We present a new approach to irreducibility in multisectoral models, based on an axiom on the vectors of total outputs and inputs. We derive new results in joint production models, including an interpretation of von Neumann's famous condition A + B > 0, where A and B are respectively matrices of input and output coefficients. Also, an application of the axiom to the commodity technology model is presented

Konijn, P.J.A.

Steenge, A.E.

English

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A New Approach to Irreducibility in Multisectoral Models with Joint Production

We present a new approach to irreducibility in multisectoral models, based on an axiom on the vectors of total outputs and inputs. We derive new results in joint production models, including an interpretation of von Neumann's famous condition A + B &gt; 0, where A and B are respectively matrices of input and output coefficients. Also, an application of the axiom to the commodity technology model is presented

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EBSCOhosthttp://web.ebscohost.com/...ost/delivery?vid=3&hid=104&sid=55b90485-289b-402f-9d56-1ce4d3028e1b%40sessionmgr113[27-1-2010 13:56:44]Database: Business Source EliteA New Approach to Irreducibility in Multisectoral Models with JointProductionABSTRACT We present a new approach to irreducibility in multisectoral models, based on an axiomon the vectors of total outputs and inputs. We derive new results in joint production models,including an interpretation of von Neumann's famous condition A + B > 0, where A and B arerespectively matrices of input and output coefficients. Also, an application of the axiom to thecommodity technology model is presented.1. IntroductionA number of authors have recently discovered the applicability of a theorem of Mangasarian (1971),which can be interpreted as a condition on input and output vectors, in multisectoral models withjoint production. We mention Punzo (1980), who applies it to the Turnpike theorem, Semmler(1984), who applies it to labour economics, and Fujimoto & Krause (1988), who apply it todistribution problems. In this paper, we apply Mangasarian's results primarily to the issue ofirreducibility.Multisectoral models with joint production can generally be described by two coefficient matrices: aninput matrix A and an output matrix B. In Section 2, we formulate the above-mentioned conditionon the vectors of total outputs and inputs which will consequently be used as an additional axiomin the model. This axiom can be seen as a generalization of a fundamental property of single-product models, namely the non-negativity of the coefficient matrix.The mathematical result of Mangasarian (1971) that we referred to above then tells us that, if theaxiom holds, there exists a third non-negative matrix which we will call M that connects A and Bthrough A=MB. This matrix M will be the fundamental tool for analysis in this paper. In Section 3we use it to derive a condition that guarantees irreducibility of the general multisectoral system (A,B).In sections 4 and 5 we apply it to two particular types of model, namely the von Neumann model(von Neumann, 1945--46) and the commodity technology model, a generalized input-output model(United Nations, 1968). In the former we establish a link between the famous von Neumanncondition( n1) A + B > 0 and the concept of irreducibility, a link that so far has not been available.2. Multisectoral ModelsTypical of a multisectoral model with homogeneous production is that it can be represented by anon-negative coefficient matrix. Examples of these models are those of Sraffa (1960) and theLeontief input-output model (Leontief, 1951). The non-negativity of the coefficients is fundamentalin these models. In Sraffa's model it is used to prove the existence of the Standard Commodity; inLeontief's model it is used for proving the existence and non-negativity of the Leontief inverse.If we denote the coefficient matrix as A (where A is an m x m matrix, with m the number ofcommodities), then non-negativity of A is equivalent tox > 0  Ax > 0provided that each row of A contains at least one non-zero element (if there is a zero row thecorresponding commodity is irrelevant and can therefore be deleted). This fundamental propertyhas the following interpretation: if we consider x to be a vector of changes in output levels, then arise in the production of all commodities implies a rise in the use of all commodities. Note that thisproperty implies nothing about the (ir)reducibility of the matrix A.A multisectoral model that explicitly incorporates joint production should distinguish betweencommodities and activities. We shall distinguish m commodities and n activities. Such a model thentypically consists of two matrices. A is the input matrix (of dimensions m x n), an element aij ofwhich represents the use of commodity i in activity j if this activity operates at unit level. B is theoutput matrix (also sized m x n), and bij represents the production of commodity i in activity jwhen operating at unit level. Examples of such models are the Sraffa model with joint production(see Sraffa, 1960), where the problem becomes the existence of the Standard Commodity, the yonNeumann model (von Neumann, 1945--46), which studies the growth rate of the economy (seeEBSCOhosthttp://web.ebscohost.com/...ost/delivery?vid=3&hid=104&sid=55b90485-289b-402f-9d56-1ce4d3028e1b%40sessionmgr113[27-1-2010 13:56:44]Section 4), and the commodity technology model (United Nations, 1968), a generalized input-output model (see Section 5).Now in these models problems arise because of the increased mathematical difficulties. In our viewthis is primarily due to the fact that the non-negativity property of the standard single-productmodels does not yet have an analogue in the extended models. We shall therefore introduce acondition that generalizes the nonnegativity property for joint production models. It is presentedhere as an axiom and we retain it throughout this paper to investigate the consequences ofimposing the axiom on our models.Axiom: x: Bx > 0  Ax > 0The vector x may contain negative elements and it should therefore be interpreted as a vector ofchanges in activity levels. The economic interpretation of the axiom is the same as theinterpretation of the fundamental property of single-product models described above: an increase inthe production of all commodities implies that the use of all commodities should be increased.Again, nothing about irreducibility is implied yet.The following proposition, which is due to Mangasarian (1971), tells us that the axiom is equivalentto the existence of a third non-negative matrix.Proposition 1. For arbitrary matrices A, *(This character cannot be converted in ASCII text) thefollowing equivalence holds, assuming that {x: Bx>0} ≠ Ø:( x: Bx>0 Ax>0)*(This character cannot be converted in ASCII text) M≥0, i: Mi.≥0: A = MBProof. See Mangasarian (1971), theorem 3.1, part (vii).Thus, a third matrix is involved. This matrix M can be seen as the generalization of the coefficientmatrix of the single-product models. It is non-negative and has no zero rows, i.e. all commoditiesare used in some process.( n2) In Section 3 we shall give an application for general multisectoralmodels concerning the irreducibility of these models.3. IrreducibilityThe system (A, B) is called reducible (see Gale, 1960) if there exist permutation matrices P(m x m)and Q(n n) such that(Multiple lines cannot be converted in ASCII text)(1 ≤ k < m, 1 ≤ l < n), while each row of B11 contains at least one positive element. If such apermutation does not exist the system is called irreducible. A reducible system has a subsystemthat can produce 'on its own'. Sraffa's notion of basic and non-basic products is related to theconcept of reducibility. If m = n, k = l and B = I, the above definition reduces to the standarddefinition of irreducibility of a square matrix.We now present our main result about the property of (ir)reducibility and the properties of M.Proposition 2. Let the axiom be fulfilled, such that there exists a matrix M ≥ 0 with A = MB. Letthis M be irreducible. Then the system (A, B) is irreducible.Proof. Suppose to the contrary that (A, B) is reducible. Then there exist permutation matrices P andQ such that[Multiple line equation(s) cannot be represented in ASCII text](1 ≤ l < n, 1 ≤ k < m) and B11, i. ≥ 0 ( i). Now take[Multiple line equation(s) cannot be represented in ASCII text]From A = MB it follows that Â = PAQ = PMBQ = PMP'PBQ = MB and thus Âx = MBx, which can bewritten as[Multiple line equation(s) cannot be represented in ASCII text]EBSCOhosthttp://web.ebscohost.com/...ost/delivery?vid=3&hid=104&sid=55b90485-289b-402f-9d56-1ce4d3028e1b%40sessionmgr113[27-1-2010 13:56:44]Because B11x1>0 this implies M21 = 0. This contradicts the assumption of the irreducibility of M.Thus (A, B) must be irreducible.If M is irreducible, it is impossible to find a system (A, B) such that the model become reducible.Thus reducibility of M is a prerequisite for the reducibility of (A, B). However, it is not sufficient. Asimple counterexample is[Multiple line equation(s) cannot be represented in ASCII text]The following proposition gives a sufficient condition for (A, B) to be reducible when M is reducible.Proposition 3. Let the axiom be fulfilled, such that there exists a matrix M ≥ 0 with A = MB. Letthis M ≥ 0 be reducible, such that[Multiple line equation(s) cannot be represented in ASCII text]with P a suitable permutation matrix. Then the system (A, B) is reducible if there exists a vector z≥ 0 such that[Multiple line equation(s) cannot be represented in ASCII text]Proof.[Multiple line equation(s) cannot be represented in ASCII text]Thus, for this z we can produce a subset of goods without the use of goods from outside thesubset. The system then must be reducible.These proofs do not make use of the structure of any model in particular, except that it isrepresented by an input matrix and an output matrix. Therefore, the results hold for all models thatcan thus be described. The reducibility concept is in particular of great importance in the yonNeumann model. We discuss this in Section 4.4. The von Neumann ConditionIn the von Neumann model we have, besides A and B, a vector x which is the activity level vectorand p' which gives the prices of the commodities. The real side of the von Neumann model can bewritten asmax αsuch that Bx ≥ αAx, x ≥ 0with α the growth factor of the economy. The dual price side ismin βsuch that p'B ≤ βp'A, p'≥0with β the profit rate of the economy.One of the central questions in the literature concerning the yon Neumann model is the equality ofthe growth rate α and the profit rate β. Two conditions ensuring this equality coexist, without anyrelation between them. yon Neumann imposed his famous condition A + B > 0 to guarantee that αwould equal β. This approach is unsatisfactory from an economic point of view, since yonNeumann's condition is clearly not plausible. Gale (1960) and Kemeny et al. (1956) gaverelaxations of von Neumann's condition. They impose the following, easily interpretable, conditions:A.j≥0 for all j, Bi.≥0 for all iThis guarantees that optimal values of α and β exist, and α ≥ β. If the system is irreducible wehave α = β.However, the yon Neumann condition does not imply irreducibility. Robinson (1973) gave thefollowing counterexample:EBSCOhosthttp://web.ebscohost.com/...ost/delivery?vid=3&hid=104&sid=55b90485-289b-402f-9d56-1ce4d3028e1b%40sessionmgr113[27-1-2010 13:56:44][Multiple line equation(s) cannot be represented in ASCII text]Clearly, this system is reducible but satisfies the yon Neumann condition. Therefore, there is so farno relation between yon Neumann's condition and the condition of irreducibility of Gale et al. Weshall show now that if our axiom is fulfilled we are able to establish a link.Proposition 4. Suppose the von Neumann model (A, B) satisfies the axiom. Then if A + B > 0, thesystem (A, B) must be irreducible.Proof. Since (A, B) satisfies Bx > 0 Ax > 0 there exists a matrix M ≥ 0, with Mi. ≥ 0 for all i, andA = MB. Now, suppose to the contrary that (A, B) is reducible. Choose P, Q and x as in the proof ofProposition 2. From A + B > 0 we have Â + B > 0 and hence B21 > 0. Then[Multiple line equation(s) cannot be represented in ASCII text]which is clearly not a positive vector. But from A = MB we have again Â = MB with M = PMP'.From. Mi≥0 and Proposition 1, we have that Bx > 0 implies Â > 0. A contradiction thus exists, and(A, B) must be irreducible.So we have established a link between the yon Neumann condition and irreducibility. However, wedo not need the yon Neumann condition to prove irreducibility in the context of the axiom we use,as was shown in the previous section. The results of Section 3 can be viewed as an alternativerelaxation of von Neumann's condition: if the axiom is accepted, the irreducibility of M is sufficientfor the equality of the rates.Another application will now be given, which deals with an upper bound to the growth rate of theeconomy. We give a relation between α and, µM, the PerronFrobenius eigenvalue of M. To savespace, we present it without proof.( n3)Proposition 5. Let the axiom be fulfilled. Let Bx ≥ αAx, with α > 0, x ≥ 0. Then α≤ 1/µM with µM ≠0 the Perron-Frobenius eigenvalue of M.5. The Commodity Technology ModelThe above theorems are very theoretical in nature; there exist only a small number of practicalapplications of yon Neumann's model (see, for example, Stolwijk, 1987). The implications ofMangasarian's theorem, however, may also be of use in the more practical world of nationalaccounting and input-output analysis.United Nations (1968) (see Stone (1984) for a survey) introduced the make-use framework, asystem of two tables, one for the production of commodities by industries and one for the(intermediate) consumption of commodities by industries. U is the use matrix: element % describesthe input of commodity i in the production of industry j. V' is the make matrix: element vijdescribes the production of commodity i by industry j. Furthermore, we have the vector e of finaldemands for commodities, the vector y of primary inputs into the industries, the vector q of totalproduction of commodities and the vector g of total production of industries.From the make matrix we can derive an output (coefficient) matrix B, an element bij of whichrepresents the output share of commodity i in the production of industry j:B = V'(g)-1 g = diag(g)From the use matrix we obtain similarly an input matrix A, an element aij of which gives the inputshare of commodity i in the production of industry j:A = U(g)-1The commodity technology model assumes that each commodity has its own characteristic inputstructure. This can be given effect by combining the two (empirical) matrices A and B to derive aninput-output matrix M as M = AB-1 (here it should be assumed that A and B are square and that Bis non-singular). Each column of M gives the characteristic input structure of the correspondingcommodity. The matrix M, connecting A and B, which we introduced for general multisectoralmodels, therefore has a well-known meaning in input-output modelling. Now, imposing our axiomEBSCOhosthttp://web.ebscohost.com/...ost/delivery?vid=3&hid=104&sid=55b90485-289b-402f-9d56-1ce4d3028e1b%40sessionmgr113[27-1-2010 13:56:44]on this model yields a generalization of the fundamental property of the classical Leontief model, asdescribed above. It yields, through Proposition 1, a non-negative matrix M (with Mi. ≥ 0 for all 0such that A = MB. It thus appears that our axiom give additional grounds for the commoditytechnology approach. Any negative element of M found in empirical work is in contradiction with theaxiom. This suggests another way of interpreting these negatives: they should not be regarded asrejecting the commodity technology assumption, but as indicators of flaws in the make and usetables. For a more detailed discussion on these issues see Steenge (1990).6. ConclusionsIn this paper we introduced an axiom for general multisectoral models with joint production. It isargued that it is necessary to impose such an axiom in order to extend results of our simple modelswith homogeneous production. Our major focus was on irreducibility, a subject of importance inmost multisectoral models. We have given a new sufficient condition for irreducibility.Furthermore, we have given a context for the yon Neumann model in which a clear connectionexists between yon Neumann's condition and irreducibility, and we have shown that the axiomgives an upper bound to the growth rate of the economy.Another application of the axiom is for the commodity technology model. Here it gives a guaranteefor the non-negativity of the coefficients. Thus the applications we have given illustrate the factthat the axiom gives new directions to solutions to long existing problems, and therefore deservesmore attention than it has received in the literature so far.Notes(n1.) The following notational conventions are used.A ≥ 0: all elements of A are non-negative A ≥ 0: all elements of A are non-negative, and someelement is positive A > 0: all elements of A are positiveSimilar conventions are used for vectors.(n2.) Punzo (1980) formulates Mangasarian's theorem in terms of the polyhedral cones defined bythe matrices A and B. This is just the geometrical way of presenting the economic condition whichwe present as an axiom. We prefer our algebraic way of putting it, because of the clearer economicunderstanding.(n3.) The proof can be obtained from the authors upon request.ReferencesFujimoto, T. & Krause, U. (1988) More theorems on joint production, Journal of Economics, 48, pp.189-196.Gale, D. (1960) The Theory of Linear Economic Models (New York, McGraw-Hill).Kemeny, J. G., Morgenstern, O. & Thompson, G. L. (1956) A generalization of the von Neumannmodel of an expanding economy, Econometrica, 24, pp. 115-135.Leontief, W. W. (1951) The Structure of the American Economy, 1919-1939 (White Plains, NY,IASP).Mangasarian, O. L. (1971) Perron-Frobenius properties of Ax = 2Bx, Journal of MathematicalAnalysis and Applications, 36, pp. 86-102.von Neumann, J. (194546) A model of general economic equilibrium, Review of Economic Studies,13 (1), pp. 1-9.Punzo, L. F. (1980) Economic applications of a generalized Perron-Frobenius problem, EconomicNotes, 9, pp. 101-116.Robinson, S. M. (1973) Irreducibility in the yon Neumann model, Econometrica, 41, pp. 569-573.Semmler, W. (1984) Competition, Monopoly and Differential Profit Rates (New York, ColumbiaUniversity Press).EBSCOhosthttp://web.ebscohost.com/...ost/delivery?vid=3&hid=104&sid=55b90485-289b-402f-9d56-1ce4d3028e1b%40sessionmgr113[27-1-2010 13:56:44]Sraffa, P. (1960) Production of Commodities by Means of Commodities (Cambridge, CambridgeUniversity Press).Steenge, A. E. (1990) The commodity technology revisited: theoretical basis and an application toerror location in the make-use framework, Economic Modelling, 7, pp. 376-387.Stolwijk, H. J. J. (1987) Options for Economic Growth in Bangladesh (Amsterdam, VU Press).Stone, R. (1984) Accounting matrices in economics and demography, in: F. van der Ploeg (Ed.),Mathematical Models in Economics (Chichester, Wiley), pp. 9-36. United Nations (1968) A Systemof National Accounts, series F, no. 2 (New York, UNSO).~~~~~~~~By A. E. Steenge and P. J. A. Konijn, University of Twente, Faculty of Public Administration andPublic Policy, PO box 217, 7500 AE Enschede, The Netherlands.

A generalization of the von Neumann model of an expanding economy,

Economic applications of a generalized Perron-Frobenius problem,

More theorems on joint production,

Perron-Frobenius properties of Ax = 2Bx,

The Structure of the American Economy,

The Theory of Linear Economic Models

https://doi.org/10.1080/09535319200000011

A New Approach to Irreducibility in Multisectoral Models with Joint Production

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