Journal
of Public
Economics
A Coasean
regulation
53 (1994) 459475.
general
North-Holland
equilibrium
model
of
Karen Clay*
Department of Economics, Stanlord University, Stanford. CA 94305, USA
Received June 1990, final version
received July 1992
Although marginal cost pricing is necessary for welfare maximization,
firms that produce under
conditions
of decreasing average cost incur losses if they price at marginal cost. In this paper,
the Coase two-part tariN is extended to several cases of common costs by assessing surplus in
order to recover the losses of the regulated
natural monopolist;
the existence of equilibria
in
these cases is demonstrated.
Three models are considered
in the context of an Arrow-Debreu
private ownership economy: a model of the regulated monopolist
who charges hookups to both
households
and iirms who use the good; a model of the regulated natural monopolist
who is
allowed to make positive economic profit; and finally a model where two or more monopolists
are under the control of a single regulator.
1. Introduction
During the marginal cost pricing controversy
of the 1930s and 194Os, the
issue of pricing in industries with declining average costs was hotly debated,
because these firms could not price at marginal cost without incurring losses.
Dominant
in the discussion of pricing of decreasing average cost goods has
been the view that prices should be set at marginal
cost and the losses
should be covered by the government
through taxation; this view is evident
in the works of Dickenson,
Lerner, Meade, Lange, Hotelling, Troxel, Reder,
and Vickrey.’
Not all of the participants
in the early debate favored government
taxation; prominent
among this group were Lewis (1941) and Coase (1946).
Coase was concerned
with two-part pricing when the overhead was composed of individually
assignable costs and vigorously attacked the notion of
Correspondence to: Karen Clay, Department
of Economics, Stanford University, Stanford, CA
94305, USA.
*I am pleased to acknowledge
the comments and remarks of Ken Arrow, Aaron Edlin, Mario
Eppelbaum,
Peter Hammond,
Chiaki
Moriguchi,
participants
in the General
Equilibrium
Seminar, and of two anonymous
referees. I am indebted
to Don Brown for many useful
discussions. This research was supported in part by the National Science Foundation.
‘This debate is excellently surveyed by Nancy Ruggles in a pair of articles entitled ‘The
welfare basis of the marginal
cost pricing principle’ (1949) and ‘Recent developments
in the
theory of marginal cost pricing’ (1950).
0047-2727/94/$07.00
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0047-2727(93)01353-X
460
K. Clay, A Coasean general equilibrium model of regulation
covering the losses through government
taxation. Indeed, all of the modern
general equilibrium
literature
on marginal
cost pricing with government
taxation has been put forth despite the damning criticism of marginal cost
pricing found in Coase’s seminal paper. Coase’s objections
to marginal cost
pricing were threefold: it distorts resource allocation;
it redistributes
income;
and it creates additional
distortions
by requiring taxation to cover the losses.
Recall the problem that Coase considered.
Assume that consumers are situated around a central market in which a
certain product is available at constant
prices. Assume that roads run
out from the central
market
but that each road passes only one
consumer of the product. Assume also that a carrier can carry on each
journey additional
units of the product at no additional
cost (at least to
a point beyond the limit of consumption
of any individual
consumer).
Assume further that the product is sold at the point of consumption.
It
is clear that the cost of supplying each individual
consumer
would be
the cost of the carrier plus the cost at the central market of the number
of units consumed
by that particular
consumer
of the product.
The
marginal cost would be equai to the cost of a unit of the product at the
central market. The average cost would be higher than the marginal cost
and would decline as the cost of the carrier was spread over an
increasing number of units [Coase (1946, p. 171)].
Coase argued that the Hotelling-Lerner
approach was subject to the above
criticisms
and suggested
instead
a two-part
pricing structure
where the
consumer
paid the assignable
fixed costs for any units he chose to buy at
marginal cost. Coase stated at the end of his article that the next step was to
examine the problem of two-part pricing when there are common costs.
The modern literature has examined existence and optimality
of marginal
cost pricing with government
taxation
and two-part
pricing in economies
where firms may exhibit decreasing average costs. Marginal cost pricing with
government
taxation
was utilized in the sophisticated
proofs of existence
found in works by Mantel (1979), Beato and Mas-Cole11 (1985), Brown et al.
(1986) and Bonniseau
and Cornet (1988). In this class of Arrow-Debreu
private ownership economies with one or more non-convex
production
sets,
the lump-sum
taxation is effected through unlimited liability holdings in the
firms with non-convex
technologies.
Guesnerie
(1975), Brown and Heal
(1979) Dierker (1986) and Quinzii (1991) considered
the issue of optimality.
Results on optimality
have been rather dismal, not only is marginal
cost
pricing not sufficient for allocative efficiency (Guesnerie,
Brown and Heal),
but there are efficient allocations
which cannot be supported
as marginal
cost pricing equilibria
without
resorting
to lump-sum
transfers
(Quinzii,
Dierker).
K. Clay, A Coasean general
equilibrium
model qfregulation
461
Another
criticism beyond Coase’s criticisms can be levelled against the
modern implementation
of marginal cost pricing where losses are covered by
lump-sum
taxation.
An allocation
may not be individually
rational
in the
sense that after the tax, the individual
may no longer be able to afford his
endowment,
if he is endowed with goods, or a zero net trade vector, if he is
endowed with production
sets2 In examples given by Guesnerie and Brown
and Heal, where marginal cost pricing is not efficient, one individual,
the one
who does not own the firm, is able to appropriate
all of the gains from
production
and part of the endowment
of the other agent by forcing him to
cover the losses of the firm. In the Brown and Heal example, we note that
lack of individual
rationality
precludes efficiency. In the production
equilibrium, by returning
the first individual,
the owner of the firm, to his
reservation
utility level as defined by his original
endowment,
we can
guarantee that the marginal cost pricing equilibrium
is efficient. An example
given in Vohra (1988) has production
equilibria
which, if actually implemented, would guarantee
negative income to one individual;
thus, it is not
surprising
that no marginal
cost pricing equilibria
are efficient. Two-part
pricing avoids the problem of individual
rationality
by only imposing
the
costs of the firm on consumers of the product; thus, consumers
always have
the option of consuming
their endowment.
More recently the literature
has turned to two-part
pricing in an effort
both to be more realistic and to avoid some of the undesirable
properties
mentioned
above. Work on existence has been quite successful. Vohra (1990)
provides a proof of existence for two-part marginal cost pricing equilibria in
a two-good model where non-convexities
are due to a fixed start-up cost.
Brown et al. (1992) prove existence
of two-part
marginal
cost pricing
equilibria in a general equilibrium
setting for general non-convexities.
Work
on optimality
was initially
far less successful.
Early work by Quinzii
demonstrated
that some Pareto-efficient
allocations
cannot be supported
as
two-part marginal
cost pricing equilibria.
Brown et al. (1992) showed that
the first and second welfare theorems fail in general, but that the second
welfare theorem holds if there is sufficient willingness
to pay. Vohra (1990)
provided a variety of examples which illustrated
important
ways in which
efficiency fails for two-part tariffs. However, recently Moriguchi
(1991) has
been able to demonstrate
that for the case where the non-convexity
is due to
pure fixed costs and where no consumer
is endowed with the monopoly
good, first best is achieved.
‘We define individual
rationality
with reference to the individual
being allowed to consume
his endowment
without taxation. We assume in our models that utilize two-part pricing and in
the following discussion
that no consumer
is endowed with the monopoly
good. If for some
reason
they are endowed
with the monopoly
good as in Vohra (1990). then individual
rationality,
as defined here, may not hold even under two-part pricing. However. for the types of
goods that we have in mind, such as electricity, sewers, postal service, and telephone service, the
assumption
that the individuals is not endowed with the monopoly good is appropriate.
462
K. Clay, A Coasean general
equilibrium
model of regulation
Despite the general lack of efficiency, except in the special case of pure
fixed cost, we present this work on existence in these models because of its
positive worth, and in the hope that subsequent
welfare analysis might build
upon it. This paper is a natural extension of Coase’s work to common costs
in a general equilibrium
framework.
Similar to Coase, we use a two-part
tariff structure
where households
and firms pay hookup fees to cover the
losses of the decreasing average cost firms. However, unlike Coase, the costs
in the models are common
costs, and thus hookup fees are based on the
surplus of each household or firm.
In order to explore a Coasean model of regulation,
we build upon earlier
work by Brown et al. (1992). The models that we examine are standard
Arrow-Debreu
private ownership
economies
where convex firms maximize
profits, and households
maximize utility subject to a budget constraint.
In
addition, we have added non-convex
firms that price at marginal cost. Nonconvex firms are assumed to be regulated natural monopolies
and individuals
are endowed with shares of the monopoly.
These shares carry limited liability
rather than the unlimited liability assumed when losses are covered by lumpsum taxation. Losses are covered by imposing assessments
on the surpluses
of the households
and the convex lirms.3 The surplus is the compensating
variation or willingness to pay for use of the monopoly
good. The definition
of consumer
surplus that we utilize is from Brown et al. (1992) and is the
same definition utilized by Roberts (1979) to define his license or hookup fee.
This concept of consumer
surplus is also implicit in the pioneering
work of
Oi (1971) on the Disneyland
monopolist.
Assessing consumer surplus in this
manner has two desirable characteristics:
it is both individually
rational, a
consumer can avoid the charge by not buying the good, and it is based on
an unalterable
characteristic
of the consumer,
consumer
surplus. Furthermore, the assessment
will be paid voluntarily
by any consumer
who has
positive consumer
surplus, because he is always left with positive surplus
after the assessment has been paid.
This paper presents existence results for three distinct models. The first is
for a single non-convex
firm that prices at marginal
cost and charges
hookups for both households and firms. The second is for a monopolist
who
is allowed to make positive economic
profits. In the limiting
case this
monopolist
behaves as an almost-perfectly-discriminating
monopolist.
The
third case is a model with two monopolies
under the control of a single
regulator. None of the foregoing analysis is captured by the earlier work on
two-part
pricing. These models were constructed
to reflect institutional
arrangements
that we observe in modern market economies of most OECD
countries.
Both households
and firms pay hookups to telephone companies
jThe word surplus here is used loosely. Surplus as we detine it below is a measure of the
willingness
to pay to use the monopoly
good and not the standard
delinition
of consumer
surplus.
K. Clay, A Coasean general
equilibrium
model 01 regulation
463
for local service; a cable television company which contracts with a locality
may be allowed to make positive economic
profits; and many areas have
electricity, water, gas and sewage under the control of a local utilities board.
Section 2 contains the models and the proofs are found in the appendix.
2. The models
The first model considers existence of a two-part marginal
cost pricing
equilibrium
(TPMCPE)
where the monopolist
is allowed to charge hookups
to both households
and firms. Hookups
are charged in proportion
to the
surplus that households and firms derive from use of the monopoly goods.
The second model allows a single regulated monopolist
to earn positive
profits. The monopolist
is regulated in two senses: first he is required to price
at marginal
cost; second he is only allowed to extract a fixed amount
of
surplus, which is used here as a proxy for rate of return. We consider the
case where the natural
monopolist
is allowed to take all but E of each
household’s
willingness
to pay. We demonstrate
the existence of an equilibrium where the monopolist
takes within 1 --F of the surplus, for all E which
are sufficiently small.
The third model extends earlier results for a single monopolist
to a case
where two monopolists
with increasing
returns operate in the market and
each monopolist
charges
households
an individualized
hookup.
In this
model, a regulator
divides
the total surplus
between
the monopolists
according to properties of willingness to pay for each good individually
and
for the two goods jointly. The model allows for production
of both goods as
long as the aggregate willingness to pay for each good, relative to its surplus,
is sufficient to cover its losses. We show that because each consumer knows
which goods she is consuming
and each monopolist
knows her surplus, an
equilibrium
exists.
2.1. Firms and households
This paper builds on results found in work by Brown et al. (1992) who
considered
existence and optimality
for a model where a single natural
monopoly
prices at marginal cost and charges only households for hookups.
We present a model that is more descriptively
accurate, because it incorporates hookups for both households and firms. The compelling reason to assess
both households
and firms is potential
welfare gains. If we assess all
consumers,
more revenue can be generated
than by assessing households
alone. Consider a case where a firm is charged a hookup fee of one dollar. If
we allow this dollar to be passed back to the household
in the form of
protits, she divides the expenditure
between private and monopoly
goods. By
assessing the household, we can recover less than one dollar, because we are
464
K. Clay, A Coasean general equilibrium
model of regulation
restricted to levying benefits due to consumption
of monopoly
goods. If the
monopoly
good is only an intermediate
good, not consumed by households,
then the only means to cover the losses of the monopoly
is to charge firms
hookup fees. Thus, monopolies
with higher losses can be sustained
in our
models by charging both firms and households
hookup fees. Furthermore,
supporting
these monopolies
may be efficient due to the fact that household
willingness to pay is always less than or equal to total surplus; so, it may be
desirable
from a welfare viewpoint
to use this additional
mechanism
to
extract more surplus.’ Moreover, depending
on how she measures welfare, a
planner may also want to utilize differential
assessment
of households
and
firms as a policy tool.
In the analysis
to this point, we have assumed that when faced with
hookups, consumers
in fact pay the hookups and consume the monopoly
good. As was noted above, if households and firms have positive surplus and
the fractions of the surplus taken from the firms and from the households are
strictly less than one, then households and firms are left with positive surplus
after paying the hookup and would strictly prefer to consume. If households
and firms have zero surplus, then they do not pay a hookup charge.
Allowing surplus to be extracted from both households and firms makes it
necessary to define surplus for the households
in two ways. First we define
household
surplus given that all of the surplus from the firms has been
extracted by the monopolist.
We use this definition in condition (S) to ensure
that there is sufficient surplus in the economy
to cover the fixed costs of
production
of the monopoly
good. Then we must define household
surplus
conditional
on the amount of surplus extracted from the firms because this
affects the profits that the consumers
receive through
their shareholdings.
These surpluses define a one-parameter
family of assessments on households
and firms which will cover any losses. The first part of the proof demonstrates that the set of all production
equilibria is non-empty.
The second part
shows that aggregate demand
is continuous
on the set of all production
equilibria.
Finally,
we use a fixed-point
argument
to demonstrate
the
existence of an equilibrium.
2.1.1. Firms
We begin by describing
the production
assumptions
found here originated
in Beato
side of the economy.
The
and Mas-Cole11 and are the
“Willingness
to pay as detined below is equivalent
to compensating
variation.
Compensating
variation has been shown to underestimate
true consumer surplus. As was rightly pointed out by
a referee, one might inadvertently
support enterprises that were not ellicient by effectively taking
too much surplus. This point is a reason to exercise caution when supporting
firms that are
making losses and is not a reason to assess only households.
K. Clap, A Coasean general equilibrium model
same assumptions
found in Bonniseau
assume that the production
possibilities
tion sets Ys satisfying:
(Fl)
Y’=Kf-Ry’,
(F2)
K* is convex
(F3)
Output
of firm 0, the monopoly
other firm.
(F4)
and Cornet and
may be described
Brown
by Ft
465
et al. We
1 produc-
where K’ is compact.
for 1 sf
5 F.
firm, cannot
be produced
by any
The pricing rule is given by @: 6Y/ + A, where A is the unit simplex in
Re+l
cone with the
4” is the intersection
of the Clarke normal
simplex, and 6Yf is the boundary of Y’.
(Fl)+F4)
(F5)
qf regulation
imply (F5).
There exists an r>O such that for all f, K* is contained
in the interior
of {-re}+Ry’,
where e=(l,l,...,l).
If y{s-r
and PEAR,
then
pk=o.
Define a production equilibrium to be a pair ( y, p), where for all f, yf E 6 Ys
and PE@“( yf). Let PE denote the set of all production
equilibria.
The single monopoly
produces at a feasible production
plan y” and sells
good 0 to households
and firms. We define the profits made by firm 0 at the
production plan y” as
UO( y”, P, 4) = .qh + w
+ P YO,
(1)
where q is the hookup fee paid by households and firms. However, the firm
is only allowed to use q to recover losses. Thus U”( y”,p, q) = max(O,p. y”)
and so we can drop the dependence on q and write lI”( yO,p).
2.1.2. Households
Now we turn to the consumption
side of the economy.
There are H
households
that are fully characterized
by their endowments
wh, shareholdings in firms Oh’, consumption
sets Xh, and utility functions
Uh. We will
make the standard neoclassical assumptions:
(Hl)
R$+’ 3 Xh is a non-empty,
closed, convex
(H2) No one needs monopoly
goods
(0, x’~) E Xh, where k 2 1 * xLh=x:.
set which is bounded
to survive.
That
below.
is, for all xh~Xh,
466
K. Clay, A Coasean general equilibrium model of regulation
(H3)
Uh is continuous.
(H4)
Uh is strictly
(H5) We assume
the natural
households
quasi-concave.
limited liability for shareholders
in all firms. Any losses of
monopoly
will be recovered
by hookup
fees paid by
and firms. Each household
has Ohf shares in firm f for
where CO hf = 1 and OhI 2 0 for all h, f.
f=O,...,F,
(H6) Households
have no endowment
of the monopoly
good.
Define A(w) as
A(w)={((P),(
yf))E17xh
x 17yCZxh~Zwh+Cyf}.
(2)
We must show that A(w) is compact in order to show that the attainable
consumption
sets, proj A(w) n Xh, are compact.
We need only show that
(ZWh + Cy”) A Re,+ l is compact since Cxh sCwh + CyJ. By definition
Yf =
KS- R/;t ‘, CKJ is compact,
and the intersection
with Ryl
is compact.
Thus, A(w) is compact and the attainable
consumption
sets, denoted Xh, are
compact.
Define Xh as Xh n Q, where Q is a large compact ‘box’ which
contains all of the attainable
consumption
sets, Xh, in the interior.
2.1.3.
Willingness
to pa)
Define the firm’s willingness to pay for use of the monopoly
good as
s’( y, p) = Z7/( y, p) -IF/*( y, p), where I7/ is the maximum profit given that the
firm may use the monopoly
good in production
and IT” is the maximum
profit when the firm does not use the monopoly
good. Define i as the
fraction of surplus taken from the firms by the monopolist,
where i E [0,1).
Now we will define the household’s willingness
to pay for the monopoly
good. Define llh as total household dividends,
P(
y, p, A) = oh0 P(
y”, p) + C@((
1 - lL)Hf( y, p) + 3JIf’(
y, p)).
(3)
Define
income
of household
h as rh( y, p, A), where
rh( y,p, A) = p. wh +
IZh( y,p, 2). Define expenditures
as qh +p’ xh, if xt >O, where qh is the hookup
budget
for the monopoly
good, and as p. xh, if xt = 0. Define the household
correspondence,
Bh( y, p, q), as follows:
Bh(y,p,ii)={~h~Xh(p.~h~p.~h+17h(y,p,I.)if~hg=0,
orqh(y,p,A)+p~xhSp.wh+Z7h(y,p,A)ifx~>0}.
The
minimum
expenditure
at
prices
p
necessary
to
(4)
reach
utility
Oh is
K. Claq’, A Coasean general equilibrium model @’regulation
467
denoted by _i?‘(p, a”), where oh is defined as utility maximized
subject to
xh E Xh, satisfaction
of the budget constraint,
and zero consumption
of the
monopoly
good. Then willingness to pay, gh, is defined as sh( y,p, 2) =
rh( y,p, A)-Eh(p, oh). The special case where all of the profits resulting from
the use of the good are taken from the firm is defined as Sh( y,p) =gh( y,p, 1).
Assume that there is always enough surplus at all production
equilibria
from households
and firms to cover any losses of the non-convex
firm,
(S)
32~ [0,1) such that for all production
y,p)>
-min(p.y,,O).
equilibria,
( y,p) E PE, CSh( y,p) +
lCs’(
Define /?( y,p, A) to be the fraction of surplus taken from the households,
where jl(y,p,i.) =( -min(p.y”,O)-i.CsJ(y,p))/Cs”h(
y,p, A). Note that a priori
j3(y,p, 3.) could be either positive or negative. However, we are going to
restrict attention
to the case where both households
and firms pay nonnegative hookups.
This assumption
constrains
the choice of ~-E[O, 1) to 2
which satisfy Es/( y, p) 2 -min(p
y”, 0), and therefore,
/?( y, p, A) E [0,1).
Assumption
(S) guarantees
that at least one such pair ()_,/I( y,p,3.)) exists.
The hookup fee ftir household
h is q”( J: p, A), where
qh( y,p, 2) =
b( y,p,2)sh( y,p,i.),
and the hookup fee for ,jrm .f’ is q’( y,p, A), where
qf( Y,P,4=@(Y>P).
Assume that surplus
(N)
is increasing
dSh/drh >O for all households
with income,
h and dsf/dUJ
>O for all firms.’
This normality
assumption
rules out the case where, if we allow the
households
or firms to keep a positive fraction of their willingness
to pay,
their surplus could fall to the point where they are no longer willing to pay
the hookup fee. This assumption
is used in the last part of the proof to
demonstrate
that, even though we treated the hookup charges as lump-sum,
consumers and firms would pay them voluntarily.
We make the following assumptions
on the set of production
equilibria:
(SA) For every production
equilibrium
( y,p) E PE,p(Cy” + Zwh) > infp. Cgh.
We assume that there is more than enough
aggregate
income to
purchase consumption
goods which will allow households to survive.
(R)
For every production
equilibrium
( y, p) E PE, rh( y, p) > inf(p zh). Here
we assume that each household has enough income to survive without
consuming
the monopoly good.
‘Thanks are due to Aaron Edlin and Mario Eppelbaum
for pointing out that this assumption
was necessary
in order to prove existence. The interested
reader is referred to Edlin and
Eppelbaum (1990), where this notion of normality was developed to deal with similar problems.
468
K. Clay, A Coasean general equilibrium model of regulation
A two-part marginal cost pricing equilibrium is defined as a vector of prices
consumption
plans and production
plans (p,(Xh),( jf)) such that,
(a) households are maximizing
utility subject to their budget constraint,
(b) firms 1,. . , F maximize profits at prices p,
(c) p E +/( y’) for all S
(d) - min(p . Y’, 0) = &“(
(e) CXh = CJf + Cwh.
Theorem 1.
there exists
Moreover,
Y, P, 3.) + -Q/t Y, P, 3.L
Given assumptions (Hl)-(H6),
(Fl)+FS),
(SA), (R), (S), and (N),
a TPMCPE
where both firms and households pay hookups.
in this equilibrium
the monopoly
Proof:
The proofs for this and
are located in the appendix.
2.2.
Monopoly
is producing.
all subsequent
theorems
and
propositions
profits
Recall that both Roberts and Oi considered partial equilibrium
models in
which a perfectly-discriminating
monopolist
extracted
all of the consumer
surplus
from each individual.
We demonstrate
that we can allow our
regulated monopolist
to act as an almost-perfectly-discriminating
monopolist
in much the same manner.
There is a subtle difference
between
our
monopolist
and 02s Disneyland
monopolist
- we cannot allow the regulated
monopolist
to extract
all of the consumer
surplus.
If the monopolist
extracted
all of the surplus, consumers
would then be indifferent
between
entering and not entering Disneyland,
and demand correspondences
would
be non-convex
which would vitiate a proof of existence for any economy
with a finite number of consumers.
The reasons to allow our regulated monopolist
to make positive profits are
both positive and normative.
We observe that natural monopolies,
such as
public utilities, are frequently
allowed to have non-zero
rates of return.
Furthermore,
if non-convex
tirms are allowed to make positive profits, the
profit level is an additional
policy tool that can be utilized to effect the
income distribution.
These policy tools are especially relevant if we are trying
to decentralize
a Pareto-optimal
allocation
without resorting to lump-sum
transfers.
Now suppose that we allow the regulated monopolist
to make a profit by
acting as an almost-perfectly-discriminating
monopolist.
For ease of exposition, we consider the case where only the households pay hookups. We must
account for the circular flow of income whereby the monopolist
returns the
profits to shareholders
which further increases the amount of surplus. First,
we show that by putting
restrictions
on the utility functions,
there are
K. Clay, A Coasean general
equilibrium
469
model of regulation
equilibrium
levels of income. Then, we prove that for every E sufficiently
small, if the monopolist
takes 1 --E of the surplus, an equilibrium
exists. If the
monopolist
is allowed to take all of the surplus, consumers
will be strictly
indifferent
between consuming
the monopoly
good and not consuming
it.
However, for every E>O the consumers
will strictly prefer to consume the
monopoly good.
Firms, households, willingness to pay
Assume (Fl)+FS),
(Hl)gH6),
(SA), and (R). Note that the household
surplus, sh( y,p), is just the case where ,I=0 from the previous model, i.e.
sh( y, p) = gh( y, p, 0). Furthermore,
assume
(N’)
dsh/drh > 0 for all households
h.
(S’)
For all production
( y, p) E PE, Cs”( y,p) > -min(py’,
equilibria
0).
Define the hookup as a fraction 2~ [a, 1) such that ish( y,p) =qh. We only
consider
hookups
over this set to guarantee
that, for any production
equilibrium
in PE that we choose, there will be sufficient surplus to ensure
production
of the monopoly
good. We demonstrate
in Proposition
1 that a
lower bound a exists.
Proposition I.
3~ E [0,1) 3 V( y, p) E PE, crIsh( y, p) 2 - min(p . y”, 0).
In order to ensure that a fixed point exists, we must define the upper limit
on the amount of profits our natural monopoly
can make. The upper bound
is defined in Proposition
2.
Proposition 2. The maximum possible profit of the natural monopoly, which
we will denote IT*. is bounded.
Theorem 2. Gioen assumptions (Hl)+H6),
(Fl)<FS),
(SA), (R), (S’), and (N’),
for all AE [cr, l), there exists a TPMCPE where the monopolist extracts ,I of
the surplus, and moreover, the monopoly is producing.
2.3. Two monopolists with a regulator
Both descriptively
and from a welfare standpoint
it is attractive to have a
model with more than one monopolist.
Consider
an economy
of convex
firms and no monopolists.
Suppose that there are two candidate monopolists
which we might permit to operate. We will define two goods as being
complements (substitutes) if the aggregate
amount
of benefits
is larger
(smaller) than the sum of benefits to each good individually.
Suppose thzt
J.P.E.
F
470
K. Clay, A Coasean general equilibrium model of regulation
the two monopolies are complements, and while the sum of benefits to each
good individually is not sufficient to cover their respective tixed costs, that
the aggregate benefit is sufficient to support production. From a welfare
standpoint, we would want to consider producing both goods. Similarly we
can imagine situations where either of two monopolies could be selfsupporting, but, because the goods are substitutes, both cannot be supported.
We consider the special case where there is sufficient surplus to produce both
goods, although the goods many be either complements or substitutes. The
model focuses on the allocation of surplus to the monopolists and the
subsequent setting of hookup fees.
The situation that we consider here is one where the natural monopolies
are both under the control of a regulator who guarantees that both goods
are produced if there is sufficient surplus to cover the fixed costs of both
firms. One typical example is that the electricity, gas and water utilities in
many communities are grouped together under the control of one agency.
Another example was the control of both mail delivery and package delivery
by the United States Postal Service prior to the entry of private firms such as
United Parcel Service, Federal Express and others.
2.3.1. Firms, households, willingness to pay
The assumptions on the production sector are analogous to above but
now there are two monopoly firms which we denote firm 01 and firm 02. The
assumptions on the consumption sector are the same as above.
Now we define willingness to pay for the household. Recall that income is
defined as rh( y”, p) =p . wh + COhoZ7’( y”, p) + COhfZI/(p), where Ill(p) is the
maximized profits of firm f. The minimum expenditure at prices p necessary
to reach utility 8:,, is denoted by Eh(p, i?:,,), where u:,, is defined as
utility maximized subject to feasibility, the budget constraint, and x&=0 for
i= 1,2. The minimum expenditure at prices p necessary to reach utility 0: is
denoted by Eh(p, u!), where ir: is defined as utility maximized subject to
feasibility, the budget constraint, and x&=0, where [i] denotes the other
good.
Define the surplus if both monopoly goods are consumed relative to
consuming neither good as s: 2( y, p) = rh( y, p) - Eh(p, V”,, 2), where sh is willingness to pay. Define individual surpluses as before, i.e. consuming one good
relative to consuming neither good as s:( y, p) = rh( y, p) - Eh(p, 0:) and
similarly, s”,(y,p) = rh( y, p) - E:(p, 0;).
Consider the following two cases:
(1) s:2(y~P)2s:(Y>P)fsh2(Y~P).
This corresponds to the case where the goods are complements.
(2) s:,(Y,P)4(Y>P)+sh,(y,P).
K. Clay, A Coasean general equilibrium model of regulation
471
This corresponds
to the case where the goods are substitutes.
We consider
the cases of complements
and substitutes
separately
and
assume that a given good is either a substitute
or a complement
for all
households and for all production
equilibria.
(MM)
Assume
that either (1) ~:~(y,p)Zs:(
y,p)+sh,(y,p),
Vh,V( y,p)ePE,
or (2) s’L( Y, PI < $ ( Y, P) +s”,( Y, P), % W Y, PI E PE.
2.3.2. Complements
Define the surplus
s$, where
that
each monopolist
s:*=s:+0.5(s:,-s:
s;* = s; + 0.5($,
can extract
in case 1 as s:. and
-&,
-s:
- s”,).
This is a continuous
division of the surplus such that each monopolist
gets at
least as much surplus as he would singly. Note that the individual
will
always consume both goods because even if neither of the goods is desirable
by itself, (e.g. As:, >s:) the goods are jointly desirable because st2 >,I$. +
AS”,.. Furthermore,
if one good is desirable (e.g. L$.Zs:),
then the second is
always desirable because, by definition, ,L$ < st 2 - s:il.
We define aggregate surplus as the sum of the individual
surpluses, so
s’ = cs:, and S2 = Cs’&. Thus we define the fraction
of the surplus that
monopolists
01 and 02 take as %(y,p) = -min(p, yy,O)/S’
and p( y,p) =
-min(p2yi,0)/S2,
respectively.
Assume (SA) and (R).
(N”) d.$/drh > 0 for i = 1,2 and ds: 2/drh > 0, for all households
(S”)
For
all production
S2(y,p)>
equilibria,
( y, p) E PE, S’( y, p) > - min(p, yy, 0) and
-min(p2y%W
Define
the
hookups
as
qh’(yTp)=~(yTp)s?4y,p)
p( y,p)s’&( y,p). Note that these hookups are continuous
tinuous functions and are therefore continuous.
2.3.3. Substitutes
This is a more difficult case
following additional
assumption.
(C)
h.
The aggregate
surpluses
than
complements.
and
qh2(y,p)=
functions
of con-
We need
S’( y,p) and S2( y,p) are continuous
to make
on ( y,p).
the
412
K. Clay, A Coasean general equilibrium model of regulation
As above, we need the aggregate surpluses to be continuous
so that the
hookups are continuous.
Above we got this simply by having an excess of
surplus. Here there are several possible ways to generate this outcome. First,
we could always have consumers consume both goods even if they are nearly
perfect substitutes.
This assumption
is counterintuitive
and poses severe
restrictions
on the amount of surplus that can be extracted, as is noted in
Edlin and Eppelbaum
(1990). Second, we could have the consumer consume
only one good but restrict the agent to consume
the same good over all
( y,p). Again this restriction
is very severe because agents might want to
switch between goods, but would not be allowed to do so. The most natural
way to motivate the assumption
is to allow consumers
to consume only one
good, but to switch between goods over ( y, p). and to assume that there is a
continuum
of agents so that switching does not induce discontinuities
in the
aggregate
surplus. Allowing consumers
to switch between goods is more
realistic than the first two assumptions.
Furthermore,
this assumption
seems
to be implicit in models such as telecommunication
demand where a switch
by the marginal
customer in a large population
is not significant.
In any
case, the hookups are well defined and are continuous.
Theorem 3. Given assumptions (H l)+H6),
(Fl)-(F5),
(SA), (R), (S”), (N”),
(MM), and (C) there exists a TPMCPE with two monopolies where the goods
are either complements or substitutes. Moreover, in this equilibrium both of the
monopolies are producing.
Appendix
This work is an extension
of Brown et al. (1992). As such, we refer the
reader to their paper for the details of the proof. We will only present details
to the extent that our proof differs substantially
from theirs.
to show that the set of all
Proof of Theorem 1. First, it is necessary
production
equilibria is non-empty.
Next one must show that aggregate
demand
is continuous
on PE.
fih( y, p, A), the household budget correspondence,
is defined for ( y, p) E PE as
Bh(y,p,~)={.xh~Xh~qh+p~xh~rh(y,p,~)).
B’(y,p,A) is a convex, compactvalued correspondence
on PE. In fact, gh( y,p, A) is a continuous
correspondence on PE. Recall that rh( y,p, A) and qh( y,p, A) are continuous
on PE.
on PE. If qh( y,p, A) =O, then the
By assumption
(R), rh(p, y) >infp.xh
standard argument
[Debreu (1959)] shows that fi’“( y,p,l) is continuous.
Let
Xh be the minimizer of the expenditure
function. If O<qh( y,p,A) <sh( y,p, A),
and Fh(y,p,A)=rh(y,p,A)-E(p,ii)
then qh(y,p,;i)+p.Zh<sh(y,p,A)+p.Xh=
d( y, p, A)+ E(p, 0). Thus, qh( y, p, A) +p . Xh < rh( y, p, A). Hence the minimizer
K. Clay, A Coasean general
equilibrium
model of regulation
473
zh is in the interior of fih( y,p,E,), and again the standard
argument
shows
that fih( y, p, 2) is a continuous
correspondence.
Now
define
the
household demand correspondence
as dh( y,p,A) =
argmax( Uh(xh): xh E 8”( y, p, A)). By the Berge Maximization
Theorem, dh is
upper hemicontinuous.
dh is a function from the strict quasi-concavity
of Uh
and convexity of fib; thus, it is a continuous
function. Define the firm demand
function as, df( y, p, 2) = argmax(pyf
-4”: yJ E Y I). Aggregate demand is the
sum of continuous
functions dl and dh and is a continuous
function on the
closed set PE. PE is a closed subset of dxnP,,GYf,
so by the Tietze
extension theorem, d has a continuous
extension 2 to all of dxnT,,GYf.
If
we utilize this demand function in place of the demand function
used by
Brown et al., their proof follows, and we have a fixed point which is an
equilibrium.
In the foregoing
analysis, consumers
are required to pay the hookups
qs( y,p, jb) and qh( y,p, L). Given that households
and firms have positive
surplus, the fraction of the surplus taken from firms and from households
is
strictly less than one, and the normality
assumption
(N) holds, households
and firms are left with positive
surplus
after paying
the hookup.
If
households
and firms do not have any surplus, they do not pay a hookup.
Thus, households and firms will pay the hookups voluntarily.
Proof of Proposition 1. - min(p. y”, O)/Csh( y, p) is a continuous
the compact set of production
equilibria,
and therefore attains
on the set which we will denote as ~1, where 0 < LX<1. Note
function on
a maximum
that for all
(Y?P)EPE,
-min(p.
y”, O)/Csh( y, p)
S ~1.
function on the compact set of
Proof of Proposition 2. p’ y” is a continuous
production
equilibria and therefore attains a maximum on the set which we
denote p.y”. There exists a ( y,p) which generates the largest value of the
economy,
p.(Cy’ +Zwh), which excludes
any losses of the monopolist.
p.(Zyf
+Cwh)
is a continuous
function on the compact set of production
equilibria,
and hence attains
a maximum
on the set which we denote
P’.(Z#+CW~).
Th ere fore, the maximum
profit of the monopolist
cannot
exceed p. y” +p' . (Cys + Zwh), which we denote as II*.
1. The
Proof of Theorem 2. The proof is similar to the proof of Theorem
PE is non-empty
by the same argument found in Brown et al. (1992), where
now the map is @n: AF+2 x [O,I7*] + AF+2 x [O,I7*]. The first F+2 equations are now functions
of Zl and the last equation
is cF+~ ((zJj,p, Iz) =
Cqh+p.yO.
Aggregate demand is continuous
by an argument similar to the one found
above.
474
K. Clay, A Coasean general equilibrium model of regulation
The Beato and Mas-Cole11 map for the fixed point is
Tj/ ((4,
w = Z{+
maX(O,Pj-ggf(l(ZJ,
P,
fl)))
1+Cf+ max(O,P~-gi(rl(zf,n)))
64.1)
(A.2)
for f =F+ 1 and Osjs/;
and
T;+2((ZJ),P,n)=C
4h+P.vo(Y0)
(A.31
h
for f = F + 2 and 0 j js 8. This map has a fixed point; thus, we are assured
that an equilibrium exists. The remaining arguments follow as in Theorem 1.
3. This proof is similar to the proof given for Theorem 1
and is left to the reader.
Proof of Theorem
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