A Monopolistic and Oligopolistic Stochastic Flow Revenue Management Model.

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OPERATIONSRESEARCH
Vol. 54, No. 6, November–December 2006, pp. 1098–1109
issn0030-364X?eissn1526-5463?06?5406?1098
informs®
doi10.1287/opre.1060.0336
©2006 INFORMS
A Monopolistic and Oligopolistic Stochastic
Flow Revenue Management Model
Xiaowei Xu
Department of Management Science and Information Systems, Rutgers University, The State University of New Jersey,
Newark, New Jersey 07102, xiaoweix@andromeda.rutgers.edu
Wallace J. Hopp
Department of Industrial Engineering and Management Sciences, Northwestern University,
Evanston, Illinois 60208, hopp@northwestern.edu
This paper studies a one-shot inventory replenishment problem with dynamic pricing. The customer arrival rate is assumed
to follow a geometric Brownian motion. Homogeneous customers have an isoelastic demand function and do not behave
strategically. We find a closed-form optimal pricing policy, which utilizes current demand information. Under this pricing
policy the inventory trajectory is deterministic, and a retailer sells all inventory. We show that dynamic pricing coordinated
with the inventory decision achieves significantly higher profits than does static pricing. Furthermore, under oligopolistic
competition we establish a weak perfect Bayesian equilibrium for the price and inventory replenishment game. We find
the pricing equilibrium to be cooperative even in a noncooperative environment, but that inventory competition results in
overstock and damages profits. Finally, we examine the trade-off between dynamic pricing and price precommitment and
find that flexible pricing is still beneficial, provided competition is not too intense.
Subject classifications: differential games; dynamic pricing; geometric Brownian motion; martingales; revenue
management.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received June 2004; revisions received December 2004, October 2005; accepted November 2005.
1. Introduction and Literature Review
Advances in information technology have made it possi-
ble to track sales and inventory, as well as adjust pro-
duction and pricing levels, more rapidly than ever. Many
industries, such as airlines, hotels, and various retailers,
use dynamic pricing to match demand with capacity or
inventory, maximize revenue, or achieve other strategic
goals (see, e.g., Talluri and Van Ryzin 2004 for more
applications). In response, recent years have witnessed the
growth of a dynamic pricing literature (see Bitran and
Caldentey 2003 and Elmaghraby and Keskinocak 2003 for
reviews). Because of the complex nature of dynamic pric-
ing (mainly due to the cumbersome recursive structure of
dynamic programming representations), much of this liter-
ature has relied heavily on numerical optimization. Closed-
form optimal pricing strategies are still rare (Gallego and
van Ryzin 1994, §2.3 provides an excellent exception).
Although numerical techniques can be adapted to many
different settings, they are not well suited to deriving gen-
eral managerial insights. Moreover, it is difficult to com-
bine numerical approaches to dynamic pricing with other
operations decisions, such as production or procurement,
or with a competitive framework. In this paper, we propose
an analytically tractable pricing model that incorporates an
inventory decision as well as competition.
We study dynamic pricing in the context of a one-shot
inventory replenishment problem, in which the customer
arrival rate is assumed to follow a geometric Brownian
motion. Most of the dynamic pricing literature has mod-
eled customer arrivals as a Poisson process (see, e.g.,
Gallego and van Ryzin 1994), which implies that demands
are independent across intervals. Under this assumption,
current demand information is not helpful for inferring
future demand, and therefore there is no need to com-
bine dynamic pricing and demand forecasting. However,
in reality, demands often show a correlation structure, and
hence demand information is valuable in dynamic pric-
ing. Clearly, the simplification of the independent incre-
ment assumption of a Poisson process presents a serious
obstacle to evaluating the merits of dynamic pricing as a
tool for hedging demand uncertainty and allocating limited
resources. Unlike a Poisson process, a geometric Brownian
motion is Markovian and partially captures demand cor-
relation, so using it in place of a Poisson process causes
pricing policies to depend on current demand information
and permits integration of demand forecasting with pricing.
Geometric Brownian motion is widely used in the
finance literature and recently has been adopted in the oper-
ations management literature. Caldentey and Wein (2006)
modeled spot prices as a bounded and piecewise-linear
version of a geometric Brownian motion. Chod and Rudi
1098

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(2003) assumed that a forecast evolution process follows
a geometric Brownian motion. There is also precedent for
using general diffusion processes to model demand flows
(see, e.g., Harrison et al. 1983 and Harrison and Taksar
1983). Raman and Chatterjee (1995) studied a dynamic
pricing problem of a monopolist, in which sales are mod-
eled by a diffusion process. Their focus was on the dif-
fusion/saturation effects of demand, and experience curve
effects of cost. Sapra and Jackson (2004) modeled buyer
demand as a diffusion process and derived closed-form
optimal capacity, production, and price trajectories. They
assumed that no agents have market power and that cus-
tomers have rational expectations. Our model assumes that
retailers have market power to determine prices, but we
avoid modeling strategic customer behavior and multiple
inventory replenishments.
In addition to assuming nonstrategic customer behavior,
we assume homogeneous customers who have an isoelastic
demand function. Because of its useful log-linearity prop-
erty, isoelastic demand functions are widely used in empir-
ical economics and marketing studies (see Monahan et al.
2004 for other merits). Coupling isoelastic demand with a
geometric Brownian motion gives us a closed-form solu-
tion of the optimal dynamic pricing policy. It is interesting
to note that this optimal pricing policy forms a martingale.
As such, our result can be considered a “random” version
of the result by Gallego and van Ryzin (1994, §3.2), in
which the optimal pricing policy is constant for a determin-
istic customer arrival process. Moreover, because isoelastic
demand has decreasing revenue in price, by applying the
optimal pricing policy the inventory path is deterministic
(i.e., all uncertainty is absorbed by pricing), and a retailer
sells all inventory.
Monahan et al. (2004) considered a multistage model
with independent customer arrivals. At every stage, a price
is set before the number of customer arrivals at that stage is
revealed. They formulated a dynamic program, which has
to be solved numerically. By comparing the value of pric-
ing flexibility (recourse) with the one-shot pricing model
of Petruzzi and Dada (1999), they demonstrated the advan-
tage of dynamic pricing. In this paper, by allowing an infi-
nite number of price changes and switching to Markovian
arrivals, we are able to obtain a closed-form pricing pol-
icy. In Xu and Hopp (2005a), we extended the framework
of Monahan et al. (2004) to allow serial correlation in
demand by switching to responsive pricing and derived a
semiclosed form optimal pricing policy. We compared it
with two heuristic pricing policies and found that sepa-
rating demand forecasting from dynamic pricing results in
systematic overpricing and a downward price trend. This
paper goes beyond Xu and Hopp (2005a) by considering
continuous-time pricing and oligopolistic competition.
In the competitive case, we assume that multiple retailers
sell a single product. Customers select a retailer according
to a stochastic process that is independent of the customer
arrival rate process and allows purchases only from retailers
with the lowest price and positive inventory. We model the
dynamic pricing competition process as a differential game
(see, e.g., Ba¸ sar and Olsder 1982). Differential games are
widely used in economics and marketing areas for model-
ing capital accumulation, R&D, and advertising (see, e.g.,
Dockner et al. 2000 and Jørgensen and Zaccour 2004). As
far as we know, however, they have rarely been applied in
operations management. By assuming that retailers know
the aggregate inventory level, we establish a weak perfect
Bayesian equilibrium that coincides with a public perfect
equilibrium (see, e.g., Fudenberg et al. 1994); that is, retail-
ers form their pricing strategies based on publicly available
signals and cooperate to achieve the system optimum. How-
ever, we find that this leads to significant overstocking in
the inventory competition, which damages expected retailer
profits. By comparing our model with a fixed-price model,
we are able to show that dynamic pricing is most beneficial
when competition is not too intense.
With price precommitment (i.e., fixed price), Lippman
and McCardle (1997), Mahajan and van Ryzin (2001), and
Netessine and Rudi (2003) showed that competition consis-
tently leads to overstock. Similarly, Netessine and Shumsky
(2004) found that horizontal competition lowers the book-
ing limit for lower-fare passengers. Without price precom-
mitment, Kreps and Scheinkman (1983) showed that a
capacity competition followed by a price competition leads
to Cournot outcomes under efficient rationing of demand.
Davidson and Deneckere (1986) demonstrated that mixed-
strategy equilibria are often likely under the proportional
rationing rule. Bikhchandani and Mamer (1993) derived
a mixed-strategy equilibrium for an oligopolistic model
and proved uniqueness for the duopoly case. All three
models were static in pricing, and deterministic. Bernstein
and Federgruen (2005, 2004) studied stochastic models in
which inventory is set after price decisions. Because they
assume that unmet demand is backlogged (and henceforth
no demand substitution), the inventory problem was solved
as a standard newsvendor model, and a reduced game in
price was obtained. For other competitive dynamic pricing
models that allow general demand functions and customer
arrival processes, see Perakis and Sood (2005a, b). In this
stream of research, our work can be viewed as extending
the results of Kreps and Scheinkman (1983) for isoelas-
tic demand to a more general environment with demand
uncertainty and pricing flexibility.
Using a modified version of the taxonomy developed by
Elmaghraby and Keskinocak (2003), our paper can be clas-
sified as C-S-NR-D-M (competition, single product, non-
replenishment, dependent demand over time, and myopic
customers), where we have added the first two dimensions
of noncompetition versus competition and single product
versus multiple products.
The remainder of this paper is organized as follows.
In §2, we study the monopoly model. We consider the
oligopoly case in §3. Conclusions and future research are
given in §4. All proofs are presented in the appendix.

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2. The Monopoly Model
We consider a retailer who purchases a quantity of a prod-
uct at the beginning of a “season” and sells to price-sensi-
tive customers over a finite time period. The problem faced
by the retailer is to determine both an optimal order quan-
tity and an optimal pricing policy.
We assume the following customer behavior model.
Identical customers have a quasilinear utility function
?
?−1x??−1?/?+sx+e
and face a budget constraint px+e ?w, where ?>1, x is
the consumption amount of the product, s is the salvage
value, e is the numeraire, p is the price, and w is the wealth
level. Hence, the customer’s demand function is x?p? =
+? if p ?s and x?p?=?p−s?−?if p >s. Without loss of
generality, we restrict price to the set p ∈?s?+??. If s =0,
the demand function x?p? has a constant price elasticity
coefficient ?.
The “season” corresponds to the finite interval ?0?T?. We
let N?t? be the stochastic arrival rate of customers, which
follows a geometric Brownian motion dNt= Nt??tdt +
?tdBt?, or equivalently, Nt=e
?tand ?tare deterministic and time dependent. We define
the history up to time t as Ht=??Ns? 0 ?s ?t? and htas
a realization of Ht. Let D?t?p?t?? = x?p?t??N?t? be the
aggregate demand rate, where p?t? is the price at time t
?∈?0?T?). Define the aggregate inverse demand function as
?N?t?
z
u?x?e?=
?t
0??s−?1/2??2
s?ds+?t
0?sdBs, where
q?N?t??z?=s +
?1/?
?
We denote the quantity ordered by the retailer by y and
the unit cost by c (>s). The retailer’s inventory at time t is
y?t? = y −?t
p = ?p?t?? t ∈ ?0?T? ? p?t? is a function of y?t? and ht}.
Denote ?p?u?z= sup?t ∈ ?u?T? ?
where z ? 0, and set p?t? = +? when t > ?p?u?z; that is,
the demand process is “turned off” when the retailer runs
out of inventory. Note that throughout this paper sup and
inf have their usual meanings because they are defined on
nonempty sets.
Define the retailer’s expected revenue from t to T
as a function of pricing policy p, inventory level ztat
time t, and history ht, by R?t?zt?p?ht? = E???p?t?zt
D?u?p?u??du ? ht?. Let R∗?t?zt?ht? = suppR?t?zt?p?ht?
be the maximal expected revenue from t to T given the
history htand inventory ztat time t. Finally, define the
expected profit at time zero as ??y?=E?R∗?0?y?h0??−cy
and let y∗∈ argmaxy??y?. Because the customer arrival
rate process Ntis Markovian, we replace the full his-
tory htby the arrival rate ntand rewrite expected rev-
enue as R?t?zt?p?nt? = E???p?t?zt
and R∗?t?zt?nt?=suppR?t?zt?p?nt?.
Define the elastic arrival forecast as
0D?u?p?u??du. A pricing policy is given by
?t
uD?l?p?l??dl ? z?,
t
p?u? ·
t
p?u?D?u?p?u??du ? nt?
at=
??T
t
e
?u
t?l−???−1?/2???2
ldldu
?1/?
n1/?
t
?
where t∈?0?T??
Note that given the customer arrival rate ntat time t,
the expected number of total customer arrivals in ?t?T?
is E??T
atwhen ? = 1. Hence, atrepresents the expected number
of customer arrivals between t and T with a calibration by
the price elasticity ? and volatility ?t.
tNudu ? nt? = nt
?T
te
?u
t?ldldu, which is equal to
Theorem 1. The optimal expected revenue from t to T
given the arrival rate ntand inventory ztat time t is
R∗?t?zt?nt?=szt+atz1−1/?
is p∗=?p∗?t??p∗?t?zt?nt?=s +atz−1/?
Note that the optimal pricing policy p∗is Markovian.
Theorem 1 gives us closed-form expressions for the opti-
mal expected revenues and prices. Gallego and van Ryzin
(1994) gave a closed-form optimal pricing policy for an
exponential demand function of price and a Poisson arrival
process. Because a Poisson process has independent incre-
ments, their optimal pricing policy is only a function of
inventory, and hence is independent of the current demand
information. In contrast, because we model arrivals with a
geometric Brownian motion, which is Markovian, our pric-
ing policy utilizes the current demand information Nt.
We can characterize the optimal inventory levels in the
following proposition.
t
, and the optimal pricing policy
, t ∈?0?T??.
t
Proposition 1. The optimal inventory level is given by
y∗
t=y∗
0
?T
?T
te
?l
?l
0?r−???−1?/2???2
rdrdl
0e
0?r−???−1?/2???2
rdrdl
?
which is deterministic and hits zero at time T.
Because an isoelastic demand function implies that rev-
enue is decreasing in price, the retailer has an incentive to
set price as low as possible and sell all inventory. More-
over, she is enabled to do so because an isoelastic demand
function takes values from 0 to +? as price decreases,
and a customer arrival rate process, governed by a geomet-
ric Brownian motion, always provides a positive customer
flow. Moreover, Proposition 1 implies that all stochasticity
in customer arrivals is absorbed into the optimal pricing
policy p∗, which results in a deterministic inventory path.
Note that the full value range (from 0 to +?) of isoelastic
demand and the positivity of a geometric Brownian motion
contribute to the results of Proposition 1. In contrast, a
Poisson arrival process does not guarantee positive cus-
tomer arrivals in a time interval. Some other demand func-
tions, such as linear and exponential, imply that a retailer
is not willing to price below the optimal price, which max-
imizes the concave revenue function and limits the range
over which demand can vary. Hence, the optimal inventory
path is often stochastic under other circumstances, such as
a Poisson arrival process coupled with exponential demand
(Gallego and van Ryzin 1994).
Figure 1(b) shows the optimal inventory trajectories y∗
for the ?tvalues given in Figure 1(a), where y∗
t
0=1, ?t=2,

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Figure 1.
(a) ?t; (b) the optimal inventory
trajectory y∗
t.
2.0
1.8
1.6
1.4
1.2
1.0
0.8
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.6
0.4
0.2
00
246810
02468 10
t
t
(a)
(b)
µt
yt*
?=2, and T =10. We see that the optimal inventory level
y∗
Now, to use our model to gain insight into consumer
behavior, we note that our model, like many revenue man-
agement models, assumes an exogenous customer arrival
process. However, customers may sometimes delay their
purchase in hopes of a lower price, and hence customer
arrivals could depend on the pricing strategy. If customers
behave strategically, they will purchase when consumer
surplus is maximized. See Mas-Colell et al. (1995) for a
discussion of using consumer surplus to quantify a simi-
lar trade-off between consuming a product versus paying
a price. Hence, examining the trend of consumer surplus
along the optimal price path provides an indication of how
strategic customers can be expected to behave.
The consumer surplus function for isoelastic demand is
defined as
tdrops quickly as the arrival rate ?treaches its peak.
v?p?=
?+?
p
x?u?du=
1
?−1?p−s?1−??
Noting that the optimal price at time t is p∗
and the consumer surplus at the optimal price p∗
t=s+at?y∗
t?−1/?
tis
v∗
t=v?p∗
t?=
1
?−1?p∗
t−s?1−??
we can characterize the dynamic behavior of the elastic
arrival forecast, optimal prices, and consumer surplus in the
following proposition.
Proposition 2. (1) ?at?Ht? is a supermartingale;
(2) ?p∗
(3) ?v∗
t?Ht? is a martingale; and
t?Ht? is a submartingale.
Proposition 2 suggests that the retailer should keep prices
“flat” ex ante. Note that if ?t= 0 for t ∈ ?0?T? (i.e., the
customer arrival rate process is deterministic), p∗
stant on ?0?T? and depends only on the realization of N0.
Hence, the martingale property of optimal prices can be
considered a stochastic version of Proposition 2 in Gallego
and van Ryzin (1994). Note that this property depends on
the nature of customer arrivals. Instead, if customer arrivals
follow a Poisson process instead of a geometric Brownian
motion, optimal prices turn out to be a submartingale (see
Xu and Hopp 2005b) because the retailer is not certain
that there will be incoming customers, and hence tends to
set price lower at the beginning of a season. In contrast,
because a geometric Brownian motion always provides a
positive customer flow, the retailer can count on future cus-
tomer arrivals. Partially consistent with our results, Sapra
and Jackson (2004) showed that the forecast process for
future price evolves as a martingale for a general diffu-
sion process, but the price process itself does not follow a
martingale in their dynamic production and pricing model.
Finally, note that the expectation of consumer surplus is
increasing. Hence, a strategic customer with full informa-
tion will wait until the last minute to make a purchase.
Of course, in many real-world settings customers cannot
delay their purchase or behave strategically. For instance,
customers may not have the information about demand and
inventory possessed by the retailer, and hence cannot antic-
ipate future prices. Furthermore, there may be nonprice
characteristics (e.g., product availability) that are not cap-
tured in consumer surplus (see Xu and Hopp 2005b for a
discussion). Therefore, even though Proposition 2 suggests
a tendency for bargain-hunting customers to delay pur-
chases of seasonal products, other factors may alter actual
behavior in the marketplace.
We can now complete the specification of an optimal
policy for the monopoly case by characterizing the opti-
mal inventory policy. To do this, note that by Theorem 1,
R∗?0?y?h0? = sy +a0y1−1/?and ??y? = E?R∗?0?y?h0??−
cy =E?a0?y1−1/?−?c−s?y? where
??T
tis con-
E?a0?=E?N1/?
0
?
0
e
?u
0?l−???−1?/2???2
ldldu
?1/?
?
We can optimize ??y? to get
Theorem 2. The optimal initial inventory level is
y∗=
?
1−1
?
??E?a0??
?c−s??

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Operations Research 54(6), pp. 1098–1109, ©2006 INFORMS
1102
and the optimal profit is
?∗=??y∗?=1
?
?
1−1
?
??−1
E?a0??
?c−s??−1?
A one-shot pricing model, in which a single price is
set for the entire season before any demand information is
revealed, has been widely studied (see, e.g., Petruzzi and
Dada 1999 for a review). Monahan et al. (2004), in Propo-
sition 5 and Corollary 1, compared the effect of dynamic
pricing (recourse) with a one-shot pricing model, and found
that pricing flexibility induces a higher initial inventory
level and larger expected profit. We obtain the same con-
clusion in the following proposition.
Proposition 3. Assume that s = 0. Denote the optimal
inventory and expected profit for one-shot pricing by ¯ y and
? ?. Let? V =maxk?0V?k?, where V?k?=k1/?−1?k−E??k−
?T
where ?f measures the value of pricing flexibility with
inventory coordination.
0Nudu?+??. Then, ?f=y∗/¯ y =?∗/? ?=?E?a0?/? V??>1,
Because the one-shot pricing model and the dynamic
pricing model generate revenues y1−1/?? V and y1−1/?E?a0?,
respectively, for fixed initial inventory y, we may use ?p=
E?a0?/? V > 1 to measure the value of pricing flexibility
without inventory coordination. Because ?>1 and ?p>1,
it follows that ?f= ??
more beneficial when coordinated with inventory procure-
ment than when inventory is not taken into consideration.
We further explore the value of pricing flexibility in the
following example.
p> ?p; that is, pricing flexibility is
Example 1. Let T = 10, ? ∈ ?1?5?4?, N0= 1, ?t= 0, and
?t= ? ∈ ?0?1?3?. We generate the contour graphs of ?f
and ?pin Figure 2. Note that the value of pricing flexibil-
ity is asymptotically stable as the uncertainty of customer
arrivals becomes large (i.e., ? → +?), but is sensitive to
price elasticity. In contrast, the value of pricing flexibility
is insensitive to ?, but is predominantly determined by ?
if the customer arrival rate is relatively certain (i.e., ? is
small).
Monahan et al. (2004, Figure 2) illustrated that the value
of pricing flexibility with inventory coordination increases
in ? and ?. However, they only allow approximately
10 price changes, while we allow an infinite number of
price changes. Hence, it is not surprising that their values
of pricing flexibility are much lower than ours. As such,
our results can be considered an upper bound on the value
of pricing flexibility.
Moreover, it is interesting to observe that the value
of pricing flexibility with inventory coordination (?f) is
increasing in price elasticity ???, but the value of pricing
flexibility without inventory coordination ??p? is decreasing
in ?. Hence, as demand becomes more elastic, pricing flex-
ibility becomes more valuable only if it is coordinated with
the inventory decision; otherwise, elastic demand deval-
ues pricing flexibility because it is difficult to adjust prices
without losing sales.
Figure 2.
(a) The value of pricing flexibility with in-
ventory coordination ?f; (b) the value of pric-
ing flexibility without inventory coordination
?p.
5
4
3
2
5
4
3
2
α
α
σ
σ
123
123
(a)
(b)
1.5
2.2
1.8
2.4
2.6
2.7
2.7
2.6
2.5
2.4
2.3
2.2
2.1
1.1
1.2
1.23
1.26
1.3
1.35
1.4
1.55
1.45
1.35
1.3
1.26
1.23
3. The Oligopoly Model
In this section, we generalize the monopoly model of
§2 into an oligopoly model, in which customers are rep-
resented by an atomic flow. That is, their instantaneous
demand is infinitesimal and can be fulfilled by the retailers
with the lowest market price and positive inventory.
To formulate a model, we assume that n retailers pro-
cure an amount of the product at a unit cost of c. Let pi?t?
and yi?t? be retailer i’s price and inventory level at time t,
where yi?0? = yi, i = 1?????n, represent initial inventory
levels. At time t, define the price vector p?t?=?p1?t??????
pn?t??, inventory vector y?t? = ?y1?t??????yn?t??, and
aggregate inventory level Y?t? =?n
set of low-cost retailers as I1?p?t?? = ?1 ? i ? n ? pi?t? ?
pj?t?? j ?= i?, the set of retailers with positive inventory
as I2?y?t?? = ?1 ? i ? n ? yi?t? > 0?, and define I?p?t??
y?t?? = I1?p?t?? ∩ I2?y?t??. Hence, only retailers in
I?p?t??y?t?? can make instantaneous sales at time t.
Denote P?p?t??y?t?? = pi?t?, where i ∈ I?p?t??y?t??, as
the instantaneous sale price. Because customers are indif-
ferent among retailers in I?p?t??y?t??, we assume the fol-
lowing splitting rule. Let ??t? = ??1?t???????n?t?? be an
n-dimensional random process on ?0?T?, which takes pos-
itive values and is independent with ?N?t?? 0 ? t ? T?.
We assume that all retailers know the distribution of ??t?.
Define ??t?p?t??y?t?? = ??1?t?p?t??y?t????????n?t?p?t??
y?t???, where ?i?t?p?t??y?t?? = 0 if i ? I?p?t??y?t??
i=1yi?t?. Denote the