شرایط برای برخی از تعادل های نظام اقتصادی: تجزیه و تحلیل ورودی خروجی با استفاده از نظریه طیفی ماتریس های غیر منفی

Twelve kinds of special semipositive matrices and their basic characters are presented. Employing these matrices and the previous results in Zeng (2008), we research the conditions for the balances between final output values and values-added, and between input multipliers and output multipliers in an economy. A necessary and sufficient condition that (i) there exists a unique vector of output adjustment coefficients such that (a) all sectoral final output values equal their respective sectoral values-added in the new output system, or (b) all sectoral input multipliers redefined by the new output system equal their respective sectoral output multipliers; or (ii) there exists a unique vector of price adjustment coefficients such that (a) all sectoral values-added equal their respective sectoral final output values in the new price system, or (b) all sectoral output multipliers redefined by the new price system equal their respective sectoral input multipliers; is the irreducibility of the matrix of intermediate output (or input) coefficients. A necessary and sufficient condition that (i) there is no vector of output adjustment coefficients which enables all sectoral final output values (or input multipliers) to equal their respective sectoral values-added (or output multipliers), or (ii) there is no vector of price adjustment coefficients which enables all sectoral values-added (or output multipliers) to equal their respective sectoral final output values (or input multipliers); is that the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class. These class relations and their equivalent conditions are summarized. The elaborate examples verify the main conclusions thoroughly.

Since an input–output table can accurately reflect the interdependence of industries or sectors in an economic system, the various input–output models are extensively used. It is known that in an economic system expressed by a monetary input–output table the sum of every sectoral final output value equals the sum of every sectoral value-added, which is called the gross national product. In general, some sectoral final output value may be unequal to this sectoral value-added. However, theoretically it is possible that all sectoral final output values equal their respective sectoral values-added. This is one kind of balance of the economic system, which can be called a balance between final output values and values-added. Besides, in an economic system shown via a monetary input–output table, a sectoral input (or supply) multiplier measures the rate of change of total input values throughout all sectors of the economy with respect to this sectoral value-added (cf. Miller and Blair (1985, Chapter 9)). Dually, a sectoral output (or demand) multiplier measures the rate of change of total output values throughout all sectors of the economy with respect to this sectoral final output value. Simply speaking, the input multiplier reflects the push influence of change of primary input on the economy; the output multiplier reflects the pull influence of change of final output on the economy. Generally, some sectoral input multiplier may be unequal to this sectoral output multiplier. Nevertheless, theoretically it is possible that all sectoral input multipliers equal their respective sectoral output multipliers. Namely, the push influence of change of every sectoral primary input on the economy is exactly equivalent to the pull influence of change of the matching sectoral final output on the economy. This is another kind of balance of the economic system, which can be called a balance between input multipliers and output multipliers. Several problems arise immediately. What is the necessary and sufficient condition for the balance between final output values and values-added? What is the necessary and sufficient condition for the balance between input multipliers and output multipliers? If the necessary conditions are satisfactory, but the necessary and sufficient conditions are not, can the necessary and sufficient conditions be satisfied via adjusting output system or price system? What is the necessary and sufficient condition that the adjustable possibility and uniqueness hold? Zeng (2008) discussed the output adjustment model and the price adjustment model and their basic properties, and analyzed the effects of changes in outputs and in prices on the economic system using the spectral theory of nonnegative matrices. In this paper, based on the results in Zeng (2008), we solve the above several problems, and deduce a set of necessary and sufficient conditions for (im)possibility and (non)uniqueness of the economic adjustment that enables all sectoral final output values to equal their respective sectoral values-added, and/or all sectoral input multipliers to equal their respective sectoral output multipliers. The paper is organized as follows. In Section 2, some formulas on output adjustment model and price adjustment model are derived, which will be applied to the next sections. Moreover, we construct twelve kinds of special semipositive matrices whose spectral radii are all equal to one, and display their fundamental properties, which are very useful and indispensable to the next sections. In Section 3, employing the results in Zeng (2008) and Section 2 of this paper, we pose and solve six input–output economic problems involving the balances between final output values and values-added, and between input multipliers and output multipliers, where Theorem 1 is the core. Compared with the above several problems, the six input–output economic problems are more concrete. Section 4 expressed by Theorem 2 and its proof is a summary of the class relations for the matrix of intermediate output (or input) coefficients and the corresponding equivalent conditions. In Section 5 the elaborate examples illustrate or verify the main conclusions thoroughly.

Using the results in Zeng (2008) and Section 3 of this paper, the following Theorem 2 thoroughly summarizes the class relations for the matrix of intermediate output (or input) coefficients and the matching equivalent conditions. Theorem 2. In an economy (a) the following conditions are equivalent (a1) View the MathML sourceA→orAAisCC-irreducible; (a2) if the column vector of output adjustment coefficients is nontrivial then the accommodation of output system enables at least one intermediate output coefficient to alter, i.e.View the MathML sourceQ≠λE⇒A→#≠A→; (a3) if the row vector of price adjustment coefficients is nontrivial then an adjustment of price system enables at least one monetary intermediate input coefficient to change, that is,P≠λEt⇒A∗≠AP≠λEt⇒A∗≠A; (b) the following conditions are equivalent (b1) View the MathML sourceA→orAAhas at least one non-final (or non-initial) class; (b2) the adaptation of output system enables some sectoral final output rates to rise (or fall) and all others to be constant; (b3) any change of output system disables all sectoral final output values to equal their respective sectoral values-added; (b4) an alteration of output system enables some sectoral input multipliers to increase (or decrease) and all others to be unchanged; (b5) any accommodation of output system disables all sectoral input multipliers to equal their respective sectoral output multipliers; (b6) an adjustment of price system enables some sectoral value-added rates to rise (or fall) and all others to be constant; (b7) any adaptation of price system disables all sectoral values-added to equal their respective sectoral final output values; (b8) an alteration of price system enables some sectoral output multipliers to increase (or decrease) and all others to be unchanged; (b9) any change of price system disables all sectoral output multipliers to equal their respective sectoral input multipliers; (c) the following conditions are equivalent (c1) View the MathML sourceA→orAAhas only one final class; (c2) if the column vector of output adjustment coefficients is nontrivial then an adjustment of output system enables at least one sectoral final output rate to alter, i.e.Q≠λE⇒Y#≠YQ≠λE⇒Y#≠Y; (c3) if the column vector of output adjustment coefficients is nontrivial then the adaptation of output system enables at least one sectoral input multiplier to change, i.e.Q≠λE⇒G#≠GQ≠λE⇒G#≠G; (d) the following conditions are equivalent (d1) AAorView the MathML sourceA→has only one initial class; (d2) if the row vector of price adjustment coefficients is nontrivial then the accommodation of price system enables at least one sectoral value-added rate to alter, i.e.P≠λEt⇒R∗≠RP≠λEt⇒R∗≠R; (d3) if the row vector of price adjustment coefficients is nontrivial then an adjustment of price system enables at least one sectoral output multiplier to change, i.e.P≠λEt⇒D∗≠DP≠λEt⇒D∗≠D; (e) the following conditions are equivalent (e1) View the MathML sourceA→orAAhas precisely one basic class, which is also the only final class; (e2) there is a unique column vector of output adjustment coefficients,Q(1)Q(1), satisfyingView the MathML sourceA→Q(1)=(1−μ)Q(1), such that every sectoral final output rate equalsμμin the new output system; (e3) there is a unique column vector of output adjustment coefficients,Q(1)Q(1), satisfyingView the MathML sourceA→Q(1)=(1−μ)Q(1), such that every sectoral input multiplier equalsμ−1μ−1in the new output system; (f) the following conditions are equivalent (f1) AAorView the MathML sourceA→has exactly one basic class, which is also the only initial class; (f2) there exists a unique row vector of price adjustment coefficients,P(1)P(1), satisfyingP(1)A=(1−μ)P(1)P(1)A=(1−μ)P(1), such that every sectoral value-added rate is equal toμμin the new price system; (f3) there exists a unique row vector of price adjustment coefficients,P(1)P(1), satisfyingP(1)A=(1−μ)P(1)P(1)A=(1−μ)P(1), such that every sectoral output multiplier equalsμ−1μ−1in the new price system; (g) the following conditions are equivalent (g1) View the MathML sourceA→orAAis irreducible, namely, the matrix has only one class; (g2) the following sub-conditions are equivalent (g2.1) the column vector of output adjustment coefficients is nontrivial, i.e.Q≠λEQ≠λE; (g2.2) the adaptation of output system enables at least one sectoral final output rate to rise and at least one sectoral final output rate to fall; (g2.3) the accommodation of output system enables at least one sectoral input multiplier to increase and at least one sectoral input multiplier to decrease; (g3) the following sub-conditions are equivalent (g3.1) the row vector of price adjustment coefficients is nontrivial, i.e.P≠λEtP≠λEt; (g3.2) a change of price system enables at least one sectoral value-added rate to rise and at least one sectoral value-added rate to fall; (g3.3) an alteration of price system enables at least one sectoral output multiplier to increase and at least one sectoral output multiplier to decrease; (g4) there is a unique column vector of output adjustment coefficients,Q(2)Q(2), satisfyingM10Q(2)=Q(2)M10Q(2)=Q(2), such that all sectoral final output values equal their respective sectoral values-added in the new output system; (g5) there is a unique column vector of output adjustment coefficients,Q(3)Q(3), satisfyingView the MathML source(Dˆ−B→)Q(3)=0, such that all sectoral input multipliers redefined by the new output system equal their respective sectoral output multipliers; (g6) there exists a unique row vector of price adjustment coefficients,P(2)P(2), satisfyingP(2)M7=P(2)P(2)M7=P(2), such that all sectoral values-added equal their respective sectoral final output values in the new price system; (g7) there exists a unique row vector of price adjustment coefficients,P(3)P(3), satisfyingView the MathML sourceP(3)(Gˆ−B)=0, such that all sectoral output multipliers redefined by the new price system equal their respective sectoral input multipliers; (g8) the following sub-conditions are equivalent (g8.1) every sectoral value-added rate is equal toμμ, i.e.R=μEtR=μEt; (g8.2) there is a unique column vector of output adjustment coefficients, Q, satisfyingView the MathML sourceA→Q=(1−μ)Q,M10Q=Q, andView the MathML source(Dˆ−B→)Q=0, simultaneously, such that every sectoral final output rate redefined by the new output system equalsμμ, which is equal to every sectoral value-added rate, and every sectoral input multiplier redefined by the new output system equalsμ−1μ−1, which is equal to every sectoral output multiplier; (g9) the following sub-conditions are equivalent (g9.1) every sectoral final output rate is equal toμμ, i.e.Y=μEY=μE; (g9.2) there is a unique row vector of price adjustment coefficients, P, satisfyingPA=(1−μ)P,PM7=PPA=(1−μ)P,PM7=P, andView the MathML sourceP(Gˆ−B)=0, simultaneously, such that every sectoral value-added rate redefined by the new price system equalsμμ, which is equal to every sectoral final output rate, and every sectoral output multiplier redefined by the new price system equalsμ−1μ−1, which is equal to every sectoral input multiplier; whereμ=1−ρ(A)μ=1−ρ(A); (h) the following conditions are equivalent (h1) View the MathML sourceA→or A is reducible and the classes of the matrix are all final or all initial; (h2) there is a nonunique column vector of output adjustment coefficients such that all sectoral final output values equal their respective sectoral values-added in the new output system; (h3) there is a nonunique column vector of output adjustment coefficients such that all sectoral input multipliers redefined by the new output system equal their respective sectoral output multipliers; (h4) there exists a nonunique row vector of price adjustment coefficients such that all sectoral values-added equal their respective sectoral final output values in the new price system; (h5) there exists a nonunique row vector of price adjustment coefficients such that all sectoral output multipliers redefined by the new price system equal their respective sectoral input multipliers; (i) the above conditions satisfy the following logical relationsView the MathML source¬(ht)⇐(ai)⇐[(ck)∨(dl)];¬(bj)⇔[(gs)∨(ht)];(ck)⇐(em);(dl)⇐(fr);[(em)∧(fr)]⇔(gs);i,k,l,m,r=1,2,3,j,s=1,2,…,9,t=1,2,…,5. Proof. Part (a) holds via part (c) in Zeng (2008, Proposition 2). Part (b) follows from part (c) in Zeng (2008, Theorem 1), and part (c) in Theorem 1 of this paper. Parts (c) and (d) hold by parts (a) and (b) in Zeng (2008, Theorem 2), respectively. Parts (e) and (f) come from parts (a) and (b) in Zeng (2008, Theorem 3), and Zeng (2008, Proposition 8). Next we prove part (g).Proof of (g1)⇔(g2). Let (g1) hold. Then, via parts (a) and (c) in Zeng (2008, Theorem 1), and part (a) in Zeng (2008, Theorem 2), we have (g2.2)⇐(g2.1)⇒(g2.3). From part (i) in Zeng (2008, Proposition 3), we have (g2.2)⇒(g2.1)⇐(g2.3). Conversely, suppose that (g2.2)⇔(g2.1)⇔(g2.3). Then, by part (c) in this theorem, (c1) holds. Hence, if (g1) does not hold then (b1) holds. This contradicts (g2.2)⇐(g2.1)⇒(g2.3) via part (b). The proof of (g1)⇔(g2) is completed.As the dual form of the proof of (g1)⇔(g2), we can similarly prove (g1)⇔(g3).From part (a.1) in Theorem 1, and parts (a), (c), (b) and (d) in Proposition 11, we can obtain (g1)⇔(g4)⇔(g5)⇔(g6)⇔(g7).Proof of (g1)⇔(g8). Let (g1) hold. Then, by part (e) and (g1)⇔(g4)⇔(g5) in this theorem, part (e) in Proposition 11, and part (b) in Zeng (2008, Proposition 7), we have (g8.1)⇒(g8.2). Via part (e) in Proposition 11, we have (g8.1)⇐(g8.2). Namely, (g1)⇒(g8) holds. Inversely, if (g1) does not hold, then M10M10 cannot has a unique positive right eigenvector by (a.1) of Theorem 1, thus (g8.2) does not hold. So (g8.1)⇒(g8.2) does not hold. That is, (g8) does not hold. The proof of (g1)⇔(g8) is completed.As the dual form of the proof of (g1)⇔(g8), we can similarly prove (g1)⇔(g9).Part (h) follows from part (b.1) in Theorem 1. Part (i) is clear. □