Analyzing the influence of government policy on building carbon emission reduction based on differential game

Model assumption and notation description

Hypothesis 1

In the daily production of construction enterprises, due to the government’s encouragement and supervision of low-carbon production, and people’s low-carbon requirements for building products are getting higher and higher, both developers and manufacturers will use a certain cost $u_h (t)$ to reduce part of the carbon emissions $e_h (t)$²⁰.

$$u_h (t) = \varepsilon_h e_h^2 (t)/2,h \in \ i,j\ \,$$

(1)

where $e_i (t)$ and $e_j (t)$ represents the CER effort level of the developer and manufacturer at time t, respectively.

Hypothesis 2

The CER of developers and manufacturers constitute the whole building’s CER, considering that the building’s CER are affected by natural degradation, according to the study²¹, in order to characterize the dynamic change process of the whole building’s CER, we propose the following differential equation.

$$\tfracdG(t)dt = e_i + e_j – \eta G(t)$$

(2)

where $G(t)$ denotes the CER per unit of the building at moment t and the initial CER $G(0) = G_0 > 0$, Meanwhile, $\eta \in (0,1)$ represents the carbon degradation rate of building products under the influence of nature.

Hypothesis 3

With the improvement of people’s environmental awareness, the higher the requirements for low-carbon building products, assuming that people’s purchasing power of products is positively correlated with the CER of building products, the unit demand for low-carbon building products can be as follows:

$$M(t) = M_0 + \lambda \textG(t)$$

(3)

where $M(t)$ represents people’s desire to buy building products, $M(0) = M_0 > 0$ denotes the initial demand of the building. $\lambda$ represents the positive correlation coefficient between people’s purchasing power and buildings’ CER, $\lambda > 0$.

Hypothesis 4

This paper assumes that both players agree to cooperate and continue the whole process of buildings’ CER. In addition, we assume that the developer as the leader In the Stackelberg game model, take the initiative to bear a certain proportion $\theta (0 \le \theta \le 1)$ of buildings’ CER costs for manufacturers to strengthen the close cooperation between the two sides of the game.

Research on buildings’ CER under government policy

As an indispensable role in the production of buildings’ CER, according to its own needs, government can formulate different policies to supervise and manage the process of buildings’ CER. which can encourage some enterprises to carry out the production activities of CER by subsidizing the cost of buildings’ CER or punish some enterprises that negatively respond to CER by levying carbon tax. There, in this section, we mainly consider three different government policies. The superscript $\left\ D,E,F \right\$ represents three types of government policies in the construction process, namely cost subsidy policy, carbon tax policy, carbon tax and cost subsidy combination policy, we assume that in all of these strategies, developers and manufacturers use feedback information structure to play the Stackelberg game.

Carbon reduction under scenario D

In this situation, to encourage developers and manufacturers develop the new technologies for low-carbon production actively, the government continuously provides certain cost subsidies of CER for both parties in the process of construction product. Meanwhile, to obtain the cost subsidies, enterprises actively carry out the efforts of CER in the production process of building products. In this case, both sides of the construction enterprise maximize their respective target value functions as follows:

$$\prod\limits_i^D = \int_0^\infty e^ – \rho t [I_i M(t) – (1 – \gamma_i )\varepsilon_i e_i^2 (t)/2 – \theta \varepsilon_j e_j^2 (t)/2]dt$$

(D1)

$$\prod\limits_j^D = \int_0^\infty e^ – \rho t [I_j M(t) – (1 – \gamma_j – \theta )\varepsilon_j e_j^2 (t)/2]dt$$

(D2)

Among them, $\rho \in (0,1)$ is the discount rate that changes with time, and in the dynamic game process between developers and manufacturers, $e_i ,e_j$ are used as the control variable and $G(t)$ is used as the state variable to satisfy the differential equation, which is the countermeasure problem of the differential game.

In the first step of the study of the above situation, it is necessary to construct the HJB equation based on the differential game model proposed by Richard Bellman in the 1950s²².

$$\beginaligned \rho V_i^D & = \mathop \max \limits_e_i ,e_j [I_i (M_0 + \lambda \textG) – (1 – \gamma_i )\varepsilon_i e_i^2 (t)/2 \\ & \quad – \theta \varepsilon_j e_j^2 (t)/2 + V_i^D^\prime (e_i + e_j – \eta G)] \\ \endaligned$$

(D3)

$$\beginaligned \rho V_j^D & = \mathop \max \limits_e_i ,e_j [I_j (M_0 + \lambda \textG) – (1 – \gamma_j – \theta )\varepsilon_j e_j^2 (t)/2 \\ & \quad + V_j^D^\prime (e_i + e_j – \eta G)] \\ \endaligned$$

(D4)

where $V_i^D^\prime = \fracdV_i^D^\prime dG,V_j^D^\prime = \fracdV_j^D^\prime dG$.

After establishing the HJB equation, the partial derivatives of $e_i$ and $e_j$ on both sides of the HJB equation can be obtained, and the equilibrium strategy of developers and manufacturers on CER can be obtained, as shown in proposition 1.

Proposition 1

For policy D, the optimal strategy of buildings’ CER for developers and manufacturers $e_h^D (t),h = i,j$ can be written as follows:

$$\left\{ \begingathered e_i^D* = \fracI_i \lambda (1 – \gamma_i )(\rho + \eta )\varepsilon_i \hfill \\ e_j^D^* = \fracI_j \lambda (1 – \gamma_j – \theta )(\rho + \eta )\varepsilon_j \hfill \\ \endgathered \right.$$

(D5)

When the optimal emission reduction strategy of construction enterprises is obtained, it is easy to obtain the CER per unit of the building $G(t)$ and the market demand for low-carbon building products $M(t)$ as shown in proposition 2.

Proposition 2

For policy D, the CER per unit of the building $G^D (t)$ and the market demand for low-carbon building products $M^D (t)$ can be written as follows:

$$G^D* (t) = \Omega^D + (G_0 – \Omega^D )e^ – \eta t$$

(D6)

$$M^D* (t) = M_0 + \lambda \textG(t) = M_0 + \lambda \Omega^D + \lambda (G_0 – \Omega^D )e^ – \eta t$$

(D7)

where $\Omega^D = \fracI_i \lambda (1 – \gamma_i )(\rho + \eta )\eta \varepsilon_i + \fracI_j \lambda (1 – \gamma_j – \theta )(\rho + \eta )\eta \varepsilon_j $.

A detailed certification process is provided in Appendix A.

Carbon reduction under scenario E

In this scenario. On the one hand, the government does not need to provide subsidies for the construction enterprises’ costs of CER; On the other hand, construction companies will also be penalized for their carbon emissions, paying a carbon tax. At the same time, enterprises have to improve their efforts for CER to avoid bearing the carbon tax brought by more carbon emissions. Therefore, according to^9,23, We assume that the carbon tax rate $\xi_\texth (h = i,j)$ formulated by the government is ideal and will not change over time. For the convenience of calculation, we believe that the carbon tax rate is positively correlated with the buildings’ CER. Suppose the total carbon produced by developers and manufacturers are $\overlinee_h (h = i,j)$. Then their carbon emission are $\overlinee_h – e_h(h=i,j)$, both sides of the game maximize their target income function respectively, as follows:

$$\begingathered \prod\limits_i^E = \int_0^\infty e^ – \rho t [I_i M(t) – \varepsilon_i e_i^2 (t)/2 – \theta \varepsilon_j e_j^2 (t)/2 \\ \beginarray*20c & \\ \endarray – \xi_i \varepsilon_i (\overlinee_i – e_i )^2 /2]dt \\ \endgathered$$

(E1)

$$\begingathered \prod\limits_j^E = \int_0^\infty e^ – \rho t [I_j M(t) – (1 – \theta )\varepsilon_j e_j^2 (t)/2 \hfill \\ \beginarray*20c & \\ \endarray – \xi_j \varepsilon_j (\overlinee_j – e_j )^2 /2]dt \hfill \\ \endgathered$$

(E2)

Similarly, the establishment of HJB equation is similar to scenario D:

$$\beginaligned \rho V_i^E & = \mathop \max \limits_e_i ,e_j [I_i (M_0 + \lambda \textG) – \varepsilon_i e_i^2 (t)/2 – \theta \varepsilon_j e_j^2 (t)/2 \\ & \quad – \xi_i \varepsilon_i (\overlinee_i – e_i )^2 /2 + V_i^E^\prime (e_i + e_j – \eta G)] \\ \endaligned$$

(E3)

$$\beginaligned \rho V_j^E & = \mathop \max \limits_e_i ,e_j [I_j (M_0 + \lambda \textG) – (1 – \theta )\varepsilon_j e_j^2 (t)/2 – \\ & \quad \xi_j \varepsilon_j (\overlinee_j – e_j )^2 /2 + V_j^E^\prime (e_i + e_j – \eta G)] \\ \endaligned$$

(E4)

where $V_i^E^\prime = \fracdV_i^E dG$ and $V_j^E^\prime = \fracdV_j^E dG$.

Proposition 3

For policy E, the optimal strategy of buildings’ CER for developers and manufacturers $e_h^E (t),h = i,j$ can be written as follows:

$$\left\{ \begingathered e_i^E* = \fracI_i \lambda + (\rho + \eta )\xi_i \varepsilon_i \overlinee_i (1 + \xi_i )(\rho + \eta )\varepsilon_i \hfill \\ e_j^E* = \fracI_j \lambda + (\rho + \eta )\xi_j \varepsilon_j \overlinee_j (1 + \xi_j – \theta )(\rho + \eta )\varepsilon_j \hfill \\ \endgathered \right.$$

(E5)

Proposition 4

For policy E, the CER per unit of the building $G^E (t)$ and the market demand for low-carbon building products $M^E (t)$ can be written as follows:

$$G^E (t) = \Omega^E + (G_0 – \Omega^E )e^ – \eta t$$

(E6)

$$M^E (t) = M_0 + \lambda \textG(t) = M_0 + \lambda \Omega^E + \lambda (G_0 – \Omega^E )e^ – \eta t$$

(E7)

where

$$\Omega^E = \fracI_i \lambda + (\rho + \eta )\xi_i \varepsilon_i \overlinee_i (1 + \xi_i )(\rho + \eta )\eta \varepsilon_i + \fracI_j \lambda + (\rho + \eta )\xi_j \varepsilon_j \overlinee_j (1 + \xi_j – \theta )(\rho + \eta )\eta \varepsilon_j $$

A detailed certification process is provided in Appendix B.

Through the comparative analysis of Propositions 1 and 3, we can get conclusion 1, as follows.

Conclusion 1

$\gamma_h \in (0,\omega_h )$ and $\xi_h \in (\Delta_h ,1),$ $e_h^D* < e_h^E* ;$

$\gamma_h \in (\omega_h ,1)$ and $\xi_h \in (0,\Delta_h ),$ $e_h^D* > e_h^E* .$where

$$\left\{ \begingathered \omega_i = \frac(\rho + \eta )\xi_i \varepsilon_i \overlinee_i – I_i \lambda \xi_i I_i \lambda + (\rho + \eta )\xi_i \varepsilon_i \overlinee_i \hfill \\ \Delta_i = \fracI_i \lambda \gamma_i (1 – \gamma_i )(\rho + \eta )\xi_i \varepsilon_i \overlinee_i – I_i \lambda \hfill \\ \endgathered \right.$$

and

$$\left\{ \begingathered \omega_j = \frac(1 – \theta )(\rho + \eta )\xi_j \varepsilon_j \overlinee_j – I_j \lambda (\xi_j + \theta )I_j \lambda + (\rho + \eta )\xi_j \varepsilon_j \overlinee_j \hfill \\ \Delta_j = \fracI_j \lambda (\gamma_j + \theta )(1 – \gamma_j – \theta )(\rho + \eta )\xi_j \varepsilon_j \overlinee_j – I_j \lambda \hfill \\ \endgathered \right.$$

Before reaching conclusion 1, we have always believed that government subsidy policy is more promising than carbon tax policy as a better choice to promote the process of building’s CER.

However, according to conclusion 1, we have gained some new understanding. We can find that when the subsidy rate satisfies $\gamma_h \in (0,\omega_h )$, and the Carbon tax rate satisfies $\xi_h \in (\Delta_h ,1)$, the effect of buildings’ CER in scenario E is better than that in scenario D. When the subsidy rate satisfies $\gamma_h \in (\omega_h ,1)$, and the Carbon tax rate satisfies $\xi_h \in (0,\Delta_h )$, the effect of buildings’ CER in scenario D is better than that in scenario E. This shows that in the management of buildings’ CER, When the cost subsidy rate is low and the carbon tax rate is high, the CER effect of the carbon tax policy is better than that of the subsidy policy. When the cost subsidy rate is high and the carbon tax rate is low, the CER effect of the subsidy policy is better than that of the carbon tax policy.

Therefore, conclusion 1 provides a theoretical support for the formulation of government policies. When the government decides on the subsidy policy, the cost subsidy rate of buildings’ CER $\gamma_h$ provided by the government for enterprises should be set at $(\omega_h ,1)$; and when the government that chooses the carbon tax policy, the carbon tax rate $\xi_h$ should be set at $(\Delta_h ,1)$.

Carbon reduction under scenario F

Although scenario D and scenario E are both conducive to the process of building’s CER to a certain extent, scenario D only provides cost subsidies without punishing the carbon emissions of construction enterprises, and scenario E only penalizes the carbon emissions of construction enterprises without providing cost subsidies for construction enterprises. Therefore, we propose scenario F. In this case, the government formulates a cost subsidy policy. It provides certain subsidies to the two construction enterprises in the production process of construction products, aiming to encourage developers and manufacturers to reduce carbon emissions actively. On the other hand, the government has formulated a carbon tax policy to levy a carbon tax on the carbon emissions generated in the process of building products, aiming to supervise and manage the carbon emissions of construction enterprises. To receive more government subsidies and bear less carbon tax for their own carbon emissions, construction enterprises will make more efforts to reduce carbon emissions. Then developers and manufacturers will maximize their target revenue functions respectively, as follows:

$$\begingathered \prod\limits_i^F = \int_0^\infty e^ – \rho t [I_i M(t) – (1 – \gamma_i )\varepsilon_i e_i^2 (t)/2 – \theta \varepsilon_j e_j^2 (t)/2 – \hfill \\ \beginarray*20c & \\ \endarray \xi_i \varepsilon_i (\overlinee_i – e_i )^2 /2]dt \hfill \\ \endgathered$$

(F1)

$$\begingathered \prod\limits_j^F = \int_0^\infty e^ – \rho t [I_j M(t) – (1 – \gamma_j – \theta )\varepsilon_j e_j^2 (t)/2 \\ \beginarray*20c & \\ \endarray – \xi_j \varepsilon_j (\overlinee_j – e_j )^2 /2]dt \\ \endgathered$$

(F2)

Similarly, the establishment of HJB equation is similar to scenario E:

$$\beginaligned \rho V_i^F & = \mathop \max \limits_e_i ,e_j [I_i (M_0 + \lambda \textG) – (1 – \gamma_i )\varepsilon_i e_i^2 (t)/2 \\ & \quad – \theta \varepsilon_j e_j^2 (t)/2 – \xi_i \varepsilon_i (\overlinee_i – e_i )^2 /2 + V_i^F^\prime (e_i + e_j – \eta G)] \\ \endaligned$$

(F3)

$$\beginaligned \rho V_j^F & = \mathop \max \limits_e_i ,e_j [I_j (M_0 + \lambda \textG) – (1 – \gamma_j – \theta )\varepsilon_j e_j^2 (t)/2 \\ & \quad – \xi_j \varepsilon_j (\overlinee_j – e_j )^2 /2 + V_j^F^\prime (e_i + e_j – \eta G)] \\ \endaligned$$

(F4)

where $V_i^F^\prime = \fracdV_i^F dG$ and $V_j^F^\prime = \fracdV_j^F dG$.

Proposition 5

For policy F, the optimal strategy of buildings’ CER for developers and manufacturers $e_h^\textF (t),h = i,j$ can be written as follows:

$$\left\{ \begingathered e_i^F* = \fracI_i \lambda + (\rho + \eta )\xi_i \varepsilon_i \overlinee_i (1 – \gamma_i + \xi_i )(\rho + \eta )\varepsilon_i \hfill \\ e_j^F* = \fracI_j \lambda + (\rho + \eta )\xi_j \varepsilon_j \overlinee_j (1 + \xi_j – \gamma_j – \theta )(\rho + \eta )\varepsilon_j \hfill \\ \endgathered \right.$$

(F5)

In order to better characterize the evolution trajectory of buildings’ CER, we find the CER per unit of the building $G(t)$ and the market demand for low-carbon building products $M(t)$ Proposition 6.

Proposition 6

For policy F, the CER per unit of the building $G^F (t)$ and the market demand for low-carbon building products $M^\textF (t)$ can be written as follows:

$$G^F (t) = \Omega^F + (G_0 – \Omega^F )e^ – \eta t$$

(F6)

$$M^F (t) = M_0 + \lambda \textG(t) = M_0 + \lambda \Omega^F + \lambda (G_0 – \Omega^F )e^ – \eta t$$

(F7)

where,

$$\beginaligned \Omega^f & = (e_i^F* + e_j^F* )/\eta = \fracI_i \lambda + (\rho + \eta )\xi_i \varepsilon_i \overlinee_i (1 – \gamma_i + \xi_i )(\rho + \eta )\eta \varepsilon_i \\ & \quad + \fracI_j \lambda + (\rho + \eta )\xi_j \varepsilon_j \overlinee_j (1 + \xi_j – \gamma_j – \theta )(\rho + \eta )\eta \varepsilon_j \\ \endaligned$$

A detailed certification process is provided in Appendix C.

Conclusion 2

$$e_i^F* > e_i^E*$$

According to Conclusion 2, no matter how the $\gamma_i ,\xi_i$ changes, there is $e_i^F* > e_i^E*$. In other words, compared with scenario E, scenario F is more conducive to reducing carbon emission, which is a better choice for the government to make. When the government chooses scenario F, compare with scenario E, although the government needs to provide the CER cost for the construction enterprise, it can encourage the CER efforts of the construction enterprise and achieve a more significant CER effect, which is acceptable for the government aiming at reducing carbon emissions.

Conclusion 3

$$\begingathered \gamma_h \in (0,\pi_h ),e_i^F* > e_i^D* ; \hfill \\ \gamma_h \in (\pi_h ,1),e_i^D* > e_i^F* . \hfill \\ \endgathered$$

where

$$\left\{ \begingathered \pi_i = \frac(\rho + \eta )\varepsilon_i \overlinee_i – I_i \lambda (\rho + \eta )\varepsilon_i \overlinee_i \hfill \\ \pi_j = \frac(1 – \theta )(\rho + \eta )\varepsilon_j \overlinee_j – I_j \lambda (\rho + \eta )\varepsilon_j \overlinee_j \hfill \\ \endgathered \right.$$

According to conclusion 3, when the subsidy rate $\gamma_h$ changes in the construction production process, scenario E and scenario F have different performance on carbon emission reduction effect, which are embodied as follows. When the subsidy rate is $\gamma_h \in (0,\pi_h ),e_h^F* > e_h^D*$ when the subsidy rate is $\gamma_h \in (\pi_h ,1),e_h^D* > e_h^F*$. This shows that when the cost subsidy is low, if the carbon emission of construction enterprises is not supervised and managed, the efforts of buildings’ CER of enterprises cannot be improved. On the other hand, when the cost subsidy is relatively high, the government’s carbon emission supervision on construction enterprises is too strict, which will also inhibit the enterprises’ efforts in buildings’ CER.

Therefore, in real life, when the government’s cost subsidy $\gamma_h$ for construction enterprises is within the range of 0 to $\pi_h$. Compared with policy E, choosing policy F can make the CER of construction enterprises reach a better level. Under this policy, the government will not only provide a certain proportion of cost subsidies for construction enterprises, but also supervise and manage the carbon emissions of construction enterprises. Meanwhile, for the construction enterprise, to obtain government subsidies and avoid higher carbon taxes, enterprises will have to actively carry out carbon emission reduction activities.

When the government’s carbon emission reduction cost subsidy $\gamma_h$ for construction enterprises is within the range of $\pi_h$ to 1, compared with policy F, choosing government policy E can make the CER of construction enterprises reach a better level. Under this policy, the supervision and management of the carbon emission of construction enterprises will inhibit the carbon emission activities of construction enterprises. So the government only needs to encourage construction enterprises to carry out the process of CER by providing a high proportion of cost subsidies for construction enterprises. At the same time, in order to obtain high cost subsidies from the government, enterprises will also actively carry out carbon emission reduction activities.

Conclusion 4

When $\pi_h > \omega_h$, $\xi_h < \Delta_h$:

$$\begingathered \gamma_h \in (0,\pi_h ),e_h^F* > e_h^E* > e_h^D* ; \hfill \\ \gamma_h \in (\pi_h ,1),e_h^D* > e_h^F* > e_h^E* . \hfill \\ \endgathered$$

(1)

According to conclusion 4, To achieve greater buildings’ CER, the subsidy rate and carbon tax rate set by the government are within a relatively straightforward range, namely $\xi_h < \Delta_h$ and $\gamma_h \in (0,\pi_h ) \cup (\pi_h ,1).$ This conclusion provides theoretical support and basis for the government to set the scope of subsidy rate and carbon tax rate.
(2)

When $\xi_h < \Delta_h$ and $\gamma_h \in (0,\pi_h )$ we can get $e_h^F* > e_h^E* > e_h^D*$. It shows that in the process of building production, when the carbon tax rate is $\xi_h < \Delta_h$ and the cost subsidy rate is $\gamma_h \in (0,\pi_h )$, scenario F is the optimal strategy for buildings’ CER, which can maximize the level of buildings’ CER. Therefore, for a government that cannot provide a large amount of cost subsidies, implementing a combination of subsidy policies and carbon tax policies is the best strategy for the government.
(3)

When $\xi_h < \Delta_h$ and $\gamma_h \in (\pi_h ,1),$ we can get $e_h^D* > e_h^F* > e_h^E*$. It shows that in the process of building production, when the carbon tax rate is $\xi_h < \Delta_h$ and the cost subsidy rate is $\gamma_h \in (\pi_h ,1),$ scenario D is the optimal strategy for buildings’ CER, which can maximize the level of buildings’ CER. It indicates that high government subsidies can maximize the enthusiasm of enterprises for buildings’ CER. At this time, if imposed the carbon emission tax on enterprises, it will weaken the enthusiasm of enterprises for buildings’ CER, and have a negative effect on the process of buildings’ CER in the entire industrial chain. Therefore, for governments aiming to reduce carbon emissions, policy D is the optimal strategy for the government.

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Source Link: https://www.nature.com/articles/s41598-024-68012-7

Model assumption and notation description Hypothesis 1 In the daily production of construction enterprises, due to the government’s encouragement and supervision of low-carbon production, and people’s low-carbon requirements for building products are getting higher and higher, both developers and manufacturers will use a certain cost $u_h (t)$ to reduce part of the carbon emissions \(e_h …

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Analyzing the influence of government policy on building carbon emission reduction based on differential game

Model assumption and notation description

Hypothesis 1

Hypothesis 2

Hypothesis 3

Hypothesis 4

Research on buildings’ CER under government policy

Carbon reduction under scenario D

Proposition 1

Proposition 2

Carbon reduction under scenario E

Proposition 3

Proposition 4

Conclusion 1

Carbon reduction under scenario F

Proposition 5

Proposition 6

Conclusion 2

Conclusion 3

Conclusion 4

Author: BLOGGER

Merchant Business News

Merchant Business News