Research on Construction Waste Recycling Subsidy Model Considering Contractor’s Environmental Awareness

Abstract

Considering the environmental awareness of contractors, this paper constructs a construction material supply chain consisting of the government, a manufacturer, and a contractor, analyzes the decision-making and influencing factors of various stakeholders under no subsidy, contractor subsidy, and manufacturer subsidy, and studies the formulation of subsidy policies for construction waste recycling from the perspective of the government. The results show that: compared with subsidizing the manufacturer, when the government subsidizes the contractor, the recycling rate is lower, and the manufacturer’s profit, consumer surplus, and social welfare are higher. The manufacturer will use the price mechanism to grab government subsidies at this time, which will increase governmental subsidy expenditure. We also show that the recycling cost and the environmental awareness of the contractor affect the decision-making of the government and the manufacturer in different ways. Therefore, the government should analyze the contractor’s environmental awareness and evaluate the manufacturer’s recycling cost before formulating subsidy policies to promote construction waste recycling.

1. Introduction

With the advancement of urbanization in China, the speed of infrastructure construction has accelerated, and a large amount of construction waste has been generated while satisfying economic development. It is estimated that China produces more than 2 billion tons of construction waste each year [1]. However, it is known that the annual production of construction waste in Germany is only about 192 million tons, and that in Japan is about 370 million tons. At present, construction waste in China is mainly disposed of by incineration and landfill, which not only occupy a large amount of land resources, but also produce toxic substances, causing pollution of soil, water, and air. However, in some developed countries and regions, such as the United States, Japan, and many European Union countries, the recycling method is mainly adopted to deal with construction waste, and remarkable results have been achieved in theory and practice. The recycling rate of construction waste in the United States has reached 75%, and the recycling rate of construction waste in European countries, such as the Netherlands and Denmark, has exceeded 90%. If the construction waste can be recycled, it can not only save resources, but also protect the environment, so it is particularly important to carry out construction waste recycling activities.

For this reason, local governments have introduced a series of subsidy policies to promote the recycling of construction waste. For example, the Measures for the Management of Financial Subsidies for Comprehensive Utilization of Construction Waste issued by Guangzhou City stipulates that the government subsidizes construction waste recycling enterprises [2], and the Regulations on Construction Waste Disposal promulgated by Shanghai City also stipulates financial subsidies for construction projects that use construction waste comprehensive utilization products [3]. However, these regulatory practices are not as effective as expected, and the recycling rate of construction waste in China is still low. The reason for this is that the previous policies were mainly determined empirically and were not targeted according to the different external conditions, and there are differences in the implementation effects of the policies due to the different subsidy models [4,5]. Therefore, it is important to study how the government should develop an effective subsidy policy to promote construction waste resourcing for the current situation of waste management in China.

The recycling of construction waste is a complex process involving multiple stakeholders, mainly including contractors, building material manufacturers (later called manufacturers), and the government. Studies have shown that there are differences in the effectiveness of subsidies to the parties involved in resource reclamation. In the past, most scholars studied manufacturers’ subsidies from the government, but there was a lack of research on contractors’ subsidies as consumers. Through a comprehensive literature review, we found two research gaps that were not mentioned in previous studies. First of all, in the previous research on the reclamation of construction waste, the government subsidy contractor has not been studied. Second, previous studies failed to consider the impact of contractors’ environmental awareness on government decision-making. The above research blank may limit the implementation effect of construction waste reclamation to a large extent. Based on this, the research problems to be solved in this study are as follows:

(1)

What is the performance of the recycling rate, manufacturer’s profit, consumer surplus, and social welfare when the government provides subsidy to the contractor?

(2)

How do recycling costs and contractors’ environmental awareness influence government and manufacturer decisions?

In order to solve these problems, this paper constructs a construction material supply chain composed of the government, the manufacturer, and the contractor. While considering the environmental awareness of the contractor, the government’s subsidy decisions are internalized, and the decision-making and influencing factors of various stakeholders under different subsidy modes are compared. In addition, we also study the formulation of construction waste resource subsidy policies from the perspective of the government. To the best of our knowledge, this paper is the first attempt to consider subsidy for contractors, compare the implementation performance of different environmental regulations, and consider the impact of contractors’ environmental awareness in the recycling process of construction waste. It is expected that this study will enrich the existing knowledge system of construction waste recycling and provide a theoretical basis for the formulation and optimization of government regulations.

The rest of this article is organized as follows. Section 2 reviews relevant literature. Section 3 describes the symbols and assumptions of the model. In Section 4, we build and resolve decision models under different environmental regulations. Section 5 compares the results of different models and analyzes the parameters. Section 6 provides the analysis, the last part ends the paper.

2. Literature Review

2.1. Construction Waste Recycling Management under Government Regulation

Given the key role of government in construction waste recycling, many scholars have studied the development of policies. For example, a study by Huang et al. pointed out that many recycling enterprises are in a loss-making situation and there is an urgent need for the government to provide appropriate economic incentives to promote waste recycling [6]. Jia et al. introduced penalty and subsidy mechanisms, used a system dynamics approach to develop a construction waste management model, and studied the reasonable range of subsidies and fines [7]. Liu T. et al. used the same method to analyze the environmental benefit model and concluded that the environmental benefit was in a better state when the unit penalty was 50–60 RMB/t, the unit landfill charge was 30 RMB/t, and the unit subsidy was 25 RMB/t [8]. Chen et al. developed a construction waste recycling policy subsidy decision model and conducted simulations. Liu et al. analyzed the costs and benefits of the whole process of construction waste treatment, simulated the impact of government subsidies, and proposed a combined “penalty and subsidy” strategy to promote construction waste recycling [9]. The strategy of “penalty and subsidy” is proposed to promote construction waste recycling [10].

2.2. Decision-Making Behaviors of Stakeholders in the Reclamation of Construction Waste

In analyzing the decision-making behavior of stakeholders in waste recycling, Begum et al. studied the factors influencing enterprises’ attitudes and behaviors toward waste management and found that positive attitudes toward waste management can lead to better recycling behaviors [11]. Wu et al. found that contractors’ behavior is most influenced by economic feasibility. If the behavior of enterprises is not regulated by the government, contractors will dump waste freely [12]. There is also some literature that demonstrates that enterprises’ recycling decisions depend heavily on their cost-benefit profile, i.e., the economics of waste recycling [13,14].

Previous literature on construction waste recycling has contributed to the promotion of construction waste recycling by providing in-depth studies on policy formulation and factors influencing recycling behavior [15,16,17,18,19,20]. Most of the aforementioned literature considers government subsidies for manufacturers or recycling enterprises, without analyzing and comparing government subsidies for contractors, and policies for subsidizing contractors are abundant in practice [4]. In addition, little literature has considered the practical factor of contractors’ environmental awareness, which has an important influence on corporate decisions when studying construction waste recycling [21,22]. Based on this, this paper constructs a construction material supply chain consisting of the government, a manufacturer, and a contractor, internalizes the government’s subsidy decision while considering the contractor’s environmental awareness, compares the decision making and influencing factors of each stakeholder under different subsidy models, and provides a reference for the government to formulate a subsidy policy for construction waste recycling.

3. Problem Description

The structure of the decision model of the construction material supply chain consisting of a construction material manufacturer (later referred to as manufacturer) and a contractor under government regulation is shown in Figure 1. The manufacturer sells construction materials to the contractor at a unit price p and decides the optimal production quantity q based on the profit maximization principle, referring to existing studies [23,24,25], and sets the inverse demand function as a linear function, i.e., $p\left(q\right)=a-bq$. The manufacturer can produce construction materials in two ways: directly using virgin materials or recycling construction waste, where the recycling rate is, and respectively denoting the amount of recycled material production and the amount of virgin material production. The contractor’s sensitivity coefficient to the manufacturer’s environmentally unfriendly behavior is noted as the contractor’s environmental awareness. In order to maximize social welfare, the government subsidizes the process of construction waste recycling, and two subsidy models are considered in this paper, one is to subsidize the purchase behavior of the contractor (i.e., contractor subsidy) and the other is to subsidize the recycling of the manufacturer (i.e., manufacturer subsidy), and the subsidy rates are r, s. The decision sequence of each party is the same in different cases, and the decisions among stakeholders will mutually influence each other. The government first determines the subsidy model (i.e., subsidy for the manufacturer, or contractor), and the government sets the subsidy rate(s/r), the manufacturer or contractor gets the subsidy. Then, the manufacturer decides the recycling rate and production quantity. At the end of the decision cycle, the manufacturer gets the profit and the contractor gets the consumer surplus. The decision sequence is shown in Figure 2.

3.1. Symbol Description

The Description of symbols is shown

Table 1. Description of symbols.

Symbols	Description
Parameters
a	the upper limit of the price function, 0 < a
b	sensitivity coefficient of sales price to quantity demanded, 0 < b
c	unit production cost of new materials, 0 < c < a
γ	cost coefficient of construction waste recycling, 0 < γ
δ	contractor’s environmental awareness, 0 < δ
Decision Variables
s	government subsidy rates to the manufacturer of construction materials, 0 < s
r	government subsidy rate to the contractor, 0 < r
τ	recycling rate, 0 ≤ τ < 1
q	manufacturer’s production quantity, 0 < q
p(q)	construction material sales price, p(q) = a − bq > 0
Objective function
Π	profit of construction material manufacturer
CS	contractor’s consumer surplus
SW	social welfare
M	production volume of recycled materials, $M=q\tau$
N	production volume of virgin materials, $N=q\left(1-\tau \right)$
E	subsidy amount
Corner marker
A	no subsidy
B	contractor subsidy
C	manufacturer subsidy

.

3.2. Model Assumptions

1.

Considering the feasibility of problem solving, this paper does not consider the competition problem, and only one manufacturer and one contractor are considered.

2.

The shortage and inventory problems of construction materials are not considered, and only a single cycle of the construction materials supply chain is considered.

3.

Referring to the existing literature, the consumer surplus is assumed to be as follows: $CS=aq-\frac{1}{2}b{q}^2-pq-\frac{1}{2}\delta Nq$, where $aq-\frac{1}{2}b{q}^2-pq$ is the traditional consumer surplus [26,27]. In addition, $\frac{1}{2}\delta Nq$ is added to reflect the environmental awareness of the contractor, and the larger the contractor’s environmental awareness, the more sensitive the contractor is to environmentally unfriendly behavior.

4.

The manufacturer’s unit production cost is $c$. The additional cost to be paid for recycling compared to virgin materials is $\frac{1}{2}\gamma M^2$ [28], where $\gamma$ is the construction waste recycling cost coefficient, the larger the $\gamma$, the higher the cost of recycling.

5.

When the government subsidizes the contractor, the amount of subsidy received by the contractor is based on the purchase price, i.e., $E_B=rp$; when the government subsidizes the manufacturer, the amount of subsidy received by the manufacturer is based on the cost of recycling, i.e., $E_C=\frac{1}{2}s\gamma M^2$.

4. Model Solving

4.1. Benchmark Model without Subsidy (Model A)

In the benchmark model, neither the manufacturer nor the contractor receives government subsidies. Based on the above assumptions, the objective functions for the manufacturer and contractor are obtained as:

$$\Pi =\left(p-c\right)q-\frac{1}{2}\gamma M^2$$

$$CS=aq-\frac{1}{2}b{q}^2-pq-\frac{1}{2}\delta Nq$$

In this paper, a dynamic game model is constructed, and since the game is a full information dynamic game and its equilibrium is a subgame refined Nash equilibrium, it can be solved by using the inverse induction method. According to the reverse induction method, let $\frac{\partial \Pi }{\partial q}=a-c-2bq-q\gamma {\tau }^2=0$ be solved to obtain $q\left(\tau \right)=\frac{a-c}{2b+\gamma {\tau }^2}$. Similarly, let $\frac{\partial \Pi }{\partial q}=-\frac{{\left(a-c\right)}^2\gamma \tau }{{\left(2b+\gamma {\tau }^2\right)}^2}=0$ be solved to obtain $\tau =0$. That is, at this time, the manufacturer has no incentive to recycling and tends to use virgin materials for production. Bringing $\tau =0$ back to the equilibrium solution under $q$, $p$, $\Pi$, $CS$, $SW$ without environmental regulation, i.e., the optimal decision and corresponding benefits for the manufacturer and contractor in this case:

$${\tau }_A=0$$

$$q_A=\frac{a-c}{2b}$$

$$p_A=\frac{a+c}{2}$$

$${\Pi }_A=\frac{{\left(a-c\right)}^2}{4b}$$

$$C{S}_A=\frac{{(a-c)}^2}{8b}$$

$$S{W}_A=\frac{3{\left(a-c\right)}^2}{8b}$$

4.2. Contractor Subsidy Model (Model B)

When the government subsidizes the contractor at the price paid per unit of construction material, the contractor’s actual purchase price of construction material will be reduced per unit by $r$. The objective functions of the manufacturer, contractor, and government under this model are:

$$\Pi =\left(p-c\right)q-\frac{1}{2}\gamma M^2$$

$$CS=aq-\frac{1}{2}b{q}^2-pq-\frac{1}{2}\delta Nq+rq$$

$$SW=\left(a-c\right)q-\frac{1}{2}b{q}^2-\frac{1}{2}\gamma M^2-\frac{1}{2}\delta Nq$$

Let $\frac{\partial \Pi }{\partial q}=0$ and solve for $q\left(\tau ,r\right)=\frac{\mathrm{a}-\mathrm{c}+\mathrm{k}}{2\mathrm{b}+2\delta -2\delta \tau +{\gamma \tau }^2}$. Bringing $q\left(\tau ,r\right)$ into $\Pi$ and finding the first-order partial derivative of $\Pi$ and making it 0, we solve for $\tau =\frac{\delta }{\gamma }$. Bringing $\tau$ into $SW$ and finding the partial derivative of it and making it 0, we obtain $\mathrm{r}=\frac{\left(\mathrm{a}-\mathrm{c}\right)\left(\mathrm{b}\gamma +\left(\gamma -\delta \right)\delta \right)}{\gamma \left(\mathrm{b}+\delta \right)}$, and bringing $r$ into $q$, $p$, $\Pi$, $CS$, $SW$, we obtain the equilibrium solution at the subsidy contractor:

$${\tau }_B=\frac{\delta }{\gamma }$$

$$q_B=\frac{a-c}{b+\delta }$$

$$p_B=a$$

$$r=\frac{\left(a-c\right)\left(b\gamma +\left(\gamma -\delta \right)\delta \right)}{\gamma \left(b+\delta \right)}$$

$${\Pi }_B=\frac{{\left(a-c\right)}^2\left(2b\gamma +\left(2\gamma -\delta \right)\delta \right)}{2\gamma {\left(b+\delta \right)}^2}$$

$$C{S}_B=\frac{{\left(a-c\right)}^2\left(b\gamma +\left(\gamma -\delta \right)\delta \right)}{2\gamma {\left(b+\delta \right)}^2}$$

$$S{W}_B=\frac{{\left(a-c\right)}^2}{2\left(b+\delta \right)}$$

4.3. Manufacturer Subsidy Model (Model C)

When the government subsidizes the manufacturer, the subsidy is based on the cost of recycling per unit of resourced product, with a unit subsidy of $s$. The resulting objective functions for the manufacturer, contractor, and government, respectively, are:

$$\Pi =\left(p-c\right)q-\frac{1}{2}\left(1-s\right)\gamma M^2$$

$$CS=aq-\frac{1}{2}b{q}^2-pq-\frac{1}{2}\delta Nq$$

$$SW=\left(a-c\right)q-\frac{1}{2}b{q}^2-\frac{1}{2}\gamma M^2-\frac{1}{2}\delta Nq$$

Similarly, the equilibrium solution at the time of the subsidy manufacturer is obtained by solving the first-order condition according to the inverse induction method:

$${\tau }_C=\frac{\delta }{\gamma \left(1-s\right)}$$

$$q_C=\frac{\left(a-c\right)\left(1-s\right)\gamma }{2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2}$$

$$p_C=\frac{\gamma \left(1-s\right)\left(b+\delta \right)a+c\left(\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}{2\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2}$$

$${\Pi }_C=\frac{{\left(a-c\right)}^2\left(1-s\right)\gamma }{2\left(2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2\right)}$$

$$C{S}_C=\frac{\gamma \left(1-s\right){\left(a-c\right)}^2\left(\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}{2{\left(2\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}^2}$$

$$S{W}_C=\frac{\gamma {\left(a-c\right)}^2\left(3\gamma \left(b+\delta \right){\left(1-s\right)}^2-\left(2-s\right){\delta }^2\right)}{2{\left(2\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}^2}$$

Using each equilibrium solution, the objective function values of M, N, and E can be derived for each of the three cases. In summary, the results of solving the decision model for all three subsidy models are summarized in

Table 2. Equilibrium solutions of the three models.

	Model A	Model B	Model C
$r\left(or s\right)$	-	$\frac{\left(a-c\right)\left(b\gamma +\left(\gamma -\delta \right)\delta \right)}{\gamma \left(b+\delta \right)}$	$\frac{{\left(a-c\right)}^{2}s\gamma {\delta }^2}{2{\left(2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2\right)}^2}$
${\tau }^{ }$	0	$\frac{\delta }{\gamma }$	$\frac{\delta }{\gamma \left(1-s\right)}$
$q$	$\frac{a-c}{2b}$	$\frac{a-c}{b+\delta }$	$\frac{\left(a-c\right)\left(1-s\right)\gamma }{2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2}$
$p$	$\frac{a+c}{2}$	$a$	$\frac{\gamma \left(1-s\right)\left(b+\delta \right)a+c\left(\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}{2\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2}$
$M$	0	$\frac{\left(a-c\right)\delta }{\gamma \left(b+\delta \right)}$	$\frac{\left(a-c\right)\delta }{2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2}$
$N^{ }$	$\frac{a-c}{2b}$	$\frac{\left(a-c\right)\left(\gamma -\delta \right)}{\gamma \left(b+\delta \right)}$	$\frac{\left(a-c\right)\left(\left(1-s\right)\gamma -\delta \right)}{2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2}$
$E$	-	$\frac{{\left(a-c\right)}^2\left(b\gamma +\left(\gamma -\delta \right)\delta \right)}{\gamma {\left(b+\delta \right)}^2}$	$\frac{{\left(a-c\right)}^{2}s\gamma {\delta }^2}{2{\left(2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2\right)}^2}$
$\Pi$	$\frac{{\left(a-c\right)}^2}{4b}$	$\frac{{\left(a-c\right)}^2\left(2b\gamma +\left(2\gamma -\delta \right)\delta \right)}{2\gamma {\left(b+\delta \right)}^2}$	$\frac{{\left(a-c\right)}^2\left(1-s\right)\gamma }{2\left(2\left(1-s\right)\left(b+\delta \right)\gamma -{\delta }^2\right)}$
$CS$	$\frac{{\left(a-c\right)}^2}{8b}$	$\frac{{\left(a-c\right)}^2\left(b\gamma +\left(\gamma -\delta \right)\delta \right)}{2\gamma {\left(b+\delta \right)}^2}$	$\frac{\gamma \left(1-s\right){\left(a-c\right)}^2\left(\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}{2{\left(2\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}^2}$
$S{W}^{ }$	$\frac{3{\left(a-c\right)}^2}{8b}$	$\frac{{\left(a-c\right)}^2}{2\left(b+\delta \right)}$	$\frac{\gamma {\left(a-c\right)}^2\left(3\gamma \left(b+\delta \right){\left(1-s\right)}^2-\left(2-s\right){\delta }^2\right)}{2{\left(2\gamma \left(1-s\right)\left(b+\delta \right)-{\delta }^2\right)}^2}$

.

5. Model Comparison and Parameter Analysis

5.1. Model Comparison

In this section, the equilibrium solutions of the model under different subsidy models are compared in terms of the research problem and the following conclusions are obtained:

Conclusion 1.

${\tau }_B<{\tau }_C$

Proof of Conclusion 1.

${\tau }_C-{\tau }_B=\frac{s\delta }{\gamma \left(1-s\right)}>0$ □

Conclusion 1 shows that the recycling rate under the contractor subsidy is lower than that under the manufacturer subsidy. Because the recycling rate when subsidizing the contractor does not depend on the subsidy itself ($\tau =\frac{\delta }{\gamma }$), the subsidy does not directly incentivize the manufacturer to engage in waste recycling, while the recycling rate when subsidizing the manufacturer is directly related to the government subsidy ($\tau =\frac{\delta }{\gamma \left(1-s\right)}$). This implies that the government subsidy promotes waste recycling more when they subsidize the manufacturer.

Conclusion 2.

$q_B>q_C,p_B>p_C$

Proof of Conclusion 2.

$q_C-q_B=\frac{\left(a-c\right)\left(b\left(1-s\right)\gamma +\delta \left(\left(1-s\right)\gamma -\delta \right)\right)}{\left(b+\delta \right)\left(2b\left(-1+s\right)\gamma +\delta \left(2\left(-1+s\right)\gamma +\delta \right)\right)}>0;p_C-p_B=\frac{\left(a-c\right)\left(b\left(1-s\right)\gamma +\delta \left(\left(1-s\right)\gamma -\delta \right)\right)}{2b\left(-1+s\right)\gamma +\delta \left(2\left(-1+s\right)\gamma +\delta \right)}>0$ holds for $\gamma >{\gamma }_1=\frac{\delta }{1-s}$ and the inequality $\gamma >{\gamma }_1$ guarantees that the equilibrium solution for the production quantity is not negative. □

Conclusion 2 shows that both the selling price and the construction materials production quantity under the contractor subsidy are higher than under the manufacturer subsidy. When the government subsidizes the contractor to purchase recycled materials, it somewhat reduces the purchase cost of the contractor, while at the same time, the manufacturer will increase the supply of materials and raise prices to take advantage of the contractor subsidy policy. Thus, the free-riding behavior of the manufacturer will result in higher sales prices and production quantity under the contractor subsidy.

Conclusion 3.

$E_B>E_C$

Proof of Conclusion 3.

$E_C-E_B=\frac{{\left(a-c\right)}^2{f}_1\left(\gamma \right)}{2\gamma {\left(b+\delta \right)}^2{\left(2b\left(1-s\right)\gamma +\delta \left(2\left(1-s\right)\gamma -\delta \right)\right)}^2}$, where $$\begin{gathered} {\mathrm{f}}_1\left(\gamma \right)=\left[8{\left(1-\mathrm{s}\right)}^2{\left(\mathrm{b}+\delta \right)}^3{\gamma }^3-{\delta }^2{\left(\mathrm{b}+\delta \right)}^2\left(8{\mathrm{s}}^2-23\mathrm{s}+16\right){\gamma }^2+2{\delta }^4\left(\mathrm{b}+\delta \right)\left(5-4\mathrm{s}\right)\gamma -2{\delta }^6\right], {\mathrm{f}}_1{}^{’}\left(\gamma \right)=2\left(\mathrm{b}+\delta \right)\left(\left(5-4\mathrm{s}\right){\delta }^4 \\ -\left(16-23\mathrm{s}+8{\mathrm{s}}^2\right){\gamma \delta }^2\left(\mathrm{b}+\delta \right)+12{\left(-1+\mathrm{s}\right)}^2{\gamma }^2{\left(\mathrm{b}+\delta \right)}^2\right) \end{gathered}$$, since $f_1{}^{\prime }\left(\gamma \right)$ is a quadratic, therefore, there exist ${\gamma }_2,{\gamma }_3$ so that $f_1{}^{\prime }\left(\gamma \right)=0$. Additionally, $\gamma >{\gamma }_1$ when $f_1{}^{\prime }\left(\gamma \right)>0$, so $\gamma >{\gamma }_1$ when $f_1\left(\gamma \right)>0$, so $E_C-E_B>0$, the proof is over. □

Conclusion 3 shows that the optimal government subsidy under the contractor subsidy is higher than under the manufacturer subsidy. The presence of free-riding behavior of the manufacturer in the government subsidy to the contractor increases the sales price and quantity under the contractor subsidy and further increases the financial burden of the government subsidy. Therefore, the government should consider the financial consequences of the subsidy before deciding which policy to implement.

Conclusion 4.

${\Pi }_B>{\Pi }_C,C{S}_B>C{S}_C$

Proof of Conclusion 4.

${\Pi }_C-{\Pi }_B=\frac{{(a-c)}^2{f}_2(\gamma )}{2\gamma {(b+\delta )}^2(2b(1-s)\gamma +\delta (2(1-s)\gamma -\delta ))},C{S}_C-C{S}_B=\frac{{(a-c)}^2{f}_3(\gamma )}{2\gamma {(b+\delta )}^2{(2b(1-s)\gamma +\delta (2(1-s)\gamma -\delta ))}^2},f_2(\gamma )=-3{\gamma }^2(1-s){(b+\delta )}^2+2\gamma \delta (2-s)(b-\delta )-{\delta }^4,f_2(\gamma )=3b^3{(-1+s)}^2{\gamma }^3+b^2(-1+s){\gamma }^2\delta (9(-1+s)\gamma +(7-4s)\delta )+b\gamma {\delta }^2(9{(-1+s)}^2{\gamma }^2-2(7-11s+4s^2)\gamma \delta +(5-4s){\delta }^2-{\delta }^3(-3{(-1+s)}^2{\gamma }^3+(7-11s+4s^2){\gamma }^2\delta +(-5+4s)\gamma {\delta }^2+{\delta }^3$ As above, we can get that ${\mathrm{f}}_2\left(\gamma \right)>0$ and $f_3\left(\gamma \right)>0$ for $\gamma >{\gamma }_1$, so ${\Pi }_C-{\Pi }_B>0$ and $C{S}_C-C{S}_B>0$, the proof is over. □

Conclusion 4 shows that both the manufacturer’s profit and the contractor’s consumer surplus are greater under the contractor subsidy than under the manufacturer subsidy. This finding reflects the preference of the manufacturer and the contractor between the two subsidy models, i.e., both the manufacturer and the contractor prefer the contractor subsidy model.

Conclusion 5.

$S{W}_B>S{W}_C$

Proof of Conclusion 5.

$S{W}_C-S{W}_B=\frac{{\left(a-c\right)}^2{f}_4\left(\gamma \right)}{2\left(b+\delta \right){\left(2b\left(1-s\right)\gamma +\delta \left(2\left(1-s\right)\gamma -\delta \right)\right)}^2}, f_4\left(\gamma \right)=\left[{\left(1-s\right)}^2{\left(b+\delta \right)}^2{\gamma }^2+\gamma {\delta }^2\left(3s-2\right)\left(b+\delta \right)+{\delta }^4\right]$, and as above, $f_4\left(\gamma \right)>0$ when $\gamma >{\gamma }_1$, so $S{W}_C-S{W}_B>0$, the proof is over. □

Conclusion 5 shows that social welfare is greater under the contractor subsidy than under the manufacturer subsidy. Since both the manufacturer profit and consumer surplus are greater under the contractor subsidy than under the manufacturer subsidy, social welfare is also greater. Combined with conclusion 3, it can be obtained that which subsidy policy the government should implement depends on the government’s specific situation. The manufacturer subsidy policy should be considered when the government has a limited fiscal budget because its fiscal expenditure is lower; the contractor subsidy policy should be used when the government is more concerned about social welfare because it can generate a higher manufacturer profit, consumer surplus, and social welfare.

5.2. Parameter Analysis

Through the expressions of each equilibrium solution in Table 2, this section analyzes and explores the effects of changes in recycling cost and the contractor’s environmental awareness on the decisions of each member in the construction materials supply chain and supply chain performance, and obtains the following inferences.

Corollary 1.

When the government subsidizes the contractor, the following relationships exist between the recycling rate, construction materials production quantity, the optimal subsidy amount, manufacturer’s profit, consumer surplus, and social welfare, on the one hand, and the recycling cost and contractor’s environmental awareness, on the other.

(1)

$\frac{\partial {\tau }_{\mathrm{B}}}{\partial \gamma }<0,\frac{\partial {\tau }_{\mathrm{B}}}{\partial \delta }>0$;

(2)

$\frac{\partial {\mathrm{q}}_{\mathrm{B}}}{\partial \gamma }=0,\frac{\partial {\mathrm{q}}_{\mathrm{B}}}{\partial \delta }<0;$

(3)

$\frac{\partial {\mathrm{E}}_{\mathrm{B}}}{\partial \gamma }>0,\frac{\partial {\mathrm{E}}_{\mathrm{B}}}{\partial \delta }<0;$

(4)

$\frac{\partial {\Pi }_{\mathrm{B}}}{\partial \gamma }>0,\frac{\partial {\Pi }_{\mathrm{B}}}{\partial \delta }<0;$

(5)

$\frac{\partial {\mathrm{CS}}_{\mathrm{B}}}{\partial \gamma }>0,\frac{\partial {\mathrm{CS}}_{\mathrm{B}}}{\partial \delta }<0;$

(6)

$\frac{\partial {\mathrm{SW}}_{\mathrm{B}}}{\partial \gamma }=0,\frac{\partial {\mathrm{SW}}_{\mathrm{B}}}{\partial \delta }<0$

Proof of Corollary 1.

$\frac{\partial {\tau }_B}{\partial \gamma }=-\frac{\delta }{{\gamma }^2}<0, \frac{\partial {\tau }_B}{\partial \delta }=\frac{1}{\gamma }>0;$;$\frac{\partial {\Pi }_B}{\partial \gamma }=\frac{{\left(a-c\right)}^2{\delta }^2}{2{\gamma }^2{\left(b+\delta \right)}^2}>0, \frac{\partial {\Pi }_B}{\partial \delta }=-\frac{{\left(a-c\right)}^2\left(\gamma \delta +b\left(\gamma +\delta \right)\right)}{\gamma {\left(b+\delta \right)}^3}<0$;$\frac{\partial C{S}_B}{\partial \gamma }=\frac{{\left(a-c\right)}^2{\delta }^2}{2{\gamma }^2{\left(b+\delta \right)}^2}>0, \frac{\partial C{S}_B}{\partial \delta }=-\frac{{\left(a-c\right)}^2\left(\gamma \delta +b\left(\gamma +2\delta \right)\right)}{2\gamma {\left(b+\delta \right)}^3}<0$;$\frac{\partial S{W}_B}{\partial \gamma }=0,\frac{\partial S{W}_B}{\partial \delta }=-\frac{{\left(a-c\right)}^2}{2{\left(b+\delta \right)}^2}<0$ □

Corollary 1. (1) shows that the higher the recycling cost under the contractor subsidy policy, the lower the incentive for the manufacturer to recycling. Corollary 1. (2) and 1. (3) show that when the manufacturer’s recycling cost is higher, the government will give more subsidy to the contractor in order to incentivize the contractor to purchase recycled products and reduce the contractor’s purchase cost, while the manufacturer has free-rider behavior at this time and will transfer the government subsidy to itself by increasing the sales price and production quantity, so both the manufacturer’s profit and consumer welfare will increase, as Corollary 1. (4) and 1. (5) show. Corollary 1. (6), on the other hand, shows that changes in the manufacturer’s recycling cost have no effect on social welfare. Corollary 1 also shows that when the contractor become more environmentally conscious, the manufacturer increases the recycling rate, when the manufacturer costs increase and the optimal government subsidy decrease, thus having a negative impact on the manufacturer’s profit, contractor surplus, and social welfare.

Corollary 2.

When the government subsidizes the manufacturer, the following relationships exist between the recycling rate, construction materials production quantity, the optimal subsidy amount, manufacturer profit, consumer surplus, and social welfare, on the one hand, and recycling cost and contractor environmental awareness, on the other.

(1)

$\frac{\partial {\tau }_C}{\partial \gamma }<0, \frac{\partial {\tau }_C}{\partial \delta }>0;$

(2)

$\frac{\partial q_C}{\partial \gamma }<0, \frac{\partial q_C}{\partial \delta }<0;$

(3)

$\frac{\partial E_C}{\partial \gamma }<0, \frac{\partial E_C}{\partial \delta }>0;$

(4)

$\frac{\partial {\Pi }_C}{\partial \gamma }<0, \frac{\partial {\Pi }_C}{\partial \delta }<0;$

(5)

$\frac{\partial C{S}_C}{\partial \gamma }>0, \frac{\partial C{S}_C}{\partial \delta }<0;$

(6)

$when s>\frac{1}{2} or \frac{2b}{4b+3\delta }<s<\frac{1}{2} and \gamma <\frac{\left(2-s\right){\delta }^2}{2\left(1-s\right)\left(2s-1\right)\left(b+\delta \right)},\frac{\partial S{W}_C}{\partial \gamma }>0,otherwise,\frac{\partial S{W}_C}{\partial \gamma }<0;$

(7)

$\frac{\partial S{W}_2}{\partial \delta }<0$

Proof of Corollary 2.

$\frac{\partial {\tau }_{\mathrm{C}}}{\partial \gamma }=-\frac{\delta }{\left(1-\mathrm{s}\right){\gamma }^2}<0,\frac{\partial {\tau }_{\mathrm{C}}}{\partial \delta }=\frac{1}{\left(1-\mathrm{s}\right)\gamma }>0;$

$\frac{\partial {\mathrm{q}}_{\mathrm{C}}}{\partial \gamma }=\frac{\left(\mathrm{a}-\mathrm{c}\right)\left(-1+\mathrm{s}\right){\delta }^2}{{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^2}<0,\frac{\partial {\mathrm{q}}_{\mathrm{C}}}{\partial \delta }=-\frac{2\left(\mathrm{a}-\mathrm{c}\right)\left(-1+\mathrm{s}\right)\gamma \left(\left(-1+\mathrm{s}\right)\gamma +\delta \right)}{{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^2}<0;$$\frac{\partial {\mathrm{E}}_{\mathrm{C}}}{\partial \gamma }=-\frac{2{\left(\mathrm{a}-\mathrm{c}\right)}^2\left(-1+\mathrm{s}\right){\mathrm{s}\gamma \delta }^2\left(\mathrm{b}+\delta \right)}{{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^3}<0,\frac{\partial {\mathrm{E}}_{\mathrm{C}}}{\partial \delta }=\frac{2{\left(\mathrm{a}-\mathrm{c}\right)}^2{\mathrm{s}\gamma \delta }^2\left(\left(1-\mathrm{s}\right)\gamma -\delta \right)}{{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^3}>0$;

$\frac{\partial {\Pi }_{\mathrm{C}}}{\partial \gamma }=\frac{{\left(\mathrm{a}-\mathrm{c}\right)}^2\left(-1+\mathrm{s}\right){\delta }^2}{2{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^2}<0,\frac{\partial {\Pi }_{\mathrm{C}}}{\partial \delta }=\frac{{\left(\mathrm{a}-\mathrm{c}\right)}^2\left(-1+\mathrm{s}\right){\delta }^2}{2{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^2}<0$;

$\frac{\partial {\mathrm{CS}}_{\mathrm{C}}}{\partial \gamma }=\frac{{\left(\mathrm{a}-\mathrm{c}\right)}^2\left(-1+\mathrm{s}\right){\delta }^4}{2{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^3}>0,$$\frac{\partial {\mathrm{CS}}_{\mathrm{C}}}{\partial \delta }=-\frac{{\left(\mathrm{a}-\mathrm{c}\right)}^2\left(-1+\mathrm{s}\right)\gamma \left(2\mathrm{b}{\left(-1+\mathrm{s}\right)}^2{\gamma }^2+\delta \left(2{\left(-1+\mathrm{s}\right)}^2{\gamma }^2+3\left(-1+\mathrm{s}\right)\gamma \delta +2{\delta }^2\right)\right)}{2{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^3}<0$;

$\frac{\partial {\mathrm{SW}}_{\mathrm{C}}}{\partial \gamma }=\frac{{\left(\mathrm{a}-\mathrm{c}\right)}^2{\delta }^2\left(2\mathrm{b}\left(1-3\mathrm{s}+2{\mathrm{s}}^2\right)\gamma +\delta \left(\left(2-6\mathrm{s}+4{\mathrm{s}}^2\right)\gamma +\left(-2+\mathrm{s}\right)\delta \right)\right)}{2{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^3},$ when $\mathrm{s}>\frac{1}{2}\mathrm{or}\frac{2\mathrm{b}}{4\mathrm{b}+3\delta }<\mathrm{s}<\frac{1}{2}\mathrm{and}\gamma <\frac{\left(2-\mathrm{s}\right){\delta }^2}{2\left(1-\mathrm{s}\right)\left(2\mathrm{s}-1\right)\left(\mathrm{b}+\delta \right)},\frac{\partial {\mathrm{SW}}_{\mathrm{C}}}{\partial \gamma }>0,\mathrm{otherwise},\frac{\partial {\mathrm{SW}}_{\mathrm{C}}}{\partial \gamma }<0$$\frac{\partial {\mathrm{SW}}_{\mathrm{C}}}{\partial \delta }=-\frac{{\left(\mathrm{a}-\mathrm{c}\right)}^2\gamma \left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma \left(3{\left(-1+\mathrm{s}\right)}^2\gamma +2\left(-1+2\mathrm{s}\right)\delta \right)+\delta \left(6{\left(-1+\mathrm{s}\right)}^3{\gamma }^2+9{\left(-1+\mathrm{s}\right)}^2\gamma \delta +2\left(-2+\mathrm{s}\right){\delta }^2\right)\right)}{2{\left(2\mathrm{b}\left(-1+\mathrm{s}\right)\gamma +\delta \left(2\left(-1+\mathrm{s}\right)\gamma +\delta \right)\right)}^3}<0$

Corollary 2. (1) and 2. (2) show that when the recycling cost is higher, the manufacturer will reduce the recycling rate and control cost by reducing the amount of material produced. Corollary 2. (3) and 2. (4) show that both the optimal government subsidy and manufacturer profit are inversely related to the manufacturer’s recycling cost, while Corollary 2. (5) shows that the consumer surplus is positively related to the manufacturer’s recycling cost. Corollary 2. (6) analyzes the effect of exogenous variables on social welfare: the change in social welfare in terms of recycling cost depends on the level of both subsidy and recycling cost, and when both are high, the recycling cost has a negative effect on social welfare, otherwise, the recycling cost has a positive effect on social welfare. Corollary 2 also shows that when the contractor is more environmentally conscious, the manufacturer’ recycling rate increases. The government subsidy will also increase due to public pressure, at which point the manufacturer profit, consumer surplus, and social welfare will all decrease.

6. Numerical Analysis

In the above analysis, a building material supply chain decision model is established, and the optimal decision of the building material manufacturer and contractor under different subsidy policies is calculated. In order to analyze and verify the model, an example analysis is carried out in this paper, and the influence of each parameter on the model output is investigated. By using Mathematica software to simulate, we verified the accuracy of the above conclusions, and discussed the relationship between the equilibrium solution of each model and the influence of exogenous variables under different government subsidy models. Parameter setting was based on field investigation and the related literature of two typical building materials manufacturers in Shenzhen [6,8].

The numerical simulation results verify conclusions 1–5 above.

As can be seen from Figure 3, the recycling rate of construction waste under the contractor subsidy mode and manufacturer subsidy mode increases with the increase of the contractor’s environmental awareness, and decreases with the increase of the cost coefficient of construction waste recycling. In addition, the recycling rate in the contractor subsidy mode is lower than that in the manufacturer subsidy mode. As the above analysis shows, the recycling rate in the contractor subsidy mode has nothing to do with the subsidy, while in the manufacturer subsidy mode, it is the opposite. Therefore, the subsidy for manufacturer can promote the improvement of the recycling rate more.

Figure 3, Figure 4 and Figure 5 respectively show that when the contractor’s environmental awareness is enhanced, the production quantity of building materials under the two subsidy modes decreases, while the selling price under the contractor’s subsidy mode has no significant change, while the selling price under the manufacturer’s subsidy mode increases and the subsidy amount increases with the enhancement of environmental awareness. In addition, the output, selling price, and subsidy amount under the contractor subsidy mode are higher than those under the manufacturer subsidy mode. This is because the subsidy to the contractor is equivalent to reducing the purchase cost of the contractor, while the free rider behavior of the manufacturer—boosting production and raising prices—increases the profit. Yet, the manufacturers’ free-riding adds to government spending.

From Figure 6, Figure 7, Figure 8 and Figure 9, we can see that the manufacturer’s profit, consumer surplus, and social welfare decrease with the increase of the contractor’s environmental awareness, and all values in the contractor’s subsidy mode are higher than those in the manufacturer’s subsidy mode. The contractor’s environmental awareness leads to a decline in the production of building materials, when higher prices cannot compensate for the decline in production, leading to a decline in manufacturers’ profits, consumer surpluses, and social welfare. When subsidizing the contractor, both output and price are higher, that is, both the manufacturer and contractor prefer the subsidy model to the contractor subsidy model. For the government, from the perspective of social welfare, the contractor subsidy model can improve social welfare more, because the manufacturer’s profit and consumer surplus are higher in the contractor subsidy model than in the manufacturer’s subsidy model. However, as can be seen from Figure 5, the government needs to pay more fiscal expenditure to achieve this effect. When the government decides on the subsidy mode, it should take into consideration the fiscal expenditure, manufacturer’s profit, consumer surplus, and social welfare comprehensively to achieve the purpose of regulation.

7. Results & Discussion

Previous studies have studied subsidy models for manufacturer or recycling enterprises [9,11,12]. Based on this, we establish a comparative study of the subsidy models of the contractor and manufacturer and analyze the impact of exogenous variables, such as contractor environmental awareness and cost coefficient, on government and manufacturer decisions. Additionally, some new findings of management significance have been obtained.

For the government, the main focus is to choose the appropriate subsidy model to maximize social welfare. Compared with the manufacturer subsidy model, the contractor subsidy model has a higher recycling rate, greater manufacturer profits, consumer surplus, and social welfare, which indicates that the contractor subsidy model can better promote construction waste recycling. However, the government will face higher fiscal costs. This finding is partly consistent with the research of Liu and Chen [7,9,10], which indicates that subsidies to contractors can better stimulate the recycling of construction waste.

For firms, we find that both contractor and manufacturer prefer the contractor subsidy model because both benefit more. Interestingly, the improvement of the contractor’s environmental awareness will lead to the decrease of production quantity, and the improvement of the contractor’s environmental awareness will promote the recycling of the manufacturer and thus increase the cost of recycling, thus leading to the decrease of production quantity.

8. Conclusions

This paper investigates how government subsidy models affect the choice of optimal decisions of government, manufacturers, and contractors in the construction material supply chain considering contractors’ environmental awareness. Three decision models of no subsidy (Model A), contractor subsidy (Model B), and manufacturer subsidy (Model C) are constructed respectively, and the equilibrium solutions under different subsidy models are obtained and the two subsidy models are compared and analyzed by the inverse induction method. It was found that.

(1)

The recycling rate under the contractor subsidy is lower than that under the manufacturer subsidy because the recycling rate when the government subsidizes the contractor does not depend on the subsidy, so the subsidy does not directly incentivize waste recycling, while the recycling under the manufacturer subsidy is directly incentivized by the government subsidy.

(2)

Manufacturer profits, consumer surplus, and social welfare are greater when the government subsidizes the contractor than when it subsidizes the manufacturer, so both manufacturers and contractors prefer the contractor subsidy model.

(3)

There is free-riding behavior by manufacturers in the contractor subsidy model, when manufacturers will capture government subsidies through the price mechanism, making government subsidy expenditures increase.

(4)

As contractors’ environmental awareness increases and manufacturers pay more attention to construction waste recycling, the waste recycling rate will increase, while manufacturers’ profits, consumer surplus, and social welfare will decrease.

(5)

As the cost of waste recycling increases, the waste recycling rate decreases, at which time the changes in manufacturer profits, consumer surplus, and social welfare are affected by the pattern of government subsidies and the level of recycling costs.

Based on the findings of this paper, the following management insights can be obtained.

(1)

For the government, there are differences in the implementation effects of different subsidy models. If the contractor is subsidized, higher social welfare can be obtained, but at this time, the resource utilization rate of waste is reduced and the government spends more on subsidies; the opposite is true if the manufacturer is subsidized. Therefore, the government should implement the subsidy policy that depends on the specific situation of the government, when the government budget is limited, it should consider the manufacturer subsidy policy, while when the government is more concerned about social welfare, the contractor subsidy policy should be used.

(2)

Resourcefulness costs and contractors’ environmental awareness influence government and manufacturers’ decisions in different ways. Therefore, the government should understand the environmental awareness of contractors and evaluate the recycling cost of manufacturers before formulating a subsidy policy to promote construction waste recycling.

(3)

The environmental awareness of contractors creates public pressure on manufacturers and will promote waste recycling by manufacturers, which also indicates that in addition to increasing the recycling rate through policy measures, the government can also draw contractors’ attention to the environment.

In this paper, considering the environmental awareness of the contractor, a comparative analysis of the decision-making and influencing factors of various stakeholders under no subsidy, subsidy to contractor, and subsidy to manufacturer, and a study on the formulation of the construction wastes subsidy policy from the perspective of the government. The results of this study inspire something worth investigating further. Firstly, from the perspective of government regulation, the strategy of setting multiple complex regulations at the same time can be studied. Further research can be extended to the information asymmetry, enterprise competition, and other situations to test the applicability of the conclusion.