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.