We develop methods for calculating profit-maximizing and socially optimal rates of partial refunds on customers' no-shows and cancellations. We demonstrate how partial refunds can be used to screen consumers according to their different probabilities of cancellation and no-shows. Finally, we show that the socially-optimal rate of partial refund exceeds the equilibrium rate of partial refund.

Oz Shy

Staffan Ringbom

English

Research Papers in Economics

Advance booking, cancellations, and partial refunds 
Staffan Ringbom Oz Shy
Swedish School of Economics and HECER University of Haifa and Swedish School of Economics
Abstract
We develop methods for calculating profit−maximizing and socially optimal rates of partial
refunds on customers' no−shows and cancellations. We demonstrate how partial refunds can
be used to screen consumers according to their different probabilities of cancellation and
no−shows. Finally, we show that the socially−optimal rate of partial refund exceeds the
equilibrium rate of partial refund.
Citation: Ringbom, Staffan and Oz Shy, (2004) "Advance booking, cancellations, and partial refunds." Economics
Bulletin, Vol. 13, No. 1 pp. 1−7
Submitted: January 11, 2004.  Accepted: January 20, 2004.
URL: http://www.economicsbulletin.com/2004/volume13/EB−04M20001A.pdf1 Introduction
Partial refunds are widely used in advance booking and reservation systems. In this note
we propose how to compute the proﬁt-maximizing rates of partial refunds. Partial refunds
are used to control for the selection of potential customers who make reservations but diﬀer
with respect to their cancellation probabilities. In addition, we explore the social welfare
consequences associated with the proﬁt-maximizing partial refund rates.
Consumers making reservations for products and services diﬀer in their probability of
showing up to collect the good or the service at the pre-agreed time of delivery. Agencies
and dealers that sell these goods and services can save on unused capacity costs, generated
by consumers’ cancellations and no-shows, by varying the degree of partial refunds oﬀered to
those consumers who cancel or do not show. Partial refunds are widely observed in almost all
privately-provided services. They are heavily used by airline companies in selling discounted
tickets where cheaper tickets allow for a very small refund (if any) on cancellations, whereas
full-fare tickets are either fully-refundable or subject to low penalty rates.
In the economics literature there are a few papers analyzing post-delivery contracts in the
form of money-back-warranties, see Mann and Wissink (1988, 1990), Moorthy and Srinivasan
(1995), and Shiou (1996). In their models, money-back-guarantees serve as a signaling device
for products’ quality which is observable only after the product is delivered. Here the quality
of the product is exogenous.
Advance booking mechanisms received a limited attention in the economic literature.
Gale and Holmes (1992, 1993) compare monopoly pricing with social optimum. In their
analysis, the price mechanism is used to shift the demand from the peak period to the oﬀ-
peak period. Gale (1993) analyzes consumers who learn their preferences after they are
oﬀered an advance purchase option. On this line, Miravete (1996) and more recently Courty
and Li (2000) further investigate how consumers who learn their valuation over time can be
screened via the introduction of refunds in their advance booking mechanism. Dana (1998)
also investigates market segmentation under advance booking of price-taking ﬁrms. There
also exist a literature dealing with supply-rationing under demand uncertainty in revenue
management which is broadly related to this paper (see McGill and van Ryzin 1999 for
a survey). Ringbom and Shy (2003) analyze the beneﬁts of fully-refundable reservation
systems without allowing for partial refunds. Xie and Shugan (2001) analyze advance sales
in the presence of spot sales for exogenously-given participation levels. In this note, we
endogenously solve for the participation rates, which can be adjusted by the rate of partial
refunds. We also show how welfare statements can be drawn from our model.
2 Two consumer types
Let β denote consumers’ basic valuation (maximal willingness to pay) for a certain service
or a product. Let p denote the market price. We assume that p is exogenously given by
the manufacturers and service providers. What we have in mind are products and services
that are sold under resale-price maintenance agreements, such as airline and train tickets,
car rental fees, books with a “recommended” price, and telecommunication services.
12.1 Consumer types and sellers
The are n potential consumers who vary according to their cancellation habits, as reﬂected
by the probability of showing up of each customer. Suppose that there are only two types of
potential customers, indexed by i = H,L. More precisely, there are αHn consumers with a
“high” probability of showing up given by σH, and αLn consumers with a “low” probability
of showing up given by σL, where αH + αL = 1. We assume that 0 < σL < σH < 1.
Suppose that a representative seller purchases the service/good at the cost of c ≥ 0. If
the customer does not show up at the delivery time, the seller’s salvage value of the product
is s ≥ 0. Given the service price p, the seller’s only choice variable is the rate of partial
refund oﬀered to those consumers who do not show up or cancel their reservations. We
denote this choice variable by r, where 0 ≤ r ≤ 1. Thus, on each reservation, the seller earns
a proﬁt of p − c if the customer shows up, and loses c + rp − s if the customer cancels, or
simply does not show up.
The expected utility of a type i consumer, i = H,L, is given by
EU =
(
σi(β − p) − (1 − σi)(1 − r)p if he makes a reservation
0 otherwise.
(1)
Thus, if the consumer shows up at the delivery time, the net beneﬁt is β − p. However,
if the consumer cancels, the consumer loses the non-refundable portion of the price, which
amounts to (1 − r)p.
2.2 Proﬁt-maximizing refund rate
Inspecting the utility functions (1) reveals that the seller can adjust the participation of
each consumer group by the decision variable r. A “high” refund rate would induce both
consumer types to make a reservation, whereas a low refund rate would make the reservation
non-beneﬁcial (in expected terms) to at least type-L consumers (i.e., those consumers who
are more likely to cancel).
Suppose that the seller sets up a “high” refund rate, so that both consumer types make
reservations. Formally, let r2 be the refund rate that makes type-L consumers indiﬀerent
between making a reservation and not consuming. From (1) we have
r2 =
p − βσL
p(1 − σL)
. (2)
Similarly, let r1 be the refund rate that would make type H consumers indiﬀerent between
making reservations and not consuming. Then,
r1 =
p − βσH
p(1 − σH)
. (3)
Clearly, r2 > r1 reﬂecting the fact that the refund rate must be higher if the seller would
like consumers with high cancellation rates (type L) to make reservations.
2The seller’s expected proﬁt levels at any refund level r ∈ [0,1] is then given by
Eπ(r) =

 
 
n(p − c) − [αLn(1 − σL) + αHn(1 − σH)](r2p − s) r ≥ r2
αHn[p − c − (1 − σH)(r1p − s)] r1 ≤ r < r2
0 r < r1.
(4)
We observe from (4) that a high refund rate is more proﬁtable than the low refund rate
which excludes type L consumers, that is Eπ(r2) ≥ Eπ(r1), if and only if
(p − c) − [αL(1 − σL) + αH(1 − σH)](r2p − s) ≥ αH [p − c − (1 − σH)(r1p − s)]. (5)
By substituting (2) and (3) into (5), with some manipulation we can state the following
proposition
Proposition 1. The proﬁt-maximizing partial refund rate is r2 if and only if
αL[σL(β − s) − c + s] ≥ αH(σH − σL)(β − p). (6)
Otherwise, the proﬁt maximizing rate is r1.
The insights to draw from Proposition 1 are summarized in the following corollary.
Corollary 1. Both consumer types are served if p is suﬃciently high, αH or σH are suﬃ-
ciently low, or σL is suﬃciently high.
3 General Distribution of Consumer Types
In this section we generalize the two-point distribution of consumers by assuming that there
is a continuum of consumers indexed by σ ∈ (0,1], who diﬀer according to their probability
of showing up. The distribution function of the consumers’ showing up probability σ is Fσ,
which is absolutely continuous with respect to σ, Fσ(0) = 0 and Fσ(1) = 1, and fσ = dFσ/dσ.
Formally, similar to (1) the expected utility of a type σ consumer is
EU(σ) =
(
σ (β − p) − (1 − σ)(1 − r)p ≥ 0 if makes reservation
0 otherwise.
(7)
3.1 Refundability and proﬁts
Let ˆ σ denote a consumer who is indiﬀerent between making reservations or not. From (7),
ˆ σ is implicitly deﬁned by ˆ σ (β − p) − (1 − ˆ σ)(1 − r)p = 0. Hence,
ˆ σ(r) =
p − rp
β − rp
. (8)
ˆ σ is the threshold showing up probability, which is controllable through the refund rate, r.
Therefore, only consumers indexed by σ ≥ ˆ σ make reservations. Next, observe that the
expected unit gross proﬁt from a type σ consumer who makes a reservation, is given by
gσ = p − c − (1 − σ)(rp − s)
| {z }
no−showcost
= p − rp + σ(rp − s) + s − c. (9)
3Consequently, the gross proﬁt of the seller is
π(r) = n
Z
(ˆ σ,1]
gσdFσ. (10)
We observe by implementing Leibnitz’ rule, that the ﬁrst order condition for an internal
proﬁt maximum, 0 < r < 1, becomes
1
n
dπ(r)
dr
=
Z
(ˆ σ,1]
∂gσ
∂r
dFσ −
dˆ σ
dr
f(ˆ σ) gσ|σ=ˆ σ = 0. (11)
From (8) we conclude that dˆ σ/dr = −p(β − p)/(β − rp)2 = −p(1 − ˆ σ)/(β − rp) < 0, and
from (9) that dgσ/dr = p(σ − 1). Therefore, (11) simpliﬁes to
1
np
dπ(r)
dr
=
Z
(ˆ σ,1]
(σ − 1)f(σ)dσ + (β − p)f(ˆ σ)
gσ|σ=ˆ σ
(β − rp)2 = 0. (12)
If an interior solution to (12) does not exist, (0 < r < 1), then proﬁt is maximized by selling
non-refundable tickets, that is, the corner solution r = 0 maximizes (10). In order to be able
to interpret our results we now simplify the distribution of σ.
3.2 Partial refunds with uniformly distributed showing up prob-
ability
Suppose that σ is uniformly distributed over the unit interval, σ ∼ U[0,1]. From (8) we
conclude that 1 − ˆ σ = (β − p)/(β − rp). In this case the ﬁrst order condition for internal
proﬁt maximum (12) can be rewritten as
1
np
dπ(r)
dr
= −(1 − ˆ σ) +
(1 − ˆ σ2)
2
+
(β − p)
(β − rp)2 gσ|σ=ˆ σ =
= −
(β − p)2
2(β − rp)2 +
(β − p)
(β − rp)2 gσ|σ=ˆ σ = 0. (13)
Therefore, (β − p)/2 = gσ|σ=ˆ σ > 0. We now solve for the unique proﬁt-maximizing refund
rate ¯ r ≥ 0. From (8), (9) and (13) we obtain
¯ r = max

0,
β(3p − 2(c − s)) − 2ps − β2
p(β − 2c + p)

, (14)
which is strictly positive for a suﬃciently high price satisfying
p > ˜ p
def = β

1 − 2
β − c
3β − 2s

. (15)
By combining (8) and (14) we observe that
ˆ σ(¯ r) = min

p
β
,
(β − p) + 2(c − s)
2(β − s)

. (16)
We conclude that:
4Proposition 2. If p > ˜ p, the proﬁt maximizing refund rate ¯ r is an increasing function of the
price p and the salvage value s, and a decreasing function of the marginal cost c. Otherwise
¯ r = 0.
Numerical examples of proﬁt maximizing refund rates are given in Table 1 below.
4 Welfare analysis
We deﬁne the social welfare as the sum the consumers’ utilities and seller’s proﬁt. Formally,
W(r) = CS + Eπ = n
Z 1
ˆ σ(r)
(σ (β − p) − (1 − σ)(1 − r)p)dFσ + n
Z 1
ˆ σ(r)
gσdFσ
= n(β − s)
Z 1
ˆ σ(r)
σdFσ − n(c − s)
Z 1
ˆ σ(r)
dFσ. (17)
Lemma 1. Let r∗ be the socially optimal refund rate that maximizes (17). Then r∗ =
1 − (c − s)(β − p)/[p(β − c)]. In addition, gσ|ˆ σ(r∗) = 0, where gσ is deﬁned in (9).
Proof. By applying Leibnitz’ rule on (17) we observe that the critical showing up probability
ˆ σ(r∗) which maximizes welfare is ˆ σ = (c − s)/(β − s). Therefore, c − s = ˆ σ(β − s), and
by (8), p − rp = (β − rp)ˆ σ. Substituting these values into (9) yields gσ = ˆ σ · 0 = 0.
ˆ σ(r∗) = (c − s)/(β − s) = (p − r∗p)/(β − r∗p) determines r∗.
Intuitively, Lemma 1 implies that the social planner increases the refund rate until the unit
gross proﬁt goes to zero. Our welfare conclusion is given in the following proposition.
Proposition 3. The socially optimal refund rate exceeds the equilibrium refund rate. Form-
ally, r∗ > ¯ r. If the salvage value s equals marginal cost (s = c), a full refund is socially
optimal, otherwise, (s < c), the socially optimal refund rate is partial, 0 < r∗ < 1.
Proof. p > c ≥ s =⇒ r∗ > 0. Substituting gσ|ˆ σ(r∗) = 0 into (12) yields dEπ/dr|r∗ < 0.
Proposition 3 implies that the seller excludes “too many” customers from a social point of
view. The proﬁt maximizing and socially optimal refund rates are compared in Table 1.
¯ r < r∗ c = 0.2, s = 0.1 c = s = 0.1 c = 0.1, s = 0 c = s = 0
p = 0.8 0.929 < 0.969 0.969 < 1 0.9375 < 0.972 0.972 < 1
p = 0.6 0.667 < 0.917 0.810 < 1 0.714 < 0.926 0.833 < 1
p = 0.5 0.364 < 0.875 0.250 < 1 0.462 < 0.889 0.357 < 1
p = 0.4 0.000 < 0.8125 0.250 < 1 0.000 < 0.833 0.357 < 1
Table 1: Simulations of proﬁt maximizing refunds ¯ r and socially optimal refunds r∗ assuming
that σ ∼ U[0,1], β = 1.
Using (17) we measure the welfare loss by
∆W = W(r
∗) − W(¯ r) = n
Z ˆ σ(¯ r)
(c−s)/(β−s)
[(β − s)σ − (c − s)]dFσ > 0, (18)
5Assuming a uniform distribution of σ, (18) becomes
∆W = n ·

ˆ σ2(¯ r)
2
(β − s) − ˆ σ(¯ r)(c − s) +
(c − s)2
2(β − s)

=
n
2(β − s)

  

min

p
β
,
(β − p) + 2(c − s)
2(β − s)

| {z }
ˆ σ(¯ r)
(β − s) − (c − s)

 


2
=
n
8
(β − p)2
(β − s)
· 1{p>˜ p} +
n
2
[p(β − s) − β(c − s)]2
β2(β − s)
· 1{p≤˜ p}, (19)
where 1{.} is the indicator function taking value 1 if the subscript condition is true and zero
otherwise, and ˜ p was given in (15). Therefore,
Proposition 4. If σ ∼ U[0,1] and p > ˜ p, the welfare loss ∆W is independent of the marginal
cost, a decreasing function of the price p, and an increasing function of the salvage value s.
Table 2 provides some numerical calculations of the welfare loss associated with an in-
suﬃciently high refund rate.
∆W c = 0.2, s = 0.1 c = s = 0.1 c = 0.1, s = 0 c = s = 0
p = 0.8 5.556 5.556 5.000 5.000
p = 0.6 22.22 22.22 20.00 20.00
p = 0.5 34.72 34.72 31.25 31.25
p = 0.4 37.56 50.00 45.00 45.00
Table 2: Simulations of welfare losses ∆W, σ ∼ U[0,1], β = 1, n = 1000.
References
Courty, P. and H. Li (2000) “Sequantial Screening” Review of Economic Studies 67:697–717.
Dana, J. D. J. (1998) “Advanced Purchase discounts and Price Discrimination in Competitive
Markets” Journal of Political Economy 106(2):395–422.
Gale, I. (1993) “Price Dispersion in a Market with Advance-Purchases” Review of Industrial
Organization 8(4):451–464.
Gale, I. L. and T. J. Holmes (1992) “The eﬃciency of advance-purchase discounts in the pres-
ence of aggregate demand uncertainty” International Journal of Industrial Organization
10(3):413–425.
Gale, I. L. and T. J. Holmes (1993) “Advance-Purchase Discounts and Monopoly Allocation
of Capacity” American Economic Review 83(1):135–146.
6Mann, D. P. and J. P. Wissink (1988) “Money-back contracts with double moral hazard”
RAND Journal of Economics 19(2):285–292.
Mann, D. P. and J. P. Wissink (1990) “Money-Back Warranties vs. Replacement Warranties:
A Simple Comparison” American Economic Review 80(2):432 – 436.
McGill, J. and G. van Ryzin (1999) “Revenue Management: Research Overview and Pro-
spects” Transportation Science 33(2):233–256.
Miravete, E. J. (1996) “Screening Consumers through Alternative Pricing Mechanisms”
Journal of Regulatory Economics 9(2):111–132.
Moorthy, S. and K. Srinivasan (1995) “Signalling Quality With a Money-Back Guarantee:
The Role of Transaction Costs” Marketing Science 14(4):442 – 466.
Ringbom, S. and O. Shy (2003) “Reservations and Refunds” Working Paper available for
downloading from www.ozshy.com.
Shiou, S. (1996) “Price and Money Back Guarantees as Signals of Product Quality” Journal
of Economics and Management Strategy 5(3):361 – 377.
Xie, J. and S. M. Shugan (2001) “Electronic Tickets, Smart Cards, and Online Payments:
When and How to Advance Sell” Marketing Science 20(3):219–243.
7

Advance-Purchase Discounts and

Electronic Tickets, Smart Cards, and Online Payments: When and How to Advance Sell”

Money-back contracts with double moral hazard”

Money-Back Warranties vs. Replacement Warranties:

Price and Money Back Guarantees as Signals of Product Quality”

Price Dispersion in a Market with

Reservations and Refunds” Working Paper available for downloading from www.ozshy.com.

Screening Consumers through Alternative Pricing Mechanisms”

Sequantial Screening” Review of Economic Studies

Signalling Quality With a Money-Back Guarantee:

The eﬃciency of advance-purchase discounts in the presence of aggregate demand uncertainty”

Advance booking, cancellations, and partial refunds

http://www.accessecon.com/pubs/EB/2004/Volume13/EB-04M20001A.pdf

Advance booking, cancellations, and partial refunds

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics