ON PUBLIC UTILITY PRICING UNDER STOCHASTIC DEMAND
| Date | 01 November 1980 |
| DOI | http://doi.org/10.1111/j.1467-9485.1980.tb00928.x |
| Author | JOHN TSCHIRHART |
| Published date | 01 November 1980 |
Srotlish
Journal
of
Polilicol Ec0nom.v.
Vol.
21,
No.
3,
Novemher
1980
0
1980 Scottish
Ecanornic Society
0036
9292
X0!0016021h
ON
PUBLIC UTILITY
PRICING
UNDER
STOCHASTIC DEMAND
JOHN
TSCHIRHART*
Department
of
Economics,
University
of’
Wyoming
The problem
of
peak-load pricing for public utilities has been thoroughly
explored under conditions of certainty.’ Brown and Johnson
(1
969) (herein-
after referred
to
as
BJ)
extended the problem to situations where demand is
subject to a random disturbance. They assumed that a producer is directed
to maximize expected social welfare and must announce a price and capacity
output before actual demand is known. The results of the maximization
problem establish that price is equal to short-run marginal cost and that
optimal capacity is greater than riskless optimal capacity. This pricing policy
prompted Turvey (1970, p. 485) to comment about the problems associated
with excess demand and rationing at times of high demand. “There is thus
a tradeoff between the sacrifice of consumers’ surplus on the one hand and
the stringency of rationing on the other hand.” Meyer (1975) and Crew and
Kleindorfer (1976) extended the
BJ
model by allowing for this tradeoff. In
the former, a chance constraint on reliability is used, while in the latter,
rationing costs are used.
There is also the problem of how the producer can set prices to avoid
deficits. In the imaginary world of certainty, a budget constraint will ensure
adequate revenues2 However, conventional budget constraints are meaning-
less if demand is stochastic. Sherman and Visscher (1978) and Tschirhart
(1975) address this problem by using different forms of
a
stochastic budget
constraint showing that a new set of tradeoffs arise.
In all of this work, there is an implicit assumption that demand is indepen-
dent of the reliability of service. Indeed, when this assumption is relaxed,
the controversial
BJ
solution can no longer be obtained. The analysis
presented here shows that for a given output capacity, an increase in price
may improve welfare via an improvement in reliability. This has not been
recognized previously. Therefore, relaxing the independency assumption
extends earlier work by capturing the idea that welfare may explicitly depend
on reliability.
The purposes of this paper are threefold: first, to summarize briefly the
above mentioned results; second, to integrate these results in a geometric
’
For
exatnples
of
this
literature, see Boiteux
(1960),
Steiner
(1957)
Williamson
(1966),
and
See Bailey and White
(1974)
for
an example of a budget constraint in a peak-load model
*
I
am grateful
for
helpful comments from Todd Sandler and an anonymous referee, and
for
Date of receipt
of
final manuscript
:
5
November
1979
Crew
and Kleindorfer
(1975).
under certainty.
editorial suggestions from Donna Lake.
216
PUBLIC UTILITY PRICING
217
example
;
and third, to relax the assumption that demand is independent of
reliability and show how this alters previous work.
I
STOCHASTIC
DEMAND
IN
A
PEAK-LOAD MODEL
Assume that there are
n
demand periods of equal d~ration.~ The pro-
ducer’s problem is to choose a capacity,
Z,
and a price vector,
p
=
(pl,.
. .
,
p,) where
pi
is the price in period
i
that will maximize expected social welfare.
Throughout, subscript
i
will run over all periods from
1
to
n.
Demand in
each period is subject to random fluctuations
so
that a unique relationship
does not exist between
pi
and the quantity demanded given by
qi.
The
producer cannot choose a particular price and be certain as to what the
demand response will be. Instead of the riskless relationship
qi
=
qi(pi),
the
producer is confronted with
qi
=
qi(pi,ui),
where
ui
is a random variable.
The producer is cognizant of the riskless demand curves, but knows only
the density functions of the random variables given byf;(ui).
Leland
(1972)
suggests that the electric power industry is a good example
of a price setter, that is, a firm which sets prices before
u
is known and then
adjusts output to meet demand. However, when demand exceeds capacity,
no further output can be produced. The utility cannot very well plan
on
adjusting capacity to meet demand since capacity adjustments require too
much time. Leland, then, was concerned with the short run.
In
the long run,
a public utility is a price-quantity setter; prices and capacity level are chosen
before the
ui’s
are known. This is essentially the situation analysed in BJ.
BJ examined
two
specific forms
of
the demand function, namely the addi-
tive and multiplicative forms. Similar results are obtained for both forms.
The following analysis will be confined to the latter. Thus,
where
Xi
is
the mean demand function for period
i.
The distribution functions
for the
ui
are given by
F,(y)
=
jGfi(ui)dui and the expected value of
ui
by
qibi
9
ui)
=
Xi(Pi)ui
j;Uifi(ui)dui
=
1.
The expected value of welfare is
E[
W]
=
E[Consumers’ Surplus (CS-
L)
+
Total Revenue (TR)
-
Total Cost
(TC)].
(1)
The term
CS
in
(1)
is the usual consumers’ surplus triangle under the demand
curve. But when demand exceeds capacity, the entire triangle is not attained.
Any portion of the triangle beyond the capacity level is lost, since demand
in this region is not satisfied. This lost portion, denoted
L,
must be subtracted
from
CS.
Figure
1
depicts
L
for a demand curve where
ui
=
iii.
The value
of
L will be positive when there is excess demand and zero otherwise. Equations
(2)
and
(3)
represent the summations over all periods of
E[CS]
and E[L]
’
The assumption of equal demand periods
is
for simplicity. The results
of
this
paper can be
converted
to
the more general case of unequal periods in the manner done
by
Williamson (1966).
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