B ondino, Gab iel; Fo a-Alcalde, Facund; Casaú Gui ao, Miguel Ángel
A icle
Measu ing o al labou p oduc i i y in he open economy:
A p oposal
PSL Qua e ly Re iew
P o ided in Coope a ion wi h:
Associazione Economia ci ile, Rome
Sugges ed Ci a ion: B ondino, Gab iel; Fo a-Alcalde, Facund; Casaú Gui ao, Miguel Ángel (2024) :
Measu ing o al labou p oduc i i y in he open economy: A p oposal, PSL Qua e ly Re iew, ISSN
2037-3643, Associazione Economia ci ile, Rome, Vol. 77, Iss. 309, pp. 173-193,
h ps://doi.o g/10.13133/2037-3643/18495
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ol. 77 n. 309 (June 2024)
Measu ing o al labou p oduc i i y
in he open economy: A p oposal
GABRIEL BRONDINO, FACUND FORA ALCALDE, MIGUEL ÁNGEL CASAÚ GUIRAO
Abs ac :
In he classical app oach, one o he main measu es o
p oduc i i y is he so-called ' o al labou p oduc i i y'
(TLP), which can be compu ed as he ecip ocal o he
e ically in eg a ed labou equi ed o p oduce one uni
o ne ou pu . In closed economies, his index changes
exclusi ely due o echnical p og ess. Howe e , in open
economies, whe e in e media e impo s a e equi ed o
he p oduc ion o na ional ou pu , changes in TLP may
e lec changes in sou cing p ac ices, which do no
necessa ily ep esen echnical p og ess bu a he he
eo ganiza ion o p oduc ion chains. To accoun o hese
new sou ces o p oduc i i y, Pasine i p oposed
conside ing he quan i y o domes ic labou embedded in
expo s ha is employed o inance impo s. Building on
his idea, his pape de elops an 'adjus ed' measu e o TLP
ha co ec s he e ec o impo s on p oduc i i y and
s udies i s p ope ies. We compa e his measu e o o he
indices ela ed o TLP and highligh i s ele ance o
empi ical analysis.
B ondino: Uni e si à Ca olica del Sac o Cuo e di Piacenza,
email: gab iel.b ondino@unica .i
Fo a Alcalde: Uni e si a Au ònoma de Ba celona and
Uni e si a de les Illes Balea s, email: acund. o [email p o ec ed]
Casaú Gui ao: Uni e sidad Au ónoma de Mad id,
email: miguel[email p o ec ed]
How o ci e his a icle:
B ondino G., Fo a Alcalde F., Casaú Gui ao M.Á. (2024),
“Measu ing o al labou p oduc i i y in he open economy: a
p oposal”, PSL Qua e ly Re iew, 77 (309), pp. 173-193.
DOI: h ps://doi.o g/10.13133/2037-3643/18495
JEL codes:
C67, O47, F15
Keywo ds:
Pasine i, open economy, o al labou p oduc i i y, inpu -ou pu
Jou nal homepage:
h p: //www.pslqua e ly e iew.in o
P oduc i i y is a a io be ween ou pu s and inpu s, bu which p oduc s and inpu s should be
conside ed is a deba ed issue. The discussion depends on he esea ch ques ion and he adop ed
heo e ical amewo k. Wi hin he classical app oach, labou p oduc i i y s ands as he mos
ele an measu e o p oduc i i y, and he e a e a ious al e na i e ways o measu ing i . The so-
called o al labou p oduc i i y is a good op ion when he analysis is linked o he quan i a i e
dimension o echnological change. I measu es he ne p oduc ion o a sec o ha a uni o
in eg a ed labou , ha is, di ec and indi ec , gene a es. A special ea u e is ha i conside s all
necessa y wo k, including ha o o he ac i i ies di ec ly o indi ec ly linked o he sec o . In his
heo e ical amewo k, labou sa ing is he ul ima e mani es a ion o echnological p og ess.
This ype o measu e pai s well wi h inpu -ou pu analysis, as i s calcula ion equi es
in o ma ion on in e -indus y ela ionships. When he uni o analysis is a coun y wi h an
economy open o in e na ional ade, some addi ional di icul ies a ise. O sho ing, ou sou cing –
in sho , he c oss-bo de leng hening o p oduc ion chains – in ol es using impo ed inpu s and
Special issue on he Solow-Pasine i deba e on p oduc i i y measu emen
174 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
capi al goods, ha is, o eign esou ces. In his scena io, bo h globalisa ion and echnological
change a e sou ces o domes ic labou sa ings.
Se e al al e na i e ways o measu ing p oduc i i y a ise, depending on how impo ed inpu s
a e ea ed. Fi s ly, hey can be ea ed as non ep oducible esou ces (equa ing hem o labou ),
and domes ic labou p oduc i i y can be calcula ed wi hou dis inguishing al e na i e sou ces o
labou sa ings. Secondly, impo ed inpu s can be ea ed as i hey we e p oduced domes ically,
and a no ional labou p oduc i i y can be calcula ed. Thi dly, in he case o ha ing in o ma ion on
in e -indus y ela ionships be ween coun ies (as in he mul i- egional inpu -ou pu
amewo k), global labou p oduc i i y can be compu ed, conside ing he domes ic and o eign
labou necessa y o inal p oduc ion. Each measu e cap u es one aspec o he labou sa ings
p ocess and lea es ou ano he .
This pape explo es he cons uc ion o a p oduc i i y measu e based on a sugges ion made
by Pasine i in his book S uc u al Change and Economic G ow h. The sugges ion is o conside he
domes ic labou necessa y o p oduce expo s ha allow he pu chase o impo ed inpu s
equi ed in p oduc ion. We elabo a e on his idea o he case o a small economy wi h balanced
ade. Addi ionally, we de elop a simple nume ical example o compa e how al e na i e measu es
o o al labou p oduc i i y espond o an inc ease in o sho ing and highligh he cha ac e is ics
o he p oposed measu e.
The s uc u e o he a icle is as ollows. The sec ion a e his in oduc ion o e s some
p elimina y obse a ions on p oduc i i y measu emen . Sec ion 2 ocuses on measu ing
p oduc i i y in he classical app oach h ough he quan i y sys em. Sec ion 3 discusses he a ious
al e na i es a ailable o measu ing p oduc i i y when coun ies ade wi h each o he . These
al e na i es a e de i ed om an ex ensi e e iew o specialised li e a u e. In he i h sec ion, we
de elop an al e na i e measu e inspi ed by Pasine i's idea, while in he ollowing sec ion we
p esen a nume ical example o compa e all measu es o p oduc i i y and highligh he salien
ea u es o ou p oposal. The las sec ion concludes and p oposes u u e lines o esea ch.
1. P elimina y ema ks
Vi ually all economis s ag ee ha p oduc i i y is a a io be ween ou pu s and inpu s. Howe e ,
which ou pu s and inpu s should be conside ed is usually a ma e o con o e sy, and his usually
depends on he esea ch ques ion and he economic heo y.
The esea ch could ocus on s udying he e olu ion o he uni s o a inal p oduc c ea ed in a gi en
sec o o g oup o sec o s by one hou o wo k. This would be a pu ely physical measu e o
p oduc i i y. Changes in his index h ough ime would e lec shi s in he echnical capabili ies
o his speci ic sec o o g oup o sec o s.
I , ins ead, he esea ch is aimed a s udying he e olu ion o he pu chasing powe c ea ed by an
inpu low (say, one hou o wo k), one could measu e how much ou pu o sec o i, o o a baske
o ou pu s can be commanded by one hou o wo k in sec o k. Since we a e alking abou
pu chasing powe , ela i e p ices a e also equi ed in his measu emen . Fu he mo e, p ice
a ia ions will a ec his measu emen oo. I he pu pose is o s udy physical changes in
p oduc i i y, his is no accep able.
Ano he key issue o con o e sy a ises in he concep ualisa ion o p oduc ion adop ed. In he
neoclassical heo y, p oduc ion is concei ed as a “one-way a enue ha leads om ‘ ac o s o
p oduc ion’ o ‘consump ion goods’”. Solow’s (1957) is one o he canonical models used o
analysing economic g ow h and echnical p og ess. A key ool wi hin his model is he agg ega e
G. B ondino, F. Fo a Alcalde, M.Á. Casaú Gui ao 175
PSL Qua e ly Re iew
p oduc ion unc ion.
1
I ep esen s al e na i e ways o e icien ly p oducing a consump ion good
Q wi h capi al C and labou L and may be w i en in he ollowing way:
𝑄=𝐴⋅𝐹(𝐶,𝐿)
𝐴 is a pa ame e ha e lec s he s a e o echnical knowledge. 𝐹 ul ils all he desi ed p ope ies,
he mos ele an being ha i is homogeneous o deg ee one. Gi en his p ope y, i is possible o
es a e he p oduc ion unc ion as ollows:
𝑞=𝐴⋅𝐹(𝑐,1)=𝐴⋅𝑓(𝑐)
whe e 𝑞 is he ou pu pe labou uni , and 𝑐 is he capi al-labou a io. As can be seen, 𝑞 is a
measu e o p oduc i i y, and i s immedia e de e minan s in his amewo k a e 𝐴 and 𝑐. The
change in he o me is e med “ o al ac o p oduc i i y” g ow h (and e lec s shi s o jumps in
he p oduc ion unc ion). In con as , he change in he la e is called “capi al deepening” (and
e lec s mo emen s along he p oduc ion unc ion). G ow h accoun ing exe cises ocus on
assessing which de e minan is mo e ele an in each case. Theo e ically, o al ac o p oduc i i y
is he key de e minan o sus ained long- un p oduc i i y g ow h because he p oduc i i y gains
om capi al deepening ( om each wo ke being employed wi h mo e capi al) a e e en ually
exhaus ed due o diminishing e u ns (ano he p ope y o he p oduc ion unc ion).
Despi e i s widesp ead use, he neoclassical p oduc ion unc ion has been a acked on so many
on s ha e iewing all he c i icisms (and hei ejoinde s) would equi e a mul i- olume
collec ion. One o he main p oblems is whe he o al ac o p oduc i i y e ec i ely cap u es
echnical p og ess. Value added is he p e e ed measu e o ou pu in empi ical analyses.
Ne e heless, alue added ep esen s income dis ibu ed o bo h wo ke s, in he o m o wages,
and capi alis s, in he o m o p o i s. The e o e, es ima ing o al ac o p oduc i i y g ow h using
alue-added da a ul ima ely e lec s changes in income dis ibu ion (Shaikh, 1974), which may
no necessa ily co ela e wi h physical changes in p oduc i i y.
2
2. The classical app oach
In he classical amewo k, p oduc ion is concei ed as a ci cula p ocess. In he simples se ing,
he p oduc ion o commodi ies in ol es he use o o he commodi ies and labou . Labou is he
sole nonp oduced ac o and, he e o e, is he c i ical inpu o measu ing p oduc i i y.
3
P oduc ion is ca ied ou h ough an ex ensi e di ision o labou and ma ked specialisa ion
(Pasine i, 1993). These wo componen s a e key de e minan s o labou p oduc i i y. As he
sepa a ion o asks pe o med by each wo ke expands, and wo ke s specialise in hei espec i e
asks, hei highe p oduc i i y spills o e and imp o es o e all e iciency. Fu he mo e,
1
Despi e i s well-known p oblems, i is s ill ex ensi ely employed in he li e a u e. In ac , Pasine i (2000) was
as onished ha agg ega e p oduc ion unc ions made a comeback in he li e a u e in he 1980s, as i he e had ne e
been any issues a ound hei use.
2
Na u ally, changes in income dis ibu ion could be due o echnical p og ess – say, gi en s able ac o sha es, eal
wages inc ease because ou pu pe wo ke has isen – bu i would be misleading o asse ha his cons i u es he only
sou ce o changes in income dis ibu ion in he long e m.
3
A c ucial di e ence be ween his app oach and he neoclassical one is ha , in he la e , he ea men o ac o s is
symme ical. We could ha e jus as well wo ked wi h “capi al’ p oduc i i y by ans o ming he p oduc ion unc ion
in o 𝑣=𝐴𝑓(𝑙), whe e 𝑣 is ou pu pe uni o capi al and 𝑙 is he labou -capi al a io.
176 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
p oduc i i y may be e e enced o a single commodi y wi hou losing sigh o his
in e dependency. Se e al asks and ac i i ies con ibu e o he p oduc ion o a inished p oduc .
The abo e is made possible h ough he me hod o subsys ems. Following S a a (1960,
Appendix C), i is possible o subdi ide he economic sys em in o many pa s, each consis ing o
one kind o commodi y (belonging o he ne ou pu ). Each pa o ms a “smalle sel - eplacing
sys em” (ibid.), encompassing he di ec and indi ec inpu and labou equi emen s o p oduce
he commodi y. Pasine i (1973) shows how o de i e hem analy ically and e ms hem
“ e ically in eg a ed sec o s”. Fu he mo e, Pasine i (1988) ex ended his de ini ion o conside
he expansion o he p oduc i e capaci y, i.e. g owing subsys ems.
2.1. The de ice o e ical in eg a ion
We will wo k wi h a simple se ing o single-ou pu indus ies and ci cula ing capi al. The
echnical me hods o p oduc ion a e ep esen ed by:
4
A squa e ma ix 𝐀=[𝑎𝑖𝑗], 𝑖,𝑗 = 1,2,… 𝑛 – 1, all 𝑎𝑖𝑗≥0, whe e 𝑎𝑖𝑗 deno es he quan i y o
commodi y 𝑖 used as an inpu o p oducing one uni o commodi y 𝑗.
A ec o 𝐚𝑛=[𝑎𝑛𝑗], 𝑗 = 1,2,… 𝑛 – 1, all 𝑎𝑛𝑗≥0, whe e 𝑎𝑛𝑗 deno es he amoun o labou
uni s (hou s, pe sons engaged, e c.) equi ed o p oducing one uni o commodi y 𝑗.
Addi ionally, le ec o 𝐪=[𝑞𝑖] deno e he ec o o g oss ou pu s, and ec o 𝐲=[𝑦𝑖] ep esen
he ec o o ne ou pu s. Wi h hese de ini ions, he physical economic sys em is ep esen ed as
ollows:
(𝐈−𝐀)𝐪=𝐲 (1)
𝐀𝐪=𝐬 (2)
𝐚𝑛
T𝐪=𝐿 (3)
whe e 𝐿 is he o al amoun o labou uni s equi ed by he economic sys em and 𝐬 is he ec o o
in e media e consump ion.
Fo a gi en ne ou pu ec o ( inal demand), his sys em o equa ions ep esen s an “open
Leon ie sys em” (Pasine i, 1977, p. 61). The solu ion is:
𝐪=(𝐈−𝐀)−1𝐲
Subs i u ing his solu ion in equa ions (2) and (3) gi es:
𝐀(𝐈−𝐀)−1𝐲=𝐬
𝐚𝑛
T(𝐈−𝐀)−1𝐲=𝐿
The columns o ma ix 𝐇≔𝐀(𝐈−𝐀)−1 ep esen he e ically in eg a ed uni s o p oduc i e
capaci ies. On he o he hand, ec o 𝐯T=𝐚𝑛
T(𝐈−𝐀)−1 con ains he e ically in eg a ed labou
coe icien s.
A e ically in eg a ed sec o may be ep esen ed as ollows (Pasine i, 1990):
4
We will adop he usual no a ion, whe eby a bold ace capi al le e ep esen s a ma ix; a bold ace lowe case le e
ep esen s a ec o ; all ec o s a e columns excep when ansposed, ep esen ed by he le e T.
G. B ondino, F. Fo a Alcalde, M.Á. Casaú Gui ao 177
PSL Qua e ly Re iew
[𝑦𝑖, 𝐡𝑖𝑦𝑖, 𝑣𝑖𝑦𝑖]
whe e 𝐡𝑖 deno es he i- h column o ma ix 𝐇. Di iding all e ms by 𝑦𝑖 gi es:
[1, 𝐡𝑖, 𝑣𝑖]
Commodi ies en e ing ec o 𝐡𝑖 a e measu ed in hei speci ic physical uni (such as kWh,
li es, and numbe s). I is possible o ea 𝐡𝑖 as a composi e commodi y and e e o i as a physical
uni o e ically in eg a ed p oduc i e capaci y, 𝑘𝑖. In equilib ium, one uni o e ically
in eg a ed p oduc i e capaci y is equi ed o p oduce one uni o inal ou pu ; hus, a uni a y
ac i i y le els, 𝑘𝑖=1. Hence, we may w i e:
[1, 1, 𝑣𝑖]
To deli e one uni o inal ou pu , one uni o e ically in eg a ed p oduc i e capaci y and 𝑣𝑖
uni s o e ically in eg a ed labou a e equi ed.
5
The ecip ocal o he la e is he so-called “ o al
labou p oduc i i y”. I gi es he quan i y o ne ou pu p oduced by one uni o e ically
in eg a ed labou . Two c ucial p ope ies o his index s and ou :
I can be de i ed by looking exclusi ely a he quan i y sys em. In o he wo ds, i can be known
o gi en echnical condi ions o p oduc ion. The e o e, any changes in he index e lec solely
echnical p og ess.
As an icipa ed, i cap u es all he ele an in e indus y ela ions wo king in he backg ound
ha in luence he p oduc ion o ne ou pu s. Any imp o emen (e.g., a mo e e icien use o
labou o ep oducible inpu s) in ups eam o downs eam ac i i ies is e lec ed in he
p oduc i i y o inished p oduc s. Conside , o example, echnical p og ess in he co on
ex ile sec o du ing he Indus ial Re olu ion. The sec o ’s success was due o echnical
inno a ions in spinning and wea ing. Fu he mo e, he c ucial b eak h ough was using s eam
engines as he powe sou ce. O he echnical inno a ions came om he bleaching, dyeing and
p in ing p ocesses. These inno a ions also spil o e o o he ela ed sec o s (see Findlay and
O’Rou ke, 2007, pp. 318-320).
F om hese p ope ies he e ollows a ema kable insigh . When 𝑣𝑖 diminishes, i is possible
o asse ha “somewhe e in he economic sys em he e has been a sa ing o labou
equi emen s” (Pasine i, 1981, p. 207).
6
The o emos implica ion is ha “ echnical p og ess is,
in he end, labou -sa ing” (ibid.). In he classical amewo k, sa ing labou is he “ul ima e
meaning o echnical p og ess” (ibid.).
7
Recalling he p e ious example o he co on ex ile sec o ,
5
“[T]he ema kable ea u e o he concep o a e ically in eg a ed sec o is ha , complex hough i may be (behind he
scenes, so o speak) in i s composi ion, i is simply educed o wo ones and o a u he single numbe ep esen ing a
physical quan i y o labou ” (Pasine i, 1990, p. 236).
6
Pasine i d aws his conclusion om analysing he dynamics o uni p ices a he in eg a ed le el. I he p ice o a
commodi y dec eases o e ime o a gi en a e o p o i , i indica es echnical p og ess in i s p oduc ion. When he a e
o p o i is se a i s 'na u al' le el, he uni p ice o a commodi y equals i s e ically hype -in eg a ed labou coe icien ,
which includes bo h di ec and indi ec labou , as well as he hype -indi ec labou necessa y o expand p oduc i e
capaci y. In his scena io, i can be unambiguously es ablished ha he educ ion o his coe icien o e ime is
associa ed wi h echnical p og ess in he p oduc ion o he commodi y. The e ec s o echnical p og ess on he
p oduc i e s uc u e may a y, depending on whe he i occu s in he p oduc ion o consump ion goods o capi al
goods. Fo ins ance, echnical p og ess may encompass physical-capi al-using echniques o physical-capi al-sa ing
cum di ec -labou -using echniques. See Pasine i (1981, p. 213) o a de ailed classi ica ion o echnical p og ess.
7
See also he so-called “law o dec easing labou con en ’ (Fa joun and Macho e , 1985; Flaschel e al., 2013).
178 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
Chapman (1972, p. 20) epo ed ha he numbe o ope a i e hou s o p ocess 100 pounds
(a ound 45 kilog ams) o co on was o e 50,000 o Indian hand spinne s in he eigh een h
cen u y. In England, i was cu o 2,000 in 1780 wi h he in oduc ion o C omp on’s mule, and i
dec eased o 300 by 1795 wi h he help o powe -assis ed mules.
3. In e na ional economic ela ions
Technological p og ess and ins i u ional change in he las qua e o he wen ie h cen u y
educed na u al and a i icial ade ba ie s, inc easing he ype and numbe o commodi ies
coun ies could exchange. Baldwin (2016) has called his p ocess “globalisa ion’s second
unbundling”.
Mul ina ional i ms began sou cing inpu s ha came om o sho e loca ions and ha we e
p oduced by o he i ms and began eloca ing se e al s ages o p oduc ion ab oad o minimise
cos s and inc ease ma ke sha e. T ade in in e media e inpu s su ged. F om he s andpoin o
na ional economies, he ade pa e ns became mo e complex han he adi ional impo -expo
o inished goods.
This p ocess has ce ainly a ec ed he domes ic condi ions o p oduc ion. When a coun y’s
economy is open o in e na ional ade, he impo o capi al goods and in e media e inpu s plays
a ele an ole in domes ic p oduc ion. The p oduc ion o ne ou pu s no longe equi es only
domes ic esou ces bu also o eign ones.
8
Le us e u n o he ini ial model o adap i o he case o an open economy. The majo change
is ha ma ix 𝐀 is spli in o wo: a ma ix o domes ic inpu coe icien s 𝐀𝑑 and a ma ix o
impo ed inpu coe icien s 𝐀𝑚 (𝐀≡𝐀𝑑+𝐀𝑚).
The physical economic sys em is now w i en as ollows:
(𝐈−𝐀𝑑)𝐪=𝐲 (4)
𝐀𝑑𝐪=𝐬𝑑 (5)
𝐀𝑚𝐪=𝐬𝑚 (6)
𝐚𝑛
T𝐪=𝐿 (7)
In his scena io, labou -sa ing is no longe s ic ly associa ed wi h echnological p og ess. I
is also possible o educe he labou con en o a e ically in eg a ed sec o by subs i u ing
domes ic inpu s o impo ed inpu s wi hou changing he me hod o p oduc ion.
To see his, le us i s de ine ma ix 𝐃=[𝑑𝑖𝑗], whe e 𝑑𝑖𝑗=𝑎𝑖𝑗
𝑑𝑎𝑖𝑗
⁄ is he sha e o inpu i ha
is sou ced domes ically in he p oduc ion o commodi y j. Ma ix 𝐀𝑑 can be es a ed as 𝐀𝑑=𝐃∘
𝐀, whe e ∘ deno es elemen -wise mul iplica ion.
9
The solu ion o equa ion (4) is 𝐪=(𝐈−𝐀𝑑)−1𝐲=(𝐈−𝐃∘𝐀)−1𝐲. The ec o o e ically
in eg a ed labou coe icien s is now:
8
This is no he only ele an ea u e o in e na ional economic ela ions. The possibili y o lea ning, copying ad anced
echniques, e c., is also c ucial.
9
Fo example, gi en ma ices 𝐀=[𝑎𝑖𝑗] and 𝐁=[𝑏𝑖𝑗], i s elemen -wise p oduc is ma ix 𝐀∘𝐁= [𝑎𝑖𝑗⋅𝑏𝑖𝑗].
G. B ondino, F. Fo a Alcalde, M.Á. Casaú Gui ao 179
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𝐯T=𝐚𝑛
T(𝐈−𝐃∘𝐀)−1 (8)
A dec ease in any elemen o ma ix D, ela ed o lowe domes ic inpu consump ion, leads o
a all in and an inc ease in domes ic p oduc i i y.
10
In o he wo ds, as p oduc ion is inc easingly
eloca ed ab oad, he domes ic labou equi ed pe uni o ne ou pu dec eases acco dingly.
The p e ious esul bea s signi icance o he p oduc i i y measu e and i s in e p e a ion. To
he bes o ou knowledge, he e a e h ee app oaches o measu ing o al labou p oduc i i y in
an open economy. The i s app oach is o compu e i using equa ion (8) abo e. We may call he
measu e domes ic o al labou p oduc i i y. In his case, impo ed inpu s a e ega ded as
nonp oduced esou ces and a e placed on an equal oo ing wi h domes ic labou (Rampa, 1981).
The main se back o his app oach is ha i is impossible o dis inguish he sou ce o labou sa ing;
ul ima ely, one is cons ained o emb ace a b oade concep o echnological p og ess ha also
encompasses changes in sou cing p ac ices.
The second app oach is cons uc ing a able o echnical coe icien s by adding up he domes ic
and impo ed inpu coe icien ma ices. In his case, he measu e o p oduc i i y is no ional, since
i compu es he o al labou equi ed unde he assump ion ha all inpu s a e sou ced
domes ically (Gup a and S eedman, 1971). Howe e , changes in sou cing p ac ices lea e
p oduc i i y una ec ed.
The hi d and inal app oach is o wo k wi hin a global inpu -ou pu amewo k and conside
all he labou equi emen s (domes ic and o eign) o global e ically in eg a ed sec o s
(Ga bellini, 2014). De eloping his measu e equi es some mino changes in ou no a ion.
Imagine ha he wo ld economy consis s o wo egions. Le 𝐀𝑟𝑠=[𝑎𝑖𝑗
𝑟𝑠] be he ma ix o
in e egional inpu coe icien s, whe e 𝑎𝑖𝑗
𝑟𝑠 deno es he quan i y o commodi y 𝑖 p oduced in
coun y 𝑟 used as an inpu o one uni o commodi y 𝑗 p oduced in coun y 𝑠. Thus, 𝐀𝑟𝑟 is he
ma ix o in a egional inpu coe icien s. The comple e ma ix o inpu coe icien s, also called he
global sou cing ma ix is:
𝚲=[𝐀11 ⋯ 𝐀1𝑝
⋮ ⋱ ⋮
𝐀𝑝1 ⋯ 𝐀𝑝𝑝] (9)
Fu he mo e, le 𝐚𝑛
𝑟 deno e he ec o o di ec labou coe icien s o coun y 𝑟. We can s ack
he ec o s o labou coe icien s o each egion o ge he comple e ec o o labou coe icien s:
𝛂𝑛=[𝐚𝑛
1
⋮
𝐚𝑛
𝑝] (10)
Wi h hese de ini ions, i is possible o compu e he ec o o e ically in eg a ed labou
coe icien s as ollows:
𝛎T=𝛂𝑛
T(𝐈−𝚲)−1 (11)
10
A dec ease in he sha e o domes ic sou cing o a leas one inpu , such ha 𝐃′≤𝐃, implies ha (𝐈−𝐃′∘𝐀)−1≤
(𝐈−𝐃∘𝐀)−1. I ollows ha 𝐯 necessa ily alls.
180 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
In his app oach, i is p e e able o alk o global e ically in eg a ed sec o s because each
sec o employs esou ces om all egions, and he no ion o domes ic and impo ed makes sense
only i explici e e ence o he p oduc ion loca ion is made. Fo example, au omo i e sec o s a e
global because hey equi e inpu s om di e en coun ies, and, i we ocus on he Ge man
au omo i e sec o , we can dis inguish domes ic om o eign esou ces. Fu he mo e, i is also
possible o dis inguish he egion in which echnical p og ess occu .
3.1. Empi ical analysis
Se e al schola s ha e o e ed an empi ical analysis o o al labou p oduc i i y wi h he help o
inpu -ou pu analysis. This is bo h an ad an age and a cu se. I is ad an ageous since i p o ides
a s aigh o wa d empi ical coun e pa o he heo e ical a iables and a cu se because i canno
be measu ed wi hou inpu -ou pu da a, which is no always a ailable.
An impo an s and o he li e a u e has used he measu e o accoun o echnical p og ess
in di e en con ex s. Pionee applica ions ha e ocused on speci ic sec o s. Fo example, Gossling
and Do ing (1966) and la e Gossling (1972) s udied ag icul u e in he US, and Pane himi akis
(1993) s udied he G eek manu ac u ing sec o . Mo eo e , he e a e se e al wo ks on coun y
case s udies, such as I aly (Rampa, 1981; Rampa and Rampa, 1982; Ga bellini and Wi kie man,
2010), he Uni ed S a es (Ochoa, 1986), Aus alia (Gowdy, 1992) and Spain (De Juan and Feb e o,
2000).
O he s udies apply his measu e in a compa a i e way. In his sense, Elmslie and Milbe g
(1992) used o al labou p oduc i i y o app oxima e he echnological di e ences be ween se en
OECD coun ies. Yea s la e , Elmslie and Milbe g (1996) ied o measu e he c oss-coun y
con e gence in labou p oduc i i y o Po ugal and Japan a a e ically in eg a ed le el.
11
S uc u al decomposi ion analysis
12
is applied in ano he g oup o s udies in o de o analyse
he e olu ion o occupa ional s uc u es, de ining he “p oduc i i y e ec ” as he a ia ion in he
decomposi ion ela ed o he ec o o o al labou equi emen s by indus y. This ool has been
used o s udy employmen in Japan (Han, 1995), he g ow h o in o ma ion wo ke s in he Uni ed
S a es (Wol , 2006), he impac s o in e na ional ade on di e en domes ic economies (Po ella-
Ca bó, 2016), he d i e s o Eu opean you h unemploymen (Ca ascal-Ince a, 2017) and he
employmen changes in he Spanish economy (Mada iaga, 2018).
Mo e ecen ly, Ga bellini and Wi kie man (2014), B ondino (2018), and Wi kie man (2022)
ha e e i ed his app oach, no only by p o iding new es ima es o o al labou p oduc i i y o
a ious economies bu also by including he analysis in e ms o e ically hype -in eg a ed
sec o s ( ollowing Pasine i, 1988), making i mo e sui able o he analysis o g owing economies.
Mos s udies ha e adop ed he measu e o domes ic o al labou p oduc i i y, lea ing aside
he ole o impo ed inpu s. Some wo ks include he impo s by building he echnical coe icien s
ma ix and analysing he no ional measu e o p oduc i i y. Gup a and S eedman (1971) applied
his app oach o he B i ish economy. La e , Ruiz and Wol (1996) compa ed he domes ic
measu e wi h he no ional one o s udy echnological change and impo leakages in he
employmen g ow h o Pue o Rico.
11
Howe e , he de ini ion o he coe icien echnical ma ices used in he calcula ion in bo h wo ks is e y ambiguous.
Whe he i is based on domes ic in e media e inpu s o o al in e media e inpu s is no de ined.
12
S uc u al decomposi ion analysis is a widely used inpu -ou pu echnique o b eak down he g ow h o a a iable
in o changes o i s de e minan s.
G. B ondino, F. Fo a Alcalde, M.Á. Casaú Gui ao 187
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Figu e 2 – E ec o ou sou cing in sec o 2
DTLP in o ms o a educ ion in he labou equi ed o p oduce commodi ies 1 and 2 because wha
was p e iously p oduced a home is now p oduced ab oad, and his index neglec s o eign inpu s.
GTLP, a i s u n, will be nega i ely o posi i ely a ec ed, depending on whe he he DTLP o he
ou sou cing sec o is highe o lowe han ha o he sec o whe e p oduc ion begins o ake place
a e ou sou cing. In his case, i egis e s an inc ease in o al labou equi emen s o p oduce
bo h commodi ies because, a e ou sou cing, he mo e labou -in ensi e inpu s ha e a bigge
sha e in he o al cos s uc u e.
NTLP is una ec ed because he labou ha is a ibu ed o o eign inpu s is ha which would
ha e been needed o p oduce hem a home; he e o e, whe e e p oduc ion akes place, i he e
is no echnical change, his index will no change. Finally, ou adjus ed measu e will inc ease
because he labou equi emen s o he commodi ies wi h which he inc eased impo s a e
inanced a e lowe han he labou equi emen s o p oducing he ou sou ced commodi y a
home. The eason o his is ha he ou sou ced commodi y is he one ha equi es mo e
domes ic labou o be p oduced a home, while expo s a e a composi e commodi y in which bo h
domes ic commodi ies ha e he same sha e. The e o e, when egion 1 ou sou ces he p oduc ion
o i s mos labou in ensi e commodi y and s a s paying o i wi h he e enue ob ained om
expo s ha a e less labou in ensi e, ATLP is inc eased.
Following he p e ious easoning, i he ela i e weigh s o each commodi y in he expo s
baske (measu ed in money e ms) change, he same will happen o ou ATLP measu e, which will
188 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
be he only index a ec ed by changes in his a iable. Mo e p ecisely, he bigge he weigh o he
mos p oduc i e sec o on he expo s baske (measu ed in money e ms), he bigge he ATLP o
his sec o and ha o he o he s o he same egion.
To see his, imagine an economy, such as ha o ou i s example, ha is cha ac e ized by he
echnology desc ibed in equa ions (17) o (21). Now, hough, hink ha he ec o o expo s
changes om equa ion (19), as i has been so a , o:
16
𝐞=[1
0.25] (23)
Each measu e o p oduc i i y will now yield he esul s p esen ed in igu e 3.
Figu e 3 – E ec o a change in he expo s baske sha e
16
Con a y o he assump ions es ablished a he beginning o his sec ion, he change in expo s shown in (19) mus
be hough o as an inc ease in he inal demand o egion 2. Tha is, i is no he esul (o he cause) o any change in
he echnical coe icien s ma ix. This is done o isola e he impac o a change in he composi ion o he baske o expo s
wi hou changing he echnology o egion 2.
G. B ondino, F. Fo a Alcalde, M.Á. Casaú Gui ao 189
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A his poin o analysis, i should be clea ha echnical p og ess will ha e di e en impac s
on each o he indica o s desc ibed in he pape ; and hese impac s will also depend on se e al
elemen s, such as he ade s uc u e o he di e en ials in p oduc i i y be ween sec o s. Fo
example, imagine ha we a e analysing, once again, he i s economy desc ibed in his sec ion
which was o malised in equa ions (17)-(21). I , ins ead o he change in he ade s uc u e
analysed be o e, we pose a echnical change en ailing a 50% educ ion in he di ec labou
coe icien s o sec o 1, in such a way ha he ec o o di ec labou coe icien s becomes:
𝛂𝑛
T=[1 3 4 5] (24)
we can see he changes in he p oduc i i y measu es shown in igu e 4.
Figu e 4 – E ec o a echnical imp o emen in sec o 1
DTLP o sec o 1 o egion 1 changes he mos because his sec o ’s weigh in he cos s conside ed
by his measu e is highe han he weigh ha his same sec o has in any o he index. Ins ead,
GTLP changes he leas because, as explained abo e, he labou equi emen s o one single sec o
ha e a low weigh in i s s uc u e o cos s, which includes he labou pe o med ab oad o p oduce
he inpu s ha a e impo ed in he egion unde conside a ion. In he case o sec o 1, NTLP and
ATLP changes a e ound be ween he o he wo measu es because, con a y o DTLP, hey ake
190 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
in o accoun ha p oduc ion in sec o 1 o egion 1 equi es inpu s p oduced ab oad, bu , in hese
indices, he labou equi ed o hei acquisi ion is no ha which is pe o med in he o eign
coun y (as in he case o GTLP).
In he case o NTLP, he labou ha is ele an o accoun o he impo ed inpu s is ha which
would ha e been needed i hose inpu s had been p oduced a home. In he case o ATLP, ins ead,
he labou ha is ele an o hei compu a ion is ha which is equi ed o p oduce he expo s
wi h which o inance he impo s om egion 2 equi ed by each domes ic sec o . Gi en ha , in
his example, he inpu s coming om sec o 1 o bo h egions ha e a highe weigh in he cos
s uc u e han hose inpu s coming om sec o 1 o egion 1 – e en when we ake in o accoun
ha hey a e needed o p oduce he expo s wi h which o impo any good – NTLP inc eases mo e
han ATLP in he ace o a echnical imp o emen in sec o 1 o egion 1. Had sec o 1 had a high
enough sha e in he baske o expo s, hen ATLP g ow h would ha e been highe han ha o
NTLP.
In he case o sec o 2, he si ua ion is di e en . DTLP s ill eaches he highes alue in absolu e
e ms; howe e , i s g ow h a e a e echnical p og ess is no he mos ele an one. This is so
because he weigh on o al cos s o he inpu s coming om sec o 1 – which is he one whe e
echnical p og ess has aken place – is lowe in his measu e han in hose measu es – NTLP and
ATLP – in which he domes ic labou used o accoun o impo ed inpu s has a bigge weigh .
Mo eo e , in he case o sec o 2 o egion 1, inpu s coming om sec o 1 o bo h egions ha e a
lowe weigh in he o al cos s uc u e han he one hey had in he case o sec o 1 o his same
egion; his di e ence is enough o make ATLP highe han NTLP in his o he sec o , and a highe
sha e o sec o 1 in he expo s baske would make his di e ence e en g ea e . Finally, GTLP s ill
yields he lowe absolu e measu e o p oduc i i y and he lowes inc ease: emembe ha o eign
inpu s a e included in i s calcula ion, weigh ed acco ding o he labou ha mus be pe o med in
he coun y o o igin, which is less echnically ad anced in bo h sec o s unde he cu en
assump ions.
All his b ings o he o e ha , while (domes ic) echnical p og ess will impac all ou
measu es, al hough in pa icula ways, changes in he ade s uc u e will a ec only DTLP, GTLP
and ATLP. The i s wo indices will be a ec ed in qui e in ui i e and s aigh o wa d manne s:
DTLP will imp o e (de e io a e) whene e he e is ou sou cing (insou cing) and GTLP will
imp o e whene e ou sou cing (insou cing) is done om a sec o wi h a lowe DTLP han he one
in which p oduc ion has been insou ced. ATLP, on he o he hand, will imp o e o de e io a e,
and will do so a a highe o lowe deg ee, due o changes in he ade s uc u e, depending on he
e ms o ade. Mo e p ecisely, ou sou cing will ha e a posi i e (nega i e) impac on ATLP
whene e he e ically in eg a ed labou equi ed o p oduce he ou sou ced commodi y is
g ea e (smalle ) han he labou equi ed o p oduce he expo s wi h which o inance he
impo s o he ou sou ced commodi ies; some hing ha is ul ima ely de e mined by he
magni udes o he expo s and impo s baske s in money e ms.
6. Concluding ema ks
This pape has examined he index o o al labou p oduc i i y as a key measu e o p oduc i i y
wi hin he classical amewo k. The measu e has se e al ad an ages o assessing echnical
p og ess, compa ed o o he commonly used indica o s, such as o al ac o p oduc i i y o alue
added pe wo ke . Howe e , as a gued, he p esence o impo ed capi al and in e media e goods
G. B ondino, F. Fo a Alcalde, M.Á. Casaú Gui ao 191
PSL Qua e ly Re iew
may pose some di icul ies in i s in e p e a ion. Speci ically, he sa ing o domes ic labou may
occu due o echnological and non echnological ac o s.
We ha e iden i ied se e al al e na i e ways o measu ing o al labou p oduc i i y in his
se ing. Fi s ly, impo ed inpu s can be ea ed as non ep oducible ac o s and domes ic
p oduc i i y can be compu ed wi hou dis inguishing he al e na i e sou ces o labou -sa ing.
Secondly, impo ed inpu s can be ea ed as i hey we e p oduced in e nally and a no ional
measu e o p oduc i i y can be compu ed. Finally, in he case o ha ing in o ma ion on in e -
indus y ela ionships amongs coun ies, a measu e o global p oduc i i y can be compu ed,
including domes ic and o eign labou . Each measu e cap u es one aspec o he labou -sa ing
p ocess and lea es ou o he s.
In his pape , ollowing a sugges ion by Pasine i, we ha e pu o wa d a ou h al e na i e,
which consis s o aking in o accoun he labou equi ed o p oduce he commodi ies gi en in
exchange o he impo s. We ha e discussed he measu e o he case o a small coun y ha does
no a ec in e na ional p ices and has balanced ade. A no el aspec o his measu e is ha mo e
indus ial in e dependencies eme ge, ela i e o he case o a closed economy. Speci ically,
expo ing sec o s play a c i ical ole o p oduc i i y g ow h in all sec o s. On he downside,
p oduc i i y is no longe exp essed in e ms o pu e labou bu in e ms o labou equi alen s,
which a e a ec ed by he e ms o ade. Fu he esea ch should in es iga e he ac o s
in luencing e ms o ade and he bidi ec ional e ec s. Fu he mo e, nonbalanced ade and
al e na i e ways o inancing impo s should be conside ed in o hcoming con ibu ions.
The measu e also has s ong po en ial o empi ical analysis. I s use may imp o e he
measu ing o commodi ies' eal cos o p oduc ion in e ms o labou uni s. Fu he mo e, i may
help o assess he d i e s o p oduc i i y g ow h in he age o globalisa ion. To his end, u u e
esea ch should ocus on de eloping he empi ical coun e pa s o he heo e ical magni udes
in ol ed in he measu e.
Appendix
Conside he ollowing p ice sys em.
𝐩T=𝑤𝐚𝑛
T+𝐩𝑚
T𝐀𝑚+𝐩T𝐀𝑑
Fo simplici y, he a e o p o i is assumed o be nil ( he esul is una ec ed by his
assump ion).
Rea anging he e ms:
𝐩T=𝑤𝐚𝑛
T(𝐈−𝐀𝑑)−1+𝐩𝑚
T𝐀𝑚(𝐈−𝐀𝑑)−1
Pos -mul iply all e ms by 𝐞:
𝐩T𝐞=𝑤𝐚𝑛
T(𝐈−𝐀𝑑)−1𝐞+𝐩𝑚
T𝐀𝑚(𝐈−𝐀𝑑)−1𝐞
The exp ession can be simpli ied o ge :
1=𝑤𝑣∗+𝜇∗
192 Measu ing o al labou p oduc i i y in he open economy: A p oposal
PSL Qua e ly Re iew
F om his esul , i ollows ha :
I wages a e g ea e han ze o, 𝜇∗<1
I 𝐀𝑚=𝐎, 𝜇∗=0
I wages a e equal o ze o, 𝜇∗=1. The se ies would no con e ge. Ne e heless, wi hin his
amewo k, wages canno each ze o. Gi en in e na ional p ices, he wage a e a ies o
ensu e ha a leas one ac i i y is in e na ionally compe i i e and p o i able (see Dea do ,
2005). No e, howe e , ha he la e does no hold i he a e o p o i is posi i e and he e is
ee capi al mobili y (see C espo e al., 2021). I he economy does no ha e any compe i i e
ac i i y, i does no expo and hus does no impo ; i ially, 𝜇∗=0. See B ondino and
D oskin (2023) o a discussion.
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