JOURNAL
OF ECONOMIC
THEORY
Optimal
45, 65-84 (1988)
Regulation
under Fixed Rules
for Income Distribution*
RAJIV VOHRA
Brown
University,
Providence,
Rhode
Island
02912
Received February 24, 1986; revised June 2, 1987
It is by now well known that if arbitrary lump sum transfers are feasible, then the
optimal regulation of firms involves marginal cost pricing. Here, we consider
optimal regulation in an economy in which lump sum taxation is possible but the
income distribution is, in some sense, fixed. We show that under certain conditions,
production efficiency and marginal cost pricing are desirable. In general, however,
this need not be the case. In an economy with increasing returns, it is possible that
the second best utility possibility frontier is strictly dominated by the Pareto
frontier and none of the marginal cost pricing equilibria lie on the second best
frontier. Journal of Economic Literature
Classification Numbers: 021, 022, 024.
(!:I 1988 Academic
Press, Inc
1. INTR~OUCTI~N
In the classical Arrow-Debreu model, marginal cost pricing is suflicient
for Pareto optimality. This follows from the First Welfare Theorem given
an appropriate definition of “marginal cost prices”.’ Moreover, according
to the Second Welfare Theorem, if lump sum transfers are feasible, any
Pareto optimal allocation can be sustained as a marginal cost pricing
equilibrium. This implies that under smoothness hypotheses, marginal cost
pricing is also necessary for Pareto optimality.’
As one would expect, under
certain conditions, marginal cost pricing is a necessary condition for Pareto
optimality
even in the presence of increasing returns. Indeed, this was
Hotelling’s
[S] argument for regulating natural monopolies to follow
marginal cost pricing. A rigorous basis for this argument is now available
in the form a generalized Second Welfare Theorem (see, for example,
* This paper is an extensive revision of [ 111 and is very heavily inspired by discussions with
Andreu Mas-Colell. I am grateful to M. Ali Khan for his encouragement and help during
various stages of this research. Thanks are also due to an Associate Editor and participants of
seminars at Brown University, the Indian Statistical Institute, and the London School of
Economics for many helpful comments. Financial support from NSF Grants SES 8410229 and
SES 8605630 is gratefully acknowledged.
* A formal definition of marginal cost prices is provided below.
2 See also Remark 4.1 below.
65
0022-0531/88 63.00
Copyright
0 1988 by Academic Press. Inc.
All rights of reproduction
m any form reserved.
66
RAJIV
VOHRA
Guesnerie [S, Theorem 1 ] and Khan and Vohra [9, Theorem 11); any
Pareto optimal allocation can be sustained through marginal cost pricing
even in economies with nonconvex production sets. In other words, if
arbitrary lump sum transfers are feasible, optimality
can be achieved by
regulating firms to follow marginal cost pricing irrespective of whether
production sets are convex or not.
The normative significance of marginal cost pricing appears not to be so
clear in an economy with increasing returns if arbitrary lump sum taxation
is not feasible. Consider a model which is identical to the Arrow-Debreu
model except that some firms have nonconvex production sets and the
government can regulate firms to follow any feasible production plans that
it chooses. As in the Arrow-Debreu model, the profits of firms are distributed to the consumers according to exogenously given rules. Consumers
behave competitively
at the given market prices and a regulated
equilibrium can be defined as an allocation and prices at which all markets
clear. Notice that since consumers have exogenously given shares in the
various firms, if a regulated firm makes losses these are effectively collected
from the consumers in a lump sum manner. But the distribution of these
taxes is predetermined;
while lump sum taxation is feasible, the
government’s ability to redistribute income is restricted. It is for this
reason, and this reason alone, that one cannot simply appeal to a general
version of the Second Welfare Theorem to assert that optimality can be
achieved by regulating firms to follow marginal cost pricing. In fact, it is
already known that with fixed rules for income distribution none of the
marginal cost pricing equilibria may be Pareto optimal (see Guesnerie [5]
and Brown and Heal [2]). While the existence of a marginal cost pricing
equilibrium
has been established under fairly general conditions (see, for
example, Vohra [ 121 and the references therein) its welfare properties have
not yet been fully explored.
The aim of this paper is to characterize the optimal form of regulation in
an economy with fixed rules for income distribution. As we shall see, the
fact that no marginal cost pricing equilibrium
may be Pareto optimal
suggests that it may be impossible to achieve Pareto optimality.
The
central issue we shall be concerned with is the characterization of second
best optimality
under distributional
constraints and its relationship to
marginal cost pricing. We consider two kinds of income distribution rules:
(1) a $xed structure of reuenues under which each consumer owns a fixed
proportion of the economy’s net outut; (2) a fixed structure of shares under
which each consumer, just as in the Arrow-Debreu model, has given shares
in the firms and endowments. Interestingly, the results do depend on which
of these rules is followed.
Section 2 specifies the basic model and related notions of equilibrium
and optimality.
OPTIMAL
REGULATION
61
In Section 3 we deal with optimality in the context of a given social
welfare function. Here, the results do not depend in any significant way on
the presence of increasing returns. It is reasonable to expect, and we show
explicitly, that due to distributional
considerations marginal cost pricing
may not be an optimal policy. What may be smewhat more surprising is
the fact that the desirability of production efficiency or marginal cost
pricing depends on the distribution
rule being followed. Given a fixed
structure of shares, production efficiency may not be desirable. However, if
the income distribution is given according to a fixed structure of revenues,
production efficiency is always desirable. In this case, any second best
equilibrium
relative to a Paretian social welfare function can be characterized as a marginal cost pricing equilibrium
if commodity taxation is
feasible.
In Section 4 we consider the situation in which distributional
considerations are not relevant. In particular, we examine the possibility of
using marginal cost pricing simply to reach the second best utility
possibility frontier. Of course, if all production sets are convex, under the
classical assumptions, there exists a marginal cost pricing equilibrium
which is Pareto optimal and hence second best. It is here that the presence
of increasing returns makes a crucial difference in the welfare analysis. As
mentioned above, none of the marginal cost pricing equilibria may be
Pareto optimal, as has been shown by Guesnerie [5], and Brown and Heal
[Z]. More recently, Beato and Mas-Cole11 [l] have shown that none of
the marginal cost pricing equilibria may satisfy even aggregate production
efficiency. But, since the Beato-Mas-Cole11 example pertains to a fixed
structure of shares, and in that case production efficiency is not necessarily
desirable, this still leaves open the possibility that some marginal cost
pricing equilibria may be on the second best utility possibility frontier.
However, we show that this may turn out not to be the case even in
economies with a single firm and a fixed structure of revenues. Moreover, it
is also possible that there exists an average cost pricing equilibrium which
is a second best equilibrium
and dominates every marginal cost pricing
equilibrium.
The main conclusions are summarized in Section 5.
2.
THE
MODEL
We consider an economy with m consumers indexed by i, i= 1, .... m, and
n firms indexed by j, j = 1, .... n. Xi E R’ and Xi E Xi refer, respectively, to the
consumption set and a consumption plan of consumer i. xjk refers to consumer i’s consumption of commodity k. The utility function of consumer i
is denoted ui: Xi H R’. We shall assume that all consumers have continuous
68
RAJIV
VOHRA
utility functions. Y, c R’ and yj~ Y, denote the production set and a
production plan of the jth firm. yjk is firm j’s production of commodity k.
Let x = CT!, xi and y = C;= I yj. The aggregate endowment is denoted w
and consumer i’s claim to the net output is denoted wi. For 5, z E R’, z $ z
means that Z,>z, for all k, k=l,...,I.
R’+={z~R’(z30}
is the nonnegative orthant of R’ and R’+ + = (z E R’[ z 9 01.
The income distribution is said to be given according to a fixed structure
of revenues if, for all i, wi = ai( y + w), where ai > 0 and ET!, ai = 1.
The income distribution is said to be given according to a fixed structure
of shares if, for all i, wi = I;=, 8, y.i + oi, where 8, > 0, I?= I 8, = 1 for all
j, wi E Xi for all i, and Cy=, oi = w.
Given market prices p E R’, consumer i maximizes utility subject to
income Ii = p . wi. The demand correspondence of consumer i is denoted
gi(p, wi). We define a regulated equilibrium as an allocation and a vector of
prices such that all consumers maximize utility subject to their budget constraints and all markets clear. No restrictions are imposed, at this stage, on
the relationship between the production plans of the firms and the market
prices, the interpretation being that the government is allowed to regulate
all the firms in any way that it chooses. If optimal regulation requires
marginal cost pricing, then it is only the firms subject to increasing returns
which actually need to be regulated. It may also be worth reemphasizing
that throughout this paper we shall be concerned with regulated equilibria
in the context of fixed income distribution rules; if redistribution through
lump sum transfers is feasible, a complete characterization
of Pareto
optimality is provided by a generalized Second Welfare Theorem.
A regulated equilibrium consists of consumption plans (Xi), production
plans ( yj), and prices p E R’, p # 0, such that
(i)
(ii)
JT,E<&, wi) for all i, i= 1, .... m,
x = Y + 0.
In the recent literature on marginal cost pricing, Clarke’s normal cone is
used as a genera1 definition of marginal cost prices. The normal cone
characterizes marginal cost prices precisely as those which satisfy the first
order conditions for profit maximization.
In the case of a firm with a
convex production set, therefore, the normal cone is identical to the set of
profit maximizing prices. For a definition of the normal cone and a more
detailed account of its properties, the reader is referred to Khan and Vohra
[9, Section 21. Let N( Yj, y,) denote the normal cone of Yj at y;~ Y,.
Henceforth we shall consider any nonzero element of N( Yj, yj) to be a
vector of marginal cost prices for firm j when it produces yj.
A marginal cost pricing equilibrium is a regulated
equilibrium
((x,), ( yj), p) such that p E N( Y,, yj) for all i.
OPTIMALREGULATION
69
An average cost pricing
equilibrium
is a regulated equilibrium
((x,), ( y,), p) such that p . y, = 0 for all j.
Whenever a social welfarefunction
W((u,)) is specified, we shall assume it
to be continuous and monotonically
increasing in all the utilities. We can
now define the two optimality notions we shall use.
A regulated equilibrium ((-xi), (y,), p) is said to be second best relahe to
a social welfare function I#‘( .) if there does not exist another regulated
equilibrium ((X,), (j,), p) such that W((u,(X,)) > W((u,(x,)).
A regulated equilibrium
((x,), (y,), p) is said to lie on the second best
utility possibility frontier
if here does not exist another regulated
equilibrium ((Xi), (jj), p) such that ui(.Ui) > ui(xi) for all i, strict inequality
holding for at least some i.
3. CHARACTERIZATION OF SECOND BEST EQUILIBRIA RELATIVE
TO A SOCIAL WELFARE FUNCTION
If the government’s objective is to maximize a social welfare function
over all the regulated equilibria that exist, distributional
considerations
may, in general, dictate production plans which do not satisfy aggregate
production efficiency and prices which are not marginal cost prices. We
begin by illustrating this for an Arrow-Debreu economy in Example 3.1. It
will be clear from the example that this phenomenon may persist even if
one of the production sets is made nonconvex. Example 3.2 is a simple
modification of Example 3.1 and shows that even if production efficiency is
desirable, due to distributional
considerations, marginal cost pricing may
not be optimal.
Theorem 3.1 shows that if the distribution is given by a fixed structure of
revenues and demand functions are continuous, aggregate production
efficiency is always desirable. Remark 3.1 points out that production
efficiency in any individual firm is always desirable irrespective of the distribution rule being followed. Remark 3.2 relates this result and its proof to
the literature on production efficiency.
Despite Example 3.2, if production efficiency is desirable, as is the case
under a fixed structure of revenues, the marginal cost pricing principle
turns out to be optimal provided the government has the added instrument
of commodity taxation. This is the content of Corollary 3.1 which follows
from Theorem 3.1 and Lemma 3.1. Lemma 3.1 guarantees that even if
production sets are nonconvex, for any productively efficient production
plan there exists a vector of prices which is a vector of marginal costs for
each firm at its production plan.
EXAMPLE 3.1.
modities,
The economy consists of two consumers, two comand two firms. x,~ refers to consumer i’s consumption
of
70
RAJIV
VOHRA
commodity k and yjk to firms j’s production
the economy are as follows:
X,=R$,
X2= R2+,
u,(-~,)=-~,l,
u2(x2) = Minh,
of commodity
WI = (10, 01,
, x22 ),
k. The data of
e,, = 1,
02 = (10, O),
e2, = 0,
e,2=0,
e22= I,
Y,={~‘,ER~I~,,~O;~‘,,+,~~~O},
Y2 = { y2 E R2 I yzl d 0; My,,
+ ~22 d 0).
This information is sketched in Figs. 3. la and 3.lb. From consumer l’s
preferences it is clear that in any regulated equilibrium
pi, the price of
commodity 1 must be positive. We can, therefore, normalize p, = 1. It is
easy to check that the only Walrasian equilibrium is one where p2 = 2/3,
y, = (0, 0), y, = (-4, 6), .Y, = (10, 0), and x2 = (6,6). However, a regulated
equilibrium
can be found where consumer 2’s utility is increased at the
expense of consumer 1 by regulating firm 1 to operate at a loss. Suppose
production
and prices are regulated as follows: y, = (-2.5, 2.5),
y,=(-3,4.5),
and p=(l,O).
Notice that p=(l,O)
is not a vector of
marginal cost prices for either firm at its given production plan. Now con-
FIGURE
3.la
OPTIMAL
REGULATION
71
i -
0
FIGURE
3.lb
sumer l’s income is 7.5 and consumer 2’s income is 7. With xi = (7.5,O)
and x2 = (7, 7) this is a regulated equilibrium in which consumer 2’s utility
is higher relative to that at the Walrasian equilibrium. We can also verify
that this regulated equilibrium
lies on the second best utility possibility
frontier. If consumer 1 is to be provided with utility level 7.5, then, for
p2 > 0, at least 2.5 units of commodity 2 must be produced by firm 1. Since
all of commodity 2 is consumed by consumer 2, given the more efficient
technology of firm 2, it is not worthwhile to have more than 2.5 units of
commodity
2 produced in firm 1. Thus, this regulated equilibrium
corresponds to maximizing u2 subject to U, = 7.5. Similarly, it can be shown
that maximizing ui subject to u2 = 7 also leads to the same regulated
equilibrium.
Since this equilibrium
is on the second best frontier there
exists a social welfare function which dictates that this equilibrium
be
established. Of course, any production in firm 1 involves aggregate production inefficiency and we have, therefore, shown that if the income distribution is given by a fixed structure of shares, aggregate production
efficiency and marginal cost pricing may not be desirable.
EXAMPLE 3.2. The only firm is firm 2 of Example 3.1, o = (20,0), and
the consumers have the same preferences as in the previous example. The
72
RAJIV VOHRA
income distribution
is given by a fixed structure of revenues and
a, = a, = l/2. It is easy to verify that the only Walrasian equilibrium is one
where p = (1,2/3), y = (-4,6),
X, = (10, 0), and x2 = (6,6). If production
is regulated to be ( - 2, 3) and p = (1,4), there is a regulated equilibrium
with xi = (15,O) and x2 = (3, 3). Thus by setting prices which are not
marginal cost prices U, can be increased at the expense of u2. Similarly, by
producing more and reducing p2 it is possible to increase u2 at the expense
of u,. This illustrates that even if production efficiency is desirable, due to
distributional
considerations, the same is not necessarily true of marginal
cost pricing.
THEOREM 3.1. Suppose the income distribution is given by a fixed structure of revenues, all utility functions are weakly monotonic, and all production sets satisfy free disposal, in the sensethat Yj- R’G Y,. Zf ((xi), (y,),p)
lies on the secondbest utility possibility frontier and ti(p, wi) are continuous
functions for all i, then y satisfies aggregate production efficiency.
Proof. Suppose y does not satisfy aggregate production
efficiency.
Given free disposal, this implies that y lies in the interior of the aggregate
production set Y, i.e., there exists E> 0 such that y + EE Y for all EE R’,
(11(1GE. By the continuity of the demand functions, there exists 6 > 0,
depending on E, such that for all SE R’ and ll8ll 6 6,
where
X=CXi
and
-fi=ti(P,ai(.J+J+Q))),
for all i.
Choose 6~ 0. Since, by weak montonicity,
p B 0 and p # 0, this ensures
that p. ai( y + S+ o) > p. a,(y + o) for all i. By weak monotonicity,
this
means that ui(Xi) > uj(xi) for all i. Let j = y + (X-x). Since IlX --XII GE,
j E Y and there exist jje Y, such that cj jj = j. Moreover, X - j =
x-y=w,
so that ((Xi), (Y,)) is feasible. By weak monotonicity,
we
also know that p.x=p.(y+o)
and p.X=p.(y+S+o).
This implies
that
p.(X--x)=p.6
and
p.j=p(y+X-x)=p.(y+S).
Thus
xi=~,(p,ai(y+S+o))=~,(p,ai(j+w))
for all i which implies that
((Xi), (jj), p) is a regulated equilibrium. Since ui(Xi) > ui(xi) for all i, this
contradicts the hypothesis that ((x,), (yj), p) lies on the second best utility
possibility frontier. 1
Remark 3.1. From the proof of Theorem 3.1 it should be clear that a
similar argument can be used to show that even if income distribution is
given by a fixed structure of shares, production efficiency in each individual
firm is always desirable.
OPTIMAL REGULATION
73
Remark 3.2. The existing literature (see, for example, Diamond and
Mirrlees [4] and Hahn [7]) shows that in an economy in which commodity taxes are used to finance public expenditure, the presence of constant returns or the availability of profit taxes in a convex economy is a
sufficient condition for production efficiency to be desirable. Theorem 3.1
shows that in our model, a fixed structure of revenues is a suffkient condition for production efficiency to be desirable even if production sets are
nonconvex. It is instructive to compare the proof of Theorem 3.1 with the
Diamond-Mirrlees
proof. The latter depends on the fact that a small
change in prices changes aggregate demand by a small amount which in
turn requires a small change in aggregate production. Under constant
returns to scale, as in Diamond and Mirrlees [4], or if profits can be
taxed, as in Hahn [7], this has no further effect on demands so that utility
improving changes in prices are feasible. In our model, not only do
demands depend on production plans but also, because of increasing
returns, a small change in aggregate production could require a large
change in individual
production
plans3 Under a fixed structure of
revenues, demands depend on aggregate production
and, therefore,
continity arguments can be made. Under a fixed structure of shares this
reasoning does not apply since an increase in aggregate production need
not increase the income, evaluated at the original prices, of all the consumers. In fact production efficiency may not be desirable, as shown by
Mirrlees [lo] for a slightly different model and confirmed by Example 3.1
for our model.
3.1. Suppose all production sets satisfy free disposal and
YIE yj f or all j. If y satisfies aggregate production efficiency, in
the sensethat y belongs to the boundary of Y, then there exists p ER’, p # 0
such that p E N( Y,, yj) for all j.
LEMMA
.Y=CjYj,
ProoJ Consider the economy with a single consumer having a
consumption
set X, and endowment w such that z = y + OE X, and
preferences such that (z} + R’+ + is the set of consumption bundles preferred to z and (z} + R’+ is set of bundles at least as good as z. Clearly,
(z, (y,)) is a Pareto optimal allocation for this economy. We can now
apply Theorem 1 of [9] to assert that there exists p E R’, p #O such that
PE N( Y,, y,) for all j. 1
It is worth emphasizing that in the presence of increasing returns, in
general, if p E N( Y, y) there is no guarantee that p E N( Y,, y,) for all j. This
is brought out very clearly in the Beato-Mas-Cole11 example [ 11. However,
Lemma 3.1 ensures that any efficient production plan can be decentralized
through some vector of marginal costs.
3 As is the case in the Beato-Mas-Cold
example [ 11.
74
RAJIV VOHRA
From Theorem 3.1 and Lemma
following corollary.
3.1 we can immediately
deduce the
COROLLARY 3.1. Suppose ((x,), (y,), p) is a regulated equilibrium which
lies on the second best utility possibility frontier and all the conditions of
Theorem 3.1 are satisfied. Then there exists a vector of marginal cost prices
p such that the allocation ((x,), (y,)) can be sustained as a marginal cost
pricing equilibrium with commodity taxes (p - p) in the sensethat:
(a) x, = ri(p, a,( y + 0))) for all i,
(b) p EN( Y,, yj) for all j,
(c)
x=y+o.
Thus, if the income distribution is given by a fixed structure of revenues,
commodity taxation can be seen as an additional instrument which restores
the desirability of marginal cost pricing. Notice that in condition (a) above,
given our definition of t,(p, w,), consumers’ profit incomes are evaluated at
consumer prices p; Ii = p . ai( y + 0). It may be preferable to consider
profits evaluated at producer prices J& in which case the income of consumer i would be Z, = a,(p . y + p . 0). However, if this changes the incomes,
the government’s budget will no longer be balanced. The revenue collected
would amount to t = (p-p). y. If r is returned to the consumers lump
sum, in accordance with the original distribution
rule, we would have
Ii = a,(p . y + p . w + r) = ai(p . y + p . w ), as in (a) above. Alternatively, we
may consider consumers’ profit incomes arising only from the competitive
sector. In this case again it is easy to check that if any surplus or deficit is
distributed to the consumers according to the original distribution rule this
is equivalent to setting Ii as indicated by condition (a). In either case,
therefore, Corollary 3.1 is applicable.
Given that in our model lump sum taxation is possible, as long as the
distribution
rule, given according to a fixed structure of revenues, is
followed, it may seem that need for commodity taxation arises solely from
distributional
considerations. However, as we shall see in Example 4.2, in
an economy with increasing returns it may not even be possible to achieve
economic efficiency without such taxation. This underscores the view of
Brown and Heal [2] that in the presence of increasing returns judgments
on elliciency and equity cannot be regarded as separable.
4. CHARACTERIZATION
OF EQUILIBRIA THAT LIE ON THE
SECOND BEST UTILITY POSSIBILITY FRONTIER
If all production
exists a Walrasian
sets are convex, under the classical assumptions there
equilibrium.
Of course, this is also a marginal cost
OPTIMAL
REGULATION
75
pricing equilibrium and, since it is Pareto optimal, it lies on the second best
utility possibility frontier. Thus, in the absence of distributional
considerations, marginal cost pricing is optimal. If all production sets are not
convex and the income distribution
is fixed, none of the marginal cost
pricing equilibria may be Pareto optimal. This was demonstrated by
Guesnerie [S] through an example with a fixed structure of revenues and
by Brown and Heal [2] through an example with a fixed structure of
shares. The fact that some marginal cost pricing equilibrium may not be
Pareto optimal is easy to see in an example with one consumer and a
production possibility curve which is not concave to the origin. However,
examples in which none of the marginal cost pricing equilibria are Pareto
optimal must necessarily be ones in which there are many consumers. This
is so because for a single consumer economy the question of income
distribution being fixed or not is irrelevant. And, given the existence of a
Pareto optimal allocation, the general version of the Second Welfare
Theorem now implies that such an allocation can be sustained as a
marginal cost pricing equilibrium.4 Given this observation, it is easy to
understand why Brown and Heal [3, Proposition 23 have also been able
to show that if the income distribution is given by a fixed structure of
revenues and all preferences are homothetic, so that, by the Eisenberg
theorem, they can be aggregated, there exists a Pareto optimal marginal
cost pricing equilibrium.
Remark 4.1. If none of the marginal cost pricing equilibria is Pareto
optimal, then, given the income distribution
rule, the ability to achieve
Pareto optimality
through regulation is severely restricted. Suppose
((x,), (,;)) is a Pareto optimal allocation. By the generalized Second
Welfare Theorem, this allocation can be supported by marginal cost prices.
If at this allocation, any consumer has smooth preferences or if any firm
has a production set with a smooth boundary, these (normalized) support
prices must be unique. Thus, marginal cost pricing is necessary for Pareto
optimality. If no marginal cost pricing equilibrium
is Pareto optimal, no
other form of regulation can achieve Pareto optimality either. In this case,
therefore, the second best utility possibility frontier is strictly dominated by
the Pareto frontier.
We shall now show that in an economy
the marginal cost pricing equilibria may
possibility frontier. Given Theorem 3.1 it
example in which the income distribution
4 See also Brown and Heal [2, Section
and Guesnerie
[6, Section III A].
61, Beato
with increasing returns, none of
lie on the second best utility
would suffice to construct an
is given by a fixed structure of
and Mas-Cole11
[l.
Remarks
2 and 31.
76
RAJIV
VOHRA
revenues and none of the marginal cost pricing equilibria is productively
efficient.’ But this would still not resolve the issue if there is a single firm in
the economy. We, therefore, present examples with a single firm so that,
given Remark 3.1, production efficiency is no longer an issue. It is clearly
preferable to construct an example with a fixed structure of revenues since
this can also be seen as one with a fixed structure of shares. We do this in
Example 4.2.6 However, we begin by considering in Example 4.1 an
economy with a fixed structure of shares since this shows in a very simple
way that not only may no marginal cost pricing equilibria lie on the second
best frontier but every one of these may actually be dominated by an
average cost pricing equilibrium which lies on the second best frontier.
EXAMPLE
4.1.
81=1,
X,=R:,
ml= (0, lo),
%(x*)=x12,
X,=R:,
8, = 0,
~,(x,)=4logxzl +x22,
02 = (20, Oh
Y={y~R~Iy,<O;y,+y,<0
if y,> -16 and lOy,+y2+14460
if y, < -16).
Given consumer l’s preferences, in looking for marginal cost pricing
equilibria we can normalize p2 = 1. Now, marginal cost prices must be such
that p, = 1 if y, < 16, pi E [l, lo] if y, = 16 and pi = 10 if y, > 16. If there
is a marginal cost pricing equilibrium
with y, > 16, we must have
I, = 10 + lOy, + y,. Given Y, this implies that I, = -134~0,
so there
cannot exist a marginal
cost pricing equilibrium
with y, > 16. If
y=(-16,16)
and p=(l,
1) w e h ave a marginal cost pricing equilibrium
where x1 = (0, 10) and x2 = (4, 16). It is easy to see that this is the only
marginal cost pricing equilibrium in this economy.
The consumption
possibilities of consumer 2 when xi = (0, 10) are
sketched in Fig. 4.1. It is clear that the marginal cost pricing equilibrium is
not Pareto optimal. Moreover, given ur = 10 we can increase consumer 2’s
utility by considering a regulated equilibrium
which is in fact an average
cost pricing equilibrium.
Let y = (- 18, 36), p = (2, l), x1 = (0, lo), and
xq = (2, 36). This is a regulated equilibrium
in which U, = 10 but
u2 = 4 log 2 + 36 > 4 log 4 + 16. It is also clear from Fig. 4.1 that, given the
5 Note that the Beato-Mas-Colell
example
[l]
on production
inelfciency
pertains
to an
economy
with a fixed structure
of shares. In [ 1 l] a similar
result was established
for an
economy
with a fixed structure
of revenues.
6 While Guesnerie’s
example [5] is one with a fixed structure
of revenues, it does not s&ice
for our puroses since in that example
all regulated
eqilibria
are also marginal
cost pricing
equilibria;
all marginal
cost pricing equilibria
lie on the second best frontier.
77
OPTIMALREGULATION
FIGURE4.1
income distribution rule, this average cost pricing equilibrium
second best utility frontier.
lies on the
EXAMPLE 4.2. There are two consumers, two commodities,
and one
firm and the income distribution is given by a fixed structure of revenues:
o = (6, 16),
X,=R:,
a, = l/4,
a2 = 314,
Ul =x11 +x,2,
78
RAJIV
VOHRA
for u2 6 30, u2 = 6x?, + xZ2
for 3O<u,<31.5,
zd2=
foruza31.5,
u2=
6.~1 + ~22,
(106 - (10/3) u2) x2, +x2? + IO@, - 30),
6.~
+
x22
3
x21 + x22 + 1.5,
ifx,,<3;
otherwise
if xl1 < 3;
otherwise.
if .)J,d -2).
The preferences of the consumers and the production possibility curve
are illustrated in Figs. 4.2a, 4.2b, and 4.2c, respectively. Notice that consumer 2 has nonhomothetic preferences. The indifference curve of consumer
2 through (3, 12) has slope - 6 and all indifference curves below it are
parallel to it. All indifference curves have slope -6 if x2! < 3. For u2 3 31.5
and xyzZ> 3 all indifference curves has slope - 1. For u2 between 30 and
31.5, and xZ2 > 3, the slopes of the indifference curves vary linearly, going
from -6 to - 1.
Since preferences are monotonic, in searching for marginal cost pricing
equilibria we can normalize p2 = 1. Marginal cost prices are as follows:
FIGURE 4.2a
OPTIMAL
REGULATION
79
0
FIGURE 4.2b
P,E[O,~O]
if y,=O, p,=O if -2<y,<O,
pI~[0,5]
if y,= -2, and
p, = 5 if y, < -2. Suppose there is a marginal cost pricing equilibrium with
y, 6 -2. It is easy to check that in this case I, +Z2 =4pl + 16 and
Z2= 3p, + 12. This means that consumer 2’s budget line must pass through
(3, 12), where the slope of the indifference curve is -6 (see Fig. 4.2b). For
p, E [0, 51, which must be the case for marginal cost pricing with y, < -2,
this implies that consumer 2 will consume only commodity 1 and we get
xal = 3 + 12/p, > 4. But, by the feasibility condition of equilibrium,
xzl > 4
implies that y, > -2, a contradiction.
Clearly then, the only possible
marginal cost pricing equilibria must involve y = (0,O) and x = (6, 16). In
this case I, + Zz = 6p, + 16 and I, = 4.5~~ + 12. This means that consumer
64?,45:1-6
80
RAJIV VOHRA
j-
I
0
4
6
-)xll+x2l
FIGURE 4.2~
2’s budget line must pass through the point (4.5, 12), where the slope of the
indifference curve is - 1 (see Fig. 4.2b). If p, < 1 both consumers consume
only commodity 1 and the aggregate demand for commodity 1, x,r + xzl =
(6p, f 16)/p,. Since p1 < 1, x,* +x1, > 6 which contradicts the fact that
x = (6, 16). If pr > 1 then consumer 1 will consume only commodity 2.
Since consumer 2’s budget line passes through (4.5, 12), consumer 2 will
consume at most 3 units of commodity 1 and spend the rest of the income
on ‘commodity 2 (see Fig. 4.2b). Thus, the aggregate demand for commodity 2, x12+xz2>6p1 + 16-3~~. Since pr > 1, .x1* + xz2 > 16, which
again is impossible given x = (6, 16). The only marginal cost pricing
equilibrium is one with y = (0,O) and p1 = 1. In this case we have I, = 5.5,
Z2= 16.5. Consumer 1 is indifferent between consuming either commodity.
Consumer 2’s budget line passes through (3, 13.5) and, for x2, >, 3 it is
identical to the indifference curve corresponding to u2 = 31.5. Thus con-
OPTIMAL
REGULATION
81
sumer 2’s utility maximization involves consuming at least 3 units of commodity 1 and spending the rest of the income on commodity 2. x, = (3,2.5)
and x2 = (3, 13.5) are two possible demand vectors. There are of course,
many other combinations of demands which are consistent with p, = 1 and
y = (0,O) but they are all equivalent in terms of U, and us. Thus all the
marginal cost pricing equilibria
in this economy yield U, = 5.5 and
2.42= 31.5.
Now consider the following regulated equilibrium
which involves
average cost pricing: let y = (- 3, 5), x = (3,21), and p = (5/3, 1). In this
case I, = 6.5, I, = 19.5 X, = (0, 6.5) and x2 = (3, 14.5) are demands for the
two consumers which add up to (3, 21). Thus, this is a regulated
equilibrium.
Moreover,
it dominates
every marginal
cost pricing
equilibrium since U, = 6.5 and u2 = 32.5.
5. CONCLUSION
In this section we conclude by summarizing the results on the welfare
analysis of marginal cost pricing. In general, marginal cost pricing cannot
be considered optimal if the income distribution is given either by a fixed
structure of revenues or by a fixed structure of shares. As we saw in Examples 3.1 and 3.2, this is the case even in convex economies if distributional
considerations are relevant. The relationship between the Pareto frontier
and the second best utility possibility frontier can, therefore, be represented
as in Fig. 5a. In economies with increasing returns none of the marginal
cost pricing equilibria may be on the Pareto frontier. Moreover, by
Remark 4.1, it may then be impossible to attain Pareto optimality. Given
Examples 4.1 and 4.2, every marginal cost pricing equilibrium
may be
dominated by an average cost pricing equilibrium which lies on the second
best frontier. This is illustrated in Fig. 5b.
There are some cases in which marginal cost pricing is desirable. If the
income distribution
is given by a fixed structure of revenues and commodity taxation is feasible, then according to Corollary 3.1 any second best
allocation can be supported through marginal cost pricing. If distributional
considerations are not relevant then again if the income distribution is
given by a fixed structure of revenues and preferences are homothetic, there
exists a Pareto optimal marginal cost pricing equilibrium,
as shown by
Brown and Heal [3]. Yet other cases in which marginal cost pricing is
desirable are discussed in a recent survey of Guesnerie [6, Section IV C].
In Table I we present a summary of the welfare results concerning production efficiency and marginal cost pricing under various assumptions about
the income distribution.
A
Marginal
Cost
Pricmg
Second
Equilibrium
Best
Frontier
Poreto
Frontier
+
UI
0
CONVEX
ECONOMY
FIGURE 5a
A
Marginal
Cost
Averoge
Prlcmg
Cost
Equllibrlum
Prlclng
Equlllbrlum
m
NONCONVEX
FIGURE
ECONOMY
5b
83
OPTIMALREGULATION
TABLE1
Summary of Welfare Results
Welfare properties of
Distribution
Rules
Aggregate production efficiency
Arbitrary lump sum
transfers
Fixed structure of
shares
Fixed structure of
revenues
Desirable (generalized second
Welfare Theorem, Guesnerie
[S], Khan and Vohra [9])
Not necessarily desirable
(Example 3.1)
Desirable (Theorem 3.1 )
Marginal cost pricing
Desirable (generalized Second
Welfare Theorem)
Not necessarily Pareto optimal“
(Brown and Heal [Z])
Not necessarily on second best
frontier’ (Example 4.1)
Not necessarily on production
possibility frontier” (Beat0
and Mas-Colell [ I])
Not necessarily Pareto optimal“
(Guesnerie [5 1). unless
preferences are homothetic
(Brown and Heal [3])
Not necessarily on second best
frontier” (Example 4.2).
unless commodity taxation is
available (Corollary 3.1)
Not necessarily on production
possibility frontier”
(Vohra [ll])
“These results depend crucially on the presence of increasing returns.
REFERENCES
1. P. BEATO AND A. MAS-COLELL. On marginal cost pricing with given tax-subsidy rules,
J. Econ. Theory 37 (1985), 356365.
2. D. J. BROWN AND G. M. HEAL, Equity, efficiency and increasing returns, Reu. Econ. Stud.
46 (1979),
3.
4.
5.
6.
7.
8.
471485.
D. J. BROWN AND G. M. HEAL, The optimality of regulated pricing: A general equilibrium
analysis, in “Lecture Notes in Economics and Mathematical Systems,” Vol. 244, “Advances in Equilibrium Theory” (C. D. Aliprantis, 0. Burkinshaw, and N. J. Rothman, Eds.),
Springer-Verlag, Berlin, 1985.
P. A. DIAMOND AND J. A. MIRRLEES, Optimal taxation and public production, I. Production etliciency, Amer. Econ. Rev. 61 (1971), S-27.
R. GUESNERIE,Pareto optimality in non convex economies, Econometrica
43 (1975), l-29.
R. GUESNERIE,First best allocation of resources with non convexities in production,
mimeo, 1984.
F. HAHN, On optimum taxation, J. Econ. Theory 6 (1973), 96106.
H. HOTELLING, The relation of prices to marginal costs in an optimum system,
Econometrica
7 (1939). 151-155.
84
RAJIV VOHRA
ALI KHAN
AND R. VOHRA.
An extension of the second welfare theorem to economies
with nonconvexities and public goods, Quart. J. Econ. 102 (1987), 223-241.
10. J. A. MIRRLEES,
On producer taxation, Rev. Econ. Bud. 39 (1972), 105-111.
11. R. VOHRA, Second best equilibria in economies with increasing returns and fixed income
distribution, Brown University mimeo, 1984.
12. R. VOHRA, On the existence of equilibria in economies with increasing returns, in “Special
Issue on Increasing Returns,” (B. Cornet, Ed.), J. Math. Econ., in press.
9. M.