Financial compensation strategy of PPP project based on game theory and intelligent optimization

Abstract

The public sector is the manager and supervisor of the PPP project, and the financial compensation for the PPP project should be considered as many factors as possible. We set up a game model of private participation in PPP project. We analyzed their influence on financial compensation from 3 aspects: the marginal cost of financial funds, the proportion of private equity of PPP project and the scale of investment, and the financial compensation of the PPP project is optimized intelligently. On this basis, the countermeasures and suggestions about the financial compensation of the PPP project are put forward.

Keywords

Financial compensation strategy PPP project game theory intelligent optimization

1 Introduction

In its paper, “On the key tasks of deepening economic reform views of 2014”, the State Council clearly demands that local government debt which can be identified as a part of the corporate debt should be stripped out in the form of PPP, and the enterprises will be compensated by the local government in tax or transfer funds for special financial compensation to make up for the loss of the enterprise. The means mentioned above provides a legal basis for accounting the PPP model for more than 80% of the quasi-public welfare projects in our local debt. However, how to concretely manage the financial compensation has not been arranged uniformly, so it is urgent to explore the paradigm to improve the efficiency of the allocation of financial funds in PPP projects. The efficiency of financial compensation for PPP projects stems from the idea of cost benefit matching under constraint conditions, its basic principle lying as following:

The financial commitment should be linked to actual cost and income. When the actual project income is equal to the expected return of private capital, government departments must pay compensation for the private sector; when the project’s actual income is less than the expected return of private capital, the government, in addition to the fixed cost of compensation, should compensate the fiscal expenditure compensation which is less the expected return; when the actual benefits of the project is greater than the expected return of private capital, government departments should tax the excess profits according to a certain proportion for the prevention of moral hazard and protect the public welfare, seeking the negative incentive on the enthusiasm of quasi-public products.

Adams considered that, the increasing concern of Chinese local governments on value for money has aroused the desire to improve the efficiency of public service supply [1]. Buse has been widely used in the global health field [2]. Chan found that the three most important risk factors of China’s PPP project were: government intervention, government corruption and public decision-making process was not good [3]. De Jong believed that the development level of PPP in China was quite different. The level of specialization of coastal metropolis was relatively high. Clear incentive mechanism and strict power restriction were the key to further development of PPP [4]. Engel believed that the government mistakenly thinks that PPP provides a way to finance infrastructure without increasing public debt in many cases, and the risk allocation under the optimal contract indicates that the public and private cooperation was closer to the public supply than the privatization [5, 6]. Flinders pointed out that the government should not ignore the long-term cost of the project while paying attention to the short-term benefits of the project [7]. McQuaid pointed out that for the PPP project, the problems of value for money, efficiency, competition, capital availability, contract design and management, regulatory mechanism and risk sharing were still critical [8]. Regan pointed out that the liquidity and risk of Australia’s capital markets had a great impact on infrastructure project financing [9]. Spackman pointed out that the practice of private participation in infrastructure construction in the UK indicates that private participation in infrastructure had created clearer contracts and more operational innovations [10]. Wu Xiaoning considered the optimal compensation strategy on the basis of public and private game, but did not take the proportion of private equity into account [11]. The above literature carefully examined the development process and influencing factors of PPP. We intend to investigate the corresponding optimal fiscal compensation policy from three aspects: the marginal cost of financial funds, the proportion of private equity in PPP project and the scale of investment.

On the condition that social value of the public goods or services in the PPP project is to be maximized, we should confirm and quantify the key indicators which influence the total value of social and private benefits, fiscal expenditure performance commitments so as to attract social capital to participate in projects to promote PPP with appropriate income, to alleviate the pressure of fiscal budget, to release financial funds, and to make the best compensation for private investments.

2 Hypothesis and construction of theoretical model

Influence of government funds allocation to the private sector which participates in the decision-making of PPP project is obvious, that is to say, two sides of the game directly determines the overall efficiency of the PPP project, the private sector cost-income efficiency and the financial expenditure efficiency. The allocation of financial funds is realized in the form of government compensation, and the government compensation includes fixed compensation and expected revenue which is not equal to the actual income when the risk is compensated. In the unit time, the government compensation can be quantified as: $U = U_{0} + \frac{R_{0} - R^{*}}{R_{0}}$ (1)

Among the formula: U stands for the overall cost of financial commitment, while U₀ stands for the settled compensation for investments such as construction senses, U₁ stands for the risk compensation when the actual income deviates from the expected profits of the private sector. R₀ bears the anticipate profit expected operating income under the contract of PPP, which is a settled statistic. R^* represents the actual income of the project during the operation period, is a random variable, therefore, $\frac{R_{0} - R^{*}}{R_{0}}$ represents the risk compensation coefficient of PPP project financial compensation. Formula (1) bases itself on the financial compensation of project operation profit.

Input and output are closely related. The expected operating income of the PPP contract can be expressed by the private sector input cost as following: $R_{0} (C) = iC / t$ (2)

R₀ represents the return on investment or the cost rate of return, and its value can be confirmed by the method of capital asset pricing model of the same type of project in the industry, while t indicates the franchise period.

On the occasion that the basic information is symmetric, once the government takes the financial compensation in formula (1), the private sector will adapt the reflective adjustment to its scale of investment in order to maximize their own profit and return the sum of financial compensation in a spontaneously. Therefore, PPP contract is not only designed for the pursuit of the maximization of the public products provided by the social welfare, but also designed to meet the requirements of the compensation scale of private capital, and to improve the financial efficiency of capital allocation, which is actually a typical Stackelberg game [12].

The scale of government compensation for specific PPP projects depends on the risk compensation coefficient $\frac{R_{0} - R^{*}}{R_{0}}$ , Cost reimbursement benefits required by the private sector R₀ is generally set by the bond, while the random variable-actual profit R^* is hard to set.

When the total investment cost of the project is C, the actual income of the project can be expressed by referring to the capital asset pricing model: $R_{p} = iC = r_{f} C + θ \partial_{r} t$ (3)

In the formula (3), r_f stands for the risk-free interest rate in the market, generally replaced by the medium and long term treasury bond interest rate, θ stands for the market risk premium within the project operating time, ∂_r stands for the standard value difference of R^* (r is the expected value of R^*), also known as the risk parameter; t represents the operating time the project lasts for.

When the government promises to compensate, the private investment profits-cost can be expressed as following: $π (U, C) = R^{*} + U - \frac{C}{t}$ (4)

π (U, C) refers to the private investment profit for a unit time, while $\frac{C}{t}$ is the private investment cost per unit time. When measuring the social net income of PPP project, we should not only consider the social total value the project will bring, private sector net investment, financial capital expenditure costs, but also consider “the deadweight loss” of bearing financial compensation brought by tax. $X (U, C) = \frac{Y (C)}{t} - [R^{*} - π (U, C)] - μ U$ (5)

In formula (5), X (U, C) represents the social net income of PPP projects, $\frac{Y (C)}{t}$ stands for the total social value produced during a unit time, R^* - π (U, C) represents the net investment in the private sector, μU is the financial commitment expenditure cost and the society deadweight loss it brought about. Which μ > 1 is the deadweight loss coefficient. However, the total value of society has a characteristic of first order increasing and marginal diminishing, then we can have another formula. $Y (C) = α C^{β}$ (6)

In the formula (6), α > 0, 0 < β < 1. According to the basic principle of choosing the best financial compensation method, the allocation of financial funds has double goal orientation: Maximizing the social welfare of the public goods provided and meeting the cost benefit constraints of private capital maximizes private sector profits and attracts them to participate. Assuming that all participants in PPP model get neutral risk, the best financial compensation method is to solve the following game model. $\underset{U_{0}, U_{1}}{Max} E [X (U, C)] = \frac{Y (C) - C}{t} + (1 - μ) [(U_{0} + U_{1}) - \int_{- \infty}^{+ \infty} \frac{U_{1}}{R_{0} (C)} rf (r) dr]$ (7) $\underset{C}{Max} E [π (U, C)] = U_{0} + U_{1} - \frac{C}{t} + \int_{- \infty}^{+ \infty} (1 - \frac{U_{1}}{R_{0} (C)}) rf (r) dr$ (8) $E [π (U, C)] \geq R_{0}$ (9)

Formula (7) and (8) are the decision making requirement that public and private sectors are to join the PPP project. Formula (9) stands for the Private sector decision cost constraints: the private sector is required to earn higher profits than the PPP contract operating income, otherwise, it will refuse to invest.

3 Model solution

If we solve the model answer, we can get the cost decision model of private sector participation (10). Assuming that PPP project game information gets symmetry, the private sector’s behavior choice can be observation and reference for government departments, and formula (9) constraints can be solved, we shall get the best financial compensation behavior selection model (10), (11), and (12), also $C_{U}^{*}$ can be realized in formula (13). $C_{U} = \frac{[(\bar{r} r_{f} U_{1})^{1 / 2} - θ \partial_{r}] t}{r_{f}}$ (10) $U_{1}^{*} = \frac{(r_{f} C_{U}^{*} + θ \partial_{r} t)^{2}}{r_{f} {\bar{rt}}^{2}} = \frac{R_{0}^{2} (C_{U}^{*})}{r_{f} \bar{r}}$ (11) $\begin{matrix} U_{0}^{*} & = & \frac{R_{0}^{2} (C_{U}^{*}) - (1 + r_{f}) {\bar{rR}}_{0} C_{U}^{*} + r_{f} (\bar{r})^{2}}{r_{f} \bar{r}} \\ + \frac{C_{U}^{*}}{t} \end{matrix}$ (12) $\begin{matrix} C_{U}^{*} & = & {[\frac{α β}{(1 + r_{f}) μ - r_{f}}]}^{\frac{1}{1 - β}} \\ = & {[\frac{α β}{μ + (μ - 1) r_{f}}]}^{\frac{1}{1 - β}} \end{matrix}$ (13)

From the models above, the cost decision of the private sector depends on the expected revenue $\bar{r}$ during the operation, operational risk compensation U₁ together with market risk premium coefficient θ, but has no relation to the government fixed compensation U₀. The allocation efficiency of government financial funds is related to marginal cost of capital, market risk-free interest rate, investment scale and so on: $\partial C_{U}^{*} / \partial μ < 0$ , $\partial C_{U}^{*} / \partial r_{f} < 0$ , $\partial C_{U}^{*} / \partial β > 0$ , $\partial C_{U}^{*} / \partial α > 0$ . The optimal financial commitment cost changes inversely with the marginal cost of capital, and positively changes with the scale of investment.

4 Pareto efficiency of fiscal compensation method

The Pareto efficiency model of financial compensation method can be obtained based on the above analysis: $U^{*} = U_{0}^{*} + (\frac{R_{0} - R^{*}}{R_{0}}) U_{1}^{*}$ (14)

The best financial compensation U^* consists of two parts: $U_{0}^{*}$ mainly depends on fixed compensation such as construction cost and fixed management cost. $U^{*} = U_{0}^{*} + (\frac{R_{0} - R^{*}}{R_{0}}) U_{1}^{*}$ consists of operational risk compensation coefficient which is got by comparing R₀ the contractual franchise income and R^* the actual profit and $U_{1}^{*}$ the unit operation risk compensation, which is a random value. In the case of adverse selection. Contractual franchise income $R_{0} (C_{U}^{*})$ is positively related with $C_{U}^{*}$ the investment cost scale, r_f risk-free interest rate and ∂_r project operation risk, inversely related with t the project duration. $R_{0} (C_{U}^{*}) = \frac{r_{f} C_{U}^{*}}{t} + θ \partial_{r}$ (15)

The Pareto efficiency of fiscal compensation method includes three aspects: firstly, the project can bring happiness to the public, $E_{W} (U^{*}, C_{U}^{*}) \geq 0$ . Secondly. The total income of a project satisfies at least the requirements of private sector participants, $E_{π} (U^{*}, C_{U}^{*}) \geq R_{0} (C_{U}^{*})$ , and thirdly the financial commitment compensation stays within the constraints of financial funds. In the case of private sector decision making, the allocation of financial capital is: ${TE}_{π} (U^{*}, C_{U}^{*}) = t \bar{r} + E [{tU}^{*}] - C_{U}^{*} = {tR}_{0} (C_{U}^{*})$ (16)

Formula (16) shows that the Pareto efficiency of the fiscal compensation method is the fixed franchise income that the private sector just gets the contractual agreement. On the other hand, the government must have a limit on the compensation of private capital in the PPP project, so as to avoid the loss of social welfare and financial resources of the load, therefore the financial compensation method of Pareto efficiency should at the same time satisfy the constraints of social welfare:

$\begin{matrix} E [U^{*}] & \leq & \frac{α (C_{U}^{*})^{β} - C_{U}^{*}}{(μ - 1) t} \\ = & α (1 - \frac{β}{μ + (μ - 1) r_{f}}) \frac{(C_{U}^{*})^{β}}{(μ - 1) t} \end{matrix}$ (17)

In formula (16), μ > 1, 0 < β < 1, meanwhile, $E [U^{*}] = \frac{C_{U}^{*}}{t} + [R_{0} (C_{U}^{*}) - \bar{r}]$ , then by using formula (17) we can conclude that, the PPP project contractual franchise income have the following characteristics: $R_{0} (C_{U}^{*}) - \bar{r} \leq α (1 - \frac{μ β}{μ + (μ - 1) r_{f}}) \frac{(C_{U}^{*})^{β}}{(μ - 1) t}$ (18)

With formula (17) into E [U^*], $E [U^{*}] = \frac{(1 + r_{f}) C_{U}^{*} - \bar{rt}}{t} + θ \partial_{r}$ , the again in formula (17), we shall get formula (19): $θ \partial_{r} - \bar{r} \leq \frac{α (1 - β) (C_{U}^{*})^{β}}{(μ - 1) t}$ (19)

In the formula (19), α > 0, 0 < β < 1. Formula (19) reveals the path of Pareto efficiency to realize fiscal compensation.

5 Result analysis

As the managers and supervisors of PPP projects, the choice of the best financial compensation methods for public sectors should focus on two aspects: the choice of the specific types of PPP model and the factors influencing the financial compensation.

Firstly, implementation path of specific type selection based on PPP project. Formula (18) presents the inspiration: According to the different degree of publicity, the cost compensation methods of specific PPP projects are different. Those purely public projects, because they can’t make profits or make little profit themselves, determine that financial commitment is the most important or even the only way to compensate for the cost of private participation in PPP projects. But for quasi-public products, financial commitment is no longer the only way of cost compensation, according to the different comparison between the actual income and the minimum benefit demand (private cost), the choice of PPP mode has the following specific types:

In some PPP projects, the future operating income is very small or the amount of compensation income required by the private sector is larger, and the government can take the BT (Build-Transfer) type. The government attracts the private sector enterprises to complete the construction link of a specific project in the form of bidding, etc., and directly returns the public sector after the completion of the construction. In such cases, the private sector plays the role of contractors and builders, the government’s financial commitment only covers the enterprise party construction costs and other fixed costs and necessary construction returns, the private sector is not involved in the operation, the operation risk burden is entirely on the government. This is one of the simplest and most direct PPP mode in which government purchases service from enterprises. Not only does it meet the social public needs, but also it does not need to make the public budget operation risk compensation arrangements, which will reduce the financial pressure of management.

In some PPP projects, we can predict that the expected operating income of the participants is basically the same as the actual income, or the difference is not big, and the government can adopt BOS (Build-Operate-Sale) and other types. In the case that the expected return and the actual income has a little difference, the operational risk is low, the operator may bear a smaller operational risk uncertainty, when adopting BOS type, government financial commitment only have to purchase cost in the concession period at the end of the commitment, under which operation types, the enterprises can corporate financing, and can achieve positive social benefits.

When some PPP projects operating income is enough to compensate for party enterprise construction cost, operation cost and satisfy the expected income, the government can choose BOT (Build-Operate-Transfer), BOO (Build-Own-Operation). In BOT mode, the private sector at the expiration of the concession project will be transferred to the government departments, government departments needn’t to make financial compensation of fixed costs and operational risks, what’s more, it can have a certain percentage of the project revenue which exceeds reverted to finance, to suppress moral risk that private sector makes for profit in public projects; For some projects, participants does not get the project operating income directly but for other projects to develop, or in other cases there is no franchises profit demands, therefore the government can consider the privatization or privatization of this quasi-public goods, of course, there are still property rights constraints in our country at present.

Secondly, the realization path of influencing factors of financial fund allocation efficiency based on PPP project. Formula (19) is the expression of the influence factors of the PPP project financial compensation method of Pareto efficiency. It enlightens us to grasp the impact factors through the financial capital allocation efficiency, such as operational risk and concession period and other factors, and also conduct the implementation of effective financial compensation design, to enhance the dual constraints of social welfare, satisfy the private and social benefits, the release of public budget pressure and the pressure of debt.

6 Conclusions

This paper focuses on operational risk perspective and concession term perspective.

Moderate reduction and control of operational risk compensation scale. In order to improve the efficiency of financial allocation of funds, the government should control and moderately reduce the operational risk compensation. Operational risk is influenced by the systematic risk of the market environment and industry development level, the management level of enterprises and the market factor etc. This has great uncertainty. Once the efficiency of the allocation of financial funds is linked to operational risk, the scale of expenditure and the pressure of capital have great uncertainty, so it is a good policy to solve the problem and reduce the operational risk moderately.

The government can remove the risk by buying commercial insurance for the project, or directly purchase the products or services produced by the project to avoid the operational risks.

For instance, for the power plant and water supply facilities, the government can buy a certain amount of power and water within a certain period of time, to ensure the power plant, water supply company market have a stable income, and to avoid the risk bound of combining financial commitment and the project operation. Purchasing the product incentives private sector participation in the PPP project, and promoting the market operation of PPP project are achieved.

Shorten the concession period reasonably. In order to improve the efficiency of the allocation of financial funds, the government should also shorten the franchise period reasonably. The scale of financial commitment expenditure and management performance are not only related to the scale of unit compensation, but also affected by the franchise period of the project. The longer the project lasts, the greater the demand for the medium and long term financial commitment and budget fund arrangement is, so is the controllability and more complex the risk. Long-term PPP projects also have the risk of premature death. Compressing PPP concession period as possible, compressing the uncontrollability and uncertainty, and getting control over the projects as soon as possible will not only improve the efficiency of financial allocation of funds, but also stabilize and expand the public welfare.

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