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Capital Expansion, Rate of Growth, and EmploymentAuthor(s): Evsey D. DomarSource: Econometrica, Vol. 14, No. 2 (Apr., 1946), pp. 137-147Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/1905364 .

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CAPITAL EXPANSION, RATE OF GROWTH,AND EMPLOYMENT'By EVSEYD. DOMAR

I. INTRODUCTIONThispaperdealswith problem hat sbotholdand new-therela-tion etweenapital ccumulationndemployment.neconomicitera-ture thasbeendiscussed number f imes,hemost otable ontribu-tionbelongingo Marx.Morerecently,twasbroughtorthy Keynesand his followers.A thoroughnalysis feconomic spects fcapital ccumulationsa tremendousob.Theonlyway n which heproblemanbeexaminedat all in a short aper ikethis s by solatingt from hegeneral co-nomic tructurend ntroducingnumberf implifyingssumptions.Some ofthem re notentirelyecessary nd,as theargument ro-gresses,hereaderwill eehow hey anbemodifiedrremoved.The followingssumptionsnd definitionshouldbe noted t theoutset: a) theres a constant eneral rice evel; b) nolagsarepres-ent; c) savingsnd nvestmenteferothe ncomef he ameperiod;(d) both renet, .e., over ndabove depreciation;e) depreciationsmeasured ot nrespect ohistoricalosts, ut to the cost ofreplace-mentof thedepreciatedsset by another ne of the ameproductivecapacity;2f)productiveapacity fan assetorofthe whole conomyis a measurable oncept.The ast assumption,nwhiche) alsodepends,s notentirelyafe.Whether certain ieceofcapital quipmentrthewhole conomysconsidered,heir roductiveapacities epend otonly nphysical ndtechnicalactors, ut on the whole nterplayf economicndinstitu-tionalforces,uch as distributionf ncome, onsumers'references,1This is a summary f a paper presented efore jointsessionof theEcono-metric ociety nd the American tatisticalAssociation nCleveland onJanuary24, 1946. It containsthe logicalessenceofthe argumentwithrelatively ittleeconomicdetail. hopeto develop the atter n a separate paperto be publishedin one of theother conomic ournals.Many thanks forhelp and criticism o to my fellowmembers fthe "LittleSeminar": Paul Baran, SvendLaursen,Lloyd A. Metzler,RichardA. Musgrave,Mary S. Painter,Melvin W. Reder,Tibor de Scitovszky,Alfred herrard,MaryWiseSmelker,Merlin melker, nd mostof all to James . Duesenberry.2 If the originalmachineworth 1,000 and producing 00 units s replacedbyanother one worth lso $1,000, but producing120 units,only $833.33 will beregarded s replacement, nd the remaining166.67 as new nvestment. simi-lar correctionsmade when henew machine osts moreor essthan theoriginalone. The treatment f depreciation, articularlywhenaccompaniedby sharptechnological nd pricechanges,presents n extremely ifficultroblem. t isquite possiblethat our approach, whileconvenient orpresentpurposes,maygiveriseto seriousdifficultiesn the future.

137

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138 EVSEY D. DOMARwagerates, elative rices, tructuref ndustry,nd so on,manyofwhich re in turn ffected y the behavior f the variables nalyzedhere.We shall neverthelessssume ll theseconditionss given ndshallmeanbytheproductiveapacity fan economyor an asset) tstotaloutputwhen ll productiveactorsrefully mployed nderheseconditions.3The economywillbe saidto be in equilibrium hen ts productivecapacity equals tsnationalncomeY. Our first ask s to discoverthe onditionsnderwhich his quilibriuman be maintained,rmoreprecisely,he ate fgrowtht which he conomy ust xpandn ordertoremainna continuoustateoffull mployment.

II. THE PROBLEM OF GROWTHThe idea that thepreservationffullemploymentn a capitalisteconomy equires growingncome oesback inoneform ranother)at leastto Marx. It has been fully ecognizedn numeroustudies(recentlymade in Washingtonnd elsewhere) f the magnitudefgrossnational productneededto maintain ull employment.utthoughhevarious uthors ome o differentumerical esults, heyall approach heir roblem rom hepointof view ofthe sizeofthelaborforce. he laborforceman-hours orked) nd itsproductivityare supposed o increase ccording o one formularanother,nd iffull mployments to be maintained,ationalncomemustgrow tthecombinedate.Forpractical elativelyhort-runurposeshis s agoodmethod,ut ts nalyticalmeritsrenothigh, ecause tpresentsa theoreticallyncompleteystem: ince n increasen laborforce rin its productivitynlyraisesproductive apacity nd doesnot byitself eneratencome similar o that produced y investment),hedemand ide of theequationsmissing. or s thedifficultyisposedofbyMr.Kalecki'smethodccordingowhich apital houldncreaseproportionallyo the ncrease n laborforce nd its productivity.4sMrs.Robinson ellremarked,Therateof ncreasenproductivityflabor s not omethingiven y Nature."6 aborproductivitys notafunctionftechnologicalrogressn theabstract, ut technologicalprogressmbodiedncapitalgoods, nd the mount f apitalgoodsn

3 It should undoubtedly e possibleto work out a more precisedefinitionfproductive apacity,but I prefero leave the matter pen, because a morepre-cise definitions not entirely ecessary n this paperand can be workedout asand whenneeded.4 See his essay,"Three Waysto Full Employment" n The Economics fFullEmployment,xford, 944,p. 47, and alsohis "Full Employment y StimulatingPrivateInvestment?"n Oxford conomic apers,March, 1945,pp. 83-92.5 See herreview f The Economics fFull Employment,conomicJournal,Vol.55, April,1945, p. 79.

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CAPITAL EXPANSION, RATE OF GROWTH, AND EMPLOYMENT 139general. ven withoutechnologicalrogress,apital ccumulationn-creases aborproductivity,t least to a certain oint,both becausemore apital s used perworkmanneach ndustrynd because hereis a shift f abor o industrieshatuse more apital nd can affordopaya higher age. o if abor roductivitysaffectedycapital ccum-ulation,heformulahat he atter hould roceedt the amerate stheformerandas the ncreasen abor orce)snot s helpfuls it ap-pears.The standard eynesianystem oes not provide s with ny toolsfor erivingheequilibriumateofgrowth.heproblemfgrowthsentirelybsent romtbecause f he xplicitssumptionhat mploy-ment sa functionfnationalncome. his ssumptionanbe ustifiedonlyover hort eriods ftime; t willresult n serious rrorsverperiod fa fewyears.Clearly, full-employmentevel of ncome ffive ears go would reate onsiderablenemploymentoday. We hallassume nsteadhatmployments a functionfthe atio fnationaln-come oproductiveapacity.While hisapproach eemsto me to besuperioro thatofKeynes,t should e lookeduponas a second p-proximationather han final olution:t doesnot llow s to separateunusedcapacity nto dlemachines nd idlemen; depending ponvarious ircumstances,he ameratio f ncomeocapacitymayyielddifferentractionsf aborforcemployed.Because nvestmentnthe Keynesian ystems merelyn instru-ment orgeneratingncome, he system oesnottake ntoaccounttheextremelyssential,lementary,ndwell-knownact hat nvest-ment lso increases roductiveapacity.6 hisdual characterf theinvestmentrocessmakes the approach o the equilibrium ate ofgrowth rom he nvestmentcapital)point fviewmore romising:finvestmentoth ncreases roductiveapacity ndgeneratesncome,it provides s with othides f he quation he olution fwhichmayyield herequired ateofgrowth.Let investmentroceed t the rate per year, ndlet theratioofthepotential etvalue added after epreciation),.e.,oftheproduc-tivecapacity f thenewprojects o capital nvestedn them, .e.,toI, be indicatedys.?The net nnualpotentialutput f hese rojectswill hen e equalto s. But theproductiveapacity f hewhole con-6 Whether verydollar nvested ncreasesproductive apaoity s essentiallymatter f definition.t can safelybe said that nvestmentaken as a wholecer-tainly does. To make this statement old in regard o residentialhousing,m-puted rent houldbe included n the national ncome.See also note 19.7The use ofthe word project" does not mply hat nvestments donebythegovernment,rthat t s alwaysmade n newundertakings. amusing project"(in the absence of a better erm)because nvestmentanmean theact of nvest-ing and the result f theact.

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140 EVSEY D. DOMARomymay ncrease y a smaller mount,ecause heoperationf hesenewprojectsmay nvolve transferf abor and other actors) romother lants,whose roductiveapacitys thereforeeduced.8Weshalldefine -,the potentialocial averagenvestmentroductivitysdP

dt(1)The followingharacteristicsfo- hould e noted:1. Its use doesnot mply hatother actors fproductionnd tech-nology emain onstant. n the contrary,ts magnitude epends o averygreat xtent ntechnologicalrogress.t wouldbe more orrectto saythat - eferso an ncreasencapacitywhich ccompaniesatherthanonewhichs causedby nvestment.2. o- eferso the ncreasen potentialapacity.Whether r notthispotentialncrease esultsn a largerncome epends n the behaviorofmoney xpenditures.3. o- s concerned ith he increasen productive apacity fthewhole ociety,nd notwith herateofreturn erivedrexpectedrom

investment.herefore-s not ffectedirectlyy changesndistribu-tion f ncome.4. s is themaximumhat -can attain.The differenceetweenhemwilldepend n the magnitude f therate of nvestmentn theonehand, ndthegrowth fother actors,uch s labor, atural esources,andtechnologicalrogressnthe other.A misdirectionf nvestmentwill lsoproduce differenceetween and o-.We shallmake heheroicssumptionhat anda are constant.From 1) it followshat(2) dP I.dt -It is importanto notethat,with given -, P/dt s a functionfI, and not of dI/dt.Whether I/dt s positive r negative, P/dt salways ositiveolong s a and arepositive.Expression2) showinghe ncreasen productiveapacity s essen-tially hesupply ideofour ystem. n the demand idewe havethemultiplierheory,oofamiliar o needany comment,xceptfor nemphasisntheobvious ut often orgottenact hatwith ny givenmarginalpropensity o save, dY/dt s a function ot of , butofdI/dt.Indicatinghemarginal ropensityo save by , and assumingtto beconstant,9ehave the imple elationshiphat8 I am disregardinghe externaleconomiesand diseconomies f the olderplants due to the operation f the new projects.' Over he period1879-1941 he averagepropensityo save (ratioofnetcapital

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CAPITAL EXPANSION, RATE OF GROWTH, AND EMPLOYMENT 141dY dI 1(3)=__ dt dt a

Let theeconomy e in an equilibriumosition o that10(4) Po = Yo.To retain heequilibriumosition, emusthave

dP dYdt dt

Substituting2) and (3) into 5) we obtain urfundamentalquationdl 1(X6) I - -,dt athesolutionfwhich ives

(7) I = Ioeaat.ao- s theequilibriumateofgrowth.o long s it remainsonstant,themaintenance ffull employmentequiresnvestmento grow t a con-

stant ompound-interestate.If, as a crude stimate, is taken t 12per cent nd a-at some30percent, heequilibriumateofgrowth illbe some3.6 per centperyear.OaThe readerwillnow ee thattheassumptionfconstant and af snotentirelyecessary,nd that hewhole robleman beworked utwith ariable and a.formation o national ncome) was fairly onstant nd approximatelyqual tosome 12 per cent. See Simon Kuznets,NationalProduct ince 1869,NationalBureau ofEconomicResearch mimeographed,945) p. II-89 and theSurvey fCurrent usiness,Vol. 22, May, 1942,and Vol. 24, April,1944. In a problem fcyclical haracter, n assumption f a constant ropensity o save would be verybad. Since weare nterested ere n a secularproblem fcontinuous ull mploy-ment, hisassumption s nottoo dangerous.10The problem an be also worked ut for hecase whenPO YO.

lOa After hispaper was sent to the printer, found very nterestingrticleby E. H. Stern, Capital Requirementsn Progressive conomies,"Economica,Vol. 12, August,1945, pp. 163-171, n which herelationbetweencapital andoutput nthe U. S. during 879-1929 sexpressed in billions fdollars) s capital=3.274 income-3.55. My estimatesgave roughly imilarresults.This wouldplace s around30 per cent, hough his figurehouldbe raised to accountfor heunderutilizationf capitalduring partof thatperiod. t is also not clearhowthe unkingprocess see p. 144) wasreflectednthesefigures.The averagerate ofgrowth f realnational ncomeovertheperiod1879-1941was some 3.3 percent. See Table V, p. 818, andAppendixB, pp. 826-827, nmypaper, The 'Burden' of heDebt and theNational ncome,"American conomicReview, ol. 34, December,1944.

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142 EVSEY D. DOMARIII. THE EFFECTS OF GROWTHOurnextproblems to explorewhathappenswhennvestmentoes

grow t some onstant ercentageater,which, owever,s notneces-sarily qualtothe quilibriumate c-.Itwillbenecessaryo ntroducetwo additional oncepts: veragepropensityo save IIY and theaverageratioof productive apacityo capitalP/K. To simplifyheproblem, eshall ssume hat1. IIY= a, so that verage ropensityo save s equaltomarginal.2. P/K=s, i.e., the ratioofproductiveapacity o capitalfor hewhole conomys equal to that of thenew nvestmentrojects.We shall considerfirst he special simple case a-=, and then themore generalcase when - 8s.11Case 1: a- s. Since = oert, apital, eing hesumof ll net nvest-ments,quals rt 10(8) K =Ko + IoJ ertdt Ko +- (ert 1).O ~~~~r

As tbecomesarge,K will pproachheexpression(9) - ert,rso thatcapitalwill lsogrow t a rate pproaching.As Y= (l/a)Ioert,heratio f ncome o capital s

1- IoertY a(10) . Ko +-(ert-1)rand

y r Clim- = -t(11 goo K aThus so long as r and a remain onstant or change n thesame

proportion) o "deepening" f capital takes place. This, roughlyspeaking, as the ituationn theUnited tatesover he astseventyyears rso prior o thiswar.11 t is also possible hat,owing o capital-savingnventionsn existing lants,a>s. Formallythis case can be excludedby fallingback on the definition fdepreciation iven n note 2. This, however, s not a very happysolution,buttheapproachused in this paper will hardly ffer better ne. I think,however,thata inoursociety s sufficientlyigh omakeof s in a continuous tateoffullemploymentmore n exception han a rule.

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CAPITAL EXPANSION, RATE OF GROWTH, AND EMPLOYMENT 143Substituting = P/s into 11) weobtain

Y r(12) lim-=t-*O P aSSince nthepresentase o=s, Y r(13) lim-=-*t- o P ao

Theexpression r(14) 0 =-maybe calledthecoefficientfutilization. hentheeconomy rowsat theequilibriumate, o thatr ao-,0= 100percent ndproductivecapacitysfully tilized. utas rfalls elow o-, fractionfcapacity(1-0) isgraduallyeft nused.12 hus he ailure f he conomyogrowatthe equiredate reatesnused apacity ndunemployment.Case 2: o-

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144 EVSEY D. DOMARAlso, Y r s(17) lim- = -.-,)K a aand Y r(18) lim-=-,t c P aowhich s exactly hesame resultwe had in (13).The second pproachonsistsntreatingheamounts(s - o)/s notas capital osses utas a special llowance or bsolescence.et nvest-mentwould henhave tobe defined otas I, but as IoIs. Other ym-bolswouldhaveto be redefinedccordingly,nd thewholeproblemcould hen e reworkedut nthe amewayas onpp. 142-143.In a sense the choicebetween hesetwo methods s a matter fbookkeeping;ependingpon the character ftheproblemnhand,oneor theother an beused, hough suspect hat he econdmethodcan easilybecomemisleading.he nature ftheprocesswillbe thesamewhichever ethods used.Thefact sthat, wing oa differencebetween and o-, heconstructionf new nvestmentrojectsmakescertainssets notexcludinghenewprojects hemselves)seless, e-causeunder henewconditionsroughtboutby changesndemand,or a rise nthewagerates, r both, heproductsf hese ssets annotbe sold.'4 s stated n p. 140thedifferenceetween and o- s createdeither y misdirectionfinvestmentrby the ackofbalancebetweenthepropensityo save onthe one hand, nd the growthf abor,dis-covery fnatural esources,nd technologicalrogressn theother.So long smistakesremadeorthis ack ofbalance xists, he unkingprocess s inevitable.From socialpointofview, he unking rocesss notnecessarilyundesirable.n thiscountry, here aving nvolvesittlehardship,tmaybe perfectlyustified. ut it maypresent serious bstacle otheachievementf fullemployment,ecausetheowners f capitalassetsheadedfor he unk pile will ry o avoid the osses. o long sthey onfinehemselveso changesn their ccounting ractices, ospecialconsequences illfollow. ut it is more ikely hattheywilltryto accumulate arger eserves ither y reducingheir wncon-

14 To be strictly rue, the statementn the textwould requireconsiderabledivisibility f capital assets. In the absence of such divisibility,he expression"junking" should not be taken too literally.The fact that these assetsmay still be operatedto some extentorthat theirproducts re sold at lowerprices or that both these conditions xist,does notinvalidateour argument, ecauseo, being xpressednreal terms,will be higherthan t wouldbe ifthe assetswere eft ompletely nused.

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CAPITAL EXPANSION, RATE OF GROWTH, AND EMPLOYMENT 145sumptionrbycharging igher rices or payingowerwages).As aresult, he totalpropensityo savemay rise.This willbe exactly heoppositemeasure romwhat s needed o avoidthe unking rocess,andwill fcourseead togreaterrouble,hough amnotprepared osay to what extent apitalownerswillsucceed n passing n theselosses.In so far s they re able to control ew nvestment,heywilltryto avoid osses ypostponingt.Consequently,herateofgrowth aywellbe depressedelow herequiredo-, nd unused apacitywillde-velop.Ourpresentmodel oesnot llowus to separate nused apacityinto dlecapital nd dlemen, houghmost ikely othwillbe present.'"Because of humanitarianonsiderations,e are more oncerned ithunemployedmen.But unemployedapital s extremelymportant,ecauseits presence nhibitsnewinvestment.'6t presents gravedanger o afull-employmentquilibriumna capitalistociety.

IV. GUARANTEED GROWTH OF INCOMEIn the precedingectionstwasshown hata stateoffull mploy-

ment an be maintainedf nvestmentnd income row t an annualrate o. The question ow rises s towhat xtent he argumentanbereversed:upposencomesguaranteedogrow t the o rate;willthatcall forth ufficientnvestmentogenerate heneeded ncome?We are concernederewith situationwhere pontaneousnvest-ment i.e., nvestmentade nresponseo changesn technique,hiftsin consumers'references,iscoveryfnewresources,tc.) s not uffi-cient, ndthereforecertain mount f nducednvestmentmade nresponse o a rise n income) s alsorequired." o simplifyhe argu-ment, et us assume hat pontaneousnvestments absent ltogether.It should lsobemade clear hattheproblems treated rom theo-retical oint fview,without onsideringhenumerous racticalues-tions hat he ncome uarantee ould aise.Ifan economytarts romn equilibriumosition,n expected isein ncome fYao will equire n investmentqual to Yao/s.As before,two caseshave to be considered.15 The presence f unemployedmenmay be obscuredby inefficienttilizationof abor,as in agriculture.16 It is true hata given apitalownermayoftenhave a hardtimedistinguish-ing between apital idle becauseofa

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146 EVSEY D. DOMAR1. If a is equal or reasonably lose to s, the resultingmount finvestmentfYa will qual the volume f savings hatwillbemadeat that evelof ncome, nd equilibriumillbe maintained.18husmereguarantee fa rise n income if taken seriously y the investors)willactually enerate noughnvestmentnd income o makethe uaranteegood without ecessarilyesortingoa governmenteficit.2. Ifu-s appreciablyelow , nvestment illprobably all hort fsavingsndequilibriumillbedestroyed.hedifficultyrises ecausea full-employmentateof nvestmentn thefaceofa u

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CAPITAL EXPANSION, RATE OF GROWTH, AND EMPLOYMENT 147I am not preparedo saywhether e already reorshall soon befacedwith serious ifferenceetween and o-, hough doubtthat

itwas an importantroblemnthepast,except erhaps or heshortboomyears.Myownguess sthatwe shallbemore oncerned ith hedisparityetweeno-and r, that swith hefailuref ncome o growat therequiredate.If,however,hedifferenceetween-ands becomes eriousnd n-hibitsnvestment,r if the unking rocess roceeds t a faster atethan s deemed ocially esirable, he ocietywillhaveat itsdisposaltwomethods otmutually xclusive: 1) thereduction f thepro-pensityosave,or 2) the peeding p of echnologicalrogress.hopethatthemain mphasiswillbe placedonthe atter.Thispaperattempted o analyze he relation etweennvestment,rate ofgrowth,nd employment.he analysiswas carried ut on avery bstractndsimplifiedevel-a procedure hichmaybe ustifiedat thebeginningf n investigation,ut whichmust e correctedateron. In general, here s no sucha thing s an absolutelyoodorbadassumption: hatmaybe safe n onekindof a problemanbecomefatal nanother. fthe everal ssumptions adehere, hatregardingdepreciations likely o causethegreatest ifficulties,ut t is bynomeans heonlyone. hope to develop hewhole ubject urthert alaterdate.The central heme fthepaperwas therateofgrowth, conceptwhichhas beenlittleused in economic heory, nd inwhich putmuchfaith s an extremelysefulnstrumentfeconomic nalysis.One doesnothaveto be a Keynesian o believe hatemploymentssomehowependentn nationalncome,ndthatnationalncome assomethingo do withnvestment.ut as soon s investmentomes n,growthannot e left ut,becausefor n individual irmnvestmentmaymeanmoreapital nd ess abor, ut for heeconomy s a whole(as a generalase)investmenteansmore apital nd not ess abor. fboth reto beprofitablymployed,growthf ncomemust akeplace.Washington,. C.