Selling or sharing: Business model selection problem for an automobile manufacturer with uncertain information

Abstract

The rapid development of car sharing raises a concerned question: is it a better alternative to car sales? This paper studies a business model selection problem where an automobile manufacturer with uncertain information chooses his strategy from three options: sale only (SO), car sharing only (CO), and both sale and car sharing (SC). A game-theoretic model is built to investigate what are the key factors influencing the introduction of car sharing and how it brings environmental impact. Our results show that both the selling price and car-sharing fee keep invariable regardless of which business model the manufacturer chooses. CO is a dominated strategy. Besides, SC dominates SO when producing cost is either sufficiently low or high, while the opposite is true for a medium producing cost. In addition, only when producing cost is sufficiently high, a win-win outcome emerges: the introduction of car sharing not only increases the manufacturer’s profit, but also reduces total vehicle quantity, which helps alleviate urban congestion. Finally, our results prove that the increase of uncertainty degree benefits the manufacturer when SC is adopted.

Keywords

Car sharing game theory pricing strategy uncertain information business model selection

1. Introduction

Recently, several automobile manufacturers such as Daimler and BMW, have initiated car-sharing programs (i.e., Car2Go and ReachNow) to consumers in the market [14]. In most developed countries, though many households own at least one car, a large amount of vehicles are not fully utilized. Therefore many customers are attracted to car sharing since it both let people enjoy the convenience of self-driving and save them an expense for buying cars [26]. With such advantages, this new business model achieves great success and keeps a high growth rate [34]. Berg Insight [18] estimates the quantity of worldwide car-sharing membership has reached 23.8 million at the end of 2017. It forecasts this number will grow to about 60.8 million at the end of 2022 and total number of car-sharing fleet will be more than approximately 705,000.

Some researchers regard car sharing as a greener and more sustainable transportation mode compared with driving private cars [13]. The development of modern automobile industry makes cars more popular and affordable. At the same time, however, it also causes environmental issues—congestion, air pollution, etc. One particular feature of car sharing is that one shared car can satisfy several customers’ needs. Because of this, car sharing is viewed as an effective way to reduce the number of car ownership, which makes it seem to be a superior alternative to car sales.

Nevertheless, on the one hand, from the perspective of automobile manufacturers, the introduction of car-sharing service is likely to expand the market as it may be attractive to some non-car owners. This market expansion effect tends to increase total number of vehicles. On the other hand, it is naturally to expect that car sharing can also cannibalize the selling market, which transfers some car owners to car-sharing program members. Consequently, it is not clear whether introducing car sharing is a better business model for automobile manufacturers and the city environment. In this paper, we are interested in finding out: which business model is a better strategy for the manufacturer? What are the key factors impacting the manufacturer’s business model selection? Does the introduction of car sharing always help reduce total vehicle quantity?

Because of the fast changing environment and existence of much unobservable information in real world, hardly can manufacturers accurately forecast demands of their products by collecting historical data. In such cases, manufacturers have to rely on the belief degrees suggested from experienced experts or managers to make their decisions. However, human beings tend to overestimate the possibilities of unlikely events [19]. Previous researches show that it would be a huge mistake if we treat belief degrees with probability theory [29]. Under this condition, uncertainty theory is proved to be more suitable to deal with belief degrees. Since manufacturers are lacking of historical samples about customers’ transportation need, valuation of driving self-owned and shared cars before they introduce car-sharing programs, these three factors are assumed to be uncertain variables in this paper.

We introduce some of our main findings and managerial insights as follows. First, the equilibrium selling price and car-sharing fee are invariable no matter which business model the manufacturer adopts. Second, the strategy car sharing only is dominated by both sale and car sharing. Consequently, the manufacturer has no incentive to thoroughly abandon their car selling business. Third, both sale and car sharing is a better strategy only if producing cost is either sufficiently low or high, otherwise sale only achieves higher profit. Besides, our results demonstrate that only when producing cost is sufficiently high, can car sharing reduce total vehicle quantity. This result may provide some suggestions to municipal governments on when it is preferable to encourage the development of car sharing. Last, the increase of uncertainty degree tends to benefit the manufacturer if he chooses both sale and car sharing.

The rest of this paper is organized as follows. Section 2 reviews related literature about car sharing and uncertainty theory. In Section 3, we discuss how the model is set up, and introduce the parameters used throughout this paper. Then equilibrium solutions are computed and presented in Section 4. Comparing the equilibrium solutions, we show several main findings of our paper and give managerial insights in Section 5. Two sets of numerical experiments are conducted in Section 6 to study the effects of uncertain variables. Finally, Section 7 summarizes main conclusions in this paper. Proofs and some preliminaries of uncertainty theory are given in Appendices A and B, respectively.

2. Literature review

2.1. Car sharing

2.1.1. User characteristics

As a new business model, the characteristics of car sharing fall between renting and owning: it is more convenient than renting but less expensive than owning if use ratio is low. A stream of literature studies what are the key factors influencing customers’ adoption of car sharing. Wilhelms et al. [37] show that economic interest ("earn"), quality of life ("enjoy"), helping others ("enrich"), and sustainability ("enhance") are the four motivational patterns influencing customers’ participation of peer-to-peer car sharing. They find self-centered motives “earn” and “enjoy” are the two main drivers of customers’ participation, while “enhance” is rather an indirect consequence. In Greece, Efthymiou et al. [11] conduct an on-line survey focusing on the young people between 18 and 35 years old, and find car sharing mainly attracts people who commute with bus, trolley or tram. By conducting a survey in a university, Zheng et al. [38] show customers’ status rather than socioeconomic factors mainly influences the acceptance of car sharing. Through direct contacts and web-based survey in North America, Burkhardt and Millard [6] find social activists, environmental protectors, innovators, economizers, or practical travelers are most likely to be attracted by car sharing.

2.1.2. Environmental impact

Some researchers investigate the environmental impact of car sharing. Through an on-line survey to car-sharing members, Martin and Shaheen [30] find car sharing reduces green house gas emissions to a large extent. Baptista et al. [1] conduct a case study to show car sharing can dramatically reduce the number of ownership. In addition, related researches show people drive less when they become car-sharing members [20 , 35]. Different from these empirical studies, we find car sharing may not always bring positive environmental impact since it increases total number of vehicles when marginal producing cost is sufficiently low.

2.1.3. Operations management

Because car-sharing service allows users to return vehicles at any permissible parking area, it causes imbalanced vehicle quantity among different parking lots. He et al. [17] study a car-sharing service provider’s geographic service region design problem by developing a mixed integer program model. Chang et al. [7] investigate location design and relocation problems considering a car-sharing firm operating a mixed fleet of vehicles under the limitation of CO₂ emissions. Several works [4 , 32] study relocation problems in one-way vehicle-sharing systems, while Weikl and Bogenberger [36] discuss the same problem in a free-floating system. The aforementioned studies solve optimization problems about choosing locations of charging points or strategies of relocation, whereas our paper focuses on an automobile manufacturer’s business model selection to investigate which factors impact the adoption of car sharing.

2.2. Business model with uncertain information

Uncertainty theory, founded by Liu [27] and refined by Liu [29], is a branch of axiomatic mathematics to study the indeterminacy when historical data is insufficient. It has been well developed and widely applied to many areas [8 , 33]. Several studies investigate pricing strategies under different business models. Cheng et al. [10] study the impact of store brands under different supply-chain power structures and find the retailer always derives higher profit while the manufacturer tends to suffer if he has more bargaining power. Ke et al. [23] study a remanufacturing problem in a closed-loop supply chain and demonstrate that when retailers are more risk averse, they earn less while manufacturers gain more. Considering channel preference and sales effort, Ke and Liu [22] investigate the impact of supply chain members’ competition on their profit under uncertain environment. They indicate that both the total profits under centralized and decentralized cases increase in expected value of sales effort elasticity. To our knowledge, no previous research studies the business model selection problem in an automobile industry under uncertain environment. Our results show that a higher uncertainty degree tends to increase the manufacturer’s profit when he provides both sale and car-sharing service.

2.3. Automobile business model

This paper also complements a stream of literature studying automobile business model selection. Based on a qualitative analysis of some main electric vehicle manufacturers to trace the evolution of their business models, Bohnsack et al. [3] find incumbent and entrepreneurial firms choose different approaches for their innovation. Landau et al. [25] conduct a case study of a German automobile manufacturer’s internationalization to India. They identify four phases of adaptation process when firms from developed markets broaden their business to emerging markets. Bellos et al. [2] study the business model choice and product line design problem of an automobile manufacturer who faces environmental regulations. Contrary to their studies, we set up our model based on a different market structure and obtain some novel managerial insights.

3. Model description

We consider a model where one monopolistic automobile manufacturer implements sale only (SO) strategy to produce cars at a constant marginal cost and sells them in a market with n customers. In addition, the manufacturer also contemplates to adopt car sharing only (CO) strategy by producing vehicles for sharing use. Besides, combining these two strategies to provide both sale and car sharing (SC) service can be the third option for the manufacturer.

In a typical car-sharing program, customers should first register on the website to become car-sharing members, then they are allowed to get access to a fleet of vehicles. Fee is charged according to driving time. Since car-sharing fleet size is limited, there exists a risk that no car is available if they are all being occupied by other users. Consequently, from the perspective of the manufacturer, he needs to decide the fleet size and per-unit-of-time sharing fee to ensure a certain service level a before introducing car-sharing business. In our model, the service level represents the chance that customers’ transportation needs can be satisfied by shared vehicles.

We assume all the customers are strategic and forward-looking. After observing strategies from the manufacturer, customers decide their own transportation modes among three options—ownership, membership, and outside option. Ownership (O) represents customers’ decisions of owning cars, and membership (M) denotes being car-sharing program members. In addition, consumers may choose outside option (∅) to take public transportation. Fig. 1 presents the strategies of the manufacturer and customers in our model.

Fig. 1.

Strategies of the manufacturer and customers.

The sequence of events is as follows. First, an automobile manufacturer decides which business model of the three (i.e., SO, CO, and SC) to adopt. Second, the manufacturer enters the market and sets the selling price and car-sharing fee to maximize his profit. Third, after observing the strategies of the manufacturer, customers strategically choose their transportation modes from three options—ownership, membership, and outside option. Accordingly, this model can be studied as a sub-game perfect Nash equilibrium problem. By backward induction, let us first examine customers’ utility functions.

Customers’ utility functions

In our model, customers are assumed to be residents who live not far from the car-sharing parking lot to ensure they do not exclude the option membership. At first, we define the utility function of ownership as: $U_{O} = (\tilde{v} - e) \tilde{t} - f .$ (1)

When a customer drives his own car, we assume he derives the valuation $\tilde{v}$ in per unit of time. In practice, people take very different attitudes toward driving: some see it as a lifestyle and enjoy it, while others may get bored of it and prefer to be passengers. For this reason, we assume customers’ valuation of driving is their private information and heterogeneous in our model. We use the notation e to denote the per-unit-of-time operating cost generated from driving vehicles, such as gasoline, maintenance, and parking, to name a few. The parameter $\tilde{t}$ represents customers’ transportation need, which is defined as a fraction of vehicles’ useful life. Without loss of generality, vehicles’ useful life is normalized to be 1. Besides, f is a decision variable of vehicle’s selling price. This utility function indicates that car owners derive $\tilde{v} \tilde{t}$ for the satisfaction of driving but have to cover the expense of total operating cost $e \tilde{t}$ and selling price f.

The utility derived from membership is defined as: $U_{M} = a (\tilde{θ} \tilde{v} - p) \tilde{t},$ (2) where $\tilde{θ}$ is the discount factor and p is the per-unit-of-time fee.

In practice, customers generally have a lower valuation about driving shared cars, due to several reasons such as cost of arriving at the car-sharing parking lot, concerns about finding a car with poor condition, or inconvenience of returning vehicles to designated parking area. Accordingly, customers are assumed to have the valuation $\tilde{θ} \tilde{v}$ when they choose membership, where $\tilde{θ}$ is the discount factor with positive value and less than 1. Different from the car renting, members in a car-sharing program only need to pay the per-unit-of-time sharing fees p according to their occupying time, while the operating cost such as gasoline is covered by the car-sharing service provider. Note that car-sharing members’ transportation needs can only be satisfied with probability a, so they attain the utility $a \tilde{θ} \tilde{v} \tilde{t}$ at the expense of $ap \tilde{t}$ . For the scenario when no car is available, customers are assumed to choose the outside option, and derive zero utility: $U_{\emptyset} = 0 .$ (3)

Generally speaking, since the valuation of driving is unobservable, it is hard to accurately estimate the real distribution of $\tilde{v}$ . For a specific customer, each time using car-sharing service, his discount factor varies according to the conditions of vehicles, distances of arriving at a car-sharing parking lot, etc. Besides, it is hard for a consumer to accurately estimate his transportation need. Hence, we assume $\tilde{v}$ , $\tilde{θ}$ , and $\tilde{t}$ are uncertain variables in our model.

In order to obtain the closed-form solutions, we assume all uncertain variables in our model are positive and mutually independent with regular uncertainty distributions. Customers and the manufacturer are risk neutral and aiming to maximize their expected utility and profit, respectively. In addition, customers’ per-unit-of-time valuation $\tilde{v}$ has a linear distribution $L (0, 1)$ . Notations used throughout this paper are summarized in Table 1.

Table 1

Notations

Parameters	Description
c	Marginal producing cost.
e	Per-unit-of-time operating cost.
a	Car-sharing service level.
n	Market size.
$\tilde{t}$	Transportation need, an uncertain variable.
$\tilde{v}$	Per-unit-of-time valuation of driving, an uncertain variable.
$\tilde{θ}$	Discount factor, an uncertain variable.
Decision Variables	Description
f	Retail price.
w	Wholesale price.
p	Per-unit-of-time fee for car-sharing service.

With the above assumptions, given a specific customer’s valuation, his expected utility of ownership and membership can be computed as follows: $\begin{matrix} E [U_{O} | \tilde{v}] \equiv & \int_{0}^{1} ((\tilde{v} - e) Φ_{t}^{- 1} (α) - f) d α \\ = & E [\tilde{t}] (\tilde{v} - e) - f, \end{matrix}$ (4)

$\begin{matrix} E [U_{M} | \tilde{v}] \equiv & \int_{0}^{1} (a (Φ_{θ}^{- 1} (α) \tilde{v} - p) Φ_{t}^{- 1} (α)) d α \\ = & aE [\tilde{θ} \tilde{t}] \tilde{v} - aE [\tilde{t}] p, \end{matrix}$ (5) where $Φ_{t}^{- 1} (α)$ and $Φ_{θ}^{- 1} (α)$ represent the inverse uncertainty distributions of $\tilde{t}$ and $\tilde{θ}$ , α ∈ [0, 1].

4. Equilibrium solutions

For the ease of expression, we use subscript i ∈ {O, M, ∅} to denote customers’ choices, and superscript j ∈ {SO, CO, SC} to represent the three strategies of the manufacturer. For example, $d_{O}^{SC}$ means the demand for owning cars when both sale and car sharing service are provided. In what follows, we check the manufacturer’s strategies individually.

4.1. Sale only

Let us first investigate the sale only strategy, when the automobile manufacturer produces vehicles at a marginal cost c and sells them at a selling price f. Hence, customers choose their transportation modes between ownership and outside option. Note that $\tilde{v}$ is the private information, so customers know their valuation before making decisions. They choose ownership if and only if $E [U_{O} | \tilde{v}] ⩾ U_{\emptyset}$ , and outside option otherwise.

Then we obtain the demand function: $d_{O}^{SO} = \begin{matrix} n (1 - f / E [\tilde{t}] - e) . \end{matrix}$ (6)

The manufacturer’s expected profit is therefore formulated as: $\begin{matrix} {\begin{matrix} max_{f} Π^{SO} = (f - c) d_{O}^{SO} \\ subject to : \\ f - c > 0 \\ f < E [\tilde{t}] (1 - e) . \end{matrix} \end{matrix}$ (7)

Since the second-order derivative $\frac{\partial^{2} Π^{SO}}{\partial f^{2}} = - \frac{2 n}{E [\tilde{t}]} < 0$ , Π^SO is concave in f. Let $\frac{\partial Π^{SO}}{\partial f} = 0$ , we have $f^{SO *} = \frac{c + E [\tilde{t}] (1 - e)}{2} .$ (8)

Note that the condition $f < E [\tilde{t}] (1 - e)$ must be satisfied, otherwise the demand for ownership is zero. To focus on the nontrivial parameter region, we assume the inequality $E [\tilde{t}] (1 - e) - c > 0$ holds in the rest of this section. For brevity, we let $A \equiv E [\tilde{t}] (1 - e) - c$ . Accordingly, the optimal demand is obtained as: $d_{O}^{SO *} = \frac{nA}{2 E [\tilde{t}]} .$ (9)

The expected profit can be obtained as: $Π^{SO *} = \frac{n A^{2}}{4 E [\tilde{t}]} .$ (10)

4.2. Car sharing only

When the manufacturer chooses car-sharing only strategy, he uses produced vehicles to gain profit through car-sharing program. In this case, customers choose their strategies between membership and outside option. They choose membership if and only if $E [U_{M} | \tilde{v}] ⩾ U_{\emptyset}$ , and outside option otherwise.

The demand function in car sharing only case is given as: $d_{M}^{CO} = \begin{matrix} n (1 - \frac{E [\tilde{t}] p}{E [\tilde{θ} \tilde{t}]}) . \end{matrix}$ (11)

The manufacturer’s expected profit is defined as:

$\begin{matrix} {\begin{matrix} max_{p} E [Π^{CO}] = E [a (p - e) \tilde{t} d_{M}^{CO} - c q_{M}^{CO}] \\ subject to : \\ p - e - c > 0 \\ p < \frac{E [\tilde{θ} \tilde{t}]}{E [\tilde{t}]}, \end{matrix} \end{matrix}$ (12) where $q_{M}^{CO}$ is the fleet size.

In the above function, the first term $a (p - e) \tilde{t} d_{M}^{CO}$ represents the revenue of operating car-sharing service, and the second term $c q_{M}^{CO}$ is the cost of producing shared vehicles. $q_{M}^{CO} = \frac{a}{1 - a} + aE [\tilde{t}] d_{M}^{CO}$ represents the relation of fleet size, service level, and quantity of car-sharing members. This formulation is initiated by Bellos et al. [2], who model the shared cars circulating process by applying the closed queuing network technique in their excellent study.

The expected profit under car sharing only case is obtained as:

$\begin{matrix} E [Π^{CO}] = & \int_{0}^{1} (an (p - e - c) Φ_{t}^{- 1} (α) (1 - \frac{E [\tilde{t}] p}{E [\tilde{θ} \tilde{t}]}) \\ - \frac{ca}{1 - a}) d α \\ = & \frac{anE [\tilde{t}] (E [\tilde{θ} \tilde{t}] - E [\tilde{t}] p) (p - e - c)}{E [\tilde{θ} \tilde{t}]} \\ - \frac{ca}{1 - a} . \end{matrix}$ (13)

Since $\frac{\partial^{2} E [Π^{CO}]}{\partial p^{2}} = - \frac{2 an E^{2} [\tilde{t}]}{E [\tilde{θ} \tilde{t}]} < 0$ , E [Π^CO] is concave in p. By solving the first order condition, we have:

$p^{CO *} = \frac{E [\tilde{t}] (c + e) + E [\tilde{θ} \tilde{t}]}{2 E [\tilde{t}]} .$ (14)

The condition $p < \frac{E [\tilde{θ} \tilde{t}]}{E [\tilde{t}]}$ must be satisfied, otherwise the demand for car sharing is zero. To focus on the nontrivial parameter region, we assume the inequality $E [\tilde{θ} \tilde{t}] - E [\tilde{t}] (c + e) > 0$ holds in the rest of this section. For brevity, let $B \equiv E [\tilde{θ} \tilde{t}] - E [\tilde{t}] (c + e)$ . We have the demand function as:

$d_{M}^{CO *} = \frac{nB}{2 E [\tilde{θ} \tilde{t}]} .$ (15)

Accordingly, the car-sharing fleet size is given as: $\begin{matrix} q_{M}^{CO *} = \frac{a}{1 - a} + \frac{anE [\tilde{t}] B}{2 E [\tilde{θ} \tilde{t}]} . \end{matrix}$ (16)

The expected profit under car sharing only case is: $\begin{matrix} E [Π^{CO *}] = \frac{an B^{2}}{4 E [\tilde{θ} \tilde{t}]} - \frac{ca}{1 - a} . \end{matrix}$ (17)

4.3. Both sale and car sharing

When it comes to both sale and car sharing, the manufacturer provides customers more choices. Under this circumstance, car owners have the valuation ${\tilde{v} | E [U_{O} | \tilde{v}] ⩾ E [U_{M} | \tilde{v}], E [U_{O} | \tilde{v}] ⩾ U_{\emptyset}}$ , car-sharing members have the valuation ${\tilde{v} | E [U_{M} |$ $\tilde{v}] > E [U_{O} | \tilde{v}], E [U_{M} | \tilde{v}] ⩾ U_{\emptyset}}$ , and ${\tilde{v} | E [U_{O} | \tilde{v}] < U_{\emptyset}, E [U_{M} | \tilde{v}] < U_{\emptyset}}$ represents the customers who choose outside option.

The demand for ownership is givens as: $d_{O}^{SC} = \begin{matrix} n (1 - \frac{f + eE [\tilde{t}] - aE [\tilde{t}] p}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]}) . \end{matrix}$ (18)

The demand for membership is given as:

$d_{M}^{SC} = \begin{matrix} n (\frac{E [\tilde{θ} \tilde{t}] f + eE [\tilde{t}] E [\tilde{θ} \tilde{t}] - E^{2} [\tilde{t}] p}{E [\tilde{θ} \tilde{t}] (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}])}) . \end{matrix}$ (19)

The manufacturer’s profits by selling cars and providing car-sharing service are given as: $Π_{O}^{SC} = (f - c) d_{O}^{SC},$ (20) $Π_{M}^{SC} = a (p - e) \tilde{t} d_{M}^{SC} - c q_{M}^{SC},$ (21) where $q_{M}^{SC} = \frac{a}{1 - a} + aE [\tilde{t}] d_{M}^{SC}$ , in accordance with what we have discussed in the car sharing only case.

When the manufacturer chooses both sale and car sharing, the expected profit he gains is as follows: $\begin{matrix} {\begin{matrix} max_{f, p} E [Π^{SC}] = E [Π_{O}^{SC} + Π_{M}^{SC}] \\ subject to : \\ f - c > 0 \\ p - e - c > 0 \\ f < E [\tilde{t}] + aE [\tilde{t}] p - aE [\tilde{θ} \tilde{t}] - eE [\tilde{t}] \\ f > \frac{E^{2} [\tilde{t}] p}{E [\tilde{θ} \tilde{t}]} - eE [\tilde{t}] . \end{matrix} \end{matrix}$ (22)

The expected profit under both sale and car sharing is given as: $\begin{matrix} E [Π^{SC}] = \int_{0}^{1} ((f - c) n (1 - \frac{f + eE [\tilde{t}] - aE [\tilde{t}] p}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]}) \\ + an (p - e - c) Φ_{t}^{- 1} (α) \frac{E [\tilde{θ} \tilde{t}] f + eE [\tilde{θ} \tilde{t}] E [\tilde{t}] - E^{2} [\tilde{t}] p}{E [\tilde{θ} \tilde{t}] (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}])} \\ - \frac{ca}{1 - a}) d α \\ = & \frac{anE [\tilde{t}] (E [\tilde{θ} \tilde{t}] f + eE [\tilde{θ} \tilde{t}] E [\tilde{t}] - E^{2} [\tilde{t}] p) (p - e - c)}{E [\tilde{θ} \tilde{t}] (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}])} \\ + n (f - c) (1 - \frac{f + eE [\tilde{t}] - aE [\tilde{t}] p}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]}) - \frac{ca}{1 - a} . \end{matrix}$ (23)

The Hessian matrix of E [Π^SC] with respect to f and p is:

$H = (\begin{matrix} \frac{- 2 n}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]} & \frac{2 anE [\tilde{t}]}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]} \\ \frac{2 anE [\tilde{t}]}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]} & \frac{- 2 an E^{3} [\tilde{t}]}{E [\tilde{θ} \tilde{t}] (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}])} \end{matrix}) .$ (24)

Since $\frac{- 2 n}{E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]} < 0$ , and $| H | = \frac{4 {an}^{2} E^{2} [\tilde{t}]}{E [\tilde{θ} \tilde{t}] (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}])}$ >0, E [Π^SC] is jointly concave in f and p. Solving the first order condition, we derive the equilibrium selling price and car-sharing fee as follows:

$f^{SC *} = \frac{c + E [\tilde{t}] (1 - e)}{2},$ (25) $p^{SC *} = \frac{E [\tilde{θ} \tilde{t}] + E [\tilde{t}] (c + e)}{2 E [\tilde{t}]} .$ (26)

To exclude the cases when demands for ownership or membership is zero, we focus on the parameter region A - aB > 0, $E [\tilde{t}] B - E [\tilde{θ} \tilde{t}] A > 0$ . For brevity, let $C \equiv E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]$ , so the demand of ownership and membership are respectively given as: $d_{O}^{SC *} = \frac{n (A - aB)}{2 C},$ (27) $d_{M}^{SC *} = \frac{n (E [\tilde{t}] B - E [\tilde{θ} \tilde{t}] A)}{2 E [\tilde{θ} \tilde{t}] C} .$ (28)

The total quantity of vehicles produced by the manufacturer is: $\begin{matrix} Q^{SC *} = d_{O}^{SC *} + q_{M}^{SC *} \\ = \frac{anE [\tilde{t}] (E [\tilde{t}] B - E [\tilde{θ} \tilde{t}] A)}{2 E [\tilde{θ} \tilde{t}] C} + \frac{n (A - aB)}{2 C} + \frac{a}{1 - a} . \end{matrix}$ (29)

The expected profit is given by: $\begin{matrix} E [Π^{SC *}] = \frac{anB (E [\tilde{t}] B - E [\tilde{θ} \tilde{t}] A)}{4 E [\tilde{θ} \tilde{t}] C} \\ + \frac{nA (A - aB)}{4 C} - \frac{ca}{1 - a} . \end{matrix}$ (30)

5. Analysis and comparison

Proposition 1. The selling prices under SO and SC are equal and per-unit-of-time car-sharing fees under CO and SC are equal. Or mathematically, f^SO* = f^SC* and p^CO* = p^SC*.

One may intuit that since providing both sale and car-sharing service brings more intense competition to the market, it may lead to lower prices. However, Proposition 1 shows that both the selling price and car-sharing fee keep invariable. The reason behind this result is that when the manufacturer chooses SC, high-valuation customers choose ownership (i.e., $(f + eE [\tilde{t}] - aE [\tilde{t}] p) / (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}]) < \tilde{v} < 1$ ), and medium-valuation customers choose membership (i.e., $E [\tilde{t}] p / E [\tilde{θ} \tilde{t}] < \tilde{v} < (f + eE [\tilde{t}] - aE [\tilde{t}] p) / (E [\tilde{t}] - aE [\tilde{θ} \tilde{t}])$ ). Under CO, customers choose membership if $E [\tilde{t}] p / E [\tilde{θ} \tilde{t}] < \tilde{v} < 1$ . As a consequence, given the equilibrium sharing fee under CO, the manufacturer does not attract extra customers when he switches to SC. Now suppose the manufacturer lowers the sharing fee under SC, it makes him suffer more since it not only decreases the car-sharing profit (because the sharing fee under CO is the optimal), but also cannibalizes the selling market, which lowers the sale profit. Same reason is also applicable for f^SO* = f^SC*. Thus, the manufacturer does not change the selling price or sharing fee regardless of which business model he adopts.

This result is also consistent with practices in reality that the selling price of the smart (a small family car manufactured by Daimler) is stable before or after the car-sharing service Car2Go is introduced to the market. As a consequence, customers gain no benefit if they don’t change their strategies when the manufacturer choose different business models.

Proposition 2.The demands $d_{O}^{SO *}$ , $d_{M}^{CO *}$ and $d_{O}^{SC *}$ decrease in c, but $d_{M}^{SC *}$ increases in c. However, $d_{O}^{SO *}$ , $d_{M}^{CO *}$ , $d_{O}^{SC *}$ and $d_{M}^{SC *}$ all decrease in e.

Since the increase of producing cost makes the manufacturer bear heavier cost burden, which tends to raise the selling price and car-sharing fee, it is easily to infer that the demands under SO and CO decrease in marginal producing cost. However, under the both sale and car sharing strategy, our result shows the demand for ownership decreases in c, but the demand for membership increases in c. To find out the reasons behind this result, let us review customers’ expected utility, $E [U_{O} | \tilde{v}] = E [\tilde{t}] (\tilde{v} - e) - f$ and $E [U_{M} | \tilde{v}] = aE [\tilde{θ} \tilde{t}] \tilde{v} - aE [\tilde{t}] p$ . As we can see, $E [U_{O} | \tilde{v}]$ is more sensitive with respect to f than $E [U_{M} | \tilde{v}]$ with respect to p. In addition, ∂f^SC/∂c = ∂p^SC/∂c = 1/2, which means every unit of increase in c makes both f^SC and p^SC 1/2 units higher. As a consequence, with the increase of c, customers’ utility of ownership declines faster than that of membership, which makes car sharing attracts more owners.

Proposition 2 also indicates that the four demands we have discussed above all decrease in e. The intuition stems from the fact that unlike the producing cost, the operating cost is generated only when driving, which is same for either car owners or members. As a result, the utilities of both owners and members have much less distinct sensitivities with respect to e in this case, so each demand decreases in operating cost.

Proposition 3.The equilibrium demand $d_{M}^{SC *} + d_{O}^{SC *} = d_{M}^{CO *}$ . However, $d_{M}^{CO *} > d_{O}^{SO *}$ if the condition $c (E [\tilde{θ} \tilde{t}] - E^{2} [\tilde{t}]) > e (E^{2} [\tilde{t}] - E [\tilde{t}] E [\tilde{θ} \tilde{t}])$ holds.

Generally speaking, providing multiple choices may help expand the market. However, Proposition 3 shows some interesting results that the aggregate demands are the same between CO and SC. The reason is as what we have discussed in Proposition 1: only customers with high valuation choose ownership under SC, and they are the same people who have to adopt membership under CO since no better choice is available. Thus, the aggregate demand under SC is equal to that under CO.

Proposition 3 also compares demands under CO and SO. Since $E [\tilde{θ} \tilde{t}] - E^{2} [\tilde{t}] > 0$ and $E^{2} [\tilde{t}] - E [\tilde{t}] E [\tilde{θ} \tilde{t}] > 0$ (for the proof of propositions, please refer to the Appendix A), our results suggest that when producing cost exceeds a threshold (Alternatively, operating cost is less than a threshold), it tends to induce a higher demand under CO compared with that under SO, and vice versa. In other words, the manufacturer can attract more customers if he switches from SO to CO when manufacturing vehicles with comparatively high producing cost and low operating cost.

Proposition 4. E [Π^SC*] > E [Π^CO*], so the strategy CO is dominated by SC.

Proposition 4 presents an interesting finding, which shows that CO is the dominated strategy, so the manufacturer does not choose it under any circumstance. The intuition behind this result is not hard to reach: remember what we have discussed in Proposition 3, customers with high valuation have to adopt membership under CO because they have no choice to buy cars. Consequently, under SC the manufacturer can gain more profit by providing differentiated services to different market segment. Note that switching from CO to SC not only enlarges the manufacturer’s profit, but also increases high-valuation customers’ net utility and satisfies their needs. That is to say, switching from CO to SC achieves Pareto improvement and increases social welfare.

Besides, Proposition 4 sheds some light on how to introduce and develop car-sharing service in practice. Since CO is a dominated strategy, car sharing can only be considered as a supplement, but not a substitute, to satisfy medium-valuation customers. Hence, though this new business model has advantages such as higher vehicle use ratio, it must exist with the traditional car selling model.

Since CO is a dominated strategy, we focus on SO and SC in the rest of this paper.

Proposition 5. If the market size is large enough, there exists threshold values c₁, c₂ and c₃, such that when c ∈ (0, c₁) ⋃ (c₃, 1), SC is the dominant strategy, whereas when c ∈ (c₁, c₃), SO is the dominant strategy. In addition, $Q^{SC *} > d_{O}^{SO *}$ if c ∈ (0, c₂), and $Q^{SC *} < d_{O}^{SO *}$ if c ∈ (c₂, 1), where c₂ = (c₁ + c₃)/2.

Fig. 2.

Impact of producing cost on business model selection and total vehicle quantity.

Proposition 5 indicates the major results in this paper, which is also illustrated in Fig. 2. When the producing cost is either sufficiently low or high, the equilibrium profit under SC is larger than that under SO, while if the producing cost is at medium value, SO is a better strategy. This result reveals that when car sharing is available, it is more profitable for the manufacturer to introduce it in the market with low-cost or high-cost vehicles. However, medium-cost automobiles are more suitable for sale-only market. In addition, the introduction of car sharing can help reduce total number of vehicles only if producing cost is large enough, otherwise the manufacturer has incentive to produce more vehicles.

Combining the results of business model selection and total vehicle quantities, we obtain another interesting finding. Firstly, when c ∈ (0, c₁), though it is of manufacturer’s interest to introduce SC, it also tends to add more vehicles to the market, which may even aggravate some severe urban problems such as traffic jam or air pollution. Consequently, it may not seem to be a good idea to let the manufacturer provide car-sharing service under this condition. Secondly, when c ∈ (c₂, c₃), though total number of vehicles under SC is lower than that under SO, the manufacturer has no incentive to introduce car sharing. Lastly, we find the most promising way for the manufacturer to introduce this new business model is when c ∈ (c₃, 1): not only the introduction of car sharing makes manufacturer attain larger profit, it also helps reduce total vehicle quantity. Accordingly, when c > c₃, car sharing can be welcomed by both the manufacturer and municipal governments. Proposition 5 presents some useful suggestions to policy makers about when it is preferable to encourage the development of car sharing.

Proposition 6. If $\tilde{θ}$ and $\tilde{t}$ have linear distributions, the expected profit E [Π^SC*] increases with respect to uncertainty degrees.

Proposition 6 shows that a more changeable environment tends to benefit the manufacturer under SC. It demonstrates an interesting finding that though the manufacturer and customers in our model are assumed to be risk neutral, both sale and car sharing business model can achieve higher profit when uncertainty degree increases. Consequently, ceteris paribus, the introduction of car sharing may be more attractive in an emerging market than in a developed market.

6. Numerical experiments

Due to the complicated forms of uncertain variables in equilibrium solutions, it is difficult to obtain extra closed-form results with respect to $\tilde{θ}$ and $\tilde{t}$ . In this section, we conduct two sets of numerical experiments to study the impacts of transportation need and discount factor. The basic parameter settings are as follows: $\tilde{θ} \sim L (0.7, 0.9)$ , $\tilde{t} \sim L (0.2, 0.35)$ , c = 0.1, e = 0.1, a = 0.7, n = 100.

6.1. The effects of transportation need $\tilde{t}$

We set uncertainty distributions of $\tilde{t}$ as $L (0.2, 0.25)$ , $L (0.2, 0.3)$ , $L (0.2, 0.35)$ , $L (0.2, 0.4)$ , and $L (0.2, 0.45)$ to investigate how transportation need impacts equilibrium solutions. Experimental results are presented in Table 2. Note that under SO, total vehicle quantity is equal to the demand for ownership, or mathematically $d_{O}^{SO} = Q^{SO}$ . We compare results (see Fig. 3) and show some main findings:

Both the selling price f and car-sharing fee p increase in $E [\tilde{t}]$ . That is to say, the higher transportation need encourages the manufacturer to set a more expensive price and fee. In addition, per-unit-of-time fee is more robust to transportation need, which implies car-sharing service provider may not implement price discrimination between different market segments.

When customers have higher transportation needs, quantity of self-owned vehicles increases rapidly, but car-sharing fleet size decreases moderately.

Consequently, total vehicle number increases in transportation need under both SO and SC business model.

Manufacturer’s profit increases in transportation need, since it leads to more demands and higher prices.

Table 2
Equilibrium results under SO and SC with different uncertainty distributions of transportation need $\tilde{t}$

Strategies $\tilde{t}$ f p d _O q _M Q E [Π]

SO (0.2,0.25) 0.151 22.778 22.778 1.167

(0.2,0.3) 0.163 25 25 1.563

(0.2,0.35) 0.174 26.818 26.818 1.978

(0.2,0.4) 0.185 28.33 28.33 2.408

(0.2,0.45) 0.196 29.615 29.615 2.850

SC (0.2,0.25) 0.151 0.502 3.768 7.655 11.423 1.566

(0.2,0.3) 0.163 0.503 8.652 7.400 16.052 1.844

(0.2,0.35) 0.174 0.505 12.683 7.138 19.821 2.165

(0.2,0.4) 0.185 0.506 16.067 6.870 22.937 2.519

(0.2,0.45) 0.196 0.506 18.947 6.599 25.546 2.897

Strategies	$\tilde{t}$	f	p	d _O	q _M	Q	E [Π]
SO	(0.2,0.25)	0.151		22.778		22.778	1.167
	(0.2,0.3)	0.163		25		25	1.563
	(0.2,0.35)	0.174		26.818		26.818	1.978
	(0.2,0.4)	0.185		28.33		28.33	2.408
	(0.2,0.45)	0.196		29.615		29.615	2.850
SC	(0.2,0.25)	0.151	0.502	3.768	7.655	11.423	1.566
	(0.2,0.3)	0.163	0.503	8.652	7.400	16.052	1.844
	(0.2,0.35)	0.174	0.505	12.683	7.138	19.821	2.165
	(0.2,0.4)	0.185	0.506	16.067	6.870	22.937	2.519
	(0.2,0.45)	0.196	0.506	18.947	6.599	25.546	2.897

Fig. 3.

Comparison of equilibrium results under SO and SC with different uncertainty distributions of transportation need $\tilde{t}$ .

6.2. The effects of discount factor

\tilde{θ}

To examine the effects of discount factor, we change the uncertainty distributions of $\tilde{θ}$ as: $L (0.5, 0.6)$ , $L (0.6, 0.7)$ , $L (0.7, 0.8)$ , $L (0.8, 0.9)$ , $L (0.9, 1)$ (see Table 3). Fig. 4 shows the comparison results:

The manufacturer tends to charge a higher car-sharing fee when discount factor increases. However, selling price is not affected by discount factor since the price is unaltered under either SO or SC, as we have discussed in Proposition 1.

When the valuation of driving shared cars becomes higher, customers tend to be reluctant to buy cars but choose membership instead. Thus, the increase of $E [\tilde{θ}]$ transfers more car owners to members, which decreases the demand in the selling market, but increases the fleet size of shared cars.

When $E [\tilde{θ}]$ is comparatively low, total vehicle quantity under SC is larger than that under SO. However, with the increase of $E [\tilde{θ}]$ , Q^SC decreases dramatically and falls far below than Q^SO.

The expected profit gained from the manufacturer under SC increases in $E [\tilde{θ}]$ . This result suggests that $\tilde{θ}$ is a critical factor impacting profit of car-sharing business.

Table 3
Equilibrium results under SO and SC with different uncertainty distributions of discount factor $\tilde{θ}$

Strategies $\tilde{θ}$ f p d _O q _M Q E [Π]

SO (0.5,0.6) 0.174 26.818 26.818 1.978

(0.6,0.7) 0.174 26.818 26.818 1.978

(0.7,0.8) 0.174 26.818 26.818 1.978

(0.8,0.9) 0.174 26.818 26.818 1.978

(0.9,1) 0.174 26.818 26.818 1.978

SC (0.5,0.6) 0.174 0.377 23.551 3.953 27.504 1.791

(0.6,0.7) 0.174 0.427 20.134 5.142 25.276 1.890

(0.7,0.8) 0.174 0.477 15.703 6.384 22.087 2.048

(0.8,0.9) 0.174 0.527 9.729 7.833 17.561 2.284

(0.9,1) 0.174 0.577 1.233 9.704 10.937 2.638

Strategies	$\tilde{θ}$	f	p	d _O	q _M	Q	E [Π]
SO	(0.5,0.6)	0.174		26.818		26.818	1.978
	(0.6,0.7)	0.174		26.818		26.818	1.978
	(0.7,0.8)	0.174		26.818		26.818	1.978
	(0.8,0.9)	0.174		26.818		26.818	1.978
	(0.9,1)	0.174		26.818		26.818	1.978
SC	(0.5,0.6)	0.174	0.377	23.551	3.953	27.504	1.791
	(0.6,0.7)	0.174	0.427	20.134	5.142	25.276	1.890
	(0.7,0.8)	0.174	0.477	15.703	6.384	22.087	2.048
	(0.8,0.9)	0.174	0.527	9.729	7.833	17.561	2.284
	(0.9,1)	0.174	0.577	1.233	9.704	10.937	2.638

Fig. 4.

Comparison of equilibrium results under SO and SC with different uncertainty distributions of discount factor $\tilde{θ}$ .

7. Conclusions

In this paper, we study a business model selection problem for an automobile manufacturer to investigate what are the key factors impacting the introduction of car sharing and whether it helps reduce total number of vehicles. Because in real world, customers’ valuation of driving, discount factor and transportation need are generally hard to estimate accurately, they are assumed to be uncertain variables. Several managerial insights are obtained based on the closed-form equilibrium solutions. Besides, two sets of numerical experiments are conducted to study the effects of uncertain variables.

Our results show that both the selling price and car-sharing fee keep invariable regardless of which business model the manufacturer chooses. Car sharing only is a dominated strategy. However, the manufacturer tends to provide both sale and car-sharing service when marginal producing cost is either sufficiently low or high, while sale only is a better choice if the producing cost is at medium value. We also find that only when producing cost is sufficiently high can both sale and car sharing be a win-win strategy—it not only increases the manufacturer’s profit but also reduces total vehicle quantity. Besides, the increase of uncertainty degree also benefits the manufacturer if he chooses both sale and car sharing.

Our work can be extended in the following directions. First, throughout this paper, we assume the manufacturer is a monopoly in the market to focus on which factors impact business model selection. However, a competitive market composed of several manufacturers may be worthwhile studying in another paper. Second, a specific customer is assumed to choose only one option in our model, while in practice he may possess more than one choice simultaneously. For instance, a car owner can also be a car-sharing member. Third, our model can be extended to a multi-stage dynamic game, where customers can make their decisions at different stages in a time line rather than only once.

Footnotes

Appendix

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.71371141) and the Fundamental Research Funds for the Central Universities.

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