Platform Pricing for Ride-Sharing

Transcription

1 Platform Platform for Ride-Sharing Jonathan Andy HBS Digital Initiative May 2016

2 Ride-Sharing Platforms Platform Dramatic growth of Didi Kuadi, Uber, Lyft, Ola. Spot market approach to transportation Platform sets price, drivers and riders self-schedule Dynamic pricing to balance demand / supply Riquelme, Banerjee, Johari (2015) Cachon, Daniel, Lobel (2015) This talk: alternative model of pricing.

3 Overview Platform Simple steady-state model Wait times and externalities Effi cient allocation (conflict w/ budget balance) Platform growth & scale economies Problems with competitive pricing Extensions: dynamic pricing, pool pricing.

4 Platform Demand: q flow of riders q = Q (p + δw) or p = P (q) δw Supply: s stock of drivers s = S (e) e = C (s) Wait time: w = W (q, s)

5 Wait Time Depends on stock of idle drivers σ w = ω (σ). Allocation of drivers across states Platform Active drivers q (w + τ), where τ is length of ride. Derive w = ω (σ) from flow balance s = σ + q (ω (σ) + τ).

6 More on Wait Time Assume ω decreasing, convex, xω (x) x Example: ω (x) = x k, for any k > 0. Platform

7 Platform Fixing p, demand is q = Q (p δw (q, s)). Positive externality from more s... W decreasing in s Negative externality from more q... W increasing in q Some useful structure, from s = σ + q (W + τ). W q = W s (W + τ)

8 Platform Objectives Platform Total Surplus Profit TS (q, s) = q 0 s P (x) dx δqw (q, s) C (z) 0 Π (q, s) = qp (q) δqw (q, s) sc (s) Optimization problem max (1 φ) TS (q, s) + φπ (q, s) q,s

9 Optimal Platform First order conditions, assuming q, s > 0 [ P (q) + φqp (q) δw + δq W ] q Written as prices C (s) + φsc (s) δq W s p = δq W q + φqp (q) e = δq W s + φsc (s) = 0 = 0

10 Effi cient vs Monopoly Platform Optimal pricing conditions re-written p φqp (q) = δq W q e + φsc (s) = δq W s Effi cient pricing => equate MV=MC p = C (s) (w + τ) Monopoly pricing => equate MR=ME MR (q) = ME (s) (W + τ)

11 Requires a Subsidy Proposition. Effi cient allocation requires a subsidy. Proof. Fix φ = 0. Total surplus maximized at p = δq W q Substituting the other FOC = δq W s (W + τ) p = C (s) (W + τ) = e (W + τ) Effi cient rider price = driver cost for the ride. Effi cient ride subsidy = driver cost for wait time. Platform γ = e σ q

12 Competitive Platform What would happen if prices were set competitively? E.g. through a real-time auction among drivers. Or competing platforms with full multi-homing. Ride price is bid down to driver opportunity cost p = C (s) (W + τ) With endogenous supply e = C (s), so σ = 0. At the competitive outcome, σ = 0 and w 0 = ω (0) Supply: s 0 = q 0 (w 0 + τ). Demand: P (q 0 ) δw 0 = C (s 0 ) (w 0 + τ)

13 Competitive Proposition. Competitive outcome is constrained ineffi cient. Identify effi cient allocation subject to balance balance max TS s.t. Π 0 q,σ KT condition for q (with φ > 0) P (q) δw C (s) (w + τ) +φ { qp (q) sc (s) (w + τ) } = 0 At q 0, s 0, w 0, first term is zero, second term is < 0. Intuition: At competitive outcome, small reduction in q has no effect on TS. But it raises Π. Extra revenue can be used to pay more drivers, raising σ, which makes inframarginal riders better off. Platform

14 Scale Economies Platform Proposition. Doubling q and s reduces wait time w: W q/q + W s/s = W σ s/s s < 0. Proof. From earlier property of wait time q W q = s W s s σ s Pretty obvious when you think about it.

15 Effi cient Growth Trajectories Suppose market size is given by a demand is P (q/a) and supply is C (s/a). Proposition. An increase in a leads to an increase in q/a, a decrease in s/q, an increase in σ and a decrease in w. At larger scale, capacity utilization is more effi cient b/c of shorter wait times => grow riders faster than drivers. Proof sketch. Consider raising q, s to keep s/a and q/a constant. New allocation has shorter wait time (scale effect), and smaller externalities => incentive to raise q further, and lower s => q/a increases and s/q decreases. However, extra rise in q => new incentive to raise s, so change in s/a is unclear. Platform

16 Managing Wait Times Platform In general, two ways to reduce wait time: (1) Reduce q, which motivates a reduction in s as well. (2) Increase s, which motivates an increase in q as well. Proposition. Increase in δ leads to either increase in s, q and decrease in w, or decrease in s, q and increase in s/q. Proposition. Increase in drive times by κ so that s = σ + κq (ω + τ) leads to either an increase in s or decrease in κq or both.

17 : Unanticipated Demand Platform Consider variation in demand: P (q/a). With unanticipated surge: s fixed Effi cient response satisfies: P ( q a ) δw = p = δq W q Result: q increases, W increases, σ decreases, p increases.

18 : Anticipated Demand Platform With anticipated surge: s adapts. Assume perfectly elastic supply: C (s) = e. Result: q increases, W decreases, σ increases, p decreases. Anticipated and unanticipated demand are very different! Further note: γ decreases with increase in a. With elastic supply, anticipated surge = larger scale. Case where effi cient ride subsidy declines with scale.

19 : Anticipated Demand Proof. Formulate optimization as choice of q, σ q ( x ) σ+q(ω(σ)+τ) max P dx δqω (σ) C (z) dz q,σ 0 a 0 Marginal returns to q, σ TS q TS σ = P ( q a ) δω (σ) C (s) [ω (σ) + τ] = δqω (σ) C (s) [ 1 + qω (σ) ] With C (s) = e, objective is spm in (q, σ, a). So increase in a increase in q, σ decrease in w. Since p = C (s) (w + τ) decrease in p. Platform

20 Pooled Rides Platform Pooled rides take w + τ + instead of w + τ. Assume falls if more pool riders. Suppose two demand segments Individual rides q 1 with p 1 = P 1 (q 1 ) δw Pooled rides q 2 with p 2 = P 2 (q 2 ) δw δ There is a new flow balance equation s = σ + q 1 (ω (σ) + τ) + q 2 (ω (σ) + τ + (q 2 )).

21 Pooled Rides Platform Effi cient pool pricing max q1 q 1,q 2,σ 0 q P 1 (x) dx δq 1 w P 2 (x) dx 2δq 2 (w + ) s 0 C (z) dz Effi cient pricing p 1 = C (s) (w + τ) p 2 = 1 2 C (s) (w + τ + ) + q 2 (q 2 ) (δ + 12 ) C (s) It is effi cient to subsidize pool riders.

22 Platform Simple model of transportation pricing Usual pricing logic but natural structure on externalities Some results so far: Subsidy required for effi cient allocation Competitive outcomes constrained ineffi cient Scale economies enable s/q decrease Effi cient dynamic prices depend on s elasticity Pooled ride price policy - subsidize pooling. Extensions: heterogeneity, geography.