- Email: [email protected]
The over-capitalization effect with diversification and cross subsidization
159
Economics Letters 16 (1984) 159-163 North-Holland
THE OVER-CAPITALIZATION AND CROSS SUBSIDIZATION
EFFECT *
WITH DIVERSIFICATION
Richard P. ROZEK Federal Trade Commission,
Washington, DC 20580, USA
Received 9 February 1984
This paper examines diversification and cross subsidization in the context of an Averch-Johnson model. Even if the allowed rate of return is less than the cost of capital, the over-capitalization effect exists provided the firm is sufficiently diversified.
1. Introduction
A basic component of the theory of regulation is the model developed by Averch and Johnson (1962). The main result is that a monopolist, subject to a rate of return constraint, will select a larger capital-labor ratio than the one that minimizes costs for the output level it decides to produce. This result is known as the over-capitalization or A-J effect. A crucial assumption in this model is that the allowed rate of return is greater than the market cost of capital. If the reverse is true, the regulated monopolist will actually under-capitalize. See Navarro (1983). One strategy for a regulated firm such as a public utility confronted with a regulatory environment in which the allowed rate of return is less than the market cost of capital is diversification into unregulated markets. For example, an electric or gas utility may diversify into markets for energy conservation products in order to reduce the need for additional investment in new capacity. Diversification by regulated firms creates new problems for regulatory agencies. The National Association of Regulatory Utility Commissioners * The views expressed in this paper are those of the author. They are not intended to reflect views or policies of the Federal Trade Commission or individual Commissioners. 0165-1765/84/$3.00
0 1984, Elsevier Science Publishers B.V. (North-Holland)
R. P. Rozek / Over-capitalization effect
160
(1982) identified some of the legal, economic, and financial issues that must be considered when utilities pursue diversification strategies. One of the main concerns is cross-subsidization; that is, the firm may include some of the costs of operating
in an unregulated
market in its rate base.
Thus, a higher price may prevail in the regulated market. Moreover, some of the firms in the unregulated market may be forced to exit as a result of the regulated firm charging operation in that market.
prices
that do not reflect
all the costs
of
This paper presents a model of a diversified firm subject to rate of return regulation in one market. The firm’s incentive to over- or undercapitalize is, then, examined. If there is a sufficient degree of diversification, the firm may over-capitalize even though the allowed rate of return is less than the market cost of capital.
2. Model The problem
solved by Averch and Johnson
maxrr=p(z)z(x,,x,)--ix,-rr,x,, x1.x2 x,,x,lO
and
is
subjectto
~(z)z(x,,x,)-~~x~ssx~,
where rr
=
Z
=
P(Z) x1 x* t-1
> 0 and ~(0, x1) = z(x,, 0) = 0, = inverse demand function, = quantity of capital input, = quantity of labor input, = unit cost of capital,
r2 s
firm’s total profit, z(xi, x2) = production
function,
zi = 6’z/ax,
> 0, z2 = az/ax,
= unit cost of labor, = allowed rate of return.
Manipulating the first order conditions function, we get
from the associated
Lagrangian
R. P. Rozek / Over-capitalization effect
161
where * denotes the value of the variables at the solution to the constrained optimization problem and X* is the Lagrange multiplier. If s>r, andO
The existence of the over-capitalization then
effect follows. However, if s < r,,
Z1* I1 ->-
4
‘2
The firm will select a smaller capital-labor ratio than that which minimizes cost for the given output level. Navarro (1983) provides evidence that the reverse A-J effect actually exists in the electric industry. Many regulated firms are obliged to provide service, so exit is not a viable strategy even if s < r,. Some firms are, thus, choosing to diversify as a means of dealing with the existing regulatory environment. Whenever there is diversification by regulated firms, the possibility of cross-subsidization is present. Therefore, consider a firm diversified in the sense that it is a monopolist in one market where it is subject to an A-J rate of return constraint, and a seller in an unregulated competitive market. The homogeneous inputs of capital and labor are required to produce both products. Furthermore, the regulatory authority does not separate the capital used in the production of the regulated product from that used to produce the unregulated product. In other words, the rate of return constraint applies to the total amount of capital used by the firm. Suppose there exists cxE (0, 1) which represents the proportion of capital used in the production of the regulated product. The firm’s decision problem is max77 =~(z)z(ax,, +qw((l
x1,.320,
ax2) -
a)~,,
P(z)z(+?
(1 -
a)x2)
-rlxl
ax2) - i-ax2 5
-
sxl,
where the notation is the same as above and
r2xz,
subject to
162
R. P. Rorek / Ouer- capitalization
effect
w = w(s, t) = production function for the competitive product, w1 = Jw/as > 0, w2 = aw/& > 0, and ~(0, t) = w(s, 0) = 0. q = price of output (in the competitive market) The Lagrangian with multiplier p and associated first order conditions for this problem are L(x,,
x2,
P)
a+)
=P(z)z(~xlT
-
r1x1
-
‘2X2
-
+qw(O
PM449,
-
a)x,,(l ax2)
-
4x2)
SXl
-
ar2x2),
~=(l-p’)az~(p’+z’~)+(l--a)~;-rl+~’s=O, 1
8L’ -= &
4
pi’
- sxi
-
arzx; )=o.
Rewriting and deleting the ’ notation for simplicity, we get
pz
- wlxl
-
(3)
ffr2x2 = (s - ar,)x,.
These equations have the usual interpretation. Eq. (1) and eq. (2) indicate that the marginal revenue product for each factor equals the respective private marginal expense, and eq. (3) indicates that the constraint is binding. Theorem. Proof. cLs>parl
Ifs > arl, then the A-J
effect still exists.
We know s > cur,. So,
(O
-ps<
-par,,
R. P. Rozek / Over-capitalization effect
rl - PS (1 -pa)‘* r1
-PS
(1 - ~13
<$
163
ButT
ces.0.
Q.E.D.
eq.(2)
The over-capitalization effect exists for a diversified firm with crosssubsidization provided s > oLr,. Certainly, if s > rl, then s > ari since (YE (0, 1). But even if s < ri, cymay be sufficiently large so that s > (~ri or s/r1 > (Y. In other words, without diversification and cross-subsidization, s < r, leads to under-capitalization, but with diversification and crosssubsidization, s < ri may still result in over-capitalization.
3. Conclusion Diversification by regulated firms, especially public utilities, is a topic of interest for both academic and policy oriented research. The results of this paper indicate that the over-capitalization effect may still exist for a diversified firm operating in a regulatory environment in which s < r,. Additional work is needed on the behavior of diversified regulated firms in order to understand the nature and extent of economies of scale and scope as well as the regulatory problems that arise due to the possibility of cross-subsidization. From a policy perspective, the issue is whether constraints on diversification will ultimately be required.
References Averch, Harvey and Leland Johtlson, 1962, Behavior of the firm under regulatory constraint, American Economic Review 52, 1053-1069. National Association of Regulatory Utility Commissioners, 1982 report of the ad hoc committee on utility diversification, 1-134. Navarro, Peter, 1983, Save now, freeze later: The real price of cheap electricity, Regulation 7, 31-36.
Economics Letters 16 (1984) 159-163 North-Holland
THE OVER-CAPITALIZATION AND CROSS SUBSIDIZATION
EFFECT *
WITH DIVERSIFICATION
Richard P. ROZEK Federal Trade Commission,
Washington, DC 20580, USA
Received 9 February 1984
This paper examines diversification and cross subsidization in the context of an Averch-Johnson model. Even if the allowed rate of return is less than the cost of capital, the over-capitalization effect exists provided the firm is sufficiently diversified.
1. Introduction
A basic component of the theory of regulation is the model developed by Averch and Johnson (1962). The main result is that a monopolist, subject to a rate of return constraint, will select a larger capital-labor ratio than the one that minimizes costs for the output level it decides to produce. This result is known as the over-capitalization or A-J effect. A crucial assumption in this model is that the allowed rate of return is greater than the market cost of capital. If the reverse is true, the regulated monopolist will actually under-capitalize. See Navarro (1983). One strategy for a regulated firm such as a public utility confronted with a regulatory environment in which the allowed rate of return is less than the market cost of capital is diversification into unregulated markets. For example, an electric or gas utility may diversify into markets for energy conservation products in order to reduce the need for additional investment in new capacity. Diversification by regulated firms creates new problems for regulatory agencies. The National Association of Regulatory Utility Commissioners * The views expressed in this paper are those of the author. They are not intended to reflect views or policies of the Federal Trade Commission or individual Commissioners. 0165-1765/84/$3.00
0 1984, Elsevier Science Publishers B.V. (North-Holland)
R. P. Rozek / Over-capitalization effect
160
(1982) identified some of the legal, economic, and financial issues that must be considered when utilities pursue diversification strategies. One of the main concerns is cross-subsidization; that is, the firm may include some of the costs of operating
in an unregulated
market in its rate base.
Thus, a higher price may prevail in the regulated market. Moreover, some of the firms in the unregulated market may be forced to exit as a result of the regulated firm charging operation in that market.
prices
that do not reflect
all the costs
of
This paper presents a model of a diversified firm subject to rate of return regulation in one market. The firm’s incentive to over- or undercapitalize is, then, examined. If there is a sufficient degree of diversification, the firm may over-capitalize even though the allowed rate of return is less than the market cost of capital.
2. Model The problem
solved by Averch and Johnson
maxrr=p(z)z(x,,x,)--ix,-rr,x,, x1.x2 x,,x,lO
and
is
subjectto
~(z)z(x,,x,)-~~x~ssx~,
where rr
=
Z
=
P(Z) x1 x* t-1
> 0 and ~(0, x1) = z(x,, 0) = 0, = inverse demand function, = quantity of capital input, = quantity of labor input, = unit cost of capital,
r2 s
firm’s total profit, z(xi, x2) = production
function,
zi = 6’z/ax,
> 0, z2 = az/ax,
= unit cost of labor, = allowed rate of return.
Manipulating the first order conditions function, we get
from the associated
Lagrangian
R. P. Rozek / Over-capitalization effect
161
where * denotes the value of the variables at the solution to the constrained optimization problem and X* is the Lagrange multiplier. If s>r, andO
The existence of the over-capitalization then
effect follows. However, if s < r,,
Z1* I1 ->-
4
‘2
The firm will select a smaller capital-labor ratio than that which minimizes cost for the given output level. Navarro (1983) provides evidence that the reverse A-J effect actually exists in the electric industry. Many regulated firms are obliged to provide service, so exit is not a viable strategy even if s < r,. Some firms are, thus, choosing to diversify as a means of dealing with the existing regulatory environment. Whenever there is diversification by regulated firms, the possibility of cross-subsidization is present. Therefore, consider a firm diversified in the sense that it is a monopolist in one market where it is subject to an A-J rate of return constraint, and a seller in an unregulated competitive market. The homogeneous inputs of capital and labor are required to produce both products. Furthermore, the regulatory authority does not separate the capital used in the production of the regulated product from that used to produce the unregulated product. In other words, the rate of return constraint applies to the total amount of capital used by the firm. Suppose there exists cxE (0, 1) which represents the proportion of capital used in the production of the regulated product. The firm’s decision problem is max77 =~(z)z(ax,, +qw((l
x1,.320,
ax2) -
a)~,,
P(z)z(+?
(1 -
a)x2)
-rlxl
ax2) - i-ax2 5
-
sxl,
where the notation is the same as above and
r2xz,
subject to
162
R. P. Rorek / Ouer- capitalization
effect
w = w(s, t) = production function for the competitive product, w1 = Jw/as > 0, w2 = aw/& > 0, and ~(0, t) = w(s, 0) = 0. q = price of output (in the competitive market) The Lagrangian with multiplier p and associated first order conditions for this problem are L(x,,
x2,
P)
a+)
=P(z)z(~xlT
-
r1x1
-
‘2X2
-
+qw(O
PM449,
-
a)x,,(l ax2)
-
4x2)
SXl
-
ar2x2),
~=(l-p’)az~(p’+z’~)+(l--a)~;-rl+~’s=O, 1
8L’ -= &
4
pi’
- sxi
-
arzx; )=o.
Rewriting and deleting the ’ notation for simplicity, we get
pz
- wlxl
-
(3)
ffr2x2 = (s - ar,)x,.
These equations have the usual interpretation. Eq. (1) and eq. (2) indicate that the marginal revenue product for each factor equals the respective private marginal expense, and eq. (3) indicates that the constraint is binding. Theorem. Proof. cLs>parl
Ifs > arl, then the A-J
effect still exists.
We know s > cur,. So,
(O
-ps<
-par,,
R. P. Rozek / Over-capitalization effect
rl - PS (1 -pa)‘* r1
-PS
(1 - ~13
<$
163
ButT
ces.0.
Q.E.D.
eq.(2)
The over-capitalization effect exists for a diversified firm with crosssubsidization provided s > oLr,. Certainly, if s > rl, then s > ari since (YE (0, 1). But even if s < ri, cymay be sufficiently large so that s > (~ri or s/r1 > (Y. In other words, without diversification and cross-subsidization, s < r, leads to under-capitalization, but with diversification and crosssubsidization, s < ri may still result in over-capitalization.
3. Conclusion Diversification by regulated firms, especially public utilities, is a topic of interest for both academic and policy oriented research. The results of this paper indicate that the over-capitalization effect may still exist for a diversified firm operating in a regulatory environment in which s < r,. Additional work is needed on the behavior of diversified regulated firms in order to understand the nature and extent of economies of scale and scope as well as the regulatory problems that arise due to the possibility of cross-subsidization. From a policy perspective, the issue is whether constraints on diversification will ultimately be required.
References Averch, Harvey and Leland Johtlson, 1962, Behavior of the firm under regulatory constraint, American Economic Review 52, 1053-1069. National Association of Regulatory Utility Commissioners, 1982 report of the ad hoc committee on utility diversification, 1-134. Navarro, Peter, 1983, Save now, freeze later: The real price of cheap electricity, Regulation 7, 31-36.