The Review of Economic Studies, Ltd.
Optimal Pricing with a Budget Constraint--The Case of the Two-part Tariff
Author(s): Yew-Kwang Ng and Mendel Weisser
Source: The Review of Economic Studies, Vol. 41, No. 3 (Jul., 1974), pp. 337-345
Published by: Oxford University Press
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Optimal
with
Pricing
Constraint
The
a
Case
Budget
of
the
Tarif2
Two-part
YEW-KWANG NG and MENDEL WEISSER
Monash Universityand Universityof New England, Australia.
I. INTRODUCTION
Two-part tariffs are well-known devices to improve the efficiency of public pricing when
average cost is decreasing.3 In such a context, marginal cost pricing results in a deficit
the financing of which by non-lump sum taxes would cause prices and factor costs affected
by the required taxation to diverge from first-best marginal values. Also, when the set
of taxpayers financing the subsidy and the set of consumers benefiting from lower prices
are not the same, marginal cost pricing entails a redistribution which conflicts with the
benefit principle and may not have sufficient merit to make it generally acceptable. Moreover, simple marginal cost pricing usually does not provide sufficient check to further
investments.4 On the other hand, average cost pricing, while avoiding all the above shortcomings, restricts consumption to a suboptimal level. With a two-part tariff, a price less
than average cost may be charged and the balance of total cost is covered by levying on
each consumer a licence fee which is not based on the amount consumed by him. Though
licence fees may vary between consumer groups according to certain criteria, we shall not
be concerned with this discriminatory possibility, but shall consider only the uniform
two-part tariff under which price does not vary with quantity bought and each consumer
pays the same licence fee.
The problem of the efficient two-part tariff is how much less than average cost should
price be, and what amount would it be necessary to levy as licence fee with such a price, given
the budget constraint that total receipts have to equal total costs plus a predetermined
amount of profit (which may be negative). If we consult the literature on the two-part
tariff in the wake of the marginal cost pricing controversy, we find authors who seem to
take it for granted that the unit price should be equal to marginal cost (e.g. [3], p. 173;
[10], p. 452; [14], pp. 237-239). Such a view may be open to criticism if two-part marginal
cost pricing keeps potential consumers out. Though an increase in the unit price above
marginal cost seems undesirable on its own, it makes it possible, with a given budget
constraint, to reduce the licence fee. This has a positive effect in reducing the number of
consumers kept out of the market. It is by no means established that this positive effect
is necessarily smaller (in absolute terms) than the negative price effect. On the other hand,
in the more recent literature on optimal pricing with a budget constraint and the related
field of optimal taxation, attention is almost exclusively focused on an approach which
implicitly equates the licence fee to zero (e.g. [1], p. 268 ff.; [2], p. 221; [9], p. 509; [12],
1 First versionreceivedDecember1972; final versionacceptedJanuary1974 (Eds.).
We are grateful to Professor James Mirrleesfor very detailed criticism and encouragement. The
shortcomingsof the final productmust be blamed on us alone.
3 Sir ArthurLewis (l8], p. 50) recordsthat a two-parttariffwas first proposed by Dr John Hopkinson
in 1892 with referenceto electricitypricingin England.
4 See, e.g. [3] and [7].
2
337
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338
REVIEW OF ECONOMIC STUDIES
pp. 696-697; [15], p. 152).1 By comparison a generalized two-part tariff could secure
gains in efficiency unless the administrative costs are prohibitive. The purpose of this
paper is to enquire into the optimal balance between the unit price and the licence fee.2
To simplify analysis, we assume a first best world apart from the single decreasing
cost industry. The main points of difference in our analysis as compared with other recent
treatments of the two-part tariff are as follows. Unlike Gabor [6] and Oi [13], we do not
have a profit-maximizing management but one that aims at Pareto optimality; unlike
Feldstein [5], we do not keep the number of consumers invariant with respect to changes
in the two-part tariff, nor do we attempt to offset inequities in the income and wealth
distribution by corrective pricing. Like all these authors, we do not necessarily equate
price to marginal cost. A central feature of our analysis is the emphasis placed on the
number of consumers as a variable.
II. THE MODEL
Consider a two-commodity economy: commodity X is produced subject to decreasing
cost by a single authority and composite commodity Z is produced competitively. Let
the production possibility frontier be defined by Z = g(X). This makes total cost of X
in terms of Z equal g(O)-g(X) and marginal cost is -g'(X). Confinement to the frontier
implicitly assumes that satiety has not been reached. The authority charges its customers
a unit price P and a licence fee L which like all values are expressed in units of Z. It is also
assumed that the collection cost of revenue is negligible and payment of the licence fee
cannot be avoided by using intermediariesbecause, e.g., it is technically costly to transfer X.
A profit target 7r is given which may be positive, zero, or negative; but if it is negative,
its absolute value must still fall short of the marginal cost pricing deficit. In other words,
the revenue required in excess of simple marginal cost pricing revenue must always be
positive, i.e.
K g(O)-g(X) + Xg'(X) +i > 0.
The problem posed by this paper does not arise if K is negative or zero.
The population consists of a continuum of individuals who are labelled by the continuous variable h. The distribution of h in the population is described by the density
function f(h) while individual incomes are denoted by y(h). The indirect utility functions
can then be written as v(P, L, y(h), h), and individual demand for X as x(P, L, y(h), h).
Perfect divisibility is assumed.
To derive the Pareto optimality conditions, we maximize the objective function
V = J'v{P, L, y(h), h}f(h)dh
. .. (1)
X=
x{P, L, y(h), h}f(h)dh
... (2)
= r
... (3)
subject to the constraints
PX+LM-g(O)+g(X)
g(O)+=
{
yf(h)dh,
.. .(4)
where M is the number of consumers.
Constraint (2) states the equality of supply and demand, (3) is the budget constraint,
and (4) specifies that total income is completely spent either on Z or X. Maximization
of the objective function (1) ensures a Pareto optimal point since, when it is maximized,
1 Diamond and Mirrlees([4], p. 272) note the possibilityof incorporatingtwo-parttariffsin a general
optimizationproblem but do not proceed to a formal analysis.
2 When this paper was in one of its later drafts, we came across Mayston's paper [11] which deals
with the same problem using a multi-periodanalysis but assumingzero-incomeeffects.
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Y. NG & WEISSER
OPTIMAL PRICING
339
we cannot increase the value of any v without reducing the value of some other. Conversely, if the utility functions are chosen suitably (e.g. through multiplying by scalars),
any Pareto optimum will maximize (1).
III. OPTIMALITY CONDITIONS
The following first-order conditions are derived from the relevant Lagrangean, using A
and p as multipliers associated with (3) and (4), and denoting P+g' -E (the excess of
price over marginal cost, or simply " excess price ").
... (5)
VP= A{X + E(XP + (Mp) + LMP}
....(6)
VL= A{M + E(XL+ XML)+ LML}
[1
=
+
for
all
infra-marginal
consumers.
... (7)
VY AExy
A marginal consumer is a consumer whose gain from consumption of X is exactly
zero (after paying for the amount consumed and the required L), so that an infinitesimal
increase in P or L drives him away from consumption but does not decrease his utility.
The notation 4 stands for the consumption of a marginal consumer which is assumed to
be the same for all such consumers. This is to simplify the proofs of the theorems below.
The case of non-uniform marginal consumption is discussed in the appendix. The number
of marginal consumers is assumed to be of measure zero. This is not really a restrictive
assumption in our model of a continuum of individuals with divisibility (see the appendix).
Division of (5) by (6) would give an optimality condition in the form VY/VL= ltPI7CL
or the equality of the marginal rate of substitution and the marginal rate of transformation
between P and L as allowed by the budget constraint. But we can further simplify the
equation by using the following substitution.
The relations between the effects on utility of changes in P, L, and y are given by the
if x>0, = 0 if x = 0; and vp = -xvy, since a marginal change in
equations VL =-vY
price is equivalent to a marginal change in income in the opposite direction multiplied by
the amount of X consumed. Substituting these equations in (7) which in turn is substituted
in (5) and (6), we get,
1{X + E
(xp + XxY)fdh+ EMp + LMp}-
A.{M+ E
,X
(XL + XY)fdh+ EXML+ LML} =-gM.
...
(8)
...(9)
x>O
Notice that f(xp + xxy)fdh is an integral of Slutsky terms for which we may write S
and which is necessarily negative. Also
(XL+
xy)fdh, as the integral of income-
compensated licence-fee derivatives of demand, is necessarily zero. This means that (8)
and (9) can be written, after eliminating A and ,u, as
E(S + {Mp)+ LMp
EXML+LML
X
M
which is the basic optimality condition. Using EX+LM = K (from the budget constraint
(3) and the definitions of E and K), we can then express optimal E and L as,
E
M(hX-eQ)Q'
whereYX=X/XMand Q _XML
QK
MS + M(X- c)Q
MS +
-MP.
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1)
. . .(1
REVIEW OF ECONOMIC STUDIES
340
If we know the relevant cost and demand conditions (including the response of the
number of consumers to changes in P and L), the optimal two-part tariff can then be
calculated.
IV. THE OPTIMAL TWO-PART TARIFF THEOREMS
We now wish to prove the following theorem.
Theorem 1. A Pareto-optimaltwo-parttariff (P, L)financing theproductionof a decreasing cost commodity with constant profit (or deficit) requirement, under the specifications
of the model satisfies thefollowing:
1. The unit price P equals marginal cost if either the price elasticity of the number
of consumersrlMequals zero, or if average consumptionX equals marginal consumption4.
2. If i7Mis not zero, P exceeds (falls short of) marginalcost if X exceeds (falls short of) ,.
3. The licencefee L is always non-negative.
Proof. To prove the theorem, we first show that MP = ML. To see this note that
MP/ML = - (aLIaP)M,i.e. the rate at which one can " substitute " L for P while keeping
M constant. With our assumption of uniform consumption by marginal consumers,'
we have to keep the utility of each marginal consumer constant in order to keep M constant
for infinitesimal changes in P and L. Since vp = XVL for a marginal consumer, the ratio
of the changes in P and L must equal - X in order to keep his utility unchanged, or
- (OLI8P)M= d. Hence MP = XML. This equality means that we can rewrite (11) and
(12) in terms of elasticities.
- 4)X;
EX
.(13)
71M(X
+
)2/
krxX r7M(.y
LM
K
where CM-Mp
1XX-11M(lXx? + qM(X4
(14)
2/.(
I and 'lx =S
X
AM
By the nature of our problem, K is necessarily positive, while X, M, and 4 are obviously
positive. Hence, from (13), a zero value of qm or (X - 4) is sufficient for a zero E. This
proves the first part of Theorem 1. If C7Mis not zero, it must be negative, as an increase
in price cannot increase the number of consumers. 11xis also necessarily negative as S
is negative. Hence E must have the same sign as (X - d) which establishes the second part
of Theorem 1.
To show the non-negativity of optimal L, assume that the contrary is true. A negative
L is in effect a subsidy. With our assumption of divisibility and given the subsidy, all
individuals will be consumers and a marginal change in P or L will induce no one to cease
consumption since he can reduce his consumption to a nominal amount and retain the
subsidy. This means that 11M= 0. Substituting this value in (14) we get L = KIM which
is positive, contrary to assumption. Hence L must be non-negative, which completes the
proof of Theorem 1.
Theorem2. Withnormal demandconditionsin the sense X > g, underthePareto-optimal
two-parttarif (P, L): (1) theproportionof Kfinanced by excess price (denotedas H) increases
with the absolute price elasticity of the number of consumersand decreases (unless 11M = 0)
with the absolute income-compensatedprice elasticity of consumption,provided that X and
4 do not change with changes in these elasticities and neither elasticity changes with changes
1 This assumption is relaxed in the appendix where it is shown that a much weaker assumption is
sufficientfor the proof that MP = eML.
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Y. NG & WEISSER
341
OPTIMAL PRICING
in the other; (2) the relative effect of the two elasticities on H is equal to the negative inverse
of their ratio.
Proof. Using equation (13) to evaluatethe partialderivativesof H(- EXIK) with
respectto im and 11x,we get,
aH!llM
-
aH/dl1=-x
q5X2(X - 0)/4 over a perfectsquare,
MX(X-
.. .(15a)
)/ over a perfect square.
...(15b)
Since Uxis necessarilynegative,(a) is negativeif X > (. If 11Mis zero, H does not change
with ijx. This is the case when we have two-partmarginalcost pricingso that H = 0
irrespectiveof the value of ilx. WhenqM is negative,(b) is positive. This provesthe first
part of Theorem2. Since the perfect squaresreferredto in both equationsabove are
identical,divisionof (a) by (b) establishesthe secondpart of Theorem2.
HI
H2
FIGURE1
If we use the horizontal and vertical axes of a figure to represent
-tlM
and - lx
respectively,and plot in the isoquants (each isoquant representsa fixed value of H),
Theorem2 impliesthat these isoquantsare positivelysloped and increasingtowardsthe
south-east. Moreoverthe second part of Theorem 2 implies that these isoquants are
are the same along a ray
homothetic as their slopes (PXt1I1M)H =- (aHla1M)l(aHlaqX)
from the origin. Combiningall these characteristicsof the isoquants,we find that these
isoquantsmustthemselvesbe raysfromthe origin,as depictedin Figure1, with the vertical
axis representingH = 0, since E = 0 as 1M = 0. The lower bound for the slope of the
isoquants is given by (X d)!X, which means that H = 1 lies below the 450 line and above
the horizontalaxis.
As K = EX+LM, if we defineJ = LM/K, i.e. the proportionof K financedby the
licencefee, (H+J) must equal unity. Then, the effectsof 1M and ix on J are the same as
those on H, exceptthat the signs are different.For example,J decreaseswith the absolute
priceelasticityof numberof consumers,and (aJ/a7M)/(aJ/8qx) also equals - qx/lM. These
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342
REVIEW OF ECONOMIC STUDIES
are easily confirmed by evaluating the partial derivatives of the left hand side of (14) with
respect to NlM and tx, and then dividing the two derivatives.
The first part of Theorem 2 is intuitively appealing if we recall that the price elasticity
of M is related to the licence fee elasticity of M, as MP = (ML. The second part of the
theorem, while not intuitively obvious, implies diminishing marginal rates of substitution
in some form. The proportion financed by either the licence fee or by the excess price is
a function of tjA and ix. But if either elasticity gets absolutely larger in relation to the
other, its relative effect declines. All these have of course to be subject to the constancy
of X and (. While the constancy may leave little room for manoeuvering, the comparison
need not be confined to a given situation.
V. CONCLUDING REMARKS
A few words may be said about the special cases of L = 0, > X, and E = 0.
From (14) it can be seen that an optimal zero licence fee requires that X> and
I MlI> INx 1. The meaning of the former inequality can easily be appreciated if we
remember that E< 0 if X < (. L must then be positive to finance K. The latter inequality
means that, for L to be zero, the number of consumers must be proportionately more
responsive to price changes than the amount of consumption is. This may occur if
individual demand curves are almost vertical and there are a large number of individuals
with demand curves close to the vertical axis. Then a reduction in P would bring in a
large number of consumers but the increase in total consumption is small.
We have spent weeks trying unsuccessfully to prove that L is necessarily positive
but are now convinced that this does not follow from our model without further restrictions.
For example, if individual demand curves, while differing from each other, are all vertical
(this is the case if the negative income effects just offset the substitution effects, or if both
effects are negligible), an increase in P does not reduce the consumption of any consumer
unless he is induced to leave the market. But with all individual demand curves vertical,
L can be reduced to an extent such that dP = -dL
where X is the consumption of the
marginal consumer. No one is then induced to leave the market, and yet revenue may
increase, as average consumers may consume more than 4. Moreover, the combined
change in (P, L) may also bring in some new consumers. Hence, all effects are favourable.
The increase in P and the associated reduction in L must therefore continue as long as all
individual demand curves are vertical and X> 4. A zero L could therefore be optimal.
In fact, if we allow for indivisibility or transaction costs, so that ML need not equal zero
when L < 0, a negative L may well be optimal.
Since the denominator of the RHS of (14) is necessarily negative, a necessary and
sufficient condition for a positive L is that the numerator is negative. This is satisfied in
either of the following three cases: (i) CiM = 0, (ii) X _ 4, (iii) I IxI> I(X - I)mx I. In
the first two cases, E is either zero or negative, so that L must be positive. In the third
case, the elasticity of consumption is not insignificant in comparison to that of the number
of consumers, thus justifying a positive L. But since I qx I may be arbitrarily close
to zero, it may well happen that Iqx I< I(X-()flMIX 1. Then L is zero unless (i) or (ii)
applies.
At one stage, we thought that L must be positive if income effects are negligible. This
however is not true since the argument of the previous section holds even if there are no
income effects. In terms of the slope of the individual demand curves, the assumption
of negligible income effects means that these slopes must be negative. But since the value
of S or Nx may be arbitrarilyclose to zero, these individual demand curves can be virtually
vertical, leading to L = 0.
Our analysis also reveals, somewhat counter-intuitively, that the optimal price may be
below marginal cost if consumers at the margin have above average consumption. This
would be the case if a large number of small consumers with very steep demand curves
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Y. NG & WEISSER
OPTIMAL PRICING
343
are inframarginaland keep averageconsumptionlow, while those with largeconsumption
have flat demandcurves. Since marginalconsumershave large consumption,they are
more responsiveto price reductionthan to licence fee reduction. Optimalitymay then
requirethe loweringof price below marginalcost to induce marginalconsumersto stay
in the marketand pay the licencefee. In a first-bestworldit cannotbe optimalto charge
such prices,but then optimaltwo-parttariffsare problemsin second-bestwherea budget
constrainthas to be satisfied. When all this is said, a positive excess price is still more
probable. It can be ensuredin a modelwheresomeregularityrequirementfor the distribution of individualdemandcurvesis postulated,e.g. that large consumersare as likely to
have steepdemandcurvesas smallconsumers.
That two-partmarginalcost pricing (i.e. L>0 and E = 0) need not necessarilybe
optimalis obvious,seeingthatthe argumentfor marginalcost pricingis basedon optimization withoutbudgetaryconstraintwhilethe two-parttariffis a pricingpolicywith a budget
constraint. Hencethe conclusionderivedfrom the formerneed not be valid for the latter.
However,as shownin Theorem1, thereare two specialcaseswhen two-partmarginalcost
=
The formeris most likely to be
pricingis optimal,i.e. when flM = 0 and when X -.
the case when all individualsderivelarge net benefitsfrom consumingthe good so that
they are all inframarginalconsumers. Then it is best to keep priceat marginalcost while
usingthe licencefee to financeK. A particularinstanceof the secondcase (whereX = 4)
is one in whichall consumersare requiredto consumethe sameamount. It is then evident
that alternativetwo-parttariffssatisfyingthe budget constraintcannot improveon twopart marginalcost pricing. (In fact, theireffectsare indistinguishable.)
It is generallyacceptedthat apart from administrativecosts two-parttariffssecure
efficiencysuperiorto that of uniform pricing. From Feldstein'sapproachwhich aims
at an optimaltrade-offbetweenefficiencyand equity one might infer that one-parttariffs
are of superiorequity since their incidenceis proportionalto consumptionwhile that of
the two-parttariffsis regressiveto consumption. Thoughthis is truethe claimfor superior
equity is not necessarilyvalid, unless the consumptionpatternsof all consumersinclude
all goods producedunderdecreasingcosts and are essentiallysimilar. If, e.g., thereis a
high-incomegood with high price elasticityproducedin one industryand a low-income
good with low price elasticityproducedin another,a one-parttariffwould entail an inequitablecross-subsidizationwhich a two-parttariff would avoid. In generaltwo-part
pricing implies adherenceto the benefit principle,irrespectiveof the number of goods
so priced.
To concludethe conclusion,we wish to stressthat the case of publicindustrieswith
decreasingcosts is quite prevalent,and hope that our results may be ultimatelyuseful
for formingpublicpricingpolicies,perhapsafter some generalizationsor extensions.
APPENDIX
The Case of Non-Uniform Marginal Consumption
The assumptionof uniform consumptionby all marginalconsumersis made mainly to
simplifythe proof of the theoremsin the text and is recognizedto be not very realistic
In this appendix,we discusshow this assumptionis relaxed.
If we assume that all individualsdiffer from each other only in one characteristic
(e.g. income) and that this characteristichas a monotonicallyincreasingeffect on their
decision to consume (e.g. people with higher incomes are more likely to become consumers),then marginalconsumersoccur only at one point on this characteristicscale.
It is then obviousthat they must be of measurezero and also consumethe same amount.
But individualsmay differfromeach otherin morethan one characteristic.Firstconsider
the case of two characteristics.
The populationis distributedaccordingto the density functionf(y, a) where y is
individualincome and a is the other distinguishingcharacteristic;both are assumedto
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344
REVIEW OF ECONOMIC STUDIES
have a monotonically increasing effect on his likelihood to consume. Given divisibility,
this last is not really a restrictive assumption as our argument can be generalized to any
number of characteristics, so that we can always define the characteristics fine enough to
have a monotonic effect.
As depicted in Figure 2, the curve B which separates the population into consumers
and non-consumers represents the locus of marginal consumers. It is of course a function
of P and L. Now suppose that there is a small reduction in P so that B is shifted to B',
and the number of consumers M increases. To keep M constant, we must have a corresponding increase in L which shifts B' to B". Since individuals of different characteristics
may react differently to changes in P and L, B" need not coincide with B, although M is
a
B"
B
B'
consutmers
\
+
R~~
non-consutmers
v
0
FIGURE
2
the same in both cases. Suppose B and B" intersect at R. Individuals at R may be called
representative marginal consumers. They are those consumers who stay indifferent after
a combined change in (P, L) which leaves M constant.
As the combined change in (P, L) becomes infinitesimal B" tends to coincide with
B. But the point R is still identifiable. What is needed for our proof that Mp = QML
is the assumption that a representative marginal consumer consumes an amount equal to
the average consumption of all marginal consumers, {. To keep M unchanged, the utility
of the representative marginal consumer(s) must be kept unchanged. The relative change
in P and L must therefore be equal to - 4. The proofs of the theorems in the text then
follow by interpreting 4 as the average consumption of marginal consumers, which, by
assumption, also equals the consumption of the representative marginal consumers.
The argument above can be easily extended to any number of characteristics. In the
case of more than four characteristics, R becomes a hypersurface. But all individuals on
this hypersurface must necessarily consume the same amount, as they are all indifferent
before and after the given change in (P, L). If they had different amounts of consumption,
they could not stay indifferent after the same change in (P, L).
It is also clear that, with the assumption of a monotonically increasing effect of each
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Y. NG & WEISSER
OPTIMAL PRICING
345
characteristic, R can only be a hypersurface and cannot expand into a hyperspace. Hence
the weight of the density function applying to the set of marginal consumers must be of
measure zero in comparison to the number of total population. In fact, the assumption of
monotonic effect is used partly to simplify the drawing of Figure 2 and partly to justify
the assumption that the set of marginal consumers is of measure zero; it is not necessary
for the proof that MP = ML.
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[15]
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