Research ArticleGlobal Stability and Dynamic Analysis for a Type ofMacroeconomic Systems

Ya-Juan Yang 12 Ru-Fei Ma1 and Jing Zhang3

1School of Business Macau University of Science and Technology Taipa Macau2City College Dongguan University of Technology Dongguan Guangdong China3Department of Continuing Education Dong Guan Polytechnic College Dong Guan China

Correspondence should be addressed to Ya-Juan Yang 1909853gbm30002studentmustedumo

Received 26 June 2020 Accepted 29 July 2020 Published 28 August 2020

Guest Editor Fuqiang Gu

Copyright copy 2020 Ya-Juan Yang et al+is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+is paper aims at the dynamic properties of the proposed globally planned economic systems named after CPE proposed by Loo-Keng Hua who is one of the worldwide famous Chinese mathematicians First we give new existence conditions of growthbalanced solution to the model Second we lead into the concept of stability for balanced output and carry out a theorem that dealswith some equivalent conditions for judging a solution of output starting from the fact that any initial input can whether approachthe existing balanced solution or not +ird a new dynamic price system related to interest factors is proposed here and it isdemonstrated that the new price equation is a much generalized one in comparison with the original price one which is only aspecial case of this new price equation Also relationships of the balanced solutions between the price and the output equation areinvestigated and the stability analysis is studied as well for the new price system Finally two examples are employed to illustratethe technical operation of input-output method and some new contributions of this article

1 Introduction

In the last century undoubtedly the most widely usedeconomic theory is the input-output method which wasproposed byWassily Leontief a winner of the Nobel Prize ineconomics in the year 1973+e seminal paper for the input-output method originated in 1953 by Leontief (see [1]) andthe dynamic analysis for these types of economic systemswas investigated in his two classical papers [2 3] A sys-tematic discussion for input-output theory can be found inLeontiefrsquos book [4] and more references therein Just as thismethodrsquos validity in analyzing a countryrsquos economic de-velopment there had been over 80 counties which preparedand reported their input-output economy statements by theinput-output theory since 1979 Meanwhile the UnitedNationrsquos social and economic sections suggest its memberthat the input-output method should be led into the nationalstatistics accounting systems

Research of the input-output method for fully freecompetition market economy FFCME in short has received

tremendous attention in the past six decades from the firstproposal by Leontief and massive literature has been pub-lished since then All these research articles of the mostimportant ones presented here focused on various sorts ofFFCME problems that could be typically classified into thefollowing two types one investigated the dynamical prop-erties for Leontiefrsquos model (see [5ndash9]) and the other focusedon the macroforecasting problems of the FFCME systems(see [10ndash13]) Status progress and application fields ofLeontiefrsquos theory can be found in the latest symposium byPeterson (see [14]) while a complete and standard treatmentto the theory here recommends to read the book by Millerand Blair [15]

Different from the FFCME system there is anothereconomic system named after centrally planned economyCPE in short characterized by central planning being chiefwhile the market readjustment being accessorial and it hasbeen executed in China over 60 years since the creation ofPRC exceeding 30 years going on from 1949 to 1980 forcomplete CPE and 37 years for socialist market economy

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 4904829 10 pageshttpsdoiorg10115520204904829

beginning from 1981 For the CPE system Leontief ac-companied President Nixon in an advisory capacity at theBeijing interview of President Nixon with Chairman Mao in1987 and he then gave a high appraisal to the policy of thesocialist market economy in his review article ldquosocialism ispracticable in Chinardquo when he came back to America +einput-output technology owns a wide application in themacroforecasting and planning of economic problems re-gardless of short term or long term as well as the ability to beapplied to forecast and plan different economic policysystems being whether the FFCME ones or CPE ones

Compared with the massive research literature to theFFCME system there has been much less research focusingon the CPE system until Professor Loo-Keng Hua a worldfamous mathematician of China starting with a seminalpaper in USA [16] published a series of articles in ChineseScience Bulletin at the middle of 80s last century see[17ndash23] Of course Professor Loo-Keng Huarsquos researchextends Leontiefrsquos theory to the CPE systemwhich is put intopractice by China and hence is more suitable for analyzingthe current economic development in China Fundamentalsof Professor Loo-Keng Huarsquos analysis are based on a hy-pothesis that Chinarsquos productivity elasticity is large enoughit is allowable as Professor Blanchard points out in [24] sothat the relationships between inputs and outputs can bethen described well for the CPE system

Although Professor Loo-Keng Hua established themathematical models for analyzing the CPE system in ar-ticles as mentioned above and also some basic concepts andanalysis such as superior limit of production capacity crisisof production system price and consuming coefficientswere introduced there there are still many problems es-pecially dynamical properties which need to be furtherinvestigated for instance the existence of this CPE systemrsquosbalanced solution and its stability and price dynamics

Motivated by this idea and in order to be in a parallel wayin studying dynamical properties of the FFCME system[2 3 5ndash9] this paper first presents the existence of growthbalanced solutions and their stabilities to the CPE systemand then a generalized dynamic price system is proposedwith key merit that Loo-Keng Huarsquos primary price system isa special case of this Meanwhile relationships of the bal-anced solution between price and output are analyzed as wellas the stability of the dynamic price system investigated too

+e remainder of this paper is organized as follows Section2 is an outline of CPE system formulation and some results ofthe study beingmade by Professor Loo-KengHua are reviewedthere +e main study of this economic system in this paper iscarried out in the next sections new existence theorem for thegrowth balanced solution is addressed in Section 3 and thestability analysis for the output balance is in Section 4 Section 5deals with the dynamic price system including the formulationand the stability of balanced price while Section 6 gives twoexamples for supporting our theoretic results Finally someconcluding remarks are drawn in Section 7

Remark 1 +e so-called FFCME also known as free en-terprise economy in which the production and sale of goodsand services are completely guided by the free price

mechanism of fully free competition market Up to nowabsolute fully free competition market economies do notexist but their markets are open wider than others +eUnited States is generally regarded as the representative ofthe fully free competition market economy for itrsquos highermarket access Typical examples of CPMs are the SovietUnion and China from 1949 to 1978 Actually in todayrsquosworld many countries conduct ldquomore than halfrdquo of fully freecompetition economies such as todayrsquos China

2 CPE Model and Leontief Model

In Leontiefrsquos macroeconomic model the production activ-ities of an economy are divided into n industrial sectors andproduct transactions between these sectors are analyzedbased on the following basic assumptions

H1 there is no joint production and each industrysector produces only one product which means there is aone-to-one correspondence between sectors and productsand so they can be substituted for each other

H2 a single product produced by each sector requires acertain amount of input from other sectors that is sector j

requires tij units of product i as input to produce a unit of thejth product here i 1 2 n and tij are called the inputcoefficients

H3 in the whole production process of an economythere is no lag in production no capital goods no foreigntrade neither involvement of government activities

Under the above assumptions let xi be the total outputof the jth product and x (x1 x2 xn)T the outputvector For production of the other n sectors the amountconsumed by xi is

1113944

n

j1tijxi (1)

and the remaining amount will be

di xi minus 1113944n

j1tijxi (2)

+is net output di is called the final demand of the ithproduct and let d (d1 d2 dn)T represent the finaldemand vector

+en as the aggregate demand must be equal to theaggregate supply we can get the following total input-outputequation systems

(I minus T)x d (3)

which describe the economy as a whole hereT (tij) isinMn(R) is called the input matrix and L I minus T

the Leontief matrix +ese input-output systems (3) arecalled open Leontief model and they are the theoretical basisfor analyzing the macroeconomic state of an economy

Leontiefrsquos method gives an important contribution in theoptimization of the entire production process since it is avery complex problem and there has not been a corre-sponding mathematical method which can fully solve thisoptimal task especially the lack of mathematical description

2 Discrete Dynamics in Nature and Society

for the large-scale economic behavior +e hypotheses(H1)(H3) of Leontiefrsquos model are very strict ones (only canbe partially meted by FFCME as mentioned in Section 1)which are far from the actual economic operation state butthis does not affect the wide application of the input-outputanalysis for an economy

Why do we say that +e most important economicactivity in an economy is the production and the domesticsales and hence what we are interested in here are theeconomyrsquos intersectoral input and the external demand thismay be the reason that foreign trade is excluded in hy-pothesis 3 Because Leontiefrsquos method just meets this pur-pose the input-output reports have been employed by asmentioned earlier more than 80 countries in the worldWhile carrying out an economyrsquos input-output report thereare many various input-output tables to be created anddetails for making an input-output table can be found inSection 6 see Example 1 therein

Although the input-output analysis has become thestandard template for much national economic planningnational development plans cover much more than eco-nomics such as population welfare and environment For acountry an economy taking these problems together boilsdown to a multiobjective decision problem of which the firststudy is made by Schinnar who analyzes developmentplanning in a Leontief input-output model in order to takeinto account both economic and demographic goals see [25]

+ereafter Chichilnisky carried out first the greengolden rule in respect of social welfare and then definessocial welfare as a weighted average between the discountedutilitarianism and the green golden rule welfare see [26 27]Up to now we believe that Leontiefrsquos model combined withthe multigoal decision-making model will be the mostpromising research topics for addressing macroeconomicplanning and forecasting for which the interested readersmay refer to the review article [28] by Cinzia et al

Different from Leontiefrsquos input-output model ProfessorLoo-Keng Hua proposed a CPE system model (see [17 18])where the national production is divided into n sectors alsowhich are interrelated with a so-called consumption coef-ficient matrix A (aij)ntimesn with aij being the quantities ofthe jth class to be consumed for producing a unit quantity ofthe ith class It is obvious that A (aij)ntimesnge 0 and aij gt 0 forsome i j

By letting Xt normally being a nminusdimensional columnvector be the output of the tminusperiod Loo-Keng Huaestablished the CPE system as follows

(CPE) Xt

AXt+1

t isin T 0 1 2 (4)

If A is an invertible matrix the previous equationequivalently can be formulated by

(CPE) Xt+1

Aminus 1

Xt t isin T 0 1 2 (5)

+is CPE system is based on the assumption that theproduction of every class grows in a fixed proportion at eachperiod t and the consumption coefficient matrix does notvary with time t It is easy to see by the inductive methodthat X(t) (1λt)X(0) will be the tminus period production

namely a solution of the CPE system (4) or (5) if the initialinput starting from an eigenvector X(0) gt 0 of A with thecorresponding eigenvalue λ

It should be noted that if we let t 0 by (4) then X0

AX1 substituting it to X1 minus X0 gives us

X1

minus X0

(I minus A)X1 (6)

which is the same as Leontiefrsquos model (3) with d X1 minus X0 x X1 and T A So in this sense Leontiefrsquos model is aspecial case of the proposed CPE model

At the same time Loo-Keng Hua also gave a priceequation coupled with equation (4) as follows

q1 q2 qn( 1113857λlowast q1 q2 qn( 1113857A (7)

where qi is the price per-unit of the ith class and λlowast alsocalled the ldquoprice changing raterdquo is the largest eigenvalue ofA According to Loo-Keng Huarsquos primary definition theprice defined here means the productrsquos basic output valueonly without the profits being contained see [20]

For the CPE system (4) Loo-Keng Hua investigated thefundamental dynamical property (see +eorem 1 of [18])where it is called ldquobasic theoremrdquo and we keep it here

Basic+eorem let A be a nonnegative irreducible squarematrix for any noneigenvector of A but positive vector xgt 0there exists a positive integer l0 such that

xAminus l

(8)

must be variable vector when lge l0 namely some entries ofxAminus l are positive and some negative

It follows from this Basic +eorem that some outputs ofproduction will be negative in several years if the initial input isa positive noneigenvector of A and thus the economic systemwill collapse So the Basic +eorem reveals a basic principle ofphilosophy that balance is temporary but imbalance is per-petual for any economic system imbalance will ultimatelyhappen and even collapse someday if the initial input x is notan eigenvector of the consumption coefficient matrix A

Of course it is far from enough to stop at this Basic+eorem for the dynamical properties of the CPE system (4)as a much deeper study with respect to the dynamicalproperty for the FFCME system is investigated by[2 3 5 25]

To proceed we need some concepts of dynamicalproperty for economic systems such as definitions of thebalanced solution and the causal indeterminacy proposed bySolow and Samuelson in [29]

Definition 1 For the CPE system (4) if there exists aconstant αgt 0 such that Xt αXtminus 1 for all t isin N 1 2 middot middot middot then Xt is called a balanced solution of (4) with the growthrate α

It is clear that if the consumption coefficient matrix A is anonnegative irreducible square matrix then by the well-known PerronndashFrobenius theorem there exists a positiveeigenvector X of A that must be a balanced solution of theCPE system (4) with the growth rate (1λlowast) here λlowast is thelargest positive eigenvalue of A and X being the

Discrete Dynamics in Nature and Society 3

corresponding eigenvector Obviously by Definition 1 theeconomic quantity of the CPE system (4) will increase as(1λlowast gt 1) and decrease as (0lt 1λlowast lt 1)

+e case occurring in the Basic +eorem is called thecausal indeterminacy for an economic system as defined inthe following definition

Definition 2 If the initial input of an economic system doesnot meet any possible balanced solution then there mustexist at least one output to be negative+e economic systemthat owns this property will be known as having the causalindeterminacy

Remark 2 By the Basic +eorem it follows that the causalindeterminacy must happen for the CPE system (4) if theconsumption coefficient matrix A is a nonnegative irre-ducible square matrix

3 Existence of the Growth Solution

As stated in Remark 2 of the previous section the CPEsystem (4) yields a balanced solution with the economicgrowth rate (1λlowast) if the consumption coefficient matrix A

is a nonnegative irreducible square matrix thus the eco-nomic system will develop well when (1λlowast gt 1) since theeconomic growth keeps on It is certain that there are someother existence conditions on the balanced growth solutionof the CPE system (4) except for the ones proposed by Loo-Keng Hua and it would be best to find these conditions forthe existence of the growth balanced solution so as to controlthe economic system developing under the way Meanwhilethis section will focus on these themes then

Different from Loo-Keng Huarsquos assumption on theconsumption coefficient matrix A being an irreduciblematrix here we try to find the existence condition for themore general matrix ones First we need two lemmas forproving the main existence theorem

Lemma 1 LetA (aij)ntimesn be a nonnegative real matrix witha nonnegative real eigenvalue λge 0 that dominates all theother eigenvalues λi(i 1 2 ) of A in absolute value thatis λge |λi| i 1 2 then for any positive vectorx (x1 x2 xn)T we have

min1leilen

1xj

1113944

n

j1aijxj

⎛⎝ ⎞⎠le λle max1leilen

1xj

1113944

n

j1aijxj

⎛⎝ ⎞⎠ (9)

Proof See [30] +eorem 8126

Lemma 2 Suppose A is the consumption coefficient matrix ofCPE system (4) and I minus A is a diagonal strictly dominantmatrix then there exist a positive real number μ 0lt 1113957μlt 1and a positive vector 1113957xge 0 such that A1113957x 1113957μ1113957x

Proof Since A is the consumption coefficient matrix of CPEsystem (4) we have Age 0+en it follows by the well-knownPerronndashFrobenius theorem that there exist a positive real

number 0le 1113957μle 1 and a positive vector 1113957xge 0 such thatA1113957x 1113957μ1113957x

By Gershgorin theorem all the eigenvalues of A belongto the set

cupiΩi μ μ minus aii

11138681113868111386811138681113868111386811138681113868le 1113944

jneiaij

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (10)

+us we have |1113957μ minus taii|le1113936jneiaijBy this and the assumption of I minus A being a diagonal

strictly dominant matrix that is 1 minus aii gt1113936jneiaij we get2aii minus 1lt 1113957μlt 1 and hence 1113957μlt 1

In addition according to Lemma 1 by lettingx (x1 x2 xn)T (1 1 1)T we get

min1leilen

1113944

n

j1aij

⎛⎝ ⎞⎠le 1113957μle max1leilen

1113944

n

j1aij

⎛⎝ ⎞⎠ (11)

So according to the nonnegative of A namely1113936

nj1 aij ge 0 as well as the economic meaning of A we get that

1113936nj1 aij ge 0 and hence 1113957μgt 0 +erefore it follows that there

exist 1113957μ 0lt 1113957μlt 1 and a vector 1113957xge 0 such that A1113957x 1113957μ1113957x

Now we can give the main result about the existence ofgrowth balanced solution for the CPE system (4)

Theorem 1 Let A (aij)ntimesn be the consumption coefficientmatrix of CPE system (4) with the assumption of1 minus aii gt1113936jneiaij it could be satisfied when I minus A is a diagonalstrictly dominant matrix3en there exists a growth balancedsolution X(t) (11113957μ)t1113957x of system (4) with the growth rate(11113957μ) here 1113957μ 0lt 1113957μlt 1 is the eigenvalue of A and 1113957xge 0 thecorresponding eigenvector

Proof First letting t 0 by system (4) we get

X0

AX1 (12)

If X0 is a balanced solution of (12) with the growth rate(1λ) then by Definition 1 we get

X1

1λX

0 (13)

It follows from (12) and (13) that

λX0

AX0 (14)

Inductively it is easy to get that Xt (1λ)tX0 will alsobe a balanced solution

Second by letting X0 1113957x and according to Lemma 2 weobtain that X(t) (11113957μ)t1113957x will be a growth balanced solutionwith growth rate (11113957μ)gt 1 +erefore +eorem 1 is provedthen

Remark 3 +eorem 1 gets rid of the restriction that theconsumption coefficient matrix should be irreducibleFurther under the condition proposed here it is impossiblethat the total costs in quantities for each class will be morethan the total outputs to be produced +is is the reason forthe occurrence of negative output then So controlling the

4 Discrete Dynamics in Nature and Society

economic system (4) in this way will be a better choice sincethere is no causal indeterminacy occurring under this di-rection for the CPE system (4)

4 Stability of the Balanced Output

As stated in the previous section a better way to control aneconomic system is to avoid the causal indeterminacy oc-curring and find the growth solution so as to keep the economygrowing +ough +eorem 1 gives a good description offinding out the growth balanced solution the conditionfor this needs a rigid initial point so as to reach the goal

If started at any point it is vitally important to inves-tigate whether this solution can approach the existinggrowth balanced solution or not namely the stability of theeconomy balance First we need a mathematical descriptionfor the stability see the following definition

Definition 3 Let Age 0 be the consumption coefficientmatrix of system (4) and assume it is an invertible matrixSuppose Xlowast(t) (1μt)Xgt 0 is a balanced solution of system(4) if for every solution 1113954X

(t) starting from any initial input1113954X

(0) ge 0 there exists a constant σ 0lt σ ltinfin such thatlimt⟶infin (1113954x

(t)i xlowast (t)

i ) σ where 1113954x(t)i xlowast (t)i is the ith entry of

1113954X(t) and Xlowast(t) respectively then the balanced solution Xlowast(t)

is called a stable balanced solutionClearly if the CPE system (4) yields a stable balanced

solution then all the solutions determined by any initialinput will eventually be greater than zero since any of thesesolutions approaches asymptotically to the positive balancedsolution Xlowast(t) gt 0 +us there has been no causal indeter-minacy happening in this case for system (4)

Next we give a necessary and sufficient condition to thestability of a balanced solution for system (4)

Theorem 2 Suppose Age 0 is the consumption coefficientmatrix of system (4) as well as an invertible matrix then wehave the following

(i) 3ere exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i

1 2 n are the eigenvalues of A and Xi the cor-responding eigenvectors

(ii) 3e balanced solution Xlowast(t) (1λt1)X1 gt 0 is stable if

and only if |λi|gt λ1 gt 0 where λi i 1 2 n arethe eigenvalues of A and Xi the correspondingeigenvectors

Proof First of all let us prove that (i) for any invertible matrixAge 0 there exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i 1 2 n are theeigenvalues of A and Xi the corresponding eigenvectors

If (Aminus 1)t gt 0 for some positive t then (Aminus 1)t gt 0 will be anirreducible matrix It follows by the well-known Frobeniustheorem that matrix ((Aminus 1)t)primes eigenvalues μi and the cor-responding eigenvectors Yi(i 1 2 n) must satisfyμ1 gt |μi| Y1 gt 0 with AY1 μ1Y1 where i 2 3 n Let λi

be the eigenvalues of A and Xi the corresponding eigenvectors

then it is easy to say μi (1λti ) i 1 2 n Y1 X1 So

by μ1 gt |μi| we get (1λt1)gt |(1λt

i)|(i 2 3 n) and hence|λi|gt λ1 gt 0 X1 gt 0

On the contrary let λi be the eigenvalues of A and Xi thecorresponding eigenvectors with |λi|gt λ1 gt 0 X1 gt 0 wherei 2 3 n First any vector V could be represented as alinear combination of Xi for the invertibility of Xi that is

V h1X1 + h2X2 + middot middot middot + hnXn (15)

LetV Ei be the column vector of the unit matrix E thatis all the entries of Ei being 0 except for the ith being 1 thenwe have

Ei

h1X1 + h2X2 + middot middot middot + hnXn (16)

Making the inner product of this Ei with X1 we get

X1 Ei

1113960 1113961 X1 h1X1 + h2X2 + middot middot middot + hnXn1113858 1113859 1113944n

i1hi X1 Xi1113858 1113859

(17)

Note that [X1 Xi] 0 for ine 1 we have[X1 Ei] h1[X1 X1] and so h1 ([X1 Ei][X1 X1]) h1must be positive since X1 gt 0 and the choice of Ei

Similarly any solution 1113954X(t) with starting initial input Ei

for system (5) can be reformulated as

1113954X(t)

Aminus 1

1113872 1113873tE

i h1

1λt1X1 + h2

1λt2X2 + middot middot middot + hn

1λt

n

Xn (18)

or equivalently

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i h1X1 + h2

λt1

λt2X2 + middot middot middot + hn

λt1

λtn

Xn

(19)

By |λi|gt λ1 gt 0 that is 0lt (λ1|λi|)lt 1 and paying at-tention to X1 gt 0 h1 gt 0 we get that for sufficient enoughlarge tgeN

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i gt 0 (20)

As λt1 gt 0 we obtain (Aminus 1)tEi gt 0 for large t Since it

holds for any Ei we get that (Aminus 1)t gt 0Secondly we prove (ii) If |λi|gt λ1 gt 0 according to

Definition 3 and (18) we have

limt⟶infin

1113954x(t)i

xlowast (t)i

h1 1λt

11113872 1113873X1i + h2 1λt21113872 1113873X2i + middot middot middot + hn 1λt

n1113872 1113873Xni

1λt11113872 1113873X1i

h1

(21)

Let σ h1 and note that h1 gt 0 we get that this Xlowast(t)

should be a stable balanced solutionOn the other hand if limt⟶infin(1113954x

(t)i xlowast (t)

i ) σ gt 0 thenfor sufficiently large tgeN we have 1113954x

(t)i gt 0 i 1 2 n

So 1113954X(t) gt 0 for sufficiently large tgeN with any initial input

vector 1113954X0 ge 0 Similar to the previous proof of (i) if we let

Discrete Dynamics in Nature and Society 5

1113954X0

Ei then we obtain 1113954X(t)

(Aminus 1)tEi gt 0 for alli 1 2 n and sufficiently large tgeN +us (Aminus 1)t gt 0+erefore by proving (i) we get that |λi|gt λ1 gt 0 X1 gt 0

Remark 4 Up to now for the CPE system (4) under theassumption of the consumption coefficient matrix Age 0being invertible we have the following equivalent resultsthe balanced solution Xlowast(t) (1λt

1)X1 gt 0 is stablehArr thereexists a positive integer t such that (Aminus 1)t gt 0hArr |λi|gtλ1 gt 0 X1 gt 0hArr causal indeterminacy cannot occurhArr eachproductrsquos output cannot be negative in the subsequent yearslater

5 Dynamic Price System

Loo-Keng Huarsquos price equation (7) is based on not con-sidering the capital costs such as the money rate At the sametime it gives an assumption that the productrsquos price changesin a fixed proportion for instance λlowast that is P(t+1) λlowastP(t)

for each period t But certainly interest rate impacts directlyon the productrsquos sale price while the productrsquos sale price maygenerally not change in a fixed proportion in practicaleconomic environment Hence we need to rebuild or extendLoo-Keng Huarsquos price system so as to match the practicaleconomic system soundly and this will be carried out in thefollowing subsection 51

In addition subsection 52 deals with the dynamicalproperties for the proposed price system such as the bal-anced price solution and its stability as well +e deep re-search on this respect will contribute to controlling theeconomic system (4) in a way of keeping the economygrowing continually

51 Formulation of the Price System Let P(t)i be the price of

the ith class per-unit product in the t-period +e pricevector for all n class products in the t-period could berepresented as P(t) (P

(t)1 P

(t)2 P(t)

n ) Assume anyproductrsquos price to be a fixed constant during each period andcosts of other products consumed for producing any productto be paid at the beginning of each period+en the per-unitcost for producing the jth class product in the t-period is

v(t)j 1113944

n

i1P

(t)i aij P

(t)aj (22)

where aj is the jth column of the consumption coefficientmatrix A

+e per-unit profit regardless of other costs for pro-ducing the jth class product in the t-period will be

π(t)j P

(t+1)j minus P

(t)aj (23)

At the same time if money to be loaned out can get aninterest rate rt during the time between t-period and

(t + 1)-period then the capital revenue for buying v(t)j will

be

R(t)j r

tv

(t)j r

tP

(t)aj (24)

By the competition arbitrage principle interest andprofit should eventually reach an equilibrium state that is

π(t)j R

(t)j (25)

So

P(t+1)j minus P

(t)aj r

tP

(t)aj (26)

or

P(t+1)j 1 + r

t1113872 1113873P

(t)aj (27)

As (27) holds for all j 1 2 n we can formulate it inthe following matrix representation

P(t+1)

1 + rt

1113872 1113873P(t)

A (28)

Letting M (1 + rt)A we obtain the dynamic priceequation corresponding to (4) as follows

P(t+1)

P(t)

M (29)

If interest rate rt 0 and the price changing rate varieswith a fixed proportion λlowast (Pt+1

i Pti) for each period t

then this dynamic price equation (29) becomes the priceequation (7) proposed by Loo-Keng Hua Next we turn toinvestigate the balanced price solution and the stability forthe price equation (29) as made previously for the outputequation (4)

52 Stability of the Balanced Price In order to simplify wesuppose the interest rate rt r to be a constant and Age 0 tobe invertible It is easy to see that M (1 + r)A is alsoinvertible and the eigenvalues of M will be (1 + r)λi if λi isthe eigenvalues of A while the corresponding eigenvectorsare the same

Definition of the balanced price solution to (29) can bedefined as Definition 1 for output equation (4) that is ifP(t) βP(tminus 1) holds for all t isin N 1 2 n andsome constant βgt 0 then P(t) will be a balanced solution of(29) with price changing rate β +e definition of stability fora balanced price solution can be found in the followingmathematical description

Definition 4 Let Age 0 be an invertible matrix andM (1 + r)A Suppose Plowast(t) ζt

1p1 gt 0 is a balanced pricesolution of system (29) if for a solution 1113954P

(t) starting fromany initial price 1113954P

(0) ge 0 there exists a constant σ 0lt σ ltinfinsuch that limt⟶infin(1113954p

(t)i plowast (t)

i ) σ where Z is the ith entryof 1113954P

(t) and Plowast(t) respectively then the balanced solutionPlowast(t) is called a stable price solution

6 Discrete Dynamics in Nature and Society

Similarly we have +eorem 3 which reveals the dy-namical properties for price equation (29) like+eorem 2 forthe output equation (4)

Theorem 3 Suppose Age 0 to be an invertible matrix and letM (1 + r)A where rgt 0 is the interest rate 3en we havethe following

(i) 3ere exists a positive integer t such that Mt gt 0 if andonly if μ1 gt |μi|ge 0 p1 gt 0 where μi and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

(ii) 3e balanced price Plowast(t) ζt1p1 gt 0 is stable if and

only if ζ1 gt |ζ i|ge 0 p1 gt 0 where ζ i and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

Proof +e proof of this theorem is similar to+eorem 2 anda survey is given here

First it is noted that Age 0 and being invertible impliesM (1 + r)Age 0 and being invertible also Second if μ is aneigenvalue of matrix M with eigenvector p then μt will bethe eigenvalue of Mt with the same eigenvector p in re-gardless of scalar times+ird inductively it is easy to obtainP(t) P(0)Mt Finally note that the general solution ofequation (29) can be written as

1113954P(t)

α1ζt1p1 + α2ζ

t2p2 + middot middot middot + αnζ

tnpn (30)

where αi are constants to be determined by the initial price1113954P

(0) All these four points give the key respects to prove+eorem 3 and the procedure of the proof is omittedhere

Remark 5 +e solutions Xlowast(t) (1λt1)X1 and Plowast(t) ζt

1p1are a pair of output balance and price balance by the cor-respondence ζ1 (1 + r)λ1 X1 p1 In fact eigenvalues λi

of A and eigenvalues ζ i of M (1 + r)A have a corre-spondence ζ i (1 + r)λi So if ζ1 gt |ζ i| then λ1 gt |λi| and if|λi|gt λ1 then |ζ i|gt ζ1 +us by +eorem 2 and +eorem 3balanced price solution Plowast(t) being stable means balancedoutput Xlowast(t) is not stable and vice versa

6 Illustrative Examples

In this section we illustrate some of themain results of this paperwith two examples one is about the input-output table for-mulation and the other is an application of+eorem 1 as well asa comparison declaration First let us give a simplified exampleto show the process of how to create an input-output table

Example 1 Assuming a hypothetical economy is composedof (1) agriculture (2) the industrial sector (manufacturing)and (3) the service provider (service) each of these de-partments produces only one type of product namely theagricultural industrial or service supply and there is in-terdependence between and among them Each departmentbuys products from the other departments and sells its ownproducts to the opposite ones but the final product and

service supply (they do not enter the production process) isused by external departments such as consumers +eproduction process and the external demand are not as-sociated with the government and there is no foreign tradethen according to the early hypotheses (H1)(H3) we cancarry out a form to summarize the product and the currentsituation of service supply as shown in Table 1 where xij

represents products (in US $) sold by sector i to sector j+e data of each row in Table 1 represents the allocation of

the total output to different departments and users while thedata of each column represents the sources of departmentinputs required for the total output For example the first rowshows that the total output of 100$ agricultural products isassigned to 15$ products for reproducing 20$ products beingsold to manufacturing 30$ products to service and the last35$ products being used tomeet external demand Similarly itcan be seen from the second column that for the total outputof 200$ the manufacturing needs to invest 20$ of agriculturalproducts 10$ of its own products and 60$ of service input

For convenience of analysis coupled with the input-outputtable the so-called technology input-output table can beconstructed to show the amount that each sector for thepurpose of producing one unit of its own product needs forconsuming the other sectorrsquos +e quantity in the technicalinput-output table represents the input coefficient tij (inLeontiefrsquos model) or consumption coefficient (in CPEmodel) ofthe economy which can be obtained from Table 1 for examplethe data of each column in Table 1 being divided by the totalagriculture output 100 gives the agriculture sectorrsquos input co-efficients and so on +us the technology input table or con-sumption coefficient table is completed and shown in Table 2

By Table 2 the input matrix T or consumption coeffi-cient matrix A should be

T A

015 010 020

030 005 030

030 030 000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

+e input-output table can comprehensively and sys-tematically reflect the input-output relationship among allsectors of the national economy and reveal the economic andtechnological relations of interdependence and mutual re-striction among all sectors in the production process On theother hand it can tell people about the output of varioussectors of the national economy and how the output of thesesectors is distributed to other sectors for production or toresidents and society for final consumption or exportabroad Furthermore it can tell people how each departmentobtains intermediate inputs and initial inputs from otherdepartments for its own production

Example 1 only provides the fundamental principle forthe compilation of input-output table with a simple case Ingeneral the actual input-output table of an economy is muchmore complicated than this one provided and it generallyincludes national table regional table sectoral table andjoint enterprise table according to different scopes as well asstatic table and dynamic table according to the modelcharacteristics In addition there are input-output tables for

Discrete Dynamics in Nature and Society 7

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

for the large-scale economic behavior +e hypotheses(H1)(H3) of Leontiefrsquos model are very strict ones (only canbe partially meted by FFCME as mentioned in Section 1)which are far from the actual economic operation state butthis does not affect the wide application of the input-outputanalysis for an economy

Why do we say that +e most important economicactivity in an economy is the production and the domesticsales and hence what we are interested in here are theeconomyrsquos intersectoral input and the external demand thismay be the reason that foreign trade is excluded in hy-pothesis 3 Because Leontiefrsquos method just meets this pur-pose the input-output reports have been employed by asmentioned earlier more than 80 countries in the worldWhile carrying out an economyrsquos input-output report thereare many various input-output tables to be created anddetails for making an input-output table can be found inSection 6 see Example 1 therein

Although the input-output analysis has become thestandard template for much national economic planningnational development plans cover much more than eco-nomics such as population welfare and environment For acountry an economy taking these problems together boilsdown to a multiobjective decision problem of which the firststudy is made by Schinnar who analyzes developmentplanning in a Leontief input-output model in order to takeinto account both economic and demographic goals see [25]

+ereafter Chichilnisky carried out first the greengolden rule in respect of social welfare and then definessocial welfare as a weighted average between the discountedutilitarianism and the green golden rule welfare see [26 27]Up to now we believe that Leontiefrsquos model combined withthe multigoal decision-making model will be the mostpromising research topics for addressing macroeconomicplanning and forecasting for which the interested readersmay refer to the review article [28] by Cinzia et al

Different from Leontiefrsquos input-output model ProfessorLoo-Keng Hua proposed a CPE system model (see [17 18])where the national production is divided into n sectors alsowhich are interrelated with a so-called consumption coef-ficient matrix A (aij)ntimesn with aij being the quantities ofthe jth class to be consumed for producing a unit quantity ofthe ith class It is obvious that A (aij)ntimesnge 0 and aij gt 0 forsome i j

By letting Xt normally being a nminusdimensional columnvector be the output of the tminusperiod Loo-Keng Huaestablished the CPE system as follows

(CPE) Xt

AXt+1

t isin T 0 1 2 (4)

If A is an invertible matrix the previous equationequivalently can be formulated by

(CPE) Xt+1

Aminus 1

Xt t isin T 0 1 2 (5)

+is CPE system is based on the assumption that theproduction of every class grows in a fixed proportion at eachperiod t and the consumption coefficient matrix does notvary with time t It is easy to see by the inductive methodthat X(t) (1λt)X(0) will be the tminus period production

namely a solution of the CPE system (4) or (5) if the initialinput starting from an eigenvector X(0) gt 0 of A with thecorresponding eigenvalue λ

It should be noted that if we let t 0 by (4) then X0

AX1 substituting it to X1 minus X0 gives us

X1

minus X0

(I minus A)X1 (6)

which is the same as Leontiefrsquos model (3) with d X1 minus X0 x X1 and T A So in this sense Leontiefrsquos model is aspecial case of the proposed CPE model

At the same time Loo-Keng Hua also gave a priceequation coupled with equation (4) as follows

q1 q2 qn( 1113857λlowast q1 q2 qn( 1113857A (7)

where qi is the price per-unit of the ith class and λlowast alsocalled the ldquoprice changing raterdquo is the largest eigenvalue ofA According to Loo-Keng Huarsquos primary definition theprice defined here means the productrsquos basic output valueonly without the profits being contained see [20]

For the CPE system (4) Loo-Keng Hua investigated thefundamental dynamical property (see +eorem 1 of [18])where it is called ldquobasic theoremrdquo and we keep it here

Basic+eorem let A be a nonnegative irreducible squarematrix for any noneigenvector of A but positive vector xgt 0there exists a positive integer l0 such that

xAminus l

(8)

must be variable vector when lge l0 namely some entries ofxAminus l are positive and some negative

It follows from this Basic +eorem that some outputs ofproduction will be negative in several years if the initial input isa positive noneigenvector of A and thus the economic systemwill collapse So the Basic +eorem reveals a basic principle ofphilosophy that balance is temporary but imbalance is per-petual for any economic system imbalance will ultimatelyhappen and even collapse someday if the initial input x is notan eigenvector of the consumption coefficient matrix A

Of course it is far from enough to stop at this Basic+eorem for the dynamical properties of the CPE system (4)as a much deeper study with respect to the dynamicalproperty for the FFCME system is investigated by[2 3 5 25]

To proceed we need some concepts of dynamicalproperty for economic systems such as definitions of thebalanced solution and the causal indeterminacy proposed bySolow and Samuelson in [29]

Definition 1 For the CPE system (4) if there exists aconstant αgt 0 such that Xt αXtminus 1 for all t isin N 1 2 middot middot middot then Xt is called a balanced solution of (4) with the growthrate α

It is clear that if the consumption coefficient matrix A is anonnegative irreducible square matrix then by the well-known PerronndashFrobenius theorem there exists a positiveeigenvector X of A that must be a balanced solution of theCPE system (4) with the growth rate (1λlowast) here λlowast is thelargest positive eigenvalue of A and X being the

Discrete Dynamics in Nature and Society 3

corresponding eigenvector Obviously by Definition 1 theeconomic quantity of the CPE system (4) will increase as(1λlowast gt 1) and decrease as (0lt 1λlowast lt 1)

+e case occurring in the Basic +eorem is called thecausal indeterminacy for an economic system as defined inthe following definition

Definition 2 If the initial input of an economic system doesnot meet any possible balanced solution then there mustexist at least one output to be negative+e economic systemthat owns this property will be known as having the causalindeterminacy

Remark 2 By the Basic +eorem it follows that the causalindeterminacy must happen for the CPE system (4) if theconsumption coefficient matrix A is a nonnegative irre-ducible square matrix

3 Existence of the Growth Solution

As stated in Remark 2 of the previous section the CPEsystem (4) yields a balanced solution with the economicgrowth rate (1λlowast) if the consumption coefficient matrix A

is a nonnegative irreducible square matrix thus the eco-nomic system will develop well when (1λlowast gt 1) since theeconomic growth keeps on It is certain that there are someother existence conditions on the balanced growth solutionof the CPE system (4) except for the ones proposed by Loo-Keng Hua and it would be best to find these conditions forthe existence of the growth balanced solution so as to controlthe economic system developing under the way Meanwhilethis section will focus on these themes then

Different from Loo-Keng Huarsquos assumption on theconsumption coefficient matrix A being an irreduciblematrix here we try to find the existence condition for themore general matrix ones First we need two lemmas forproving the main existence theorem

Lemma 1 LetA (aij)ntimesn be a nonnegative real matrix witha nonnegative real eigenvalue λge 0 that dominates all theother eigenvalues λi(i 1 2 ) of A in absolute value thatis λge |λi| i 1 2 then for any positive vectorx (x1 x2 xn)T we have

min1leilen

1xj

1113944

n

j1aijxj

⎛⎝ ⎞⎠le λle max1leilen

1xj

1113944

n

j1aijxj

⎛⎝ ⎞⎠ (9)

Proof See [30] +eorem 8126

Lemma 2 Suppose A is the consumption coefficient matrix ofCPE system (4) and I minus A is a diagonal strictly dominantmatrix then there exist a positive real number μ 0lt 1113957μlt 1and a positive vector 1113957xge 0 such that A1113957x 1113957μ1113957x

Proof Since A is the consumption coefficient matrix of CPEsystem (4) we have Age 0+en it follows by the well-knownPerronndashFrobenius theorem that there exist a positive real

number 0le 1113957μle 1 and a positive vector 1113957xge 0 such thatA1113957x 1113957μ1113957x

By Gershgorin theorem all the eigenvalues of A belongto the set

cupiΩi μ μ minus aii

11138681113868111386811138681113868111386811138681113868le 1113944

jneiaij

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (10)

+us we have |1113957μ minus taii|le1113936jneiaijBy this and the assumption of I minus A being a diagonal

strictly dominant matrix that is 1 minus aii gt1113936jneiaij we get2aii minus 1lt 1113957μlt 1 and hence 1113957μlt 1

In addition according to Lemma 1 by lettingx (x1 x2 xn)T (1 1 1)T we get

min1leilen

1113944

n

j1aij

⎛⎝ ⎞⎠le 1113957μle max1leilen

1113944

n

j1aij

⎛⎝ ⎞⎠ (11)

So according to the nonnegative of A namely1113936

nj1 aij ge 0 as well as the economic meaning of A we get that

1113936nj1 aij ge 0 and hence 1113957μgt 0 +erefore it follows that there

exist 1113957μ 0lt 1113957μlt 1 and a vector 1113957xge 0 such that A1113957x 1113957μ1113957x

Now we can give the main result about the existence ofgrowth balanced solution for the CPE system (4)

Theorem 1 Let A (aij)ntimesn be the consumption coefficientmatrix of CPE system (4) with the assumption of1 minus aii gt1113936jneiaij it could be satisfied when I minus A is a diagonalstrictly dominant matrix3en there exists a growth balancedsolution X(t) (11113957μ)t1113957x of system (4) with the growth rate(11113957μ) here 1113957μ 0lt 1113957μlt 1 is the eigenvalue of A and 1113957xge 0 thecorresponding eigenvector

Proof First letting t 0 by system (4) we get

X0

AX1 (12)

If X0 is a balanced solution of (12) with the growth rate(1λ) then by Definition 1 we get

X1

1λX

0 (13)

It follows from (12) and (13) that

λX0

AX0 (14)

Inductively it is easy to get that Xt (1λ)tX0 will alsobe a balanced solution

Second by letting X0 1113957x and according to Lemma 2 weobtain that X(t) (11113957μ)t1113957x will be a growth balanced solutionwith growth rate (11113957μ)gt 1 +erefore +eorem 1 is provedthen

Remark 3 +eorem 1 gets rid of the restriction that theconsumption coefficient matrix should be irreducibleFurther under the condition proposed here it is impossiblethat the total costs in quantities for each class will be morethan the total outputs to be produced +is is the reason forthe occurrence of negative output then So controlling the

4 Discrete Dynamics in Nature and Society

economic system (4) in this way will be a better choice sincethere is no causal indeterminacy occurring under this di-rection for the CPE system (4)

4 Stability of the Balanced Output

As stated in the previous section a better way to control aneconomic system is to avoid the causal indeterminacy oc-curring and find the growth solution so as to keep the economygrowing +ough +eorem 1 gives a good description offinding out the growth balanced solution the conditionfor this needs a rigid initial point so as to reach the goal

If started at any point it is vitally important to inves-tigate whether this solution can approach the existinggrowth balanced solution or not namely the stability of theeconomy balance First we need a mathematical descriptionfor the stability see the following definition

Definition 3 Let Age 0 be the consumption coefficientmatrix of system (4) and assume it is an invertible matrixSuppose Xlowast(t) (1μt)Xgt 0 is a balanced solution of system(4) if for every solution 1113954X

(t) starting from any initial input1113954X

(0) ge 0 there exists a constant σ 0lt σ ltinfin such thatlimt⟶infin (1113954x

(t)i xlowast (t)

i ) σ where 1113954x(t)i xlowast (t)i is the ith entry of

1113954X(t) and Xlowast(t) respectively then the balanced solution Xlowast(t)

is called a stable balanced solutionClearly if the CPE system (4) yields a stable balanced

solution then all the solutions determined by any initialinput will eventually be greater than zero since any of thesesolutions approaches asymptotically to the positive balancedsolution Xlowast(t) gt 0 +us there has been no causal indeter-minacy happening in this case for system (4)

Next we give a necessary and sufficient condition to thestability of a balanced solution for system (4)

Theorem 2 Suppose Age 0 is the consumption coefficientmatrix of system (4) as well as an invertible matrix then wehave the following

(i) 3ere exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i

1 2 n are the eigenvalues of A and Xi the cor-responding eigenvectors

(ii) 3e balanced solution Xlowast(t) (1λt1)X1 gt 0 is stable if

and only if |λi|gt λ1 gt 0 where λi i 1 2 n arethe eigenvalues of A and Xi the correspondingeigenvectors

Proof First of all let us prove that (i) for any invertible matrixAge 0 there exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i 1 2 n are theeigenvalues of A and Xi the corresponding eigenvectors

If (Aminus 1)t gt 0 for some positive t then (Aminus 1)t gt 0 will be anirreducible matrix It follows by the well-known Frobeniustheorem that matrix ((Aminus 1)t)primes eigenvalues μi and the cor-responding eigenvectors Yi(i 1 2 n) must satisfyμ1 gt |μi| Y1 gt 0 with AY1 μ1Y1 where i 2 3 n Let λi

be the eigenvalues of A and Xi the corresponding eigenvectors

then it is easy to say μi (1λti ) i 1 2 n Y1 X1 So

by μ1 gt |μi| we get (1λt1)gt |(1λt

i)|(i 2 3 n) and hence|λi|gt λ1 gt 0 X1 gt 0

On the contrary let λi be the eigenvalues of A and Xi thecorresponding eigenvectors with |λi|gt λ1 gt 0 X1 gt 0 wherei 2 3 n First any vector V could be represented as alinear combination of Xi for the invertibility of Xi that is

V h1X1 + h2X2 + middot middot middot + hnXn (15)

LetV Ei be the column vector of the unit matrix E thatis all the entries of Ei being 0 except for the ith being 1 thenwe have

Ei

h1X1 + h2X2 + middot middot middot + hnXn (16)

Making the inner product of this Ei with X1 we get

X1 Ei

1113960 1113961 X1 h1X1 + h2X2 + middot middot middot + hnXn1113858 1113859 1113944n

i1hi X1 Xi1113858 1113859

(17)

Note that [X1 Xi] 0 for ine 1 we have[X1 Ei] h1[X1 X1] and so h1 ([X1 Ei][X1 X1]) h1must be positive since X1 gt 0 and the choice of Ei

Similarly any solution 1113954X(t) with starting initial input Ei

for system (5) can be reformulated as

1113954X(t)

Aminus 1

1113872 1113873tE

i h1

1λt1X1 + h2

1λt2X2 + middot middot middot + hn

1λt

n

Xn (18)

or equivalently

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i h1X1 + h2

λt1

λt2X2 + middot middot middot + hn

λt1

λtn

Xn

(19)

By |λi|gt λ1 gt 0 that is 0lt (λ1|λi|)lt 1 and paying at-tention to X1 gt 0 h1 gt 0 we get that for sufficient enoughlarge tgeN

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i gt 0 (20)

As λt1 gt 0 we obtain (Aminus 1)tEi gt 0 for large t Since it

holds for any Ei we get that (Aminus 1)t gt 0Secondly we prove (ii) If |λi|gt λ1 gt 0 according to

Definition 3 and (18) we have

limt⟶infin

1113954x(t)i

xlowast (t)i

h1 1λt

11113872 1113873X1i + h2 1λt21113872 1113873X2i + middot middot middot + hn 1λt

n1113872 1113873Xni

1λt11113872 1113873X1i

h1

(21)

Let σ h1 and note that h1 gt 0 we get that this Xlowast(t)

should be a stable balanced solutionOn the other hand if limt⟶infin(1113954x

(t)i xlowast (t)

i ) σ gt 0 thenfor sufficiently large tgeN we have 1113954x

(t)i gt 0 i 1 2 n

So 1113954X(t) gt 0 for sufficiently large tgeN with any initial input

vector 1113954X0 ge 0 Similar to the previous proof of (i) if we let

Discrete Dynamics in Nature and Society 5

1113954X0

Ei then we obtain 1113954X(t)

(Aminus 1)tEi gt 0 for alli 1 2 n and sufficiently large tgeN +us (Aminus 1)t gt 0+erefore by proving (i) we get that |λi|gt λ1 gt 0 X1 gt 0

Remark 4 Up to now for the CPE system (4) under theassumption of the consumption coefficient matrix Age 0being invertible we have the following equivalent resultsthe balanced solution Xlowast(t) (1λt

1)X1 gt 0 is stablehArr thereexists a positive integer t such that (Aminus 1)t gt 0hArr |λi|gtλ1 gt 0 X1 gt 0hArr causal indeterminacy cannot occurhArr eachproductrsquos output cannot be negative in the subsequent yearslater

5 Dynamic Price System

Loo-Keng Huarsquos price equation (7) is based on not con-sidering the capital costs such as the money rate At the sametime it gives an assumption that the productrsquos price changesin a fixed proportion for instance λlowast that is P(t+1) λlowastP(t)

for each period t But certainly interest rate impacts directlyon the productrsquos sale price while the productrsquos sale price maygenerally not change in a fixed proportion in practicaleconomic environment Hence we need to rebuild or extendLoo-Keng Huarsquos price system so as to match the practicaleconomic system soundly and this will be carried out in thefollowing subsection 51

In addition subsection 52 deals with the dynamicalproperties for the proposed price system such as the bal-anced price solution and its stability as well +e deep re-search on this respect will contribute to controlling theeconomic system (4) in a way of keeping the economygrowing continually

51 Formulation of the Price System Let P(t)i be the price of

the ith class per-unit product in the t-period +e pricevector for all n class products in the t-period could berepresented as P(t) (P

(t)1 P

(t)2 P(t)

n ) Assume anyproductrsquos price to be a fixed constant during each period andcosts of other products consumed for producing any productto be paid at the beginning of each period+en the per-unitcost for producing the jth class product in the t-period is

v(t)j 1113944

n

i1P

(t)i aij P

(t)aj (22)

where aj is the jth column of the consumption coefficientmatrix A

+e per-unit profit regardless of other costs for pro-ducing the jth class product in the t-period will be

π(t)j P

(t+1)j minus P

(t)aj (23)

At the same time if money to be loaned out can get aninterest rate rt during the time between t-period and

(t + 1)-period then the capital revenue for buying v(t)j will

be

R(t)j r

tv

(t)j r

tP

(t)aj (24)

By the competition arbitrage principle interest andprofit should eventually reach an equilibrium state that is

π(t)j R

(t)j (25)

So

P(t+1)j minus P

(t)aj r

tP

(t)aj (26)

or

P(t+1)j 1 + r

t1113872 1113873P

(t)aj (27)

As (27) holds for all j 1 2 n we can formulate it inthe following matrix representation

P(t+1)

1 + rt

1113872 1113873P(t)

A (28)

Letting M (1 + rt)A we obtain the dynamic priceequation corresponding to (4) as follows

P(t+1)

P(t)

M (29)

If interest rate rt 0 and the price changing rate varieswith a fixed proportion λlowast (Pt+1

i Pti) for each period t

then this dynamic price equation (29) becomes the priceequation (7) proposed by Loo-Keng Hua Next we turn toinvestigate the balanced price solution and the stability forthe price equation (29) as made previously for the outputequation (4)

52 Stability of the Balanced Price In order to simplify wesuppose the interest rate rt r to be a constant and Age 0 tobe invertible It is easy to see that M (1 + r)A is alsoinvertible and the eigenvalues of M will be (1 + r)λi if λi isthe eigenvalues of A while the corresponding eigenvectorsare the same

Definition of the balanced price solution to (29) can bedefined as Definition 1 for output equation (4) that is ifP(t) βP(tminus 1) holds for all t isin N 1 2 n andsome constant βgt 0 then P(t) will be a balanced solution of(29) with price changing rate β +e definition of stability fora balanced price solution can be found in the followingmathematical description

Definition 4 Let Age 0 be an invertible matrix andM (1 + r)A Suppose Plowast(t) ζt

1p1 gt 0 is a balanced pricesolution of system (29) if for a solution 1113954P

(t) starting fromany initial price 1113954P

(0) ge 0 there exists a constant σ 0lt σ ltinfinsuch that limt⟶infin(1113954p

(t)i plowast (t)

i ) σ where Z is the ith entryof 1113954P

(t) and Plowast(t) respectively then the balanced solutionPlowast(t) is called a stable price solution

6 Discrete Dynamics in Nature and Society

Similarly we have +eorem 3 which reveals the dy-namical properties for price equation (29) like+eorem 2 forthe output equation (4)

Theorem 3 Suppose Age 0 to be an invertible matrix and letM (1 + r)A where rgt 0 is the interest rate 3en we havethe following

(i) 3ere exists a positive integer t such that Mt gt 0 if andonly if μ1 gt |μi|ge 0 p1 gt 0 where μi and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

(ii) 3e balanced price Plowast(t) ζt1p1 gt 0 is stable if and

only if ζ1 gt |ζ i|ge 0 p1 gt 0 where ζ i and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

Proof +e proof of this theorem is similar to+eorem 2 anda survey is given here

First it is noted that Age 0 and being invertible impliesM (1 + r)Age 0 and being invertible also Second if μ is aneigenvalue of matrix M with eigenvector p then μt will bethe eigenvalue of Mt with the same eigenvector p in re-gardless of scalar times+ird inductively it is easy to obtainP(t) P(0)Mt Finally note that the general solution ofequation (29) can be written as

1113954P(t)

α1ζt1p1 + α2ζ

t2p2 + middot middot middot + αnζ

tnpn (30)

where αi are constants to be determined by the initial price1113954P

(0) All these four points give the key respects to prove+eorem 3 and the procedure of the proof is omittedhere

Remark 5 +e solutions Xlowast(t) (1λt1)X1 and Plowast(t) ζt

1p1are a pair of output balance and price balance by the cor-respondence ζ1 (1 + r)λ1 X1 p1 In fact eigenvalues λi

of A and eigenvalues ζ i of M (1 + r)A have a corre-spondence ζ i (1 + r)λi So if ζ1 gt |ζ i| then λ1 gt |λi| and if|λi|gt λ1 then |ζ i|gt ζ1 +us by +eorem 2 and +eorem 3balanced price solution Plowast(t) being stable means balancedoutput Xlowast(t) is not stable and vice versa

6 Illustrative Examples

In this section we illustrate some of themain results of this paperwith two examples one is about the input-output table for-mulation and the other is an application of+eorem 1 as well asa comparison declaration First let us give a simplified exampleto show the process of how to create an input-output table

Example 1 Assuming a hypothetical economy is composedof (1) agriculture (2) the industrial sector (manufacturing)and (3) the service provider (service) each of these de-partments produces only one type of product namely theagricultural industrial or service supply and there is in-terdependence between and among them Each departmentbuys products from the other departments and sells its ownproducts to the opposite ones but the final product and

service supply (they do not enter the production process) isused by external departments such as consumers +eproduction process and the external demand are not as-sociated with the government and there is no foreign tradethen according to the early hypotheses (H1)(H3) we cancarry out a form to summarize the product and the currentsituation of service supply as shown in Table 1 where xij

represents products (in US $) sold by sector i to sector j+e data of each row in Table 1 represents the allocation of

the total output to different departments and users while thedata of each column represents the sources of departmentinputs required for the total output For example the first rowshows that the total output of 100$ agricultural products isassigned to 15$ products for reproducing 20$ products beingsold to manufacturing 30$ products to service and the last35$ products being used tomeet external demand Similarly itcan be seen from the second column that for the total outputof 200$ the manufacturing needs to invest 20$ of agriculturalproducts 10$ of its own products and 60$ of service input

For convenience of analysis coupled with the input-outputtable the so-called technology input-output table can beconstructed to show the amount that each sector for thepurpose of producing one unit of its own product needs forconsuming the other sectorrsquos +e quantity in the technicalinput-output table represents the input coefficient tij (inLeontiefrsquos model) or consumption coefficient (in CPEmodel) ofthe economy which can be obtained from Table 1 for examplethe data of each column in Table 1 being divided by the totalagriculture output 100 gives the agriculture sectorrsquos input co-efficients and so on +us the technology input table or con-sumption coefficient table is completed and shown in Table 2

By Table 2 the input matrix T or consumption coeffi-cient matrix A should be

T A

015 010 020

030 005 030

030 030 000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

+e input-output table can comprehensively and sys-tematically reflect the input-output relationship among allsectors of the national economy and reveal the economic andtechnological relations of interdependence and mutual re-striction among all sectors in the production process On theother hand it can tell people about the output of varioussectors of the national economy and how the output of thesesectors is distributed to other sectors for production or toresidents and society for final consumption or exportabroad Furthermore it can tell people how each departmentobtains intermediate inputs and initial inputs from otherdepartments for its own production

Example 1 only provides the fundamental principle forthe compilation of input-output table with a simple case Ingeneral the actual input-output table of an economy is muchmore complicated than this one provided and it generallyincludes national table regional table sectoral table andjoint enterprise table according to different scopes as well asstatic table and dynamic table according to the modelcharacteristics In addition there are input-output tables for

Discrete Dynamics in Nature and Society 7

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

corresponding eigenvector Obviously by Definition 1 theeconomic quantity of the CPE system (4) will increase as(1λlowast gt 1) and decrease as (0lt 1λlowast lt 1)

+e case occurring in the Basic +eorem is called thecausal indeterminacy for an economic system as defined inthe following definition

Definition 2 If the initial input of an economic system doesnot meet any possible balanced solution then there mustexist at least one output to be negative+e economic systemthat owns this property will be known as having the causalindeterminacy

Remark 2 By the Basic +eorem it follows that the causalindeterminacy must happen for the CPE system (4) if theconsumption coefficient matrix A is a nonnegative irre-ducible square matrix

3 Existence of the Growth Solution

As stated in Remark 2 of the previous section the CPEsystem (4) yields a balanced solution with the economicgrowth rate (1λlowast) if the consumption coefficient matrix A

is a nonnegative irreducible square matrix thus the eco-nomic system will develop well when (1λlowast gt 1) since theeconomic growth keeps on It is certain that there are someother existence conditions on the balanced growth solutionof the CPE system (4) except for the ones proposed by Loo-Keng Hua and it would be best to find these conditions forthe existence of the growth balanced solution so as to controlthe economic system developing under the way Meanwhilethis section will focus on these themes then

Different from Loo-Keng Huarsquos assumption on theconsumption coefficient matrix A being an irreduciblematrix here we try to find the existence condition for themore general matrix ones First we need two lemmas forproving the main existence theorem

Lemma 1 LetA (aij)ntimesn be a nonnegative real matrix witha nonnegative real eigenvalue λge 0 that dominates all theother eigenvalues λi(i 1 2 ) of A in absolute value thatis λge |λi| i 1 2 then for any positive vectorx (x1 x2 xn)T we have

min1leilen

1xj

1113944

n

j1aijxj

⎛⎝ ⎞⎠le λle max1leilen

1xj

1113944

n

j1aijxj

⎛⎝ ⎞⎠ (9)

Proof See [30] +eorem 8126

Lemma 2 Suppose A is the consumption coefficient matrix ofCPE system (4) and I minus A is a diagonal strictly dominantmatrix then there exist a positive real number μ 0lt 1113957μlt 1and a positive vector 1113957xge 0 such that A1113957x 1113957μ1113957x

Proof Since A is the consumption coefficient matrix of CPEsystem (4) we have Age 0+en it follows by the well-knownPerronndashFrobenius theorem that there exist a positive real

number 0le 1113957μle 1 and a positive vector 1113957xge 0 such thatA1113957x 1113957μ1113957x

By Gershgorin theorem all the eigenvalues of A belongto the set

cupiΩi μ μ minus aii

11138681113868111386811138681113868111386811138681113868le 1113944

jneiaij

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (10)

+us we have |1113957μ minus taii|le1113936jneiaijBy this and the assumption of I minus A being a diagonal

strictly dominant matrix that is 1 minus aii gt1113936jneiaij we get2aii minus 1lt 1113957μlt 1 and hence 1113957μlt 1

In addition according to Lemma 1 by lettingx (x1 x2 xn)T (1 1 1)T we get

min1leilen

1113944

n

j1aij

⎛⎝ ⎞⎠le 1113957μle max1leilen

1113944

n

j1aij

⎛⎝ ⎞⎠ (11)

So according to the nonnegative of A namely1113936

nj1 aij ge 0 as well as the economic meaning of A we get that

1113936nj1 aij ge 0 and hence 1113957μgt 0 +erefore it follows that there

exist 1113957μ 0lt 1113957μlt 1 and a vector 1113957xge 0 such that A1113957x 1113957μ1113957x

Now we can give the main result about the existence ofgrowth balanced solution for the CPE system (4)

Theorem 1 Let A (aij)ntimesn be the consumption coefficientmatrix of CPE system (4) with the assumption of1 minus aii gt1113936jneiaij it could be satisfied when I minus A is a diagonalstrictly dominant matrix3en there exists a growth balancedsolution X(t) (11113957μ)t1113957x of system (4) with the growth rate(11113957μ) here 1113957μ 0lt 1113957μlt 1 is the eigenvalue of A and 1113957xge 0 thecorresponding eigenvector

Proof First letting t 0 by system (4) we get

X0

AX1 (12)

If X0 is a balanced solution of (12) with the growth rate(1λ) then by Definition 1 we get

X1

1λX

0 (13)

It follows from (12) and (13) that

λX0

AX0 (14)

Inductively it is easy to get that Xt (1λ)tX0 will alsobe a balanced solution

Second by letting X0 1113957x and according to Lemma 2 weobtain that X(t) (11113957μ)t1113957x will be a growth balanced solutionwith growth rate (11113957μ)gt 1 +erefore +eorem 1 is provedthen

Remark 3 +eorem 1 gets rid of the restriction that theconsumption coefficient matrix should be irreducibleFurther under the condition proposed here it is impossiblethat the total costs in quantities for each class will be morethan the total outputs to be produced +is is the reason forthe occurrence of negative output then So controlling the

4 Discrete Dynamics in Nature and Society

economic system (4) in this way will be a better choice sincethere is no causal indeterminacy occurring under this di-rection for the CPE system (4)

4 Stability of the Balanced Output

As stated in the previous section a better way to control aneconomic system is to avoid the causal indeterminacy oc-curring and find the growth solution so as to keep the economygrowing +ough +eorem 1 gives a good description offinding out the growth balanced solution the conditionfor this needs a rigid initial point so as to reach the goal

If started at any point it is vitally important to inves-tigate whether this solution can approach the existinggrowth balanced solution or not namely the stability of theeconomy balance First we need a mathematical descriptionfor the stability see the following definition

Definition 3 Let Age 0 be the consumption coefficientmatrix of system (4) and assume it is an invertible matrixSuppose Xlowast(t) (1μt)Xgt 0 is a balanced solution of system(4) if for every solution 1113954X

(t) starting from any initial input1113954X

(0) ge 0 there exists a constant σ 0lt σ ltinfin such thatlimt⟶infin (1113954x

(t)i xlowast (t)

i ) σ where 1113954x(t)i xlowast (t)i is the ith entry of

1113954X(t) and Xlowast(t) respectively then the balanced solution Xlowast(t)

is called a stable balanced solutionClearly if the CPE system (4) yields a stable balanced

solution then all the solutions determined by any initialinput will eventually be greater than zero since any of thesesolutions approaches asymptotically to the positive balancedsolution Xlowast(t) gt 0 +us there has been no causal indeter-minacy happening in this case for system (4)

Next we give a necessary and sufficient condition to thestability of a balanced solution for system (4)

Theorem 2 Suppose Age 0 is the consumption coefficientmatrix of system (4) as well as an invertible matrix then wehave the following

(i) 3ere exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i

1 2 n are the eigenvalues of A and Xi the cor-responding eigenvectors

(ii) 3e balanced solution Xlowast(t) (1λt1)X1 gt 0 is stable if

and only if |λi|gt λ1 gt 0 where λi i 1 2 n arethe eigenvalues of A and Xi the correspondingeigenvectors

Proof First of all let us prove that (i) for any invertible matrixAge 0 there exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i 1 2 n are theeigenvalues of A and Xi the corresponding eigenvectors

If (Aminus 1)t gt 0 for some positive t then (Aminus 1)t gt 0 will be anirreducible matrix It follows by the well-known Frobeniustheorem that matrix ((Aminus 1)t)primes eigenvalues μi and the cor-responding eigenvectors Yi(i 1 2 n) must satisfyμ1 gt |μi| Y1 gt 0 with AY1 μ1Y1 where i 2 3 n Let λi

be the eigenvalues of A and Xi the corresponding eigenvectors

then it is easy to say μi (1λti ) i 1 2 n Y1 X1 So

by μ1 gt |μi| we get (1λt1)gt |(1λt

i)|(i 2 3 n) and hence|λi|gt λ1 gt 0 X1 gt 0

On the contrary let λi be the eigenvalues of A and Xi thecorresponding eigenvectors with |λi|gt λ1 gt 0 X1 gt 0 wherei 2 3 n First any vector V could be represented as alinear combination of Xi for the invertibility of Xi that is

V h1X1 + h2X2 + middot middot middot + hnXn (15)

LetV Ei be the column vector of the unit matrix E thatis all the entries of Ei being 0 except for the ith being 1 thenwe have

Ei

h1X1 + h2X2 + middot middot middot + hnXn (16)

Making the inner product of this Ei with X1 we get

X1 Ei

1113960 1113961 X1 h1X1 + h2X2 + middot middot middot + hnXn1113858 1113859 1113944n

i1hi X1 Xi1113858 1113859

(17)

Note that [X1 Xi] 0 for ine 1 we have[X1 Ei] h1[X1 X1] and so h1 ([X1 Ei][X1 X1]) h1must be positive since X1 gt 0 and the choice of Ei

Similarly any solution 1113954X(t) with starting initial input Ei

for system (5) can be reformulated as

1113954X(t)

Aminus 1

1113872 1113873tE

i h1

1λt1X1 + h2

1λt2X2 + middot middot middot + hn

1λt

n

Xn (18)

or equivalently

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i h1X1 + h2

λt1

λt2X2 + middot middot middot + hn

λt1

λtn

Xn

(19)

By |λi|gt λ1 gt 0 that is 0lt (λ1|λi|)lt 1 and paying at-tention to X1 gt 0 h1 gt 0 we get that for sufficient enoughlarge tgeN

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i gt 0 (20)

As λt1 gt 0 we obtain (Aminus 1)tEi gt 0 for large t Since it

holds for any Ei we get that (Aminus 1)t gt 0Secondly we prove (ii) If |λi|gt λ1 gt 0 according to

Definition 3 and (18) we have

limt⟶infin

1113954x(t)i

xlowast (t)i

h1 1λt

11113872 1113873X1i + h2 1λt21113872 1113873X2i + middot middot middot + hn 1λt

n1113872 1113873Xni

1λt11113872 1113873X1i

h1

(21)

Let σ h1 and note that h1 gt 0 we get that this Xlowast(t)

should be a stable balanced solutionOn the other hand if limt⟶infin(1113954x

(t)i xlowast (t)

i ) σ gt 0 thenfor sufficiently large tgeN we have 1113954x

(t)i gt 0 i 1 2 n

So 1113954X(t) gt 0 for sufficiently large tgeN with any initial input

vector 1113954X0 ge 0 Similar to the previous proof of (i) if we let

Discrete Dynamics in Nature and Society 5

1113954X0

Ei then we obtain 1113954X(t)

(Aminus 1)tEi gt 0 for alli 1 2 n and sufficiently large tgeN +us (Aminus 1)t gt 0+erefore by proving (i) we get that |λi|gt λ1 gt 0 X1 gt 0

Remark 4 Up to now for the CPE system (4) under theassumption of the consumption coefficient matrix Age 0being invertible we have the following equivalent resultsthe balanced solution Xlowast(t) (1λt

1)X1 gt 0 is stablehArr thereexists a positive integer t such that (Aminus 1)t gt 0hArr |λi|gtλ1 gt 0 X1 gt 0hArr causal indeterminacy cannot occurhArr eachproductrsquos output cannot be negative in the subsequent yearslater

5 Dynamic Price System

Loo-Keng Huarsquos price equation (7) is based on not con-sidering the capital costs such as the money rate At the sametime it gives an assumption that the productrsquos price changesin a fixed proportion for instance λlowast that is P(t+1) λlowastP(t)

for each period t But certainly interest rate impacts directlyon the productrsquos sale price while the productrsquos sale price maygenerally not change in a fixed proportion in practicaleconomic environment Hence we need to rebuild or extendLoo-Keng Huarsquos price system so as to match the practicaleconomic system soundly and this will be carried out in thefollowing subsection 51

In addition subsection 52 deals with the dynamicalproperties for the proposed price system such as the bal-anced price solution and its stability as well +e deep re-search on this respect will contribute to controlling theeconomic system (4) in a way of keeping the economygrowing continually

51 Formulation of the Price System Let P(t)i be the price of

the ith class per-unit product in the t-period +e pricevector for all n class products in the t-period could berepresented as P(t) (P

(t)1 P

(t)2 P(t)

n ) Assume anyproductrsquos price to be a fixed constant during each period andcosts of other products consumed for producing any productto be paid at the beginning of each period+en the per-unitcost for producing the jth class product in the t-period is

v(t)j 1113944

n

i1P

(t)i aij P

(t)aj (22)

where aj is the jth column of the consumption coefficientmatrix A

+e per-unit profit regardless of other costs for pro-ducing the jth class product in the t-period will be

π(t)j P

(t+1)j minus P

(t)aj (23)

At the same time if money to be loaned out can get aninterest rate rt during the time between t-period and

(t + 1)-period then the capital revenue for buying v(t)j will

be

R(t)j r

tv

(t)j r

tP

(t)aj (24)

By the competition arbitrage principle interest andprofit should eventually reach an equilibrium state that is

π(t)j R

(t)j (25)

So

P(t+1)j minus P

(t)aj r

tP

(t)aj (26)

or

P(t+1)j 1 + r

t1113872 1113873P

(t)aj (27)

As (27) holds for all j 1 2 n we can formulate it inthe following matrix representation

P(t+1)

1 + rt

1113872 1113873P(t)

A (28)

Letting M (1 + rt)A we obtain the dynamic priceequation corresponding to (4) as follows

P(t+1)

P(t)

M (29)

If interest rate rt 0 and the price changing rate varieswith a fixed proportion λlowast (Pt+1

i Pti) for each period t

then this dynamic price equation (29) becomes the priceequation (7) proposed by Loo-Keng Hua Next we turn toinvestigate the balanced price solution and the stability forthe price equation (29) as made previously for the outputequation (4)

52 Stability of the Balanced Price In order to simplify wesuppose the interest rate rt r to be a constant and Age 0 tobe invertible It is easy to see that M (1 + r)A is alsoinvertible and the eigenvalues of M will be (1 + r)λi if λi isthe eigenvalues of A while the corresponding eigenvectorsare the same

Definition of the balanced price solution to (29) can bedefined as Definition 1 for output equation (4) that is ifP(t) βP(tminus 1) holds for all t isin N 1 2 n andsome constant βgt 0 then P(t) will be a balanced solution of(29) with price changing rate β +e definition of stability fora balanced price solution can be found in the followingmathematical description

Definition 4 Let Age 0 be an invertible matrix andM (1 + r)A Suppose Plowast(t) ζt

1p1 gt 0 is a balanced pricesolution of system (29) if for a solution 1113954P

(t) starting fromany initial price 1113954P

(0) ge 0 there exists a constant σ 0lt σ ltinfinsuch that limt⟶infin(1113954p

(t)i plowast (t)

i ) σ where Z is the ith entryof 1113954P

(t) and Plowast(t) respectively then the balanced solutionPlowast(t) is called a stable price solution

6 Discrete Dynamics in Nature and Society

Similarly we have +eorem 3 which reveals the dy-namical properties for price equation (29) like+eorem 2 forthe output equation (4)

Theorem 3 Suppose Age 0 to be an invertible matrix and letM (1 + r)A where rgt 0 is the interest rate 3en we havethe following

(i) 3ere exists a positive integer t such that Mt gt 0 if andonly if μ1 gt |μi|ge 0 p1 gt 0 where μi and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

(ii) 3e balanced price Plowast(t) ζt1p1 gt 0 is stable if and

only if ζ1 gt |ζ i|ge 0 p1 gt 0 where ζ i and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

Proof +e proof of this theorem is similar to+eorem 2 anda survey is given here

First it is noted that Age 0 and being invertible impliesM (1 + r)Age 0 and being invertible also Second if μ is aneigenvalue of matrix M with eigenvector p then μt will bethe eigenvalue of Mt with the same eigenvector p in re-gardless of scalar times+ird inductively it is easy to obtainP(t) P(0)Mt Finally note that the general solution ofequation (29) can be written as

1113954P(t)

α1ζt1p1 + α2ζ

t2p2 + middot middot middot + αnζ

tnpn (30)

where αi are constants to be determined by the initial price1113954P

(0) All these four points give the key respects to prove+eorem 3 and the procedure of the proof is omittedhere

Remark 5 +e solutions Xlowast(t) (1λt1)X1 and Plowast(t) ζt

1p1are a pair of output balance and price balance by the cor-respondence ζ1 (1 + r)λ1 X1 p1 In fact eigenvalues λi

of A and eigenvalues ζ i of M (1 + r)A have a corre-spondence ζ i (1 + r)λi So if ζ1 gt |ζ i| then λ1 gt |λi| and if|λi|gt λ1 then |ζ i|gt ζ1 +us by +eorem 2 and +eorem 3balanced price solution Plowast(t) being stable means balancedoutput Xlowast(t) is not stable and vice versa

6 Illustrative Examples

In this section we illustrate some of themain results of this paperwith two examples one is about the input-output table for-mulation and the other is an application of+eorem 1 as well asa comparison declaration First let us give a simplified exampleto show the process of how to create an input-output table

Example 1 Assuming a hypothetical economy is composedof (1) agriculture (2) the industrial sector (manufacturing)and (3) the service provider (service) each of these de-partments produces only one type of product namely theagricultural industrial or service supply and there is in-terdependence between and among them Each departmentbuys products from the other departments and sells its ownproducts to the opposite ones but the final product and

service supply (they do not enter the production process) isused by external departments such as consumers +eproduction process and the external demand are not as-sociated with the government and there is no foreign tradethen according to the early hypotheses (H1)(H3) we cancarry out a form to summarize the product and the currentsituation of service supply as shown in Table 1 where xij

represents products (in US $) sold by sector i to sector j+e data of each row in Table 1 represents the allocation of

the total output to different departments and users while thedata of each column represents the sources of departmentinputs required for the total output For example the first rowshows that the total output of 100$ agricultural products isassigned to 15$ products for reproducing 20$ products beingsold to manufacturing 30$ products to service and the last35$ products being used tomeet external demand Similarly itcan be seen from the second column that for the total outputof 200$ the manufacturing needs to invest 20$ of agriculturalproducts 10$ of its own products and 60$ of service input

For convenience of analysis coupled with the input-outputtable the so-called technology input-output table can beconstructed to show the amount that each sector for thepurpose of producing one unit of its own product needs forconsuming the other sectorrsquos +e quantity in the technicalinput-output table represents the input coefficient tij (inLeontiefrsquos model) or consumption coefficient (in CPEmodel) ofthe economy which can be obtained from Table 1 for examplethe data of each column in Table 1 being divided by the totalagriculture output 100 gives the agriculture sectorrsquos input co-efficients and so on +us the technology input table or con-sumption coefficient table is completed and shown in Table 2

By Table 2 the input matrix T or consumption coeffi-cient matrix A should be

T A

015 010 020

030 005 030

030 030 000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

+e input-output table can comprehensively and sys-tematically reflect the input-output relationship among allsectors of the national economy and reveal the economic andtechnological relations of interdependence and mutual re-striction among all sectors in the production process On theother hand it can tell people about the output of varioussectors of the national economy and how the output of thesesectors is distributed to other sectors for production or toresidents and society for final consumption or exportabroad Furthermore it can tell people how each departmentobtains intermediate inputs and initial inputs from otherdepartments for its own production

Example 1 only provides the fundamental principle forthe compilation of input-output table with a simple case Ingeneral the actual input-output table of an economy is muchmore complicated than this one provided and it generallyincludes national table regional table sectoral table andjoint enterprise table according to different scopes as well asstatic table and dynamic table according to the modelcharacteristics In addition there are input-output tables for

Discrete Dynamics in Nature and Society 7

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

economic system (4) in this way will be a better choice sincethere is no causal indeterminacy occurring under this di-rection for the CPE system (4)

4 Stability of the Balanced Output

As stated in the previous section a better way to control aneconomic system is to avoid the causal indeterminacy oc-curring and find the growth solution so as to keep the economygrowing +ough +eorem 1 gives a good description offinding out the growth balanced solution the conditionfor this needs a rigid initial point so as to reach the goal

If started at any point it is vitally important to inves-tigate whether this solution can approach the existinggrowth balanced solution or not namely the stability of theeconomy balance First we need a mathematical descriptionfor the stability see the following definition

Definition 3 Let Age 0 be the consumption coefficientmatrix of system (4) and assume it is an invertible matrixSuppose Xlowast(t) (1μt)Xgt 0 is a balanced solution of system(4) if for every solution 1113954X

(t) starting from any initial input1113954X

(0) ge 0 there exists a constant σ 0lt σ ltinfin such thatlimt⟶infin (1113954x

(t)i xlowast (t)

i ) σ where 1113954x(t)i xlowast (t)i is the ith entry of

1113954X(t) and Xlowast(t) respectively then the balanced solution Xlowast(t)

is called a stable balanced solutionClearly if the CPE system (4) yields a stable balanced

solution then all the solutions determined by any initialinput will eventually be greater than zero since any of thesesolutions approaches asymptotically to the positive balancedsolution Xlowast(t) gt 0 +us there has been no causal indeter-minacy happening in this case for system (4)

Next we give a necessary and sufficient condition to thestability of a balanced solution for system (4)

Theorem 2 Suppose Age 0 is the consumption coefficientmatrix of system (4) as well as an invertible matrix then wehave the following

(i) 3ere exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i

1 2 n are the eigenvalues of A and Xi the cor-responding eigenvectors

(ii) 3e balanced solution Xlowast(t) (1λt1)X1 gt 0 is stable if

and only if |λi|gt λ1 gt 0 where λi i 1 2 n arethe eigenvalues of A and Xi the correspondingeigenvectors

Proof First of all let us prove that (i) for any invertible matrixAge 0 there exists a positive integer t such that (Aminus 1)t gt 0 ifand only if |λi|gt λ1 gt 0 X1 gt 0 where λi i 1 2 n are theeigenvalues of A and Xi the corresponding eigenvectors

If (Aminus 1)t gt 0 for some positive t then (Aminus 1)t gt 0 will be anirreducible matrix It follows by the well-known Frobeniustheorem that matrix ((Aminus 1)t)primes eigenvalues μi and the cor-responding eigenvectors Yi(i 1 2 n) must satisfyμ1 gt |μi| Y1 gt 0 with AY1 μ1Y1 where i 2 3 n Let λi

be the eigenvalues of A and Xi the corresponding eigenvectors

then it is easy to say μi (1λti ) i 1 2 n Y1 X1 So

by μ1 gt |μi| we get (1λt1)gt |(1λt

i)|(i 2 3 n) and hence|λi|gt λ1 gt 0 X1 gt 0

On the contrary let λi be the eigenvalues of A and Xi thecorresponding eigenvectors with |λi|gt λ1 gt 0 X1 gt 0 wherei 2 3 n First any vector V could be represented as alinear combination of Xi for the invertibility of Xi that is

V h1X1 + h2X2 + middot middot middot + hnXn (15)

LetV Ei be the column vector of the unit matrix E thatis all the entries of Ei being 0 except for the ith being 1 thenwe have

Ei

h1X1 + h2X2 + middot middot middot + hnXn (16)

Making the inner product of this Ei with X1 we get

X1 Ei

1113960 1113961 X1 h1X1 + h2X2 + middot middot middot + hnXn1113858 1113859 1113944n

i1hi X1 Xi1113858 1113859

(17)

Note that [X1 Xi] 0 for ine 1 we have[X1 Ei] h1[X1 X1] and so h1 ([X1 Ei][X1 X1]) h1must be positive since X1 gt 0 and the choice of Ei

Similarly any solution 1113954X(t) with starting initial input Ei

for system (5) can be reformulated as

1113954X(t)

Aminus 1

1113872 1113873tE

i h1

1λt1X1 + h2

1λt2X2 + middot middot middot + hn

1λt

n

Xn (18)

or equivalently

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i h1X1 + h2

λt1

λt2X2 + middot middot middot + hn

λt1

λtn

Xn

(19)

By |λi|gt λ1 gt 0 that is 0lt (λ1|λi|)lt 1 and paying at-tention to X1 gt 0 h1 gt 0 we get that for sufficient enoughlarge tgeN

λt1

1113954X(t)

λt1 A

minus 11113872 1113873

tE

i gt 0 (20)

As λt1 gt 0 we obtain (Aminus 1)tEi gt 0 for large t Since it

holds for any Ei we get that (Aminus 1)t gt 0Secondly we prove (ii) If |λi|gt λ1 gt 0 according to

Definition 3 and (18) we have

limt⟶infin

1113954x(t)i

xlowast (t)i

h1 1λt

11113872 1113873X1i + h2 1λt21113872 1113873X2i + middot middot middot + hn 1λt

n1113872 1113873Xni

1λt11113872 1113873X1i

h1

(21)

Let σ h1 and note that h1 gt 0 we get that this Xlowast(t)

should be a stable balanced solutionOn the other hand if limt⟶infin(1113954x

(t)i xlowast (t)

i ) σ gt 0 thenfor sufficiently large tgeN we have 1113954x

(t)i gt 0 i 1 2 n

So 1113954X(t) gt 0 for sufficiently large tgeN with any initial input

vector 1113954X0 ge 0 Similar to the previous proof of (i) if we let

Discrete Dynamics in Nature and Society 5

1113954X0

Ei then we obtain 1113954X(t)

(Aminus 1)tEi gt 0 for alli 1 2 n and sufficiently large tgeN +us (Aminus 1)t gt 0+erefore by proving (i) we get that |λi|gt λ1 gt 0 X1 gt 0

Remark 4 Up to now for the CPE system (4) under theassumption of the consumption coefficient matrix Age 0being invertible we have the following equivalent resultsthe balanced solution Xlowast(t) (1λt

1)X1 gt 0 is stablehArr thereexists a positive integer t such that (Aminus 1)t gt 0hArr |λi|gtλ1 gt 0 X1 gt 0hArr causal indeterminacy cannot occurhArr eachproductrsquos output cannot be negative in the subsequent yearslater

5 Dynamic Price System

Loo-Keng Huarsquos price equation (7) is based on not con-sidering the capital costs such as the money rate At the sametime it gives an assumption that the productrsquos price changesin a fixed proportion for instance λlowast that is P(t+1) λlowastP(t)

for each period t But certainly interest rate impacts directlyon the productrsquos sale price while the productrsquos sale price maygenerally not change in a fixed proportion in practicaleconomic environment Hence we need to rebuild or extendLoo-Keng Huarsquos price system so as to match the practicaleconomic system soundly and this will be carried out in thefollowing subsection 51

In addition subsection 52 deals with the dynamicalproperties for the proposed price system such as the bal-anced price solution and its stability as well +e deep re-search on this respect will contribute to controlling theeconomic system (4) in a way of keeping the economygrowing continually

51 Formulation of the Price System Let P(t)i be the price of

the ith class per-unit product in the t-period +e pricevector for all n class products in the t-period could berepresented as P(t) (P

(t)1 P

(t)2 P(t)

n ) Assume anyproductrsquos price to be a fixed constant during each period andcosts of other products consumed for producing any productto be paid at the beginning of each period+en the per-unitcost for producing the jth class product in the t-period is

v(t)j 1113944

n

i1P

(t)i aij P

(t)aj (22)

where aj is the jth column of the consumption coefficientmatrix A

+e per-unit profit regardless of other costs for pro-ducing the jth class product in the t-period will be

π(t)j P

(t+1)j minus P

(t)aj (23)

At the same time if money to be loaned out can get aninterest rate rt during the time between t-period and

(t + 1)-period then the capital revenue for buying v(t)j will

be

R(t)j r

tv

(t)j r

tP

(t)aj (24)

By the competition arbitrage principle interest andprofit should eventually reach an equilibrium state that is

π(t)j R

(t)j (25)

So

P(t+1)j minus P

(t)aj r

tP

(t)aj (26)

or

P(t+1)j 1 + r

t1113872 1113873P

(t)aj (27)

As (27) holds for all j 1 2 n we can formulate it inthe following matrix representation

P(t+1)

1 + rt

1113872 1113873P(t)

A (28)

Letting M (1 + rt)A we obtain the dynamic priceequation corresponding to (4) as follows

P(t+1)

P(t)

M (29)

If interest rate rt 0 and the price changing rate varieswith a fixed proportion λlowast (Pt+1

i Pti) for each period t

then this dynamic price equation (29) becomes the priceequation (7) proposed by Loo-Keng Hua Next we turn toinvestigate the balanced price solution and the stability forthe price equation (29) as made previously for the outputequation (4)

52 Stability of the Balanced Price In order to simplify wesuppose the interest rate rt r to be a constant and Age 0 tobe invertible It is easy to see that M (1 + r)A is alsoinvertible and the eigenvalues of M will be (1 + r)λi if λi isthe eigenvalues of A while the corresponding eigenvectorsare the same

Definition of the balanced price solution to (29) can bedefined as Definition 1 for output equation (4) that is ifP(t) βP(tminus 1) holds for all t isin N 1 2 n andsome constant βgt 0 then P(t) will be a balanced solution of(29) with price changing rate β +e definition of stability fora balanced price solution can be found in the followingmathematical description

Definition 4 Let Age 0 be an invertible matrix andM (1 + r)A Suppose Plowast(t) ζt

1p1 gt 0 is a balanced pricesolution of system (29) if for a solution 1113954P

(t) starting fromany initial price 1113954P

(0) ge 0 there exists a constant σ 0lt σ ltinfinsuch that limt⟶infin(1113954p

(t)i plowast (t)

i ) σ where Z is the ith entryof 1113954P

(t) and Plowast(t) respectively then the balanced solutionPlowast(t) is called a stable price solution

6 Discrete Dynamics in Nature and Society

Similarly we have +eorem 3 which reveals the dy-namical properties for price equation (29) like+eorem 2 forthe output equation (4)

Theorem 3 Suppose Age 0 to be an invertible matrix and letM (1 + r)A where rgt 0 is the interest rate 3en we havethe following

(i) 3ere exists a positive integer t such that Mt gt 0 if andonly if μ1 gt |μi|ge 0 p1 gt 0 where μi and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

(ii) 3e balanced price Plowast(t) ζt1p1 gt 0 is stable if and

only if ζ1 gt |ζ i|ge 0 p1 gt 0 where ζ i and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

Proof +e proof of this theorem is similar to+eorem 2 anda survey is given here

First it is noted that Age 0 and being invertible impliesM (1 + r)Age 0 and being invertible also Second if μ is aneigenvalue of matrix M with eigenvector p then μt will bethe eigenvalue of Mt with the same eigenvector p in re-gardless of scalar times+ird inductively it is easy to obtainP(t) P(0)Mt Finally note that the general solution ofequation (29) can be written as

1113954P(t)

α1ζt1p1 + α2ζ

t2p2 + middot middot middot + αnζ

tnpn (30)

where αi are constants to be determined by the initial price1113954P

(0) All these four points give the key respects to prove+eorem 3 and the procedure of the proof is omittedhere

Remark 5 +e solutions Xlowast(t) (1λt1)X1 and Plowast(t) ζt

1p1are a pair of output balance and price balance by the cor-respondence ζ1 (1 + r)λ1 X1 p1 In fact eigenvalues λi

of A and eigenvalues ζ i of M (1 + r)A have a corre-spondence ζ i (1 + r)λi So if ζ1 gt |ζ i| then λ1 gt |λi| and if|λi|gt λ1 then |ζ i|gt ζ1 +us by +eorem 2 and +eorem 3balanced price solution Plowast(t) being stable means balancedoutput Xlowast(t) is not stable and vice versa

6 Illustrative Examples

In this section we illustrate some of themain results of this paperwith two examples one is about the input-output table for-mulation and the other is an application of+eorem 1 as well asa comparison declaration First let us give a simplified exampleto show the process of how to create an input-output table

Example 1 Assuming a hypothetical economy is composedof (1) agriculture (2) the industrial sector (manufacturing)and (3) the service provider (service) each of these de-partments produces only one type of product namely theagricultural industrial or service supply and there is in-terdependence between and among them Each departmentbuys products from the other departments and sells its ownproducts to the opposite ones but the final product and

service supply (they do not enter the production process) isused by external departments such as consumers +eproduction process and the external demand are not as-sociated with the government and there is no foreign tradethen according to the early hypotheses (H1)(H3) we cancarry out a form to summarize the product and the currentsituation of service supply as shown in Table 1 where xij

represents products (in US $) sold by sector i to sector j+e data of each row in Table 1 represents the allocation of

the total output to different departments and users while thedata of each column represents the sources of departmentinputs required for the total output For example the first rowshows that the total output of 100$ agricultural products isassigned to 15$ products for reproducing 20$ products beingsold to manufacturing 30$ products to service and the last35$ products being used tomeet external demand Similarly itcan be seen from the second column that for the total outputof 200$ the manufacturing needs to invest 20$ of agriculturalproducts 10$ of its own products and 60$ of service input

For convenience of analysis coupled with the input-outputtable the so-called technology input-output table can beconstructed to show the amount that each sector for thepurpose of producing one unit of its own product needs forconsuming the other sectorrsquos +e quantity in the technicalinput-output table represents the input coefficient tij (inLeontiefrsquos model) or consumption coefficient (in CPEmodel) ofthe economy which can be obtained from Table 1 for examplethe data of each column in Table 1 being divided by the totalagriculture output 100 gives the agriculture sectorrsquos input co-efficients and so on +us the technology input table or con-sumption coefficient table is completed and shown in Table 2

By Table 2 the input matrix T or consumption coeffi-cient matrix A should be

T A

015 010 020

030 005 030

030 030 000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

+e input-output table can comprehensively and sys-tematically reflect the input-output relationship among allsectors of the national economy and reveal the economic andtechnological relations of interdependence and mutual re-striction among all sectors in the production process On theother hand it can tell people about the output of varioussectors of the national economy and how the output of thesesectors is distributed to other sectors for production or toresidents and society for final consumption or exportabroad Furthermore it can tell people how each departmentobtains intermediate inputs and initial inputs from otherdepartments for its own production

Example 1 only provides the fundamental principle forthe compilation of input-output table with a simple case Ingeneral the actual input-output table of an economy is muchmore complicated than this one provided and it generallyincludes national table regional table sectoral table andjoint enterprise table according to different scopes as well asstatic table and dynamic table according to the modelcharacteristics In addition there are input-output tables for

Discrete Dynamics in Nature and Society 7

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

1113954X0

Ei then we obtain 1113954X(t)

(Aminus 1)tEi gt 0 for alli 1 2 n and sufficiently large tgeN +us (Aminus 1)t gt 0+erefore by proving (i) we get that |λi|gt λ1 gt 0 X1 gt 0

Remark 4 Up to now for the CPE system (4) under theassumption of the consumption coefficient matrix Age 0being invertible we have the following equivalent resultsthe balanced solution Xlowast(t) (1λt

1)X1 gt 0 is stablehArr thereexists a positive integer t such that (Aminus 1)t gt 0hArr |λi|gtλ1 gt 0 X1 gt 0hArr causal indeterminacy cannot occurhArr eachproductrsquos output cannot be negative in the subsequent yearslater

5 Dynamic Price System

Loo-Keng Huarsquos price equation (7) is based on not con-sidering the capital costs such as the money rate At the sametime it gives an assumption that the productrsquos price changesin a fixed proportion for instance λlowast that is P(t+1) λlowastP(t)

for each period t But certainly interest rate impacts directlyon the productrsquos sale price while the productrsquos sale price maygenerally not change in a fixed proportion in practicaleconomic environment Hence we need to rebuild or extendLoo-Keng Huarsquos price system so as to match the practicaleconomic system soundly and this will be carried out in thefollowing subsection 51

In addition subsection 52 deals with the dynamicalproperties for the proposed price system such as the bal-anced price solution and its stability as well +e deep re-search on this respect will contribute to controlling theeconomic system (4) in a way of keeping the economygrowing continually

51 Formulation of the Price System Let P(t)i be the price of

the ith class per-unit product in the t-period +e pricevector for all n class products in the t-period could berepresented as P(t) (P

(t)1 P

(t)2 P(t)

n ) Assume anyproductrsquos price to be a fixed constant during each period andcosts of other products consumed for producing any productto be paid at the beginning of each period+en the per-unitcost for producing the jth class product in the t-period is

v(t)j 1113944

n

i1P

(t)i aij P

(t)aj (22)

where aj is the jth column of the consumption coefficientmatrix A

+e per-unit profit regardless of other costs for pro-ducing the jth class product in the t-period will be

π(t)j P

(t+1)j minus P

(t)aj (23)

At the same time if money to be loaned out can get aninterest rate rt during the time between t-period and

(t + 1)-period then the capital revenue for buying v(t)j will

be

R(t)j r

tv

(t)j r

tP

(t)aj (24)

By the competition arbitrage principle interest andprofit should eventually reach an equilibrium state that is

π(t)j R

(t)j (25)

So

P(t+1)j minus P

(t)aj r

tP

(t)aj (26)

or

P(t+1)j 1 + r

t1113872 1113873P

(t)aj (27)

As (27) holds for all j 1 2 n we can formulate it inthe following matrix representation

P(t+1)

1 + rt

1113872 1113873P(t)

A (28)

Letting M (1 + rt)A we obtain the dynamic priceequation corresponding to (4) as follows

P(t+1)

P(t)

M (29)

If interest rate rt 0 and the price changing rate varieswith a fixed proportion λlowast (Pt+1

i Pti) for each period t

then this dynamic price equation (29) becomes the priceequation (7) proposed by Loo-Keng Hua Next we turn toinvestigate the balanced price solution and the stability forthe price equation (29) as made previously for the outputequation (4)

52 Stability of the Balanced Price In order to simplify wesuppose the interest rate rt r to be a constant and Age 0 tobe invertible It is easy to see that M (1 + r)A is alsoinvertible and the eigenvalues of M will be (1 + r)λi if λi isthe eigenvalues of A while the corresponding eigenvectorsare the same

Definition of the balanced price solution to (29) can bedefined as Definition 1 for output equation (4) that is ifP(t) βP(tminus 1) holds for all t isin N 1 2 n andsome constant βgt 0 then P(t) will be a balanced solution of(29) with price changing rate β +e definition of stability fora balanced price solution can be found in the followingmathematical description

Definition 4 Let Age 0 be an invertible matrix andM (1 + r)A Suppose Plowast(t) ζt

1p1 gt 0 is a balanced pricesolution of system (29) if for a solution 1113954P

(t) starting fromany initial price 1113954P

(0) ge 0 there exists a constant σ 0lt σ ltinfinsuch that limt⟶infin(1113954p

(t)i plowast (t)

i ) σ where Z is the ith entryof 1113954P

(t) and Plowast(t) respectively then the balanced solutionPlowast(t) is called a stable price solution

6 Discrete Dynamics in Nature and Society

Similarly we have +eorem 3 which reveals the dy-namical properties for price equation (29) like+eorem 2 forthe output equation (4)

Theorem 3 Suppose Age 0 to be an invertible matrix and letM (1 + r)A where rgt 0 is the interest rate 3en we havethe following

(i) 3ere exists a positive integer t such that Mt gt 0 if andonly if μ1 gt |μi|ge 0 p1 gt 0 where μi and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

(ii) 3e balanced price Plowast(t) ζt1p1 gt 0 is stable if and

only if ζ1 gt |ζ i|ge 0 p1 gt 0 where ζ i and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

Proof +e proof of this theorem is similar to+eorem 2 anda survey is given here

First it is noted that Age 0 and being invertible impliesM (1 + r)Age 0 and being invertible also Second if μ is aneigenvalue of matrix M with eigenvector p then μt will bethe eigenvalue of Mt with the same eigenvector p in re-gardless of scalar times+ird inductively it is easy to obtainP(t) P(0)Mt Finally note that the general solution ofequation (29) can be written as

1113954P(t)

α1ζt1p1 + α2ζ

t2p2 + middot middot middot + αnζ

tnpn (30)

where αi are constants to be determined by the initial price1113954P

(0) All these four points give the key respects to prove+eorem 3 and the procedure of the proof is omittedhere

Remark 5 +e solutions Xlowast(t) (1λt1)X1 and Plowast(t) ζt

1p1are a pair of output balance and price balance by the cor-respondence ζ1 (1 + r)λ1 X1 p1 In fact eigenvalues λi

of A and eigenvalues ζ i of M (1 + r)A have a corre-spondence ζ i (1 + r)λi So if ζ1 gt |ζ i| then λ1 gt |λi| and if|λi|gt λ1 then |ζ i|gt ζ1 +us by +eorem 2 and +eorem 3balanced price solution Plowast(t) being stable means balancedoutput Xlowast(t) is not stable and vice versa

6 Illustrative Examples

In this section we illustrate some of themain results of this paperwith two examples one is about the input-output table for-mulation and the other is an application of+eorem 1 as well asa comparison declaration First let us give a simplified exampleto show the process of how to create an input-output table

Example 1 Assuming a hypothetical economy is composedof (1) agriculture (2) the industrial sector (manufacturing)and (3) the service provider (service) each of these de-partments produces only one type of product namely theagricultural industrial or service supply and there is in-terdependence between and among them Each departmentbuys products from the other departments and sells its ownproducts to the opposite ones but the final product and

service supply (they do not enter the production process) isused by external departments such as consumers +eproduction process and the external demand are not as-sociated with the government and there is no foreign tradethen according to the early hypotheses (H1)(H3) we cancarry out a form to summarize the product and the currentsituation of service supply as shown in Table 1 where xij

represents products (in US $) sold by sector i to sector j+e data of each row in Table 1 represents the allocation of

the total output to different departments and users while thedata of each column represents the sources of departmentinputs required for the total output For example the first rowshows that the total output of 100$ agricultural products isassigned to 15$ products for reproducing 20$ products beingsold to manufacturing 30$ products to service and the last35$ products being used tomeet external demand Similarly itcan be seen from the second column that for the total outputof 200$ the manufacturing needs to invest 20$ of agriculturalproducts 10$ of its own products and 60$ of service input

For convenience of analysis coupled with the input-outputtable the so-called technology input-output table can beconstructed to show the amount that each sector for thepurpose of producing one unit of its own product needs forconsuming the other sectorrsquos +e quantity in the technicalinput-output table represents the input coefficient tij (inLeontiefrsquos model) or consumption coefficient (in CPEmodel) ofthe economy which can be obtained from Table 1 for examplethe data of each column in Table 1 being divided by the totalagriculture output 100 gives the agriculture sectorrsquos input co-efficients and so on +us the technology input table or con-sumption coefficient table is completed and shown in Table 2

By Table 2 the input matrix T or consumption coeffi-cient matrix A should be

T A

015 010 020

030 005 030

030 030 000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

+e input-output table can comprehensively and sys-tematically reflect the input-output relationship among allsectors of the national economy and reveal the economic andtechnological relations of interdependence and mutual re-striction among all sectors in the production process On theother hand it can tell people about the output of varioussectors of the national economy and how the output of thesesectors is distributed to other sectors for production or toresidents and society for final consumption or exportabroad Furthermore it can tell people how each departmentobtains intermediate inputs and initial inputs from otherdepartments for its own production

Example 1 only provides the fundamental principle forthe compilation of input-output table with a simple case Ingeneral the actual input-output table of an economy is muchmore complicated than this one provided and it generallyincludes national table regional table sectoral table andjoint enterprise table according to different scopes as well asstatic table and dynamic table according to the modelcharacteristics In addition there are input-output tables for

Discrete Dynamics in Nature and Society 7

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

Similarly we have +eorem 3 which reveals the dy-namical properties for price equation (29) like+eorem 2 forthe output equation (4)

Theorem 3 Suppose Age 0 to be an invertible matrix and letM (1 + r)A where rgt 0 is the interest rate 3en we havethe following

(i) 3ere exists a positive integer t such that Mt gt 0 if andonly if μ1 gt |μi|ge 0 p1 gt 0 where μi and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

(ii) 3e balanced price Plowast(t) ζt1p1 gt 0 is stable if and

only if ζ1 gt |ζ i|ge 0 p1 gt 0 where ζ i and pii 1 2 n are the eigenvalues of M and thecorresponding eigenvectors

Proof +e proof of this theorem is similar to+eorem 2 anda survey is given here

First it is noted that Age 0 and being invertible impliesM (1 + r)Age 0 and being invertible also Second if μ is aneigenvalue of matrix M with eigenvector p then μt will bethe eigenvalue of Mt with the same eigenvector p in re-gardless of scalar times+ird inductively it is easy to obtainP(t) P(0)Mt Finally note that the general solution ofequation (29) can be written as

1113954P(t)

α1ζt1p1 + α2ζ

t2p2 + middot middot middot + αnζ

tnpn (30)

where αi are constants to be determined by the initial price1113954P

(0) All these four points give the key respects to prove+eorem 3 and the procedure of the proof is omittedhere

Remark 5 +e solutions Xlowast(t) (1λt1)X1 and Plowast(t) ζt

1p1are a pair of output balance and price balance by the cor-respondence ζ1 (1 + r)λ1 X1 p1 In fact eigenvalues λi

of A and eigenvalues ζ i of M (1 + r)A have a corre-spondence ζ i (1 + r)λi So if ζ1 gt |ζ i| then λ1 gt |λi| and if|λi|gt λ1 then |ζ i|gt ζ1 +us by +eorem 2 and +eorem 3balanced price solution Plowast(t) being stable means balancedoutput Xlowast(t) is not stable and vice versa

6 Illustrative Examples

In this section we illustrate some of themain results of this paperwith two examples one is about the input-output table for-mulation and the other is an application of+eorem 1 as well asa comparison declaration First let us give a simplified exampleto show the process of how to create an input-output table

Example 1 Assuming a hypothetical economy is composedof (1) agriculture (2) the industrial sector (manufacturing)and (3) the service provider (service) each of these de-partments produces only one type of product namely theagricultural industrial or service supply and there is in-terdependence between and among them Each departmentbuys products from the other departments and sells its ownproducts to the opposite ones but the final product and

service supply (they do not enter the production process) isused by external departments such as consumers +eproduction process and the external demand are not as-sociated with the government and there is no foreign tradethen according to the early hypotheses (H1)(H3) we cancarry out a form to summarize the product and the currentsituation of service supply as shown in Table 1 where xij

represents products (in US $) sold by sector i to sector j+e data of each row in Table 1 represents the allocation of

the total output to different departments and users while thedata of each column represents the sources of departmentinputs required for the total output For example the first rowshows that the total output of 100$ agricultural products isassigned to 15$ products for reproducing 20$ products beingsold to manufacturing 30$ products to service and the last35$ products being used tomeet external demand Similarly itcan be seen from the second column that for the total outputof 200$ the manufacturing needs to invest 20$ of agriculturalproducts 10$ of its own products and 60$ of service input

For convenience of analysis coupled with the input-outputtable the so-called technology input-output table can beconstructed to show the amount that each sector for thepurpose of producing one unit of its own product needs forconsuming the other sectorrsquos +e quantity in the technicalinput-output table represents the input coefficient tij (inLeontiefrsquos model) or consumption coefficient (in CPEmodel) ofthe economy which can be obtained from Table 1 for examplethe data of each column in Table 1 being divided by the totalagriculture output 100 gives the agriculture sectorrsquos input co-efficients and so on +us the technology input table or con-sumption coefficient table is completed and shown in Table 2

By Table 2 the input matrix T or consumption coeffi-cient matrix A should be

T A

015 010 020

030 005 030

030 030 000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

+e input-output table can comprehensively and sys-tematically reflect the input-output relationship among allsectors of the national economy and reveal the economic andtechnological relations of interdependence and mutual re-striction among all sectors in the production process On theother hand it can tell people about the output of varioussectors of the national economy and how the output of thesesectors is distributed to other sectors for production or toresidents and society for final consumption or exportabroad Furthermore it can tell people how each departmentobtains intermediate inputs and initial inputs from otherdepartments for its own production

Example 1 only provides the fundamental principle forthe compilation of input-output table with a simple case Ingeneral the actual input-output table of an economy is muchmore complicated than this one provided and it generallyincludes national table regional table sectoral table andjoint enterprise table according to different scopes as well asstatic table and dynamic table according to the modelcharacteristics In addition there are input-output tables for

Discrete Dynamics in Nature and Society 7

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

studying special issues such as environmental protectionpopulation and resources

+e function of a nationrsquos input-output report is not onlyto reflect the direct and obvious economic and technologicalrelations among various departments in the productionprocess but also to reveal the indirect hidden and evenneglected economic and technological relations among var-ious departments +e input-output table provides the basisfor studying the industrial structure especially for makingand checking the national economic plan studying the pricedecision and conducting various quantitative analyses

Example 2 Let A be a matrix like the following

A

110

025

015

0

0 025

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

and it is easy to find the inverse of A as follows

B Aminus1

10 0 minus10

0 5 0

0 052

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

A simple computation gives us the following eigenvectorsX and the corresponding eigenvalues λi(i 1 2 3) of A

X

1 045

0 1 0

0 035

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ λ1 λ2 λ3( 1113857 110

1525

1113874 1113875

(34)

It is easy to check that 1 minus aii gt 1113936jnei

aij i j 1 2 3 this

means the conditions of+eorem 1 are satisfied and then by+eorem 1 CPE system (4) with consumption coefficientmatrix A must have a growth balanced solution One thingthat needs special attention here is that the matrix A isreducible which is different from the fact that most of theexisting dynamic properties for CPE and FFCME are basedon the hypothesis of matrix A to be irreducible

7 Conclusion

Other than the ldquoplanned economicrdquo policy that had been putinto practice in China for nearly 30 years from 1949 to 1978due to the first industrial revolution which started at thesixties of the 18th century there is the ldquofully free competitionmarketing economicrdquo policy being conducted by the Westfor over one hundred years +ere have been a tremendousnumber of research papers investigating various economicbehaviors of the fully free competition marketing systems inthe past hundred years but few studies investigated theplanned economic systems till Professor Loo-Keng Hua aworld famous Chinese mathematician presented a series ofresearch papers in the 80s of the last century

Table 1 Input-output table

InputOutput

Agriculture Manufacturing Service External demand Total outputAgriculture 15 (x11) 20 (x12) 30 (x13) 35 (d1) 100 (x1)Manufacturing 30 (x21) 10 (x22) 45 (x23) 115 (d2) 200 (x2)Service 20 (x31) 60 (x32) 0 (x33) 70 (d3) 150 (x3)

Table 2 Technology input-output table

InputOutput

Agriculture Manufacturing ServiceAgriculture 015 (t11) 010 (t12) 020 (t13)Manufacturing 030 (t21) 005 (t22) 030 (t23)Service 020 (t31) 030 (t32) 000 (t33)

8 Discrete Dynamics in Nature and Society

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

In parallel with the extensive investigations being donefor various FFCME systems and motivated by studyingalong the direction of dynamical properties of a system wehave carried out some important dynamical properties forCPE system which were first proposed by the famousChinese mathematician Loo-Keng Hua in this paper

+e main results obtained here include the followingsfirst we obtain new conditions for the existence of growthbalanced solution to the CPE system and investigate thestability of the balanced output as well second we propose amore generalized price equation and study the stability ofthe balanced price as well All these results provide effectiveaspects for controlling the CPE system so as to keep theeconomy grow and develop stably

Certainly further study of the CPE system should beencouraged in two directions the first one is thatmore suitablemodel is needed tomatch the economy developing state underthe varying economy policy for instance each classrsquos pro-ducing rate may be different but here in case of simplicitywith a hypothesis being the same growing rate the second isthat a deeper study especially the simulation of any economicmodel with the practical economy situation will be morecompetitive than others Just for this reason we will aim atthese problems in our future study on the economic systems

Data Availability

No data were used for this paper

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e research was supported by Grant GD17XYJ29 from theOffice of Philosophy and Social Science Research Project ofGuang Dong Province China

References

[1] W Leontief ldquoDynamic Analysis in Studies in the Structureofthe American Economy +eoretical and EmpiricalExplorationsInpinput-Output Analysisrdquo in Isard and HelenKistin W Leontief H B Chenery P G Clark et al Eds pp53ndash90 Oxford University Press Oxford UK 1953

[2] W Leontief ldquoLags and the stability of dynamic systemsrdquoEconometrica vol 29 no 4 pp 659ndash669 1961

[3] W Leontief ldquoLags and the stability of dynamic systems arejoinderrdquo Econometrica vol 29 no 4 pp 674-675 1961

[4] W Leontief Input-output Economics Oxford UniversityPress Oxford UK 1966

[5] J D Sargan ldquo+e instability of the Leontief dynamic modelrdquoEconometrica vol 26 no 3 pp 381ndash392 1958

[6] W Leontief ldquo+e dynamic inverserdquo in Proceedings of theContributions to input-output analysis proceedings of thefourth international conference on input-output techniquesNorth Holland Geneva Amsterdam January 1970

[7] A Brody and A P Carter ldquoInputcoutput techniquesrdquo inProceedings of the Fifth International Conference on

inputCoutput Techniques North Holland Geneva Amster-dam January 1972

[8] A P Schinnar ldquo+e Leontief dynamic generalized inverserdquo3eQuarterly Journal of Economics vol 92 no 4 pp 641ndash6521978

[9] J Tsukui and Y Murakami Turnpike Optimality in Input-Output Systems North Holland Geneva Amsterdam 1979

[10] R S Preston ldquo+e wharton long-term model input-outputwithin the context of a macro forecastingmodelrdquo EconometricModel Performance Comparative Simulations of the USeconomy pp 271ndash287 University of Pennsylvania PressPhiladelphia PA USA 1976

[11] W Leontief A P Carter and P A Petri 3e Future of theWorld Economy A United Nations Study Oxford UniversityPress Oxford UK 1977

[12] V Bulmer-+omas InputCoutput Analysis in DevelopingCountries Sources Methods and Applications Wiley Hobo-ken NJ USA 1982

[13] W Leontief and F Duchin3e Future Impact of Automationon Workers Oxford University Press Oxford UK 1985

[14] W Peterson Advances in Input-Output analysisTechnologyPlanning and Development Oxford University Press OxfordUK 1991

[15] R E Miller and P D Blair Input-output Analysis Founda-tions and Extensions Prentice-Hall Englewood Cliffs NJUSA 1985

[16] L-K Hua ldquoOn the mathematical theory of globally optimalplanned economic systemsrdquo Proceedings of the NationalAcademy of Science USA vol 81 no 20 pp 6549ndash6553 1984

[17] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (I)rdquo Chinese Science Bulletin vol 29no 12 pp 6ndash11 1984 in Chinese

[18] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (II-III)rdquo Chinese Science Bulletin vol 29no 13 pp 769ndash772 1984 in Chinese

[19] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IV-VI)rdquo Chinese Science Bulletinvol 29 no 16 pp 961ndash965 1984 in Chinese

[20] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VII)rdquo Chinese Science Bulletin vol 29no 18 pp 1089ndash1092 1984 in Chinese

[21] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (VIII)rdquo Chinese Science Bulletin vol 29no 21 pp 1281-1282 1984 in Chinese

[22] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (IX)rdquo Chinese Science Bulletin vol 30no 1 pp 1-2 1985 in Chinese

[23] L-K Hua ldquoMathematical theory of large-scale optimizationin planned economy (X)rdquo Chinese Science Bulletin vol 30no 9 pp 641ndash645 1985 in Chinese

[24] O Blanchard3eNeed for Different Classes ofMacroeconomicModels Blog Post Peterson Institute for International EconomicsWashington DC USA 2017 httpspiiecomblogs

[25] A P Schinnar ldquoA multidimensional accounting model fordemographic and economic planning interactionsrdquo Envi-ronment and Planning A Economy and Space vol 8 no 4pp 455ndash475 1976

[26] G Chinchilnisky G Heal and A Beltratti ldquo+e green goldenrulerdquo Economics Letters vol 49 pp 174ndash179 1995

[27] G Chinchilnisky ldquoWhat is sustainable developmentrdquo LandEconomics vol 73 pp 476ndash491 1997

[28] C Cinzia J Raja and M Simone ldquoMulti-criteria decisionanalysis with goal programming in engineering management

Discrete Dynamics in Nature and Society 9

and social sciences a state-of-the art reviewrdquo Annals ofOperations Research vol 251 2015

[29] R M Solow and P A Samuelson ldquoBalanced growth underconstant returns to scalerdquo Econometrica vol 21 no 3pp 412ndash424 1953

[30] A H Roger and R Johnson Matrix Analysis CambridgeUniversity Press London UK 2nd edition 2013

10 Discrete Dynamics in Nature and Society