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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
1
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 2 / 35
Long Queues
MSOM Conference 2008 ‐‐ RH Smith School of Business, University of Maryland
1
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 3 / 35
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
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 4 / 35
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.
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 5 / 35
The basic mechanism
Inventories convey information about other consumer’s decisions,
which influence their buying decision
Consumers observe stock-outs
SO
Inventories impact customers’ purchasing decisions
Customers’ purchasing decisions impact inventories
Equilibrium outcome determines realized sales
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 6 / 35
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?
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 7 / 35
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
Consumer purchasing behavior
Inventory Investment
5 Conclusion
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 8 / 35
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
Consumer purchasing behavior
Inventory Investment
5 Conclusion
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 8 / 35
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
Consumer purchasing behavior
Inventory Investment
5 Conclusion
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 8 / 35
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
Consumer purchasing behavior
Inventory Investment
5 Conclusion
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 8 / 35
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
Consumer purchasing behavior
Inventory Investment
5 Conclusion
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 8 / 35
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)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 9 / 35
The Model: Key ingredients
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
λ
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
Q1
Q2
RetailerPotential Market
m= number of stock-outs
Firm
I=Inventory
r= revenuec= cost
u=E[vω]=utility
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 11 / 35
The Model: Key ingredients
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)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 12 / 35
Analysis of a Single Retailer
Q = 0 (no small retailer)
∆ is total inventory at large retailer
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 13 / 35
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.
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 14 / 35
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
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 15 / 35
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:
λω
(s0,∆) =∆
Pω (s0).
Probability of observing m stock-outs:
pω (0, s0,∆) = min
(1,λω
(s0,∆)
λ
), pω (1, s0,∆) = 1−pω (0, s0,∆)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 16 / 35
Informed Consumers
Purchasing threshold: purchase if private signal is larger than
s∗0(∆), which solves:
θl (s0) =p` (0, s0,∆)
ph (0, s0,∆). (1)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 17 / 35
Equilibrium Purchasing Threshold s∗0
p0 = 1/2→ θ = 1
l(s0)
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(∆))
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 19 / 35
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
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 20 / 35
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
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 21 / 35
Two Retailers
Two retailers: small and large
Q1 = Q and Q2 = Q + ∆
Stock-out signal m ∈ {0,1,2}
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 22 / 35
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).
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 23 / 35
Inventory Depletion
Q+∆
Q
λω
Pω(s0)/2
λω λ
Inventory
Market potential
Pω(s0)/2
Pω(s1)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 24 / 35
Uninformed and Informed Consumers
Probability of observing m stock-outs:
pω (0, s0, I) = min(
1,λω (s0, I)
λ
)and
pω (1,s, I) = min
(1,λω
(s, I)λ
)− λω (s0, I)
λ
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)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 25 / 35
Small Retailer: End-of-Season-Inventory
(κ = 0.125,p0 = 0.45, α = 0.25)
Stoc
k-ou
t for
Hig
h an
d L
ow Q
ualit
y Pr
oduc
ts
Stoc
k-ou
t for
Hig
h Q
ualit
y Pr
oduc
ts o
nly
No
Stoc
k-O
ut
Small Retailer
(Small Retailer Inventory)
(Inv
ento
ry D
iffer
ence
)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 26 / 35
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)
(Inv
ento
ry D
iffer
ence
)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 27 / 35
Inventory Investment Problem
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 ∆
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 28 / 35
Inventory Investment Problem
Expected profits:
Π(I ,s) = rS(I ,s)− c (2Q + ∆)
Optimal Inventory:
maxI∈R2
+
Π(I ,s∗(I))
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 29 / 35
Optimal Inventory: Total Inventory Investment
(κ = 0.125,p = 0.45, α = 0.25)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 30 / 35
Optimal Inventory: Inventory Allocation
(κ = 0.125,p = 0.45, α = 0.25)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 31 / 35
Optimal Inventory: Profits (κ = 0.125,p = 0.45, α = 0.25)
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 32 / 35
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).
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 33 / 35
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).
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 34 / 35
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
Debo & van Ryzin (Chicago, Columbia) Herding and Stock-Outs Winter OM Conf., Jan. 2010 35 / 35