# Incentive Model Based on Cooperative Relationship in Sustainable Construction Projects

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stakeholders and Factors Surrounding Sustainable Construction

## 3. Model Description and Solution

#### 3.1. Model Description

**Hypothesis 1**:

_{1}, e

_{2}), while e > 0. The cooperative behaviors between the owner and contractor can generate a synergistic effect, thus adding benefits to the sustainable construction. The added benefits can be shared by the owner and contractor. Assuming that the cooperative behavior began at time t, the project outcome function at s (t > s) is [64]:

_{1}and A

_{2}are the output coefficients of the levels of the efforts of both parties, i.e., the integrated technical level and comprehensive management ability of the owner and contractor. The output coefficient manifests the stakeholders’ capabilities to transform the input resources into project outcomes, which is associated with the levels of operational management, qualifications and competencies. The effort levels of e

_{1}and e

_{2}are difficult to observe by the other parties, while can be verified by the input resources and the degree of the achievement of the project’s objectives. $\vartheta $ denotes the external random factors affecting outcome function, and is mutually independent of e

_{1}and e

_{2}. To ensure the Hyers–Ulan stability of the outcome function, $\vartheta $ is set to follow a normal distribution (0,${\delta}^{2}$). k presents the cooperative relationship between the owner and contractor. K = 0 (0 ≤ k ≤ 1) means that the owner and contractor do not adopt cooperative behaviors, and so, one party may chase their own benefit maximization, thereby damaging the other party’s benefits. K = 1 means that the owner and contractor adopt seamless cooperative behavior, and so both parties are willing to select their behavior in accordance with the overall project benefits. This study aims to investigate the effects of a cooperative relationship (0 < k < 1) on overall project benefits, namely the effects of cooperation on project outcomes. When 0 < k ≤ 1, the cooperative behavior between the owner and contractor results in synergistic effects. Under this condition, when one party increases its input resources, the other party’s marginal benefit progressively increases with a given level of effort. ke

_{1}e

_{2}refers to the added benefits created by synergistic effects, and e

_{1}e

_{2}represents the cooperative willingness between the owner and contractor. When e

_{i}= 0 (i = 1, 2), the owner and contractor cannot generate synergistic effects, because neither of them is willing to adopt cooperative behavior. Therefore, k can also be named as the coefficient of synergistic effects, and can be explained as the ability to create added benefits under a given condition of input resources.

**Hypothesis 2**:

_{i}) of the owner and contractor is a strictly monotonic increasing function of e

_{i}(i = 1, 2) [37,65]:

_{1}= A

_{2}= 1 in the outcome function; thus, the effort cost coefficient manifests as the output efficiency ($\raisebox{1ex}{${A}_{i}$}\!\left/ \!\raisebox{-1ex}{${\eta}_{i}$}\right.)$ of the owner and contractor [66]. In other words, the greater the effort cost coefficient, the lower is the output efficiency.

**Hypothesis 3**:

_{1}and β

_{2}(0 ≤ β

_{1}, β

_{2}≤ 1), respectively, and meet the requirement of β

_{1}+ β

_{2}= 1. The constant payments to the owner and contractor are ω

_{1}and ω

_{2}, respectively, and meet the requirement of ω

_{1}+ ω

_{2}= 0.

#### 3.2. Model Solution

#### 3.2.1. Solution under Non-Moral Hazard

_{1}and e

_{2}were calculated respectively, then set the values to zero. Thus, the researchers obtain:

_{1}and e

_{2}, the researchers obtain:

_{1}and e

_{2}to obtain:

#### 3.2.2. Solution under Moral Hazard

_{1}and e

_{2}, the researchers obtain:

_{1}+ β

_{2}= 1 into Equation (6) and obtain:

_{1}and e

_{2}, the researchers order ${F}^{\text{'}}{}_{{e}_{1}}=0$ and ${F}^{\text{'}}{}_{{e}_{2}}=0$, then make a simplification:

_{2}= 1 − β

_{1}into Equation (11) transforms the latter into a simple cubic equation about β

_{1}as follows:

_{1}+ β

_{2}= 1 into consideration, the researchers obtain:

## 4. Model Analysis and Simulations

#### 4.1. Model Analysis

**Proposition 1**:

**Proof.**

**Proposition 2**:

**Proof.**

_{1}is a monotonic increasing function about ${\eta}_{2}$, and a monotonic decreasing function about ${\eta}_{1}$. The function of β

_{2}is a monotonic increasing function about ${\eta}_{1}$, and a monotonic decreasing function about ${\eta}_{2}$.

**Corollary 1**:

**Proof.**

**Corollary 2**:

_{1}* = β

_{2}* = 0.5). If the effort cost of the owner approaches positive infinity (${\eta}_{1}\to +\infty $), the benefit allocation coefficient approaches zero (β

_{1}* → 0). If the effort cost of the contractor approaches positive infinity (${\eta}_{2}\to +\infty $), the owner is likely to have all the project benefits (β

_{1}* → 1). In other words, if the owner and contractor possess the same output efficiency, they will obtain the share ratio. If one party has no efficiency, the other party will obtain all the benefits.

**Proof.**

_{1}* = β

_{2}* = 0.5. If k > 0, q = 0 can be obtained. Substituting ${\eta}_{1}={\eta}_{2}$ and q = 0 into Equation (15), x = 0 can be obtained. Thus, there still has β

_{1}* = β

_{2}* = 0.5. When ${\mathsf{\eta}}_{1}\to +\infty $ , the following equation can be obtained:

_{1}

^{*}→ 0 can be obtained according to Equation (16). The proof of ${\eta}_{2}\to +\infty $ can be aligned with the former method.

**Proposition 3**:

^{*}) do not produce incentive effects. This is consistent with the reference [52,66], meaning that there are no distinctions amongst the traditional construction projects. In sustainable construction, the constant payments are jointly negotiated by the owner and contractor, and documented in a specific contract. Therefore, in the majority of construction contracts, emphasis should be placed on the incentive terms, and constant payment should be arranged in accordance with a bargaining agreement.

**Proposition 4**:

**Proof.**

**Proposition 5**:

**Proof.**

**Corollary 3**:

#### 4.2. Model Simulations

#### 4.2.1. The Effects of Cooperative Relationship (k) on Benefit Allocation Coefficient (β_{1})

_{1}= ω

_{2}= 0, then the researchers simulate the relationships between the benefits allocation coefficient (β

_{1}) of the owner and his/her effort cost coefficient (${\eta}_{1}$) under different k values. The results are shown in Figure 2. The researchers set ${\eta}_{1}=1$ and ω

_{1}= ω

_{2}= 0, then simulate the relationships between the benefits allocation coefficient (β

_{1}) of the owner and the contractor’s effort cost coefficient (${\eta}_{2}$) under different k values. The results are shown in Figure 3. The researchers set ${\eta}_{1}=1$, ${\eta}_{2}=2$ and ω

_{1}= ω

_{2}= 0, then simulate the relationships between the benefits allocation coefficient (β

_{1}) of the owner and the cooperative relationship (k). The result is shown in Figure 4.

_{1}is a decreasing function of ${\eta}_{1}$, as well as an increasing function of ${\eta}_{2}$. It shows that the lower the effort cost coefficient of the owner, the higher is the output efficiency, and the greater is the benefit allocation coefficient (β

_{1}). The greater the effort cost coefficient of the contractor, the lower is the output efficiency, and the greater is the benefit allocation coefficient (β

_{1}). Therefore, Proposition 2 was verified, which means that the owner and contractor could control their share ratio of project benefits through adjusting their effort cost and output efficiency. The researchers investigated the effects of ${\eta}_{1}$ and ${\eta}_{2}$ under different k values, and found that the variation tendency of the function (β

_{1}) curve of k = 0 is consistent with k ≠ 0. Thus, Corollaries 1 and 2 were verified, which means that the ideal situation for the cooperative relationship is a non-moral hazard situation between the owner and contractor. Meanwhile, the function curves under different k values intersect at one point, as shown in Figure 2 and Figure 3. This means that the benefit allocation coefficient (β

_{1}) is also affected by the ratio of the effort cost coefficients of the owner and contractor (${\eta}_{1/}{\eta}_{2}$). When the ratio of the effort cost coefficients is greater than a specific value, the increasing or decreasing trends of the benefit allocation coefficient will slow down. The researchers explored the effects of ${\eta}_{1/}{\eta}_{2}$ on the benefit allocation coefficient (β

_{1}) under different ${\eta}_{1}$ and ${\eta}_{2}$ values, and find that ${\eta}_{1/}{\eta}_{2}=0.5$ is the threshold value. Then, the researchers simulated the relationship between the cooperative relationship (k) and the benefit allocation coefficient (β

_{1}) when ${\eta}_{1/}{\eta}_{2}=0.5$. The result shows that the benefit allocation coefficient is an increasing function of the cooperative relationship. The higher the level of cooperation, the greater is the benefit allocation coefficient. This means that the owner is willing to promote cooperation, because greater benefits can be gained from the synergistic effects of cooperation.

#### 4.2.2. The Effects of Cooperative Relationship (k) on Effort Levels (e)

_{1}= ω

_{2}= 0, then simulate the relationships between the cooperative relationship (k) and effort levels (e) of the owner and contractor. The results are shown in Figure 5 and Figure 6. The non-moral and moral hazard situations are distinguished in the figures.

#### 4.2.3. The Effects of Cooperative Relationship (k) on Project Benefit (Eπ)

_{1}= ω

_{2}= 0, then substitute Equations (3), (16) and (17) into the formulae of U

_{i}and Eπ accordingly. Thus, the researchers can simulate the relationship between the owner’s expected benefit (U

_{1}) and the cooperative relationship (k), the relationship between the contractor’s expected benefit (U

_{2}) and the cooperative relationship (k), and the relationship between the net project benefit (Eπ) and the cooperative relationship (k), respectively. The results are shown in Figure 7, Figure 8 and Figure 9. The different situations of non-moral hazard and moral hazard are distinguished in the figures.

#### 4.2.4. The Numerical Example of the Incentive Model

## 5. Conclusions and Implications

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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K = 0 | K = 0.1 | K = 0.2 | K = 0.3 | K = 0.4 | |
---|---|---|---|---|---|

$({e}_{1}^{*},{e}_{2}^{*})$ | (0.333,0.333) | (0.189,0.659) | (0.210,0.656) | (0.231,0.657) | (0.252,0.663) |

$({e}_{1}^{**},{e}_{2}^{**})$ | (0.500, 1.000) | (0.553,1.055) | (0.612,1.122) | (0.681,1.204) | (0.761,1.304) |

$({\beta}_{1}^{*},{\beta}_{2}^{*})$ | (0.333, 0.667) | (0.354,0.646) | (0.371,0.629) | (0.385,0.615) | (0.398,0.602) |

$({\beta}_{1}^{**},{\beta}_{2}^{**})$ | (0.167,0.833) | (0.183,0.817) | (0.201,0.799) | (0.217,0.783) | (0.235,0.765) |

(U_{1}^{*},U_{2}^{*}) | (0.333,0.167) | (0.268,0.339) | (0.287,0.347) | (0.306,0.358) | (0.327,0.371) |

(U_{1}^{**},U_{2}^{**}) | (0.125,0.625) | (0.147,0.657) | (0.174,0.693) | (0.204,0.738) | (0.243,0.790) |

(Eπ^{*},Eπ^{**}) | (0.500,0.750) | (0.607,0.804) | (0.634,0.867) | (0.664,0.942) | (0.698,1.033) |

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## Share and Cite

**MDPI and ACS Style**

Wu, G.; Zuo, J.; Zhao, X.
Incentive Model Based on Cooperative Relationship in Sustainable Construction Projects. *Sustainability* **2017**, *9*, 1191.
https://doi.org/10.3390/su9071191

**AMA Style**

Wu G, Zuo J, Zhao X.
Incentive Model Based on Cooperative Relationship in Sustainable Construction Projects. *Sustainability*. 2017; 9(7):1191.
https://doi.org/10.3390/su9071191

**Chicago/Turabian Style**

Wu, Guangdong, Jian Zuo, and Xianbo Zhao.
2017. "Incentive Model Based on Cooperative Relationship in Sustainable Construction Projects" *Sustainability* 9, no. 7: 1191.
https://doi.org/10.3390/su9071191