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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment

Shi Yin
/ Baizhou Li
Published Online: 2018-06-14 | DOI: https://doi.org/10.1515/math-2018-0056

Abstract

Considering the fact that the development of low carbon economy calls for the low carbon technology sharing between interested enterprises, this paper study a stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises. In the paper, we consider the random interference factors that include the uncertain external environment and the internal understanding limitations of decision maker. In the model, superior enterprises and inferior enterprises are separated entities, and they play Stacklberg master-slave game, Nash non-cooperative game, and cooperative game, respectively. We discuss the feedback equilibrium strategies of superior enterprises and inferior enterprises, and it is found that some random interference factors in sharing system can make the variance of improvement degree of low carbon technology level in the cooperation game higher than the variance in the Stackelberg game, and the result of Stackelberg game is similar to the result of Nash game. Additionally, a government subsidy incentive and a special subsidy that inferior enterprises give to superior enterprises are proposed.

MSC 2010: 90B50; 91A23; 92D25

1 Introduction

Global environment is an indivisible whole ecosystem. There seems to be rather compelling evidence that environmental pollution, resource depletion and global warming are issues that we seriously need to be concerned about today. Against this background, the development of low carbon technology has become an important support for global social and economic power. Responding to the development, low carbon technological innovation is playing a vital role in development of low carbon technology. How to achieve the low carbon technological innovation in enterprises is not only an important factor affecting regional development of low carbon economy, but also the decisive factor for enterprises to acquire sustainable competitiveness and adapt to the competitive environment of future market. However, the implementation of low carbon technological innovation requires greater cost, and enterprises are faced with a great deal of pressure on capital investment. Therefore, low carbon technology sharing has become a vital role in the development of low carbon technology. Promotion of low carbon technology sharing calls for cooperation between interested enterprises. In this paper, we present a stochastic differential game of low carbon technology sharing in innovation system of superior enterprises and inferior enterprises under uncertain environment. Our objective is to find the optimal strategy of low carbon technology sharing and explore the key factors and mechanism of low carbon technology sharing.

Game theory has been used as an effective tool to study knowledge, information and technology sharing. For example, Koessler [1] provided a simple Bayesian game model for the study of knowledge sharing; the study shows that their equilibrium is always a sequential equilibrium of the associated extensive form game with communication. In 2006, Cress and Martin [2] extended the model of Koessler to study knowledge sharing and rewards based on a game-theoretical perspective. It has been found that rewarding contributions with a cost-compensating bonus can be an effective solution at the group level. Furthermore, Bandyopadhyay and Pathak [3] modelled a game of the knowledge interaction between two teams in two separate firms; it has been found that when the degree of complementarity of knowledge is higher enough, better payoffs can be achieved if the top management enforces cooperation between the employees. Wu et al. [4] established a evolutionary game model of information sharing in network organization to analyze its dynamic evolutionary procedure. Their study showed that the key factors that affect the system’s evolution, cooperation profit, initial cost of the cooperation, are obtained and researched. Ou et al. [5] modelled a game theory model to analyze the impact of important factors for low carbon international technology transfer. Their study showed that reduction of the control fees and taxes and increases of domestic subsidies all effectively promote transfers. Xu and Xu [6] used prospect theory into evolutionary game theory to construct a perceived benefit matrix to explore the internal mechanism of low carbon technology innovation diffusion under environmental regulation; theoretical study and numerical simulation showed that increasing subsidy factor, carbon tax rate and regulatory effort can all induce enterprises to adopt low carbon technology innovation, and carbon tax rate has the strongest sensitivity. Gong and Xue [7] studied a game model of cooperative innovation between ICT low carbon developers and industrial enterprises. The authors considered that the sharing proportion played a key role in the cooperation, and the preferential tax policy of the government can coordinate the conflict in their cooperation, and the government incentive and regulatory penalty can promote both sides to improve input in both sharing arrangement modes.

Form the analysis of above studies, it is a mainstream trend that many scholars use game theory to study the sharing of low carbon technology. These studies have laid the method foundation for this article but we find that most of the game models established in the literature are based on the static framework. In fact, with the rapid development of science, technology and information, the frequency and speed of low carbon technology upgrading have also improved dramatically. It means that the dynamic behavior of decision maker should be considered in the study of low carbon technology sharing in the same spatio-temporal region. In addition, Gao and Zhong [8] used differential game approach to study the dynamic strategies for information sharing. Their study showed that the superior enterprises benefits most when both firms fully cooperate, but the inferior enterprises enjoys the highest integral profit when both firms only cooperate in information sharing and the lowest integral profit. Meanwhile, low carbon technology related research have been widely studied recently due to their potential applications. Zhao et al. [9] derive the optimal solutions of the Nash equilibrium without cost sharing contract and the Stackelberg equilibrium with the integrator as the leader who partially shares the cost of the efforts of the supplier. Their study showed that cost sharing contract is an effective coordination mechanism. Yu and Shi [10] used a stochastic differential game model to study knowledge sharing between enterprise and university. However, stochastic differential game model is seldom used in low carbon technology sharing and most of studies do not consider some random interference factors in sharing system. In fact, the process of decision making is often subject to various random interference factors that include the uncertain external environment and the internal understanding limitations of decision maker [11]. The random interference factors can lead to a great uncertainty in equilibrium results because they are difficult to capture by the decision makers [12]. In this paper, we study the low carbon technology sharing in innovation system of superior enterprises and inferior enterprises under uncertain environment in the case of stochastic intervention.

The structure of this paper is organized as follows. In Section 2, stochastic differential game formulation is provided. In Section 3, we resolve models of Stacklberg master-slave game. In Section 4, we resolve models of Nash non-cooperative game. Section 5 is devoted to models of cooperative game, and comparative analysis of equilibrium results are presented in Section 6. Section 7 summarizes the paper.

2 Stochastic differential game formulation

For the sake of simplicity, enterprises of low carbon technology sharing can be divided into two interest groups: superior enterprises and inferior enterprises, which store, respectively, large quantities of low carbon technologies and heterogeneous resources of low carbon technologies. In the paper, we study a low carbon technology innovation system that consists of a single superior enterprise C and a single inferior enterprise E. In order to clarify the above problem, we further assume that decision makers are completely rational, full information, and aim to maximize their return.

Let LC(t) denote the effort level of superior enterprises at time t, and let LE(t) denote the effort level of inferior enterprises in the sharing process of low carbon technology. For further consideration, the sharing cost of low carbon technology can be denoted by CC(t) and CE(t) which are the quadratic functions of the effort level of superior enterprises and inferior enterprises at time t, respectively. Consider

$CCLCt,t=cCt2LCt2,$(1)

$CELEt,t=cEt2LEt2,$(2)

where cC(t) and cE(t) are the cost coefficients of superior enterprises and inferior enterprises at time t, respectively.

Let K(t) denote the technology level of low carbon in collaborative innovation system of superior enterprises and inferior enterprises at time t. In the sharing process of low carbon technology, the collaborative innovation between superior enterprises and inferior enterprises can improve the technology level of low carbon. Let σC(t) and ηE(t) denote the influence of the effort level of low carbon technology sharing on collaborative innovation between superior enterprises and inferior enterprises, respectively, at time t, namely, innovation capability coefficient of low carbon technology. The dynamics of technology level of low carbon are governed by the stochastic differential equation

$K˙t=dKtdt=σtLCt+ηtLEt−δKtK0=K0≥0.$(3)

Hence

$dKt=σtLCt+ηtLEt−δKtdt+εKtdztK0=K0≥0,$(4)

where δ is the attenuation coefficient of low carbon technology, δ ∈ (0, 1]; z(t) and εK(t) are the standard Wiener process and random interference factors of superior enterprises and inferior enterprises at time t, respectively.

Let π(t) denote the total payoff of low carbon in collaborative innovation system at time t. Let α(t) and β(t) denote the influence of the effort level of low carbon technology sharing on the total income of superior enterprises and inferior enterprises, respectively, at time t, namely, the marginal return coefficient of low carbon technology. Total payoff function can be expressed as

$πt=αtLCt+βtLEt+γ+λKt,$(5)

where γ is the influence of the technology innovation of low carbon on total revenue, namely, innovation influence coefficient of low carbon technology, γ ∈ (0, 1]; λ is the government subsidy coefficient of low carbon technology based on increments of low carbon technology level in collaborative innovation, λ ∈ (0, 1].

We further assume that the total revenue is allocated between two participants, and θ(t) is the payoff distribution coefficient of superior enterprises at time t, θ(t)∈ [0, 1]. Although inferior enterprises have heterogeneous resources of low carbon technologies, superior enterprises store large quantities of low carbon technologies. Many practical low carbon technologies can be acquired by inferior enterprises in the sharing process of low carbon technology. Therefore, inferior enterprises need to pay much more extra sharing cost of low carbon technology. Let ω(t) denote the subsidy of low carbon technology, which inferior enterprises give to superior enterprises. The objective function of superior enterprises and inferior enterprises satisfy the following partial differential equations

$maxLCJCK0=E∫0∞e−ρtθtαtLCt+βtLEt+γ+λKt−1−ωtcCt2LCt2dt,$(6)

$maxLE,ωt⁡{JEK0=E∫0∞e−ρt[1−θtαtLCt+βtLEt+γ+λKt−cEt2LEt2−cCt2ωtLCt2]dt}.$(7)

where ρ is the discount rate of low carbon technology of superior enterprises and inferior enterprises, ρ ∈ (0, 1].

There are three control variables, LC(t)≥ 0, LE(t)≥ 0, ω(t) ∈ (0, 1), and a state variable K(t)≥ 0 in the sharing model of low carbon technology. Feedback control has been used more and more widely in analysis of information and economic systems [13]. Moreover, feedback control strategy has better control effect, compared with open-loop control strategy. Therefore, we use feedback control strategy to analyze sharing model of low carbon technology.

3 Resolving models of Stacklberg master-slave game

In the sharing process of low carbon technology between superior enterprises and inferior enterprises, inferior enterprises can acquire many practical low carbon technologies from superior enterprises, and then inferior enterprises need to pay much more extra sharing cost of low carbon technology. In order to promote the technology sharing of low carbon, the inferior enterprises (the leaders) determine an optimal sharing effort level and an optimal subsidy of low carbon technology sharing, and then the superior enterprises (the followers) choose their optimal sharing effort level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium.

3.1 Stacklberg master-slave solutions

Proposition 3.1

If above conditions are satisfied, the feedback Stacklberg master-slave equilibria are

$LCS=2−θαρ+δ+σγ+λ2cCρ+δ,$(8a)

$LES=1−θβρ+δ+ηγ+λcEρ+δ,$(8b)

$ω=2−3θ2−θ,0≤θ≤230,otherwise,$(8c)

where $\begin{array}{}{L}_{C}^{S}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{E}^{S}\end{array}$ are the optimal effort level of low carbon technology sharing of superior enterprises and inferior enterprises, respectively,

$VCSK=θγ+λρ+δK+θ1−θϕ2ρcEρ+δ2+θ2−θϕ14ρcCρ+δ2,$(9a)

$VESK=1−θγ+λρ+δK+1−θ2ϕ22ρcEρ+δ2+2−θ2ϕ18ρcCρ+δ,$(9b)

where $\begin{array}{}{V}_{C}^{S}\left(K\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{V}_{E}^{S}\left(K\right)\end{array}$ are the optimal sharing payoff function of low carbon technology of superior enterprises and inferior enterprises respectively, ϕ1 = [α(ρ + δ) + σ (γ + λ)]2, ϕ2 = [β(ρ + δ) + η(γ + λ)]2.

Proof

In order to obtain the Stacklberg equilibrium, there exists a optimal sharing revenue function of low carbon technology, $\begin{array}{}{V}_{C}^{S}\left(K\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{V}_{E}^{S}\left(K\right),\end{array}$ which is a continuous differentiable function. First, we use backward induction to solve optimal control problem. The optimal sharing revenue function, VC(K), satisfies the following Hamilton-Jacobi-Bellman equation

$ρVCK=maxLC≥0{θαLC+βLE+γ+λK−1−ωcC2LC2+VC′KσLC+ηLE−δK+ε2K2VC″K}.$(10)

For solving formula (10), using extreme conditions and searching for the optimal value of LC by setting the first partial derivative equal to zero, we can get

$LC=θα+σVC′K1−ωcC.$(11)

Second, the optimal sharing revenue function, VE(K), satisfies the following Hamilton-Jacobi-Bellman equation

$ρVEK=maxLE≥0⁡{1−θαLC+βLE+γ+λK−cE2LE2−cL2ωLC2+VE′KσLC+ηLE−δK+ε2K2VE″K}.$(12)

Substituting the result of (11) into (12), we can obtain

$ρVEK=maxLE≥01−θαθα+σVC′K1−ωcC+βLE+γ+λK−cE2LE2−cC2ωθα+σVC′K1−ωcC2+VE′Kσθα+σVC′K1−ωcC+ηLE−δK+ε2K2VE″K.$(13)

Performing the indicated maximization in (13) and searching for the optimal value of LE and ω by setting the first partial derivative equal to zero, we can get

$LE=1−θβ+ηVE′KcE,$(14a)

$ω=α2−3θ+σ2VE′K−VC′Kα2−θ+σ2VE′K+VC′K.$(14b)

Substituting the results of (11), (14a) and (14b) into (10) and (12), we can get

$ρVCK=θγ+λ−δVC′KK+θα+σVC′KLC−1−ωcC2LC2+θβ+ηVC′KLE+ε2K2VC″K,$(15)

$ρVEK=1−θγ+λ−δVE′KK+1−θα+σVE′KLC−cC2ωLC2+1−θβ+ηVE′KLE−cE2LE2+ε2K2VE″K.$(16)

The solution of the HJB equation is a unary function with K as independent variable. As [11], we have

$VEK=a1K+b1,VCK=a2K+b2.$(17)

where a1, b1, a2 and b2 are the constants to be solved.

Setting the first partial derivative to formula (17), we can get

$VE′K=dVEKdVEKdK=dK=a1,VC′K=dVCKdVCKdK=dK=a2.$(18)

Substituting the results of (17) and (18) into (15) and (16), we can get

$a1=θγ+λρ+δ,b1=θ2αρ+δ+σγ+λ221−ωρρ+δ2cC+θ1−θβρ+δ+ηγ+λ2ρρ+δ2cE,a2=1−θγ+λρ+δ,$(19)

$b2=1−θ2βρ+δ+ηγ+λ22ρρ+δ2cE+θ1−θαρ+δ+σγ+λ21−ωρρ+δ2cC−ωθ2αρ+δ+σγ+λ221−ω2ρρ+δ2cC.$(20)

Substituting the results of a1 and a2 into (11), (14a) and (14b), we can further get

$LCS=2−θαρ+δ+σγ+λ2cCρ+δ,$(21a)

$LES=1−θβρ+δ+ηγ+λcEρ+δ,$(21b)

$ω=2−3θ2−θ,0<θ≤230,otherwise.$(21c)

Substituting the results of (17) and (18) into (7), we can get

$VCSK=θγ+λρ+δK+θ2αρ+δ+σγ+λ22ρ1−ωρ+δ2cC+θ1−θβρ+δ+ηγ+λ2ρρ+δ2cE=θγ+λρ+δK+θ1−θϕ2ρcEρ+δ2+θ2−θϕ14ρcCρ+δ2,$(22a)

$VESK=1−θγ+λρ+δK+1−θ2βρ+δ+ηγ+λ22ρρ+δ2cE+θ1−θαρ+δ+σγ+λ21−ωρρ+δ2cC−ωθ2αρ+δ+σγ+λ221−ω2ρρ+δ2cC=1−θγ+λρ+δK+1−θ2ϕ22ρcEρ+δ2+2−θ2ϕ18ρcCρ+δ,$(22b)

where ϕ1 = [α (ρ + δ) + σ(γ + λ)]2, ϕ2 = [β(ρ + δ) + η(γ + λ)]2.

Hence, the optimal total payoff of low carbon technology sharing can be expressed as follows

$VSK=VCSK+VESK.$(23)

□

Equations (21)-(22) indicate that, under model of Stacklberg game, the effort level of superior enterprises and inferior enterprises is proportional to the government subsidy of low carbon technological innovation and the innovation capability of low carbon technology; the effort level of superior enterprises and inferior enterprises is inversely proportional to the sharing cost and the discount rate of low carbon technology; the sharing payoff of low carbon technology is proportional to the marginal return of low carbon technology.

3.2 The limit of expectation and variance

From Proposition 3.1, the payoff of superior enterprises and inferior enterprises is related to the improvement degree of low carbon technical level, whose possible values are numerical outcomes of a random phenomenon by various random interference factors. Therefore, under Stacklberg game equilibrium, it is necessary to study the limit of expectation and variance.

Substituting the results of (8a) and (8b) into (4), we can get

$dKt=μ1+μ2−δKtdt+εKtdztK0=K0≥0,$(24)

where $\begin{array}{}{\mu }_{1}=\frac{\sigma \left(2-\theta \right)\left[\alpha \left(\rho +\delta \right)+\sigma \left(\gamma +\lambda \right)\right]}{2{c}_{C}\left(\rho +\delta \right)},{\mu }_{2}=\frac{\eta \left(1-\theta \right)\left[\beta \left(\rho +\delta \right)+\eta \left(\gamma +\lambda \right)\right]}{{c}_{E}\left(\rho +\delta \right)}.\end{array}$

For further analysis, let ε (K(t))dz(t) = $\begin{array}{}\epsilon \sqrt{K}dz\left(t\right),\end{array}$ and then we can get Proposition 3.2 as follows.

Proposition 3.2

The limit of expectation and variance in Stackelberg game feedback equilibrium satisfy

$EKt=μ1+μ2δ+e−δtK0−μ1+μ2δ,limt→∞⁡EKt=μ1+μ2δ,$(25a)

$DKt=ε2μ1+μ2−2μ1+μ2−δK0e−δt+μ1+μ2−2δK0e−2δt2δ2,limt→∞⁡DKt=ε2μ1+μ22δ2.$(25b)

Proof

According to formula (24), using Itô equation, we can get

$dKt2=2μ1+μ2+ε2K−2δK2dt+2KεKdztK02=K02.$(27)

We can derive the expectation value for both sides of (24) and (27), and then E(K(t)) and E(K(t))2 satisfy the following set of non-homogeneous linear differential equations

$dEKt=μ1+μ2−δEKdtEK0=K0,$(28a)

$dEKt2=2μ1+μ2+ε2EK−2δEK2dtEK02=K02.$(28b)

Solving the above non-homogeneous linear differential equation leads to

$EKt=μ1+μ2δ+e−δtK0−μ1+μ2δ,limt→∞⁡EKt=μ1+μ2δ,$(29a)

$DKt=ε2μ1+μ2−2μ1+μ2−δK0e−δt+μ1+μ2−2δK0e−2δt2δ2,limt→∞DKt=ε2μ1+μ22δ2,$(29b)

where $\begin{array}{}{\mu }_{1}=\frac{\sigma \left(2-\theta \right)\left[\alpha \left(\rho +\delta \right)+\sigma \left(\gamma +\lambda \right)\right]}{2{c}_{C}\left(\rho +\delta \right)},{\mu }_{2}=\frac{\eta \left(1-\theta \right)\left[\beta \left(\rho +\delta \right)+\eta \left(\gamma +\lambda \right)\right]}{{c}_{E}\left(\rho +\delta \right)}.\end{array}$ □

4 Resolving models of Nash non-cooperative game

Under Nash non-cooperative game, superior enterprises and inferior enterprises will simultaneously and independently choose their optimal effort levels of low carbon technology sharing based on maximization of their profits.

4.1 Nash non-cooperative game solutions

Proposition 4.1

If above conditions are satisfied, the feedback non-cooperative game Nash equilibria are

$LCN=θαρ+δ+σγ+λcCρ+δ,$(30a)

$LEN=1−θβρ+δ+ηγ+λcEρ+δ,$(30b)

where $\begin{array}{}{L}_{C}^{N}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{E}^{N}\end{array}$ are the optimal effort level of low carbon technology sharing of superior enterprises and inferior enterprises, respectively,

$VCNK=θγ+λρ+δK+θ2ϕ12ρcCρ+δ2+θ1−θϕ2ρcEρ+δ2,$(31a)

$VENK=1−θγ+λρ+δK+1−θ2ϕ22ρcEρ+δ2+θ1−θϕ1ρcCρ+δ,$(31b)

where $\begin{array}{}{V}_{C}^{N}\left(K\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{V}_{E}^{N}\left(K\right)\end{array}$ are the optimal sharing payoff functions of low carbon technology of superior enterprises and inferior enterprises, respectively, ϕ1 = [α(ρ + δ) + σ (γ + λ)]2, ϕ2 = [β (ρ + δ) + η (γ + λ)]2.

Proof

According to sufficient conditions for static feedback equalization, there exists an optimal sharing revenue function of low carbon technology, which is a continuous differentiable function. The optimal sharing revenue function satisfies the following Hamilton-Jacobi-Bellman equation

$ρVCK=maxLC≥0{θαLC+βLE+γ+λK−1−ωcC2LC2+VC′KσLC+ηLE−δK+ε2K2VC″K},$(32a)

$ρVEK=maxLE≥0⁡{1−θαLC+βLE+γ+λK−cE2LE2−cL2ωLC2+VE′KσLC+ηLE−δK+ε2K2VE″K}.$(32b)

In order to maximize their profits, the inferior enterprises are so rational that they cannot accept the optimal subsidy of low carbon technology sharing, ω = 0. For solving formula (32a) and (32b), using extreme conditions and searching for the optimal value of LC by setting the first partial derivative equal to zero, we can get

$LCN=θα+σVC′KcC,$(33a)

$LEN=1−θβ+ηVE′KcE.$(33b)

Substituting the results of (33a) and (33b) into (32a) and (32b), we can obtain

$ρVCK=maxLC≥0θαLC+βLE+γ+λK−cCLC22+VC′KσLC+ηLE−δK+ε2K2VC″K=maxLC≥0θγ+λ−δVC′KK+θα+σVC′K22cC+θβ+ηVC′K1−θβ+ηVE′KcE+ε2K2VC″K,$(34)

$ρVEK=maxLE≥01−θαLC+βLE+γ+λK−cELE22+VE′KσLC+ηLE−δK+ε2K2VE″K=maxLE≥0⁡{1−θγ+λ−δVE′KK+1−θβ+ηVE′K22cE+1−θα+σVE′Kθα+σVC′KcC+ε2K2VE″K.$(35)

The solution of the HJB equation is a unary function with K as independent variable. As [11], we have

$VEK=a1K+b1,VCK=a2K+b2,$(36)

where a1, b1, a2 and b2 are the constants to be solved.

Substituting the result of (36) into (34) and (35), we can get

$ρa1K+b1=θαLC+βLE+γ+λK−cCLC22+VC′KσLC+ηLE−δK=θγ+λ−δa1K+θα+σa122cC+θβ+ηa11−θβ+ηa2cE,$(37)

$ρa2K+b2=1−θαLC+βLE+γ+λK−cELE22+VE′KσLC+ηLE−δK=1−θγ+λ−δa2K+1−θβ+ηa222cE+1−θα+σa2θα+σa1cC.$(38)

Using the K ≥ 0 to (37) and (38), parameter values of the optimal value function can be expressed as follows

$a1=θγ+λρ+δ,b1=θ2αρ+δ+σγ+λ22ρρ+δ2cC+θ1−θβρ+δ+ηγ+λ2ρρ+δ2cE,$(39)

$a2=1−θγ+λρ+δ,b2=1−θ2βρ+δ+ηγ+λ22ρρ+δ2cE+θ1−θαρ+δ+σγ+λ2ρρ+δ2cC−ωθ2αρ+δ+σγ+λ221−ω2ρρ+δ2cC.$(40)

Substituting the results of a1, b1, a2 and b2 into (33a), (33b) and (36), we can get the optimal effort level of low carbon technology sharing and the optimal sharing payoff function of low carbon technology of superior enterprises and inferior enterprises, respectively. □

4.2 The limit of expectation and variance

From Proposition 4.1, the payoff of superior enterprises and inferior enterprises is related to the improvement degree of low carbon technical level, whose possible values are numerical outcomes of a random phenomenon by various random interference factors. Therefore, under Nash equilibrium, it is necessary to study the limit of expectation and variance.

Substituting the results of (30a) and (30b) into (4), we can get

$dKt=ϑ1+ϑ2−δKtdt+εKtdztK0=K0≥0,$(41)

where $\begin{array}{}{\vartheta }_{1}=\frac{\sigma \theta \left[\alpha \left(\rho +\delta \right)+\sigma \left(\gamma +\lambda \right)\right]}{{c}_{C}\left(\rho +\delta \right)},{\vartheta }_{2}=\frac{\eta \left(1-\theta \right)\left[\beta \left(\rho +\delta \right)+\eta \left(\gamma +\lambda \right)\right]}{{c}_{E}\left(\rho +\delta \right)}.\end{array}$

For further analysis, let ε(K(t))dz(t) = $\begin{array}{}\epsilon \sqrt{K}dz\left(t\right),\end{array}$ and then we can get Proposition 4.2 as follows.

Proposition 4.2

The limit of expectation and variance in Nash non-cooperative game feedback equilibrium satisfy

$EK¯t=ϑ1+ϑ2δ+e−δtK0−ϑ1+ϑ2δ,limt→∞⁡EK¯t=ϑ1+ϑ2δ,$(42a)

$DK¯t=ε2ϑ1+ϑ2−2ϑ1+ϑ2−δK0e−δt+ϑ1+ϑ2−2δK0e−2δt2δ2,limt→∞⁡DK¯t=ε2ϑ1+ϑ22δ2,$(42b)

where $\begin{array}{}{\vartheta }_{1}=\frac{\sigma \theta \left[\alpha \left(\rho +\delta \right)+\sigma \left(\gamma +\lambda \right)\right]}{{c}_{C}\left(\rho +\delta \right)},{\vartheta }_{2}=\frac{\eta \left(1-\theta \right)\left[\beta \left(\rho +\delta \right)+\eta \left(\gamma +\lambda \right)\right]}{{c}_{E}\left(\rho +\delta \right)}.\end{array}$

Proof

The proof of Proposition 4.2 is similarly to Proposition 3.2, so we do not repeat it here. □

5 Resolving models of cooperative game

Under cooperative game, superior enterprises and inferior enterprises will choose their optimal effort levels and sharing payoff function of low carbon technology sharing based on maximization of their total payoff. Thus, low carbon technology level can be further improved through cooperation between superior enterprises and inferior enterprises.

5.1 Cooperative game solutions

Proposition 5.1

If the above conditions are satisfied, the feedback cooperative game equilibria are

$LCC=αρ+δ+σγ+λcCρ+δ,$(43a)

$LEC=βρ+δ+ηγ+λcEρ+δ,$(43b)

where $\begin{array}{}{L}_{C}^{C}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{E}^{C}\end{array}$ are the optimal effort level of low carbon technology sharing of superior enterprises and inferior enterprises, respectively,

$VCK=γ+λρ+δK+ϕ12ρcCρ+δ2+ϕ22ρcEρ+δ,$(44)

where VC(K) is the optimal sharing payoff function of low carbon technology of superior enterprises and inferior enterprises, ϕ1 = [α (ρ + δ) + σ (γ + λ)]2, ϕ2 = [β (ρ + δ) + η (γ + λ)]2.

Proof

Under cooperative game, the sharing revenue function satisfies the following equation

$maxLC,LE⁡{JK0=JC+JE=E∫0∞e−ρt[αtLCt+βtLEt+γ+λKt−cEt2LEt2−cCt2LCt2]dt}.$(45)

In order to obtain the cooperative equilibrium state in this case, we assume that sharing revenue function of low carbon technology is a continuous differentiable function. The optimal sharing revenue function satisfies the following Hamilton-Jacobi-Bellman equation

$ρVK=maxLC,LE⁡{αLC+βLE+γ+λK−cE2LE2−cL2LC2+V′KσLC+ηLE−δK+ε2K2V″K}.$(46)

For solving formula (46), using extreme conditions and searching for the optimal value of LC by setting the first partial derivative equal to zero, we can get

$LCC=α+σV′KcC,$(47a)

$LEC=β+ηV′KcE.$(47b)

Substituting the results of (47a) and (47b) into (46), we can obtain

$ρVK=maxLC,LEαα+σV′KcC+βLE+γ+λK−cE2β+ηV′KcE2−cL2α+σV′KcC2+V′Kσα+σV′KcC+ηβ+ηV′KcE−δK+ε2K2V″K=maxLC,LEγ+λ−δV′KK+α+σV′K22cC+β+ηV′K22cE+ε2K2V″K.$(48)

The solution of the HJB equation is a unary function with K as independent variable. As [11], we have

$VK=a1K+b1,$(49)

where a1 and b1 are the constants to be solved.

Substituting the result of (49) into (48), we can get

$ρa1K+b1=γ+λ−δV′KK+α+σV′K22cC+β+ηV′K22cE=γ+λ−δa1K+α+σa122cC+β+ηa122cE.$(50)

Using the K ≥ 0 to (50), parameter values of the optimal value function can be expressed as follows

$a1=γ+λρ+δ,b1=αρ+δ+σγ+λ22ρρ+δ2cC+βρ+δ+ηγ+λ22ρρ+δ2cE.$(51)

Substituting the results of a1 and b1 into (47a), (47b) and (49), we can get the optimal effort level of low carbon technology sharing and the optimal sharing payoff function of low carbon technology of superior enterprises and inferior enterprises, respectively. □

5.2 The limit of expectation and variance

From Proposition 5.1, the payoff of superior enterprises and inferior enterprises is related to the improvement degree of low carbon technical level, whose possible values are numerical outcomes of a random phenomenon by various random interference factors. Therefore, it is necessary to study the limit of expectation and variance.

Substituting the results of (43a) and (43b) into (4), we can get

$dKt=τ1+τ2−δKtdt+εKtdztK0=K0≥0,$(52)

where $\begin{array}{}{\tau }_{1}=\frac{\sigma \left[\alpha \left(\rho +\delta \right)+\sigma \left(\gamma +\lambda \right)\right]}{{c}_{C}\left(\rho +\delta \right)},{\tau }_{2}=\frac{\eta \left[\beta \left(\rho +\delta \right)+\eta \left(\gamma +\lambda \right)\right]}{{c}_{E}\left(\rho +\delta \right)}.\end{array}$

For further analysis, let ε (K(t))dz(t) = $\begin{array}{}\epsilon \sqrt{K}dz\left(t\right),\end{array}$ and then we can get Proposition 5.2 as follows.

Proposition 5.2

The limit of expectation and variance in cooperative game feedback equilibrium satisfy

$EK¯¯t=τ1+τ2δ+e−δtK0−τ1+τ2δ,limt→∞⁡EK¯¯t=τ1+τ2δ,$(53a)

$DK¯¯t=ε2τ1+τ2−2τ1+τ2−δK0e−δt+τ1+τ2−2δK0e−2δt2δ2,limt→∞DK¯¯t=ε2τ1+τ22δ2,$(53b)

where $\begin{array}{}{\tau }_{1}=\frac{\sigma \left[\alpha \left(\rho +\delta \right)+\sigma \left(\gamma +\lambda \right)\right]}{{c}_{C}\left(\rho +\delta \right)},{\tau }_{2}=\frac{\eta \left[\beta \left(\rho +\delta \right)+\eta \left(\gamma +\lambda \right)\right]}{{c}_{E}\left(\rho +\delta \right)}.\end{array}$

Proof

The proof of Proposition 5.2 is similarly to Proposition 3.2, so we do not repeat it here. □

6 Comparative analysis of equilibrium results

Proposition 6.1

Superior enterprises can share more low carbon technologies under the condition that inferior enterprises pay much more extra cost of low carbon technology sharing. Under cooperation between superior enterprises and inferior enterprises, superior enterprises and inferior enterprises can share more low carbon technology than the other two situations. That is to say, there exist $\begin{array}{}{L}_{C}^{C}\ge {L}_{C}^{S}\ge {L}_{C}^{N}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{E}^{C}\ge {L}_{E}^{S}={L}_{E}^{N}.\end{array}$

Proof

From Proposition 3.1 and Proposition 4.1, inferior enterprises have the same strategy of low carbon technology sharing in both cases. However, superior enterprises have the different strategies of low carbon technology sharing. Therefore, we can get

$LCC−LCS=αρ+δ+σγ+λcCρ+δ−2−θαρ+δ+σγ+λ2cCρ+δ=θαρ+δ+σγ+λ2cCρ+δ,$(54a)

$LCS−LCN=2−θαρ+δ+σγ+λ2cCρ+δ−θαρ+δ+σγ+λcCρ+δ=2−3θαρ+δ+σγ+λ2cCρ+δ,$(54b)

$LEC−LES=βρ+δ+ηγ+λcEρ+δ−1−θβρ+δ+ηγ+λcEρ+δ=θβρ+δ+ηγ+λcEρ+δ.$(54c)

According to the 0 ≤ θ$\begin{array}{}\frac{2}{3}\end{array}$, we can get $\begin{array}{}{L}_{C}^{C}-{L}_{C}^{S}\ge 0,{L}_{C}^{S}-{L}_{C}^{N}\ge 0,{L}_{E}^{C}-{L}_{E}^{S}\ge 0.\end{array}$ □

Proposition 6.1 indicates that the government subsidy of low carbon technology is a long-term incentive mechanism which can promote low carbon technology sharing. Superior enterprises and inferior enterprises can share more low carbon technologies through this mechanism.

Proposition 6.2

For any K ≥ 0, under the condition that inferior enterprises pay much more extra cost of low carbon technology sharing, the optimal sharing payoff of low carbon technology of superior enterprises reaches higher than the optimal sharing payoff under the condition that inferior enterprises do not provide extra cost. Similarly, the optimal sharing payoff of low carbon technology of inferior enterprises reaches higher than the optimal sharing payoff under the condition that inferior enterprises do not provide extra cost. That is to say, there exist $\begin{array}{}{V}_{C}^{S}\left(K\right)\ge {V}_{C}^{N}\left(K\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{V}_{E}^{S}\left(K\right)\ge {V}_{E}^{N}\left(K\right).\end{array}$

Proof

From Proposition 3.1 and Proposition 4.1, we can get

$ΔVCK=VCSK−VCNK=θγ+λρ+δK+θ1−θϕ2ρcEρ+δ2+θ2−θϕ14ρcCρ+δ2−θγ+λρ+δK−θ2ϕ12ρcCρ+δ2−θ1−θϕ2ρcEρ+δ2=θ2−3θαρ+δ+σγ+λ24ρcCρ+δ2,$(55a)

$ΔVEK=VESK−VENK=1−θγ+λρ+δK+1−θ2ϕ22ρcEρ+δ2+2−θ2ϕ18ρcCρ+δ−1−θγ+λρ+δK−1−θ2ϕ22ρcEρ+δ2−θ1−θϕ1ρcCρ+δ=2−3θ2αρ+δ+σγ+λ2ρcCρ+δ2.$(55b)

According to the $\begin{array}{}0\le \theta \le \frac{2}{3},\end{array}$ we can further get $\begin{array}{}{V}_{C}^{S}\left(K\right)-{V}_{C}^{N}\left(K\right)\ge 0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{V}_{E}^{S}\left(K\right)-{V}_{E}^{N}\left(K\right)\ge 0.\end{array}$ □

Proposition 6.2 indicates that, under the condition that inferior enterprises give a subsidy to superior enterprises, the subsidy of low carbon technology is an incentive mechanism which can promote low carbon technology sharing between superior enterprises and inferior enterprises. Superior enterprises and inferior enterprises can share more low carbon technologies through this mechanism.

Proposition 6.3

Under cooperative game, the total payoff exceeds the total payoff of Stacklberg master-slave game, and the total payoff of Stacklberg master-slave game exceeds the total payoff of Nash non-cooperative game in collaborative innovation system. That is to say, there exist VC (K) ≥ VS (K) ≥ VN (K).

Proof

According to Proposition 3.1, Proposition 4.1 and Proposition 5.1, we can get

$VSK=VCSK+VESK=θγ+λρ+δK+θ1−θϕ2ρcEρ+δ2+θ2−θϕ14ρcCρ+δ2+1−θγ+λρ+δK+1−θ2ϕ22ρcEρ+δ2+2−θ2ϕ18ρcCρ+δ=γ+λρ+δK+4−θ2αρ+δ+σγ+λ28ρcCρ+δ2+1−θ2βρ+δ+ηγ+λ22ρcEρ+δ2,$(56a)

$VCK−VSK=γ+λρ+δK+ϕ12ρcCρ+δ2+ϕ22ρcEρ+δ−γ+λρ+δK−4−θ2αρ+δ+σγ+λ28ρcCρ+δ2−1−θ2βρ+δ+ηγ+λ22ρcEρ+δ2=θ2αρ+δ+σγ+λ28ρcCρ+δ2+θ2βρ+δ+ηγ+λ22ρcEρ+δ2.$(56b)

According to the $\begin{array}{}0\le \theta \le \frac{2}{3},\end{array}$ we can get VC (K) ≥ VS (K) and

$VSK−VNK=VCSK+VESK−VCNK+VENK=VCSK−VCNK+VESK−VENK.$(57)

□

From Proposition 6.2, we can further get VS (K) ≥ VN (K).

Proposition 6.4

Under cooperative game, the stability of the improvement degree of low carbon technical level is better than the stability of Stacklberg master-slave game, and the stability of Stacklberg master-slave game is better than the stability of Nash non-cooperative game. That is to say, there exists

$EK¯¯t>EKt>EK¯t,limt→∞⁡EK¯¯t>limt→∞⁡EKt>limt→∞⁡EK¯tDK¯¯t>DKt>DK¯t,limt→∞⁡DK¯¯t>limt→∞⁡DKt>limt→∞⁡DK¯t.$(58)

Proof

According to Proposition 3.2, Proposition 4.2 and Proposition 5.2, we can get

$EK¯¯t−EKt=τ1+τ2δ+e−δtK0−τ1+τ2δ−μ1+μ2δ−e−δtK0−μ1+μ2δ=τ1+τ2−μ1+μ2δ1−e−δt>0.$(59a)

Similarly, we can get

$EKt−EK¯t>0,$(59b)

$limt→∞⁡EK¯¯t−limt→∞⁡EKt=τ1+τ2δ−μ1+μ2δ=τ1+τ2−μ1+μ2δ>0,$(60a)

$limt→∞⁡EKt−limt→∞⁡EK¯t>0,$(60b)

$limt→∞⁡DK¯¯t−limt→∞⁡DKt=ε2τ1+τ22δ2−ε2μ1+μ22δ2=ε2τ1+τ2−μ1+μ22δ2>0,$(61a)

$limt→∞⁡DKt−limt→∞⁡DK¯t>0,$(61b)

$DK¯¯t−DKt=ε2τ1+τ2−2τ1+τ2−δK0e−δt+τ1+τ2−2δK0e−2δt2δ2−ε2μ1+μ2−2μ1+μ2−δK0e−δt+μ1+μ2−2δK0e−2δt2δ2=ε2τ1+τ2−μ1+μ2δ1−2e−δt+e−2δt.$(62a)

The first derivative of 1 – 2eδt + e–2δt function of t is greater than 0 for t ∈ (0, ∞). When t → 0, we have 1 – 2eδt + e–2δt = 0, and then we can get D(K̿(t)) – D(K(t)) > 0. Similarly, we can get

$DKt−DK¯t>0.$(62b)

□

Proposition 6.4 indicates that enterprises can create and bring new low carbon technologies better than in case of the Stackelberg master slave game. However, some random interference factors in sharing system can make the variance of the improvement degree of cooperation game higher than the variance of the Stackelberg master slave game. That is to say, enterprises need to bear more risk to achieve higher payoff in sharing system under the cooperative game. Similarly, the result of Stackelberg game is similar to the result of Nash game. Therefore, different game modes are chosen by enterprises with different risk preferences. Cooperative game may be chosen by some enterprises with high risk preference, while Stackelberg game may be chosen by enterprises with moderate risk preference. The risk averse entity may choose Nash non-cooperative game.

7 Conclusions

In this paper, we have shown a stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment. In our model, we use the limit of expectation and variance of the improvement degree to identify the influence of random factors. According to Hamilton-Jacobi-Bellman equation, we get the optimal effort level of low carbon technology sharing, the subsidy of low carbon technology, the optimal sharing payoff and the total payoff of low carbon in collaborative innovation system of superior enterprises and inferior enterprises, respectively in the above game models. By comparing and analyzing of equilibrium results, we have shown that the effort level of superior enterprises and inferior enterprises is proportional to the government subsidy of low carbon technological innovation and the innovation capability of low carbon technology; the effort level of superior enterprises and inferior enterprises is inversely proportional to the sharing cost and the discount rate of low carbon technology; the sharing payoff of low carbon technology is proportional to the marginal return of low carbon technology. Moreover, we have shown that some random interference factors in sharing system can make the variance of the improvement degree of cooperation game higher than the variance of the Stackelberg master slave game. Similarly, the result of Stackelberg game is similar to the result of Nash game. By analyzing this stochastic differential game models, we have also provided a government subsidy incentive and a subsidy that inferior enterprises give to superior enterprises.

Acknowledgement

The authors would like to thank the anonymous referee for valuable corrections and comments. We thank my friend, Yuanjie Tan who is a maths major, for the support of mathematical derivation and examining mathematical formulas. This research was supported by the Fundamental Research Funds for the Central Universities (HEUCFW170901).

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Accepted: 2018-02-13

Published Online: 2018-06-14

Conflict of interestConflict of interests: The authors declare that there is no conflict of interests regarding the publication of this paper.

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 607–622, ISSN (Online) 2391-5455,

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