Pasinetti s Structural Change and Economic Growth: a conceptual excursus

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1 Pasnett s Structural Change and Economc Growth: a conceptual excursus Nada Garbelln and Arel Lus Wrkerman Unverstà Cattolca del Sacro Cuore Largo Gemell, Mlano (Italy) (Prelmnary draft, please do not cte) Abstract A clear and organc exposton of Pasnett s theoretcal framework of Structural Change and Economc Growth s often complcated by msunderstandngs and ambgutes concernng the basc categores and termnology. The pre-nsttutonal character of the approach, the nature of ts equlbrum paths and the sgnfcance of the natural economc system together wth ts normatve character are some of the most controversal ssues. In partcular, there seems to be a need for a clearcut dstncton between the general dynamc analyss of the prce and quantty systems and the specfc dynamcs they follow when the sectoral proportons and levels of producton exactly satsfy dynamc equlbrum condtons, and a partcular closure of the prce system s adopted, provdng for specfc functonal ncome dstrbuton and theory of value. The am of the present paper s therefore that of attemptng at a conceptual excursus of the model, n order to establsh a sold ground on the bass of whch dscussons wth other Classcal approaches can be frutfully held. Keywords Vertcally ntegrated sectors, Vertcally hyper-ntegrated sectors, functonal ncome dstrbuton, natural economc rates of proft, natural economc system, pure labour theory of value. JEL classfcaton B51,O41 We wsh to thank Professor Pasnett for all useful dscussons and encouragement. garbnada@hotmal.com arwrkerman@gmal.com 1

2 2 The pre-nsttutonal analyss of an ndustral system 1 Introducton Pasnett s Structural Change and Economc Growth has been, snce ts publcaton n 1981, the object of many revews and comments, and t s one of the most cted works as regards the topc of structural change (see for example Slva & Texera 2008). However, many aspects of the book, both conceptual and analytcal, have not been grasped, or have been grasped only partally, thus preventng from a complete understandng of the mplcatons, and potentaltes, of Pasnett s approach. The frst stumblng block has usually been the pre-nsttutonal character of the model, sometmes msnterpreted as a pre-ndustral one. A second problem s the often mssed dstncton between the general dynamc analyss of the prce and quantty systems, the dynamc equlbrum paths one for each possble exogenous combnaton of dstrbutve varables and the natural economc system, resultng from a partcular closure of the prce system. A thrd ssue of mportance s the vertcally hyper-ntegrated character of the framework, on whch we partcularly nsst n the paper, n order for the model and some of ts most far reachng nsghts to be fully understood. The present paper conssts of a conceptual excursus of the model, and s organsed as follows. Secton 2 deals wth the pre-nsttutonal character of Pasnett s (1981) model. Secton 3 presents a synthetc exposton of the model, before the ntroducton of the natural economc system. Secton 4, then, gves the ratonale for the partcular closure of the prce system adopted by Pasnett (1981), hghlghtng some of the man nsghts. Secton 5 s a methodologcal note on the noton of equlbrum and ts role throughout the analyss. Fnally, secton 6 goes through the stages of development of the concept and analytcal devce of vertcal hyper-ntegraton. 2 The pre-nsttutonal analyss of an ndustral system Before gong nto the detals of Pasnett s (1981) analytcal formulaton, t s worth puttng forward a bref methodologcal ntroducton, n order for ths theoretcal framework [... ] [to be] approprately understood and correctly used. It s a basc framework, a skeleton, so to speak, whch s meant to reman at a pre-nsttutonal level of nvestgaton (Pasnett 1985, p. 274, talcs added). The above excerpt comes from the reply Pasnett gave to a revew, by Nna Shapro (1984), of Structural Change and Economc Growth. The pont he rased s crucal: the analytcal framework he developed can be understood and correctly used only f ts pre-nsttutonal character s constantly and clearly kept n mnd. 2

3 2 The pre-nsttutonal analyss of an ndustral system Therefore, even f the deepest mplcatons of Pasnett s (1981) methodology wll be drawn later on, n secton 5 below, a general hnt must be gven here, before gong nto the analytcal descrpton of the model, n order for the latter to be properly understood. The frst thng that should be made clear s the meanng Pasnett attaches to the word captalstc, as opposed to captalst. Whle the latter refers to the set of socal relatons of producton typcal of a captalst economc system, n contrast to those typcal, for example, of a centrally planned one, the former term descrbes the very physcal-technologcal nature of the producton process n any ndustral system, to be ntended as the producton of commodtes by means of commodtes,.e. the employment of captal goods as ntermedate commodtes, to be used together wth labour, and accumulated, for the producton process to take place. Pasnett s focus has always been, n all hs works, on ndustral socetes, wth ther tendency towards change and towards an evolvng structure, as aganst the more statc condtons of pre-ndustral socetes (Pasnett 2005, p. 247). Nonetheless, the pre-nsttutonal analyss he puts forward has sometmes been (ms)nterpreted as a pre-captalst, or pre-captalstc, one, n spte of the fact that he has never used such expressons, and that he has always made explct, and repeated, reference to pure producton systems. Anyway, Pasnett s am s that of analysng the workng of a captalstc, and not of a captalst, economc system. As wll be further dscussed n secton 5 below, the framework he develops s by no means an attempt at descrbng the functonng of an actual captalst system. Nor t s an attempt at descrbng the functonng of a centrally planned economy, as someone could have been nduced to conclude. What does t mean, therefore, that Pasnett s (1981) analyss has been carred out at the pre-nsttutonal level? The ssue s not a trval one. Qute apparently, many commentators dd not succeed n graspng the meanng of such a statement, thereby falng to grasp the very nature of Pasnett s framework. It s our contenton that many actual or pretended ambgutes n Pasnett s (1981) expostons are due to ths msunderstandng. As Pasnett states n the Introducton of Structural Change and Economc Growth, hs approach to economc theory starts from a very precse standpont: It s my purpose [... ] to develop frst of all a theory whch remans neutral wth respect to the nsttutonal organsaton of socety. My preoccupaton wll be that of snglng out, to resume Rcardo s termnology, the prmary and natural features of a pure producton system. (Pasnett 1981, p. 25) 3

4 2 The pre-nsttutonal analyss of an ndustral system A separaton as Pasnett called t later, n hs most recent book 1 s therefore needed between two stages of analyss, each concernng a very specfc knd of economc nvestgaton. The ratonale of ths separaton emerges very clearly from the Preface to the 1981 book: There s [... ] a sharp dscrmnaton between those economc problems that have to be solved on the ground of logc alone for whch economc theory s entrely autonomous and those economc problems that arse n connecton wth partcular nsttutons, or wth partcular groups or ndvduals behavour for whch economc theory s no longer autonomous and needs to be ntegrated wth further hypotheses, whch may well come from other socal scences. It s wth the frst type of problems that the present work s bascally concerned. (Pasnett 1981, p. x) Of course, Pasnett s (1981) clam for the logcal prorty of the frst stage the pre-nsttutonal one wth respect to the second stage the nsttutonal one by no means mples that he s dsregardng the role of nsttutons. On the contrary: ther role s regarded as one of prmary mportance, as nsttutons are the means through whch t s possble to shape the real world: All these consderatons only come to confrm how mportant s to keep the logcal problems concernng the natural economc system qute separate from those concernng the nsttutons, and to consder the nsttutons they really are means, and not ends n themselves. Once ther nstrumental role s properly understood and recognsed, t becomes much easer also to operate on them n as detached a way as s possble; to treat them as nstruments susceptble to be contnually mproved and changed, n relaton to ther sutablty (or unsutablty) to ensure tendences, or near-tendences, towards agreed ends. (Pasnett 1981, p. 155) Insttutons are means, not ends, but n order for them to be used to drve socety towards agreed ends t s frst of all necessary to know the fundamental mechansms they are called upon to counteract, or to favour, or smply to take advantage of. Wthout ths knowledge, nsttutons cannot pursue any nstrumental role. Pasnett s vson s that the prmary and natural features of an economc system have to be studed ndependently of a partcular nsttutonal set-up. Nonetheless, the task of descrbng an economc system wthout reference to a partcular nsttutonal set-up s not a trval one. It s very dffcult to realse how an economc system can be thought of wthout strong reference to the nsttutons whch 1 Pasnett (2007), where he stresses n a much sharper way such a dscrmnaton. 4

5 shape t, snce no actual economc system could have been brought nto exstence wthout them. Ths task can be accomplshed by lookng for those physcal requrements necessary for an ndustral system to carry out ts producton process, and grow. The way n whch Pasnett puts ths dea nto practce shall become clear by readng sectons 3 and 4, below. Pasnett s (1981) Structural Change and Economc Growth provdes us wth a model of economc growth startng from a complete descrpton of an economc system n a sngle-perod equlbrum, defned as a stuaton n whch there s full employment of the labour force and full utlsaton of the exstng productve capacty (Pasnett 1981, pp ). Ths stuaton can be thought of as the ntal condton of a general mult-sector dynamc model, whch has been developed for the purpose of detectng the permanent causes movng an economc system, rrespectve of any accdental or transtory devaton whch may temporarly occur (Pasnett 1981, p. 127). We wll now ntroduce a synthetc exposton of the model. 2 For basc notaton see Table Formulaton of quantty and prce systems Ths sngle-perod descrpton conssts of a physcal quantty system and a commodty prce system, each composed by 2m + 1 equatons, where m s the number of fnal consumpton commodtes produced n the system. The producton of each consumpton commodty requres a specfc captal good k. As to the physcal quantty system, ths means that the equaton concernng consumpton commodty and the equaton concernng the correspondng captal good k together descrbe the total quanttes produced by the vertcally hyper-ntegrated sector 3 (x (t) and x k (t), = 1, 2,..., m). The last equaton establshes the condton for the full employment of the total labour avalable n the system: x (t) a n (t)x n (t) = 0 for = 1, 2,..., m T 1 x (t) x k (t) a k n(t)x n (t) = 0 for = 1, 2,..., m a n(t)x (t) + a nk (t)x k (t) x n (t) = 0 (3.1) 2 We wll expose here the specfcaton of the model that consders captal goods produced by means of labour alone, for t s the man case Pasnett (1981) deals wth. 3 The concept of vertcally hyper-ntegraton s already present n Pasnett (1981), even though not always explctly. For a rgorous statement and development of ths concept, and of ts analytcal propertes, see Pasnett (1988). 5

6 Table 1: Basc notaton n Pasnett s Structural Change and Economc Growth x (t) x k (t) number of unts of fnal consumpton commodty produced durng tme perod t n the vertcally hyper-ntegrated sector ; gross nvestment n the vertcally hyper-ntegrated sector,.e. number of unts of productve capacty for fnal consumpton commodty produced durng tme perod t; x n (t) total unts of labour avalable at the begnnng of tme perod t; p (t) prce of a unt of fnal consumpton commodty durng tme perod t; p k (t) prce of a unt of productve capacty for fnal consumpton commodty durng tme perod t; w(t) wage rate durng tme perod t; π (t) a n (t) a kn(t) a n (t) a nk (t) T x k (t) x k (t) k (t) χ (t) proft rate of the ndustry producng fnal consumpton good durng tme perod t; average per capta demand for fnal consumpton commodty durng tme perod t; average per capta demand for unts of productve capacty for fnal consumpton commodty durng tme perod t; drect labour requrements for the producton of one unt of fnal consumpton commodty durng tme perod t; drect labour requrements for the producton of one unt of productve capacty for fnal consumpton commodty durng tme perod t; recprocal of the coeffcent of wear and tear of one unt of productve capacty for fnal consumpton commodty ; demand for unts of productve capacty for fnal consumpton commodty for replacement of worn out capacty durng tme perod t; net nvestment n the vertcally hyper-ntegrated sector,.e. new nvestment demand for unts of productve capacty for fnal consumpton commodty durng tme perod t; stock of unts of productve capacty for the vertcally hyper-ntegrated sector avalable at the begnnng of tme perod t; captal/output rato at current prces n tme perod t for vertcally hyperntegrated sector ; 6

7 As to the prce system, there wll be a prce for each fnal consumpton commodty and a prce for each captal good k assocated to t (p (t) and p k (t), = 1, 2,..., m). The last equaton establshes the condton for the full expendture of (full employment) natonal ncome: ( ) p (t) + π (t) k (t) x + 1 n(t) T p k (t) + a n w = 0 for = 1, 2,..., m p k (t) + a nk (t)w(t) = 0 for = 1, 2,..., m a n(t)p (t) + ( a k n(t) π (t) k (t) x n(t) ) p k (t) w(t) = Vertcally hyper-ntegrated productve capacty (3.2) In general, the means of producton requred to obtan one unt of a fnal consumpton good are a sector-specfc composte commodty n whch the same ntermedate nputs enter n techncally gven proportons. Ths motvates the defnton of a partcular unt of measurement one for each vertcally hyperntegrated sector for ths physcal composte commodty, a unt of vertcally hyper-ntegrated productve capacty. Each of these unts s the sum of three components: drect requrements for the producton of one unt of fnal consumpton commodty ; drect requrements for the replacement of worn-out drect and ndrect captal goods needed for the producton of one unt of fnal consumpton commodty ; and drect requrements for the producton of all ntermedate commodtes drectly and ndrectly needed for the expanson of productve capacty n lne wth the growth of fnal demand for consumpton commodty. In Pasnett s (1981) theoretcal scheme, the unts of productve capacty used as unts of measurement for captal goods are actually unts of drect productve capacty for the producton of fnal consumpton commodtes. Ths becomes clear when lookng at the most complex case, n whch captal goods are produced by means of labour and captal goods (see Pasnett 1981, pp ). However, t s our contenton not only that unts of vertcally hyper-ntegrated productve capacty are the most approprate unts of measurement for captal goods, but also that Pasnett hmself, n 1981, had already begun to argue n terms of vertcally hyperntegraton, even f the complete analytcal mplcatons were stll to be reached (many of them fnally reached ther rgorous formulaton wth the publcaton of Pasnett (1988)). 4 Anyway, n the present, smpler, case, no analytcal dfference can be found between drect, vertcally ntegrated and vertcally hyper-ntegrated productve ca- 4 Ths s partcularly clear when matchng the chapters of the book whch have been almost entrely re-wrtten (Pasnett 1981, p. xv) snce the tme of hs PhD Thess wth the entres n the ndex concernng vertcal hyper-ntegraton. We shall come back to ths pont later on, n secton 6. 7

8 pacty, snce fnal consumpton commodtes are the only ones produced by means of captal goods. Therefore we can nterpret unts of productve capacty as beng vertcally hyper-ntegrated, wthout havng to reformulate the analytcal model n Pasnett (1981). For the sake of smplcty, from now on, we wll smply say sectors nstead of vertcally hyper-ntegrated sectors ; and productve capacty nstead of vertcally hyper-ntegrated productve capacty, except where the complete expressons shall be consdered more approprate. A unt of productve capacty wll refer to the specfc fnal commodty that requres t, and therefore to the specfc sector n whch t s produced. In ths way, the analyss opens up for the possblty of separatng the pace of accumulaton of the means of producton (the number of unts of productve capacty) from ts physcal composton. In order to smplfy exposton, Pasnett (1981) regards these composte commodtes as partcular captal goods, specfc to each consumpton good. Therefore, as hnted at above, n the present context, a vertcally hyper-ntegrated sector s made up by two ndustres: one producng the fnal consumpton good, and the other one producng the correspondng captal good. These two ndustres play an asymmetrc role, snce the physcal quanttes of the means of producton appear as playng a sort of ancllary role wth respect to the physcal quanttes of fnal demand [for consumpton goods]; the former beng, so to speak, at the servce of the latter (Pasnett 1988, pp ). 3.3 Condtons for flow and stock-equlbrum solutons Both physcal quantty and commodty prce systems are formulated as sets of 2m + 1 lnear and homogeneous equatons whch have non-trval solutons f the coeffcent matrx s sngular,.e. f ts determnat s zero. The condton for ths to be true s the same for both equaton systems, and can be wrtten as: a n (t)a n (t) + T 1 a n (t)a nk (t) + a k n(t)a nk (t) = 1 (3.3) If ths condton s not satsfed, the systems are contradctory,.e. each set of 2m + 1 equatons cannot smultaneously hold. However, because of the partcular mathematcal structure of the problem, 5 we can stll get meanngful solutons for quanttes and prces, but the last equaton n each system wll not be satsfed,.e. we shall not be n a stuaton of full employment of the labour force and full expendture of total ncome. On the contrary, f ths sngle condton s satsfed, the solutons wll correspond to a stuaton of full employment and full expendture of ncome. To ths stuaton we shall refer as a flow-equlbrum stuaton. 5 For detals, see Pasnett (1981, pp ). 8

9 Assumng that condton (3.3) holds, we get two ndetermnate lnear homogeneous systems, whch means that we have solutons for relatve quanttes and relatve prces correspondng to a stuaton of flow-equlbrum, but we have to choose a scale factor for each system. For the quantty system ths scale factor s total labour avalable, snce ths s an exogenous varable. Therefore, Pasnett sets x n (t) = x n (t). For the prce system, the choce of a scale factor s n fact an arbtrary one. Followng Pasnett (1981, pp ), we choose, for convenence, the wage rate, and therefore we take t as gven both at a specfc pont n tme and through tme,.e. we set w(t) = w. Therefore, for a gven perod t, the solutons for physcal quanttes and commodty prces n a flow equlbrum are, respectvely: 6 { x (t) = a n (t)x n (t) x k (t) = T 1 (3.4) a n (t)x n (t) + a k n(t)x n (t) and ( p (t) = a n (t) + a nk (t) k (t) x (t) p k (t) = a nk (t)w for = 1, 2,..., m (T 1 ) + π (t)) w (3.5) Expressons (3.4) and (3.5) are made up by m pars of equatons each. The frst equaton of each par concerns consumpton commodty ( = 1, 2,..., m); the second concerns the correspondng unt of vertcally hyper-ntegrated productve capacty k ( = 1, 2,..., m). Expressons (3.4) represent solutons for total physcal quanttes produced n the vertcally hyper-ntegrated sectors = 1, 2,..., m. Frst, x s determned by the average per capta demand for fnal consumpton commodty multpled by total populaton. Second, x k s the sum of two components: T 1 a n (t)x n (t) = x k (t),.e. the number of unts of productve capacty for consumpton good necessary for the replacement of worn out productve capacty, and a k n(t)x n (t) = x k (t),.e. the number of unts of productve capacty demanded as new nvestment. The captal-producng ndustry has to produce not only those unts of productve 6 In the soluton for p (t) gven n Pasnett (1981, p. 41) t s mplctly assumed that x (t) = k (t): p (t) = ( a n (t) + a nk (t)(t 1 + π (t)) ) w Ths amounts to statng that the productve capacty avalable at the begnnng of tme perod t s exactly used up. In order to make the formulaton as general as possble, we have decded not to make ths assumpton at ths stage of the analyss. 9

10 capacty necessary for keepng the ntal stock of (unts of) productve capacty ntact, but also those unts requred to expand t. 7 Expressons (3.5) are the solutons for commodty producton prces. Snce we are assumng that captal goods are produced by labour alone, each prce p k s determned by ts drect labour requrements multpled by the wage rate, whle n the expressons for prces p the wage rate also multples a gross proft mark-up proportonal to the drect labour requred to produce a unt of the correspondng productve capacty. In prncple, there s no dfference between producton prces obtaned when the prce system s formulated n terms of ndustres and when t s formulated n terms of sectors. The technque n use and the dstrbutve varables do not change as a consequence of adoptng the procedure of vertcal hyper-ntegraton, whch s smply a way of re-classfyng and parttonng actvtes n order to explctly acknowledge for the relatonshp between each actvty producng a fnal consumpton commodty and those actvtes producng the means of producton for self-replacement and expanson of the correspondng productve capacty. 8 The dfference s ntroduced as a consequence of the specfc adopton of the unts of productve capacty for fnal consumpton commodtes as the unts of measurement of captal goods. In ths way, each prce p k does not stand for the prce of one ordnary unt of commodty k, but for the prce of one unt of productve capacty for consumpton good. As stated above (secton 3, page 5), an equlbrum poston entals full employment of the total labour avalable whch mples a sngle condton concernng flows and full utlsaton of the exstng productve capacty n each vertcally hyper ntegrated sector ( = 1, 2,..., m) a seres of sectoral condtons concernng stocks. The condton concernng the flows of the economc system has already emerged as the condton for non-trval solutons to the quantty and prce systems,.e. expresson (3.3), whch s a macroeconomc condton, snce t refers to the economc system as a whole, no matter how many sectors there are. Moreover, t emerges from a model whch has been developed on a mult-sector bass, thereby revealng ts truly macro-economc nature (Pasnett 1981, p. 35). As concerns stocks, we have a seres of sectoral condtons, sayng that n each sector, the number of unts of productve capacty avalable at the begnnng of the perod as the captal stock endowment must be exactly equal to the number of 7 Ths s a crucal dfference between the noton of vertcally ntegrated sectors and that of vertcally hyper-ntegrated sectors (see Pasnett 1973, Pasnett 1988). 8 See Pasnett (1973, p. 7, secton 5) and Pasnett (1988, p. 130, secton 4). 10

11 unts of fnal consumpton good to be produced durng the same tme perod,.e.: k (t) = x (t), = 1, 2,..., m (3.6) Expressons (3.4) and (3.5) together wth condtons (3.3) and (3.6) exhaust the descrpton of sngle-perod equlbrum whch s the startng pont for the development of the general mult-sector dynamc model of growth. For the analyss we are gong to perform, we shall assume, from now on, that the economc system starts from a stuaton of both flow and stock equlbrum,.e. we shall assume that, for tme t = 0, both condton (3.3) and the seres of condtons (3.6) hold true. 3.4 General dynamc analyss The dynamc method Pasnett (1981) adopts s that of specfyng exponental laws of movement for the coeffcents n (3.1) and (3.2) concernng total avalable labour (x n ), average per capta demand (a n ), and labour nput requrements (a n and a nk ), accordng to: 9 x n (t) = x n (0)e gt a n (t) = a n (0)e r t a n (t) = a n (0)e ϱ t a nk (t) = a nk (0)e ϱ k t = 1, 2,..., m (3.7) Solutons (3.4) and (3.5) are therefore lnear structures whose components follow exponental dynamcs. Takng expressons (3.4) and (3.5) evaluated at tme perod t = 0, and nsertng the dynamcs descrbed n (3.7) we obtan the followng 9 For the sake of smplcty, we are here assumng steady rates of change of the relevant varables, though ths s not the procedure adopted by Pasnett (1981), at least for the rate of change of fnal demand for consumpton commodtes (See Pasnett 1981, p. 82). Ths s a crude smplfcaton, though t s not possble accordng to the authors to take full advantage of the ncreasng realsm of workng wth non-steady rates of change f the model s specfed n contnuous tme. For the scope of the present work, moreover, the smplfcaton adopted does not compromse the conclusons to be reached. 11

12 solutons for physcal quanttes and commodty prces: { x (t) = a n (0)x n (0)e (g+r )t x k (t) = T 1 a n (0)x n (0)e (g+r)t + a k n(t)x n (0)e gt (3.8) and { p (t) = ( for a n (0)e ϱ t + a nk (0) k (t) x (t) p k (t) = a nk (0)e ϱ k t w = 1, 2,..., m (T 1 + π (t))e ϱ k t ) w (3.9) The dynamc movements n (3.8) and (3.9) do not mply full employment and full utlsaton of productve capacty after tme perod t = 0. In partcular, full utlsaton of productve capacty depends on a seres of stock condtons. In the present model, the stocks of the economc system change accordng to the flow of demand for new nvestment, lnkng one perod to the followng one. The accountng dentty that descrbes ths process of captal accumulaton n each sector s: k (t) x k (t), = 1, 2,..., m (3.10) Gven that x k (t) = a k n(t)x n (t), we obtan k (t) = a k n(t)x n (t). Therefore, the seres of coeffcents a k n(t) s the only one that affects the stocks of the economc system,.e. productve capacty n each sector; hence t cannot be taken as gven from outsde (Pasnett 1981, p. 85). Ths opens up for the possblty to perform a general dynamc analyss by specfyng a law of movement for the level of per capta new nvestment demand (a k n), allowng for the dscrepancy between productve capacty avalable at the begnnng of perod t (k ) and the unts of productve capacty actually used up durng perod t (x ). The specfcaton of the dynamcs of nvestment s a degree of freedom that, once closed, allows for performng an nsttutonal analyss of dfferent theores of captal accumulaton. Another degree of freedom can be opened by changng the last equaton of both the physcal quantty and the commodty prce systems, n order to explctly allow for the possblty of flow-dsequlbrum, e.g. by wrtng: a n (t)x (t) + a nk (t)x k (t) αx n (t) = 0 (3.11) and a n (t)p (t) + ( a k n(t) π (t) k ) (t) p k (t) αw(t) = 0 (3.12) x n (t) 12

13 where α 1. Ths wll accordngly modfy the condton for non-trval solutons to exst, whch wll become: a n (t)a n (t) + T 1 a n (t)a nk (t) + a k n(t)a nk (t) = α 1 (3.13) meanng that macroeconomc condton (3.3) s not satsfed f α Dynamc equlbrum condtons and vertcal hyper-ntegraton Snce Pasnett s (1981) theoretcal scheme ams at descrbng the prmary and natural features of a pure producton system [... ] [whch] wll smply emerge as necessary requrements for equlbrum growth (Pasnett 1981, p. 25), he s concerned, on the one hand, wth the condton for keepng full-employment through tme (flow-equlbrum), and on the other hand, wth the condton for mantanng full utlsaton of productve capacty through tme (stock-equlbrum). As regards the flow-equlbrum, by nsertng (3.7) nto (3.3) we get: a n (0)a n (0)e (r ϱ )t + 1 a n (0)a nk (0)e (r ϱ k )t + a k n(t)a nk (0)e ϱ k t = 1 T (3.14) It can be notced that the demand coeffcents for new nvestment are stll taken as exogenously gven, whle ther specfcaton wll be the subject of the followng few paragraphs. As regards the stock-equlbrum, the laws of moton of average per capta sectoral demands for new nvestment must be such as to be compatble wth the process of economc growth and wll therefore themselves be determned as part of the equlbrum condtons (Pasnett 1981, p. 85). Therefore, as a k n(t)x n (t) = k (t) and, n stock equlbrum, k (t) = x (t), demand for new nvestment must exactly satsfy the growth requrements of productve capacty n each sector, as determned by the growth of demand for each fnal consumpton commodty (ẋ (t) = (g + r )a n (t)x n (t)) n all perods beyond t = 0,.e. the followng set of sectoral captal accumulaton condtons must be satsfed: a k n(t) = (g + r )a n (t), = 1, 2,..., m; t 0 (3.15) whch are the dynamc counterpart of stock-equlbrum condtons (3.6), statng the sectoral equlbrum rates of new nvestment (g+r ) defned as the number of unts of productve capacty, per unt of fnal demand for each consumpton 10 For a hnt at dfferent cases that can occur as a consequence of flow and stock dsequlbra, see Pasnett (1981, pp ). 13

14 commodty, necessary as new nvestment for the expanson of the correspondng productve capacty. The seres of condtons (3.15) can also be expressed as ratos of sectoral new nvestment to producton, at current prces: p k (t)x k (t) = (g + r ) p k (t)k (t) p (t)x (t) p (t)x (t) (g + r )χ (t) (3.16) When wrtten n ths way, sectoral condtons (3.15) tell us that, n stockequlbrum, the rato of new nvestments to the level of producton must be equal, n each sector, to the technologcally determned captal/output rato [χ (t)] multpled by [the sum of] the rate of populaton growth [and the rate of growth of per capta demand] (Pasnett 1981, pp ). In order to fully acknowledge for the mportance of condtons (3.16), we have frst to notce the vertcally hyper-ntegrated character of captal/output rato χ (t). In a tradtonal nter-ndustry scheme, the net output of the economy s the set of commodtes produced for fnal consumpton and new nvestment. However, n a vertcally hyper-ntegrated framework, the net output of the system s made up only of the set of commodtes for fnal consumpton, as new nvestment demand s part of the means of producton requred to expand productve capacty. Therefore, when thnkng of the captal ntensty of a sector, ts denomnator (the net output) wll be the value of fnal consumpton commodty produced n the system, whle ts numerator (the value of captal) wll be the value of the unts of productve capacty specfc to each fnal consumpton commodty requred to self-replace and expand the productve capacty durng perod t. In the lght of ths, the specfcaton of an equlbrum schedule of captal accumulaton n vertcally hyper-ntegrated terms reflects, on the one sde, the nterdependent nature of the producton process, as n the most general case a sngle ndustry producng a basc commodty utlsed as a captal good would partcpate n dfferent sectors wth a dfferent captal ntensty n each of them; and on the other sde, t hghlghts the potental of workng wth vertcally hyperntegrated sectors, as the noton of a physcal unt of productve capacty, by beng defned wth reference to the commodty that s produced, contnues to make sense, as a physcal unt, whatever complcatons techncal change may cause to ts composton n terms of ordnary commodtes (Pasnett 1973, p. 24). 11 Ths s the most remarkable property of the chosen unt of measurement: whatever the tme perod, whatever the stage of techncal progress, whatever the tech- 11 In Pasnett (1981), as each captal goods-producng ndustry s specfc to each consumpton goods-producng one, t s the second aspect that s emphassed, though the framework allows for further generalsaton to reflect also the frst one. See Pasnett (1988). 14

15 nque actually n use, captal goods can always be measured n unts of productve capacty, and the accumulaton of captal can always be studed by evaluatng the number of unts of productve capacty that have to be produced durng tme perod t to mantan stock-equlbrum at the begnnng of tme perod t + 1. In ths way, we can lnk the stocks of dfferent tme perods through the smple captal accumulaton (equlbrum) condtons (3.15) or, equvalently, (3.16). Complementarly, the problem of the change n the physcal composton of these unts can be studed separately by explotng the one-to-one correspondence between vertcally hyper-ntegrated and nter-ndustry relatons as the producton coeffcents of a vertcally [hyper-]ntegrated model turn out to be a lnear combnaton of the producton coeffcents of the correspondng nput-output model (Pasnett 1981, p. 111). By substtutng captal accumulaton condtons (3.15) nto the macroeconomc condton (3.14) and wrtng t as follows: a n (0)a n (0)e (r ϱ )t + ( g + r + T 1 ) ank (0)a n (0)e (r ϱ k )t = 1 (3.17) we can notce that the two addenda dstrbute total labour of the system between the labour requrements of fnal consumpton commodtes and the labour requrements of equlbrum gross nvestments. By lookng at expresson (3.17), t s mmedately clear that, for any specfc composton of fnal demand for consumpton, the equlbrum amount of gross nvestment s unvocally determned by the technque n use and by the dynamcs of populaton and of fnal consumpton demand tself. Hence, the left-hand sde of (3.17) stands for the sze of per-capta total effectve demand n tme perod t. Therefore, (3.17) may be called the effectve demand condton for keepng full employment (Pasnett 1981, p. 54), snce t establshes whether a gven composton of fnal demand for consumpton s compatble wth flow-equlbrum,.e wth full-employment of the labour force. It therefore follows that the dffculty of ncreasng total effectve demand s one of fndng out, and achevng, at a suffcent speed, ts approprate structural composton, and not one of reachng any absolute level (Pasnett 1981, p. 242), hghlghtng the multsectoral foundaton of an effectve demand theory of output. 3.6 Vertcally hyper-ntegrated labour The unts of productve capacty are one of the two consttuent components of the technque of a vertcally hyper-ntegrated sector, the other one beng the vertcally hyper-ntegrated labour coeffcents. In order to defne them, we shall start from the full-employment macroeconomc condton for flow-equlbrum. By nsertng 15

16 (3.15) nto (3.14) and rearrangng, we get: ( a n (0)e r t a n (0)e ϱt + 1 ) a nk (0)e ϱ k t + (g + r )a nk (0)e ϱ k t = 1 (3.18) T whch, by defnng can be wrtten as: l (t) a n (t) + 1 T a nk (t) + (g + r )a nk (t) (3.19) a n (t)l (t) = 1 (3.20) l (t) the vertcally hyper-ntegrated labour coeffcent for sector s the sum of three components: a n (t),.e. drect labour for the producton of one unt of fnal consumpton commodty (drect labour); T 1 a nk (t),.e. drect labour for the replacement of worn-out unts of productve capacty for vertcally hyperntegrated sector (ndrect labour); and (g + r )a nk (t),.e. drect labour requred for the expanson of productve capacty of sector accordng to the growth of fnal demand for consumpton good (hyper-ndrect labour). 12 We can now mmedately take advantage of the just gven defnton n order to express prces n terms of vertcally hyper-ntegrated labour. When condtons (3.15) hold, the prces of fnal consumpton commodtes n (3.5) can be wrtten as: ( p (t) = a n (t) + 1 ) a nk (t) + (g + r )a nk (t) w + (π (t) (g + r ))p k (t) T or p (t) = l (t)w + (π (t) (g + r ))p k (t) (3.21) for = 1, 2,..., m It s nterestng to notce that expresson (3.21) establshes the producton prce of each fnal consumpton commodty as the sum of two components: the cost of vertcally hyper-ntegrated labour emboded n t l (t)w and a proftdfferental wth respect to the sectoral equlbrum rate of new nvestment (π (t) (g + r )) computed on the value of equlbrum productve capacty at current producton prces p k (t). Ths second component s not the (dual) value counterpart of necessary physcal quantty requrements of (re-)producton and expanson, but emerges as an amount of purchasng power created n excess to these requrements, that goes nto the owners of the means of producton through 12 For detals, see Pasnett (1981, p. 102). 16

17 the process of ncome dstrbuton. The amount of ths magntude s a drect consequence of the theory of the ncome dstrbuton that shall be adopted to close the prce system, and t wll nfluence the whole process of structural dynamcs, va ts effect on the pattern of expendture of real ncome. Another mportant magntude whch we shall ntroduce nto the analyss s the level of equlbrum employment n each sector, gven by the product of the correspondng vertcally hyper-ntegrated labour coeffcent and the physcal quantty of fnal consumpton commodty produced durng the tme perod: L (t) = l (t)x (t), = 1, 2,..., m (3.22) It s relevant to stress the vertcally hyper-ntegrated character of L (t): n the most general specfcaton of technology, a fracton of the total labour employed by a sngle ndustry producng a basc commodty would enter nto the employment of all vertcally hyper-ntegrated sectors, ether drectly and/or (hyper-)ndrectly. The comparson wth the vertcally hyper-ntegrated nature of the sectoral captal/output ratos χ (t) n (3.16) s straghtforward. The composton of sectoral employment reflects not only the change n labour requrements of the ndustry producng the fnal consumpton commodty concerned, but also the changng physcal composton of the correspondng unt of productve capacty, and therefore the changes n labour requrements of all the ndustres composng the sector. It s for ths reason that evaluatng only the change n drect labour requrements cannot account for the nterdependent and systemc nature of productvty changes. Ths opens up the possblty of performng emprcal nvestgatons on the dynamcs of productvty takng vertcally hyper-ntegrated sectors as the unt of analyss. 13 We can now specfy the dynamcs of l (t). By defnng, for any varable y(t) n the system, ẏ(t) dy(t)/dt, we can wrte: l (t) l (t) a n (t) ϱ (t) = ϱ l (t) + ϱ T 1 a nk (t) k l (t) + ϱ k (g + r )a nk (t) l (t) (3.23) ϱ (t) s the rate of growth of vertcally hyper-ntegrated labour productvty of sector, gven by the weghted average of the rates of growth of drect, ndrect and hyper-ndrect labour productvty, the weghts beng the proportons of the three knds of labour to total labour employed n vertcally hyper-ntegrated sector, respectvely. 13 For an emprcal study takng ths drecton, see Garbelln & Wrkerman (2010). 17

18 3.7 Dynamc equlbrum path for relatve quanttes, sectoral employment and relatve prces At ths pont, t s possble to descrbe the equlbrum path of relatve quanttes, sectoral employment and relatve prces. Let us start from the general dynamc movements for relatve quanttes and prces gven by expressons (3.8) and (3.9). As we have assumed that the captal accumulaton equlbrum condtons (3.15), as well as the effectve demand condton (3.20), hold, we can now specfy the equlbrum path of relatve quanttes and prces, and the evoluton of sectoral employment gven by expressons (3.22). If we choose to reckon prces n terms of Classcal labour commanded (Pasnett 1981, p. 99), the wage rate stll beng the bass for the prce system, we set w = 1. Hence, the equlbrum dynamc path of relatve physcal quanttes, sectoral employment and commodty prces s gven by: 14 { x (t) = a n (0)x n (0)e (g+r )t ( ) x k (t) = 1 T + g + r a n (0)x n (0)e (g+r )t { (3.24) L (t) = l (t)a n (0)x n (0)e (g+r )t (3.25) and { for p (w) p (w) k (t) = l (t) + (π (t) (g + r )) a nk (0)e ϱ k t (t) = a nk (0)e ϱ k t = 1, 2,..., m (3.26) where the expresson for p (w) (t) s already wrtten n the same form as n (3.21). 15 The equlbrum solutons for physcal quanttes, gven by expressons (3.24), together wth equlbrum sectoral employment, gven by expressons (3.25), represent a set of growng subsystems, one for each fnal consumpton commodty. Each growng subsystem or, equvalently, hyper-subsystem conssts of three components: x (t), x k (t) and L (t). The frst one represents the producton of one sngle consumpton good, expandng through tme at ts partcular rate of growth (g + r ) (Pasnett 1988, p. 127). The second one, represents the physcal 14 For a complete analyss of the equlbrum structural dynamcs of a growng economc system, see Pasnett (1981, pp ). 15 In what follows, whenever a nomnal magntude has a letter n brackets as a superscrpt, that letter wll ndcate the numérare commodty adopted. Therefore, p (w) (t) ndcates the prce of commodty when the numérare of the prce system s the wage rate. 18

19 quanttes for the mantenance of a crcular producton process that both reproduces all the means of producton whch are absorbed by the producton process for [each] consumpton good [... ] and also produces those means of producton that are strctly necessary to expand such a crcular process at a rate of growth (g +r ) (Pasnett 1988, p. 127). Fnally, the thrd one represents the absorpton of a physcal quantty of labour L (t) (Pasnett 1988, p. 127) requred to produce physcal quanttes x (t) and x k (t). To see the mpled structural dynamcs, t s worth computng the rates of change of relatve quanttes, sectoral employment and relatve prces: ẋ (t) x (t) = ẋk (t) x k (t) = g + r (3.27) L (t) L (t) = g + r ϱ (t) (3.28) ṗ (w) (t) p (w) (t) σ(w) (t) = ϱ (t) l ( ) (t) p (w) (t) + π (t) (π π (t) (g + r ) ϱ (t) (g + r ))p (w) k (t) k p (w) (t) (3.29) ṗ (w) k (t) p (w) k (t) σ(w) k (t) = ϱ k (3.30) where σ (w) (t) s the rate of change of the relatve prce of commodty when the numérare of the prce system s the wage rate. 16 As regards sectoral physcal quanttes, ther equlbrum evoluton, gven by expressons (3.27), s completely determned by the evoluton of effectve demand for the correspondng fnal consumpton commodty on whch each growng subsystem s bult. Ths holds true for both x and x k, due to the adopton of the unts of productve capacty as the partcular unts of measurement for captal goods. Hence, the rate of change of physcal quanttes s gven by the sum of two components: the rate of growth of populaton, g, common to all sectors; and the rate of change of sectoral per-capta demands for fnal consumpton commodtes, r, specfc to each sector. Snce these are dfferent from sector to sector, the whole structure of relatve quanttes s changng through tme. As regards sectoral employment, ts equlbrum evoluton, gven by expressons (3.28), s determned both by the equlbrum evoluton of relatve quanttes, and by the dynamcs of vertcally hyper-ntegrated labour productvtes. Snce r s dfferent from ϱ (t), and both are dfferent from sector to sector, the whole 16 It must be carefully notced that expressons (3.29) have a fnte value for π (t) g + r, = 1, 2,..., m. 19

20 structure of employment,.e. the dvson of labour wthn the economc system, s contnuously changng through tme. Ths makes clear how a sectoral reallocaton of employment s an essental requrement for the system to follow a full-employment path. It s worth notcng that, at the most general descrpton of technology, a change n labour productvty n a sngle ndustry producng a basc commodty would be enough to change the whole structure of sectoral employment. As regards relatve prces, let us frst notce that the equlbrum dynamcs for the prce of a unt of productve capacty for fnal consumpton commodty, gven by expressons (3.30), s a partcularly smple one, due to the smplfyng assumpton made,.e. captal goods are produced by means of labour alone. As a consequence, prces are completely determned by labour costs, and therefore ther equlbrum evoluton only depends on the changes of labour productvty n the ndustry producng the specfc captal good for sector. The equlbrum dynamcs of commodty prces, gven by expressons (3.29), reveals the process of change of a producton prce brought about by the nteracton of techncal progress and changes n the dstrbuton of ncome. The rates of change of commodty prces are gven by the weghted average of the rates of change of ther two components. The frst addendum shows how an ncrease n vertcally hyper-ntegrated labour productvty exerts a unvocally negatve effect on producton prces. The second addendum quantfes the effect of a change n ncome dstrbuton through a varaton n the sectoral proft rate on the labour commanded producton prces. As we have already sad, the frst component reflects a necessary, physcal selfreplacement and expanson requrement; accordngly, ts rate of change s completely determned by technology and equlbrum new nvestment,.e. by the rate of change of vertcally hyper-ntegrated labour ϱ (t). On the contrary, the second component of producton prce also reflects ncome dstrbuton. Accordngly, ts rate of change depends not only on labour productvy n the captal goods producng ndustry.e. the rate of change of the prce of the unt of productve capacty on whch profts are computed but also on the varaton through tme of the sectoral rate of proft.e. on the rate of change of the proft dfferental wth respect to the sectoral equlbrum rate of new nvestment. Ths analyss completes the descrpton of the structural equlbrum dynamcs of a growng economc system. In fact, we have descrbed a set of equlbrum paths, one for each possble realsaton of the sequence of sectoral rates of proft, so far consdered as exogenous magntudes. 20

21 4 The natural economc system 4 The natural economc system We are now n a poston to ntroduce what Pasnett calls the natural economc system,.e. that partcular equlbrum path assocated to one specfc sequence of sectoral rates of proft, that, wthout recourse any longer to any exogenously gven economc magntude, now come to complete and close the whole relatve prce system of our theoretcal scheme (Pasnett 1981, p. 131), due to the adopton of a partcular theory of the rate of proft. 4.1 The natural rates of proft As we have explaned n secton 2, the am of Pasnett s (1981) book s that of developng a framework explanng the prmary and natural features of a growng economc system, ndependently of a partcular nsttutonal set-up. A reasonng n these terms, when comng to the ssue of the dstrbuton of ncome, would seem, at frst sght, counterntutve, snce the way n whch ncome s dstrbuted crucally depends on the character of the socal relatons of producton, no less than on cultural, ethc, legal consderatons, that s to say, precsely on the nsttutonal set-up of socety. In fact, those analyses takng ncome dstrbuton as exogenous, are clearly embedded n a specfc nsttutonal set-up. Then, how can a theory of the rate of proft be conceved that s ndependent of t? As Pasnett states, the natural economc system deals wth logcal relatons, based on magntudes gven from outsde economc analyss (and therefore taken as exogenous), and emergng from the physcal growth requrements of the system tself. The problem must be therefore faced from ths perspectve: s there a natural rate of proft (... ) already logcally mpled n the prevous theoretcal framework because the economc system consdered s a growng one? (Pasnett 1981, p. 128, talcs added) The answer to ths queston s: yes. The crucal pont s that at a pre-nsttutonal stage of the analyss, a theory of the rate of proft s not a theory of ncome dstrbuton among ncome recpents,.e. ndvduals or groups of ndvduals. Ths s because the very defnton of the categores among whch the purchasng power generated n the process of producton s to be dstrbuted essentally depends on the socal relatons of producton of a partcular nsttutonal set-up. However, the very nature of an ndustral system requres to perform a separaton between the means of producton that enter a crcular process, and the set of commodtes that are left out from the crcular flow, once they are produced. Moreover, when the system s a growng one, the new nvestment requrements become a necessary expanson of the means of producton. 21

22 4 The natural economc system Now, hence, prces of producton must, on the one hand, be precsely those exchange ratos that satsfy the condtons of re-producton and growth.e. thatnclude the growth of the means of producton at the equlbrum rate of accumulaton. But gven that the equlbrum requrement to expand productve capacty dffers among vertcally hyper-ntegrated sectors, the surplus factor n the prce of producton of each consumpton commodty must reflect ths dfference. On the other hand, prces of producton provde for the purchasng power both to self-replace and expand productve capacty and to consume those commodtes not re-enterng the crcular flow. Consder that profts and wages just establsh the amount of purchasng power that must be channeled to demand for means of producton to expand productve capacty and to demand for fnal consumpton commodtes, respectvely. In ths sense, profts and wages would establsh a truly functonal dstrbuton of ncome, as they stand for categores that channel purchasng power to two dfferent economc functons. These functons arse from the condtons of producton of physcal quanttes, n partcular, from the need to separate what enters the crcular flow (and s used as means of producton) from what t does not (and s consumed). As a consequence, from the reasonng stated above, t follows that profts must correspond to the purchasng power necessary for the equlbrum expanson of productve capacty n each vertcally hyper-ntegrated sector to take place. In formal terms: π (t)p k (t)k (t) = (g + r )p k (t)x (t), = 1, 2,..., m; t 0 (4.1) and therefore, snce n equlbrum x (t) = k (t): π (t) = π = g + r, = 1, 2,..., m (4.2) The equlbrum confguraton correspondng to the just gven structure of rates of proft s the only one that keeps the analyss at a strctly pre-nsttutonal level. 4.2 A pure labour theory of value The rates of proft n (4.2) are the natural rates of proft. When nserted nto the equlbrum solutons for consumpton commodty prces n (3.26), these become: ther rate of change through tme beng: p (w) (t) = l (t), = 1, 2,..., m (4.3) ṗ (w) (t) p (w) (t) = ϱ (t), = 1, 2,..., m (4.4) 22

23 4 The natural economc system Expressons (4.3) hghlght the man result of the present formulaton: when labour s the numérare of the prce system, and the rates of proft are the natural ones, prces.e. labour commanded prces come to be exactly equal to labour emboded. Therefore, ths theoretcal scheme mples a generalsaton of a pure labour theory of value, where the equalty of labour commanded and labour emboded s acheved thanks to a re-defnton of the concept of labour emboded, whch must be ntended as the quantty of labour requred drectly, ndrectly and hyper-ndrectly to obtan the correspondng commodty as a consumpton good (Pasnett 1988, pp ). Wth the ntroducton of the natural rates of proft, both the value of productve capacty for self-replacement and the profts computed on the value of exstng productve capacty have the same functon of computng amounts of labour ndrectly requred elsewhere n the economc system for the equlbrum producton of consumpton good (Pasnett 1981, p. 132). Not less mportantly, ths result holds not at the level of the economc system as a whole, but n each sngle sector. Each growng subsystem followng an equlbrum path of accumulaton has a dual value sde that ascrbes to natural prces a straghtforward foundaton, based on the basc prncple of equal rewards for equal amounts of homogeneous labour. (Pasnett 1981, p. 133). Equally nterestng, at the most general specfcaton of technology, the labour emboded n the basc commodtes produced by a sngle ndustry partcpate n profts of all vertcally hyper-ntegrated sectors. In ths way, changes n the productvy of labour n an ndustry alter the value of profts on captal of all sectors. Thus, t becomes clear that t s not the productvty of captal, or of any commodty, that turns out to be the rason d être of the rate of proft. It s the growth, and the ncreasng productvty, of labour! (Pasnett 1981, p. 133). 4.3 Natural profts, wages, new nvestments and consumpton There s a very clear asymmetrcal relaton between total natural profts and wages. The total natonal ncome produced n a specfc tme perod,.e. the value of total producton, net of replacements, at current prces, s dstrbuted among total (natural) profts and total wages. Whle the former emerge from the physcal condtons for equlbrum growth as a necessty, f full employment and full capacty utlsaton are to be mantaned through tme, the latter can be seen as a surplus, absorbng all the remanng natonal ncome. To produce, and to contnually ncrease ths surplus, through techncal progress, s precsely the purpose of the whole producton process (Pasnett 1981, p. 144). In the same way, there s an asymmetrc relaton between total new nvestments and consumpton. The total quanttes produced n a specfc tme perod, net of 23