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Giandemetrio Marangoni. The Leontief Model and Economic Theory. Quaderno n

Gindemetio Mngoni The Leontief Mode nd Economic Theoy Qudeno n -6 Decnto de Fcotà di Scienze economiche Vi Oede, 3 C-69 Lugno Te / 8 F The Leontief mode nd economic theoy by Gindemetio
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Gindemetio Mngoni The Leontief Mode nd Economic Theoy Qudeno n -6 Decnto de Fcotà di Scienze economiche Vi Oede, 3 C-69 Lugno Te / 8 F - - The Leontief mode nd economic theoy by Gindemetio Mngoni Univeity of Veon Abtct: Thi tice emine the etionhi between cic, mginit nd Keynein economic nd the Leontief mode nd how how the nyi of oductive nd ditibution intedeendencie my ovide n oite concetu fmewo fo coming the diffeent nytic oche Keywod: Gene Equiibium, Inut-Outut Mode, Cic Economic, Mginit Economic, Keynein Economic JEL cifiction: C6, C67, B4 Pof Gindemetio Mngoni Univeità di Veon Ditimento di Scienze Economiche Vi de Atigiee, 9 I Veon - Ity te f e- mi : - 3- - 4- Intoduction The im of thi tice i to nye the etionhi between cic, mginit nd Keynein economic nd the Leontief mode, nd to how how the nyi of oductive nd ditibution intedeendencie my ovide n oite concetu fmewo fo coming the viou choo of economic thought nd deveoing thei emie to thei ogic concuion A comion between diffeent choo of thought e difficut becue economic etionhi e come nd ech choo de with thi comeity in diffeent wy Cic nd mginit economit hod tht fu emoyment nd the fu utiition of cit wi be chieved if the economy i fee to wo untmmeed by etiction, whie Keynein economit deny the eitence of utomtic nd efficient mechnim fo the equiibium of the fcto met Thee ooing concuion e ined to the ccetnce o othewie of Sy' Lw ( oi de débouché), ccoding to which uy cete it own demnd [Sy, 87] Moeove, cic economit cim tht, on the bi of the wge fund theoy (the fied quntity of good vibe in ociety t ny given time to dvnce wge to woe invoved in the oduction oce), the imefect woing of Sy' Lw cn be coected by eduction in e wge Fo thei t, Keynein economit beieve tht eduction in wge cue f in the demnd fo conume good, but h no ignificnt ubtitution effect between bou nd cit [Pinetti, 98; Bnzini nd Sczziei, 99; Moihim, 99] At the nytic-concetu eve, the diffeent choo of economic thought hod ooing view, conideing economic eithe e - 5- cience, ituted in hitoic contet nd concened with the ognition of ociety, o fom cience uneted to time nd bed on tion incie [Schumete, 954; Cn, 935] In thee tem, ny comion eem imoibe The Leontief mode h eement in common with thee diffeent oche nd cn thu eve n goithm fo tntion, owing u to move fom one concetu fme of efeence to nothe Secto intedeendence i be to eeent, t et in the ti dimenion, the come ognition of ociety founded on the diviion of bou, the ditibution of income, the ccumution of cit nd the echnge of good on the met At the me time, ecto intedeendence offe owefu nytic too fo the oution of obem of equiibium nd of mimum nd minimum tyic of tion ccuu uing the fom ocedue of the geb of mtice In hi intoduction to The Stuctue of Ameicn Economy 99-99, the fit ytemtic eenttion of wht w to become inut-outut nyi, Leontief decibed hi wo n ttemt to y the economic theoy of gene equiibium - o bette, gene intedeendence - to n emiic tudy of inteetion mong the diffeent t of ntion economy eveed though covition of ice, outut, invetment, nd income [Leontief, 94] Leontief' oigin ide tuned out to be etemey fuitfu nd the inut-outut och etended it fied of iction beyond the imit of ticty defined economic ocee The mjoity of emiic iction, howeve, wee imited to the ue of the oen veion of the mode, which doe not ow comete oution of obem eted to the etionhi between the oductive t of the economic ytem nd thoe t concened with conumtion nd ccumution [Mie nd Bi, 985] If we wih the inut-outut och to eeent the gene intedeendence of n economic ytem, we need to coe the mode, - 6- tht i to me the eve of conumtion nd invetment endogenou o, in du tem, to me the eve of wge nd ofit endogenou [Cot nd Mngoni, 995] In ode to coe the mode we need to intoduce new umtion ongide the bic hyothei of the inut-outut mode, which i tht of contnt coefficient of oduction It i t thi tge tht we cn te into conidetion the diffeent wy in which the viou nytic oche beieve economic ytem to wo The ccetnce of one umtion the thn nothe ed to ticu coue of the inut-outut mode, which te on cic, mginit o Keynein chcteitic, whie emining gene fme of efeence fo eeenting n economic ytem nd eoning bout it The tice h been tuctued foow Section eent the inut-outut mode homogeneou ytem of ine eqution Thi fomution ow u to bing degee of feedom bout nd conequenty to intoduce umtion tyic of the diffeent nytic oche The homogeneou ytem o ed to viety of Leontief invee Section 3, 4 nd 5 dicu the mginit, Keynein nd cic eding of the Leontief mode, ech with it own ditinctive fetue Finy, ou concuding em highight the neutity of the mode with egd to the viou method of eoning, but o it ubtntiy cic chcteitic - 7- The inut-outut mode nd the viety of Leontief invee Let u conide n economic ytem mde u of two oductive ecto The fow of good between the ecto cn be chemtied by the foowing inut-outut tbe: d V V d V D The ignificnce of the ymbo i foow ( i, j,) : i ice of commodity i, ij quntity of commodity i ued inut by ecto j; d i quntity of commodity i detined fo the fin demnd (conumtion nd invetment), i outut of ecto i, V i vue dded of ecto i (wge nd ofit), V tot vue dded, D tot fin demnd Let u intoduce the hyothei of contnt etionhi between inut nd outut nd define the coefficient of oduction: ij ij j The inut-outut tbe cn then be witten in the fom: d V V d V D If we conide the ow nd coumn tot, the foowing etion hod: d + d + v + v whee v i indicte the vue dded of ecto i e unit of outut In the tdition oen mode, d i nd v i e conideed given eogenou vibe, whie i e i e the endogenou vibe to be detemined If we define the vecto nd the mtice: d A d d v the mode cn be witten in comct fom : A + d ( I A) d o A + v ( I A) v nd the oution i: ( I A) d v( I A) [ ] [ v v ] The Leontief invee ( I A) i the oeto which ow u to ove the two cic obem fced by the inut-outut mode: the ech fo vecto of ecto oduction cbe of tifying the fin demnd d nd the ech fo vecto of ecto ice conitent with given vecto of ecto coefficient of vue dded v The Leontief invee i, howeve, ony one of the invee which cn be ued fo the oution of the oen mode The inut-outut mode cn be conveted into homogeneou ytem of ine eqution which h moe degee of feedom thn the tdition mode [Cot nd Mngoni, 995] The quntity nd ice eqution cn be witten : - - 9 d d v v o, in mti fom, : d d v v If we et: d d d d the quntity mode my be witten foow: o: + nd my be oved fo : ) ( - - In thi ce the fin demnd fo the two ecto e conideed to be the eogenou vibe whoe vue i given, whie the oduction of the two ecto e the unnown endogenou vibe to be detemined oweve, the oe of eogenou vibe nd endogenou vibe cn be echnged: we cn, fo eme, conide the oduction of the fit ecto given, nd im to detemine the fin demnd necey to bob thi oduction; o even conide oduction e of both ecto given nd im to detemine the eve of fin demnd necey to bob thee oduction Fom fom oint of view, it wi uffice to wite, in the fit ce: d d d d nd, in the econd ce: d d d d In both ce the oution i: ) ( Thi ocedue i not uey mthemtic-fom eecie, but i bed on ecie hioohy of the oe yed by demnd nd - - uy in n economic ytem If, in Keynein economic, we ecognie tht demnd h n utonomou nd diving oe, it become the eogenou conto vibe nd eve of oduction ign themeve ccodingy On the othe hnd, when, in cic economic, emhi i ced on the uy of good nd evice, o, in the mginit mode, the emhi i on the tentive ue of cce eouce, it i the eve of oduction which detemine the efomnce of the whoe economic ytem nd cete ufficient eve of demnd It i woth ointing out tht the Leontief mode i imited to detemining the theoetic eve of oduction necey to tify cetin fin demnd o, in the veion eented hee, the theoetic eve of fin demnd necey to bob cetin oduction [Pinetti, 977] Indeed, the mode doe not guntee tht the economic ytem wi efom ccodingy: hotge of bou nd cit fcto my cue oduction to be inufficient to tify cetin eve of demnd, nd the imefection of the mechnim of income ditibution my event the otenti demnd fom becoming effective demnd Tuning to the ytem of ice, we cn, uing n nogou nottion, wite: hence: o: v v v v - - + If we ove it fo we hve: ) ( The ytem detemine ice comtibe with given eve of vue dded; tht i, ice which enbe comnie to y cetin monety wge fo wo nd to ech tifctoy ofit eve In thi ce, too, it i oibe to ume the inveion of oe between eogenou vibe (vue dded e unit of outut) nd endogenou vibe (ice) The inveion of oe coeond to the hyothei (cetiny dtic in thi contet) of fied ice, to which wge nd ofit mut confom Thu we woud hve: v v v v with the identic fom oution: ) ( - 3-3 Endogenou fin demnd nd the mginit theoy Let u etun to the ime economic ytem with two oductive ecto eented bove And et u uoe tht, of the two oductive ecto, the fit oduce conume good, the econd cit good We wi divide income ene into two ce: the woe, who eceive wge, nd the citit, who eceive ofit The woe end thei tot income on the uche of conume good, whie the citit ue of thei ofit to uche cit good Thee two itution, but ticuy the econd, imy the fu ccetnce, tyic of the mginit choo, of Sy' Lw, in tht both of them guntee the comete botion of the uy by the demnd Agin, both itution, but ticuy the fit, imy the intoduction of mutiie mechnim which futhe boot the oduction of conume good nd invetment nd when income ie [Miyzw, 976] Ech woe eceive e wge w, o given he of conume good The monety wge i equ to w In ode to oduce conume nd cit good it i necey to emoy bou nd cit in fied ootion:,, nd indicte the eective technic coefficient Pofit e ccuted on cit emoyed, ccoding to given te of ofit The tot numbe of woe emoyed i equ to L, whie the tot quntity of cit emoyed i equ to K The foowing inut-outut tbe how the fow of good between the two ecto, the ditibution of income between the two ce, nd the ue of income fo the uche of good: - 4- wl K w w wl K wl K The mode which eut fom thi tbe i: - coed with eect to conumtion, which deend on the woe' income (in fct it coincide with thi income, ince we hve et the oenity to conume equ to one); - coed with eect to invetment, which deend on nd coincide with the citit' income; - coed with eect to wge, in tht the monety wge w dt utomticy to chnge in the ice of conume good in ode to ee the woe' uching owe the me; - coed with eect to ofit, which e ccuted on the bi of cit whoe vue i utomticy evued when ice vy ence we hve: + L + + K wl + K + w + + w + If we ue mti nottion, we cn wite the inut-outut mode homogeneou ytem of ine eqution: - 5- wl K L K w The fit thing we obeve i tht the quntity ytem nd the ice ytem e now ined by e wge w nd the ofit te The oution of the two ytem cn be eched in two tge, by oving fit the quntity ytem nd then the ice ytem The quntity ytem i homogeneou ine ytem of fou eqution in i unnown, which cn be oved fte the vue of two unnown i fied The oution of the ytem detemine, diecty o indiecty, the vue of i unnown, incuding w nd If we ece the w nd vue in the ice ytem, thi become homogeneou ine ytem of two eqution in two unnown One eqution i iney deendent on the othe, o it my be eiminted nd it i necey to fi the vue of one of the two vibe o Thi oetion i efecty egitimte fom the economic oint of view in tht, in evey economic ytem, ice e wy etive nd deend on the commodity choen numéie But et u etun to the quntity ytem Autonomou fin demnd doe not e mong the vibe, becue of Sy' Lw, which guntee comete botion of oduction If we wih to be conitent with the mginit mode, the ytem mut guntee fu emoyment of the fcto We cn chieve thi eut by ming - - 6 ue of the two degee of feedom which the ytem offe: in othe wod, we cn fi the vue of L nd K t the eve of fu emoyment The quntity ytem become: K L w K L with the oution: K L K L w If the oution me economic ene, tht i, if the vibe te on non-negtive vue, the ytem detemine, mong othe thing, the vue of w which, t thi oint, we e entited to conide comtibe with fu emoyment Thi itution i iewie conitent with the mginit theoy, which ow fo wge feibiity If we ce the w nd vue in the ice ytem we get: w w The ytem i oved, we hve id, by conideing the ice of the commodity choen numéie to be equ to one - 7-4 The oen mode with eect to invetment: utonomou invetment nd the Keynein equiibium of undeemoyment The utomtic mechnim which ee the bou nd cit met in equiibium do not wo if the invetment deciion of comnie e indeendent of ofit eve (o, ince ony citit ve, of eve of ving) The invetment I e n utonomou comonent of the fin demnd Fom the modified tbe: wl I w w wl K wl I we cn wite the foowing: + L + + K wl + I + w + + w + The inut-outut mode my be et in the fom of homogeneou ytem of ine eqution: - - 8 K L I wl w In comion with the mginit mode, the Keynein mode i now oen with eect to invetment The quntity ytem, with it fou eqution nd i unnown, eent ony one degee of feedom, ince the vue of I i given We my now fi the vue of the econd vibe, e wge w Thi choice incoote the Keynein concet of the non-feibiity of wge: ccoding to thi mode, eduction in wge doe not ed to the ubtitution of fcto of oduction, but imy deee conumtion The choice we hve mde enbe u to wite the quntity ytem in the fom: I K L w The ytem cn be oved thu: - - 9 I w K L If the oution me economic ene, the mode detemine the eve of oduction nd of ue of the fcto comtibe with given e wge nd invetment The vue ten on by L e K my be, nd in gene e, diffeent fom the vue of fu emoyment Afte fiing (o detemining) e wge, the ice ytem become: w w Thi i homogeneou ine ytem of two eqution in thee unnown We need to fi one of the unnown It i oite to fi the ice of one commodity, choen numéie, which in thi ce my be the econd commodity If we et, the ice ytem become: w w The ytem my be oved foow: w w thu detemining the ice of the fit commodity nd the ofit te which me ene, obviouy, ony on condition tht the eut i nonnegtive - - 5 The oen mode with eect to conumtion: oduction g nd the cic wge fund theoy The cic och to economic eeent the woing of the economic ytem cicu oce Poduction te ce by uing good which wee oduced in n eie ound Thee e ued w mtei nd men of ubitence fo the woe At the end of the oduction oce, if the ytem i vibe, thee i uu equ to the quntity oduced ove nd bove the quntity ued men of oduction In ode to coe the inut-outut mode uing the cic fmewo, two chcteitic ect houd be ten into ccount [Moihim, 989]: - the eitence of time n necey fo the oduction of conume good, tyicy the ye coveed by one fu cyce of gicutu oduction (hyothei of oduction g); - the imit imoed by the fied quntity of conume good oduced nd eft ove in the eviou eiod, which the woe cn uche with the wge they eceive (wge fund theoy) Thee two hyothee chnge the inut-outut tbe quite ignificnty, which theefoe become: C I I w w wl + A K + A wl K + A - - Let u nye the ow nd coumn The fit ow efe to the oduction of conume good One t of the oduction ( + ) i e-emoyed in the oduction oce, nothe t (C) i ued to te the wge fund bc to the eviou eve fo ue in the net eiod of oduction; whie the t t ( I ) goe to incee the wge fund fo the net eiod, o tht ge numbe of woe my be emoyed o e wge my be inceed the foowing ye The deciion to incee the wge fund i mde by the citit, who et ide ootion of the ofit fo thi uoe The econd ow concen the oduction of cit good The citit ue u the ofit, ecet fo thoe needed to incee the wge fund, fo new invetment I Thee i theefoe tde-off between I nd I ; eut, the oce of ccumution te ce t the eene of emoyment nd e wge The thid ow how the numbe of woe nd tot wge, whie the fouth how tot ofit nd the ue of cit Since the oduction of conume good cove eiod of time equ to ye, buinemen hve to ovide befoehnd the men of oduction necey fo the eiod hed They eceive inteet on the money they me vibe, which i ccuted t the me te the ofit te The vue of the dvnce i equ to: A w nd thu the inteet (ofit) i equ to A No dvnce i equied fo the oduction of cit good, thi oduction i umed to be intntneou The quntity nd ice eqution e foow: + L + + K + + C + I + I - - ) )( ( w w Witten in mti fom, the inut-outut mode my be eeented foow: C K L I I w w The quntity nd ice ytem e to be indeendent but e not ey o, ince the e wge (w) nd the eve of emoyment (L) e ined to the (wge fund) etionhi: wl C A to the ice ytem, we my fi ditibutive vibe, fo eme the ofit te, nd the ice of commodity choen numéie, fo eme Thi yied: w with the oution: w The vue of w cn be ccuted immeditey by dividing w by The oution me ene economicy ony if the eut i not negtive If we now move on to the quntity ytem, we obeve tht the vibe L i now deteminte nd i equ to: w C L Moeove, the econd eqution i in fct n identity, nd cn be eiminted The quntity ytem my be educed to: L C K I The quntity ytem offe one moe degee of feedom nd the vue of one of the fou vibe,, I o K my be fied bitiy Let u fi the vue of I If we me thi choice the cic veion of the inut-outut mode i oen with eect to conumtion, in tht the fin demnd fo conume good i cometey utonomou The quntity ytem become: + L I C K with the oution: L I C K A we now the vue of ecto oduction, the new cit invetment e equ to: I The oution e economicy ccetbe ony if they e not negtive - 5-6 Concuion In the eceding gh we nyed the etionhi between cic, mginit nd Keynein economic nd the Leontief mode Leontief nyi doe not de with the digm of tdition economic It i concened gmticy ony with the obem of the imct of cetin eogenou vibe on the economic ytem in gene, nd on the oduction ytem in ticu [Roe nd Mieniny, 989] Indeed, in it oen veion, in which conumtion nd invetment (nd, in du tem, wge nd ofit) y the oe of eogenou vibe, the mode i neut with eect to tdition fme of efeence o, if we efe, it contitute one ticu imifiction of thee If we wih to coe the mode, tht i, if we conide the feedbc of the eve of oduction on conumtion nd invetment (nd, in du tem, of ice on wge nd ofit), we hve to chooe theoy of conumtion nd theoy of invetment At thi oint the Leontief mode te on chcteitic tht my ign it viouy, deending on whethe we tnd in the cic, the mginit o the Keynein cm In ticu, we hve een how the cic wge fund umtion ed to n inut-outut mode which i coed with eect to invetment, but oen with eect to conumtion The mginit ccetnce of Sy' Lw ed to cometey coed mode, with eect to both conumtion nd invetment, whie the nonccetnce of Sy' Lw end Keynein-tye connottion to the mode, which thu become coed with eect to conumtion, but oen with eect to invetment - 6- The obem of coue highight the infuence of the diffeent nytic oche on the bic tuctue of the mode itef We houd oint out, fit of, tht in the fit edition of hi wo [Leontief, 94], Leontief eented inut-outut nyi mode tht w coed with eect to fin demnd, in which fmiie wee imited to ny oductive ecto Fmiie eeent induty ( n +), which equie fin good n inut, nd which ovide wo evice n outut Thi coue, in which fin demnd nd vue dded e teted eidu quntitie, i uey n ccounting coue It diffe fom the coue eented in the eviou gh in t et two wy: ) the fin demnd i not diggegted into it inci comonent; ) theoie which jutify the deendence of conumtion nd invetment on eve of oduction e not emined In the econd edition of The Stuctue of the Ameicn Economy [Leontief, 95], Leontief me eicit efeence to n oen mode, in which fin demnd i utonomou e quote Keyne nd eie tht ctic iction of the mode e imed t eing the imct of vition in fin demnd on the economic ytem Muti-ecto nyi deveo the Keynein mutiie incie in moe deti If the Leontief mode cn be defined Keynein in it efomnce, it tuctue e, t fit ight, to be mginit Leontief ccet, on the one hnd, the Win imifiction of fied coefficient of oduction [W, 874] ( ctic hyothei, but not contituent t of the mode, given tht thi i o fuy comtibe with the oibiity of thee being vibe coefficient, hown by Cte nd Peti, [989]) On the othe hnd, he ie Gutv Ce' ide of deiving economic etion fom emiic obevtion, the thn deducing them fom the incie of mimum utiity [Ce, 93; Aow nd Debeu, 954] The - 7- Leontief mode i one of gene economic equiibium, nd it ototye cn ony be Leon W' mthemtic mode It my be gued, howeve, tht the cic t of the Leontief mode goe beyond Keynein nd mginit connottion nd ugget concet of c
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