Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3241973, 12 pages

http://dx.doi.org/10.1155/2016/3241973

## A Game of Two Elderly Care Facilities: Competition, Mothballing Options, and Policy Implications

^{1}School of Finance, Zhejiang University of Finance & Economics, Hangzhou 310018, China^{2}China Academy of Financial Research, Zhejiang University of Finance & Economics, Hangzhou 310018, China

Received 14 April 2016; Accepted 14 July 2016

Academic Editor: Xiaodong Lin

Copyright © 2016 Congcong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This article develops a model to investigate the entry strategies of private investors to the elderly care service market, with the purpose of explaining the reasons behind dilemma of low signing rate plaguing China’s Public-Private Partnership projects. We focus on the competition between two private investors with or without mothballing options under price uncertainty. After the derivation of equilibria of entry strategies, we employ numerical examples to analyze the dependencies of entry thresholds on market parameters, cost parameters, subsidy, and possession of mothballing option. Conclusions are drawn and some policy implications are given with the intention to alleviate the problem of low signing rate.

#### 1. Introduction

In recent years, the looming challenges of China’s aging population draw more and more attentions. According to the disclosed data of National Bureau of Statistics of People’s Republic of China, senior citizens (citizens over 60 years old) are expected to take up more than 30% of the total population in 2050, compared with only 18% in 2015. Since 2014, the State Council of China has issued a series of policies to promote the Public-Private Partnerships (PPP) projects aiming to address the imminent challenges of aging population, motivating private investors to support the infrastructure, facilities, and human resources for elderly care services. More than 16% of all the PPP projects promoted by State Council are directly related to elderly care facilities. However, the private investors seem more conservative than expected. The signing rate of these PPP projects has remained at low level since the beginning. In April, 2015, National Development and Reform Commission (NDRC) estimated the signing rate to be between 10% and 20%. With the implementation of simulation packages, polices including reduced financing costs, tax abatement, and government subsidies, the signing rate rose slowly to 31.5%.

There are several reasons behind the low signing rate, and one of them is the market uncertainty. When private investors consider the participation of PPP projects, they face opportunities of irreversible investments. Just as the purchaser of the call option in financial derivative market, the potential private investor of PPP project holds a real option. The exercise of the real option, or signing of the PPP contract, depends on the prevailing market price (e.g., for elderly care service), as well as market volatility.

Competition is another important reason. Most contracts of elderly care PPP projects promoted do not contain clauses of exclusive licenses. The leader who provides elderly care service in a local community market cannot prevent the entry of follower and therefore has to consider the probability with which the monopoly market turns to a duopoly one and the resulting profit reduction.

Because of the aforementioned reasons, we use the framework of real option games to analyze the dilemma of low signing rate of PPP projects with the background of elder care industry. Smets first introduced game theory to real option analysis [1]. Based on Smets’ work, Grenadier considered market uncertainty and used option games approach to analyze seemingly irrational real estate development cascades [2]. Huisman and Kort later considered an investment timing problem in a duopoly market [3]. Based on the assumption of symmetric duopoly, Grenadier developed Cournot Nash equilibrium option exercise strategies [4]. Huang and Wu further developed the duopoly option games to incorporate two sources of uncertainties, namely, market uncertainty and emergencies [5]. Qiu and Yu expanded the option games of only two firms to games in oligopoly setting [6]. Folta and O’Brien analyzed the tension between the option to defer and the option to grow and recognized the nonmonotonic effect [7]. Pawlina and Kort analyzed the impact of cost asymmetry on the optimal real option exercise strategies in duopoly setting [8]. Smit and Trigeorgis provided a valuation method to assess whether investors overpaid for infrastructure assets or a premium for the strategic growth option value was justified due to their operating flexibility [9]. Suttinon et al. proposed using option games to evaluate trade-offs between flexibility and strategic commitment in industrial water infrastructure projects [10]. Martzoukosa and Zachariasb demonstrated optimal exercise conditions for the real option to make costly strategic preinvestment R&D decisions in the presence of spillover effects with analytic tractability [11].

Among various types of real options, mothballing options have been discussed by many scholars recently. Mothballing option allows its holder to suspend operation if the market conditions are unfavorable. Takashima et al. analyzed the optimal strategies of two firms, with asymmetries in fixed cost, variable cost, and possession of mothballing options, to enter the electricity market under competition and uncertain price [12]. Zhang et al. expanded the model by incorporating the parameter of government subsidy [13]. Gao et al. developed a valuation model for investment of low-carbon power plant with mothballing options to calculate the investment threshold and its sensitivity to various market parameters [14]. Lv and Shao presented an asymmetric option game in an oligopoly market and derived investment thresholds for firms in the market [15].

The contribution of the paper is twofold. First, although there are many studies conducted using option game approach to analyze the competitions in several industries, elder care industry is not among them. We therefore use the real options approach and game theory to consider both uncertainty and competition in elderly care industry, which is particularly concerned by China’s central government because of the looming challenge of the vast aging population. Second, among all the real options, such as options to expand, to contract, and to abandon, mothballing option is largely ignored in spite of its great potential. We focus our study on the mothballing option adopted by private elderly care facilities and analyze the effects of mothballing option on the entry thresholds of private investors in the elderly care industry.

We consider two symmetric potential private investors of PPP projects who compete with each other to enter elderly care service market. Neither of the two investors is in possession of mothballing options in the first case, while both of them have mothballing options in the second case. The dependencies of market entry thresholds on market parameters, cost parameters, subsidy, and possession of mothballing option are then analyzed.

We found that the entry thresholds for the private investors of PPP projects in elderly care industry decrease with the increase of expected growth rate and government subsidy and increase with the increase of volatility, discount rate, fixed cost, and variable cost. By comparing the dependencies of thresholds on different parameters, we found that volatility had more significant effect on thresholds than expected growth rate and that variable cost had more significant effect than fixed cost. Based on the results, we propose that the government should allow the inclusion of mothballing option in PPP contracts, reduce financing costs, and subsidize the facilities for their variable cost instead of fixed cost in elderly care PPP projects to increase the signing rate.

Our paper is organized as follows. Section 2 introduces the model and basic settings. In Section 3, we derive the entry thresholds and the optimal strategies of investors. In Section 4, we use numerical examples with realistic parameter specifications to demonstrate our results and policy implications. Section 5 concludes.

#### 2. Model

We consider two potential private investors of PPP projects, labeled as 1 and 2. Index 1 represents an arbitrary investor, and index 2 represents the other. We assume that they are willing to enter the elderly care market if the entry thresholds are reached, and these investors’ decisions affect each other.

The profit function of investor at time satisfieswhere is the demand parameter for investors given , the number of investors that have entered the elderly care market. is the price per unit of elderly care service, and is the variable cost per unit. If investor has mothballing option, the profit function is given by

Furthermore, we assume the price at time follows a geometric Brownian motionwhere denotes the instantaneous expected growth rate of , denotes the instantaneous volatility of , and denotes a standard Wiener process.

Then, the value function of investor is given by where denotes the stopping time for investor to enter the market, denotes the collection of admissible stopping times, denotes a discount rate, denotes the fixed investment cost for the investor , and denotes the government subsidy for investor .

We assume that the private investors are price-takers, and the entries of these investors have no effect on prices for elderly care services. These are reasonable assumptions, because the prices for care services for senior citizens in China are regulated by government and fluctuate randomly due to expansion of the market, changing policies and other sources of uncertainty. However, the two elderly care facilities constructed by private investors in our models can be assumed to have impact on the sales volume of each other, since they are the competitors who provide the services of the same quality in the same local community. There are three patterns of entry. If , investor enters the market first at time and earns a higher profit (temporary monopoly profit) until investor enters the market at . In this case, investor is called the* leader*, and . If , investor (leader) enters the market first at , and investor waits to enter the market and earns no profit until . In this case, investor is called the* follower*, and . If , both investors earn lower profit because they enter the market simultaneously and share the market from time . This case is called* simultaneous investment*, and in this case . We assume that each investor strives to maximize their investment value (present value), and therefore we maximize the value function equation (4) in the next section at the moment the leader has entered, , after Grenadier [2] and Takashima et al. [12].

#### 3. Equilibria

##### 3.1. Case 1: Two Investors without Mothballing Option

In this case, the two investors are identical and willing to enter the elderly care market. First, we obtain the value function of simultaneous investment

Since neither of the investors is in possession of mothballing option, we havewhere .

We then consider the follower’s value. Since the leader has already entered the market, the value function of the follower is obtained bywhere is the stopping time for the follower to enter and is the collection of admissible stopping times. We assume that the constant threshold of the follower is given; then satisfies

According to the Bellman equation and Ito’s lemma, we calculate the value function of the follower by the following ordinary differential equation:with boundary conditions

Among the three boundary conditions, the second condition is called the value-matching condition and the third is called the smooth-pasting condition. From (9) and (10), we have and the optimal entry price for the follower iswhere is the positive root of the following quadratic equation:

We now consider the leader’s side. Suppose that the follower chooses the optimal policy; we obtain the value function of the leader as follows:

Similarly, according to the Bellman equation and Ito’s lemma, the value function of the leader satisfies the following ordinary differential equation:with boundary conditions

Here, the smooth-pasting condition is not necessary, because (14) is not a maximum problem. From (14) and (15), we have

Proposition 1. *There exists a unique value for , labeled as , such that**The stopping time is defined as*

does not have the analytical solution and therefore has to be numerically determined by the following equation:

We use the strategy space and equilibrium concept defined by Huisman and Kort [3] due to Proposition 1 and obtain the equilibrium in this case.

Proposition 2. *Depending on the value of , there exist three types of equilibria:*(1)*If , there are two possible scenarios. In the first one, investor is the leader and enters the elderly care market at time , and investor is the follower and enters at time with probability . The second is the symmetric counterpart, and the probability that both investors enter simultaneously is zero.*(2)*If , there are three possible scenarios. In the first, investor is the leader and enters the market at time , and investor is the follower and enters at time with probability . The second is the symmetric counterpart. In the third, both investors enter simultaneously at time with probability .*(3)*If , then both investors enter at time with probability .*

*The proof is presented by Huisman and Kort [3]. Proposition 2 states that there is no simultaneous investment if the game starts with low price for elderly care service.*

*3.2. Case 2: Two Investors with Mothballing Options*

*In Case , the two investors are identical and willing to enter the elderly care market as in Case . The difference is that the elderly care facilities can now temporarily suspend their operation if the price for care service is unfavorable (less than the variable cost).*

*We first obtain the value function of simultaneous investmentAccording to the Bellman equation and Ito’s lemma, we calculate the value function of simultaneous investment by the following ordinary differential equation:with the value-matching condition and the smoothing-pasting condition at similar to Case ; we havewhere is the negative root of (13).*

*We then consider the value function of the follower. Since the leader has already entered the market, we havewhere denotes the stopping time for the follower to enter. Assuming that the threshold of the follower is given, takes the form of*

*As in Case , we haveThe optimal threshold does not have analytical solution. From the smooth-pasting condition at , it can be numerically determined by the following equation:By solving (27), we have the solution of .*

*Assuming that the follower chooses the optimal strategy, the value function of the leader is given byFrom Case , we obtain the value function of the leaderwhere*

*Proposition 3. There exists a unique value for , labeled as , such that The stopping time is defined as*

* has to be numerically determined by the following equation: Again, we use the strategy space and equilibrium concept defined by Huisman and Kort [3] based on Proposition 3.*

*Proposition 4. Depending on the value of , there exist three types of equilibria:(1)If , there are two possible scenarios. In the first one, investor is the leader and enters the elder care market at time , and investor is the follower and enters at time with probability . The second is the symmetric counterpart, and the probability that both investors enter simultaneously is zero.(2)If , there are three possible scenarios. In the first, investor is the leader and enters the market at time , and investor is the follower and enters at time with probability . The second is the symmetric counterpart. In the third, both investors enter simultaneously at time with probability .(3)If , then both investors enter at time with probability .*

*The proof is the same as Huisman and Kort [3].*

*4. Numerical Examples*

*4. Numerical Examples*

*In Sections 2 and 3, we developed models which enable us to analyze the entry strategies of potential private investors to the elderly care service market and then derived equilibria in two duopoly market cases. In the first case, neither of the two potential investors is in possession of mothballing option, while in the second case, both of them have the mothballing options.*

*In this section, we use specific values for parameters, which have realistic meanings, to calculate the entry thresholds of leader and follower in a local elderly care service market.*

*4.1. Parameter Specifications*

*4.1. Parameter Specifications*

*Table 1 shows the parameter specifications for our models. The discount rate is set to 0.05, the prevailing interest rate for government supported projects in China. The growth rate of Chinese senior citizens is predicted to be 1.3% per year, which is calculated based on the published data of National Bureau of Statistics of People’s Republic of China in April, 2015, and the elderly care service market is expected to grow at the rate of 4.1%. The expected growth rate in our model is therefore set to 0.04. We estimate the volatility of the elderly care service market by calculating the average standard deviation of annual earnings per share of listed companies which provide care services for senior citizens in China, and the result is 0.2.*