Conference on Develoin Quaniaive Marxism Aril 2008, Universiy of Brisol Simulain exended reroducion Povery reducion and class dynamics in Bolivia Jore Buzalo a and Alvaro Calzadilla b a Dearmen of Economics, Göebor Universiy. b ZMAW, Research Uni Susainabiliy and Global Chane Hambur Universiy. ABSTRACT: Dynamic inu-ouu can be seen (Oskar Lane, András Bródy) as a develomen of Marx s exended reroducion model. Soluion of an emirical dynamic inu-ouu sysem yically ives he (equal, consan) rowh raes of secoral ouus, a lon run, equilibrium roorions. In olicy oriened alicaions, a more flexible, simulaion aroach may be useful. Our model resonds o he need of evaluain he effecs of alernaive overy reducion sraeies. Three olicy variables are inroduced, namely, (de)indebedness olicy, invesmen olicy and income disribuion olicy, conribuin resecively o he objecives of olicy auonomy, srucural chane and social jusice. The Millennium Develomen Goal of halvin exreme overy by 2015 seems o be a difficul, bu aainable oal for Bolivia. Given he execed deb reducion areed wih inernaional crediors, he oal can be aained by a combinaion of invesmen and redisribuion olicies. The model shows also he effecs of overy reducion sraeies on he differen social classes. Conens 1. Inroducion 3 2. Reroducion schemes and inu-ouu analysis 5 2.1. Simle reroducion 8 2.2. Exended reroducion 8 2.3. Exended reroducion and he Harrod-Domar equaion 9 3. A simulaion model for exended reroducion 10 3.1. The equaion of exended reroducion 11 We hank research rojec colleaues Rolando Morales and Irene Loiseau, and aricians a seminars and conferences in which revious ars of his sudy were discussed. The sudy has received suor from he Swedish Inernaional Develomen Aency. Corresondin auhor: Tel.: 46 8 283594. j.buzalo@elia.com. 1
3.2. Income disribuion 14 3.3. Savin and consumion 16 3.4. Excess demands 18 4. Simulain exended reroducion sraeies 19 4.1. Reroducin saus quo 21 4.2. Povery minimisin srucural chane 23 4.3. Exanded reroducion wih srucural chane and income redisribuion 26 4.4. Povery reducin vs. emloymen increasin reroducion 28 5. Povery reducion sraeies and social classes 30 5.1. An aroximaion o overy and class in Bolivia 30 5.2. Srucural chane and social classes 33 5.3. Povery reducion and social classes 35 References 43 Aendices A: Deailed model srucure 45 B: Model daa 52 C: Social Accounin Marix of he Model 57 I is enerally aken for raned ha Marxian economics and recen achievemens in mahemaical economics wih larescale diial comuaion are worlds aar. Cerainly his need no be he case. András Bródy (1970,. 163) 1. Inroducion In his Preface o Bródy s (1970) book, Wassily Leonief holds ha economics does no roress in a sraih line, as a yical naural science does. Economic houh advances in curves and loos, like a broad river slowly windin is way across a fla lain. I urns lef and rih and divides from ime o ime ino searae branches, some of which end u in sanan ools, while ohers unie aain ino a sinle sream. Abou hree decades ao, he mainsream of economics briskly urned rih, o ener he inellecual oorahy characerisic of lon-eriod deression. I enered he raher serile landscae of commodiy and marke feishism. Eonomics could his broad branch of he river more roerly be called, based as i is on he consriced ersecive of he homunculus oeconomicus. Afer raversin he dark valley of he lon deressive hase, i seems o have ended u in a bi, sanan ool, incaable of ivin meaninful answers o he many roblems of senile, lobal caialism. The several beaches on he sides of he ool such as monearism, suly side economics, and oher varians of he neo-liberal/neo-conservaive credo are raidly becomin ou of fashion and desered. 2
The Marxian branches of he river almos disaeared under he round ha was oened as a resul of he owerful seismic movemens of he eriod. They were oblied o follow difficul and rocky underround ways ha hoefully urified and clarified hem. Similar faes corresonded o close sandin branches, such as neo-ricardian, neo-keynesian and neosrucuralis economics. Will hese branches be able o surface aain? Alon wih he new branch of ecoloical economics, will hey unie ino a sinle lively sream, as in Leonief s meahor? Will hey exloi he achievemens in mahemaical economics and he hue advances in diial comuaion, as Bródy suesed? Is he resen Conference on Develoin Quaniaive Marxism a sin ha i is so? The model consruced for he resen sudy resonds o real and uren needs how o reduce overy in a oor counry. I was no inended as a conribuion o Marxian economics, bu as an aroximaion o he analysis of a aricular, hihly relevan roblem. Wha is ineresin wih Marxian economics is ha he mos meaninful and useful analyical insrumen o aroach he roblem of fihin overy ha can be found, was already drafed by Marx. The choice of he heoreical srucure of our sudy was no he consequence of an ideoloical ari ris, bu he resul of he sric alicaion of he scienific mehod he choice of he bes available mehod of analysis alicable o a concree se of observaions in order o obain esable deducions. As we see i, his is scienific socialism, in he sense of uin reason and science in he service of he free flourishin of human bein and human sociey. The source of he aroach is Marx s reroducion schemes. As we show below, Marx s aroach was refined by several auhors, before ein is mos develoed form from Leonief. Ye only Marxis auhors such as Lane and Bródy dared o refer o he aaren Marxian roos of inu-ouu analysis. Mos ohers referred no o be associaed wih such a hereic. Quesnay and Walras were more resenable forefahers even if Walras canno honesly be associaed wih he aroach. Our reroducion simulaion model is, like he dynamic inu-ouu model, a disareaed Harrod-Domar model. Growh deends on he maniude of social surlus (savin) and he effeciveness of invesmen (incremenal caial ouu raios). Wha rincially disinuishes he model from he dynamic inu-ouu model is he characer of he soluion. The soluion of he dynamic inu-ouu model ives he uniform, equiroorional rae of exansion for all secors of he economy, and he ouu comosiion consisen wih ha rae of rowh. This soluion can be undersood as he ideal, inheren exansion caabiliies and equilibrium roorions belonin o he echnoloical infrasrucure (inu and caial marices) of he economy, for iven disribuional srucures and savin/consumion behaviours. The simulaion aroach is an alied, olicy oriened aroach. I dears from condiions exisin a he sar of he eriod of analysis, i.e. iniial ouu levels and roorions. I sudies differen olicies, which resul in differen ouu rowh rajecories, iven iniial condiions, and iven echnoloies and behaviours in he economy. A key olicy variable is invesmen olicy, which influences srucural chane in he economy. Anoher imoran olicy variable, which affecs he deree of olicy auonomy, is he level of forein indebedness, or accumulaed forein savins. The hird key olicy elemen is income disribuion olicy, aricularly crucial in he conex of hihly unequal economies. The deree of disribuive equiy or social jusice direcly affecs he overy rae, whose reducion is he main focus of our sudy. 3
The frame of our sraey simulaions is he UN Millennium Develomen Goal of halvin exreme overy by 2015. Differen ouu rajecories resul in differen raes of overy reducion. A saus quo invesmen olicy of mainainin he as emhasis in naural resource roducion and exor is less effecive in reducin overy han a olicy which ries o enerae overy minimisin ouu rajecories. Our sudy also shows ha a olicy which simulaes increasin emloymen is very similar in is effecs o a overy minimisin olicy. Given in Bolivia a very hih overy rae of 70 ercen of he oulaion and a very unequal income disribuion, chanes in he srucure of ouu are no sufficien for achievin he Millennium Goal. The sudy shows wha are he minimal income redisribuion effors, which in addiion o ro-oor srucural chane are necessary in order o halve exreme overy by 2015. A more advanced redisribuion olicy is also simulaed, which oally eliminaes exreme overy. The oores counry in Souh America, wih an averae income no much above he overy line of 2 dollars a day, all social classes are more or less affeced by overy in Bolivia. Bu caialiss, workers and easans benefi differenly from differen sraeies. Saus quo ends o benefi hih income classes, and redisribuion olicies benefi low income classes. Dualism and heeroeneiy in he economy and ossibly also saisical limiaions ive rise o some counerinuiive resuls, such as for insance ha he workers do no direcly benefi of overy reducin olicies. These are main hemes of he followin secions. 2. Reroducion schemes and inu-ouu analysis A ermanen rai of he classical heory of economic rowh, from Quesnay o Marx, is he idea ha he exansion of socieies roduc deends on invesmen of he non consumed surlus roduc. There is rowh when he roduc of sociey s work is more han wha is necessary o saisfy consumion needs, and when his social surlus ouu akes he form of increased means of roducion u o work wih increased labour force. Two cenral ideas are hen, ha oal ouu mus be larer han consumion ha is, osiive surlus roduc or savins, and ha surlus roduc increases he exisin means of roducion ha is, i is roducively invesed. Quesnay s Tableau Économique (1766) was, in he words of Marx, he firs sysemaic conceion of caialisic roducion an aem o orray he whole roducion rocess of caial as a rocess of reroducion he mos brillian idea of which oliical economy had hihero been uily (1963,. 344). The Tableau models he circular flow amon of he hree main social classes of he inciien caialism of he ime easans, manufacurers/merchans and landlords and he inerdeendence beween income and exendiure in consumion and roducion. A he same ime, he classes reresen differen secors of economic aciviy. Quesnay refers o he sequence surlus-invesmen-rowh in oher wriins, bu he Tableau deics an economy in simle reroducion, or saionary equilibrium. 1 Marx s analysis of reroducion esablishes he condiions for equilibrium in an economy wih wo secors (roducin means of roducion and consumion oods) and wo social classes, 1 Ealy (1969) consrucs a dynamic version of he Tableau. 4
caialiss and workers. The seady reroducion of he circular flow of he economy requires ha he comonens of he social ouu ha is, used means of roducion and labour ower, and surlus kee definie roorions. In aricular, he ouu of he secor roducin means of roducion mus equal he use (demand) of means of roducion by boh secors. In an exandin economy no all he surlus is consumed; ar of he surlus is emloyed in increasin he amoun of labour and means of roducion. These condiions can be inerreed as inu-ouu equaions, which resul in aricular condiions of equilibrium beween he wo dearmens. For Marx, he imorance of he reroducion schemes ress no only in he inersecoral consisency of equilibrium quaniies, bu also in he eneral framework required for a coheren deerminaion of labour values and roducion rices. As shown by Kurz and Salvadori (2000), afer Marx, he mos imoran roos of inu-ouu analysis are o be found in he works of auhors invesiain he deerminaion of values and/or roducion rices wihin a comlee reresenaion of he circular flow of he economy as an inerrelaed sysem. Amon he mos imoran hey include Dmiriev, Borkiewicz and Charasoff. 2 Anoher imoran source, no oo ofen recalled oday, is he aem o consruc a naional economic balance in he Sovie Union of he 1920s. This inended ool of lannin for indusrialisaion and rowh, was an exlici emirical imlemenaion of he ideas underlyin he reroducion schemes of Marx. 3 Sill a suden a Leninrad Universiy, Leonief wroe abou his work in 1925: Wha is essenially new in his balance is he aem o embrace in fiures no only he ouu bu also he disribuion of he naional roduc, so as o obain in his way a comrehensive icure of he whole rocess of reroducion in he form of a kind of 'Tableau Économique'. 4 For auhors conizan of he Marxian radiion, such as Oskar Lane and András Bródy, i was easy oo see he direc connexion beween Marx s reroducion schemes and Leonief s inuouu models. For Lane (1969,. 47), []he srucure of roducion inu equaions is he same as ha of Marx s schemes I can be seen ha he roducion inu equaions are an exension of he division of he Marxian schemes ino n branches. Or also, Marx u forward he eneral idea ha a balanced exchane of roducs amon he various subdivisions of he naional economy was essenial if he rocesses of roducion and reroducion were o coninue smoohly; in inu-ouu analysis his idea is alied o he relaionshis arisin amon a lare number of secors of he naional economy. (Lane, 1964,. 192) 2. 1. Simle reroducion 2 An ineresin deail is ha von Borkiewicz was Leonief s disseraion adviser a Berlin Universiy in he 1920s (Kurz and Salvadori, 2000,. 169). 3 Accordin o Jasny (1962), he idea was fahered by V.G.Groman, who roduced he firs draf of a balance of he naional economy in 1923 a he Goslan. Salin houh his kind of work was a ame wih fiures (see references in Sulber and Dadkhah, 1975). The ask which Groman had houh could have been accomlished in a maer of several weeks was ulimaely done 38 years laer. Early in 1961 Sovie saisicians comleed The Inerbranch Balance of Producion and Disribuion of Ouu in he Naional Economy of he USSR for 1959 The seed of accomlishmen is cerainly amazin. (Jasny, 1962,. 79) 4 Welwirschafliches Archiv, Ocober 1925 (Jasny, 1962,. 79). 5
Wih he owerful ools of he eienvalue marix alebra and Leonief s inu-ouu reresenaion, Bródy (1970) achieves a mos comac and lucid formulaion of he heory of reroducion. Given he marix A of inu coefficiens (denoin amouns of roduc i used o roduce one uni of roduc j), and vecors v reresenin he inus of labour force and c consumion needs, we can form (Bródy, 1970,. 23) he comlee marix A: A A = v c 0. (1) The economy is in simle reroducion when ouus x are jus enouh o cover (inermediae and final) consumion needs: Ax = x. (2) This is he same as sayin ha he maximal eienvalue of marix A is equal o one. Tha is, iven he marix A, and asked which are he scalars α (eienvalues) and vecors x (eienvecors) ha saisfy he eienequaion Ax = α x, we find in simle reroducion ha he maximal eienvalue α = 1. 5 The ross ouu vecor x, as an eienvecor, ives only he roorions which saisfy he eienequaion Ax = α x any scalar mulile of x is also an eienvecor; only ouu roorions are deermined by A, no he absolue scale of roducion. 2.2. Exended reroducion In simle reroducion hen, final and inermediae consumion Ax equal oal ouu x no surlus is lef. Exended reroducion is ossible when no all ouu is consumed, when Ax < x; ha is, when here is a osiive surlus roduc in every secor: x Ax = (I A) x > 0. Each secor of he economy roduces a osiive surlus. Ouu rowh deends on surlus bein invesed in exandin roducive caaciies. A marix B is inroduced, is coefficiens indicain he quaniy of ouu of secor i which mus be invesed in secor j in order o increase by one uni secor j s ouu in he nex eriod. Balanced exanded reroducion requires ha he surlus roducs of he differen roducion secors on he lef hand of equaion (3) mach he invesmen needs eneraed by he uniform rae of rowh λ on he rih hand: (I A) x = λ Bx, (3) ha is, ouu should be so srucured as o make ossible a balanced rowh in all secors. Or also, if we wrie his as x = Ax + λ Bx, (4) ouu sulies x equal consumion lus invesmen demands associaed wih rowh a he rae λ. 5 The eienequaion Ax = α x may also be wrien: α x -Ax = (αi A) x = 0. The eienvalues are hose values, α, ha make he deerminan of he marix (αi A) sinular. The deerminan is an equaion of deree n in α, wih n, no necessarily disinc roos, associaed wih heir resecive eienvecors (for more deails see e.. Bródy, 1970, Aendix I, or Wilkinson, 1965). 6
The soluion of his equaion for x ives ouu roorions ha, afer coverin (inermediae and final) consumion Ax for simle reroducion, allow for rowh in every secor a rae λ. 6 Bródy (1970,. 47) shows ha he numerical examles of exended reroducion se ou in he second volume of Caial are based on he same assumions as hose imlici in equaion (4). The ideas underlyin his model and hose exressed in Marx s schemaa are he same hey are an exension o n secors of he exended reroducion scheme. 2.3. Exended reroducion and he Harrod-Domar equaion Several auhors have seen Marx s exanded reroducion scheme as he oriinal source of he Harrod-Domar ye of rowh model. Bródy (1970,. 100) shows his in a very sraihforward manner, by definin roducion rices wihin he exandin reroducion sysem of equaion (4). 7 Prices of roducion are hose rices which cover inu and labour coss A, lus an averae rae of rofi λ on caial invesed B: (5) = A + λb. 8 If we now exress equaion (3) of exanded relicaion usin he rice sysem and he ouu srucure which corresond o he balanced soluion eienvecors, we will have: (I A) x = λ Bx. (6) The rae of rowh (and rofi) λ can hus be exressed as he ne roduc of sociey divided by oal caial emloyed: λ = (I A) x / Bx. (7) Mulilyin numeraor and denominaor by he oal value of roducion x, we e: λ = [(I A) x / x]. [x / Bx] (8) in which he firs facor is he savin raio, and he second facor is wha Bródy calls caial roduciviy, he recirocal of he caial/ouu raio. This is he Harrod-Domar rowh equaion. In his illuminain aricle of 1957 Lane arrived, wih a differen (more consrucivis) mahemaical aroach, o he same conclusion: he rae of increase of ross naional roduc is he roduc of he overall rae of invesmen and of he averae ouu-oulay raio (Lane, 1964,. 217). Lane s ouu-oulay raios indicae he effec of a uni increase in invesmen oulay in he various secors of he economy on naional ross ouu. 6 The corresondin eienequaion now is: [I/λ (I A) 1 B] x = 0. The viable rowh rae λ is (he recirocal of) he maximal eienvalue of he marix (I A) 1 B, o which corresonds a osiive eienvecor x. Bródy (1970,. 113) calls his soluion saionary sae i is also called urnike or von Neumann ah. 7 For Bródy, he main oins of he Harrod-Domar model were already imlici in Kalecki s heory of he business cycle dain from he early 1930s, and also in he Sovie economis Feldmann s wo-secor models from he 1920s. 8 The eienequaion of his sysem is : [I λ B (I A) 1 ] = 0. 7
3. A simulaion model for exended reroducion Bródy (1970,. 112) reconised he lack of realism of a soluion ivin a uniform exansion rae wih fixed ouu roorions as one of he major limiaions of he dynamic inu-ouu model. However, a eneral soluion of he eienequaion for a iven economy, ha is, a soluion for he rices, ouu srucure and raes of rowh and rofi ha corresond o a lon run, closed, balanced rowh ah may ive many ineresin insihs. 9 In racical alicaion, however, in aricular when analysin olicies for rowh and overy reducion, a differen soluion aroach seems necessary. Insead of definin oimal ouu srucures for a closed economy, our aroach is o sar from exisin ouus in an oen economy, and o look a he effecs of differen ouu rajecories resulin from differen invesmen aerns. Tha is, we ado he simulaion aroach. Oimisaion eners he icure when we aem o selec he ouu rajecory ha maximises cerain objecive, e.. overy reducion. Two oher differences of our model refer o he inclusion of forein savin and deb, and income disribuion. The model reflecs he ossibiliy of enlarin he economy s savin and invesmen caaciy by borrowin from abroad (i.e. by increasin forein indebedness) or also he ossibiliy of aumenin economic olicy auonomy by reducin exernal indebedness, alon wih he condiionaliies imosed by forein crediors. The model includes also a deailed reresenaion of income disribuion and consumion/savin. Vecor v reresenin inus of labour force and vecor c describin consumion in equaion (1) are disareaed by income level and social class, in order o race he effecs of alernaive sraeies on savin, consumion, invesmen and rowh. 3.1. The equaion of exended reroducion: he dynamic link As shown above, exanded inu-ouu reroducion can be seen as a disareaed Harrod- Domar model, in which rowh deends on averae caial ouu raios and oal savin. As shown in equaion (8), he rowh rae equals he ouu caial raio imes he savin raio. Or, mulilyin by ouu, he incremen of ouu equals he ouu caial raio imes savins. Accordinly, he equaion of moion of he model is: x +1 x = ˆ α 1 d. (9) Tha is, he incremen of secoral ouus x (an n-vecor) equal heir resecive secor s incremenal caial ouu raios αˆ (a diaonal n-marix), imes he secoral invesmens (an n-vecor). d ˆ 1 Wriin his as: x + 1 = α d + x (10) we see ha he ime-ah of secoral ouus deends on he (inverses of) secoral incremenal caial ouu raios and secoral invesmens. (The inverse elemens ˆ α 1 reflec secoral ouu resonses o invesmen; we will call hem invesmen efficiency coefficiens. ) In he words of Lane (1964,. 269): The invesmen done in one eriod adds o he amoun of 9 See e.. Bródy (1970, Par 3) on USA and Hunary and Tsukui and Murakami (1979) on Jaan. 8
means of roducion in oeraion in he nex eriod. In consequence, a larer ouu is obained in he nex eriod. The ouus of successive eriods are linked u in a chain hrouh he invesmens underaken in each eriod. Thus, roducive invesmen eneraes a rocess of rowh of ouu. The rocess follows he simle loic illusraed in Fiure 1. Gross ouus x Invesmen efficiency ˆ 1 α Secoral invesmens d Fi. 1. Flow diaram of he dynamic law of he model We can now inroduce our firs olicy insrumen, invesmen olicy. Invesmen olicy is he olicy insrumen which allows for influencin secoral rowh and ouu srucure. In order o secify an invesmen olicy in he model of equaion (9), le us disinuish beween rivae invesmens d, and ublic invesmen d, of which secoral invesmen d is he sum. 10 Toal invesmen equals oal savins, and we assume for simliciy ha he overall equaliy beween savins and invesmen also alies for he rivae and ublic secor aken searaely. Then, in he conex of he model, iven endoenously deermined ublic savin s * (a scalar), ublic invesmen is deermined by invesmen olicy: d = z s *, (10) where z is a disribuion vecor of ublic invesmen allocaion shares. An invesmen olicy is a ime sequence { z } of ublic invesmen allocaion shares. A { z } sequence can be exoenously iven, as for insance in a hisorical simulaion, or when some exoenously siulaed olicy is esed. A { z } sequence can be also deermined by oimisaion of some exression of social welfare e.. minimisaion of he share of he oor in year 2015. We see hen ha we are osulain he ossibiliy of any allocaion of invesmen we are no ryin o find balanced ouu ahs as in exanded reroducion (equaion (3)), in which invesmen demands mus be saisfyied by domesic ouus. We assume an oen economy, where excess demands and sulies can be inernaionally raded e.. excess sulies of oil can be exored, and excess demands of invesmen oods can be imored. (We are now describin secoral sulies; secoral demands will ve described in shor.) Also, rivae invesmen behaviour differs from Marx s schemes, where caialiss inves in heir own secor. In he resen model, caialiss allocae invesmens ha is, available savin/invesmen funds accordin o he differen secors as rowh and heir resecive 10 Public invesmen is defined in a very wide sense, as he cos for he ublic secor of roduciviy-increasin chanes. Public invesmen includes invesmens in infrasrucure and ublic and mixed enerrises, and also he coss of exlici or imlici subsidisaion of rivae invesmen (such as he differen forms of indusrial olicy ). Theoreically, i should also include invesmen in he social secors ( human caial ), and in research, bu his is difficul o imlemen saisically. 9
caial densiy. In he followin equaion, rivae invesmens,, equal rivae savins s *, endoenously allocaed accordin o he so called acceleraor funcion z : d d = z s *, (11) in which z ˆ( α x x = ι' ˆ( α x x 1 1 ), (12) ) where ι is a summin vecor (1,1,,1)'. Tha is, oal rivae savins are allocaed accordin o as rowh and caial/ouu raios in he secor. 11 3.2. Income disribuion We have hus far described he invesmen of rivae and ublic savins and heir effecs on ouu rowh, iven invesmen efficiency arameers, as shown in he lower and lef sides of Fi. 2. Le us now comlee he loo, describin how savins are deermined from incomes eneraed in roducion, for iven income disribuion and savins coefficiens, as shown in he uer and rih sides of he fiure. Gross ouus x Invesmen efficiency ˆ 1 α x Income disribuion V V = Vi Vx Public consumion c Public income y Forein Savin Φ Privae income y Savin roensiies } I ι ' Γ ( I Vˆ ) d Secoral invesmens d z Invesmen olicy z Privae invesmen allocaion ˆ α = ι' α ( x x 1) ˆ( x x ) 1 Public savin s * Privae savin s * Fi. 2. Flow diaram of he dynamic core of he model In he simles form of reroducion analysis workers receive subsisence waes, and caialiss aroriae all surlus value or rofis. In sandard inu-ouu, and even in he more elaboraed analyses of Lane and Bródy he srucure of income disribuion is no 11 The esimaed funcion is a disribued acceleraor includin hree revious eriods, see Aendix A, eq.(11). 10
secified. Wihin he inu ouu framework, however, a quie naural exension is o assume income disribuion coefficiens analoue o roducion inu coefficiens, hus eneralizin o he disribuion of value added he roorionaliy assumion made in relaion o roducion inus. Insired by Kaldor (1956), Miyasawa and Masei (1963) inroduced his aroach, and defined secoral income disribuion coefficiens by income class (e.. caialiss and workers), wih aricular consumion and savin behaviors. 12 In line wih his aroach, he y vecor of incomes of he k income classes, lus he overnmen, lus exernal roducion facors deends linearly on secoral ross ouus x : y = V x, (13) in which V is a (k + 2) n marix of shares of income (value added) by income class (lus he overnmen and exernal facors), secific o each roducion secor. The V marix is comosed of a (k n) V marix of rivae incomes, an (1 n) V i marix of secoral coefficiens of indirec axes and oerain surlus of domesic enerrises, and an (1 n) V x marix of ne flows of rivae exernal facors. 13 The model hus direcly disribues value added eneraed in roducion amon households, he overnmen and he res of he world a simlificaion of he usual social accounin marix (SAM) framework, in which he disribuion amon facors of roducion and insiuions (firms, overnmen, ec.) is also included. 14 The V marix of secoral income disribuion may assume differen secificaions. The Kaldor-Miyasawa-Masei secificaion analyses he secoral disribuion of income amon social classes. Analysis of overy and overy reducion olicy also requires a reresenaion of he size disribuion of incomes. Unless defined very narrowly, a socioeconomic classs may include boh oor and non-oor households. Thus, a (100 n) marix V is defined, describin secoral income shares by erceniles. The (100 n) able: c w x V (14) deics he disribuion of incomes by erceniles in each secor, wih is cells showin oal incomes by secor and ercenile. Given he execed (oal and secoral) oulaion over ime, er caia secoral incomes by erceniles are obained. For a iven overy line income y (a scalar), he sum of ersons belonin o cells wih er caia incomes below of he overy line ives he number of he oor in eriod. 15 line 12 Baey and Rose (1998) survey he lieraure on his class of exended inu-ouu models. 13 In addiion o indirec axes and oerain surluses, he ublic secor s income includes imor duies, ransacion axes (and oher indirec axes), direc axes, ne unilaeral ransfers and ne deb service (see equaion (2) in Aendix A). 14 Aendix C shows he srucure of he simlified SAM of he model, and he SAM eneraed by he model for he iniial year 2000 (he model eneraes SAMs for every year of he simulaion). See Round (2003) for a survey of SAM lieraure, an aroach closely relaed o exended inu-ouu. 15 See eqs. 24-30, Aendix A, on he deerminaion of oulaion, secoral emloymen, er caia incomes, and overy measures. 11
The second olicy dimension of he model, in addiion o invesmen olicy, is income disribuion olicy. In he conex of he resen model, an income disribuion olicy is a sequence { V } of income disribuion marices. In Secion 4.3, for examle, we solve for an income disribuion olicy saisfyin he Millennium Goal of halvin exreme overy by 2015. 3.3. Savin and consumion Le us now close he loo of Fiure 2, briefly describin he deerminaion of savin and consumion from incomes eneraed in roducion. Our reresenaion of consumion and savin behaviour is already resen in Lane s (1964) ahbreakin analysis. Lane inroduces (. 207) demand equaions, based in behavioural daa, which relae final consumion by roduced iem o value added. Bu while in Lane savin/invesmen is an exoenously deermined share of oal income, in he resen model domesic savins are endoenously deermined, non-consumed incomes. Economic olicy can sill influence he rae of invesmen by influencin income disribuion olicy { V } in aricular, he share of ublic income in oal income. This ossibiliy is no ried in he resen sudy. Public savins (a scalar) is defined as he difference beween ublic income and ublic consumion: s = y c. (15) In order o deermine rivae savins, le us firs define rivae consumion by ye of ouu: c ( I Vˆ ) y = Γ, (16) d where c is an n-vecor of consumion demands, Γ is an (n k) marix of consumion roensiies, V d is a (1 n) marix of coefficiens of direc axes and I is he ideniy marix. Privae consumion demands by income class (a k-vecor) are: c * = } Γ ' ι ( V ) y d I ), } in which ι is a (n 1) summin vecor (1, 1,, 1) and ι ' Γ a (k k) diaonal marix of oal consumion roensiies by income class. Hence, rivae savins by income class (a k-vecor) are: s } = I ' ( I Vˆ d ) y ι Γ. (17) 12
Our hird olicy dimension, alon wih invesmen and disribuion olicies, is forein savin and indebedness. The rae of ne inflow of forein caial, which adds o exernal indebedness, is an exoenous variable in he conex of he model. I may be undersood as a olicy variable, which works hrouh olicies relaed o inernaional caial flows and currency exchane. The level of exernal indebedness reflecs he economy s deree of olicy auonomy, as hihly indebed counries are consrained in heir olicy choices by exernal crediors olicy references. Exoenously deermined (neaive or osiive) ne forein savins Φ added o rivae and ublic savin (for simliciy, a an equal rae), hus increasin (or decreasin) he volume of resources available for domesic invesmen. A aricular ime sequence of { Φ } forein savin consiues an indebedness olicy in he conex of he model. In Fiure 2, s * and * s reresen hen oal (domesic lus forein) savins i. e. hey include he effecs of addin forein savins o domesic savins s and s (see eqs. 8 o 10 in Aendix A). 16 To recaiulae. The model simulaes he reroducion of he economic sysem rouh ime under differen olicy assumions. Sarin from a iven iniial siuaion, he model describes he evoluion of he economy under differen olicy sequences. Given: (a) iniial ouu values, (b) values of model arameers and exoenous variables, and (c) olicy arameer sequences for invesmen olicy{ z }, income disribuion olicy { V }, and indebedness olicy { Φ }, he model can be recursively solved forward in ime, so as o numerically deermine he rajecories of secoral ouus and oher endoenous variables. Endoenous income levels (by erceniles) deermine in urn overy incidence (for a iven overy line) and oher welfare indicaors such as he Gini coefficien of inequaliy. Policy sequences { z }, { V } and { Φ } can be exoenously iven, or hey can also be deermined by oimisaion of some exression of social welfare. Policy sequences are exoenously iven, for insance, when as develomen is simulaed, or when a aricular sraey is esed, as in he case of he official overy reducion sraey in he nex secion. In Secion 4.2, he model is solved for he invesmen olicy { } z ha minimises overy by 2015, for iven income disribuion { V } and indebedness { Φ } olicy sequences. 3.4. Excess demands We have hus far described he evoluion of ouus/sulies over ime under differen olicy assumions (oal suly is comosed of secoral ross ouus x, and includes also secoral imor duies, ransacion axes and oher indirec axes n-vecor o in equaion (18) below). Le us finally inroduce secoral demands, in order o enquire ino he horizonal balance beween secoral sulies and demands. 17 Toal secoral demand is comosed of: (i) inermediae demands, (ii) consumion demands, and (iii) invesmen demands. 16 Forein savins add also o he exernal deb of he eriod; see equaion (7) in Aendix A. 17 In Aendix A.9, a flow diaram describes he whole model, includin suly/demand balances. 13
Inermediae demands resul immediaely as he roduc Ax, iven he (n n) A marix of echnical coefficiens. Consumion demands by ye of ood or service where described in equaion (16) above. Invesmen demands are relaed o invesmen by desinaion d hrouh he caial coefficiens (n n) marix H of secoral comosiion by secor of oriin of invesmens by desinaion. 18 Then, he n-vecor of secoral excess demands q is he difference beween secoral sulies and demands: q = x + o Ax c c H d. (18) When all ouu is inernaionally radable and world rices revail in he economy, q reresens secoral rade balances osiive elemens are ne exors and neaive elemens are ne imors. Tha is, he model is aimed a deicin a small oen economy, in which exoenous and for he uroses of he analysis, fixed relaive rices are assumed o revail. The q vecor hus reflecs he effec of he sraey on inernaional rade secialisaion. In he framework of his model, a balanced rowh ah such as he urnike soluion of Bródy s equaion (3) for exended reroducion in racice, an invesmen olicy ha minimises secoral excess demands (in absolue value) is one of many differen ossible sraeies. I would be he case of a aricular aern of srucural chane, from cerain iniial secialisaion rofile oward a more self-sufficien economy. 4. Simulain exended reroducion sraeies The above reroducion model was firs conceived for simulain alernaive develomen sraeies in Mexico (Buzalo, 1984). The by ha ime curren official sraey of erolisaion, ha is, of dramaically exandin roducion caaciies in he oil secor, was comared o an alernaive basic needs oriened sraey, in which ariculure and oher wae oods roducin secors were develoed. The basic needs sraey included also an income disribuion olicy which siulaed imrovemens in he incomes of low-income classes (ariculural day labourers, oor easans and he urban workin classes, formal and informal). Simulaion wih he model showed how erolisaion would resul in increasin imors of foodsuffs. I showed also he crucial imorance of increasin invesmen efficiency in ariculure, esecially in he case of a basic needs-based sraey. In he Mexican sudy invesmen olicies are exoenously siulaed; he lanned erolisaion invesmen olicy of he overnmen is simulaed, and comared wih a ublic invesmen sequence in which he weih of oil-relaed secors is reduced, and ha of essenial oods roducin secors is increased (also, he weih of invesmen oods roducin secors is increased in he medium erm). In Buzalo (1991), alied o Arenina, some invesmen olicies are obained by oimisaion, insead of bein exoenously siulaed. Livin off our means, a olicy sloan a a ime of hih indebedness and widesread caial flih, is simulaed as a sraey combinin (a) an invesmen olicy renderin rade balances (secoral excess demands) as close o zero as ossible (i.e. maximisin imor- and exor subsiuion), (b) a saus-quo, 18 Bródy s marix B is equal o our marix H mulilied by incremenal caial ouu raios αˆ (see e.. Dervis e al. 1982, Chaer 2). 14
hih inequaliy income disribuion olicy, and (c) an exernal deb moraorium reducin he sream of fuure forein dissavin. In an alernaive ye of oimisaion, an Ideal oenin is exlored, in which invesmen olicy searches he ouu aern ha maximises rowh. Growh maximisaion is sulemened by an advanced redisribuion olicy, and a saus-quo, non moraorium indebedness olicy. The resen sudy exlores he viabiliy for Bolivia of achievin he Millennium Develomen Goal of halvin exreme overy by 2015, and he effecs ha his would imly for he socioeconomy, in aricular for he differen social classes. Oimisaion is used in he selecion of boh invesmen and income disribuion olicies. Invesmen olicies are deermined such as o induce overy minimisin (and for comarison, emloymen maximisin) srucural chane and rowh. A comlemenary income disribuion olicy is exlored, which would be needed in order o achieve he oal of halvin exreme overy by 2015 for comarison, a olicy oally eliminain exreme overy is also obained. A furher difference wih he revious sudies, in addiion o he reaer deail in he ublic secor s accouns, is he roducion of annual SAMs of ar of he model simulaion ouu (see Aendix C). All sraeies share he same forein indebedness olicy. In all cases alies he forein deb areemen aained wih inernaional crediors in he framework of he Deb Iniiaive for Hihly Indebed Poor Counries, which reduces he execed ouflow of savins in conce of deb reaymens. Povery reducion sraeies will hen only differ in heir invesmen and income disribuion olicies. In wha follows we succincly describe five differen sraeies, (a) Base scenario, (b) Millennium invesmen sraey, (c) wo differen Millennium invesmen lus redisribuion sraeies, (d) Millennium Emloymen sraey. 4.1. Reroducin saus quo The Base scenario is he now suerseded, official Esraeia Boliviana de Reducción de la Pobreza (EBRP). The olicies roosed in he former official sraey aer (Bolivia, 2001), in sie of a deailed and accurae sudy of he naure and causes of widesread overy, were a mere coninuaion of revious olicies. The naural resource inensiy of he as aern of rowh, wih is concenraion on oil, as and exor cros, is mainained. The EBRP invesmen olicy mainains he exisin secializaion and ouu aerns, coninuin he invesmen olicy of he as. I mainains he focus of he as aern of rowh in caial inensive, rimary secors such as as, oil, minerals and soybeans, and in eneral, i mainains he aern of secoral rioriies of he as. The EBRP had no saemens of olicies addressin he hihly unequal disribuion of resources and incomes. Tax olicy, a key insrumen of redisribuion olicy and an indicaor of disribuional references, was no o be acivaed as an insrumen for overy reducion. The simulaed EBRP disribuion olicy is hen an unchaned income disribuion marix. 19 Le us now breafly commen on he resuls of his Base, EBRP reroducion scenario. Fiure 3 shows he resuls of he differen simulaions on he share of he exremely oor in oal oulaion in 2000-2015. The EBRP iniial value for exreme overy (50 ercen) corresonds o he absolue overy line of 1 US dollar a day, he hreshold commonly used in inernaional comarisons. The fiure shows ha under he assumions of our model he 19 For he assumed values of invesmen olicy { } z, disribuion olicy{ } V, indebedness olicy { Φ } oher model daa, see Aendix B. They are consan for 2000-2015. The model is formulaed and solved wihin he eneral alebraic modellin sysem GAMS (see e.. Brooke e al., 1992)., and 15
EBRP does no succeed in halvin he incidence of exreme overy by 2015. The reducion in he share of he exremely oor is 5 ercenae oins, ha is, 20 oins below he oal. 60 50 40 Percenae 30 20 10 0 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 1. EBRP 2. Millennium invesmen olicy 3. Millennium ax reform 4. Millenium emloymen sraey 5. Rawlsian ax reform Fi. 3. Share of he oulaion in exreme overy The simulaed effecs of he EBRP sraey on secoral ouus are shown in Fiure 4. The saus-quo invesmen olicy of he EBRP induces coninued exansion of a naural resource and caial inensive secor such as Peroleum, as and minin. This secor raidly increases is weih in oal ouu, alhouh is rowh rae diminishes wih ime. I becomes he lares secor for he mos ar of he eriod. Modern, Exor cros ariculure also follows a ah of raid exansion. Food cro, radiional ariculure, on he oher hand, rolons is as sanain rend, and is share in oal ouu is furher reduced. 25,000 20,000 Millions of bolivianos 15,000 10,000 5,000 0 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 1. Food sales ariculure 2. Exors cros ariculure 3. Peroleum, as and minin 4. Bi indusry 5. Small and medium indusry 6. Peroleum rocessin 7. Consrucion 8. Commerce 9. Transor 10. Infrasrucure and services 11. Public adminisraion 12. Finances Fi. 4. Ouu aern of he EBRP 16
In synhesis, he EBRP fails o sinificanly reduce overy, due o in addiion o a low overall rowh caaciy a concenraed income disribuion srucure, and an unhelful secoral rowh aern. A more helful secoral rowh aern, and a less concenraed income disribuion srucure should imly hiher overy reducion oucomes for equivalen raes of overall rowh. These are he subjecs of discussion of he followin secions. 4.2. Povery minimisin srucural chane We invesiae now he ossibiliy for invesmen olicy alone o induce such chanes in he level and srucure of ouus as o reduce exreme overy by a half by 2015. Tha is, we solve for he { z } allocaion vecor sequence of available ublic invesmen funds i.e. ublic savins available for financin he use of invesmen olicy insrumens such ha he secoral rowh aern of he economy is mos effecive in reducin exreme overy. In order o searaely analyse he effecs of invesmen and income disribuion olicies, we assume in his secion a saus quo disribuion olicy. Tha is, income disribuion (corresondin o he laes income survey of year 2000) remains unchaned hrouhou he sraey horizon. 20 The effec of his overy minimizin Millennium invesmen olicy on he evoluion of exreme overy can be seen in Fiure 3. The Millennium invesmen olicy does no succeed in halvin overy by 2015. Pro-oor srucural chane achieves only wo fifhs of he reducion in overy needed o aain he Millennium Goal. The Millennium invesmen olicy is wice as effecive as he EBRP in reducin overy, bu i is sill a bi more han halfway from he are. The chanes in ouu srucure obained by he Millennium invesmen olicy are raher lare (see Fiure 5). In order o minimize overy, invesmen olicy favours secors wih more equal income disribuions and/or hiher dynamic (invesmen) efficiency. Aciviy secors in which he oor accoun for a relaively lare income share, and/or where he ouu resonse o invesmen is relaively hih, end o e hiher weihs in he invesmen olicy vecor. 20 For simliciy, we solve for a consan { z } sequence for he whole 2000-2015 eriod. A resricion is imosed on he excess demands of non-radables secors (7 h o 12 h ), so as o mainain he iniial equilibria. One hird of he ublic invesmen bude is allocaed equally o all secors. The resulin z vecor is: (0.029 0.029 0.029 0.029 0.238 0.029 0.078 0.029 0.374 0.054 0.043 0.037). 17
40,000 35,000 30,000 Millions of bolivianos 25,000 20,000 15,000 10,000 5,000 0 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 1. Food sales ariculure 2. Exors cros ariculure 3. Peroleum, as and minin 4. Bi indusry 5. Small and medium indusry 6. Peroleum rocessin 7. Consrucion 8. Commerce 9. Transor 10. Infrasrucure and services 11. Public adminisraion 12. Finances Fi. 5. Ouu aern minimizin exreme overy by 2015 Thus, secors such as Peroleum, as and minin, favoured by as invesmen olicy and enjoyin hih rowh raes, end o radually lose is rivileed osiion, due o heir relaively unequal disribuional srucure and low dynamic efficiency. Bi indusry, wih he lares iniial ouu share is o some exen a similar case arially similar, as he efficiency of invesmen in he secor is no aricularly low. Bi indusry loses is osiion of a relaively imoran conribuor o oal ouu, o become a more ordinary secor. Similarly, he ouu share of Exor cros ariculure radually declines alon he eriod. The Millennium invesmen olicy favours he exansion of Small and medium indusries, due o heir aricular income disribuion srucure. While his secor s invesmen efficiency is abou he same as Bi indusry, is more ro-oor income disribuion srucure makes i suerior from he overy reducion efficiency ersecive, and increases is weih in invesmen olicy. This, in urn, acceleraes rowh in he secor, and increases is share in oal ouu and emloymen. The case of Food sales ariculure deserves a secial commen. Susained rowh of ouu and incomes in his secor has been sinled ou as he key for disribuionally roressive rowh. Kalecki s (1954) heoreical insihs have been larely confirmed by emirical sudies (see e.. Lion and Ravaillon, 1995). The overy minimizin Millennium invesmen olicy resuls in a disaoininly low rowh for Food sales ariculure. Income disribuion srucure in he secor should qualify i for a hih weih in invesmen olicy i.e., mos easans roducin Food sales are oor. The roblem is he secor s very low dynamic efficiency. A eso invesed in Food sales ariculure ives rise o a very low increase in he secor s ouu he second lowes afer Transor, a very caial-inensive secor. 21 Food sales ariculure mosly occuies very oor easans in he hihlands 21 Incremenal caial ouu raios α in our sudy were deermined by hisorical simulaion wih he model, so as o rack as ouus (1990-1997) as accuraely as ossible (see Aendix B). 18
(Alilano), roducin in very difficul soil and climacic condiions, and wihou sinifican infrasrucure, or echnical and credi suor. A any rae, resen knowlede seems o sues a careful aroach o olicy reform in he Food sales ariculural secor. Deailed sudy and exerimenaion should be required o arrive o effecive olicy reforms. Also, land enure reform should be considered amon he efficiency increasin reforms (de Janvry e al., 2001). Ye he mos widely shared imlicaion of he analyses is ha ariculural olicy and land reform need o be embedded in comrehensive olicy and insiuional reforms (de Janvry e al., 2001,. 23). 22 The resen sudy of oimal overy reducin invesmen olicies suess also he ossibiliy and convenience of combinin ariculural reform and develomen wih romoion of small and medium indusry (and services) in rural areas. The exisence of radiional, communal forms of roery and roducion in he Alilano hihlands and oher ariculural reions mih resemble he condiions in he Chinese counryside wo or hree decades ao. The Chinese exerience in recen decades shows he vas oenial caaciies exisen in rural areas for exandin non-ariculural roducion. For insance, he ouu of China s rural indusry secor increased in 1978-2000 a he asonishin rae of 22 ercen er year in averae (Kwon and Lee, 2005). 4.3. Exanded reroducion wih srucural chane and income redisribuion Le us ax he rich o subsidize he oor. Jean Paul Mara (Thomson, 1989,. 170) We have seen in he revious secion ha he overy minimizin Millennium invesmen olicy did no succeed in aainin he Millennium Goal of halvin exreme overy by 2015. The ouu srucure induced by his olicy imlies a reducion in he overy rae of abou a half of wha is needed o reach he Goal, while he Base scenario aained one enh of he needed reducion. Given his overy minimisin invesmen olicy, we ask now if here are, iven he consumion and savin behaviours of he differen income classes in he model, viable chanes in he srucure of income disribuion ha would accomlish he remainin reducion needed. We ask if here is any { } V income disribuion olicy sequence which alon wih he overy minimisin invesmen olicy of he revious secion halves exreme overy by 2015. We desin he simles conceivable redisribuive olicy, consisin of a ax alied a a consan rae on all incomes above wice he overy line (of 2 dollars a day). The fund hus colleced is equally disribued o all eole below he line of exreme overy (1 dollar a day). This is of course no inended as a realisic descriion of wha an income redisribuion olicy mih be, bu a firs feasibiliy check of income redisribuion as a means of achievin he Millennium Goal. 23 I is also inended o rouhly quanify he maniude of he redisribuion effor required. The oimisaion exercise of his secion consiss hus in solvin he model for he income ax rae which would make he share of he exremely oor in 2015 o aroach 25 ercen as 22 The overnmen eleced in 2006 has launched an ambiious land redisribuion roram, based rimarily on he disribuion amon oor easans of sae, idle, and illeally occuied land. 23 A realisic descriion of redisribuion olicies would also include reform olicies such as asse redisribuions (e.. he already iniiaed land and naural resource reforms) which are no easy o quanify. 19
much as i is ossible i.e. a half of wha i was in 2000. The Millennium invesmen olicy of he revious secion is now sulemened by a sraihforward redisribuion scheme. All oher hins iniial condiions, behavioral coefficiens and in aricular, invesmen olicy arameers are equal o hose of he revious secion. The Millennium ax [and exendiure] reform ransforms he { } and subsidies sequence { * } V saus quo income disribuion ino a new, os-ax V which achieves he Millennium overy oal. Fiure 3 above shows he evoluion of overy in wha resuls o be an indusrialisin rowhcum-redisribuion overy reducion sraey a overy minimisin olicy focusin aricularly on indusrial rowh based on small and medium enerrises, combined wih redisribuive reforms. Fiure 3 shows how redisribuive reforms, immediaely from is inceion reduce exreme overy by 10 ercenae oins, and coninue o roressively cu i unil he Millennium Goal is aained in 2015. The Millennium ax rae ha obains his resul is 8.1 ercen. A no unrealisically hih rae, if i is ke in mind ha, as for mos Lain American counries, Bolivia s ax revenues are relaively low. Bolivia s ax revenues/gdp raio is low in comarison o wha is normal iven he counry s level of develomen, ha is, below he reression line relain his raio o GDP er caia. As calculaed by Perry e al. (2006, Table 5.7) Bolivia is undercollecin i.e., collecin under wha would be execed iven is GDP er caia level a 3.6 ercen of GDP. As a share of GDP, he Millennium ax reresens 4.6 ercen he required redisribuion would hus involve an addiional effor of one ercen of GDP above he averae. Fiure 3 shows also he effecs of a more ambiious, fully Rawlsian sraey. A fully Rawlsian sraey would robably ado he more ambiious oal of oally eliminain exreme overy by 2015, ivin absolue rioriy o he needs of he exremely oor. The Rawlsian ax obained by solvin he model in exacly he same way as in he Millennium ax above exce ha he objecive is zero exremely oor in 2015 is 16.7 ercen. 4.4. Povery reducin vs. emloymen increasin reroducion The secoral ouu elasiciy of overy, ha is, he effec of secoral rowh on overy reducion, has recenly been invesiaed in a wide emirical sudy (Perry e al., 2006). Accordin o his cross counry sudy, relaive labour inensiy deermines a secor s influence on overy alleviaion. Boh he size of rowh and he deree of labour inensiy in ha rowh are relevan for exlainin overy reducion. [A]riculure, he mos labour-inensive secor, resens he lares rowh elasiciy of overy, while minin [includin oil] and uiliies carry he lowes elasiciies for overy reducion (Perry e al., 2006,. 91). I is ineresin o check his ideas in he conex of our simulaion model, no only o es he emirical lausibiliy of he model, bu also because of he oenial ineres of is olicy inferences. The objec of his secion is o ceeris aribus comare he revious Millennium invesmen olicy which minimises overy, wih an invesmen olicy which induces emloymen maximisin srucural chane. In erms of our analyical model, such an emloymen focus is reresened by he search of he invesmen olicy ha obains he hihes emloymen rowh. In order o evaluae emloymen creaion under differen sraeies, a marix of secoral emloymen coefficiens by ye of labour is defined, and a vecor of emloymen by ye of labour is so deermined 20