Price Discrimination in Broadway Theater
Author(s): Phillip Leslie
Source: The RAND Journal of Economics, Vol. 35, No. 3 (Autumn, 2004), pp. 520-541
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RANDJournalof Economics
Vol.35,No. 3, Autumn2004
pp.520-541
Price discrimination in Broadway theater
Phillip Leslie*
A commonthreadin the theoryliteratureon price discriminationhas been the ambiguouswelfare
effectsfor consumersand the rise in profitfor firms, relative to uniformpricing. In this study
I resolve the ambiguityfor consumers and quantifythe benefitfor a firm. I describe a model
of price discriminationthat includes both second-degreeand third-degreeprice discrimination.
Usingdatafrom a Broadwayplay, I estimatethe structuralmodeland conductvariousexperiments
to investigatethe implicationsof alternativepricing policies. The observedprice discrimination
may improvethefirm'sprofitby approximately5%, relativeto uniformpricing,while the difference
for aggregate consumerwelfare is negligible.Also, I show that the gain from changingprices in
theface offluctuatingdemandis small underthe observedprice discrimination.
1. Introduction
*
Price discriminationallows firms to increase their revenue above what may be obtained
fromuniformpricing.The impacton consumersis, in general,ambiguous.Althoughnonuniform
pricingis common,its welfareimplicationsareanempiricalissue aboutwhich little is known.The
goal of this study is to undertakean analysis of the welfare implicationsof price discrimination,
using the example of Broadwaytheater.I present a model of individualconsumerbehaviorand
monopoly price discrimination,which is then estimatedwith data from a Broadwayplay. Using
the estimateddemandsystem, a rangeof counterfactualexperimentsareconductedto analyzethe
effects on welfare fromprice discriminationin this market.
The theoreticalframeworkis a utility-basedmodel of consumerbehaviorthat incorporates
characteristicssuggested by the data and institutionaldetails of the Broadwaytheaterindustry.
The demandmodelis designedto be consistentwith the observedbehaviorof the firmandincludes
both second-degreeand third-degreeprice discrimination.1Setting differentprices for different
seat qualitiesis an exampleof second-degreepricediscrimination,or nonlinearpricing.2Discount
mail coupons aretargetedto consumerswith lower willingness to pay, which providesan example
* Stanford
University;[email protected].
I thank Lanier Benkard, Dirk Bergemann,Steve Berry, Moshe Buchinsky, Greg Crawford,Ron Goettler, Tom
Hubbard,Guido Imbens, and Ariel Pakes for valuable advice and assistance. I am also thankfulto the Editor and two
anonymousreferees for many helpful suggestions. I am especially gratefulto Benjamin Mordecaifor sharingwith me
his insights into theaterticket pricingand for his interestin the study from the very beginning,and to ChristinaMills for
helping me to obtain the data. Finally, I thank the Cowles Foundationfor financialsupportin the form of a Carl Arvid
AndersonPrize Fellowship.
1 Tirole(1988)
gives a thoroughdiscussionof the differentkindsof price discrimination.The analysisin this study
would be the same regardlessof whetherI call it price discrimination,nonuniformpricing, or multiproductmonopoly
pricing.
2 Wilson
(1993) provides a detailed account of the theory of nonlinearpricing. In this article, I use the terms
nonlinearpricing and second-degreeprice discriminationinterchangeably.
520
Copyright? 2004, RAND.
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LESLIE / 521
of third-degreepricediscrimination,ormarketsegmentation.The sale of day-of-performancehalfprice tickets sold at a discount booth is modelled as a damagedgood that furtherdiscriminates
among self-selecting consumers.
The data consist of price and quantity sold for all 17 different ticket categories for all
199 performancesof Seven Guitars, a play that ran on Broadway in 1996. The econometric
specificationof thebehavioralmodel is a random-utilitydiscrete-choicemodelwith endogenously
randomchoice sets. A virtueof using a structuraleconometricframeworkin this case is thata range
of experimentscan be performedusing the estimateddemandsystem.An empiricalinvestigationof
the welfareimplicationsof price discriminationmustrely uponan abilityto analyzebehaviorwith
and without price discrimination,and to compute appropriatewelfare measures.In the absence
of data exhibiting both price discriminationand uniform pricing, it would not be possible to
identifythe differencein social surplusbetweenuniformpricingandprice discriminationwithout
a behavioral model to form predictions. The experimentsinclude uniform pricing, nonsticky
prices over time, and abolishing the discount booth. In each case, comparisonsare drawnwith
the benchmarkscenarioof the actualbehaviorof the firmand consumers.
Among the results,I find thatthe observedprice discriminationincreasesthe firm'sprofitby
to a policy of optimaluniformpricing.Thegain frompricediscriminationsignificantly
relative
5%,
on
the magnitude of the discount offered for tickets sold at the day-of-performance
depends
discount booth. In particular,if the booth discount were 30% instead of 50%, firm profitwould
rise by 7%.Fromthe point of view of consumers,the changein aggregateconsumersurplusunder
price discriminationrelative to uniformpricing is insignificant,though there is a redistribution
of surplusamong consumers. I also show the increase in profit from reoptimizingprices in the
face of changing demandis less when a menu of several price alternativesis used than when a
single price is used each period, which may help explain the presence of rigid price policies in
this market.3
The empiricalliteratureon pricediscriminationhas evolved since theearly 1990s. Borenstein
(1991) and Shepard(1991) identifythe presenceof pricediscriminationfrompossible cost-based
explanationsfor the observedprice dispersion.BorensteinandRose (1994) quantifya high degree
of price dispersiondue to price discriminationthatis all the more interestinggiven the somewhat
competitivenatureof the industrythey study (airline travel).A few more recent studies employ
structuralmethodsto investigatea varietyof issues in relationto price discrimination-see Ivaldi
and Martimort(1994), Bousquet and Ivaldi (1997), Cohen (2000), McManus (2001), Miravete
(2002), Gary-Bobo and Larribeau(2004), and Verboven(2002). In the culturaleconomics literature,severalresearchershave analyzed theaterdemandand pricing, and two of these studies
focus on the presence of multipleticket prices. Huntington(1993) investigateswhetherrevenue
differs for theaterscharginga range of ticket prices, over theatersthat charge a single price for
all tickets. In a theoreticalstudy,Rosen and Rosenfield(1997) describe a model of ticketpricing
thatinvolves second-degreeprice discrimination.Finally,severalpreviousstudieshave estimated
price and income demandelasticities for the performingarts,as I do here also.4
The remainderof the article is organizedas follows. Section 2 summarizesthe data, with
particularattentiongiven to aspects that are incorporatedin the model and the sources of price
variationthat serve to identify the demandsystem. Section 3 presentsthe behavioralmodel and
the econometric model. Section 4 contains the results of the estimation, including the implied
demandelasticities. Based on the estimateddemandmodel, an arrayof experimentsareexplained
and the resultspresentedin Section 5. Section 6 concludes.
2. Summaryof the data
*
"Broadway theater"refers to all plays and musicals performedin theatersin the Times
Squareregion of Manhattan,New YorkCity, with seating capacities in excess of 499. Typically,
3 Otherindustriesthat share similarfeaturesfrom this
point of view include airlines,hotel accommodation,and
sportingevents.
4 See Moore
(1966), Felton (1992) and Levy-Garbouaand Montmarquette(1996).
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522 / THERANDJOURNAL
OFECONOMICS
the owner of a Broadwaytheaterrents the theaterto a show producerwho decides, among other
things, the ticket prices for the show. In contrast to most performingarts organizations,the
objectiveof Broadwaytheaterproducersis to maximizeprofit.Ticketprices arenot subjectto any
specific regulation.The majorityof tickets are sold over the phone.5 Some tickets are also sold
at the box offices located at the theaters,or througha discount booth known as the TKTS. The
Broadwayplay thatis the focus of this studyis Seven Guitars.Each Broadwayshow develops its
own idiosyncraticapproachto the marketingof the product;with respectto price discrimination,
however,Seven Guitarsis a typical example of behaviorin the industry.6
SevenGuitarsprovidesa good exampleof discriminatorypricing.Thereis littledoubtthatthe
price differencesthatareevidentin a given performancecannotbe explainedby differingcoststhe marginalcost of every ticket sold for a given performanceis effectively zero. Moreover,
while the numberof seats in the theaterpresentsa capacityconstraintfor the firm,suggestingthe
presence of a variableshadowcost of capacity,this constraintis rarelybinding for Seven Guitars.
With a maximum seating capacity of 947, the show sold out for 12 of the 199 performances,
achieving an average attendanceof 75% of capacity, or 707 people per performance(with a
standarddeviation of 157.15). Balcony seating is the only individualticket category to be sold
out in more than 12 performances,in that case selling out 23 times. Hence, congestion is not
a significantissue in the data, althoughex ante it might have been. The primarysource of data
for this study is the box office reportfor Seven Guitars,from which I observeprice and quantity
sold in each mutually exclusive sales category for every performance.In a single performance
therecould be attendeesfrom all 17 categories,thoughon averageonly 8.7 of the categoriesare
representedin a given performance.A total of 140,782 people saw the Broadwayproductionof
Seven Guitars.
An importantdistinctionamongticketcategoriesis between full-pricetickets and discountprice tickets. Full-price tickets are for a specific area of seating, namely orchestra,mezzanine,
rear-mezzanine,balcony,boxes, and standingroom. These regions are differentiatedby the average quality of the seating, or view, that is offered. All full-price options are available to all
potentialconsumersand aresold via telephone.Discount-pricetickets are availableundervarious
conditions. Some discount-pricetickets are only availableto individualswho receive a coupon
in the mail or happento come across one in a restaurantor some otherchosen location. Another
kind of discount, while availableto all potentialconsumers,requiresconsumersto incur a nonpecuniarycost of having to wait in line at a discountbooth. For discount-pricetickets, the buyers
are seated in the high-qualityregion of the theater,such as the orchestra,thoughgenerallynot in
the best seats within thatregion.
Table1presentssummarystatisticsforprices,quantities,andrevenuesof each ticketcategory.
The meanpricefor all ticketsales in all performancesis $36.43 with a standarddeviationof $15.11.
The averageGinicoefficientis .201 (standarddeviation.040), indicatingthatthe expectedabsolute
differencebetween any two ticket prices selected at randomis 40% of the mean price. By way
of comparison,in a study of prices in the airline industry,Borensteinand Rose (1994) find an
average(across flights) Gini coefficient of .181 (standarddeviation .063), which implies thatthe
expected absolutedifferencebetween any two fares selected at randomis 36% of the mean fare.
Price variation and demand identification. Table 1 also indicates the differentkinds of
o
price variationthat serve to identify demand.There are two types of price variation.First,prices
vary across the differentticket categories.The mean price for orchestratickets is $55.08. Other
categories have lower averageprices, such as the balcony with an averageprice of $16.93. The
second type of price variationis changes over time (across performances)in the prices of each
ticket category.This variationis reflectedin the second column of numbersin Table 1, showing
the standarddeviationof prices for each ticket category.7
5 Phone sales are
throughone of two phone-sales companies:Tele-chargeand Ticketmaster.
6 Based on conversationswith severaltheater
producers.
7 The ManhattanTheatreClub (MTC)categoryhas no price variationover time. This is a subscriberorganization
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LESLIE / 523
TABLE 1
Summary of Attendance and Revenues for Each Sales Category of Seven
Guitars
Price($)
Fullprice
Orchestra
Frontmezzanine
Rearmezzanine
Balcony
Boxes
room
Standing
Discountprice
10%off
Two-ferone
TKTS
MTC
AENY
Directmail
Group
Student
TDF
Wheelchair
Complimentary
Attendance
Revenue($)
Mean
Standard
Deviation
Mean
Standard
Deviation
55.08
55.08
29.20
16.93
55.76
22.27
4.22
4.23
1.85
4.91
4.17
2.55
162.74
40.04
34.80
38.60
4.97
6.14
77.22
41.70
18.91
17.26
4.88
4.50
9,112.29
2,262.27
1,007.10
679.26
281.36
134.77
4,765.14
2,462.55
533.49
421.85
279.95
96.24
49.40
27.23
3.88
2.06
5.55
20.17
335.74
467.28
591.53
27.53
22.00
50.36
39.51
36.26
26.21
16.46
26.94
0
2.11
0
1.81
2.28
10.80
2.01
5.81
2.23
0
6.71
16.65
158.87
258.99
3.81
48.43
89.91
68.35
153.72
2.02
38.91
71.29
60.28
2.46
36.80
63.84
56.38
90.67
0.66
75.57
4,358.12
5,697.71
193.07
1,956.91
1,326.18
128.17
1,925.78
3,309.46
1,775.98
2,306.93
54.56
0
1,461.92
2,688.23
1,440.68
1,163.35
17.67
0
Mean
Standard
Deviation
286.61
one"aretwo-for-one
Notes:"Two-fer
discountbooths."MTC"
stands
couponsales.'TKTS"areticketssoldviatheday-of-performance
forManhattan
TheatreClub,whichis a subscriber
"AENY"
standsforArtsEntertainment
New York,whichis a private
organization.
ticketsto its customers.
"TDF"standsforTheaterDevelopment
firmspecializingin providinghigh-quality
Fund,whichis a nonprofit
thatprovides
ticketstoschoolchildren
andso forth.Inthemodel,"coupons"
aretheaggregation
of alldiscount-price
organization
categories
andcomplimentary
tickets.
exceptTKTS,wheelchair,
As usual with demandestimation,thereis a question aboutthe endogeneityof prices. What
explains the observed variationin prices, and is it sufficient to identify the effect of price on
demand?Consider first the price variationacross the full-price ticket categories. The price of
an orchestraticket is higher than a balcony ticket because seat quality is higher in the orchestra
than in the balcony.As explainedin the next section, I estimatethe differentseat qualitiesin the
demandsystem. Seat qualityis thereforenot containedin an errorterm,precludingthis particular
source of correlationbetween prices and a residual. But functional-formassumptionsare now
relied on to utilize this source of variationto identify the effect of price on demand.
Next, considerthe time-seriesvariationin prices for the full-pricetickets.Prices for full-price
ticketcategoriesvaryacrossperformancesdue to predeterminedpeak-loadpricing-performances
for differenttimes in the week are priced differently.For example, Saturdayevening orchestra
tickets are priced higher than Sunday matinee orchestratickets. This price variationis decided
by the producerpriorto the firstperformanceand is not changedover the life of the show. Analogous to the price variationacross seat qualities, in this case I estimate time-of-week effects as
partof the demandspecification.Again, this precludesthe usual endogeneityconcern but limits
identificationof the price effect from this source of variationto rely on functionalform.
While every seat in the theatermay differin quality,the firmused only threeseat-qualitycategories for the purposeof settingdifferentprices (at full price). But for most of the performances,
thatformed an agreementwith the producerof Seven Guitarsfor tickets to be availableat $22 for certainperformances.
In the analysis, this categoryis aggregatedwith othercoupon categories,as explainedbelow.
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OFECONOMICS
524 / THERANDJOURNAL
only two qualitycategorieswere used. The medium-qualityregionwas offeredat a differentprice
in only 50 of the 199 performances,the firsttime in the 133rdperformance.8This providesa useful
source of price variation.For a given seat quality,for given time-of-week,thereis variationin the
ticket price. Furthermore,althoughthe introductionof the medium-qualitytickets is correlated
with the time trend(presumablyit was done as a response to dwindling demandover time), the
time trendis a smoothprocess, while this variationis discrete.For these reasons,variationin the
availabilityof medium-qualitytickets provides a useful source of price variationto identify the
demandsystem.
There is also price variationin the discount-priceticket categories. Ratherthan attemptto
estimate demand for all ten of the discount ticket categories, I distinguish only two types of
discounts: coupon and booth. I define the coupon category as the aggregationof all discount
categories except TKTS.9The key featureof the coupon categoryis thatit includes all discount
categoriesthatare restrictedto individualswho eitherreceived an actualcoupon or are members
of specific organizationsor groups. That is to say, these categories are interpretedas a form of
third-degreeprice discrimination.10Given this aggregation,the mean coupon ticket price in the
data is $31.01, with a standarddeviation of $8.71. The mean numberof coupon tickets sold is
252.63, with a standarddeviationof 167.62.
What explains the time-series variationin the coupon price? It is partlydrivenby time-ofweek peak-loadpricing, as with the time-seriesvariationfor the full-pricetickets. But unlike the
full-pricetickets,day-of-weekdummiesexplainrelativelylittle of thevariationin the couponprice.
The coupon price variationis due mainly to the firm trying differentways of offering targeted
discounts.11For example, targeteddirect-mailcoupons were used in the early performances,
while two-fer one tickets were not introduceduntil midway in the show's run.12According to
the producerof Seven Guitars, this variationin coupon availabilityreflects standardmarketing
practices for Broadway shows, ratherthan any deliberatetechnique for changing price in the
face of fluctuatingdemand. In other words, there is some reason to view this source of price
variationas being exogenous. To the extent that the time-series variationin the coupon price is
not exogenous, the demandspecificationincludestime-of-weekdummies,a quadratictime trend,
and a dummy for afterthe TonyAwards,which took place in the middle of Seven Guitars'srun.
These variablesshould control for many of the obvious explanationsof this price variation.The
remainingvariationin the coupon price over time may be an exogenous component.
The second type of discount is the booth ticket category,which correspondsto the TKTS
category in the data. Below I explain the interpretationof this category. TKTS tickets were
availablefor Seven Guitarsin 197 of the 199 performances.The numberof tickets made available
at the TKTS booth varied from day to day, based on the numberof unsold tickets up until the
morningof the performance.In termsof price variation,note thatbooth tickets are sold at a 50%
discount off the top full price (plus a $2.50 service charge).Consequently,the same issues arise
as with the price variationin the full-priceticket categories,describedabove.
In summary,there are several kinds of price variationin the data that serve to identify
the demand system. Some of this variationis helpful only in conjunctionwith functional-form
assumptions,which will be detailedin the next section. To what extent the particularfunctionalformassumptionsused can be motivatedby economic considerationswill also be discussed in the
next section. Meanwhile, there are other componentsof the price variationthat should provide
identificationindependentlyof functionalform.
8 Forthe other 149
performances,these seats were includedin the high-qualityregion.The medium-qualitytickets
are for the seats in the rear-mezzanine.
9 I also exclude the wheelchairand complimentarytickets.
10An alternative
interpretationis thatcoupons area form of second-degreeprice discrimination.In thatcase, some
consumers who see the show and had received a coupon, don't use the coupon. See Shaffer and Zhang (1995) for an
analysis of coupon targeting.
11Most
types of coupons are not limited to usage on certaindays of the week. An exceptionis the MTC member
coupons, which could not be used for Saturdayevenings.
12Two-ferone coupons were placed in targetedlocations, such as studentcafeterias.
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LESLIE / 525
D Day-of-performance booth ticket sales as a damaged good. The definingcharacteristic
of booth ticket sales that I seek to incorporatein the demand model is that consumers must
physically attendthe booth on the day of the performanceto purchasetickets.It is interestingthat
firmschoose to sell ticketsvia this method.Despite havinganeffective telephonesales mechanism
alreadyavailablefor the sale of tickets, the theaterproducerchooses not to use this mechanism
for discountsales on the day of the performance,insteadforcingconsumersto incurthe disutility
associatedwith purchasinga ticket at the booth. I interpretthis ticket category as a deliberately
damagedversion of the product.The firmchooses to make the good less attractivethanit would
otherwise be. This would be an example of a damaged good, as modelled by Deneckere and
McAfee (1996).
However,the conventionaldamaged-goodsexplanationis not entirelyadequatein this case,
for the following reason.Thereis actuallysignificantvariationin seat qualitywithin the orchestra
region of the theater,which I discuss below. Despite this quality variation,there is no variation
in price for full-price tickets in the orchestra(for a given performance).13One consumer may
buy a full-price ticket over the phone for the orchestra,pay $55 and get the best seat in the
orchestra.Another consumer may also buy a full-price ticket over the phone for the orchestra,
pay $55 andget the worst seat in the orchestra.While this may seem puzzling, note thatdiscount
ticket purchaserswill generally obtain a seat in the orchestra.In particular,booth ticket buyers
(i.e., TKTS sales) tend to be seated in the orchestra,typically in the lower-qualityseats within
the section. Hence, there is variationin prices paid by people seated in the orchestraat a given
performance,but this is due to the pricingof discountticketcategoriesratherthanvariationin the
price of full-pricetickets.
An alternativeto selling booth discounttickets and seatingthe buyersin the orchestrawould
be to subdivide the orchestrainto high- and low-quality seats, set two differentprices, and sell
tickets over the phone.14This would accomplishthe same goal in termsof having multipleprices
for orchestraseats and using quality differences to facilitate sorting of consumers.Moreover,
this alternativemay be preferred,since consumers are not requiredto incur any disutility from
having to line up at a booth on the day of performance,providinga greateroverall surplus.To
providea justificationfor the firmpreferringbooth ticket sales over orchestrasubdivision,I allow
for the possibility that the disutility of attendingthe booth depends on consumers' willingness
to pay for seat quality (i.e., their income).15By incorporatingtype-dependentdisutility into the
damaged-goodsframework,thereis now the potentialfor the firmto preferbooth ticketsales over
orchestrasbudivision.While booth ticket sales and orchestrasubdivisionwould provide tickets
in the lower-qualityseats within the orchestra,the booth has the addedadvantageof being even
less attractiveto high-incomepeople.16
Other data. In additionto the price and quantitydatadescribedabove, I observe variables
o
that help to capture shifts in demand for Seven Guitars. This includes advertising,the Tony
Awards,and the numberof otherBroadwayshows, which I now explain.
A total of $878,337 was spent on advertisingSeven Guitars, amounting to 20% of the
show's running costs, or an average of more than $30,000 per week during the 25 weeks of
performances.The disaggregateddatacoversadvertisingin newspapers,magazines,travelguides,
and theaterguides; on billboards and bus shelters; on radio; and on cards in restaurants.The
majorityof advertisingexpenditureswas with TheNew Yorklimes (63%), mostly as graphical
display advertisementsin the Sunday edition (30%). For the purpose of this study the data are
13 Prices do
varyfrom performanceto performance.
There could be multiplesubdivisionsof the orchestra,and multipleprices, the point is the same.
15It is
importantthat the model incorporatean explanationfor why the firm would prefer booth ticket sales,
otherwiseit would be inconsistentwith a key institutionalfeatureof the industry.
16Another
possible reason to preferbooth sales is because the numberof tickets made availableat the booth is
determinedon a daily basis and depends on the numberof full-price sales priorto the day of performance.In this way,
the booth ticket categoryprovides a fairly low-cost and effective way of determiningwhen to increase or decrease the
numberof discounttickets to offer for a given performance.
14
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526 / THERANDJOURNAL
aggregatedinto a scalar variableintended to measure the daily level of advertising.I take the
level of advertisingfor each separateform of advertisingto be equal to the dollarexpenditure.17
To correct for the fact that a particularadvertisementis seen by people on days other than the
firstday it appears,each advertisingexpenditureis uniformlydistributedover the durationof the
publication.
Once an individualhas been exposed to an advertisement,or several advertisements,there
may be a time lag until the individualcontacts the box office, or more correctly Tele-charge,
which handledtelephone bookings for Seven Guitars.There will typically be a furthertime lag
between the time of booking and the time this person attendsa performance.This suggests that
attendancesduringa given week may dependupon advertisingover the previousmonth,say. For
this reason, the advertisingvariablethatis used in the empiricalanalysis is a moving averageof
the daily advertisingexpendituresover the last 28 days. Thereis a questionaboutthe endogenous
natureof advertising.This is analogousto the discussion of the price variationabove, in which
the variationmay be explainedby day-of-week dummies and a time trend.Hence, the variation
in the advertisingvariable may add explanatorypower when combined with functional-form
assumptions.
Seven Guitarswas performedin the WalterKerrtheater,and I use informationon the seating
in this theaterfor the estimation.In particular,the managerof the box office at the WalterKerr
assigned every seat in the theatera ratingfrom one to ten, based upon his experienceof people's
preferences when buying tickets, to reflect the quality of the view from each seat. From this
procedureit is apparentthatthereis significantvariationin seat qualitywithinthe areadeemedas
thehigh-qualityregion,while for the medium-qualityandlow-qualityregionsthereis insignificant
variationin seat quality across seats within each of the regions. The capacity of the high-quality
region is 755; for the medium-qualityregion it is 126; and for the low-qualityregion it is 66.
From VarietymagazineI have weekly informationon the attendancesfor every otherBroadshow
that was performingduringthe same period as Seven Guitars.Such data may capture
way
shocks from tourism,the weather,and otherpeculiaritiesthatmight affect Broadwayattendance.
For the 25-week periodthatSevenGuitarswas performed,therewere an averageof 15.6 musicals
and 9.0 plays performingon Broadway.During the same period, the averageweekly total attendance for all Broadwayshows was 200,839.87 (standarddeviationof 20,256.48). With almost all
shows being performedeight times a week, this amountsto an averageof 25,105 people attending
Broadwaytheatereach night in New York.
Incomeis animportantdimensionof consumerheterogeneityin thedemandsystempresented
in the next section. The producersof Seven Guitarschose not to advertisenationallyin orderto
encouragemoretouriststo see theirshow,becausesuch aninvestmentis believedto be worthwhile
only for longer-runningshows. I consider the potentialconsumersin this case to be a subset of
all people in the New Yorkmetropolitanregion, including touriststo the area.In particular,it is
assumed the potential consumers for a given performanceof Seven Guitars are all people who
attend Broadway theater at the same time. The League of American Theatresand Producers
(2001) surveyedBroadwayaudiencesin the 1990-91 season, obtainingdataon family income for
7,281 people attendinga sample of performancesacross 12 differentshows.18From that survey
the proportionof people whose annualfamily income lies within certainintervalsis known, as
reportedin Table2.
17
This reflects the principleof the marginaldollar spent on each form of advertisingon any given day having an
equal effective advertisingvalue, while ignoring the advertisingvalue of inframarginalexpenditures(and the fact that
some categorieswill be zero on some days).
18 As far as I
know, the survey was done by providingall Broadwaypatrons,over a period of a about a month,
with a questionnaireandreturnmail envelope. No informationwas availableon the responserate.The main advantageof
the survey is that it providesinformationon the people who attendBroadwaytheater.As explainedin the next section,
in my analysis I consider the behaviorof people who attendBroadwaytheater,not of New Yorkmore generally.Other
alternativedata,such as the distributionof income in Manhattan,may not be very relevantfor Broadway.
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LESLIE
TABLE 2
/
527
Distribution of Annual Family Income and
Estimated Parameters for the Log-Normal
Distribution of Income
Percent
Income Interval
Less than$25,000
$25,000 to $34,999
$35,000 to $49,999
$50,000 to $74,999
$75,000 to $99,999
$100,000 to $149,999
$150,000 and above
5.7
10.3
14.3
26.7
16.2
15.2
11.6
Numberof respondents
7,281
Estimated parameters
Mean
Standarddeviation
11.3483
.7937
(.8248)
(.7092)
Notes:Dataare fromThe Leagueof AmericanTheatresandProducers
(1991).Incomeis
are for the underlyingnormal
measuredin November1990 dollars.Estimatedparameters
of income.Standard
errorsarein parentheses.
distribution
of thelog-normal
distribution
3. Structuraleconometric model
*
Behavioral model. Consumersare presentedwith a menu of different ticket options for
seeing a play, or not. There are various tickets for specific seat qualities or views within the
theater,and thereare variousdiscounts.Differentprices for differentseat qualitiesis an example
of second-degreeprice discrimination,and differentprices for individualswith coupons is an
exampleof third-degreeprice discrimination.It is assumedthateach consumercan choose among
the ticket options for a single performanceonly, which rules out intertemporalsubstitution.19
Individualsare differentiatedalong two dimensions: income and their taste for this play
relative to the outside alternative.Let Yi > 0 denote the income of consumer i, and let ji > 0
denote consumeri's relativetaste for the outside alternative(higher4i means a higher valuation
of the play). As a matterof interpretation,ijcan also be thoughtof as the individual'sperception
of the quality of the show and may depend on numerousaspects.20Hence, each individualis
characterizedby the pair (yi, i). Both yi and 4i are known to the individualbut unobservableto
the firm.Incomeis distributedaccordingto the cumulativedistributionfunctionF(y), andtaste is
distributedaccordingto the cumulativedistributionfunction G(4). Both distributionsare known
to the firm.For simplicity, F and G are assumedto be independent.21
Consistent with the data, I allow for three quality-differentiatedfull-price ticket options.
The three regions are labelled high quality, medium quality, and low quality. All individuals
preferhigher-qualityseats but differ in their willingness to pay for higher quality.As previously
discussed, all seats in the high-qualityregion of the theaterdo not provideequivalentseat quality.
Indeed, there appearsto be fairly significantvariationin quality within the high-qualityregion,
which is likely to play an importantrole in consumers'decision making.For the low-qualityand
medium-qualitycategories, the assumptionof equal seat qualities within each region is a good
approximation.Each qualityregion also has a capacity constraint.
19This is
mainlydone for computationaltractabilitypurposes.However,the restrictionalso enhancesidentification
of the demand system by providing variationat the performancelevel. An alternativewould be to allow consumersto
choose over performancesin a whole week, in which case variationis limited to the weekly level.
20 For a detailed discussion on the
perceptionof show quality in relationto the demandfor theater,see Throsby
(1990).
21 As discussed
below, the outside alternativeis seeing anotherBroadwayshow. The independenceassumption
means thatincome and the taste for this particularplay are uncorrelated.
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528 / THERANDJOURNAL
OFECONOMICS
The presence of both capacityconstraintsand qualityheterogeneitywithin the high-quality
region lead me to incorporaterationing in the model by arrangingconsumers in a random
sequence.Specifically,M potentialconsumersarein a randomsequence{(yi, 41);... ; (YM,TM)}.
Following the orderof the sequence, consumersare individuallypresentedwith their choice set,
and depending on their decision the choice set for the next consumer in the sequence may be
modified-an option may be removedbecause a capacityconstrainthas been reached,or the best
availableseat in the high-qualityregion may have a lower quality.Consequently,the seat quality
that is offered to individuali in the high-qualityregion depends on the numberof seats in the
high-qualityregion that are sold to individualsahead of individuali in the sequence.22Let qih
denote the qualityof seat, or view, thatis associatedwith the high-quality,full-priceticket option
for individuali, and let qim = qm and qi = qe denote the seat quality for medium-qualityand
low-qualityfull-priceticketbuyers.
Subjectto availability,the net utility to individuali fromchoosing a full-priceticket for seat
quality j E {t, m, h} is given by
Uij = qij[B(yi)
- pj]
(1)
in which B(yi) < yi is individuali's budget for entertainmentexpenditures,pj is the price of
the ticket, and Y is a parameter.23
With this formulation,consumers' marginalutility from seat
qualitydependson theirlevel of income, leading to a self-selectionprocess in which high-income
individualschoose high-qualityseats and low-income individualschoose low-qualityseats. The
function B contains parametersthat allow me to estimate the appropriateproportionof income
that is relevantfor individuals' entertainmentexpendituredecisions.24I use a specificationthat
allows for wealthierpeople to spend a greaterabsolute amountof income on entertainment,but
a lower proportionof their total income thanless-wealthy individuals.Specifically,
B(yi)= 8ly82,
where 81 > 0 and82 E (0, 1] areparameters.Individualsfirstdecide how muchincome to allocate
to entertainmentand perhapsothercategoriessuch as clothing, travel,savings, and so forth,and
then subsequentlydecide how to spendtheirentertainmentbudgeton the variouspossibilities that
include this play.25I assumethe 8 parametersare the same for all individuals(i.e., conditionalon
income thereis no unobservedheterogeneityin budgeting).26
In additionto the abovefull-priceticketoptions,withprobabilityX(yiIy) consumeri receives
a coupon thatcan be used to purchasea ticket for a high-qualityseat at price Pc < Ph and obtain
utility
Uic = qih[B(yi) -pc]1.
The densityX(y, |y) is the outcomeof some coupontechnologyavailableto thefirm.Theparameter
y correspondsto the efficiencyof thecoupontechnology(i.e., how accuratelycouponsaretargeted
to low-income individuals).27
22 I assume seats are allocatedin orderof best to
worst.
An alternativeto the functional form in (1) is an additivelyseparablespecification.However,for such utility
functions,in the presence of a continuousdistributionof consumertypes, a revenue-maximizingfirmtypically prefersto
offer only the high quality good. With this specification,in general, the firm optimally chooses to offer many different
qualitylevels. Note also the model does not allow consumersto decide when they will see the play.
24
An alternativeapproachis to include a coefficient on price that would increasethe disutility of price.
25
One set of assumptionsthat would permittwo-stage budgetingin this context is if the overall utility function
were additivelyseparablein the utility from consumingentertainment,andif the prices of all goods in the entertainment
categorymoved in proportionto one another.See Deaton and Muellbauer(1980).
26
This may not be a restrictiveassumptionfor the following reason. Income is already a form of unobserved
heterogeneityin the model and I use separatedata to estimate the distributionof income. The budget equation then
transformsthe income distributionto help explainobservedattendances.Allowing for additionalunobservedheterogeneity
in the budgetequationwould providegreaterflexibility,but the currentfunctionis alreadya quite flexible transformation.
27
Althougheachindividuals'level of income is privateinformation,thefirmcontractsa thirdparty,whichpossesses
privatesignals about each individual'slevel of income, to disseminatecoupons.
23
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LESLIE / 529
Consumersalso have the choice of going to a discount booth where they can purchasea
ticket for one of the high-qualityseats thatremain after all individualshave had an opportunity
to purchasea full-price seat. The booth ticket quality is denoted as qib. I assume there is a time
cost for having to physically attendthe booth.28In particular,the utility frompurchasinga booth
ticket is given by
Uib= qib[B(yi)- Pb - (yi)]1,
where r(yj) > 0 is an increasingfunctionthatrepresentsthe time-cost of attendingthe booth. I
adopta simple linearspecificationfor the cost of attendingthe booth. As describedin Section 2,
the essential featureis for this cost to depend on individuals'income levels:
(Yi) = tlYi + 2,
where rl > 0 and T2areparametersto be estimated.29
To complete the choice set I specify the utility from the outside alternative.30The goal here
is to provide a specificationthat acts as a reduced-formvaluationof the highest utility that may
be obtainedfrom seeing anothershow, and thatalso gives rise to sensible substitutionpatterns.I
assume the utility from the outside option is given by
Uio= i-'[B(yi) - po]71,
where po is the price of the outside option and qrois a parameter.Individualconsumers' tastes
for Seven Guitarsrelative to seeing anothershow are capturedby s. I divide the outside utility
by ij(higher values of ij imply lower willingness to see a show other than Seven Guitars),but
I could equally have written 4j to multiply the utilities of each inside alternative.By including
heterogeneousvaluationsof the outside alternative,the aim is to partiallycapturethe presence of
competingfirms.31Doing so requiresnot only thatthe valueof the outsidealternativebe correlated
with the value of the inside alternatives,but that individualsalso vary in their relative taste for
the outside option-as if the outside alternativerepresentsa set of vertically differentiatedseat
qualities for a horizontallydifferentiatedshow.32
To summarize,the utility for individuali fromproductj, assumingthis individualreceived
a coupon, is given by
Uij =
qij [B(yi) - pj]
for j E { , m, h}
qih[B(yi) - pj]1
for j = c
qij[B(yi) - pj - t(yi)]'
for j = b
ei-'[B(y) - pj],o
for j = o.cr
28
Fromthe data,qm < qb for all i, while pb = pm. The time cost is neededto providean incentivefor individuals
to choose medium-quality,full-pricetickets over booth tickets that are the same price and for higher-qualityseating.
29
As with the budget equation above, I assume there is no additionalunobserved heterogeneity in this cost
specification.For the same reasons I would arguethatthis is of little consequence.
30
As previously noted, the outside alternativeis to see anotherBroadwayshow. The main reasonfor interpreting
the outside alternativein this relativelynarrowway is that I have data on the numberof people who attendedBroadway
theaterat the same time that Seven Guitarswas on Broadway.Hence, given this interpretation,I observe the numberof
individualswho in fact chose the outside alternative.
31
The common practice in discrete-choicedemand models is to normalizethe utility of the outside alternative
to zero. For vertically differentiatedproducts,however, zero-outsideutility implies unreasonablesubstitutionpatterns.
For example, if the relative value of the outside alternativeincreases by a small amount, then a zero-outside utility
normalizationimplies thatconsumersswitch out from the low-quality-insidealternativeonly. In reality,consumerswould
switch from the high-quality-insideoption to the outside option also.
32 this
In
respectI drawon the approachof Stole (1995), who providesa theoreticalanalysis of oligopoly nonlinear
pricingin which individualssimilarlypossess both horizontaland verticalpreferenceheterogeneity.
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530 / THERANDJOURNAL
OFECONOMICS
If an individualdid not receive a coupon, his utility functionis the same except thatit excludes
choice j = c.33The expected demandfor tickets in categoryj is equal to
Sj()=M
I
dF(y)dG(4),
,(y)c-Aj
where Aj is the set of consumertypes who most preferoption j and, as noted above, I assume F
and G are independent.More formally,
Aj = [(Yi,4i): Uij > Uik,Vk E 2] C
2,
where Q2= {(, m, h, c, b, o}.
I defer the explanationof the firm side of the the model to the section on counterfactual
experiments(Section 5), where it is used. I make no assumptionson the firm'sbehaviorthat are
incorporatedin the estimationof the demandsystem. This helps to limit the possible sources of
misspecificationin the estimatedmodel, with a potentialefficiency loss.
Econometric model. The behavioralmodel describedabove is an example of a discreteO
choice random-utilitymodel with endogenously randomchoice sets.34The model is estimated
using a maximum-likelihoodestimator.I now detail distributionaland otherassumptionsthatare
needed for estimationof the econometricspecificationof the behavioralmodel.
The distributionof potential consumers' income is estimated based on the survey by the
Leagueof AmericanTheatresandProducers.The dataconsist of theproportionof people attending
Broadway theaterin 1990-91 with annual family income within n intervals.The intervals are
inflated to 1996 levels using the ConsumerPrice Index, and a log-normaldistributionis fitted
using a minimumdistanceestimator.The results arereportedin Table2. Estimatedmean annual
family income is $116,225 (in 1996 dollars).
The distributionof individuals' tastes for the show may change from performance to
performance.I assume an exponentialdistribution,
tit
- exp(Xtp),
in which the vector Xt = {constant, advertising,dummy for before the Tony Awards, various
day-of-performancedummies,numberof otherBroadwayshows in the same week, time}, and f
is a parametervector.35
For the probabilitydensity of receiving a coupon I assume the following specification:
=
exp(ayi - Zty)
1 + exp(ayi - Zty)'
where Zt = {constant, dummy for performanceswhen ManhattanTheatreClub memberswere
allowed to attend,variousday-of-performancedummies,time, time-squared}is a vector of data.
This density has the appearanceof a backward"s"for a > 0, with the probabilityof receiving a
coupon decreasingin yi. The vector y is a parametervectorrepresentingthe numberof coupons
the firm sends out. I interpretthe scalara as a coupon efficiency parameter-higher values of a
33Or equivalently,assume Uic = 0.
34The choice sets are randombecause an individualhas a
coupon with only a certainprobability.In addition,the
choice sets are endogenousbecause the choices availableto an individualdepend on the optimal choices of individuals
aheadin the sequence.
35For
simplicity,the income and taste distributionsare independent.Since the outside alternativeis to see another
Broadwayshow, it seems reasonableto believe that income and the taste for this particularplay would be uncorrelated.
Note also, the advertisingvariableis a moving average of the past 28 days' advertisingexpenditures,as explained in
Section 2.
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LESLIE / 531
imply a less-efficientcoupontechnology(thatis, a greaterprobabilityof wealthypeople receiving
a coupon).
Thecapacitiesof the threeseatingregionsaredenotedby Ce, Cm,andCh. Once thecapacityof
any region is reachedwithin a sequence of simulatedconsumers,the optionis no longer available
for subsequentindividualsin the sequence. Let kijt denote the numberof tickets purchasedby
consumersaheadof individuali (in the sequence) for region j in performancet. Then tickets for
categoryj are only in the choice set if kijt < Cj. To computethe seat qualityin the high-quality
region of the theaterthatis offered to individuali, I use the distributionof rankingsand assume
the differencebetween consecutivelyrankedseats within the high-qualityregion is the same, no
matterwhat their rankings.Since the high-qualityregion adjoins the medium-qualityregion, I
also assume the medium quality is uniformly different from the worst seat in the high-quality
region. For the low-quality seats, I allow for an arbitrarydifferencein quality,since these seats
are physically separatein the upperbalcony. Given these assumptions,I thereforeestimate the
qualityof the best seat in the house (Qmax),the medium-qualitylevel (qm),andthe lowest-quality
seat (qe).
Conditionalon an individualreceiving a coupon, the utility specificationthatis the basis for
estimationis given by
qijt(8Yi
- Pjt)1
for kijt < Cj and
for kijt < Ch and j = c
qiht(81Yi2 - pjt)l
Uijt =
qijt(81
it l(8lyi2
-2
-
jt - ri
j E {e,m, h}
for kijt < Ch and j = b
- r2)n
for j = o.
pj)7lo
If the individualdoes not receive a coupon, the utility functiondoes not include the choice j = c.
The choice set for each individualis random,dependingon the exogenousprobabilityof receiving
a coupon and the endogenousbehaviorof otherindividualsin the market.
The set of parametersto be estimatedis
()= {q, qm,,Qmax,81, 82, rl, t2, r1o,
,
Po, a, }, y}.
Noting that the distributionof individuals'income is separatelyestimated,the predictedmarket
shareof productj E 2, in periodt E {1,..., T}, conditionalon all parameters,is computedby
sit(Pt, Xt, Z, @) =
dF(y)dG(
|fJ
IlXt),
(2)
,(y )Ajt
where
Ajt = [(Yi, i) : Uijt(pt, Xt, Zt; O) > Uikt(pt, Xt, Zt; 0), Vk E 2].
For notationalease, the specification shown in (2) suppressesthe integrationover realizations
of the coupon distributionand availablecapacity.Denote the actualnumberof individualswho
choose optionj in periodt as Nit, wherethe numberof individualschoosing the outsidealternative
is determinedby
Not = M-
E
Njt.
jE2\{o}
The marketsize, Mt, is the total numberof people attendingBroadwaytheaterin the same week
dividedby eight (theweekly numberof performancesfor all Broadwayshows).The log-likelihood
? RAND 2004.
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OFECONOMICS
532 / THERANDJOURNAL
functioncan now be stated:
T
.
) = ,E E Njtlogs
(,
).
(3)
t=l jE2
The vector of estimatedparametersis the value of 0 thatmaximizes (3).
In Section 2 I described the different kinds of price variationcontained in the dataset. I
arguedthat some of the price variationhelps to identify the demandsystem without relying on
functional-formassumptions.Meanwhile, other kinds of price variationprovide identification
only when combinedwith functional-formassumptions.Considerfull-pricehigh-qualitytickets,
for example. The price of a high-qualityseat, Pht, varies across days of the week, and day-ofperformancedummies are included in both Xt and Zt. The effects of each of these three timevaryingcomponentsareidentifiedseparatelybecauseeach entersnonlinearlyin differentpartsof
the model. Two factorssupportthis approach.First, the functionalformhas been motivatedby a
behavioralmodel. Inparticular,the separatenonlinearcomponentsaremotivatedby distinguishing
the utility function from roles played by the distributionsof individual heterogeneity (tastes
and coupon availability).36Second, there still remains some variationin the data that provides
identificationwithoutrelying on functionalform.
In the econometricmodel, as in thebehavioralmodel, the only sourcesof uncertaintyarefrom
the individuals'unobservedheterogeneity.Thereis no additionallogit, probit,or othersuch error
term.All of the stochasticelements in the econometricmodel have specific interpretationwithin
the behavioralmodel. This limits the model's ability to explain discrepanciesbetween predicted
behaviorand actualbehavior.37To compute the optimalparametervalues, I use a nonderivative
simplex search algorithm.The random sequences are simulated,which introducesa source of
bias to the maximum-likelihoodestimator.38
4. Empiricalresults
*
To estimate the model, several normalizationsare imposed. The quality level of the lowquality seats is set equal to one (qe = 1). It appearsfrom the estimation that ir and r/o are not
separatelyidentified,while the differencebetween the two parametersis identified.I thereforeset
/o = 1. Since I have no data on the numberof coupons that were sent out, I set the value of a to
.01 (robustnesschecks are discussed below). Lastly, the price of the outside alternativeappears
to be poorly identifiedby the data, so this is set to zero, po = 0.39The remaining28 parameters
areestimated.Observationsfor the openingnight of Seven Guitarsare not used in the estimation,
since the large numberof complimentarytickets on thatnight (572) suggests an aberration.This
leaves 198 performances.
The estimatedparametersareshownin Table3. The qualityparameters,qmand Qmax,should
36 Of
course, thereremainsa degree of arbitrarinesseven in the behavioralmodel. The point is, the model offers a
clear context and explanationfor the specific functionalform.
37 An unlikelydatapointin the currentmodelcan be explainedby eitherthe uncertaintyof income or the
uncertainty
of the taste for the play. Adding an additional errorterm would be a generalizationthat provides anothersource of
explanationfor an unlikely datapoint and will have the effect of smoothingthe likelihood function. Whetheradditional
errorterms would improve the model is not clear.An implicationof the limited sources of errorin the model is the low
standarderrorsthat arise from estimationin the next section.
38The bias is
mitigated by using a large number of simulation draws of the randomsequences. The reported
estimates were computed based on 1,000 draws. See Pakes and Pollard (1989) and McFadden(1989) on the use of
simulationmethodswith extremumestimatorssuch as thatused here.
39Since I have no comparabledata on the price of the outside alternative,I must estimate or normalize po. If
the actual price of the outside alternativeis correlatedwith the ticket prices for this show, as one might expect, then
normalizingthe price of the outside alternativewill lead me to underestimatethe sensitivityof aggregatedemandto price
changes. It may be possible to allow po to depend on the day of the week, which could alleviate this concern.However,
note that the distributionof 4, which affects the utility of the outside alternative,does alreadydepend on the day of the
week, helping to mitigatethis potentialproblem.
? RAND 2004.
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LESLIE
TABLE 3
533
Estimated Parameters
qm
Qmax
81
1.6921
3.3314
2.5199
(.0064)
(.0244)
(.0163)
82
.4414
(.0007)
TI
T2
7r
.0067
2.7365
1.0316
(.0000)
(.0305)
(.0022)
:Constant
.0180
.0100
.0008
.0307
.0080
.0237
.0045
.0040
.0050
.0094
.0525
(.0006)
(.0005)
(.0002)
(.0015)
(.0010)
(.0038)
(.0016)
(.0004)
(.0011)
(.0001)
(.0008)
21.2021
-.8105
-3.1797
-1.9682
.6080
-.2090
-.1995
-.3824
-4.4849
(.1045)
(.0406)
(.1322)
(.1144)
(.6669)
(.0879)
(.0820)
(.1284)
(.0635)
-.0653
(.0076)
($'00,000)
Advertising
TonyAwards
Saturday
evening
Fridayevening
Sundayevening
Sundaymatinee
matinee
Saturday
Thursday
evening
Numberof othershows
t/100
y:
/
Constant
Manhattan
TheatreClub
Saturday
evening
Fridayevening
Sundayevening
Sundaymatinee
matinee
Saturday
Thursday
evening
t/100
t2/10, 000
Numberof observations
Log-likelihood
4,886,572
-776,703.44
Notes:Standard
errorsarein parentheses.
Thefollowingnormalizations
wereapplied:qi = 1,
is a movingaverageovertheprevious28 days.
rio= 1, po = 0, anda = .01.Advertising
be gauged with respect to the normalizationfor qe. The estimates imply that the best seat in
the house is 3.3 times betterthan the worst seat in the house. With the typical prices of $15 for
low-qualityseats and $55 for high-qualityseats, highest-qualitybuyerspay 3.66 times more for a
seat, thatis, 3.3 times higher quality.At the mean weekly income level of $2,227.10, the implied
budget for entertainmentexpendituresis $75.69, or 3% of weekly income. The cost of attending
the booth is implied by the parametersT1 and r2. The type-dependentcomponent of the cost
depends on rl, with the estimatedparameterimplying that the cost of attendingthe booth rises
by almost .7 of a cent for every additionaldollar of weekly income. Thus, for example, the cost
of attendingthe booth for an individualwith the mean level of family income is estimatedto be
$12.11, which seems reasonable.40
The estimatedparametersfor the distributionof individuals'taste for this play are given in
Table3 underthe headingof the P parameters.Recall thatI use an exponentialdistribution;hence,
for example, the dummy for the Tony Awardsincreases the mean (variance)of the distribution
by .01 (.0001). Though the positive time trend was not anticipated,the advertisingand Tony
40The booth ticket
price variableincludes the service charge of $2.50. Consequently,the estimate of f2 = 2.74
ought not be gauged by its closeness to 2.50.
? RAND 2004.
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534 /
THE RAND JOURNALOF ECONOMICS
FIGURE1
PERCENTAGE
DIFFERENCE:
PREDICTEDAND ACTUALFULL-PRICE,
HIGH-QUALITY
TICKETSALES
21
25
00
U.
1
5I0
I1
nn
I
-100
0
100
n
200
I
300
n
Inm
400
rh
500
n
600
Percentdifference
variableswould explainthe dropin attendancesafterthe TonyAwards.The y coefficientsexplain
movementsin coupon sales. While almost all parametersare extremelyaccuratelyestimatedand
significant,several of the y parametersare not.
As a check on how informativethe X and Z variablesarefor explainingvariationin demand,
I reestimatethe model withoutthese variables.Thatis, I impose the restrictionthatthe coefficients
on these variablesare all zero. The likelihood-ratiotest overwhelminglyrejectsthe hypothesisof
zero coefficients.As an indicationof the differencesin the fit of the model with covariatesversus
the fit of the model withoutcovariates,the meanabsolutepercentagedifferencesin predictedsales
for each ticketcategoryare:26%(low quality), 154%(mediumquality),23% (high quality),64%
(coupons),and49% (boothtickets).Tohelp assess the fit of the model (with covariates),Figures 1
to 4 presentthe differencesbetweenpredictedand actualbehavior.In Figure 1 arethe percentage
differencesbetween the numberof predictedhigh-qualityticket sales and the actual numberof
high-qualitysales for each performance.It is apparentthat the model predictshigh-qualitysales
to within roughly 50% of the actual numberin a large proportionof performances,while also
exhibiting a tendency to overpredict.The equivalentfigures for low-quality,booth, and coupon
sales appearas Figures 2, 3, and 4, respectively.Of these cases, booth and coupon sales appear
to be the best-predictedticket categories, with the majorityof performancespredictedto within
roughly 50% of the actualnumberof sales.
The reason why the model tends to overestimatedemand is related to the presence of the
capacityconstraints.For some observations,the estimatedmodel overpredictsdemand,while for
others it underpredictsdemand.For the cases where predicteddemandis too low, the estimator
FIGURE2
PERCENTAGE
DIFFERENCE:
PREDICTEDAND ACTUALFULL-PRICE,
LOW-QUALITY
TICKETSALES
v
35
30
525
I15
10
5
tV
-200
rTkl
m
I
IfllJ if, f
IrOmmmmm
II m
0
III
I
II LEmwm
200
_
n
400
n
800
I
800
rercentdifference
? RAND 2004.
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n
1,000
LESLIE /
535
FIGURE3
DIFFERENCE:
PREDICTEDAND ACTUALBOOTHTICKETSALES
PERCENTAGE
ia
302520| 1510 -
LL
5-
0
-200
0
I
m
II
I
I
m
II
200
0
IIII
I
I
I
I
400
I
I
I
600
800
I
- I
!
I?I
I
1,000
1,200
Percentdifference
seeks to increasepredicteddemand,at the potentialcost of exacerbatingoverestimatesof demand
in other cases. But the capacity constraintsserve to limit the magnitude of overestimatesof
demand.Consequently,the estimatorexploits the capacityconstraintsto limit the overestimates,
giving rise to above-averagelevels of demand for most observations.Some verificationof this
comes from the resultthatdemandfor the low-qualityseats is overestimatedthe most frequently,
and this categoryhas the smallest capacityconstraint.
Own-price,cross-price,and income elasticities arepresentedin Table4. These are obtained
by computingpredictedmarketsharesunderthe empiricalpricesandcomparingthemto predicted
market shares following a 1% increase of the relevant ticket price. The price elasticities are
computedwith and without capacity constraints.41Considerfirst the capacity-constrainedprice
elasticities. As expected for a monopolist, the own-price elasticities are almost all greaterthan
one. The exceptionto this is low-qualityseats, which is due to poorlypredictedlow-qualityticket
sales reachingthe capacityconstraintin 182 (out of 199) performances.Actual low-qualityticket
sales sell out in only 23 performances.High-qualitytickets are the largestrevenue category of
sales, and the estimate of the own-priceelasticity for these tickets is -2.5.
The columnon the farrightof Table4 takesinto accountthe factthatboothtickets arealways
sold at 50%off the price of high-qualitytickets. Thus, elasticitiesin this column arebased on 1%
increasesin both high-qualityand booth ticketprices. As expected, this reducesthe sensitivityof
both high-qualityticket and booth ticket buyers, since there is a diminishedincentive to switch
FIGURE4
PERCENTAGE
DIFFERENCE:
PREDICTEDAND ACTUALCOUPONTICKETSALES
I
100
200
300
400
500
600
700
800
900
Percentdifference
41 When there are no
capacity constraints,a consumers' choice set is not affected by the choices of othersahead
in the sequence.
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OFECONOMICS
536 / THERANDJOURNAL
TABLE 4
Demand Elasticities
Low
Medium
High
Booth
Coupon
High and Booth
-.0390
2.8448
-2.5142
1.3204
.9841
.0008
.2194
5.1226
.9310
-2.8745
.5510
.0009
-.1920
.4904
1.1274
.7304
-1.5202
.0014
.2175
7.3497
-1.5859
-1.5051
1.5352
.0019
0
5.9170
-4.0583
4.0942
0
.0950
10.5382
6.2060
.4150
-8.4894
.0634
.0370
3.0179
.0301
0
10.5382
12.1230
-3.6523
-4.4629
.0634
.1332
Price elasticities with capacity constraints
Low
Medium
High
Booth
Coupon
Outside
-.2468
-.0087
-.0668
.1619
-.0320
.0004
-.0032
-4.3119
.1141
.3623
.0170
-.0000
Price elasticities without capacity constraints
Low
Medium
High
Booth
Coupon
Outside
-5.4539
0
0
.1007
.0036
.0013
0
-9.2957
.1967
2.4070
0
.0085
Income elasticities with capacity constraints
.1224
Low
Medium
.4707
1.2209
High
Booth
Coupon
Outside
.6894
-2.1766
.1129
-.1659
-.9125
-.0009
Notes: Cells for price elasticities are the percentagechange in demandfor the row product,in response to a 1% increase in the price of
the column product(s).The column for "high and booth"is for simultaneousprice increasesof high-quality,full-pricetickets and booth
tickets (because booth tickets are always sold at 50% off the price of high-quality,full-pricetickets).
fromhigh qualityto booth, or frombooth to high quality.In addition,the magnitudeof the crossprice elasticities on the othercategoriesincreases;for example,the elasticityof coupon sales with
respectto the high-qualityprice rises from .98 to 1.53, and with respectto the booth price it rises
from .55 to 1.53.
A strikingfeatureof thecross-priceelasticitieswith capacityconstraintsis thatseveralof them
are negative.In the behavioralmodel, all tickets are substitutesfor one another,which ordinarily
implies positive cross-price elasticities. The reason for the negative cross-price elasticities is
that the capacity constraints cause some consumers to select their second- or lower-ranked
alternatives.For example, increasing the price of low-quality tickets causes some individuals
to no longer purchase a low-quality ticket, making the ticket available for anotherindividual
who may have purchaseda medium-qualityticket only because therewere no low-qualitytickets
availablepreviously.In this way, increasingthe price of a capacity-constrainedcategorycan lead
to fewer sales in other categories. To confirm this, I compute price elasticities for the demand
system with no capacity constraints,as also reportedin Table 4. In this case, all cross-price
elasticities are indeed positive.
Table4 also reportsthe implied income elasticities for the estimateddemandsystem. Rather
than interpretthe negative income elasticities on the booth and coupon tickets as evidence of
inferiority,we againsee the capacityconstraintscausingsubstitutionswhichunderliethese income
elasticities.Note thattheelasticitiesarecomputedbasedon a 1%increasein weekly familyincome,
and thatthis will translateinto a less than 1%increasein each individual'sentertainmentbudget.
In any event,high-qualityticket sales appearto be highly sensitiveto income, while lower-quality
sales are less so.
It was noted at the startof this section thatI appliedthe normalizationa = .01. As described
in Section 3, this variable captures the effectiveness of the coupon technology. To check the
robustnessof the analysis to this normalization,I reestimatethe model with the normalization
a = .02. The implied price elasticities were all of the same sign, and similar magnitude,to the
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LESLIE / 537
estimatesbased on a = .01. In the next section I examine a varietyof counterfactualexperiments
and welfare comparisons.
5. Counterfactualexperiments and welfare analysis
I now describeseveralcounterfactualexperimentsbasedupon the estimateddemandsystem.
*
The experimentsinvolve reoptimizingpricesunderdifferentrestrictions,such as uniformpricing,
and examining the effect on consumer welfare. No supply-side assumptionswere used as part
of the demandestimation.42However,to performthe experimentsin this section I will requirea
model of how the firmsets prices.I assumethe firmchooses prices, pt = {Pet, Pmt,Pht, Pbt,Pct},
to maximize expected revenue:
T
R=
C
E
t=l jc2\{o}
pjtqjt(Pt, )
where qjt = Mtsjt(pt, Xt, Zt, 0) comes from the estimated demand model (0 denotes the
estimatedparameters).43The following experimentsdifferin the extent of price flexibility thatis
allowed-from allowing all prices to vary in every performance,to having all prices equal and
constantacross all performancesand all categories.
By not allowingthe firmto choose the qualitylevels, qe, qm,andqh,the firmside of the model
differsfrommodels of second-degreepricediscriminationin whichthe firmchooses bothqualities
andprices. The restrictionis motivatedby the fact thatthe producerof a Broadwayshow rentsthe
As with
theaterwherethe show is performedandmakes no physical changes to the auditorium.44
most multiproductmonopoly problemsin which the demandfor each productis interdependent,
it is hopeless to solve for an analytic solution to the firm's optimizationproblem, except in
unrealisticallysimple cases such as discrete consumertypes or a uniformdensity of consumer
types. The problem is furthercomplicatedby the stochastic value of the outside alternative.45
Nevertheless,it is straightforwardto solve for optimalprices using numericalmethods.
The outcomesfromvariousexperimentsarecomparedwith a benchmark,andI considertwo
benchmarkcases:
Base-A. Using the empiricalprices, the model providesa predictionof consumerbehavior,
o
which yields a measurementof total net utility for all performances.As indicatedin Table5, the
measureof total utility in this case is 3.59 (units are meaningless).The associatedpredictedtotal
revenue is $6.27 million, and predictedaverageattendanceis 906.9. Due to the discrepancyin
predictedattendancesfromactualattendances,relativecomparisonswill be the most meaningful.
In Table 5, the prices shown for Base-A are the average(across performances)prices in each of
the categories.
O Base-B. While Base-A uses empiricalprices, Base-B is based on predictedoptimalprices.
Optimalprices arecomputedbasedon restrictionsintendedto resemblethe actualdecision making
of the firm. Specifically, I assume the firm is constrainedto using the same price menu for
all performances.Since the actual pricing policy of the firm does not change over time, this
assumptionis a reasonableapproximationfor a benchmarkscenario. Also, to match the actual
behaviorof the firm,I assume the booth ticketprice is 50% off the full-price,high-qualityticket
42
Supply-side assumptionscan enhance the accuracy of demand-sideestimates, but at the risk of introducing
misspecificationbias.
43For simplicity,I assume T is known and fixed.
44The problemI analyze is equivalentto the firststage of the nonlinearpricingproblemaddressedin Rosen and
Rosenfield (1997).
45 See Rochet and Stole
(2002) for an analysis of monopoly nonlinearpricing where consumers have random
participationconstraints.
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538 / THERANDJOURNAL
OFECONOMICS
TABLE 5
Experiment
Actual
Base-A
Base-B
Uniform
No-booth-A
No-booth-B
Boothnot50%
Nonsticky
Results of Counterfactual Experiments
Revenue($ million)
Utility
Average
Attendance
Pt
Pm
Ph
Pb
Pc
4.6951
6.2698
7.8965
8.0204
6.7301
8.3495
8.4516
8.0194
NA
3.5859
3.5775
3.6039
3.5837
3.5925
3.5900
3.5800
661.56
906.86
864.11
809.57
873.01
873.73
850.30
887.37
16.93
16.93
23.90
50.04
16.93
22.28
24.47
24.11
29.20
29.20
29.80
50.04
29.20
38.33
40.86
30.11
55.08
55.08
60.22
50.04
55.08
51.53
54.21
59.73
27.53
27.53
30.11
NA
NA
NA
38.05
29.87
31.01
31.01
45.26
NA
31.01
43.23
46.32
46.03
of eachexperiment.
Thepricesshownarethe averagepricesacrossall performances.
Notes:See Section5 forexplanations
Forsome
to performance,
forotherstheydo.Thefigureforaverageactualattendance
doesnot
experiments,
pricesdonotchangefromperformance
tickets.If thesecategoriesareincluded,the averageactualattendance
includewheelchair
is
tickets,standingroom,andcomplimentary
707.
price (Pb = .5ph).46 It is necessary to make an assumptionabout the firm's knowledge of the
explanatoryvariables,Xt and Zt, and the total numberof performances,T. For simplicity,it is
assumedthe firmhas perfect foresightof each of these.
In additionto having a well-specified model of demand,it is importantthat the firm side of
the model also be well specified.A good test of whetherthe above optimizationproblemfor the
firm is a well-specified descriptionof actualbehavioris to see if Base-B yields predictedprices
thatareclose to the observedprices.47It is worthemphasizingthatthe estimationprocedurein no
way "forces"predictedprices to equal observedprices. In this respect,it is a fairlystringenttest to
expect the estimateddemandsystem, combinedwith the abovemodel of price-settingbehavior,to
yield predictedprices thatapproximatethe observedprices. Table5 shows the predictedoptimal
prices for Base-B, which shouldbe comparedto the top row for actualprices (or,equivalently,the
row for Base-A). It is strikinghow close the predictedprices areto the actualprices. Forexample,
predictedph equals $60.22, while the averageactualph equals $55.08.48The coupon price is the
least well predicted.Overall, the predictedprices underBase-B provide a degree of confidence
that the model is well specified in orderto performthe following counterfactuals.
These counterfactualsconcern alternativeassumptionsas to how much flexibility the firm
has in determiningthe price menu. In calculatingthe welfare effects for consumers,I take into
account the utility obtainedby those consumerswho choose the outside alternative.For some
experimentsBase-A is the meaningfulbenchmark,while for othersBase-B is the most relevant.
o
Uniform. In this experimentthe firm is restrictedto selling all tickets for all performances
at a single price. In particular,thereare no booth sales and no coupons. As shown in Table5, the
optimalprice in this case is $50.04. Since the experimentinvolves reoptimizingprice, it is most
relevantto comparethe outcomewith Base-B, which also uses optimalprices. Relativeto Base-B,
attendancedrops by 6.3% and utility rises by a trivial amount.Apparentlythe improvementin
utility for people who were paying higherprices before marginallyoutweighs the loss for people
who either pay more or switch to the outside alternative.In additionto these effects there is a
new allocationof seat qualitiesin the theater,which is good for some people and bad for others.
The best seats in the house are now more attractivedue to the lower price. The worst seats in the
house are likely to be filled only by people who have a high taste level for the play.
46The service
chargeimposed by the booth of $2.50 per ticket is also included.
47This is actually a joint test of both the demandand supply specifications.
48
Prototypeversionsof the demandmodel gave rise to predictedpricesthatwere higherby severalhundreddollars,
which I took as compellingevidenceof misspecification.Note thatpredictedprices arebased on the estimatedcoefficients
from the demandmodel. Hence, in principle,it should be possible to determinestandarderrorsfor the predictedprices
shown in Table 5 (indeed, for all predictedoutcomes in Table 5). But as the predictedprices are solved via numerical
optimization,computingstandarderrorsfor predictedprices is a nontrivialcomputationaltask.
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LESLIE / 539
A surprisingresult,however,is thattotal revenueis higher,albeitby only .6%.How is it that
uniformpricingleads to higherrevenuethandiscriminatorypricing?The answeris relatedto the
booth ticket category.In this experimentthere is no booth ticket category.When thereis one, it
could be thata good numberof consumerssubstituteaway frombuying a full-priceticket toward
a booth discount ticket. This substitutioneffect may be harmfulto the firm'sprofit,especially if
the booth discountis restrictedto equal 50%.The next threeexperimentsexaminethis issue more
closely.
O No-booth-A. Inthisexperimentthe firmuses the empiricalprices,with theonly modification
that no booth tickets are sold. Note thatthe prices in Table5 for this experimentare the same as
for Base-A. Comparingthe resultswith Base-A, revenuerises by 7.3%, and attendancedecreases
by 3.7%. The differencein utility is negligible. This providessome confirmationof the argument
that selling tickets at the booth is harmfuldue to the negative effect it has on the demand for
full-pricetickets.
No-booth-B. As with the experimentNo-Booth-A, there are no tickets sold via a discount
o
booth. In this case, however,the firmreoptimizesthe prices of the remainingcategoriesbut must
applythe samemenuin everyperformance.The useful comparisonis with Base-B. Again, revenue
rises, this time by 5.7%. Unlike the previous experiment,attendancerises by just over 1%. In
effect, the absence of the booth causes the firmto lower the price on the expensive tickets, which
has a positive effect on attendance.On the basis of these two experiments,we may conclude that
the 50% discount booth tickets are more damaging than beneficial to firm profits. Comparing
these results with the uniformpricingexperimentreveals thatrevenueis 4.1% higherunderprice
discrimination(with no booth ticket sales) than with a single price policy. A question remains
as to the possible gain of changing the booth discount from 50%, which is addressedin the next
experiment.
Booth-not-50%. In this experimentthe firm optimallychooses all ticketprices, including
o
the booth ticketprice. In particular,the firmis not restrictedto selling booth ticketsat 50% off the
high-qualityprice. As with the above experiments,I maintainthe restrictionthat the firm must
offer the same menu of prices for all performances.As indicated in Table 5, the firm chooses
a higher price for booth tickets. The booth price is now approximately70% of the high-quality
ticketprice. Revenueis now 7%higherthanin Base-B, and attendanceis 1.6%lower.In addition,
revenue is also now higher than under uniform pricing, by approximately5%. Therefore, in
principlethe booth is an optimalmechanismfor selling tickets.A 50%discountappearsto be too
high from the firm'spoint of view. For consumersthereis also a benefit fromraising the price of
booth tickets. Since the firmnow lowers the high-qualityprice, the net effect on consumersis not
detrimental.Indeed, total utility rises by less than .5%. One may note, however,that the change
amountsto a transferfrom less wealthypeople to wealthy people.
o
Nonsticky. A curious featureof behaviorin the Broadwaytheaterindustryis the presence
of stickyprices-firms do not changetheirpricingpolicy over time, despitefluctuatingdemand.49
While understandingthis phenomenonis a researchagendain itself, it is interestingto see how
much better off the firm would be in this model if prices could be costlessly adjustedfor each
performance.In this experimentI returnto the restrictionthat booth tickets are sold at 50% off
the price of a high-qualityticket,but I allow the firmto optimallychoose the remainingprices on
a performance-by-performance
basis. The appropriatebenchmarkis Base-B. The prices shown
in Table5 are the averageprices for each of the ticket categories.Surprisingly,revenueincreases
by only 1.6%,as shown in the table. The increasein revenuefromimplementingnonstickyprices
is less than the gain from alteringthe booth discount. However,there are reasons to expect that
this estimate may either under- or overstatethe true impact of nonstickyprices on revenue. On
the one hand, if the model allowed for intertemporalsubstitutionby consumers,the revenuegain
49As an aside, morethan one Broadwayproducerclaimed to me thatprices are not loweredonce demandfalls for
a show because it would send a signal that it is not a good show.
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OFECONOMICS
540 / THERANDJOURNAL
from nonstickypricing would be even lower. On the otherhand, if relativedemandfor different
ticket alternativesis changing over time, the potentialincrease in revenue from nonstickyprices
would be greaterthan 1.6%.50Note also thatif there are costs to reoptimizingprices over time,
this would reduce the gain to the firm'sprofitfrom implementingnonstickyprices.
From these simulations,it appearsthat the firm stands to gain more by carefully setting a
time-invariantprice menuthanby reoptimizingpricesovertime. The improvementfromnonsticky
pricing may be small because the price menu is "robust"to demandfluctuations.In otherwords,
as demandfluctuates,consumerssubstitutearoundthe price menuinsteadof going to the outside
alternative.For example, when demandshifts down, people shift frombuying high-quality,fullprice tickets to low-quality,full-pricetickets or booth tickets. Of course, costless reoptimization
of prices in the face of fluctuatingdemandshould always benefit a firm. But the benefit may be
smaller for a firm thatoffers a menu of prices than for a firmwith a single price in each period.
6. Conclusion
*
The data in this study highlight the lengths that a firm can go to in orderto sell its product
at different prices to different people. By incorporatingseveral kinds of price discrimination
into a single framework,designed to representthe example of Broadwaytheaterticket sales, the
theoreticalmodel formalizeshow the firm in question is able to sustain such an arrayof prices.
The main results stem from experimentswith alternativepricing policies. I find that uniform
pricing,relativeto the existing price-discriminationpolicy, implies lower overall attendancesfor
the play withoutsignificantlyalteringthe total consumersurplus.This suggests thatthe apparent
lack of concernby antitrustenforcementagencies for price discriminationin final goods may be
well founded.
The estimateddemandsystem allows for the calculationof priceelasticitiesforeach category
of ticket sales. The presence of capacity constraintscauses some consumersto buy tickets that
would nothavebeen theirfirstchoice in the absenceof capacityconstraints,andit leadsto theresult
that raising the price of one ticket category may cause fewer tickets to be sold in another.Such
behavioris apparentfromthe impliedcross-pricedemandelasticities, some of which arenegative
for this reason.It seems likely thatsimilarsubstitutionpatternsmay arisein otherindustries(such
as airlines and hotels) and would be an importantfactorto be taken into accountby the firms in
these cases.
The common practice of Broadwayproducershas been to sell tickets throughthe TKTS
booth at a 50% discount.The counterfactualexperimentsindicate that the discountbooth ticket
category draws some consumers away from buying full-price tickets for the show. With a 50%
booth discount, it appearsthat so many consumerssubstituteaway from full-price sales that it
would be more profitableto not offer any booth tickets at all. However, if the booth discount
were reducedto 30%,thereis less substitutionaway from full-pricesales, and the firmcan profit
from selling tickets at the booth. As a measureof the gain to the firm fromprice discrimination,
based on a 30%booth discount,revenueis approximately5%higherthanunderoptimaluniform
pricing.Alternatively,the firmmight offer a 50%booth discountbut limit the availabilityof these
tickets to fewer performances,refrainingfrom offering booth tickets in relatively high-demand
performanceseven thoughthe show is not selling out.
The second-degreeprice-discriminationcomponentof ticketsales for SevenGuitarsinvolves
only three seat-qualitydivisions.51An obvious question is, why not use more quality divisions
thanthis?Thereexists significantqualityheterogeneitywithinthe high-qualityregion.An answer
to this puzzle may be relatedto the use of otherdiscountticket categories.52Due to the coupons,
thereare many moreprices paid by consumersthanthereare seat-qualitydivisions in the theater.
50The model assumesthat aggregatedemandfluctuatesfor the show. It is conceivablethatthe relativedemandfor
differentqualitiesof seating is also changingover time.
51In fact, for the
majorityof performancesonly two seat-qualityregions were used for ticket sales.
52Another
possible explanationmay be the presence of a direct cost for adding furtherquality divisions. Wilson
(1993) shows thatthe marginalincreasein profitfrom addingmorequalitiesto the price menuis decreasingin the number
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LESLIE
/
541
It is conceivablethatthe use of couponseliminatesanygainsfromaddingfurtherseat-quality
divisions.Itwouldbe interesting
toinvestigatethecircumstances
underwhicha firmwouldprefer
to extendonetypeof pricediscrimination
ratherthananother.
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