The Review of Economic Studies, Ltd.
On the Inefficiency of Two-Part Tariffs
Author(s): Rajiv Vohra
Reviewed work(s):
Source: The Review of Economic Studies, Vol. 57, No. 3 (Jul., 1990), pp. 415-438
Published by: Oxford University Press
Stable URL: http://www.jstor.org/stable/2298022 .
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Review of Economic Studies (1990) 57, 415-438
? 1990 The Review of Economic Studies Limited
On
the
0034-6527/90/00260415$02.00
Inefficiency
Two-Part
of
Tariffs
RAJIV VOHRA
Brown University
First version received October 1988; final version accepted January 1990 (Eds.)
The partial equilibrium literature on two-part tariffs suggests that if a commodity is produced
under increasing returns, efficiency can be achieved through marginal cost pricing and a suitable
choice of "entry fees" (fixed charge) which may vary from consumer to consumer. We show that
this partial equilibrium intuition cannot,be extended beyond some special cases. Even with a
consumer-specific fixed charge, it is possible that none of the equilibria yield Pareto efficiency.
Furthermore, it may be impossible to achieve Pareto efficiency through any specification of taxes
which are levied solely to cover losses.
1. INTRODUCTION
Hotelling (1939) and Lerner (1944) among others (see for example the surveys by Ruggles
(1949, 1950)) argued that firms operating under increasing returns to scale should be
regulated to equate their marginal cost to the price of the output and the resulting losses
financed through lump-sum taxes. Being concerned primarily with efficiency issues, the
proponents of the marginal cost pricing principle did not advocate any particular distribution of the lump-sum taxes among the consumers. After the ensuing debate, see for
example Ruggles (1950), there seems to have been a general agreement that marginal
cost pricing is necessary for efficiency. While admitting this, Coase (1946) presented an
illuminating analysis of the drawbacks of what he called the Hotelling-Lerner solution
and also proposed an alternative solution.'
It will be useful to begin by briefly recalling Coase's criticism of the Hotelling-Lerner
solution. Apart from the practical problem of collecting lump-sum taxes, he emphasized
two distinct drawbacks of the Hotelling-Lerner scheme. He claimed that this solution
would not necessarily guarantee efficiency in the sense that it might lead to a marginal
cost pricing equilibrium in which a positive amount of the commodity was produced
whereas it would be better not to produce the commodity at all. This can be restated
simply as the argument that in the presence of non-convexities marginal conditions need
not guarantee global optimality; there may be several marginal cost pricing equilibria,
some efficient and some inefficient. As such, this phenomenon can hardly be ruled out.
Coase's second objection to the Hotelling-Lerner solution was that it could effectively
alter the income distribution. If, for instance, a lump-sum tax was imposed on all the
consumers but only some of them consumed the commodity produced by the firm operating
under increasing returns to scale, then there would effectively be a redistribution in favour
of those who consumed this subsidized commodity. It was natural to keep the issue of
efficiency separate from that of equity and Coase argued that the Hotelling-Lerner solution
1. As Coase himself pointed out, various parts of his arguments had been presented earlier. His paper,
however, seems to cover all the significant objections to the Hotelling-Lerner solution along with a remarkably
simple alternative which too was not new.
415
416
REVIEW OF ECONOMIC STUDIES
would invariably not be neutral with respect to income distribution; efficiency without
income redistribution would call for an alternative solution.2
Coase proposed an alternative solution which he claimed would not suffer from the
above mentioned defects. He argued for a system of two-part tariffs in which consumers
of the commodity produced under increasing returns would be required to pay a fixed
amount for the privilege of buying the commodity at marginal cost. The fixed entry fees
were required to equal the loss of the firm as a result of marginal cost pricing. If each
commodity is consumed by a single consumer it is clear that this solution is neutral with
respect to income distribution. If many consumers buy a commodity being produced
under increasing returns there is the question of how the total loss is distributed among
the consumers of the commodity through the fixed parts of the tariff. Notice also that
in this case it is not clear what one means by a tariff which leaves the income distribution
unchanged. An obvious, non-discriminatory way of distributing the total loss is to set a
fixed part which is the same for all consumers. This kind of tariff we shall refer to as a
uniform two-part tariff. Another alternative is to consider for each consumer a personalized fixed part, such that the total fixed parts of all the consumers who buy the commodity
cover the total loss. We shall refer to such a tariff as a personalized two-part tariff, or
simply as a two-part tariff.
Coase also argued that, unlike the Hotelling-Lerner solution, this solution would
not lead to an inefficient equilibrium. In the next section we shall consider the specific
model used by Coase and show that this argument does not apply to more general cases.
Recently there has been a renewed interest in the marginal cost pricing principle;
see in particular the Symposium on Increasing Returns in the Journal of Mathematical
Economics edited by Cornet (1988). Formally, this recent literature analyses the consequences of dropping the assumption of convex production sets in the Arrow-Debreu
model. For the purpose of this paper, what is important is that this literature has dealt
primarily with a model in which the rules for income distribution are fixed.3 It is assumed
that either each consumer has an exogenously given share in every endowment and in
every firm, including the increasing returns firm or, more generally, there is an exogenously
given function of prices and production plans which determines the distribution of
aggregate income among the consumers. Thus, if a firm makes losses as a result of
marginal cost pricing these are covered in a lump-sum manner and the distribution of
this liability among the consumers is predetermined. No distinction is made between
shareholdings in firms which never make losses and those which might. Private ownership
extends not only to endowments and profitable firms but also, with unlimited liability,
to firms which may make losses. One may question why, in a world of complete certainty,
agents hold shares of firms which may make losses but can never make profits. This is
a serious drawback of this approach4 and, in this respect, two-part tariffs are clearly
preferable; the distribution of losses among consumers depends on whether or not they
consume the commodity, not on some exogenous rule. This difference in the two literatures
makes it difficult to revisit the older controversy with the new results.
Naturally, questions of existence and optimality have been studied. With an appropriate generalization of the notion of marginal costs, which is valid for non-smooth and
2. One important contribution of the more recent literature is precisely to show that, in a certain sense,
this separation may be impossible in a world with increasing returns. We shall elaborate on this later.
3. We shall follow this literature in using the term marginal cost pricing equilibrium to refer specifically
to an equilibrium with marginal cost pricing in a model with fixed rules for income distribution.
4. For recent work which focuses on the distribution of the liability of the public sector see Mas-Colell
and Silvestre (1989) and Roemer and Silvestre (1989).
VOHRA
TWO-PART TARIFFS
417
non-convex technologies, a generalized second welfare theorem is now available. The
first such result was provided by Guesnerie (1975) and has since been further extended
(see Bonnisseau and Cornet (1988), Khan (1987), Khan and Vohra.(1987, 1988), Quinzii
(1986) and Yun (1985)). This result states that any Pareto optimal allocation can be
sustained as a marginal cost pricing equilibrium with some redistribution5 and can be
seen as a rigorous and general version of the assertion that marginal cost pricing is
necessary for efficiency.
Some of the results in the new literature were clearly not foreseen earlier. It has
been shown that if the rules for income distribution are fixed, none of the marginal cost
pricing equilibria may be Pareto optimal (Guesnerie (1975), Brown and Heal (1979)).
Indeed, none of them may even satisfy aggregate production efficiency (Beato and
Mas-Colell (1985)), or second-best optimality (Vohra (1988)). In light of the generalized
second welfare theorem, it should be clear that this phenomenon cannot arise in a
single-consumer economy6 and must, therefore, depend on the given rule for income
distribution. As we mentioned earlier, there seems to be no compelling reason for treating
the shares of consumers in the increasing returns firms as exogenously fixed. It is natural
to examine the possibilities of restoring the efficiency of at least one marginal cost pricing
equilibrium by suitably choosing (non-negative) shares in the increasing returns firm
(which is equivalent to choosing a vector of lump-sum taxes which exactly cover any
possible losses of the increasing returns firm). This issue is one of the primary concerns
of the present paper. Notice also that it is closely related, though not identical, to the
question of the efficiency of personalized two-part tariffs.
The fact that two-part tariffs allow a consumer to effectively avoid the tax by not
consuming the commodity means that taxes might not be lump-sum.7 This can certainly
lead to inefficiency. For uniform two-part tariffs, this was first noticed by Vickrey (1948),
and since then it is well recognized in the literature; see, for example, Oi (1971), Ng and
Weisser (1974), Brown and Sibley (1986), pp. 65-68 and Example 3.2 below.8 On the
other hand, there seems to have been a general acceptance of the view that efficiency can
be achieved through personalized two-part tariffs; see, for example, Brown and Sibley
(1986), pp. 67-68 and Oi (1971), p. 81. Under the scheme of personalized two-part tariffs,
where the fixed parts may vary from consumer to consumer, there is clearly some additional
flexibility in income distribution compared to cases in which the rules for income
distribution are exogenously fixed. But whether this flexibility is, in general, enough to
restore the efficiency of at least one equilibrium is an issue which has so far remained
unexamined. A central concern of this paper is to settle this issue.
Some related issues have already been analysed in the literature. Brown and Heal
(1980) consider the possibilities of sustaining an efficient marginal cost pricing equilibrium
as a two-part tariff equilibrium. We will illustrate their result in Section 3 below. Quinzii
(1986) provides an example of a single-consumer economy in which a Pareto efficient
allocation cannot be sustained as a two-part tariff equilibrium. More recently, a similar
5. In the case of convex economies this result reduces to the familiar second welfare theorem.
6. For other sufficient conditions under which there always exists an efficient marginal cost pricing
equilibrium see Dierker (1986) and Quinzii (1988).
7. However, for commodities such as electricity which all consumers do buy, this may not be a source
of inefficiency. In that case it should also be clear that uniform two-part tariff equilibrium is identical to a
marginal cost pricing equilibrium with all consumers having an equal share in the increasing returns firm.
8. This source of inefficiency is bound to arise whenever the consumer preferences are so diverse that
there is always a consumer who is not willing to pay the fixed part and enter the market for this commodity
but has a marginal rate of substitution lower than the marginal rate of transformation. This was pointed out
to me by Debraj Ray. An early discussion of this phenomenon can be found in Vickrey (1948), p. 222.
418
REVIEW OF ECONOMIC STUDIES
illustration of the failure of the second welfare theorem in terms of two-part tariffs is
provided by Brown, Heller and Starr (1989). They also provide sufficient conditions for
the existence of a two-part tariff equilibrium:9 see also Vohra (1989).
A comparison of the efficiency properties of two-part tariffs and marginal cost pricing
under fixed rules for income distribution does not yield simple answers. Our main results
are as follows:
(i) there may exist an efficient equilibrium with uniform two-part tariffs but no
efficient marginal cost pricing equilibrium; Example 3.1.
(ii) there may exist an efficient marginal cost pricing equilibrium but no efficient
equilibrium with two-part tariffs; Examples 3.2 and 4.2.
(iii) there may exist no efficient equilibrium either with two-part tariffs or with
marginal cost pricing for any distribution of shares in the increasing returns
firm; Example 4.1. The fact that freedom in choosing the shares in the increasing
returns firm may not be enough to restore the efficiency of marginal cost pricing
had not been established by any of the existing examples. Our example shows
that, in general, explicit redistribution (in the sense of redistributing endowments
or shares in the convex sector) may be required to find an efficient marginal
cost pricing equilibrium. In other words, it may not be possible to achieve
first-best efficiency through any system of lump-sum taxes, provided these taxes
are used only for financing the increasing returns firm and not for redistributing
income. This example also shows that a balanced linear cost-share equilibrium,
as defined in Mas-Colell and Silvestre (1989), may not exist in the presence of
increasing returns.
(iv) in a simple model with two commodities where the only source of non-convexity
is fixed costs, there always exists an efficient equilibrium with two-part tariffs
while there may not exist any efficient marginal cost pricing equilibrium;
Theorem 5.1 and Example 3.1.
The outline of the paper is as follows. In the next section we briefly discuss Coase's
arguments in favour of two-part tariffs and observe that even in the simplest of models
there may exist inefficient two-part tariff equilibria. Therefore, the outstanding issue in
tems of efficiency concerns the existence of at least one efficient equilibrium. Section 3
presents a formal comparison of two-part tariff equilibria, marginal cost pricing equilibria
under fixed rules for income distribution and marginal cost pricing equilibria under
general lump-sum tax schemes. We also relate these equilibrium concepts to the notion
of two-part tariffs used by Brown and Heal (1980). In Section 4 we provide an example
with three commodities to show that even with personalized two-part tariffs it may be
impossible to achieve efficiency. In Section 5 we present a simple model in which there
always exists an efficient equilibrium with personalized two-part tariffs. This result serves
to explain why in partial equilibrium analysis the inefficiency of personalized two-part
tariffs cannot be detected. It also shows that, in this simple setting, the additional flexibility
(in distributing losses among consumers) of personalized two-part tariffs (compared to
marginal cost pricing equilibria under fixed rules for income distribution) is enough to
guarantee the existence of at least one efficient equilibrium.
2. TWO-PART TARIFFS
As mentioned above, Coase claimed that a two-part tariff equilibrium would avoid the
problems of inefficiency and redistribution which were inherent in the Hotelling-Lerner
9. As our results show, there is no guarantee that any such equilibrium will yield Pareto efficiency.
VOHRA
TWO-PART TARIFFS
419
solution. To be sure, his proposal was tailored for a very specific model and he clearly
pointed out that in a more general context his proposal might need to be re-evaluated.
The specific model he considered was one in which a commodity is delivered to each
consumer's doorstep. And each trip can serve only one consumer. The transport cost is
independent of the amount delivered to a consumer. If the commodity itself is produced
under constant returns to scale we have a situation in which the total cost of the commodity
delivered to a consumer is made up of a fixed cost and a variable cost which is proportional
to quantity delivered. The fact that costs can be attributed unambiguously to particular
consumers means that formally this is a model in which each consumer consumes a
different commodity. The solution proposed by Coase is to give each consumer the choice
of buying the commodity by paying the fixed part of the cost (the transport cost) along
with the marginal cost for every unit bought. Clearly, this will involve the marginal units
being sold at marginal cost and the total losses (transport costs) covered by payments
from the consumers in accordance with their shares in the total costs. It is easy to see
that in this simple model such a solution is efficient.
Clearly, this solution is neutral with respect to distribution. Moreover, if it is inefficient
to have the commodity delivered to some consumer, that consumer will not be willing
to buy a positive amount after paying the fixed costs plus the variable costs and this
system of two-part tariffs will therefore, not lead to production when it is inefficient to
do so. We illustrate this, along with a comparison with the Hotelling-Lerner solution in
Figure 2.1.
Consider an economy with a single consumer and two commodities. The endowment
consists only of commodity 1, and commodity 2 is the only produced commodity.
Production requires a fixed cost after which commodity 2 can be produced at a constant
.Y,
D
xI
B
FIGURE
A
2.1
420
REVIEW OF ECONOMIC STUDIES
marginal cost. The consumption possibilities of the consumer are represented by the area
on and below the curve ABC in the positive quadrant. A is the consumer's endowment.
Paying the fixed cost and marginal cost corresponds to consumption along the line segment
BC. Thus ABC also represents the frontier of the consumption possibilities made available
by a system of two-part tariffs. D the point of tangency between the production possibility
set and an indifference curve corresponds to a marginal cost pricing equilibrium. So does
the allocation A (given an appropriate definition of "marginal costs"'1). Clearly, A is
efficient while D is not. However, A is the only one which corresponds to a two-part
tariff equilibrium. Here, Coase's solution, unlike the Hotelling-Lerner solution, precludes
the emergence of an inefficient equilibrium.
Unfortunately, except in the simplest of cases, such as the one illustrated in Figure
2.1, the selection of a "right" marginal cost pricing equilibrium is not an advantage that
can be claimed for Coase's solution over the Hotelling-Lerner solution. This is illustrated
in Figure 2.2, where there are two two-part tariff equilibria corresponding to the allocations
D and E. E is Pareto optimal while D is not. In general, therefore, we have to accept
the fact that with Coase's solution, as with the Hotelling-Lerner solution, there may be
many equilibria and some of them may not be efficient. Consequently, we shall henceforth
be concerned only with the issue of the existence of an efficient equilibrium.
It should be clear that the phenomenon illustrated in Figure 2.2 stems from the
existence of multiple equilibria with positive production, and is quite different from the
problem of setting personalized tariffs to ensure that the correct number of consumers
participate in the market. Moreover, it would apply even to the more general case in
X2
D
E
A
FiGURE
X
2.2
10. A formal definition of marginal costs in terms of Clarke's normal cone is given in the next section.
VOHRA
TWO-PART TARIFFS
421
which the losses of the firm cannot be unambiguously attributed to different consumers,
as for instance, is the case when many consumers consume the same commodity which
is produced under increasing returns to scale. While Coase himself was quite explicit in
pointing out that his arguments were based on a very specific model where costs could
be attributed to particular consumers, the notion of two-part tariffs has since then been
applied to the more general situation with the losses being assigned to consumers in the
form of a fixed part of two-part tariffs. And even in the general case this solution concept
is often attributed to Coase (see for example Brown and Sibley (1986), p. 66). In the
following section we shall formally define the notion of a two-part tariff equilibrium in
a general model.
3. TWO-PART TARIFF EQUILIBRIA AND MARGINAL COST
PRICING EQUILIBRIA
We consider an economy with m consumers, indexed i E {1, . . ., m}, each consumer i
having a consumption set Xi and a utility function ui. There are n firms indexed
jc {1, .. ., n}, with production sets Yj. The aggregate endowment is denoted cl). A consumption plan is x = (xl, . . ., xm) c X = HIi Xi and a production plan is y = (yi, . . *,Yn)E
Y = j1Yj. There are I commodities indexed he {1,..., l} and all consumption and
production sets are subsets of R', the 1-dimensional Euclidean space. For any vector
z E R we shall use Zh to denote the h-th coordinate of z; Xih will denote consumer i's
consumption of commodity h. The positive orthant of R' is denoted R' = {x E R' Ix ' O}.
For x, y e R', x y denotes the scalar product. Let the unit simplex of R' be denoted
S={se R' |E=1 Sh = 1-}
We shall use Clarke's normal cone to formalize the notion of marginal cost prices.
The normal cone to the set Yj at the point yj E Yj is denoted N( Yj, yj). To formally define
the normal cone we begin with a definition of Clarke's tangent cone.
For a non-empty set Yc R' and ye Y, the tangent cone of Y at y is
T( Y, y) = {x E R' | for every sequence yk E y yk - y and every
sequence tk E (0, oo), tk -+0, there exists a sequence
x e R,x _>x, such that y +t x E Y for all k}.
The normal cone to Y at y E Y is defined as
N( Y, y) = ( T(Y, y))+,
where (T(Y, y))+ is the polar of T(Y, y). For any set A c R, A+ = {z E R'j z x c0, for
all xeA}.
The set of (normalized) marginal cost prices for firmj at production plan yj is defined
as N(Yj,y) r) S.
It will be useful to recall a few basic properties of N( Yj, yj) (these can be found in
Clarke (1983)).
Proposition 3.1. Suppose Yc R' and ye Y
(1) If p - y _ p - y' for all y'e Y, then p e N( Y, y).
(2) If Y is convex and p E N(Y, y), then p *y ' p *y' for all y'E Y.
(3) If Y is a C1 manifold and y E a Y, then N( Y, y) coincides with the usual space of
normals at y.
(4) N( Y, y) is non-empty and convex-valued. Moreover, iffor all y E Y, {y}-Rc
Y,
then the mapping N(Y,.): Y~-4R' is upper hemicontinuous and for ye a(Y),
N(Y,y) n S${O}.
REVIEW OF ECONOMIC STUDIES
422
For the sake of simplicity we shall assume that only firm n has a non-convex
production set and that it produces a single commodity I which is not produced by any
other firm and which no consumer possesses as an initial endowment. We shall also
assume that Xi = lRl for all i. These assumptions are certainly not necessary as far as the
notion of marginal cost pricing equilibrium is concerned. It does, however, simplify
considerably the notion of a two-part tariff equilibrium which we shall define shortly.
Moreover, all the issues we shall cover can be dealt with in this simple framework.
As in the classical Arrow-Debreu model, we shall consider a private ownership
economy in which the consumers have exogenously given endowments ci and shares in
the convex firms (0ij)J7T. Convex firms are assumed to maximize profits, which, by
Proposition 3.1 (1) and (2), is identical to following the marginal cost pricing rule.
Marginal cost pricing by firm n need not correspond to maximizing profits and may, in
fact, lead to a loss. The manner in which the losses or profits of firm n are to be distributed
among the consumers is essentially what distinguishes the notion of two-part tariff
equilibrium, or equilibrium notions with other kinds of tax schemes, from the (now)
standard notion of a marginal cost pricing equilibrium.
The recent literature on marginal cost pricing considers a model in which the rules
for income distribution are fixed in the sense that each consumer is also assigned,
exogenously, a share Ofi in firm n. Thus, if we denote by ri(p, y) the income of consumer
i at prices p and production plans y,
ri(p, y) = p *wi +J1
Oijp
*yj.
The budget set of i can then be defined in the usual way as yi(p, Y) =
{xi E Xi Ip*xi ' ri(p, y)}. A marginal cost pricing equilibrium is then defined as an allocation and a vector of prices such that, given these prices and the rule for income distribution,
consumers maximize utility and for each firm the market prices correspond to marginal
cost prices.
A Marginal Cost Pricing Equilibrium,MCPE, is defined as (x, y, p) E X x Y x S such
that"1
(i) For all i xi E yi(p,jy) and
(ii) For all j, p E N(YYj9yj),
...i 1lsnm
=En j + @
i=l x=j=l
ui(Zj)
u
u(xi)
for all xi
Y (P,Y),
Notice that (Oin) is simply a rule for distributing the loss or profit of firm n in a
lump-sum manner among the consumers; we can interpret OinP Ynas the lump-sum tax
(subsidy) imposed on consumer i if p Yn is negative (positive). We shall follow the
convention of using the term MCPE to refer specifically to an equilibrium in a model
with fixed rules for income distribution. As we argued in Section 1, it is natural to consider
a system of lump-sum taxes (or subsidies) the distribution of which is not exogenously
fixed. Formally, this involves studying MCPE corresponding to different specifications
of (Oin)and we can define a corresponding equilibrium notion as follows.
A Marginal Cost Pricing Equilibrium with Variable Shares, MCPEVS, is defined as
Oin' 0 for all i Ei Oin= 1and (x, y, p) is a MCPE
(x Y A (C)) E X x Y x S x Rm such thatI
for the economy in which consumers have shares Oinin firm n.
Notice that any equilibrium notion involving marginal cost pricing and some scheme
_0 for all i implies
for lump-sum taxes can be seen as a MCPEVS. The restriction that O_in
that if firm n makes a loss no consumer gets a subsidy, i.e. taxes are levied solely to
11. We shall sometimes find it more convenient to normalize the price of some commodity to be 1 rather
than to consider the price vector in the simplex.
VOHRA
TWO-PART TARIFFS
423
finance firm n and not to redistribute income. There are, of course, many well-known
tax schemes which could be incorporated in this framework; for example, wealth taxes
or benefit taxes.
Two-part tariffs are closely related to benefit taxes. However, they do not involve
lump-sum taxation and for that reason alone cannot be studied as a MCPEVS. We shall
now formally define an equilibrium notion which involves marginal cost pricing and
two-part tariffs. In the case of two-part tariffs with personalized fixed parts, instead of
exogenously given shares (Oin), fixed parts (ti) are specified. If firm n makes a loss, all
ti are required to be non-negative. If firm n makes a profit they are all required to be
non-positive. The restriction that ti all have the same sign is important. It rules out
arbitrary redistribution of income in the guise of fixed parts. In equilibrium, the sum of
the fixed parts must equal the loss of firm n. Consumers are offered the choice of buying
1 at the marginal cost after paying the corresponding fixed part ti or not paying ti and
not consuming 1. The income of consumer i, net of any possible tax, can now be defined
as
,
ti, Xi)
=
{
i +Zj=l oijp*Y
if x1 =0O
otherwise
P* +Ej= Ojjp ty *xi_A(p, y, t,, xi)}.
and the budget set is 'y(p,y,
A Two Part Tariff Equilibrium, TPTE, is defined as (x, y, (ti), p) E X x Y x Rm x S
such that
t) and ui(xi) _ui (xi) for all xi Ei(P,Y, i ),
(i) For all i xEY(P,y,
(ii) Ii Ix,,>o4ti = -p* Ynand either, ti '_-0 for all i or ti _?0 for all i,
(iii) For all j, p E N( Yj,yj),
(iV)
_
E=m1
Enj
+.
In the above definition, the actions of firm n may be interpreted as follows. The
firm uses two-part tariffs to break even and the variable part of the tariff corresponds to
marginal cost prices. Since the firm has no incentive to behave in this manner, it is
interpreted to be regulated so as to follow this pricing scheme (for the same reason, the
notion of a MCPE should also be seen in the context of regulation). It is well known,
at least since Oi (1971), that there is a close connection between two-part tariffs and
perfect price discrimination. A perfectly discriminating profit maximizing monopolist
can achieve the same result by setting two-part tariffs which extract the entire consumers'
surplus. In a partial equilibrium setting, the only difference between an equilibrium
involving perfect price discrimination and a TPTE will be in terms of income distribution.
While a complete analysis of the profit maximizing monopolist is beyond the scope of
the present paper, our examples will be simple enough to make such a comparison. In
particular, it is possible that efficiency may be achieved (or may be impossible to achieve)
either through two-part tariffs which yield zero profits or through two-part tariffs which
maximize profits; see Examples 3.1 and 4.1 below.
There are two significant differences between a MCPE and a TPTE:
(a) to the extent that the distribution of ti's can be changed without disturbing
condition (ii) of a TPTE there is some degree of freedom in determining the
income distribution, which is not possible in a MCPE.
(b) since a consumer can avoid paying the tax by not buying commodity 1,the fixed
part in a TPTE is not a lump-sum tax.
The additional flexibility mentioned in (a), however, is not afforded by a system of
uniform two-part tariffs, which we can now define formally.
REVIEW OF ECONOMIC STUDIES
424
A Uniform Two Part Tariff Equilibrium, UTPTE, is a TPTE (x, y, (Ti),p) such that
ti = t for all i.
A comparison of the efficiency properties of MCPE, TPTE and UTPTE is of fundamental importance. We have already seen in Section 2 that none of these equilibrium
notions can guarantee that every equilibrium is efficient. Consequently, we shall concentrate on the issue of the existence of at least one efficient equilibrium. Clearly, TPTE,
allowing for additional flexibility, are better suited than UTPTE for attaining efficiency.
As the recent literature emphasizes, exogenously given rules for income distribution can
impose a serious constraint on the efficiency of marginal cost pricing; for some values
of shares
(fij)
there may not exist any Pareto optimal (or even second-best)
marginal
cost pricing equilibria while for some other values there may. In view of this, and property
(a), it would seem thatkTPTE may be more compatible with efficiency than MCPE. In
Example 3.1 we show that this is possible; the rules for income distribution are such that
no marginal cost pricing equilibrium is efficient but there exists an efficient TPTE which
is in fact a UTPTE. However, this does not imply a clear advantage of TPTE over MCPE.
Property (b) may turn out to be crucial; the fact that fixed -parts of a two-part tariff are
not necessarily lump-sum may preclude the existence of an efficient UTPTE. Through
Example 3.2 we establish the possibility that there may exist no efficient UTPTE even if
there does exist an efficient MCPE.12 This example also serves to illustrate the difference
between our notion of a TPTE and that of Brown and Heal (1980).
Finally, note that, in terms of efficiency, the best one can hope to do is consider
MCPEVS. The only reason such an equilibrium may not be a TPTE is because taxes on
consumers are not conditional on consuming commodity 1; it shares with TPTE property
(a) but not (b). As such, this equilibrium notion may perform better, in terms of efficiency,
than either MCPE or TPTE. In Example 4.1 we show that there may exist no efficient
MCPEVS. This shows that it may be impossible to achieve efficiency through any scheme
of lump-sum taxes, or more generally, through any specification of (0j,,). This also shows
that a balanced linear cost share equilibrium, as defined by Mas-Colell and Silvestre
(1989), may not exist in the presence of increasing returns. This follows from the fact
that a balanced linear cost share equilibrium is Pareto efficient and, in our context, such
an equilibrium can be viewed as a MCPE with an endogenous determination of (0i,).13
Example 3.1. There are two consumers, two commodities and a single firm. The
data of the economy are as follows:
X1= R+,
X2=R 2,
u1(x1) = xi1,
w1= (4, 0),
u2(x2)= 4 log(x21 + 1) + X22,
Y =J{Y ER2Iyll'-
ty{y1E R I1
Y12
0}
0, Y11+Y12+5?'-0}
02 =(20, 0),
if yl l-5
otherwise.
The production set is sketched in Figure 3.1 (a). The direction of the vectors in the normal
cone at various production plans is indicated by arrows or the shaded area between
12. Example 4.2 shows that even an efficient TPTE may not exist while there does exist an efficient MCPE.
13. They show in Proposition 6 and Section IV.2, that in a convex, private goods economy a balanced
linear cost share equilibrium is a Walrasian equilibrium with endogenous (Oin). If firm n does not have a
convex technology, their arguments imply that it is a MCPE with endogenous shares in firm n. Unlike their
model, we also have a convex competitive sector, but that does not affect this observation.
VOHRA
TWO-PART TARIFFS
425
Y12
YiI
=1ad01=.I
rules.. ar specified.
by.
the income ditiuin
Suppose
x
inom
is amarginal cost prcigeqilbiu
f.onuer1oldb
. .. .. .. .wt Y.2 0, th
a
be emty i.e..thr.
antb
= -1and consumer l'sbugesewul
. .. . ..
with Y12>0. n fact, it iseasy to.chec tha th onlMCEisoe.hr
12>4log21,consmer2
tonfolmlow
2=
qiiru
Y (,)
i beter ffseeFig;URE3.1(b))wiecnurlsutiyrmas
two-part tarif
parteferthes,i
.Nw
cost
etinuto-at
marginal
pricinead
tarro5ws. thvencomsmonfixe
fir
hr
Inrother wors,pregaing theliri
tarifsuwhch yieldpzer profits
n 010ftettin
disrbtiwonuldeaso brespossfibedtcivefcec
by
iSucopatibe wthe eficincy.e
1
profi maximiingcs firmiwillextractrthe entir consumter' surplus ofro consumer 2,which,b
with
Y1
=
wilbistributeds
1-Ifc,
to cnuerk
1.athe indifereCPEcurv ofe cosuere
2 going
poItseeof
thgrough1(20,0 whaseae sloetc-1e aetxY =3 and passe thonughr(3, prefereThus.
ic
is x1
maxsimiedrb
shetigt=133Thcorsndg =1allocation
theocfirmnar
0),
129x,=33,
X=(,1)
3.1(b).hlecnue Isuiit
efcin;see
12= (3
better whoiflsfart
46lo63) aondumYi2 s1~3
Figure
o
ean
REVIEW OF ECONOMIC STUDIES
426
XI'
x2
x--12,
*X w
*
tn
t
x
x
1-
IaxS
x
*
n
Xt
x
x
'X
X
x
_
x
X
fl
x
x
*t
X
a
,
,
fi%
"
XI
X
X *"
12
x
Inmthen.1.ext
Tit
TpTEduexists.
3 1,s
x
:
x
n
aX
s owh
x:
XX
"~~~~
~~
ps
x
x
"
:u x
XN
X"
n
x
X
U
3X Xan
*.
x
N
x
x
It
nX
"
...
X
_
U
x
x
x
X
*
1 I$xax
u\t
i
xx
t
xo
4X1X
x
Jt x :6x
d
:
x
x
x
X
x
Do
.
x
n
f
FGURE
xl
x
x
x
t
K
x
x
X21
xx
r
3.1(b
an efficient MCPE
myes
examplew
turs
southat
x
no beficisen
casnn be suppoted as aTPTe
thfferentfficnthPe
in
terms of our definition. But, according to a result of Brown and Heal (1980), it can be
supported as a two-part tariff equilibrium in terms of their definition. We shall use this
example to bring out the difference between their notion of two-part tariff equilibria and
the one used here, which we believe is also the notion discussed in the literature following
Coase (1946).
Example 3.2. There are two consumers with identical endowments. Their preferences are homothetic but not identical. The single firm produces commodity 2 using
commodity 1 and has a technology similar to that of the firm in Example 3.1. There is
a fixed cost of 10 units of commodity 1 after which every additional unit of commodity
1 can be transformed into one unit of commodity 2.
XI =X2= R4,
XX
1(x1)
X12X
=I
# # X l(x)
##
W- W2-
(20, 0),
XXll
if
1/4)+ X12
otherwise
TWO-PART TARIFFS
VOHRA
U2(X2)=
Yf{Y
if
TFX2l
x22?'
427
X21
| (3x2,/4) + x22 otherwise
0
if Y, _-10
O}
Y1 +Yy12+10?C0O}otherwise.
leR2 IY1
I{yI1ERYIY1
Y12
The preferences of the consumers are sketched in Figures 3.2(a) and 3.2(b).
Given the preferences, we can normalize Pi = 1 in looking for an equilibrium. As in
Example 3A, any non-negative P2 is consistent with marginal cost pricing at Yi = (0, 0).
Clearly then, there exists a trivial UTPTE with Yi = 0 and P2-4. As we shall see, this is
not Pareto optimal since consumer 1 is better off making use of the technology and
consuming commodity 2. In computing a UTPTE with positive production it suffices to
consider, for each consumer, a fixed part equal to either 10 or 5. From consumer l's
preferences (see Figure 3.2(a)), it follows that this consumer would buy commodity 2
even if the fixed part was 10. In view of consumer 2's preferences, see Figure 3.2(b), it
is clear that this consumer would not buy commodity 2 under a system of two-part tariffs
even if the fixed part is 5. Thus there is only one UTPTE with positive production,
where consumer 1 finances the loss of the firm completely and is the only consumer of
commodity 2.
y,
(1
p= (l,
5, 5),
x,=(5,5),
tl =t2 =10
1),
x2=(20,0).
cannot be
Notice that uI(5, 5)> u1(20, 0) and hence the UTPTE with x = x2=(20,0)
Pareto optimal. Moreover, even the UTPTE with positive production is not Pareto optimal.
This is simply because the marginal cost of production at y, = (-15, 5) is 1 and consumer
2 would be better off substituting commodity 1 for commodity 2. This phenomenon can
be viewed as an illustration of property (b) mentioned earlier. While there is no efficient
UTPTE, in this example, efficiency can be restored by setting personalized two-part tariffs.
For example, with t, = 1 and t2 = 0, there exists an efficient TPTE.
Let us now consider MCPE in the same economy as above with the consumer shares
in the firm fixed as
01l=
021=1/2.
Since preferences are homothetic and income distribution is given by a fixed structure
of revenues, it follows from a result of Brown and Heal (1985) that, in this example,
there exists an efficient MCPE. Some computation yields the following MCPE to be
efficient.
y, = (-25, 15),
p=
(1, 1),
xI = x2
= (7 557-5).
The efficiency of this allocation is illustrated in Figure 3.2(c) where we sketch the
production possibility curve, abc, and the Scitovsky indifference curve, defg. In this case
too, consumer 2 would prefer to consume the initial endowment (but that is no longer a
feasible choice in a MCPE, given 021 = 1/2). It therefore follows that the allocation
corresponding to this MCPE cannot be supported as a TPTE.
On the other hand, Brown and Heal (1980) provide a result which seems to indicate
that any efficient MCPE can be supported through two-part tariffs. This is due to the
fact that they employ a somewhat different notion of two-part tariffs.14 For our example,
14. Their general result is expressed in terms of a more complicated budget set, but for our example it
has the usual form.
REVIEW OF ECONOMIC STUDIES
428
xt2
15
A
10
7.5~~~~~~~~~~~~~~~~~~~l
5
7.5 10
20 22.5 25
15
3.2(a)
Consumer 1
FIGURE
x22
15- A
7.5
B
I
7.5
Dw2
15 17.5 20
3.2(b)
Consumer 2
FIGURE
x21
VOHRA
TWO-PART TARIFFS
429
X12+X22
d
30
a
e
15
7.5
-
I
_
f
Xxi
15
25
30
I
40
-
I +X21
FIGURE 3.2(c)
their result says the following: In terms of Figures 3.2(a) and 3.2(b), it is possible to
support the above allocation by considering the budget lines of the two consumers to be
ABC and ABD respectively, where BC + BD = 10. This however, is not a two-part tariff
in the usual sense. These budget lines do have the "spikes" associated with two-part
tariffs. But they would imply that by not buying commodity 2, consumer 1 can consume
22*5 units of commodity 1, which is greater than w,,. On the other hand, consumer 2
can only consume a maximum of 17-5 units of commodity 1, which is less than ?021. Only
with a reallocation of initial endowments can this allocation be supported in terms of a
TPTE as defined above. For example, with a redistribution of endowments to (22-5, 0)
and (17-5, 0) for consumers 1 and 2 respectively, this can be seen as a TPTE with t1 = 7-5
and t2= 2-5. Quinzii (1986) and Brown, Heller and Starr (1989) have shown that, in
general, even this may not be possible; there exist single consumer economies in which
an efficient allocation cannot be supported as a TPTE.
4. AN EXAMPLE WITH NO EFFICIENT TPTE OR MCPEVS
In Example 4.1 we consider an economy with three commodities, two consumers and
two firms where there does not exist any efficient TPTE. In this example there also does
not exist an efficient MCPEVS. In other words, efficiency of marginal cost pricing cannot
be restored without explicit redistribution. Example 4.2 is a modification of 4.1 to show
that there may exist an efficient MCPE while no TPTE is efficient.
Example 4.1. Both consumers own 20 units of commodity 1 and for both of them
commodities 2 and 3 are perfect substitutes. Firm 1 uses commodity 1 to produce
commodity 2 and has a convex production set. This firm is owned completely by consumer
REVIEW OF ECONOMIC STUDIES
430
2. Firm 2 uses commodity 1 to produce commodity 3. The technology involves a fixed
cost and constant marginal cost for positive output levels.
1 =-co2 = (20, 0,),?
X1= X2= R3+,
021
I
if xI2 + x312
otherwise
0 5x1l+x12+x13
O-5x, + 12
(
O9l = 0,
3x21 + x22 + x23
if x21? 19-5
58-5 + x22+ x23 otherwise
Y J{y ER3 IY_O, Y13 OY;1 5y, I + Y12 O} if y, I--4
Yi E RIY11 ?0,Y13-C0; Y11+Y12-2O}
otherwise
ifY21?-5
{Y2,ER31Y21O, Y22-0;Y23'O}
otherwise.
+
Y23
2Y2
1
E
+1O?<O}
l{Y2R|Y21_?, Y22-CO;
The preferences of the two consumers are sketched in Figures 4.1(a) and 4.1(b) respectively. The indifference curves have been drawn between commodity 1 and the sum of
commodities 2 and 3. The production sets are depicted in Figures 4.1(c) and 4.1(d).
Notice that since commodities 2 and 3 are perfect substitutes in consumption,
efficiency requires that only firm 1 be used for producing low levels of aggregate output
and only firm 2 be used for producing high levels of output; see Figures 4.1(c) and (d).
In particular, efficiency demands that either
Y
Y23=
O and Y12_ 14
or
Y12= 0
and
Y23-14.
We shall now show that no TPTE can satisfy these conditions.
(a) Suppose Y23= 0 and Y12- 14:
Given the preferences of consumer 2 we can normalize P2= 1 in searching for TPTE.
From the technology of firm 1 and the marginal cost pricing rule it follows that, in
equilibrium, p i 1. Since firm 2 is not active there are no losses to be distributed.
Therefore, consumer l's income is at least 20. Since Y23= 0, it now follows that consumer
1 will consume at least 12 units of commodity 2 (see Figure 4.1(a)). Thus, in equilibrium,
Y12>12. Given the technology of firm 1, and the marginal cost pricing rule, this implies
that in equilibrium, Pi = 1 and x12= 12. Since Y12> 6, the profit of firm 1 is 2 and consumer
2's income is 22. This implies tht x22 = 2 5; see Figure 4.1(b). Thus, Y12= 14 5 > 14 which
is a contradiction.
(b) Suppose Y12= 0 and Y23-~ 14:
In this case consider the normalization P3= 1. Since Y23> Othe marginal cost pricing rule
implies that Pi = 2. The total loss of the firm is 10 and tl + t2 = 10. We know that x13- 12.
Consumer 2 will consume 19-5 units of commodity 1 since (20, 0, 0) is feasible (see Figure
4.1(b)). Any remaining income is then spent on commodity 3. Since income cannot be
greater than 40 and 19 5 units of commodity 1 cost 39, x23 1. Thus x13+ x23 13 and
Y23 13, a contradiction.
A similar argument can also be used to establish that no MCPEVS is efficient in this
example. Again, for an equilibrium to be efficient one of the two firms must be inactive.
If this is firm 2, then the argument in (a) above remains valid to show that no MCPE is
efficient. If firm 1 is inactive we need to modify the argument in (b) above. We still have
x13?12. Consumer 2 may no longer have the choice of consuming at least 19-5 units of
TWO-PART TARIFFS
VOHRA
X12+
431
X1
208
12?
8
20XI
FIGURE 4.1(a)
Consumer 1
X22 +X23
A
~~~~T
I
\
.
\
4.1(b)
Consumer 2
FIGURE
(2
W
, x21
432
REVIEW OF ECONOMIC STUDIES
Y12
~~~~~~~~14
?-
---6----6
.
.
. . .
.
.
.
.
.
.
\. . . . . . .
remains\unchanged.
Finally,
it is possible
to use
commodity1. Nevertheless
it
thi
example
and
show
s ..sil
. th cas .tha
2.rs.f
th
1
h
h
irrelevat. The Nearlerthargumentcannowl
bhe uasetose X3.Theredost
that
rgmn
nothexs
arumny
efficient two-part tariff equilibrium even ifthereiobekee constrath
ae nwintho
the firm.i
Example 4.2. We shall now consider an economy which is identical to that specified
in Example 4.1 except for the preferences of consumer 2. These are modified so that this
consumer now consumes only commodity 2 or 3 if income is low and P1/P3 = 2. We take
care to ensure that this results in no change with respect to arguments made above,
regarding the non-existence of an efficient TPTE. However, if consumer 2 is also the sole
owner of firm 2, there does exist an efficient MCPE.
X2=i4,
W2=120,O0,O),
021 =1I,
022=1I
585+
u
X22
23
X21+X22+X23
13x2,+[81/(8U2-387)](X22+
X21 +18(X22
+ X23)
X23)
if X21
195
if x21
19.5
andU u2?
if 54?<-U2
?<-58*5
if u2
54
58.5
433
TWO-PART TARIFFS
VOHRA
Y23
X X
x\
x x
I
4
__________________
x x14
x-?xx
x
x \
x
X X\
xX
.11XXXr.1.
x
,tX . X X X . Xx
X ItX X X w1
VX X X\X
* It x x X x x x
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x , Itx x F X X X
It 11 xX X X X IX X X X X
. .
. x x x I.
Xn x X"
x
is
ec
x
1
...
,
I X
th xxsame
X
kr
X
:
X X X X
Xx
x
:
asXinx
map is
exactly
same~s
the
In Ithi
tariffeqildibferi
.
t
x
X
\
x X
X X X\-
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-5
::
X.
41
x
x
x
x
Y2
I
xt
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x "1
"x
xx
, X X
x
ItX X
X
11 t X X
X X
X X
X Xt
XX
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X X
.It
tariffequilibria,~
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.
X X X
XXXX
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X X
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41
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TPItE
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is
al
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tha
is
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reevn
is
fowopr
no
efiin
is noef
TP
TE.
4.2(a
However,5 in this example,, unlike Example 4.1, for 18?--X21? 19-5 the indifference
curves vary in slope. The one going through (19-5, 0) has slope -3 while the one through
(18, 0) has slope -12. Thus if the consumer'sincome is 30, pi = 2,P2~ - 13=, (0, 0, 30) is
a demand. Some computation yields (and Figures 4.1 and 4.2(a) can be used to verify)
that the following is a MCPE:
Xl= (14, 0, 12),
x2 = (0,0,30),
Yi = 0,
Y2
=(-26,0,42),
p = (1, 2, 2).
The allocation corresponding to this MCPE is Pareto optimal. This is illustrated in Figure
4.2(b), where abc is the production possibility curve and defg the Scitovsky indifference
curve through (14, 42). Both have been drawn-between the aggregate consumption of
commodity 1 and the aggregate consumption of the sum of commodities 2 and 3.
Remark 4.1. Given the negative implications of the above examples, it would be
natural to examine the second-best efficiency (in the sense of Vohra (1988)) of two-part
tariffs. While this remains an issue for further research (except in the context of the
special model of the following section), our results seem to suggest that it would be
surprising if second-best efficiency could generally be attained through marginal cost
pricing and two-part tariffs.
REVIEW OF ECONOMIC STUDIES
434
X22 +
X23
30-
19.5
+
X21
18
FIGURE
4.2(a)
X12+X13+X22+X23
d
42
-
-
42~
12 -1
14
6
-
_
--
_
~~_
_
_
I
_
b
e
-14
28 3236
Consumer 2
FIGURE
4.2(b)
XI1I+X21
VOHRA
435
TWO-PART TARIFFS
5. A RESULT ON THE EXISTENCE OF EFFICIENT TPTE
In this section we provide sufficient conditions under which there always exists an efficient
TPTE. Our result will concern an economy with only two commodities and one in which
the only source of non-convexity in the technology is the presence of fixed or set-up
costs. As Example 4.1 shows, the possibilities of extending this positive result are quite
limited.15 The fact that even in the simple setting of this section, there may not exist any
efficient MCPE will follow from Example 3.1.
There are only two commodities. The endowment consists only of commodity 1
which can also be used to produce commodity 2. There is a single firm and its technology
involves a fixed cost and non-decreasing marginal cost at all positive levels of output;
i.e. fixed costs are the only source of non-convexity in production. Since endowments
are not strictly positive, we shall also assume that no indifference curve becomes tangent
to any axis; without such an assumption only the existence of a "quasi" equilibrium will
be guaranteed. The other assumptions are standard.16
Al. For all i, Xi =R; wil > 0 and wi2 =0; ui(*) is continuous, quasi concave and
strictly monotonic.
A2. Y1= Cu R2, where C is closed and convex. If zE C then z1<0 and z'-z
implies z' E C.
A3. For all iif xi 0 and p- x>p -xi for all x such that ui(x')?ui(xi), then
ui(z) > ui(xi) implies that p - z > p - xi.
Theorem 5.1.
If Al, A2 and A3 are satisfied, there exists a Pareto optimal TPTE.
Notice that Example 3.1 satisfies all the assumptions of this theorem'7 but has no
Pareto optimal MCPE.
Proof of Theorem 5.1. Consider the allocation ((wi), 0). Suppose this allocation is
Pareto optimal. Then by the generalized second welfare theorem (for example, Khan
and Vohra (1987), Theorem 1) there exists p e R2 such that p $ 0, p E N( Y1, 0) and for
all i if ui(x) _ ui(wi), then p - x opp wi. Notice that by free disposal p 0> and there is
no loss of generality in assuming that p E S. Moreover, using A3 and the assumption that
wi> 0 for all i we can claim that all consumers are maximizing utility given incomes
(p *i). It is now easy to see that ((wi), 0, (ti), p) is a TPTE, where ti = 0 for all i. Thus,
if ((wi), 0) is Pareto optimal there is nothing more to be proved. Suppose, therefore, that
autarky is not Pareto optimal. Letting i denote the set of Pareto optimal allocations,
we shall, henceforth, assume that
((wi), 0)7
?
Y'
(1)
The remainder of the proof is a variation on the familiar Negishi (1960) argument
for proving existence. Consider the normalization ui(wi) = 0 for all i. Let
U = {u E Rm [there exist (xi)
Ei xiyY, +
3
i
E IIi
Xi, Y' E Y1such that
and ui=ui(xi)
for all i}
15. It is possible to view Examples 4.1 and 4.2 as examples with two commodities. However, the aggregate
technology would not satisfy the assumption we shall make in this section. I am grateful to an anonymous
referee for pointing this out.
16. This model is the same as the one used by Brown and Sibley to illustrate that the inefficiency which
may result from uniform two-part tariffs can be eliminated by setting personalized two-part tariffs.
17. Although preferences in Example 3.1 are not strictly monotonic, that can be easily changed without
altering the results.
436
REVIEW OF ECONOMIC STUDIES
and
U=aUr)R+.
By A2, Co( Y1) n R2 = {O},where Co( Y1) denotes the convex hull of Y1. Since n = 1, it
follows from 5.4 (2) of Debreu (1959) that the set of attainable allocations is compact.
Given (1) and the monotonicity of preferences, it can be shown, using an argument similar
to that of Arrow and Hahn (1971, Lemma 5.3), that U is homeomorphic to Sm, the unit
simplex in Rm';moreover, the homeomorphism can be defined by a function v: Smh-4 U,
where v(s) = qs for some q > 0.
"->$P be defined as
Let 4f:SS
+(s) = {((xi), YI)E -OIvi(s) = ui(xi) for all i}.
Since preferences are monotonic, it follows that if there is an attainable allocation of the
form ((xi), 0) and ui(xi) ' 0, then (xi) = (wi). Since autarky is not Pareto optimal, recalling
the definition of U, this yields
if ((xi), Yi) EE+(s), then Yi E C.
Since C is convex and the utility functions are quasi concave it is easy to see that
convex-valued.
Let 7r:OY--4-Sbe defined as
v((xi), y) ={pES I for all i, if ui(x') 'ui(xi) then p -x'i' p - xi
(2)
4' is
and p E N( Yl, Yl)}.
By the generalized second welfare theorem, X is non-empty. Since by Proposition 3.1
(4) the normal cone is convex-valued, it is easy to see that so is V. By Proposition 3.1
(4), N( Y, *) is also upper hemicontinuous. Given the continuity of the utility functions
it is certainly the case that for every i, the correspondence xi H->{p E S Ip x'-p
xi for
all x< such that ui(x')? ui(xi)} is upper hemicontinuous. Since v is the intersection of
these correspondences with S, and is non-empty, it too is upper hemicontinuous.
From (2), and Proposition 3.1 (2) it follows that
if p E ((xi), y1), then p - y 'p * y for all y' E C.
(3)
Suppose ((xi), Yi) and ((x'), y') belong to +'(s). Let p E ir((xi), Yi). Given (2) and
=E x' -y=y
E w
E Wi, this must
i and E xi -Yi
mean that p E. xi -=p
xi. However, as ui(xi) ui(x9) for all i, from the definition of
p
v, we also know that p- xi_p- xi for all i. Hence, we must have p- x' =p xi for all i
and p - Yi = p - y'. Moreover, since ui(xi) = ui(x') for all i, this means that p E v ((x'), y').
As this argument is valid for any supporting price p, we can now claim that
(3), we know that p yl 'P - Yi. Since
if ((xi), Yi) E +i(s) and ((x'), y') EE+(s), then w((xi), Yi) = w((xD),yD)
and for any p E v((xi), y,), p - xi =p - x' for all i.
(4)
Let r:S"' -R ' be defined as
r(s) = {(p - i - p - xi) Ithere exist Yi such that ((xi), Yi) E +'(s) and p E V((xi), Yi)}.
Since the attainable sets are compact, r(S') is a compact set in R'. Let T= Co(r(S")).
Since 4' and v are non-empty, so is r. Since we have already established the convexvaluedness of v, we can use (4) to assert that r is convex-valued. We have already argued
that 4'is continuous and that v is upper hemicontinuous. It now follows that r: S' H-> T
is non-empty, upper hemicontinuous and convex-valued.
TWO-PART TARIFFS
VOHRA
437
Let h: Sm x TH->S'"be defined as
hi(s, t)
=
si+max(ti,
0)
1+E' lmax(tk,
0M).
Certainly, h is a well-defined, continuous function.
Now consider the mapping a: Sm x TZ->Sm x T, where
a (s, t) = h(s, t) x r(s).
By Kakutani's fixed-point theorem, we can find (s, T)E a (s, T). Since Tier(&), there exist
yj)) such that (ti) = (pl wi-p *xi). Since s E h(s, t),
((Xi), Y1)EE4'() and pEE (
Si (Z
=1 max
(tk, 0)) = max (ti, 0) . for all i.
(5)
We shall now show that ti has the same sign for all i. Let I = {i Isi > 01 and J = {i Isi = 01.
There are two cases to consider: (i) Suppose Ti 0 for some i E L Then, from (5), this
means that Ei max (ti, 0) = 0 and, therefore, [i 0 for all i. (ii) Suppose ti>0 for some
and ti >0 for all i I. On the
i I. Then, from (5), this implies that Eimax(ti,0)>0
other hand, for all j E J, uj(Xj) = uj(w3) and it follows from the definition of vT that t3> 0.
Thus, ti ' 0 for all i.
Notice that E =ij ji (f - i -p *xi) = -p - Y9. As we have shown, either ti '-0 for all
i or ti ' 0 for all i. We now assert that ((xi), (y9), (ti), fi) is a Pareto optimal TPTE. The
allocation is Pareto optimal by construction. We have already established that (1i) satisfy
condition (ii) of equilibrium. From the construction of X we also know that p E N( Y1, y9),
which is condition (iii) of a TPTE. To complete the proof, it only remains to verify that
condition (i) of equilibrium is satisfied for all consumers. From the construction of 'Tr,
we know that for all i, ui(x ui) u.(Zi) implies that fp x'i-fp- xi. Appealing to A3, we can
claim that for all i, ui(z) > ui(xi) implies that p* z > pl*xi. Moreover, p *xi = p. i - t,i
Thus, if consumer i pays the fixed part ti, he cannot get higher utility than ui(xi). The
only other possibility for doing better is not paying the fixed part and not consuming
commodity 2. But that allows, at best, consuming wi. Since ui(Zi) ' ui(w) by construction,
this cannot be a better alternative. Thus, condition (i) of a TPTE is also satisfied. 11
Remark 5.1. A fixed point argument is not necessary if the technology involves
fixed cost and constant marginal cost. In that case we can simply compute for each
consumer, the willingness to pay (compensating variation in income) for the privilege of
buying at marginal cost. Let this be denoted (,3i). If the sum of these is less than the
fixed cost, autarky is Pareto optimal. Otherwise, it is possible to construct an efficient
two-part tariff equilibrium such that ti '-,8i for all i.
Acknowledgement. I am indebted to Debraj Ray for raising some of the issues discussed in this paper,
and for his many insightful comments. Thanks are also due to two anonymous referees, to Vernon Henderson,
Andreu Mas-Colell, Joaquim Silvestre, Jerome Stein and the participants of the Public Enterprise Conference
of the University of California, Davis held in Cambridge Mass. (1989) and seminars at the University of Toronto,
University of California, Los Angeles, University of California, San Diego, Harvard University and Johns
Hopkins University. I am grateful for the hospitality of the Indian Statistical Institute, New Delhi and the
Universitat Aut6noma de Barcelona, where much of this work was done. This research has been supported in
part by NSF grants SES-8605630 and SES-8646400.
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