter component is found to contain the pathology. These two components can in turn be decomposed to isolate the source of the pathology. This decomposition is effected by constructing the regulated cost-minimizing demand for a nonrate-base input from the unregulated conditional cost-minimizing demand function. The latter is conditioned on the physical rate of output and the employment of the rate-base input. Substituting the regulated profit-maximizing supply function for output and the regulated profit-maximizing demand function for the rate-base input for the conditional values, the conditional cost-minimizing demand function becomes identically equal to the regulated profit-maximizing demand function.4 Using this identity, the effect of an input price change on profitmaximizing input demand can be decomposed into the familiar substitution effect, holding the rate of output constant, and the profit-maximizing output effect. These two effects can in turn be decomposed. When the rate of employment of the rate-base input is held constant, the resulting conditional substitution and output effects are well-behaved; however, the second components of the substitution effect and the output effect, which result from the adjustment of the rate-base input to satisfy the rate-of-return constraint, are not necessarily well-behaved. 3There arc many papers which have applied the conditional (or equivalently the variable or restrlcted) profit function to the problem of rate-of-return regulation. See, for example, Atkinson and Halvorsen (1976), Diewert (1981), and Nelson (1985). 4This result has also been demonstrated by Diewert (1981).
J.P. Hughes, Profit-maximning
input demand
81
The adjustment of the rate-base input to satisfy the rate-of-return constraint introduces a pervasive pathology which upsets the a priori negativity of both the own-price substitution effects and outputs effects and destroys the symmetry of the cross effects. When the adjustment of the rate-base input is precluded, the consequent conditional effects are well-behaved. This finding suggests that the regulated firm’s short-run input demand is well-behaved, regardless of the effectiveness of regulation; however, in the long run effective regulation will cause adjustments in the rate-base input which upset the usual predictions which hold in the short run and in the absence of effective regulation. In the sections which follow, these assertions are demonstrated and the productive relationships among inputs which lead to this seemingly perverse behavior are explored.
2. The profit-maximizing
technology
Letting y denote the rate of output and x =(x1,. . . ,x,) 20 the vector of rates of employment of inputs, the set of feasible production plans (y,x) is defined by T. The maximum y, given x, is indicated by the production function f(x), which is assumed to be twice continuously differentiable and strictly quasi-concave. Let x- =(x1,. ..,.K,-i) denote the nonrate-base inputs and x, the nondepreciating rate-base input. The flow prices of the nonrate-base inputs are w-=(&I,,..., w,_ ,)>>O, and the stock price of the nth input is u’,,>O. (It is often assumed that w,,= 1.) The value of the rate base is thus w,x,,, and the opportunity cost is rw,x,, where r > 0 is the relevant market rate of interest. The firm’s price of output is given by the demand function p(y, u) >O where ap/lJy < 0 and, for the shift parameter a, c7p/& > 0. Thus, the revenue function is defined by R(y,a) =p(y, a)y, where R is twice continuously differentiable and aR2/iiy2 < 0. In the absence of regulation, the firm’s profit function is defined by n( w, r, a) = max [R(y, a) - wmx- - rw,x,: Y.i
(y, x) E T],
(1)
where w-x- is the dot product of the two vectors and where the maximizing arguments are the unregulated input demand functions x~(w, r, a), and from the production function, the optimal output is y”( w, r, u) E f(x”( w, r, a)). Regulation is constituted by establishing an allowable rate of return s, consisting of r and a premium v, where v>O, so that s=r +v and s> r. Thus, the firm’s profit must yield a rate of return consonant with the constraint [R(y,a)-WxR(y, a) - w x- - sw,x, 5 0. - rw,x,]/w,x, 5 u, or equivalently, The set of production plans which satisfy the regulatory constraint is
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J.P. Hughes, Profit-maximizing
input demand
(2)
Q(~,S,a)-{(y,x):R(y,a)-w-x--sw,x,~0}, so that the set of plans regulation is $(w,s,u)=
Tn
which
satisfy
the constraints
of technology
and
Q(w,s,u).
(3)
If the firm is viewed as either setting the price of output or proposing to the regulators an acceptable price, that is, one which yields the maximum profit consistent with the allowable rate of return, the firm’s profit function is defined as (4)
where the maximizing arguments are the regulated input demand functions P(w, I, s, a) and, substituting them into the production function, the regulated output, y( w, r, s, a). The regulated profit function can also be obtained from the unregulated profit function when it is made conditional on the rate-base input and is then maximized with respect to the conditioning argument, subject to the rate-ofreturn constraint. The unregulated conditional profit function is defined by it(w,r,u,xz)
=max
[R(y,u)-
W-X- -rw,,x,]:x,=x~
and
(y,x) E T,
Y-X-
(5) where the maximizing arguments are the unregulated conditional input 1, and from the production demand functions $( W-, a, xz), i = 1,...,nfunction, the conditional output function y”“(w-, a, x,“). Solving the problem max 7?(W,r, a, x,) Xn
s.t.
E(w,r,u,x,)-(s-r)w,x,50
(6)
yields the regulated demand function for the rate-base input 2;( W,r, s, a). Since both the two-stage maximization problem of (6) and the single-stage problem of (4) share tha same objective function and feasible choice set, the following identities hold: _CE( W,r, s, a) = 5$ w, r, s. a) so that
(7)
J.P. Hughes.
Pro~t~m~x~m~zing
~r(w,r,s,a)-x”:(w-,a,~~(w,r,s,u)),
input demand
i=l,...,
n-l,
~(W,Y,S,a)~y”“(w-,a,X,X(w,r,s,a)),
83
(8) (9)
and A(W,T,S,a) = qw, r, a, n,n(W,r, s, a)).
(10)
Thus, when the regulated demand for the rate-base input is substitute for x,, the unregulated conditional profit function is identicaiIy equal to the regulated profit function.
3. Applying Hotelling’s lemma Hotelling’s lemma cannot be applied directly to the regulated profit function. Consider a regulated production plan (j’,i”) which is optimal for (wo7r”, so, a”) and an arbitrary profit function, R(F”, a’) - W-A?- -rw,$, defined for any values of (w, r, s) which satisfy the regulatory constraint. The difference between the arbitrary and optimal profit functions for any value (w,F) which satisfies the regulatory constraint must be nonpositive and for (@,rolsa) zero. Thus, the solution to the maximization problem max [RfjP,#)-W-Pw9r.s
-r*w,?,z--n(~,r,s,u~)f
is (IV’,I’, so) and is defined by the first-order conditions -(i -(r-
-d)i_P45/f3w,=O, as>aX-
i= I ,...,n-I,
a~jaw~ =o,
-R(jO,uO)+w-P-
+sw,g=o,
(124 (12bj
(1-w
where the constraint is assumed binding and A denotes the Lagrangean multiplier. Conditions (12a) and (12b) indicate that differentiation of the profit function with respect to the factor prices fails to yieid the factor demands. On the other hand, applying Hotelling’s lemma to the ~onditiona1 profit
84
J.P. Hughes, Profit-maximizing
input demand
function, evaluated at the regulated optimum, Y?(w,r,a,Zz), yields the regulated values of the nonrate-base inputs: a?$ w, I, a, :,“)/aw‘ = - Z;(W-, a, 2;) = - .?F( w, r, s, a),
i=l,...,n-1,
(13)
by the identity (8). Thus, the absence of the regulatory constraint in determining the conditional profit function permits the application of Hotelling’s lemma while setting the value of x, equal to its regulated optimum 2:: allows the recovery of the regulated values of the nonrate-base inputs. 4. Quasi-convexity of the regulated profit function The sufficient second-order condition for the maximization problem (11) requires that the bordered Hessian of the objective function exhibit negative definite form. Letting 72ij= a2S/aw, awi and fi,, = a2fi/as &, the bordered Hessian matrix is 1
-7L11
...
-=1.-1
...
..
-7t_1,
...
-=t,-ln-
-fin, -it,, --jl,l
. . . -%-I . . . -fir,-1 . . . -%,-I
a;
... a;_
I:. 1
1
(14)
The signs of the border-preserving principal minors formed from the first (n- 1) rows and columns are indicated by
w
I
-jlij
a:
I
aT=(-I)‘, 0
(15)
where t is the order of the minor and to preserve the border, t 2 2, and since
(16) then
J.P. Hughes, Profit-maximizing
mput demand
icij a; < 0.
I I a:
85
(17)
0
It is evident that border-preserving principal minors involving the additional rows and columns also must conform to this pattern. Thus, the second-order conditions require that the bordered Hessian (17) of the regulated profit function exhibit positive definite form. Hence, it cannot be concluded that the regulated profit function is necessarily convex in input prices. By contrast, when the regulatory constraint is nonbinding, the border is eliminated and the principal minors of the Hessian exhibit positive definite form [i.e., the expression in (17) becomes ISijl >O], which yields the usual convexity. It does not appear that the second-order conditions imply even quasiconvexity. Quasi-convexity requires that the Hessian bordered by the gradient of the function conform to the above sign restrictions on the borderpreserving principal minors. However, the border of the Hessian (14) is not strictly the gradient of the profit function. Fare and Logan (1983b) have provided an example of a two-input, rate-of-return-regulated cost function which is not quasi-concave in input prices.
5. Restrictions
on the demand functions for input
In the absence of regulation the direct application of Hotelling’s lemma to the profit function permits the retrieval of the demands for inputs; thus, the Hessian matrix of the profit function shows the effect of variations in the parameters (w,r) on the demands, while the convexity of the profit function in (w, r) generates important restrictions. Effective regulation upsets the neatness of these derivations. The Hessian matrix no longer has a straightforward interpretation as the response of demands for inputs to variations in prices, nor do the maximum properties of the problem place any restrictions on the signs of the principal minors of the unbordered Hessian. The restrictions on input demand can be obtained relatively easily from the identity (8). Differentiating (8) with respect to a variation in w, results in the decomposition of the response of demand into its conditional components: a.?y/aw, = Z;(w-,
a, Z:,“)/awi + [iX;(w-,
a,
i;)/ax,l [(a.qawi)].(18)
The first term on the right-hand side indicates the direct price effect when the rate of employment of the rate-base input is held constant. It can be derived from the unregulated, conditional profit function,
a3qW, r,a, n;)/aw:= -X;(W-,a, q/awi,
J.P. Hughes, Pro~t-maximizing
86
input demand
and from the convexity of the unregulated function, its negative sign is evident: X~/Jw, < 0. The second term of the direct price effect is the result of the adjustment of the rate-base input to satisfy the regulatory constraint. The sign of the first term in the product depends on whether the ith and nth inputs are gross complements or substitutes in the absence of regulation. This assertion is easily demonstrated. Consider the demand for the ith input, conditional on the ~~~eg~~ufe~pro~t-maximizing demand for the nth input [see (111, x’f(w-, a, xE(w, i; a)) =
.qw, F,a),
(19)
where i is a synthetic rate of interest whose value is established such that the values of the unregulated demands evaluated at (w,i) are equal to those under regulation at market prices (w, r), xX( w, i, a) = in( w, r, s, a).
69)
If the unregulated conditional profit function ~(w, i, a, x,) is maximized over x, without imposing the regulatory constraint and if the maximizing argument is .%;(rv,i,a), the value of F is set such that ~~(~,~,s,~~ =Zf(~.i,u).~ Thus, it is evident that a:( W,i-,a) =i,“(w,r,s, u) and that the equalities (20) hold. Then using (20) and (7) and differentiating (19) with respect to WI,,
WY w - , u, Wdx, = (ax;( w, i, a)/3w,)/( t?x,x(w, i, ayaw,) >
as
i?x%jSw,f50,
ifn,
(21)
which is to say, the conditional demand for the ith input varies directly (inversely) with the employment of the nth input when they are gross complements (substitutes). The sign of the final component of (18) is easily determined by differentiating the regulatory constraint in (6), evaluated at the optimum, with respect to w,:
5The synthettc rate of interest respect to (I’. n) for gtven values (I-I.)[R,f,-wp]=O R(y, a’) - W’X-
F can also be determined
(d, r”, so).The tirst-order Vr#n,
from the maxtmtzation conditions are
of
(I 1) wtth
(l--I)R,S,-(r”-Iso)~‘~=O.
-s”w,0x,=O.
In the absence of regulation, the vector of input demands will be identical to their regulated values if the firm faces the vector of parameters (w”,i,a”), where ?=(r’-i*s’)/( 1 -A*) and A* is the opttmal value of the Lagrange multiplier. because the synthetic price vector yields the same first-order conditions as under regulatron with market prtces.
J.P. Hughes, Profit-maxwnizing
Fg.
a3;/awi = sy[ayax,
input demand
87
1
-(s - r) w,]
co,
(22)
since &Y(.)/ax,
as
ax:tw,
i, a)/iiw,
<
as
aq(t++,a)/aw,>o,
0,
Vifn.
(23)
If the price of the ith input rises, cost increases and profit falls so that the regulatory constraint is no longer binding. Thus, less of the rate-base input is needed to satisfy the constraint. When the ith and nth inputs are gross complements, the reduced employment of the rate-base input will tend to reduce the employment of the ith input as well. However, in the case when they are gross substitutes, the decrease in the nth input tends to increase the ith input, even though its price has risen. The sign of the first term in the decomposition ( 18) the conditional direct-price effect, is unambiguously negative. Thus, in the case of substitutes, the sign of the sum of the two components is ambiguous. An examination of the cross-price effects shows that regulation upsets the symmetry of the unregulated case. Differentiating the identity (8) with respect t0 Wj,
88
J.P. Hughes, Profit-maximizing
V
i#j
and
input demand
i,j#n,
(24)
and for xj, with respect to wi, (25) The good behavior of the conditional profit function symmetry relation E,(w, r, a, $) = Eji(w, r, a, 2:) so that
aqw, a,z;)/awi=aq~-, a,~;)/awj;
(5) results
in the
(26)
however, the second terms in the decompositions (24) and (25) are not in general equal so that the symmetry of the cross-price effects is upset by regulation. Finally, the role in production created for the rate-base input is evident when the effect on its demand of variations in r and s are considered. When s>r, regulation influences the choice of production plans through w and s, given technology and the demand for output, while the influence of r is limited to the inframarginal determination of profit. This assertion is apparent when the two-stage maximization problem is examined. In the first stage, r and x, determine the fixed cost while profit is maximized by the variable inputs x . In the second stage, x, is adjusted to achieve the maximum profit, defined as the difference [R(y,a)-w-x-], consistent with s and w,. As long as s> r, s will override r at the margin in the determination of the optimal value of x, in the second stage. Hence, an infinitesimal variation in r, holding s constant will have no effect on the optimal values P; however, profit will be varied as indicated by (12~) %/Fr= - w,$;. This claim can be demonstrated by differentiating the regulatory constant (6), evaluated at the optimum, with respect to r and solving for aR,“/dr: di;/ar
= ( - ~,a:
- aiilaiy(a7?/ax,
=o
-(s - r) w,) (27)
by (12~). In the first stage (5) of the two-stage problem, a; is parametric so that a variation in r does not affect the solution .?r(w-, a,.?:) V ifn. Thus, &?;I& = 0 V i. The effect of a variation in the allowable rate of return s on the rate-base input can be ascertained in a similar manner. Differentiating the constraint in (6) evaluated at the optimum, with respect to s and solving for &it/& X;/&
= w,i;pi7jax,
-(s - r) WJ (28)
J.P. Hughes, Profit-maximizing
input demand
89
since %/ax,<0 when the regulatory constraint is binding (see fig. 1). Hence, a rise in the allowable rate of return reduces the rate of employment of the rate-base input since less of the input is required for any given level of profit to satisfy the regulatory constraint. Thus, it is evident now that the special role given to the rate-base input under rate-of-return regulation introduces a pervasive pathology to the usual comparative-static properties of the unregulated optimum. The workings of this pathology can be detailed by decomposing the profit-maximizing price effects into substitution and output effects, using the conditional cost function and its conditional input demand functions, evaluated at the regulated optimum.
6. The cost-minimizing
technology
In the absence of a regulatory constraint, the firm would choose to produce the rate of output y using the least costly input vector from the input requirement set X(y) = {x:.04
ZY)?
(29)
and the resulting cost function is defined by C(y, w, r) = min [wx + rw,x,:x E X(y)], 1
(30)
where the minimizing arguments are the constant-output input demand functions x*(y, w, r). Under regulation the firm’s production plan must obey the regulatory regime. In defining the least cost of any rate of output, the regulated cost function must account for the demand schedule of output as well as the allowable rate of return. Thus, the input employment vector must be a member of the subset of Q(w, s, a),
-sw,x,~q,
Z(y,w,s,a)=(x:R(y,a)-w-x
(31)
so that the firm is constrained to choose its production plan from the set RY,
w,s,4 = {xE X(Y) n
Z(Y,
w,s,4).
(32)
Consequently, the regulated cost function is defined by e(y,W,r,s,a)-min[w-x-+rw,x,:xE~(y,w,s,a)], x
(33)
90
J.P. Hughes, Profit-maximizing
input demand
where the minimizing arguments are the regulated input demand functions f(Y, w, s, a).
The conditional cost function is defined as c(y, W,r, xt) = min [w-xx-
+ rw,x,]:x,
=x,0
and
x E X(y),
where the minimizing arguments are the conditional li(y, W-, XH), i= 1, . . . , n - 1. Solving the problem
input
(34)
demands
min C(y, w, r, x,) X” s.t.
R(y,u)-Z;(y,w,r,x,)-(s-r)w,x,sO
(35)
yields the rate-base demand function Z,(y, W,r, s, a) which, when substituted into c(y, W,r, x,,), results in the identity Qy, w, r, Z,(y, w, r, s, u)) = Qy, w, r, s, u),
(36)
since the choice sets for the two problems are identical, i.e. x~_%(y, w,s,~). Hence, it also follows that i,(y,
IV, S,
U)= -Ci(yjWe, IQ,
W, r, s, a))
vi Z 4
(37)
and a,(y, W,r, s, a) = %(y, w, r, s, a).
(38)
It is a straightforward matter to relate cost-minimizing demands to profit-maximizing demand. Since maximizing profit subject to the regulatory constraint implies minimizing cost subject to the same constraint, the conditional cost-minimizing input demand functions, evaluated at the prolitmaximizing regulated optimum, are identically equal to the regulated profitmaximizing input demand functions: ay(w,r,s,u)rKi(w-,~(w-,U,~~(w,r,s,U)),I~(w,r,s,u))
and from (7)
Vi+%
(39)
J.P. Hughes, Profit-maximizing input demand
91
Using the identity (39), the profit-maximizing direct-price effect can be decomposed into substitution and output effects, each containing a component conditional on x, and a component resulting from the adjustment of X, to satisfy the regulatory constraint. The sign of the former is negative; the latter, ambiguous. Differentiating (39) with respect to M’i,
x
faqaw,} V i #
n,
which is an extended decomposition
=ani(W~,~n,~~)/dWi+[a~i(w-,
(41) of the components of (18),
~,X~)/ay]Cirq;"(w-,~:)/aWill (41’)
and
The components of (41’) are the unregulated, conditional substitution and output effects, evaluated at the regulated optimum; thus, their signs are both negative.6 The response of the ith input to the adjustment of the rate-base input to satisfy the regulatory constraint is decomposed in (41”) into a substitution effect, holding the rate of output constant, and an output effect, resulting from the response of output to the adjustment in the nth input. As will be shown below, neither component’s sign can be determined a priori. 6This assertton ts proved by Sakai for the case of a firm which is compettttve in the output market. The proof for a monopoly IS straightforward. The negativity of the condittonal substitution effect follows from the convexity of the condittonal profit function. The negattvity of the output effect follows from the relatronship of the response of marginal cost to changes m input prtces. In parttcular. when the ith input is normal, i.e. F%,(.)/ii~>O, marginal cost vartes directly with M’,. Thus, assuming the second-order conditions for a maximum of protit are satisfied, the profit-maximizing rate of output must vary inversely with w,, ie. ?y(.)/?w,
RE
D
92
J.P. Hughes, Profit-maximuing
input demand
Therefore, neither the aggregate substitution effect nor the aggregate output effect has a refutable restriction on its sign. The sign of the substitution component of (41”) can be shown to depend on whether the ith and nth inputs are net complements or substitutes in the absence of regulation. When the unregulated conditional cost-minimizing demands for inputs are evaluated at the synthetic prices (lo,?) which are established such that
and .f,(jT, w. r, s, a) = x,*(F, w, i, a).
(43)
Substituting (43) into (42), differentiating with respect to u’,, and rearranging,
5
o
as
?xT(j”,
w, t,
a)/8w, 2 0
Q i#n.
(44)
Thus, from (24),
so that the sign of the substitution effect is unambiguously negative only when the ith and nth inputs are net complements. When the price of the ith input rises, profits decrease and the regulatory constraint is no longer binding. Thus, at any given rate of output, less of the rate-base input is needed to satisfy the regulatory constraint. When the ith and nth inputs are net complements, the decrease in the employment of the rate-base input to satisfy the constraint would tend to reduce the employment of the ith input as well, thereby reinforcing the conditional substitution component. On the other hand, when the ith and nth inputs are net substitutes, the reduced employment of the nth input wouid tend to raise the employment of the now-more-expensive ith input and calls into question the sign of the aggregate substitution effect. In the absence of regulation, the sign of the output effect is unambiguously negative. Since the conditional output effect is derived from the unregulated conditional input demand, evaluated at the regulated optimum, it too is unambiguously negative. However, when the conditioning value of the ratebase input’s employment is adjusted to satisfy the regulatory constraint, the resulting component of the output effect is found to have no a priori sign restriction so that the sign of the aggregate output effect is not necessarily negative. From (41”) the regulation-induced component of the output effect
J.P. Hughes, Profit-maximizing
input demand
93
is [ii~i(‘)/ay] [aF( .)/ax,] [Z~(.)/C?W,]. The sign of the third term has been shown negative in (24) and Zi( *)/c?Y~ 0 as the conditional demand for the ith input is normal or inferior. The sign of the second term can be derived from the equality (46) that is to say, the synthetic prices (IV,?) yield an optimal unregulated rate of output equal to the optimal regulated rate for the market prices (IV,r) and the allowable rate of return s. Since additionally, 5?:(w, r, s, a) = x,“( w, i, a), differentiating (46) with respect to w, and rearranging yields the relationship
ayyw- , U, wax, = c~Y=(w,i, 4iaw,ii[ax;(w, 2 0
as
i, 4iaw,i
apyw, f, a)/?~, ><0.
(47)
It is well established that ay”/aw n 50 as iix,*/dy2 0.’ Thus, the sign of the regulation-induced component of the output effect depends on whether the demand for the nth input and the conditional demand for the ith input are normal or inferior. When both are normal (or inferior), the sign of this component of the output effect is unambiguously negative and, hence, so is the sign of the aggregate output effect. However, when one or the other, but not both, is inferior, the sign of the regulation-induced component is positive and that of the aggregate effect is ambiguous. Differentiating (39) with respect to wjr the cross-price effect is r, S, u)/?w~
an;(w,
= d~i( .)/aw, + [fXi( .)/Zy] +
[ajn( w-, ?~)/aw,]
v i,jfn. (ami(')/SX,+[c?~,(')/ay][a~K(W-,~~)/i)X,]} (azyawj) (48)
Without writing the expression for %2,7(.)/awi, it is evident that the first and second terms on the right-hand side of (48) are equal, respectively, to the first and second terms of axzj”/awi.8 However, the regulation-induced components of the substitution and output effects, the third and fourth terms, are not ‘The argument supporting this claim is identical to that of the previous footnote. ‘Comparing the cross-prxe effects m (24) and in (48). it IS evident that the first term m (24), the conditional ‘gross’ cross-price effect, can be decomposed mto the lirst and second terms of (48), the conditional ‘net’ cross-price effect and the conditional output effect. Since the gross effects and the net effects are symmetric between pairs of inputs, the output effects must also be symmetric.
94
J.P. Hughes, Profit-maximizing
input demand
necessarily equal to their respective counterparts. Thus, neither the aggregate cross substitution nor the aggregate cross output effects are necessarily equal between pairs of inputs, although their conditional components are equal. 7. Conclusions Both the cost-minimizing and the protit-maximizing demand for inputs are well behaved when the rate of employment of the rate-base input is held constant. The conditional cost and profit functions are also well behaved. However, when the rate of employment of the rate-base input is allowed to adjust to parametric variations in order to satisfy the rate-of-return constraint, the usual properties of the input demand functions and cost and profit functions will not necessarily be obtained. Thus, the regulated firm’s short-run behavior should conform to the usual restrictions of cost minimization and profit maximization while its long-run behavior under a regime of effective regulation may be qualitatively different from that of an unregulated regime. References Atkinson, Scott E. and Robert Halvorsen, 1976, Interfuel substitution in steam electric power generation, Journal of Political Economy 84, no. 5, 959-978. Baumol, W.J. and A.K. Klevorick, 1970. Input choices and rate-of-return regulation: An overview of the discussion, Bell Journal of Economics 1, Autumn, 162-190. Cowing, Thomas G., 1978, The effectiveness of rate-of-return regulation: An empirical test using profit functions, in: Melvyn Fuss and Daniel McFadden, eds., Production economics: A dual approach to theory and applicaitons, Vol. 2 (North-Holland, Amsterdam). Cowing, Thomas G., 1979. Duality and the estimation of a restricted technology, Working paper 79-l (Department of Economics, State University of New York, Binghamton, NY). Diewert. W. Erwin, 1981, The theory of total factor productivity measurement m regulated industries, in: Thomas G. Cowing and Rodney E. Stevenson, eds., Productivity measurement m regulated industries (Academic Press, New York). Fare, Rolf and James Logan, 1983a, The rate-of-return regulated lirm: Cost and production duality, Bell Journal of Economics 14, no. 2, 405414. Fare, Rolf and James Logan. 1983b, Shephard’s lemma and rate-of-return regulation, Economics Letters 12, 121-125. Fare, Rolf and James Logan, 1983~. The rate-of-return regulated version of Shephard’s lemma, Economics Letters 13. 297-302. Fuss, Melvyn and Leonard Waverman, 1981, Regulation and the multiproduct firm: The case of telecommunications in Canada, in: Gary Fromm, ed., Studies m public regulation (MIT Press, Cambridge, MA). Hughes. Joseph P., 1981, Giffen mputs and the theory of mulitple production. Journal of Economic Theory 25, 287-301. Hughes. Joseph P., 1984, Pathological substitution effects under rate-of-return regulation, Working paper (Department of Economics, Rutgers Umversity, New Brunswick, NJ). Hughes, Joseph P., 1986, Input demand under rate-of-return regulation: Applying Shephard’s lemma, Working paper (Department of Economics, Rutgers University, New Brunswick, NJ). McNicol, David L., 1973, The comparative statics properties of the theory of the regulated firm, Bell Journal of Economics 4. Autumn, 428453. Nelson, R.A., 1985, Returns to scale from variable and total cost functions, Economics Letters 18, 271-276.
Nelson, Randy A. and Mark E. Wohar, 1983, Regulation. scale economies, and productivity in steam-electric generation, lnternationsl Economic Review 24, no. I, 57-79. Nelson, Randy A. and Mark E. Wohar, 1987, A reply to ‘Regulatron, scale and productivity: A comment’, International Economic Review 28, no. 2, 535-539. Gfori-Mensa, Charles, 1982, Theory and estimation of input demand for the regulated firm. Ph.D. dissertation (Rutgers University, New Brunswick, NJ). Silberberg, Eugene, 1978, The structure of economics (McGraw-Hill, New York). Smith, V. Kerry, 1985, Elasticities of substitution for a regulated cost function, Economics Letters 7. 215-219.
PROFIT-MAXIMIZING INPUT DEMAND UNDER RATE-OF-RETURN REGULATION Pathological Substitution and Output Effects Joseph P. HUGHES Rutgers
U~tversit~~,
New
~ru~s~~ck.
NJ
~89~3, USA
Received March 1988, final verston recetved May 1989 Cost-mmimrzing and profit-maximtzing input demand for the firm subJect to rate-of-return regulation are examined. Unregulated cost and profit functtons which are condittonal on the rate of employment of the rate-base input are shown to be Identically equal to the regulated cost and profit functions, evaluated at the regulated opttmum. Shephard’s lemma and Hotelling’s lemma applied to the conditional cost and profit functions, evaluated at the regulated opttmum, yield the regulated input demands. For both the profit-maxtmizing and cost-mimmizing demands, own-prtce effects are not necessartly negative nor are the cross-price effects in general equal. Moreover, the profit-maximizing output effect is not necessartly negative. Various decompositions of input demand are explored to explain these conclustons.
1. Introduction Rate-of-return regulation introduces a pervasive pathology to the profit function and to the profit-maximizing input demands. When the regulatory constraint is binding, the profit function is not necessarily convex. Hotelling’s lemma cannot be applied directly.’ The own-price effect on profit-maximizing input demand is not necessarily negative. Neither component of the ownprice effect, the substitution effect and the profit-maximizing output effect, is necessarily negative.2 Moreover, the cross-price effects between pairs of inputs are not necessarily equal. ‘See Cowmg (1978) and for the equivalent result for Shephar~s lemma, Cowmg ( 1979). Fuss and Waverman (1981). Smith (1981). and Nelson and Wohar f 1983). ‘For analyses of the comparative-static restrictions on the demand for inputs, see. for example, Baumol and Klevonck (1970) and McNicoi (1973). who assume that there are only two inputs, and Ofori-Mensa (1982) who treats the n-input case. Ofori-Mensa shows that the results of the two-input case do not generalize to the n-input case for cost minimrzation. He does not explore the comparative-static restricttons for profit maxtmization nor does he investigate the productive relationshtps whtch lead to pathologtcal results. None of these papers decomposes the profit-maximizing response of input demand to prtce changes into substitutton and output effects. Hughes (1984. 1986) examines the n-input case of cost minimization and, usmg the conditional (or restricted) cost function, investigates the productive relationships which lead to pathological substitutton effects. 0165-0572/90/%3.50 8 1990, Elsevier Science Publishers B.V. (North-Holland)
80
J.P. Hughes, Profit-maximizing
input demand
The workings of this pathology can be uncovered by formulating an unregulated profit function which is conditional on the rate of employment of the rate-base input. When this unregulated conditional profit function is maximized subject to the rate-of-return constraint and the resulting demand function for the rate-base input is substituted into the conditional profit function, the conditional profit function becomes identically equal to the unconditional regulated profit function3 Applying Hotelling’s lemma directly to the unconditional regulated profit function fails to yield these demands. However, Hotelling’s lemma can be directly applied to the conditional profit function, evaluated at the regulated optimum, to yield the conditional rates of employment of the nonrate-base inputs, which at the regulated optimum, are equal to the regulated values of input demand. When the regulated demand function for the rate-base input is substituted into these unregulated conditional demand functions, the conditional demand functions are identically equal to the unconditional regulated demand functions. Using this identity, the effect on the demand for a nonrate-base input of a change in an input’s price can be decomposed into two components, one the result of holding the rate of employment of the rate-base input constant and the other the result of adjusting the rate base to satisfy the rate-of-return constraint. The latter component is found to contain the pathology. These two components can in turn be decomposed to isolate the source of the pathology. This decomposition is effected by constructing the regulated cost-minimizing demand for a nonrate-base input from the unregulated conditional cost-minimizing demand function. The latter is conditioned on the physical rate of output and the employment of the rate-base input. Substituting the regulated profit-maximizing supply function for output and the regulated profit-maximizing demand function for the rate-base input for the conditional values, the conditional cost-minimizing demand function becomes identically equal to the regulated profit-maximizing demand function.4 Using this identity, the effect of an input price change on profitmaximizing input demand can be decomposed into the familiar substitution effect, holding the rate of output constant, and the profit-maximizing output effect. These two effects can in turn be decomposed. When the rate of employment of the rate-base input is held constant, the resulting conditional substitution and output effects are well-behaved; however, the second components of the substitution effect and the output effect, which result from the adjustment of the rate-base input to satisfy the rate-of-return constraint, are not necessarily well-behaved. 3There arc many papers which have applied the conditional (or equivalently the variable or restrlcted) profit function to the problem of rate-of-return regulation. See, for example, Atkinson and Halvorsen (1976), Diewert (1981), and Nelson (1985). 4This result has also been demonstrated by Diewert (1981).
J.P. Hughes, Profit-maximning
input demand
81
The adjustment of the rate-base input to satisfy the rate-of-return constraint introduces a pervasive pathology which upsets the a priori negativity of both the own-price substitution effects and outputs effects and destroys the symmetry of the cross effects. When the adjustment of the rate-base input is precluded, the consequent conditional effects are well-behaved. This finding suggests that the regulated firm’s short-run input demand is well-behaved, regardless of the effectiveness of regulation; however, in the long run effective regulation will cause adjustments in the rate-base input which upset the usual predictions which hold in the short run and in the absence of effective regulation. In the sections which follow, these assertions are demonstrated and the productive relationships among inputs which lead to this seemingly perverse behavior are explored.
2. The profit-maximizing
technology
Letting y denote the rate of output and x =(x1,. . . ,x,) 20 the vector of rates of employment of inputs, the set of feasible production plans (y,x) is defined by T. The maximum y, given x, is indicated by the production function f(x), which is assumed to be twice continuously differentiable and strictly quasi-concave. Let x- =(x1,. ..,.K,-i) denote the nonrate-base inputs and x, the nondepreciating rate-base input. The flow prices of the nonrate-base inputs are w-=(&I,,..., w,_ ,)>>O, and the stock price of the nth input is u’,,>O. (It is often assumed that w,,= 1.) The value of the rate base is thus w,x,,, and the opportunity cost is rw,x,, where r > 0 is the relevant market rate of interest. The firm’s price of output is given by the demand function p(y, u) >O where ap/lJy < 0 and, for the shift parameter a, c7p/& > 0. Thus, the revenue function is defined by R(y,a) =p(y, a)y, where R is twice continuously differentiable and aR2/iiy2 < 0. In the absence of regulation, the firm’s profit function is defined by n( w, r, a) = max [R(y, a) - wmx- - rw,x,: Y.i
(y, x) E T],
(1)
where w-x- is the dot product of the two vectors and where the maximizing arguments are the unregulated input demand functions x~(w, r, a), and from the production function, the optimal output is y”( w, r, u) E f(x”( w, r, a)). Regulation is constituted by establishing an allowable rate of return s, consisting of r and a premium v, where v>O, so that s=r +v and s> r. Thus, the firm’s profit must yield a rate of return consonant with the constraint [R(y,a)-WxR(y, a) - w x- - sw,x, 5 0. - rw,x,]/w,x, 5 u, or equivalently, The set of production plans which satisfy the regulatory constraint is
82
J.P. Hughes, Profit-maximizing
input demand
(2)
Q(~,S,a)-{(y,x):R(y,a)-w-x--sw,x,~0}, so that the set of plans regulation is $(w,s,u)=
Tn
which
satisfy
the constraints
of technology
and
Q(w,s,u).
(3)
If the firm is viewed as either setting the price of output or proposing to the regulators an acceptable price, that is, one which yields the maximum profit consistent with the allowable rate of return, the firm’s profit function is defined as (4)
where the maximizing arguments are the regulated input demand functions P(w, I, s, a) and, substituting them into the production function, the regulated output, y( w, r, s, a). The regulated profit function can also be obtained from the unregulated profit function when it is made conditional on the rate-base input and is then maximized with respect to the conditioning argument, subject to the rate-ofreturn constraint. The unregulated conditional profit function is defined by it(w,r,u,xz)
=max
[R(y,u)-
W-X- -rw,,x,]:x,=x~
and
(y,x) E T,
Y-X-
(5) where the maximizing arguments are the unregulated conditional input 1, and from the production demand functions $( W-, a, xz), i = 1,...,nfunction, the conditional output function y”“(w-, a, x,“). Solving the problem max 7?(W,r, a, x,) Xn
s.t.
E(w,r,u,x,)-(s-r)w,x,50
(6)
yields the regulated demand function for the rate-base input 2;( W,r, s, a). Since both the two-stage maximization problem of (6) and the single-stage problem of (4) share tha same objective function and feasible choice set, the following identities hold: _CE( W,r, s, a) = 5$ w, r, s. a) so that
(7)
J.P. Hughes.
Pro~t~m~x~m~zing
~r(w,r,s,a)-x”:(w-,a,~~(w,r,s,u)),
input demand
i=l,...,
n-l,
~(W,Y,S,a)~y”“(w-,a,X,X(w,r,s,a)),
83
(8) (9)
and A(W,T,S,a) = qw, r, a, n,n(W,r, s, a)).
(10)
Thus, when the regulated demand for the rate-base input is substitute for x,, the unregulated conditional profit function is identicaiIy equal to the regulated profit function.
3. Applying Hotelling’s lemma Hotelling’s lemma cannot be applied directly to the regulated profit function. Consider a regulated production plan (j’,i”) which is optimal for (wo7r”, so, a”) and an arbitrary profit function, R(F”, a’) - W-A?- -rw,$, defined for any values of (w, r, s) which satisfy the regulatory constraint. The difference between the arbitrary and optimal profit functions for any value (w,F) which satisfies the regulatory constraint must be nonpositive and for (@,rolsa) zero. Thus, the solution to the maximization problem max [RfjP,#)-W-Pw9r.s
-r*w,?,z--n(~,r,s,u~)f
is (IV’,I’, so) and is defined by the first-order conditions -(i -(r-
-d)i_P45/f3w,=O, as>aX-
i= I ,...,n-I,
a~jaw~ =o,
-R(jO,uO)+w-P-
+sw,g=o,
(124 (12bj
(1-w
where the constraint is assumed binding and A denotes the Lagrangean multiplier. Conditions (12a) and (12b) indicate that differentiation of the profit function with respect to the factor prices fails to yieid the factor demands. On the other hand, applying Hotelling’s lemma to the ~onditiona1 profit
84
J.P. Hughes, Profit-maximizing
input demand
function, evaluated at the regulated optimum, Y?(w,r,a,Zz), yields the regulated values of the nonrate-base inputs: a?$ w, I, a, :,“)/aw‘ = - Z;(W-, a, 2;) = - .?F( w, r, s, a),
i=l,...,n-1,
(13)
by the identity (8). Thus, the absence of the regulatory constraint in determining the conditional profit function permits the application of Hotelling’s lemma while setting the value of x, equal to its regulated optimum 2:: allows the recovery of the regulated values of the nonrate-base inputs. 4. Quasi-convexity of the regulated profit function The sufficient second-order condition for the maximization problem (11) requires that the bordered Hessian of the objective function exhibit negative definite form. Letting 72ij= a2S/aw, awi and fi,, = a2fi/as &, the bordered Hessian matrix is 1
-7L11
...
-=1.-1
...
..
-7t_1,
...
-=t,-ln-
-fin, -it,, --jl,l
. . . -%-I . . . -fir,-1 . . . -%,-I
a;
... a;_
I:. 1
1
(14)
The signs of the border-preserving principal minors formed from the first (n- 1) rows and columns are indicated by
w
I
-jlij
a:
I
aT=(-I)‘, 0
(15)
where t is the order of the minor and to preserve the border, t 2 2, and since
(16) then
J.P. Hughes, Profit-maximizing
mput demand
icij a; < 0.
I I a:
85
(17)
0
It is evident that border-preserving principal minors involving the additional rows and columns also must conform to this pattern. Thus, the second-order conditions require that the bordered Hessian (17) of the regulated profit function exhibit positive definite form. Hence, it cannot be concluded that the regulated profit function is necessarily convex in input prices. By contrast, when the regulatory constraint is nonbinding, the border is eliminated and the principal minors of the Hessian exhibit positive definite form [i.e., the expression in (17) becomes ISijl >O], which yields the usual convexity. It does not appear that the second-order conditions imply even quasiconvexity. Quasi-convexity requires that the Hessian bordered by the gradient of the function conform to the above sign restrictions on the borderpreserving principal minors. However, the border of the Hessian (14) is not strictly the gradient of the profit function. Fare and Logan (1983b) have provided an example of a two-input, rate-of-return-regulated cost function which is not quasi-concave in input prices.
5. Restrictions
on the demand functions for input
In the absence of regulation the direct application of Hotelling’s lemma to the profit function permits the retrieval of the demands for inputs; thus, the Hessian matrix of the profit function shows the effect of variations in the parameters (w,r) on the demands, while the convexity of the profit function in (w, r) generates important restrictions. Effective regulation upsets the neatness of these derivations. The Hessian matrix no longer has a straightforward interpretation as the response of demands for inputs to variations in prices, nor do the maximum properties of the problem place any restrictions on the signs of the principal minors of the unbordered Hessian. The restrictions on input demand can be obtained relatively easily from the identity (8). Differentiating (8) with respect to a variation in w, results in the decomposition of the response of demand into its conditional components: a.?y/aw, = Z;(w-,
a, Z:,“)/awi + [iX;(w-,
a,
i;)/ax,l [(a.qawi)].(18)
The first term on the right-hand side indicates the direct price effect when the rate of employment of the rate-base input is held constant. It can be derived from the unregulated, conditional profit function,
a3qW, r,a, n;)/aw:= -X;(W-,a, q/awi,
J.P. Hughes, Pro~t-maximizing
86
input demand
and from the convexity of the unregulated function, its negative sign is evident: X~/Jw, < 0. The second term of the direct price effect is the result of the adjustment of the rate-base input to satisfy the regulatory constraint. The sign of the first term in the product depends on whether the ith and nth inputs are gross complements or substitutes in the absence of regulation. This assertion is easily demonstrated. Consider the demand for the ith input, conditional on the ~~~eg~~ufe~pro~t-maximizing demand for the nth input [see (111, x’f(w-, a, xE(w, i; a)) =
.qw, F,a),
(19)
where i is a synthetic rate of interest whose value is established such that the values of the unregulated demands evaluated at (w,i) are equal to those under regulation at market prices (w, r), xX( w, i, a) = in( w, r, s, a).
69)
If the unregulated conditional profit function ~(w, i, a, x,) is maximized over x, without imposing the regulatory constraint and if the maximizing argument is .%;(rv,i,a), the value of F is set such that ~~(~,~,s,~~ =Zf(~.i,u).~ Thus, it is evident that a:( W,i-,a) =i,“(w,r,s, u) and that the equalities (20) hold. Then using (20) and (7) and differentiating (19) with respect to WI,,
WY w - , u, Wdx, = (ax;( w, i, a)/3w,)/( t?x,x(w, i, ayaw,) >
as
i?x%jSw,f50,
ifn,
(21)
which is to say, the conditional demand for the ith input varies directly (inversely) with the employment of the nth input when they are gross complements (substitutes). The sign of the final component of (18) is easily determined by differentiating the regulatory constraint in (6), evaluated at the optimum, with respect to w,:
5The synthettc rate of interest respect to (I’. n) for gtven values (I-I.)[R,f,-wp]=O R(y, a’) - W’X-
F can also be determined
(d, r”, so).The tirst-order Vr#n,
from the maxtmtzation conditions are
of
(I 1) wtth
(l--I)R,S,-(r”-Iso)~‘~=O.
-s”w,0x,=O.
In the absence of regulation, the vector of input demands will be identical to their regulated values if the firm faces the vector of parameters (w”,i,a”), where ?=(r’-i*s’)/( 1 -A*) and A* is the opttmal value of the Lagrange multiplier. because the synthetic price vector yields the same first-order conditions as under regulatron with market prtces.
J.P. Hughes, Profit-maxwnizing
Fg.
a3;/awi = sy[ayax,
input demand
87
1
-(s - r) w,]
co,
(22)
since &Y(.)/ax,
as
ax:tw,
i, a)/iiw,
<
as
aq(t++,a)/aw,>o,
0,
Vifn.
(23)
If the price of the ith input rises, cost increases and profit falls so that the regulatory constraint is no longer binding. Thus, less of the rate-base input is needed to satisfy the constraint. When the ith and nth inputs are gross complements, the reduced employment of the rate-base input will tend to reduce the employment of the ith input as well. However, in the case when they are gross substitutes, the decrease in the nth input tends to increase the ith input, even though its price has risen. The sign of the first term in the decomposition ( 18) the conditional direct-price effect, is unambiguously negative. Thus, in the case of substitutes, the sign of the sum of the two components is ambiguous. An examination of the cross-price effects shows that regulation upsets the symmetry of the unregulated case. Differentiating the identity (8) with respect t0 Wj,
88
J.P. Hughes, Profit-maximizing
V
i#j
and
input demand
i,j#n,
(24)
and for xj, with respect to wi, (25) The good behavior of the conditional profit function symmetry relation E,(w, r, a, $) = Eji(w, r, a, 2:) so that
aqw, a,z;)/awi=aq~-, a,~;)/awj;
(5) results
in the
(26)
however, the second terms in the decompositions (24) and (25) are not in general equal so that the symmetry of the cross-price effects is upset by regulation. Finally, the role in production created for the rate-base input is evident when the effect on its demand of variations in r and s are considered. When s>r, regulation influences the choice of production plans through w and s, given technology and the demand for output, while the influence of r is limited to the inframarginal determination of profit. This assertion is apparent when the two-stage maximization problem is examined. In the first stage, r and x, determine the fixed cost while profit is maximized by the variable inputs x . In the second stage, x, is adjusted to achieve the maximum profit, defined as the difference [R(y,a)-w-x-], consistent with s and w,. As long as s> r, s will override r at the margin in the determination of the optimal value of x, in the second stage. Hence, an infinitesimal variation in r, holding s constant will have no effect on the optimal values P; however, profit will be varied as indicated by (12~) %/Fr= - w,$;. This claim can be demonstrated by differentiating the regulatory constant (6), evaluated at the optimum, with respect to r and solving for aR,“/dr: di;/ar
= ( - ~,a:
- aiilaiy(a7?/ax,
=o
-(s - r) w,) (27)
by (12~). In the first stage (5) of the two-stage problem, a; is parametric so that a variation in r does not affect the solution .?r(w-, a,.?:) V ifn. Thus, &?;I& = 0 V i. The effect of a variation in the allowable rate of return s on the rate-base input can be ascertained in a similar manner. Differentiating the constraint in (6) evaluated at the optimum, with respect to s and solving for &it/& X;/&
= w,i;pi7jax,
-(s - r) WJ (28)
J.P. Hughes, Profit-maximizing
input demand
89
since %/ax,<0 when the regulatory constraint is binding (see fig. 1). Hence, a rise in the allowable rate of return reduces the rate of employment of the rate-base input since less of the input is required for any given level of profit to satisfy the regulatory constraint. Thus, it is evident now that the special role given to the rate-base input under rate-of-return regulation introduces a pervasive pathology to the usual comparative-static properties of the unregulated optimum. The workings of this pathology can be detailed by decomposing the profit-maximizing price effects into substitution and output effects, using the conditional cost function and its conditional input demand functions, evaluated at the regulated optimum.
6. The cost-minimizing
technology
In the absence of a regulatory constraint, the firm would choose to produce the rate of output y using the least costly input vector from the input requirement set X(y) = {x:.04
ZY)?
(29)
and the resulting cost function is defined by C(y, w, r) = min [wx + rw,x,:x E X(y)], 1
(30)
where the minimizing arguments are the constant-output input demand functions x*(y, w, r). Under regulation the firm’s production plan must obey the regulatory regime. In defining the least cost of any rate of output, the regulated cost function must account for the demand schedule of output as well as the allowable rate of return. Thus, the input employment vector must be a member of the subset of Q(w, s, a),
-sw,x,~q,
Z(y,w,s,a)=(x:R(y,a)-w-x
(31)
so that the firm is constrained to choose its production plan from the set RY,
w,s,4 = {xE X(Y) n
Z(Y,
w,s,4).
(32)
Consequently, the regulated cost function is defined by e(y,W,r,s,a)-min[w-x-+rw,x,:xE~(y,w,s,a)], x
(33)
90
J.P. Hughes, Profit-maximizing
input demand
where the minimizing arguments are the regulated input demand functions f(Y, w, s, a).
The conditional cost function is defined as c(y, W,r, xt) = min [w-xx-
+ rw,x,]:x,
=x,0
and
x E X(y),
where the minimizing arguments are the conditional li(y, W-, XH), i= 1, . . . , n - 1. Solving the problem
input
(34)
demands
min C(y, w, r, x,) X” s.t.
R(y,u)-Z;(y,w,r,x,)-(s-r)w,x,sO
(35)
yields the rate-base demand function Z,(y, W,r, s, a) which, when substituted into c(y, W,r, x,,), results in the identity Qy, w, r, Z,(y, w, r, s, u)) = Qy, w, r, s, u),
(36)
since the choice sets for the two problems are identical, i.e. x~_%(y, w,s,~). Hence, it also follows that i,(y,
IV, S,
U)= -Ci(yjWe, IQ,
W, r, s, a))
vi Z 4
(37)
and a,(y, W,r, s, a) = %(y, w, r, s, a).
(38)
It is a straightforward matter to relate cost-minimizing demands to profit-maximizing demand. Since maximizing profit subject to the regulatory constraint implies minimizing cost subject to the same constraint, the conditional cost-minimizing input demand functions, evaluated at the prolitmaximizing regulated optimum, are identically equal to the regulated profitmaximizing input demand functions: ay(w,r,s,u)rKi(w-,~(w-,U,~~(w,r,s,U)),I~(w,r,s,u))
and from (7)
Vi+%
(39)
J.P. Hughes, Profit-maximizing input demand
91
Using the identity (39), the profit-maximizing direct-price effect can be decomposed into substitution and output effects, each containing a component conditional on x, and a component resulting from the adjustment of X, to satisfy the regulatory constraint. The sign of the former is negative; the latter, ambiguous. Differentiating (39) with respect to M’i,
x
faqaw,} V i #
n,
which is an extended decomposition
=ani(W~,~n,~~)/dWi+[a~i(w-,
(41) of the components of (18),
~,X~)/ay]Cirq;"(w-,~:)/aWill (41’)
and
The components of (41’) are the unregulated, conditional substitution and output effects, evaluated at the regulated optimum; thus, their signs are both negative.6 The response of the ith input to the adjustment of the rate-base input to satisfy the regulatory constraint is decomposed in (41”) into a substitution effect, holding the rate of output constant, and an output effect, resulting from the response of output to the adjustment in the nth input. As will be shown below, neither component’s sign can be determined a priori. 6This assertton ts proved by Sakai for the case of a firm which is compettttve in the output market. The proof for a monopoly IS straightforward. The negativity of the condittonal substitution effect follows from the convexity of the condittonal profit function. The negattvity of the output effect follows from the relatronship of the response of marginal cost to changes m input prtces. In parttcular. when the ith input is normal, i.e. F%,(.)/ii~>O, marginal cost vartes directly with M’,. Thus, assuming the second-order conditions for a maximum of protit are satisfied, the profit-maximizing rate of output must vary inversely with w,, ie. ?y(.)/?w,
RE
D
92
J.P. Hughes, Profit-maximuing
input demand
Therefore, neither the aggregate substitution effect nor the aggregate output effect has a refutable restriction on its sign. The sign of the substitution component of (41”) can be shown to depend on whether the ith and nth inputs are net complements or substitutes in the absence of regulation. When the unregulated conditional cost-minimizing demands for inputs are evaluated at the synthetic prices (lo,?) which are established such that
and .f,(jT, w. r, s, a) = x,*(F, w, i, a).
(43)
Substituting (43) into (42), differentiating with respect to u’,, and rearranging,
5
o
as
?xT(j”,
w, t,
a)/8w, 2 0
Q i#n.
(44)
Thus, from (24),
so that the sign of the substitution effect is unambiguously negative only when the ith and nth inputs are net complements. When the price of the ith input rises, profits decrease and the regulatory constraint is no longer binding. Thus, at any given rate of output, less of the rate-base input is needed to satisfy the regulatory constraint. When the ith and nth inputs are net complements, the decrease in the employment of the rate-base input to satisfy the constraint would tend to reduce the employment of the ith input as well, thereby reinforcing the conditional substitution component. On the other hand, when the ith and nth inputs are net substitutes, the reduced employment of the nth input wouid tend to raise the employment of the now-more-expensive ith input and calls into question the sign of the aggregate substitution effect. In the absence of regulation, the sign of the output effect is unambiguously negative. Since the conditional output effect is derived from the unregulated conditional input demand, evaluated at the regulated optimum, it too is unambiguously negative. However, when the conditioning value of the ratebase input’s employment is adjusted to satisfy the regulatory constraint, the resulting component of the output effect is found to have no a priori sign restriction so that the sign of the aggregate output effect is not necessarily negative. From (41”) the regulation-induced component of the output effect
J.P. Hughes, Profit-maximizing
input demand
93
is [ii~i(‘)/ay] [aF( .)/ax,] [Z~(.)/C?W,]. The sign of the third term has been shown negative in (24) and Zi( *)/c?Y~ 0 as the conditional demand for the ith input is normal or inferior. The sign of the second term can be derived from the equality (46) that is to say, the synthetic prices (IV,?) yield an optimal unregulated rate of output equal to the optimal regulated rate for the market prices (IV,r) and the allowable rate of return s. Since additionally, 5?:(w, r, s, a) = x,“( w, i, a), differentiating (46) with respect to w, and rearranging yields the relationship
ayyw- , U, wax, = c~Y=(w,i, 4iaw,ii[ax;(w, 2 0
as
i, 4iaw,i
apyw, f, a)/?~, ><0.
(47)
It is well established that ay”/aw n 50 as iix,*/dy2 0.’ Thus, the sign of the regulation-induced component of the output effect depends on whether the demand for the nth input and the conditional demand for the ith input are normal or inferior. When both are normal (or inferior), the sign of this component of the output effect is unambiguously negative and, hence, so is the sign of the aggregate output effect. However, when one or the other, but not both, is inferior, the sign of the regulation-induced component is positive and that of the aggregate effect is ambiguous. Differentiating (39) with respect to wjr the cross-price effect is r, S, u)/?w~
an;(w,
= d~i( .)/aw, + [fXi( .)/Zy] +
[ajn( w-, ?~)/aw,]
v i,jfn. (ami(')/SX,+[c?~,(')/ay][a~K(W-,~~)/i)X,]} (azyawj) (48)
Without writing the expression for %2,7(.)/awi, it is evident that the first and second terms on the right-hand side of (48) are equal, respectively, to the first and second terms of axzj”/awi.8 However, the regulation-induced components of the substitution and output effects, the third and fourth terms, are not ‘The argument supporting this claim is identical to that of the previous footnote. ‘Comparing the cross-prxe effects m (24) and in (48). it IS evident that the first term m (24), the conditional ‘gross’ cross-price effect, can be decomposed mto the lirst and second terms of (48), the conditional ‘net’ cross-price effect and the conditional output effect. Since the gross effects and the net effects are symmetric between pairs of inputs, the output effects must also be symmetric.
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J.P. Hughes, Profit-maximizing
input demand
necessarily equal to their respective counterparts. Thus, neither the aggregate cross substitution nor the aggregate cross output effects are necessarily equal between pairs of inputs, although their conditional components are equal. 7. Conclusions Both the cost-minimizing and the protit-maximizing demand for inputs are well behaved when the rate of employment of the rate-base input is held constant. The conditional cost and profit functions are also well behaved. However, when the rate of employment of the rate-base input is allowed to adjust to parametric variations in order to satisfy the rate-of-return constraint, the usual properties of the input demand functions and cost and profit functions will not necessarily be obtained. Thus, the regulated firm’s short-run behavior should conform to the usual restrictions of cost minimization and profit maximization while its long-run behavior under a regime of effective regulation may be qualitatively different from that of an unregulated regime. References Atkinson, Scott E. and Robert Halvorsen, 1976, Interfuel substitution in steam electric power generation, Journal of Political Economy 84, no. 5, 959-978. Baumol, W.J. and A.K. Klevorick, 1970. Input choices and rate-of-return regulation: An overview of the discussion, Bell Journal of Economics 1, Autumn, 162-190. Cowing, Thomas G., 1978, The effectiveness of rate-of-return regulation: An empirical test using profit functions, in: Melvyn Fuss and Daniel McFadden, eds., Production economics: A dual approach to theory and applicaitons, Vol. 2 (North-Holland, Amsterdam). Cowing, Thomas G., 1979. Duality and the estimation of a restricted technology, Working paper 79-l (Department of Economics, State University of New York, Binghamton, NY). Diewert. W. Erwin, 1981, The theory of total factor productivity measurement m regulated industries, in: Thomas G. Cowing and Rodney E. Stevenson, eds., Productivity measurement m regulated industries (Academic Press, New York). Fare, Rolf and James Logan, 1983a, The rate-of-return regulated lirm: Cost and production duality, Bell Journal of Economics 14, no. 2, 405414. Fare, Rolf and James Logan. 1983b, Shephard’s lemma and rate-of-return regulation, Economics Letters 12, 121-125. Fare, Rolf and James Logan, 1983~. The rate-of-return regulated version of Shephard’s lemma, Economics Letters 13. 297-302. Fuss, Melvyn and Leonard Waverman, 1981, Regulation and the multiproduct firm: The case of telecommunications in Canada, in: Gary Fromm, ed., Studies m public regulation (MIT Press, Cambridge, MA). Hughes. Joseph P., 1981, Giffen mputs and the theory of mulitple production. Journal of Economic Theory 25, 287-301. Hughes. Joseph P., 1984, Pathological substitution effects under rate-of-return regulation, Working paper (Department of Economics, Rutgers Umversity, New Brunswick, NJ). Hughes, Joseph P., 1986, Input demand under rate-of-return regulation: Applying Shephard’s lemma, Working paper (Department of Economics, Rutgers University, New Brunswick, NJ). McNicol, David L., 1973, The comparative statics properties of the theory of the regulated firm, Bell Journal of Economics 4. Autumn, 428453. Nelson, R.A., 1985, Returns to scale from variable and total cost functions, Economics Letters 18, 271-276.
Nelson, Randy A. and Mark E. Wohar, 1983, Regulation. scale economies, and productivity in steam-electric generation, lnternationsl Economic Review 24, no. I, 57-79. Nelson, Randy A. and Mark E. Wohar, 1987, A reply to ‘Regulatron, scale and productivity: A comment’, International Economic Review 28, no. 2, 535-539. Gfori-Mensa, Charles, 1982, Theory and estimation of input demand for the regulated firm. Ph.D. dissertation (Rutgers University, New Brunswick, NJ). Silberberg, Eugene, 1978, The structure of economics (McGraw-Hill, New York). Smith, V. Kerry, 1985, Elasticities of substitution for a regulated cost function, Economics Letters 7. 215-219.