Stock-Outs and Customer Purchasing Behavior when Product...

Stock-Outs and Customer Purchasing Behavior

when Product Quality is Uncertain

Laurens G. Debo

Garrett J. van Ryzin

University of Chicago, Columbia University

Winter OM Conf., Jan. 2010

Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 1 / 35

Stock-outs

Long Queues

MSOM Conference 2008 ‐‐ RH Smith School of Business, University of Maryland

More Stories of Shortages

WSJ, Dec 2, 2005, Why shortages of Hot gifts endure as a

Christmas ritual

1983 Cabbage Patch Kids 2000 Play station 2

1990s Beanie Babies 2002 Nike Ainforce1

1998 Tickle me Elmo 2004 iPod mini, Nitendo DS

1999 Pokeman 2005 iPod nano

Mighty Morphin Power Rangers, shortage in 1994

What explains these shortages?

“Newsvendor logic” – High demand uncertainty combined with

risks of overage and underage; firms simply can’t meet all demand

under all conditions. These are just examples of “losing the bet”

on the upside.

They’re deliberately created – Shortages create buzz, hype, a

sense of urgency. . . .shortages boost demand, so firms have an

incentive to deliberately limit supply.

The basic mechanism

Inventories convey information about other consumer’s decisions,

which influence their buying decision

Consumers observe stock-outs

Inventories impact customers’ purchasing decisions

Customers’ purchasing decisions impact inventories

Equilibrium outcome determines realized sales

Key Questions

How do customers learn from stock-outs, and how does that

learning influence total sales?

How does the inventory investment and allocation to retailers

impact their purchasing behavior?

How should a firm take this consumer behavior into account when

investing in inventory and allocating it to retailers?

Outline

1 Introduction and Motivation

2 The Model

3 Analysis of a Single Retailer

Consumer purchasing behavior

Inventory investment

4 Analysis of a Two retailers Retailer

Inventory Investment

5 Conclusion

Outline

2 The Model

5 Conclusion

Outline

2 The Model

5 Conclusion

Outline

2 The Model

5 Conclusion

Outline

2 The Model

5 Conclusion

The Model: Key ingredients

The product is short-lived, and its true (common) quality is not

revealed during the season.

Two retailers; consumers arrive randomly at these two retailers;

sales are lost if there is no inventory.

Consumer quality-related information:

Common prior on product quality

Private: private signal about realized quality

Some consumers observe how many retailers are out of stock

(informed)

Some consumers do not observe how many retailers are out of

stock (uninformed)

Quality vω, where ω ∈ {`,h} is unknown and v` < 0 < vh, where

vh = −v` = v with prior Pr (ω = h) = p0.

Private a signal s ∼ Gω(s). s ∈ [s, s], gh(s)g`(s) is increasing,

gh(s) = g`(s) = 0.

Retailer inventory Q1 = Q and Q2 = Q + ∆. I = (Q,∆).

Mass λ of potential consumers.

A fraction α ∈ (0,1) observes only a private signal, a fraction 1−α

also observe the number of stocked out retailers, m ∈ {0,1,2}.

All consumers can switch at no cost from one retailer to the other

retailer.Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 10 / 35

Overview of the model

ω=l or hvl>0>vl

α: au=1 iff uu(s,I)>0 & m<2

s= private signal~ω

1−α: ai=1 iff ui(s,m,a,I)>0 & m<2

RetailerPotential Market

m= number of stock-outs

I=Inventory

r= revenuec= cost

u=E[vω]=utility

Equilibrium Conditions:

1 uu (s) is uninformed consumer utility

uu (s, I) > 0⇒ buy

2 ui (m, s,s, I) is informed consumer utility

ui (m, s,s∗, I) > 0⇒ buy

3 Π(s, I) is the firm’s expected profit

Π∗ = maxI∈R2

Π(s∗(I), I)

Analysis of a Single Retailer

Q = 0 (no small retailer)

∆ is total inventory at large retailer

Uninformed Consumers

Updated utility:

uu (s) = p′ (s) v +(1− p′ (s)

)(−v)

Updated prior on product quality:

p′ (s) =gh (s) p0

gh (s) p0 + (1− p0) g` (s)

Purchasing threshold: purchase if private signal is larger than s:

l(s)θ = 1,

where l (s) = gh(s)g`(s) and θ = p0

1−p0.

Informed Consumers

Updated utility:

ui (m, s, s0,∆) = p′′ (m, s, s0,∆) v +(1− p′′ (m, s, s0,∆)

)(−v)

Updated prior on product quality:

p′′ (m, s, s0,∆) =gh (s) p0ph (m, s0,∆)

gh (s) p0ph (m, s0,∆) + (1− p0) g` (s) p` (m, s0,∆).

where pω (m, s0,∆) is the probability of observing m stock-outs

Informed Consumers

Probability of observing m stock-outs: pω (m, s0,∆).

Quality-dependent purchasing probability :

Pω (s) = αGω (s) + (1− α)Gω (s) .

Volume of consumers observing no retailer out of stock:

(s0,∆) =∆

Pω (s0).

Probability of observing m stock-outs:

pω (0, s0,∆) = min

(1,λω

(s0,∆)

), pω (1, s0,∆) = 1−pω (0, s0,∆)

Informed Consumers

Purchasing threshold: purchase if private signal is larger than

s∗0(∆), which solves:

θl (s0) =p` (0, s0,∆)

ph (0, s0,∆). (1)

Equilibrium Purchasing Threshold s∗0

p0 = 1/2→ θ = 1

Alwaysstock-out

Neverstock-out

Stock-outfor high qualityproducts

pl(0,s0,∆)/ph(0,s0,∆)

s0Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 18 / 35

Inventory Investment Problem

Expected satisfied sales:

S(∆, s0) = Eω [min (∆,Pω (s0)λ)]

Expected profits:

Π(∆, s0) = rS(∆, s0)− c∆

Optimal Inventory:

max∆∈[0,λ]

Π(∆, s∗0(∆))

Baseline: Inventory Investment for All Uniformed

Consumers

max∆∈[0,λ] rS(∆, s)− c∆

Ph (s)λ is optimal when p0 >cr . High margin: stock out if high

quality only

P` (s)λ is optimal when p0 <cr . Low margin: stock out if both low

and high quality

Inventory Investment Problem: Informed Consumers

There exist a c > c and s > s:

Ph (s)λ is optimal when p0 >cr .

P` (s)λ < P` (s) is optimal when p0 <cr .

Conclusion:

Expanded range of ‘high margin’ products

Low-quality realization has less demand

Low-margin inventory investment is lower

Two Retailers

Two retailers: small and large

Q1 = Q and Q2 = Q + ∆

Stock-out signal m ∈ {0,1,2}

Uninformed and Informed Consumers

Uninformed Consumers: same as for the Single Retailer Case.

Informed Consumers: Key is pω (m,s, I) is the probability of

observing m stock-outs.

Volume of consumers observing m retailers out of stock:

λω (s, I) =Q

12Pω (s0)

and λω (s, I) =Q

12Pω (s0)

Pω (s1).

Inventory Depletion

Pω(s0)/2

λω λ

Inventory

Market potential

Pω(s0)/2

Pω(s1)

Uninformed and Informed Consumers

Probability of observing m stock-outs:

pω (0, s0, I) = min(

1,λω (s0, I)

pω (1,s, I) = min

(1,λω

(s, I)λ

)− λω (s0, I)

for 1 ≥ λω(s0,I)λ (and 0 otherwise).

Purchasing threshold: purchase if private signal is larger than

s∗(I):

θl (s∗m) =p` (m,s∗, I)ph (m,s∗, I)

, m ∈ {0,1}. (2)

Small Retailer: End-of-Season-Inventory

(κ = 0.125,p0 = 0.45, α = 0.25)

Small Retailer

(Small Retailer Inventory)

Large Retailer: End-of-Season-Inventory

(κ = 0.125,p0 = 0.45, α = 0.25)

Always Excess Inventory

Always Stock-out

LargeRetailer stocks out for

High Quality Products

Large Retailer

(Small Retailer Inventory)

Expected satisfied sales:

S(I ,s) = Eω[2 min(Q,12

Pω (s0)λ)

+ min(∆,Pω (s1) (λ−Q/(12

Pωs0))+)]

The impact of Inventory on satisfied sales:

∂IS(I ,s∗(I))︸︷︷︸

direct effect

∂s0S(I ,s∗(I))

∂s∗0∂I

∂s1S(I ,s∗(I))

∂s∗1∂I︸︷︷︸

informed consumer effect

for I = Q or ∆

Expected profits:

Π(I ,s) = rS(I ,s)− c (2Q + ∆)

Optimal Inventory:

maxI∈R2

Π(I ,s∗(I))

Optimal Inventory: Total Inventory Investment

(κ = 0.125,p = 0.45, α = 0.25)

Optimal Inventory: Inventory Allocation

(κ = 0.125,p = 0.45, α = 0.25)

Optimal Inventory: Profits (κ = 0.125,p = 0.45, α = 0.25)

Inventory Investment Problem: Insights

For high margin products with a low prior, the optimal inventory

investment is higher than the uninformed inventory (capture the

herd!)

Mostly, small small retailers are optimal (no perfect high quality

communication through stock-outs).

Inventory Investment Problem: Insights

The profit increase with respect to the uninformed profits can be

substantial (30%), especially for high margin products with a low

prior and noisy private signals.

For very low margin products it is optimal to reduce the inventory

investment below the uninformed inventory (forgo high demand

option, waste less informed consumers).

Conclusion

Increase sales with stock-outs?

Requires asymmetric inventory allocation.

Example: Apple introduces iPod at BestBuy and Apple Stores

Extensions

Stochastic demand

Search costs

Different retailer margins

Laboratory or empirical research

Stock-Outs and Customer Purchasing Behavior when Product...

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