LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
[ii^ASS. 'iNOT.
p
7^
ALFRED
P.
A
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Dynamic
ilodpl
of the
Regulated Firm under Uncertainty*
by
Hart
I
n.
fiulirahmanyain
**•
-[VX'Ti
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
INSTITUTE OF
TECH.
KOV 2
74
DEvVtY LIBRARY
A Dynamic Model
RG,'^ulntG(i
of the
Firm under Uncor ta
i
nty->v
by
Marti
""
There are no pages numbered
**
niibrdhnianyain
n.
1
through
^
m-T^
6.
wish to acknov/l odno tlio guidance and helpful comments
of my thesis committee Professors Paul MacAvoy, Robert
Merton and ote\;art Myers.
I
PACE
A Dynar^.Ic Model
the Regulate'-!
of
^
\
7
rm
Under Uncertainty
1.
Introduction
Much of the work done In
the regulated
firm has
Averch and L.L.
employs
model)
producln;^
a
recent years on the theory of
been influenced by
the work
Johnson. The Averch-Johnson model
static framev/ork
a
homogeneous
single
for
good,
and facing
markets for the two Inputs (labour and capital)
monopoly firm
uses
a
is
subjected
capl tal -labour
to
ratio
(the A-J
monopoly
a
major proposition emerging from this model,
of H.
firm
perfect
The
It uses.
that when the
is
rate-o^-return regulation.
higher
than
the
one
It
that
minimizes cost for the level of output It elects to produce.
This result and
several
corollaries have been
clarified by several authors.
The A-J model has been
analytical
simple.
of
However,
omitting some
practice.
1.
Insights,
In
In
providing
added advantage
of being
extremely fruitful
and has the
the simplicity Is
of the
extended and
2
Important
achieved at the expense
aspects of
regulatory
Many of these abstractions from reality have been
(1).
of the
See P-aumol and Klevorick
(6) for
a good survey
literature.
A
more complete and up to date exposition Is
provided by Hilley (2).
2.
PAGE
onrllcr and
polntorj out
analysis
the
regulatory
such
hy
lap:,"
researchers
other
maximization of
maximization, and
problems
e.r;.
soc al -wel
I
f
Implications
the
Into
of
for
Important
several
are.' Hov;evor,
the repiulatory problem have
aspects of
Incorporator!
the choice of a different objective function
revenue
as
few have boon
a
8
received relatively
limited attention In the literature.
The A-J model and Its
Leland,
the exception of
With
uncertainty.
the received
on the subject.
The firm In the A-J model
demand
the
problem
the work of
Is
Myers and
theory has surprlslnrjy little
problems of uncertain Input
V/hen
extensions Ip;nore the problem of
of adjusting
does not face the
prices or uiipredl
uncertain,
the firm
Its combination
satisfy demand, provided, of course.
cta!^
Is
1
e
demand.
with the
faced
Is
of inputs
It
to say
so as
to
profitable to do
This would require us to dlstlnr;ulsh between relatively
so.
flexible
more
capital;
this,
factor,
in
turn,
labour,
has
and
the
Implications for
durable
tlie
Input,
allocation
7
dccl
s
Ion.
The static view of the
Ignores an
r'^rulatory problem taken by A-J
Important regulatory Issue.
In
the
A-J model.
See Ralloy and Coleman (3).
See Bailey and Ma lone (4).
For Instance, Klevorick (10) and SheshlnskI (15).
6.
In (13) and (14), and (12) respectively.
7. The effect of the
greater flexibility of labour relative
to
capital, on
the allocation
decisions of the monopoly
firm, will be the subject of a separate paper.
3.
h.
5.
PAHF
the firm rGmnlns continuously
It
There
no scope
Is
related
A
chan^f^/
Issue
constraint Is hlndin^T/
nature
the
Is
This
behaviour of
Is
clearly
the
rer;ulatory
regulated firms,
action by
earns exactly
hase at all
inconsistent
with
the
which do
some of
points In
observed
earn more
causing regulators to
than the allowed rate-of-return,
appropriate
of
Implies that the firm
rate-of-return on its rate
the fair
throujrh
The A-J assumption that the fair rate-of -return
constraint.
time.
demand conditions
for chanr;Ing
time and technological
once
and consequently output and price.
Inputs
has chosen Its
the optlmuin position,
In
9
o
pricrv.,
r(?dijcing
v/h
i
1
e
others
talce
earn
less than the fair rate and consequently petition for a rate
Increase.
An Important
feature of the actual
captured by the
not
price
the
firm
A-.J
version.
can charge
based
criteria, rather than specifying
constraint.
Adjustments of the
are undertalcen on
the basis of
of the firm I.e., whether or not
than
the
allowed
Is
regulatory process
that regulators fix the
on
the
an explicit
ratc-of-return
rate-of-return
price downv/ards or upwards,
the
It
rate-of-return.
past profit performance
has earned more or less
Thus,
regulators
act
Is provided by Klevorick (11).
The
observed practice.
of
styllzation
This
Is
a
"reduction" process consists of denials of furtlier proposed
rate Increases.
In a period of rising factor costs, this In
effect reduces price in real terms.
8.
9.
A detailed discussion
PACE 10
specification of
through the
rather than
charge,
all
points
rate-of- return
that this mechanism Is
should be noted
fact that
the
the price
time,
unlike
the regulated
here,
firm In
a
revision.
specifying
by
Uncertainty
Is
Is
are as many
of
1
deal
1
."re'!,
dynamic
a
explicitly
since
contingencies,
securities
or,
today,
particular
for
futiire
oversimplification
a
unit
of
it
Is
assumed
Thus,
the number
alternatively,
of
proposed
the
this
's
an
It
Is
uncertainty.
be gained of the behaviour
uncertain setting.
the A-J theorem
Is
In
problem
In an
and
regulated
a
received
of
and output decisions
uncertainty.
exists
there
VJhlle
of
are
contingency
consumption
This note studies the Implications
analogue of
there
future
case,
the
tiiere
tiiat,
of nature.
of the regulated firm.
rate
for
states
hoped that some undertanding can
decisions
formula
employed here.
is
In either
the
used
model
Introduced through the
possible
every
for
(Arrow-Deb reu securities).
the Input
The
Independent securities available as
future
price
context.
price-setting function of the
state-preference frnmev/ork that
uncertainty
binding at
Is not
study the behaviour of
to
Is
dynamic
attempts to capture the
regulators,
It
the A-J model.
In
The objective of this note
'
can earn..
It
consistent with the
the rate-of -return constraint
In
firm can
that the
In
of such
of the regulated
the dynamic
proved.
and
a
model
for
firm.
An
context, under
The capital
unregulated
firms
stock
are
PAHH 11
compareo, when ref?;u1atlon
by
the
re^^ulators
no
Is
"effective," I.e. the price set
Is
more
than what
unre/rulated
an
monopolist would have chosen, under the same demand and cost
conditions.
The
ex post efficiency of
allocation of
Inputs of the two firms are compared.
Section
describes
2
firm's environment
the
with a characterization of the uncertainty.
actions
are
modelled
In
Section
The regulator's
The
3.
together
section
next
describes the regulated firm's problem and characterizes its
optimal
policy.
optimal
policy
Section
5
considers the unregulated firm's
and compares It
with that of
the regulated
firm.
2.
The Firm's rnvlronment
the
In
describe
state-preference framework
conditions
uncertainty,
characterized
by listing
employed
the
In
the possible
here
future
"states of
to
are
nature"
that can occur In the future.
Out of the
can occur
random
set of possible states of
at a point
variables are
In
be
described by
10.
For
a
The realized values
time.
contingent
occuring, and uncertainty
listing
of
all
nature, only one
on
the state
of
the variables under
possible contingent
detailed discussion, see Hlrshlelfer (7).
of the
nature
study can
values.
PAGE 12
that a particular
given
Thus,
relevant variables are known
values of the
By considering all
future periods
decision making, and
obtain
In
ranges from
1
the future are
to M^
regulated
The
producing
a
period
the
t,
and
,
firm
(&,t)
time
t
Is
q
p
(
The revenue
obtained by
occurs
.q
is
D
,
If
The firm's output
p.
It
).
period
regarding the labour to be
and can be
.
monopolist
a
beginning of
a
price
given that
the firm satisfies
period L^
be
assumed that
Is
t
has at the beginning of the period,
hires during the
where
It
Is
q'
In
state
(p^XO.
state (O,
t)
the entire demand.
depends on
K,
on
p,
allowed to
the firm Is
that
the firm,
In
The states
the demand obtaining
/
the
(f),t)
At the
regulatory commission fixes
Given the price
In
to
single homogeneous good.
charge.
of
to T.
1
assumed
of past experience,
the basis
under study.
Indicated by
Is
nature In each
complete picture
a
from
t
the
relevant for purposes of
uncertainty regarding the variables
of nature
occurs,
with certainty.
the possible states of
one can
period,
time
state of nature
,
the stock
It
and the labour
It
assumed that decisions
employed are completely flexible
made once the state of nature
firm's production function remains
Is
revealed.
stable through
The
tl mev-'-The
11.
be straightforward
Introduce a variable
It would
to
corresponding to technology (analogous to capital stock), as
proposed by Klevorick (11), wl'-ich results from tnvostnont in
research (analogous to investment).
PAGE 13
production function
01 ^
Incorporating
-f
oppor tin
1
1
les.
In
^O
L
assumptions
marginal productivity
convenient
more
^O
K
.
positive,
of
but
productivity of both factors, as well as
other factor Is Increased.
Is
as
(K,L)
usual
the
declining/ marginal
Increasing
Is v/rltten
to
For
factor, when
of one
purposes of our analysis.
express
It
production
firm's
the
the
terms of the labour-requirements function.
12
(2)
The
perfect markets
firm faces
adding to its stock of capital.
at the beginning
the firm
Investment
l^
of period (t-1)
at the
;
it
hiring
In
of period
t,
12.
The properties
follov/ from those of
It
Is
its
is
stock of
which can be Increased,
t.
In
The decision to be made by
has a capital
beginning of period
labour, and
K^._,
level
of
at the end
but not decreased,
assumed
that capital
labour-requirements function
the
of
For example.
the production function.
PAGE Ih
1
ri
does not depreclater^Hence/
and
a
price
speclflerj hy the
.at the given
If
)
for given
may
It
required to
rate w.
price.
and
the demand
firm
for the
This will happen
This level
Is
a
when ^-^
QdC^.
,
p^
q
)
X
t"
('^c'
However,
(p
),
It
labour
at the
fixed wage
Thus,
the maximum
and
)
(p^
q
hire the
>p/w.
function of K^
defined to be
)
Is
K,
The state
fully.
to
(p^
q
.
hire labour
at high enough values of
produce the quantity
Is
-^^
re;;ulators of p
satisfy the demand
profitable
profitable output
w.
p
stock
capital
a
The firm would have to
wished to
K
not be
with
t
revealed to be (©-/t),
of nature Is
/Kj-
period
starts !n
The firm
>0.
I,
p
/for given
.^5
13.
It
relatively simple
to
Incorporate
fIxe'^
Is
a
depreciation rate for capital
which v/ould modify the
Investment
constraint from one on net
to one
on gross
Insights would be
Investment.
However,
additional
no
forthcomi ng.
model that
between
14. It
In this
the period
Is assumed
the prcHuctlon
period
regulatory reviews Is
the same
as
I.e., the period In which the capital Investment decision Is
made.
In many cases, the production period Is shorter (e.g.
between
reviews
three
one year)
than
the period
(e.g.
years).
In such a situation, the qualitative conclusions of
regulated firm
The
the present
model would bo
preserved.
price as given , between
acts as a profit maximlzor, taking
fact that
the regulatory
In view tlie
reviews, but keeping
affect
Interim will
capital
stock decisions made In
tlie
The
next review.
price specified by
the regulators at the
relative to
longer the period between
regulatory reviews,
the production period, the less
the regulated firm deviates
with
from
tlie behaviour
firm that optimizes value,
of a
the period becomes
given price.
the limiting case when
In
production period,
the
Infinitely
to the
large, compared
firm takes price as given and optimizes value. It should be
to the result of Hal ley
noted that this is somev/hat similar
and Coleman
extent of overcapitalization of
(3), that the
the regulated firm varies inversely with the regulatory lag.
PAGE 15
The possible states
dIvMed
Into
demand
Is
of nature
one containing
two sets,
satisfied and
fully
\/here excess demand exists.
are
define
a
number M^
which demand
function
demand
Is
an^
all
profitable
states
output
satisfied In
Increases.
Kj.
As
,p^
)
In
price p
the
nature
of
Increases
demand/ one
falls,
Hemand
can
stock
fij-
Kt-
Is
thus
the
given by
If
demand
satisfied
^
y/
,.
\
ft-Vc- ^^'^%,r^'<C')
,
be
'
The "operating profit" In state (O/t)
„
Is a
t
maximum
the
and the number
the capital
In
Increases/,
and
Increases.
maximum profitable output Increases.
can
the state of nature
to
more states of nature
For
those
ncl udl ng
The number H^ In period
.
Q(Kj.
I
that states of nature
Increasing
corresponding
just met.
of p^
In
of
can thus be
t
those states" where
other
the
Assuming
the order
arranged In
In period
(0,t)
^
Is
I.e.
f-V
15.
It Is
assumed that_ the wage rate w Is constant and
certain through time.
0.( !',
function of K
p ) Is a concave
and
At
0, dX(^,k.)/^a.=p/\^ ^
p.
price
ncroases, '^^5-*>^
As
Increases,
but since c'-^''('^^^k)/.>& >q^ o
Increases less
than '^'^
dX(6i,K} /<.Hi
proportionately.
As K Increases,
decreases, but
si nee ^^^''^'^''^)/'-''^^^'^<0, the
Increase In n to compensate for the
Increase In K Is less than proportionate.
16. It Is
assumed that the deman'-' functions
in the various
states of nature do not
Intersect i.e. demand functions are
monotoni cal y
Increasing In
the Index
the
state
of
of
nature, 0, at every price.
17. As
K Increases, dX/do, decreases
In every state
of the
OX./dts
world,
as O'X /'>^ck <,o.
As
Increases with
and
consequently v/i th the number of the state of the world, the
value of M^ Increases so as to make DX/Jaj
=p/w
.
1
I
PAGE 16
p^SCKt^.pt) -uj
(3)
X
C^C^t,Pf),
if demand
I.e.
/<-(.)
met
defined to
Is
heglnning of the
he the market value as of
first period, of a contract
unit of consumption In state O^t).
horl zon
I
gl
s
period
Is
t
ir
(5)
time 0,
the
The value of contlnj^ent
the decision
ven by
The present market value of
In
not
that pays one
the beginning of
state (0/t) as of
Income In
Is
(0^t)>l'i
the firm's net operating Income
given by
0o.t Lft
V
-""
^ (u.t^Kt)i
^;t
where KL
demand.
stock Is
of nature with
represents the state
It
Is
unity.
assumed that the price of
Thus, all
prices
a
the highest
unit of capital
and costs are
scaled to
units of the price of capital.
3.
The Regulatory nnvlronmont
The
regulatory commission
uses
Information on costs and demand and
firm's
current rate-of- return
-
the
latest
available
sets prices so that the
based
on current
capital
PAHH 17
Stock/
level.
demand and cost conditions
prices fixed for period (t + 1)
t's output at
capital
earning exactly the
In period
stock
mechanism,
and as
achieved at any
to Implement
costs, output,
reduced to the "fair"
Is
followed by the regulators
The rule
the firm's
-
t.
such,
fair
-^Thls
the
the
and capital
v/I
1
result In
1
rato-of-return
Is
only
chance.
regulators examine
stock In period
t
on the
adjustment
an
fair rate-of-return
point In time, except by
this rule,
that period
Is
Is
In
not
order
the firm's
and
set the
price for period (t + 1) according to the followlnfj formula.
the period of
This Is a CvTse of lagged regulation v/I th
Dalley and Colenan (3) show that
the lag being one period.
Is diminished when the
the A-J effect of overcapitalization
extent
of
the
regulatory
Introduced;
lag
Is
overcapitalization varies Inversely with the length of the
Baumol and
Klevorick (6) sketch out the implications
lag.
of regulatory lag. In a model where the firm makes decisions
investment in research, to
labour, and
regarding capital,
the
model, where
Increase
the stock
of knowledge.
In a
I'levorlck
(11)
period between
revieivs
Is
stochastic,
Investigates the relation between the investments In capital
and
the
expected period between
stock and
research
,
detailed
Me
without mnlclng
reviews.
concludes
that
production and regulatory
assumptions regarding the demand,
predict
the
possible
to
reaction functions.
It
Is not
the firm's
lag, on
direction of the effect of regulatory
this
references on
For
other
allocation decisions.
Also
question, see "onhright (7), Baupiol (5) and V/eln (16).
see footnote (I't).
Baumol and
the views of
19. This
spirit to
Is similar In
regulation.
To
of
the working
Klevorick
(R),
regarding
the latest
by taking
procep'ls, roughly
quote, "regulation
and
.Information
available on company costs an^ demands,
curmnt
firm's
resetting
that
the
prices
so
stock,
current capital
its
rate-of-return - based on
technology, demand conditions, and so forth - is reduced to
the "fair" rate level."
See also Klevorick (11), p. 53. and
Davis (8), p. 271. for similar opinions.
modelling the action of
that,
Apart from the fact
18.
PAGF.
(6)
p
where
^^
i^yt^KQ
the regulator's price for period
!s
p,
5«ir ^
^
18
(t+1),
s
Is
tt-i
the fair rate-of-return,
and wX(Qj,/'K
t.
Thus
earn
the actual
represents the actual
output In period
exactly
proportion
a
course,
s
of
the distribution of
the price p
the firm to
stock K^
Its capital
that the output
turns out to bo
q
.
for (f^,T)s^M,
ft.
^K,4-ujm(Kt,r^J<^^^^^ probability
where
,
by
Is given
sK^^ajX(%i-,Kt) ^.^^
probahllityTT
(7)
t
labour costs In period
represents the price that allov/s
p,
provided, of
Thus,
)
Is
q
TTq
^,
Is
the
probability of occurence of
1
-
f.'
tt
state (f>,t).
regulators as setting the price the firm can charge , rather
than
specifying
rate-of-return constraint.
Is
more
a
realistic. In the context of uncertainty, there
Is another
argument In favour of
Kx ante
price-setting regulation.
the regulators cannot
set the "fair" rate-of-return
,
since
the
rate-of-return
is
stochastic variable.
a
/
Setting the central
the mean)
does
parameter
not
(say,
precisely constrain the firm's actions; there Is no v/ay of
determining ej< post , whether the firm followed the
regulator's rate-of-return formula.
The firm can "cheat" by
actually aiming ex ante for a higher rate-of-return,
and claiming ex. post that this was the result
of an
outcome on the positive side of the distribution of returns.
This Is similar
to the "cheating" described
In Myers (I'O,
earn more or
price sotting, tho firm can still
p. 31'i.
In
by
loss
However,
than
rate-of-return.
the
"fair"
constraining price explicitly, the regulators take away the
r\nf^Ton of
freedom the firm neofls. In order to be able to
"cheat."
In
addition
"effective,"
the
greater than
what an
I
t
wM
1
^'
Is
re;^u1atIon
If
firm's
re/^ulatpd
Is
should
price
unregulated monopolist
n
to
be
not
be
would charge.
horizon of the firm
he assumed that the
Its decision
'
formula.
this
to
PACE
In
making
T periods.
The Firm's decision
At
beginning
the
decision
to
Is
of
optimize the
"operating
profits," net
Investment
costs of
first
the
period,
present market
present
of the
the programme
In
the
firm's
value of
Its
market value
succeeding
of
periods.
The firm's objective function Is
Max
(8)
where
such that
'^rs'-
l!9^..i
T
^
^i~^\:-\
[
I
> o
^^JL^J-±hLl,lLL
with probab
-^ii-L^!?a,,ra.^jJ_
^^i^,^
probabi-
'^^<v^f\)
lity
('>_:i:;,,c
1
-
not one
of
Js
20.
The relationship between JT(j and ''/o t
supply of
Is determined by the
strict proportionality, but
securities and the preferences of the various participants
In the market.
i-
PAGE 20
Define
^
iSt,('<t,rt)
(9)
•^
q CKupi) -
ft
i cscKt,FO^
Assuming th?t the
the above.
In
nature ^rn finely enough divided/
replaced by Intep;ral
The solution
to
beginning of period
the beginning of
the above
We define
future
of
states of
the sunmatlon sipns can be
sifjins.
dynamic prorrnmnlng.
value
^O
—
~:
b+i
and substitute
present
vo
t,
cash
problem can bo
obtalnofi by
as
the maximum
Vj.( K^,
p^
flows
discounted
)
the
to
given that the state of the system at
period
t
Is fK|,^,p^l
.
V
(K^^^^Pj.
)
I
s
the
maximum present value across states of nature, discounted to
the beginning of
time
t
period
t
with the price set for
capital stock at the end of
V
(K
of an optimal
,p
)
is
policy starting at
the period being p^
,
and
the previous period being
the
K^.
.
^
defined recursively by the following equation.
Is
solution for an optimum In the discrete case
21. The
Hov/ever,
the
continuous case.
In the
to that
analogous
necessary conditions in the latter case are less tedious to
the "continuous"
internret. While
calculate and easl-^r to
of
the delineation
of
in view
assumption
Is unrealistic
for finely divided
large
is not
states,
the discrepency
relations obtained In
be noted that the
It should
states.
case though
into the discrete
this case
can be translated
the calculations are messy.
,/ /,.
^^^^
(10)
Is
/TS
a
(
.
"^
PAGE 21
r
ct)
^''-'
'
the beginning of period
the market value as of
contract that delivers one unit (of consumption)
I.e.,
in
state
value as
of nature
period
t.
<^,^
of period
1
of a
In
of the beginning
delivers one unit with certainty
at the beginning of period
t.
one unit In
(0,t)
The first two
I.e.,
terms
of
t
the above
in
market values as of the beginning
Income across
a
the market
Is
contract that
contract that delivers
of nature
In
expression,
represent the
of period
of nature In
states
all
(P,t),
^^^ market value
'^
/'^t-i
state
In
of
(In every state of nature)
'Pq ^
as of the beginning of period
in
1
period
t.
of operating
t,
time period
t;
the
third term represents the Investment costs Incurred, and the
two terms
last
beginning
of
are
period
market
the present
t,
optimal
of
values
policies
as of
from
the
(t+1)
onwards, given that the state of the system at the beginning
of period
(t+1).
decision, and
it
should be
present market
Is
Inherited as
that
Vj.^^
re^^ l^t-\
to the beginning of period
(0,t)
After
substituting
/Pet
^
the beginning of
value as of
and weighting It by
(Kj,
the
the present
result of
the uncertainty
the outcome of
noted
a
discounts
It
In
period
represents
the
period (t+1),
back from state
t.
reaction
function
regulators, the functional equation can be written as
^C-
t.
of
the
PAGE
(11)
-(kr^t-,)
Deflnin/;
X,
m^
4-
=K,
,
Vt,,
CkV,Pet.,>oL°
Sii- V^^, Ck,.
^
dropping subscripts
and
2 2
P(,^.)ci0
except where
t
needed for purpose of clarity, we have
Define
(12b)
K.CiC.p)
'"'
Thus,
HC»vP)
'
(13)
If
Vt CK,r)
the optimum
K
Mc^x
-
then K
solves the problem Max
In
making its
expression for Vt;(K,p),
not nyopic.
period
t,
as
(.)
Vj-,.1
depends on
"
r)N
Hence,
=X.
X
f
K/=Max {x,
V,
happens
? 0,
K'
where
j
it
K'
and so
1^
=
of the
clear that the firm's decision
is
knov\'
selects
an inspection
From
(K,p).
its
its
optimal
on.
he solved usin/r: dynamic pro,";ramni
investment
investment in period
enters the optimization problem.
(.),
I
the firm
decision,
To determine
the firm must
V^^ ^
Ha^ KtC^.P)
f\-(fC,p).
optimal
to maximize
Kj.-Kj_,
( X + Mt (K,
without the constraint that
less than X,
to be
Is
^ r^[6,(^c,p)]o
J-^'L6eCK.p)J
.
V.'hile
nr.
V^^.,
an actual
methods,
it
(.)
for
(t + 1),
in
turn
problem can
is
difficult
PAHE 23
assumptions.
firm In period
.
t,
all
In
the optimal
decision of the
nature i.e.,
Kfc,.-^
and
K.
I
In
(t+1)
Is
(t+1), except
the
that price
problem of optimization
of optimization
between
the extent
affected by investment In period
Thus,
one
to
(.)
V,^.
of
\\^
t
-.
from
period
(X,p) becomes
from the
relation
and H,,, (.), we can v/rlte
ilk)
By definition,
SubstI
Into H
tutln,*!
do
\
(K,p),
=
'^f
and collecting terms, we have
Kck^p)
For a maximum, we must have
(5k
(16a)
<3<
^,..
%(.,
•-
K,.,
t.
of V^
(K,p), while
In
and
t
K,
This assumption del nks the decision made In period
that
that
Is
both periods
Investment In
states of
further
nj^
assumption made here
the specific
will undertake
the firm
(t+1).
order to study
In
without maki
results
analytical
derive any
to
^
1-t-.
ife
PAGE 2h
[%. l.^^ii^:^Ai!^' -_£!!l^^_^[/V,,C<„,r,,)l
4-
-
Multiplying by
throughout
'<^_^
and P^(K,p)=B(K,p),
we have
=p
p
and
,
since q
MCK,r)
^(n,K)=p/w
using
,
=Q(K,p) for 0=M,
rj
(16b)
„here
i,
sufficiency
=
||
this
of
specifying
without
||
relation
-
the demand
marginal unit of capital
H
=
functions,
present value
represents the
^"
difficult
Is
the uncertainty In
possibilities and
)
'
^^
,
stock In
establish
to
production
the
greater detail.
the cost
of
period
t,
^he
•
('<t~<t--i
of adding
a
for one period.
The second and third terms represent the reduction In labour
costs discounted to
The
stock.
capital
present,
due to
terms
last two
values of changes In operating
resulting from
turn result
period
,5..
a
from
change
a
In
marginal
a
marginal
Increase In
Indicate the
present
cash flows In future periods
the
regulated price,
Increase
In
capital
which In
stock In
t,
Th e l)nre":n atod FIrf''s
1
The unregulated firm
nc c ision
is
free to choose both
prices
p
PAGE 25
and Investment for period
given
t/
possible
the
conditions,
L
states
on the
price
myopic fashion,
(17)
i
Max
need not
take
thus,
periods
t
^.V^
^.
7- ^o,i
-^
C
\
Cf<t, P.>]
As
In
(18)
^t-'V,
-
I..]
<=^t.,
<
^/O
the case of the regulated firm,
Max
t
[
(
- -<„
where
4e
K^O^^^r^^V'
^
-^r
'^^
this
(\,
Is
I. ^
</o,
'v.t
such that
k
-'
^t -'Hi
^"
''-^^
.
written as
[^(K^n)]^^
It
and (t+l).
such that
It-
a
make
that
Is
t
Into
periods as
firm can,
In
demand
making Its
provided, of course,
[iV,,, Kr(Kt-,Pc)J
where
In
future
In
would undertake positive Investments
The unregulated firm's problem
the
possibilities.
The unregulated
regulated firm would.
a
nature,
of
the unregulated firm
Its effect
decisions In
the beginning of period
at
/
and the production
optimal decision,
account
t,
PAHE 26
This
Is
dynamic
a
recursively.
define
Vie.
V/^ ( K^^,,
as
)
problem
nf^
period
t,
of period (t-1)
Is
the benlnnlnj; of
nature discounted to
that
;^Iven
K^.
solved
be
to
the maximum present value
flows across states of
of future cash
the end
profirarml
functional equation
The
.
stock at
capital
the
for this problem Is defined by
(19)
/tvit
PC
'^t
where, as In equations (3),
In
this
capital
the
case,
stock alone,
state of
the system
jK^^_^l.,
rather than
Is
defined
p^ ^
as
In the
free
to
choose
,
J!'^^,^
ref^ulated case,
since the unror;ulated
both p
.
t
and K
and rewrl te
17
(
We define
K
.
^t=f^t,
f
I
^^"^
rn
Is
'^''°^
^^^ subscripts
),
(20)
Defining Jj,(K,p) as
to
'^(^r)
(21)
Vx/,
we can wr te
I
'.'^.
as
(O
do
by the
PAGE 27
(22) w^(x]. Ma. ^x.j;(K,r)V- XV wf^>
{j,u,p)^^
X. r,iK%^)
^
p
If
the optlnun K without tho
less than
to be
then K* =X.
X,
solves the prohlen
constraint that
Thus,
[
^
happens
K* =Max{x, K'U/hore
K'
J^(K,p).
flax
I'
As
In
decision
the case
Is
of
myopic,
not
regulated firm,
the
due
to
Investment has to be positive.
(t + 1),
so that the problem of
opt ml
I
one of optimization of J (K,p) and
(23)
w^^, (K)
By definition,
.
p4
i<^
"^^
t
restriction
11,
In both
-tat
perio'^s
ion of
(K)
\J(^^(X)
t
can be written as
°^t
The firm's optimum strategy can be obtained by solving
^•'
(AM
The first order conditions for
25a)
-
o
'
a
and
becomes
j;„rCri)
~
that
the assumption
aj^aln make
Vie
firm undertakes investment
that the
the
firm's
the
maximum are v/rltton as
"^
PAGE 28
(25b)
-
M.^^i'il
{ qM_'
^l^b^^l
d.0
^
{
^liL^S^
^_.
o
^i^
or.
order condition for
The first
marginal
the present value" of the cost of addin?^ a
capital
In
period
for one period,
There
capital stoci;
rej^ardlng
(except Insofar
as a result
Implications
no
are
decisions
for
as capital
unit of
In
to
of this
decision
the
of
future
cannot be
stock
that
has to be set equal
decrease In labour costs
the marpjnal
decision.
t,
Implies,
stock
capital
periods
reduced).
The
second equation Indicates that the present value of marginal
cost and marginal
Mov/ever,
demand
periods whore
In
Is
unregulated firms
first order
firms, for
conditions for
.mCkp)
.^y^
(2Ra)
(2Eb)
Increases and
satisfied,
hence,
price
and output decisions of the
in
order to
capital
stock,
compare them.
of the
two
given price, are v/ritten as
a
Regulated Firm
Unrcgul ated
not fully
cost.
now examine the capital
regulated and
The
demand Is
Insensitive to small
equals marginal
\-ie
revenue have to be equated, across states.
F
I
^^"^'C^
'
*
-
ap
1—
^c,,
r
rm
(^t-Kt.,1
-
f
p^^[.o4Cv'^)]'^-
k[--C'sC^^r),K)jX^--
PAGE 29
The last term !n the equation
for
regulatod firm can be
the
wr tten as
I
(s/Q-sKQ,
But
=
^^^Sp)
>^
r<^
^^
'^^'''''^
(27)
/a'")>0
implied by
is
V
/,^
J
follows from
Q,^
Q, /Q-wXQ, /Q" )>0 since
(w/,
^^(^>0.
=
J""^
^'
<S
since 0,>0,K, which
Similarly,
<0.
which
c^--^
y,
Q
Hence the last terr
dK
can be wr tten as
i
v/here
cJC
=
The expression for the regulated firm can be written as
/^^^'f'-'^t'-,r-'""-^^^'^''°
dp
K-^..,)"
.
(28)
_
7<^
M
'^
The
terms of
stocl<
In
decrease
anove expression
present value, of
period,
In
the
together with
t,
have
a
marginal
for one period
has
the
increment
Increase
the present value
two
The changes
of
in
components, both
in
In capital
to be equated
to the
In period
present value of labour costs
succeeding periods.
periods,
Indicates that
'>./
a p
t,
changes In
revenue in
revenue
succeeding
arising
In
through
tfie
PAHE 30
regulatory price
mechanism
1)
decrease
price
the
through
In
a
posslhle
the
decrease
In
lahour
In
of nature,
states
Increase In
Is
dependent on the signs
and,
of
^^'^*'(Ktn
provided the regulator's
what the monopoly
period
will
the
and
/Pp.tti
)
formula Implies
vice
versa.
combination chosen
In general,
not.
t.
direction
the demand curves
equal
the
of
precise knowledge
of the
it
faces
by
the A-J
be
price
shift
Is
result In
the
g
)
t-^,
In
positive
less than
while
the
the regulated
firm,
unregulated firm,
Indeterminate,
firm's production
in
by
at the beginning
Thus,
that of the
without
possi bll
I
ti
os,
the various states of nature,
and the description of uncertainty.
from
will
a
an
to
of the change
,p
firm would have charged
(t+1)
capl tal -labour
period
all
turn,
In
a
,
In
t.
(.^j
^^+ti(K
th
p
another due
2)
The direction
In
v/i
period,
period
In
of
the various states of nature
of
costs
capital costs.
an
sign
succeeding
price In period (t+1), caused.
Increase In the
The
associated
change
a
This Is quite different
static case
and Its
dynamic
extens Ion.
The main reason why the behaviour of the regulated firm
In
tlie
above
discussion
Is
Indeterminate
application of the regulatory formula
In
is
that
the
equation (7) could
the stationary
27..
See
Davis (8) for
a demonstration that
exhibit'",
rep;ulatory
process
point
the
dynamic
of
overcapitalization in the A-J sense.
PAGE 31
lead to prices that are "too hlf^h" I.e.,
set
never
yields
formula
regulatory
would
circumstances.
It
can
be
that the
shov/n
chooses an optimal capital stock, at
circumstances,
under such
ratio
Is
greater for
the
the
so that
an
though
the
Under
sucli
regulated
firm
price.
a
we
If
what
than
even
charge
such
chosen by the unregulated firm,
"effective/"
higher
price
a
monopolist
unconstrained
Is
than v/hat an
However,
charge.
regulatory process
assume that the
regulators
would
monopolist
unconstrained
h!r:hor
^
least as large as that
given the same price. Also,
ex.
nost capl tal -labour
regulated
than
firm,
for
the
unregulated firm, for every state of nature that may obtain.
Under "effective
regulation,
^Au(..)
0,
for
all
0.
or
Equation (28) can be written as
(29)
-
(/^
[-/.(^f(^'.r).K)]['-^i'^^'
^
."
-
Mtl-vP)
where
b^
"
r.
-J-
"
"^^
^<fi,,i
C'<ch
j
ly^t,,;
;^
q
.f
In
the
the assumption
similar
to
23.
This
Is somewhat
regulation Is "hinding," I.o.,
standard A-.l model, that the
less than
regulators Is
allowe'^ by tlie
the rate-of -return
the monopoly rnte-of- re turn that the firm could have earno-!,
were It not regulate'^.
PAGf:
The relation In the ahove equation
first order condition
and K^
for
unregulated firms
can he comparer! with the
the unregulated firm.
optimal capital
as the
32
stock
regulated and
of the
respectively/ we can
Defining K^
write for
the same
price/
(30)
M(K.^;r)
ThI
Impl
s
i
es
that
(31)
MKwrt
For this to he true,
has
to he greater
Kt^
.
\!e
not so.
the capital
stock of the regulated firm
than that for the unregulated firm i.e.,
can prove this hy contradiction.
I.e.
the optimum
capital
firm exceeds that of the regulated
the prices specified are equal.
the
last
greater
state of
for
tlie
nature
It
where
unregul .ited firm
stock of
firm or,
Suppose this
K^t
is
the unregulated
K^
>K/'
.
Since
follows that the index of
demand is
than
for
satisfied
the
Is
regulated
PACE 33
firm.
M(K^
,
p)>M(K,., p)
quantity
producod
I.e.
maximum
?;reater.
I.e.
QCK
^^
,
p) >n.(
sinco
,
by
as
t^
V7e
for f><-fUK^,p).
,
Since
Also,
the
firm
Is
Since
>0.
o",
For states of nature where demand
Q(K^,p)
>0.
unregulated
the
,p)
K^^
<2ilX^i:>
>0,
fully satisfied,
Is
^^^ <0,
^^^
q<
^T^Cc^/lv)^
(n(t's^,,p),K^).
can wr te
I
(32)
/UX^7-..'<-)]^
/^/['ll^>cokj]dx)
^
•^(•^or)
^^(n /K^)
since,
(310
Rut,
'
.N
^.,„r)
.
\^f^[t.i^(<^>V)^^^)\d-0
I
?/j:o( K^
'^^'^'r)
N
,
p),
K;C'.-^,r^
K^).
We con write
.
^
PAGE
,
3lj
,
MCK,,f)
^
OK
since
Hence,
^,
(
^,
;>u.
(a (K,p),K))>0.
[XJ^^ A'JJ
'-^
^
j/^,
[
X.(c^(Ka,p\Kjcix3
j
J
This
contradicts equation
conditions for the two firms.
false.
^^(
Hence
Y.^
yK^
^/u(a (K,p),K))>0,
Thus,
the
ex.
,or
It
derived
(31)
Hence the assumption made Is
K^ ^ K^^
.
Given that K^?
K^^
and
,
follows that
post capl tal
-1
ahour
ratio Is at least as
large for the regulated firm as for
all
optimum
from the
the unregulated firm.
In
states of nature.
The regulatory process
In
most cases
Is
asymmetr
I
c
I. c.
2'».
It can
he shown that
( ^. (n( K, p) ,
K)
>n for fairly
^^]^
general production
functions. See Appendix
for
the proofs
in
the
case
of
the
Cobh-Pouglas and
CFS
production
f unct ions
25. This term Is defined and
discussed In detail In Hailey
PAGE 35
productive
the
of
ono
ml sal
regulatory
the
of
(labour)
the constraint celling.
entirely from
aspect
variables
result of
locat on
I
In fact.
mechanism,
A-J
the
certainty context, as well as
In
theorem In
Is
this
causes
the
It
that
the model
efficiency
regulated
the
of
unregulated firm, at the same price.
the static A-J model where
static,
the
discussed here.
The theorem proved above compares the capital
Input
excluHed
Is
firm
stock and
with
that
of
This Is In contrast to
the capl tal
-
1
ahour decisions are
compared at the output the regulated firm elects to produce.
comparison
The
Is
present model, since
output
Is
known only
made with
price
ex.
specified
Is
post
constant.
price held
ex.
In
the
ante , but
.
Conclusion
5,
This paper has presented
In an
uncertain context.
-
model
of
the regulated firm.
attempt was made
An
to
overcome
the Averch- Johnson model
some of the shortcomings of
regulated firm
a
rate-of-return
of the
regulation, problems due to
uncertainty, and the static nature
of the model.
The model
presented here, while more realistic than the Averch- Johnson
version.
Is
still
not
regulation in practice.
(
I
)
pp.
52-r'7.
a
complete
characterization
of
PAGE
behaviour
optimal
The
examined, and contrasted
with
tfiat
firm
Is
unrej^ulated firm
of the
employs the
The model
describe
framework to
preference
repulatod
the
similar conditions.
operating under
state
of
36
uncertainty,
and
attempts to capture the price-setting function of regulators
In a dynamic
Is
The rule followed
context.
the price fixed for
that
rate-of-return,
stock.
It
more "quickly,"
than It can
firm decides
Its
time,
on
adjiist
Investment In
and adjusts "labour," to
nature
state of
capital
that of
order
to answer
rather than numerically,
The
firm.
under study.
was
rogulntinn,
optimal
stock:
the
aheaH of
level,
after the
conditions,
is
the
compared to
Input
post
also examined.
questions,
to
analytically,
assume that the
Investments In
the period
including specific production and
prorramnes.
shown that
'^nse'^
as fuel)
numerical solutions can be developed for
demand conditions,
the firms'
qx_
its
stock
capital
necessary
it v/as
by
Hov/ever,
is
above
tlie
compared, undertook
firms being
It
capital
stock employed by the regulated firm,
the unregulated
the
capital
can adjust
Under those
efficiency of the regulated firm
In
its
its optimal
revealed.
Is
period's
that the firm
to
earning the
"variable" Inputs such
labour (or other
input of
previous
on the
further assumed,
Is
period, when applied
a
results in the firm's
previous period's output
"fair"
by the regulators
on
In
a
dynamic context,
past outcomes of demand
under price
and cost,
the
PAGE
re/^u1ated firm uses an
any
given price
inefficient conblnntlon of Inputs for
Specifically,
level.
"effective," (i.e. the regulators set
unregulated monopolist
what an
employs
a
efficient.
higher level
Is
unregulated firm
post
meant
at
of capital
the same
capi tal -labour ratio
Is
a
when regulation
than
the
is
conHinatlon
price level.
at
Is
price no higher than
v/ould),
the Input
efficient combination.
37
least as
regulated firm
efficient.
used
Also,
large
by
Py
the
the ex
the
PAGE 38
Append?
For
(AD
i<\^'
a
the
At
the Cobb-Douglas
,
when
point
It
x
production function.
X,-
Is
^'^k-^^J'
profitable
just
to
Increase
production.
7-
(A2)
Thus,
the maximum production,
a
(A3)
-.
derivative
The
(a.)
-'k
of the
0.
,
Is
given by,
^
labour-requirements
function
with
respect to capital stock Is given by
(AU)
Substituting
1
aliour-roqui roment
at
Into
(A3)
s
(A'l),
derivative
function with respect
the point of rnaximum production.
(A5)
the
Is
X,(«0.-,r),K). _-(,._r.)'\>"f
to
capital
given by
'^/'-i'
of
the
stock,
J
The quostlon now.
the expression
positive
respect to
(A6)
i
(A5) with
In
or at
whether the sl^n of
Is
least,
respect
to capital
stock/
Differentiating^
non-nep;at ve.
i
Is
K,
with
we have
K,
^-^fctl^ (If]
[..(«..),.)]
since
confirms
which
the derivative of
-c
«-f
$
-^
Cj<4-y-0 /!-):'
^ o
K
I
assumption
the
/.-y
the
In
text,
for
the
Cohb-Douglas case.
case
the
In
general)
(more
of the
CE5
proHuctlon
function, we can write
(A7)
C«^-''.-^L-''j'^J'
^
,
Differentiating
respect to Q
,
the
1
.L-l^-LaK-r]"'''
;;._
abour-requi roments
and setting It equal
maximum profitable production,
in
to p/w,
Q
-
[-1 "
v/I
th
we can write the
this case as
^
(A8)
function
'JL
'
f/uj<
where
C =
[A
J-
Differentiating
the
labour
respect to capital stock,
K,
requirements
function
with
PAHE
(A9)
4.
fL^_-
.
I;0
- a. k-i^-aK-n"^"''^^^^-^-?)
Substituting the maximum output
(AlO)
-
Into (A9), we have
(AB),
In
--^[-fe--]
.
K
\K
b
To see the
sign of
X^~^
at
the
of maximum profitable
point
output, we differentiate (AlC) with respect to
(All)
Thus,
1^ [x.r^a,F)'»-)J
aK
in
production
the
case
of
functions,
labour-requirements
both
the
function,
K
°
-'
Cobb
the
second
at
profitable output. Is non-negative.
the
Douglas
derivative
point
of
and
CES
of
the
maximum
,
Referonces
and Leland L. Johnson.
"Pohavlour of the
1. Averch, Mnrvey
Constraint,"
Anor can
under
Rep:ulatory
Firm
Revi ew
Vol.
No. 5.
EcononI c
52.
(December
1962) pp. 1053-69.
i
.
Elizabeth
2. Bailey,
Regulatory Constraint
Lexington, 1973.
Theory of
Econoni c
Lexington,
E.
.
flass.
:
Bailey, Elizabeth E. and Roger P. Coleman. "The Effect of
an
Averch-Johnson
Model," The
In
Lagged Regulation
Rel
of
EcononI cs and flanagement
Science Vol. 2. tlo. 1. (Spring 1971). pp. 278-92.
3.
1
.
dohn
Malone.
"Resource
C.
E. and
Pel
Regulated
Firm," The
Elizabeth
Ralley,
h.
Allocation
and
Journal
the
o_f.
-^anagement S c
n c p*
1970) pp. 129-'t2.
I
f
<"^
.
1
EconomI cs
Vol. 1, Mo.
and
(Spring
1.
Baumol, IJIllInm vJ. "Reasonable Rules for Rate Regulation:
V/.
MacAvoy,
Plausible Policies for an Imperfect \.'orld," P.
Pqf^mI tor y
ed., ThJI Crisis of
the
Com.ml ssions
N'ew
York: l-J.VJ.
fJorton,
1970.
pp.
187-206.
5.
r>
.
Baumol, V/Illlam J. and Alvln K. Klevorick. "Input Choices
Overview
of
tne
Rate-of -Return
Regulation:
An
Pel
Journal
of
Discussion,"
The
EconomI cs and
f^anagemen t
ci ence
Vol. 1. r.'o. 1. (Autumn 1970) pp. 162-90.
6.
and
1
.S
Ronbrlght,
7.
Pub
l
Ic
Ut
i
1
i
t
ies
University Press,
.
PrI ncl pies
James
C.
Rates
New York: Columbia
of
.
1961.
Davis, Eric. "A Pynamlc Model
of the Regulated Firm with
J o urn a
Price Adjustment Mechanism,"
Th-C ^'^^l
Mana'Tonon t
of
Economi cs
and
in73) pp.
270-82.
Science.
Vol. U.
(Spring
No. 1.
(Spring 1973) pp. 270-82.
8.
a
1
HI
9.
ilDii
19
:
-h
''
i
ntprr's t
n v-s tmrnt
Jac!<.
Pren t ce-!'a
Englewood Cliffs, M.J.
1
el Per,
t
a
1
.
1
,
I
I
:
i
1
1
7 0".
10.
Klevorick,
Alvln,
K.
"The
'Optimal'
Fair
Rate
of
e
t
.
PAGE
Return/" The "o
1
1
>!ot.jr nr^
of '^cnnoni c^
Vol.
2. ^!o.
1
and r^nnnp-.onAn t S c oner'
(Spring 1971) pjp. 122-53.
.
i
Ij2
1.
J
of a -irm Sutijcct to
11. Klevorlck, Alvln K. "The ^.ehavlour
Journal
Stochastic Re.Tulatory Review," The i^el
1
anH
r^ana,":en':^n
Econoni cs
Science Vol. k. No. 1. (Spring 1973) pp.
of
.
57-33.
12. Leiand/ flayne. "'^.er;ul at Ion of Natural Monopolies and the
Journni
Return," The Rel
of
Rate of
Fair
Economl cs and Mana.f^onen t S c e n c
Vol. 5. No. 1 (Spring 197'0 pp. 3-15.
1
.
i
Myers, Stev^art C. "The Application of Finance Theory to
dournal
Puhllc Utility Rate
Cases," The Pell
and
flanaren^nt
Economl rs
of
Science Vol. 3. No. 1. (Spring 1972) pp. 58-97.
13.
.
Stewart
Regulation
flyers,
Ih.
Under
Jour al
of
ri
ManagcTK'nt
1973") "pp."
15.
Simple
"A
C.
and
Science. Vol.
'.'o.
nsl'si
,
i
1.
r>ty, "
anri
/
''
Pehavlour
of Firm
Tho
"ell
(Spring
'
30/1-15.
Sheshl
tlodol
I'ncer ta
Fcono ml rs
"i/elfare
Eytan.
Aspects
-of
a
Regulatory
Am^r cnn
'^cnnomi c
(f'arch 1971) pp. 175-8.
f'ote,"
Constraint:
Review Vol. 61. No. 1.
i
.
Incentives Harold '1. "Pair Rate of Return and
16. IJeIn,
A.
Considerations," with comments hy Carl
Some General
Trehing,
ed
R.
Hughes
in
H.,M.
Conrad
and William
Performance U nder Regii a t on
'^ast
Lansing,: Michigan State University, 1968.
1
i
.
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