On Dynamic Pricing - Ilia Krasikov

On Dynamic Pricing Ilia Krasikov Rohit Lamba Pennsylvania State University January 21, 2019 1

Motivation I There are many reasons firms can price discriminate on the basis of timing of purchases. I Focus here on sequential arrival of private info. ”In this case the monopolist faces the problem of structuring his pricing so that consumers ’self-select’ into appropriate categories.” — Varian [1989] I Applications: I unit demand: refundable and non-refundable airfares. I repeated demand: options and forwards on electricity. 2

In this paper I I I Build a model of dynamic price discrimination: I single seller. I one buyer who’s value for trade can change at random times. Look at two settings: I unit demand. I repeated demand. Construct a simple dynamic pricing instrument: American option on forwards (AOF) 3

Results I I I In the unit demand setting: I Characterize the best AOF. I Show that the best AOF implements the dynamic optimum. I Build an alternative implementation with refunds. In the repeated demand setting: I Construct a pricing strategy with AOFs. I Show that the pricing strategy is approximately optimal. Technical insight: deal with global incentive constraints. 4

Plan I I Unit demand: I Example. I Continuous time model. I Refund contracts. Repeated demand: I Dynamics of sales. I Approximate optimality. 5

Unit demand 6

Example I Two periods: today and tomorrow. I The seller has one unit to sell tomorrow. I Today: buyer’s value is v1 U[0, 1]. I Tomorrow: buyer’s value v2 might be different from v1 : I wp 1/2, the value is unchanged, ie v2 v1 . I wp 1/2, the value is redrawn, ie v1 v2 with v2 U[0, 1]. I Preferences are linear, production is costless. I The seller wants to max her profits. 7

Example: spot pricing I Suppose that the seller offers the good for a spot price α. I The buyer buys the good if and only if v2 α. I Seller’s expected profit is αP(v2 α) α(1 α) I The optimal price is α 1/2 which yields 1/4. 8

Example: European option. I Buyer’s expected payoff from the spot contract: 0 E v2 α v1 1/2 v1 1/2 1/2 E v2 1/2 {z } 1/8 I The seller leaves too much surplus on the table. I Consider European option {α, M} where I α is a spot price. I M is a flat upfront payment (option premium). 9

Example: buyer’s problem Today v1 drawn Tomorrow pay M B don’t pay M v2 drawn B don’t exercise M exercise 0 v2 α M 10

Example: optimal European option I What is the optimal combination of α and M? I Choose the largest M st the buyer always takes the option: M(α) E v2 α v1 0 I Seller’s profit is then M(α) α(1 α) {z } {z } premium spot pricing I The optimal offer is α 1/3, M 1/9 which yields 1/3. I Can not be improved by choosing larger M. 11

Example: spot pricing vs European option v2 v2 sold sold 1/2 1/3 unsold unsold v1 0 (a) spot pricing v1 0 (b) European option 12

Example: American option on forwards I I Can the seller sort the buyer today and increase her profit? Consider American option α, M, y where I α is a spot (strike) price. I M is a flat upfront fee (option premium). I y is a forward price. 13

Example: buyer’s problem Today v1 drawn Tomorrow pay M B don’t pay M B exercise 0 v2 drawn don’t exercise M exercise E v2 v1 y M B v2 α M 14

Example: screening by AOF I As before, let M(α) st the buyer always takes the option. I Choose y st the buyer self-selects into 2 groups today: I v1 threshold: exercise the forward today. I v1 threshold: wait for the spot price tomorrow. I Tomorrow, the buyer will buy iff v2 α. I Set threshold spot price, y solves: E v2 v1 α y (α) E v2 α v1 α {z } {z } net value of taking the forward for v1 α net value of waiting for v1 α 15

Example: screening by AOF v2 spot price forward price α unsold 0 α v1 16

Example: optimal AOF I Seller’s profit can be written as Z M(α) α(1 α) 1/2 (1 α) {z } {z } {z premium spot pricing 0 α vdv } extra due to forward pricing I The optimal AOF is {α 7/18, M 121/1296, y 455/1296} which yields approximately 0.387. 17

Example: direct mechanisms I Will show that the best AOF implements the optimum. I Invoking revelation principle, the seller offers a contract n o q(v̂2 ), p(v̂2 ) where v̂2 (v̂1 , v̂2 ) is a history of reports, and I q(v̂2 ) {0, 1} is an allocation. I p(v̂2 ) R is a payment. 18

Example: buyer’s problem Today v1 drawn B Tomorrow report v̂1 v2 drawn B report v̂2 Buyer’s payoff v2 q(v̂2 ) p(v̂2 ) I IC tomorrow: v̂2 v2 is optimal v2 given v̂1 v1 . I Note IC tomorrow truthtelling tomorrow even if v̂1 6 v1 . I IC today: v̂1 v1 is optimal given truthtelling tomorrow v1 . 19

Example: IC tomorrow I IC tomorrow can be written as u(v2 ) : v2 q(v2 ) p(v2 ) v2 q(v1 , v̂2 ) p(v1 , v̂2 ) I Note u(v1 , .) must be convex as the max of linear functions. I By the Envelope theorem: u(v2 ) q(v2 ) v2 I Also, can show that convexity envelope IC tomorrow 20

Example: IC today I IC tomorrow can be written as U(v1 ) : E u(v2 ) v1 1/2 u(v1 , v1 ) 1/2 1 Z u(v1 , v2 )dv2 0 1 Z 1/2 u(v̂1 , v1 ) 1/2 u(v̂1 , v2 )dv2 0 I Note U must be convex as the max of convex functions. I By the Envelope theorem: U 0 (v1 ) 1/2 u(v1 , v2 ) v2 IC tomorrow v2 v1 1/2 q(v1 , v1 ) 21

Example: integral monotonicity I Unfortunately, convexity envelope 6 IC today I Subtract U(vˆ1 ) from the both sides of IC today: h i U(v1 ) U(vˆ1 ) 1/2 u(v̂1 , v1 ) u(v̂1 , v̂1 ) I Equivalently, in the integral form: Z v1 Z v1 1/2 u(v̂1 , v ) dv U 0 (v ) dv {z } v2 v̂1 v̂1 {z } 1/2 q(v ,v ) envelope today 1/2 q(v̂1 ,v ) envelope tomorrow 22

Example: integral monotonicity Z v1 Z v1 q(v , v )dv v̂1 q(v̂1 , v )dv v̂1 v2 1 di ag on al projection v1 v̂1 0 v̂1 v1 1 v1 23

Example: seller’s profit I By linearity of preferences: profit surplus buyer’s payoff I surplus E v2 q(v2 ) is given by 1 Z 1/2 Z v1 q(v1 , v1 )dv1 1/2 I v2 q(v2 )dv2 [0,1]2 0 Using integration by parts, buyer’s payoff E [U(v1 )] is Z U(0) {z } 0 by IR 1 (1 v1 ) 0 U 0 (v1 ) {z } dv1 1/2 q(v1 ,v1 ) envelope today 24

Example: standard approach (FOA) I I The common approach is the first-order approach (FOA) ignores the constraints: I Integral monotonicity. I Convexity tomorrow, ie q(v1 , .) is . Then, we maximize profit pointwise and obtain: q2 (v2 ) 1 v1 v2 1/2 v1 6 v2 25

Example: FOA fails so ld v2 1 un so ld 1/2 0 1/2 1 v1 Figure: blue- q2 (v2 ) 0, red- q2 (v2 ) 1. 26

Example: FOA needs extra info I The FOA allocation can be implemented iff event v2 6 v1 is publicly observed I Implementation in the hypothetical setting: I Offer the good for 1/2 whenever v1 v2 I Give the good for free whenever v1 6 v2 Charge upfront 1/2 E v2 v1 6 v2 1/2 1/2. I I In the original setting, the buyer will always claim v̂1 6 v̂2 . I Then, trade happens wp one and seller’s profit is 1/4 0.387. 27

Example: optimum I Seller’s problem is to maximize her profit subject to I Integral monotonicity. I Convexity tomorrow, ie q(v1 , .) is . Proposition Suppose v1 U(0, 1), v2 v1 wp 1/2 or v2 is independently drawn from U(0, 1). The optimal allocation q is given by q (v2 ) 1 max {v1 , v2 } 7/18 28

Example: sketch of the proof U is convex, U 0 (v1 ) 1/2 q(v1 , v1 ) α : q(v1 , v1 ) 1 v1 α so ld v2 1 un so l d α 0 α 1 v1 29

Example: sketch of the proof q(v1 , .) is q(v2 ) 6 q(v1 , v1 ) 0 v2 6 v1 6 α so ld v2 1 un so ld α 0 α 1 v1 30

Example: sketch of the proof Z v2 Z v2 q(v , v )dv 0 v1 6 v2 6 α q(v1 , v )dv 6 IM v1 v1 so ld v2 1 un so ld α 0 α 1 v1 31

Example: sketch of the proof profit is the highest for q(v2 ) 1 max{v1 , v2 } α so ld v2 1 un so l d α 0 α 1 v1 32

Example: summary v2 sold 7/18 unsold 0 7/18 v1 33

Example: summary v2 spot price forward price 7/18 unsold 0 7/18 v1 34

Related literature I Screening/price discrimination with fixed population: Stokey [1979], Baron & Besanko [1984], Courty & Li [2000], Battaglini [2005], Eso & Szentes [2007], Boleslavsky & Said [2013], Deb [2014], Bergemann & Strack [2015] I Screening/price discrimination with changing population: Conlisk, Gerstner & Sobel [1984], Pesendorfer [2002], Board[2008], Gershkov, Moldovanu & Strack [2014], Garrett [2016], Board & Skrzypacz[2016] I Theory of dynamic mechanism design: Pavan, Segal & Toikka [2014], Eso & Szentes [2017], Battaglini & Lamba [2018], Garrett, Pavan, & Toikka [2018] 35

Model for the unit demand I Time is continuous, t [0, T ]. I The seller has one unit to sell at T . I Buyer’s value {Vt } follows renewal process: Vt XNt where I {Nt } is the Poisson process with intensity λ 0. I {Xn }n 0 is a sequence of independent random draws from F . 36

American option on forwards I American option on forwards is defined by n o α, M, y α, M, yt t T I α is a spot price. I M is an flat upfront fee (option premium). I yt is a forward price at t 6 T , yT α. 37

AOF as an optimal stopping. I I After paying M buyer faces a stopping problem: I Stop (exercise) at t: buyer’s payoff is E VT Vt yt M. I Never: buyer’s payoff is M. Need to select the AOF and the optimal stopping time. 38

Optimal AOF Proposition such that the optimal AOF is α , M , y There exists αT T T T where MT E VT αT V0 0 h i Z αT λ(T t) yt,T αT 1 e F (v )dv 0 {z } forward discount In this AOF: I The buyer always pays MT . I The buyer exercises the option at time t iff Vt αT . 39

Max-contract I A direct mechanism is defined by {Q, P} Q(V̂T ), P(V̂T Corollary The optimal AOF implements the following mechanism: Q (VT ) 1 iff max Vt αT t6T P (V ) MT price at the exercise date(VT ) I T We will refer to {Q , P } as the max-contract. 40

Optimality of the max-contract I Formulate the problem using dynamic direct mechanisms. I Seller’s problem is to find the optimal mechanism subject to I Incentive compatibility. I Individual rationality. Proposition The max-contract is the seller’s optimal contract. 41

Gains from randomization I Allow for randomization: q or Q [0, 1]. I The seller can in general do better. Proposition 1. For two periods, there is no gain from randomization. 2. For more than three periods and continuous time, there are gains from randomization. 42

Alternative implementation: refunds I Non-refundable sales (simple futures): xtnon E VT Vt αT I Refundable sales (callable futures): xtref E max{VT , αT } Vt αT I The buyer takes the callable futures at t 0 for x0ref . I The first time t when Vt αT : I The buyer refunds xtref . I He pays xtnon for the simple futures. 43

Alternative implementation: refunds 0.5 price of simple futures price of callable futures 0.45 0.4 0.35 0.3 0.25 0.2 0 0.5 1 1.5 2 2.5 3 t Figure: F (v ) v , λ 1 and T 3 with αT 0.2532. 44

Repeated demand 45

Model for the repeated demand I Suppose that the seller has one unit to sell at each date. I The good is perishable. I The common discount rate is r . I Signals in the previous model are now payoff relevant values. 46

AOF/Max-contract for repeated sales I Paste the previous contracts together I Each t-good is sold by the means of the best t-AOF: αt , Mt , yt I Direct max-contract: Qt (Vt ) 1 iff max Vs αt s6t t rt Pt (V ) e Mt price at the exercise date(Vt ) dT 47

Two period example revisited spot for 2-good 7/18 both forward for 2-good v2 spot for 1-good forward for 2-good unsold 0 7/18 1/2 v1 48

Dynamics of sales Proposition The threshold function t 7 αt is continuous: I α0 equals to the optimal static fixed price. I It is strictly decreasing with lim αt 0. I To describe dynamics, define t stock, St measure of available goods at t 49

Dynamics of sales 1 0 t

Dynamics of sales 1 αt 0 t

Dynamics of sales 1 αt V0 0 t

Dynamics of sales 1 αt V0 t 0 sold at 0

Dynamics of sales 1 αt V0 t 0 sold at 0 S0

Dynamics of sales 1 αt Vt1 V0 0 t t1 sold at 0 S0

Dynamics of sales 1 αt Vt1 V0 0 t1 t sold at t1 sold at 0 S0

Dynamics of sales 1 αt Vt1 V0 0 t1 t sold at t1 St1 t1 sold at 0 S0

Dynamics of sales 1 αt max Vs Vt2 s [0,t] Vt1 V0 0 t1 t2 t sold at t1 St1 t1 sold at 0 S0

Dynamics of sales 1 αt all goods are sold, St2 0 max Vs Vt2 s [0,t] Vt1 V0 0 t1 t2 sold at t2 sold at t1 St1 t1 t sold at 0 S0 50