External Diseconomies in Consumption and Monopoly Pricing
Author(s): Israel Luski and Rafael Lusky
Source: Econometrica, Vol. 43, No. 2 (Mar., 1975), pp. 223-229
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1913582
Accessed: 24/01/2010 07:16
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=econosoc.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.
http://www.jstor.org
Econometrica,Vol. 43, No. 2 (March, 1975)
EXTERNAL DISECONOMIES IN CONSUMPTION AND
MONOPOLY PRICING
BY ISRAEL LUSKI
AND RAFAEL LUSKY1
This paper deals with the relationship between the monopoly price and the socially
optimal price in the presence of consumption diseconomies. The above relationship is
derived from characteristics of the utility function.
1. INTRODUCTION
external diseconomies asserts that the socially optimal price of a
good causing a diseconomy should be above its marginal cost. We also know
that a monopolist will charge a price above its marginal cost. The purpose of this
paper is to determine the relationship between these two prices. In addition, we
demonstrate that knowledge of consumer preferences permits one to infer whether
the monopoly price would be higher or lower than the socially optimal price. This
relationship is of both theoretical and applied interest.
Baumol and Oates [1] suggested some practical ways to measure the marginal
harm to the environment in the presence of external diseconomies, using certain
environmental standards. In general, the determination of the socially optimal
price is impossible because the elements determining the price are unobserved.
We will prove a theorem based on elasticity characteristics of the utility function,
which yields policy information necessary to achieve a social optimum. We shall
also prove that a large class of utility functions will yield the same optimal prices;
in particular, the socially optimal price is equal to the monopoly price.
In the past, it has been commonly asserted that the monopolistic price would
be higher than the socially optimal price in the presence of consumption diseconomies. This view is illustrated by Naor [8], and in a more general case by
Knudsen [7]. They do not deal with a general model of external diseconomies, but
rather with a specific case of a queuing model with a waiting line. Within this
context, including some restrictions on the utility function, they derive the above
relationship between the monopolistic and the socially optimal prices. Buchanan
[2] also discusses the possibility of having a socially optimal price higher than the
monopolistic price. Examples of such possibilities in the context of diseconomies
arising from congestion are supplied by Edelson [6].2 Similar problems were
discussed by Diamond and Mirrlees [5] in dealing with the relationship between
the Pareto optimal situation and the competitive equilibrium. In addition, a
general discussion of optimal surcharge in cases of consumption externalities is
given in Diamond [4].
THE THEORY of
' Research for this paper was done while I. Luski was Visiting Assistant Professor of Economics,
University of Florida. We are indebted to Milton Z. Kafoglis and David Levhari for helpful comments
and suggestions.
2 It can be shown that the results of his examples can be inferred from our theorem.
223
224
I. LUSKI AND R. LUSKY
This paper states general conditions upon the ronsumer's utility function which
will enable us to predict the precise relationship between the socially optimal
price and the monopoly price in the case of external diseconomies. Thus, the
condition derived here narrows the range of uncertainty about the socially optimal
price. We will assume that the externalities are in terms of aggregates; i.e., the
marginal utility of an increased demand arising from someone else is independent
of which other person is demanding. The second assumption is that consumers,
when determining nct demand, ignore the effect upon themselves caused by thcir
demand, which is equivalent to the effect caused by the net demand of others.
An example of a good for which these assumptions may be reasonable is
automobile use, which gives rise to either congestion or smog. Thus, when Ui is
the utility function of the ith consumer, Ui = Ui(Xi, W) where W enters only as
a parameter. In particular, we assume W = W(IJ= 1 X). 3
2. THE MODEL
Consumption
For simplicity we shall assume that there are n consumers, each consuming
one commodity which gives rise to the external diseconomy. Let Xi be the amount
consumed by consumer i. The level of the external diseconomy, W, is a function of
the overall quantity of X consumed.
(1)
W(
W=
Xi)
assuming ax> >
Following Diamond and Mirrlees [5], and Buchanan and Kafoglis [3], let the ith
consumer maximize his net utility, which is linear in income:
(2)
Vi(Xi, W) = Ui(X1, W)
-
pXi
where p is the market price of the commodity. We also assume:
au.
ax, >
0,
au.
aw < 0,
and
a2U.
X<O
ax2
(= 1,...,n).
The first order condition which maximizes utility for the ith consumer is:
(3)
a
-
(i
=
1,..
n)
Solving the demand equations over the n consumers, we express the utility function
as a function of the market price p. Thus the consumer's indirect utility function is:
(4)
qi(p) = Ui[Xi(p), W(p)] - pXi(p).
3 As indicated by Diamond [4], under these assumptions uniform corrective taxation does restore
efficiency.
225
EXTERNAL DISECONOMIES
It is more convenient to use the properties of qi(p) for our theorem, i.e., using the
optimality condition (3):
u
OU8W
aOWOp -Xi-p
Oqi(p)_u OU OXj
Op
OXi Op
OX
Xi?-
OUx+
OW
Also from (1),
OW
EO
j= 1 0P
Op
Therefore:
()
(5)
Oqi(p)
Op
a
Ia.n X
W
+
a
=-Xi
oa E1api
The Social Optimum
By assuming an additive welfare function (R) we can consider:
n
(6)
R = E Uj(Xj, W).
j-1
The conditions for a social optimum are:
(7)
aox +
n a
(i =
=
n)
with IRijibeing negative definite. Thus, if the consumers face the optimal price p*
defined as:
(8)
p*
(,
asW'
-W' EO
=
the resulting equilibrium will represent the social optimum.
Monopoly Pricing
Assuming that there are no variable costs in producing X, the monopolist then
maximizes his profits:
n
(9)
(10)
maxn(p) = p E
p
O7r(p)
a
Xj(p);
=
n
I
= MR(p)=
X(p) + p
n2
ax%c
i
=
0
and a
< 0.
Let p-be the solution to the above problem. Our major task is to investigate the
relationship between the socially optimal price, p*, and the monopoly price, p.
226
I. LUSKI AND R. LUSKY
3.
THE GENERAL SOLUTION
The socially optimal price (p*) will be higher than the monopoly price (fi) if
MR(p*) < 0. From the second order condition of the monopoly maximization
we know that dMR(p)/dp < 0; thus, for MR(-) = 0, p* must be higher than -.4
Therefore, the question of the relationship of p to p* depends on the sign of
MR(p*). We will first demonstrate a relationship between marginal revenue and
marginal utility (with respect to the price p).
LEMMA:
MR(p*)-
-
Z aqj(p)
j=1
PROOF:
a
p*
By substituting p* from (8) into (10) we get:
MR(p*)-= E Xj(p*)- W'Z aw L ap '
and by using (5)
(11)
MR(p*) =
-
E
Q.E.D.
ap P*
MR(p)
F~~~~
FIGURE 1
4 Care must be taken here since MR(p) is a function of the price and not the usual variable, quantity.
From the second order condition for the monopolist maximization, MR(p) is a monotonic decreasing
function of p; thus, for p* > p, MR(p*) < 0. Figure 1 helps to clarify the matter.
227
EXTERNAL DISECONOMIES
This lemma asserts that the case of p*
>
-
is the uncommon one, since we usually
expectaqi/ap< 0.
For the remainder of our discussion we will assume that there are identical
consumers and therefore we can consider a representative one. Thus, the above
equations become:
(5')
ax
a-
(S1)
Op
(7')u au
Xp=
(let Xi-X),
+
=-X
au
op,
nowW
(T)
,0+
ax
n,awwW =0,
(8')
p* =
au
aw W'n,
(10')
MR(p)= X + p a
(11')
MR(p*)=
n, and
aq
nap
P*
Note the following relationships: From (3'),
(12)
ax
I
_
<0
ap (02u/aX2)? (a2u/aXa
W)nW'
(if > 0,
aq < O always)
ap<0awy)
By substitutingin (5')we see that aq/ap> 0 and thereforeMR(p*)< 0 if:
(13)
? a
u+)
-
?+naW'
u.
<
Let us now use the definition of the elasticity of the marginal utility (see, for example,
Rader [9, p. 212]) and define:
X(a2
U/aX2)
X(a2 UIaXaW)
ad
and ,
au/aX
aua
where a is the elasticity with respect to X of the marginal utility of X, and ,uis the
elasticity with respect to X of the marginal utility of W.
THEOREM:
,U- a < 1 (>
PROOF:
A necessary and sufficient condition for p* > p (p* < p) is that
1) a and ,uare evaluated at the socially optimal point.
In order for p*
d
dX
>
p, (13) must be negative, or
au
au
ax aX
au
aw
I. LUSKI AND R. LUSKY
228
or, using (7'),
(14)
dX aOU> 0
dX' aX)
at the social optimum;
d
OU
dXl' aX),
=
02
au0
au
+ X
X
+ X
O
axXa?
2
au (
X(a2 U/aX2)\
aX
au/aX
au(1
a) -
au
nW'
aaW '
x(a2 U/axaw)n(aU/aW)W
/ ?
w
au/aW
by using (7').
Thus, for p* > p (<) a necessary and sufficient condition is that 1 + a
(<0), or ,u- a < 1 (> 1).
-
i > 0
Q.E.D.
To prove sufficiency we reverse the order of the proof.
At the socially optimal point, a necessary and sufficientconditionfor
COROLLARY:
p* > p (<) is thatd/dX(sux)> 0 (< 0) where ux = (aU/aX)(X/U).
PROOF:In the proof of the theorem we had, for p* > p,
(14)
dx
ax; > 0 or d (USUX)> 0.
But as at the socially optimum dU/dX = 0, d(Usux)/dX
the condition for p* > p is d(sux)/dX > 0.
Ud(sux)/dX; therefore,
Q.E.D.
Note that the condition of the theorem does not include the term aWIaX
explicitly. Thus we can predict the price relationship only from the utility function.
Examples
Example (i): From the corollary we see immediately that for any utility function
with constant elasticity sux, the monopoly price will always be equal to the socially
optimal price. An example of such a function is U = X'W-f.
Example (ii): From the theorem, when we have unitary a, and assume (a2U/
aXaW) < 0, we will always have the monQpoly price higher than the socially
optimal price. An example of such a function is U = h(W) ln X. In this case
a = -1 and y > 0; therefore, it will always be true that,u > 1.
Example (iii): For a separable utility function of the form U = (X1- /(l - a)) +
h(W), where 0 < a < 1, the condition for the socially optimal price to be higher
EXTERNAL DISECONOMIES
229
than the monopoly price is always satisfied. Here p = 0, and a = -a; therefore,
-
a<
1.
University of the Negev, Israel
and
University of Florida, Gainesville
Manuscript received September, 1973; last revision received March, 1974.
REFERENCES
[1] BAUMOL,W. J., AND W. E. OATES:"The Use of Standards and Prices for Protection of the Environment," Swedish Journal of Economics, 73 (1971), 42-54.
[2] BUCHANAN,J. M.: "External Diseconomies, Corrective Taxes, and Market Structure," American
Economic Review, 59 (1969), 174-177.
[3] BUCHANAN,J. M., AND M. Z. KAFOGLIS:"A Note on Public Goods Supply," American Economic
Review, 53 (1963), 403-414.
[4] DIAMOND,P. A.: "Consumption Externalities and Imperfect Corrective Pricing," The Bell Journal
of Economics and Management Science, 4 (1973), 526-538.
[5] DIAMOND, P. A., AND J. A. MIRRLEES:"Aggregate Production with Consumption Externalities,"
QuarterlyJournal of Economics, 87 (1973) 1-24.
[6] EDELSON,N. M.: "Congestion Tolls Under Monopoly," American Economic Review, 61 (1971),
873-882..
[7] KNUDSEN, N. G.: "Individual and Social Optimization in a Multiserver Queue with a General
Cost-Benefit Structure," Econometrica, 40 (1972), 515-528.
[8] NAOR, P.: "On the Regulation of Queues Size by Levying Tolls," Econometrica, 37 (1969), 15-24.
[9] RADER, T.: Theory of Microeconomics. New York: Academic Press, 1972.