Marginal Analysis, Multi-Plant Firms, and Business Practice: An Example
Author(s): Fred M. Westfield
Source: The Quarterly Journal of Economics, Vol. 69, No. 2 (May, 1955), pp. 253-268
Published by: Oxford University Press
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MARGINAL ANALYSIS, MULTI-PLANT FIRMS,
AND BUSINESS PRACTICE: AN EXAMPLE*
By FRED M. WESTFIELD
I. Introduction,
253. - II. Allocationof outputamongplants,neglecting
transmission
losses,254.- III. Practicalsolutionof this problem,258.- IV.
losses, 261.- V.
Allocationof output, makingallowancesfor transmission
losses,265.- VI. ConPracticalsolution,makingallowancesfortransmission
clusion,268.
I
Theoreticaleconomistsare oftencriticizedforfailingto demonstrate the practical value of theirresearch. Economic literatureis
almost completelybarrenof expositionsrelatinghow even some of
the most basic principlesof economiclogic findactual use in specific
businesssituations. There existsin facta widespreadbeliefthat the
techniquesand resultsof traditionaleconomictheoryhave little or
nothingto contributeto the practical solutionof the businessman's
concreteproblems. It seems appropriate,therefore,to report an
interesting
exampleofhowthe generalprinciplesofmarginalanalysis,
so fundamentalin theoreticaldiscussionsof resourceallocation,are
actually profitablyapplied by men of affairsin the electricpower
generatingindustryfaced witha complexallocationproblem.
Typical of the modernelectricpower generatingindustryare
large integratedcompaniesthat operateseveral geographicallyseparated generatingplants and service customersover a wide region
frequentlyencompassingparts of severalstates. The plants of such
a systemare interconnected
by a complexweb of transmissionlines
so that electricitycan be "transported"fromany plant to every
part of the network.
Over a twenty-four
hour period the demand forelectricityfluctuatessharply. Periodsduringwhichdemandis less than40 per cent
of peak demand are not uncommonfor even the largest systems.
These demand patternsvary somewhatfromday to day. In the
shortrun the numberand types of available plants are fixed,the
price structureis given,and the electricitydemandedmustbe delivexpensesare
ered to consumersregardlessof whetherout-of-pocket
covered by revenues. From the economist'sas well as fromthe
* I am indebtedto ProfessorPaul A. Samuelsonforfirststimulating
my
interestin thisinquiryand forsubsequenthelpand encouragement.
253
254
QUARTERLY JOURNAL OF ECONOMICS
operator'spoint of view, the importantshort-rundecisionis how to
allocate the demandedoutputamongthe alternativeplant capacities
so as to minimizecosts. The stakes are high. Even a small percentagesaving in costs averaging,say, $20 per hour amountsto an
annual saving of about $175,000.
Untilrecentlylosses ofelectricityduringtransmission(transport
costs) were generallyignoredin the solutionof the problem,except
in isolated cases where they were obviouslygreat. During World
War II, however,practical methods of determiningthe losses for
routineoperatingpurposesweredevelopedby electricalengineers;and
in 1951 the AmericanGas and ElectricCompany,a systemoperating
over fortyinterconnected
plants and servingcustomersin some 2300
communitiesof seven states,was pioneeringin using these loss data
to determineleast costproductionschedulesfortheirplants. Taking
explicit cognizance of these losses, equivalent to abandoning the
economist'sfamiliar"no-transport-costs"
assumption,leads to estimated additional cost savings of close to $150,000 per year in the
operationof this system.
Paradoxically,engineers,not economists,were called upon to
analyzeand solve thisresourceallocationproblem. They workedout
the correctsolutions long known and understoodby economists.
Showingno familiaritywith economicliterature,they "discovered"
marginalanalysisaftersome stumblingand thenproceededto apply
its principles. Theirtheoreticalexpositionslack the rigorand sophistication to which economistsare accustomed; but theirtechniques
and devices formakingthe discoveredoptimalsolutionsoperational
withinthe firmsof the industryexhibitadroitnessand expediency
to whicheconomistsare unaccustomed.
We shall report,in turn, the solutions of the problem,first
neglectingtransmission
losses,thenintroducingthemexplicitly.For
each case therewillbe developed: (i) the economist'stheoreticalsolution which corresponds,except for detail, to the engineer'sconclusions, and (ii) the industry'smethods of utilizingthe theoretical
results.
II
How a multi-plantfirmmust allocate output among plants,in
the absence of transportlosses and transportcosts,so as to minimize
costswas carefullydevelopedsomeyearsago in an articleby Professor
Patinkinand a commentthereonby ProfessorLeontief.' They con1. D. Patinkin,"Multi-PlantFirms,Cartels,and Competition,"
thisJournal,
LXI (Feb. 1947), 173-205. W. Leontief,"Multiple-PlantFirms:Comment,"
MULTI-PLANT FIRMS AND BUSINESS PRACTICE
255
sidered a firmwith n plants, each with a total cost curve of the
generalform
ci(xi) + Fi,
Ci=
whereCi are the totalcostsofplanti, ci(xi) its variablecostsas a functionofits outputxi,and Fi its fixedcosts. The total costsofthe firm
(1)
n
z
i =1
Ci+
G = C,
whereG are fixedexpensesnot attributableto the operationsof any
oneplant,mustbe a minimumforeach leveloftotaloutputdemanded.
This is one of those constrainedminimumproblemsso familiarto
economists.2The constraintis simply
(2)
n
z
i =1
Xi-D,
D beingthe total outputdemanded.
Making the customarycontinuityassumptionsand employing
the techniquesof the calculus,one can deduce the desiredtheoretical'
requirementsfor minimumcosts. The firstorder conditionsfor a
minimumtell us that if more than one of the firm'splants are in
operation,each must be made to produce at a rate such that its
marginalcosts are equal to the marginalcosts of everyotherplant
in operation. And a plant is to stand idle onlyif its marginalcosts
when idle are higherthan those of the plants in operation. These
conditionsmustbe satisfiedforeverylevel of the firm'stotal output.
It is usefulforwhat followsto summarizethese resultsby the
expression
(3)
Ci'
X
(i = IY22... ,n),
whereC,' stands for the marginalcosts of the ith plant, and X is a
Langrangeanmultiplierwhosevalue is determinedby the amountof
thisJournal,LXI (Aug. 1947), 650-51. How a monopolistshouldallocate his
outputamongseveralseparatedmarketsin the absenceof transportcosts- a
similarproblem- was carefully
workedout byT. 0. Yntema,"The Influence
of Dumpingon MonopolyPrice," Journalof PoliticalEconomy(Dec. 1928),
XXXVI, 686-98. Almosteverymoderneconomicstextmightbe citedforthe
solutionofproblemsthathave muchin commonwiththatunderdiscussionhere.
2. Cf., forexample,P. A. Samuelson,Foundationsof EconomicAnalysis,
pp. 57 ff.
QUARTERLY JOURNAL OF ECONOMICS
256
total output demanded. It is to be understoodthat the equality
signsmustbe replacedwitha "greaterthan" signforidle plants.
In general,these conditionsare not enough. The second order
conditionsrequirethat (i) not more than one plant with "forwardfalling"marginalcosts should ever be in operation. Further,(ii) if
one of the severalplants in operationis producingon a fallingpart
ofits marginalcost curve,then,at theseoutputs,the rate of decrease
of its marginalcost curve must be smallerin absolute value than
the rate of increaseof the horizontallysummedmarginalcost curves
- a combined marginal cost curve- of all the other plants not
standingidle.3
The mathematicallymindedreaderwill note that these results
are readilyobtained by discoveringthe conditionsunder whichthe
Lagrangeanexpression
C - (xi -D)
subject to the constraint,takes on a relativeminimum. By setting
the firstderivativeswithrespectto each xi equal to zero one obtains
the set of equations (3). Boundary minima will introduce the
inequalities.
The second orderconditionsrequirethat the expression
ki
H ciff 2
(4)
i=1
i=1
1l
i
(k = 2,3, . .. ., m < n),
, > 0
whereCi" refersto the slope of the marginalcost curve forplant i
evaluated at the optimumoutput, be satisfiedfor all k, regardless
how we numberthe m (< n) plants in operation.4 Part (i) of the
second orderconditionsstated in words above is deduced fromthe
3. The next two paragraphsmay be omittedwithoutloss of continuity
in themathematical
derivationofthe results.
by thosenotinterested
4. The principalminorsofthe determinant
A Ci"ij
1i
0ij
foritrSj
ofones,
1j 0
10j= columnand row,respectively,
as well as the determinant
itselfmustbe negative(cf.Samuelson,op. cit.,pp.
362 ff.). Evaluating A by the Laplace Development:A = -2
m
i=1
Aii, where
each Ass is a nonvanishing
cofactorof an element(i.e., of a diagonalelement)
in the determinant
lCii''iii, and notingthat
C11 C2,
_if
A..~~~~~~C
,,
.. **W
A
.
..
***E
{
,,
MULTI-PLANT FIRMS AND BUSINESS PRACTICE
257
tact that this expressionmust hold for k 2; part (ii), fromthe
fact that it must hold fork = m.5
There may be several allocationssatisfyingour conditions. To
determinewhich one of these is "best," it is necessaryto compare
total variable costs at the variousallocationsand selectthe one with
the lowest. The optimumallocationsare, of course,independentof
all fixedcosts.
(I
X
FIGURE
I
The outputofplant1 (xi) is readofffromleftto right,thatofplant2
(x2)
fromright to left. Total output to be delivered is representedby D, equal in
the absence of transportlosses to the sum of xi and x2. The output allo cation
among plants correspondingto point A, where the marginal costs of the two
plants are equal (Cl' = C2' = X) is the least cost allocation. The combined
variable costs forthe two plants, representedby the sum of the areas under their
respective marginal cost curves, is seen to be greater for every other allocation.
it is readily seen that the expression (4) is a statement of the conventional
secondary conditionsfora constrainedminimum.
5. If, say, Ci" < O. then in (4) II Ci" < 0; but I
i=1
i =1i
i
A < 0 must then be
satisfiedfor k=m, which assures that it will'also be satisfiedfor all other (lower)
k=m
1
-i, which
k. This condition on the summation may be written 7, 11
-a, < impliesI
- < |
>C".
k|Hencem
But theleft-hand
sideofthis
2 Ci
inequality is the slope of the horizontal sum of the risingmarginal cost curves of
theplantsin operation;the right-hand
side,the absolutevalue of the slope of
the fallingmarginalcostcurve.
258
QUARTERLY JOURNAL OF ECONOMICS
If a firmhas onlytwo plants it is simpleto show the theoretical
conclusionson a familiardiagram(Figure I). Experimentationwith
different
shaped marginalcost curveswill convincethe readerof the
validityof all the conditionsforminimumcosts that we have stated.
III
So much for the theoreticaleconomics. How is the problem
solved in practice?
In the productionof electricpower, the output at each plant
varies, in the short run, primarilywith the energy (heat) input.6
These relationshipsbetweenthe amount of heat energyprovidedby
maximumamount of electricenergy
the fuel and the corresponding
producedby the equipmentin a plant,the productionfunctions,are
obtained fromengineeringdata. With the technologicalcharacteristicsand the priceoffuelforeach plant known,(short-run)marginal
cost functionsare readilycalculated.7 They may be writtenas
dxi/dvi
where wi is the price of fuelin plant i and dxi/dviits marginalproductivity. The unitsof vtare stated in millionsof B.T.U. per hour;
of wi in dollars per millionB.T.U.; and of xi in megawatts. The
B.T.U. contentand thereforethe effective
price of fuelare obtained
fromroutineanalyses.
In orderto facilitatethe actual job of allocatingthe demanded
electricityor "load" amongplantsin conformancewith ourtheoretical principles,a Station-LoadingSliderule was developed.8 This
mechanicaldevicepermitsa centrallylocated dispatcher,in constant
communicationwith the plants, to calculate quickly the optimum
allocationsforeach demandsituation.
The essentialsofthe sliderule,shownschematicallyin FigureII,
are two unmovable scales, the "marginalfuel rate scale" and the
"relative fuel price scale," and as many movable "plant output
slides" as thereare plantsin the system- one foreach plant. Marginal-fuelratesare theadditionalamountsoffuelrequired(in millions
of B.T.U.) in a plant to produceextraunitsof output (in megawatt6. In thispaperwe shall ignorethe complications
that may ariseif some
ofthe system'selectricity
is generatedin hydroelectric
plants.
7. M. J. Steinberg
and T. H. Smith,EconomyLoadingofPowerPlantsand
ElectricSystems(New York,1943),pp. 141 ff.
8. Ibid.,pp. 189 ff.
259
MULTI-PLANT FIRMS AND BUSINESS PRACTICE
0
,. a. w
I-
2~
=
0
Cn
135
C
-ll
30 -
0
S 28o
r-c
a.
135
i26-
~~~~
~~~~~110 225
36
3 24-
170
CP2
170100
10
o 355
25
85
a20
10-820
170
C6
225
0120
365
o
4
0
~~~~~~~0220
15503
0
140
265
4- 1730
~
~
0l >5
(I)
9
20120
-
013
14
0~
0
~~~~~~~2
265
0__
180
14
__
7
0~~~~~~~
*(i 10
(1
-
D 40
W9
135
U
10U
0
0
0~~~
75.~~~~~1.0
0
Q
~~~~~~~~~~~~~~~~1
1.20
0~~~~~~~~~~~~~
7j;
70
0
1.10
*
1.00
W-
1.05 .
FIGURE II
Sliderule. MarginalFuel Rates and RelativeFuel
The Station-Loading
scales. Plant 1 is the nume'raire
Prices are on logarithmic
plant. Outputis
demandedat therateof360 megawatts.The optimumallocationsto theplants
are readoffalongtheruler.
QUARTERLY JOURNAL OF ECONOMICS
260
hours). They are reciprocalsof marginalproductivities. Relative
fuelpricesare the pricesof fuelat each plant relativeto the priceof
fuel at the plant with the lowestfuelprice. We may call the latter
plant(s) the numeraire plant(s).
The marginalfuelrate scale is calibratedwiththe logarithmsof
marginalfuel rates (log dv/dx),startingat the bottomwith values
somewhatbelowthoseactuallyto be encounteredand increasingcontinuouslyto some point at the top slightlygreaterthan the largest
value to be encounteredin the system. Relative fuelpricesare also
on a logarithmicscale. This scale beginswith log 1 - the index
and runsalongsidethemarginalfuelrate scale. Each outputslide is
calibratedin such a way that whenthe bottomofthe slide is linedup
withtheindexon thefuelpricescale,its markingsshowthemaximum
output rates in the plant that the slide representscorresponding
to
everymarginalfuelrate.
In using the sliderulethe dispatcherdisplaces each of the plant
outputslides (except that of the num6raireplant) fromthe index to
the appropriatepoint on the relativefuel price scale. Thus, if the
price of fuel at the numeraire plant is 25s?per millionB.T.U. and at
plant 2 the priceis 30/, the bottomof the output slide of plant 2 is
moved to 1.2 on the relativefuelprice scale. To divide the output
among plants so as to "equalize" marginalcost, he merelyhas to
move a "ruler,"crossingthe outputslides horizontally,to that position wherethe sums of the outputsmarked on the slides along the
ruler add up to the total demand- whatever it may be at the
moment. These outputsoftheindividualplantssatisfythefirstorder
conditionsfora minimumwhichwe discussedabove.9
Instead ofequatingmarginalcostsofthe plantsin operation,the
dispatcherusingthis device actually equates
(5a)
log
I
- = log
+ log
I + log W11
w
w1
dx2/dv2
dx1/dV1
W2
....
But thisis identically
(5b)
logC1 = log
W1
2
W1
=.
oftheoneintroduced
by Edgeworth(Papers
9. This slideruleis reminiscent
Relatingto PoliticalEconomy,II, 52-58) to illustrateMangoldt'scontribution
trade theory. Cf. J. Viner,
to the comparativecost doctrinein international
Trade,pp. 458-67.
Studiesin theTheoryofInternational
MULTI-PLANT FIRMS AND BUSINESS PRACTICE
261
A moment'sreflectionwill convincethe readerthat these equations
have the same solutionsas set (3).1 Also, as inspectionof Figure II
willshow,the requiredinequalitieswillhold forall idle plants. Using
logarithmsof marginalcosts is convenientwhen thereis only one
variable factor(input) because the marginalcost functionis simply
shiftedwithoutchange in shape in responseto price changesof the
factor.
Fortunately,the complexitiesintroducedby decreasingmarginal
costs can be ignored. The very nature of plant equipmentinsures
that marginalfuelratesand hencemarginalcostsare increasingfunctionsofoutput. Properuse ofthe sliderulewillsatisfythe firstorder
necessaryconditionsand risingmarginalcost curvesat everyplant
make these conditionsalso sufficient
forminimumcost allocationof
output.
It should be noted that no essentialcomplicationsarise, either
in the theoreticalsolutionor in its application,ifmarginalcost curves
oftheplantsare discontinuous.2In fact,in manysystemssomeplants
are always
(the most modernand favorablylocated fuel-pricewise)
used up to capacity.
IV
The foregoingsolutionto the allocationproblemis optimalonly
ifno variablecostsor losses ofelectricityresultfromtransporting
the
generatedelectricityfromplants to consumers. Typicallythereare
no variable transportcosts as such once the systemis built; but electricalenergyis lost duringthe processof transmission. Such losses
fora system,it was shownas recentlyas twelveyears ago,3can be
expressedin termsof a quadraticform
n
y =Y
(XlYX2y . . . Y Xn)
=; z
i=1
n
j=1
Bijxixj ,
wherey representsthe loss in megawattsand the x's, as previously,
are theoutputsoftheplants. Subsequently,methodsweredeveloped
1. Justas the equilibriumconditionsin consumertheoryare independent
of monotonicstretchings
of the "utilityindex,"so the equilibriumconditions
fora multi-plant
firmare independent
ofsucha "costindex."
2. Since,corresponding
to each marginalfuelrate,themaximumoutputis
markedon theplant'soutputslide,a discontinuity
meansthatscheduledoutput
fortheplantin theneighborhood
ofthediscontinuity
willremainunchangedfor
severaldifferent
values of the marginalfuelrate. Also, cf.Samuelson,op. cit.,
pp. 70 ff.
3. E. E. George,"Intrasystem
Transmission
Losses,"ElectricalEngineering
(AIEE Transactions),
Vol. 62 (March 1943),pp. 153-58.
262
QUARTERLY JOURNAL OF ECONOMICS
forderivingthe loss formulacoefficients,
Bij, by means of network
analyzers- miniaturerepresentations(electricalanalogues) of the
entireintegratedgeneratingand transmissionsystem.4
assumptionsare made in the developA numberof simplifying
ment of the formulaand in the estimationof the coefficients.The
most significantof these,fromthe economist'spoint of view, is that
duringthe normalcourseofdemandvariations,the demandsforelectricityat everypoint in the systemare assumed to vary proportionately. This means that we may thinkof all the electricityactually
sold to many consumersall along the systemas if it were delivered
to a singleconsumerlocated at a fixeddeliverypoint. Should this
assumptionbe unrealisticfora particularsystem,additionalvariables
set of coefficients
may be introducedinto the formula,or a different
may have to be used for each patternof demand variations. But
these complicationswill not concernus in this paper.'
The essence of the theoreticalsolutionwhichmakes allowances
for transmissionlosses was workedout quite recentlyby electrical
engineers. Again the problemis to minimizethe firm'stotal costs,
representedby equation (1), for each level of demanded output.
constraint"(equation (2))
But the simple "conservation-of-product
is no longerappropriate. There are losses whichdepend upon how
each plantis producing.The new constraintbecomes
muchelectricity
(2')
n
z
i=1
xi = D + y(x1,x2 ... Xn).
It simplysays that the total electricitygeneratedby all the plants
togethermustadd up to the total quantitydemandedby consumers
plus the quantitylost in transmission.
Proceedingas before,we can readilydiscoverthe necessaryand
conditionsfor minimumcost allocation.7 The firstorder
sufficient
4. J. B. Ward,J. R. Eaton, H. W. Hale, "Total and IncrementalLosses
Vol. 69 (Part I, 1950),
Networks,"AIEE Transactions,
in PowerTransmission
pp. 626-31.
5. Ibid.,p. 629.
6. E. E. George,H. W. Page, and J. B. Ward,"CoordinationofFuel Cost
Loss by Use of the NetworkAnalyzerto DeterminePlant
and Transmission
Vol. 68 (Part II, 1949),pp. 1152-60.
LoadingSchedules,"AIEE Transactions,
and G. W. Stagg,"Evaluationof Methodsof Co-ordinating
L. K. Kirchmayer
IncrementalFuel Costs and IncrementalTransmissionLosses," AIEE Transactions,Vol. 71 (Part III), 1952,pp. 513-20.
is
underconstraint
to be minimized
7. The Lagrangeanexpression
C - ,(2;x
-
y-
D).
MULTI-PLANT FIRMS AND BUSINESS
PRACTICE
263
conditionscan be summarizedby
(7a)
i(1
Ci'-
-
yj) = 0,
whereysis the marginaltransmissionloss fromplant i (= 9y/axf);
it is the percentageofan extraunit ofpowerproducedat plant i that
is lost duringtransmissionif the output of all otherplants remains
the same. By rewritingthese relationsin the formanalogous to (3)
(7b)
(-
ci,
T =
whereit is again understoodthattheequal signmustbe replacedwith
a "greaterthan" sign if the ith plant is idle, we see, since 1 representsthe marginalpercentageof the powerproducedin plant i
deliveredto the consumer,that the new LagrangeanMultiplier, is
- the marginalc.i.f. cost,
the marginaldeliveredcost of electricity
so to speak.
It is now simpleto state these resultsin words. If more than
one plant is being operated,whateverthe demandedoutputmay be,
each mustbe operatedat a rate such that the marginaldeliveredcost
ofits output(the marginalcost ofthe outputgeneratedby it actually
reachingthe consumer)is equal to the marginaldeliveredcost of the
output of each and everyotherplant in operation. A plant should
be keptidle onlyifthemarginaldeliveredcost ofits outputwhenidle
is greaterthanthemarginaldeliveredcostofthoseactuallyoperating.
The necessary and sufficientsecond order conditionsare not
easily stated in words.8 However,if we know that all the marginal
productioncost curves are risingfunctionsof output,the marginal
to assure us of a regularminimum
conditionsabove are sufficient
solution. As we have indicated,the marginalproductioncost curves
of electricgeneratingplants are by their very nature this type of
function.
Marginal transmissionlosses can also be expressedas marginal
transmission(transport) costs. Rewritingequations (7b) in the
equivalentform
8. Lettingd9y/oxj(xj= yijandusingthenotationofnote4, p. 256,thenecesare thatthebordereddeterminant
saryand sufficient
conditions
forthe minimum
Ci"Sij + Ayij
1i - Yi
0
lj - yj
and all its principalminorsbe less thanzero. As the quadraticformy (theloss
do holdifall Ci"
formula)is intrinsically
positive,it is seenthattheseconditions
are non-negative.
264
QUARTERLY JOURNAL OF ECONOMICS
(7c)
Ci, + gys= q,
the marginaldeliveredcost, q, is seen to be the sum of the marginal
productioncost, Cj', and the marginaltransmissionloss, gyj-the
marginaltransmissionloss "priced" at the marginaldeliveredcost.
Beforeproceedingwiththe explanationof how these resultsare
broughtto bear in actual practice in schedulingproductionof the
generatingplants,it may be helpfulto show our theoreticalsolution
graphicallyfora two-plantfirm. For thispurposea diagramdifferent
fromthat used to illustratethe no-transmission-loss-case
is convenient. In Figure III the electricityproducedby plant 1 is plottedon
the horizontalaxis, that producedby plant 2 on the vertical. The
familyof curves labeled CaCbCC
etc., are what I call Constant
Outlay Curves. For a given rate of total expenditureby the firm,
say Cb, the curvelabeled Cb showsthe maximumrate of outputthat
o
0
K
K
Outputof Plant I
Xi
FIGURE III
The pointsK, L, M, represent
least-costallocationsof demandedoutput
rates De, Df, Da, respectively.No allocationscan be foundforthese demand
situationswhichlie on lowerConstantOutlayCurves.
can be producedin plant 2 foreach specifiedoutput rate of plant 1.
The furthersuch curvesare away fromthe origin,the greateris the
outlay (total expenditure)identifiedwith them. Their slopes are
- C1'/C2'- the ratio of marginalproductioncosts in plant 1 to those
in plant 2; and if we know that marginalproductioncosts.in the
plants are increasingfunctionsof output, we can be sure that the
ConstantOutlay Curves are concave to the origin. The otherset of
curves,D eDf D&, etc., are what we may call Allocation Possibility
Curves. If consumersdemand electricityat the rate, say De, the
MULTI-PLANT FIRMS AND BUSINESS PRACTICE
265
De curverepresentsall the possibleways of dividingtotal production
ofthe firm(greaterbecause oflosses than the rate of demand) among
plants such that in fact the rate De reaches the consumers. The
greaterthe consumerdemand,the furtheraway fromthe originis the
relevantAllocationPossibilityCurve. If thereare transmission
losses,
all of these curves must be convex to the origin. Their slopes are
- (1 - y1)/(l
Y2) - the ratio at the marginof the percentageof
electricityproduced in plant 1 reachingthe consumerto the percentageof electricityproducedin plant 2 reachingthe consumer. If
transmissionlosses are neglected,Yi = Y2 = 0 and the Allocation
PossibilityCurves are negativelysloped 450 lines.
The optimumallocationsare all thosepointson the graphwhere
the AllocationPossibilityCurvestouchbut do not crossthe Constant
Outlay Curves. All otherallocationsforspecifieddemandsituations
representedby otherpointson the AllocationPossibilityCurves will
lie on higherConstant Outlay Curves; hence such allocations cost
more. It is seen that the optimumpointsare preciselythe ones specifiedby the previouslystated marginalconditions(equations (7b)).
V
Attemptsto put the theoreticalresults of this more complex
problem into practical operation without modificationswould be
fraughtwith difficulties.There are n + 1 nonlinearrelations (i.e.,
equations(2') and (7c)) whichwouldhave to be solvedsimultaneously
forthe n plant outputs and for,u,foreach demand situation. The
costsofsuch computationsprimarilybecause ofthe nonlinearities
are
likelyto be verylarge.
In actual practicemarginalproductioncost curvesof generating
plants can be adequately approximatedup to maximumoutput by
straightlines
Ci'
as +
bi xi
ai,bi >
O.
Nordin9showedthisby a carefulstatisticalstudyand engineers'have
independently
cometo thesame conclusion. But substitutionofthese
linear relationsinto equations (7c) will show that this good fortune
does not eliminatethe difficulties.There are still n + 1 equations,
and theyare stillnonlinear.
Two approximatesolutionshave been suggestedby the engineers.
9. "Note on a LightPlant's Cost Curves,"Econometrica
(July1947),Vol.
15, pp. 231-35. The findings
implythata risinglinearfunction
of outputgives
significantly
betterestimatesof marginalcostsin the plantthat Nordinstudied
thanpolynomials
or a constant.
1. George,Page, and Ward,op. cit.,p. 1153. See also thediscussionofthis
paper,pp. 1160-61.
266
QUARTERLY JOURNAL OF ECONOMICS
Both have been verifiedby actual sample calculationsto yieldallocationsextremelyclose to the theoreticaloptimum. The first,a linear
approximation,2
is obtainedby replacingon the leftof each equation
(7c) the variable ,uwitha constanty:
(8a)
CZ + Yyi= i
Making use of the linear estimatesof the marginalcost curvesand
substitutingfor yi the partial derivativesof the loss formulathis
becomes
(8b)
n
ai + bitx+ y(2 2 Bijxj)
=
j=1
In
The new constantlyis a weightedaveragevalue ofmarginaldelivered
costs, and is used to "price" the transmission
losses. Note thatthese
are linear relationsin the unknowns. Even if inequalitiesreplace
some equalitiesforcertainvalues ofdemandedoutput,theallocations
correspondingto each u, hence by equation (2') for each level of
deliveredoutput,may be obtainedwithoutdifficulties
by the use of
modernhigh-speedelectroniccomputingequipment.
The other approximation,now actually used in the American
Gas and Electric Company System,is called the "Penalty Factor
Method."3 Ratherthan "pricing"transmissionlosses at actual marginal deliveredcosts (g), as in the exact solution,or at some average
value of marginaldeliveredcosts (-y),as in the linearapproximation,
thismethodchargesforthemat themarginalproductioncostsofeach
plant in question (Ci'). The equations (7c) thenbecome
(9a)
CZ + Ci'yi = CZ'(l + yi) = 4;
or substitutingforyi,
(9b)
Cil(1 +
n
2 2 Bijxj) = j,
j=1
The term 1 + yj is called the "penalty factor"forplant j.4 These
penalty factorscan be efficiently
calculated for each plant under
numerousoperatingconditionsof the system.5
An advantage of this methodis that it permitscontinueduse of
the Station-LoadingSliderule,which, as will be recalled, does not
requirelinear approximationsor, for that matter,continuityof the
2. Loc. cit.
3. L. K. Kirchmayerand G. H. McDaniel, "TransmissionLosses and
EconomicLoading of Power Systems,"GeneralElectricReview,Vol. 54 (Oct.
1951),pp. 39-46. Also,Kirchmayer
and Stagg,op. cit.
4. The "penaltyfactor"consistsof the firsttwo termsof the expansion
of 1/1 - y in theexactformulation,
equations(7b).
5. The Wall StreetJournal,August12, 1953,p. 18, reportedthe purchase
of a $110,000electroniccomputerby the AmericanGas and ElectricCompany
forcalculating"penaltyfactors."
MULTI-PLANT FIRMS AND BUSINESS PRACTICE
267
marginalproductioncost curves. Taking logarithmsof the set of
equations(9a) and writingit in the formof equations(5a):
(10)
log
-log
+ log Wi + log (1 + y)
dxl/dv1
Wi
+ log W2+ 109(I + Y2)
dx2/dv2
Wi
one sees that the penaltyfactors,once known,play a rolemuchlike
relativefuelpricesand can be handledby the dispatcherin a similar
manner.
He proceedsby firstobtaininga "trialallocationschedule"which
lossesin preciselythe mannerdescribedearlier.
neglectstransmission
The outputslideforeach plantis thenmovedupwardon therelative
to
fuelpricescale by the value of its penaltyfactorcorresponding
thistrialallocation. A new allocationscheduleis read offand each
output slide is now displacedby the change in its penaltyfactor
fromthechangein allocations. This cycleis repeateduntil
resulting
foreveryplantthe penaltyfactorsused on thefuelpricescale correspond to thosepenaltyfactorscalled forby the allocationschedule
obtainedwith theiruse. This iterativeprocessconvergesrapidly
becausepenaltyfactorsare generallysmallcomparedto relativefuel
allocationson an hour-to-hour
prices. Moreover,in determining
small changesin total demandare usually
basis onlycomparatively
involved. By startingwiththe allocationscheduleof the previous
period ratherthan with the trial scheduleobtainedby neglecting
losses entirely,the changesin penaltyfactorsare even
transmission
smaller.'
Whiletheallocationsobtainedby eitherofthetwoapproximate
methodsdo not vary forpracticalpurposesfromthoseobtainedby
markedlyfrom
solvingtheexactnonlinearequations,theymaydiffer
losses are neglected. Results of
those obtainedwhen transmission
eightsamplecalculations,reproducedin FigureIV, fora simplified
of the AmericanGas and Electric Company's
plant representation
plantsshowthe estimatedadditionalsavings
networkofalmostfifty
to thefirmunderthevariousdemandconditionswhenin determining
losses. It is estimated
allocationscognizanceis takenoftransmission
in
that savingsin productioncosts,or,perhapsmoreappropriately,
(1 + yi) is calculated
6. It is notclearto theauthorwhytheapproximate
as the penaltyfactorratherthanthe exact (1/1 - ye). The lattercouldalso
approximation
be usedon thesliderule.Its use wouldinvolveno mathematical
withtheAmerican
andshouldbe no moredifficult
to calculate. Correspondence
Gas and ElectricServiceCompanyindicatesthatthissystemis now (whilethis
articleis in proof)usingthisexactpenaltyfactormethod.
268
QUARTERLY JOURNAL OF ECONOMICS
transportationcosts will average about $150,000 per year. These
benefitsare apparentlysufficiently
large to justifythe costs of the
additionalcomputations.
50
-
40
-
10
0
6
8
0
C120 -
20~~~~~~~~~~~~~~
0~~~~~~~~~~~~~
C
E 1i0
at
h
v
a
n
t
e
I~~~~~V
0
Fctiiu
V
80
ilutain
1000
1200
1400
1600
DemandedElectricity
in Megawatts
ftetertclcntutfclsia
1800
FIGURE IV
Cost reductionsfromtakingaccountof transmission
losses in model of
AmericanGas and ElectricCompanySystem.
Source:Kirchmayer
and Stagg,op. cit.,p. 519.
VI
Fictitious illustrationsof the theoreticalconstructsof classical
economics abound; authentic applications and tests are rare. In
thesepages we have cometo gripswitha specificproblemin marginal
analysis that has a veryreal counterpartin the day-to-daymanagement of firmsin the electricpowerindustry.
ofthe classicalmodel ofthe
We proceededwithinthe framework
profitmotivatedfirm. Using the hypothesisof cost minimizationwe
wereable to specifyhow such a model firmwould apportionproduction amongthe severalplantsunderits control. Two cases weredistinguished. For each, themodelby any reasonablestandardsproved
itselfnot onlyas a predictivedevice,but also showeditselfof fundamentalimportanceforsolvingthe problemin practice.
FRED M. WESTFIELD.
NORTHWESTERN UNIVERSITY