Analysis of two-tier public service systems under a government subsidy policy

Computers & Industrial Engineering 90 (2015) 146–157

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Analysis of two-tier public service systems under a government subsidy policy Wuhua Chen a, Zhe George Zhang b,c,⇑, Zhongsheng Hua d a

School of Management, University of Science and Technology of China, Hefei, Anhui 230036, China Beedie School of Business, Simon Fraser University, Burnaby, BC V5A 1S6, Canada c Department of Decision Sciences, Western Washington University, Bellingham, WA 98225, USA d School of Management, Zhejiang University, Hangzhou, Zhejiang 310058, China b

a r t i c l e

i n f o

Article history: Received 23 January 2015 Received in revised form 25 June 2015 Accepted 18 August 2015 Available online 24 August 2015 Keywords: Two-tier service system Subsidy policy Regulated price Nash equilibrium

a b s t r a c t Some public service systems such as healthcare systems consist of both free public service provider with a long wait time and paid private service provider with a short wait time. Such service systems are often called a two-tier service system. In general, more customers will choose the free service provider (SP). To reduce the congestion in the free system, the government may encourage customers to use the pay system by offering them a subsidy. This paper studies whether such a subsidy can reduce the free system’s waiting time and improves the social welfare. While the objective of the free system is to maximize its own total customers’ utility, the objective of the pay system is to maximize its profit. We develop a mixed duopoly game to analyze the Nash equilibrium for the competition between the free and toll systems. Two scenarios with unregulated and regulated prices are considered. When the pay system price is unregulated (the private SP can set prices freely), we find that if customers are less sensitive to the service delay, the subsidy policy can effectively reduce the expected waiting time for the free system and increase the customer utility surplus of the entire two-tier system. However, if customers are more sensitive to the service delay, the subsidy policy may have the opposite effect. When the pay system price is regulated (the price determined by government), the subsidy policy can effectively reduce the expected waiting time for the free system and improve the social welfare of the two-tier system. And there exists an optimal regulated price to maximize the social welfare of the entire public service system. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The healthcare systems in Australia and some European countries consist of both free (or almost free) public hospitals and paid private hospitals. Customers have two options of getting either free or paid service. Such a system is called a two-tier service system and has been studied in Tuohy, Flood, and Stabile (2004), Guo, Lindsey, and Zhang (2014) and Hua, Chen, and Zhang (2012). These studies are motivated by over-congestion problems in free public service systems due to the growing demand and limited government budget. For example, some elective surgeries scheduled in the public hospitals have an average waiting time of 6 months (Hogg, Gibson, Helou, DeGabriele, & Farrell, 2012). Therefore, how to coordinate the free and toll systems to improve the social welfare of public service is an important issue.

⇑ Corresponding author at: Department of Decision Sciences, Western Washington University, Bellingham, WA 98225, USA. http://dx.doi.org/10.1016/j.cie.2015.08.009 0360-8352/Ó 2015 Elsevier Ltd. All rights reserved.

Hua et al. (2012) first study a tax coordination mechanism in the two-tier public service system. In their model, the free system tries to maximize its own total customers’ utility and the toll system wants to maximize its profit. The government subsidizes the free system by using the government tax collected from the toll system to maximize total social welfare. They show that the public service can be delivered more efficiently via customer choice and service provider competition, and a relatively low tax rate can almost perfectly coordinate the two service providers and achieve most of the benefit of the two-tier service system. However, when the competitive advantage of the free system is strong, the toll (pay) system may not be able to enter the market and the tax coordination mechanism in favor of free system is not feasible in the two-tier service system. To alleviate the overcrowding problem and improve the social welfare in the two-tier healthcare system, Australian government offers a subsidy to the customers choosing private hospitals to reduce the service demand for the public hospitals (Duckett, 2005). That motivates us to discuss another way of coordinating two-tier system which cuts some free system budget

W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157

to subsidize the toll system customers. The goal is to encourage customers to use the toll system and reduce the waiting time of the free system. However, government’s subsidizing the toll system users will reduce the resource (capacity) for the free system. If this subsidy policy cannot attract enough customers to go to toll system, then it may not reduce the waiting time of the free system. Therefore, the policy-maker (or government) and service system managers are interested in the answers to the following questions: (1) Is the policy of subsidizing toll system users effective in reducing the expected waiting time of the free system and improving the social welfare of the two-tier service system? (2) If the subsidy policy can improve the social welfare, then what is the optimal subsidy price and how does it affect the performances of the two-tier service system? We analyze the two-tier system by formulating a mixed duopoly game where the free and toll systems compete for serving the same market with their own objectives. While the free system funded by the government seeks to maximize its own total customer utility surplus, the toll system as a private firm tries to maximize its profit. The government may cut some budget of free system and subsidizes toll system customers to improve the social welfare. We find that there exists a Nash equilibrium under such a subsidy policy. Furthermore, we study the performance of the twotier system under both unregulated and regulated prices. In the unregulated price situation, the toll system can freely set the price to maximize its profit. It is found that the system performance depends on the customer’s delay sensitivity. For the case with less delay sensitive customers, the subsidy policy can be effective in reducing the expected waiting time of the free system and increasing the total customer utility surplus. In contrast, for the case with more delay sensitive customers, the subsidy policy may worsen the system performance. In the regulated price situation, the government determines the price of the toll system. The regulated price can make the subsidy policy more effective in improving the performance of the two-tier system. We also show that there exists an optimal regulated price to maximize the social welfare. The focus of this study is on characterizing how the subsidy for toll system customer affects the waiting time of the free service system and the social welfare of the entire two-tier service system. The results are insightful to both governments and service providers (SPs). We show that the two-tier service system can gain more social welfare under a subsidy policy with an appropriate regulated price. It should be noted that if the government subsidizes the private service provider directly, the service provider just obtains an additional revenue from the government. We find that it often cannot effectively improve the customer service with the method of subsidy to customers directly. So we only discuss the method of subsidy to the toll system customers in this paper. The balance of the paper is organized as follows. Section 2 provides a literature review. Some preliminaries are given in Section 3. Sections 4 and 5 analyze the behaviors of the two-tier service system under unregulated and regulated prices, respectively. Both qualitative properties and quantitative analysis of the system performance are obtained. Finally, Section 6 concludes with a summary. All proofs are provided in Appendix A.

2. Literature review Our work is related to some recent work on healthcare systems. Dobrzykowski, Saboori Deilami, Hong, and Kim (2014) using structured analysis to discuss the primary topics, methodological approaches and future research directions in the operations management and supply chain management research in healthcare.

147

Andritsos and Tang (2013) study a question that motivated by a hot discussion in the European Union about European patients can be freedom to choose the country they receive treatment. With the assumption that each patient will prefer to receive treatment at home when the waiting time is below her individual tolerance level, they show that cross-border patient movement can increase patient welfare due to increased access to care in the long run. Chen, Qian, and Zhang (2010) conduct an empirical study on the positive effect of allowing private SP to exist in the public healthcare system using joint replacement surgery data in Canada. Guo et al. (2014) analyze the two-tier system with a free system with a fixed service capacity and demonstrate that the social welfare can be maximized under the self-financing condition for the toll system. de Véricourt and Lobo (2009) study multi-period capacity and pricing decisions for a non-profit firm operating both free and pay hospitals to treat blindness. Using a newsvendor model, they find the sufficient conditions for the optimal capacity and pricing decision to be of threshold type. Motivated by de Véricourt and Lobo (2009) and Hua et al. (2012) consider two-tier public service systems in which the government subsidizes the free system by using tax collected from the toll system. However, when the toll system is small and vulnerable, the government may subsidize the toll system by offering a subsidy to its customers like in the Australia healthcare system. This paper focuses on investigating such a situation and can be considered as a complement to Hua et al. (2012). Another stream of research related to our work addresses the price and capacity competition between service providers. Li and Lee (1994) address the competition issue on price, quality, and service speed between two SPs and establish the necessary and sufficient conditions for the existence of the Nash equilibrium. They assume that the customers can observe the queue lengths and have the option of jockeying. Lederer and Li (1997) analyze the competition between two servers in terms of price and service capacity. In their paper, customers are differentiated by their sensitivities to the price and delivery time. Armony and Haviv (2003) consider the situation where two firms offer the identical service with different prices and response times to two types of time-valued customers. Chen and Wan (2003) examine the simultaneous price competition between two firms with different service values and firm-dependent unit costs of waiting. Given the fact that customers select a specific SP based on multi-attributes, Allon and Federgruen (2008) investigate how competing SPs differentiate themselves by offering appropriate price and service-level guarantees. Afanasyev and Mendelson (2010) investigate the competition between the two SPs with different capacities or operating costs. Although the competition issue has been well studied in these past studies, there is no work addressing the issue on both competition and coordination between the two types (free and toll) of SPs under a policy of subsidizing the toll system users. There has been some studies on subsidizing private healthcare insurance/hospital in a two-tier health care system. Duckett and Jackson (2000) argue that a public hospital may be more efficient than a private hospital, and subsidizing the private hospital insurance results in a social welfare loss. Vaithianathan (2002) argue that subsidizing the private healthcare provider rather than the healthcare insurance is a more effective way of reducing the demand for public healthcare service provider. Frech and Hopkins (2004) show that the optimal subsidy is negative (that is a tax on private health insurance premiums) based on shortrun demand estimates but positive from a long-run dynamic perspective. However, these studies are empirical type. In contrast, our work is the first attempt to provide a quantitative analysis based on a queuing model. The coordination method of subsidizing to customers can also be found in other related research areas. For example, to improve the social welfare, governments often subsi-

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dize agriculture producers to increase production or purchase machines. See Kirwan (2009), Huang, Wang, Zhi, Huang, and Rozelle (2011) and Yang, Huang, Zhang, and Reardon (2013). Another example is in two-sided market, because the crossnetwork effect the platform may subsidize one side to attract another side and improve the profit ( Armstrong, 2006; Hagiu & Spulber, 2013). The difference is in two-sided market, the platform subsidizes customers to improve profit while in a healthcare system the government subsidizes customers to improve customer service. Our work is also related to the research on the market price regulation. Brekke, Nuscheler, and Rune Straume (2006) study the competition among firms choosing their optimal locations and product qualities under a regulated price. Grimm and Zöttl (2010) provide the taxonomy of price cap regulation in oligopoly under demand uncertainty. More studies on the pricing regulation in a market are done by Doherty and Garven (1986) and Vogelsang (2001). In this paper, we analyze a new situation where the government determines the toll system price while the SPs determine the optimal capacity under their own objectives. 3. Preliminaries Consider a market with customers who are heterogeneous in delay sensitivity. There are two SPs offering free and toll services, respectively, to these customers. The free system tries to maximize its own expected total customer utility while the toll system tries to maximize its expected profit. To improve the customers’ satisfactions or social welfare objective, the government may cut part of the free system budget and utilize it to subsidize every toll system customer by a reimbursement.

assume that the subsidy value ps satisfies ps 6 p, that is the waiting time of the toll system is not more than the free system’s(Otherwise, if Wðkv ; lv Þ > Wðkf ; lf Þ we can verify that the net utility of free system will always be higher than the toll system’s and kv ¼ 0). In equilibrium, there exists a waiting cost parameter
such that the customer’s decentralized choice behavior threshold h results in the following relations:



p  ps þ hWðk v ; lv Þ ¼ hWðkf ; lf Þ;

ð1Þ

kf þ kv 6 k: where kf ¼

R h2Hf

kdh, kv ¼

R h2Hv

kdh, Hf ¼ fh 2 ½0; 1 : U  hWðkf ; lf Þ

P maxfU  p  hWðkv ; lv Þ þ ps ; 0gg, Hv ¼ fh 2 ½0; 1 : U  p
¼ h0 þ
hWðkv ; lv Þ þ ps > maxfU  hWðkf ; lf Þ; 0gg and h hh1 . In Eq. (1), Hf ;v represent the sets of customer who will go to the free system and toll system, respectively. Correspondingly, the integrals of the total market shares over these two sets are the arrival rates to the free system and toll system, respectively. The
makes the customers to split two streams. Because threshold h p  ps P 0, we have Wðkv ; lv Þ 6 Wðkf ; lf Þ and
h ¼ kf =k in the equilibrium. That is to maximize his or her net utility, a customer with h lower than kf =k goes to the free system and a customer with h higher than kf =k goes to the toll system or balk. It should be noted that there may exist two theoretically possible cases for the customer choice equilibrium: p  ps þ h0 Wðkv ; lv Þ < h0 Wð0; lf Þ (i.e.

kf ¼ 0) and p  ps þ ðh0 þ kf h1 =kÞWð0; lv Þ > ðh0 þ kf h1 =kÞWðkf ; lf Þ (i.e. kv ¼ 0). However, for the first case the toll SP can improve the price to p0 such that p0  ps þ h0 Wðkv ; lv Þ ¼ h0 Wð0; lf Þ and obtain a higher profit, and it is clear that the toll SP will obtain a negative profit in the second case. Thus we can discard these two cases and only focus on Eq. (1) in the following analysis.

3.1. Service value and customer choice 3.2. The toll system We assume that the service benefit is measured by a service utility. The service utility represents the achievement of the service goal. For example, for a healthcare SP, the knee replacement is the service goal. Assume that customers have the same service utility, denoted by U. Customers arrive according to a Poisson process with rate k. By self-interest choice, customers split into two flows entering the free and toll systems with equilibrium rates kf and kv , respectively. Service times are assumed to be independent and identically distributed (i.i.d.) exponential random variables with the service rates li ði ¼ f ; v Þ, which represent the service capacities. A customer’s waiting cost per time unit is assumed to be given by h ¼ h0 þ hh1 ; h0 ; h1 are constants and h is assumed to be uniformly distributed over [0, 1]. This represents that customers are heterogeneous in delay sensitivity and the waiting cost h is uniformly distributed over ½h0 ; h0 þ h1 , where h0 and h1 are the minimum value and the length of interval for the uniformly distribution. Normalizing h to one can simplify the analysis on the customer choice equilibrium. The customer’s expected waiting cost is computed as hWðki ; li Þ, where Wðki ; li Þ ¼ 1=ðli  ki Þ is the expected waiting time (including service time) in an M/M/1 queue with arrival rate of ki and service rate of li ði ¼ f ; v Þ. Therefore, the two-tier service system is modeled as two parallel M/M/1 queues. To encourage customers to choose the toll system, the government subsidizes every customer of toll system with a reimbursement ps . Thus the actual price paid by a user is p  ps and the toll system still receives p for each service. And a toll system customer’s net utility is U  p þ ps  hWðkv ; lv Þ. Similarly, a customer’s net utility for the free system is U  hWðkf ; lf Þ. A customer joins the toll system if and only if U  p þ ps  hWðkv ; lv Þ > maxfU  hWðkf ; lf Þ; 0g, otherwise, he or she will choose the free system or balk. To avoid the trivial case,

The toll system (private firm) serves customers on a FCFS basis and is modeled as an M/M/1 queue. There is a service rate cost of cv . Usually, in a competitive environment, a toll system can deliver the same service more cost-efficiently than a free system. Thus, we assume cv 6 cf . However, for the case of cv > cf , our model remains valid. The decision problem for the toll system is to determine the optimal capacity lv and price p to maximize its expected profit, denoted by pðp; lv Þ. That is

max

pPps ;lv P0

pðp; lv Þ ¼ pkv  cv lv kf þ kv 6 k:

We define pðp; 0Þ ¼ 0. Thus lv ¼ 0 represents the case that the toll system does not enter the market. Equivalently, we can say that the toll system will exit the market if it cannot generate a positive profit. 3.3. The free system The free system also serves customers on a first-come-firstserved (FCFS) basis and is modeled as an M/M/1 queue with arrival rate of kf and service rate of lf . There is a service rate cost of cf . A budget, B, is imposed exogenously by the government. To improve the total customer utility surplus, the government may split the budget into two parts. One part is to subsidize the toll system customers ðDÞ and the other part is to support the free system ðB  DÞ. As a non-profit organization, the free system tries to maximize its own total customer net utility (abbreviated as NU) with the funding rate of B  D. Thus the decision problem for the free system is

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@ pðp; kv Þ ¼ kv  cv @p

max NUðlf Þ ¼ kEh2Hf ½U  hWðkf ; lf Þ

lf >kf :

c f l f 6 B  D; We do not require ps kv ¼ D at the beginning. Instead, we will first prove that the duopoly game has a Nash equilibrium for given ps and D, and then discuss the problem that whether there exists a ~s kv ¼ D in the Nash equilibrium. That is because we ~s such that p p will find that there may not exist any ps such that ps kv ¼ D in the Nash equilibrium for some special case. Such an analysis will help the government make budget plan given the competitive behaviors of both SPs. 4. Unregulated price case We now analyze the duopoly competition and coordination in the two-tier service system when the toll system can set any price. 4.1. The mixed duopoly competition

h0 þ ðk  kv Þh1 =k ;   2 ½ðh0 þ ðk  kv Þh1 =kÞW k  kv ; lf þ ps  p

@ 2 pðp; kv Þ h0 þ ðk  kv Þh1 =k ¼ 2cv h i3 < 0:   @p2 ðh0 þ ðk  kv Þh1 =kÞW k  kv ; lf þ ps  p Thus, pðp; kv Þ is concave in p and the first-order condition of @ pðp; kv Þ=@p ¼ 0 yields

h0 þ ðk  kv Þh1 =k p ðkv Þ ¼  lf þ kv  k 1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h0 þ ðk  kv Þh1 =k þ ps : cv kv

Using (1), we obtain the corresponding l1 v ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ^ðkv Þ, If p ðkv Þ P p then kv ðh0 þ ðk  kv Þh1 =kÞ=cv þ kv . ^ðkv Þ. To simplify the analysis, we p ¼ p1 ðkv Þ; otherwise, p ¼ p h i ^1þ ; k , where ^ðkv Þ for any kv 2 k assume that p1 ðkv Þ P p v   ^ ^1 k1þ v ¼ max 0; kv . Thus we make the following assumption.

We first obtain the best response strategy of the free system for a given strategy of the toll system.

Assumption. The service utility U has a lower bound

Proposition 1. Given lv and p of the toll system, the optimal strategy for the free system is to invest all capital of B  D in its capacity or

U P max ðh0 þ ðk  kv Þh1 =kÞ=ððB  DÞ=cf þ kv  kÞ

l ¼ ðB  DÞ=cf :  f

capacity strategies the toll system adopts, we only need to consider the best response strategy of the toll system under lf . Finding such a response function also determines the Nash equilibrium and provides the conditions for a two-tier system to exist. For the given free system strategy lf , the best response strategies of the toll system have the following property.

l of the free system, the toll system either  f

pðp; kv Þ ¼ ðp  cv Þkv  cv h0 þ ðk  kv Þh1 =k   : ðh0 þ ðk  kv Þh1 =kÞW k  kv ; lf þ ps  p

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cv kv =ðh0 þ ðk  kv Þh1 =kÞ=k:

This is a realistic assumption because health service is one of the most important services for customers and often has a sufficiently high service utility that even the most delay sensitive customers have a positive net utility when a free system exists in the market. Substituting p1 ðkv Þ into (3) yields

p^ ðkv Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh0 þ ðk  kv Þh1 =kÞkv  2 cv ðh0 þ ðk  kv Þh1 =kÞkv  lf þ k v  k  ðcv  ps Þkv :

ð4Þ

The assumption of p P ps requires kv to satisfy k1 6 kv 6 k2 , where

enters the market to maximize its profit at kf þ kv ¼ k or leaves the market because it cannot obtain a positive profit. This proposition implies that we only need to discuss the optimal policy for the toll system when no balking behavior occurs in the two-tier system. Assume that the toll system enters the market, based on (1) and Proposition 2, the profit function can be written as



þ h1

ð2Þ

Eq. (2) is intuitive as for a non-profit system, covering its operating cost is the only requirement. Thus (2) gives the maximum service capacity from the total capital which maximizes the throughput or the total customer utility of the free system. Since the free system will always uses the strategy lf no matter what price and

Proposition 2. Given

kv 2½^k1þ v ;k

" #   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi  2 k1;2 ¼ h0 þ h1  2 lf  k  ðh0 þ h1  2ðlf  kÞÞ  4cv ðcv þ h1 =kÞ lf  k

ð2cv þ 2h1 =kÞ; i ¼ 1; 2. Therefore, the domain of the profit function p^ ðkv Þ should be X ¼ ½k1 ; k2 , where k1 ¼ maxðk1 ; ^k1þ v Þ and

k2 ¼ minðk; k2 Þ. Taking the first and second order derivatives of (4) with respect to kv , we have 2

^ ðkv Þ h1 kv =k þ ðh0 þ ðk  2kv Þh1 =kÞ @p ¼  2 @kv lf þ kv  k

sensitive customer type ðh ¼ 1 or h ¼ h0 þ h1 Þ, we have U  p ðh0 þ h1 Þ½ðh0 þ ðk  kv Þh1 =kÞWðk  kv ; lf Þ þ ps  p=ðh0 þ ðk  kv Þh1 =kÞþ ps P 0. Thus the optimal response price policy should ^ðkv Þ, where p ^ðkv Þ ¼ ðh0 þ ðk  kv Þh1 =kÞ½ðh0 þ h1 Þ satisfy p P p ðWðk  kv ; lf Þ þ ps =ðh0 þ ðk  kv Þh1 =kÞÞ  ps  U=ðkv h1 =kÞ. Taking the first and second order derivatives of (3) with respect to p, we have



lf  k

cv ðh0 þ ðk  2kv Þh1 =kÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  cv þ ps cv ðh0 þ ðk  kv Þh1 =kÞkv

ð3Þ

From kf þ kv ¼ k, the arrival rate to the free system is k  kv and   holds. That is U  ½h0 þ ðk  kv Þh1 =kW k  kv ; lf P 0 h  i. 1  kv ¼ h0 þ h1  U lf  k ðU þ h1 =kÞ. For the most delay kv P ^



^ ðkv Þ @ p

ðh0 þ h1 Þ

2

@k2v

¼ 2

þ





ð5Þ

2  k



lf  k þ h1 lf  k 

3

lf þ kv  k

c2v ðh0 þ h1 Þ

2

2½cv ðh0 þ ðk  kv Þh1 =kÞkv  



ð6Þ

3=2





2 . 2=3 k cv h1 =k,

4=3 4=3 Letting a ¼ c4=3 ðh0 þ h1 Þ lf  k þ h1 lf  k v ðh0 þ h1 Þ þ 2

" , #2=3      2 4=3 4=3 k b ¼ 2 lf  k cv ðh0 þ h1 Þ  24=3 cv ðh0 þ h1 Þ lf  k þ h1 lf  k

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ðh0 þ h1 Þ,

and

4=3

4=3

c ¼ cv ðh0 þ h1 Þ



2

lf  k .

^ ðkv Þ=@k2v ¼ 0 @2p

can

be

written as

ak2v þ bkv þ c ¼ 0:

ð7Þ

pffiffiffiffiffiffiffiffiffiffiffi 2 2 It is clear that if b  4ac > 0, (7) has two roots
k1v ¼ bþ 2ab 4ac and pffiffiffiffiffiffiffiffiffiffiffi 2
k2v ¼ b 2ab 4ac for the second order condition. ^ ðkv Þ=@kv ¼ 0 has at most three Lemma 1. The first order condition @ p solutions over X. Denote these solutions (if exist) by
k1v ;
k2v and
k3v with 1 2 3 1 3 2





kv < kv < kv , then kv and kv are the minimum points, and kv is the maximum point. The optimal kv can be achieved at either an endpoint of X or one ^ ðkv Þ=@kv ¼ 0. Denote X0 as the boundary of the solutions to @ p points of X. If the toll system enters the market, the optimal kv can be achieved at an element of X0 or
k2v (if it exists and satisfies
k2v 2 X). Otherwise, kv ¼ 0, which means that the toll system will not enter the market as it cannot cover its operating cost. The following proposition describes the optimal response strategy of the toll system and the condition of for the existence of a two-tier system. Proposition 3. Given lf of the free system, the optimal strategies for the toll system are as follows:

lf > l
f , then kv ¼ 0; ^ ðkv Þ; Specially, if lf > l
f ; kv ¼ maxkv 2k ;
k2 \X p ~ f or (b) if lf 6 l 1;2 v (a) if

h

pffiffiffiffiffiffiffiffiffiffiffiffi

i

l2 cv ðh0  h1 Þ= cv h0 k þ cv  ps  ðh0  h1 Þlf > h0 k, f both the toll system and free system co-exist in the market  0 < kv < k .

 ðh0 þðkxÞh1 =kÞx ~f ¼
f ¼ maxx2X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þkx , l where l 2

pffiffiffiffiffiffiffiffihffi0 k

2

cv h0 kþðcv ps Þk

cv ðh0 þðkxÞh1 =kÞxþðcv ps Þx

sented are sufficient ones and can be satisfied in most of the numerical examples presented in this paper. It is well known that the necessary and sufficient condition of Nash equilibrium is at the intersection of the two best response functions. Such a intersection of the two best response functions ^ ðkv Þ. Note that the optimal strategy of satisfies kv ¼ maxkv 2X;0 p   ^ pðkv Þ can be obtained from Proposition 3. Therefore, lf ; kv is the Nash equilibrium and the following result is immediate. Proposition 4. For a given budget reduction D and a subsidy price ps ,   the two SP game has a Nash equilibrium lf ; kv . Note that for a given 

lf , if @ p^ ðkv Þ=@kv ¼ 0 has three different

solutions, the optimal kv of the toll system may take multiple different values and the Nash equilibrium strategy may not be unique. We will discuss the Nash equilibrium point that has the smallest capacity for the pay system, which have the highest return on investment (ROI) that is often more concerned by private hospital administrators (Goetzel, Ozminkowski, Villagra, & Duffy, 2005). In the above model, the total practical subsidy ps kv can be less or more than the budget reduction D. An interesting question for ~s kv ¼ D in the ~s such that p policy maker is whether there exist a p ~s must satisfy k2v . It is clear that p Nash equilibrium. Denoting k3 ¼
one of the three equations: ps ki ¼ D; i 2 f1; 2; 3g. We can obtain ~s by verifying these solutions corresponding market shares the p ðki ; i 2 f1; 2; 3gÞ to find which one is the Nash equilibrium. In the following, we provide a sufficient condition for the policy maker ~s in practice. to check whether there exists a p Proposition 5. For a given subsidy budget D, if kv ðps Þ p ¼D=k ¼ ki and 2 s kv ðps Þ p ¼D=k ðD=k Þ ¼ ki , where i 2 f1; 2; 3g, then there must exist a 2 s v ~s ; Particularly, if k ðp Þ ~s ¼ D=k2 . unique p ¼ k2 , it must be p v

s

ps ¼D=k2

.

We have several observations from Proposition 3. First, there
f for the free system capacity. As long as exists a threshold l lf > l
f or the free system capacity is sufficiently large, the toll system will not enter the market. Otherwise or

lf 6 l
f , the toll sys-

tem will either monopolize the market or compete with the free system. The second part of (b) in Proposition 3 gives a sufficient condition for the existence of a two tier system. It is easy to see ~ f Þ for the free SP’s capacity sat
f ; l that the two threshold values ðl ~ f from the definition of l
f and the expression of l ~ f . That
f P l isfy l ~ f 6 lf 6 l
f ), both the toll and is when lf is moderately large (or l the free systems will exist in the market. The condition .pffiffiffiffiffiffiffiffiffiffiffiffi h i l2 cv ðh0  h1 Þ cv h0 k þ cv  ps  ðh0  h1 Þlf > h0 k is complif .pffiffiffiffiffiffiffiffiffiffiffiffi cv h0 k þ cv  ps . cated and depends on the values of cv ðh0  h1 Þ .pffiffiffiffiffiffiffiffiffiffiffiffi However, if cv ðh0  h1 Þ cv h0 k þ cv  ps < 0, it is easy to verify  
f ), that when lf is in a middle range (i.e., l1f 6 lf 6 min l2f ; l the two-tier service system must exist in the market, where  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffiffiffi 2 l1;2 ¼  ðh0  h1 Þ  4h0 k cv ðh0  h1 Þ= cv h0 k þ cv  ps þ f .h i pffiffiffiffiffiffiffiffiffiffiffiffi 2cv ðh0  h1 Þ= cv h0 k þ 2cv  2ps . Note that the conh0  h1 ditions for the existence of a two-tier system in Proposition 3 are quite complex and depend on multiple factors including the competitors’ capacity, cost efficiency, the customers’ unit waiting cost, and the overall market demand. In addition, the conditions pre-

Proposition 5 reveals that under certain conditions there ~s which is consistent with the submust exist a subsidy price p sidy D in the equilibrium (i.e. using up the subsidy). This is because if the equilibrium strategy kv ðps Þ the same type strategy as at subsidy prices of p0s ¼ D=k2 and p00s ¼ D=kv ðD=k2 Þ, then k ðp Þ must be an increasing continuous function of ps over

v 0 s 00  ~s that uses up the ps ; ps . Thus there exists a subsidy price p ~s kv ¼ D may not hold in some special subsidy D. However, p case. That is because that kv ðps Þ can be a piecewise continuous function, and there may not be an intersection between ps kv ðps Þ ~s ) and the case and D. The case of with an intersection (exists p ~s ) are showed in the following of without intersection (no p Fig. 1(a) and (b) respectively. ~s often with customer It should be noted that the case of no p low delay sensitivity and it is rare in practice. The government can adjust budget reduction to avoid it. Therefore, in the following, we will discuss the effect of D on the system performance when ~s kv ¼ D in the Nash equilibrium. ~s such that p there exist a p 4.2. The effect of the subsidy coordination Generally, the purpose of the government’s subsidizing the toll system users is to improve customers’ service. Customers’ service level can be evaluated by the expected waiting time of the free system or the total customer utility surplus. Now we discuss whether implementing a subsidy policy without price regulation can reduce the expected waiting time of the free system or improve the total customer utility.

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The total customer utility denoted by TU is defined as the sum of the expected net utilities of customers of both free and pay systems which is given by

TU ¼ kEh2Hf ½U  hWðkf ; lf Þ þ kEh2Hv ½U  p  hWðkv ; lv Þ þ ps  ¼ Uk  ðp  ps Þkv  ðh0 kf þ h1 k2f =ð2k2 ÞÞ=ðlf  kf Þ  ðh0 kv þ h1 kv ðk þ kf Þ=ð2k2 ÞÞ=ðlv  kv Þ

ð8Þ

As we cannot obtain the closed-form solution of Nash equilibrium, we examine these performance measures numerically. The impacts of subsidy ps on the free system’s expected waiting time, the TU, and the arrival rate to the free system are shown in Figs. 1–3 for three levels of customer delay sensitivity. For the low delay sensitivity case of h 2 ½3; 8 shown in Fig. 2(a), the toll system will not enter the market when D is smaller than 0.06. The arrival rate to the free system is first keep constant, then has a convex shape and achieves its minimum at D ¼ 0:22. While

0.8

0.7 ps λv

Investment

Investment

Δ

0.6

0.5

0.4

0.3

0.2

0.5 0.4 0.3 0.2

0.1

0

ps λv

0.7

Δ

0.6

0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0

0.2

0.4

0.6

0.8

1

ps

1.2

1.4

1.6

1.8

2

ps

(a) Exists an intersection (Δ = 0.4)

(b) No intersection (Δ = 0.3)

~s kv ¼ D with customer low delay sensitivity ðk ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 1; h1 ¼ 3; B ¼ 1:6; U ¼ 200Þ. Fig. 1. Impact of D on p

2.8 2.6 2.4 2.2 2 1.8 1.6 1.4

0

0.05 0.1

0.15 0.2

0.25 0.3

194

1

192

0.95

190

0.9

188

0.85

186

0.8

184

0.75

182

0.35 0.4

f

3

Total Customer Surplus

Waiting time of the free system

3.2

0

(a) Expected waiting time of the free system

0.05 0.1

0.15 0.2

0.25 0.3

0.7

0.35 0.4

0

(b) Total customer utility

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

(c) Arrival rate of the free system

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

(a) Expected waiting time of the free system

192

0.77

190

0.76 0.75

188

0.74

186

f

3.2

Total Customer Surplus

Waiting time of the free system

Fig. 2. Impact of D on customer service with low delay sensitive customers (k ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 3; h1 ¼ 5; B ¼ 1:5; U ¼ 200).

184

0.73 0.72

182

0.71

180

0.7

178

0.69

0

0.05 0.1

0.15 0.2

0.25 0.3

(b) Total customer utility

0.35 0.4

0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

(c) Arrival rate of the free system

Fig. 3. Impact of D on customer service with medium delay sensitivity customers (k ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 4; h1 ¼ 6; B ¼ 1:5; U ¼ 200).

W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157 0.76

190

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1

0.74

185

0.72 180

0.7

175

f

3 2.8

Total Customer Surplus

Waiting time of the free system

152

0.66

170

0.64 165

0.62 0.6

160 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

(a) Expected waiting time of the free system

0.68

0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

(b)Total customer utility

0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

(c) Arrival rate of the free system

Fig. 4. Impact of D on customer service with high delay sensitivity customers (k ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 8; h1 ¼ 10; B ¼ 1:5; U ¼ 200).

Wðkf ; lf Þ first remain unchanged and has a sudden drop at D ¼ 0:06 (as the toll system enters the market) and achieves its minimum, then increases with D. In contrast, the TU has an opposite pattern and achieves the maximum at D ¼ 0:06. Thus it is revealed that when customers are not highly sensitive to the delay (in terms of both the mean and the variance of h), subsidizing the toll system users can reduce Wðkf ; lf Þ and increase the TU. Moreover, the optimal subsidy amount for maximizing the total customer utility is the minimum level that makes the toll system to enter the market. The reasons are as the follows. When customers are less sensitive to the service delay, customers are only willing to pay a relatively low price to go to the toll system. This means that the toll system cannot obtain a positive profit and will not enter the market if without or too low subsidy. When the subsidy amount is high enough, the toll system will enter the market and the customer service will be improved. Furthermore, if the subsidy amount is increasing, the increase of toll system customers is not significant enough so that the customers’ service level will become worse. The implication is when customer’s delay sensitivity is low, to improve the customer service, the government can subsidize the toll system customers so that the toll system will stay in the market. For the medium delay sensitivity case of h 2 ½4; 10 shown in Fig. 3, the arrival rate to the free system has a convex shape and achieves its minimum at D ¼ 0:16. However, Wðkf ; lf Þ becomes an increasing function of D, and TU becomes a decreasing function of D. Fig. 3 indicates that as the customer’s delay sensitivity increases, kf can be reduced but Wðkf ; lf Þ and TU may not be improved by subsidizing the toll system users. This is mainly due to the fact that the budget cut causes the free system capacity to reduce too much, resulting longer waiting time in free system and lower TU despite the toll system serves some customers (see Fig. 4). When customer’s delay sensitivity is relatively high, i.e. h 2 ½8; 18, we find that Wðkf ; lf Þ and kf will increase and the TU will decrease in the subsidy D. This implies that the customer service is not improved by subsidizing the toll system customers. This is because that the toll system charges a higher price to serve only a small number of elite customers or highly delay sensitive customers when the government adopts subsidy policy. Therefore, regulating the toll price is necessary and we will discuss this issue in the next section.

systems. Then, we examine the effect of the subsidy-regulation policy in terms of social welfare maximization. 5.1. Capacity competition We assume that the government determines the price of toll system p0 and the toll system only uses the capacity decision to compete with free system. Like Proposition 1, we can prove that ~ f ¼ ðB  DÞ=cf . And for the the optimal strategy of free system is l   ~ ~ given lf , denote lv as the optimal capacity of the toll system. In the regulated price case, we can prove that the toll system also maximizes its profit at kf þ kv ¼ k or does not enter the market. ~ f of the free system and a regulated price Proposition 6. Given the l p0 , the toll system maximizes its profit at kf þ kv ¼ k or does not enter the market. Thus, it follows from (1) that

lv ¼

h0 þ ðk  kv Þh1 =k þ kv : ~ f Þ  p0 þ ps ðh0 þ ðk  kv Þh1 =kÞWðkf ; l

The profit function of the toll system becomes

p~ ðkv Þ ¼ ðp0  cv Þkv  cv

h0 þ ðk  kv Þh1 =k   : ~ f  p0 þ ps ðh0 þ ðk  kv Þh1 =kÞW kf ; l

Similarly to the analysis in Section 4, we need kv to satisfy ^ k1þ v 6 kv 6 k. In addition, from kf þ kv ¼ k, for the most delay sensiwe have tive customer type ðh ¼ 1 or h ¼ h0 þ h1 Þ, ~ f Þ þ ps  p0 =ðh0 þ U  p0  ðh0 þ h1 Þ½ðh0 þ ðk  kv Þh1 =kÞWðk  kv ; l
1 6 kv 6 k
2 , ðk  kv Þh1 =kÞ þ ps P 0. That is kv should satisfies k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi.  i 2
where k ¼ e1  e1  4e2 e0 ð2e2 Þ, e2 ¼ h1 ðU  p0 þ ps Þ, ~ f  kÞh1  ðh0 þ h1 Þh1 , e0 ¼ e1 ¼ Uðh0 þ h1 Þk þ ðU  p0 þ ps Þðl  ~ ðh0 þ h1 Þðh0 þ h1  Uðlf  kÞkÞ; i ¼ 1; 2. Therefore, the domain ~ ðkv Þ should be Z ¼ ½k_ 1 ; k_ 2 , where of the profit function p
1 _
2 k_ 1 ¼ maxð^ k1þ v ; k Þ and k2 ¼ minðk; k Þ. Denoting the denominator of the second term of (9) by ~ f Þ  p0 þ ps and taking the first Eðkv Þ ¼ ðh0 þ ðk  kv Þh1 =kÞWðkf ; l order derivative of (9) with respect to kv , we have

5. Regulated price case This section focuses on the situation where the price is regulated or determined by the government. We first establish the Nash equilibrium of the capacity competition between the free and toll

ð9Þ

 2 ~ f ~ f =kÞW kf ; l h1 Eðkv Þ=k  ðh0 þ ðk  kv Þh1 =kÞðh0 þ h1 l ~ ðkv Þ @p ¼ cv @kv Eðkv Þ2 þ p0  cv :

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W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157



tinuous function in some special cases, and there is no intersection ~ ðp Þ and D. We just consider the p ~ ¼ D case which ~s k between p k

2

equivalent to gðkv Þ ¼ 0. Rewriting gðkv Þ ¼ 0 as A2 kv þ A1 kv þ A0 ¼ 0, where A2 ¼ ½ðp0  cv Þðh1 =k þ p0  ps Þ  cv h1 =kðh1 =kþ p0  ps Þ,    ~ f  k ðh1 =k þ p0  ps Þ þ2cv h1 A1 ¼ 2ðcv  p0 Þ h0 þ h1  ðp0  ps Þ l    ~ f  k =k, h0 þ h1 þ ðps  p0 Þ l A0 ¼ ðp0  cv Þðh0 þ h1 þ ðps  p0 Þ   h    2 i l~ f  k Þ2 þ cv h1 ðh0 þ h1 Þ l~ f  k þ ðps  p0 Þ l~ f  k =k   ~ f =k . Therefore, @ p ~ ðkv Þ=@kv ¼ 0 has at most cv ðh0 þ h1 Þ h0 þ h1 l two ~ k1;2 v



l~ f ; ~kv for any given D and ps . Also ~kv ðps Þ may be a piecewise con-

Defining gðkv Þ ¼ ðp0  cv ÞEðkv Þ2 þ cv h1 Eðkv Þ=k  cv ðh0 þ ðk  kv Þh1 = 2    ~ f =k W kf ; l ~ f , it is obvious that @ p ~ ðkv Þ=@kv ¼ 0 is kÞ h0 þ h1 l

s

s

v

5.2. The benefit of subsidy-regulation coordination Now we examine the coordination effect of using the subsidy~ f ¼ ðB  DÞ=cf , it follows from 5.1 that regulation policy. Given l ~ is either a boundary point of Z or root ~ ~ ðkv Þ=@kv ¼ 0. In k k2 of @ p v

v

the following, we present some monotonic properties for the optimal strategy of the toll system and waiting time of free system k2 case. with respect to regulated price for ~ k ¼ ~

positive roots which can be denoted by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ A1  A1  4A0 A2 ð2A2 Þ. To simplify the analysis,

v

we assume p0 > 2cv , which is realistic because the toll system often serves fewer customers and needs to charge a higher price to obtain a positive profit. Thus we can easily verify that A2 < 0 and ~ k2 is the only maximum point. In such a case, the profit maxi-

v

^0 such that Proposition 7. If there exists a threshold p  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2   ~ ~ ~ ^0  cv Þ þ cv h1 =k þ p ^0  cv , then ~ kf h0 þ kf h1 =k W kv ; lv ¼ ðp   ~ f are decreasing in p0 when p0 smaller than p ^0 ; ~ and W ~ kf ; l kf and   ~ f are increasing in p0 when p0 larger than p ^0 ; and ~ kf and W ~ kf ; l   ^0 . ~ f archive their minimums at p W ~ kf ; l

v

mization ~ kv is at either ~ k2v or a boundary point of the domain Z. From the above analysis, the optimal policy of the toll system is ~ in the regulated price case. Similarly to Proposidetermined by k v

tion 3, we can show that this game can reach a Nash equilibrium

3.5

0.9 No price regulation p0=15

3

No price regulation p0=15

0.8

p0=9

p0=9

0.7

2.5

0.6

f

2

0.5 1.5

0.4

1

0.5

0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.2

0.4

(a) Expected waiting time of the free system.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(b) Arrival rate to the free system

Fig. 5. The effects of D on the free system at different p0 values ðk ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 5; h1 ¼ 25; B ¼ 1:5; U ¼ 200Þ.

14

0.55 No price regulation p0=15

12

No price regulation p0=15

p0=9

Waiting time of the toll system

Waiting time of the free system

v

has more practical significance in this section.

10

8

6

4

2

0

0.05

0.1

0.15

0.2

0.25

0.3

(a) Profit of the toll system

0.35

0.4

p0=9

0.5

0.45

0.4

0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(b) Expected waiting time of the toll system

Fig. 6. The effects of D on the toll system at different p0 values ðk ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 5; h1 ¼ 25; B ¼ 1:5; U ¼ 200Þ.

0.4

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W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157

objective function of the two-tier system, denoted by SW, is defined as

93.3

122.6

93.25

122.4

93.2

122.2

93.15

122 121.8 121.6 121.4 121.2

94.6 =1/2

94.4

Social Welfare

94.2 94 93.8 93.6 93.4 93.2 93

2

4

6

0

0.1

0.2

0.3

0.4

0.5

(a) Large =2/3

0.6

0.7

8

10

12

14

16

18

20

22

p0 Fig. 8. The effects of regulated price p0 on social welfare value (k ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 5; h1 ¼ 20; B ¼ 1:5; b ¼ 1=2; U ¼ 200.)

65.5 65

93.1 93.05 93 92.95

64.5 64 =1/3

63.5

92.85

=2/3

ð10Þ

where 0 6 b 6 1 is the weight parameter. The social planner needs to choose optimal D and p0 to maximize the SW. When b ¼ 1, this optimization becomes the solely customer utility maximization problem; when b ¼ 0; SW is equal to the solely toll system profit maximization problem. Fig. 7 shows, for a given p0 ¼ 15, how the social welfare of a two-tier service system changes with D for three different b values. When b is large or the toll system profit is less weighted in social welfare, SWðDÞ is always decreasing in D as shown in Fig. 7(a). When b is moderate, SWðDÞ first increases and then decreases with D as shown in Fig. 7(b). This observation indicates that for a moderate b, which is more likely to occur in practice, there may exist a unique optimal D that maximizes the social welfare of the two-tier system. When b is small or the toll system profit is more weighted in social welfare, SWðDÞ becomes an increasing function of D as shown in Fig. 7(c). Furthermore, we can find the optimal regulated price to maximize the social welfare as shown in Fig. 8. In Table 1, we compare three cases (non-coordination, subsidy coordination, subsidy with price regulation coordination) in terms of social welfare under low and high delay sensitivities. The example presented is based on a set of base parameters of

92.9

121 120.8

SW ¼ bTU þ ð1  bÞp;

Social Welfare

122.8

Social Welfare

Social Welfare

Proposition 7 reveals the impact of price regulation on the free system’s arrival rate and waiting time. Such an impact is because that the toll system implements an aggressive market-share expansion policy by increasing its service capacity to make a profit ^0 , and adopts a conservative when the regulated price less than p market-share policy by decreasing its service capacity to serve a small number of highly delay sensitive customers when the regu^0 . The implication is that a moderate regulated price more than p lated price may result in the less congestion in the free system. Due to the complexity of the best response function of the toll system, we utilize the numerical approach to investigating the coordination effects of implementing a subsidy-regulation policy. Under different p0 values, we examine the impacts of subsidy D on the waiting time and arrival rate for the free system, and the profit and waiting time for the toll system. We also provide a case without price regulation case for comparison. Figs. 5 and 6 show that the price regulation affects the performances of the two-tier service system. For p0 ¼ 15; and 9 (it can be verified that these p0 are smaller than the corresponding p ), Figs. 5(a) and 6(b) show that no subsidy or small subsidy policies can reduce both the free and toll systems’ waiting time compare for the unregulated price case. In addition, both the free and toll systems’ waiting times are increasing in the subsidy D. Therefore, a relatively small subsidy combined with a lower regulated price can effectively improve customer service. The effect of the subsidy on the arrival rate to the free system is intuitive as shown in Fig. 5 (b). Fig. 6(a) shows that the toll system’s profit first jumps to a low level when the government sets a regulated price. But it is increasing in D and can become larger than the no coordination case for sufficiently large D. Intuitively, Fig. 6(a) shows that the higher the regulated price and subsidy budget makes the toll system’s profit higher. Based on Figs. 5 and 6, to reduce the waiting time, customers would like the government to put a lower regulated price and a smaller subsidy. On the other hand, to make more profit, the toll system prefers a higher regulated price and a greater subsidy. For the government, an important question is how to balance the objectives of customers and toll system to maximize the social welfare. To answer this question, we introduce a general social objective function. In the two-tier service system, the free system or non-profit firm does not retain any surplus, so we only consider the customers service utility and the profit of the toll system. Thus the weighted sum of the customer utility (consumer surplus) and the toll SP’s utility (producer surplus) is an appropriate social welfare function for evaluating the performance of the service system. Such a definition for social welfare is standard in the economics literature (Anagnostopoulos, Carceles-Poveda, & Tauman, 2010; Baldwin, 1987; Ma & Burgess, 1993). Thus the balanced

=1/2

0

0.1

0.2

0.3

0.4

0.5

(b) Medium =1/2

0.6

0.7

63

0

0.1

0.2

0.3

0.4

0.5

(c) Small =1/3

Fig. 7. Impact of D on social objective for different b values ðk ¼ 1; cf ¼ 1; cv ¼ 0:8; h0 ¼ 5; h1 ¼ 25; B ¼ 1:5; p0 ¼ 15; U ¼ 200Þ.

0.6

0.7

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W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157 Table 1 Comparison of no coordination, subsidy and subsidy with price regulation cases. Experiment

p

TU

SW

Low delay sensitivity

High delay sensitivity

Non

Subsidy

Subsidy PR

Non

Subsidy

Subsidy PR

0 188 94

0.177 191.690 95.933

0.749 192.080 96.414

3.5939 180.4181 92.0060

3.594 180.418 92.006

1.913 186.012 93.963

k ¼ 1; cf ¼ 1; cv ¼ 0:8; B ¼ 1:5; b ¼ 1=2; U ¼ 200. In the low delay sensitivity case ðh0 ¼ 2; h1 ¼ 8Þ, the toll system will not enter market if no subsidy is provided. And both toll system profit and TU can be improved if government offers the subsidy or subsidy with regulated price to the toll system. Specially, the government’s subsidy with regulated price leads to the largest profit, TU and social welfare. In the high delay sensitivity case ðh0 ¼ 5; h1 ¼ 25Þ, the subsidy policy cannot improve the social welfare compare to non-coordination case as discussed in Section 4. However, the government may utilize the subsidy with regulated price to improve the total customer utility, but will worsen the toll system’s profit. Define customers cost as the sum of waiting cost and service payment (i.e. it is equal to Uk  TU). We can compute that the customers costs in the low delay sensitivity situation for nonsubsidy, subsidy, and subsidy with price regulation cases as 12, 8.31 and 7.92, respectively. Thus we know that the two subsidy methods can reduce the customers cost by 30.75% and 34% respectively. Similarly, in the high delay sensitivity case, we can computationally show that the two subsidy methods can reduce the customers cost by 0% and 28.57%, respectively. These results indicate that the subsidy with regulated price policy is an effective way to improve social welfare of the two-tier system. It should be noted that when the customers’ waiting cost is high enough, the toll system often can obtain a high profit. In this case, the government can use another coordination method in which the government subsidizes the free system by using the government tax collected from the toll system to improve the customer service, which has been discussed in Hua et al. (2012).

6. Conclusion To find a solution to the overcrowding problems in public service systems such as healthcare, we study the two-tier system with government’s subsidizing toll system users (the healthcare systems in Australia and some European countries). A mixed duopoly game model is developed for analyzing the competition between the free and toll system with delay sensitive heterogeneous customers. To coordinate the free and toll systems, the government subsidizes the toll system customers to improve the total customer utility and the social welfare. We consider both the unregulated and regulated price cases and examine how the subsidy and price regulation affect the major performances of the two-tier service system. It is observed that in the unregulated price case, if customers are less delay sensitive, subsidizing toll system customers can reduce the free system’s customer waiting time and improve the total customer utility. However, if customers are more delay sensitive, subsidizing toll system customers may increase the free system’s waiting time and reduce the total customer utility. To overcome the negative effect of a subsidy policy, we also investigate the regulated price case where the toll system price is determined by the government. We find that such a price regulation can improve customer service and the social welfare of the two-tier service system and there exists an optimal subsidy level and an optimal regulated price to achieve the maximum social welfare. However, our analysis should be viewed as the first step to quantitatively study the coordination issues in two-tier service systems

by government subsidy and price regulation policies. Some further empirical analysis to confirm these results can be a future research direction. Appendix A

Proof of Proposition 1. Given lv and p of the toll system, we first prove that kf is increasing in lf and Wðkf ; lf Þ is decreasing in lf . When the capacity of the free system is l0f , the customer’s selfinterest choice behavior results in

       p  ps þ h0 þ k0f h1 =k W k0v ; lv ¼ h0 þ k0f h1 =k W k0f ; l0f : Suppose the free system increases its capacity to

ðA1Þ

l00f > l0f , we have

       p  ps þ h0 þ k00f h1 =k W k00v ; lv ¼ h0 þ k00f h1 =k W k00f ; l00f :

ðA2Þ      p  ps þ h0 þ k0f h1 =k W k0v ; lv > h0 þ k0f h1 =k

And have   0 W kf ; l00f under given

lv and p. We next prove the monotonicity

of kf and Wðkf ; lf Þ in two cases.

Case (1). k0f þ k0v ¼ k. For the customer of type h that h > k0f =k, we    have U  ðh0 þ hh1 ÞW k0f ; l00f > U  p  ðh0 þ hh1 ÞW k0v ; lv þ ps , which indicates that a part of customers in the toll system will    transfer to the free system until U  h0 þ k00f h1 = kÞW k00f ; l00f ¼    U  p  h0 þ k00f h1 =k W k00v ; lv þ ps . Thus we have k00f > k0f ; k00v <     and W k00f ; l00f < k0v ; k00f þ k00v ¼ k0f þ k0v ; W k00v ; lv < W k0v ; lv   W k0f ; l0f . That is, kf is increasing in lf ; Wðkf ; lf Þ is decreasing in

lf .

Case (2). k0f þ k0v < k. When the capacity of the free system is l0f , it is easily verified that utility surplus of customers of type  .  . k is zero, and customers of type h > k0f þ k0v k h ¼ k0f þ k0v will not enter the two-tier system because their utility surplus will be negative if they join the system. When the free system increases     its capacity to l00f > l0f , since U  h0 þ k0f h1 =k W k0f ; l00f >    U  p  h0 þ k0f h1 =k W k0v ; lv þ ps , a part of customers of type  . k in the toll system will transfer to the free system, h 6 k0f þ k0v and a part of previously balking customers of type  . i h2 k0f þ k0v k; 1 will join the free system (because   U  ðh0 þ hh1 ÞW k0f ; l00f > 0) or the toll system (because some customers have transferred from the toll system to the free     system) until U  h0 þ k00f h1 =k W k00f ; l00f ¼ U  p    h0 þ k00f h1 =k W k00v ; lv þ ps . Therefore we have k00f > k0f and     k00f þ k00v > k0f þ k0v . Further, since U  p  h0 þ k00f þ k00v h1 =k

156

W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157

      W k00v ; lv þ ps P 0 > U  p  h0 þ k00f þ k00v h1 =k W k0v ; lv þ ps ,   we have W k00v ; lv < W k0v ; lv , which indicates   00 0 00 0 W kf ; lf < Wðkf ; lf Þ from (A1) and (A2). That is, kf is increasing in

lf , and Wðkf ; lf Þ is decreasing in lf . It is easy to verify that

h  i Z kEh2H00f U  hW k00f ; l00f ¼

k00f =k

0

Z

   U  ðh0 þ hh1 =kÞW k00f ; l00f kdh

 U  ðh0 þ hh1 =kÞWðk0f ; l0f Þ kdh 0 h  i ¼ kEh2H0f U  hW k0f ; l0f

>



k0f =k



    since k00f > k0f and W k00f ; l00f < W k0f ; l0f .

Proof of Proposition 2. We will prove it by contradiction. For a given lf , if the equilibrium customer choice behavior result in kf þ kv < k. Then we have

v

v

v

are at most three monoh i  i
k1 ;
k2 ; k . Because k2 ;
v

v

v

there is at most one solution in each interval, the first order condi^ ðkv Þ=@kv ¼ 0 has at most three positive solutions. Denote tion @ p these solutions (if they exist) by
k1v ;
k2v and
k3v with
k1v <
k2v <
k3v . ^ ðkv Þ=@kv ¼ 1. Thus,
k1v and
k3v It is easy to verify that limkv !0 @ p must be the minimum points, and
k2v is the maximum point. h

lf , the toll system will choose the optimal strategy kv to maximize the profit (where lv ; p are correv Þh1 =kÞkv spondingly determined). Defining p^ 1 ðkv Þ ¼ ðh0 þðkk  lf þkv k Proof of Proposition 3. Given

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 1 ðkv Þ ¼ 0, then 2 cv ðh0 þ ðk  kv Þh1 =kÞkv  ðcv  ps Þkv and let p . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we have lf ¼ ðh0 þ ðk  kv Þh1 =kÞkv 2 cv ðh0 þ ðk  kv Þh1 =kÞkv þ  ^ 1 ðkv Þ=@ lf < 0. Hence when ðcv  ps Þkv . It is easy to verify that @ p .h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lf > l
f ¼ maxx2X ððh0 þ ðk  xÞh1 =kÞx 2 cv ðh0 þ ðk  xÞh1 =kÞxþ i ^ ðkv Þ ¼ p ^ 1 ðkv Þjl ¼l < 0 for any kv 2 X, and the ðcv  ps Þx  x þ kÞ; p f f

U  p þ ps  ðh0 þ ðkf þ kv Þh1 =kÞWðkv ; lv Þ ¼ 0

ðA3Þ

  p  ps þ ðh0 þ kf h1 =kÞWðkv ; lv Þ ¼ ðh0 þ kf h1 =kÞW kf ; lf

ðA4Þ

Taking the partial derivative of (A3) and (A4) with respect to p, from simultaneous equations we can obtain

@kf 1 ¼ h þh l =k ðh0 þh1 kf =kÞðh0 þh1 lf =kÞ 0 1 f h1 @p

2  kðlv kv Þ þ

2

Since

v

^ ðkv Þ=@kv , respectively. There points of @ p h  ^ ðkv Þ=@kv : k1 ;
tone intervals for @ p k1v ;



That is, the utility surplus of customers in the free system, kEh2Hf ½U  hWðkf ; lf Þ, is increasing in lf , thus we have lf ¼ ðB  DÞ=cf . h

lf kf

be a minimum point. When b < 0, Eq. (7) at most has two real roots, denoting them as
k1v and
k2v (if they exist). It is easy to verify



that k1 ; k2 > 0, and k1 ; k2 are the maximum point and minimum

h1 lv

lf kf

=k

h þh1 lf =k

lf > kf ; p P ps , we have 0

lf kf

2 >

k

h1

lf kf

1  P kðl hk

v

vÞ

and

@kf =@p > 0. Thus, the toll system can replace p by kf as the decision. We can obtain U  kv Wðkv ; lv Þh1 =k ¼ ðh0 þ kf h1 =kÞWðkf ; lf Þ from (A3) + (A4). Taking the partial derivative of the equation with respect to lv , we have @kv =@ lv ¼ kv =lv > 0. So the toll system can replace lv by kv as the decision. That is _

pðp; lv Þ ¼ pðkf ; kv Þ

h .  i kv ¼ U  cv þ ps  cv h1 Uk  ðh0 k þ kf h1 ÞW kf ; lf  ðh0 þ ðkf þ kv Þh1 =kÞ h  . h1  Uk  ðh0 k þ kf h1 ÞW kf ; lf

 For the given kf , if there exists a k0v 0 6 k0v 6 k  kf such that

  p kf ; k0v ¼ ðh0 þ kf h1 =kÞW kf ; lf þ ps  ðh0 þ kf h1 =kÞðU  ðh0 þ

  then we must have kf h1 =kÞW kf ; lf Þk= h1 k0v > ps , pðkf ; k  kf Þ > ps . It is easy to verify that @ 2 pðkf ; kv Þ=@k2v ¼ 0. Thus, the toll system obtains its maximum profit at kv ¼ k  kf or kv ¼ 0 (cannot obtain positive profit and does not enter the market). h Proof of Lemma 1. It is clear that a; c > 0. If b P 0, the right item

^ ðkv Þ=@k2v P 0. of Eq. (7) is non-negative and corresponding to @ 2 p ^ ðkv Þ=@kv ¼ 0 at most has a real root in the interval X and it must @p

optimal strategy of the toll system is kv ¼ 0. We next discuss the optimal strategy kv when the toll system enters the market. ^ ðkv Þ at most has a unique From the proof of Lemma 1, p
f , the optimal kv can be maximum point
k2v over X. When lf 6 l k2 (if it exists and satisfies
k2 2 X), achieved at an element of X0 or
v

v

^ ðkv Þ. Assume that kv ¼ k, then k is the that is kv ¼ maxkv 2X0 ;
k2v \X p ^ ðkÞ P 0 and right endpoint of X. Thus, it must have p ^ ðkv Þ=@kv jkv ¼k P 0 (otherwise, there must exist a @p ^ ðk  eÞ > p ^ ðkÞ). Consequently, if kv ¼ k  e; e ! 0þ , such that p pffiffiffiffiffiffiffiffiffiffiffiffiffi p^ ðkÞ < 0 (that is h0 k=lf < 2 cv h0 k þ ðcv  ps Þk) or @ p^ ðkv Þ= h i pffiffiffiffiffiffiffiffiffiffiffiffiffi (that is l2 cv ðh0  h1 Þ= cv h0 k þ cv  ps  @kv jkv ¼k < 0 f ðh0  h1 Þlf > h0 k), it must be kv – k (i.e. 0 < kv < k). h

Proof of Proposition 5. For a given proposed total subsidy D, when lf ¼ ðB  DÞ=cf ; kv may be one of the value k1;2;3 from the ^ ðkv Þ=@ps ¼ kv , it is easy to verify proof of proposition 3. Because @ p that kv ðps Þ must be an increasing function with respect to ps . Denoting Pi ¼ fps jkv ðps Þ ¼ ki ; ps > 0g; i ¼ 1; 2; 3. Then kv ðps Þ is a continuous function of ps in each subfield Pi ði ¼ 1; 2; 3Þ. It is clear that uðps Þ ¼ ps kv ðps Þ  D also is an increasing continuous function of ps in each subfield Pi . If kv ðps Þ p ¼D=k ¼ ki and 2 s kv ðps Þ p ¼D=k ðD=k Þ ¼ ki , where i 2 f1; 2; 3g. uðps Þ is continuous func2 s v

 tion with respect to ps over D=k2 ; D=kv ðD=k2 Þ . Since kv ðps Þ is increasing in ps , it must be uðD=k2 Þ ¼ D=k2  kv ðD=k2 Þ  D < 0, and uðD=kv ðD=k2 ÞÞ ¼ D=kv ðD=k2 Þ  kv ðD=kv ðD=k2 ÞÞ  D > 0. From the intermediate value theorem of the continuous function, there must ~s kv ¼ D in the equilibrium. h ~s such that p exists a unique p Proof of Proposition 6. We will prove it by contradiction. For a ~ f , if the equilibrium customer choice behavior result in given l kf þ kv < k. Then we have

U  p0 þ ps  ðh0 þ ðkf þ kv Þh1 =kÞWðkv ; lv Þ ¼ 0

ðA5Þ

  ~ f p0  ps þ ðh0 þ kf h1 =kÞWðkv ; lv Þ ¼ ðh0 þ kf h1 =kÞW kf ; l

ðA6Þ

W. Chen et al. / Computers & Industrial Engineering 90 (2015) 146–157

from (A6) we have that

lv ¼

References

h0 þ kf h1 =k þ kv : ~ f Þ  p0 þ ps ðh0 þ kf h1 =kÞWðkf ; l

The profit function of the toll system can be rewritten as

pðlv Þ ¼ p~ 1 ðkv ; kf Þ ¼ ðp0  cv Þkv  cv

h0 þ kf h1 =k   : ~ f  p0 þ ps ðh0 þ kf h1 =kÞW kf ; l

~ For a given kf , we have 0 6 kv < k  kf , and because @@kpv1 ¼ p0  cv ,

the maximization profit should be at the kv ¼ k  kf or the toll system does not enters the market. h Proof gð~ k Þ v

of

ps ¼D=~ kv

Proposition

~ kv ¼ ~ k2v ,

7. If

taking

derivative

of

¼ 0 with respect to p0 , we have

    1  . ~k @g k~v  2ðp  c ÞE h k þ c v v 1 =k D 0 v ~ ps ¼ps B C @ ~kv  @ A ~k2 @p @ k~v 0 v   @g ~kv ps ¼D=~kv þ ¼ 0: @p0 0

@g ð~ kv Þj ps ¼~ ps Since ~ kv ¼ ~ k2v is the maximum point, it must be < 0. There@~ kv  ~     @gðkv Þj 2 ps ¼D=~ k v ¼ E ~ kv  2ðp0  cv ÞE ~ kv  cv h1 =k > 0 (or fore, when @p     0    ~ v ¼ h0 þ ~ ~ f  p0 þ D=~ h0 þ ~ kf h1 =k W ~ kv ; l kf h1 =k W ~ kf ; l kv > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp0  cv Þ2 þ cv h1 =k  cv þ p0 ), we have  ~  @g ð~ kv Þj  ~ @g k   ð Þj ~ ~ v ps ¼~ps ps ¼D=k ð2ðp0 cv ÞEðkv Þ=kþcv h1 =kÞD @ kv v ¼  >0 and ~ @p0 @p0 @~ kv k2 v    ~ @ kf ~ f with respect to p0 , we can < 0. Taking derivative of W ~ kf ; l @p0

 @W

obtain that

~ ~ kf ;l f @p0

¼ 

1

2

@~ kv @p0

< 0. On the contrary, if

l~ f þ~kv k

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ~ v 6 ðp0  cv Þ2 þ cv h1 =k  cv þ p0 , h0 þ ~ kf h1 =k W ~ kv ; l

 @~ kf

@p0

@W

P 0 and

we

~ ~ kf ;l f

P 0. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi It is clear that ðp0  cv Þ2 þ cv h1 =k þ p0  cv is increasing in p0 .   ~ f  p0 þ D=~ Taking the derivation of ðh0 þ ~ kf h1 =kÞW ~ kf ; l kv with    ~ =kþh0 h1 l kf h1 =k respect to ~ kf , we can obtain f 2 þ D~ 2 > 0, i.e. h0 þ ~

have

157

@p0

l~ f ~kf

ðkkf Þ

  ~ v is increasing in ~ ^0 such kf . Therefore, if there exists a p W ~ kv ; l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2 ^0  cv Þ þ cv h1 =k  cv þ p ~ v ¼ ^0 , ðp kf h1 =k W ~ kv ; l that h0 þ ~     ~ v > kf h1 =k W ~ kv ; l then there must have h0 þ ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^0 , when p0 < p and ðp0  cv Þ2 þ cv h1 =k  cv þ p0  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 ~ v < ðp0  cv Þ þ cv h1 =k  cv þ p0 h0 þ ~ kf h1 =k W ~ kv ; l when ^0 . Base on the above property, we know that both the p0 > p waiting time and arrival rate of free system reach the minimums at ^0 . h p

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