ppled Mathematcal Scences, Vol. 4, 200, no. 60, 2955-296 Norm Bounds for a ransformed ctvty Level Vector n Sraffan Systems: Dual Exercse Nkolaos Rodousaks Department of Publc dmnstraton, Panteon Unversty 36 Syngrou ve, 767 thens, Greece nrodousaks@gmal.com George Sokls Department of Publc dmnstraton, Panteon Unversty 36 Syngrou ve, 767 thens, Greece gsok@hotmal.gr bstract hs paper gves lower and upper bounds for the largest and the smallest element of a transformed actvty level vector n Sraffan systems. he bounds appear as dual to those of the prce system, ndcated by Marols n 200: () they are expressed n terms of the maxmum row sum matrx norm ; and () depend on the vertcally ntegrated coeffcents and the rato of the unform rate of growth to the maxmum rate of growth. Mathematcs Subject Classfcaton: 9B38, 9B66 Keywords: Norm bounds, actvty level vector, dual systems, Standard prces, row stochastc matrx, vertcal ntegraton, relatve rate of growth Introducton In a lnear producton system, the relatonshp between the wage rate and the rate of proft can be obtaned from the prce system, whlst the relatonshp between consumpton per head and the growth rate can be obtaned from the system of physcal quanttes. s s t has been ponted out, these two relatonshps (systems) have exactly the same mathematcal form (see, e.g., [6] and [2], p. 4). hs fact s referred to the lterature as the dualty of the two relatonshps (systems).
2956 N. Rodousaks and G. Sokls It s well known that, wthn the Sraffan framework, the long-run relatve prces (the structure of outputs) can change n a complcated way as the rate of proft (growth rate) changes (see, e.g., [7, chs 3 and 6] and [6], respectvely). In a recent paper n ths Journal, Marols [3] has shown that, when the prce system can be transformed (va a dagonal smlarty matrx formed from the elements of the left-hand sde Perron-Frobenus egenvector of the techncal coeffcents matrx) nto a vertcally ntegrated system n whch the techncal coeffcents matrx s a column stochastc matrx, the largest and the smallest element of the transformed (and normalzed wth Sraffa s Standard commodty ) prce vector admt norm bounds that depend on the soco-techncal condtons of producton. he purpose of ths paper s to carry out (for reasons of symmetry and completeness) a smlar elaboraton on the system of physcal quanttes. Snce the quantty sde of a Sraffan system s formally dual to the prce system, t s reasonable to expect that, by followng the route suggested by Marols [3], one could obtan a sutably transformed vector of actvty levels that admts norm bounds for ts largest and smallest element. 2 he remander of the paper s structured as follows. Secton 2 deals wth the usual crculatng captal model and gves norm bounds for a transformed actvty level vector. Secton 3 concludes the paper. 2 Norm Bounds Consder a closed lnear system, nvolvng only sngle products, basc commodtes (n the sense of Sraffa [7, 6]) and crculatng captal. Furthermore, assume that () the nput-output coeffcents are fxed; () the system s vable,.e., the Perron-Frobenus (P-F hereafter) egenvalue of the rreducble n n matrx of nput-output coeffcents,, s less than ; 3 () the rate of growth, g, s unform; and (v) commodtes are consumed n proporton to the entres of the n vector of consumpton bundle, b ( 0 ), whch serves as the unt of consumpton. It should be noted, however, that Steedman [9] has detected a rule that the prce (net output) vector follows as the rate of proft (growth rate) vares. 2 It s should be noted, however, that the formal symmetry between the prce and physcal quantty systems does not necessarly mples a symmetry n a substantal sense (see [6]). 3 Matrces (and vectors) are denoted by boldface letters. he transpose of an n vector a [ a ] s denoted by a. denotes the P-F egenvalue of a (sem-) postve matrx [ a j ], ( q, y ) the correspondng egenvectors, and q ( y) the dagonal matrx formed from the elements of q ( y ). Fnally, e denotes the summaton vector,.e., e [,,...,].
Norm bounds for a transformed actvty level vector 2957 On the bass of these assumptons, the quantty sde of the system s descrbed by the followng relaton x ( g) x cb () where x denotes the vector of actvty levels and c the ndex of consumpton. Relaton () after rearrangement gves: x ggx cf (2) where G [ I ] denotes the vertcally ntegrated matrx ([4]) the j th column of whch represents the vector of actvty levels whose operaton would produce, as a net product, just the captal stock requred (drectly) to support the operaton of the j th process at unt level, and f [ I ] b ( 0) denotes the actvty vector requred to support one unt of consumpton, when g 0. 4 If we evaluate the physcal quanttes n terms of Standard prces ([6]),.e., y[ I ] x, wth yb, then () mples that 5 c ( g / g max ) or c (3) where gmax (/ ) ( / G ) represents the maxmum rate of growth and g/ g max, 0, the relatve rate of growth. Substtutng (3) n (2) yelds x Zx ( ) f (4) or, f, ( )[ ] ( )( k k x I Z f Z ) f (5) where Z gmaxg and Z. Gven that [ q ] Zq e q Zq q q e t follows that Z s smlar to the row stochastc matrx M [ ] q Zq ( 0 ), the elements of whch are ndependent of the choce of mj physcal measurement unts (and the normalzaton of q ). Substtutng Z q Mq, wth y [ I ] q, n (4) yelds φ Mφ ( ) ζ (6) k 0 4 It may be noted that the matrx G s dual to the well-known vertcally ntegrated coeffcents matrx ([4]), labour coeffcents (bd.), H [ I ], whlst f s dual to the vector of the vertcally ntegrated v ( l [ I ] ) (where l ( 0 ) s the vector of drect labour coeffcents). Furthermore, for the sgnfcance of the matrx G n economc theory, see [8]. 5 It s known that the vector of Standard prces s dual to vector of physcal quanttes that represent Sraffa s Standard commodty (see [6]). Furthermore, the vector of Standard prces, y, corresponds to a partcular prce vector whch expresses a pure captal theory of value ([5], pp. 76-78), as opposed to the well known pure labour theory of value.
2958 N. Rodousaks and G. Sokls or, 6 f, where φ q x, stochastc matrx, snce φ [ B( )] ζ (7) ζ q f, [ B( )] B( ) ( ) [ I M ], and 0 and [ ( )] B s a row [ B( )] e ( )( ) e e From relatons (6)-(7), and the normalzaton condtons, we derve the followng: (). φ ζ at 0, and φ e at. In the trval case n whch ζ e, then φ e. 7 Furthermore, snce ( yq)( φ ζ ) 0 for each, t follows that ( y q )( e ζ ) 0, whch n ts turn mples mn{ } max{ } (8) (f ζ e, then both nequaltes n (8) are strct). (). Relaton (7) mples that φ,,2,..., n, s a convex combnaton of the elements of ζ. hus, we may wrte or and where mn{ } max{ } φ max{ } ζ max{ } (9) mn{ } / ζ mn{ } / φ (0) denotes the maxmum row sum matrx norm. (). Relaton (6) can be restated as ( ) ζ [ I M] φ () akng norms of (), and usng the Hölder s nequalty, we obtan or, gven that n ( ) ζ φ (max{ m m }),,2,..., n (2) n j j m m, j j j j 6 Relaton (6) represents a transformed vertcally ntegrated physcal quantty system n whch the techncal coeffcents matrx, M, s a row stochastc matrx. Note that M s dual to the column stochastc techncal coeffcents matrx, K ( y Jy ), (where J RH and R represents the maxmum rate of proft) of the transformed vertcally ntegrated prce system obtaned by Marols ([3], relaton 6), whlst ζ s dual to the transformed vector of vertcally ntegrated labour coeffcents, ω ( v y ), that also appears n the transformed prce system (bd.). 7 hs s the case n whch the proportons of the economc system are those of Sraffa s Standard system (see, e.g., [6], p. 27).
Norm bounds for a transformed actvty level vector 2959 or or ( ) ζ φ (max{ ( 2 m )}) ( ) ζ φ [ ( 2 )] f ( ) φ / ζ (3) where mn{ m }, 0, and f ( ) ( )/[ ( 2 )], 0 f ( ), a strctly decreasng functon of, whch s strctly convex to the orgn for 0.5 and tends to as tends to. 8 (v). Pre-multplyng (6) by φ ζ ( ζ φ ) gves ζ e φ ζ Mφ ( ) φ e akng norms, and recallng M, we obtan or, dvdng both sdes by where ζ φ ζ φ ( ) φ φ and recallng (9) and (0), ζ / φ ( ) h( ) (4) ζ φ ( ) (4a) ( ) ( ) () () / () (snce φ e at ), and ( ) ( ) ζ φ ζ φ ζ (4b) h ζ ζ ζ ζ (4c) Combnng (9) and (3) gves f ( ) φ / ζ (5) whlst combnng (0) and (4) gves ζ / φ h( ) (6) We therefore conclude that the upper (lower) bound for φ / ζ (for ζ / φ ) equals and the lower (upper) bound decreases (ncreases) wth ncreasng. 9 Furthermore, these bounds are perfectly dual to those of the prce system (see [3]). More specfcally, we observe the followng dualtes between the bounds of the two systems: () the bounds of the prce system are expressed n terms of the maxmum column sum matrx norm, whlst those of the physcal quantty 8 It may be noted that the condton number (see, e.g., [], pp. 399-400) of Β ( ), defned as Β( ) [ Β ( )], equals / ( ) f. 9 he monotoncty of ( ) s a pror unknown.
2960 N. Rodousaks and G. Sokls system are expressed n terms of the maxmum row sum matrx norm ; and () the bounds of the prce system depend on the vertcally ntegrated coeffcents that correspond to the transformed vertcally ntegrated prce system (.e., the elements of K and ω ) and the relatve rate of proft, whlst those of the physcal quantty system depend on the vertcally ntegrated coeffcents that correspond to the transformed vertcally ntegrated physcal quantty system (.e., the elements of M and ζ ) and the relatve rate of growth. 3 Concludng Remarks It has been shown that, when the quantty sde of a Sraffan producton system can be transformed (va a dagonal smlarty matrx formed from the elements of the rght-hand sde Perron-Frobenus egenvector of the techncal coeffcents matrx) nto a vertcally ntegrated system n whch the techncal coeffcents matrx s a row stochastc matrx, the largest and smallest element of the transformed (and expressed n terms of Standard prces ([6])) actvty level vector admt lower and upper bounds. he bounds appear, not qute unexpectedly, as perfectly dual to those of Marols [3] regardng the prce system: () they are expressed n terms of the maxmum row sum matrx norm ; and () depend on the vertcally ntegrated coeffcents and the relatve rate of growth. cknowledgements. frst draft of ths paper was presented at a Workshop of the Study Group on Sraffan Economcs at the Panteon Unversty, n September 2009: We are ndebted to Elefthera Rodousak and, n partcular, heodore Marols for extremely helpful dscussons and comments. he usual dsclamer apples. References [] S. Barnett, Matrces. Methods and pplcatons, Oxford Unversty Press, Oxford, 990. [2] H.D. Kurz and N. Salvador, heory of Producton. Long-Perod nalyss, Cambrdge Unversty Press, Cambrdge, 995. [3]. Marols, Norm Bounds for a ransformed Prce Vector n Sraffan Systems, ppled Mathematcal Scences, 2 (200), 55-574. [4] L. Pasnett, he noton of vertcal ntegraton n economc analyss, Metroeconomca, 25 (973), -29. [5] L. Pasnett, Lectures on the heory of Producton, Columba Unversty Press, New York, 977.
Norm bounds for a transformed actvty level vector 296 [6] L. Pasnett, Standard prces and a lnear consumpton/growth-rate relaton, n: L. Pasnett (ed.), Italan Economc Papers, vol., Oxford Unversty Press, Oxford, 992. [7] P. Sraffa, Producton of Commodtes by Means of Commodtes. Prelude to a Crtque of Economc heory, Cambrdge Unversty Press, Cambrdge, 960. [8] I. Steedman, On Pasnett s G matrx, Metroeconomca, 40 (989), 3-5. [9] I. Steedman, Values Do Follow a Smple Rule!, Economc Systems Research, (999), 5-4. Receved: May, 200