THE FOUNDATIONS OF DYNAMIC INPUT-OUTPUT REVISITED: DOES DYNAMIC INPUT-OUTPUT BELONG TO GROWTH THEORY?

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1 THE FOUNDATIONS OF DYNAMIC INPUT-OUTPUT REVISITED: DOES DYNAMIC INPUT-OUTPUT BELONG TO GROWTH THEORY? Marian Blanc Díaz 1 Carmen Rams Carvajal 2 Miembrs del grup de investigación en Análisis Input-Output. Departament de Ecnmía Aplicada. Facultad de Ciencias Ecnómicas y Empresariales 1 mblanc@vd.gdfsa.cm 2 crams@crre.univi.es 1

2 THE FOUNDATIONS OF DYNAMIC INPUT-OUTPUT REVISITED: DOES DYNAMIC INPUT-OUTPUT BELONG TO GROWTH THEORY? TABLE OF CONTENTS ABSTRACT INTRODUCTION MATHEMATICAL BACKGROUND ON THE CONSISTENCY OF THE STATIC AND DYNAMIC INPUT- OUTPUT MODELS Case B = 0 Case X (t) = 0 Case B 0 and X (t) 0, with cnstant final demand Case B 0 and X (t) 0, with steadily grwing final demand ON THE STABILITY OF LEONTIEF S DYNAMIC MODEL CONCLUSIONS DIRECTIONS FOR FUTURE EMPIRICAL RESEARCH Appendix A Appendix B Appendix C REFERENCES 2

3 INTRODUCTION Classical input-utput cmes in tw flavrs: static and dynamic. The dynamic versin was develped as an extensin f the static ne, t cpe with time. But nt with any influence f time: specifically with the effect f capital accumulatin. Thus dynamic input-utput mdels are cnsidered amng the early members f the family f grwth mdels. Frm mathematical cnsideratins, we cnclude in this paper that their lng time accepted interpretatin can hardly be sustained. Under certain assumptins, they d represent an ecnmic reality but nt grwth. Sme directins fr future empirical research are extracted frm the analysis. In the rest f the paper, we shall use capital letter S, as synnymus f Lentief static mdel and letter D as a substitute fr dynamic input-utput mdel. MATHEMATICAL BACKGROUND Static frmulae. The characteristic balance equatin f S is X = A X + Y E.1.1. where X is a vectr f utput, Y is a vectr f final demands and A is a square matrix f interindustry cefficients. 3

4 The slutin f E.1.1. is: X = [ I A ] 1 Y E.1.2. Dynamic frmulae The characteristic balance equatin f D is: X(t) = A X(t) + Y(t) + B X (t) E.2.1. where B is a square matrix f capital cefficients. It represents the willingness f the ecnmy t invest (1). E.2.1 can als be written: X (t) = B 1 [ I A ] X(t) B 1 Y(t) E.2.2. Naming M = B 1 [ I A ] and N =- B 1, E.2.2. becmes: X (t) = M X(t) + N Y(t) E.2.3. Its slutin is: t X(t) = e Mt X(0) + e M (t - τ) N Y(τ) d τ E.2.4. (1) The rigrus technical definitin is certainly different. It measures the invlvement f each sectr in the capital accumulatin f the rest. Fr simplicity and better understanding we assciate invlvement with willingness. 4

5 ON THE CONSISTENCE OF S AND D The prcess f understanding under which cnditins S may be cnsidered a particular slutin f D, allws t surface sme nt s evident characteristics f D s behavir. There is mre than ne way t see the slutins f S and D cincide. Let us review different appraches: 1. Apprach 1. Case 1. Assuming that B = 0, in E.2.1. This is generally cnsidered the bvius apprach (2). This situatin wuld crrespnd t an ecnmy where prductin fluctuates but the willingness t invest is zer. Cnsequently investment itself is als zer. In fact, a situatin like this can be depicted by S but nt by D. This is clear when we cnsider D under its frm E.2.2. If B is singular, B -1 des nt exist and D is nt peratinal. We d nt have then tw wrking mdels which merge, but a mdel which wrks and anther which des nt. Therefre, this apprach tries t eliminate the distance between the tw mdels by annihilating ne f the tw terms f cmparisn. This is nt acceptable as means t establish a parenthd relatinship between the tw mdels. (2) See fr instance this viewpint in: Linear prgramming and ecnmic analysis Rbert Drfman, Paul A. Samuelsn and Rbert M. Slw. Mc Graw Hill

6 2. Apprach 2. Case 2. An alternative way is t assume that B 0 but X (t) = 0 in E.2.1. This situatin wuld crrespnd t an ecnmy with willingness t invest but where prductin needs d nt grw. Therefre investment is again zer. Under thse assumptins, D truly satisfies the equatin: X(t) = [I A] 1 Y(t) and S and D describe equally the same reality frm tw different view pints. Bth mdels are peratinal and independent. It is als true that the apprach des nt generate any infrmatin f interest abut the dynamics f the ecnmy: E.3 nly wrks if X(t)=C since therwise X (t) wuld be 0, which is against the assumptins; it als necessarily implies Y(t) = K. We end up cncluding that E.3 is in reality: C = [I A] 1 K an expressin which adds nthing t what we knw frm E Apprach 3. Different cases. Appraches 1 and 2 have smething in cmmn: they bth try t apprximate the balance equatins f S (E.1.1) and D (E.2.1) by remving the term BX (t) frm E

7 But it is nt necessary t make BX (t)=0 t have S and D cnfirm each ther. T prve it, let us slve E.2.4 fr different shapes f Y(t). Case 3 T allw an easier cmparisn with the previus apprach, let us slve first the case Y(t) = K. This situatin wuld represent an ecnmy with willingness t invest, where prductin is allwed t fluctuate while final demand is cnstant. One must nt take fr granted that cnstant final demands lead t cnstant prductins. As we shall see, under Lentief s dynamic frmulae, prductin can grw in an explsive way even when final demands remain unchanged ver time. Under this assumptin, Appendix A): E.2.4. becmes (demnstratin in X(t) = [I A] 1 K + e Mt [ X(0) [I A] 1 K ] E.4. If the secnd term f the right hand side f E.4 can be made zer, S and D d prvide an identical slutin: This can happen in tw ways: 3.1 When X (0) = [ I A ] 1 K E.5 that is when the ecnmy is already wrking at time zer at the regime crrespnding t the lng term steady state. 7

8 This case is a redefinitin f E.3. All assumptins cincide: B 0 Y(t) = K X (t) = M e Mt [ 0] = 0, therefre X(t) = C. Nevertheless apprach 3 has bviusly enriched, with respect t apprach 2, ur infrmatin abut the dynamic equilibrium. We knw nw that the validity f E.3., which rests n the cnstancy f prductins, implies tw cnditins: nt nly cnstant final demands but als very specific requirements abut the initial situatin f the ecnmy. 3.2 Additinally, apprach 3 tells us that even if E.5. des nt hld and therefre X (t) 0, E.4. can still reprduce the slutin f the static mdel when (and nly when) the system is stable: the secnd term will fade away as time passes and becme zer fr practical purpses after a while. Case 4. Appendix B slves again D fr steadily grwing final demands : Y(t) = Kt. In this case, B 0, final demands are nt cnstant and X (t) is never zer. N restrictins are impsed n initial cnditins. Nevertheless, the cnclusin is again the same: if the system is stable, the slutin f D cnverges t that f S and the mdels prvide cnsistent results (see Appendix B). On the cntrary, if the system is unstable, the static and dynamic input-utput mdels will prvide slutins that diverge cntinuusly ver time, even when identical final demands are applied. 8

9 A minr difference in initial cnditins frm the required state, leads als withut real justificatin t cntradictry slutins f the tw mdels. As mentined befre, stability requests that all eigenvalues f matrix M = B -1 [I A] have negative real parts. ON THE STABILITY OF LEONTIEF S DYNAMIC MODEL Appendix C prves fr the tw sectr case that if cefficients f matrix B are prevented frm being negative, D is unstable. It can nly exhibit the fllwing types f behavirs, crrespnding t a system with nly this variety f singular pints: unstable fcus, unstable ndes r saddle pints Thse behavirs wuld be: Explsive diversin frm the steady state with scillatins fr unstable fcus. Explsive diversin, in either directin, withut scillatins fr unstable ndes. Strange nn scillatry trajectries fr saddle pints, which start cming clser t the steady state, t depart always in the ppsite directin in an explsive divergence befre ever reaching it (3). The situatin described abve ccurs always under Lentief s hypthesis since the very definitin f capital cefficients request frm them t be psitive r zer (althugh we pint in this paper, in case f zer cefficients, they must nt be situated in psitins which make B singular). 9

10 If the cefficients f B are negative, it is still pssible t have the same type f instabilities r under additinal restrictins (see again Appendix C fr the details), ne can find the nly cases f stability and cnsistency f the tw mdels. In Lentief s interpretatin negative cefficients may have n meaning. But alternative interpretatins are pssible: when prductin decays, X (t) is negative and a negative B wuld turn int psitive the cmpnent f utput BX (t), thus meaning a cmpensatin f the slw dwn taking place. In case f grwing prductin, X (t) is psitive and B<0 makes BX (t) negative, thus representing a crrectin effect t the expansin ccurring. Negative cefficients cannt represent grwth but yes perhaps sme kind f shrt term cuntercyclical plicy. (3) The saddle pints include ne case f stability. Out f the infinite numbers f pssible trajectries, all f them are unstable except ne, which happens when the system mves alng ne f the tw separatrice lines. This requests such stringent cnditins n the prprtins f the cefficients and their maintenance ver time that the prbability f such things ccurring is nt small r very small but abslutely negligible. The situatin, if ccurring, wuld be mre a cnsequence f a successful lttery draw than a case representative f the behavir f the system. One specific case deserves attentin: when B is the negative identity matrix. Case 5. Assumptins: B = -I, Y(t) = K and X(t) is allwed t fluctuate. The balance equatin f D, 2.2.1, becmes: 10

11 X (t) = [A I] X(t) + Y(t) E.6 r X (t) = A X(t) + Y(t) X(t) E.7 In the right hand side f E.7 : The term AX(t) + Y(t) represents ttal demand f the ecnmy. The term X(t) represents ttal utput f the ecnmy. Therefre E.7 is the representatin f an ecnmy that intrduces shrt term charges in its prductin levels (X (t) is change f utput ver time) fllwing the infrmatin received abut the excess r default f demand ver supply ccurred in the previus perid. It is the general case f the static mdel whse equatin E.1.1. can be rewritten: 0 = A X + Y X E.8. The static mdel assumes perfect equilibrium f supply and demand instant by instant. D with B = -I, cvers that pssibility but als mre general situatins f temprary unbalances. The applicatin f the stability criteria develped in Appendix C applied t the case B = -I, cnfirm fully that it is ne f the situatins f stability and therefre identical slutin f S and D. 11

12 CONCLUSIONS 1. The largely accepted ntin that S is equivalent t D with B = 0 is incrrect. 2. If the cefficients f B are nt negative, D is unstable: It prvides unrealistic descriptins f the wrld: the ecnmy culd grw indefinitely under unchanged final demands and minr differences in initial cnditins lead t explsive grwth whether scillatry r nt. In nne f these cases S and D wuld prvide cherent slutins. Thus S and D can be cnsidered incnsistent mdels if B > 0 3. D with negative B cefficients is the nly apprach which prduces behavirs cnsistent with the static mdel, whatever the initial cnditins and the final demands applied. Therefre B cefficients might be reinterpreted as an expressin f shrt term cuntercyclical plicy and nt as lng term grwth agents. 4. D is the general versin f S, when B is the negative identity matrix. 5. Frm the previus bservatins, it must be cncluded that matrix B cannt be interpreted as a capital cefficient matrix. 12

13 6-DIRECTIONS FOR FUTURE EMPIRICAL RESEARCH 1. The equivalence f the static mdel and the dynamic mdel with B = - I, allws t tackle all the research typical f the static mdel by numerical cmputatin f equatin E.7 instead if ging thrugh the inversin f matrix [I A]. Numerical cmputatin f E.7 allws the intrductin f a number f nn-linearities. In particular it is pssible t substitute cnstant interindustry cefficients by nes variable with the level f ccupatin, reprducing sme law f diminishing returns r any ther behavir determined by parameter estimatin techniques. 2. Equatin E.2.4 reminds us that the utput f ne perid des nt depend nly f the final demand f the perid but als f the initial lad f backlgs. In the real wrld the ecnmy exhibits at any time sme inertia. The influence f the riginal mmentum vanishes if the perid selected is very lng since, after a while, the weight f the new flwing final demand plays the majr rle in the determinatin f prductin levels. Nevertheless, if the perid is ne year, the influence f the initial cnditins certainly distrts the utput calculated by the mdel if nly final demand f the perid is taken int accunt by the mdel. In empirical research sme attentin shuld be paid t the tpic. Again, the use f numerical cmputatin may make easier t intrduce the influence f the initial lads. 13

14 3. The cnventinal dynamic mdel represented by E.2.1. assumes naively that the utput f a perid crrespnds t the final demand f the same perid, whether the perid is lng r shrt. Since prductin cnsumes time, it is unrealistic t maintain such simultaneity, if the perid cnsidered is very shrt. It is necessary t intrduce prductin lags s that the research can trace the prpagatin ver time f effects demands and crrespnding utputs thrugh the structure f the ecnmy. It is pssible t intrduce easily such a treatment, starting frm equatin E.7. but we leave fr a further paper this presentatin. The inclusin f prductin delays wuld cmplete the timing picture: while the utput f a year is influenced by situatins inherited frm the previus year, part f the cnsequences f the final demand f ne year will be filtered thrugh the industrial system in later perids. Much f the difficulties attributed t an inapprpriate selectin f the level f aggregatin r t practical prblems in the cllectin f statistical data, culd prbably be transferred t an insufficient understanding and treatment f these timing factrs, cmpletely ignred in the static analysis. 14

15 APPENDIX A Slutin f Lentief s dynamic mdel fr cnstant final demands: Y(t) = K E.2.4 becmes t Mt X(t) = e Mt X(0) + e e -Mτ N K d τ A.1 t t but e -Mτ N K dτ = -M -1 e Mt N K A.2 We prve, befre any further prgress, that M -1 N = - [I A] -1 A.3 Prf f A.3. M -1 N = [ B -1 (I A) ] -1. [- B -1 ] = [I A] -1 B [- B -1 ] = - [I A] -1 Therefre, the expressin inside the parenthesis in A.2 is: - M -1 e Mt N K = - M 1 e Mt [- M]. [- M 1 ] N K = - M 1 [- M] e Mt [ - M 1 N K ] = e Mt [ I - A ] -1 K Therefre the slutin f A.2 is: t e -Mτ N K d τ = e -Mt. [I - A] -1 K e -M0 [I A] -1 K= 15

16 = e Mt [I A] 1 K [I A] 1 K A.4 If we replace in A.1 the expressin A.2 by A.4, we btain: X(t) = e Mt X(0) + e Mt. e Mt [I - A] 1 K [I A] 1 K = e Mt X(0) [I A] 1 K + I A -1. K 16

17 APPENDIX B Slutin f D when final demand grws steadily: Y(t) = Kt t Mt Equatin E.2 becmes: X(t) = e Mt X(0) + e e -Mτ N K τ d τ We slve this equatin in three steps: 1. Integratin by parts f: t e -Mτ N K τ d τ t btain [ - M -1 e -Mt t M -2 e -Mt M -2 ]. N. K 2. Multiplicatin by e Mt t btain: - M -1 N Kt M -2 N K - e Mt M -2 N K 3. Additin f e Mt X(0) and rerdering: [I A] -1 Kt M -1 [I A] -1 K + e Mt [ X(0) M -1 [I A] -1 K ] B.1 (remember frm Appendix A: - M -1 N = [I A] -1 ) If B = - I, then X(t) = [I A] -1 K t - [I A] -2 K + e Mt [ X(0) [I A] -2 K ] B.2 In B.1 and B.2: 17

18 1. The first term cincides with the slutin f the static mdel. 2. the secnd term represents a steady state errr. It is cnstant. When the first term grws this cmpnent lses its weight. 3. The third term fades away t becme zer if the system is stable. Therefre the slutin f D cnverges t that f S when t again if the system is stable. 18

19 APPENDIX C (4) Determinatin f the stability cnditins f D fr the tw sectr case. Determinatin f the sign f the eigenvalues f matrix M = B -1 [I A] We define: b 11 b 12 b 22 -b 12 B B * B = therefre B -1 = - b 21 b 11 b 21 b 22 B B where B = b 11 b 22 b 12 b 21 * [ I A ] = (1-a 11 ) - a 12 -a 21 (1-a 22 ) therefre m 11 m 12 M = = B -1 [ I A ] = m 21 m 22 = b 22 (1-a 11 )+ b 12 a 21 - b 22 a 12 - b 12 (1-a 22 ) B B - b 21 (1-a 11 ) a 21 b 11 b 21 a 12 + b 11 (1 a 22 ) B B 19

20 We calculate the eigenvalues λ f M defined by: λ I M = 0 that is: λ - m 11 - m 12 and therefre: - m 21 λ - m 22 λ 1 = (m 11 + m 22 ) + 2 and λ 2 = (m 11 + m 22 ) 2 where = ( m 11 + m 22 ) 2-4 M where M = m 11 m 22 m 12 m 21 We d nt develp the fllwing demnstratins which are mechanical: b 22 (1-a 11 ) + b 12 a 21 + b 21 a 12 + b 11 (1 a 22 ) m 11 + m 22 = b 11 b 22 b 12 b 21 and M = (1-a 11 ) (1-a 22 ) a 12 a 21 b 11 b 22 b 12 b 21 We have t assume that the system cmplies with the tw Hawkin-Simns cnditins necessary fr the existence f the underlying static mdel. 20

21 ( 1 a 11 ) ( 1 a 22 ) a 12 a 11 > 0 ( 1 a 11 ) > 0 and ( 1 a 22 ) > 0 We analyze the fllwing pssible cases: (A) (B) (C) All cefficients b ij are psitive. All cefficients b ij are psitive r zer. All cefficients b ij are negative. (A) All b ij psitive leads t tw pssibilities: A 1 ) When b 12 b 21 > b 11 b 22 m 11 + m 22 < 0 since its numeratr is psitive and the denminatr is negative. But M < 0 fr the same reasn Therefre is the additin f tw psitive quantities and is nt cmplex. Bth eigenvalues are real. Since is larger then m 11 + m 22, λ 1 is real and psitive. λ 2 is real and negative. Therefre the system exhibits a saddle pint. A 2 ) When b 12 b 21 < b 11 b 22, m 11 + m 22 > 0, since bth numeratr and denminatr are psitive. M > 0 fr the same reasn Therefre is the additin f ne psitive quantity ( m 11 +m 22 ) 2 and a negative quantity: - 4 M 21

22 Tw cases are pssible: A.2.1. When (m 11 + m 22 ) 2 > 4 M, is psitive and is nt cmplex. Bth eigenvalues are real. but since < m 11 + m 22, bth λ 1 and λ 2 are psitive. The system has an unstable nde. A.2.2. When ( m 11 + m 12 ) 2 < 4 M, < 0, is cmplex, and bth eigenvalues are cmplex with real parts [ ( m 11 + m 22 ) ] psitive. The mdel shws an unstable fcus. (B) Cefficients b ij psitive r zer. B 1 ) At least all the cefficients f ne f the diagnals must be nn zer. Otherwise B wuld be singular and D wuld nt exist. B 2 ) When ne r tw cefficients f the main diagnal are zer and the rest psitive: m 11 + m 22 < 0 since numeratr is psitive and denminatr negative but M < 0 fr the same reasn. We fall in case A1. B 3 ) When ne r tw cefficients f the secnd diagnal are zer and the rest psitive. 22

23 m 11 + m 12 > 0 since numeratr and denminatr are psitive but M > 0 fr the same reasn We fall in case A2. (C) All cefficients b ij are negative Tw alternatives: C 1 ) When b 11 b 22 < b 12 b 21, then m 11 + m 22 > 0 since bth the denminatr and the numeratr are negative but M < 0 since the numeratr is psitive and the denminatr negative therefre = ( m 11 + m 22 ) 2 4 M is the additin f tw psitive quantities is then real and bth eigenvalues are real. Since > m 11 + m 22, λ 1 is real psitive λ 2 is real negative Therefre we have a saddle pint. C 2 ) When b 11 b 22 > b 12 b 21 m 11 + m 22 < 0, since its numeratr is negative and its denminatr is psitive but M > 0 since bth numeratr and denminatr are psitive 23

24 Tw cases are pssible: C.2.1. ( m 11 + m 22 ) 2 > 4 M and therefre is real. Bth eigenvalues are real and since > m 11 + m 22, λ 1 is real psitive and λ 2 is real negative. We again have a saddle pint as in A1. C.2.2. ( m 11 + m 22 ) 2 < 4 M and therefre is cmplex The tw eigenvalues are cmplex with negative real parts. Therefre the system shws a stable fcus. C.2.3. ( m 11 + m 22 ) 2 = 4 M and therefre = 0. The tw eigenvalues are identical real and negative. The system shws a stable nde. The specific cases C.2.2. and C.2.3. f B with negative cefficients are the nly nes which prevent incnsistency f the static and the dynamic mdel, by representing a stable system where dynamic statinary behavir and static prvide cnsistent representatins f the same reality. (4) Fr the analysis included n the nature f the singular pints we suggest cntrast with Mdern Cntrl Engineering Katsuhik Ogata Prentice Hall, 1970, Pages

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