platform pricing for ride-sharing
TRANSCRIPT
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Platform Pricing for Ride-Sharing
Jonathan Levin and Andy Skrzypacz
HBS Digital Initiative
May 2016
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Ride-Sharing Platforms
I Dramatic growth of Didi Kuadi, Uber, Lyft, Ola.I Spot market approach to transportationI Platform sets price, drivers and riders “self-schedule”I Dynamic pricing to balance demand / supply
I Riquelme, Banerjee, Johari (2015)I Cachon, Daniel, Lobel (2015)
I This talk: alternative model of pricing.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Overview
I Simple steady-state modelI Wait times and externalitiesI Effi cient allocation (conflict w/ budget balance)I Platform growth & scale economiesI Problems with “competitive”pricingI Extensions: dynamic pricing, pool pricing.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Model
I Demand: q flow of ridersI q = Q (p + δw) orI p = P (q)− δw
I Supply: s stock of driversI s = S (e)I e = C (s)
I Wait time: w = W (q, s)
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Wait Time
I Depends on stock of idle drivers σ ⇒ w = ω (σ).
I Allocation of drivers across states
I Active drivers q (w + τ), where τ is length of ride.
I Derive w = ω (σ) from flow balance
s = σ+ q (ω (σ) + τ) .
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
More on Wait Time
I Assume ω decreasing, convex, xω′ (x) ↗ x
I Example: ω (x) = x−k , for any k > 0.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Externalities
I Fixing p, demand is q = Q (p − δW (q, s)).I Positive externality from more s ... W decreasing in sI Negative externality from more q ... W increasing in q
I Some useful structure, from s = σ+ q (W + τ).
∂W∂q
= −∂W∂s
(W + τ)
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Platform Objectives
I Total Surplus
TS (q, s) =∫ q
0P (x) dx − δqW (q, s)−
∫ s
0C (z)
I Profit
Π (q, s) = qP (q)− δqW (q, s)− sC (s)
I Optimization problem
maxq,s(1− φ)TS (q, s) + φΠ (q, s)
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Optimal Pricing
I First order conditions, assuming q, s > 0
P (q) + φqP ′ (q)−[
δW + δq∂W∂q
]= 0
−C (s) + φsC ′ (s)− δq∂W∂s
= 0
I Written as prices
p = δq∂W∂q
+ φqP ′ (q)
e = δq−∂W
∂s+ φsC ′ (s)
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Effi cient vs Monopoly Pricing
I Optimal pricing conditions re-written
p − φqP ′ (q) = δq∂W∂q
e + φsC ′ (s) = δq−∂W
∂s
I Effi cient pricing => equate MV=MC
p = C (s) (w + τ)
I Monopoly pricing => equate MR=ME
MR (q) = ME (s) (W + τ)
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Effi cient Allocation Requires a Subsidy
Proposition. Effi cient allocation requires a subsidy.
I Proof. Fix φ = 0. Total surplus maximized at
p = δq∂W∂q
= δq−∂W
∂s(W + τ)
I Substituting the other FOC
p = C (s) (W + τ) = e (W + τ)
I Effi cient rider price = driver cost for the ride.I Effi cient ride subsidy = driver cost for wait time.
γ = eσ
q
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Competitive Pricing
I What would happen if prices were set competitively?I E.g. through a real-time auction among drivers.I Or competing platforms with full multi-homing.
I Ride price is bid down to driver opportunity cost
p = C (s) · (W + τ)
I With endogenous supply e = C (s), so σ = 0.I At the competitive outcome, σ = 0 and w0 = ω (0)
I Supply: s0 = q0 (w0 + τ).I Demand: P (q0)− δw0 = C (s0) (w0 + τ)
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Competitive Pricing
Proposition. Competitive outcome is constrained ineffi cient.
I Identify effi cient allocation subject to balance balance
maxq,σ
TS s.t. Π ≥ 0
I KT condition for q (with φ∗ > 0)
P (q)− δw − C (s) (w + τ)
+φ∗{qP ′ (q)− sC ′ (s) (w + τ)
}= 0
I At q0, s0,w0, first term is zero, second term is < 0.I Intuition: At competitive outcome, small reduction in qhas no effect on TS . But it raises Π. Extra revenue canbe used to pay more drivers, raising σ, which makesinframarginal riders better off.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Scale Economies
Proposition. Doubling q and s reduces wait time w :
∂W∂q/q
+∂W∂s/s
=∂W∂s/s
σ
s< 0.
I Proof. From earlier property of wait time
q∂W∂q
= −s ∂W∂s
s − σ
s
I Pretty obvious when you think about it.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Effi cient Growth Trajectories
I Suppose market size is given by aI demand is P (q/a) and supply is C (s/a).
Proposition. An increase in a leads to an increase in q/a, adecrease in s/q, an increase in σ and a decrease in w .
I At larger scale, capacity utilization is more effi cient b/cof shorter wait times => grow riders faster than drivers.
I Proof sketch. Consider raising q, s to keep s/a and q/aconstant. New allocation has shorter wait time (scaleeffect), and smaller externalities => incentive to raise qfurther, and lower s => q/a increases and s/qdecreases. However, extra rise in q => new incentiveto raise s, so change in s/a is unclear.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Managing Wait Times
I In general, two ways to reduce wait time:I (1) Reduce q, which motivates a reduction in s as well.I (2) Increase s, which motivates an increase in q as well.
Proposition. Increase in δ leads to either increase in s, qand decrease in w , or decrease in s, q and increase in s/q.
Proposition. Increase in drive times by κ so thats = σ+ κq (ω+ τ) leads to either an increase in s ordecrease in κq or both.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Dynamic Pricing: Unanticipated Demand
I Consider variation in demand: P (q/a).
I With unanticipated surge: s fixed
I Effi cient response satisfies:
P(qa
)− δW = p = δq
∂W∂q
I Result: q increases, W increases, σ decreases, pincreases.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Dynamic Pricing: Anticipated Demand
I With anticipated surge: s adapts.
I Assume perfectly elastic supply: C (s) = e.
I Result: q increases, W decreases, σ increases, pdecreases.
I Anticipated and unanticipated demand are verydifferent!
I Further note: γ decreases with increase in a.I With elastic supply, anticipated surge = larger scale.I Case where effi cient ride subsidy declines with scale.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Dynamic Pricing: Anticipated Demand
I Proof. Formulate optimization as choice of q, σ
maxq,σ
∫ q
0P(xa
)dx − δqω (σ)−
∫ σ+q(ω(σ)+τ)
0C (z) dz
I Marginal returns to q, σ
∂TS∂q
= P(qa
)− δω (σ)− C (s) [ω (σ) + τ]
∂TS∂σ
= −δqω′ (σ)− C (s)[1+ qω′ (σ)
]I With C (s) = e, objective is spm in (q, σ, a).I So increase in a ⇒ increase in q, σ⇒ decrease in w .I Since p = C (s) (w + τ) ⇒ decrease in p.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Pooled Rides
I Pooled rides take w + τ + ∆ instead of w + τ.I Assume ∆ falls if more pool riders.
I Suppose two demand segmentsI Individual rides q1 with p1 = P1 (q1)− δwI Pooled rides q2 with p2 = P2 (q2)− δw − δ∆
I There is a new flow balance equation
s = σ+ q1 (ω (σ) + τ) + q2 (ω (σ) + τ + ∆ (q2)) .
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Pooled Rides
I Effi cient pool pricing
maxq1,q2,σ
∫ q1
0P1 (x) dx − δq1w
+2∫ q2
0P2 (x) dx − 2δq2 (w + ∆)−
∫ s
0C (z) dz
I Effi cient pricing
p1 = C (s) (w + τ)
p2 =12C (s) (w + τ + ∆) + q2∆′ (q2)
(δ+
12C (s)
)I It is effi cient to subsidize pool riders.
Platform Pricing
Levin andSkrzypacz
Introduction
Model
Externalities
Pricing
Effi cient Allocation
Competition
Scale and Growth
Dynamic Pricing
Pool Pricing
Conclusion
Conclusion
I Simple model of transportation pricing
I Usual pricing logic but natural structure on externalities
I Some results so far:I Subsidy required for effi cient allocationI Competitive outcomes constrained ineffi cientI Scale economies enable s/q decreaseI Effi cient dynamic prices depend on s elasticityI Pooled ride price policy - subsidize pooling.
I Extensions: heterogeneity, geography.