# Dynamic Incentive Contract of Government for Port Enterprises to Reduce Emissions in the Blockchain Era: Considering Carbon Trading Policy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Blockchain Technology in the Shipping Industry

#### 2.2. Port Emission Reduction and Government Subsidy

#### 2.3. Incentive Contract Design

#### 2.4. Research Gap

## 3. Problem Formulation and Assumptions

_{g}(t) in terms of reducing carbon emissions and improving the environment.

- We assume that the investment level of the port enterprise in emission reduction in the t period is I(t), which indicates the investment in energy-saving and emission reducing technologies such as “oil-to-electricity” shore-side power and LNG terminals by port enterprises to build green ports, and assume that the carbon emission reductions (CERs) of the port in period t is:$$G(t)={\displaystyle \underset{0}{\overset{t}{\int}}(\beta \cdot I(t)-\sigma \cdot G(t))dt}+v$$
^{2}, which reflects the uncertainty of port emission reduction market. - Referring to existing relevant literature (e.g., Hong and Guo [55], Chai et al. [56]), we assume that the social benefit brought by the port enterprise to the government through efforts to improve carbon emission reductions in the period t is R
_{g}(t) = h⋅G(t), and h is the monetary expression of the social benefit generated by unit port emission reduction. In addition, based on the dynamic changes of the port’s CERs, we propose a linear dynamic incentive contract for the government to implement subsidy to the port enterprise in the t period is:S(t) = s_{0}(t) + s_{1}(t) ⋅ G(t)_{0}(t) and s_{1}(t), respectively, refer to the fixed subsidy and unit subsidy paid by the government to the port enterprise. In addition, we assume that the government will incur a contract execution cost c (such as information cost and supervision cost) in determining contract terms, fulfilling contracts, and resolving disputes during the t period. - As port emission reduction helps to attract green customers and promote shipping demand, it is assumed that the shipping customer demand caused by the port enterprise’s CERs in the t period is D(t) = γ⋅G(t). γ > 0 is the impact of port emission reduction on shipping customer demand, which depends on shipping customers’ awareness of green environmental protection. In addition, we also assume that the revenue of the port enterprise consists of two parts: one part is the revenue R
_{p}(t) = ξ ⋅D(t), and ξ > 0 is the service price of the port. The other part is the revenue W(g_{0}, S(t)) = g_{0}⋅S(t) brought to the port enterprise by the government’s dynamic incentive contract S(t), and 0 < g_{0}< 1 is the execution efficiency of the contract, which reflects the sensitivity of port enterprise to the government contract. - Considering that both the government and port enterprise are risk averse to their revenues [57], we define the degree of risk aversion θ using the Arrow–Pratt absolute risk aversion measure, and assume that the government’s risk avoidance cost for social benefit R
_{g}(t) is CR_{g}(t) = θ⋅Var(R_{g}(t))/2 = θ⋅h^{2}δ^{2}/2, and the port enterprise’s risk avoidance cost for its revenue is CR_{p}(t) = θ⋅Var(R_{p}(t) + W(t))/2 = θ⋅(g_{0}⋅s_{1}(t) + γξ)^{2}δ^{2}/2. - Suppose that the investment cost function of emission reduction for the port enterprise is η⋅I
^{2}(t)/2 (such as the cost of equipment purchase, human input, technological innovation, and shore power maintenance), where η > 0 is the corresponding cost coefficient. The setting of the cost function meets the general convexity assumption in economics, and the economic implication behind it is that the investment cost of port emission reduction meets the law of marginal cost increase. In addition, we assume that the revenue of the port enterprise without emission reduction investment while maintaining the traditional operation mode is Φ, which reflects the opportunity cost of the port enterprise’s green transformation investment in shore power and other emission reduction technologies. In addition, it is assumed that the government will provide dynamic incentives to the port enterprise’s emission reduction within time t ∈ [0, +∞), ρ is used to express the discount rate of the port service market.

- (1)
- According to the value of blockchain smart contracts, it is assumed that the execution efficiency of government dynamic incentive contracts under blockchain technology is g
_{1}and meets 0 < g_{0}< g_{1}< 1. At the same time, without losing generality, let the contract execution cost c = 0. - (2)
- According to the value of blockchain green certification, this paper introduces the green trust coefficient r of customers on the port’s ERI level, which is reflected in the accumulation of the port enterprise’s effort to reduce emissions. Therefore, it is assumed that the port’s CERs in the t period under blockchain technology is:$$G(t)={\displaystyle \underset{0}{\overset{t}{\int}}(\beta \cdot I(t)-\sigma \cdot G(t)+r\cdot I(t))dt}+{v}^{*}$$
- (3)
- According to the value of blockchain disclosure of market information on green energy efficiency and emission reduction in ports, this paper introduces the disclosure degree ω of blockchain for port emission reduction uncertain information [58]. We assume that the random disturbance factor of port emission reduction under blockchain technology is v* (see Equation (3)) with mean 0 and variance (1 − ω) ⋅ δ
^{2}, where 0 < ω < 1 denotes the degree of information disclosure. It reduces the variance of random disturbance factor without blockchain, and reduces the uncertainty of the port emission reduction market.

_{b}, while the unit operating cost for the port enterprise to participate in and apply blockchain technology is C

_{b}.

## 4. Models

#### 4.1. No Blockchain (Case N)

^{N}(t) is:

_{g}

^{N}in the objective function formula (6) represents the discounted value of the total expected benefit of the government in the time of [0, +∞] under the case of N, which includes four parts: E(R

_{g}

^{N}(t)) denotes the expected social benefit to the government from the port’s CERs in period t, E(S

^{N}(t)) refers to the dynamic incentive contract expected to be paid by the government to port enterprise in period t, CR

_{g}

^{N}(t) refers to the risk aversion cost of the government to social benefit, and c⋅E(S

^{N}(t)) refers to the contract execution cost expected by the government in the period t. In Equation (7), IR refers to the individual rational constraint of port enterprise, which ensures that the expected return of the port enterprise’s green transformation and active emission reduction investment is not lower than that of the traditional operation mode. IC is the incentive compatibility constraint of port enterprise; that is, after signing the contract, the port enterprise determines its ERI level I

^{N}(t) based on maximizing its own expected revenue π

_{p}

^{N}. SE is the state equation of port VERs.

_{p}

^{N}(EG

^{N}), which represents the discounted value of the total expected revenue in the period from time t to +∞. V

_{p}

^{N}′(EG

^{N}), be the first derivative of the optimal expected value function with respect to the port EG

^{N}, which represents the marginal contribution of the unit port VERs to the discounted value of the total expected revenue of the port enterprise. According to the continuous dynamic optimal control theory, the optimal expected value function V

_{p}

^{N}(EG

^{N}) satisfies the following Hamilton–Jacobi–Bellman (HJB) equation:

**Lemma**

**1.**

^{N*}(t) when maximizing its own interests. Therefore, according to Lemma 1, the incentive compatibility constraint condition IC of port enterprise can be expressed equivalently by Equation (10). For port enterprise’s individual rationality constraint IR, it is a tight constraint when the government objective function is maximized. The reason is that if IR is not equal, the government in the model will always increase its expected benefit by reducing the fixed subsidy s

_{0}

^{N}(t) without affecting the establishment of the port enterprise’s IR condition. Therefore, the government dynamic incentive contract model in the N case can be transformed into the following optimal control problem:

_{g}

^{N}(EG

^{N}), which satisfies the following HJB equation:

**Theorem**

**1.**

**Corollary**

**1.**

#### 4.2. Blockchain Adoption (Case B)

_{g}

^{B}(t) = θ⋅Var(R

_{g}

^{B}(t))/2 =θ ⋅ h

^{2}(1 − ω)δ

^{2}/2. In the individual rational constraint IR and incentive compatibility constraint IC, the revenue that the government dynamic incentive contract S

^{B}(t) brings to port enterprise is updated to W

^{B}(t) = g

_{1}⋅S

^{B}(t), the risk aversion cost of port enterprise’s revenue is updated to CR

_{p}

^{B}(t) = θ⋅Var(R

_{p}

^{B}(t) + W

^{B}(t))/2 = θ⋅(g

_{1}⋅ s

_{1}

^{B}(t) + γξ)

^{2}(1 − ω)δ

^{2}/2, and the port enterprise needs to bear certain operating costs of blockchain C

_{b}⋅E(G

^{B}(t)).

^{B}(t), is designed. As mentioned above, blockchain green certification helps the port enterprise to prove the authenticity and credibility of its green energy conservation and emission reduction to customers, and relevant customers will generate an additional green trust item for the emission reduction process of blockchain-supported port enterprise out of low-carbon preference. Moreover, the cumulative change of port EG

^{B}(t) will also directly affect the social benefit E(R

_{g}

^{B}(t)) = h⋅EG

^{B}(t) of the government and the income E(R

_{p}

^{B}(t)) = γξ⋅EG

^{B}(t) of port enterprise⋅

**Lemma**

**2.**

^{B*}(t) into the personal rational constraint IR and tightening the constraint, the optimal fixed subsidy for port enterprise in the government dynamic incentive contract can be obtained as follows:

^{B*}(t) and s

_{0}

^{B*}(t), the objective function and state change equation of the government are updated, and the optimization problem of the original model can be transformed into:

_{1}

^{B}(t), Theorem 2 can be obtained.

**Theorem**

**2.**

**Corollary**

**2.**

#### 4.3. Blockchain Adoption When Considering Carbon Trading Policy (Case TB)

_{2}) as commodities, which is a market mechanism used to reduce global greenhouse gas emissions. This subsection will further discuss the dynamic incentive contract of the government for the port enterprise’s emission reduction when the carbon trading market mechanism is introduced under blockchain technology.

**Lemma**

**3.**

**Theorem**

**3.**

**Corollary**

**3.**

## 5. Model Analysis

#### 5.1. Government’s Dynamic Incentive Strategy for Port Emission Reduction

_{0}*(t), incentive contract S*(t), and the government’s expected benefit discount value V

_{g}*(t). This subsection will analyze the optimal dynamic trajectory change rules of the government’s incentive strategy. The purpose is to study how the government should dynamically adjust the incentive contract for port enterprises to reduce emissions, and how the port VERs and the government’s expected benefit discount value will evolve under the influence of the government’s optimal contract. At the same time, it provides a reference and theoretical basis for relevant government decision-makers to adjust and control the optimal state of dynamic changes in port enterprises’ emission reduction incentive strategies before and after the adoption of blockchain and the launch of the carbon trading market.

**Proposition**

**1.**

- (i)
- When t < t
_{th}^{N}, then $\frac{\partial E{{G}^{N}}^{*}(t)}{\partial t}>0$, $\frac{\partial {s}_{0}{{}^{N}}^{*}(t)}{\partial t}<0$, $\frac{\partial {{S}^{N}}^{*}(t)}{\partial t}<0$, $\frac{\partial {V}_{g}{{}^{N}}^{*}(t)}{\partial t}>0$; - (ii)
- When t ≥ t
_{th}^{N}, then $\frac{\partial E{{G}^{N}}^{*}(t)}{\partial t}=0$, $\frac{\partial {s}_{0}{{}^{N}}^{*}(t)}{\partial t}=0$, $\frac{\partial {{S}^{N}}^{*}(t)}{\partial t}=0$, $\frac{\partial {V}_{g}{{}^{N}}^{*}(t)}{\partial t}=0$;

_{th}

^{N}is the time threshold in (0, +∞).

_{th}

^{N}, the port VERs EG

^{N*}(t) and the government’s expected benefit discount value V

_{g}

^{N*}(t) monotonically increased with time t, while the government’s fixed subsidy s

_{0}

^{N*}and incentive contract S

^{N*}(t) monotonically decreased with time t. This means that the government dynamic incentive contract not only promotes the emission reduction of the port enterprise, but also improves the expected benefit of the government. This is because the utilization rate of green equipment such as port electricity increases with the increase in emission reduction, while the carbon emissions decrease over the same period. The government has achieved economic growth and social benefit in the process of carbon emission reduction and environmental improvement. At the same time, the government’s investment in incentive subsidies for port enterprise in the early stage is large, and with the passage of time and the improvement of the emission reduction incentive effect, the government’s incentive subsidies for port enterprise will gradually decrease. When the government’s dynamic incentive time exceeds a critical point, that is, t ≥ t

_{th}

^{N}, the optimal dynamic trajectories of the government’s emission reduction incentive strategy will not change with time. At this time, EG

^{N*}(t), V

_{g}

^{N*}(t), s

_{0}

^{N*}(t), and S

^{N*}(t) all reach steady-state values.

**Proposition**

**2.**

- (i)
- When t < t
_{th}^{B}, then $\frac{\partial E{G}^{B}(t)}{\partial t}>0$, $\left\{\begin{array}{l}if{C}_{b}=0,then\frac{\partial {s}_{0}{}^{B}(t)}{\partial t}0\\ if{C}_{b}0,\left\{\begin{array}{l}and{C}_{b}{C}_{b}{}^{th1},then\frac{\partial {s}_{0}{}^{B}(t)}{\partial t}0\\ and{C}_{b}\ge {C}_{b}{}^{th1},then\frac{\partial {s}_{0}{}^{B}(t)}{\partial t}\ge 0\end{array}\right.\end{array}\right.$,$\left\{\begin{array}{l}if{C}_{b}=0,then\frac{\partial {S}^{B}(t)}{\partial t}0\\ if{C}_{b}0,\left\{\begin{array}{l}and{C}_{b}\gamma \xi ,then\frac{\partial {S}^{B}(t)}{\partial t}0\\ and{C}_{b}\ge \gamma \xi ,then\frac{\partial {S}^{B}(t)}{\partial t}\ge 0\end{array}\right.\end{array}\right.$, $\left\{\begin{array}{l}if{C}_{b}=0,then\frac{\partial {V}_{g}{}^{B}(t)}{\partial t}0\\ if{C}_{b}0,\left\{\begin{array}{l}and{C}_{b}{g}_{1}h+\gamma \xi ,then\frac{\partial {V}_{g}{}^{B}(t)}{\partial t}0\\ and{C}_{b}\ge {g}_{1}h+\gamma \xi ,then\frac{\partial {V}_{g}{}^{B}(t)}{\partial t}\le 0\end{array}\right.\end{array}\right.$; - (ii)
- When t ≥ t
_{th}^{B}, then$\frac{\partial E{G}^{B}(t)}{\partial t}=0$,$\frac{\partial {s}_{0}{}^{B}(t)}{\partial t}=0$,$\frac{\partial {S}^{B}(t)}{\partial t}=0$,$\frac{\partial {V}_{g}{}^{B}(t)}{\partial t}=0$;

_{th}

^{B}is the time threshold in (0, +∞).

_{b}= 0 [10,59], the optimal dynamic trajectory change rules of the government incentive strategy are the same as the traditional model without blockchain, which is only related to time t. However, when C

_{b}> 0, the change rules of the fixed subsidy s

_{0}

^{B*}(t), incentive contract S

^{B*}(t) and the government expected benefit discount value V

_{g}

^{B*}(t) are not only related to time t, but also related to cost C

_{b}. Specifically, even in the early stage, that is, when t < t

_{th}

^{B}, the government’s fixed subsidy s

_{0}

^{B*}(t) and incentive contract S

^{B*}(t) may increase over time, while the government’s expected benefit discount value V

_{g}

^{B*}(t) may decrease over time, which is different from Proposition 1. The reason is that if the cost C

_{b}of the blockchain is large, the port enterprise will have to bear large costs when participating in the implementation of the blockchain, and its enthusiasm for emission reduction investment may be reduced. The government will have to provide more incentive subsidies to promote port emission reduction, which is often unfavorable to the expected benefit of the government. Therefore, Proposition 2 means that the implementation of blockchain will not affect the optimal dynamic trajectory change rule of port VERs EG

^{B*}(t). However, in a period of time before the government’s emission reduction incentive strategy reaches a steady state, the size of the blockchain’s unit operating cost will directly affect the optimal dynamic trajectory change rules of the government’s fixed subsidy s

_{0}

^{B*}(t), incentive contract S

^{B*}(t) and the government’s expected benefit discount value V

_{g}

^{B*}(t).

**Proposition**

**3.**

- (i)
- When t < t
_{th}^{TB}, then $\frac{\partial E{G}^{TB}(t)}{\partial t}>0$,$\left\{\begin{array}{l}if0\le {C}_{b}{C}_{b}{}^{th1},then\frac{\partial {s}_{0}{}^{TB}(t)}{\partial t}0\\ if{C}_{b}\ge {C}_{b}{}^{th1},\left\{\begin{array}{l}and\tau \le \left({C}_{b}-{C}_{b}{}^{th1}\right),then\frac{\partial {s}_{0}{}^{TB}(t)}{\partial t}\ge 0\\ and\tau \left({C}_{b}-{C}_{b}{}^{th1}\right),then\frac{\partial {s}_{0}{}^{TB}(t)}{\partial t}0\end{array}\right.\end{array}\right.$,$\left\{\begin{array}{l}if0\le {C}_{b}\gamma \xi ,then\frac{\partial {S}^{TB}(t)}{\partial t}0\\ \\ if{C}_{b}\ge \gamma \xi ,\\ \left\{\begin{array}{l}and\tau \le \left({C}_{b}-\gamma \xi \right),then\frac{\partial {S}^{TB}(t)}{\partial t}\ge 0\\ and\tau \left({C}_{b}-\gamma \xi \right),then\frac{\partial {S}^{TB}(t)}{\partial t}0\end{array}\right.\end{array}\right.$,$\left\{\begin{array}{l}if0\le {C}_{b}{g}_{1}h+\gamma \xi ,then\frac{\partial {V}_{g}{}^{TB}(t)}{\partial t}0\\ \\ if{C}_{b}\ge {g}_{1}h+\gamma \xi ,\\ \left\{\begin{array}{l}and\tau \le \left({C}_{b}-{g}_{1}h-\gamma \xi \right),then\frac{\partial {V}_{g}{}^{TB}(t)}{\partial t}\le 0\\ and\tau \left({C}_{b}-{g}_{1}h-\gamma \xi \right),then\frac{\partial {V}_{g}{}^{TB}(t)}{\partial t}0\end{array}\right.\end{array}\right.$; - (ii)
- When t ≥ t
_{th}^{TB}, then$\frac{\partial E{G}^{TB}(t)}{\partial t}=0$,$\frac{\partial {s}_{0}{}^{TB}(t)}{\partial t}=0$,$\frac{\partial {S}^{TB}(t)}{\partial t}=0$,$\frac{\partial {V}_{g}{}^{TB}(t)}{\partial t}=0$;

_{th}

^{TB}is the time threshold in (0, +∞).

_{b}of the blockchain and the market price τ of carbon trading. Compared with Proposition 2, when the government starts the carbon trading market, in the early stage of the government dynamic incentive, that is, t < t

_{th}

^{TB}, even if the unit operating cost C

_{b}of the blockchain is large, and if the carbon trading market price τ is also large, the fixed subsidy s

_{0}

^{TB*}(t) and incentive contract S

^{B*}(t) in the government incentive strategy for port emission reduction will still decrease with time, while the government expected benefit discount value V

_{g}

^{TB*}(t) will still increase with time. This means that carbon trading policy can weaken the impact of the unit operating cost brought by the implementation of blockchain on the change rules of the government’s dynamic incentive strategy for port emission reduction to a certain extent. The reason is that the government’s opening of the carbon trading market helps to improve the motivation of the port enterprise’s effort to invest in emission reduction, especially the size of the carbon trading market price directly affects the enthusiasm of the port enterprise’s investment in emission reduction. Therefore, even if the unit operation cost of port enterprise participating in the implementation of blockchain is large, under the positive effect of carbon trading, the government may not need to increase the incentive subsidies for port enterprise, the port emission reduction can still be improved, and the expected benefit of the government will also be continuously improved.

#### 5.2. Parameter Analysis

^{2}, θ), contract execution efficiency (g

_{0}, g

_{1}) and execution cost c, and blockchain and carbon trading-related parameters (C

_{b}, r, ω, τ) on the port enterprise’s optimal ERI level, the government’s optimal unit subsidy, and the port VERs. The results of the study are summarized in Propositions 4 to 7. According to the analysis in Section 5.1, the port VERs under different cases converge to steady state and no longer vary over time. Therefore, this subsection provides a parametric analysis of port VERs at steady state, which helps to provide some insights into the long-term dynamics of government incentives for port enterprises to reduce emissions.

**Proposition**

**4.**

- (i)
- $\frac{\partial {I}^{{i}^{*}}}{\partial \beta}>0$,$\frac{\partial {s}_{1}{i}^{*}}{\partial \beta}>0$,$\frac{\partial E{G}^{{i}^{*}}}{\partial \beta}>0$;
- (ii)
- $\frac{\partial {I}^{{i}^{*}}}{\partial h}>0$,$\frac{\partial {s}_{1}{i}^{*}}{\partial h}>0$,$\frac{\partial E{G}^{{i}^{*}}}{\partial h}>0$;
- (iii)
- $\frac{\partial {I}^{{i}^{*}}}{\partial \xi}>0$,$\frac{\partial {s}_{1}{i}^{*}}{\partial \xi}<0$,$\frac{\partial E{G}^{{i}^{*}}}{\partial \xi}>0$;
- (iv)
- $\frac{\partial {I}^{{i}^{*}}}{\partial \gamma}>0$,$\frac{\partial {s}_{1}{i}^{*}}{\partial \gamma}<0$,$\frac{\partial E{G}^{{i}^{*}}}{\partial \gamma}>0$;

^{i*}of port enterprise, the optimal unit subsidy s

_{1}

^{i*}of government and port VERs EG

^{i*}will increase. This is consistent with intuition. The reason is that the increase in parameter β will help port enterprises improve their ERI level to significantly promote port emission reduction, and the government will also enhance the unit subsidy incentives for port enterprises. According to Proposition 4 (ii), the optimal solutions I

^{i*}, s

_{1}

^{i*}, and EG

^{i*}in different cases are positively correlated with parameter h. This implies that when port emission reduction becomes more important and has greater social benefits, port enterprises will have greater motivation to improve their ERI enthusiasm, the government will also increase unit subsidies to port enterprises, and port emission reduction will also be positively affected. Proposition 4 (iii) and (iv) show that the optimal solutions I

^{i*}and EG

^{i*}in different cases are both positively related to the parameters ξ and γ, while s

_{1}

^{i*}is negatively related to the parameters ξ and γ. This means that when the service price of the port increases or customers become more aware of environmental protection, port enterprises will make more efforts to improve the ERI level of the port, and the government will appropriately reduce the unit subsidy incentives for port enterprises in order to balance the income of port enterprises.

**Proposition**

**5.**

^{2}, θ), contract execution efficiency (i.e., g

_{0}, g

_{1}), and execution cost (i.e., c) on the optimal decisions of government and port enterprise are analyzed as follows:

- (i)
- $\frac{\partial {I}^{{i}^{*}}}{\partial {\delta}^{2}}<0$,$\frac{\partial {s}_{1}{i}^{*}}{\partial {\delta}^{2}}<0$,$\frac{\partial E{G}^{{i}^{*}}}{\partial {\delta}^{2}}<0$;
- (ii)
- $\frac{\partial {I}^{{i}^{*}}}{\partial \theta}<0$,$\frac{\partial {s}_{1}{i}^{*}}{\partial \theta}<0$,$\frac{\partial E{G}^{{i}^{*}}}{\partial \theta}<0$;
- (iii)
- $\frac{\partial {I}^{{N}^{*}}}{\partial {g}_{0}}>0$,$\frac{\partial {s}_{1}{N}^{*}}{\partial {g}_{0}}>0$,$\frac{\partial E{G}^{{N}^{*}}}{\partial {g}_{0}}>0$,$\frac{\partial {I}^{{j}^{*}}}{\partial {g}_{1}}>0$,$\frac{\partial {s}_{1}{j}^{*}}{\partial {g}_{1}}>0$,$\frac{\partial E{G}^{{j}^{*}}}{\partial {g}_{1}}>0$;
- (iv)
- $\frac{\partial {I}^{{N}^{*}}}{\partial c}<0$,$\frac{\partial {s}_{1}{N}^{*}}{\partial c}<0$,$\frac{\partial E{G}^{{N}^{*}}}{\partial c}<0$;

^{2}of market random disturbance factor v and the degree θ of risk aversion of the government and port enterprise to their respective benefits due to the uncertainty of emission reduction. From Proposition 5 (i) and (ii), it can be seen that the optimal ERI level I

^{i*}of port enterprise, the optimal government unit subsidy s

_{1}

^{i*}, and port VERs EG

^{i*}are negatively correlated with parameters δ

^{2}and θ. This implies that when the uncertainty of port emission reduction increases, such as the difference in customers’ preferences for green ports, the competition between traditional facility ports, and emission reduction investment facility ports, the enthusiasm of port enterprises’ emission reduction investment will be hit, port VERs will also be reduced, and the government’s unit subsidy incentive to port enterprises will also be weakened. At the same time, when the degree of risk aversion increases, the cost of risk aversion of the government and port enterprises to their respective benefits increases. At this time, port enterprises are unwilling to make more efforts to reduce costs, and the government is also unwilling to provide more unit subsidies and incentives. Accordingly, the VERs of port enterprises will also be reduced. Proposition 5 (iii) uncovers that when the contract execution efficiency increases, the optimal solutions in different cases will increase. This means that blockchain smart contracts to improve the efficiency of contract execution will help improve the emission reduction investment ability of port enterprises, promote port emission reduction, and enhance the government’s subsidy incentives for port enterprises. Proposition 5 (iv) confirms that the increase in contract execution cost c is not conducive to the emission reduction investment of port enterprises, and will reduce the unit subsidy incentive of the government and the VERs of ports.

**Proposition**

**6.**

_{b}, r, ω) on the optimal decisions of government and port enterprise are analyzed as follows:

- (i)
- $\frac{\partial {I}^{{j}^{*}}}{\partial {C}_{b}}<0$,$\frac{\partial {s}_{1}{j}^{*}}{\partial {C}_{b}}=0$,$\frac{\partial E{G}^{{j}^{*}}}{\partial {C}_{b}}<0$;
- (ii)
- $\frac{\partial {I}^{{j}^{*}}}{\partial r}>0$,$\frac{\partial {s}^{{j}^{*}}}{\partial r}>0$,$\frac{\partial E{G}^{{j}^{*}}}{\partial r}>0$;
- (iii)
- $\frac{\partial {I}^{{j}^{*}}}{\partial \omega}>0$,$\frac{\partial {s}_{1}{j}^{*}}{\partial \omega}>0$,$\frac{\partial E{G}^{{j}^{*}}}{\partial \omega}>0$;

^{j*}and VERs EG

^{j*}of port enterprise are negatively correlated with the unit operating cost C

_{b}of blockchain, but the optimal unit subsidy s

_{1}

^{j*}of government is independent of the parameter C

_{b}. This implies that the increase in the unit operating cost of the blockchain is unfavorable to the emission reduction of port enterprises, but the government’s unit subsidy incentive has not been reduced. The reason may be that the government is optimizing its dynamic incentive contract by adjusting the fixed subsidies to port enterprises at this time while keeping the unit subsidies unchanged helps to maintain the enthusiasm of port enterprises’ emission reduction investment to a certain extent. From Proposition 6, it is clear that the optimal solutions for both the government and port enterprise are positively related to the parameters r and ω. This means that if blockchain technology enhances customers’ green trust in port ERI, port enterprises’ enthusiasm to participate in the implementation of blockchain and invest in emission reduction will increase. The more port emission reductions are expected to be achieved, the more motivated the government will be to increase subsidies and incentives for port enterprises. At the same time, the higher the disclosure of port emission reduction information by the blockchain, the lower the uncertainty of the market, and the greater the optimal solutions of the government and port enterprises.

**Proposition**

**7.**

^{TB*}of port enterprise and port VERs EG

^{TB*}are positively correlated with the parameter τ, while the optimal unit subsidy s

_{1}

^{TB*}of the government to port enterprise is independent of the parameter τ. This implies that when the government starts the carbon emissions trading market for port enterprises, the increase in the carbon trading market price will help to stimulate the port enterprises’ emission reduction efforts and improve the VERs of ports, but the unit subsidy in the government dynamic incentive contract should remain unchanged. The reason may be that the government is optimizing its dynamic incentive contract by adjusting the fixed subsidies to port enterprises. At the same time, port enterprises have benefited from the carbon trading market, and the government does not need to provide additional unit subsidy incentives for their emission reductions.

#### 5.3. Effects of Blockchain Technology and Carbon Trading Policy

**Proposition**

**8.**

- (i)
- $\left\{\begin{array}{l}if{C}_{b}=0,then{I}^{{B}^{*}}{I}^{{N}^{*}}\\ if{C}_{b}0,\left\{\begin{array}{l}and{C}_{b}\le {C}_{b}{}^{th2},then{I}^{{B}^{*}}\ge {I}^{{N}^{*}}\\ and{C}_{b}{}^{th2}{C}_{b},then{I}^{{B}^{*}}{I}^{{N}^{*}}\end{array}\right.\end{array}\right.$;
- (ii)
- s
^{B*}> s^{N*}; - (iii)
- $\left\{\begin{array}{l}if{C}_{b}=0,thenE{G}^{{B}^{*}}E{G}^{{N}^{*}}\\ if{C}_{b}0,\left\{\begin{array}{l}and{C}_{b}\le {C}_{b}{}^{th3},thenE{G}^{{B}^{*}}\ge E{G}^{{N}^{*}}\\ and{C}_{b}{}^{th3}{C}_{b},thenE{G}^{{B}^{*}}E{G}^{{N}^{*}}\end{array}\right.\end{array}\right.$;

^{B*}> s

^{N*}). This implies that after the implementation of blockchain, the government needs to improve the unit subsidy incentives for port enterprises to increase the marginal income of port enterprises, and indirectly share a certain unit operating cost of blockchain for port enterprises, so as to encourage port enterprises to participate in the implementation of blockchain. Proposition 8 also shows that when the government’s dynamic incentive contract for the port enterprise is changed from the traditional mode to the blockchain technology mode, if the unit operating cost of the blockchain is ignored, that is, C

_{b}= 0, the optimal ERI level of port enterprise and port VERs will increase. At this time, the implementation of blockchain is completely beneficial to the government’s incentive for the port enterprise’s emission reduction. However, if the unit operation cost of blockchain is considered, that is, C

_{b}> 0, the implementation of blockchain is not necessarily beneficial to the port emission reduction investment. When C

_{b}is large, the marginal income of port enterprises participating in the implementation of blockchain will be reduced, its ERI level will be reduced, and the expected port emission reduction will also be reduced. At this time, the implementation of blockchain is unfavorable.

**Proposition**

**9.**

^{TB*}> I

^{B*}; s

^{TB*}= s

^{B*}; EG

^{TB*}> EG

^{B*}.

#### 5.4. Values of Blockchain Technology and Carbon Trading Policy

**Proposition**

**10.**

- (i)
- When ignoring the fixed cost of establishing the blockchain (as a sunk cost), i.e., F
_{b}= 0, we have$$\left\{\begin{array}{l}if\Phi \le \frac{\left(Y(\cdot )-X(\cdot )\right){g}_{0}{g}_{1}\rho}{\left({g}_{1}\left(1+c\right)-{g}_{0}\right)},then{V}_{g}{B}^{*}\le {V}_{g}{N}^{*}\\ if\Phi \frac{\left(Y(\cdot )-X(\cdot )\right){g}_{0}{g}_{1}\rho}{\left({g}_{1}\left(1+c\right)-{g}_{0}\right)},then{V}_{g}{B}^{*}{V}_{g}{N}^{*}\end{array}\right.;$$ - (ii)
- when considering the fixed cost of establishing the blockchain, that is, F
_{b}> 0, we have$$\left\{\begin{array}{l}if\Phi \le \frac{\left(Y(\cdot )-X(\cdot )\right){g}_{0}{g}_{1}\rho}{\left({g}_{1}\left(1+c\right)-{g}_{0}\right)},then{V}_{g}{B}^{*}{V}_{g}{N}^{*}\\ if\Phi \frac{\left(Y(\cdot )-X(\cdot )\right){g}_{0}{g}_{1}\rho}{\left({g}_{1}\left(1+c\right)-{g}_{0}\right)},\left\{\begin{array}{l}and{F}_{b}\le \frac{\left({g}_{1}\left(1+c\right)-{g}_{0}\right)\Phi}{{g}_{0}{g}_{1}\rho}-\left(Y(\cdot )-X(\cdot )\right),then{V}_{g}{B}^{*}\ge {V}_{g}{N}^{*}\\ and{F}_{b}\frac{\left({g}_{1}\left(1+c\right)-{g}_{0}\right)\Phi}{{g}_{0}{g}_{1}\rho}-\left(Y(\cdot )-X(\cdot )\right),then{V}_{g}{B}^{*}{V}_{g}{N}^{*}\end{array}\right.\end{array}\right.;$$

_{0}of the port enterprise ERI is greater than a certain critical point (corresponding to Region II in Figure 2a), the government’s expected benefit under blockchain technology is higher than that without blockchain; Otherwise, the result is the opposite. This implies that if the fixed cost of blockchain is taken as a sunk cost and not considered in the relevant decisions of the government, the opportunity cost of port enterprises to invest in emission reduction and actively build green ports determines whether the establishment of blockchain in the government’s dynamic incentive contract is beneficial. This finding is interesting and non-intuitive, and the reason can be explained as follows: according to the optimal solution of the model, it is known that there exists ∂V

_{g}

^{N}/∂Φ = −(1 + c)/g

_{0}⋅ ρ < 0 in the absence of blockchain, while there exists ∂V

_{g}

^{B}/∂Φ = −1/g

_{1}⋅ ρ < 0 under blockchain technology, and the system satisfies |∂V

_{g}

^{N}/∂Φ| > |∂V

_{g}

^{B}/∂Φ|. Thus, although the parameter Φ is detrimental to the government’s expected benefit when the government dynamically incentivizes the port enterprise to reduce emissions, interestingly, blockchain can reduce this detrimental effect. Thus, the implementation of blockchain technology is beneficial when Φ is greater than a certain threshold.

_{b}is less than a critical point, corresponding to Region II in Figure 2b, the expected benefit of the government under blockchain technology is higher than that without blockchain; Otherwise, the result is opposite. This means that if the fixed cost of blockchain is taken into account in the government dynamic incentive contract, the conditions for the government to implement blockchain to improve the expected benefit are related not only to the opportunity cost of the ERI of port enterprises, but also to the fixed cost of blockchain.

**Proposition**

**11.**

- (i)
- When the unit operation cost of implementing blockchain is ignored, that is, C
_{b}= 0, we have$$\left\{\begin{array}{l}if\psi \le {g}_{1}\rho Z(\cdot ),then{V}_{g}{TB}^{*}\ge {V}_{g}{B}^{*}\\ if\psi {g}_{1}\rho Z(\cdot ),then{V}_{g}{TB}^{*}{V}_{g}{B}^{*}\end{array}\right.;$$ - (ii)
- When considering the unit operation cost of implementing the blockchain, that is, C
_{b}> 0, we have$$\left\{\begin{array}{l}if\psi \le {g}_{1}\rho Z(\cdot ),\left\{\begin{array}{l}and{C}_{b}\le \frac{({g}_{1}\rho Z(\cdot )-\psi ){g}_{1}\eta \rho \sigma {\left(\rho +\sigma \right)}^{2}}{{g}_{1}\rho \left(2\rho +\sigma \right){\left(r+\beta \right)}^{2}},then{V}_{g}{TB}^{*}\ge {V}_{g}{B}^{*}\\ and{C}_{b}\frac{({g}_{1}\rho Z(\cdot )-\psi ){g}_{1}\eta \rho \sigma {\left(\rho +\sigma \right)}^{2}}{{g}_{1}\rho \left(2\rho +\sigma \right){\left(r+\beta \right)}^{2}},then{V}_{g}{TB}^{*}{V}_{g}{B}^{*}\end{array}\right.\\ \\ if\psi {g}_{1}\rho Z(\cdot ),then{V}_{g}{TB}^{*}{V}_{g}{B}^{*}\\ \end{array}\right.;$$

_{b}= 0, if the critical value ψ of carbon emission reduction of the port enterprise is small, corresponding to Region I in Figure 3a, the expected benefit of the government under the carbon trading policy is higher than that without carbon trading. If ψ is large, it corresponds to Region II in Figure 3a, and the result is opposite. This implies that after the launch of the carbon trading market, the government can appropriately increase the carbon quota of port enterprises, reduce the critical value of carbon emission reduction of port enterprises so that port enterprises have more opportunities to sell excess carbon quota in the carbon trading market, and drive port enterprises to actively invest in emission reduction, so as to improve social benefits and realize the positive value of carbon trading policy. According to Proposition 11, when considering the unit operation cost of implementing the blockchain, that is, C

_{b}> 0, as shown in Figure 3b, if and only if the parameter ψ is small and C

_{b}is also small, corresponding to Region I in Figure 3b, the expected benefit of the government under the carbon trading policy is higher than that of the carbon-free trading; Otherwise, the result is opposite. This finding is non-intuitive. When the carbon trading market is launched under blockchain technology, the unit operating cost C

_{b}of the blockchain will affect the value of carbon trading policy. The reason is that the parameter C

_{b}directly affects the ERI level of port enterprises. If the parameter C

_{b}is large, it is difficult for port enterprises to make more efforts to improve the carbon emission reduction generated by emission reduction investment, and port enterprises will have little opportunity to sell carbon credits from carbon trading. At this time, the government’s carbon trading policy will not be conducive to encouraging port enterprises to reduce emissions.

## 6. Numerical Analysis

^{2}= 1; η = 0.15; θ = 0.2; σ = 0.3; ρ = 0.1; β = 0.4; g

_{0}= 0.6; h = 1.501; γ = 1.2; ξ = 1; Φ = 23.1; g

_{1}= 0.8; c = 0.02; C

_{b}= 0.72; F

_{b}= 1.9; r = 0.1; ω = 0.2; ψ = 5.1; τ = 0.004.

#### 6.1. The Optimal Dynamic Trajectories of Government Incentive Strategy

_{0}*(t) and dynamic incentive contract S*(t) both decrease with time and then tend to be stable. At the same time, in a period of time after the implementation of the government dynamic incentive contract, the discount value of the government expected benefit V

_{g}*(t) under different cases increases with time and then tends to be stable. This is consistent with the relevant conclusions of Propositions 1–3 in the previous sections.

#### 6.2. Impacts of Key Parameters on the Government Dynamic Incentive Strategy

^{2}, θ), contract execution efficiency (g

_{0}, g

_{1}) and execution cost c, and blockchain and carbon trading-related parameters (C

_{b}, r, ω, τ).

^{2}of market random disturbance factor v and the degree θ of risk aversion of the government and port enterprise to their respective revenues increase, the port VERs (i.e., data in Rows 3, 7, and 11 of Table 4) and the discount value of government expected benefit (i.e., data in Rows 6, 10 and 14 of Table 4) in different cases decrease, while the government fixed subsidy (i.e., data in Rows 4, 8 and 12 of Table 4) and government dynamic incentive contract (i.e., data in Rows 5, 9 and 13 of Table 4) increase. This implies that market uncertainty is unfavorable for port emission reduction and the expected benefit of the government. At the same time, in order to reduce the impact of market uncertainty on port enterprises, the government will increase the dynamic incentive contract for port enterprises, so as to enhance the anti-risk ability of port enterprises and promote port emission reduction.

_{0}and unit execution cost c on government incentive strategy for the case N of no blockchain. As can be seen from Figure 7, the port VERs and the discount value of the government’s expected benefit are both positively correlated with the parameter g

_{0}, while the government’s fixed subsidy and dynamic incentive contract are both negatively correlated with the parameter g

_{0}. This indicates that the government should actively improve the contract execution efficiency of port enterprises, which will not only reduce the government’s contract expenditure, but also promote the emission reduction power of port enterprises and improve the government’s expected benefit. However, according to the observation in Figure 8, we find that the port’s VERs and the discount value of government expected benefit are negatively correlated with parameter c, while government fixed subsidy is positively correlated with parameter c, and parameter c has a weak impact on government dynamic incentive contract. This is consistent with intuition. When the government’s information cost and supervision cost to implement the contract increase, the government will increase the fixed subsidy to port enterprise and maintain the dynamic incentive contract unchanged. However, according to Proposition 5, the government’s unit subsidy and the investment level of port enterprise’s emission reduction will both decrease, which is unfavorable for port emission reduction and government benefit.

_{b}of the blockchain is generally unfavorable to the emission reduction of port enterprise and the government.

_{1}on the government’s dynamic incentive strategy in Case B of blockchain technology and in the case TB of considering carbon trading policy under blockchain technology. The result is similar to the effect of parameter g

_{0}in Figure 7, and will not be repeated here. In addition, it can be seen from Figure 10 that the effect of blockchain on the degree ω of disclosure of port emission reduction information on the government incentive strategy is similar to the effect of parameter r. The reason is that the information disclosure effect of blockchain reduces the uncertainty of the market, reduces the risk avoidance cost of the government and port enterprises, promotes the emission reduction of port enterprises, and improves the expected benefit of the government.

^{TB*}in Figure 11a and V

_{g}

^{TB*}in Figure 11d increase, while both s

_{0}

^{TB*}in Figure 11b and S

^{TB*}in Figure 11c decrease, but the relative decrease in STB* is not significant. This finding confirms that an increase in the price τ is usually beneficial to the reduction of emissions by port enterprises and the desired benefits to the government, and therefore, it is beneficial for the government to further exploit the value of carbon trading policy by regulating and appropriately increasing the price τ.

## 7. Conclusions

#### 7.1. Key Findings

- (1)
- This paper determines the optimal dynamic trajectory change rules of the government’s incentive strategy for port emission reduction under different cases (see Propositions 1–3). We find that under the government dynamic incentive contract, the optimal dynamic trajectory of port VERs in different cases will first monotonously increase and then tend to steady state with the passage of time. However, the optimal dynamic trajectories of the government’s fixed subsidy for port enterprises, the incentive contract, and the discount value of the government’s expected benefit are different in different cases. Especially after the implementation of the blockchain, the government’s dynamic incentive strategy for port emission reduction is related to the unit operating cost of the blockchain. After the carbon trading policy is launched, the government’s dynamic incentive strategy is not only related to the unit operating cost of the blockchain, but also related to the price of the carbon trading market.
- (2)
- This paper reveals the impacts of relevant parameters on the equilibrium solutions of the government and port enterprise in different cases (see Propositions 4–7). We find that the equilibrium solutions in different cases are positively correlated with the influencing factor of the investment level of port emission reduction on its emission reductions, the monetary expression of the social benefit generated by unit emission reduction, and the contract execution efficiency, while they are negatively correlated with the variance of market random disturbance factor, the degree of risk aversion and the contract execution cost. In addition, when the port service price increases, the optimal ERI level and VERs of port enterprises in different cases will increase, while the optimal unit subsidy in the government dynamic incentive contract will decrease. Under blockchain technology, the equilibrium solutions of the government and port enterprise are positively correlated with the green trust coefficient of shipping customers to port ERI level and the disclosure degree of blockchain to market information. Moreover, the optimal ERI level and VERs of port enterprises are negatively correlated with the unit operating cost of blockchain, while the government’s optimal unit subsidy has nothing to do with the unit operating cost of blockchain. When considering carbon trading policy, the increase in carbon trading market price will positively affect the optimal ERI level and VERs of port enterprises, but will not affect the optimal unit subsidy of the government.
- (3)
- This paper compares and analyzes the equilibrium solutions in different cases, and gives the effects of blockchain and carbon trading policy on the optimal decisions of the government and port enterprises (see Propositions 8–9). We find that compared with the traditional mode without blockchain, the government’s optimal unit subsidy under blockchain technology will increase. Moreover, if the unit operation cost of blockchain is ignored, the optimal ERI level and VERs of port enterprises will also increase. However, if the unit operation cost of blockchain is considered, the optimal ERI level and VERs of port enterprises will not necessarily increase. In addition, compared with Case B of blockchain technology, the optimal ERI level and VERs of port enterprises will increase under the TB case of considering carbon trading under blockchain technology, while the optimal unit subsidy of the government will remain unchanged.
- (4)
- This paper determines the influencing factors and specific conditions for the government to implement the blockchain and start the carbon trading policy (see Propositions 10–11). We find that when the fixed cost of establishing blockchain is ignored, only if the opportunity cost Φ of port enterprise is greater than a critical point, the expected benefit of the government under blockchain technology will be higher than that under the traditional mode without blockchain. However, when the fixed cost of blockchain is considered, only if the opportunity cost Φ is greater than a certain threshold and the fixed cost of blockchain is less than a certain threshold are simultaneously satisfied, the expected benefit of the government under blockchain technology will be higher than that under the traditional mode without blockchain. In addition, when the unit operation cost of implementing the blockchain is ignored, only if the critical value of carbon emission reduction of port enterprises is small, the expected benefit of the government under the carbon trading policy is higher than that under the carbon-free trading policy. However, when the unit operation cost of blockchain is considered, only if the critical value is small and the unit operation cost of the blockchain is small, the expected benefit of the government under the carbon trading policy is higher than that of the carbon-free trading.
- (5)
- Through the numerical simulation, we confirm that the reasonable implementation of blockchain technology and carbon trading policy will help to improve the VERs of port enterprise and the expected benefit of the government, and the government’s incentive strategy for port emission reduction is sensitive to the changes of relevant parameters under different cases, especially the increase in carbon trading market price is usually conducive to promoting the enthusiasm of port emission reduction and improving the expected benefit of the government.

#### 7.2. Managerial Insights

#### 7.3. Limitations and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Description |

ERI | Emission reduction investment |

CERs | Carbon emission reductions |

VERs | Verified emission reductions |

v | Market random disturbance factor, v~N(0, δ^{2}) |

δ^{2} | Variance of random disturbance factor v |

η | The investment cost coefficient of port enterprise |

θ | Degree of risk aversion |

σ | Attenuation rate of port emission reduction |

ρ | Discount rate |

β | Influence factor of port ERI on CERs |

g | Contract execution efficiency |

c | Unit execution cost of contract |

C_{b} | Unit operation cost of blockchain |

F_{b} | Fixed cost of blockchain |

r | Green trust coefficient of customers on port ERI |

h | Monetary expression of social benefits generated by unit CERs |

γ | Impact of port’s CERs on shipping customer demand |

ξ | Service price of port |

Φ | Opportunity cost of ERI in port enterprise |

τ | Carbon trading market price |

ψ | A critical value of carbon emission reduction to achieve carbon trading |

ω | Disclosure degree of blockchain for port emission reduction market information |

I | The port’s ERI level (decision variable) |

s_{0} | Fixed subsidy paid by the government to port enterprise (decision variable) |

s_{1} | Unit subsidy paid by the government to port enterprise (decision variable) |

G | CERs of port enterprise |

EG | VERs of port enterprise |

S | Dynamic incentive contract of government (S = s_{0} + s_{1}⋅G) |

π_{p} | Expected revenue of port enterprise |

π_{g} | Government expected benefit |

V_{p} | Expected discounted profit of port enterprise |

V_{g} | Discount value of government expected benefit |

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

^{N}, we obtain:

_{p}

^{N}(EG

^{N}) = l

_{p}

_{1}

^{N}⋅ EG

^{N}+ l

_{p}

_{2}

^{N}, where l

_{p}

_{1}

^{N}and l

_{p}

_{2}

^{N}are undetermined constant coefficients. In order to obtain the undetermined constant coefficient, the optimal expected value function V

_{p}

^{N}(EG

^{N}) and its derivative V

_{p}

^{N}

^{′}(EG

^{N}) = l

_{p}

_{1}

^{N}are substituted into the HJB Equation (A2), and according to the identity relationship, l

_{p}

_{1}

^{N}= (s

_{1}⋅ g

_{0}+ γξ)/(ρ + σ) can be obtained, which is substituted into Equation (A1), and Equation (10) in Lemma 1 can be obtained. □

**Proof**

**of**

**Theorem**

**1.**

_{1}

^{N*}in Theorem 1 can be obtained. By introducing it into Equation (10) in Lemma 1, Equation (16) in Theorem 1 can be obtained. □

**Proof**

**of**

**Corollary**

**1.**

^{N}* and s

_{1}

^{N}* of the government and port enterprise into the port VERs state equation of Equation (5), we can obtain:

^{N*}(t) of port VERs, that is, Equation (18) in Corollary 1. Further, EG

^{N*}(t) and Equation (13) are combined with Equation (15) to deduce Equations (18)–(20) in Corollary 1. □

**Proof**

**of**

**Lemma**

**2.**

**Proof**

**of**

**Theorem**

**2.**

_{1}

^{B}(t) in Case B. □

**Proof**

**of**

**Corollary**

**2.**

**Proof**

**of**

**Lemma**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Corollary**

**3.**

**Proof**

**of**

**Proposition**

**1.**

^{N*}(t) of port enterprise approved by the government, the fixed subsidy b

_{0}

^{N*}(t) of the government for port enterprises, the discount value of government expected benefit V

_{g}

^{N*}(t) and the government dynamic incentive contract S

^{N*}(t) = s

_{0}

^{N*}(t) + s

_{1}

^{N*}⋅ EG

^{N*}(t), we can derive that when t → +∞, e

^{−σ}

^{⋅t}→ 0, then EG

^{N*}(t) = EG

^{N*}= β

^{4}(cγξ + g

_{0}h + γξ)/(σ(1 + c)η(ρ + σ)(β

^{2}+ δ

^{2}ηθ (ρ + σ)

^{2})), s

_{0}

^{N*}(t) = s

_{0}

^{N*}(EG

^{N*}), S

^{N*}(t) = S

^{N*}(EG

^{N*}), V

_{g}

^{N*}(t) = V

^{N*}(EG

^{N*}). Thus, there is t = t

_{th}

^{N}(t

_{th}

^{N}→ +∞), and when t ≥ t

_{th}

^{N}, then ∂EG

^{N*}(t)/∂t = 0, ∂s

_{0}

^{N*}(t)/∂t = 0, ∂S

^{N*}(t)/∂t = 0, ∂V

_{g}

^{N*}(t)/∂t = 0; when t < t

_{th}

^{N}, EG

^{N*}(t), s

_{0}

^{N*}(t), S

^{N*}(t) and V

_{g}

^{N*}(t) are respectively calculated for the first derivative of time t and judged positive or negative, then Proposition 1 can be obtained. □

**Proofs**

**of**

**Propositions**

**2**

**and**

**3.**

**Proofs**

**of**

**Propositions**

**4**–

**7.**

**Proof**

**of**

**Proposition**

**8.**

**Proof**

**of**

**Proposition**

**9.**

**Proof**

**of**

**Proposition**

**10.**

_{g}

^{B*}from V

_{g}

^{N*}at steady state (i.e., t → +∞) and judging the positive or negative according to whether or not to consider the fixed cost of the blockchain (i.e., F

_{b}= 0 and F

_{b}> 0), we can obtain Proposition 10. □

**Proof**

**of**

**Proposition**

**11.**

_{g}

^{TB*}from V

_{g}

^{B*}at steady state (i.e., t → +∞) and judging the positive and negative depending on whether the unit operating cost of the blockchain (i.e., C

_{b}= 0 and C

_{b}> 0) is considered, we can obtain Proposition 11. □

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**Figure 1.**Model structure of government’s dynamic incentives for port emission reduction under blockchain technology.

**Figure 2.**Comparison of expected benefit of the government before and after the adoption of blockchain: (

**a**) F

_{b}= 0; (

**b**) F

_{b}> 0.

**Figure 3.**Comparison of government expected benefit before and after the start of carbon trading policy under blockchain technology: (

**a**) C

_{b}= 0; (

**b**) C

_{b}> 0.

**Figure 5.**Time variation trajectories of government’s optimal dynamic incentive contract under different cases: (

**a**) s

_{0}

^{N*}, s

_{0}

^{B*}and s

_{0}

^{TB*}; (

**b**) S

^{N*}, S

^{B*}and S

^{TB*}.

**Figure 6.**Optimal dynamic trajectories of the discount value of government expected benefit over time under different cases.

**Figure 7.**Effect of contract execution efficiency g

_{0}on the government’s dynamic incentive strategy in Case N.

**Figure 8.**Effect of the unit execution cost c of the contract on the government’s dynamic incentive strategy in Case N.

**Figure 9.**Effects of the parameters r and C

_{b}on the government’s dynamic incentive strategy in the B and TB cases: (

**a**) EG

^{B*}and EG

^{TB*}; (

**b**) s

_{0}

^{B*}and s

_{0}

^{TB*}; (

**c**) S

^{B*}and S

^{TB*}; (

**d**) V

^{B*}and V

_{g}

^{TB*}.

**Figure 10.**Effects of the parameters g

_{1}and ω on the government’s dynamic incentive strategy in the B and TB cases: (

**a**) EG

^{B*}and EG

^{TB*}; (

**b**) s

_{0}

^{B*}and s

_{0}

^{TB*}; (

**c**) S

^{B*}and S

^{TB*}; (

**d**) V

^{B*}and V

_{g}

^{TB*}.

**Figure 11.**Effect of the carbon trading market price τ on the government’s dynamic incentive strategy in Case B.

References | Blockchain Adoption | Shipping Industry | Port Emission Reduction | Government Subsidy | Carbon Trading Policy | |
---|---|---|---|---|---|---|

Dynamic Incentive | Static Incentive | |||||

[27,28] | No | No | Yes | No | No | No |

[29,33] | No | No | Yes | No | Yes | No |

[20,21,22,23] | Yes | Yes | No | No | No | No |

[10,11,12,13,14] | Yes | No | No | No | No | No |

[51,52] | No | No | No | Yes | No | No |

[39] | No | No | Yes | No | Yes | Yes |

[36,37,40] | No | No | Yes | No | No | Yes |

[31,32] | No | No | Yes | Yes | No | No |

This paper | Yes | Yes | Yes | Yes | Yes | Yes |

Parameters | β | 0.39925 | 0.3995 | 0.39975 | 0.4 | 0.4 | 0.4 | 0.4 |
---|---|---|---|---|---|---|---|---|

h | 1.495 | 1.495 | 1.495 | 1.495 | 1.4975 | 1.5 | 1.5025 | |

Case N | EG^{N*} | 17.876 | 17.899 | 17.922 | 17.945 | 17.958 | 17.971 | 17.983 |

s_{0}^{N*} | 1.590 | 1.541 | 1.491 | 1.441 | 1.389 | 1.336 | 1.284 | |

S^{N*} | 25.980 | 25.964 | 25.948 | 25.932 | 25.939 | 25.947 | 25.955 | |

V_{g}^{N*} | 0.015 | 0.525 | 1.034 | 1.544 | 2.097 | 2.651 | 3.206 | |

Case B | EG^{B*} | 22.705 | 22.728 | 22.751 | 22.774 | 22.802 | 22.829 | 22.856 |

s_{0}^{B*} | 0.487 | 0.457 | 0.426 | 0.396 | 0.327 | 0.258 | 0.189 | |

S^{B*} | 33.399 | 33.403 | 33.407 | 33.412 | 33.438 | 33.465 | 33.492 | |

V_{g}^{B*} | 1.312 | 1.617 | 1.921 | 2.226 | 2.929 | 3.633 | 4.338 | |

Case TB | EG^{TB*} | 22.760 | 22.783 | 22.807 | 22.830 | 22.857 | 22.885 | 22.912 |

s_{0}^{TB*} | 0.370 | 0.340 | 0.309 | 0.279 | 0.210 | 0.140 | 0.071 | |

S^{TB*} | 33.362 | 33.367 | 33.371 | 33.375 | 33.402 | 33.429 | 33.456 | |

V_{g}^{TB*} | 2.503 | 2.809 | 3.115 | 3.421 | 4.126 | 4.832 | 5.538 |

Parameters | ξ | 0.9975 | 0.998 | 0.9985 | 0.9990 | 0.9990 | 0.9990 | 0.9990 |
---|---|---|---|---|---|---|---|---|

γ | 1.1970 | 1.1970 | 1.1970 | 1.1970 | 1.1975 | 1.198 | 1.1985 | |

Case N | EG^{N*} | 17.924 | 17.929 | 17.934 | 17.940 | 17.944 | 17.948 | 17.952 |

s_{0}^{N*} | 1.529 | 1.507 | 1.486 | 1.465 | 1.447 | 1.429 | 1.411 | |

S^{N*} | 26.098 | 26.083 | 26.068 | 26.054 | 26.041 | 26.029 | 26.017 | |

V_{g}^{N*} | 0.587 | 0.815 | 1.043 | 1.271 | 1.462 | 1.652 | 1.843 | |

Case B | EG^{B*} | 22.758 | 22.766 | 22.774 | 22.783 | 22.789 | 22.796 | 22.803 |

s_{0}^{B*} | 0.437 | 0.416 | 0.396 | 0.375 | 0.358 | 0.341 | 0.323 | |

S^{B*} | 33.566 | 33.557 | 33.548 | 33.539 | 33.532 | 33.524 | 33.517 | |

V_{g}^{B*} | 1.784 | 1.997 | 2.209 | 2.422 | 2.599 | 2.777 | 2.955 | |

Case TB | EG^{TB*} | 22.814 | 22.822 | 22.830 | 22.838 | 22.845 | 22.852 | 22.859 |

s_{0}^{TB*} | 0.320 | 0.299 | 0.279 | 0.258 | 0.241 | 0.223 | 0.206 | |

S^{TB*} | 33.530 | 33.521 | 33.512 | 33.503 | 33.495 | 33.488 | 33.480 | |

V_{g}^{TB*} | 2.979 | 3.192 | 3.405 | 3.618 | 3.796 | 3.974 | 4.152 |

Parameters | δ^{2} | 1.025^{2} | 1.05^{2} | 1.075^{2} | 1.1^{2} | 1.1^{2} | 1.1^{2} | 1.1^{2} |
---|---|---|---|---|---|---|---|---|

θ | 0.19 | 0.19 | 0.19 | 0.19 | 0.1925 | 0.195 | 0.1975 | |

Case N | EG^{N*} | 17.977 | 17.951 | 17.925 | 17.898 | 17.890 | 17.882 | 17.874 |

s_{0}^{N*} | 1.310 | 1.450 | 1.592 | 1.738 | 1.780 | 1.823 | 1.865 | |

S^{N*} | 25.949 | 25.967 | 25.986 | 26.004 | 26.010 | 26.015 | 26.021 | |

^{VgN*} | 2.899 | 2.218 | 1.522 | 0.811 | 0.604 | 0.396 | 0.189 | |

Case B | EG^{B*} | 22.841 | 22.817 | 22.791 | 22.766 | 22.758 | 22.751 | 22.743 |

s_{0}^{B*} | 0.226 | 0.323 | 0.421 | 0.521 | 0.550 | 0.580 | 0.609 | |

S^{B*} | 33.4757 | 33.4860 | 33.4966 | 33.507 | 33.510 | 33.514 | 33.517 | |

V_{g}^{B*} | 3.937 | 3.356 | 2.761 | 2.153 | 1.975 | 1.798 | 1.620 | |

Case TB | EGT^{B*} | 22.897 | 22.872 | 22.847 | 22.821 | 22.814 | 22.807 | 22.799 |

s_{0}T^{B*} | 0.109 | 0.205 | 0.304 | 0.404 | 0.433 | 0.463 | 0.492 | |

S^{TB*} | 33.4391 | 33.4495 | 33.4601 | 33.471 | 33.474 | 33.477 | 33.480 | |

V_{g}T^{B*} | 5.137 | 4.555 | 3.960 | 3.352 | 3.174 | 2.996 | 2.818 |

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## Share and Cite

**MDPI and ACS Style**

Sun, Z.; Xu, Q.; Liu, J.
Dynamic Incentive Contract of Government for Port Enterprises to Reduce Emissions in the Blockchain Era: Considering Carbon Trading Policy. *Sustainability* **2023**, *15*, 12148.
https://doi.org/10.3390/su151612148

**AMA Style**

Sun Z, Xu Q, Liu J.
Dynamic Incentive Contract of Government for Port Enterprises to Reduce Emissions in the Blockchain Era: Considering Carbon Trading Policy. *Sustainability*. 2023; 15(16):12148.
https://doi.org/10.3390/su151612148

**Chicago/Turabian Style**

Sun, Zhongmiao, Qi Xu, and Jinrong Liu.
2023. "Dynamic Incentive Contract of Government for Port Enterprises to Reduce Emissions in the Blockchain Era: Considering Carbon Trading Policy" *Sustainability* 15, no. 16: 12148.
https://doi.org/10.3390/su151612148