Maxim Afanasyev(2010)Service Provider Competition- Delay Cost Structure-Segmentation and Cost Advantage

8/2/2019 Maxim Afanasyev(2010)Service Provider Competition- Delay Cost Structure-Segmentation and Cost Advantage

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MANUFACTURING & SERVICE

OPERATIONS MANAGEMENT

Vol. 12, No. 2, Spring 2010, pp. 213235issn 1523-4614 eissn 1526-5498 10 1202 0213

informs

doi 10.1287/msom.1090.0266 2010 INFORMS

Service Provider Competition: Delay Cost Structure,Segmentation, and Cost Advantage

Maxim Afanasyev, Haim MendelsonGraduate School of Business, Stanford University, Stanford, California 94305

{[email protected], [email protected]}

We model competition between two providers who serve delay-sensitive customers. We compare a gen-eralized delay cost structure, where a customers delay cost depends on her service valuation, with thetraditional additive delay cost structure, where the delay cost is independent of the customers service valua-tion. Under the additive delay cost structure, service providers offer different prices and expected delays, butcustomers are indifferent between the providers. Under the generalized delay cost structure, when the providershave different capacity or operating costs, we obtain value-based market segmentation, whereby higher-value

customers choose one provider and lower-value customers choose the other. We study how the delay costparameters, the market size, and the service providers costs affect the structure of the equilibrium.

Key words : delay cost structure; value-based market segmentation; service competitionHistory : Received: May 23, 2008; accepted: March 22, 2009. Published online in Articles in Advance

September 14, 2009.

1. IntroductionIn this paper we model competition between service

providers when their customers are sensitive to delay.

Our primary focus is on how the customers delay

cost structure and the asymmetry in the providers

costs affect the market equilibria. In most previousresearch, customers were assumed to have an addi-

tive delay cost structure. In reality, however, we often

observe interdependence between customers service

valuations and their delay costs. We argue that this

interdependence gives rise to major changes both in

the structure of the market equilibrium and in the

levels of arrival rates, prices, and service capacities.

We show how the delay cost structure, market size,

and differences between the providers costs (capacity

costs or variable costs of providing the service) affect

service differentiation and market segmentation.

The assumption of an additive delay cost structure,

which is common in the literature on the economics

of congestion and delay, is reasonable in a variety of

settings. This assumption means that the cost of delay

is independent of the customers service valuation.

This would be the case, for example, if the delay cost

reflects merely productivity losses or the value of lost

time, regardless of the nature of the service provided.

Consider, for example, the case of auto repair. While a

customer is waiting for her car to be repaired, she may

be using a rental car or public transportation. Thus,

her expected delay cost may be approximated by the

product of the expected number of days needed to

repair the car by the daily loss, the latter reflect-ing the value of lost time and the costs of alterna-

tive transportation.1 Assume that this daily loss is

not related to the value of the repair service. 2 Then,

if a repair facility repairs the car a day faster (on

average) than its competitor but charges a premium

equal to the expected daily loss, all customers will be

indifferent between the competing facilities. Although

the repair facilities are differentiated (offering differ-

ent prices and expected delays), there is no market

segmentation.

However, there are numerous situations where the

delay cost and the value of service are interdepen-dent. In electronic brokerage, for example, a delay in

1 Other opportunity costs of time may also be included in the daily

loss.

2 Across customers, the value of the repair service could be pos-

itively correlated with the daily loss, because higher-income cus-

tomers tend to have both higher opportunity costs and more

expensive cars.

213


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Afanasyev and Mendelson: Service Provider Competition214 Manufacturing & Service Operations Management 12(2), pp. 213235, 2010 INFORMS

trade execution deflates the investors expected profit,

and customers who trade frequently are willing to

pay a larger premium for fast execution. Thus, there is

a positive relationship among an investors frequencyof trading, the total amount traded, and the delay cost

she incurs over, for example, a year, which creates an

interdependence between the value of the brokerage

service to the customer and her delay cost. A similar

relationship holds when the cost of execution delay is

proportional to the trading volume at the transaction

level (see Dewan and Mendelson 1998, 2001). Indeed,

major electronic brokerage firms (e.g., Ameritrade,

E*TRADE, and Scottrade) prominently advertise their

average or median execution speeds, and brokerages

that target frequent traders bear the additional costs

of colocating their servers inside the exchanges toreduce delays (Martin and Malykhina 2007). In hospi-

tals, patients may incur a delay of weeks waiting for

surgery, and the longer the surgical delay, the higher

the mortality rate (Weller et al. 2005). We expect that

for patients in poor health, surgery is particularly

critical and they will suffer from a delay more than

patients whose condition is not as severe, implying

that the value of service and the delay cost are inter-

dependent. In online video streaming, transmission

delays affect the quality of the content viewed by the

customer. The higher the content quality, the more a

customer suffers if there are data transmission delays.

Thus, the delay cost is interdependent with the value

of the content to the customer.

Product development provides another example

of interdependence between service valuations and

delay costs. Adler et al. (1995) show that product

development can be modeled as a queue of projects

sharing common resources. While waiting in the

queue, new products become less attractive because

of changes in the marketplace and the competition.

A longer development lead time may result in a

loss of market leadership, a decrease in potential

revenue, and lower sales, as customers needs change

over time (see Takeuchi and Nonaka 1986, Stalk

1988, Gupta and Wilemon 1990, Mabert et al. 1992,

Millson et al. 1992). Gupta and Wilemon (1990) find

that a six-month delay in the market introduction of

a high-tech product reduces average profit by 33%

over five years. In the auto industry, the larger the

market potential for a new car model, the larger

the revenue loss resulting from model introduction

delays. Some well-known examples of companies

that achieved competitive advantage by reducing

their new product development times includeToyotas introduction of the Prius, Sun Microsystems

in the workstation market, Sony in the CD market,

and Samsung in memory chips and LCD televisions

in the 2000s (Stalk and Hout 1990, Clark et al. 1987,

Smith and Reinertsen 1998, Sun et al. 2004).

In this paper we study how the delay cost structure

affects the competition between service providers and

the resulting market structure. In particular, we dis-

tinguish between service differentiation and market

segmentation (see Smith 1956, Fradera 1986). Whereas

the additive delay cost structure results in differen-

tiated services, interdependence between service val-uations and delay costs leads to value-based market

segmentation. We give the following definitions.

Service differentiation occurs when at least one

firms services are different from those offered by its

competitor(s).

Market segmentation requires that, in addition, the

market is divided into customer segments so that

the customers in at least one segment strictly pre-

fer the services offered by one firm over the services

offered by its competitor(s).

Under service differentiation, (some) firms offer dif-

ferent service variants, typically at different prices,

but customers may be indifferent between these

alternative offerings. Under market segmentation,

different service variants attract different customer

segments. This means that some customers have a

strict preference for one firms offering. The segmen-

tation is value based if the customers are segmented

based on their valuations of the service.

We consider two competing service providers

whose customers are sensitive to delay. We model

interdependent service valuations and delay costs

using the generalized delay cost structure proposed

by Afeche and Mendelson (2004), which allows for

both additive and multiplicative delay cost compo-

nents. Earlier work assumed that customers util-

ity was additive in value and delay cost, namely

uUt = U Dt, where U is the gross utility absent

delay, t is the delay in the system, equal to the sum

of the waiting time in the queue and the service

time, Dt = d t is the expected delay cost, and u is


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Afanasyev and Mendelson: Service Provider CompetitionManufacturing & Service Operations Management 12(2), pp. 213235, 2010 INFORMS 215

the customers expected net utility. Under the gen-

eralized delay cost structure, the expected utility is

uUt = t U Dt, where t is a decreasing

function that deflates the customers value for theservice. Each customer is self-interested and maxi-

mizes her own expected utility, which she forecasts

using the distribution of the steady-state delay; i.e.,

EuUt = Et U Dt. In the linear case,

Dt = d t and t = 1 v t, so the expected utility

equals EuUt = U 1 vW dW, where W is the

expected delay in the system.

We find that the equilibrium is characterized by ser-

vice differentiation. Say one firm provides fast ser-

vice and charges a higher price and the other offers

slow service at a lower price. In addition, when the

service valuations and delay costs are interdependentand the providers costs (capacity costs or operat-

ing costs) are asymmetric, we find value-based mar-

ket segmentation into a high- and low-end customer

segment. We study how the delay sensitivity param-

eters and the total market size affect measures of

service differentiation and market segmentation. We

find that an increase in the multiplicative delay sen-

sitivity affects service differentiation nonmonotoni-

cally and increases the level of market segmentation.

Growth in the total market size decreases both the

degrees of service differentiation and market segmen-

tation until they disappear at the limit.

Our results on value-based market segmentation

are consistent with what we observe in a number of

markets where customers are sensitive to delay. One

example is public transportation in the San Francisco

Bay Area, where Viton (1981) finds that the value of

service and the delay cost are interdependent. Con-

sidering the competition between bus service and the

Bay Area Rapid Transit system (BART), Viton (1981)

finds that the systems differentiate their services in

terms of delay, with BART having a lower average

delay than the bus system and charging a higher

fare. Viton (1981) further finds value-based market

segmentation, with BART focusing on wealthier cus-

tomers and the bus system targeting the lower-income

population.

Mortgage loan origination is another service ex-

hibiting both an interdependence between delay cost

and service value and value-based market segmen-

tation. The longer a customer waits for mortgage

approval, the more the home price and interest rates

fluctuate and the higher the probability that the deal

will fall through before the loan is approved. Also,

the higher the home price, the greater the loss, sothe delay cost and the value of the origination ser-

vice are interdependent. In 1986, Citicorp launched

MortgagePower, a computer-based loan origination

system that dramatically reduced Citicorps mortgage

origination delay. In turn, Citicorp charged customers

higher processing fees, so its service was differenti-

ated by less delay and a higher price. Citicorp tar-

geted customers who borrowed larger amounts and

provided faster service in return for higher fees (Hess

and Kremer 1994, Stalk and Hout 1990). In con-

trast, traditional loan originators charged lower fees

and served lower-end customers whose loans were

smaller. A similar segmentation relating morgtgage

loan amounts, service fees, and average delay is

found by Guttentag (2001).

In the decorative laminates industry, customers

service values and delay costs are interdependent

because the disutility of construction delay is larger

for residential cabinetmakers and commercial speci-

fication customers, who typically work on expensive

projects, than for original equipment manufacturer

(OEM) direct purchasers. Accordingly, the former

are willing to pay a premium for fast delivery.The industry has two major product offeringsfaster

but more expensive and slower but cheaperand is

also characterized by value-based market segmenta-

tion, with Ralph Wilsonart providing faster service to

residential cabinetmakers and commercial specifica-

tion customers, and slower Formica dominating the

price-sensitive OEM segment (Stalk and Hout 1990,

Hamilton 2007, Feldman 2007).

In work under way, Afanasyev and Mendelson

(2009) consider movie production in Hollywood as a

special case of new product development. They esti-

mate the delay costs associated with this process and

show that value and delay cost are interdependent.

They then show how the market for studios is seg-

mentated based on movies market potential, with the

faster studios producing movies with higher expected

box office revenues. They also show that the lead time

in introducing movies into foreign markets exhibits

similar characteristics.


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The remainder of this paper is organized as follows.

We review the literature in 2. Section 3 considers

our benchmark, additive delay cost model. In 4 we

study the effects of the generalized delay cost struc-ture. In 5 and 6 we study the effects of market size

and differences in operating costs, and 7 offers our

concluding remarks.

2. Literature ReviewIn this section, we review the literature on delay sensi-

tivity and service provider (S) competition. Numerous

papers study markets with delay-sensitive customers

(see Hassin and Haviv 2002). Most of these papers

assume a discrete number of classes and consider the

optimal pricing or scheduling of services, or both,

among the classes, sometimes subject to incentivecompatibility constraints. Table 1 summarizes the lit-

erature on the effects of competition on queueing sys-

tems; we focus here on the first four papers, which are

closest to ours. Unlike our model, capacity is exoge-

nous and service valuations are constant in Chen

and Wan (2003) and So (2000), service providers are

symmetric in Chen and Wan (2005), and in all four

models the service providers engage in price compe-

tition. Chen and Wan (2003) study how the market

structure changes as a function of market size. For a

small total arrival rate, the market is dominated by

one of the providers. As the arrival rate increases,

there is a unique equilibrium with both firms in the

market, and then no equilibrium, followed by a con-

tinuum of equilibria and finally, a noncompetitive

market. In a model with endogenous service capaci-

ties, Chen and Wan (2005) find that although some of

the irregular equilibria in Chen and Wan (2003) are

eliminated, for a large market size there is a contin-

uum of equilibria. In contrast, we find a more regular

effect of market size on market structure under our

model.

Cachon and Harker (2002) derive conditions underwhich two competing service providers exist in the

market and study how scale economies affect the mar-

ket equilibrium. They find that the lower-cost S may

have a higher market share as well as a higher price.

They also show that the full price (service fee plus

delay cost) is an increasing function of the operating

and capacity costs. To answer their research questions,

Cachon and Harker (2002) need not explicitly model

customer choice as we do. Unlike Cachon and Harker

(2002), where the firms choose prices and waiting

times and the capacities adjust in equilibrium, in our

model firms choose capacities and arrival rates andprices adjust in equilibrium. This enables us to prove

the existence of an equilibrium.

In So (2000), heterogenous firms with exogenous

capacities compete by announcing their prices and

delivery time guarantees, again unlike our model.

The market has a fixed size and all customers are

served, so each firm serves a portion of the market.

The lower a providers price and delivery time guar-

antee, the higher its market share. So (2000) finds that

providers exploit their relative advantage to differen-

tiate their services; e.g., the high-capacity S will offer

better time guarantees, whereas the lower-operating-cost S charges a lower price.

The equilibria in Chen and Wan (2003) and So

(2000) are characterized by service differentiation but

no market segmentation, with one S offering shorter

delays and charging a price premium and the other

offering a longer delay and a lower price that exactly

offsets each customers delay cost savings. In Chen

and Wan (2005), there is no service differentiation (the

providers are symmetric), and in Cachon and Harker

(2002), there is service differentiation, but the underly-

ing customer choices are not specified in the demand

model.

With the exception of this paper, all the papers in

Table 1 that specify a delay cost structure assume

the additive cost structure, which (as we show) is

an important driver of the equilibria they obtain. To

date, few papers have considered a delay cost struc-

ture that is not additive, and none of them (to our

knowledge) studied the effects of competition. Afeche

and Mendelson (2004) present a generalized delay

cost structure and apply it to a single service facility

that has a fixed capacity. They compare the revenue-

maximizing and socially optimal equilibria for a pri-ority queue and show that some classical results may

not hold under the generalized delay cost structure. 3

They also compare the effects of bidding strategies

to uniform pricing under the generalized delay cost

3 For example, the classical results that the revenue-maximizing

admission price is higher than the socially optimal price, and the

revenue-maximizing use is lower than the socially optimal use.


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Table1

ComparisonofThis

PapertotheRelevantLiterature

Study

Serv

iceproviders

Serviceproviderdecisions

Servicevaluations

Delayco

ststructure

Researchquestions/Results

So(2000)

N

asym

metric

Price,

timeguarantee

Demandbasedonprice,

timeguarantees

Notapplicab

le(time

guarantee

model)

Findsthatprovidersexploitthe

ir

capacityadvantagetodiffere

ntiate

theirpricesandservices,

andstudies

determinantsofequilibrium

prices

CachonandHarker

(2002)

2asym

metric

Price,

delay

Demandfunction

Additive

Findthatscaleeconomiesincrease

incentivestooutsourceandthe

lower-costShashighermar

ketshare

andhigherprice

ChenandWan(2003)

2asym

metric

Price

Constant

Additive

Studyhowmarketsizeaffectsmarket

structure

ChenandWan(2005)

2symm

etric

Price,

capacity

Constant

Additive

Studyhowmarketsizeaffectsmarket

structure

Luski(1976),Levhari

andLuski(1978)

2symm

etric

Price

Constant

Additive/het

erogeneous

Conditionsforservicedifferent

iation

Reitman(1991)

N

symmetric

Price,

capacity

Constant

Additive/het

erogeneous

Conditionsforservicedifferent

iation

Loch(1991)

2symm

etric

Twomodels:(1)price;(2)price

andcapacity

Heterogeneous

Additive

Conditionsforasymmetricequilibria

DeneckereandPeck

(1995)

N

symmetric

Price,

capacity

Constant

Notspecifie

d

Studyexistenceofequilibrium

LedererandLi(1997)

LargeN,

asymmetric

Schedulingpolicy

Demandfunction

Notspecifie

d

Showthatlower-costand

lower-variabilityShavelarge

rmarket

share,

highercapacityutiliza

tion,

and

higherprofits

Gibbensetal.(2000)

2symm

etric

Price,

optiontocreatesubnetwork

Constant

Additive/het

erogeneous

Findthatundercompetitionwithtwo

classes,

eachSfocusesonly

onone

class

ArmonyandHaviv

(2003)

2symm

etric

Price

Constantwithineachof

twocustomerclasses

Additivewithineachclass

Analyzecompetitionandfindservice

differentiation

AllonandFedergruen

(2006)

2,

diffe

rentmarginal

costs

Threemodels:(1)price,

(2)capacity,(3)simultaneo

us

priceandcapacity

Demandfunction

Notspecifie

d

Findthatoutsourcing(servicepooling)

isbeneficial

AllonandFedergruen

(2007)

2asym

metric

Threemodels:(1)delayfollow

ed

byprice,

(2)pricefollowed

by

delay,(3)simultaneousprice

anddelay

Demandfunction

Notspecifie

d

Comparethethreemodels,

sho

wingthat

model1leadstohighestwaitingtime

andlowestprices

Pekgunetal.(2008)

2asym

metric

Price,

leadtime(centralizedvs.

decentralized)

Demandfunction

Additive

Studyeffectsofdecentralizationonprice

andlead-timedecisions

Thispaper

2asym

metric

Arrivalrateandcapacity

Heterogeneous/demand

function

Generalized

Effectsofgeneralizeddelaycos

t

structure;conditionsforand

implicationsofmarketsegm

entation


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structure. Katta and Sethuraman (2005) consider a

queue with priorities and customers with a gener-

alized delay cost structure. They propose an algo-

rithm that sorts customers into priority groups tomaximize the revenues of a monopoly. Kalvenes and

Keon (2007) consider an M/M/m loss system and

customers with a multiplicative delay cost structure

and propose a mechanism to reallocate customer ser-

vice times from periods when the system is highly

congested to periods when it is less congested. Our

paper studies the effects of interdependence between

customers service valuations and delay costs on S

competition. We show that when the delay cost and

service valuations are interdependent and the firms

costs are different, the equilibria must be character-

ized by value-based market segmentation.

3. Basic ModelWe start with a traditional model of competition

between two Ss who serve customers with heteroge-

neous service valuations that are subject to an addi-

tive delay cost. We first present our model of user

behavior and then proceed with a description of the

firms and the competition between them. This model

forms a foundation for the analyses that follow.

3.1. User Behavior

We first present the underlying user behavior model

for a single system with no delays. Potential cus-

tomers arrive following a Poisson process with rate .

The values they derive from receiving the ser-

vice when there is no delay are modeled as an

independent and identically distributed (i.i.d.) sam-

ple from a random variable U with cumulative dis-

tribution function (c.d.f.) F. The probability that an

arriving customer values the service at R or higher isF R = 1 F R. If only customers who value the ser-

vice at R or higher join the system, the effective arrival

rate is = FR, which gives rise to the inverse

demand function R = F1/, or V = F1/

(see Mendelson 1985, Dewan and Mendelson 1990),

where V is the expected total value created for

users per unit of time when the (effective) arrival rate

is , and V is the corresponding marginal value,

which is also equal to the cutoff value R. Given the

c.d.f. F, we can calculate V and V, the demand

curve when there is no delay.

Example. Let the distribution F of consumer values

U be uniform on 0 100 and the total arrival rate be

= 300. Then the expected number of arriving cus-

tomers with service valuations above R per unit oftime equals 300 PrU R, which gives rise to the

linear demand curve

V = 100 /3 (1)

We use this demand curve in our numerical examples

throughout the paper.

We assume that V is strictly decreasing, cor-

responding to the usual assumption of a monotone

decreasing demand curve, i.e.,

V < 0 for 0 (2)

and that

V1 + 2 + 1V1 + 2 < 0 for 1 2 0 (3)

Assumption (3) is needed to ensure that the firms

profit functions are supermodular and is satisfied

by the commonly used linear V = ( > 0,

> 0) and constant elasticity V = k (k > 0,

0 < < 1) demand curves.

Customers are sensitive to delay, and in this sectionwe follow the common assumption that the delay cost

is d per customer per unit of time. We assume that

the queue length is unobservable and that customers

make their decisions based on their expected through-

put time W. Let P be the price charged per job. It is

well known that the analogue of the demand curve

when delay costs are taken into account then becomes

P = V d W (4)

This modeling approach is common in the economics

of queues literature. Note that the model can be spec-

ified in terms of the distribution F of user valuations

or the inverse demand curve V. If the model is

specified in terms of probabilities, the effective arrival

rate is PrU d W P , reflecting the fact that as

the expected delay W increases, customers are willing

to pay less for the service, hence the expected number

of customers decreases.


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3.2. Service Providers

We consider a market with two competing Ss, S1and S2, with respective constant marginal capacity

costs g1 and g2. We assume g1 g2, so S1 is the lower-cost S. Each service provider Sj is modeled as an

M/G/1 queue with arrival rate j. We assume that a

customers service time is B/j, where B is a random

variable with coefficient of variation r (the same for

both firms) and mean 1, and j is the service capac-

ity of Sj. Thus, larger investments in service capacity

stochastically decrease each providers service time.

We further assume the first-come, first-served queue

discipline and unobservable customer service valua-

tions. The expected delay at Sj with capacity j expe-

riencing arrival rate j is given by the Pollaczek-

Khinchin formula for the expected total time in thesystem,4

Wj =

jr

2 + j

j j+ 1

1

2j (5)

The expected delay at each Sj increases in j and r

and decreases in j. We further assume that there is a

direct operating cost c of serving each customer.5 We

assume that each S charges a single price and that the

demand is high enough for both Ss to be profitable at

zero delay cost; i.e.,

V0 > c + maxg1 g2 (6)

To ensure internal solutions, we also assume that for

a high enough arrival rate , it is unprofitable to serve

additional customers; i.e., there is a satisfying

V 0. In this case, there is obviously

a U such that all customers with U > U choose to

receive service, and all customers with U U decline

the service. Because customer choice must satisfy the

individual rationality constraint, Sjs profit maximiza-

tion problem in the first stage is

maxj j

jPj cj gjj

s.t. Vj d Wj = Pj

(8)

Next, consider the customer choice when twoproviders are in the market. Now each customer has

three options to choose from:

(a) join S1 (net utility equals U d W1 P1),

(b) join S2 (net utility equals U d W2 P2), or

(c) do not join (net utility equals 0).

The customer decides among these alternatives by

comparing the net utilities in (a), (b), and (c) above.When both providers serve customers, all cus-

tomers must be indifferent between S1 and S2.7

Indeed, if there exists a customer who strictly prefers

Sj (j= 1 2), namely, for some valuation U,

U d Wj Pj > U d Wj Pj (9)

then all customers will strictly prefer Sj (because

(9) will hold for all U), so Sj will serve no cus-

tomers, which is a contradiction. This means that the

6 If there is no firm in the market, the solution is trivial.7 We will show later that this is the direct result of the assumed

additive delay cost structure.


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customer choice problem is degenerate (as all cus-

tomers are indifferent between the two firms). This

is often modeled by randomizing the choice between

the providers. Thus, there exists a service valua-tion threshold U such that the customers with ser-

vice valuations above U are randomly allocated and

those below U are not served (because their utilities

upon being served by either provider would be nega-

tive). Although customers are indifferent between the

two providers, consistency with the first-stage deci-

sions requires that the probability of joining Sj equals

j/1 + 2 j= 1 2.

To complete the specification of the game, recall

that each Sj, j = 1 2 decides on j and j, building

on the customers decision criteria (which, as shown

above, are degenerate). Hence, a necessary conditionfor an equilibrium is that each Sj solves

maxj j

jPj cj gjj

s.t. V1 + 2 d Wj = Pj

(10)

where Wj is the expected delay given by Equation (5).

If Sj decides to stay out of the market, it sets j =

j = 0, and Sj solves (8). In a mixed-strategy equilib-

rium, customers with valuations above some thresh-

old are randomly allocated between the two firms,

and customers with valuations below this threshold

are not served. It is clear from the above discussionthat under the additive delay cost structure, the only

possible equilibria are mixed-strategy equilibria.

The Cournot game is often presented as a

game with explicit or implicit probabilistic customer

choices. In our game, the probability that a customer

with service valuation U is served by Sj equals

pjU =

j

1 + 2if U > U j = 1 2

0 otherwise.

(11)

This also translates directly to probabilistic choicemodels often used in the marketing literature (see

Currim 1982, Kamakura and Srivastava 1984, Grover

and Srinivasan 1987, Kamakura and Russell 1989).

The marketing literature also allows for more gen-

eral forms than (11), which is implied by our model.

This suggests that a more general model specification

might lead to a generalization of (11), such as that

corresponding to the multinomial logistic model.

3.4. Equilibrium

Intuitively, an increase in the delay sensitivity d in-

creases customers delay costs and lowers the effective

total demand. As a result, for small d, both providersare in the market, but as d increases, only one S sur-

vives. For larger values of d, there is no S in the mar-

ket. Proposition 1 makes this intuition precise. Proofs

of propositions and corollaries are in the appendix.

Proposition 1. There exist constants 0 < d1 d2

d3 d4 such that the following applies.

(1) Duopoly solution: For d < d1, there is a mixed-

strategy equilibrium with both S1 and S2 in the market,

where the capacity j and market size j ofSj are given by

the solution to the equations8

jr2 +jjjj

2+ jr

2

jj+1 1

22j= gj

jd j= 12

V1 +2+jV1 +2dWj

djr2 +1

2jj2

= c j= 12

(12)

where Wj are given by (5).

(2) Single monopoly solution: For d d1 d2 d3 d4,

there is an equilibrium with only the low-cost provider, S1,

in the market. The equilibrium capacity 1 and market size

1 are given by the solution to the equations

1r2 + 111 1

2+

1r2

1 1+ 1

1221

=g1

d1

V1 + 1V1 dW1 d1

r2 + 1

21 12

= c

(13)

where W1 is given by (5) (with j= 1).

(3) Multiple monopoly solutions: For d d2 d3, there

are two equilibria: one with only S1 in the market and one

with only S2 in the market. In the equilibrium with Sm in

the market, the capacity m of Sm and its market size mare given by the solutions to the equations

mr2 + mmm m

2+ mr

2

m m+ 1 1

22m= gm

dm

Vm + mVm dWm dm

r2 + 1

2m m2

= c

(14)

where Wm is given by (5) (with j= m).

8 For d = 0, the equations are j = j and V1 + 2 +

jV1 + 2 = c.


9/23


(4) Market is not served: For d d4, there is no S in

the market, and 1 = 2 = 1 = 2 = 0.

The sequence of market structures in Proposition 1

shows that for small delay sensitivities, both firms

are in the market. As the delay sensitivity increases,

the higher-cost S leaves the market. This happens

because, as a monopoly, S2 can profitably serve fewer

customers than S1; hence the entry barrier for S1 is

lower than for S2. As the delay sensitivity increases

further, only one S can survive. As the delay sensitiv-

ity increases further, S2 cannot obtain positive profits.

Because of its cost advantage, S1 stays in the market

longer but eventually, for larger values of d, leaves the

market as well. Proposition 1 allows for multiple equi-

libria, so the constants d1

through d4

in Proposition 1

need not be unique. Furthermore, the interval 0 d1

consists of up to three subintervals:

(i) 0 dD1 , where only duopoly equilibria exist;

(ii) dD1 dM1 , where duopoly and S1 monopoly

equilibria exist; and

(iii) dM1 d1, where duopoly and two, S1 or S2,

monopoly equilibria exist.

In 5 we study the progression of the different equi-

libria as a function of the market size and discuss

these subintervals in more detail.

Proposition 1 also shows that when the two firms

are in the market, they may offer differentiated ser-vices, but customers are always indifferent between

them; i.e., there is a mixed strategy equilibrium. If the

providers capacity costs are equal, the expected delay

and price are determined, so there is neither service

differentiation nor market segmentation. If g1 < g2,

S1 has a cost advantage: it sets a lower expected delay

than S2 but in equilibrium charges a price premium

that exactly equals the delay cost savings of each cus-

tomer; i.e.,

P1 P2 = d W2 W1 (15)

This means that while the Ss offer differentiated ser-

vices, there is no market segmentation.

4. Generalized Delay Cost StructureIn 3 we found that under the additive delay cost

structure, there is no market segmentation. In this

section we show that under the generalized delay

cost structure, we may obtain value-based market seg-

mentation, where one S serves customers with higher

service valuations (the high-end segment) and the

other S serves customers with lower service valua-

tions (the low-end segment). In this equilibrium, the

high-end S offers a smaller delay but at a higherprice than the low-end S. Unlike in 3, high-value

customers strictly prefer the high-end S and low-

value customers strictly prefer the low-end S. More-

over, if the firms have different capacity costs, only an

equilibrium with value-based market segmentation is

possible.

The generalized delay cost specification allows the

delay cost of a customer to depend on his or her

service valuation. As described in the introduction,

such dependence is common in a variety of settings.

Under the generalized delay cost structure, the utility

derived by a customer with service valuation U anddelay t is t U Dt, where t is a decreasing

function that deflates the customers service valuation

U, and Dt is the additive component of the delay

cost. This means that the customers service valua-

tion and delay cost are interdependent. Each customer

with service valuation U maximizes his or her own

expected utility; i.e., EuUt = Et U Dt.

We assume for the most part that t is linear and

decreasing, t = 1 v t. Under the linear specifi-

cation, a customer with service valuation U has util-

ity EuUt = U 1 vW dW, where W is the

expected delay in the system. Hence, the customerincurs a delay cost of vU + d W, where v > 0 is

a multiplicative coefficient and d 0 is an additive

coefficient of the delay cost. The demand curve (4)

becomes

P = V 1 v W d W (16)

Similar to Equation (8), if only Sj is in the market, it

solves

maxj j

jPj cj gjj

s.t. Vj 1 v Wj d Wj = Pj

(17)

When both firms are in the market, we structure the

game as a Cournot model, as before, where in the first

stage Sj decides on j and j (j = 1 2), and in the

second stage the customers decide which S to join,

if any. In 3.3 we showed that under the additive

delay cost structure, there was only a mixed-strategy

equilibrium; i.e., there was a service valuation U such


10/23


that customers with service valuations above U were

randomly allocated between the two firms, and those

below U were not served. Another type of equilib-

rium is the value-based market segmentation equilibrium,where customers are divided into three groups basedon their service valuations. The lowest value group

receives no service, the highest value group is served

by one S, and the intermediate group is served by theother S. Formally, this means that U1 > U2 exist such

that a customer with service valuation U (a) joins Sj,

if U U1, (b) joins Sj, if U U2 U1, or (c) does not

join either S, if U < U2.

Hence, all customers with service valuations aboveU1 form the high-end segment, served by Sj, and

all customers with service valuations in the interval

U2 U1 form the low-end segment, served by Sj.In our model, the only equilibrium structures pos-

sible are the mixed-strategy and value-based market

segmentation equilibria that we identified. We havealready shown in 3.3 that for v = 0, we obtain only

the mixed-strategy equilibrium under a duopoly. If

for v > 0 some customers with different valuations UAand UB are indifferent between S1 and S2, then

UA vUA +dW1 P1

= UA vUA +dW2 P2 and

UB vUB +dW1 P1 = UB vUB +dW2 P2

(18)

Then, by subtracting the two equations, we obtain that

W1 = W2, which also implies that P1 = P2. Thus, any

customer UC will also be indifferent between the twoproviders. Similarly, if customers with valuations UAand UB strictly prefer Sj, then any customer with valu-

ation UC UA UB will also prefer Sj. Indeed, if UC

UA UB, then there exists an 0 < < 1 such that UC =

UA + 1 UB. Because UA and UB strictly prefer Sj,

UA vUA +dWjPj

> UA vUA +dWjPj and

UB vUB +dWjPj>UB vUB +dWjPj

(19)

Giving the two equations weights and 1 ,

respectively, and summing up, we obtain that

UCvUC+dWjPj>UCvUC+dWjPj (20)

We thus conclude that only our mixed-strategy

or value-based market segmentation equilibria arepossible.

We next derive the structure of the segmented equi-libria for a market with asymmetric providers and a

generalized delay cost structure with v > 0 (symmet-

ric providers are considered later in this section). Weshow that in such a market, there is no equilibriumwithout value-based market segmentation; i.e., among

the customers who are served, those with service val-uations above a threshold U1 choose the fast S, andthose with service valuations below that threshold but

above another threshold U2 choose the slow S. Wefocus on the equilibrium in which the lower-cost S1serves the high-value segment9 and show that for suf-

ficiently small delay sensitivities, both Ss are in themarket and the market is segmented.

Proposition 2. There exist nested sets, with 0 0

1, 1 2 3 4 in vd space such that the fol-lowing applies.

(1) Duopoly solution: For vd 1 and v > 0, thereis an equilibrium with both firms in the market, where S1serves the high-end customers and S2 serves the low-end

customers. The capacity j and the arrival rates j to eachSj are given by the solution to the following equations.

1r2 + 11

1 12

+1r

2

1 1+ 1

1

221

=g1

1V1v + d

V1 + 1V1vW2 W1

+ V1 + 2 + 1V1 + 21 vW2

dW1 1V1v + d

r2 + 1

21 12

= c

(21)

and 2r

2 + 222 2

2+

2r2

2 2+ 1

1

222

=g2

2V1 + 2v + d

V1 + 2 + 2V1 + 21 vW2 dW2

V1 + 2v + dr2 + 1

22 22

= c

(22)


9 Under certain conditions, an equilibrium exists in which the

higher-cost S, S2, serves the high-value segment. It can be shown

that whenever such an equilibrium exists, there always exists an

equilibrium in which the lower-cost S serves the high-end segment.


11/23


(2) Single monopoly solution: For vd 2\1

4\3, there is an equilibrium with only S1 in the mar-

ket. The equilibrium capacity 1 and market size 1 of S1

are given by the solution to the following equations.1r

2 + 111 1

2+

1r2

1 1+ 1

1

221

=g1

V1v + d1

V1 + 1V11 vW1 dW1

1V1v + d

r2 + 1

21 12

= c

(23)

where Wm is given by (5).

(3) Multiple monopoly solutions: For vd 3\2,

there are two equilibria: one with only S1 in the market,and one with only S2 in the market. In the equilibrium with

Sm in the market (m = 1 2), the equilibrium capacity mand market size m of Sm are given by the solution to the

following equations.mr

2 + mmm m

2+

mr2

m m+ 1

1

22m

=g2

Vmv + dm

Vm + mVm1 vWm dWm

mVmv + d r2

+ 12m m

2= c

(24)

where Wm is given by (5).

(4) Market is not served: For vd 4, there is no S

in the market and 1 = 2 = 1 = 2 = 0.

Proposition 2 shows that for small values of the

delay sensitivitiesi.e., in the set 1 where we obtain

the duopoly solutionthere is a value-based mar-

ket segmentation equilibrium, where S1 serves the

high-end segment and S2 serves the low-end segment,

and customers are not indifferent between the two

providers. Market segmentation requires delay coststhat are low enough to support two firms in the mar-

ket. As the delay sensitivities increase, so vd is in

the set 2\1 (the single monopoly solution region),

the only possible equilibrium is one with S1 being

a monopoly. In this set, the lower-cost S enters and

crowds out the higher-cost S. As the delay sensitiv-

ities grow furtheri.e., in the set 3\2 (the multi-

ple monopoly solutions region)two equilibria exist

with either of the Ss being a monopoly. This hap-

pens because as the delay sensitivities increase, Sj(j = 1 2) cannot be profitable if it enters the market

with Sj already there. As v and d increase further, S2cannot survive even as a monopoly. Because of its cost

advantage, S1 can survive for larger values of v and d,

but ultimately it cannot be profitable and leaves the

market, too, so the market is not served. Note that

Proposition 2 allows for multiple equilibria, e.g., for

vd 1, there may also be monopoly solutions, and

for vd 3\2 there are two monopoly equilibria.

Corollary 1. For v > 0 and g1 < g2, any duopoly

equilibrium must be a value-based market segmentation

equilibrium.

Corollary 1 shows that market segmentation is

driven by the generalized delay cost structure as

well as by the firms cost asymmetry. Because the

providers capacity costs are different, we expect them

to choose different delays, and, because v > 0, differ-

ent customers value the delays differently.

Similar to what we found in Proposition 1, the set

1 consists of up to three subsets:

(i) D1 , where only duopoly equilibria exist;

(ii) M1 \D

1 , where duopoly and S1 monopoly equi-

libria exist; and

(iii) 1\M1 , where duopoly and two, S1 or S2,monopoly equilibria exist.

We may also have in 1 several duopoly equilibria.

For example, for sufficiently small g2 g1, there is

an equilibrium where S2 serves the high-end segment

and S1 focuses on the low-end segment. It can be

shown that if we have an equilibrium with the higher-

cost S serving the high-end segment, there is always

an equilibrium with the lower-cost S serving the high-

end segment.

Importantly, the structure of the equilibria obtained

under the generalized delay cost structure (Proposi-

tion 2) is substantively different from the structure

we obtained under the additive delay cost struc-

ture. In particular, the framework of an additive

delay cost structure does not adequately explain the

prevalence of value-based segmentation in markets

ranging from transportation to mortgage loan origi-

nation. With generalized delay costs, one of the firms

serves the higher-end segment and the other serves


12/23


Figure 1 Distortions in Total Industry Profits (Solid Line, Left Axis)

and the Difference Between the Capacity of the High-End

and Low-End Service Providers (Broken Line, Right Axis) in

the Duopoly Equilibrium as Functions of the Multiplicative

Delay Sensitivity v

0 1 2 3 4

15

15

10

5

0

Distortionintotalprofits(%)

Distortionincapacitydifference(%)

20

20

25

25

30

35

40

V

Notes. The distortions are caused by assuming that the delay cost is addi-

tive and the average delay cost remains the same. The distortion in total

profits is the change in total profit, 1 + 2, caused by not accounting forthe multiplicative structure of the delay sensitivity relative to the correct total

profits. The distortion in capacity differences is the change in capacity dif-

ference (1 2) caused by not accounting for the multiplicative structure

of the delay sensitivity relative to the correct capacity difference. For both

line graphs, the service valuations U are uniformly distributed over the inter-

val 0 100 and the market size is = 300 (which corresponds to V =100 /3), g1 = 29, g2 = 31, c = 5, and d= 0.

the lower-end segment, resulting in a natural value-

based segmentation. In a market characterized bymultiplicative delay sensitivity, assuming an additive

delay cost structure leads to significant deviations not

only in qualitative structure but also in the quanti-

tative results. Clearly, these deviations are increasing

in v. However, as shown in Figure 1, the distortion in

the capacity differences between the service providers

tends to plateau as v increases, whereas the distor-

tion in total industry profit increases as long as both

service providers are in the market.

Our results link the structure of delay costs to the

existence of service differentiation or market segmen-

tation. Quantitatively, they show how service differ-

entiation and market segmentation depend on the

additive (d) and multiplicative (v) components of

delay cost. Our analysis shows how increasing each

of the two components of delay cost affects market

behavior. To translate this into empirically meaningful

results, we need concrete measures of service differ-

entiation and market segmentation.

In the context of our model, service differentiation

means that customers experience different expected

delays and pay different prices, depending on the

provider that serves them. Thus, two natural mea-sures of service differentiation are the price difference,

P = P1 P2, and the differences in expected delays,

W = W2 W1, between the two providers. Both of

these measures can be empirically estimated in the

marketplace, and each exhibits different behavior as

a function of the delay sensitivities.

Consider first P = P1 P2. Figure 2(a) shows that

P is an increasing function of v. The monotone

relationship between P and v is not surprising: as

v increases, the firms both suffer from lower demand,

but S1 also benefits from higher differentiation, which

gives it greater market power. One might expect P tobe an increasing function of d (as would indeed be the

case when v = 0); however, as shown in Figure 2(b), it

is a quasiconvex. This is the net result of two opposite

effects: on the one hand, P decreases in d because its

component V1vW2 W1 is a decreasing function

of d; on the other hand, its component dW2 W1 is

an increasing function of d.

Next consider W= W2 W1. Although a decreas-

ing relationship between W and d is not surprising

(an increase in d affects customers proportionally to

the expected delay), it is hard to predict the shape of

the relationship between W and v. Numerical exper-

iments show that W is a quasiconvex function of v.

As v increases, all customers are willing to pay less

for the service, so the returns on investments in excess

capacity declinethe services become more similar

and W tends to decrease. In contrast, an increase in

v makes customers more sensitive to the difference in

delay between the two providers, which is beneficial

to the high-end S. It is straightforward to prove that

for sufficiently small v, W is an increasing function

of v, and for sufficiently large v, W is a decreasing

function of v.

Empirically, this means that although W is a

monotone increasing function of d, P is quasicon-

vex, and though P is a monotone increasing function

of v, W is a quasiconcave function of d.

We next measure market segmentation by how

much the average customer in the high-value segment

loses if she switches from S1 to S2. The average con-

sumer of S1 has surplus V1/21 vWj dWj Pj


13/23


Figure 2 The Effects of Increasing the Multiplicative (a) and

Additive (b) Delay Sensitivities on the Difference in

Service Fees

0 0.04 0.08 0.12 0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

S1

S2

S1

S2

v

(a) Difference in service feesP

1 P

2as a function ofv for d= 1

0 5 10 150.42

0.43

0.44

0.45

0.46

0.47

d

(b) Difference in service fees P1 P2as a function ofd for v = 0.1

Notes. In these figures, v is the multiplicative delay sensitivity, d is the addi-

tive delay sensitivity, and P1 P2 is the difference in service fees betweenthe providers, which is an alternative measure of service differentiation. In

both figures, the service valuations Uare uniformly distributed over the inter-val 0 100 and the market size is = 300 (which corresponds to V =100 /3), g1 = 29, g2 = 31, and c = 5.

if served by Sj. The reduction in consumer surplusif she switches to S2 equals V

1/2v + dW2 W1

P1 P2. Because the marginal customer (at 1) is

indifferent between the two providers, V1v + d

W2 W1 P1 P2 = 0, which means that the loss

for the average customer is

= v

V

12

V1

W2 W1 (25)

Incentive compatibility means that 0. Indeed,

the loss from switching is commonly used in stud-

ies of incentive compatibility in queueing systems

(see Mendelson and Whang 1990, Yahalom et al.2006), and similar differences are used in the mar-

keting context (e.g., Kalish and Nelson 1991, Werten-

broch and Skiera 2002, Chung and Rao 2003, Jedidi

et al. 2003, Wang et al. 2003). Value-based market seg-

mentation and service differentiation are not indepen-

dent, as is proportional to W. However, when

there is no interdependence between the service val-

uations and delay costs, i.e., when v = 0, we have

W > 0 and = 0. Thus, positive service differenti-

ation is a necessary but not sufficient condition for

positive market segmentation.

The level of market segmentation is an increas-ing function of v. For small v, an increase in v both

increases customers sensitivity to delay and the level

of service differentiation; hence increases as well.

For large v, service differentiation starts declining

but this effect is still weaker than that of the increase

in customers sensitivity to delay (because for large v,

as v increases the number of customers served by S1decreases; hence the service valuation of the average

customer, 1/2, increases).

We next consider symmetric firms and the resulting

market structure for sufficiently small values of the

delay sensitivities.

Proposition 3. When v > 0 and g1 = g2 = g, there

exists a set in (v d) space10 in which there are threeequilibria with both firms in the market.

(1) Mixed strategy: Each Sj chooses capacity j and

market size j satisfyingjr

2 + jj

j j2

+jr

2

j j+ 1

1

22j

=g

jV

1+

2v+

dV2 + V21 vW dW

V2v + dr2 + 1

2 2= c

(26)


10 The set is analogous to 1 in Proposition 2.


14/23


(2) Equilibria with value-based market segmentation:

There are two equilibria where Sj serves the high-end seg-

ment and Sj serves the low-end segment, where j and j

satisfy jr

2 + jj

j j2

+jr

2

j j+ 1

1

22j

=g

jVjv + d

Vj + jVjvWj Wj

+ Vj + j + jVj + j1 vWj

dWj jVjv + d

r2 + 1

2j j2

= c

(27)

and jr

2 + jj

j j2

+jr

2

j j+ 1

1

22j

=g

jVj + jv + d

Vj + j + jVj + j1 vWj dWj

Vj + jv + dr2 + 1

2j j2

= c

(28)

where Wj and Wj are given by (5).

Proposition 3 shows that when the capacity costsare equal, there are three equilibria: two equilib-

ria with value-based market segmentation and one

mixed-strategy equilibrium. In the last equilibrium,

the firms have the same capacity and delay, and cus-

tomers are indifferent between them.

5. Effect of Market SizeIn this section we study how the size of the market,measured by the total arrival rate , affects the mar-

ket structure and the level of competition between the

two firms. As discussed in the literature review, Chen

and Wan (2003, 2005) found that the effect of marketsize on the market structure is quite surprising andcounter-intuitive. It is thus interesting to study the

effect of market size in our model vis--vis the mod-

els of Chen and Wan. In addition, we are interested

in the effects of increasing market size on service dif-

ferentiation and market segmentation.The following proposition studies the effects of

increasing on the market structure.

Proposition 4. For any pair vd, there exist con-

stants 0 1 2 3 4 5 such that the follow-

ing applies.

(1) No S in the market: For < 1, no S serves themarket.

(2) S1 monopoly: For 1 2, there exists only a

monopoly equilibrium with S1 in the market.

(3) Two monopoly equilibria: For 2 3, there

exist two monopoly equilibria with either S in the market.

(4) S1 monopoly or duopoly equilibria and two monopoly

equilibria: For 3 4, there exists a threshold value

g such that (i) if g2 g1 > g, there exists only an

equilibrium with S1 being a monopoly; and (ii) ifg2 g1

g, there exist duopoly equilibria with both firms in the

market as well as two monopoly equilibria with either S in

the market.(5) Duopoly equilibria and S1 monopoly: For

4 5, there exist duopoly equilibria with both firms

in the market as well as an equilibrium with S1 being a

monopoly.

(6) Duopoly equilibria: For 5, there exist only

duopoly equilibria with both firms in the market. The

arrival rates and S capacities are given by Proposition 1

( for v = 0) and 2 ( for v > 0).

In Proposition 4 we find an interesting sequence

of market equilibria, which is illustrated in Figure 3.

Potentially, there are seven regions:

Region (1): The market is so small that it cannot

support any S, so the market is not served.

Region (2): is so small that the market can sup-

port only the lower-cost S, S1. Thus, the only equilib-

rium has S1 as a monopoly.

Region (3): The market is large enough to support

either of the firms as a monopoly but too small to

allow another entrant to have nonpositive profits,

so there are two monopoly equilibria.

Region (4a): The market is large enough to support

either of the firms as a monopoly, but it is sufficiently

large that when S2 is in the market, S1 can enter andmake positive profits. However, when S1 enters, it

crowds out the higher-cost S and the only equilibrium

is S1 as a monopoly. Note that this is possible only

when S1s capacity cost is low enough in relation to

S2s.

Region (4b): The market is large enough for both

firms to be in the market. However, if one of the

firms is in the market as a monopoly, the other firm


15/23


Figure 3 The Effect of Increasing the Total Market Size for

Different Levels of the Unit Capacity Cost Differential

on the Market Structure

50 100 150 200 250 3000

5

10

15

g2

g1

(6)(4b)

(5)(3)

(4a)

(1)

(2)

Notes. Region (1): The market is not served. Region (2): Only S1 is in themarket, as a monopoly. Region (3): Two monopoly equilibria, one with S1 in

the market and one with S2 in the market. Region (4a): Only S1 is in the mar-ket, as a monopoly. Region (4b): Duopoly and two monopoly equilibria, one

with S1 and the other with S2 in the market. Region (5): Duopoly equilibriaand a monopoly equilibrium with S1 in the market. Region (6): Only duopolyequilibria. In the figure, the service valuations U are uniformly distributed

over the interval 0 100, which corresponds to V = 100 100/,g1 = 30, c = 5, v= 03, and d= 100.

cannot make positive profit. Hence, this region sup-

ports either a duopoly or two monopoly equilibria.

This region exists only if the capacity cost difference

g2 g1 is small.

Region (5): The market is large enough for the

firms to earn positive profits if both are in the mar-

ket. Only S1 can earn positive profit as a monopoly

and deter S2 from entering the market. However, if S2attempts to operate as a monopoly, S1 enters the mar-

ket. Thus, the only equilibria are duopoly and S1 is a

monopoly.

Region (6): The market is large enough that both

firms can make positive profits. Hence, only duopoly

equilibria exist.Overall, market segmentation occurs when the mar-

ket is large enough and the difference between the

capacity costs of the two firms is not too large. Value-

based market segmentation is the only equilibrium

in Region (6) of Proposition 4, which corresponds to

larger values of and smaller values of g2 g1, and

it is one of the equilibria in Regions (4b) and (5),

where both and g2 g1 have intermediate values.

In summary, larger market sizes and smaller differ-

ences in the firms capacity costs are conducive to

value-based market segmentation.

Next, we show that an increase in the market sizedecreases both service differentiation and value-based

market segmentation, and at the limit both disap-

pear. Figure 4 shows the effects of increasing the

market size on the market share and profit share

of the high-end provider, S1, and on our measures

of service differentiation and market segmentation.

First, an increase in market size benefits the low-

end provider, S2, more than it benefits the high-end

provider, S1, in terms of both market share (Fig-

ure 4(a)) and profit share (Figure 4(b)). This is because

of scale economies in queueing systems: A facility

with few customers benefits from an additional cus-tomer more than a facility with many customers. As

S2 serves fewer customers than S1, an increase in the

total market size decreases the cost of S2 more than it

decreases the cost of S1. Hence, S2 benefits more from

an increase in than S1 does. Figure 4(c) shows that

market segmentation decreases as increases. Ser-

vice differentiation decreases because a given increase

in has a larger incremental effect on S2 than on

S1; hence the difference between the two decreases.

Because market segmentation is increasing in our dif-

ferentiation measure W = W2 W1, our measure of

segmentation (25) decreases. At the limit of a verylarge market, both service differentiation and mar-

ket segmentation disappear. This result is driven by

the fact that as the number of customers increases,

the excess capacity increases at a slower rate than

the equilibrium demand rate j. Because the marginal

cost of excess capacity decreases, the firms reduce the

waiting times to zero at the limit. Formally, we have

as follows.

Proposition 5. lim Wj = 0 for j= 1 2.

Corollary 2. As , the measures of both ser-

vice differentiation and market segmentation tend to zero.

6. Different Unit Operating CostsSo far, we have assumed that the service providers

differ only in their unit capacity costs but not in

their unit operating costs. Do different unit operat-

ing costs lead to value-based market segmentation?

In this section we allow the unit operating costs of the


16/23


Figure 4 Effect of Increasing the Total Market Size on the Market Share of S1 (a), the Profit Share of S1 (b), and Market Segmentation (c)

500 600 700 800 900 1,0000.5258

0.5260

0.5262

0.5264

0.5266

0.5268

0.5270

0.5272

(a) Market share ofS1 as a function of

the market size

500 600 700 800 900 1,0000.5530

0.5535

0.5540

0.5545

0.5550

0.5555

0.5560

0.5565

1/(1

+

2)

1/(1

+

2)

(b) Profit share ofS1 as a function of

the market size

500 600 700 800 900 1,000

0.034

0.036

0.038

0.040

0.042

0.044

0.046

0.048

(c) Market segmentation as a function

of the market size

Notes. In these figures, is the total arrival rate of customers, 1/1 + 2 is the proportion of customers served by S1, 1/1 + 2 is the proportionof profits received by S1, and is our measure of market segmentation. The service valuations U are uniformly distributed over the interval [0,100] (whichcorresponds to V = 100 100/), g1 = 29, g2 = 31, c = 5, v = 01, and d= 1.

two firms, c1 and c2, to be different. We also change

assumptions (6) and (7) so that V0 > maxc1 c2 + g

and there exists a such that V < minc1 c2 + g,

respectively.

First, we show that even if the firms capacity costsare equal, differences in unit operating costs lead to

value-based market segmentation. Then, we numeri-

cally analyze how the differences in the capacity and

operating costs affect the levels of service differentia-

tion and market segmentation.

Proposition 6. Let v > 0, g1 = g2 = g, and c1 < c2,

and let both firms be in the market. Then in equilibrium,

S1 serves the high-end customers and S2 serves the low-end

customers. The capacity j and arrival rates j to each Sjsatisfy

1r2 + 111 1

2+ 1r

2

1 1+ 1 1

221= g

1V1v + d

V1 + 1V1vW2 W1

+ V1 + 2 + 1V1 + 21 vW2

dW1 1V1v + d

r2 + 1

21 12

= c1

(29)


17/23


and 2r

2 + 222 2

2+

2r2

2 2+ 1

1

222

=g

2V1 + 2v + d

V1 + 2 + 2V1 + 21 vW2 dW2

V1 + 2v + dr2 + 1

22 22

= c2

(30)

where Wj are given by Equation (5).

Corollary 3. For v > 0, g1 = g2 = g, and c1 < c2,

there is no duopoly equilibrium without value-based market

segmentation.

Proposition 7. For a 1-unit increase in the capac-ity cost gj of Sj (j = 1 2), to keep profit constant, the

marginal operating cost cj should decrease by / units

for a monopoly and by 1/ units for a duopoly.

Proposition 7 shows that a unit change in capac-

ity costs impacts each service providers profits more

than a unit change in its operating cost. From the

proof of the proposition, for a monopoly /c = ,

i.e., the change in c affects the profits proportionally to

the arrival rate, whereas /g = , i.e., the change

in g affects profits proportionally to capacity, which

is always larger than the arrival rate. Proposition 6implies that other things being equal, a unit reduction

in capacity cost is worth to each S more than a unit

reduction in its operating cost.

7. Concluding RemarksRecent years have seen an increase in the impor-

tance of service industriesand with it, an increased

focus on timeliness. As the value of customers

time increases, prompt service becomes an important

lever for gaining competitive advantage. This calls

for a more detailed examination of the relationship

between the delays and competitive strategies of ser-

vice firms. Previous research on the economics of

queues often took an additive delay cost structure for

granted. We show that when customers service val-

uations and their delay costs are interdependent, and

the service providers have different costs, the market

exhibits value-based segmentation: one of the service

providers focuses on fast service for the high-value

Table 2 How the Equilibrium Market Structures Depend on the

Multiplicative Delay Sensitivity v and Capacity Costsgj j= 1 2

g1 = g2 g1 < g2

v = 0 No service differentiation Service differentiation No market segmentation No market segmentation

v > 0 Multiple equilibria: No service differentiation Service differentiation

No market segmentation Market segmentation

or

Service differentiation

Market segmentation

segment, and the other targets the low-value segment

with a lower price and slower service. Table 2 sum-

marizes our key results: market segmentation (value-based) occurs only under the generalized delay cost

structure, and for service providers with asymmetric

costs this is the only possible equilibrium.

We find an interesting progression of market struc-

tures as a function of market size. In our model, for

a small arrival rate, the market is not served; for a

moderate total arrival rate, the market supports only

the lower-cost S being a monopoly; for a larger mar-

ket size, either S can be a monopoly; and as the mar-

ket size increases further, the only equilibrium has the

lower-cost S as being a monopoly or there are the

duopoly and two monopoly equilibria. As the arrivalrate increases further, the only equilibria possible are

the duopoly equilibria and the lower-cost S being a

monopoly. Finally, for a larger market size, the only

equilibria have both firms competing in the market.

A natural extension of our model is to consider pri-

ority queues. A service provider may assign a higher

priority to customers who choose to pay a higher ser-

vice fee and thus, these customers would be served

faster than those who pay less. We expect our qualita-

tive results (in particular, the existence of value-based

market segmentation) to hold in such a setting. For

a monopoly with two priority classes, all customers

with service valuations above some threshold would

prefer the higher-priority class, and all served cus-

tomers below this threshold would opt for the lower-

priority class. Yet the study of the duopoly case raises

the issue of how the number of priority classes affects

entry and exit decisions, which, to our knowledge,

has not been addressed in the literature.


18/23


Our analytical results were derived for only two

firms with the same coefficient of variation of the ser-

vice times. The duopoly assumption allowed us to use

the properties of supermodular games, and extendingthe results to the oligopoly case requires for numeri-

cal analysis. Numerical explorations suggest that with

multiple firms and different coefficients of variation

of service times, a duopoly market is still character-

ized by value-based market segmentation, with each

of the firms serving an interval of customers with a

range of service valuations.

Our results show that the delay cost structure

makes both a structural and a practical difference in

the analysis of competing congestion-prone service

providers. Markets where the delay cost and the value

of service are interdependent are substantively differ-ent from markets with an additive delay structure,

leading to different competitive strategies, customer

behavior, and industrial structures.

AcknowledgmentsThe authors thank the editor, associate editor, and anony-mous reviewers for their suggestions, which helped toimprove the substance and the exposition of the paper.

Appendix. ProofsProof of Proposition 1. The proof proceeds along the

following steps: (1) We derive necessary and sufficient con-

ditions for the existence of an equilibrium with both firmsin the market for sufficiently small d. (2) We find the largestd (d1) such that there is a duopoly solution in the entireinterval 0 d1. (3) We show that for d slightly above d1,only a monopoly solution is possible, and we derive nec-essary and sufficient conditions for the monopoly equilib-rium. (4) We show that there exist d2, d3, and d4 such thatin equilibrium, any of the firms may be a monopoly for d d2 d3 and only S1 is a monopoly for d d1 d2 d3 d4and that for d d4, none of the firms can earn positiveprofits.

Step 1. By (2) and (6), for d = 0, which reduces our modelto a standard duopoly, both firms earn positive profits inequilibrium. Now consider some d slightly above zero.

We first show that if we suppress the positive profitrequirement, a duopoly equilibrium exists. We then showthat for a small enough d, each duopolist makes posi-tive profit. As shown in 3.3, the customers are indifferent

between the firms and each firm solves the problem:

maxj j

jPj cj gjj

s.t. V1 + 2 d Wj = Pj

(31)


Let jj j j = jV1 + 2 d Wj c gjj

be the profit function of player j. From assumption (7)and the nonnegative cross-partial derivatives of the profitfunctions, game (31) is supermodular in 1 1 2 2and a duopoly equilibrium exists (by Theorems 4 and 5 inMilgrom and Roberts 1990).

We next show that for a small enough d, both duopolistsmake positive profits. Theorem 2 in Milgrom and Segal(2002) shows that j (j= 1 2) are continuous functions of d.By the continuity of jd, j0 > 0 implies that for suffi-ciently small d, jd> 0 as well. Summing up, if d is smallenough, there is a duopoly equilibrium with both firmsmaking positive profits. We denote the equilibrium arrivalrate to Sj when the delay sensitivity is d by jd.

11

According to Theorem 6 in Milgrom and Roberts(1990) with exogenous parameter g1, 1 2 and 1 2.This implies that in equilibrium, 1d 2d when

both firms are in the market. By our choice of d, bothfirms are in the market. Because profit goes to asj , there is an equilibrium where j and j satisfythe first-order conditions (FOC) of (31). From Equa-tion (5), Wj/j = r

2 + 1/2j j2 and Wj/j =

jr2 + jj/j j

2 + jr2/j j + 11/2

2j,

which together with the FOC of (31) give the equilibriumcondition (12) for the duopoly.

Step 2. Because of (7), there exists at least one finite dsuch that 2d = 0. Let d1 = infd 2d = 0. Because1d 2d, S1 is still in the market at d = d1 and in 0 d1,we have a duopoly solution with positive profits for bothfirms.

Step 3. Consider d slightly above d1. Then, only one S,

which we call Sm, can survive in the market (the other S willearn negative profits if it enters the market). Then Sm mustset its capacity m and service rate m to maximize its profit:

Mm d = maxm m

mPm cm gmm

s.t. Vm dWm Pm

(32)

By the choice of d and because profit goes to asm , there is an interior solution that satisfies the FOC.From (5), Wm/m = r

2 + 1/2m m2 and Wm/m =

mr2 + mm/m m

2 + mr2/m m + 1

1/22m which together with FOC give the equilibriumcondition (14) for the monopoly.

Step 4. Let jdj (j = 1 2) be the optimal profitin (31) for fixed d and given j. Let d2 be the smallest solu-tion to 1d argmax

M2 d = 0. Because when both

firms are in the market, 1d 2d, d1 d2. Let d3 be thesmallest solution to M2 d = 0, and let d4 be the smallestsolution to M1 d = 0. Because

1d 0 = max

M1 d,

d2 d3. Thus, in the interval d2 d3 there are two equilibria,

11 If there are several equilibria, we choose the equilibrium with the

maximum number of customers for S2.


19/23


with either S being a monopoly, and in the interval d1 d2there is only a monopoly S1 solution. Because g1 < g2, forany value of d, M1 d

M2 d. Hence, d3 d4. Because the

monopoly profit is a decreasing function of d, for d d3 d4only S1 can be a monopoly, and for d d4, the market is notserved.

Proof of Proposition 2. We first show in the Prelimi-nary Step below that when both firms are in the market,only an equilibrium with value-based market segmentationis possible. The proof then follows the same steps as theproof of Proposition 1.

Preliminary Step. Consider a duopoly equilibrium andassume, by contradiction, that it follows our mixed-strategystructure. We show that this assumption leads to the contra-diction that both 1 2 and 1 < 2 hold simultaneously.In our mixed-strategy equilibrium, each Sj, j= 1 2 solves

maxj j

jPj cj gjj

s.t. V1 + 21 v Wj d Wj Pj = 0

(33)

Let jj j j = jV1 + 21 v Wj d Wj c

gjj be the profit function of player j given j. Game (33)is supermodular in 1 1 2 2 by Theorem 4 inMilgrom and Roberts (1990) and the nonnegative cross-partial derivatives of the profit functions. Next, the cross-partial derivatives of j with respect to the strategyvariables of Sj, j = 1 2, and g1 are nonnegative. Hence, byTheorem 6 in Milgrom and Roberts (1990), with exogenousparameter g1,

1 2 (34)

We next show that 1 < 2. As shown in 4, in a mixed-strategy equilibrium, W1 = W2 = W. Let j = j/j. By (5),

112

1r

2 + 1

1 1+ 1

= 2

22

2r

2 + 1

1 2+ 1

= W (35)

By the FOC of Sj (j= 1 2) for (33), we have

V1 + 2v + d

W +

2j

j

r2 + 1

21 j2

=

gj

j (36)

hence for j= 1 2,

j = 3j

V1 + 2v + d

gj jV1 + 2v + dW

r2 + 1

21 j2

(37)

Equation (37) together with (36) leads to

W =

j

V1 + 2v + d

gj V1 + 2v + dWj

2j

r2 + 1

41 j2

j

jr2 + 1

1 j+ j

(38)

W in expression (38) is the product of three functions thatare increasing in j. Thus, g1 < g2 implies 1 < 2. For the

M/G/1 queue, the expected number of customers in thesystem depends only on its utilization and increases in .Therefore, by Littles Law, 2W = 2W2 > 2W1 = 1W, or

2 > 1 (39)

which contradicts (34).Given the value-based market segmentation structure,

the firms determine j and j by solving

max1 1

1P1 c1 g11

s.t. V1v + dW2 W1 P1 P2

(40)

andmax2 2

2P2 c2 g22

s.t. V1 + 21 vW2 dW2 P2 0

(41)

where Wj are given by Equation (5). Note that the constraintin Equation (40) follows from Equations (19) and (20) andthe Preliminary Step, and that Equation (41) is analogousto (10) for v > 0.

We now proceed to prove the core of Proposition 2.Step 1. By Assumptions (2) and (6), for v = d = 0 our

model is reduced to a standard duopoly with both firmsearning positive profits. Now consider some v slightlyabove zero and d slightly above or equal to zero.

We will first show that if we suppress the positive profitrequirement, a duopoly equilibrium exists and we will thenshow that for a small enough d, each duopolist makes a pos-itive profit. Each Sj (j= 1 2) decides on its capacity j andservice rate j, which maximize its profit given the strat-

egy of the other S (Sj). The Preliminary Step shows that aduopoly equilibrium must be characterized by value-basedsegmentationthat is, one of the providers, S1 or S2, servesthe higher-end segment and the other serves the lower-endsegment. The incentive compatibility and individual ratio-nality constraints determine the service fees as

Pj =

VjvWj Wj + Pj if Wj < Wj

V1 + 21 vWj dWj if Wj Wj(42)

From (3), the cross-partial derivatives of the payoff func-tions in (40)(41) are nonnegative. Therefore, by Theorems 4and 5 in Milgrom and Roberts (1990), the game is super-modular in 1 1 2 2 and an equilibrium exists.

We now show that for v and d small enough, bothduopolists earn positive profits. Theorem 2 in Milgrom andSegal (2002) shows that j (j = 1 2) are continuous func-tions of vd. By the continuity of jvd, j0 0 > 0implies that for sufficiently small vd, jvd > 0 aswell. Therefore, an equilibrium with both firms in the mar-ket exists. We denote the equilibrium arrival rate to Sj byjvd.

According to Theorem 6 in Milgrom and Roberts (1990)with exogenous parameter g1, 1 2, and 1 2. This


20/23


implies that in equilibrium, 1vd 2vd when bothfirms are in the market. Indeed, if for some vd, 1vd 0. Because 1vd 2vd, S1 is stillin the market at such vd, and in 1 we have a duopolysolution with positive profits for both firms.

Step 3. Consider vd slightly outside of 1. Then, onlyone S, which we call Sm, can be in the market (because theother S will earn nonpositive profits if it enters the market).Then, Sm must set its capacity m and probabilities m tomaximize its profit:

Mm vd = maxm m

mPm cm gmm

s.t. Vm 1 vWm dWm Pm

(44)

Because we have chosen vd slightly outside of 1 andbecause profit goes to as m , there is an interiorsolution that satisfies the FOC. From Equation (5), Wm/m= r2 + 1/2m m

2 and Wm/m = mr2 + mm/

m m2 + mr

2/m m + 11/22m, which together

with the FOC gives the equilibrium condition (24).Step 4. Let jvdj (j = 1 2) be the optimal profit

determined as a solution to FOC for fixed vd and j. Let2 = vd

1vd = argmax

M2 vd 0. Because

then both firms are in the market, 1vd 2vd,1 2.

Let 3 = vd M2 vd 0. Because

1vd0 =

max M

1

vd, 2 3. Thus, in the set 3\2 there aretwo equilibria with either S being a monopoly. Becauseg1 < g2, for any value of vd,

M1 vd = max

V1 vW dW c g1

max

V1 vW dW c g2

= M2 vd (45)

Hence, 3 4. Because the monopoly profit is a decreas-ing function of v and d, in the set 4\3, only S1 can be amonopoly, and for vd 4, the market is not served.

Proof of Corollary 1. Corollary 1 follows from the Pre-liminary Step in the proof of Proposition 2.

Proof of Proposition 3. We first show that there isa symmetric mixed-strategy equilibrium. Then taking thelimits of the results in Proposition 2 shows that thereare two asymmetric equilibria with value-based marketsegmentation.

To prove the existence of a mixed-strategy equilibrium,it is sufficient to prove the existence of an equilibrium forthe game

maxj j

jPj cj gj

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