anempiricalstudyofamathematicalmodelforinfluenceof...

Research ArticleAn Empirical Study of a Mathematical Model for Influence ofGovernment Tax on the Price Behavior and the Stability ofMarket Price

Fenglian Wang1 Chia-Huei Wu 2 and Sang-Bing Tsai 3

1School of Management Engineering Anhui Polytechnic University Anhui Wuhu 241000 China2Institute of Service Industries and Management Minghsin University of Science and Technology Hsinchu 304 Taiwan3Regional Green Economy Development Research Center School of Business Wuyi University Wuyishan 354300 China

Correspondence should be addressed to Chia-Huei Wu chiahuei530gmailcom and Sang-Bing Tsai sangbinghotmailcom

Received 21 July 2020 Accepted 18 August 2020 Published 29 September 2020

Guest Editor Yi-Zhang Jiang

Copyright copy 2020 FenglianWang et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e fierce competition among enterprises and disordered market price competition have affected the development of the wholeregional economy Government tax subsidies have always been regarded as the advantages of enterprise development but themechanism of the influence of government tax on regional economic development especially the impact on market price has notbeen explored Based on the Bertrand competition applying chaos control theory considering the government revenue rate theinfluence of government tax on the price behavior and market price stability is analyzed and then the numerical simulation iscarried out e research shows that the price adjustment of enterprises is a complicated process and government tax is helpful tothe stability of market price e government tax rate increasing can increase the stable region of price equilibrium and reduce thebifurcation and chaos of the price game system

1 Introduction

Chinarsquos economy is developing rapidly However in theprocess of the development some problems have also beenexposed [1] One concern is the plight at is enterprisecluster competition is bad and the price disorder is seriouswhich reduces the profit of the whole economy weakens theinnovation ability of the enterprise reduces the loyalty of theconsumer and then affects the development of economy [1]e market price mechanism is the soul of the market It canmaintain the stability of the market price help to curbinflation impact the real economy stabilize the marketorder and maintain the stability of the whole market Somarket price stability is a major goal of macroeconomicpolicy erefore it is imperative to study the price com-petition behavior and establish the market price stabilitymechanism

e phenomenon of economy has always been the focusof various economic schools However the theoretical

research on the price behavior and the price stability ofmarket is still weak e current research results mainlyfocus on the price level and influencing factors of the market[2ndash4] the behavioral characteristics of price monopoly[5 6 7] and the chaotic state of price competition behaviorin enterprises [8ndash12] Related studies suggest that in theprice competition the enterprise starts the game based onthe product differentiation the complex market environ-ment and the unpredictable strategy of competitors oftenlead to the unstable evolution in the price adjustmentprocess and eventually leads to the chaos of the price marketChaos theory is an important theory and method to studythe complex changes of economic evolution Some scholarsuse this theory to explore the price competition behavior andprice stability mechanism of enterprises e dynamicproperty and chaotic complexity of the price competitionmodel under different backgrounds are studied from theaspects of price adjustment mechanism cost function pricefunction price expectation method and so on

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8097402 9 pageshttpsdoiorg10115520208097402

At present there are mainly two aspects in this field efirst aspect is the study of price competition behavior of en-terprises of equal status For example Zhang et al [13] studiedtwo using the same finite rational price adjustment strategies ofoligopoly enterprises price competition process and pointedout that the change of the price adjustment speed will changethe stability of Nash equilibrium Long and Zhao [14] con-structed a duopoly price competition game model with clusterspillover in the enterprise cluster analyzed the enterprise pricecompetition behavior in the enterprise cluster and studied theimpact of cluster spillover on the price equilibrium of thediscrete dynamic system Liao [15] established a numericalsimulation model of the price competition chaotic dynamicalsystem and analyzed the complex system of price behavioresecond aspect is the study of the price competition behavior ofthe principal subordinate enterprises For example Xin andChen [16] established a Stackelberg game system based on theprice competition relationship between tap water and purifiedwater enterprises studied the price evolution process of thesystem and proposed that the improved stable linearizationmethod can control the occurrence of chaos Lu [17] studiedthe dynamic nature of the Stackelberg game model based onthe price adjustment mechanism of bounded rationalitystrategy and adaptive strategy for oligopoly enterprises andaffirmed the important role of price adjustment speed to theprice equilibrium stability Gao and Liu [18] applied chaoscontrol theory to realize adaptive fuzzy optimal control of pricemarket

Looking at the existing literature the current researchstudies on the equilibrium and stability of price competitionare mostly based on the oligarch environment in the pancompetitive market And the conclusions are relativelysimple most of them only emphasize the influence of priceadjustment speed on price stability lack of the research onthe impact of other parameters on the market price equi-librium stability and more lack of the study on the systemmechanism establishment of the market price stability ispaper studies the competitive behavior of market price andstability can reveal the process mechanism of price game toachieve Nash equilibrium in the intensive market finds thestrategy to realize the stability of market price and betterpromote the development of the economy

In order to promote economic development variouscountries have adopted government tax policies in succes-sion However the tax rate of different countries is differentSince Chinarsquos reform and opening up the tax rate changesshow a V-shaped trajectory [19 20] So it is necessary tostudy the government policy to explore the appropriate levelof the current tax rate At present the research on gov-ernment tax policy is mainly focused on the role of tax policyand how to establish a reasonable tax policy For example Jiaand Ying [21] pointed out how to stimulate enterprise de-velopment and promote the kinetic energy of the rapidgrowth of the enterprise is to present China ldquosupply sidereformsrdquo and the core of fiscal policy transformation thegovernment tax policy is a very important policy choice andtax policy has good flexibility and very direct role in thisprocess Cleeve [22] Chang and Choi [23] Choi et al [24]Wang and Yu [25] and so on proved that fiscal subsidies or

tax incentives can promote the RampD innovation of enter-prises Alonso-Carrera and Raurich [26] examined thechange of industrial structure by introducing the minimumconsumption demand analyzed a multisectoral endogenousgrowth model under the government tax policy and pointedout that the price effect drives the upgrading of the industrialstructure Room and Cisneros Ornberg [27] by studying thelessons of marijuana from the experience of alcohol mo-nopoly concluded that public monopoly is generally a betterchoice for public health and welfare Abbasian and Souri[28] studied the inefficient behavior of government policiescaused by the rise of energy prices in Iran quantified theconsumer subsidies by using the price gap method andstudied the impact of energy price rise Chen et al [29]studied the incentive mechanism of carbon emission taxpolicy for multinational or cross regional platform enter-prises and proposed that a product with high price elasticityand carbon emission intensity will not only hinder enter-prises to obtain higher income but also reduce the fairness ofthe system under the unchanged emission control policies

Combining the existing literature we find the currentresearch on the governmentrsquos tax policy lack of in-depth studyon the appropriate level of tax rates and lack of the researchon the influence of government tax on market stability ispaper will study the tax influence on enterprise competitionprice and the market price stability mechanism and processexplore the government tax on the important role of themarket price stability find the realization of market pricestability strategy and explore on the basis of the governmentthe appropriate tax rate e paper wish provides theoreticalreference for the government to formulate relevant policiesand tax policies to stabilize the market price

2 The Model

We consider a Bertrand-type duopoly market where twooligopolies choose different prices for their heterogeneousproducts Players can decide the prices according to theadjustment rules [30]

Hypothesis 1 Let pi represent the price of firm i at discretetime periods t 0 1 2 and qi represent the outputFollowing Zhang et al [13] suppose the market demandfunction of the players is

qi a minus bpi + dpj (1)

where agt 0 bgt 0 dgt 0 i j 1 2 ine j e parameter d

measures the degree of substitution of the two productsLarge d represents big degree of substitution

Hypothesis 2 Positive parameter ci is the marginal cost offirm i e cost function of enterprise i is

Ci ciqi (2)

Hypothesis 3 e government revenue is based on the salesrevenue of enterprisesrsquo products and the tax rate isr(0le rlt 1)

2 Mathematical Problems in Engineering

en the profit function of enterprise i is

πi piqi minus ciqi minus rpiqi (3)

From the profit maximization by player i the marginalprofits in period t are obtained as follows

zπi

zpi

1113888 1113889 d(1 minus r)pj minus 2b(1 minus r)pi +(1 minus r)a + bci (4)

en the optimal price response function of firm i canbe given by

pi d(1 minus r)pj +(1 minus r)a + bci

2b(1 minus r) (5)

Information in the market usually is incomplete Sup-pose players use different expectations to adjust the pricesFollowing Zhang et al [13] suppose player 1 is boundedlyrational [13] and player 2 is naive

Boundedly rational player 1 makes its price decisionbased on an estimate of the marginal profit zπ1zp1 [11]Namely it decides to increase its price p1 if it has a positivemarginal profit or decreases its price when the marginalprofit is negative en the dynamical equation of player 1can be given by

p1(t + 1) p1(t) + kp1(t) d(1 minus r)p2(t) minus 2b(1 minus r)p1(t)1113858

+(1 minus r)a + bc11113859

(6)

where k is a positive parameter which reflects the speed ofprice adjustment

Naive player 2 makes its price decision according to thenaive expectations rule [8] e player 2 decides its priceswith his reaction function Hence the dynamic equation ofthe naive expectation player 2 can be given by

p2(t + 1) d(1 minus r)p1(t) +(1 minus r)a + bc2

2b(1 minus r) (7)

With the above assumptions the duopoly game withheterogeneous players is formed from combining equations(6) and (7) en the dynamical system of the heteroge-neous players is described as

p1(t + 1) p1(t) + kp1(t) d(1 minus r)p2(t)1113858

minus 2b(1 minus r)p1(t) +(1 minus r)a + bc11113859


2b(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

3 Model Analysis

31 Nash Equilibrium and Local Stability In this part theequilibria points of the dynamic system will be first studied(8) and then the stability will be discussed

e dynamic duopoly game will achieve a Nash Equi-librium at laste possible equilibrium point of map (8) can

be obtained as nonnegative solution of the nonlinear alge-braic system [30]

kp1 d(1 minus r)p2 minus 2b(1 minus r)p1 +(1 minus r)a + bc11113858 1113859 0

p2 d(1 minus r)p1 +(1 minus r)a + bc2

2b(1 minus r)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)

After the calculation of the system it was found that themap has two equilibrium points

E1 0 p021113872 1113873

E2 plowast1 plowast2( 1113857

(10)

where

p02

(1 minus r)a + bc22b(1 minus r)

plowast1

d (1 minus r)a + bc21113858 1113859 + 2b (1 minus r)a + bc11113858 1113859

(1 minus r) 4b2

minus d2

1113872 1113873

plowast2


(1 minus r) 4b2

minus d2

1113872 1113873

(11)

In the traditional economic view nonnegative equilib-rium is meaningful Obviously E1 is a boundary equilibria(p0

2 gt 0) E2is the unique Nash equilibrium point and haseconomic meaning provided that plowast1 gt 0 plowast2 gt 0 namely

4b2

minus d2 gt 0 (12)

In order to study the local stability of equilibrium theJacobian matrix of map (8) should be consideredematrixform is as follows

1+ k d(1 minus r)p2 minus 4b(1 minus r)p1 +(1 minus r)a + bc11113858 1113859 kd(1 minus r)p1

d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(13)

e equilibrium point is stable only when all eigenvaluesϕi(i 1 2) of the Jacobian matrix satisfy |ϕi|gt 0 Accordingto this theory the following result about E1can be received

Proposition 1 2e equilibrium point E1 of system (8) is asaddle point

Proof 1 e Jacobian matrix of E1 has the form

J E1( 1113857

1 + k d(1 minus r)p02 +(1 minus r)a + bc11113960 1113961 0

d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

Its eigenvalues are

ϕ1 1 + k d(1 minus r)p02 +(1 minus r)a + bc11113960 1113961 ϕ2 0 (15)

Mathematical Problems in Engineering 3

For the condition that a b d (1 minus r) are all positiveparameters ϕ1 gt 1 is workable en the equilibrium pointE1 is a saddle node

e proof of the proposition is completedNext the local stability of the Nash equilibrium point E2

will be studied e Jacobian matrix of E2 E2 is

J E2( 1113857

1 + k d(1 minus r)plowast2 minus 4b(1 minus r)p

lowast1 +(1 minus r)a + bc11113858 1113859 kd(1 minus r)p

lowast1

d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)

where the trace of J(E2) is

T Tr J E2( 1113857( 1113857


lowast1 +(1 minus r)a + bc11113858 1113859

(17)

e determinant of J(E2) is

D Det J E2( 1113857( 1113857 minuskd

2

2b1113888 1113889(1 minus r)p

lowast1 (18)

e characteristic equation of J(E2) is

P(ϕ) ϕ2 minus Tϕ + D (19)

e discriminant is

Δ T2

minus 4 D (20)

Since Δ T2 + (2kd2b)(1 minus r)plowast1 gt 0 the eigenvaluesof Nash equilibrium E2 are real

Necessary and sufficient conditions for local stability ofthe Nash equilibrium E2 are Juryrsquos condition which is givenby

1 + T + Dgt 0

1 minus T + Dgt 0

1 minus Dgt 0

⎧⎪⎪⎨

⎪⎪⎩(21)

Since 1 minus Dgt 0 and 1 minus T + D (2kb[(1 minus r)a + bc1] +

kd[(1 minus r)a + bc2]2b)gt 0 are always satisfied then the localstability condition of is 1 + T + Dgt 0

Since 1+T+D (4b(4b2 minus d2) minus 2kb(4b2 +d2)[(1 minus r)a+

bc1] minus kd(4b2 +d2)[(1 minus r)a+bc2] 2b(4b2 minus d2))gt0 namely

4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2

1113872 1113873 (1 minus r)a + bc11113858 1113859 minus k

d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859gt 0(22)

We can draw the following conclusion

Proposition 2 2e Nash equilibrium at E2 is stable if andonly if inequality (22) holds

Proposition 2 characterizes the stability region in whichthe Nash equilibrium E2 is local stable e violation ofinequality (22) will lead to a flip bifurcation

32Analysis of the Influence ofVariables onEquilibriumPointStability

3212e Influence of Price Adjustment Speed on EquilibriumPoint Stability rough (21) the condition about adjust-ment speed k can be derived

klt4b 4b

2minus d

21113872 1113873

2b 4b2

+ d2

1113872 1113873 (1 minus r)a + bc11113858 1113859 + d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859

(23)

Proposition 3 2e evolution of price system (8) is in a stablestate and E2 is the Nash equilibrium point when klt k0 2eprice system (8) undergoes a flip bifurcation at E2 whenk k0 While the price system bifurcates from E2 E2 whenkgt k0 where k0 (4b(4b2 minus d2)2b(4b2 + d2)

[(1 minus r)a + bc1] + d(4b2 + d2)[(1 minus r)a + bc2])

Proof 2 According to the stability theory of Juryrsquos condi-tion the flip bifurcation occurs when 1 + T + D 0Namely

4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(24)

en k k0So the Nash equilibrium point will lose stability when

k k0 e system is in stable when klt k0 and bifurcateswhen kgt k0

From Proposition 3 it is known that it will cause thebifurcation and chaotic of the price evolution to take a largeprice adjustment speed for boundedly rational player 1 Sokeeping low price adjustment speed is beneficial to obtain asteady state and the Nash equilibrium profit

Noticing that the stability region is associated with rIn the same way the similar propositions can be given

about the tax rate re influence of price adjustment speed on the discrete

price system is investigated and (22) can be transformedinto a price discrete system

rgt 1 minus4b 4b

2minus d

21113872 1113873 minus kb 2bc1 + dc2( 1113857 4b

2+ d

21113872 1113873

ka(2b + d)d 4b2

+ d2

1113872 1113873 (25)


Proposition 4 When the government tax r meets condition(25) the price evolution will be in a stable state which is theequilibrium price Otherwise the evolution will be in chaos orbifurcation

Proof According to the stability theory of Juryrsquos conditionthe flip bifurcation occurs when 1 + T + D 0 Namely

4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(26)

en r 1 minus (4b(4b2 minus d2) minus kb (2bc1 + dc2)(4b2 + d2)ka(2b + d)d(4b2 + d2))

at is to say the government tax is conducive to thestability of the market price and the higher tax rate is moreconducive to the realization of the stable and balancedmarket price

From the above description and Propositions 3-4 it canbe concluded that high tax rate is beneficial to obtain a steadystate and the Nash equilibrium profit It can expand thestable region and enhance the stability of the product price ofmarket to increase the tax rate

4 Numerical Simulation and Analysis

Because the discrete dynamic system does not have analyticsolution this section will study the evolutionary charac-teristics of duopoly game dynamic system (8) by numericalsimulation and provide some numerical evidences to proveabove results In MATLAB programming given thenumber of iterations is N 500 the other parameters aresatisfied under (12) conditions e influence of priceadjustment coefficient and government tax ratio on theprice behavior of enterprises in market is investigated[30ndash34]

41 2e Market Price Evolution Situation about PriceAdjustment Speed

411 2e Influence of Price Adjustment Speed Change on theStability of Market Price When a 2 b 11 d 2 c1 1

c2 2 and r 02 the price game system is (27) and theevolution of price game is shown in Figure 1

p1(t + 1) p1(t) + kp1(t) 16p2(t) minus 176p1(t) + 271113858 1113859

p2(t + 1) 16p1(t) + 38

176

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)

Figure 1 is a dynamic evolution diagram of discretedynamical systems with respect to price adjustment speed Itis not difficult to find that when 0le klt 0031 the productprice of oligopoly enterprises is in the stable state and tendsto equilibrium (plowast1 plowast2 ) (20149 20476) Whenkge 0031 the price is in the period doubling bifurcation andchaos and the market price is in the unstable state

412 2e Stability of Market Price under a High Price Ad-justment Speed Let a 2 b 11 d 2 c1 1 c2 2 and

and r 02 and the price evolution of the chaotic attractordiagram is shown in Figure 2

Figure 2 is a chaotic attractor when the price discretesystem passes through 50 iterations when k 004

At this point the price evolution is in chaos and themarket price is in the unstable state

42 2e Market Price Evolution about the Government TaxRate

421 2e Influence of the Government Tax Ratio on theStability of Market Price e price game system is (28) andthe evolution of price game is shown in Figure 3

Figure 3 is the dynamic evolution diagram of the discretedynamic system with respect to the government tax rateWhen 0568lt rlt 1 the product price of oligopoly enterpriseis in the stable state and tends to equilibrium(plowast1 p lowast2 ) (28775 29381) When 0le rle 0568 the priceis in period doubling bifurcation and chaos

p1(t + 1) p1(t) + 004p1(t) 2(1 minus r)p2(t)1113858

minus 22(1 minus r)p1(t) + 2(1 minus r) + 111113859

p2(t + 1) p1(t) + 1

11+

1(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

422 e Stability of Market Price under a smallGovernment Tax Rate

Let a 2 b 11 d 2 c1 1 c2 2 k 004 the priceevolution chaotic attractor graph when r 02 is shown inFigure 4

Figure 4 is a chaotic attractor when the price discretesystem passes through 50 iterations in r 02 At this pointthe price evolution is in chaos

In addition the price evolution curve under differentinitial values of r 02 is shown in Figure 5 Figure 5 is theevolutionary curve of the firmrsquos 1 price at initial values(10 15) and (10001 15)

It is not difficult to find that the price evolution is verysensitive to the initial value Small initial value differencemakes the curve obviously separate e enterprises 2 and 1are similar and the whole price evolution system is in chaos

432eEffect of Government TaxRatio on the Stable Region ofPrice Adjustment Speed Make a 2 b 11 d 2 c1 1

and c2 2 a stable regional map of price adjustment speedon government tax rates as shown in Figure 6 In Figure 6with the increase of the government tax rate the stableregion of price adjustment gradually increases and themarket price is easier to achieve stability

In addition compared with Figures 1 and 3 in Figure 1under the numerical example r 02 the price discretesystem is in bifurcation or chaos when kge 0031 and theprice system is in bifurcation state at k 004


30

28

26

24

22

20

18

16

14

120

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

0005 001 0015 002 0025k

003 0035 004

p1

p2

Figure 1 Bifurcation diagram with respect to k

10 12 14 16 18k = 004p1

20 22 24

p2

20

22

24

18

16

14

12

10

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

Figure 2 Strange attractor when k 004

r02 03 04 05 06 07 08

55

50

45

40

35

30

25

20

15

10

p1

p2

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004

Figure 3 Bifurcation diagram with respect r when a 2 b 11 d 2 c1 1 c2 2 and k 004


r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02

Figure 5 Price evolution curve of player 1 on difference initial conditions

r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2

Figure 6 Region of stability in the plane (r k)

r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004

Figure 4 Strange attractor when r 02


However in Figure 3 as long as the government tax ratesto meet the stability conditions of 0568lt rlt 1 k 004 is inthe stable region and the price dispersion system is in thesteady state

Based on the theoretical analysis of the upper section andthe numerical simulation results in this section it is notdifficult to find that the government tax ratio plays animportant role in the chaotic control of the discrete pricedynamic system By increasing the government tax rate thestable region can be increased and the original chaotic orbifurcation price system tends to be stable at is to say thegovernment tax is conducive to the stability of the marketprice

5 Conclusion

In the market there is price competition between the du-opoly enterprises which produce substitutable products edynamic price game strategy may lead to the chaotic state ofthe price market and the government tax plays an importantrole in improving the stability of the price game systemResearch shows that (1) when the price adjustment speed ofenterprises with GD strategy is less than a critical value theprice game system of oligopoly enterprises will be in a stablestate and the market price will be stable (2) when thegovernment tax rate is greater than a critical value the pricegame system of oligopoly enterprises will be in a stable stateand the market price will be stable (3) the government taxcan effectively improve the stability of the price system andcontrol the chaotic phenomenon of the price market at isto say the higher government tax rate is conducive to obtaina larger stable region and realize the stable equilibrium of themarket price

e results of this paper have important theoreticalsignificance on how to stabilize the market price First thesmaller price adjustment speed is more favorable to stabilizethe market price so the government can limit the priceadjustment speed of enterprises Second the government taxis conducive to the stability of the market price so thegovernment can use the government tax policy lever toimprove the stability of the market price

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors have declared that they no conflicts of interestregarding the publication of the article

Acknowledgments

is study was supported by National Social Science Fund ofChina (16BGL201) Youth Project of Natural ScienceFoundation of Anhui Province (2008085QG346) GeneralProject of Philosophy and Social Sciences Planning in AnhuiProvince (AHSKY2019D022) and Pre-research project ofNational Natural Science Foundation of China (2019yyzr07)

References

[1] L Dong S Chen Y Cheng Z Wu C Li and H WuldquoMeasuring economic activity in China with mobile big datardquoEPJ Data Science vol 6 p 29 2017

[2] K Sakurama ldquoDistributed flow network control with demandresponse via price adjustmentrdquo Neurocomputing vol 270no SI pp 34ndash42 2017

[3] F Wang Z Yin and J Gan ldquoExchange-rate fluctuation andpricing behavior in Chinarsquos wood-based panel exportersevidence from panel datardquo Canadian Journal of Forest Re-search vol 47 no 10 pp 1392ndash1404 2017

[4] M-K Lee and J-H Kim ldquoPricing of defaultable options withmultiscale generalized Hestonrsquos stochastic volatilityrdquo Math-ematics and Computers in Simulation vol 144 pp 235ndash2462018

[5] G Zhang and Y Chen ldquoResearch on the vicious pricecompetition among enterprises in industrial clustersrdquo Eco-nomic Mathematics vol 30 no 1 pp 12ndash16 2013

[6] S Chae and J Song ldquoPrice competition between random andassortive matchmakersrdquoMathematical Social Sciences vol 90no SI pp 63ndash72 2017

[7] D Crapis B Ifrach C Maglaras and M Scarsini ldquoMonopolypricing in the presence of social learningrdquo ManagementScience vol 63 no 11 pp 3586ndash3608 2017

[8] A K Naimzada and F Tramontana ldquoDynamic properties of aCournot-Bertrand duopoly game with differentiated prod-uctsrdquo Economic Modelling vol 29 no 4 pp 1436ndash1439 2012

[9] A A Elsadany and A E Matouk ldquoDynamic cournot duopolygame with delayrdquo Journal of Complex Systems vol 2014Article ID 384843 7 pages 2014

[10] X Zhu W Zhu and L Yu ldquoAnalysis of a nonlinear mixedCournot game with boundedly rational playersrdquo ChaosSolitons amp Fractals vol 59 pp 82ndash88 2014

[11] L Gori and M Sodini ldquoPrice competition in a nonlineardifferentiated duopolyrdquo Chaos Solitons amp Fractals vol 104pp 557ndash567 2017

[12] Z Hong C Chu L L Zhang and Y Yu ldquoOptimizing anemission trading scheme for local governments a Stackelberggame model and hybrid algorithmrdquo International Journal ofProduction Economics vol 193 pp 172ndash182 2017

[13] J Zhang Q Da and Y Wang ldquoe dynamics of Bertrandmodel with bounded rationalityrdquo Chaos Solitons amp Fractalsvol 39 no 5 pp 2048ndash2055 2009

[14] J Long and H Zhao ldquoAnalysis of the impact of clusterspillovers on duopoly Bertrand competitive price equilib-riumrdquo Scientific Research Management vol 32 no 2pp 145ndash151 2015

[15] S-J Liao ldquoOn the clean numerical simulation (CNS) ofchaotic dynamic systemsrdquo Journal of Hydrodynamics vol 29no 5 pp 729ndash747 2017

[16] B Xin and T Chen ldquoOn amaster-slave Bertrand gamemodelrdquoEconomic Modelling vol 28 no 4 pp 1864ndash1870 2011

[17] Y Lu ldquoPrincipal subordinate Bertrand price game model andits dynamics analysisrdquo System Engineering vol 1 pp 91ndash942012

[18] Y Gao and Y-J Liu ldquoAdaptive fuzzy optimal control usingdirect heuristic dynamic programming for chaotic discrete-time systemrdquo Journal of Vibration and Control vol 22 no 2pp 595ndash603 2016

[19] B Xin ldquoe rational choice of tax burden rate in China fromthe perspective of effective governmentrdquoManagement Worldvol 1 no 12 pp 24ndash33 2005


[20] J Jia ldquoTax incentives corporate effective average tax rates andfirm entryrdquo Economic Research vol 1 no 7 pp 94ndash109 2014

[21] J Jia and S Ying ldquoFiscal decentralization and corporate taxincentives an analysis based on the perspective of localgovernment competitionrdquo Industrial Economy of Chinavol 1 no 10 pp 23ndash39 2016

[22] E Cleeve ldquoHow effective are fiscal incentives to attract FDI toSub- Saharan Africardquo 2e Journal of Developing Areasvol 42 no 1 pp 135ndash153 2008

[23] J Y Chang and J N Choi ldquoe dynamic relation betweenorganizational and professional commitment of highly edu-cated research and development (RampD) professionalsrdquo 2eJournal of Social Psychology vol 147 no 3 pp 299ndash315 2007

[24] H Choi Q Han and J Yang ldquoGovernment pressure taxincentives and corporate RampD investmentrdquo Studies in Scienceof Science vol 33 no 12 pp 1828ndash1838 2015

[25] X Wang and H Yu ldquoGovernment subsidies tax preferencesand enterprise RampD investmentmdashbased on dynamic panelsystem GMM analysisrdquo Technoeconomics amp ManagementResearch vol 1 no 4 pp 92ndash96 2017

[26] J Alonso-Carrera and X Raurich ldquoDemand-based structuralchange and balanced economic growth Raurichdemandmdashbased structural change and balanced economicgrowthrdquo Journal of Macroeconomics vol 46 pp 359ndash3742015

[27] R Room and J Cisneros Ornberg ldquoGovernment monopoly asan instrument for public health and welfare lessons forcannabis from experience with alcohol monopoliesrdquo Inter-national Journal of Drug Policy vol 74 pp 223ndash228 2019

[28] E Abbasian and A Souri ldquoe inefficiency of energy pricingpolicy the case of Iranrdquo International Journal of Environ-mental Research vol 13 no 6 pp 943ndash950 2019

[29] D Chen J Ignatius D Sun M Goh and S Zhan ldquoPricingand equity in cross-regional green supply chainsrdquo EuropeanJournal of Operational Research vol 280 no 3 pp 970ndash9872020

[30] F Wang B Wang and R Xie ldquoChaotic dynamics in Bertrandmodel with technological innovationrdquo VibroengineeringPROCEDIA vol 15 pp 134ndash140 2017

[31] G Khatwani and P R Srivastava ldquoImpact of informationtechnology on information search channel selection forconsumersrdquo Journal of Organizational and End User Com-puting vol 30 no 3 pp 63ndash80 2018

[32] L Fabisiak ldquoWeb service usability analysis based on userpreferencesrdquo Journal of Organizational and End User Com-puting vol 30 no 4 pp 1ndash13 2018

[33] A Shahri M Hosseini K Phalp J Taylor and R Ali ldquoHow toengineer gamificationrdquo Journal of Organizational and EndUser Computing vol 31 no 1 pp 39ndash60 2019

[34] T Grubljesic P S Coelho and J Jaklic ldquoe shift to socio-organizational drivers of business intelligence and analyticsacceptancerdquo Journal of Organizational and End User Com-puting (JOEUC) vol 31 no 2 pp 37ndash64 2019


At present there are mainly two aspects in this field efirst aspect is the study of price competition behavior of en-terprises of equal status For example Zhang et al [13] studiedtwo using the same finite rational price adjustment strategies ofoligopoly enterprises price competition process and pointedout that the change of the price adjustment speed will changethe stability of Nash equilibrium Long and Zhao [14] con-structed a duopoly price competition game model with clusterspillover in the enterprise cluster analyzed the enterprise pricecompetition behavior in the enterprise cluster and studied theimpact of cluster spillover on the price equilibrium of thediscrete dynamic system Liao [15] established a numericalsimulation model of the price competition chaotic dynamicalsystem and analyzed the complex system of price behavioresecond aspect is the study of the price competition behavior ofthe principal subordinate enterprises For example Xin andChen [16] established a Stackelberg game system based on theprice competition relationship between tap water and purifiedwater enterprises studied the price evolution process of thesystem and proposed that the improved stable linearizationmethod can control the occurrence of chaos Lu [17] studiedthe dynamic nature of the Stackelberg game model based onthe price adjustment mechanism of bounded rationalitystrategy and adaptive strategy for oligopoly enterprises andaffirmed the important role of price adjustment speed to theprice equilibrium stability Gao and Liu [18] applied chaoscontrol theory to realize adaptive fuzzy optimal control of pricemarket

Looking at the existing literature the current researchstudies on the equilibrium and stability of price competitionare mostly based on the oligarch environment in the pancompetitive market And the conclusions are relativelysimple most of them only emphasize the influence of priceadjustment speed on price stability lack of the research onthe impact of other parameters on the market price equi-librium stability and more lack of the study on the systemmechanism establishment of the market price stability ispaper studies the competitive behavior of market price andstability can reveal the process mechanism of price game toachieve Nash equilibrium in the intensive market finds thestrategy to realize the stability of market price and betterpromote the development of the economy

In order to promote economic development variouscountries have adopted government tax policies in succes-sion However the tax rate of different countries is differentSince Chinarsquos reform and opening up the tax rate changesshow a V-shaped trajectory [19 20] So it is necessary tostudy the government policy to explore the appropriate levelof the current tax rate At present the research on gov-ernment tax policy is mainly focused on the role of tax policyand how to establish a reasonable tax policy For example Jiaand Ying [21] pointed out how to stimulate enterprise de-velopment and promote the kinetic energy of the rapidgrowth of the enterprise is to present China ldquosupply sidereformsrdquo and the core of fiscal policy transformation thegovernment tax policy is a very important policy choice andtax policy has good flexibility and very direct role in thisprocess Cleeve [22] Chang and Choi [23] Choi et al [24]Wang and Yu [25] and so on proved that fiscal subsidies or

tax incentives can promote the RampD innovation of enter-prises Alonso-Carrera and Raurich [26] examined thechange of industrial structure by introducing the minimumconsumption demand analyzed a multisectoral endogenousgrowth model under the government tax policy and pointedout that the price effect drives the upgrading of the industrialstructure Room and Cisneros Ornberg [27] by studying thelessons of marijuana from the experience of alcohol mo-nopoly concluded that public monopoly is generally a betterchoice for public health and welfare Abbasian and Souri[28] studied the inefficient behavior of government policiescaused by the rise of energy prices in Iran quantified theconsumer subsidies by using the price gap method andstudied the impact of energy price rise Chen et al [29]studied the incentive mechanism of carbon emission taxpolicy for multinational or cross regional platform enter-prises and proposed that a product with high price elasticityand carbon emission intensity will not only hinder enter-prises to obtain higher income but also reduce the fairness ofthe system under the unchanged emission control policies

Combining the existing literature we find the currentresearch on the governmentrsquos tax policy lack of in-depth studyon the appropriate level of tax rates and lack of the researchon the influence of government tax on market stability ispaper will study the tax influence on enterprise competitionprice and the market price stability mechanism and processexplore the government tax on the important role of themarket price stability find the realization of market pricestability strategy and explore on the basis of the governmentthe appropriate tax rate e paper wish provides theoreticalreference for the government to formulate relevant policiesand tax policies to stabilize the market price

2 The Model

We consider a Bertrand-type duopoly market where twooligopolies choose different prices for their heterogeneousproducts Players can decide the prices according to theadjustment rules [30]

Hypothesis 1 Let pi represent the price of firm i at discretetime periods t 0 1 2 and qi represent the outputFollowing Zhang et al [13] suppose the market demandfunction of the players is

qi a minus bpi + dpj (1)

where agt 0 bgt 0 dgt 0 i j 1 2 ine j e parameter d

measures the degree of substitution of the two productsLarge d represents big degree of substitution

Hypothesis 2 Positive parameter ci is the marginal cost offirm i e cost function of enterprise i is

Ci ciqi (2)

Hypothesis 3 e government revenue is based on the salesrevenue of enterprisesrsquo products and the tax rate isr(0le rlt 1)





zπi

zpi




2b(1 minus r) (5)




+(1 minus r)a + bc11113859

(6)




2b(1 minus r) (7)





2b(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

3 Model Analysis






2b(1 minus r)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)


E1 0 p021113872 1113873


(10)

where

p02


plowast1


(1 minus r) 4b2

minus d2

1113872 1113873

plowast2


(1 minus r) 4b2

minus d2

1113872 1113873

(11)



4b2

minus d2 gt 0 (12)



d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(13)




J E1( 1113857


d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

Its eigenvalues are






J E2( 1113857



lowast1

d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


T Tr J E2( 1113857( 1113857



(17)


D Det J E2( 1113857( 1113857 minuskd

2

2b1113888 1113889(1 minus r)p

lowast1 (18)



e discriminant is

Δ T2

minus 4 D (20)



1 + T + Dgt 0

1 minus T + Dgt 0

1 minus Dgt 0

⎧⎪⎪⎨

⎪⎪⎩(21)





4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859gt 0(22)






klt4b 4b

2minus d

21113872 1113873

2b 4b2

+ d2

1113872 1113873 (1 minus r)a + bc11113858 1113859 + d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859

(23)




4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(24)







rgt 1 minus4b 4b

2minus d

21113872 1113873 minus kb 2bc1 + dc2( 1113857 4b

2+ d

21113872 1113873

ka(2b + d)d 4b2

+ d2

1113872 1113873 (25)




4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(26)










p2(t + 1) 16p1(t) + 38

176

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)









p1(t + 1) p1(t) + 004p1(t) 2(1 minus r)p2(t)1113858


p2(t + 1) p1(t) + 1

11+

1(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)










30

28

26

24

22

20

18

16

14

120

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

0005 001 0015 002 0025k

003 0035 004

p1

p2


10 12 14 16 18k = 004p1

20 22 24

p2

20

22

24

18

16

14

12

10

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02


r02 03 04 05 06 07 08

55

50

45

40

35

30

25

20

15

10

p1

p2

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004



r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02


r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2


r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004





5 Conclusion



Data Availability




Acknowledgments


References








































zπi

zpi




2b(1 minus r) (5)




+(1 minus r)a + bc11113859

(6)




2b(1 minus r) (7)





2b(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

3 Model Analysis






2b(1 minus r)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)


E1 0 p021113872 1113873


(10)

where

p02


plowast1


(1 minus r) 4b2

minus d2

1113872 1113873

plowast2


(1 minus r) 4b2

minus d2

1113872 1113873

(11)



4b2

minus d2 gt 0 (12)



d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(13)




J E1( 1113857


d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (14)

Its eigenvalues are






J E2( 1113857



lowast1

d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


T Tr J E2( 1113857( 1113857



(17)


D Det J E2( 1113857( 1113857 minuskd

2

2b1113888 1113889(1 minus r)p

lowast1 (18)



e discriminant is

Δ T2

minus 4 D (20)



1 + T + Dgt 0

1 minus T + Dgt 0

1 minus Dgt 0

⎧⎪⎪⎨

⎪⎪⎩(21)





4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859gt 0(22)






klt4b 4b

2minus d

21113872 1113873

2b 4b2

+ d2

1113872 1113873 (1 minus r)a + bc11113858 1113859 + d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859

(23)




4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(24)







rgt 1 minus4b 4b

2minus d

21113872 1113873 minus kb 2bc1 + dc2( 1113857 4b

2+ d

21113872 1113873

ka(2b + d)d 4b2

+ d2

1113872 1113873 (25)




4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(26)










p2(t + 1) 16p1(t) + 38

176

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)









p1(t + 1) p1(t) + 004p1(t) 2(1 minus r)p2(t)1113858


p2(t + 1) p1(t) + 1

11+

1(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)










30

28

26

24

22

20

18

16

14

120

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

0005 001 0015 002 0025k

003 0035 004

p1

p2


10 12 14 16 18k = 004p1

20 22 24

p2

20

22

24

18

16

14

12

10

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02


r02 03 04 05 06 07 08

55

50

45

40

35

30

25

20

15

10

p1

p2

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004



r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02


r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2


r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004





5 Conclusion



Data Availability




Acknowledgments


References








































J E2( 1113857



lowast1

d

2b0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


T Tr J E2( 1113857( 1113857



(17)


D Det J E2( 1113857( 1113857 minuskd

2

2b1113888 1113889(1 minus r)p

lowast1 (18)



e discriminant is

Δ T2

minus 4 D (20)



1 + T + Dgt 0

1 minus T + Dgt 0

1 minus Dgt 0

⎧⎪⎪⎨

⎪⎪⎩(21)





4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859gt 0(22)






klt4b 4b

2minus d

21113872 1113873

2b 4b2

+ d2

1113872 1113873 (1 minus r)a + bc11113858 1113859 + d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859

(23)




4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(24)







rgt 1 minus4b 4b

2minus d

21113872 1113873 minus kb 2bc1 + dc2( 1113857 4b

2+ d

21113872 1113873

ka(2b + d)d 4b2

+ d2

1113872 1113873 (25)




4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(26)










p2(t + 1) 16p1(t) + 38

176

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)









p1(t + 1) p1(t) + 004p1(t) 2(1 minus r)p2(t)1113858


p2(t + 1) p1(t) + 1

11+

1(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)










30

28

26

24

22

20

18

16

14

120

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

0005 001 0015 002 0025k

003 0035 004

p1

p2


10 12 14 16 18k = 004p1

20 22 24

p2

20

22

24

18

16

14

12

10

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02


r02 03 04 05 06 07 08

55

50

45

40

35

30

25

20

15

10

p1

p2

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004



r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02


r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2


r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004





5 Conclusion



Data Availability




Acknowledgments


References







































4b 4b2

minus d2

1113872 1113873 minus 2kb 4b2

+ d2


d 4b2

+ d2

1113872 1113873 (1 minus r)a + bc21113858 1113859 0(26)










p2(t + 1) 16p1(t) + 38

176

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(27)









p1(t + 1) p1(t) + 004p1(t) 2(1 minus r)p2(t)1113858


p2(t + 1) p1(t) + 1

11+

1(1 minus r)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)










30

28

26

24

22

20

18

16

14

120

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

0005 001 0015 002 0025k

003 0035 004

p1

p2


10 12 14 16 18k = 004p1

20 22 24

p2

20

22

24

18

16

14

12

10

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02


r02 03 04 05 06 07 08

55

50

45

40

35

30

25

20

15

10

p1

p2

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004



r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02


r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2


r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004





5 Conclusion



Data Availability




Acknowledgments


References





































30

28

26

24

22

20

18

16

14

120

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02

0005 001 0015 002 0025k

003 0035 004

p1

p2


10 12 14 16 18k = 004p1

20 22 24

p2

20

22

24

18

16

14

12

10

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 r = 02


r02 03 04 05 06 07 08

55

50

45

40

35

30

25

20

15

10

p1

p2

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004



r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02


r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2


r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004





5 Conclusion



Data Availability




Acknowledgments


References





































r = 02N1 10 20 30

20

19

18

17p1

16

15

14

13

p1(0) = 10p1(0) = 10001

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004 r = 02


r0 01 03 05 07 102 04 06 08 09

007

006

005

004

003k

002

001

0

a = 2 b = 11 d = 2 c1 = 1 c2 = 2


r = 01p1

12 13 14 15 16 17 18 19

p2

20

19

18

17

16

15

14

13

a = 2 b = 11 d = 2 c1 = 1 c2 = 2 k = 004





5 Conclusion



Data Availability




Acknowledgments


References







































5 Conclusion



Data Availability




Acknowledgments


References





































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