# Determining the Operator for the Public Toll Road

Authors: Bin Shang; Qiang Sun; Hao Feng; Jiancong Chang
Journal: Journal of Advanced Transportation
 (2021)

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## Abstract

After the BOT road operation contract expires, generally, the road will be transferred to the government, and then the government operates the road independently without charging costs from its users. Facing the huge amount of the operation cost, Chinese government tends to continue to charge the road users to guarantee the high quality of road operation. Then, the government will have to decide whether a private firm or government itself would be suitable to operate the road. A model is presented for decision-making through balancing interests between the government and the private firm with an introduction of an intermediate variable, i.e., bidding price. Three scenarios are investigated in the model, including the optimization of government operation, the optimization of private firm operation, and government operation with an improper decision of the intermediate variable. Improper intermediate variable will result in a higher toll charged by the government than by a private firm. The method of avoiding an improper decision is investigated. The result shows that the intermediate variable should be determined to be the government operation cost, based on which the private operator could be chosen, if available. With consideration of the private operator’s profit to be guaranteed by the government, the maximum subsidy should be equal to the minimum private operator’s profit to be disclosed when the contract is signed.


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## Body

## 1. Introduction

Due to the tight government budget, a lot of transportation facilities, especially highways, were built in the build-operate-transfer (BOT) mode in the past forty years. At present, more and more BOT contracts for the operation of roads have expired or will be going to expire.When the roads previously operated in the BOT mode are transferred to the government, it is usually assumed that they will become public roads, and the government will take over the responsibility of operating and maintaining these roads with its own budget. In fact, the operation cost (OC) including but not limited to the maintenance cost is much less than the initial construction cost of the roads. In general, no toll will be charged for the public roads. In fact, in order to guarantee the standard of the service relating to the road operations, the government needs to invest heavily to cover the OC. Due to the tight fiscal budget and for decreasing the government financial burden, continuing tolling after the transfer of the roads becomes a logical decision and, to some extent, a universal operation mode. In fact, Chinese government intends to charge the fees for these roads [1]. Chinese government could decide either to charge in its own name or entrust a private firm with this task.Based on the above description, the road outside the BOT concession period is referred to as the public toll road (PTR) hereinafter, and the toll will be charged continuously to make up the OC. Tan et al. [2] and Niu and Zhang [3] pointed out that normally, a private firm would define the whole road life cycle as the BOT concession period in the BOT contract. However, it is not the case in China. All the BOT roads will have to be transferred to the government when the contracts expire. In this paper, we assume that PTR begins a new period after the road has been transferred to the government. The road life cycle, including the BOT concession period and PTR period, is illustrated in Figure 1.Figure 1
Relationship of the BOT concession period and PTC period.There are numerous studies on the BOT road construction and operation [2–12]; however, few researchers consider the case after the roads are transferred to the government. To our best knowledge, the influence of OC during the PTR period has not been intensively studied yet. This paper aims to investigate whether the government or a private firm should operate the road under the charge of road toll if the government decides to charge to compensate the OC. This paper comes up with a new concept compared with previous studies concerning the BOT road. Previous research studies generally focus on the selection of the road operator among private firms in the BOT mode. This paper will explicitly examine the selection mechanism to determine whether the government or a private firm will operate the road under the charge of road toll during the PTR period. Note that the revenue collected from road users will be used to compensate the OC, which is different from traditional pricing scenarios such as tolling for the return of the loan (BOT and public-private-partnership).Referring to Shi et al.’s study [13], we regard that OC is composed of two parts: the demand-related OC and capacity-related OC. The demand-related OC denotes the cost relating to traffic demand; the capacity-related OC denotes the cost relating to the capacity of the road, including but not limited to the maintenance cost. Then, we introduced an intermediate variable presented by the government in the bidding document, which serves as perfect information to both government and private firms. The government and private firms are fully aware of their respective OCs, while the government is not clear about the potential profit to be earned by private firms. We consider the intermediate variable as crucial, based on which we studied the case by analyzing the relationship of the intermediate variable, OC of the private firm, and that of the government. The government may determine an improper intermediate variable based on the limited information. In contrast, we provide an optimal solution that the government sets its anticipated operation cost as the intermediate variable.The remainder of the paper is organized as follows: Section2 reviews the previous studies relating to toll road operation. Section 3 introduces the basic notations and assumptions necessary to our analysis model. Section 4 looks into the issue whether the government or a private firm is proper to operate the PTR under the condition that the toll is charged if the government decides to collect the toll to cover the OC. Section 5 investigates the optimal solution to be adopted by the government. Finally, Section 6 concludes this paper.


## 2. Literature Review

Literature on auction mechanisms of the BOT project is closely related to our research. These studies can let the government select the operator from several private firms efficiently. The least-present-value-of-revenue (LPRV) auction mechanism is used on highway franchise. According to auction theory [14], LPRV mechanism equaled to a type of auction: the first-price sealed-bid where the winner was the firm who offered the lowest bid for total revenue. Engel et al. [5] presented a framework to evaluate different awarding mechanisms for the franchise contract, and based on the LPRV auction mechanism, they presented a simple auction. Furthermore, Engel et al. [15] showed that the commonly used fixed-term contract of franchise highway did not allocate the demand risk optimally and then proposed a new auction mechanism for the franchise highway based on the LPRV mechanism.On the variable-term-based concept [5], Rus and Nombela [16] pointed out that if operation cost was not negligible, the LPVR mechanism would not guarantee the best concessionaire; then, they proposed the least-present-value-of-net-revenue (LPRNV) mechanism which required private firm making offers on total revenue, maintenance costs, annual operation and routine maintenance cost, and so on; and this mechanism eliminated the demand risk effectively. Based on a flexible-term contract and bidimensional bids for maintenance costs and total net revenue, Nombela and de Rus [17] proposed a new franchising mechanism, comparing with fixed-term concessions, which could promote the selection of efficient concessionaires through eliminating the traffic risk. Verhoef [18] investigated welfare impacts of franchise regimes for congestible toll roads and found that patronage-maximizing auction was not optimal when considering the second-best network aspects. Ubbels and Verhoef [19] looked into how to get the desired welfare through analyzing the auction rules which were set by the government in the process of bidding and found that a bid with minimal price on the toll road would lead to satisfying results for unsubsidy auction; however, considering subsidies, the best bid would appear to involve the minimum generalized travel costs. van den Berg [20] investigated private supply of two congestible infrastructures under four market structures including a monopoly and three duopolies that differed in how private firms interact with each other and found that a duopoly could lead to a different rule of capacity than the first-best one which differed from the duopoly with the facilities.However, all the above studies focus on the BOT road operation as well as how to choose a private road operator. Any existing selection mechanisms decide a private road operator without considering the government as one of the potential road operators. Therefore, compared to the previous studies, we investigate a mechanism to select the best party to operate the road during the PTR period among the government and private firms.


## 3. Notation and Assumptions

As mentioned above, we are aware that Chinese government will have to decide whether to collect the toll in its own name or to entrust a private firm with this task. Furthermore, we assume that the PTR is the unique one connecting two neighboring cities, and the private firms are able to operate the road in a more efficient and profitable way than the government. Besides, we further assume that the social welfare for the public and the private firm’s profit derived from operating the PTR are functions of the toll pricing and the road capacity.Letv≥0 and c≥c1 be the travel demand and road capacity, respectively, which are measured by the number of vehicles per unit time; c1 is the road capacity when it is transferred to the government. Let Bv be the inverse demand function (or a function of the marginal benefit) and tv,c be the travel time function. Moreover, the demand-supply equilibrium condition holds, i.e., Bv=p+βtv,c, where p is the toll charged by the operator and β is the value of time that converts time into an equivalent monetary cost. The above condition means that the travel demand of a new road is determined by the total cost (including travel time and toll collected from road users) of a trip. In this paper, we only consider homogeneous users. Subsequently, we get a price function:(1)pv,c=Bv−β⋅tv,c.Equation (1) implies that the toll p is determined by travel demand v uniquely for a given road capacity c. Hereinafter, v and c are equivalent to p and c.LetMsv and Mgv be the demand-related OC of a private firm and the government, respectively. Let Mdv be an intermediate demand-related OC which is a function of the intermediate variable. Then, we have(2)Msv=msv,Mgv=mgv,Mdv=mdv,where ms and mg are the unit demand-related OC of the private firm and the government, respectively, and md is the intermediate variable.LetJsv,c and Jgv,c be the OC of a private firm and the government, respectively. Let Jdv,c be an intermediate OC. Then, we have(3)Jsv,c=Ic+Msv=Ic+msv,Jgv,c=Ic+Mgv=Ic+mgv,Jdv,c=Ic+Mdv=Ic+mdv,where Ic is the capacity-related OC of the toll road. The construction material information of a road is known in the market, so we assume that Ic is the perfect information and identical for a private firm and the government in the paper. Given the same Ic, we can regard md as the intermediate variable that is set by the government in the bidding document. In practice, based on md, the government will evaluate the road OC including the management cost and construction cost before determining the road operator. Throughout this paper, we make the following assumptions on Bv,tv,c,Msv,Mgv,andIc.Assumption 1.
(a)
The inverse demand function,Bv, is a strictly continuously decreasing and differentiable function of v for any v≥0.(b)
The travel time functiontv,c is a continuously differentiable and strictly convex function of v and c for v≥0 and c≥0. tv,c strictly decreases with c for v>0 and strictly increases with v for c≥c1.(c)
The demand-related OC function of the government, a private firm, and the intermediate variable,Mgv,Msv, and Mdv, are continuously increasing and differentiable with respect to v for v≥0; the road construction cost function Ic is continuously increasing and differentiable with respect to c for c≥c1.(d)
For any givenc≥c1, the revenue collected from road users is given by Rv,c=v⋅pv,c, which is a strictly concave function of v for v≥0, i.e., ∂R2/∂v2<0.
It is assumed that the government knows the OC of a private firm, but is not clear of its profit; so, the government will select a private firm through a determined intermediate variablemd. With the given intermediate variable md, the problem in the following cases can be analyzed. If private firms take part in the bidding process and obtain the right to operate the PTR with md, the selection procedure will terminate. If the government sets the value of md too high, there will be no private firms participating in the bidding. The government has to take the right approaches to avoid this situation. In order to offer proposals for the government to make a proper decision, two solution approaches are proposed: set the right value of the intermediate variable; provide guarantee for the private firms to ensure their profit. If not selecting a private firm to operate the PTR, the government will operate the road by itself.
We first consider the profit of both roles. LetPgv,c, Psv,c, and Pscv,c be the governmental profit, private firm profit without governmental guarantee, and private profit with governmental guarantee, respectively.(4)Pgv,c=Rv,c−Jgv,c=v⋅Bv−vβtv,c−Ic+mgv,Psv,c=Rv,c−Jsv,c=v⋅Bv−vβtv,c−Ic+msv,(5)Pscv,c,n=Rv,c−Jsv,c+n=Psv,c+n,where Rv,c is the total revenue obtained from the road users; the last term of equation (5) is the government guarantee that the government will offer subsidy for the private firm operating the PTR. Equation (5) shows that private firm profit includes two parts: one is profit from the project Psv,c, and the other is the government subsidy n. Then, the revenue is(6)Rv,c=v⋅pv,c=v⋅Bv−vβtv,c.
Next, we define the function of social welfare. When the government operates the road, the social welfare function is formulated as(7)Wgv,c=Us+Pgv,c=∫0vBwdw−βvtv,c−Jgv,c,where Us is the total social surplus.
With equation (1), the total social surplus is defined by(8)Us=∫0vBwdw−vBv=∫0vBwdw−vpv,c+βtv,c.
When the private firm operates the toll road without the government guarantee, the social welfare is given by(9)Wsv,c=Us+Psv,c=∫0vBwdw−βvtv,c−Jsv,c.
When the private firm operates the toll road with the government guarantee, the social welfare can be computed as(10)Wscv,c=Us+δ⋅Pscv,c−n=∫0vBwdw−1−δvBv−βvtv,c−δJsv,c−1−δn,where δ is a weighting factor, 0<δ<1. The first term on the right side of equations (7)–(10) is the total social surplus of road users; the second term of equation (7) is the government profit; the second term of equation (9) is the private profit; the second term of equation (10) is the weighted provider’s surplus [21]; and the last term of equation (10) is the government guarantee.Assumption 2.
(a)
tv,c is homogeneous of degree zero in v and c, i.e., tαv,αc=tv,c for any α>0(b)
Constant return to scale in the construction and capacity-related cost, namely,Ic=kc, where k denotes the constant cost per unit capacity(c)
vBv is a concave function of v for any v≥0
Part (a) of Assumption2 shows that vehicle travel time on the road is only related with the volume-capacity ratio, r=v/c. For example, the BPR (Bureau of Public Roads) travel time function satisfies this assumption. In equation (1), the toll p can be uniquely determined by the travel demand v for a given road capacity c. In other words, social welfare and profit, created by both the government and a private firm, are determined by the travel demand v, road capacity c, and the government subsidy n.


## 4. The First Selection of the PTR Operator by the Government

The government has the right to set an intermediate variablemd as the threshold value of the public toll road for private firms. The government will not select md>mg; otherwise, it will operate the road by itself. So, when the government aims to choose a private firm to build/rebuild and operate the road, obviously, it will set the intermediate variable md≤mg.


### 4.1. Under the Condition ofms>mg with the Government Operation

The government and private firm know the information of OC for each other. When the unit demand-related OC of the private firm is greater than the government’s, namely,ms>mg, the government will operate the PTR by itself to maximize social welfare. In order to distinguish from the analysis of Section 4.3, the superscript “s>g” is used to represent the case ms>mg. Then, if ms>mg, the unconstraint social welfare maximization problem is formulated as(11)maxv,cWgs>gv,c=∫0vBwdw−βvtv,c−Jgs>gv,c,where the first-order optimal conditions of equation (11) can be found in Tan et al. [2].There are many highways operated by the government in China. The government needs the revenue collected from the road users at least to cover the road investment including operation and construction cost. So, when analyzing the operating PTR, it is necessary to analyze the social welfare maximization under the condition that the total investment including but not limited to operation costs is equal to the total revenueRv,c, namely, the road operator obtains positive profit or zero profit from the project. Then, if the government operates the road on its own, the constrained problem is defined as(12)maxv,cWgs>gv,c=∫0vBwdw−βvtv,c−Jgs>gv,c,subject to(13)Pgs>gv,c=Rgs>gv,c−Jgs>gv,c≥0,(14)v≥0,c≥c1,where equation (13) implies nonnegative profit; Pgs>gv,c is the profit of the government; and Rgs>gv,c is the governmental revenue.


### 4.2. Under the Condition ofms≤mg with the Private Firm Operation

In practice, the OC of a private firm, with stronger execution and more flexibility, is usually lower than that of the government. If the government makes a correct decision onmd, the private firm will be willing to operate the PTR with proper profit. In this section, we first propose the condition that the private firm operates the road under the condition of ms≤mg; then, based on the unconstrained and constrained maximization private profit models, the definitions of the first-best and second-best problems for the private firm are provided, respectively. For simplicity, the subscript “s” was used to denote ms≤mg.First, the following proposition provides the condition for the private firm to operate the road. We assume the acceptable profit to bePa.Proposition 1.
If the sum of the unit demand-related OC and unit benefit of the private firm is less thanmd, namely, ms+Pa/v≤md, and the government sets md in the interval ms+Pa/v,mg, then the contract between the government and a private firm will be signed.Proof.
If allowed to operate the PTR, the private firm will aim to make maximal profit from the project; and the government will certainly not let the private firm get so much profit. In other words, the private profit will be less than the difference between the intermediate OC and the OC of the private firm. Then, we have(15)v⋅pv,c−Jsv,c≥Pa,v⋅pv,c−Jsv,c≤Jdv,c−Jsv,csuch that(16)Jdv,c≥Pa+Jsv,c.
SinceIc is the perfect information,(17)Mdv≥Pa+Msv.
The proof is completed.
Although Proposition1 includes the OC of the private firm and the intermediate OC, it does not mean that the government knows the amount of the private profit. In reality, only when the government sets the intermediate variable md, the private firm will decide whether or not to operate the road according to its OC and profit. However, the government usually evaluates the trade secret Pa of private firms to determine md.
Getting the road franchising, the private firm will maximize its profit by selecting the road capacityc and toll pc. In the extreme case of the private profit maximization without any constraints, the unconstraint optimal problem can be formulated as(18)maxv,cPsv,c=vpv,c−Jsv,c.
Then, we investigate the profit maximization under the constraint that the private profit is not larger than the difference between the intermediate OC and the private firm OC. The constrained optimization problem can be formulated as(19)maxv,cPsv,c=vpv,c−Jsc,subject to(20)vpv,c−Jsv,c≥Pa,(21)vpv,c−Jsv,c≤Jdv,c−Jsv,c,(22)ms<md≤mg,v≥0,c≥c1,Pa≥0,where Pa≥0 is the minimum profit of the private firm. Even if md>ms, the government is still willing to let a private firm operate the road, which meets constraint (20). Alternatively, only when the private profit is larger than the profit margin Pa, the private firm will be willing to operate the road. Constraint (21) means that the private firm profit will not be larger than the difference between the intermediate OC and the private firm OC, and the government will permit the private firm to operate the PTR.
Based on the aforementioned problems, the definition of the first-best and second-best solutions of the private firm is introduced.Definition 1.
(first-best and second-best profit solutions of the private firm). Letvs,cs be an optimal solution to optimal problems (19)–(22); it will be the first-best solution if it maximizes problem (18); otherwise, it will be the second-best solution.
Definition1 is based on the profit maximization that could classify the private profit problems. It is different from the definition of the first-best and second-best contracts based on the social welfare [22]. In fact, the private firm will easily reach the second-best solution with constraints (20) and (21). By comparing first-best problem (18) and second-best problems (19)–(22), the following observation can be obtained based on Definition 1.Observation 1.
The first-best solution of the private firm exists if there only exists a solutionvs∗,cs∗, which solves the first-best problem and satisfies(23)v⋅pvs∗,cs∗−Jsvs∗,cs∗≥Pa,(24)vs∗⋅pvs∗,c−Jdvs∗,cs∗≤0.
The above observation is intuitive. If the total revenue generated by the first-best profit solution can cover the OC and create an acceptable profit threshold for the private firm, then the first-best profit solution can be archived.
From Definition1 and Observation 1, it is clear that if the government permits the private firm to determine the optimal profit by itself, the private firm will choose either the first-best or second-best solution, but not both. The first-best solution exists only if conditions (23) and (24) are satisfied; otherwise, the private firm will obtain the second-best solution. Therefore, to solve private profit problems (19)–(22), it is necessary to solve problem (18) and check whether its solution satisfies constraints (23) and (24). If these conditions hold, it is unnecessary to consider problems (19)–(22).
The Kuhn–Tucker conditions can be applied to obtain the first-order conditions (Appendix A). Note that letPssb be the optimal value of the objective function of problems (19)–(22), and it distinguishes from the value of P¯s that is a monopoly optimal solution of problem (18).The relationship between Pssb and P¯s is that Pssb≤P¯s, and Pssb is on the left side of P¯s in Figure 2. The following observation about Pssb can be obtained through analyzing the constraints.Figure 2
A geometric illustration ofPa,P¯s,Ps,andPssb.Observation 2.
(a)
If constraints (20) and (21) are effective, the acceptable private firm profit is equal to the maximal profit Jd−Js, namely, Pssb=Pa=Jd−Js=Md−Ms(b)
If constraint (20) is effective but constraint (21) is inactive, the private firm profit is Pssb=Pa(c)
If constraint (21) is effective but constraint (20) is inactive, the private firm profit is Pssb=Jdv,c−Jsv,c(d)
If both constraints (20) and (21) are ineffective and the second-best problem becomes the first-best problem, then the private firm profit is Pssb=P¯s
From the aforementioned observations, we investigate the case of the first-best and second-best private profit problem, which is presented on the left side of the monopoly optimum in Figure2. However, when will the monopoly optimum case happen? As shown in Figure 2, we can see that, responding to the same private profit, social welfare on the right side of the monopoly optimum is higher than that on the left side. Additionally, the government can make some rules in the contract to make the private firm set the social welfare maximization as the goal [23]. So, it is necessary to analyze the following two problems with private firm management: one is the unconstrained social welfare maximization problem; the other is the constrained social welfare maximization problem.
Base on equation (7), the unconstrained social welfare maximization problem can be formulated by(25)maxv,c∈ΩWsv,c=∫0vBwdw−βvtv,c−Jsv,c.
The constrained social problem is given by(26)maxv,c∈ΩWsv,c=∫0vBwdw−βvtv,c−Jsv,c,subject to(27)vpv,c−Jsv,c≥Pa,(28)vpv,c−Jsv,c≤Jdv,c−Jsv,c,(29)ms<md≤mg,v≥0,c≥c1,Pa≥0,where Ω=v,c:v≥0,c≥0.The constrained conditions are the same as those of the second-best problem of the private firm, i.e., (20)–(22), which guarantee the private firm to operate the road.
Since this kind of constrained social welfare problem has more constraints than the traditional Pareto optimal problem, the win-win optimal problem is defined as follows.Definition 2.
(win-win optimal problem). A pairvsww,csww∈Ω is called the win-win solution if there are no other feasible solutions v,c∈Ω such that Wv,c≥Wvsww,csww and Pv,c≥Pvsww,csww with at least one strict inequality.
The definition of the win-win optimal problem is a meaningful and important concept which meets all the conditions of the Pareto problem, but with more constraints, and its solution is a part of the Pareto optimal solution.
In general, more than one constrained condition in problems (27)–(29) will be effective. In response to the case on the right side of monopoly optimum in Figure 2, similar to Observation 2, the following observation is provided through analyzing the constrained conditions (Appendix B).Observation 3.
(a)
If constraints (27) and (28) are effective, the acceptable private firm profit is the same as its maximal profit, which is equal to Jd−Js, namely, Pa=Md−Ms(b)
If constraint (27) is effective and constraint (28) is inactive, the private firm profit is equal to Pa(c)
If constraint (28) is effective and constraint (27) is inactive, the private firm profit is equal to Md−Ms
Furthermore, letv˜s and v¯s be the social welfare optimality solution of problem (18) and the monopoly optimal solution of problem (25), respectively. The following proposition is obtained in terms of the above second-best solution of the private problem and win-win optimal problem (see more details in Appendix C).Proposition 2.
It holds thatvssb<v¯s≤vsww≤v˜s for the win-win optimal solution vsww,csww and the second-best solution of the private firm vssb,cssb when the private firm operates the road.
Proposition2 means that travel demand of the win-win solution should not be strictly less than the level of the profit maximization and less than the level of the social welfare maximization. Travel demand of the second-best optimal solution is smaller than that of any other optimal solutions. Figure 2 implies that the government must set the regulation to gain the win-win optimal solution, namely, make Md−Ms on the right side of the monopoly optimal solution. Otherwise, the private firm will optimize the travel demand vssb.
Based on the second-best solution and the Win-Win optimal solution, Proposition3 can be derived:Proposition 3.
For the second-best solution of the private firm, the biggerMd−Ms is, the more possible it is that the government and a private firm have the same objective; For the win-win solution, the smaller Md−Ms is, the more possible it is that the government and private firm have the same objective.
Refer to the proof in Appendix D. Proposition3 tells us why the private firm wants to know the value of Md and the government also wants to evaluate the demand-related OC of the private firm to set a better Md.The result is consistent with Figure 2, which implies the objective of the government and private firm will be realized at the same time on the left side of monopoly optimality, namely, the private firm obtains the maximum solution, while the government achieves the maximal social welfare. If the difference Md−Ms is bigger, the private firm will obtain more profit. On the right side of monopoly optimality in Figure 2, the government and private firm will have different objectives, namely, when the private firm chooses the profit maximization, the government will reach the minimum social welfare. If the difference Md−Ms is smaller, the private firm will obtain more profit, and the government can also gain more social welfare. However, welfare on the right side that belongs to the win-win optimal solution is bigger than that on the left side.
LetPsw be the private profit corresponding to the social optimal welfare. Based on Proposition 3, an extreme case is analyzed: if private profit satisfies Pa≤Psw, when the intermediate variable set by the government meets Md−Ms≥Psw, namely, the variable relationship meets Pa≤Psw≤Md−Ms; under the win-win optimal regulation in the contract, not only that the government may gain the optimal social welfare but also the private firm meets its profit. So, the ideal condition is Pa≤Psw=Md−Ms, which is also the best case for the win-win optimal solution.
Some private firms usually earn huge profit from the project, which happened in China such as Beijing Airport Expressway and Guangzhou-Shenzhen Highway. Although these roads were built with the largest road capacity (allowed by the design code), there is oversaturated traffic flow every day. In contrast with the traditional contract, we can regard constraint (21) as a regulation to allow the private firm to make the decision. Then,(30)vpv,c−Jsv,c>Jdv,c−Jsv,c.
The above condition implies that if the private firm profit is bigger than the difference between the intermediate OC and private firm OC, the government will operate the road by itself. Condition (30) can be simplified by(31)vpv,c−Ic+mdv>0.
If there is heavy traffic pressure, then the private firm will get profit from the project quickly. Meanwhile, condition (31) implies that if the road is operated, all terms of (31) are the symmetric information for the government and a private firm, namely, the road capacity has been determined when the road is operating; the government can get the information of road capacity, toll, and travel demand from real data. Then, the government knows the profit of the private firm based on the intermediate variable even if it knows nothing about the private firm operation. When condition (31) is satisfied, the government can operate the road by itself, namely, condition (31) points out that the difference between profit vpv,c and intermediate variable md determines the road operation right of the private firm.
In the last part of this section, the private firm zero profit will be analyzed. If private firms get the road franchise, they will have the right to determine locations of on-ramps and off-ramps that have an impact on the land development, and so on. With the above description, the private firm will be willing to operate the toll road projects with zero profit or negative profit to earn more profit from other aspects. So, it is necessary to analyze the zero or negative profit case. However, in the light of knowing nothing about how much loss the private firm can bear, the zero profit is analyzed as follows.Proposition 4.
Under the condition of zero profit, the unit demand-related OC of the private firmms can approach the biggest value, namely, ms is equal to mg.Proof.
From constraints (20) and (21),(32)Jdv,c−Jsv,c≥Pa.
If the private profit is equal to zero, namely,Pa=0, then(33)Mdv,c−Msv,c=0.
IfMdv,c=Mgv,c, then(34)Msv,c=Mgv,c.
The proof is completed.


### 4.3. Under the Condition ofms≤mg with the Government Operation

Different from Section4.1, it is assumed that the government knows nothing about the private profit, which may make the government set an improper decision on the intermediate variable md. The private firm will not operate the road if it cannot gain the satisfied profit from the project. The following two cases are related with an improper decision of md.In contrast with Section4.2, we provide the condition that the private firm cannot gain enough profit. Using constraints (20) through (22),(35)Jdv,c−Jsv,c≥Pa,which implies that only when the difference between the intermediate OC and the private OC is larger than the private firm profit, the private firm will be willing to operate the toll road.On the contrary,(36)Jdv,c−Jsv,c<Pa,which implies that the government will be reluctant to operate the road by itself.Ifmd given by the government is less than the private OC, namely, md≤ms<mg, the private firm will not operate the road due to the negative profit.In Section4.1, we analyze the case that the government is willing to operate the road. In this section, there will be no contract between the government and a private firm. The superscript “s<g” is used to represent ms≤mg. Then, the problem can be formulated as(37)maxv,cWgs<gv,c=∫0vBwdw−βvtv,c−Jgs<gv,c.In the above analysis, two kinds of government operation scenarios were considered: the first one was analyzed in Section4.1 given ms>mg; the second one was analyzed in this section under the condition of ms≤mg. If the government does not make an improper decision on md, we assume that the government will choose to maximize the social welfare without loss, so it is not necessary to analyze this case. Edelson [24] showed that a private firm may charge a lower toll than the government for a given and fixed capacity. Let cd∗ be the given road capacity, and all analyses of this section are based on this capacity. Then, we compare the toll charged by the government and the private firm based on the assumption that the government makes an improper decision on the variable md under the condition of ms≤mg.Note that the toll charged by the government is the solution at the zero profit, namely, the governmental revenue just covers its capital cost.Then, we define the average social cost (ASC) of the government and private firm, respectively:(38)ASCgs<g=βvtv,cd∗+Jgs<gv,cd∗v=βtv,cd∗+Jgs<gv,cd∗v,(39)ASCs=βvtv,cd∗+Jsv,cd∗v=βtv,cd∗+Jsv,cd∗v.The first term of equations (38) and (39) on the right side is the average travel time in monetary units; the second term is the cost of operation per trip. Intuitively, with Assumptions 1 and 2 and the condition of ms≤mg, the average social cost of the government will be bigger than the private firm OC for given road capacity and travel demand.If considering the average social profit in the cost of the private firm, equation (39) can be rewritten as(40)ASCsp=βtv,cd∗+Jsv,cd∗v+Psv.Then, combining equations (38) and (40) for the given travel demand and road capacity cs∗,(41)ASCgs<g−ASCsp=Mgs<gv−Msv+Psv,where Ps is in the interval Pa,Md−Ms.From equation (41), though Ps=Md−Ms, the average social cost of the government will be bigger than that of private firm operation under Md≤Mg. We know that the government may charge a higher toll than the private firm under the condition that the government operates the road without loss.Letp¯s,p˜s,andp˜gs<g be the monopoly price of the private firm, the price of social optimal welfare of the private firm, and the price of social optimal welfare of the government, respectively. Then, the following proposition can be obtained (Appendix E).Proposition 5.
With a given road capacitycd∗, the toll v¯s,v˜s,v˜gs<g has the following relationship: p˜s≤p˜gs<g<p¯s.
Proposition5 shows that if the social welfare maximum is reached, the toll charged by a private firm is less than the government, which is derived from the improper decision of the government on the intermediate variable. In order to avoid the circumstance that the government operates the road with higher charge than the private firm, the best option for the government is investigated to get a satisfactory result.


## 4.1. Under the Condition ofms>mg with the Government Operation

The government and private firm know the information of OC for each other. When the unit demand-related OC of the private firm is greater than the government’s, namely,ms>mg, the government will operate the PTR by itself to maximize social welfare. In order to distinguish from the analysis of Section 4.3, the superscript “s>g” is used to represent the case ms>mg. Then, if ms>mg, the unconstraint social welfare maximization problem is formulated as(11)maxv,cWgs>gv,c=∫0vBwdw−βvtv,c−Jgs>gv,c,where the first-order optimal conditions of equation (11) can be found in Tan et al. [2].There are many highways operated by the government in China. The government needs the revenue collected from the road users at least to cover the road investment including operation and construction cost. So, when analyzing the operating PTR, it is necessary to analyze the social welfare maximization under the condition that the total investment including but not limited to operation costs is equal to the total revenueRv,c, namely, the road operator obtains positive profit or zero profit from the project. Then, if the government operates the road on its own, the constrained problem is defined as(12)maxv,cWgs>gv,c=∫0vBwdw−βvtv,c−Jgs>gv,c,subject to(13)Pgs>gv,c=Rgs>gv,c−Jgs>gv,c≥0,(14)v≥0,c≥c1,where equation (13) implies nonnegative profit; Pgs>gv,c is the profit of the government; and Rgs>gv,c is the governmental revenue.


## 4.2. Under the Condition ofms≤mg with the Private Firm Operation

In practice, the OC of a private firm, with stronger execution and more flexibility, is usually lower than that of the government. If the government makes a correct decision onmd, the private firm will be willing to operate the PTR with proper profit. In this section, we first propose the condition that the private firm operates the road under the condition of ms≤mg; then, based on the unconstrained and constrained maximization private profit models, the definitions of the first-best and second-best problems for the private firm are provided, respectively. For simplicity, the subscript “s” was used to denote ms≤mg.First, the following proposition provides the condition for the private firm to operate the road. We assume the acceptable profit to bePa.Proposition 1.
If the sum of the unit demand-related OC and unit benefit of the private firm is less thanmd, namely, ms+Pa/v≤md, and the government sets md in the interval ms+Pa/v,mg, then the contract between the government and a private firm will be signed.Proof.
If allowed to operate the PTR, the private firm will aim to make maximal profit from the project; and the government will certainly not let the private firm get so much profit. In other words, the private profit will be less than the difference between the intermediate OC and the OC of the private firm. Then, we have(15)v⋅pv,c−Jsv,c≥Pa,v⋅pv,c−Jsv,c≤Jdv,c−Jsv,csuch that(16)Jdv,c≥Pa+Jsv,c.
SinceIc is the perfect information,(17)Mdv≥Pa+Msv.
The proof is completed.
Although Proposition1 includes the OC of the private firm and the intermediate OC, it does not mean that the government knows the amount of the private profit. In reality, only when the government sets the intermediate variable md, the private firm will decide whether or not to operate the road according to its OC and profit. However, the government usually evaluates the trade secret Pa of private firms to determine md.
Getting the road franchising, the private firm will maximize its profit by selecting the road capacityc and toll pc. In the extreme case of the private profit maximization without any constraints, the unconstraint optimal problem can be formulated as(18)maxv,cPsv,c=vpv,c−Jsv,c.
Then, we investigate the profit maximization under the constraint that the private profit is not larger than the difference between the intermediate OC and the private firm OC. The constrained optimization problem can be formulated as(19)maxv,cPsv,c=vpv,c−Jsc,subject to(20)vpv,c−Jsv,c≥Pa,(21)vpv,c−Jsv,c≤Jdv,c−Jsv,c,(22)ms<md≤mg,v≥0,c≥c1,Pa≥0,where Pa≥0 is the minimum profit of the private firm. Even if md>ms, the government is still willing to let a private firm operate the road, which meets constraint (20). Alternatively, only when the private profit is larger than the profit margin Pa, the private firm will be willing to operate the road. Constraint (21) means that the private firm profit will not be larger than the difference between the intermediate OC and the private firm OC, and the government will permit the private firm to operate the PTR.
Based on the aforementioned problems, the definition of the first-best and second-best solutions of the private firm is introduced.Definition 1.
(first-best and second-best profit solutions of the private firm). Letvs,cs be an optimal solution to optimal problems (19)–(22); it will be the first-best solution if it maximizes problem (18); otherwise, it will be the second-best solution.
Definition1 is based on the profit maximization that could classify the private profit problems. It is different from the definition of the first-best and second-best contracts based on the social welfare [22]. In fact, the private firm will easily reach the second-best solution with constraints (20) and (21). By comparing first-best problem (18) and second-best problems (19)–(22), the following observation can be obtained based on Definition 1.Observation 1.
The first-best solution of the private firm exists if there only exists a solutionvs∗,cs∗, which solves the first-best problem and satisfies(23)v⋅pvs∗,cs∗−Jsvs∗,cs∗≥Pa,(24)vs∗⋅pvs∗,c−Jdvs∗,cs∗≤0.
The above observation is intuitive. If the total revenue generated by the first-best profit solution can cover the OC and create an acceptable profit threshold for the private firm, then the first-best profit solution can be archived.
From Definition1 and Observation 1, it is clear that if the government permits the private firm to determine the optimal profit by itself, the private firm will choose either the first-best or second-best solution, but not both. The first-best solution exists only if conditions (23) and (24) are satisfied; otherwise, the private firm will obtain the second-best solution. Therefore, to solve private profit problems (19)–(22), it is necessary to solve problem (18) and check whether its solution satisfies constraints (23) and (24). If these conditions hold, it is unnecessary to consider problems (19)–(22).
The Kuhn–Tucker conditions can be applied to obtain the first-order conditions (Appendix A). Note that letPssb be the optimal value of the objective function of problems (19)–(22), and it distinguishes from the value of P¯s that is a monopoly optimal solution of problem (18).The relationship between Pssb and P¯s is that Pssb≤P¯s, and Pssb is on the left side of P¯s in Figure 2. The following observation about Pssb can be obtained through analyzing the constraints.Figure 2
A geometric illustration ofPa,P¯s,Ps,andPssb.Observation 2.
(a)
If constraints (20) and (21) are effective, the acceptable private firm profit is equal to the maximal profit Jd−Js, namely, Pssb=Pa=Jd−Js=Md−Ms(b)
If constraint (20) is effective but constraint (21) is inactive, the private firm profit is Pssb=Pa(c)
If constraint (21) is effective but constraint (20) is inactive, the private firm profit is Pssb=Jdv,c−Jsv,c(d)
If both constraints (20) and (21) are ineffective and the second-best problem becomes the first-best problem, then the private firm profit is Pssb=P¯s
From the aforementioned observations, we investigate the case of the first-best and second-best private profit problem, which is presented on the left side of the monopoly optimum in Figure2. However, when will the monopoly optimum case happen? As shown in Figure 2, we can see that, responding to the same private profit, social welfare on the right side of the monopoly optimum is higher than that on the left side. Additionally, the government can make some rules in the contract to make the private firm set the social welfare maximization as the goal [23]. So, it is necessary to analyze the following two problems with private firm management: one is the unconstrained social welfare maximization problem; the other is the constrained social welfare maximization problem.
Base on equation (7), the unconstrained social welfare maximization problem can be formulated by(25)maxv,c∈ΩWsv,c=∫0vBwdw−βvtv,c−Jsv,c.
The constrained social problem is given by(26)maxv,c∈ΩWsv,c=∫0vBwdw−βvtv,c−Jsv,c,subject to(27)vpv,c−Jsv,c≥Pa,(28)vpv,c−Jsv,c≤Jdv,c−Jsv,c,(29)ms<md≤mg,v≥0,c≥c1,Pa≥0,where Ω=v,c:v≥0,c≥0.The constrained conditions are the same as those of the second-best problem of the private firm, i.e., (20)–(22), which guarantee the private firm to operate the road.
Since this kind of constrained social welfare problem has more constraints than the traditional Pareto optimal problem, the win-win optimal problem is defined as follows.Definition 2.
(win-win optimal problem). A pairvsww,csww∈Ω is called the win-win solution if there are no other feasible solutions v,c∈Ω such that Wv,c≥Wvsww,csww and Pv,c≥Pvsww,csww with at least one strict inequality.
The definition of the win-win optimal problem is a meaningful and important concept which meets all the conditions of the Pareto problem, but with more constraints, and its solution is a part of the Pareto optimal solution.
In general, more than one constrained condition in problems (27)–(29) will be effective. In response to the case on the right side of monopoly optimum in Figure 2, similar to Observation 2, the following observation is provided through analyzing the constrained conditions (Appendix B).Observation 3.
(a)
If constraints (27) and (28) are effective, the acceptable private firm profit is the same as its maximal profit, which is equal to Jd−Js, namely, Pa=Md−Ms(b)
If constraint (27) is effective and constraint (28) is inactive, the private firm profit is equal to Pa(c)
If constraint (28) is effective and constraint (27) is inactive, the private firm profit is equal to Md−Ms
Furthermore, letv˜s and v¯s be the social welfare optimality solution of problem (18) and the monopoly optimal solution of problem (25), respectively. The following proposition is obtained in terms of the above second-best solution of the private problem and win-win optimal problem (see more details in Appendix C).Proposition 2.
It holds thatvssb<v¯s≤vsww≤v˜s for the win-win optimal solution vsww,csww and the second-best solution of the private firm vssb,cssb when the private firm operates the road.
Proposition2 means that travel demand of the win-win solution should not be strictly less than the level of the profit maximization and less than the level of the social welfare maximization. Travel demand of the second-best optimal solution is smaller than that of any other optimal solutions. Figure 2 implies that the government must set the regulation to gain the win-win optimal solution, namely, make Md−Ms on the right side of the monopoly optimal solution. Otherwise, the private firm will optimize the travel demand vssb.
Based on the second-best solution and the Win-Win optimal solution, Proposition3 can be derived:Proposition 3.
For the second-best solution of the private firm, the biggerMd−Ms is, the more possible it is that the government and a private firm have the same objective; For the win-win solution, the smaller Md−Ms is, the more possible it is that the government and private firm have the same objective.
Refer to the proof in Appendix D. Proposition3 tells us why the private firm wants to know the value of Md and the government also wants to evaluate the demand-related OC of the private firm to set a better Md.The result is consistent with Figure 2, which implies the objective of the government and private firm will be realized at the same time on the left side of monopoly optimality, namely, the private firm obtains the maximum solution, while the government achieves the maximal social welfare. If the difference Md−Ms is bigger, the private firm will obtain more profit. On the right side of monopoly optimality in Figure 2, the government and private firm will have different objectives, namely, when the private firm chooses the profit maximization, the government will reach the minimum social welfare. If the difference Md−Ms is smaller, the private firm will obtain more profit, and the government can also gain more social welfare. However, welfare on the right side that belongs to the win-win optimal solution is bigger than that on the left side.
LetPsw be the private profit corresponding to the social optimal welfare. Based on Proposition 3, an extreme case is analyzed: if private profit satisfies Pa≤Psw, when the intermediate variable set by the government meets Md−Ms≥Psw, namely, the variable relationship meets Pa≤Psw≤Md−Ms; under the win-win optimal regulation in the contract, not only that the government may gain the optimal social welfare but also the private firm meets its profit. So, the ideal condition is Pa≤Psw=Md−Ms, which is also the best case for the win-win optimal solution.
Some private firms usually earn huge profit from the project, which happened in China such as Beijing Airport Expressway and Guangzhou-Shenzhen Highway. Although these roads were built with the largest road capacity (allowed by the design code), there is oversaturated traffic flow every day. In contrast with the traditional contract, we can regard constraint (21) as a regulation to allow the private firm to make the decision. Then,(30)vpv,c−Jsv,c>Jdv,c−Jsv,c.
The above condition implies that if the private firm profit is bigger than the difference between the intermediate OC and private firm OC, the government will operate the road by itself. Condition (30) can be simplified by(31)vpv,c−Ic+mdv>0.
If there is heavy traffic pressure, then the private firm will get profit from the project quickly. Meanwhile, condition (31) implies that if the road is operated, all terms of (31) are the symmetric information for the government and a private firm, namely, the road capacity has been determined when the road is operating; the government can get the information of road capacity, toll, and travel demand from real data. Then, the government knows the profit of the private firm based on the intermediate variable even if it knows nothing about the private firm operation. When condition (31) is satisfied, the government can operate the road by itself, namely, condition (31) points out that the difference between profit vpv,c and intermediate variable md determines the road operation right of the private firm.
In the last part of this section, the private firm zero profit will be analyzed. If private firms get the road franchise, they will have the right to determine locations of on-ramps and off-ramps that have an impact on the land development, and so on. With the above description, the private firm will be willing to operate the toll road projects with zero profit or negative profit to earn more profit from other aspects. So, it is necessary to analyze the zero or negative profit case. However, in the light of knowing nothing about how much loss the private firm can bear, the zero profit is analyzed as follows.Proposition 4.
Under the condition of zero profit, the unit demand-related OC of the private firmms can approach the biggest value, namely, ms is equal to mg.Proof.
From constraints (20) and (21),(32)Jdv,c−Jsv,c≥Pa.
If the private profit is equal to zero, namely,Pa=0, then(33)Mdv,c−Msv,c=0.
IfMdv,c=Mgv,c, then(34)Msv,c=Mgv,c.
The proof is completed.


## 4.3. Under the Condition ofms≤mg with the Government Operation

Different from Section4.1, it is assumed that the government knows nothing about the private profit, which may make the government set an improper decision on the intermediate variable md. The private firm will not operate the road if it cannot gain the satisfied profit from the project. The following two cases are related with an improper decision of md.In contrast with Section4.2, we provide the condition that the private firm cannot gain enough profit. Using constraints (20) through (22),(35)Jdv,c−Jsv,c≥Pa,which implies that only when the difference between the intermediate OC and the private OC is larger than the private firm profit, the private firm will be willing to operate the toll road.On the contrary,(36)Jdv,c−Jsv,c<Pa,which implies that the government will be reluctant to operate the road by itself.Ifmd given by the government is less than the private OC, namely, md≤ms<mg, the private firm will not operate the road due to the negative profit.In Section4.1, we analyze the case that the government is willing to operate the road. In this section, there will be no contract between the government and a private firm. The superscript “s<g” is used to represent ms≤mg. Then, the problem can be formulated as(37)maxv,cWgs<gv,c=∫0vBwdw−βvtv,c−Jgs<gv,c.In the above analysis, two kinds of government operation scenarios were considered: the first one was analyzed in Section4.1 given ms>mg; the second one was analyzed in this section under the condition of ms≤mg. If the government does not make an improper decision on md, we assume that the government will choose to maximize the social welfare without loss, so it is not necessary to analyze this case. Edelson [24] showed that a private firm may charge a lower toll than the government for a given and fixed capacity. Let cd∗ be the given road capacity, and all analyses of this section are based on this capacity. Then, we compare the toll charged by the government and the private firm based on the assumption that the government makes an improper decision on the variable md under the condition of ms≤mg.Note that the toll charged by the government is the solution at the zero profit, namely, the governmental revenue just covers its capital cost.Then, we define the average social cost (ASC) of the government and private firm, respectively:(38)ASCgs<g=βvtv,cd∗+Jgs<gv,cd∗v=βtv,cd∗+Jgs<gv,cd∗v,(39)ASCs=βvtv,cd∗+Jsv,cd∗v=βtv,cd∗+Jsv,cd∗v.The first term of equations (38) and (39) on the right side is the average travel time in monetary units; the second term is the cost of operation per trip. Intuitively, with Assumptions 1 and 2 and the condition of ms≤mg, the average social cost of the government will be bigger than the private firm OC for given road capacity and travel demand.If considering the average social profit in the cost of the private firm, equation (39) can be rewritten as(40)ASCsp=βtv,cd∗+Jsv,cd∗v+Psv.Then, combining equations (38) and (40) for the given travel demand and road capacity cs∗,(41)ASCgs<g−ASCsp=Mgs<gv−Msv+Psv,where Ps is in the interval Pa,Md−Ms.From equation (41), though Ps=Md−Ms, the average social cost of the government will be bigger than that of private firm operation under Md≤Mg. We know that the government may charge a higher toll than the private firm under the condition that the government operates the road without loss.Letp¯s,p˜s,andp˜gs<g be the monopoly price of the private firm, the price of social optimal welfare of the private firm, and the price of social optimal welfare of the government, respectively. Then, the following proposition can be obtained (Appendix E).Proposition 5.
With a given road capacitycd∗, the toll v¯s,v˜s,v˜gs<g has the following relationship: p˜s≤p˜gs<g<p¯s.
Proposition5 shows that if the social welfare maximum is reached, the toll charged by a private firm is less than the government, which is derived from the improper decision of the government on the intermediate variable. In order to avoid the circumstance that the government operates the road with higher charge than the private firm, the best option for the government is investigated to get a satisfactory result.


## 5. The Optimal Intermediate Variable

The case discussed in Section4.3 occurs because of the improper intermediate variable md given by the government. With the governmental operation, road users will pay higher toll that is analyzed in Section 4.3 for ms≤mg. So, in order to find a proper private firm to operate the road and not let the road users undertake higher toll, the government will make a change on the selection tactics. In the following, two modes will be studied to avoid the improper decision on md: one is without guarantee, and the other is considering guarantee provided by the government.


### 5.1. The Selection Mode without the Government Guarantee

At first, we investigate the mode that the government will not provide guarantee for the private firm. In Section4, we discussed the case that the private firm and the government were willing to operate the road; we also discussed the case that the government was reluctant to operate the road under the improper decision. Under the condition of knowing nothing about private profit Pa, the government will try its best to make a correct decision. In Section 4, we pointed out that the case of md>mg was impossible in that the government would operate the road by itself. So, in this section, we only investigate the value of md which can result in a satisfactory result for the government under md≤mg. We firstly provide the satisfactory result and unsatisfactory result under md≤mg and Pa≥0.The definition of satisfactory results:(a)Ifmd=mg and ms>md, the government will operate the road by itself(b)
Ifms≤md=mg and ms+Pa/v>md, the government will operate the road by itself(c)
Ifms≤md≤mg and ms+Pa/v>md, a private firm will operate the road(d)
Ifmd<ms≤mg and ms+Pa/v>md, the government will operate the road by itselfThe definition of unsatisfactory results:(e)
Ifms≤md<mg and ms+Pa/v>mg, the government will be reluctant to operate the road by itself(f)
Ifmd<ms≤mg and ms+Pa/v>mg, the government will be reluctant to operate the road by itselfWith the definition of satisfactory results and unsatisfactory results of the government, we have the following proposition. We know that there are two constraints:ms+Ps/v>md and Ps≥Pa. Then, we make another assumption that the government must obtain the satisfactory result.Proposition 6.
If the government wants to obtain a satisfied result, the intermediate variablemd should be equal to the demand-related OC of the government mg, namely, md=mg.
At the beginning of this section, we pointed out that the government is not clear about profit of the private firm. The governmental objective is to obtain the satisfied result with the private firm. If the government determines the value of the intermediate variable so that the private firms, after their independent calculation, believe that they will be able to make necessary profit from running the project, the private firms may decide to take part in the bid. The government does not need to know the OC of the private firm. So, when the government setsmd=mg, the government can easily choose a proper road operator under the condition of satisfactory results.Whenmd=mg, the problem can be transformed to the following:(42)maxv,cWsv,c=∫0vBwdw−βvtv,c−Jsv,c,subject to(43)v⋅pv,c−Jsv,c≥Pa,v⋅pv,c−Jsv,c≤Jgv,c−Jsv,c,ms≤mg,v≥0,c≥c1,Pa≥0.Then, the government can make the decision to obtain satisfactory results through comparing formulas (39)–(42) with the social optimal welfare problem in Section 4.3. Private firm can obtain its profit through setting the toll and road capacity based on the solution of problems (39)–(42).


### 5.2. The Selection Mode with the Government Guarantee

In this section, the government guarantee will be considered to make the government reach a satisfactory result. In order to obtain satisfactory results, like in Section4.3, the government can make proper decisions through providing subsidy for the private firm.We firstly give the condition that the government may provide the guarantee for a private firm:(44)v⋅pv,c−Jsv,c=Psv,c<Pa.From equation (44), we know that the private firm will not operate the road in that it will not get satisfactory profit from the project or get negative profit. If the government wants the private firm to operate the road, it must provide guarantee, namely, additional profit for the private firm, which makes sure the private profit is not less than a threshold value Pa [25]. Then, we have(45)Pscv,c=Psv,c+n=v⋅pv,c−Jsv,c+n≥Pa,where the first term on the left side of the inequality is the private profit and n is the government subsidy. Therefore, private profit is composed of a private firm profit and the government subsidy.Even if providing subsidy for the private firm, the government will be reluctant that the total amount of profits, the OC of the private firm, and subsidies are bigger than the governmental OC. According to Proposition6, we point out that md=mg is a better option for the government. Then, the second constraint becomes(46)v⋅pv,c−Jsv,c+n≤Jgv,c−Jsv,c.Equation (46) can be simply written as v⋅pv,c+n≤Jgv,c. According to equation (46), the government can control the biggest subsidy which is n=Jgv,c−v⋅pv,c. With this constraint, even if nothing is known about private profit, the government can determine the minimal subsidy. An extreme case is that the minimal private profit is equal to the maximum of guarantee, namely, Pa=Jgc−Jsc. From Proposition 6, as long as the subsidies do not exceed the maximum subsidies, the government will regard it as a satisfactory result.In the bidding process, many private companies have put forward different subsidy requirements, which can further determine subsidies. If the government provides subsidies for the private firm, it must require a private firm to select the win-win optimal solution. With constraints (45) and (46) and objective function (10), the optimal problem with the government guarantee can be formulated by(47)maxv,cWscv,c,n=∫0vBwdw−vBv+δv⋅pv,c−Jsc−1−δn,subject to equations (45) and (46), and(48)ms≤mg,v≥0,c>c1,0<δ<1,Pa≥0.Based on the above problem, we have the following proposition (Appendix F).Proposition 7.
If the government provides guarantee for the private firm, the minimum private profit is equal to the maximum subsidies.
The above proposition shows that if the government wants to provide guarantee for the private firm, the private firm will choose the maximum subsides. And the government will not permit the total amount of toll sum charged by the private firm and guarantee larger than the governmental OC on the PTR.


## 5.1. The Selection Mode without the Government Guarantee

At first, we investigate the mode that the government will not provide guarantee for the private firm. In Section4, we discussed the case that the private firm and the government were willing to operate the road; we also discussed the case that the government was reluctant to operate the road under the improper decision. Under the condition of knowing nothing about private profit Pa, the government will try its best to make a correct decision. In Section 4, we pointed out that the case of md>mg was impossible in that the government would operate the road by itself. So, in this section, we only investigate the value of md which can result in a satisfactory result for the government under md≤mg. We firstly provide the satisfactory result and unsatisfactory result under md≤mg and Pa≥0.The definition of satisfactory results:(a)Ifmd=mg and ms>md, the government will operate the road by itself(b)
Ifms≤md=mg and ms+Pa/v>md, the government will operate the road by itself(c)
Ifms≤md≤mg and ms+Pa/v>md, a private firm will operate the road(d)
Ifmd<ms≤mg and ms+Pa/v>md, the government will operate the road by itselfThe definition of unsatisfactory results:(e)
Ifms≤md<mg and ms+Pa/v>mg, the government will be reluctant to operate the road by itself(f)
Ifmd<ms≤mg and ms+Pa/v>mg, the government will be reluctant to operate the road by itselfWith the definition of satisfactory results and unsatisfactory results of the government, we have the following proposition. We know that there are two constraints:ms+Ps/v>md and Ps≥Pa. Then, we make another assumption that the government must obtain the satisfactory result.Proposition 6.
If the government wants to obtain a satisfied result, the intermediate variablemd should be equal to the demand-related OC of the government mg, namely, md=mg.
At the beginning of this section, we pointed out that the government is not clear about profit of the private firm. The governmental objective is to obtain the satisfied result with the private firm. If the government determines the value of the intermediate variable so that the private firms, after their independent calculation, believe that they will be able to make necessary profit from running the project, the private firms may decide to take part in the bid. The government does not need to know the OC of the private firm. So, when the government setsmd=mg, the government can easily choose a proper road operator under the condition of satisfactory results.Whenmd=mg, the problem can be transformed to the following:(42)maxv,cWsv,c=∫0vBwdw−βvtv,c−Jsv,c,subject to(43)v⋅pv,c−Jsv,c≥Pa,v⋅pv,c−Jsv,c≤Jgv,c−Jsv,c,ms≤mg,v≥0,c≥c1,Pa≥0.Then, the government can make the decision to obtain satisfactory results through comparing formulas (39)–(42) with the social optimal welfare problem in Section 4.3. Private firm can obtain its profit through setting the toll and road capacity based on the solution of problems (39)–(42).


## 5.2. The Selection Mode with the Government Guarantee

In this section, the government guarantee will be considered to make the government reach a satisfactory result. In order to obtain satisfactory results, like in Section4.3, the government can make proper decisions through providing subsidy for the private firm.We firstly give the condition that the government may provide the guarantee for a private firm:(44)v⋅pv,c−Jsv,c=Psv,c<Pa.From equation (44), we know that the private firm will not operate the road in that it will not get satisfactory profit from the project or get negative profit. If the government wants the private firm to operate the road, it must provide guarantee, namely, additional profit for the private firm, which makes sure the private profit is not less than a threshold value Pa [25]. Then, we have(45)Pscv,c=Psv,c+n=v⋅pv,c−Jsv,c+n≥Pa,where the first term on the left side of the inequality is the private profit and n is the government subsidy. Therefore, private profit is composed of a private firm profit and the government subsidy.Even if providing subsidy for the private firm, the government will be reluctant that the total amount of profits, the OC of the private firm, and subsidies are bigger than the governmental OC. According to Proposition6, we point out that md=mg is a better option for the government. Then, the second constraint becomes(46)v⋅pv,c−Jsv,c+n≤Jgv,c−Jsv,c.Equation (46) can be simply written as v⋅pv,c+n≤Jgv,c. According to equation (46), the government can control the biggest subsidy which is n=Jgv,c−v⋅pv,c. With this constraint, even if nothing is known about private profit, the government can determine the minimal subsidy. An extreme case is that the minimal private profit is equal to the maximum of guarantee, namely, Pa=Jgc−Jsc. From Proposition 6, as long as the subsidies do not exceed the maximum subsidies, the government will regard it as a satisfactory result.In the bidding process, many private companies have put forward different subsidy requirements, which can further determine subsidies. If the government provides subsidies for the private firm, it must require a private firm to select the win-win optimal solution. With constraints (45) and (46) and objective function (10), the optimal problem with the government guarantee can be formulated by(47)maxv,cWscv,c,n=∫0vBwdw−vBv+δv⋅pv,c−Jsc−1−δn,subject to equations (45) and (46), and(48)ms≤mg,v≥0,c>c1,0<δ<1,Pa≥0.Based on the above problem, we have the following proposition (Appendix F).Proposition 7.
If the government provides guarantee for the private firm, the minimum private profit is equal to the maximum subsidies.
The above proposition shows that if the government wants to provide guarantee for the private firm, the private firm will choose the maximum subsides. And the government will not permit the total amount of toll sum charged by the private firm and guarantee larger than the governmental OC on the PTR.


## 6. Conclusions

The operation of the PTR becomes a significant issue of the government. This study offers valuable insights into the selection mechanism of PTR operators by comparing the OC of the government and the private firms. Previous studies conducted few research on the PTR operation. Considering a combination of the toll charge, road capacity, and guarantee, this paper analyses the operation problem through an intermediate variable. If the government sets the right intermediate variable in the bidding document, private firms will be willing to operate the road actively with the profit maximization. The first-best and second-best optimal solutions for the private firm and the win-win solution of the private operation were investigated. The average social cost and the toll charged by operators were compared; in some cases, the toll charged by the government is higher than private firms. The case that toll collected by the private firm from the road users is lower than that of the government will happen. Analytical results show that the social optimal pricing set by the private firm is less than that of the government, and then two approaches are proposed to solve the problem given a biased decision of the intermediate variable. The first approach sets the intermediate variable equal to the governmental OC without guarantee; the second one is that the government provides profit guarantee for the private firm.To optimize the succeeding operation of different kinds of facilities developed in the BOT mode after the concession period expires, future research can be carried out under information asymmetry to investigate whether and how to charge the toll and to set regulations to be followed by the private firms when operating these facilities. Even if the government is in a position to build/rebuild and operate its transportation facilities, it is still necessary for it to seriously consider how to operate them in a sustainable way to maximize social welfare.



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*Source: 6652278-2021-01-18.xml*