A Game-Theory-Based Interaction Mechanism between Central and Local Governments on Financing Model Selection in China

Abstract

Local financing platforms and (public–private partnerships) PPPs have received extensive attention, but there are few studies on the interaction mechanism of financing model selection. This paper presents a game-theory-based interaction mechanism of local financing platforms and a PPP model based on the government heterogeneity objective function. The study results found that the central government’s tolerance of local governments participating directly in municipal projects with financing platforms or PPP models mainly depends on land price premiums. When the premium is small, the collusion between local governments and financing platforms does not violate the objectives of the central government. Then, local and central governments prefer financing platforms to participate directly in municipal projects. In contrast, the local government prefers the financing platform model when the premium is significant. The central government no longer tolerates the financing platform model and prefers to complete municipal projects with the PPP model. This study believes that promoting the PPP model is a critical way to moderately resolve the debt risk of local government financing platforms and reduce the financial pressure on local governments.

1. Introduction

Urbanization is one of the essential paths towards achieving high-quality economic development. It is the long-term driving force to expand domestic demand and the vehicle to promote sustained and healthy economic development. Urbanization can promote the upgrading of residents’ consumption level, stimulate domestic demand potential, and promote the aggregation of technology, talent, capital, and other factors [1]. It can also promote the optimization and upgrading of industry, an important power source for long-term and healthy economic development.

Solving the investment and financing problems of urban infrastructure construction is a vital prerequisite for urbanization. With the advancement of urbanization, local governments have launched a fierce competition to obtain financial resources and promote economic and social development, thus giving birth to the financing platform and promoting its development. For some time, urban investment and financing platforms have been the pillar of urban development and construction. After 2013, the central government began actively trying to improve the PPP model, and social capital participated in the investment and operation of urban infrastructure through the franchise [2]. How do we understand the transformation of urbanization financing from a local financing platform to a PPP model?

Local financing platform debt financing expansion leads to the accumulation of local government debt risk. In developing countries such as Brazil, Argentina, Mexico, and India, and developed countries such as Germany, Spain, Italy, and other cooperative federal economies, the expansion of local government debt financing is a lingering “growth worry”, which has experienced the runaway scale of local government debt and forced the central government to come to the rescue afterward [3]. There are significant risks in developing local financing platforms in China [4,5,6]. The main reason lies in the following two points: first, the local government financing platform asset yield is low, the corporate governance structure is not perfect, the responsibility is not clear, the operation procedure is not standardized, and the government and enterprises are not divided [7]. Second, the local financing platform is generally a short-term financing long-term investment model. Although driving the project’s construction, this may bury a vast risk [8]. Lou Jiwei pointed out that the key to preventing and resolving the hidden debt risk of local governments is to clean up and standardize the financing platform and its related debt and realize the market transformation of the financing platform as soon as possible

Much of the scientific literature utilizes the PPP model to solve the debt risk of local financing platforms and the funding gap of urban infrastructure construction. Xu Lei studied the debt problem of many local financing platforms after the 2008 financial crisis and the contradiction between the capital demand for new urbanization and the shortage of government monetary funds. The PPP model is believed to alleviate the financing pressure caused by many public projects and solve the local government debt problem [9]. Zhang Wei and Luo Liping pointed out that promoting the PPP model is a crucial way to improve the efficiency of public service supply, alleviate the financial pressure on local governments, and moderately resolve the debt risk of local government financing platforms [10]. Zhang Qiming and Guo Xuemeng believe that the current state’s vigorous implementation of the PPP model helps alleviate the soft constraints of local government finance and solve the local government debt problem [11]. Zheng Ai believes that if the PPP model is implemented correctly, it can solve the debt risk and construction fund gap local governments face [12]. Yu Yang and Jiao Yong believe that investment and financing behavior in urban development and construction is an essential source of local debt risk. Only by regulating the operation of the PPP model can the original intention of policy design be resolved for local debt risks [13].

The local government debt problem results from the game between the local government and the central government. The local government financing platform is vital in helping “China’s infrastructure miracle”, but the central government has not always recognized the “heroes” in developing this infrastructure. Whether the relevant policies encourage or deter is governed mainly by the dual governance logic of economic growth and risk aversion [14]. Based on the hypothesis of rational man in economics, some literature studies the formation and expansion of local government debt from the perspective of policy and mechanism and believes that institutional and policy factors are the main reasons for the formation and continued growth of local government debt in China [15,16,17,18]. Some other literature studied the risk evolution and expansion of local government debt from the perspective of behavioral preference in behavioral economics and proposed that the deviation of behavioral preference should be corrected, the debt repayment mechanism should be changed, and the risk management approach of establishing an early warning system for risks should be established [19,20,21,22]. Dixit et al. [23], Bowdler et al. [24], and Pitchford et al. [25] applied game models to explain local government debt default and central government rescue behavior. Inman [26] constructed a game matrix to prove that intergovernmental cooperation can solve the prisoner’s dilemma in local government debt. For most existing studies, the payoff function is given exogenously, and there is no numerical model verification of the game results. Additionally, little literature discusses the formation and governance of debt risk. This paper explores the causes of debt risk and debt risk governance from a macro perspective. The payment function is an endogenous variable based on heterogeneous government goals and land premiums. A numerical game simulation is conducted to make the research more rigorous and practical.

The sustainability of the PPP model is achieved by integrating efficiency, technological innovation, and risk sharing into the project. PPP has become one of the main tools used to provide many public services [27,28], occupying an important position in municipal infrastructure construction and public transport, and playing a vital role in the global economy [29,30]. The main reason is that the interaction between public and private elements produces advantages and synergies [31,32,33,34]. In this model, the two sides play their respective advantages to help promote urbanization construction projects efficiently [35,36]. Specifically, one is to effectively mobilize private capital, reducing the lack of public sector funding. The second tactic is to effectively reduce the cost of infrastructure construction and operation, play the particular advantages of government and private enterprises. The third is to provide kinetic energy for the financing of infrastructure construction, stimulate the enthusiasm of social capital to participate, reduce the pressure of local government debt repayment, and resolve the debt risk of local governments. The PPP model restrains the expansion of local government debt through a public–private cooperation mechanism to reduce the government’s dependence on tax, prevent local government debt risk [37], and become an effective strategy for infrastructure construction procurement [38].

The existing studies mainly carried out the normative analysis but did not explore the choice behavior between the local financing platforms and the PPP model. This paper presents a model to analyze the choice behavior of local government from the perspective of mathematic analysis, which can better understand the government decision-making behavior between financing platforms and the PPP model. The contributions of this paper are summarized as follows:

• This paper designs a game framework between the central and local governments by land price premium. It then compares the operation mechanism of local financing platforms and PPP models, which provides a new perspective for understanding the dependence of local economic development on land finance.
• The results of this paper enhance the understanding of the impact of land price premiums on local economic development. When the land price premium is not high, land finance plays a role in promoting economic development; when the land price premium is high, the model of land finance promoting economic development is not sustainable. The PPP model can moderately resolve the debt risk of local government financing platforms and alleviate the financial pressure on local governments.
• This paper utilizes the endogenous payment function and discusses the causes of debt risk and the governance of debt risk by comparing the proposed approach with the existing game literature. The numerical simulation of behavior decision making verified the proposed model.

The rest of the paper is organized as follows: Section 2 presents the related work. Section 3 introduces the speculative investment and financing model, analyzes the operation mechanism of the traditional financing platforms and the PPP model. Section 4 analyzes government behavior. Section 5 concludes the paper and discusses some future research directions.

2. Related Work

China’s urbanization significantly promotes economic growth. The empirical results of Yang Meicheng show that China’s urbanization development can have a positive economic growth effect [39]. Chen Junliang et al. empirically tested the positive effect of urbanization on economic growth mainly through physical capital, human capital, land capital, technology, and consumption [40]. Zhu et al. analyzed the effects of population urbanization and land urbanization on economic growth and found that land urbanization and population urbanization promoted economic growth [39]. Mao Yanbing and Yuan Yunke show that green new urbanization significantly promotes China’s economic growth [41]. Chen Yao and Zhou Hongxia believe that China’s urbanization level is positively correlated with economic growth, and urbanization has become a powerful engine for maintaining sustained and healthy economic development [42].

The financing platform promotes economic growth and increases local debt risk. On the one hand, the financing platform promotes economic growth [43,44,45]; on the other hand, it brings debt risk [46,47]. Further analysis shows that the core of the financing platform investment and financing mode operation is the “land financing” mode combined with “land transfer income” and “land mortgage loan” [48]. Although it provides an important funding source for urban infrastructure investment, it also leads to widespread short-term behavior of local governments, which increases the debt risk of local financing platforms. It is urgent to upgrade the investment and financing mode of urban construction.

One essential way to transform and develop local investment and financing platforms is to carry out government and social capital cooperation (PPP) projects. Based on the analysis of the problems and risks of local financing platforms, it is considered that one of the transformation modes of local financing platforms is the PPP model [49,50,51,52]. Han Gang et al. obtained the same results by comparing the United States, Japan, Germany, France, and other developed countries’ urbanization construction investment and financing modes [53]. Luo Guilian pointed out that the financing platform should be standardized, PPP should be prudently and steadily promoted, and the advantages of the two modes should be coordinated to gradually realize the transformation and upgrade of the investment and financing system [54].

The characteristics of the PPP model help change the current urban infrastructure investment over-reliance on financing platforms. The connotation of the PPP model should include at least three core elements of financing, property rights, and risk sharing, which also determines that the PPP model has the following advantages, namely, the responsibility integration of construction and operation [55,56,57,58,59,60,61] and relative advantage complementarity [62,63,64,65].

The central and local game is essential for transforming the financing model from a local financing platform to a PPP model. Sun Ninghua analyzed the economic game between the central government and local governments during the economic transition period, mainly focusing on the game of institutional change, economic growth rate, and fiscal and tax surrender [66]. Chen Kang and Gu Qingyang constructed a game model between the central and local governments. They analyzed the incentive mechanism affecting the behavior of local governments from the perspective of the financial relationship between the central and local governments [67]. Based on the perspective of fiscal decentralization, Tao Yuanlei explores the tripartite evolutionary game model of local government debt risk control between the central government, local government, and financial institutions [68]. Refs. [69,70] argue that local economic development and the “tournament” of officials’ promotions have led to inconsistent central and local objectives in the area of financial regulation, leading to significant behavioral differences.

It can be seen from

Table 1. Summary and Comparison of Existing Studies with the Proposed Study.

Article	Game Subjects	Payment Matrix	Reasons	Governance	Methods	Simulation
Wang Junqiang, Zuo Ting [15]	Central Government Local Government	Yes	Yes	No	Yes	No
Li Jingwei, Tang Xin [16]	Central Government Local Government	Yes	Yes	No	Yes	No
Zhou Lili, Cao Honghui [17]	Central Government Local Governments	Yes	Yes	No	Yes	No
Liu Hao, Chen Gong [18]	Central Government Local Government	Yes	Yes	No	Yes	No
Peng Wangxian, Ye Shujun [19]	Local governments financial institutions	Yes	No	Yes	Yes	No
Ma Jinhua, Yang Juan [20]	Central Government Local Government Financial Institutions	No	No	Yes	No	No
Han Wenyan, Gao Cheng [21]	Central Government Local Government	Yes	No	Yes	Yes	No
Su Ying and Liu Xing [22]	Central Government Local Government	Yes	No	Yes	Yes	No
Tao Yuanlei [68]	Central Government Local Government Financial Institutions	Yes	No	Yes	Yes	No
Zhao Bin, Wang Chaocai, Ke Xie [69]	Central Government Local Government	No	Yes	No	No	No
Hu Jiye, Dong Yawei [70]	Central Government Local Government	Yes	No	Yes	Yes	No
Proposed Study	Central Government Local Government	Yes	Yes	Yes	Yes	Yes

Note: Reasons: this paper mainly discusses the formation mechanism of debt risk; Governance: this study mainly discusses debt risk governance; Methods: using mathematical or quantitative methods; Simulation: the game results are verified by numerical simulation.

that for most existing literature, the participants in the game are local governments and central governments. However, there is no numerical simulation of the game results. Furthermore, the causes of debt risk and debt risk governance are not discussed in a unified framework. On the one hand, the central government should prevent systemic financial risks and maintain financial stability; on the other hand, local government officials pursuing local economic growth will promote their own financial risk-taking opportunistic behavior. This makes local financial regulators conflict between the goals of solving local economic development and preventing local financial risks. The central and local regulatory game focuses on how to prevent local financial risks. Overall, the game between the central and local governments mainly comes from the game between the central and local financial rights. It is necessary to straighten out the financial and administrative rights of central and local governments to ensure the central fiscal and tax revenue and actively guide local economic development. The existing literature ignores the role of the game between the central and local governments in the two types of investment and financing models and does not study the financing platform and PPP model under the framework of a government game, which cannot better explain the intrinsic motivation of the transformation of the financing platform into the PPP model.

3. Theoretical Model of Investment and Financing

3.1. Traditional Financing Platform Model and Land Finance

Local governments need adequate financing capacity. Hence, a financing platform is established and colluded to raise land prices. This model forms local government dependence on land finance. Referring to the method of Yang et al. [71], the output function of the financing platform as a state-owned enterprise is set as:

$$Y^s={(AK)}^{\alpha }{L}^{1-\alpha }$$

(1)

The cost function is:

$$C^s\left(K,L,p\right)=(1+r)K+\left(1+{\tau }_L\right)pL$$

(2)

Output is determined by capital and land input; the cost includes capital and land, $K$, $L$, $p$, $r$, ${\tau }_Y$, ${\tau }_L$ represent the capital, land, land price premium, interest rate, total tax, and the land tax levied by the government on land, respectively. After deducting gross tax and land tax, the total profit function can be expressed as:

$${\pi }^s=\left(1-{\tau }_Y\right){(AK)}^{\alpha }{L}^{1-\alpha }-\left((1+r)K+\left(1+{\tau }_L\right)pL\right)$$

Referring to the method of Yang Jidong (2016), the objective function of the financing platform is defined as follows:

$$U^s={\theta }_1{\pi }^S+{\theta }_2{Y}^S$$

(3)

where ${\theta }_1$ and ${\theta }_2$ the weight added to the objective function of the financing platform, respectively, in its profit level and total output.

Given the tax rate, and assuming that the land supplied by the municipal projects determined by the local government is limited $L\le \overline{L}$, the optimization problem of the financing platform is defined as follows:

$$\begin{array}{l}\underset{K,L}{\max }\left\{{\theta }_1\left(1-{\tau }_Y\right){(AK)}^{\alpha }{L}^{1-\alpha }-{\theta }_1\left((1+r)K+\left(1+{\tau }_L\right)pL\right)+{\theta }_2{(AK)}^{\alpha }{L}^{1-\alpha }\right\}\\ s.t.L\le \overline{L}\end{array}$$

The Lagrange function is:

$$l(K,L,\lambda )=\underset{K,L}{\max }\left\{\left(\left(1-{\tau }_Y\right){\theta }_1+{\theta }_2\right){(AK)}^{\alpha }{L}^{1-\alpha }-\left((1+r)K+\left(1+{\tau }_L\right)pL\right)+\lambda (\overline{L}-L)\right\}$$

The following three conditions (4)–(6) can be obtained using the Kuhn–Tuck theorem:

$$\frac{\partial l}{\partial K}=\left(\left(1-{\tau }_Y\right){\theta }_1+{\theta }_2\right)\alpha A^{\alpha }{K}^{\alpha -1}{L}^{1-\alpha }-{\theta }_1(1+r)=0$$

(4)

$$\frac{\partial l}{\partial L}=\left(\left(1-{\tau }_Y\right){\theta }_1+{\theta }_2\right)\left(1-\alpha \right){(AK)}^{\alpha }{L}^{-\alpha }-{\theta }_1\left(1+{\tau }_L\right)p-\lambda =0$$

(5)

$$\lambda \ge 0,\overline{L}-L\ge 0,\mathrm{and}\lambda (\overline{L}-L)=0$$

(6)

(i) When $\lambda =0,\mathrm{then}\overline{L}-L\ge 0$. Solving (4) and (5), two first-order conditions, the optimal capital investment is:

$$K=\frac{\alpha }{1-\alpha }\left(\frac{1+{\tau }_L}{1+r}\right)pL$$

(7)

$$p=\frac{1-\alpha }{1+{\tau }_L}{\left(\frac{\alpha A}{\left(1+r\right){\theta }_1}\right)}^{\frac{\alpha }{1-\alpha }}{\left(\left(1-{\tau }_Y\right){\theta }_1+{\theta }_2\right)}^{\frac{1}{1-\alpha }}$$

(8)

It can be seen from Equation (7) that capital investment is an increasing function of the land price premium. In other words, where the land price premium is higher, the investment in infrastructure is often higher. In the early stage of economic development, local governments and financing platforms colluded to improve land price premiums and promote infrastructure construction, which was helpful for local economic growth. Substituting (8) for (7) shows that land taxes ${\tau }_L$ do not affect capital demand, while increasing the total tax ${\tau }_Y$ and interest rate will reduce capital demand. It can be seen from Equation (8) that the land price premium is an increasing function of $A$ (capital-increasing technology) and a decreasing function of the total tax ${\tau }_Y$, land tax ${\tau }_L$, and interest rate. That is to say, the land price premium is determined by capital increase technology, aggregate tax, land tax, weight ${\theta }_1,{\theta }_2$, and interest rate. Technological progress, tax-rate reduction, and low interest rates lead to land price premiums.

(ii) If $\lambda >0,\mathrm{then}\overline{L}-L=0$. by Equation (4):

$$K={\left(\frac{\left(\left(1-{\tau }_Y\right){\theta }_1+{\theta }_2\right)\alpha }{\left(1+r\right){\theta }_1}\right)}^{\frac{1}{1-\alpha }}{A}^{\frac{\alpha }{1-\alpha }}\overline{L}$$

It can be seen that when the land price premium is not significant, especially in the early stage of economic development, capital investment in public goods is helpful for the local economy. At this time, the government can reduce the total tax and interest rate to promote the investment in productive capital and ultimately accelerate the development of the local economy. If the total output weight increases, it will also promote the capital investment of the financing platform. In addition, it can also increase the capital investment in public goods by reducing the land supply restrictions and increasing the land supply, thereby promoting local economic development.

3.2. PPP Model

Private enterprises are encouraged to participate in municipal projects in the PPP model, since they are usually small in scale and need cooperation with multiple participating enterprises to complete projects. Meanwhile, participating enterprises have technical complementarity. Therefore, the investment level of each participating enterprise is essential for the completion of projects. For participating enterprises $i$, suppose investment capital, and produce intermediate inputs through a capital increase technology. This technique can be expressed as the following intermediate product production function: $m_i^e={\left(A_i{K}_i\right)}^{\alpha }$, $A_i$ a technique that represents a capital increase. All participating enterprises cooperate to complete the project, and a function can express the way of cooperation $F(\cdot )$. In particular, we adopt the Bryant [72] model and set this function as determined by the minimum $\min (\cdot )$ input of intermediate goods. In other words, the production completed by the project is determined by the minimum intermediate production enterprise in all participating enterprises:

$$Y^e={\displaystyle \sum _{i=1,\dots ,I}\min \left(m_1^e,\cdots ,m_I^e\right)}{L}_i^{1-\alpha }$$

(9)

Considering the total profit of all participating enterprises $I$ to cooperate in the completion of the project, the total profit function can be expressed as follows:

$${\pi }^e=\left(1-{\tau }_Y\right){\displaystyle \sum _{i=1,\cdots ,I}\min \left(m_1^e,\cdots ,m_I^e\right)}{L}_i^{1-\alpha }-{\displaystyle \sum _{i=1,\cdots ,I}\left((1+r)K_i+\left(1+{\tau }_L\right)L_i\right)}$$

Similar to the financing platform method, the objective function of the PPP project is set as follows:

$$U^e={\phi }_1{\pi }^e+{\phi }_2{Y}^e$$

(10)

where ${\phi }_1$ and ${\phi }_2$ the weight added to the objective function of the financing platform, respectively, in its profit level ${\pi }^e$ and total output $Y^e$. The weight parameters of PPP and financing platforms are generally not equal. Because of the addition of private enterprises, the PPP model pays more attention to profitability than the financing platform.

Hypothesis 1 (H1):

The model has an interior or angular point equilibrium. Interior equilibrium: The optimal amount of capital invested by all participating enterprises is determined by the following equation:

$$\begin{gathered} K_i=\frac{\alpha }{1-\alpha }\frac{(1+{\tau }_L)}{(1+r)(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})}L \\ {K}_j=\frac{\alpha }{1-\alpha }\frac{(1+{\tau }_L)}{(1+r)(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})}\frac{A_i}{A_j}L,\begin{array}{cc}& j\ne i\end{array} \end{gathered}$$

Total social capital investment of municipal projects:

$$K^e=\frac{\alpha }{1-\alpha }\left(\frac{1+{\tau }_L}{1+r}\right)L$$

Corner equilibrium: The optimal amount of capital invested by all participating enterprises is determined by the following equation:

$$K_i={\left(I{A}_i\right)}^{\frac{\alpha }{1-\alpha }}{\left(\frac{\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha }{{\phi }_1\left(1+r\right)\left(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j}\right)}\right)}^{\frac{1}{1-\alpha }}\overline{L}$$

$$K_j=\frac{{\left(I\right)}^{\frac{\alpha }{1-\alpha }}}{A_j}{\left(\frac{\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha A_i}{{\phi }_1\left(1+r\right)\left(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j}\right)}\right)}^{\frac{1}{1-\alpha }}\overline{L},j\ne i$$

Total social capital investment of municipal projects:

$$K={\left(\frac{I}{{\displaystyle \sum _{i=1}^{I}1/A_i}}\right)}^{\frac{\alpha }{1-\alpha }}{\left(\frac{\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha }{{\phi }_1\left(1+r\right)}\right)}^{\frac{1}{1-\alpha }}\overline{L}$$

where$L={\displaystyle \sum _{i=1,2\dots ,I}{L}_i}$.

Proof of Hypothesis 1:

It is possible to assume that there is $m_i^e={\left(A_i{K}_i\right)}^{\alpha }\le {\left(A_j{K}_j\right)}^{\alpha }=m_j^e,\forall i\ne j$ any given $i$, so the Lagrange function is:

$$\begin{array}{l}l({\left\{K_i,L_i\right\}}_{i=1}^I,\lambda ,{\left\{{\lambda }_j\right\}}_{i\ne j}^{})=\\ \underset{{\left\{K_i,L_i\right\}}_{i=1}^I,\lambda ,{\left\{{\lambda }_j\right\}}_{i\ne j}^{}}{\max }\left\{\begin{array}{l}\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right){\displaystyle \sum _{i=1,\dots ,I}\min \left({\left(A_1{K}_1\right)}^{\alpha },\dots ,{\left(A_I{K}_I\right)}^{\alpha }\right)}{L}_i{}^{1-\alpha }\\ -{\displaystyle \sum _{i=1,\dots ,I}{\phi }_1\left((1+r)K_i+\left(1+{\tau }_L\right)L_i\right)+\lambda (\overline{L}-{\displaystyle \sum _{i=1,\dots ,I}{L}_i)+{\displaystyle \sum _{j\ne i}{\lambda }_j\left({\left(A_j{K}_j\right)}^{\alpha }-{\left(A_i{K}_i\right)}^{\alpha }\right)}}}\end{array}\right\}\end{array}$$

Then

$$\frac{\partial l}{\partial K_i}=\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha A_i{}^{\alpha }{K}_i{}^{\alpha -1}{\displaystyle \sum _{i=1}^I{L}_i{}^{1-\alpha }}-{\phi }_1(1+r)-{\displaystyle \sum _{j\ne i}{\lambda }_j\left(\alpha A_i{}^{\alpha }{K}_i{}^{\alpha -1}\right)}=0$$

(11)

$$\frac{\partial l}{\partial K_j}=-{\phi }_1(1+r)+{\lambda }_j\left(\alpha A_j{}^{\alpha }{K}_j{}^{\alpha -1}\right)=0,j\ne i$$

(12)

$$\frac{\partial l}{\partial L_i}=\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\left(1-\alpha \right){\left(A_i{K}_i\right)}^{\alpha }{L}_i{}^{-\alpha }-{\phi }_1\left(1+{\tau }_L\right)-\lambda =0i=1,\dots ,I$$

(13)

$$\lambda \ge 0,\overline{L}-{\displaystyle \sum _{i=1,2\dots ,I}{L}_i}\ge 0,\mathrm{and}\lambda (\overline{L}-{\displaystyle \sum _{i=1,2\dots ,I}{L}_i})=0$$

(14)

$${\lambda }_j\ge 0,{\left(A_j{K}_j\right)}^{\alpha }-{\left(A_i{K}_i\right)}^{\alpha }\ge 0,\mathrm{and}{\lambda }_j\left({\left(A_j{K}_j\right)}^{\alpha }-{\left(A_i{K}_i\right)}^{\alpha }\right)=0,\forall j\ne i$$

(15)

From Equations (12) and (15), we can get:

$${\lambda }_j>0,\mathrm{and}{\left(A_j{K}_j\right)}^{\alpha }-{\left(A_i{K}_i\right)}^{\alpha }=0,\forall j\ne i$$

(16)

From (11), (12), and (16):

$$\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha A_i{}^{\alpha }{K}_i{}^{\alpha -1}{\displaystyle \sum _{i=1}^I{L}_i{}^{1-\alpha }}-{\phi }_1(1+r)(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})=0$$

(17)

From (13), we can get: $L_i=L_j,\forall j\ne i$, combined with (13), (14), and (17),

(1) When $\lambda =0,\overline{L}-{\displaystyle \sum _{i=1,2\dots ,I}{l}_i}\ge 0$, by (13) and (17), the optimal capital investment is:

$$K_i=\frac{\alpha }{1-\alpha }\frac{(1+{\tau }_L)}{(1+r)(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})}I{L}_i=\frac{\alpha }{1-\alpha }\frac{(1+{\tau }_L)}{(1+r)(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})}L$$

(18)

$$K^e={\displaystyle \sum _{j\ne i}{K}_j+K_i}=(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})K_i=(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})\frac{\alpha }{1-\alpha }\frac{(1+{\tau }_L)}{(1+r)(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j})}L$$

$$\Rightarrow K^e=\frac{\alpha }{1-\alpha }\frac{(1+{\tau }_L)}{(1+r)}L$$

(19)

where $L={\displaystyle \sum _{i=1,2\dots ,I}{L}_i}$.

(2) When $\lambda >0,$ then $\overline{L}-L=0$.

$$K_i={\left(I{A}_i\right)}^{\frac{\alpha }{1-\alpha }}{\left(\frac{\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha }{{\phi }_1\left(1+r\right)\left(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j}\right)}\right)}^{\frac{1}{1-\alpha }}\overline{L}$$

$$K_j=\frac{{\left(I\right)}^{\frac{\alpha }{1-\alpha }}}{A_j}{\left(\frac{\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha A_i}{{\phi }_1\left(1+r\right)\left(1+{\displaystyle \sum _{j\ne i}{A}_i/A_j}\right)}\right)}^{\frac{1}{1-\alpha }}\overline{L},j\ne i$$

Finally, local governments will cooperate with participating enterprises in development projects, so the total social financing is expressed as follows:

$$K={\left(\frac{I}{{\displaystyle \sum _{i=1}^{I}1/A_i}}\right)}^{\frac{\alpha }{1-\alpha }}{\left(\frac{\left(\left(1-{\tau }_Y\right){\phi }_1+{\phi }_2\right)\alpha }{{\phi }_1\left(1+r\right)}\right)}^{\frac{1}{1-\alpha }}\overline{L}$$

(20)

Local government municipal projects need to develop land for: ${\displaystyle \sum _{i=1,\dots ,I}{L}_i}=\overline{L}$. □

It can be seen from Equation (19) that capital input is a function of land tax and interest rate. In other words, with the development of the economy, the capital investment of public goods in the PPP model should be realized through an increase in land tax and a decrease in interest rate.

4. Analysis of Government Decision Making

4.1. Government Decision-Making Model

The objectives of local governments are not consistent with those of the central government, and local governments have a certain degree of autonomy in implementing reform measures. Consequentially, the results of their implementation may deviate from the policy intention of the central government, which will affect the pace of reform and the coordination of economic development. It is challenging to promptly facilitate China’s economic restructuring through reform, which may increase the leverage level of local governments, increase bank risks and increase the contingent sovereign debt of the Chinese government.

From the perspective of the local government’s financial structure, the local government’s budgetary income cannot cover the current total fiscal expenditure. Especially when the local government faces the pressure of public infrastructure construction funds shortage, the local government has the power to capitalize on land resources to expand the fiscal space and increase the government’s disposable financial resources. Land transfer fees can be another important source of local government revenue. From 2007 to 2010, the proportion of national state-owned land use transfer and local public finance revenue increased from 18% to 41%. Public goods are generally provided by the government, which can meet the needs of people and promote economic development. Therefore, local fiscal revenue, public goods provision, and land transfer income are considered the objective functions of local governments. If the financing platform is adopted, the goal of local governments can be set as follows: $g_1^s=G\left({\tau }_Y,Y^s,{\tau }_L,pL\right)=\left(1+{\tau }_Y\right)Y^s+\left(1+{\tau }_L\right)pL$; if the PPP model is adopted, the goal of local governments can be set as follows: $g_1^e=G\left({\tau }_Y,Y^e,{\tau }_L,L\right)=\left(1+{\tau }_Y\right)Y^e+\left(1+{\tau }_L\right)L$.

The audit announcement of the National Audit Office (2013) shows that the debt repayment amount committed to using land transfer income as the source of debt repayment accounts for about 40–50% of the debt repayment amount. The tests of [73] on the credit risk of local financing platforms and their market pressure scenarios show that when the asset value of local financing platforms jumps by more than 30%, about CNY 2 trillion of the debt of local financing platforms of CNY 7.66 trillion is in a state of default risk, which is the theoretical boundary of the universal default of “problem” financing platform loans. The source of debt repayment of local financing platforms usually includes the profitability of local financing platform financing projects and local financing platform land assets. Local financing platform financing funds are mainly invested in government (quasi) public welfare projects, and the project’s profitability is generally low. Once housing prices plummet, such a scale of mortgage assets depreciation will lead to an unimaginable financial tsunami. Widespread bankruptcy will not only destroy local government credit, but will sweep every corner of the economy. Therefore, the central government’s objectives include adequate income taxes and adequate public goods while avoiding the negative impact of local government land finance on long-term economic development. Therefore, using the financing platform model, the central government’s goal can be set in the following forms: $g_2^s=G\left({\tau }_Y,Y^s,{\tau }_L,pL\right)=\left(1+{\tau }_Y\right)Y^s-\left(1+{\tau }_L\right)pL$; with the PPP model, the objectives of the central government can be set as follows: $g_2^e=G\left({\tau }_Y,Y^e,{\tau }_L,L\right)=\left(1+{\tau }_Y\right)Y^e-\left(1+{\tau }_L\right)L$.

Using the objective functions of the local and central governments illustrated above, the game payment matrix is presented in

Table 2. Game Payment Matrix between Central Government and Local Government.

		Local Government
		Financing Platform	PPP
Central Government	Financing Platform	$g_2^s$$,g_1^s$	$g_2^s$$,g_1^e$
	PPP	$g_2^e$$,g_1^s$	$g_2^e$$,g_1^e$

.

$g_2^s$$,g_1^s$$g_2^s$$,g_1^e$$g_2^e$$,g_1^s$$g_2^e$$,g_1^e$ The government’s decision-making behavior is discussed based on the difference in social returns constructed by the payment matrix. For the convenience of analysis, it is assumed that the optimal land quantity selected by the two models is the same. The difference in social returns of local government between the two models is defined in Equation (21):

$$∆g_1\equiv g_1^e-g_1^s=\left(1+{\tau }_Y\right)\left(Y^e-Y^s\right)+\left(1+{\tau }_L\right)L\left(1-p\right)$$

(21)

The difference in social returns of central government between the two models is defined in Equation (22):

$$∆g_2\equiv g_2^e-g_{{}^2}^s=\left(1+{\tau }_Y\right)\left(Y^e-Y^s\right)+\left(1+{\tau }_L\right)L\left(p-1\right)$$

(22)

Hypothesis 2 (H2):

If the land price premium expands $p>\overline{p}$, local governments prefer to participate in municipal projects directly in the financing platform mode; if the land price premium is low $p\le \overline{p}$, local governments prefer to complete municipal projects in the PPP model.

Proof of Hypothesis 2:

The preference of local governments is determined by the difference in social returns. We can bring the output of the two modes into (21) and obtain the following formula:

$$\mathsf{\Delta }{g}_1=L\left\{\left(1+{\tau }_Y\right){\left[\frac{\alpha }{1-\alpha }\left(\frac{1+{\tau }_L}{1+r}\right)\right]}^{\alpha }\left({\left(\frac{I}{{\displaystyle \sum _{i=1}^I}1/A_i}\right)}^{\alpha }-A^{\alpha }{p}^{\alpha }\right)+\left(1+{\tau }_L\right)(1-p)\right\}$$

Therefore, the choice of local governments can be judged by the positive and negative differences in social returns. Since $\frac{d∆g_1}{dp}<0$, $∆g_1$ is a strictly decreasing function, $∆g_1(0)>0,∆g_1(+\infty )<0,$ there is only one $\overline{p}$ satisfying $∆g_1(\overline{p})=0$, for which $\overline{p}$ is determined by the following equation:

$$\frac{{\overline{\mathrm{A}}}^{\alpha }-A^{\alpha }{p}^{\alpha }}{p-1}=\frac{1+{\tau }_L}{1+{\tau }_Y}{\omega }^{-\alpha }$$

(23)

where $\overline{\mathrm{A}}=I/{\displaystyle \sum _{i=1}^{I}1/A_i},\omega =\frac{\alpha }{1-\alpha }\left(\frac{1+{\tau }_L}{1+r}\right)$, (23) the threshold of available land price premium value is a function of parameters $\overline{p}=P\left({\tau }_Y,{\tau }_L,\alpha ,L,A,A_i,A_j\right)$. When $p<\overline{p},∆g_1(0)>0$, local governments prefer to complete municipal projects in the PPP model; when $p>\overline{p},∆g_1(0)<0$, local governments would prefer to participate in municipal projects in the financing platform mode directly. □

Before discussing the behavior decision-making of the central government, the following assumptions need to be made:

$$\begin{array}{l}(\mathrm{i})\overline{\mathrm{A}}>A\\ (\mathrm{ii})\left(1+{\tau }_Y\right){\omega }^{\alpha }{\overline{\mathrm{A}}}^{\alpha }-\left(1+{\tau }_L\right)>0\\ (\mathrm{iii}){\left(\frac{\alpha A^{\alpha }\left(1+{\tau }_Y\right){\omega }^{\alpha }}{1+{\tau }_L}\right)}^{\frac{1}{1-\alpha }}>\frac{\left(1+{\tau }_Y\right){\omega }^{\alpha }{\overline{\mathrm{A}}}^{\alpha }-\left(1+{\tau }_L\right)}{\left(1+{\tau }_L\right)}\frac{\alpha }{1-\alpha }\end{array}$$

Hypothesis 3 (H3):

When the parameter satisfies (i) (ii) (iii), if the land price premium is low,$p\le {\overline{\overline{p}}}_1$, the central government will prefer to complete municipal projects in the PPP model; if the land price premium is medium,${\overline{\overline{p}}}_1<p<{\overline{\overline{p}}}_2$, the central government will prefer to participate in municipal projects in the financing platform mode directly; if the land price premium is significant,$p>{\overline{\overline{p}}}_2$, the central government no longer tolerates direct participation in municipal projects in the financing platform mode, and prefers to complete municipal projects in the PPP model.

Proof of Hypothesis 3:

The preference of the central government is determined by the difference in social returns. We can bring the output of the two modes into the (22) formula and obtain the following derivation:

$$∆g_2=L\left\{\left(1+{\tau }_Y\right){\omega }^{\alpha }\left({\overline{\mathrm{A}}}^{\alpha }-A^{\alpha }{p}^{\alpha }\right)+\left(1+{\tau }_L\right)\left(p-1\right)\right\}$$

$$\frac{d∆g_2(p)}{dp}=L\left(\left(1+{\tau }_Y\right){\omega }^{\alpha }\left(-\alpha A^{\alpha }{p}^{\alpha -1}\right)+\left(1+{\tau }_L\right)\right)$$

$\frac{d∆g_2{}^2(p)}{d{p}^2}=L\left(\left(1+{\tau }_Y\right){\omega }^{\alpha }\left(\left(1-\alpha \right)\alpha A^{\alpha }{p}^{\alpha -2}\right)\right)>0$; it can be seen that $∆g_2(p)$ is a convex function.

$$\frac{d∆g_2(p)}{dp}=0\Rightarrow p={\left(\frac{1+{\tau }_L}{\alpha A^{\alpha }\left(1+{\tau }_Y\right)}\right)}^{\frac{1}{\alpha -1}}{\omega }^{\frac{\alpha }{1-\alpha }}\triangleq p_0$$

$$A^{\alpha }{p}_0{}^{\alpha }=\frac{\left(1+{\tau }_L\right)}{\alpha \left(1+{\tau }_Y\right)}{\omega }^{-\alpha }{p}_0$$

Substituting the above equation into $∆g_2(p_0)$, we can obtain:

$$∆g_2(p_0)=L\left\{\left(1+{\tau }_Y\right){\omega }^{\alpha }\left({\overline{\mathrm{A}}}^{\alpha }-\frac{\left(1+{\tau }_L\right)}{\alpha \left(1+{\tau }_Y\right)}{\omega }^{-\alpha }{p}_0\right)+\left(1+{\tau }_L\right)\left(p_0-1\right)\right\}$$

Via condition (iii): $∆g_2(p_0)<0$

There is ${\overline{\overline{p}}}_1<{\overline{\overline{p}}}_2$ such that $∆g_2({\overline{\overline{p}}}_1)=∆g_2({\overline{\overline{p}}}_2)=0$, where ${\overline{\overline{p}}}_1,{\overline{\overline{p}}}_2$ is determined by the following equation:

$$\frac{{\overline{\mathrm{A}}}^{\alpha }-A^{\alpha }{p}^{\alpha }}{1-p}=\frac{1+{\tau }_L}{1+{\tau }_Y}{\omega }^{-\alpha }$$

(24)

Note that equation (24) land price premium value critical value ${\overline{\overline{p}}}_1,{\overline{\overline{p}}}_2$ is a function of the parameter ${\tau }_Y,{\tau }_L,\alpha ,L,A,A_i,A_j$. □

If the land price premium is low ($p\le {\overline{\overline{p}}}_1$), the central government will prefer to complete municipal projects in the PPP model; if the land price premium is medium (${\overline{\overline{p}}}_1<p<{\overline{\overline{p}}}_2$), the central government will prefer to participate in municipal projects in the financing platform mode directly; if the premium is high ($p>{\overline{\overline{p}}}_2$), the central government no longer tolerates direct participation in municipal projects in the financing platform model, and prefers to complete municipal projects in the PPP model.

Hypothesis 4 (H4):

When$p<\overline{p}$, local governments and central governments prefer to complete municipal projects in the PPP model; when$\overline{p}<p<{\overline{\overline{p}}}_1$, the local government prefers to participate in municipal projects directly via financing platform mode, while the central government prefers to complete municipal projects in the PPP model. When${\overline{\overline{p}}}_1<p<{\overline{\overline{p}}}_2$, both local governments and central governments prefer to participate in municipal projects in the financing platform mode directly; when$p>{\overline{\overline{p}}}_2$, local governments prefer to participate in municipal projects directly by financing platform mode, while the central government prefers to complete municipal projects using the PPP model.

Proof of Hypothesis 4:

Equations (23) and (24) show that $1<\overline{p}<{\overline{\overline{p}}}_1<{\overline{\overline{p}}}_2$, which can be proved by combining Hypothesis 2 and 3. □

Table 3. Government Behavior Strategy.

	Land Price Premium	$\mathit{p}<\overline{\mathit{p}}$	$\overline{\mathit{p}}<\mathit{p}<{\overline{\overline{\mathit{p}}}}_1$	${\overline{\overline{\mathit{p}}}}_1<\mathit{p}<{\overline{\overline{\mathit{p}}}}_2$	$\mathit{p}>{\overline{\overline{\mathit{p}}}}_2$
Hypothesis 2	Local government	PPP	Financing platform	Financing platform	Financing platform
Hypothesis 3	Central government	PPP	PPP	Financing platform	PPP
Hypothesis 4	Local government	PPP	Financing platform	Financing platform	Financing platform
	Central government	PPP	PPP	Financing platform	PPP

summarizes the decision-making behavior of local and central governments for different land price premiums of Hypothesis 2–4.

Hypothesis 2 shows that when the land price premium reaches a certain height $\overline{p}$, the local government and the financing platform collude to increase the land price premium, which is more conducive to improving the local government’s fiscal revenue. However, this model forms the local government’s dependence on land finance. Hypothesis 4 shows that when the premium is small (${\overline{\overline{p}}}_1<p<{\overline{\overline{p}}}_2$), revealing the local government and financing platform collusion does not violate the objectives of the central government. Central and local governments prefer the financing platform model to participate directly in municipal projects. However, when the premium is significant ($p>{\overline{\overline{p}}}_2$), local governments prefer to participate directly in municipal projects in the financing platform mode. In contrast, the central government no longer tolerates participating directly in municipal projects in the financing platform mode and prefers to complete municipal projects in the government and social mode.

4.2. Numerical Simulation Experiments

In order to demonstrate the validity and feasibility of the proposed model, we used MATLAB software (MATLAB 12) for numerical simulation. The computer operating system was Windows 7 (Microsoft Ltd., Redmond, Washington, DC, USA) with an Intel Core i3-3217U CPU @1.80GHz 1.80GHz processor (Intel, Santa Clara, CA, USA) and 4G memory. The software was MATLAB version R2012a, developed by MathWorks. The simulation experiment environment is shown in

Table 4. Simulation Environment.

Component	Environment
Operating system	Windows 7
CPU	Intel Core i3-3217U CPU @1.80GHz 1.80GHz
Memory	4.0 GB
MATLAB	version R2012a (7.14.0.739)

.

Let ${\tau }_Y=0.06,{\tau }_L=0.09,\alpha =0.6,r=0.02,A=4,\overline{A}=4.5$, the social return difference between the two models of local governments: $∆g1=\left(1+{\tau }_Y\right){\omega }^{\alpha }\left({\overline{\mathrm{A}}}^{\alpha }-A^{\alpha }{p}^{\alpha }\right)+\left(1+{\tau }_L\right)\left(1-p\right)$, the social return difference between the two models of the central government: $∆g2=\left(1+{\tau }_Y\right){\omega }^{\alpha }\left({\overline{\mathrm{A}}}^{\alpha }-A^{\alpha }{p}^{\alpha }\right)+\left(1+{\tau }_L\right)\left(p-1\right)$. Numerical simulation results are shown in Figure 1.

The numerical simulation results are consistent with those of Hypothesis 4, which verifies that the model setting is reasonable. Production technology is crucial to economic growth. The following experiment is an expanded analysis of the model. In general, the technical level of the financing platform is lower than the PPP model, and the technical level of the two is not significant. In order to discuss the risk-sharing mechanism of the PPP model, we first discuss the situation when the technical level of the financing platform is higher than that of the PPP model, and then discuss the situation when the technical level of the financing platform is lower than that of the PPP model and the difference is significant.

Firstly, considering the influence of the technology of the PPP model is lower than that of the financing platform on the behavior strategy, the following parameters are taken: ${\tau }_Y=0.06,{\tau }_L=0.09,\alpha =0.6,r=0.02,A=4,\overline{A}=3$; numerical simulation results are shown in Figure 2.

Figure 2 shows that when the land price premium reaches a certain height, the local government and the financing platform’s “collusion” does not violate the central government’s objectives. Both local and central governments prefer the financing platform model to participate directly in municipal projects. However, when the premium is significant, the local government prefers to participate directly in municipal projects in the financing platform mode. In contrast, the central government no longer tolerates participating directly in municipal projects in the financing platform mode and prefers to complete municipal projects in the government and social mode. This shows that the risk-sharing mechanism of the PPP model plays a role. When the land price premium is high, the cumulative risk gets higher. Then, the central government should adopt the PPP model even at the expense of production efficiency. Although it cannot improve production efficiency, it can play a role in preventing risks. Once risk overflows, it has a long-term negative impact on the economy, which the central government cannot tolerate.

Secondly, considering the influence of technology difference between the PPP model and financing platform on behavior strategy, the following parameters are used: ${\tau }_Y=0.06,{\tau }_L=0.09,\alpha =0.6,r=0.02$, left graph parameter: $A=4,\overline{A}=7$, right graph parameter: $A=2,\overline{A}=5$; numerical simulation results are shown in Figure 3.

Figure 3 shows that when the technology of the PPP model is higher than that of the financing platform and the difference is significant, the behavior strategy of the central government is to use the PPP model to participate in municipal projects directly; when the land price premium is not high, the local government acts in line with the central government. When the land price premium exceeds a particular value, the local government directly participates in municipal projects in the financing platform mode. In addition to the private sector’s capital, PPP projects also use the production and management techniques of the private sector. At the same time, in PPP model financing, more consideration is given to the risk of both sides, and the overall risk is minimized. PPP project risk-sharing mechanisms and production efficiency advantages are also important reasons for the central government to adopt the PPP model. One of the main reasons why local governments are keen on financing platforms is that they face the pressure of a shortage of funds for public infrastructure construction. Local governments must capitalize on land resources to expand fiscal space and increase government disposable financial resources.

5. Conclusions

5.1. Research Findings and Suggestions

This paper presents a model for local financing platforms and PPP models from the game perspective between the central and local governments. The introduction of land price premiums analyzes the choice of central and local governments to participate in municipal projects. The model results show that whether local and central governments prefer financing platforms or directly participate in municipal projects depends on the land price premium. When the land price premium is low, local governments and central governments prefer the financing platform model to participate directly in municipal projects; when the land price premium is significant, local governments prefer the financing platform mode, while the central government prefers the PPP model.

This paper analyzes the dynamic mechanism of its implementation from the game perspective between the central and local governments. It is believed that the development mode of relying on financing platforms should be changed, the ecological environment suitable for PPP development should be actively created, and the PPP model should be vigorously promoted. First, the operation mode of the local government financing platform should be standardized, the dependence on financing platform debt should be reduced, and local fiscal and financial risks should be prevented. According to the model results, the local government has a strong incentive to set up financing platform companies and land as collateral for financing. However, due to the opaque operation of the local financing platform, management is not standardized, there are soft budget constraints, among other issues, and the efficiency of capital use is low; at the same time, local financing platforms are generally financed through a land mortgage, and changes in land prices contain financial risks. Therefore, it is necessary to regulate the operation mode of local government financing platforms and reduce the dependence on financing platform debt. Secondly, it is necessary to change the mode of economic development, change the assessment of local officials, dilute the growth of GDP, and strengthen the construction of a peaceful environment for economic development. Examples of this include accelerating the construction of PPP laws and regulations system, cultivating the spirit of the contract [74], improving the market mechanism [75], improving the institutional environment of private investment, and attracting social capital into the infrastructure and public service supply field.

5.2. Limitations and Future Research Directions

This study mainly analyzes and studies the two types of investment and financing modes using mathematics models without empirical research to verify the model. Future work will collect relevant data and empirical research on this model. First, the transmission mechanism of local debt risk caused by the financing platform will be verified. Additionally, the alternative relationship between PPP models and financing platforms will be explored. Furthermore, the PPP model will be validated to determine whether it can mitigate local debt risks.