A Generalized Production Function and its Special Cases

Roskamp, Ka l W. A icle A Gene alized P oduc ion Func ion and i s Special Cases K edi und Kapi al P o ided in Coope a ion wi h: Duncke & Humblo , Be lin Sugges ed Ci a ion: Roskamp, Ka l W. (1977) : A Gene alized P oduc ion Func ion and i s Special Cases, K edi und Kapi al, ISSN 0023-4591, Duncke & Humblo , Be lin, Vol. 10, Iss. 3, pp. 336-343, h ps://doi.o g/10.3790/ccm.10.3.336 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/292795 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ A Gene alized P oduc ion Func ion and i s Special Cases By Ka l W. Roskamp, De oi I. In oduc ion In a pionee ing e o o de e mine o wha ex en capi al and labo a e subs i u able o each o he in p oduc ion, A ow, Chene y, Minhas and Solow de i ed he cons an elas ici y o subs i u ion p oduc ion unc ion.1 This unc ion, usually e e ed o as CES p oduc ion unc ion, en iched economic heo y conside ably. I s p ope ies we e subsequen - ly explo ed in a numbe o a icles and some ex ensions o i ha e been o e ed.2 1 K. J .A ow, H. B. Chene y, B. S. Minhas and R. M. Solow, "Capi al-Labo Subs i u ion and Economic E iciency", The Re iew o Economics and S a- is ics, Augus 1961, Numbe 3, p. 225 o 250. The oo no e 7 in his a icle s a es ha T e o Swan independen ly deduced he cons an -elas ici y-o - subs i u ion p ope y and ha Solow used he unc ion i sel as an illus a ion in a Qua e ly Jou nal o Economics a icle in 1956. 2 See o ins ance: Mu ay, B own and John S. deCani, "Technological Change and he Dis ibu ion o Income", In e na ional Economic Re iew, Vol. 4, No. 3, Sep embe 1963, p. 289 - 309. Gio a Hanoch, "CRESH P o- duc ion Func ions", Econome ica, Vol. 39, No. 5, Sep . 1971. McFadden, D. "Cons an Elas ici y o Subs i u ion P oduc ion Func ions", Re iew o Eco- nomic S udies, 1963, p. 73 - 83. Jacob Pa oush, "A No e on he CES P o- duc ion Func ion", Econome ica, Vol. 32, No. 1-2, Janua y - Ap il 1964, p. 213. W. M. Go man, "P oduc ion Func ions in which he Elas ici ies o Subs i u ion s and in Fixed P opo ions o each o he ", Re iew o Economic S udies 1965, p. 217 - 224. J. K. Whi acke , "A No e on he CES P oduc ion Func ion", Re iew o Economic S udies, 1964, p. 166 - 167. G. C. Ha cou , "Biases in Empi ical Es ima es o he Elas ici ies o Subs i u ion o he C. E. S. P oduc ion Func ions", Re iew o Economic S udies, 1966, p. 227 - 233. H. Uzawa, "P oduc ion Func ions wi h Cons an Elas ici ies o Subs i u- ion", Re iew o Economic S udies, 1962, p. 291 - 299.. V. Muke ji, "A Gen- e alized SMAC Func ion wi h Cons an Ra ios o Elas ici ies o Sub- s i u ion", Re iew o Economic S udies, 1963, p. 233 - 236 (whe e an ex ension o he CES unc ion wi h mo e han wo ac o s o p oduc ion is a emp ed). Mu ay, B own, "A Measu e o he Change in Rela i e Exploi a ion o Capi al and Labo ", Re iew o Economics and S a is ics, Vol. 48, 1966, p. 182 - 192. P. J. Dh ymes, "Some Ex ensions and Tes s o he CES Class o P o- duc ion Func ions", Re iew o Economics and S a is ics, No . 1965, p. 357 - 366. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 A Gene alized P oduc ion Func ion and i s Special Cases 337 In his pape an e o is made o show ha he CES unc ion is a special case o a s ill wide class o i s deg ee homogeneous p oduc ion unc ions. The e exis mo e gene al p oduc ion unc ions which can allow o depa u es om compe i i e ma ginal cos p icing and ad- jus men s (o be e di icul ies in adjus men s) o he capi al-labo a io in esponse o changes in ac o p ices. Be o e hese unc ions a e de i ed i is help ul o desc ibe sho ly how he CES p oduc ion unc ion, which we shall use as a con enien s a ing poin , was o i- ginally ound. II. The De i a ion o he CES P oduc ion Func ion The de i a ion o he CES p oduc ion unc ion in ol ed wo dis inc s eps. The i s one was an empi ical s udy o he ela ionship be ween labo p oduc i i y and wages. I was obse ed ha he alue added pe uni o labo used in a gi en indus y a ies ac oss coun ies wi h he wage a e. A ow, Chene y, Minhas and Solow epo ed ha a eg ession o labo p oduc i i y on he wage a e, in a s udy co e ing 24 manu ac u ing indus ies in 19 coun ies, showed a highly signi ican co ela ion in all indus ies and also a conside able a ia ion in he eg ession coe icien s. The hypo hesis was ha wages change labo p oduc i i y, he causal low being om he o me o he la e . A posi i e co ela ion mean highe wages had an incen i e e ec causing mo e and be e wo k o be o hcoming. In his i s s ep o he s udy no assump ions we e made as o how wages we e de e mined. The empi ical s udy was a gene al one. I could ha e been applied o ma ke as well as non-ma ke economies. Es ima ed, and used o u he analysis, was he equa ion V (1) log— = log a + b Log W+ s L whe e he symbols ha e he ollowing meaning: V = alue added in cu en p ices, in dolla s L = labo -inpu in man-yea s W = money wage a e pe man-yea , in dolla s a, b = coe icien s o be es ima ed s = andom dis u bance e m The second s ep in he de i a ion o he CES unc ion is based on he assump ion ha he obse ed con igu a ion o labo p oduc i i y OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 338 Ka l W. Roskamp and wages is he ou come o a compe i i e neoclassical ma ke p ocess. I is assumed ha he e is an unde lying p oduc ion unc ion o he gene al o m: (2) V = F (K, L) whe e L deno es labo , and K capi al. The unc ion emains a i s unspeci ied, excep o he equi emen ha i shall be homogeneous o deg ee one. V K Se ing— = y—= x and deno ing wages wi h w, i can be shown JLi Li ha y = F (x) and u he : (3) w = F (x) - xF' (x) o dy (4) w = Subs i u ing equa ion (4) in o (1) and aking an iloga i hms on bo h sides yields he basic di e en ial equa ion (5) yJ = °b{y-*il) In eg a ion o (5) leads o he CES p oduc ion unc ion:3 i (6) V = (PK-p+a*L-p) P whe e: i is a cons an o in eg a ion a* = a b b is he elas ici y o subs i u ion. III. The Gene alized P oduc ion Func ion The unc ional o m o equa ion (1) is based on he hypo hesis ha labo p oduc i i y ( alue added pe uni o labo inpu ) is de e mined 3 Equa ion (6) is a o m o he CES p oduc ion unc ion gi en in A ow, Chene y, Minhas, Solow, op. ci ., o mula (11) p. 230. We eplaced a in hei no a ion by ou a*. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 A Gene alized P oduc ion Func ion and i s Special Cases 339 by he wage a e. In he ensuing analysis he disco e e s o he CES unc ion hen assumed ha he obse ed wage a es a e compe i i e ones. We p opose o change abo e app oach in wo espec s. The i s one conce ns he wage a e. We e ain he hypo hesis ha labo p oduc- i i y depends on he wage a e, ye we shall no insis ha i is com- pe i i ely de e mined. Depa u es om he compe i i e wage a e shall be possible because he ac o s o p oduc ion (labo and capi al) may possess some deg ee o ma ke powe . We w i e equa ion (4) he e o e as: dy (7) w* =y — x ; ax whe e: w* s ands o he non-compe i i e wage and he coe icien indica es a depa u e om he pe ec compe i ion ma ginal p oduc- i i y o capi al, he eal en al o capi al. Th ee cases can be dis inguished: (a) i = 1 w* = w (b) i > 1 w* <w (c) i < 1 w*> w Case (a) is he s anda d neoclassical case o pe ec compe i ion. Case (b) indica es ha capi al succeeds in ob aining a emune a ion in ex- cess o i s ma ginal p oduc . In case (c) wages a e highe han he ma ginal p oduc o labo . The second change conce ns he adjus men o he capi al-labo a io. The A ow, Chene y, Minhas, Solow analysis is in na u e a neo-classical long- un one. I assumes ha he capi al-labo a io has adjus ed in line wi h ac o p ices de e mined by long- un ma ginal p oduc i i y. An equilib ium is eached. The ul ima e long- un capi al-labo a io may howe e no be equal o obse ed sho - un ones. A disc epancy may a ise because capi al in place o en canno be adjus ed quickly and (o ) apid changes in labo inpu s may no be easible. The obse ed capi al- labo a io shall ha e a di ec e ec on p oduc i i y. We ew i e he e o e equa ion (1) as V K (8) log— = log a + b log w + c log — + j 22 K edi und Kapi al 3/1977 OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 340 Ka l W. Roskamp The symbols ha e he same meaning as in (1). The addi ional a iable K deno es capi al. The addi ional pa ame e o be es ima ed is c and j is he new andom dis u bance. I c should u n ou o be signi ican — and some p elimina y s udies indica e ha i will — he capi al- labo a io should be kep as an explana o y a iable. In e ms o ou equa ion (6) a disc epancy be ween he sho - un and long- un capi al- labo a io implies ha in he sho - un c 4= 0. I adjus men s in he capi al-labo a io ake place o e ime, c should dec ease. In he long- un, when all adjus men a e achie ed, c is equal o ze o. Subs i u ing (7) in o (8) we ob ain dyx (9) log y= lo ga + b log (y — x —-) + c log x This di e en ial equa ion is basic o he ollowing analysis. IV. Fi s Deg ee, Homogeneous P oduc ion Func ions Th ough in eg a ion o (10) a gene al, sho - un p oduc ion unc ion is ob ained. The wage a e shall be non-compe i i e ( =(= 1) and he capi al-labo a io shall no ye ha e adjus ed o his wage a e (c 4= 0). In his si ua ion he p oduc ion unc ion is: whe e y = and x = ^ as be o e. Taking an iloga i hms on bo h sides o equa ion (9) we ge : (10) y = a x c(y - % x V K b Lb * +ßK T L P 1 1 e c VP (ID whe e: p = — - 1 b ß is a cons an o in eg a ion The i s special case o (10) a ises i wages a e de e mined com- pe i i ely. In his case = 1 and (10) educes o OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 A Gene alized P oduc ion Func ion and i s Special Cases 341 (12) V2 = b K + ßK The p oblem o he capi al-labo a io adjus men emains and c= =o. The second special case a ises i c = o bu =j= 1. In his si ua ion he capi al-labo a io did adjus in line wi h a non-compe i i e wage a e. The p oduc ion unc ion becomes: (13) V,= l ~b + ßK The hi d special case is he con en ional A ow, Chene y, Solow, Minhas CES p oduc ion unc ion. I a ises i wages a e compe i i e ( = 1) and he capi al-labo a io has adjus ed in line wi h he wage a e (c = 0). The p oduc ion unc ion is now: (14) V4 = 1 -P — L + l - IK In all cases so a i was assumed ha he elas ici y o subs i u ion b alls be ween he ollowing alues: 0 < b < oo because p = — 1 his implies oc >p> - 1 This elas ici y assump ion unde lies he CES unc ion. Fo he sake o comple eness h ee special cases o he CES unc ion may be sho ly men ioned because hey yield widely used p oduc ion unc ions. I T = 1, c = 0, and b = 1, (p = 0), di e en ial equa ion (10) simpli ies o: (14a) y = a{y-xÌc) 22• OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 342 Ka l W. Roskamp In eg a ing (14a) yields he well-known Cobb-Douglas p oduc ion unc ion (15) = I x = 1, c = 0, b = oo, (p = — 1) a pe ec subs i u ion p oduc ion unc ion esul s. I s o m can be easily ound by subs i u ion — 1 o p in (14). One ob ains: (16) VQ = L + iK Finally, i = 1, c = 0, b = 0 (p = oo) he Leon ie ixed p opo ion p oduc ion unc ion can be ob aind om (14) h ough a limi ing p o- cess. I s gene al o m is: (17) V7 = y min [K, L] whe e y is a cons an . Zusammen assung Eine e allgemeine e P oduk ions unk ion und ih e Sonde älle Vo e wa sechzehn Jah en lei e en A ow, Chene y, Minhas und Solow eine P oduk ions unk ion mi kons an e Fak o subs i u ionselas izi ä ab (CES p oduc ion unc ion). Die Ablei ung e olg e in zwei Sch i en. Zunächs wu de in eine s ochas ischen Gleichung die A bei sp oduk i i ä du ch den Lohn- sa z e klä . In dem zwei en Sch i wu de dann eine Ve bindung zwischen diese empi ischen Gleichung und de adi ionellen neoklassischen Theo ie he ges ell : Es wu de angenommen, daß de Lohnsa z das E gebnis eine ollkommenen Konku enz und eine Funk ion de capi al-labo - a io is . In diesem Au sa z wu de das obige Ablei ungs e ah en modi izie . Es wu de zunächs angenommen, daß die A bei sp oduk i i ä om Lohnsa z und de capi al-labo - a io bes imm wi d. Fe ne wu de angenommen, daß de Lohnsa z un e Bedingungen eine un ollkommenen Konku enz zus ande komm . Mi diesen Ände ungen e häl man eine iel allgemeine e Klasse on linea homogenen P oduk ions unk ionen. Diese schließ als Sonde - älle die „Va iable Subs i u ionselas izi ä s-P oduk ions unk ion (VES p o- duc ion unc ion) und die CES Funk ion ein. Es is wohlbekann , daß die le z e e wiede um als Sonde älle (a) die pe ek e Subs i u ionselas izi ä s- P oduk ions unk ion, (b) die Cobb-Douglas-P oduk ions unk ion und (c) die Leon ie ( ixed p opo ion) P oduk ions unk ion ha . OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07 A Gene alized P oduc ion Func ion and i s Special Cases 343 Summa y A Gene alized P oduc ion Func ion and i s Special Cases Six een yea s ago A ow, Chene y, Minhas and Solow de i ed he Cons an Elas ici y o Subs i u ion (CES) p oduc ion unc ion. I s de i a ion p oceeded in wo s eps. The i s one was o es ima e a s ochas ic equa ion in which labo p oduc i i y is a unc ion o he wage a e. The second s ep in ol ed a linkage o his empi ically de e mined equa ion wi h he body o es ablished neoclassical heo y: he wage a e was assumed o be de e mined com- pe i i ely and a unc ion o he capi al-labo a ion. In his pape abo e p ocedu e o he de i a ion o he CES unc ion was modi ied. Fi s , he empi ical p oposi ion is ha labo p oduc i i y is de e mined by he wage a e and he capi al-labo a io. Second, he wage a e is assumed o be de e mined in a non-compe i i e manne .. Wi h hese wo changes a much wide class o linea homogeneous p oduc ion unc ions is ob ained. I includes as special cases he Va iable Elas ici y o Subs i u ion (VES) p oduc ion unc ion and he CES unc ion. As is wellknown, he la e in u n includes as special cases (a) he pe ec elas ici y o subs i u ion p oduc ion unc ion, (b) he Cobb-Douglas p oduc ion unc ion and (c) he Leon ie ixed p opo ion p oduc ion unc ion. Résumé Une onc ion géné alisée de p oduc ion e ses cas excep ionnels Il y a quelque seize années, A ow, Chene y, Minhas e Solow induisaien une onc ion de p oduc ion a ec un ac eu cons an d'élas ici é de subs i u- ion (CES p oduc ion unc ion). Ce e dé i a ion s'e ec ua en deux é apes. D'abo d une équa ion s ochas ique expliqua la p oduc i i é du a ail pa le aux de salai e. Ensui e u é ablie une ela ion en e ce e équa ion empi ique e la héo ie néoclassique adi ionnelle: l'on a supposé que le aux de salai e é ai le ui d'une concu ence pa ai e e une onc ion de la ela ion capi al- a ail. Le p ésen a icle modi ie le p océdé d'induc ion p éci é. L'on adme que la p oduc i i é du a ail es dé e minée pa le aux de salai e e pa la ela ion capi al - a ail. E l'on adme égalemen que le aux de salai e s'é abli dans les condi ions d'une concu ence impa ai e. Ces modi ica ions pe me en d'ob eni une classe plus géné ale de onc ions de p oduc ion linéai es homogènes. Ce e classe inclu comme cas excep ionnels la « onc ion de p oduc ion a iable de subs i u ion élas ique» (VES p oduc ion unc ion) e la onc ion CES. Il es bien connu que ce e de niè e a pa ailleu s comme cas d'excep ion (a) la onc ion pa ai e de p oduc ion de subs i u ion élas ique, (b) la onc ion de p oduc ion Cobb-Douglas e (c) la onc ion de p oduc ion Leon ie ( ixed p opo ion). OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/ccm.10.3.336 | Gene a ed on 2023-01-16 12:42:07