When are Advance Purchase
Discounts Profitable?
James Dana
Kellogg School of Management
Motives for Advance Purchase
Discounts
1) Traditional Price Discrimination
•
Courty and Li, 2000, Deneckere and McAfee,1996,
Anderson and Dana, 2005
2) Reducing Monopoly Distortion - Homogeneization
•
Courty, 2003, Shugan and Xie, 2001,
Gale and Holmes 1992, 1993
3) Information and Customization of Capacity (or
Price)
•
Tang, et. al., 2000, Arrow-Debreu
4) Screening and Capacity Utilization
•
Dana, 1998, 1999, Gale and Holmes 1992, 1993
Advance Purchase Discounts
= Damaged Goods?
•
Purchasing in advance is an
inconvenience
–
•
If consumers who are more inconvenienced also have
higher valuations, then advance purchases requirements
can segment high and low value consumers
Analogy seems good
Yet Advance Purchase Discounts are
Quite Different
•
Consumers are learning over time about their own
preferences
–
–
•
The inconvenience is (at least in part) derived from
uncertainty about future preferences
Consumer’s behavior is changing over time
Aggregate uncertainty
–
–
–
Capacity must be planned in advance
Market prices must be planned in advance
Consumers face greater expected costs of stockouts
(which are especially costly for high value consumers)
A Simple Model of
Price Discrimination
• A model in which the inconvenience of advance purchase is due to
uncertainty about preferences (e.g., departure time or destination)
• Two Types: Business and Leisure
• Time 0 (advance) and time 1 (spot)
• Consumers forecast their preferred product with probability i
• Ex post valuations are V(i,0) = vi for the preferred and vi – xi
• Ex ante valuations for the preferred product are
V B, 0    B vB  1   B vB  x B 
V L, 0    L vL  1   L vL  x L 
• Ex ante valuations for the alternative
 B vB  xB  1   B vB
 L vL  xL  1   L vL
Three Pricing Options
• Sell to the business travelers only in the spot
market at the business traveler’s valuation
• Sell to both leisure travelers and business travelers
in the spot market at the leisure travelers valuation
• Sell to leisure travelers in advance at their
valuation and sell to business travelers in the spot
market at the highest possible price.
The Advance Purchase Option
• Advance purchase discounts are optimal when
V B,1  V B,0  V L,1  V L,0 

V B,0   c
V L,0   c
or
1   B xB  1   L xL
vB  1   B xB  c vL  1   L xL  c
If V(B,0) < V(L,0), advance purchase discounts are clearly
profitable. But if V(B,0) < V(L,0) this is not automatically
satisfied.
The Advance Purchase Option
• Note that
1   B xB  1   L xL
is not sufficient. That is, correlation between v and
the expected cost of consuming the wrong product is
not sufficient for advance purchases to be profitable.
A General Model
• Assume a continuum of consumers with uncertain
valuations and a single firm with unit costs of capacity k,
and usage costs c, but who must produce in advance.
• Notation
– Let  denote the aggregate state with pdf g()
– Let  denote a consumer’s type with distribution f(,)
– Let v1 denote a consumer’s valuation for product 1 with
distribution 1(v,,). 1 is increasing in .
– Let v2 denote a consumer’s valuation for product 2 with
distribution 2(v,,). 2 is increasing in .
– Let h() denote the distribution of  given .
• f, g, h, and  are common knowledge
The General Timing
• The firm make an initial production decision
• The firm offers its products at the advance purchase price
– Offers may or may not include a refund option
• Consumers learn their types 
– And update their beliefs, h(), about the aggregate state
•
•
•
•
•
•
•
Consumers make purchase decisions
The firm update its beliefs about the aggregate state
The firm revises its capacity/inventory decision
The firm offers its products at the spot price
Consumers learn their realized valuations
Consumers make purchase decisions
Consumers exercise their return options, if available
Special Cases
• Ex ante homogeneous consumers
– No aggregate uncertainty, k = c = 0
• The firm will sell in advance at p = E[v] (Courty, 2003 and
Shugan and Xie, 2001)
– No aggregate uncertainty, E[v] < c + k
• The firm will sell in the spot market at the monopoly price
(Courty 2003 and Shugan and Xie, 2001)
– Aggregate uncertainty, k > 0, no consumer learning
(v =  = )
• The firm will sell a limited amount in advance to learn  and
then adjust its inventory for the spot market (Tang, et. al. 2000)
Special Cases
• Ex ante heterogeneous consumers and no
aggregate uncertainty
– Two products, x = v1 – v2, max {v1, v2} = v(), min
{v1, v2} = v() – x, k > 0.
• This is the example we solved above.
– Refunds feasible, k = 0, c > 0.
• The firm will segment consumers with a menu of advancepurchase, partially-refundable contracts (Courty and Li, 2000)
– Two products,  = v1 – v2, max {v1, v2} = constant,
refunds not feasible, k > 0, same capacity is used for
both products.
• If more consumer will prefer product 1 to product 2, advance
purchase discounts is the cheapest way to move low 
consumers to product 2 (Gale and Holmes, 1993).
Research Agenda
• Ex ante heterogeneous consumers with aggregate
uncertainty
– Related work by Gale and Holmes, 1992, Dana, 1998.
• ??
Four Effects
Firms
With
Market
Power
All
Firms
1)
A firm with market power may find it profitable to extract more
surplus from high value consumers by making its lower priced
products available only in advance, an option which is less
attractive to high value consumers.
2)
A firm with market power may be able to extract more surplus
from consumers by selling in advance; particularly if consumers
are more homogeneous in advance.
3)
A firm can use consumers’ advance purchases as information to
economize on capacity or inventory in the spot market and to
adjust prices in the spot market.
4)
A firm can use advance purchases to improve capacity or
inventory utilization and better allocate capacity, particularly
when the spot market is not a “perfect” market.