Airline overbooking : firm behavior under penalty scheme

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Faculty Working Papers

AIRLINE OVERBOOKING: FIRM BEHAVIOR
UNDER A PENALTY SCHEME

Jan K. Brueckner

#370

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign

January 20, 1977

AIRLINE OVERBOOKING: FIRM BEHAVIOR
UNDER A PENALTY SCHEME

Jan K. Brueckner

#370

Airline Overbooking: Firm Behavior
Under a Penalty Scheme

Jan K. Erueckner

Department of Economics

University of Illinois, Urban a- Champaign

Urbana, IL 61801

January, 1977

Abstract

This paper investigates firm responses to institution of an
explicit penalty for bumping passengers from an overbooked airline flight
It is assumed that this change can be represented by an increase in the
implicit fine that airlines are viewed as charging themselves for each
bumped passenger under current practice. Monopoly and duopoly models
are developed and analysed.

Airline Overbooking: Firm Behavior
Under a Penalty Scheme

by
Jan K. Brueckner*
University of Illinois, Urban a- Champaign

In order to achieve higher rates of capacity utilization,
airlines routinely sell or reserve more seats on a given flight than are
available. Since passengers holding reservations or tickets frequently
do not show up to claim their seats, failure to overbook would result
in more empty seats than necessary. Occasionally, however, the number of
passengers who show up exceeds tne number of seats available, requiring
that some travelers be "bumped" from the flight. Under current practice,
if the airline cannor arrange for the bumped passenger to arrive at his
destination within two hours of his scheduled arrival (within four hours
on international flights) , the airline must pay up to $200 of the
ticket price.

Eetween January and September of 1976, Trans World Airlines
overbooked 512,800 passengers and bumped 7,800. If these figures i_an
be taken as representative for the entire industry, they indicate that,
while overbooking results in a small percentage. of overbooked passengers
being bumped, the absolute number is far from insignificant . In addition,

*Assistant Professor of Economics. I am indebted to Julian Simo.-1 for
introducing me to this problem and for several helpful discussions.

This information, as well as that in the previous paragraph, was 'caken
from a Hew York Tier as article entitled "Airlines to Pay for Overbooking?" ,
which appeared November 7, 1976.

-2-

since airlines do no" explicitly consider the welfare loss of th^ bumped
passengei in making booking decisions, the level of overbooking is probably
non-optimal from society's point of view.

Jclian Simon (1968) has developed an auction scheme which would
force the airlines to face the true social cost of bumping a passenger
as well as identify!', g the "best" passengers to bump. Under this scheme,
when too many people show up for a flight, each would be asked to write
down the lowest dollar amount he would be willing to accept to wait for
the next flight. The airline would then bump the people with the lowest
bids and pay them the amount of their bids. Since people with lov time
valuations would enter low bids, for a given level of overbooking the
airline would automatically minimize the value of passenger time lost in
attempting to minimize cost. In addition, the airline would adjust the
level of overbooking toward the socially optimal level.

The purpose of this paper is to analyse the airlines' responses
to the ii.sLitMtaon of such a scheme. For simplicity, it is assumed that
all indj riduals have the same tine valuation. Three models are developed:
the first is a one-flight model, where a monopolistic airline chooses the
numbe?: oi seats sold and the flight capacity, the second is a mu.ii.l- flight
monopoly n.Mel, v/>e::e the choice variable is flight frequency, aid the
thiru is a duopoly model where the choice variables again are flight
frequency.

The One-Flj ght >onopo".y Mo^a 1

"uppocc; there is a monopolistic airline operating a flight at a
giver ho ur and that when more pts'sengers show up than can be accomodated,

-3-

the excess passengers <>re bumped. Since no other airline serves the route,
the bumped passengers must be flown on the monopolist's subsequent f3.igb.ts.
Similarl> , people bumped from previous flights will be among the passengers
for the given flight. In actuality, the demand for seats on any flight
depends en the overbooking policy for all previous flights during the rele-
vant period. T-Thilc a correct model of the. overbooking decision should
incorporate this interdependence, the model developed here portrays the
airline as ignoring the presence of the bumped passengers for decision-
making ^crpeses after they are bumped. A model with interdependent is
more complex than the one presented below by several orders of magnitude,
and it was felt tnat the simpler model was a sufficiently close approxi-
mation to the Eiore complex one to provide some insight into the problem.

We also assume that currently, the airlines can be viewed as
charging :hemselves an implicit penalty for each passenger bumped. They
incur a goodwill cost when a passenger is bumped in addition to facing the
potential "lability for his ticket price (or some fraction of it^ . It
is our belief that the explicit penalty would exceed, on the average, the
implicit penalty the airlines can be viewed as charging themselves. De Vai .
(1974) has estimated that the average value of tjme spent in air travel is
arounc $c, pet hour. If we include nuisances such as missed connections or
the nee' to rearrange the schedules of people who might be meeting a passen-
ger at his destination, it is easy to imagine the nuisance cost of a two
hour celay at $75 or $30. It seems that the implicit cost to the airline
of bur-ping a passenger would fall below this range, especially if the
frequency with which the airline is obliged to absorb the ticket price is
low. Further work is needed to establish the merit of this conjecture.

-4-

Central to the problem is the random nature of the arrival at
the terminal of passengers with tickets or reservations. Suppose the
airline lias a subjective probability density for show-ups which is condi-
tional on the number of seats sold or reserved. Let this function be
f (x, S) , vh^re x is the number of show-ups and S is the number of seats
sold or reserved. Clearly, f is zero for x < 0 and x > S. For analytical

convenience, we assume this function is twice dif ferentiable for x < S,

2
ruling out distributions with mass points. The results of the analysis

depend on how the density shifts when S changes. The only assumption

which yields solutions is that the density shifts to the right without

changing its shape by an amount equal to the increase in S. The total

differential of the density is

£ (x, S)dx + f2(x, S)dS. (1)

Setting dx = dS and requiring that the height of che density is unchanged
after changing x and S, we get

f^x, S) + f?(x, S) - 0 (2)

for S > 0 and 0 < x < S. Since f is a density, we must have

If f(S. S) > 0, then f^S, S) and f2(S,S) do not exist since f(x,S) = 0
for x > S. If what follows, we define f,(S, S) to be the left-hand partial
derivative of f with respect to x at x =~S and define f„(S, S) to be the
right-hand partial derivative of f with respect to S at x = S, which we
assume exist and are dif ferentiable.

-5-

S

(x, S)dx = 1, (3)

r

i

0

and differentiation with respect to S yields

f(S, S) + f.,(x, S)dx - 0. (4)

in c

Substituting (2) in (4) we have

fS
f(S, S) - ! f_(x, S)dx = f(0, S) = 0, (5)

J0 L

which says that the height of the density at x = 0 must be zero for all
S if (2) and (3) both hold. If (5) were not true, shifting the density
to the right would increase the area under it above unity. Differentiating
the expected numler of show ups with respect to S yields

(6)

Sf(S, S^> + I xf0(x, S)dx = Sf(S, S) - J xf7(x, S)dx.

Jn l Jn L

Integrating the second term in (6) by parts yields

c Is
-xf(x, ?)|; + f(x, 5)dx, '

J }0

which reduces (6) to unity in view of (3). Hence, the mean number of
show-ups increases by the increase in the number of seats sold or reserved
when the deasity shifts according to (2).

-6-

Let C be the seat capacity of the flight. The function Kx)
gives revenues as a function of show-ups :

*(x) =

p(S)x

0 < x < C

p(S)C - q(x - C) C < x < S,

where p(S) is the ticket price, which depends on the number of seats sold
because the airline is a monopolist, and q is the bumping penalty. We
assume the airline is risk-neutral, maximizing the expected value of profits
from the flight. Letting K(C) represent the cost function for seat
capacity, expected profits are

-I

4(x)f(x. S)dx - K(C)

I
p(S) xf(x, S)dx + p(S)C) f(x, S)dx - q
; 0 ' C i

(7)

(x - C)f(x, S)dx - K(C)

For the moment imagine that the airline is unregulated; the regulated case
is treated below. Its choice variables are S and C, which it determines
by solving the following first-order conditions (the airline will always
choose S > C, as the formulation of the problem implies) :

*g = p'

Xf dX + p
0

xf dx + p' C fdx + pC

s

f2dx + pCf(S, S)

q(S - C)f (S, S) - q (x - C)f0dx = 0

(8)

*c = (p + q)j -dx - k' =o

(9)

-7-

Beckmann (1958) analysed a problem related to this one, although his
model was considerably more complicated. He derived a simple approximation
for his first-order conditions and constructed a numerical example using
specific density functions. Our interest lies in analysing how the
solution changes when q, the bumping penalty, increases. We have

-*Sq V

as

1

3q

A

_1Tcq ^CC

(10)

3C = 1
3q A

SS Sq

^CS _7rCq

where A is the determinant of the Hessian of the profit function evaluated

at the solution, which is positive by the second order condition. From

(8) and (9), -tt

Cq

fdx, and

-TT

Sq

S

I
C

s

(S - C)f(S, S) • (x -

(S - C)f(S, S) + (x

J,

C)f2dx

C)f dx

(ID

fdx,

where (2) and integration by parts have been used. Hence 3S/3q and 3C/3q

are proportional to tt + tt and -(n + it ) respectively, the proportion-

ality factor being k = fdx/A, which is positive. Now tt = -(p + q)f(C, S)
rS Jc C

- K

CS

- P

fdx + (p + q)f(C, S) , yielding

-8-

3C

9q

= k[p«

h

fdx - K" J

(12)

Also, tf equals

j-C fS j-C

p"[ xfdx + C fdx] + 2p'[
; n J r i

xf dx + C

f2dx + Cf(S, S')J

-S

+ p[C

,f22dx

+ Cf.(S, S) + xf„dx] (13)

1 Jo zi

- q[f(S, S) + (S -C)f2(S, S) +

(x - C)f22dx]

fC

Using (2) and integrating by parts, the coefficient of p' in (13) becomes 2
Noting that (2) implies

fdx.

f12(x, S) + f2Z(x, 5) = 0

(14)

we nave

,f22dx "

f21dx = c(f2(C, S) - f2(S, S))

and

xf22dx = -

xf21dx= ~xf2!0- j

f.dx = -Cf,(C, S) - f(C, S),

where the last two steps use (2) and (5). Hence the coefficient of p in
(13) is -f(C, S). Similar manipulations establish that the coefficient of

q is f(C, S). So irss + ttcs equals

-9-

C

S

rC rS

fdx] + 2p'
g J

p" t xfdx + C

fi is

p"[ xfdx + C fdx] + p'[l+
in J r J

0

i;

fdx + t>' fdx =
C JC

fdx] , (15)

0

ana

H=k[a0p" + alP»3 (16)

where a and a., represent the terms in brackets in (15) , and are both positive.

Suppose for the moment that the airline faces a linear demand
curve. This means 3S/3q < 0 in view of p' < 0. From (12), if K" _> 0, then
3C/3q < 0 as well. However, casual observation suggests that marginal
capacity costs are declining in the airline industry, suggesting that K" < 0
and that the sign of 3C/3q is ambiguous.

While (12) and (16) do not permit unambiguous statements, the
introduction of regulation gives clear-cut results. With regulation, p
is fixed at some p*. The airline now may sell any number of seats between
0 and S(p*) at price p*, where S(p) is the inverse demand function. The
airline may choose to satisfy the market demand at the regulated price, or
it may turn away customers. In the latter case the airline achieves en
interior solution on its flat, truncated "demand curve," and the above
analysis applies. Since p' = 0 at the solution, 3S/3q = 0 and 3C/3q = -K".
If the airline is turning away customers, then an increase in q (institution
of the penalty scheme) will not change the number of seats sold but will
increase the seat capacity of the flight as long as marginal costs are
decreasing. It follows directly that, if the airline's subjective density
f is the true density for show- up s , as it should be given considerable

■10-

airline experience in the market, then the expected rate of capacity
rC rS

utilization, (] xfdx + C fdx)/C, falls and the expected number of people

bumped,

S J0 JC

xfdx, falls as well.

C
If the airline achieves a corner solution for S at S(p*), then

we can treat S as fixed for small changes in q, and .we have 3C/3q

rS

•IS- /ir„„ > 0, since from above ir_
Cq CC Cq

fdx > 0 and it < 0 by the second
C LC

order conditions. Since TTg will be positive in the comer solution

fS
situation, and since ti„ = - fdx < 0 from above, increases in q decrease

Sq Jc

irg . Eventually irg may become zero at S(p*), but then an interior solution

obtains and further decreases in q will not change S. Clearly, the results

on expected capacity utilization and expected number bumped hold here as

well.

If capacity is fixed and S < S(p*), then 3S/3q = — it /rr < 0

since it < 0 by the second order condition. However, if the airline meets

the demand at p*, 3S/3q = 0, but S may decrease for large changes in q.

We have

Proposition 1: Under the assumptions of the model, a regulated monopolistic
airline with decreasing marginal capacity costs will Increase flight
capacity but will not change the number of seats sold when the bumping
penalty increases (when the explicit penalty scheme is introduced). If
capacity is fixed and the airline turns away some customers, the number
of seats sold decreases when q increases. If the airline meets the
market demand, an increase in q may or may not result in a decrease in
the number of seats sold.

The Multi-Flight Monopoly Model

In this model, we postulate a demand per period for flights on a
monopolized route instead of postulating a demand for each Individual
flight. Aricraft capacity is fixed and the choice variable is flight

-11-

frequency, which is currently unregulated in the U.S. Let the aircraft seat
capacity equal one by choice of units and let the cost of operating each
flight be a, which does not depend on the number of flights, an assumption
which is not crucial in the analysis. The airline divides its passengers
equally among flights, an assumption which corresponds to a uniform distribution
of desired departure times among passengers and an even spacing of flights by
the airline. The revenue function is:

lPx 0 < x < 1

<Kx) =) (17)

( p - (x - l)q 1 < x <_ z

where p is the regulated price, z is the number of seats sold or reserved
per flight, and x is the number of show-ups. Both t and X are measured in
units of aircraft capacity.

First we show that the airline will never turn away customers.
Suppose the market demand at p is S seats, but suppose the airline finds
it optimal to sell or reserve S seats,' S < S. Suppose it also chooses
t, the number of f lights , optimally, selling or reserving S/t seats per
flight (we ignore the fact that t is an integer) . Then expected profits are

v. = t[

S/t

ij,(x)f(x, S/t)dx - a], (18)

0

since the density f pertains to each flight. If the airline increases S

and t by the same proportion, the expression in brackets in (18) is unchanged

while t increases. Thus tt increases when t«e number of seats sold or reserved

increases above S, indicating S was not optimal. Since tkjis argument holds

for any S < S, the airline does not turn away customers, selling S/t tickets

-12-

per flight when t flights are operated.

Replacing S by S in (18) and making a change of variable in the
integral, we get

u =

iKx/t)f(x/t, S/t)dx - at = ^f(x/t, S/t)dx + pf(x/t, S/t)

J0 't ■

dx

q(* °f(x/t, S/t)dx -

at

(19)

Setting ir = 0 yields

fC fxx + f2S f

- px( = + — j)dx -

Jo t t

C

-s

p <-S t x + r S f
-jUjX + f2S)dx + qx(— ~- + -j)

t •'C t t

-^■(f x + f S)dx - a = 0
Ct i ^

(20)

Since 3t/3q = -ir /u evaluated at the solution and n < 0 by the second-
tq tt tt }

order condition, the sign of n determines the sign of 3t/8q. Using (20)
and (2),

tq

' [S(X~ H)(X- S)f + *5f ]dx.
C t t

(21)

Integrating by parts repeatedly and cancelling and gathering terms, this
reduces to

(t + S) |'S . , . (t - 2) fS CJ

^ fdx + 3 — ' xf dx.

t jt t Jc

(22)

-13-

Let us choose a time period long enough so that t > 2. Since
(22) then exceeds

S

xfdx > t
t

S
fdx,

t + S + (t - 2)t fS ,, t + S - t r

? Jcfdx = — T3 —

fdx > 0, (23)

t

since S > t. Thus it > 0 and hence 3t/3q > 0. We have established

Proposition 2: Under the assumptions of the model, a regulated monopolistic
airline will increase its flight frequency when the bumping penalty increases
(when the explicit penalty scheme is introduced).

It may also be shown that the expected capacity utilization rate,

1 fS/t

xf(x, S/t)dx+ f(x, S/t)dx,

0 h

and the expected number of passengers bumped,

rS/t

xf(x, S/t)dx,

1

both decline with an increase in q.

Multi- Flight Duopoly Models

The general framework for the duopoly problem is the same as that
for the multi-flight monopolist: aircraft capacity is fixed and the airline
choice variable is flight frequency. The airline with the larger number of
flights per period captures a larger share of the market. Before formalizing
these ideas, it should be noted that a more convincing rationale for airline
myopia about bumped passengers (airlines ignore their presence for decision-
making purposes) can be provided in this model. We can imagine that the airline

-14-

assumes its bumped passengers go to the other airline for service.
Completely eliminating the bumped passengers from the decision problem also
requires that each airline believes that the other airline takes care of its
own bumped passengers. These assumptions, while logically defensible,
are fairly unrealistic, but they reduce the complexity of the problem
considerably.

At the regulated price p, the demand per period for seats on
the route is S. We assume there exists a function D(t.., t„) which impli-
citly depends on S and which gives the number of seats demanded on an
airline when it operates t flights and its competitor operates t„ flights.
Clearly ,

D(tl5 t2) + D(t2, tx) = S, (24)

and setting t = t = t in (24) , we have

D(t, t) = S/2, (25)

which says that when the airlines operate the same number of flights,
they divide the market. We assume

D(t , t2) > S/2 for t1 > t2, (26)

which says that the airline operating the larger number of flights captures
more than half of the market.

-15-

A rationale for the existence of the function D can be provided
by imagining that the "visibility" of the airline and hence demand for its
tickets depends on flight frequency. As above, we assume that the
airline allocates its customers equally among flights. It is not possible
to justify the existence of the D function by postulating a distribution
of desired departure times among passengers, with each customer dealing
with the airline that has a departure closest to his desired time. In
this situation both the number and spacing of flights of both airlines
determine the market share of each airline. This is really a complex
problem in spatial competition, where analysis of equilibrium is pro-
hibitively difficult.

Assuming both airlines have the same flight cost a and the same
subjective density f, expected profits when airlines 1 and 2 operate t-
and t flights respectively are, using (19) ,

1
ir =

i-D(t.., t.)

1 l i|>(x/t1)f(x/t1, D(t , t2)/t )dx- at1

0

(27)

2 fD(t0, t.)

v = 2 * iKx/t2)f(x/t2, D(t2, t1)/t2)dx - at2

More generally, profits can ba written it = ir(t , D(t1 , t_)) and

2
tt = 7t(t , D(t? , t ,))• The Cournot or Nash duopoly equilibrium is

characterized by the solution to the following system:

(2 8)

-16-
- = r1(t1, D(tr t2)) + t2dtv DCt^ t2))D1(t1, t2) = 0

H = ir1(t2, D(t2, tx)) + 7v2(t2, P(t2, t1))D1(t2, tx) = 0

By symmetry, t ■ t- - t is a solution for some t, which is found by
setting c = t and solving either equation in (27). We have been unable
to rule out solutions where t ¥ t • Analysing the sensitivity of the
symmetric solution to variations in q requires totally differentiating

^(t, S/2) + TT2(t, S/2)D1(t, t)

with respect to t and q, a calculation which yields ambiguous results.

Although we are not able to do comparative statics for the
Nash equilibrium, another appealing behavioral concept yields immediate
results. If it is assumed that each airline believes that its competitor
will exactly match its own flight frequency, then in view of (25), the
profits of each airline, are

tt =

S/2

iKx/t)f(x/t, S/2t)dx - at. (29)

0

Each airline optimizes over t, believing its competitor will choose the
same number of flights. It should be noted that unless the airlines are
identical, there is no equilibrium in this model. If their subjective
densities or flight costs differ, then tne solution results in different
flight frequencies , violating the behavioral premise of the model. Since

-17-

(29) is formally identical to (19) , the comparative statics results

are identical to those for the monopoly case. We have

Proposition 3: Under the assumptions of the model, two identic^ . regu-
lated airlines in a duopoly market where each believes the other mimics
its own flighc frequency choice will operate the same number of flights
and will botl increase fligh': frequency when the bumping penalty increases
(when the explicit penalty scheme is instituted).

As before, the expected rate of capacity utilization and the expected

number of passengers bumped both fall when q increases.

Conclusion,

The results in this paper conform to intuition. When the penalty
for bumping a passenger increases, airlines reduce the likelihood that
passengers will he bumped by increasing the number of flights in the
multi-flight models or by increasing capacity, or reducing seats sold when
capacity is fixed, in trie one-flight model: Further work could be directed
toward attempting to increase the generality of the change in the f
density when its upper limit increases and toward eliminating th i myopia
assumption concerning bumped passengers. That much improvement is possible
in these areas seems doubtful, however.

If one dis^rees with the conjecture that instituting the
explicit pei.*lty scheme is equivalent to increasing the bumping penalty,
believing insteid that it would amount to a decrease in the penalty,
then the results are exactly reversed: capacity or flight freqv-a-cy de-
crease, and the expected rate of capacity utilization and the expected
number of passengers bumped both increase with the institution of the
explicit penalty.

References

Beckmann, M. "Decision and Team Problems in Airline Reservations."
Econometrica, 26, 1 (January 1958).

De Vany, A. "The Revealed Value of Time in Air Travel." Review of Economics
and Statistics, LVI, 1 (February 1974).

Simon, J. "An Almost Practical Solution to Airline Overbooking." Journal
of Transport Economics and Policy, II, 2 (May 1968).

OUNDsV