The Car Sharing Economy: Interaction of Business Model Choice and Product Line Design

Transcription

1 The Car Sharing Economy: Interaction of Business Model Choice and Product Line Design Ioannis Bellos School of Business, George Mason University, Fairfax, VA 22030, Mark Ferguson Moore School of Business, University of South Carolina, Columbia, SC 29208, L Beril Toktay Scheller College of Business, Georgia Institute of Technology, Atlanta, GA 30308, In what follows we provide details on the proofs of our results The analytical expressions are explicitly given unless they hinder manuscript readability in which case only the shorthand notation is provided For instance, instead of providing the complete expression of the optimal profit under the O, M equilibrium we use Π O,M to denote it The complete forms are available from the authors upon request When comparing across different equilibria we utilize the notation h, l with h {O} and l {O, M, } For instance, e H O, M denotes the optimal fuel efficiency of the vehicles sold to the High segment under the O, M equilibrium Proof of Remark 1 Assume that each customer requests a vehicle according to a Poisson process with rate λ and that the mean duration of each vehicle use is τ Set λ τ = d, where d denotes customers transportation needs We model the service time at the /M/1 using a Poisson process with rate λ = n i λ, where n i is the size of the segment served through Membership Given that the idle time of the node represents the time when customer requests cannot be satisfied because no cars are available, the utilization ρ /M/1 provides the probability that a customer finds a vehicle available We assume that customer usage times are independent and identically distributed according to a general probability distribution G with mean τ Therefore, we capture the service process through the /G/ node whose service time distribution is G 1

2 2 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy The OEM guarantees an exogenously determined service level a, by choosing S such that a = ρ /M/1 Based on the FPM approximation Whitt 1984, we construct the open counterpart of the closed queueing network and equate the expected equilibrium population of the open network to S The open network comprises an M/M/1 and an M/G/ node The M/M/1 node is characterized by external Poisson arrivals with rate λ c and exponential service times with parameter λ The service rate of each server at the M/G/ node is 1/τ and the expected number of jobs ie, vehicles in the open network is E [N c ] = E [ N M/M/1 ] +E [ NM/G/ ] = λ c λ λ c +λ c τ For the open network to be equivalent to the closed network, the external rate λ c must satisfy E [N c ] = S In this case, ρ M/M/1 = λc λ determines the service level, which must equal a Substituting λ c = aλ in E [N c ] = S results in S a a + aλτ Given that λ = n 1 a i λ and d = λ τ, the fleet size that achieves an availability level a is S a a 1 a + an id Proof of Propositions 1 and 2 For each possible equilibrium we first determine the optimal prices and then the optimal fuel efficiencies Under O, and for given efficiencies, the OEM determines the selling price F based on max F Π O, = F c w 1 e H 2 c e e H 2 subject to the individual rationality constraints d ν + 1 e H θ H g F 0, and d ν 1 e H g θ L F 0 which can be rewritten as d ν 1 e H g θ L F d ν + 1 e H θ H g The profit Π O, is linear increasing in F Therefore, for a given fuel efficiency the optimal selling price is F = d ν + 1 e H θ H g Define Π O, = ΠO, F = F Then, the OEM determines the optimal fuel efficiency based on max eh ΠO, such that 0 e H 1 ΠO, is concave in the fuel efficiency because 2 ΠO, / 2 e H = 2 c w + c e < 0, therefore, after solving Π O, / e H = 0 we obtain e H = 2cw dθ H g 2c w+c e It is straightforward to show that e H 0 iff c w c w = dθ H g > 0 and e 2 H 1 iff c e c e = dθ H g, which is always true 2 as c e < 0 Following Chen 2001 we focus on interior values of e H 0, 1 Hence, we assume that c w > c w Based on e H we also calculate F = dν + dθ H g 2c e+dθ H g 2c w+c e and Π O, = dν + dθ H g 4c e+dθ H g 4c wc e F L 4c w+c e Under O, O and for given efficiencies, the OEM determines the selling prices F H, and we use the subscript i = H and i = L to indicate the selling price charged to the High and Low segment, respectively based on max FH,F L Π O,O = F H c w 1 e H 2 c e e H 2 + F L c w 1 e L 2 c e e L 2 n L subject to the individual rationality constraints

3 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy 3 d ν + 1 e H θ H g F H 0 and d ν + 1 e L θ L g F L 0, and the incentive compatibility constraints d ν +1 e H θ H g F H d ν +1 e L θ H g F L and d ν + 1 e L θ L g F L d ν + 1 e H θ L g F H The individual rationality constraints can be rewritten as F H d ν + 1 e H θ H g and F L d ν + 1 e L θ L g Similarly, the incentive compatibility constraints are rewritten as F L + d e H e L g θ L F H F L d e H e L θ H g, which hold iff e L e H we show below that at optimality e L > e H holds The profit Π O,O is linear increasing in both F H and F L Therefore, for given vehicle efficiencies, the optimal selling prices are F L = d ν + 1 e L θ L g and F H = min { FL + d e L e H θ H g, d ν + 1 e H θ H g }, from which is simple to show that FH = d ν + 1 e L θ L + e L e H θ H 1 e H g Define Π O,O = ΠO,O F H = F H, F L = F L Then the OEM determines the optimal vehicle efficiencies based on max eh,e L ΠO,O such that 0 e H 1 and 0 e L 1 The profit Π O,O is jointly concave in the vehicle efficiencies because 2 ΠO,O / 2 e H = 2 c w + c e < 0, 2 ΠO,O / 2 e L = 2 c w + c e n L < 0, and 2 ΠO,O / 2 e H 2 ΠO,O / 2 e L 2 2 ΠO,O / e H e L = 4 cw + c e 2 n L > 0 Therefore, after solving Π O,O / e H = 0 and gn L + θ H θ L Π O,O / e L = 0 we obtain e H = 2cw dθ H g 2c w+c e and e L = 2cwn L d 2c w+c en L The efficiency e H 0 iff c w c w = dθ H g > 0 and e 2 H 1 iff c e c e = dθ H g, which is 2 always true as c e < 0 Similarly, e d θ H θ L +n L g θ L L 0 iff c w c w = 2n L, which is always true as c w < 0 and e d θ H θ L +n L g θ L L 1 iff c e c e = > 0 Once again, we focus on interior values e H, e L 0, 1 and for that reason we assume that c w > c w and c e > c e It is straightforward to show that e L e H = d+n L θ H θ L 2c w+c en L > 0, which means that e L > e H holds Based on the optimal efficiencies, we also obtain F L = dν dg θ L 2c en L d θ H θ L +g θ L n L 2c w+c en L, FH = F L + d2 θ H gθ H θ L 2c w+c en L, and Π O,O = + n L dν 4cwcen L d dn L g θ L 2 + θ H θ L 2 +4c en L g θ L 4c w+c en L Under O, M and for given efficiencies, the OEM determines the selling price F and the per-unit-of-time price p based on max F,p Π O,M = F c w 1 e H 2 c e e H 2 + a p g 1 e L n L d c w 1 e L 2 + c e e L 2 a 1 a + an Ld subject to the individual rationality constraints d ν + θ H g 1 e H F 0 and ad ν + θ L 1 e L p 0, and the incentive compatibility constraints d ν + θ H g 1 e H F ad ν + θ H 1 e L p and ad ν +θ L 1 e L p d ν +θ L g 1 e H F The individual rationality constraints can be rewritten as F d ν + θ H g 1 e H and p ν + θ L 1 e L Similarly, the 2n L

4 4 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy incentive compatibility constraints are rewritten as d ap + 1 a ν 1 e H g θ L a 1 e L θ L F d ap + 1 a ν + 1 e H θ H g a 1 e L θ H, which hold iff e L e H we show below that at optimality e L > e H holds The profit Π O,M is linear increasing in both F and p Therefore, for given vehicle efficiencies the optimal per-unitof-time price is p = ν + 1 e L θ L and the optimal selling price is F = d a p + 1 a ν + 1 e H θ H g a 1 e L θ H Define Π O,M = ΠO,M F = F, p = p Then, the OEM determines the optimal vehicle efficiencies based on max eh,e L ΠO,M such that 0 e H 1 and 0 e L 1 The profit Π O,M is jointly concave in the vehicle efficiencies because 2 ΠO,M / 2 e H = 2 c w + c e < 0, 2 ΠO,M / 2 e L = 2 c w + c e a dn L + 1 a 1 < 0, and 2 2 ΠO,M / 2 e H 2 ΠO,M / 2 e L 2 4c w+c e ΠO,M / e H e L = 2 a 1+1 adn L > 0 1 a Therefore, after solving Π O,M / e H = 0 and Π O,M / e L = 0 we obtain e H = 2cw dθ H g 2c w+c e and e L = 2cw 1+1 adn L d1 a g θ L n L +θ H θ L 2c w+c e1+1 adn L The efficiency e H 0 iff c w c w = dθ H g > 0 and e 2 H 1 iff c e c e = dθ H g, which is always true as c 2 e < 0 With respect to e L, in this case we have that e 1 ad g θ L n L +θ H θ L L 0 iff c w ĉ w = 21+1 adn L, which is always true as ĉ w < 0 In addition, e 1 ad g θ L n L +θ H θ L L 1 iff c e ĉ e = > 0 As 21+1 adn L before, our focus is on interior values e H, e L 0, 1 and for that reason we assume that c w > c w and c e > ĉ e Based on the optimal efficiencies we also obtain the optimal prices F and p along with the optimal profit Π O,M analytical expressions available upon request Note that the difference ĉ e c e = d 1+1 an L 1 d g θ L n L +θ H θ L 2n L 1+1 adn L > 0 due to n L > n L = 1 > 1 d1 a 01 This implies that e i h, l 0, 1 for all i {H, L}, h {O}, and l {O, M} when c w > c w and c e > max {ĉ e, c e } = ĉ e By comparing the fuel efficiencies under the different market equilibria we find that e H O, = e H O, O = e H O, M = 2cw dθ H g 2c w+c e and e L O, O e H O, O = d θ H θ L 2c w+c en L > 0, therefore, e L O, O > e H O, O Similarly, given that n L > n L, e L O, M e H O, M = d θ H 1+dn L + adn L + 0 and e L O, M e L O, O = d 1+1 dn L 1 a +g 1 a1 dn L 1 1 aθ L 2c w+c e1+1 adn L > g θ L n L +θ H θ L θ H 2c w+c en L 1+1 adn L > 0, hence, e L O, M > e H O, M and e L O, M > e L O, O In terms of comparative statics we obtain: e H / θ H = d < 0, e 2c w+c e H / d = θ H g < 0, e 2c w+c e L dn O, O / θ H = H 2c w+c en L > 0, e L O, O / d = θ H θ L +g θ L n L 2c w+c en L > 0, 1 As we have already stated in 42 of the main paper, throughout the analysis we assume that n L > n L, otherwise O, M is always dominated the derivation of the condition is provided in the proof of Proposition 4

5 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy 5 e L O, O / = dθ H θ L 2c w+c en L > 0, e L O, O / n L = dθ H θ L 2c w+c en < 0, e 2 L O, M / a = L d θ H θ L +g θ L n L < 0, e dn 2c w+c e1+dn L 1 a 2 L O, M / θ H = H 1 a > 0, e 2c w+c e1+dn L 1 a L O, M / d = 1 a θ H θ L +g θ L n L > 0, e 2c w+c e1+dn L 1 a 2 L O, M / = e L O, M / n L = d1 a g θ L d 1 aθ H θ L d1 aθ H θ L 2c w+c e1+dn L 1 a > 0, and g θ L d1 aθ H θ L > 0 iff n 2c w+c e1+dn L 1 a 2 H < Using a similar notation to distinguish between the prices under different equilibria we have, FH O, O F L O, O = d2 θ H gθ H θ L 2c w+c en L > 0, FH O, M FH O, O = dθ H θ L 21 ac en L 1+1 adn L +d 1 an L a d 1 θ H θ L +g θ L n L 2n L c w+c e1 adn L +1 > 0 due to c e > ĉ e and n L > n L, and FH O, F H O, M = daθ H θ L 2c e 1+1 adn L d1 a θ H θ L +g θ L n L 2c w+c e1+1 adn L > 0 due to c e > ĉ e Therefore, FH O, > F H O, M > F H O, O > F L O, O Proof of Proposition 3 We start by calculating the CAFE level under the different equilibria Specifically, r O, = e H = 2cw dθ H g 2c w+c e, r O, O = e H + + a 1 a +an Ld e H a 1 a + +an Ld + a 1 a +an Ld e L = 2c wa+1 aadn L + +1 ad aθ H θ L +g θ H +an L g θ L 2c w+c ea+1 aadn L + We continue by calculating n L e L = 2cw dg θ L 2c w+c e, and r O, M = the difference r O, O r O, = dθ H θ L 2c w+c e > 0, which implies that r O, O > r O, With respect to the CAFE under O, O and O, M we have r O, O r O, M = dag1 1 a1 dn L aθ L n L a 11 d+1+1 a 2 θ H θ L 2c w+c ea+1 aadn L +, which can be positive or negative However, r O, O r O, M / θ H = d 1 a 2 2c w+c ea+1 a +adn L > 0, therefore, r O, O > r O, M for all θ H > θ H = {θh : r O, O r O, M = 0} = θ L a 1 aa dn L + +ag 1+1 an L 1 d ag θ L 1+1 dn L 1 a 1 a 2 1 a 2, where θh > θ L because θh θ L = > 0 for all ν > ν Proof of Proposition 4 We observe that the OEM s profit always increases in ν as Π O, / = d > 0, Π O,O / = d + n L > 0 and Π O,M / = d + an L > 0, with Π O, / > Π O,M / > Π O,O / Define ν = } {ν : Π O,O ΠO, = 0 = 4c wc en 2 L +dg θ Ln L +θ H θ L 4c en L g θ L n L +θ H θ L 4c w+c edn > 0 for c 2 e > c e and ˆν = L } {ν : Π O, = 0 = 4cwce d 4c e+dθ H g θ H g 4c w+c ed Given that Π O, / > Π O,O/ and that ν ˆν = +n L θ H θ L 4c en L +d θ L θ H +n L θ H +θ L 2g > 0 for c e > c e, if focused only on 4c w+c en 2 L selling, the OEM prefers to induce O, O for all ν ν, O, for all ν [ˆν +, ν, and, for all ν [0, ˆν + Define ν = } {ν : Π O,M ΠO,O = 0 and ν = } {ν : Π O,M ΠO, = 0

6 6 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy analytical expressions available upon requestwe also calculate dπ O,M Π O, = adn dν L > 0 and dπ O,M Π O,O = 1 a dn dν L < 0 Therefore, with car sharing, the OEM prefers to induce O, M over O, for all ν ν and O, M over O, O for all ν ν For O, M to exist it is necessary that ν ν = a 1+1 a1 dn L 4c wc en L + 1 ad2 g θ L n L +θ H θ L adn L > 0 41 a 2 dn 2 L cw+ce and ν ν = 1+1 a1 dn L 4cwcen L + d2 g θ L n L +θ H θ L 2 1 a 1+1 adn L 4dn 2 L cw+ce > 0, both of which are true iff n L > n L = 1 1 d1 a When n L > n L, the thresholds ν, and ν are guaranteed to exist as ν, ν > 0 for c e > ĉ e Hence, the OEM induces i, for all ν [0, ˆν +, ii O, for all ν [ˆν +, ν, iii O, M for all ν [ν, ν] and iv O, O for all ν > ν Proof of Corollary 1 For all ν [ν, ν the OEM replaces O, with O, M We have E O, = ζ p + ζ u e H d and E O, M = ζ p nh + a + an 1 a Ld + ζ u e H d + ζ u e L adn L, where ζ u e L < ζ u e H because e L > e H The change in the impact is 1 given by E O, M E O, = a ζ 1 a p + d ζ p + ζ u e L n L > 0, therefore, E O, M > E O, For all ν [ ν, ν] the OEM replaces O, O with O, M We have E O, O = ζ p + n L + ζ u e H + ζ u el O, O n L d and E O, M = ζ p nh + a + an 1 a Ld + ζ u e H d + ζ u el O, M adn L, where ζ u el O, M < ζ u el O, O < ζ u e H because e L O, M > e L O, O > e H The change in the impact is given by E O, M E O, O = ζ a p 1 ad n 1 a L + aζ u el O, M ζ u el O, O dn L < 0 for n L > n L, hence, E O, M < E O, O The majority of the generated insights throughout the paper are not contingent on the OEM selecting a specific equilibrium The reason is that the optimal prices and efficiencies are characterized for given equilibria and compared across them Including more Membership equilibria may only impact the presentation of Proposition 4 and Corollary 1 without however, changing any of the major points we made based on these results Extending the proofs of Proportions 1 and 2 to the M, M and M, equilibria, we obtain: i e L M, M = 2cw1+1 adn L+1 adθ H θ L +n L g 2c w+c e1+1 adn L, e H M, M = 2c w1+1 ad +1 ad g θ H 2c w+c e1+1 ad, e H M, = 2cw+g g 1+dn L 1 a 2c w+c e ii p M, = ν + θ H g 2c w+g θ 1+dn L 1 a H 2c w+c e, ph M, M, p L M, M and iii Π M,M, and Π M, analytical expressions for ph M, M, p L M, M, Π M,M, and Π M, are available upon request Based on these derivations, it is straightforward to obtain the following: Π O, = d, Π O,O Π M,M = d + n L > Π O,, Π O,M = d + an L < Π O,O, Π M, = ad + n L < Π O,M, and Π M,M > Π M, The comparisons of Π, = adn L < Π O,M, under the

7 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy 7 different equilibria indicate that for moderate values of n L ie, when the Low segment is not very large or very small compared to the High segment the OEM may prefer to induce the following equilibria from smaller to larger values of ν: M, O, M, M O, M O, O; in contrast to O, O, M O, O, which is what Proposition 4 indicates However, the inclusion of more Membership equilibria does not refute any of the insights provided in Proposition 4 and Corollary 1 In particular, Proposition 4 makes two major points: i In markets with high valuation of vehicle use, the OEM prefers to sell to both segments; ii Car sharing is not necessarily associated with low valuations of vehicle use On the contrary, car sharing can be the optimal choice in a medium-valuation market Our additional analysis indicates that these two conclusions continue to hold regardless of whether we include Membership only equilibria These insights remain valid even when accounting for M, and M, M Along the same lines, the primary contribution of Corollary 1 is to show that environmental impact reduction and CAFE level compliance may be at odds, and that there are market conditions where environmental impact increases while CAFE compliance improves and vice versa After including M, M and M, we obtain the following: rm,m ro,o d1 ag θ n L = L > 0 and EM,M EO,O c w+c e2+d 1 a 2 n L = 1 ad ζ p d ζ u e L O, O aζ u e L M, M < 0 it is straightforward to show that ζ u e L O, O > ζ u e L M, M as e L O, O < e L M, M implying that for ˆn L < n L < ñ L, where ˆn L = {nl : r M, M r O, O = 0} and ñ L = {nl : E M, M E O, O = 0}, r M, M > r O, O and E M, M > E O, O Similarly, rm,m ro, d1 ag θ n L = L > 0 c w+c e2+d 1 a 2 and EM,M EO, = ad ζ p + ζ u e H O, > 0 imply that for n L > max { n L, ń L }, where n L n L = {nl : r M, M r O, = 0} and ń L = {nl : E M, M E O, = 0}, r M, M > r O, O and E M, M > E O, The above analysis indicates that the tension between environmental impact and CAFE level continues to exist even when we account for additional Membership equilibria Proof of Proposition 5 In the calculation of r O, M we now include an incentive n multiplier m as follows: r O, M, m = H +m a 1 a +adn L e H + m a 1 a +adn L +m a 1 a +adn L e L O, M It is straightforward to show that r O, M, m / m = e L O,M e H a > +m a 1 a +adn L 2 0 as e L O, M > e H Hence, for any m > m = {m : r O, M, m r O, O = 0}, where m = 1 a +adn L 1 aθ H θ L ag1 1 dn L 1 a θ L +1 aθ L dn L + θ H > 1 for any n L >

8 8 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy n L and θ H > θ H, we obtain r O, M, m > r O, O Similarly, we obtain 1+1 dn L 1 a1 aθ H θ L m/ g = < 0 for any n ag 1+1 dn L 1 a+θ L 1 a θ H + +1 dn L θ L 2 L > 1 dnh1 a n L, m/ n L = 2 po θ L θ H θ L < 0, whereas ag 1+1 dn L 1 a+θ L 1 a θ H + +1 dn L θ L 2 n m/ θ H = H 1+1 dn L 1 a1 ag θ L > 0 for any n ag 1+1 dn L 1 a+θ L 1 a θ H + +1 dn L θ L 2 L > n n L, m/ d = H n L 1 a 2 g θ L θ H θ L > 0, m/ n ag 1+1 dn L 1 a+θ L 1 a θ H + +1 dn L θ L 2 H = 1+1 dn L 1 a1 ag θ L θ H θ L > 0 for any n ag 1+1 dn L 1 a+θ L 1 a θ H + +1 dn L θ L 2 L > n L, and m/ a = θ H θ L g 1+1 dn L 1 a 2 +θ L +1 a 2 θ H θ L 1 dn L θ L > 0 iff n a 2 g 1+1 dn L 1 a+θ L 1 a θ H + +1 dn L θ L 2 H < 1 1 dn L 1 a 2 g θ L 1 a 2 θ H θ L Proof of Proposition 6 The range of ν values for which the OEM prefers to induce O, over O, O increases in θ H because ν/ θ H = 2c en L d g θ L n L +θ H θ L 2c w+c en 2 L for c e > c e Similarly, the range of ν values for which the OEM induces O, M increases in θ H because ν ν / θ H = d 1+1 dn L 1 a g θ L n L +θ H θ L n H > 0 for n L > n L 2c w+c en 2 L 1+dn L 1 a Proof of Proposition 7 It follows the proof of Propositions 1 and 2 with the difference that when optimizing Π h,l the CAFE constraint r h, l R is binding ie, the regulation R exceeds the CAFE levels calculated in Proposition 3 As before, we focus on interior values of fuel efficiencies Under O, the OEM determines the optimal efficiency based on max R eh 1 Π O, The profit Π O, is concave in e H, therefore, if R is larger than the unconstrained optimal efficiency calculated in Proposition 1, the OEM chooses e H = R, which results in F H = d θ H g 1 R + ν and F H / R = d θ H g < 0 Under O, O the OEM determines the optimal efficiencies based on max eh,e L ΠO,O such that 0 e H 1, 0 e L 1 and e H + n L e L R We form the Lagrangean L = Π n O,O + ψ H e H + n L e L R Given that ΠO,O is jointly concave in e H and e L we solve L/e H = 0, L/e L = 0 and L/ψ = 0 to obtain e H = R dθ H θ L, e 2c w+c e L = R + dθ H θ L 2c w+c en L, and ψ = + n L 2 c w + c e R 2c w d g θ L > 0 for all R > r O, O Based on these values we also obtain F H = d g θ L R + 2cw+cen Lθ L g+ν+dθ H θ L θ H θ L +θ H gn L, F L = F H 2n L c w+c e d 2 θ H gθ H θ L 2c w+c en L and FH O, O / R = F L O, O / R = d g θ L > 0 Under O, M the OEM determines the optimal efficiencies based on max eh,e L ΠO,M such that 0 e H 1, 0 e L 1 and Lagrangean is given by L = Π O,M + ψ 1 a + a 1 a +an Ld e H + + a 1 a +an Ld e H + a 1 a +an Ld + a > 0 1 a +an Ld e L R The Once a 1 a +an Ld + a 1 a +an Ld e L R

9 Bellos, Ferguson, and Toktay: OS: The Car Sharing Economy 9 again, given that Π O,M is jointly concave in e H and e L we solve L/e H = 0, L/e L = 0 and L/ψ = 0 to obtain e H = R adθ H adn L + +dn L + +1+g1 a1 dn L 1 1 aθ L 2c w+c ea+1 aadn L +, e L = R + 1 adθ H 1 adn L +1 a +1 gn L a ad+d aθ L 2c w+c e1 adn L +1a+1 aadn L +, and ψ = 2c w1 Ra 1adn L + a 1 adan L 2c er θ L +g+ a 1θ H aθ L +g+2c er1 a +a 1 a > 0 for all R > r O, M The analytical expressions for F H and p are available upon request By differentiating we obtain p / R = θ L < 0, and F H O, M / R = d g 1 a θ H aθ L < 0 iff a < θ H g θ H θ L 0, 1 Proof of Proposition 8 For brevity define Π O,, Π O,O, and Π O,M to be the unconstrained optimal profits we developed in the Proof of Propositions 1 and 2 for the O,, O, O, and O, M equilibrium, respectively Using the optimal prices and fuel efficiencies we calculated in Proposition 7 we obtain the constrained optimal profits Π O, = Π O, 2 2c w 2c w+c er dθ H g 2 2c w 2c w+c er dg θ L 4c w+c e, Π O,O = Π O,O +n L 4c w+c e, and Π O,M = 2 Π O,M 2c w1 R1 aadn L + +a+1 adan L 2c er θ L +g+ a 1θ H aθ L +g 2c er1 a +a 41 ac w+c ea+1 aadn L + } {ν : Π O,O ΠO, = 0 Paralleling Proposition 4, we calculate ν } {ν : Π O,M ΠO, = 0 and ν = =, ν = } {ν : Π O,O ΠO,M = 0 When selling only r O, O > r O, see Proposition 3, hence, the CAFE standard is binding for either equilibrium when R r O, O ν is convex in R because 2 ν/ R2 = 2cw+ce > 0 In addition, d ν/ R R=e L O,O = 0 Therefore, ν/ R > 0 for all R e L O, O and ν/ R < 0 for all R [ r O, O, e L O, O With car sharing r O, M < r O, O for θ H > θ H r O, M > r O, O for θ H < θ H see Proposition 3 Hence, for θ H > θ H the CAFE standard is binding when R r O, O The range ν ν is convex in R because 2 ν ν / 2 R = 2cw+ce 1+1 dn L 1 a > 0 for n dn L 1 a 2 L > n L Furthermore, ν ν / R R=rO,O = dg θ L 1+1 dn L 1 a 21 a 2 dn L Similarly, for θ H < θ H case, ν ν / R R=rO,M = and > 0 for n L > n L Therefore, ν ν / R > 0 for all R > r O, O the CAFE standard is binding when for R r O, M In this 1+1 an L 1 d aθ H θ L θ H g 1 an L 1 aadn L + +a n L > n L and θ H < θ H Therefore, 2 ν ν / 2 R = 2cw+ce 1+1 dn L 1 a dn L 1 a 2 +an L g θ L ν ν / R > 0 for all R > r O, M Thus, ν ν / R > 0 for all θ H References Chen, C 2001 > 0 for > 0 implies that Design for the Environment: A Quality-Based Model for Green Product Development Management Sci Whitt, W 1984 Open and Closed Models for Networks of Queues AT&T Bell Lab Tech J