Prefabrication decisions of the construction supply chain under government subsidies

Abstract

Prefabrication has been generating increasing interest as a cleaner production strategy to promote sustainable development. Alongside this trend, various subsidies have been set to improve prefabrication levels. This paper evaluates the prefabrication levels of buildings through the assembly rate. A series of models are established to investigate the optimal assembly rate under various government subsidies. The optimal assembly rate and subsidy revenue-sharing coefficient are analyzed in both decentralized and centralized scenarios. By comparing the optimal decisions in these two scenarios, a transfer payment contract is proposed that enables the overall coordination of the prefabricated construction supply chain (PCSC). The results show that the optimal assembly rate in the centralized scenario is higher than that in the decentralized one. When the revenue-sharing coefficient is 100%, the subsidy revenue-sharing contract can coordinate the PCSC system and realize the Pareto improvement. When certain conditions are satisfied in the transfer payment contract, business profits can achieve Pareto optimality. This research provides a reference for construction enterprises making decisions to promote the development of PCSC.

Appendices

Appendix 1 Proof of Proposition 1

By computing the partial derivative of Eq. (2) with respect to the variable $g$, the first-order condition by equating this derivative to zero, that is:

$$\frac{{\partial \pi_{c} }}{\partial g} = \gamma \beta k - kg = 0$$

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Solving this equation, we can obtain

$$g = \gamma \beta$$

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And then, by computing the second-order partial derivatives of Eq. (2) with respect to $g$, we can obtain:

$$\frac{{\partial^{2} \pi_{c} }}{{\partial^{2} g}} = - k$$

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It is well known that the equations are negative. That is, $\pi_{c}$ is a concave function of $g$.

Case 1: $0 \leq g_{0} \leq \gamma \beta$. When the optimal assembly rate is not lower than the assembly rate benchmark, there exist unique values $g^{ * }$ of $g$ that maximize $\pi_{c}$.

Based on the decision on the assembly rate $g = \gamma \beta$, the developer decides the incentives $\gamma$ to optimize his profit. Substituting $\pi_{d}$ with $g = \gamma \beta$, the optimal incentive $\gamma$ is obtained from the first-order necessary condition.

$$\frac{{\partial \pi_{d} }}{\partial r} = - 2\,\beta^{2} \,k\,\gamma + \left( {\beta g_{0} + \beta^{2} } \right)\,k - B_{0} { = }0$$

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Such that, the optimal incentive can be determined as follows:

$$\gamma_{1}^{ * } = \frac{{\beta^{2} {\mathrm{k} + }\beta k{\mathrm{g}}_{{0}} - B_{0} }}{{2\beta^{2} {\mathrm{k}}}} = \frac{1}{2} + \frac{{{\mathrm{g}}_{{0}} }}{2\beta } - \frac{{B_{0} }}{{2\beta^{2} {\mathrm{k}}}}$$

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And then by respectively computing the second-order partial derivatives of Eq. (1) with respect to $\gamma$, we can obtain as follows:

$$\frac{{\partial^{2} \pi_{d} }}{{\partial^{2} r}} = - {2}\beta^{2} k < 0$$

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That is, $\pi_{d}$ is a concave function of $\gamma$, which shows that there is a unique value $\gamma *$ that maximizes $\pi_{d}$.

Furthermore, the assembly rate is obtained.

$$g_{1}^{ * } = \frac{{\beta^{2} {\mathrm{k} + }\beta k{\mathrm{g}}_{{0}} - B_{0} }}{{2\beta {\mathrm{k}}}}{ = }\frac{{\beta {+ \mathrm{g}}_{{0}} }}{2} - \frac{{B_{0} }}{{2\beta {\mathrm{k}}}}$$

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The optimal profits of the developer and the general contractor are, respectively, represented as follows:

$$\begin{gathered} \pi_{d}^{1} = \left( {\alpha R - P - C_{d} } \right) - \frac{{g_{0}^{2} k}}{2}{ + }\frac{{\left( {\beta^{2} g_{0}^{2} + 2\beta^{3} g_{0} + \beta^{4} } \right)k^{2} + \left( {2\beta^{2} B_{0} - 2\beta B_{0} g_{0} } \right)k + B_{0}^{2} }}{{4\beta^{2} k}} \\ \, = \alpha R - P - C_{d} - \frac{{kg_{0}^{2} }}{4} + \frac{{k\beta_{{~}}^{2} }}{4} + \frac{{k\beta g_{0}}}{2} + \frac{{B_{0} }}{2} - \frac{{B_{0} g_{0} }}{2\beta } + \frac{{B_{0}^{2} }}{{4\beta^{2} k}} \\ \end{gathered}$$

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$$\begin{gathered} \pi_{c}^{1} = P - C_{c} + \frac{{\beta^{2} g_{0}^{2} k^{2} { + }\beta^{4} k^{2} - 2\beta^{3} g_{0} k^{2} + 2\beta^{2} B_{0} k + 6\beta B_{0} g_{0} k - 3B_{0}^{2} }}{{8\beta^{2} k}} \\ \, = P - C_{c} + \frac{{kg_{0}^{2} }}{8} + \frac{{k\beta_{{~}}^{2} }}{8} - \frac{{k\beta g_{0}}}{4} + \frac{{B_{0} }}{4}{ + }\frac{{3B_{0} g_{0} }}{4\beta } - \frac{{3B_{0}^{2} }}{{8\beta^{2} k}} \\ \end{gathered}$$

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$$\pi^{1} { = }\alpha {\mathrm{R}} - {\mathrm{C}}_{d} - {\mathrm{C}}_{c} { + }\frac{{k(3\beta_{{~}}^{2} - g_{0}^{2} + 2\beta g_{0})}}{8} + \frac{{3B_{0} }}{4}{ + }\frac{{B_{0} g_{0} }}{4\beta } - \frac{{B_{0}^{2} }}{{8\beta^{2} k}}$$

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Case 2: $g_{0} > \gamma \beta$. When the optimal assembly rate is lower than the assembly rate benchmark, the assembly rate must reach the benchmark $g_{2}^{ * } = g_{0}$.

Based on the decision on the assembly rate $g_{2}^{ * } = g_{0}$, the developer makes the decision on the incentives $\gamma$ to optimize his profit. Substituting $\pi_{d}$ with $g_{2}^{ * }$, the optimal incentive $\gamma$ is obtained from the first-order necessary condition.

$$\frac{{\partial \pi_{d} }}{\partial r} = - B_{0} < 0$$

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That is, $\pi_{d}$ is a unimodal decreasing function of $\gamma$, which shows that there is a unique value $\gamma_{2}^{ * } = 0$ that maximizes $\pi_{d}$.

The optimal profits of the developer and the general contractor are, respectively, represented as follows:

$$\pi_{d}^{2} { = }\alpha {\mathrm{R}} - {\mathrm{C}}_{d} - {\mathrm{P}} - \frac{1}{2}kg_{0}^{2} { + }\beta kg_{0} + B_{0}$$

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$$\pi_{c}^{2} {= \mathrm{P}} - {\mathrm{C}}_{c}$$

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$$\pi^{2} { = }\alpha {\mathrm{R}} - {\mathrm{C}}_{d} - {\mathrm{C}}_{c} - \frac{1}{2}kg_{0}^{2} { + }\beta kg_{0} { + }B_{0}$$

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Appendix 2 Proof of Proposition 2

By computing the partial derivative of Eq. (3) with respect to the variable $g$, the first-order condition by equating this derivative to zero, that is:

$$\frac{\partial \pi }{{\partial g}} = \beta k - kg = 0$$

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Solving this equation, we can obtain $g_{I} = \beta$. And then, by computing the second-order partial derivatives of Eq. (3) with respect to $g$, we can obtain:

$$\frac{{\partial^{2} \pi }}{{\partial^{2} g}} = - k$$

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It is well known that the equation is negative. That is, $\pi$ is a concave function of $g$.

Case 1: $0 \leq g_{0} \leq \beta$. When the optimal assembly rate is not lower than the assembly rate benchmark, there exist unique values $g_{3}^{ * } = \beta$ of $g$ that maximize $\pi$.

By substituting $g_{3}^{ * } = \beta$ into Eq. (3), the optimal entire supply chain profits are represented as follows:

$$\pi_{{~}}^{3} = \alpha {\mathrm{R}} - {\mathrm{C}}_{c} - {\mathrm{C}}_{d} { + }\frac{1}{2}k\beta^{2} { + }B_{0}$$

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Case 2: $g_{0} > \gamma \beta$. When the optimal assembly rate is lower than the assembly rate benchmark, the assembly rate must reach the benchmark $g_{4}^{ * } = g_{0}$.

By substituting $g_{4}^{ * } = g_{0}$ into Eq. (3), the optimal entire supply chain profits are represented as follows:

$$\pi_{{~}}^{4} { = }\alpha {\mathrm{R}} - {\mathrm{C}}_{c} - {\mathrm{C}}_{d} { + }\beta kg_{0} { + }B_{0} - \frac{1}{2}kg_{0}^{2}$$

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Appendix 3. Proof of Proposition 3

Case 1: When the assembly is not lower than the assembly rate benchmark, there is a constraint that

$$0 \leq g_{0} \leq \frac{{\beta {+ \mathrm{g}}_{{0}} }}{2} - \frac{{B_{0} }}{{2\beta {\mathrm{k}}}}$$

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Shifting the inequality, we can obtain

$$0 \leq g_{0} \leq \beta - \frac{{B_{0} }}{\beta k}$$

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In addition

$$g_{3}^{ * } - g_{1}^{ * } = \beta - (\frac{{\beta {+ \mathrm{g}}_{{0}} }}{2} - \frac{{B_{0} }}{{2\beta {\mathrm{k}}}}) = \frac{\beta }{2} - \frac{{{\mathrm{g}}_{{0}} }}{2} + \frac{{B_{0} }}{{2\beta {\mathrm{k}}}} \geq \frac{{B_{0} }}{{\beta {\mathrm{k}}}} > 0$$

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$${\pi }^{3}-{\pi }^{1}=\frac{1}{{8}}k{\left(\beta -{g}_{0}\right)}^{2} + \frac{{B}_{0}}{4} + \frac{{B}_{0}{g}_{0}}{{4}\beta } + \frac{{B}_{0}^{2}}{{8}{\beta }^{2}k}-\frac{1}{8}k{\left(\frac{{B}_{0}}{\beta k}\right)}^{2}+\frac{{3B}_{0}^{2}}{8{\beta }^{2}k}>0$$

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Case 2: When the assembly is lower than the assembly rate benchmark, the results in Table 3 lead to the equations: ${g}_{2}^{*}={g}_{4}^{*},{\pi }^{2}={\pi }^{4}$.

Appendix 4. Proof of Proposition 4.

In order to coordinate the PCSC system, we must ensure $g(\hat{\gamma }) = g_{3}^{ * }$ In Eq. (2), by replacing $g$ with $g_{3}^{ * } = \beta$ and then solving $\hat{\gamma }$, we get the expression of $\hat{\gamma }{ = }1$.

Then consider the developer’s problem, substituting $\pi_{d}$ with $g = \beta$ $\hat{\gamma }{ = }1$, we have

$$\pi_{d} (\hat{\gamma }){ = }\alpha {\mathrm{R}} - {\mathrm{C}}_{d} - P - \frac{1}{2}kg_{0}^{2} { + }\beta kg_{0}$$

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$$\pi_{d} (\hat{\gamma }) - \pi_{d}^{1} { = } - \frac{1}{4}(\beta - g_{0} )^{2} - \frac{{B_{0} (\beta - g_{0} )}}{2\beta } - \frac{{B_{0}^{2} }}{{4\beta^{2} k}} < 0$$

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Then consider the general contractor’s problem, substituting $\pi_{c}$ with $g = \beta$ $\hat{\gamma }{ = }1$, we have

$$\pi_{c} (\hat{\gamma }){= \mathrm{P}} - {\mathrm{C}}_{c} - \frac{1}{2}k\beta^{2} + \frac{1}{2}kg_{0}^{2} { + }\beta^{2} k - \beta kg_{0} { + }B_{0}$$

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$$\pi_{c} (\hat{\gamma }) - \pi_{c}^{1} { = }\frac{3}{8}(\beta - g_{0} )^{2} + \frac{{3B_{0} (\beta - g_{0} )}}{4\beta } + \frac{{3B_{0}^{2} }}{{8\beta^{2} k}} > 0$$

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Appendix 5. Proof of Proposition 5

According to Proposition 4, when the subsidy revenue-sharing coefficient $\gamma =1$, there is the situation (${\mathrm{g}}_{1} = {\mathrm{g}}_{3} \, {\mathrm{g}}_{2} = {\mathrm{g}}_{4}$). Moreover, according to Cachon’s definition of supply chain coordination, only when the profits of all members of the PCSC increase, the PCSC can achieve overall coordination. Therefore, the transfer amount from the general contractor to the developer in the transfer payment contract needs to satisfy the following inequality

$$\frac{1}{4}\left(\beta-g_0\right)^2 + \frac{B_0\left(\beta-g_0\right)}{2\beta} + \frac{\mathrm{B}_0^2}{4{\beta}^2\mathrm{k}} \le{\mathrm{T}}_1 \leq \frac{3}{8}\left(B-{\mathrm{g}}_0\right)^2 + \frac{3{\mathrm{B}}_0\left(\beta-g_0\right)}{4{\beta}} + \frac{3\mathrm{B}_0^2}{8{\beta}^2\mathrm{k}}$$

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$${\mathrm{T}}_{2} = {\mathrm{B}}_{0}$$

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