Price, capacity and concession period decisions of Pareto-efficient BOT contracts with demand uncertainty

Price, capacity and concession period decisions of Pareto-efficient BOT contracts with demand uncertainty

Transportation Research Part E 53 (2013) 1–14 Contents lists available at SciVerse ScienceDirect Transportation Research Part E journal homepage: ww...

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Transportation Research Part E 53 (2013) 1–14

Contents lists available at SciVerse ScienceDirect

Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Price, capacity and concession period decisions of Pareto-efficient BOT contracts with demand uncertainty Baozhuang Niu a, Jie Zhang b,⇑ a b

Lingnan College, Sun Yat-Sen University, Guangzhou 510275, China International School, Guangdong University of Business Studies, Guangzhou 510320, China

a r t i c l e

i n f o

Article history: Received 30 May 2012 Received in revised form 4 December 2012 Accepted 19 January 2013

Keywords: BOT contract Demand uncertainty Infrastructure privatization Economic efficiency

a b s t r a c t In this paper, we study the impact of demand uncertainty on the build-operate-transfer (BOT) contract design by optimizing a bi-objective problem via three critical decisions: toll, capacity and concession period. We derive the optimums and identify the public and private sector’s economic incentives. We find that the optimal length of concession period and the service quality of the infrastructure depend on the two parties’ operational costs and negotiation powers. Under mild conditions, we prove that the government will build a larger capacity but charge less than the private sector. Furthermore, the efficiency of BOT contract is improved with demand uncertainty. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Since 1980s, the private sector has been increasingly involved in logistics infrastructure (e.g., ports, airports, dams, highways and mass transportation projects) construction and operations, especially in those developing countries and regions. The main reason for privatization is to solve the shortage of capital and limited access to technology. In the recent years, it is reported that many governments, such as Philippines, Malaysia and Thailand, have been on the way of privatizing some or all of their major shipping ports (Tongzon, 2006) and a number of countries in Africa, South America and Asia are also privatizing their airports (Oum et al., 2004). Hong Kong is among the pioneers of privatization: its Cross-Harbour Tunnel was conceptualized in the 1960s and built in the 1970s by a private developer (Tam, 1999). Even in the developed countries, there is an increasing trend of logistics infrastructure privatization due to the regulation of public investments. For example, in Canada, any port investment over $5 million has to be approved by the national port corporation (Slack et al., 1996). In US, only some states and local governments allocate appropriate money from their budgets to support port constructions (Ybarra, 2009). Facing these issues, many local governments decide to partner with the third party investors to help finance the projects. For example, the Maryland Port Authority would like to invite a private investor to fund a new 50-foot berth and increase the capacity of Seagirt Marine Terminal’s waterborne containers (Ybarra, 2009). The Transport Canada also signed a contract with a private sector to develop and operate a terminal at Toronto Pearson Airport (Padova, 2007). The private sectors generally participate the projects under a build-operate-transfer (BOT) contract. The private firm will receive the concession from the public sector (government) to finance, design, construct, and operate a logistics infrastructure for a certain period of time. After that, the infrastructure will be handed over to the host government. In the concession period, the private sector receives the revenue of charged tolls (Tam, 1999). The sequence of BOT contract is illustrated in Fig. 1. ⇑ Corresponding author. Tel.: +86 20 84094677. E-mail addresses: [email protected] (B. Niu), [email protected] (J. Zhang). 1366-5545/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tre.2013.01.012

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Private sector is authorized to build the infrastructure at a construction cost

Private sector transfers the infrastructure to the government End of the infrastructure´s life

Private sector operates the infrastructure at an operational cost Private sector period

The government operates the infrastructure at another operational cost Public sector (government) period

Fig. 1. Sequence of BOT contract.

There are many successful examples of BOT practice. For instance, in North Africa, Morocco awarded a BOT contract to a private consortium for the development of a new container facility at the port of Tangiers (Sommer, 1999). In India, the Jawaharlal Nehru Port Trust (JNPT) plans to develop a marine chemical terminal at an estimated cost of $335 million on a BOT basis. JNPT has also approved a $34 million expansion plan and invited a private Australian company to build a new container terminal under a BOT contract (Bajpai and Shastri, 1999). In Canada, the terminal 3 of Toronto Pearson Airport was constructed by a private sector on a BOT contract, which costs $433 million and was completed in 1991 (Qiu and Wang, 2011). The advantages of BOT contract can be summarized as follows (Tam, 1999; Lin, 2003): (1) the private sectors do not have to ‘‘justify their investments to the public’’, and hence can invest in a project with more flexibility than the government; (2) the private sectors are self-interest-seeking, so they tend to quickly fulfill the goals of the project and start serving the customers and collecting revenues; (3) the private sectors are normally more efficient than the government in constructing and operating the project. It is reported that, with private sectors, the operational cost is reduced, and the port productivity is increased between 15% and 20% (Peters, 1995); and (4) the host government need not be constrained by the budget but still can serve the public with the logistics infrastructures. However, infrastructure privatization also raises an important issue on pricing and capacity decisions. On the capacity aspect, private sectors usually build smaller capacity to save the sunk cost. It is frequently reported that long waiting time is suffered by the customers due to inadequate infrastructure capacities. On the price aspect, the private sectors may charge a very high toll which eventually hurts the whole industry. For example, Yantian International Container Terminals Ltd. (mainly operated by Hong Kong Hutchison, a private sector; see Barling, 2005) has charged container trailer drivers additional ‘‘gate fee’’ since 1998 (Han, 2010). In the airport industry, privatized airports may increase aeronautical charges to maximize their profits (Bilotkach et al., 2012). In general, the goal for the government is to maximize the social welfare (the sum of surplus of all the participants of the infrastructure, i.e., the customers and the infrastructure’s operators) while the private sectors only care about their own profits. Thus, how to coordinate with the private sector on pricing and capacity decisions under a BOT contract is important. Besides, the government need consider the length of the concession period when designing a BOT contract. Scholars have only recently started to study the concession period issues. On one hand, a private sector aims to maximize its profit in the concession period, so it tends to set the concession period as long as possible. On the other hand, the government aims to maximize the social welfare in the whole lifetime of an infrastructure, so a long concession period may hurt the general social welfare. Then, a natural question is how to set the concession period appropriately, when a government has to rely on a private sector to build an infrastructure? Intuitively, a shorter concession period may result in a higher social welfare. However, setting a very short concession period may lead to the private sector’s short-sighted behavior, who may build a small capacity but charge a very high toll to collect the profit as soon as possible. Therefore, there exist both conflicts and coordination chances for the government and the private sector to determine the optimal concession period under a BOT contract. Being aware of these issues, in this paper, we model the problem of BOT contract design by considering the decisions on the length of the concession period, the capacity and the toll charges. Similar approaches have been found in some recent literature, such as Guo and Yang (2009a), Tan et al. (2010), and Tan and Yang (2012a,b). We note that the previous literature addresses the BOT contract design issues mainly based on deterministic demand models. However, in practice, the investment need to be done based on the forecasted demand, which is often inaccurate and has large uncertainties. The mismatch between logistics infrastructure capacity and the demand has always been a significant problem confronted by both the private sectors and the local governments. For example, the American Association of Port Authorities (AAPA) find that 35 of the 85 ports in US have to invest in infrastructure improvements between 2007 and 2011 (Gilroy, 2009) due to demand fluctuations. Another survey in airline industry also pointed out that the quick growth in air travel has made the current airport capacity inadequate in North America (Padova, 2007). There are also some numerical and empirical studies pointing out that demand uncertainty strongly influences the decisions and outcomes in the contract

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design of an infrastructure and hence should not be ignored (Chen and Subprasom, 2007; Albalate and Bel, 2009; Tan and Yang, 2012b). In this paper, we consider demand uncertainty and try to answer the following research questions: (1) how should the infrastructure operators design a Pareto-efficient BOT contract via the decisions on the concession period, the capacity and the toll charges, in the presence of demand uncertainty? (2) what is the impact of demand uncertainty on the contract decisions and the optimal outcomes? (3) how is the BOT contract’s performance? To answer these three research questions, we propose a model with several non-restrictive assumptions which are widely adopted in operations literature. We adopt a general price-sensitive demand model in multiplicative form. After discussing the basic properties of our model and providing some preliminary results, we start with two extreme cases in which the private sector or the government dominates the decision-making process. We obtain close-formed analytical results and completely characterize their equilibrium solutions. In particular, we find that the private sector’s profit is increasing in the length of the concession period, and thus it is willing to build and operate the infrastructure till the end of the infrastructure’s life. However, from the government’s aspect, we find that if its operational cost of the infrastructure is lower than that of the private sector, it would not privatize the infrastructure. Assuming that the decisions of the government and the private sector could be coordinated according to their relative negotiation powers, we generalize the model and further prove the quasi-convexity of the two parties’ coordinated profit with respect to the length of the concession period. We provide a general condition to determine the optimal length of the concession period, where the tradeoff between the long-run discounted gains and the immediate capacity-building costs hold the key. These results differ our work from Tan et al. (2010), in which setting the concession period to be the whole lifetime of an infrastructure arises as the Pareto-optimal choice of both the private sector and the government. The policy implication behind this result is, it is not always the best choice for the government to delegate the infrastructure to a private sector throughout the lifetime of the infrastructure. The government should balance their budget, long-run gains and operational efficiency to make its decision. We also introduce the inflation rate into our model and show that, when a great inflation in the future may arise, the decision makers tend to build a small capacity and charge a high toll. As a result, the service quality of the infrastructure will be lowered. Towards the optimal decisions of the capacity and the toll, there is a classic result that the government tends to build a larger capacity but charge a lower toll than the private sector, with deterministic demand (Guo and Yang, 2009a; Tan et al., 2010). We verify these results in the presence of demand uncertainty under mild conditions. We also show that the government and the private sector are capable of coordinating with each other in the concession period, and the optimal decisions form a Pareto-optimal frontier. Interestingly, we find that the volume–capacity ratio is increasing in the relative negotiation power of the private sector, which is a very different result from Tan et al. (2010). It implies that a powerful government may help improve the service quality of an infrastructure (a small volume–capacity ratio stands for a high service quality, see Xiao et al., 2007 and Tan et al., 2010). Finally, we examine the economic efficiency of BOT contracts and the impact of the demand uncertainty on the optimal decisions and outcomes. We observe that with demand uncertainty the efficiency of BOT contracts is generally improved, compared to Tan et al.’s (2010) results. By adopting BOT contracts, the total social welfare is inevitably reduced, but the reduction is not too large and thus may be acceptable. BOT contracts are still much better than completely privatized schemes for the government since the social welfare is increased with the government’s influence, which provides supports to the recent trend of project privatization via BOT contracts. We further demonstrate that with larger demand variance (larger average demand size), both the parties’ profits will decrease (increase). When the potential demand is large, investors tend to build a large capacity, regardless of the possible demand risk. Towards the optimal toll, we observe that it will increase (decrease) along with larger demand variance (larger average demand size). The rest of this paper is organized as follows: Section 2 reviews the related literature. Section 3 presents our model settings, assumptions and preliminary results. Section 4 studies the optimization problem of designing a BOT contract. The optimal decisions on the capacity, the toll and the length of the concession period are derived and analyzed. Section 5 compensates the analytical findings by numerical experiments. We conclude the paper in Section 6. All the proofs are placed in the Appendix.

2. Literature review Literature on infrastructure investment and management under BOT contracts is closely related to our research. Yang and Meng (2000) study a road network with elastic demand and BOT contracts, and investigate the private sector’s profit and total social welfare under several capacity-price combinations. Tsai and Chu (2003) examine the impact of a BOT contract on the traffic flows, traveling costs, toll, capacity and social welfare and then conduct a simulation-based research. Subprasom and Chen (2007) provide a bi-level programming model in which the low level program determines the route choice of the customers for a given price-capacity constraint and the upper level program measures the private sector’s profit or the government’s social welfare. Chen and Subprasom (2007) further conduct a numerical study using real data from the case of the Ban Pong-Kanchananburi Motorway in Thailand. Verhoef (2007) discusses the design of auctions for road concessions to make the private sector behave more closely to the price and capacity settings which guarantee the social welfare

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maximization. Ubbels and Verhoef (2008) analyze how the government organizes the bidding process among private sectors (they decide on their capacities and prices) to obtain the maximum welfare gains. Guo and Yang (2009a) study the joint decisions of the optimal concession period, the road capacity and the charged toll under social-welfare-maximizing BOT contracts. They consider bilateral negotiation and competitive auction methods to reach the optimal BOT contracts. Bel and Fageda (2010) focus on the airport’s pricing behavior, and use data from 100 European airports to examine the impact of privatization, traffic volume, airline negotiation power, airport competition and so on. Bilotkach et al. (2012) investigate the impacts of privatization and government regulations on the aeronautical charges by employing the panel data of 61 European airports over 18 years. Qiu and Wang (2011) assume demand is increasing in quality and decreasing in price, and then study the incentives and efficiency of BOT contracts. Tan and Yang (2012a) investigate the Pareto-efficient BOT contracts by assuming heterogenous road users with different value of time (VOT). They find that the optimal volume–capacity ratio (measured by demand/capacity) significantly depends on the curvature of the mean residual VOT function. Specifically, Tan et al. (2010) adopt a bi-objective programming approach to study the Pareto-optimal frontier of the social welfare and the private sector’s profit. The length of the concession period, the traveling demand and the road capacity are the three decision variables. The price is assumed to be a function of the demand, which is deterministic. They show that the length of the concession period should be equal to the whole life of the road, and the ratio of demand versus capacity is identical to the social optimum given certain conditions. Following this stream of literature, researchers have developed different models and solving algorithms. The traveling demand are always assumed to be deterministic, and can be a function of price (e.g., Tsai and Chu, 2003; Ubbels and Verhoef, 2008; Tan et al., 2010; Qiu and Wang, 2011; Tan and Yang, 2012a) or the network equilibrium problem with variable demand which can be predicted (e.g., Sheffi, 1985; Yang and Meng, 2000; Subprasom and Chen, 2007; Guo and Yang, 2009a). In a recent work, Tan and Yang (2012b) incorporate demand uncertainty into BOT contracts but assume the demand uncertainty will be resolved after the construction of the infrastructure. In contrast, we assume the demand uncertainty cannot be resolved and is always present during the whole lifetime of the infrastructure. This appears new and realistic. We note that Guo and Niu (2010) generally discuss a similar idea in a technical report, but in this paper we go further by studying the endogenous concession period issue and considering the discounted profits of the government and the private sector with potential future inflations. We further explore the volume–capacity ratio under the optimal decisions and find that this ratio varies when the government has different negotiation powers over the private sector, which differs our work from the previous literature, such as Tan et al. (2010) and Tan and Yang (2012a). Operations literature on price-sensitive demand and capacity investment is also related. There have been substantial research papers in this field, see the review by Petruzzi and Dada (1999) and references therein for further discussion. In particular, Chen et al. (2004) consider a joint pricing/inventory model and show the existence and uniqueness of Nash equilibrium. Wu et al. (2007) study the retailer and manufacturers’ pricing and fill rate (capacity/demand) decisions in a Bertrand competition environment. de Véricourt and Lobo (2009) develop a multi-period dynamic programming model to study the pricing and capacity decision in a nonprofit organization. Kocabiyikog˘lu and Popescu (2011) provide some conditions such that a newsvendor model is jointly concave in price and capacity. Xu et al. (2011) study three applications of the price-sensitive newsvendor models. Under mild assumptions they show the unimodality of the profit functions.

3. Model setting and preliminaries We now present our model setting as well as some assumptions and preliminary results for further analysis. In the following sections, we denote the parameters and decision variables of the private sector and the government by subscripts s and g respectively.

3.1. Model setting As described in Section 1, the planning horizon (the lifetime of the infrastructure) can be divided into two periods: the concession period and the public sector period. The government cares about the social welfare, defined as ‘‘the sum of consumers’ and producers’ surplus’’ (Yang and Meng, 2000), while the private sector cares about its own profit. Under a BOT contract, the private sector builds the infrastructure with a unit construction cost ck, and the total capacity to be built is y P 0. We assume all of the capacity building cost cky occurs at the beginning of the planning horizon and is sunk. The private sector then operates the infrastructure with a unit cost cs, and it obtains returns by charging a toll denoted by ps. After the infrastructure is transferred, the government operates the infrastructure at a unit cost cg and makes the pricing decision b denote the lifetime of the to maximize the social welfare. Let pg denote the toll in the public sector period and T infrastructure. During the planning horizon, customers arrive continuously with price-sensitive and uncertain demand size of each arrival. Given the toll p, we assume the demand size per unit time is D(p) = d(p)e, decreasing and twice differentiable in p. The random factor e is continuous, i.i.d. among different arrivals and having price-independent cumulative density function (cdf) U() and probability density function (pdf) /(). It can be viewed as a measurement of demand risk, or sales driver, which cannot be perfectly controlled (Kocabiyikog˘lu and Popescu, 2011). Note that the life of an infrastructure is often very long,

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so the discount on the future profit and/or the social welfare due to possible inflations may be critical in the analysis and should not be ignored. For simplicity, we assume a system with continuous timing and a inflation rate c P 0. Consider the private sector’s decisions first. In the concession period, the net revenue from one demand arrival is (ps  cs) min{D(ps), y}, where min{D(ps), y} is the satisfiable part of the demand constrained by the capacity. To calculate the long-run profits, we take the expectation E[psmin{D(ps), y}  csy], and hence, it arises as a general price-sensitive newsvendor problem. Considering the private sector’s expected return when it invests, we assume the inflation rate works following the form ect, which will be multiplied to the expected profits. Similar setting has been widely used in the literature, see Chen and SimchiLevi (2006) and the reference therein for further discussion. Thus, the expected total profit of the private sector during the concession period is

Ps ðT; ps ; yÞ ¼ ck y þ  ¼

Z

T

E½ps min fDðps Þ; yg  cs yect dt

0

ð1  ecT Þðps  cs Þ

c

 Z ð1  ecT Þps  ck y 

c

0

y=dðps Þ

½y  dðps Þv /ðv Þdv :

ð1Þ

The first part of Eq. (1) captures the revenues when the capacity is fully utilized, while the second part captures the revenue loss due to inadequate demand and the ‘‘wasted’’ capacity y  d(ps)v. We then consider the government’s decisions. The government’s objective is to maximize the expected total social welfare, which includes the customers’ surplus, the private sector’s profit in the concession period and its own profit in the pub   R bT  lic sector period, T E pg min Dðpg Þ; y  cg y ect dt. Customers coming to this infrastructure receive a value of service R(p), an increasing function of p, and their surplus of using this service is R(p)  p. In practice, high price usually signals high-end service, high quality and/or prestigious brand. Many marketing papers have illustrated this phenomenon, e.g., Bagwell and Riordan (1991) and Voss et al. (1998). Furthermore, the economic and marketing literature also points out that the increasing speed of R(p) will become slower as p increases, i.e., R(p) is concave in p. Without loss of generality, we assume consumers’   surplus R(p)  p > 0 for any feasible price p. Then, the expected customers’ surplus during the whole planning horizon is   RT R bT  E 0 ðRðps Þ  ps Þ min fDðps Þ; ygect dt þ T Rðpg Þ  pg min Dðpg Þ; y ect dt and the expected total social welfare is

Pg ðT; ps ; pg ; yÞ ¼ ck y þ

Z

T

0 cT

¼

ð1  e

c

E½Rðps Þ min fDðps Þ; yg  cs yect dt þ ÞRðps Þ

Z

y=dðps Þ

½y  dðps Þv /ðv Þdv 

0

Z bT     E Rðpg Þ min Dðpg Þ; y  cg y ect dt

ðe

T

cT

Z b  ec T ÞRðpg Þ

c

2 3 b ð1  ecT ÞðRðps Þ  cs Þ þ ðecT  ec T ÞðRðpg Þ  cg Þ 4 þ  ck 5y:

c

y=dðpg Þ

½y  dðpg Þv /ðv Þdv

0

ð2Þ

3.2. Assumptions To facilitate our analysis, we propose several non-restrictive assumptions. First, we assume that the random factor e has an increasing failure rate distribution (IFR). This assumption has been widely used in the operations literature, and can be satisfied by most of the commonly used probability distributions, including uniform, exponential, normal, truncated normal, log-normal as well as the Gamma and Weibull distributions, subject to some parameter restrictions (Bagnoli and Bergstrom, 2005; Lariviere, 2006). Furthermore, U() and /() will not change after the infrastructure is transferred to the government. Similar to Kocabiyikog˘lu and Popescu (2011), we also assume that the riskless unconstrained revenue pd(p) and social welfare R(p)d(p) are strictly concave in p. If we denote partial derivatives by the superscripts, the foregoing assumption corresponds to 0

00

2d ðpÞ þ pd ðpÞ < 0;

0

00

R00 ðpÞdðpÞ þ 2R0 ðpÞd ðpÞ þ RðpÞd ðpÞ < 0:

This assumption is not restrictive and can be satisfied by commonly used demand functions. We further assume R(p)/ p > R0 (p), i.e., R(p)/p decreases in p, which is also general and can be satisfied by a large number of increasing concave functions such as polynomial functions and log functions. 3.3. Bi-objective problem Note that the private sector and the government have distinct objectives in a BOT contract. Therefore, how to coordinate their objectives and select an appropriate combination of (T, p, y) = (T, ps, pg, y) is the major task and difficulty in BOT contract design. According to Tan et al. (2010), this problem can be formulated as a bi-objective optimization problem as follows:

Max

ðT;p;yÞ2X



Ps ðT; p; yÞ Pg ðT; p; yÞ



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where X is the feasible space guaranteeing Ps(T, p, y) and Pg(T, p, y) are both positive. Then, the two parties negotiate to achieve a Pareto-efficient contract, under which neither of them could be better off without hurting the other one’s profit. Similar definition has been also used in Guo and Yang (2009b) and Tan and Yang (2012a,b). We then propose a method to coordinate the private sector and the government under the BOT contract. Miettinen (1999) have pointed out that the convex multi-objective minimization problem can be transformed to a single objective by using the weighting method, i.e., minimizing the weighted sum of the objective functions. The weight reflects the private sector or the government’s relative power in the negotiation. Following this result, we transform our bi-objective maximization problem into a single-objective maximization problem:

Max Pw ðT; p; yÞ ¼ aPs ðT; p; yÞ þ ð1  aÞPg ðT; p; yÞ;

ðT;p;yÞ2X

ð3Þ

where a 2 [0, 1] represents the private sector’s relative negotiation power towards the government. The extreme case a = 1 (a = 0) means the private sector (the government) takes full control over the project and dominates the decision-making process. 4. Characterization of the optimal BOT contract To design a BOT contract, three critical parameters need to be considered: the length of the concession period, the infrastructure’s capacity and the toll. In this section, we aim to derive and characterize these three optimal decisions, and then we can correspondingly analyze the service quality level of a BOT contract and the economic efficiency of a BOT contract over the privatized project. We first analyze two extreme cases as the benchmarks: (1) the private sector dominates the decision-making process and (2) the government dominates the decision-making process. Then, we analyze the more general case where the government and the private sector coordinate with each other according to their negotiation powers. The decision sequence are as follows throughout all the three cases: (1) The optimal length of the concession period T is negotiated and determined. (2) The capacity of the infrastructure is determined. (3) The holding party/parties determine the optimal toll. In practice, this sequential price-capacity interaction has been widely adopted. For example, the first stage of the BOT contract for Hong Kong Cross-Harbour Tunnel was to decide the construction time. After the tunnel was finished, the private sector started the tunnel management and set the charged toll (Tam, 1999). We solve the problem by backward induction. To facilitate the following analysis, we first study the government’s decision on the optimal toll in the public sector period b  T, denoted by pg(y), a function of y. T Lemma 1. Given y > 0, there exists a unique optimal toll pg(y) which maximizes Pg(ps, pg, y). Both pg(y) and the volume–capacity ratio d(pg(y))/y are decreasing in y. Lemma 1 indicates that there is a one-to-one relationship between the optimal toll in the public sector period and the capacity of the infrastructure. It would be easy to understand pg(y) is decreasing in y, since a high supply (y in our context) often yields a low selling price (pg(y) in our context). The volume–capacity ratio d(pg(y))/y, defined as the ratio between the mean demand and the capacity, refers to a measurement of the service quality of an infrastructure (Xiao et al., 2007; Tan et al., 2010). We show that a larger capacity of the infrastructure results in a better service quality. The underlying logic is intuitive: a large volume–capacity ratio indicates a tight capacity of the infrastructure, which therefore makes customers wait and experience poor service quality. By Lemma 1, we can simplify the expression of the expected social welfare by letting Pg(ps, y) = Pg(ps, pg(y), y). We then ignore the subscript of ps if the discussion is limited in the concession period. 4.1. Analysis of two extreme cases If the private sector dominates the contract, i.e., a = 1 in the bi-objective problem Eq. (3), the private sector solely determines T, y and p. On the other hand, if the government dominates the contract, i.e., a = 0, the government makes all the three decisions. For these two extreme cases, we have the following proposition. Proposition 1. Suppose that either the private sector or the government dominates the contract. 1. For any given y > 0, there exists a unique optimal price pi(y) maximizing Pi(p, y), i = s, g. Both pi(y) and d(pi(y))/y decrease in y. Furthermore, ps(y) > pg(y). 2. There exists a unique optimal capacity yi maximizing Pi(pi(y), y), i = s,g.

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Proposition 1 provides close-formed solutions for the two extreme cases respectively. It also reveals some properties of these optimal solutions, explained as follows. 1. The uniqueness of the optimal toll and capacity is guaranteed. As Cachon and Netessine (2004) declared, ‘‘if there is only one equilibrium, then you can characterize the actions in that equilibrium and claim with some confidence that those actions should indeed be observed in practice’’, thus our results have practical and applicable meanings as well as theoretic ones. 2. A classic result that the private sector charges a higher toll than the government, is verified with random demand in our model. Similar result has been derived by Tan et al. (2010) and Tan and Yang (2012a), with deterministic demand. Here we extend their results to a more general environment with demand uncertainty. 3. Again, similar to Lemma 1, a larger infrastructure capacity results in a lower toll and a higher service quality level. The underlying incentives are also similar and hence omitted. Next, we solve the optimal concession period issues and have the following proposition. b , while the government’s Proposition 2. The private sector’s optimal decision on the length of the concession period is to set T s ¼ T b ; otherwise, Tg = 0. optimal decision depends on the two parties’ operational costs: if cg P cs, T g ¼ T Consistent to traditional wisdom, the private sector wants to operate the infrastructure as long as possible, since the private sector can guarantee its positive profit margin by choosing appropriate capacity and toll. As long as the private sector has a positive profit margin, it is willing to extend the concession period. In practice, many private sectors have successfully persuaded the government to sign a very long concession period. For example, the private sector is awarded a concession period of 99 years in the Highway 407 project in Toronto, Canada. Such decisions are mostly seen in Asia, where shortage of capital and limited access to technology generally exist (Tongzon, 2006). When the government dominates the contract, we find an interesting result that the government’s decision on T actually depends on the operational cost difference between the private sector and the government. When the government has a lower operational cost, i.e., cg < cs, the government should build and operate the infrastructure by itself, since it is more efficient to b always holds without do so. This result differs our work from Guo and Yang (2009a) and Tan et al. (2010), in which T g ¼ T the consideration of the operational costs. Corollary 1. The private sector’s optimal capacity ys is increasing in T. The incentives that explained by Guo and Yang (2009a) and Tan et al. (2010) can also be applied to our model. On one hand, in Corollary 1, we show that the optimal infrastructure capacity determined by the private sector (i.e., ys ) is increasing in T. Therefore, setting a long concession period inspires the private sector to build a large capacity at the beginning of the project, and may increase the social welfare. On the other hand, a long concession period obliviously reduces the public sector period, in which the government operates the infrastructure. This may result in a reduction of the social welfare. Besides the aforementioned tradeoff, we find that there is another tradeoff between the profit margins in the two periods, due to the difference between the operational costs of the government and the private sector. These two tradeoffs work together and yield the result on the optimal T when the government dominates the contract. If we ignore the operational costs and assume b is always the Pareto-optimal decision of the government and the private sector, which is consistent cg = cs = 0, then T g ¼ T with the findings of Guo and Yang (2009a) and Tan et al. (2010). Note that, if T = 0, there is no BOT contract and the government will build and operate the infrastructure by itself, or the government has to provide sufficient subsidies to the private sector to hold the contract. In other words, the government should never choose a private sector with too high operational cost as its partner in the building of an infrastructure. We further notice that the inflation rate c is an important parameter that can influence the private sector’s decision, and then we conduct sensitive analysis with respect to c. Corollary 2. For a privatized infrastructure, the optimal capacity ys decreases in c, the optimal toll ps increases in c, and the volume–capacity ratio dðps Þ=ys increases in c. If the private sector expects that there will be a great inflation in the future, i.e., c is large, it will build a small capacity. The logic is intuitive: since future profit will greatly depreciate with a large c, the return on the investment to the infrastructure becomes low. Thus, the private sector tends to charge a high toll to cover its investment and hence, reduces the service quality of the infrastructure to improve the efficiency. 4.2. Analysis of coordinated BOT contract Having characterized the two different extreme cases, we now consider a more general case in which the government and the private sector coordinate with each other under a BOT contract. They make the decisions of T, y and p, based on their relative negotiation powers, which can be estimated via some transactional or financial data. For example, Chen et al. (2008) design an estimation model to find two parties’ negotiation powers using data from American automobile industry. In this subsection, we denote the coordinated decisions by subscript w and we focus on the results with a 2 (0, 1). Recall that, the transformed objective function stated in Section 3.3 is

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Pw ðp; yÞ ¼ aPs ðp; yÞ þ ð1  aÞPg ðp; yÞ; which is a linear combination of Ps(p, y) and Pg(p, y). Therefore, it is not difficult for us to guarantee the uniqueness of the solutions, if Ps(p, y) and Pg(p, y) are both concave. Similar to the previous analysis, we characterize the coordinated decisions under a BOT contract in the following proposition. Proposition 3. Under a BOT contract, 1. for any given y > 0, there exists a unique pw(y) which maximizes Pw(p, y); pw(y) decreases in y but increases in a, and d(pw(y))/y decreases in both y and a; 2. both Pw(pw(y), y) and Pg(pw(y), y) are concave in y. Thus, there exists a unique pair of optimal decisions ðpw ; yw Þ which solves the BOT contract. Proposition 3 shows that the main results of the two extreme cases in Proposition 1 can be generalized to the case under a BOT contract. We also conduct sensitive analysis with respect to the negotiation power a. Recall that we have shown that ps(y) > pg(y) in Proposition 1. Thus, as the negotiation power of the private sector becomes higher, the coordinated toll becomes more favorable to the private sector and hence higher. Then, for a given y > 0, d(pw(y))/y decreases in a, because a large a indicates a higher toll which reduces the number of arriving customers and improves the service quality. Therefore, for a given infrastructure capacity, a higher negotiation power of the private sector leads to a higher service quality level. Now we move to analyze the Pareto-efficient decision on the length of the concession period under a BOT contract. This decision, denoted by Tw, obviously depends on the private sector’s negotiation power a and the difference between the social welfare in the concession period and the public sector period. We show that, the government should either authorize the b , or operate the infrastructure by itself, i.e., Tw = 0. The main reason is, with whole period to the private sector, i.e., T w ¼ T demand uncertainty, the total profit under a Pareto-efficient BOT is quasi-convex in the concession period T. Define







b mg ¼ Rðp ðy ÞÞE minfdðp ðy ÞÞe; y g  cg y P g g w w w w b ms ¼ P b sm P





Rðpw ÞE

minfdðpw Þ

; yw g





 T¼0

;

cs yw b; T¼ T

e    ¼ pw E minfdðpw Þe; yw g  cs yw b: T¼ T 

b sm and P b ms represent the profit margins of the private sector and the government, respectively, when the Given a 2 (0, 1), P b mg represents the government’s profit margin private sector is authorized to operate the infrastructure in the whole period. P when the government builds and operates the infrastructure by itself. The detail characterization of Tw are summarized in the following proposition. Proposition 4. Under a Pareto-efficient BOT contract, Pw ðpw ; yw Þ is convex in T. The optimal concession period is determined by the following rule: if

h i i b h b  b  b b ð1  ec T Þ a P sm þ ð1  aÞ P ms  ð1  aÞ P mg > c k yw ð T Þ  yw ð0Þ ; b ; otherwise, Tw = 0. then T w ¼ T By examining the proofs, we find that the three parameters cg, cs and a hold the key. When the government is less efficient b sustains as the Pareto-optimal decision of the two parties, regardless of the negotiain operations, i.e., cg P cs, then T w ¼ T tion power. However, when the government is more efficient, then the private sector and the government have different decisions on T and the coordinated one depends on a. Our numerical experiment further demonstrates that there exists a b, threshold value of a, above which the coordinated decision will be more favorable to the private sector and thus T w ¼ T and below which Tw = 0. As a result, yw is not continuous in a, as shown in Fig. 2. We then take a close look at the condition in Proposition 4:

2

3

2

3

6 ^ 6  ^ 7  ^ ^ 7 ð1  e Þ 4aP sm þ ð1  aÞPms  ð1  aÞPmg 5 > c k 4yw ðTÞ  yw ð0Þ5: |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} cT^

ð1Þ

ð2Þ

ð3Þ

Item (1) represents the profit discount due to the inflation rate, which plays an important role because investing to an infrastructure is usually a long-run strategic decision. Item (2) can be rewritten as

b sm þ ð1  aÞ P b ms   ½a  0 þ ð1  aÞ P b mg ; ½a P b and the second part represents that when Tw = 0. in which the first part represents the weighted total welfare when T w ¼ T When the government builds and operates the infrastructure by itself, the private sector’s profit margin is zero. Thus, item b and 0. Then, item (1) mul(2) acts as the difference of the weighted total welfare between the two extreme values of Tw, T tiplied by item (2) represents the discounted welfare gains by authorizing the private sector to operate an infrastructure throughout the lifetime of the infrastructure. Item (3) multiplied by ck represents the difference of the total capacity-building

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Fig. 2. Optimal capacity with optimal concession period when cg < cs.

cost between the two options. Combining the foregoing analysis, we find that the tradeoff between the long-run discounted gains and the immediate capacity-building costs determines the optimal concession period under a BOT contract and the b and Tw = 0. decision maker just compares the weighted welfare when T w ¼ T In practice, the government often tends to find an efficient private sector to participate, so the concession period is more b , we are able to obtain some more likely to be authorized to be the whole life of the infrastructure. Providing that T w ¼ T properties on the optimal decisions on the capacity and the toll. Corollary 3. Under a Pareto-efficient BOT contract, 1. the optimal capacity yw decreases in c, the optimal toll pw increases in c and the volume–capacity ratio dðpw Þ=yw increases in c; 2. if R(pg(y)) > ps(y) for any y > 0, then yg > ys and pg < ps . Here the first part generalizes Corollary 2 to a coordinated BOT contract case, showing that a large inflation rate always reduces either the private sector, or the government’s incentive to build a large infrastructure capacity. Otherwise, the return on the investment cannot be guaranteed. The toll will be thus raised to cover the sunk cost, and the service quality of the infrastructure becomes poor. Neither a small capacity nor a high toll would benefit the customers. The second part of Corollary 3 shows that the government tends to build a larger capacity and charge a lower toll if it obtains a higher (social) profit than the private sector for any given capacity level. The condition is not restrictive and can be satisfied with various demand functions d(p) and service value functions R(p). We provide a corollary for the linear demand function as follows. Corollary 4. Assume d(p) = a  bp > 0, a > 0, b > 0. In this case, the government always builds a larger capacity and charges a lower toll than the private sector. The assumption that D(p) = (a  bp)e has been widely used in operations and marketing literature (Petruzzi and Dada, 1999; Huang et al., 2012). Under this mild assumption, we show the classic results that yg > ys and pg < ps hold with demand uncertainty. The corresponding insights demonstrated by Guo and Yang (2009a) and Tan et al. (2010) then apply to our model. Finally, we analyze the optimal volume–capacity dðpw Þ=yw with respect to a. With a deterministic demand, Tan et al. (2010) show that the volume–capacity ratio is a constant. However, with a random demand, we find a completely different result that this ratio is not a constant. In Proposition 3 we show that, for any given y, this ratio is decreasing in a. Interestingly, when the infrastructure capacity is endogenously determined, the reverse result holds: it is the government who leads to a higher service quality level since yg > ys indicates yw decreases in a. Fig. 3 illustrate this finding. As a becomes larger, the private sector becomes more powerful in a BOT contract and it tends to better utilize the capacity of the infrastructure, which lowers the service quality and sacrifices the customer satisfaction. On the other hand, when the government dominates the contract, it prefers a high service quality level of the infrastructure to help the customers, and then the utilization of the infrastructure will be low. 5. Numerical studies To obtain more insights about the optimal decisions with demand uncertainty, we conduct several numerical studies in this section. We test manypcombinations of parameters and functions and find similar results. As a typical example, we ffiffiffiffiffiffi choose d(p) = 5  p, RðpÞ ¼ 5p, and e following a truncated normal distribution N(l,r2) between [0, 1] for our illustrations.

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Fig. 3. Impact of a on the volume–capacity ratio.

5.1. The efficiency of BOT contracts We first obtain the set of all Pareto-optimal solutions by changing a from 0 to 1, referred as the Pareto-optimal frontier, see Fig. 4. The left part reflects the Pareto-efficient profits of the private sector and the government, and the right part represents the curve of the coordinated capacity and toll. As the negotiation power changes, all efficient negotiation outcomes of the pair ðpw ; yw Þ will be located on the curve. It is not surprising to find that the coordinated toll pw is increasing in a and the capacity yw is decreasing in a. We explain this finding in the following way: on one hand, as the private sector’s negotiation power becomes relatively larger, i.e., a increases, the coordinated decisions become more preferable to the private sector. The optimums thus go closer to the private sector’s optimal decisions (complete privatization); on the other hand, when a becomes small, the coordinated decisions become more preferable to the government, whose decision of capacity is larger and that of toll charge is lower. We then investigate the efficiency of a BOT contract by analyzing the ratio of the social welfare, defined as Pw/Pg. See Fig. 5 for the representative curve. Clearly, this ratio is decreasing in a, since the contract is more preferable to the private sector when it has larger negotiation power. It is worthy noting that this ratio has a relatively large lower bound (above 0.85), which is larger than that with deterministic demand (Tan et al. (2010) find a lower bound 0.75). Therefore, a BOT

Fig. 4. The Pareto optimal frontier.

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Fig. 5. Impact of a on the economic efficiency of a BOT contract.

contract does not significantly reduce the total social welfare, but simplifies the government’s operations. Our observation therefore indicates that demand uncertainty improves the economic efficiency of BOT contracts. Specifically, when a = 0.5, i.e., the government and the private sector are equally powerful, this ratio is above 0.95, also larger than that with deterministic demand, which is 8/9 (Tan et al., 2010). Therefore, with demand uncertainty, a BOT contract is generally more efficient to the government and hence more applicable in practice. 5.2. The impact of uncertainty By introducing demand uncertainty into the optimization problem of a BOT contract in the infrastructure building, we have obtained some properties of the optimal capacity and toll in the previous sections. Now, we want to further investigate how the magnitude of the uncertainty affects the optimal decisions and outcomes by varying either the mean l or the standard deviation r or both parameters. We let a = 0.5 in this subsection. First, by fixing l = 0.5 and varying r from 0.25 to 0.75, we have our results illustrated in Fig. 6. It is not surprising that both the government’s and the private sector’s profits are decreasing in r, due to the higher risk of insufficient capacity. To mitigate this risk, the resulting optimal capacity will be larger. We also observe higher toll charges, which reduces the customers’ utility and hence reduces the probability of demand loss. Second, by fixing r = 0.5 and varying l from 0.25 to 0.75 we have Fig. 7. As the mean of the demand becomes larger, both parties’ expected profits increase. The resulting optimal capacity will also be larger, in order to satisfy the larger customer demand. We also observe lower toll charges in this case, which attract more customers. Finally, we vary l and r simultaneously by keeping a constant coefficient of variation (CV) cv = r/l. We conduct two sets of numerical experiments for different levels of CV: cv = 0.5 (as shown in Fig. 8) and cv = 1.5 (as shown in Fig. 9). When the mean and the standard deviation of the demand randomness increase at the same speed, it is observed that parties’ profits as

Fig. 6. Impact of r on the optimal decisions and outcomes.

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Fig. 7. Impact of l on the optimal decisions and outcomes.

Fig. 8. The optimums with cv = 0.5.

Fig. 9. The optimums with cv = 1.5.

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well as the optimal capacity are still increasing. However, we find that, the optimal toll decreases as l and r increase with a constant cv. This indicates that the impact of l dominates the impact of r. A large CV weakens such a dominant effect: the increasing/decreasing rates are smaller and the curves in Fig. 9 are flatter than that in Fig. 8.

6. Concluding remarks BOT contracts are widely adopted in practice. When demand is deterministic, the properties of Pareto-efficient BOT contracts have been thoroughly studied in the previous literature. However, when the government and/or the private sector take charge of building and operating a logistics infrastructure, demand uncertainty may significantly influence their price, capacity and concession period decisions. We contribute to the literature by the following findings. First, we examine the impact of demand uncertainty by assuming a random and price-sensitive demand. We characterize the infrastructure investment and operations problem with the newsvendor model with price decisions. Note that the private sector and the government have conflicting objectives: while the private sector cares its profit in the concession period, the government cares the social welfare in the whole life time of the infrastructure. By assuming a specific but reasonable decision sequence on the three key variables (T first, then y, and p last), we solve the bi-objective optimization problem so as to coordinate the optimal decisions of the private sector and the government. In particular, we transform the bi-objective problem to a single-objective one, using the weighting method. We show that all the optimums can be analytically derived based on a few non-restrictive assumptions, such as increasing failure rate distribution of the random factor and concave unconstrained revenues. Second, we show the consistence and inconsistence of our results with respect to the optimal price, capacity and concession period to Tan et al. (2010). We find that, in most cases, setting the concession period to be the whole life of the infrastructure arises to be the Pareto-optimal decision of the government and the private sector. However, when the government is more efficient to operate the infrastructure than the private sector, infrastructure privatization may not occur. Although in practice the government is more likely to privatize the infrastructure due to some budget issues, our finding points out that it is not always the best choice. We contribute by proving the quasi-convexity of the coordinated profits of the two parties and providing the analytical conditions for their preference of concession period when facing random and price-sensitive demand. Third, we find that the optimal volume–capacity ratio, which indicates the inverse of service quality of an infrastructure, is increasing in the private sector’s relative negotiation power over the government. Thus, when the government holds larger power to influence the optimal decisions of the infrastructure, the customer satisfaction can be improved. This finding also differs our work from Tan et al. (2010), in which the volume–capacity ratio is proven to be a constant and equals to that at the social optimum. Besides, we study the impact of inflation rate on the contractors’ decisions, which appears new in this stream of literature. Finally, we conduct numerical experiments to further study the impact of demand uncertainty on the decisions under a BOT contract. We first analyze the economic efficiency ratio of a BOT contract, i.e., the government’s profit (total social welfare) under a BOT contract over that without privatization. We find that this ratio is decreasing in the private sector’s relative negotiation power, but still large enough, especially when a is small. Compared to Tan et al. (2010), we find the economic efficiency of a BOT contract is improved with demand uncertainty. Finally, we vary the parameters of random demand and examine the corresponding changes in the optimums. With a fixed coefficient of variation (CV), we find the impact of the mean dominates that of the standard deviation, but a large CV can reduce such dominant effect. By considering demand uncertainty, many research questions arise, since it is generally neglected in the previous literature. We discuss two possible future research directions here. First is the regulation designed by the government to control the private sector. There are various regulations such as rate-of-return regulation, price-cap regulation and capacity-regulation, see Tan et al. (2010) and Tan and Yang (2012a) for further discussion. Although they have been studied with deterministic demand previously, it would be interesting to examine their effectiveness and efficiency with demand uncertainty, through both analytical and empirical studies. Another interesting topic is to study the optimal BOT contract in a dynamic environment. Since the whole lifetime of the infrastructure is long, we can divide it into multiple sub-periods. In each subperiod, the operators may change their decisions on (T, p, y). In that case, a dynamic programming model will be more suitable, which deserves further investigation but is beyond the scope of this paper.

Acknowledgements The authors are grateful to Dr. Pengfei Guo from the Hong Kong Polytechnic University, the two anonymous reviewers, and the managing editor Prof. Wayne K. Talley, for their constructive suggestions on the earlier version of this paper. This project is supported in part by National Natural Science Foundation of China under Grant 71201175 and Guangdong Natural Science Foundation under Grants S2012040008081 and S2011040001069.

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