Özçam, Ahme
A icle
The economic ma ke ou comes and income dis ibu ion
when capi al is no homogeneous: Limi s o echnology
Jou nal o Indus y-Uni e si y Collabo a ion (JIUC)
P o ided in Coope a ion wi h:
Resea ch Cen e o China Indus y-Uni e si y-Resea ch Ins i u e Collabo a ion, Wuhan Uni e si y
Sugges ed Ci a ion: Özçam, Ahme (2019) : The economic ma ke ou comes and income dis ibu ion
when capi al is no homogeneous: Limi s o echnology, Jou nal o Indus y-Uni e si y Collabo a ion
(JIUC), ISSN 2631-357X, Eme ald, Leeds, Vol. 1, Iss. 1, pp. 38-56,
h ps://doi.o g/10.1108/JIUC-03-2019-004
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/319845
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The economic ma ke ou comes
and income dis ibu ion when
capi al is no homogeneous
Limi s o echnology
Ahme Özçam
Depa men o Economics, Yedi epe Uni e si y, Is anbul, Tu key
Abs ac
Pu pose –An agg ega e p oduc ion unc ion has been used in mac oeconomic analysis o a long ime, e en
hough i seems ha i is concep ually con using and p oblema ic. The pu pose o his pape is o a gue ha
he measu emen p oblem ela ed o he he e ogenous capi al inpu ha exis s in mac oeconomics is also
ele an o mic oeconomic ma ke si ua ions.
Design/me hodology/app oach –The au ho cons uc ed a mic oeconomic ma ke model o add ess bo h
he p oblems o he measu emen o he physical capi al and o subs i u abili y be ween labo and capi al in
he sho un using wo ypes o echnologies: labo neu al and labo educing. The au ho p oposed ha
labo and physical capi al inpu s a e complemen a y in he sho un and can become subs i u es only in he
long un when he echnology ad ances.
Findings –The au ho ound ha e en i he echnology imp o es a a as a e o e ime, he e a e hen
diminishing e u ns o p o i s o echnology and an uppe limi o p o i s. Mo eo e , he au ho showed ha
unde he labo - educing echnology, labo class ea ns mo e ini ially as echnology imp o es, bu hei
incomes s a declining a e some h eshold le el o passage o ime.
O iginali y/ alue –The au ho cau ioned he applied esea che ha he es ima ed labo and capi al
coe icien s o gene alized Cobb–Douglas and cons an elas ici y o subs i u ion o ypes o p oduc ion
unc ions could no be in e p e ed as pa ial elas ici ies o labo and capi al i in eali y he da a come om
ixed-p opo ions ypes o p ocesses.
Keywo ds High echnology, Diminishing a e o p o i s o echnology,
Fixed-p opo ions p oduc ion echnology, Speed as an economic esou ce,
Cobb–Douglas p oduc ion unc ions
Pape ype Resea ch pape
1. In oduc ion
The e exis s a as economics li e a u e on he agg ega ion o p oduc ion unc ions. An
agg ega e p oduc ion unc ion, like GDP ¼F(K,L), links agg ega e inpu s in o agg ega e
ou pu and is belie ed o desc ibe he echnology a he agg ega e le el. I is ypically
a gued ha he p oduc ion unc ion in he heo e ical mac o model is a mic oeconomic
p oduc ion unc ion. Howe e , he empi ical e idence and examples a e om
mac oeconomics. Such a cons uc has been used in mac oeconomic analysis o a long
ime, bu i seems ha i is concep ually con using and p oblema ic, since he echnical
condi ions which speci y a well-beha ed agg ega e p oduc ion unc ion ha can be de i ed
om mic o p oduc ion unc ions a e so s ingen ha i is di icul o belie e ha in applied
wo k one can ha e hose equi emen s me . Mo eo e , a ious a emp s by many au ho s o
speci y hose condi ions ha e each had hei own se o assump ions (Leon ie , 1947;
Go man, 1953; Fishe , 1969).
Jou nal o Indus y-Uni e si y
Collabo a ion
Vol. 1 No. 1, 2019
pp. 38-56
Eme ald Publishing Limi ed
2631-357X
DOI 10.1108/JIUC-03-2019-004
Recei ed 25 Decembe 2018
Re ised 6 Janua y 2019
Accep ed 10 Janua y 2019
The cu en issue and ull ex a chi e o his jou nal is a ailable on Eme ald Insigh a :
www.eme aldinsigh .com/2631-357X.h m
© Ahme Özçam. Published in Jou nal o Indus y-Uni e si y Collabo a ion. Published by Eme ald
Publishing Limi ed. This a icle is published unde he C ea i e Commons A ibu ion (CC BY 4.0) licence.
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Wi h he ise o he new neoclassical endogenous g ow h li e a u e in he 1980s, a
cu iosi y o es he old and new heo ies o g ow h and p oduc i i y a oused, especially
when la ge da a se s became a ailable. Howe e , hese new g ow h models depended
hea ily on he old neoclassical agg ega e mac o p oduc ion unc ion which was hough o
be ob ained by adding mic o p oduc ion unc ions. Since new g ow h models li e a u e did
no seem o be bo he ed wi h he old agg ega ion p oblem, i.e. wi hou p ope agg ega ion,
we canno s udy he p ope ies o an agg ega e p oduc ion unc ion (Felipe and Fishe ,
2003). This is known as he agg ega ion p oblem. Temple (1999) a gued ha he p oduc ion
unc ion is he leas sa is ac o y elemen o mac oeconomics, ye many economis s conside
his inep cons uc as a undamen al basis o unde s anding na ional ou pu s and g ow h
a es. So, i seems ha he agg ega e p oduc ion unc ion does no ha e a sound heo e ical
basis and he e o e is decei ing.
Why do economis s keep using agg ega e p oduc ion unc ions in spi e o hese
di icul ies? Since hese p oblems a e qui e incon enien o an impo an pa o
neoclassical mac oeconomics, he p o ession has simply chosen o igno e hem and jus i y
i s use by eso ing o some s e eo ypes. Fo example, since agg ega e p oduc ion unc ions
appea o gi e empi ically sa is ac o y esul s, why should no hey be used? O , he e is no
o he choice, and agg ega e p oduc ion unc ions a e seen as pa ables.
In he economics li e a u e, capi al has adi ionally been hough as a und o esou ces
o a se o p oduc i e ac o s (he e ogeneous machines) speci ied in physical uni s. O hese
wo concep s, whe eas und o money could easily be measu ed, eal capi al was no .
Robinson (1953–1954) asked he cen al ques ion ha igge ed he so-called Camb idge
capi al con o e sies: In wha uni is “capi al” o be measu ed? Since capi al goods we e a
se ies o he e ogeneous commodi ies (in es men goods), each ha ing a pa icula ype o
echnical cha ac e is ic embedded in hem, i was no possible o exp ess he s ock o capi al
goods as a homogeneous physical en i y. The e o e, only hei alues could be agg ega ed.
Such a alue agg ega e, howe e , could no be independen o he a e o p o i and he e o e
o income dis ibu ion. So, i was di icul o concei e ime se ies o quan i ies o capi al
goods o be combined in o a single numbe ha may be used as he amoun o eal capi al in
he economy as a whole (Felipe and Fishe , 2003).
The agg ega e p oduc ion unc ion includes wo ypes o agg ega ion. The i s has o do
wi h he meaning o his ic i ious cons uc ob ained by agg ega ing he he e ogeneous
inpu s and ou pu s (di e en ypes o labo , capi al and ou pu in o one labo , one capi al
and one ou pu ). Al hough labo is no a homogeneous inpu , i can na u ally be measu ed in
man hou s o wo k[1]. The same is ue o land: so many ac es o land o a gi en quali y.
Since hese a e na u al uni s, he ma ginal p oduc s o land and labo can be de ined
independen ly o he equilib ium ac o p ices. Howe e , he measu emen o he capi al
inpu is p oblema ic. The s a is ics o capi al used a e in dolla s and no ma e how hey a e
de la ed o con e hem in o cons an dolla s, hey con inue being a mone a y alue (Felipe
and Fishe , 2003). The o he agg ega ion ype is summing o e i ms and in ce ain con ex s
o e indus ies in a coun y o he same indus y o e di e en na ions.
Fishe (1965) ema ked ha he assump ion ha echnology is embodied in capi al, i.e. ha
i is i m speci ic, causes di icul ies wi h espec o he exis ence o a capi al agg ega e. He
a gued ha unde ce ain echnical condi ions, capi al-augmen ing echnical di e ences among
i ms, i.e. ha he embodimen o new echnology can be w i en as he p oduc o he amoun
o capi al and a coe icien , seem o be he only si ua ion whe e a capi al agg ega e exis s. Unde
hese ci cums ances, a uni o one ype o new capi al equipmen is he exac duplica e o a ixed
numbe o old capi al equipmen . The agg ega e s ock o capi al can be cons uc ed wi h capi al
measu ed in e iciency uni s (Fishe , 1965). This las commen o Fishe ha capi al being
measu ed in e iciency u ns ou wha we exac ly ha e in mind o do he e. In his pape , we will
in oduce he concep o “speed”as a measu emen o he capaci y o eal physical capi al.
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In his pape , we will i s a gue ha he measu emen p oblem ela ed o he capi al
inpu ha exis s in mac oeconomics is also p esen in mic oeconomics. Simila o he
si ua ion in mac oeconomics, assume, o example, ha he capi al in es men s o some
i ms in a mic o ma ke a e made a di e en poin s in ime and ha he p o i s o hese
i ms decline, due o pe haps a le wa d shi in he ma ke demand o he p oduc hey sell.
This is likely o pu a downwa d p essu e on he ini ial pu chasing p ice (cos ) o capi al. So,
his means ha we end up wi h dissimila dolla alues o cos s o in es men made a
di e en pe iods o ime (pe haps a ew yea s apa om each o he ) and i i ually
becomes impossible o measu e he eal quan i y o physical capi al used in he p oduc ions
o hese i ms unless a capi al good p ice index is eadily a ailable[2]. No ice ha in his
example we did no e en pe mi he possibili y o he e ogeneous capi al goods ye ha he
i ms may ha e used o e ime due o a possible echnological ad ance.
Mo eo e , we will u he a gue ha he e is also a p oblem wi h he use o p oduc ion
unc ions in mic oeconomic models which easily allow subs i u ion be ween labo and capi al,
whe e in eali y he ype o echnology may possibly be o ixed-p opo ions and hese
p opo ions may a y among i ms o he da a collec ed o e ime and i ms may be o such
na u e ha he echnology may ha e signi ican ly changed so ha again he assumed easy
subs i u ion be ween capi al and labo may no be so ele an . In applied wo k, i is common
p ac ice in he li e a u e o assume a single ype o p oduc ion unc ion (e.g. Cobb–Douglas
and cons an elas ici y o subs i u ion (CES)) o all he di e en i ms in an indus y a a poin
in ime o o he same i ms o e ime while he ype o echnology may e y well change in
he mean ime. In Sec ion 5, in o de o assess he econome ic implica ions o such a s ingen
assump ion which assumes smoo h subs i u ion be ween labo and capi al, we will i bo h
gene alized Cobb–Douglas and CES p oduc ion unc ions o a hypo he ical da a se consis ing
o ou pu s cons uc ed unde ou di e en se s o ixed-p opo ions p oduc ion unc ions
whe e he capi al/labo a ios di e ( he i s ep esen ing he oldes echnology (mos labo -
in ensi e) and he ou h ep esen ing he newes echnology (mos capi al-in ensi e)). As
expec ed, when we es ima e such da a wi h hese wo p oduc ion unc ions, some da a ou pu
le els p oduced unde di e en echnologies (pe haps also wi h dissimila physical capi al
uni s) will be concei ed as i hey we e ob ained wi h inpu s subs i u ed. Pu i di e en ly,
hese ypical p oduc ion unc ions (Cobb–Douglas and CES) will ea he da a ou pu poin s
ob ained wi h he e ogeneous ypes o capi al amoun s as i hey all lied on he same isoquan ,
i.e. as i he i ms we e able o swi ch om one ixed-p opo ions echnology o ano he
quickly in he sho un. Clea ly, such a subs i u ion can only come abou in he long un, and
no in he sho un when ixed-p opo ions echnologies seem o be mo e app op ia e.
In e es ingly, e en hough he non-linea es ima ion p ocedu e will cap u e he cons an
e u ns o scale (CRS) p ope ies o hese ou di e en ypes o echnologies p ope ly using
he gene alized Cobb–Douglas unc ion, i will no be app op ia e o in e p e he es ima ed
pa ial ou pu elas ici ies o inpu s he usual way, since hese elas ici ies a e 0 wi h
ixed-p opo ions p oduc ion unc ions[3].
In his pape , we will a emp o cons uc a mic oeconomic ma ke model (see Sec ion 3)
o add ess bo h he p oblems o he measu emen o he physical capi al and o
subs i u abili y be ween labo and capi al as discussed abo e. In ou model, he i ms will be
eplacing hei old machine y/equipmen wi h mo e p oduc i e machine y/equipmen
embodying impo an echnological ad ances. The e seems o exis only a e y limi ed
numbe o cases/examples, i any, ha we can enume a e whe e labo and capi al can be
subs i u es in he sho un. Fo example, i he dish-washing machine b eaks down in a
es au an , hen o cou se, an employee in he ki chen can wash he dishes. Howe e ,
assume ha a la ge pan is missing in he ki chen o a es au an . Can labo do he wo k o
such a pan? I seems like ha capi al and labo a e mos likely o be complemen s in he
sho un, and hey can become subs i u es in he long un only i a echnological
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imp o emen occu s. Gi en an app op ia e ans o ma ion on he capaci y o physical
capi al inpu (machine y) ha we will in oduce (speed), we will show how we will be able o
compa e he e ogeneous ypes o capi al as he echnological imp o emen o e ime and he
sense in which he capi al inpu can be subs i u ed o he labo inpu in he long un. We
will dis inguish be ween wo ypes o echnologies: labo neu al and labo educing. Gi en
he as echnological ad ances o oday, i is clea ha he subs i u ion is unidi ec ional in
ou e a: eplacemen o capi al o labo , and no he o he way a ound.
In Sec ion 2, we will b ie ly explain he po k p oduc ion as an example o a simple
p oduc ion p ocess and discuss he ex en o which he subs i u abili y be ween capi al and
labo is possible and when. In Sec ion 3, we will in oduce wo ypes o echnologies: labo
neu al and labo educing and ou economic esou ce which is “speed.”In he same
sec ion, we will also p esen he sho - un and he long- un ma ke ou comes unde
ixed-p opo ions p oduc ion unc ions. In Sec ion 4, we will discuss he ou pu and
p o i -maximiza ion e ec s o he new echnology. In Sec ion 5, using hypo he ical da a, we
will show he s a is ical implica ions o es ima ing he echnical p oduc ion pa ame e s o
Cobb–Douglas and CES ypes o p oduc ion unc ion which allow subs i u ion whe e by
con as , he da a a e pu posely cons uc ed unde ixed-p opo ions p oduc ion. Finally,
Sec ion 6 concludes he pape .
2. An example o a simple p oduc ion unc ion whe e labo and capi al a e
hough o be subs i u able: Swine p oduc ion and i s in e p e a ion
We will b ie ly explain he po k p oduc ion as an example o a simple p oduc ion p ocess
and discuss o which ex en he subs i u abili y be ween capi al and labo is possible and
when. Ea ly man, by in es ing oday’s labo in he c ea ion o c ude ools, made
omo ow’s labo easie and/o mo e p oduc i e. Swine con inemen , he p ocess o
subs i u ing capi al o labo , began when pas u es we e i s enced o educe he labo
needed o ha es he po k (Lines, 1986). The p oduc ion sys ems used in po k p oduc ion
can be b oadly desc ibed as:
(1) high capi al/low labo –high-in es men con inemen ;
(2) medium capi al/medium labo –low-in es men con inemen ; and
(3) low capi al/high labo –pas u e sys ems.
In gene al, he p ices o capi al and labo a ec he combina ion o quan i ies o labo and
physical capi al p e e ed by en ep eneu s in any indus y. Howe e , he e may be some
o he ac o s as well:
(1) quan i y and ype (equi y o deb ) o money capi al a ailable;
(2) he cos o capi al (in e es a e and/o oppo uni y cos );
(3) he quan i y, quali y and iming o a ailable labo besides i s cos ;
(4) animal pe o mance ( eed and b eeding e iciency, li e sizes and su i al a es);
(5) p oduc p ices;
(6) di e si ica ion; and
(7) isk a e sion and isk-bea ing abili y o he ope a o .
As a as he low- echnology po k p oduc ion is conce ned, he o al cos pe uni o
p oduc ion may no a y g ea ly om one sys em o ano he among he h ee ypes
o combina ions o inpu s conside ed abo e.Thismay henpe mi ope a o s,each
wi h a di e en se o capi al, labo and managemen esou ces, o compe e e ec i ely in
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Swine p oduc ion. The young en ep eneu , gene ally long on labo and sho on money
capi al, can p o i ably build a small uni and educe isk. A he same ime, a highly
skilled well- inanced ope a o can p o i ably cons uc la ge specialized isky
en e p ises. The subs i u abili y o capi al and labo in he Swine indus y may
con inue o pe mi he p o i able co-exis ence o di e se p oduc ion sys ems (Lines, 1986).
Howe e , his almos equali y o p o i s ha may ha e pe sis ed in he example o po k
p oduc ion needs no be he case i a new echnology subs an ially educing he cos o
p oduc ion is in oduced.
The c ucial issue he e is, o cou se, he subs i u abili y o capi al and labo . We will
a gue, in his pape , ha such an oppo uni y is only possible in he long un and is no
eadily a ailable in he sho un. E en in he echnically ela i ely simple p oduc ion
se ing o po k p ocessing, we canno assume ha en ep eneu s can swi ch om a
combina ion o a speci ic ype o physical capi al and labo o ano he combina ion o a
di e en ype o machine y and labo in he sho un, o example, a quick swi ch om
pas u es sys ems o high-in es men con inemen . This impossibili y which cons ain s he
en ep eneu s is e en mo e p onounced in he si ua ion o oday’s high- echnology
indus ial p oduc ion, whe e he de elopmen o au oma ed assembly ope a ions and he
inc eased use o indus ial obo s ha e emendously inc eased he p oduc i e e iciency,
bu wi h emendous e o s o R&D, ingenui y and a a subs an ial cos o in es men
which can only be achie ed gi en enough ime.
E en i we allow enough ime, he e is addi ionally he di icul y o compa ing he
echnologically di e en quan i ies (uni s) o physical capi al. Many au ho s in
mic oeconomics assume ha hese in insically he e ogeneous quan i ies can all be shown
on he e ical axis o a ypical labo -capi al diag am. The quick eachabili y om one inpu
combina ion o labo and capi al o ano he combina ion allows one o de i e he well-known
smoo h downwa d sloping and con ex isoquan s. Howe e , in hese combina ions o labo
and capi al, i s o all, he capi al is no homogeneous. Mo eo e , a di e en sys em
( echnology) may equi e labo wi h di e en ypes o skills/educa ion/ aining, and he e o e
we canno ea labo homogeneous ei he . This is simply a subs i u ion o one ype o labo
wi h ano he ype. Fo example, he younge gene a ions will be mo e ap o lea n and
unde s and he new echnology. So, i seems ha we a e subs i u ing non-homogeneous
capi al uni s and non-homogeneous amoun s o labo wi hin hemsel es, and no labo wi h
capi al. Ne e heless, i is ue ha in gene al when echnologically mo e ad anced equipmen
is used, he e is need o less labo in he long un.
The p ice o capi al inpu can o cou se be measu ed in e ms o mone a y expendi u es
made on machine y pe hou . No e, howe e , ha a change in he p ice o aw ma e ial
which is essen ial o machine y manu ac u ing may a ec he capi al expendi u es. Fo
example, in he case o po k p oduc ion explained abo e, a ise in he p ice o encing may
ende low-capi al pas u e sys ems mo e expensi e, and hence make i look like mo e
capi al-in ensi e han medium-capi al in es men con inemen . This is an example o a
p oblem associa ed wi h measu ing physical capi al inpu in mone a y e ms. Mo eo e ,
agg ega ing such mone a y alues o e ime may no be so meaning ul.
As an example o capi al measu ed in mone a y e ms, A ow e al. (1961), in a sample o
19 coun ies co e ing 24 manu ac u ing indus ies, es ima ed he a es o e u n on capi al
om balance shee s o di e en indus ies. Capi al was measu ed by ne ixed asse s
(including land) plus cash and wo king capi al. All inancial in es men s we e excluded.
To al e u ns o capi al we e aken o be equal o g oss p o i om ope a ions minus
dep ecia ion. Again, he e we ha e he p oblem associa ed wi h capi al measu ed in
mone a y e ms.
We now s a discussing ou ma ke model o ixed-p opo ions p oduc ions wi h
“speed”as a sca ce economic esou ce.
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3. A ma ke model o ixed-p opo ions p oduc ion unc ions wi h speed as
an economic esou ce
In his pape , we dis inguish be ween wo ypes o echnologies: labo neu al and
labo educing.
3.1 The labo -neu al and labo - educing echnologies
We a e in e es ed in he si ua ion whe e he i ms eplace hei old machine y/equipmen
wi h mo e p oduc i e machine y/equipmen embodying impo an echnological ad ances.
Fo example in s anda d assembly lines, wo k assignmen s, numbe s o machines, and
p oduc ion a es a e p og ammed so ha all ope a ions along he line a e compa ible.
Au oma ed assembly lines, on he o he hand, consis en i ely o machines un wi h li le o
no human supe ision. In such con inuous-p ocess indus ies as pe oleum e ining and
chemical manu ac u e and in many mode n au omobile-engine plan s, assembly lines a e
comple ely mechanized and consis almos en i ely o au oma ic, sel - egula ing equipmen .
Some p oduc s, howe e , a e s ill assembled by hand because many componen pa s a e
no easily handled by machines. Ne e heless, he de elopmen o e sa ile machine y and
he inc eased use o indus ial obo s and a i icial in elligence ha e imp o ed he e iciency
o ully au oma ed assembly ope a ions. Expensi e and somewha in lexible, au oma ic
assembly machines a e economical only i hey p oduce a high le el o ou pu wi hin a gi en
pe iod o ime (www.b i annica.com/ echnology/assembly-line). To be success ul, a new
echnology mus no only deli e highe speed o p oduc ion, bu also do so a an accep able
cos (p ice o machine y).
In he ecen pas , he e has been an ex ao dina y dec ease in p ices o pe sonal
compu e s and equally imp essi e ad ance in he compu ing powe and o he asks
pe o med wi h he ypical pe sonal compu e . The demand o wo ke s depended on he
ype o p oduc ion unc ion. I has inc eased o some ypes o wo ke s, while i has
dec eased o o he s. The demand o so wa e compu e p og amme s and compu e -
assembly wo ke s has soa ed apidly. Howe e , he demand o o ice wo ke s has been
mixed (McConnell and B ue, 1999). In some o ices, he highe speed o he pe sonal
compu e s has inc eased he demand o compu e enginee s, since in his case, he
compu e s and he enginee s a e complemen a y inpu s, whe eas in some o he cases, he
i ms subs i u ed compu e s o labo and dec eased hei demand o labo .
In his pape , we dis inguish be ween wo ypes o echnologies: labo neu al and
labo educing. In Table I, we display a nume ical example o a labo -neu al
echnology. Technology A is he i m’s cu en p ocess o p oduc ion. I uses 1 man-hou
o labo and 4 machine-hou s o capi al pe uni o ou pu . Assume ha a new echnology
o p oduc ion, B, is in oduced esul ing in p oduc i e e iciency which uses he same
man-hou s o labo (hence labo neu al) bu uses 3 machine-hou s o capi al pe uni o
ou pu (p ocess inno a ion a he han p oduc inno a ion). Technology C cap u es e en
mo e oppo uni ies in ha i s ill employs 1 man-hou bu only 2 machine-hou s. We
assume ha he capabili y o he physical capi al inpu de e mines he du a ion o he
p oduc ion o one uni o ou pu and he i m adjus s i s demand o labo acco dingly. In
o he wo ds, machine y is he mo e impo an ac o o p oduc ion. Fo example, wi h
he Technology A, he i m pu chases ou machines and uses 16 machine-hou s and
Labo equi emen Capi al equi emen Speed o capi al
Technology A 1 man-hou 4 machine-hou s 0.25
Technology B 1 man-hou 3 machine-hou s 0.333
Technology C 1 man-hou 2 machine-hou s 0.5
Table I.
Labo -neu al
echnologies ( o
p oduce 1 uni
o ou pu )
43
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ou comes
4 man-hou s o labo o p oduce 4 uni s o ou pu a he end o he ou h hou , so ha no
inpu s ays idle.
In Table II, we show an example o a labo - educing echnology. Cu en echnology is
again A and uses 1 man-hou o labo and 4 machine-hou s o capi al pe uni o ou pu , as
shown in Table I. Howe e , he new echnology o p oduc ion D now equi es only 0.95
man-hou s, while he physical capi al equi emen s ays he same as be o e (3 machine-hou s
pe ou pu ). Technology E employs e en less o labo , only 0.90 man-hou s.
3.2 The sho un s he long un and speed as an economic esou ce
In s anda d mic oeconomics ex books, he p oduc ion unc ion is d awn whe e he
amoun o labo is pu on he ho izon al axis and he quan i y o physical capi al o
di e en echnologies on he e ical axis as i he capi al we e a homogeneous inpu .
Pe lo (2012, p. 189), o example, discussed he isoquan s o semiconduc o ci cui s
(o “chips”) in o de o gi e a eal-wo ld example. He discussed h ee di e en ypes
o echnology:
(1) aligne s: leas capi al-in ensi e p ocess;
(2) s eppe s: mo e capi al-in ensi e p ocess; and
(3) s eppe s wi h wa e -handling equipmen : mos capi al-in ensi e p ocess.
In all o hese h ee cases, he ou pu is supposed o be 25 en-laye chips. Capi al lies on he
e ical axis and labo on he ho izon al axis, as usual. Pe lo hen p oceeds o ob aining
downwa d sloping piecewise-linea isoquan s by combining di e en combina ions o
ixed-p opo ions o labo and capi al. I mo e combina ions a e conside ed, hen in he limi ,
he amilia smoo h downwa d sloping con ex isoquan s ep esen ing pe ec ly
subs i u abili y among inpu s as asse ed in he heo y o mic oeconomics a e p e ended o
p e ail. Howe e , “amoun s o capi al”do no ha e a common uni o measu emen in di e en
echnological p ocesses and hence canno be shown on he same axis. Uni s o apples and
o anges canno be compa ed. Mo eo e , an a emp o d aw an isocos line would also be no
well g ounded since echnological uni s unlike in o m canno be combined wi h a single p ice.
We now in oduce ou concep o “speed”which can be used o compa e echnologically
di e en uni s o physical capi al:
De ini ion 1. The “speed o physical capi al”is he in e se o he machine-hou s o
p oduce 1 uni o ou pu .
In o he wo ds, i is he numbe o ou pu a machine can p oduce in a uni o ime (e.g. an
hou ). The speeds o Technologies A–E a e shown in he ou h column o Tables I and II.
P oduc i e e iciency wi h espec o p ocess inno a ion in economics is de ined as he
abili y o p oduce he same amoun o ou pu using ewe amoun s o inpu s (o o p oduce
mo e wi h he same quan i y o inpu s). Un o una ely, his de ini ion o e iciency does no
seem o be app op ia e in he case o a echnological ad ance. When some echnological
ad ance occu s so ha e iciency occu s, he na u e o some o he inpu s is expec ed o
change (i.e. a mo e au oma ed p ocess), so ha he le els o inpu s be o e and a e
he echnological change canno be compa ed. This is clea ly a con adic ion. Wi h he
Labo equi emen Capi al equi emen Speed o capi al
Technology A 1 man-hou 4 machine-hou s 0.25
Technology D 0.95 man-hou s 3 machine-hou s 0.333
Technology E 0.90 man-hou s 2 machine-hou s 0.5
Table II.
Labo - educing
echnologies ( o
p oduce 1 uni
o ou pu )
44
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in oduc ion o ou e m “speed o capi al,”we a e de ining p oduc i e e iciency as he
capabili y o physical capi al (including obo s and a i icial in elligence) o p oduce
mo e pe uni o ime (i.e. an hou ) and, he e o e, bo h ypes o physical capi al, old and
new, a e eplaced wi h he same a iable (speed).
Based on all he eal-wo ld examples discussed abo e, we will p opose he ollowing:
P1. In gene al, labo and physical capi al inpu s a e complemen a y in he sho un and
can become subs i u es only in he long un when he echnology ad ances.
In Sec ion 5, we will discuss he econome ic consequences o es ima ing p oduc ion unc ions
whe e he applied esea che would assume pe ec subs i u abili y be ween inpu s, whe eas
in eali y, we cons uc ed he hypo he ical da a acco ding o ixed-p opo ions echnology.
3.3 The sho - un ma ke equilib ium wi h ixed-p opo ions p oduc ion unc ion
Many cha ac e is ics o he oligopolis ic ma ke s uc u e help echnological ad ance.
Fi s o all, oligopolis s ealize economic p o i s, some o which may be e ained. This
undis ibu ed p o i may se e as a majo sou ce o unding hei R&D expendi u es.
Second, he exis ence o possible en y ba ie s gi es he oligopolis s an oppo uni y o
con inue any economic p o i hey ecei e om inno a ion. Thi d, la ge sales olumes
allow hem o ha e smalle R&D expendi u es pe uni o ou pu . Finally, he b oad na u e
o R&D ac i i y wi hin oligopolis ic i ms helps hem coun e balance he ine i able
unsuccess ul inno a i e a emp s wi h he success ul ones. The e o e, in gene al, he
oligopolis s ha e he means and incen i e o inno a e e en hough in oducing cos ly new
echnology may ende some o i s cu en equipmen obsole e. On he o he hand, a pu e
monopolis p o ec ed wi h high en y ba ie s may end o b eed complacency (McConnell
and B ue, 1999).
We s a wi h he Cou no ’s model wi h N i ms p oducing a homogeneous good.
Assuming ha he ma ke demand cu e is linea wi h p¼a−bQ (whe e pis he ma ke
p ice, Qis ma ke ou pu and aand ba e some posi i e eal numbe s). We u he
assume ha each i m i(i¼1, 2, …,N) is using a p oduc ion p ocess cha ac e ized wi h
ixed-p opo ions o labo , capi al and aw ma e ial/in e media y goods in he sho un
based on ou P1. Concen a ing on he p oduc ion o 1 uni o ou pu , he cons an ma ginal
cos o he i h i m is:
mci¼waþpK1
SþpRAW ;(1)
whe e Sis he speed o physical capi al, αand βa e man-hou s and machine-hou s
equi ed o p oduce 1 uni o ou pu , espec i ely, and S¼(1/β). w,p
K
and p
RAW
a e he
wage a e, en al p ice o capi al and cos o aw ma e ial (o in e media y inpu s) all used
in p oducing 1 uni o ou pu , espec i ely. No ice ha he p ices o inpu s a e de ined wi h
espec o he p oduc ion o 1 uni o ou pu and no pe pe iod o ime. This seems o be
necessa y since all p ices mus ha e a common dimension in he calcula ion o any ype o
cos unc ion. E en hough he labo and capi al a e pe haps en ed, he e is a ixed
amoun o cos o aw ma e ial (o in e media y inpu s) o uni o ou pu ha is p oduced.
Fo example, i he capi al is en ed o $3,200 a week, and he p oduc ion se ing o he
plan is such ha 200 uni s a e planned o be p oduced pe week, hen he p ice o capi al
is $16 pe uni o ou pu [4]:
P2. I seems ha i makes mo e sense o calcula e he cos s o inpu s pe uni o ou pu
a he han pe pe iod o ime. A e all, in he case when all inpu s a e
complemen a y in he sho un and some inpu s s ay idle, his does no help any
u he p oduc ion.
45
Economic
ma ke
ou comes
co espond o he same mul iples o he capi al and labo amoun s keeping capi al/labo
a ios cons an in each g oup o ha e a ixed-p opo ions p oduc ion unc ion. Fo example,
in he i s g oup ( he i s i e obse a ions), he capi al-labo a io is ixed, K/L¼10/
5¼24/12 ¼32/16 …50/25 ¼2 and he ou pu le els a e 1, 2.4, 3.2, 4 and 5 uni s.
Obs. No. Labo (L) Capi al (K) Ou pu (Q)
1 5 10 1
2 12 24 2.4
3 16 32 3.2
420404
525505
6 4 11 1
7 9 24.75 2.25
8 11 30.25 2.75
916444
10 20 55 5
11 3 12 1
12 7 28 2.33
13 9 36 3
14 13 52 4.33
15 15 60 5
16 2 15 1
17 3 22.5 1.5
18 6 45 3
19 9 67.5 4.5
20 10 75 5
Table III.
Hypo he ical da a on
ou ixed-p opo ions
p oduc ion unc ions
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16 18
LABOR
T1
T2
T3
T4
ISOQ_1_COBB–DOUGLAS
ISOQUANT1_CES
ISOQ_2_COBB–DOUGLAS
ISOQUANT2_CES
T1
T2
T3
T4
ISOQ_1_CES
ISOQ_1_COBB–DOUGLAS
ISOQ_2_CES
CAPITAL
ISOQ_2_COBB–DOUGLAS
Figu e 7.
Fixed-p opo ions
echnologies and he
es ima ion o Cobb–
Douglas and CES
p oduc ion unc ions
52
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Hence, whe he he da a e e o ixed-p opo ions p oduc ion unc ions changing o e ime,
which is a echnological ad ance (case 1 abo e) o 4 di e en ypes o echnologies exis ing
concu en ly a a poin in ime (case 2 abo e), he e is clea ly no possibili y o subs i u ion
be ween labo and capi al inpu s in he sho un.
We also suppose ha he e is no p oblem o measu emen o physical capi al, since
ou aim is o in es iga e he econome ic consequences o es ima ing 4 di e en se s o
ixed-p opo ions o inpu combina ions ep esen ing echnologically dissimila p ocesses,
using wo ypical p oduc ion unc ions, gene alized Cobb–Douglas and CES which do
allow subs i u ion among inpu s. These wo p oduc ion unc ions a e shown in he
ollowing equa ions:
Q¼ALabo aCapi albGene alized CobbDouglasðÞ;(10)
Q¼AdCapi al þ1dðÞLabo 
ðÞ
1= CESðÞ;(11)
whe e αand βa e coe icien s o labo and capi al, espec i ely, in Equa ion (10), and δand ρ
a e he dis ibu ion and subs i u ion pa ame e s, espec i ely, in Equa ion (11). Ais he
scale o e iciency pa ame e . The es ima ion esul s a e gi en in Equa ions (12) and (13),
using non-linea leas squa es es ima ion me hod in E iews, whe e all coe icien s a e highly
signi ican (p- alues almos 0):
Q¼0:1249 Labo 0:304 Capi al0:695 Gene alized CobbDouglas
ðÞ
;(12)
R2¼0:99;
Q¼0:1145 0:8258 Capi al0:525 þ10:8285
ðÞ
Labo 0:525

1=0:525 CES
ðÞ
;(13)
R2¼0:99:
The i s in bo h equa ions a e almos pe ec (R
2
¼0.99), since we cons uc ed he da a in
such a way ha he ou pu le els co espond o he same mul iples o he capi al and labo
amoun s keeping K/L cons an , as men ioned be o e and shown in Table III.
The isoquan s de i ed om hese wo ypes o p oduc ion unc ions a e as ollows:
Capi al ¼Q
ALabo a
!
1=b
Gene alized CobbDouglasðÞ;(14)
Capi al ¼
Q=A


1dðÞLabo 
d
0
@1
A
1=
CESðÞ:(15)
In Figu e 7, T1, T2, T3 and T4 a e some s aigh lines displaying he ixed inpu a ios
ep esen ing 4 di e en ypes o p ocesses discussed abo e, and he ones ha ing ela i ely
highe capi al/labo (K/L) a ios p esumably ep esen ing mo e ad anced echnologies. The
es ima ed isoquan s co esponding o he ou pu le els o Q¼1 and Q¼2 a e also shown in
he same igu e.
53
Economic
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Ou indings in his example a e as ollows:
(1) The gene alized Cobb–Douglas unc ion co ec ly es ima es ha he e is CRS, since
he sum o he coe icien s o labo and capi al adds up o one (0.304+0.695 ¼0.999).
This needed no o be he case since we did no impose any es ic ions on hese
elas ici y pa ame e s (hence, he name gene alized Cobb–Douglas unc ion). So a so
good! Howe e , hese labo and capi al coe icien s canno be in e p e ed as pa ial
elas ici ies o labo and capi al, since in ou example, hei ma ginal p oduc s o inpu s
a e p ecisely ze o by he cons uc ion o ixed-p opo ions p oduc ion unc ions.
(2) The CES unc ion, which is inhe en ly cha ac e ized by CRS, assigns di e en
coe icien s o labo and capi al om hose ob ained wi h he Cobb–Douglas case abo e
in pa (1): 0.174 and 0.826, espec i ely. This esul is a consequence o he ac ha
he elas ici ies o subs i u ions o hese wo di e en ypes o unc ions di e . Whe eas
he Cobb–Douglas is cha ac e ized by uni a y elas ici y o subs i u ion, he elas ici y o
subs i u ion o he CES is es ima ed o be 0.656 ((1/1+ρ)¼1/(1+0.525)). Again his
elas ici y o subs i u ion does no make sense in ou p esen con ex , since we canno
speak o any subs i u ion a all due o he ixed-p opo ions na u e o he da a.
(3) As expec ed, hese wo well-known p oduc ion unc ions concei e a ious cons an
ou pu le els (e.g. Isoquan s 1 and 2 in Figu e 7) p oduced wi h dissimila physical
capi al-labo a ios p esumably ep esen ing di e en p ocesses as i subs i u ion
we e possible, i.e. as i he i ms we e able o swi ch om one echnology o ano he
quickly in he sho un. In e es ingly enough, we ind a qui e a success ul i as he
isoquan s o Q¼1 and Q¼2 nea ly pass h ough he poin s o 1 uni and 2 uni s o
ou pu s indica ed by T1, T2, T3 and T4 lines s a ing om he o igin and
ep esen ing ixed inpu combina ions (see Figu e 7).
Ou main conclusion is hen he ollowing. I , in da a, dissimila p oduc ion p ocesses a e
mixed, hen he isoquan s shown in Figu e 7 can, pe haps a bes , be in e p e ed as a
subs i u ion be ween he capi al and he labo in he long un. Bu hen, again we ace he
p oblem o measu emen o capi al. He e ogeneous uni s o capi al depic ing dissimila
echnologies canno be measu ed on he same axis. I hey a e measu ed anyways in mone a y
e ms (in dolla s) and we accep , o example, $1,000 as a nume ai e as i is equen ly done in
applied wo k, $1,000’s wo h o capi al o old echnology is no he same hing as $1,000’s
wo h o capi al o new echnology. Tha was ou a ionale o in oducing he concep o
“speed”as a uni o measu emen o physical capi al ep esen ing machine y in Sec ion 3.2.
6. Conclusion
An agg ega e p oduc ion unc ion has been used in mac oeconomic analysis o a long ime,
e en hough i seems ha i is concep ually con using and p oblema ic. In his pape , we a gued
ha he measu emen p oblem ela ed o he he e ogenous capi al inpu ha exis s in
mac oeconomics is also ele an o mic oeconomic ma ke si ua ions. Mo eo e , we u he
a gued ha he e was also a p oblem in using in applied wo k: he ypical p oduc ion unc ions
in mic oeconomic heo y which allowed subs i u ion be ween labo and capi al in a case whe e
he da a migh ep esen ou pu s ob ained unde di e en ixed-p opo ions echnologies
adop ed in he long un. We cons uc ed a mic oeconomic ma ke model whe e he i ms
eplaced hei old machine y/equipmen wi h mo e p oduc i e machine y/equipmen embodying
impo an echnological ad ances. We in oduced he e m “speed”and ied o show how we
ha e been able o compa e he e ogeneous ypes o physical capi al as he echnological
imp o emen o e ime. We dis inguished be ween wo ypes o echnologies: labo neu al and
labo educing. Gi en he as echnological ad ances o oday, i is clea ha he subs i u ion is
unidi ec ional in ou e a: eplacemen o capi al o labo , and no he o he way a ound.
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We made he ollowing se en p oposi ions:
P1. In gene al, labo and physical capi al inpu s a e complemen a y in he sho un and
can become subs i u es only in he long un when he echnology ad ances.
P2. I seems ha i makes mo e sense o calcula e he cos s o inpu s pe uni o
ou pu a he han pe pe iod o ime. A e all, in he case when all inpu s a e
complemen a y in he sho un and some inpu s s ay idle, his does no help any
u he p oduc ion.
P3. Unde he labo -neu al echnology, e en i he echnology imp o es a a as a e
o e ime, he e is a lowe limi o he ma ginal cos o p oducing and he e o e
a lowe limi o he p ice o he p oduc and an uppe limi o he ma ke quan i y
o ou pu .
P4. Unde he labo -neu al echnology, e en i he echnology imp o es a a as a e
o e ime, he e a e posi i e diminishing e u ns o p o i s o echnology and an
uppe limi o p o i s de e mined by o he cha ac e is ics/pa ame e s o he ma ke ,
i.e. he ma ke demand cu e and p ices o labo , aw ma e ial inpu s.
P5. Unde he labo - educing echnology, labo class ea ns mo e ini ially as echnology
imp o es, bu hei incomes s a declining a e some h eshold le el o passage
o ime.
P6. Unde he labo - educing echnology, he demand o labo inc eases i s bu s a s
dec easing e en ually, i.e. he labo demand bends backwa d.
P7. Fi ms ha e mo e ma ke powe unde he labo - educing echnologies compa ed o
hose unde labo -neu al echnologies.
Finally, we es ima ed he pa ame e s o gene alized Cobb–Douglas and CES o ypes o
p oduc ion unc ions using a hypo he ical da a se ep esen ing ou di e en ixed-p opo ions
p oduc ion unc ions. We cau ioned he applied esea che ha hese labo and capi al
coe icien s could no be in e p e ed as pa ial elas ici ies o labo and capi al, since he
ma ginal p oduc s o inpu s we e 0 unde ixed-p opo ions p oduc ion unc ions.
No es
1. Howe e , he e a e ce ainly some ins ances whe e di e en ypes o labo can subs i u e each
o he and he e o e ep esen examples whe e labo is no homogeneous ei he . Fo example, a
c a ed ca pen e who is paid $20 pe hou inishes a ask in 8 hou s using $100 in ma e ials
cos ing $260 o he day o wo k. I migh ake an unskilled labo e 20 hou s o comple e he same
job wo king o $10 pe hou and using $150 in ma e ials cos ing $350 in o al. Cos minimizing
equi es hi ing he skilled labo e who does he job as e using less cos ly ma e ial.
2. Some o he ac o s on he supply side may as well a ec he ini ial pu chasing p ice (cos )
o capi al.
3. On he o he hand, he CES ypes o p oduc ion unc ions display CRS by cons uc ion.
4. O cou se, i is much mo e di icul o calcula e he cos o capi al in a p oduc ion con ex when he
capi al is owned.
5. Using he new echnology and disca ding he old one a e economical since hei ma ginal cos will
be lowe . Howe e , enewing all he machines may ake some ime and in he sho un, he i m
may con inue o use i s old echnology machines because o possible inancing cons ain s. I he
machines ha he i m a e employing a e en ed, a he han owned, hen enewing can be
accomplished a he end o he lease pe iod and he di usion o he new echnology is expec ed o
be much as e .
55
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ou comes
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Web e e ence
Assembly Line, Indus ial Enginee ing, a ailable a : www.b i annica.com/ echnology/assembly-line
(accessed Augus 25, 2018).
Fu he eading
Chiang, A.C. (2005), Fundamen al Me hods o Ma hema ical Economics, McG aw-Hill Book.
Co esponding au ho
Ahme Özçam can be con ac ed a : [email p o ec ed]
Fo ins uc ions on how o o de ep in s o his a icle, please isi ou websi e:
www.eme aldg ouppublishing.com/licensing/ ep in s.h m
O con ac us o u he de ails: pe missions@eme aldinsigh .com
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