temporal distance and price elasticity: empirical ... abstract.pdf · airline-based conventional...

Temporal Distance and Price Elasticity: Empirical Investigation

of the Cruise Industry

by

Mingyu JooFisher College of BusinessOhio State University

[email protected]

Kenneth C. WilburRady School of Management

University of California, San [email protected]

Dinesh K. GauriWalton College of BusinessUniversity of [email protected]

September 9, 2017

The authors thank Greg Allenby, Javier Donna, Michaela Draganska, Rex Du, Guofang Huang, Dong SooKim, Gurhan Kok, Hyojin Lee, Selin Malkoc, Kanishka Misra, Vincent Nijs, Hyoduk Shin and numerousseminar audiences for helpful comments.

Temporal Distance and Price Elasticity: Empirical Investigation of the Cruise Industry

Abstract

The conventional view of advance-sales industries is that aggregate demand becomes less price-

elastic as the advance sales period proceeds. This view has substantial empirical support but

is based on analyses of data from a single industry: air travel. We explore how the response of

demand to price changes with temporal distance in a large, proprietary dataset of Florida cruise

prices, bookings, product attributes and advertising. We offer the first evidence that, unlike the

airline-based conventional wisdom, cruise demand becomes more sensitive to price during the

advance sales period. The pattern is large enough to appear in data visualizations and replicates

across models, parameterizations and partitions of the data.

Keywords: Advance Sales, Cruise, Price Elasticity, Temporal Distance

1 Introduction

Many service industries – including airlines, cruises, hotels, rental cars, and entertainment events

– sell in advance to price discriminate among consumers and to hedge against demand uncer-

tainty for their perishable products. When consumers buy in advance, the time between purchase

and consumption is called temporal distance. For example, some consumers buy vacation tickets

three months in advance, while others buy the same tickets just three weeks in advance. Con-

sumers’ self-selected purchase timing is often used for intertemporal price discrimination, as it

often correlates with product valuation and price sensitivity.

It is normally thought that demand becomes less responsive to price as the advance sales

period proceeds. Consumers who purchase early have more options, including the option to

wait longer before purchasing, so they should be less willing to pay a high price. On the

other hand, consumers who arrive late or wait to purchase have fewer options due to binding

capacity constraints, so they should be relatively less sensitive to price. The evidence supporting

this capacity-driven intertemporal segmentation is substantial, but it comes exclusively from

the airline industry (e.g., Li et al. 2014). Although this phenomenon has rarely been studied

empirically outside of the airline industry, many service industries with advance sales and fixed

capacity offer early-bird discounts or weakly increasing prices as the advance sales window

proceeds. This strategy is believed to be profitable because low prices when demand is elastic

ensure capacity utilization, whereas high prices when demand is inelastic enhance unit margins.1

The purpose of the current article is to investigate how the price elasticity of demand changes

with temporal distance in the Florida cruise market. The cruise industry is understudied rel-

ative to its size: at $21.2 billion in US revenues ($39.6 billion worldwide)2, it is substantially

larger than such frequently-researched markets as movies or bath tissue. Cruises offer an inter-

esting setting to study how price elasticity changes with temporal distance because, unlike air

travel, cruise products compete against larger numbers of functional substitutes. For example,

a businesswoman seeking to fly from New York City to Los Angeles is unlikely to consider the

1Bitran and Caldentey (2003) and Kimes (2010) discuss industry practices to balance the tradeoffs betweencapacity utilization and unit margin.

2http://www.cruisemarketwatch.com, https://www.statista.com

1

NYC-Seattle city-pair as a functionally equivalent substitute if the NYC-LA ticket is too ex-

pensive. But, the same person considering a cruise to the Bahamas may be willing to consider

a cruise to Jamaica (among many other possible destinations) as a functional substitute if the

cruise to the Bahamas is too expensive. Such functional substitutability suggests that capacity

constraints of individual cruise products may be weaker drivers of intertemporal segmentation

in cruise demand than in airlines.

We provide the first field evidence that, unlike the standard airline result, demand can be-

come more responsive to price as the advance sales period proceeds. This finding comes from

a proprietary dataset of nearly 400,000 bookings of 805 distinct cruise products departing from

ports in Florida operated by a market-leading cruise line. Each cruise’s price changed weekly

over a 40-week advance sales period. Within each product, price increases as temporal distance

between purchase and consumption falls, a pattern that is consistent with the airline-based

conventional wisdom. Across products, higher-priced cruises tend to be booked farther in ad-

vance, indicating that demand is heterogeneous over the advance sales window. The association

between temporal distance and the response of demand to price is large enough to be seen in

simple data visualizations.

To achieve point identification and estimate causal effects, we specify a flexible discrete

choice model, controlling for main effects of temporal distance, cruise characteristics, seasonality,

advertising by the focal firm and its competitors, and unobserved heterogeneity. Price sensitivity

is modeled as a function of temporal distance as well as product attributes and unobserved

heterogeneity. The model allows consumers to defer purchase and re-enter the market if the

price of their preferred cruise is expected to fall in the following week, in a style similar to Ching

et al. (2009). Crude oil prices are used as instruments to separate demand response to cruise

price from unobserved demand shocks that may correlate with price.

The response of cruise bookings to price increases steadily in the final 20 weeks of ticket

availability when 87% of tickets are booked. The finding replicates across many alternate pa-

rameterizations, model specifications, and partitions of the data. Supplemental data analyses

show no evidence that sellouts or competitive pricing can explain the observed price elasticity

patterns.

2

At first glance, it may appear counterintuitive that prices were highest late in the advance

sales period when demand was most price elastic. However, recall that cruise demand is influ-

enced by temporal distance in two ways: one is a main effect and the other is an interaction with

cruise price. The main effect of temporal distance generally increases as temporal distance falls,

meaning that demand is higher in later reservation weeks than in earlier periods. For most of the

advance sales period, the increase in total demand offsets the increasing response of demand to

price, helping to explain the firm’s observed policy of increasing prices prior to cruise departure.

We report a series of policy experiments based on the empirical model to further explore the

cruise line’s dynamic pricing incentives.

2 Relationship to Prior Literature

Advance selling has attracted significant theoretical attention, especially in the context of travel

and entertainment industries (Fay and Xie 2010). A large literature has documented that

the profitability of advance selling often comes from price discrimination during the advance

sales window. For example, Dana (1998) showed that advance purchase discounts allow firms

to profit from greater sales to low-valuation customers, consistent with second-degree price

discrimination. Gale and Holmes (1993) found that advance pricing can be designed to maximize

capacity-constrained profits by diverting demand from peak to off-peak periods. Theoretical

models typically start with the assumption that demand becomes less responsive to price as the

advance sales window proceeds (as in the airline case) and recommend increasing price paths to

balance demand discovery with capacity utilization.3

The empirical literature investigates airline data and generally supports these assumptions.

For example, Garrow et al. (2006) reported that leisure travelers are about twice as price-sensitive

as business travelers. Li et al. (2014) found that airline demand sensitivity to price increases as

temporal distance falls because of differing times of arrival among leisure and business travelers.

3A handful of papers have characterized optimal dynamic pricing policies without assuming that price sensi-tivity diminishes as the advance sales window proceeds. These papers show that, even without diminishing pricesensitivity, advance selling can contribute to profitability by providing capacity assurance to risk-averse buyers(Png 1989, Xie and Shugan 2001), protecting the seller against low realizations of stochastic demand (Png 1991),and altering consumers’ incentives to time purchases strategically (Cachon and Feldman 2010).

3

There are several behavioral literatures that study how temporal distance affects consumer

preferences; such effects may partially drive the main finding in the current paper. Experimental

studies find that savoring and other forms of anticipation motivate deferral of hedonic goods

consumption (e.g., Loewenstein 1987, Chan and Mukhopadhyay 2010). Research in construal

level theory has shown that consumers planning an event in the distant future focus more on

the core benefits of the event, whereas those considering an event in the near future focus

more on costs and logistics (Liberman and Trope 1998, Trope and Liberman 2000, Bornemann

and Homburg 2011, Lee and Zhao 2014). Work on perceived substitutability suggests that

substitution is often driven by consumers’ commitment to the category (Hamilton et al. 2014),

with highly committed consumers tending to be less responsive to price and more likely to buy

farther in advance. We mention these results in passing because the available data do not offer

any means to distinguish between or among supply-side and demand-side drivers of the main

result.

To the best of our knowledge, the current paper is the first analysis of market data showing

that aggregate demand responsiveness to price can increase as temporal distance between pur-

chase and consumption falls.4 Many firms rely on the airline-based conventional wisdom when

designing pricing policies, so this finding may offer an important consideration for managers in

addition to enriching our academic understanding of advance sales and pricing.

3 Intertemporal Patterns in Demand and Pricing

3.1 Data

Booking data were obtained from a leading cruise company for 2004. The data include all two-

person cabin sales. Each sales record specifies the purchase date, price paid, departure date,

itinerary and cabin type. The empirical analysis focuses on 11 itineraries and three cabin types

4Sweeting (2012) estimated time-varying price response parameters using data from baseball game ticket resalemarkets; this is a substantially different industry structure as baseball tickets were resold by numerous atomisticsellers and ticket prices fell throughout the advance sales window. Sweeting’s time-varying price responsivenessparameter estimates are directionally consistent with the results reported in this paper, but the differences betweenthose estimates were not statistically significant.

4

that collectively accounted for 397,498 bookings, more than 90% of all tickets sold by the firm.5

The product set consists of 27 unique combinations of itinerary and cabin type, as shown

in Table 1. This set was stable with no product introductions or deletions observed during the

sample period. Each itinerary provided a constant duration between four and eight days. On

average, each of the 11 itineraries departed 30 times during 2004. In total, there are 805 different

combinations of cabin types, itineraries, and departure dates.

Table 1: Mean prices and sales by cruise itineraries and cabin types

Interior Ocean View Balcony All Cabins # Tickets Sold

Itinerary 19 $288.32 $339.05 $313.81 28,865

Itinerary 23 $524.91 $652.94 $812.70 $664.75 66,922

Itinerary 28 $494.39 $601.77 $1,018.58 $576.08 16,100

Itinerary 30 $553.57 $678.85 $860.31 $696.45 24,774

Itinerary 31 $557.76 $661.16 $1,057.31 $635.25 35,066

Itinerary 34 $321.72 $368.43 $345.08 29,260

Itinerary 37 $400.70 $472.90 $436.58 32,765

Itinerary 38 $439.80 $520.31 $479.55 21,980

Itinerary 39 $261.27 $308.49 $284.92 36,436

Itinerary 43 $234.57 $277.72 $256.19 65,603

Itinerary 55 $536.59 $638.88 $810.64 $663.31 39,727

All Itineraries $403.79 $479.51 $834.11 $494.96 397,498

# Tickets Sold 180,213 167,036 50,249

Cruise prices and bookings varied widely across product attributes. The average price for

a Balcony cabin was $834, more than double that of an Interior cabin ($404). Interior cabins

were booked the most frequently, and Balcony cabins least often, as some ships did not offer

the highest class of cabin. The 11 different itineraries included various destinations, such as

multiple ports in the Caribbean. The highest-priced itinerary ($696) was more than twice as

expensive as the lowest-priced ($256). Itinerary 23 was the most popular itinerary with about

67,000 tickets sold, whereas itinerary 28 was the least popular with about 16,000 tickets sold.

The booking data were supplemented with advertising expenditures from TNS Media Intel-

ligence. The focal cruise line spent $1.67 million per week on advertising during 2004, about

13% of its national revenue. The firm’s six largest competitors were aggregated into an index

5All 11 itineraries departed from Florida, the firm’s most active cruise departure market. The data do notreport distribution channel, but all tickets were sold directly by the cruise line or by travel agents; no ticketswere sold through consolidators. Price did not vary between distribution channels. Cancellations carried heavyfinancial penalties and were extremely rare.

5

of competitive spending which averaged $3.7 million per week. These top seven cruise brands

collectively accounted for 80% of product category advertising; 91% of their expenditures were

directed to TV and newspapers. Figure 1 shows that advertising spending fluctuated widely

over time but did not display any clear or regular pattern. The focal firm’s advertising was not

significantly correlated (.06) with competitors’ weekly ad spending.

Figure 1: Weekly focal and competitive advertising expenditure

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To understand the nature of advertising content, we conducted an informal but comprehen-

sive review of several hundred television advertisements aired in 2004 for the seven largest cruise

lines, including the focal firm. It showed that advertising copy almost always promoted cruise

consumption and brand attributes. Large cruise lines did not advertise specific itineraries, cabin

types or prices to consumers. Category incidence and brand recognition seemed to be the main

advertising objectives, rather than providing specific cruise attribute or pricing information,

suggesting that competition between cruise lines may have occurred more at the brand level

than at the specific product level.6

6Further supplementing this observation, we reviewed competitors’ cruise offerings in the market. Competi-tors almost never offered directly comparable products (defined as identical combinations of departure port anditinerary attributes).

6

3.2 Demand variation over temporal distance

Cruise tickets were offered for purchase up to one year prior to departure. Cruise prices changed

weekly, so the temporal distance in each purchase occasion is defined as the number of weeks

between the purchase date and the departure date.

Figure 2 shows that total bookings increased throughout the advance sales period until the

final three weeks prior to departure. 99.4% of bookings occurred during the final 40 weeks, with

the final 20 weeks accounting for 87% of all tickets sold. Consequently, we analyze an advance

sales window covering the 40 reservation weeks prior to each cruise’s departure.

Figure 2: Temporal distance and tickets sold

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The aggregate ticket sales distribution in Figure 2 masks substantial product-specific het-

erogeneity. Figure 3 shows how the three highest-selling itineraries’ (23, 43 and 55) shares of

total bookings varied throughout the advance sales period. Each point in the graph is the

sales of an itinerary in that week of temporal distance divided by all tickets sold in that week

(e.g., Itinerary 23 tickets sold 20 weeks prior to departure / all tickets sold 20 weeks prior to

departure). The two higher-priced itineraries (23 and 55) show larger purchase shares early in

the advance purchase window, more than 20 weeks prior to departure. On the other hand, the

lower-priced itinerary (43) sold fewer tickets early in the advance sales window, with increasing

purchase shares in the final 20 weeks. These patterns show that consumers buying high-price

itineraries often book tickets earlier, whereas those booking low-price itineraries tend to buy

later. Similar trends hold for other itineraries.

7

Figure 3: Temporal distance and booking shares of top 3 itineraries

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Itinerary 23 ($665 on average)



A similar tendency is observed in sales of different cabin types. Figure 4 shows that the

purchase share of Balcony cabins ($834 average price) peaked at 30% of cabins sold 39 weeks

before departure, but this share decreases over time until the week of departure. Meanwhile,

purchase shares of the other two cabins (Interior and Ocean View) increase throughout the

advance sales period, again showing that higher-priced tickets are disproportionately likely to

sell early in the advance sales period.

Figure 4: Temporal distance and booking shares of Balcony cabin ($834 on average)

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The descriptive summaries show that consumers who buy early are more likely to buy higher-

priced cruises, whereas late buyers tend to book cheaper cruises, suggesting possible heterogene-

8

ity in demand corresponding to consumers’ time of arrival. Although there is no leisure/business

distinction in the cruise context, the patterns suggest that segmentation based on temporal dis-

tance may be a feasible strategy.

3.3 Descriptive price regression

The goal here is to describe how the cruise line changed prices within products over the advance

sales window, controlling for product characteristics and seasonality in departure weeks. A

cruise product j is uniquely defined by a cabin type and an itinerary (a unique combination of

cruise ship, departure port, list of destination ports and duration). We define time subscripts as

follows: reservation weeks are indexed by t, the temporal distance in weeks between reservation

week and departure week is indexed by r, and the week of cruise departure is therefore t + r.

For example, suppose a consumer buys a ticket in week 5 for cruise product j departing in week

15. This purchase has reservation week t = 5, temporal distance r = 10, and departure week

t+ r = 15.

We regress log price pjtr, observed for cruise product j in reservation week t with a departure

date r weeks later, on unsold capacity and four sets of fixed effects:

pjtr = Wjtrζ + Cjφ1 + Ijφ2 +Kt+rη1 +Krη2 + εjtr (1)

where Wjtr is the number of unsold cabins on cruise j departing in week t + r in reservation

week t; Cj and Ij are vectors of indicator variables describing cruise product j’s cabin type and

itinerary, respectively; Kt+r is a vector of indicator variables for departure weeks to control for

seasonality in cruise departures; Kr is a vector of indicator variables for temporal distance, i.e.,

the number of weeks between booking and departure, to estimate intertemporal price variation

within products; and εjtr is an error term.

Figure 5 displays the point estimates in the vector of temporal distance fixed effects η2 and

the associated 95% confidence intervals. Average price changed little between 40 and 25 weeks

prior to departure, but cruise prices rose by about 20% in the final 25 reservation weeks. The

increasing pattern is consistent with the airline-based conventional wisdom that prices should

9

rise as temporal distance between purchase and consumption falls.

Figure 5: Temporal distance fixed effects in log price regression

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Figure 6 presents point estimates and the 95% confidence intervals of departure week fixed

effects to control for the price variation across cruise departure seasons. The averages conform

to intuition: average prices are greatest for cruises departing in summer (weeks 24-32 cover June

to August), and major holiday weeks.

Figure 6: Departure week fixed effects in log price regression

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Table 2 presents correlational estimates between log price and unsold capacity, itinerary fixed

effects, and cabin type fixed effects. Unsold capacity was not strongly associated with ticket

prices when controlling for other factors.7 The itinerary and cabin type fixed effect estimates

are consistent with the descriptive statistics presented in Table 1. The variables in Equation (1)

7Puller et al. (2015) found a similar result in airline data: scarcity had “virtually no effect on fare dispersion.”

10

Table 2: Log price variation across products and by unsold capacity

Point Est. Std Err.

Intercept 6.066 0.065 **

Unsold tickets 1.02e-5 8.83e-6

Itineraries - Base: Itinerary 19

Itinerary 23 0.680 0.006 **

Itinerary 28 0.464 0.009 **

Itinerary 30 0.727 0.008 **

Itinerary 31 0.619 0.007 **

Itinerary 34 0.079 0.007 **

Itinerary 37 0.335 0.008 **

Itinerary 38 0.336 0.009 **

Itinerary 39 -0.038 0.007 **

Itinerary 43 -0.168 0.007 **

Itinerary 55 0.631 0.008 **

Cabin Types - Base: Balcony

Interior Cabin -0.457 0.005 **

Ocean View Cabin -0.266 0.005 **

Adjusted R-Squared 0.854

** Significant at the 99% confidence level.

explain more than 85% of the variation in log price.

3.4 Expected price changes

Figure 5 shows that average cruise price rose 20% during the final 25 weeks of the advance

sales period. However, prices of individual cruise products did not rise uniformly. Recent

empirical work by Li et al. (2014) found that a small yet significant number of airline consumers

strategically defer their purchases when prices are expected to fall. To control for such behavior

in the demand model, the fitted values from the descriptive log price regression are used to proxy

for consumer expectations about impending price changes. Section 4.1.5 explains in more detail

how this information enters the demand model.

The variable Expected Price Change (EPC) is defined as the expectation in week t of the

change in price level between reservation weeks t and t + 1 for a given cruise product j and

departure week t+ r:

EPCjtr ≡ Et[Pj(t+1)(r−1)]− Pjtr (2)

11

Figure 7: Distributions of expected future price changes over temporal distance

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

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Temporal distance between booking and consumption

where Pjtr is the price level observed in reservation week t of cruise j departing r weeks later,

and Et[Pj(t+1)(r−1)] is the expectation (in reservation week t) of the same cruise price in the

following reservation week t+ 1, as predicted by Equation (1).

Figure 7 presents box-and-whiskers plots of EPCjtr for each temporal distance r. 25% of

the observations of EPCjtr are negative. Although most negative values during the final 27

weeks of temporal distance are near zero, accounting for strategic purchase deferral may be an

important issue.

4 Temporal Distance and Price Elasticity

The primary interest of this paper is to investigate how the price elasticity of demand changes

during the advance sales period. The causal modeling framework developed below is complicated,

in an effort to control for multiple competing explanations. Therefore, we begin with data

visualizations to show that the main result is evident in the raw data, and to show the variation

that identifies the main result.

We calculate approximate derivatives of demand with respect to price for every itinerary/

departure week combination in every pair of consecutive reservation weeks. The approximate

derivatives are given by Hjtr =Tktjtr−Tktj(t−1)(r+1)

Pjtr−Pj(t−1)(r+1)where Tktjtr is the number of tickets sold

and Pjtr is the price level. The denominator is the change in price of the cruise product between

12

consecutive reservation weeks t − 1 and t and the numerator is the corresponding change in

bookings. We also consider approximate elasticities, Elasjtr = Hjtr ∗ Pjtr/Tktjtr.

Figure 8 shows that the mean of Hjtr and Elasjtr among all j and t within each week of

temporal distance r.8 Unlike the airline-based conventional wisdom, the approximate demand

derivatives and elasticities increase (in absolute value) as the advance sales period proceeds.

Figure 8: Approximate derivatives and elasticities by temporal distance

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To provide more context, Figure 9 shows how the mass of positive, negative and zero values of

Hjtr change with temporal distance. As purchase dates approach consumption dates, increasing

numbers of cruise products exhibit negative responses of demand to price.

Although the data illustrate the paper’s main finding, we stress that these results are correla-

tional and do not control for many important factors such as seasonality and price endogeneity.

We proceed by developing an empirical model to isolate the causal effect of temporal distance

on price elasticity.

4.1 Empirical model of cruise demand

A baseline random coefficient logit model is specified to estimate the price elasticity of cruise

demand, controlling for other factors. We supplement the standard model with three components

8In Figures 8 and 9, we retain only the 88.5% of observations with |Pjtr − Pj(t−1)(r+1)| ≥ $5 to avoid givingtoo much importance to minimal price changes. The $5 threshold is approximately 1% of the average sale price.The patterns are robust to large and small changes in the $5 threshold.

13

Figure 9: Frequency of negative, zero, and positive approximate derivatives

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to address important features of the market.

First, we specify a utility function that includes a main effect of temporal distance as well

as interactions between temporal distance and cruise price. Fixed cruise attributes (itinerary

and cabin type) are treated similarly with both main effects and interactions with price. We

also allow demand responsiveness to advertising by the focal cruise line and its competitors to

vary with temporal distance. The model nests the standard assumptions of temporal-distance-

invariant price and advertising sensitivity as a special case.

Second, the model allows for strategic deferral based on expected future price changes. We

adapt the Price Consideration Model (“PCM” hereafter; Ching et al. 2009, 2014a,b) for the

cruise purchase setting.9 A simulated consumer motivated to choose a particular cruise may

9Ching et al. (2009, 2014a,b) developed the PCM to describe consumers’ dynamic response to temporaryprice promotions in package goods categories, so consumer choice is conditional on purchase opportunities (storevisits). In our setting, a consumer first determines the best cruise available, then may decide to defer choice ifthe cruise price is expected to fall in the following week. The key distinction is that, in the consumer packagedgoods setting studied by Ching et al. (2009, 2014a,b), the purchase occasion (i.e., store visit) nests the brandchoice probability. In the current application, the choice probability nests the purchase decision. This is becausea consumer whose best available cruise features a price that is expected to fall faces little downside to purchasedeferral. The consumer expects to save money by waiting one week and preserves flexibility in case the followingweek’s new price set makes some other cruise more attractive.

14

decide to defer purchase if the cruise price is expected to fall in the following reservation week.

The purchase deferral decision is probabilistic, depending on the size of the expected price change

and a behavioral parameter to be estimated. If the consumer defers, she re-enters the market in

the following week, observes the new set of cruise prices and may select from the same choice set

as before: non-purchase, optimal cruise selection and purchase, or optimal cruise selection and

another deferral. There is no limit to the number of times a simulated consumer can re-enter the

market and no commitment implied by a deferral. For example, a simulated consumer may most

prefer cruise j with departure week r in reservation week t, defer purchase because EPCjtr < 0,

observe new prices for all cruises in the following reservation week t+ 1 and then most prefer a

different cruise j′(t + 1)r′ (for j′ 6= j and/or r′ 6= r). Such behavior could be motivated by an

unexpectedly high realization of pj(t+1)(r−1) (the price of the originally preferred cruise observed

in the following reservation week) or by an unexpectedly low realization of pj′(t+1)r′ (the price

of some other cruise observed in the following reservation week). The market size assumption

in each week is adjusted according to the number of simulated consumers who were predicted

to defer action in the previous week.

Third, crude oil price is used as an instrument to separate price response from unobserved

demand shocks. Crude oil is a direct cost shifter for cruise products and therefore offers a

strong exogenous time-varying instrument for price, as discussed further in section 4.1.6 and

Online Appendix A. We view the instrument as conservative, in the sense that if the exogeneity

condition does not hold, the price response estimates are likely to be biased toward zero.10

4.1.1 Individual-level consumer utility and purchase decision

For a simulated consumer i, the perceived quality in reservation week t of cruise product j

departing r weeks later in week t+ r is specified in Equation (3):

µijtr = Cjδi + Ijα+Kt+rω1 +Krω2 +Atr +Btr (3)

10The results of primary interest are robust to estimation without instrumental variables.

15

where Cj , Ij , Kt+r, and Kr are the same vectors of indicator variables of cabin types, itineraries,

departure week, and temporal distance introduced previously. δi, α, ω1, and ω2 are parameter

vectors to be estimated. ω2, in particular, represents the utility value of the degree of flexibility

afforded by the week of temporal distance in which the consumer purchases. Atr denotes the

focal cruise line’s advertising stock and Btr represents its competitors’ cumulative advertising

stock in week t; as defined formally below, adstock variables are specified with r subscripts to

allow consumer response to vary with temporal distance.

Given a set Rt that includes all combinations of cruise products (j) and departure weeks

(t+r for r = 0, 1, 2, . . . , 40) available for booking in week t, and standard Logit model errors, the

probability that a simulated consumer i chooses a cabin in reservation week t on cruise product

j departing r weeks in the future is:

πijtr =exp (βijrpjtr + µijtr)

1 +∑

(j′,r′)∈Rt exp(βij′r′pj′tr′ + µij′tr′

) . (4)

where pjtr is log price for the focal cruise product and βijr represents the price sensitivity of

simulated consumer i.

4.1.2 Time-varying price responsiveness

The price coefficient βijr varies with product attributes and r, the temporal distance in weeks

between ticket purchase and cruise departure, as specified in Equation (5):

βijr = β0i + Cjβ

C + IjβI + f (r|βi) (5)

where β0i is the baseline price responsiveness parameter of simulated consumer i. Cj and Ij are

the vectors of cabin and itinerary indicator variables, respectively, included in Equations (1)

and (3). Parameter vectors βC and βI capture differences in purchasers’ price sensitivity across

cabin types and itineraries to allow price sensitivity to vary across products in different price

tiers; a price change for a high-priced good may affect demand differently from that of a lower-

priced good. Including the product attribute indicator variables in Equation 5 helps to ensure

that the temporal distance interactions with price are not confounded with other observable

16

characteristics that correlate with temporal distance.

The time-varying component of the price coefficient, f (r|βi), represents the part of simulated

consumer i’s price sensitivity that may vary with temporal distance between ticket purchase

and cruise departure. As a preliminary investigation of the data, we estimated a multinomial

logit (MNL) model specifying f (r|βi) as a maximally flexible function of fixed effects for each

individual value of temporal distance r. Figure 10 shows that the estimated price sensitivity

over weeks in the advance sales window appears to be approximately linear within each 10-week

period. Fixed effect estimates were mostly flat between 40 and 20 weeks of temporal distance,

then increasing (in absolute value) at an increasing rate thereafter. The baseline price coefficient

β0 is −0.66, so price responsiveness is negative in all ranges of temporal distance.

Figure 10: Price coefficients with temporal distance fixed effects in MNL model

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

Temporal Distance between Booking and Consumption (Baseline Price Coefficient: -0.66)

Other things equal, we prefer to specify a parsimonious and continuous function f (r| ·).

Therefore, we report a piecewise linear function f (r|βi) with four possible slopes over the advance

sales window that can reflect the pattern observed in Figure 10.

f (r|βi) =

β1i · (40− r) , if r ≥ 30

β1i · 10 + β2i · (30− r) , if 20 ≤ r < 30

β1i · 10 + β2i · 10 + β3i · (20− r) , if 10 ≤ r < 20

β1i · 10 + β2i · 10 + β3i · 10 + β4i · (10− r) , if r < 10

(6)

17

Four parameters estimate marginal effects of temporal distance r on price sensitivity within each

range of ten weeks of temporal distance. β1i is the marginal impact of temporal distance on

price responsiveness when r is greater than or equal to 30 weeks, β2i is the marginal impact for

20 ≤ r ≤ 29, β3i is the marginal impact for 10 ≤ r ≤ 19, and β4i is the marginal impact for r less

than ten weeks. The additive terms in the second, third and fourth ranges of temporal distance

in Equation (6) ensure continuity of f (r|βi) at the break points between ranges of weeks. We

refer to β1i, ..., β4i as “slope parameters.”

The time-varying coefficients are interaction effects between price and temporal distance r.

They are identified by changes in the covariation in market shares and prices corresponding to

temporal distance between purchase and consumption, as depicted in Figures 8 and 9. In other

words, the degree to which individual cruise demand responds negatively to price changes, and

how this negative response changes with temporal distance after controlling for other factors,

will determine the degree to which we find β1i 6= β2i 6= β3i 6= β4i.

4.1.3 Advertising stock variables

Own and competitive advertising stock variables are specified using distributed lags:

Atr =

ρ∑τ=0

γτr ln at−τ (7)

Btr =

ρ∑τ=0

λτr ln bt−τ (8)

where at is the focal cruise line’s advertising expenditure in week t, bt is its rivals’ cumulative

advertising expenditure in week t, and ρ = 2 is the number of advertising lags included in the

model.11 Consistent with the specification of price responsiveness parameters, advertising effects

on consumer demand (γτr and λτr) are modeled as functions of temporal distance to avoid any

possible confounds.12 For example, γτr is the effect of the τ th lag of own advertising on the

11Using simpler multinomial logit models, the number of lags was increased until the last lag became statisticallyindistinguishable from zero, resulting in ρ = 2. The Bayesian Information Criterion is minimized at two lags.

12We do not believe that advertising response estimates are likely to be biased by unobserved product-specificdemand shocks, for two reasons. Advertising variables vary across reservation weeks, but not across productswithin each reservation week. Model observables in the model include departure week fixed effects and temporaldistance fixed effects; together, they perfectly correlate with reservation week fixed effects. Therefore, any potentialcorrelation between week-specific demand shocks and advertising variables exists between observed covariates,

18

advertising stock effect in the purchase of a cruise departing in r weeks. The time-varying

advertising coefficients of the τ th lag are specified as piecewise linear functions of temporal

distance between booking and departure r for consistency with Equation (6):

γτr = γ0τ + f (r|γτ ) (9)

λτr = λ0τ + f (r|λτ ) (10)

where γ0τ and λ0τ capture the baseline effects of own and competitive advertising, respec-

tively, and f (r|γτ ) and f (r|λτ ) are specified as piecewise linear functions with time-varying

slopes of temporal distance.13 The time-varying components control for potential covariation

in advertising response and temporal distance.14 Equations (9) and (10) nest the standard

temporal-distance-invariant advertising response function as a special case.

4.1.4 Taste heterogeneity

The price coefficients and cabin-type preference vector θi = {β0i, β1i, ..., β4i, δi} are specified as

the sum of mean responses θ = {β0, β1, ..., β4, δ} and an unobserved heterogeneity vector Di,

where θi = θ + Di and Di follows a mean-zero multivariate normal distribution denoted F (D)

with a diagonal covariance matrix Σ to be estimated.

The aggregate-level choice probability πjtr of cruise product j in reservation week t departing

in week t + r is obtained by integrating the simulated individual choice probabilities πijtr over

the distribution of unobserved heterogeneity F (D):

πjtr =

∫πijtrdF (D) (11)

rather than between observed adspend and an error. Second, advertising did not promote specific cruise productsand many products were offered in each reservation week, so we think it unlikely that adspend variables correlatewith product-specific unobservables. We lack good advertising instruments to test those beliefs, but estimatingthe model without advertising variables shows that their exclusion does not alter the results of primary interest.

13γτ and λτ are vectors of four time-varying slopes, corresponding to the same ranges (0 − 9 weeks, 10 − 19weeks, 20 − 29 weeks, and 30+ weeks) used in Equation (6). However, unlike βi, γτ and λτ do not vary acrosssimulated individuals.

14The time-varying slopes are identified by the ways in which the covariation between advertising stock variablesand all cruise bookings change with temporal distance r.

19

4.1.5 Purchase deferral and aggregate market share

Consumers may book their preferred product in reservation week t or they may decide to defer

action if they expect a lower price for their most preferred cruise in the following reservation

week t + 1. Deferral means that the consumer takes no action and re-enters the market in the

following reservation week without any commitment. The purchase/deferral decision taken in

reservation week t is binary following Ching et al. (2009, 2014a,b), and conditions on the most

preferred cruise (defined by j and r) at the observed price set in week t. Purchase occurs with

the following probability:

Φjtr =exp (κ0 + κ1EPCjtr])

1 + exp (κ0 + κ1EPCjtr)(12)

where EPCjtr is the expected price change of the most preferred cruise between reservation

weeks t and t+ 1, κ0 captures the baseline likelihood of booking in reservation week t, and κ1 is

a sensitivity parameter estimating the tendency to defer action when price is expected to drop

in the following week.15 The aggregate market share of cruise j departing in r weeks booked

in week t is defined as the proportion of simulated consumers who most prefer that cruise and

decide to purchase:

sjtr = πjtrΦjtr. (13)

The outside share in week t is defined as

s0tr = 1−∑

(j,r)∈Rt

πjtrΦjtr. =

1−∑

(j,r)∈Rt

πjtr

+∑

(j,r)∈Rt

πjtr (1− Φjtr). (14)

The first term(

1−∑

(j,r)∈Rt πjtr

)represents the proportion of consumers who choose to buy

tickets from other cruise lines, and the second term∑

(j,r)∈Rt πjtr (1− Φjtr) represents the total

proportion of consumers who defer action. The mass of simulated consumers described by

15When κ1 > 0, a negative expected price change makes deferral more likely. The model nests a myopic choicemodel when κ1 = 0. We estimated another version of the model allowing for asymmetric effects of positive andnegative EPC in Equation (12), but the results did not change meaningfully.

20

∑(j,r)∈Rt πjtr (1− Φjtr) is added to the market size assumption in the following reservation

week, thereby allowing each simulated consumer to appear in multiple reservation weeks.

This representation of consumer behavior is intended to incorporate dynamic market re-

sponse to expected price changes, in a tractable manner and without requiring consumers to

form expectations about how all prices in the choice set change in subsequent weeks. Deferral

decisions are based on the expectation of the price change of the most preferred cruise, using

information available at the time the deferral decision is made. The expectation is rational ex

ante but may not always turn out to be correct ex post, as unanticipated shocks may lead actual

price changes to differ from expected price changes.

The model allows for a flexible series of actions. A simulated consumer may enter and make

a purchase or non-purchase decision on the spot, then exit. A consumer may enter, defer and

re-enter once, and then make a purchase or non-purchase decision before exiting. This pattern

of deferral and re-entry may repeat any arbitrary number of times, depending on the expected

price change of the consumer’s most preferred cruise in each reservation week. Each re-entry into

the market comes without commitment, in the sense that a consumer’s most preferred product

after a deferral may or may not match the most preferred product before the deferral.

This model is intended as a simplified representation of the consumer’s optimal purchase

timing problem. Ultimately the current paper’s intended contribution is to document a pattern

that exists in the data we analyze, as opposed to focusing on a particular model to point-identify

that pattern. Section 5 below shows that the paper’s main result is robust to a wide variety of

models, parameterizations and partitions of the data.

4.1.6 Instrumental Variables

Instrumental variables are required because of the standard endogeneity concern that price may

be correlated with unobserved demand shocks. The primary instrument we use is the log price

of West Texas Intermediate crude oil, as this is the standard benchmark price for the light, sweet

crude oil used to operate most cruise ships. Crude oil price is exogenously set by an international

commodity market and affects the focal cruise line and its competitors equally. We construct

moments by interacting demand shocks with crude oil price and interactions between crude oil

21

price and observed cruise attributes: departure week indicators, temporal distance indicators,

and cabin type and itinerary fixed effects.

Crude oil is a necessary input for cruise provision and correlates strongly with cruise prices.

Online Appendix A shows that crude oil price meaningfully increases explanatory power in a

linear model of cruise price on exogenous product characteristics and timing variables. Further,

interaction between crude oil price and itinerary dummies increase cruise price much more for

itineraries with longer durations, as one would expect, given that longer cruises travel farther

and burn more crude oil.

The focal cruise line did not assess crude oil surcharges during the sample period. Still,

this instrument comes with an important caveat that it may correlate with the price of cruise

product complements. For example, if crude oil price correlates with aviation fuel prices, airline

ticket prices may be higher in periods where crude oil price is greater, indirectly depressing

cruise demand beyond the main depressive effect of crude oil price on cruise ticket sales (which

operates through a higher cruise price). In such cases, the instrument will likely over-correct

price endogeneity in the demand model by underestimating the direct effect of cruise price on

cruise demand. However, we do not see any reason this bias should apply to the estimates of

focal interest, namely the interactions between cruise prices and ranges of temporal distance.

4.2 Estimation

Demand is estimated by applying the Generalized Method of Moments (GMM) using the stan-

dard procedure described by Greene (2008, p.333). Moments are constructed by interacting

market share residuals with a vector of instruments:16

(Θ, Σ

)= argmin

Θ,Σ

1

2{[S− s (Θ,Σ,p)]′ Z}

(Z′Z

)−1 {Z′ [S− s (Θ,Σ,p)]} (15)

16The objective function in Equation (15) allows for measurement error in observed market shares (Fox et al.2011).

22

where S = {Sjtr} is a vector of the observed market shares of all cruise products j,17 reservation

weeks t, and departure weeks t+ r; s (Θ,Σ,p) = {sjtr (Θ,Σ,p)} is a vector of predicted market

shares based on a set of all demand parameters Θ, variance parameters of random coefficients Σ,

and observed price vector p = {pjtr} for all j, t, and t+r; and Z is a vector of instruments. The

instrument vector contains crude oil price Zt, fixed product attributes (cabin type and itinerary

dummies), departure week dummies, temporal distance dummies, interactions between crude

oil price and fixed product attributes, and interactions between crude oil price and temporal

distance dummies:

Z = {[Cj Ij Kr Kt+r Zt (Cj · Zt) (Ij · Zt) (Kr · Zt)]}. (16)

Interactions are included to meet necessary conditions for identification and to capture exogenous

variation across products and time periods (Rossi 2014).

4.3 Empirical results

The discussion focuses on the main findings – the relationship between price responsiveness

and temporal distance – from the preferred PCM model, considering results from a Random

Coefficient Logit model (RCL) for purposes of comparison. We also briefly discuss other predic-

tors of price responsiveness, advertising elasticities and purchase deferral probability estimates.

Remaining results are covered in the Online Appendices.

4.3.1 Temporal distance and price elasticity

Table 3 presents the slope parameters in the price responsiveness specification. For example,

β4i is the mean effect of one less week of temporal distance on price responsiveness within the

last range of temporal distance (0-9 weeks). So, β4i < 0 indicates that price response becomes

stronger (more negative) as temporal distance between purchase and consumption decreases.

17The observed market share in reservation week t is defined as the number of tickets sold in week t for Floridacruises in the focal cruise line divided by weekly market size in that week. Weekly market size has a fixedcomponent and a variable component. The fixed component is the number of Florida cruise bookings in 2004divided by 52. The variable component is the number of simulated consumers who were predicted to defer theirpurchase in the previous week.

23

Table 3: Price coefficients by temporal distance from consumption

PCM Myopic RCL

Point Est. Std. Err. Point Est. Std. Err.

Baseline β0i -0.552 0.006 ** -0.552 0.006 **

Time-varying Slope β1i (r ≥ 30) 0.008 0.024 0.008 0.024

Time-varying Slope β2i (20 ≤ r < 30) 0.007 0.037 0.007 0.037

Time-varying Slope β3i (10 ≤ r < 20) -0.054 0.027 * -0.054 0.027 *

Time-varying Slope β4i (r < 10) -0.065 0.025 ** -0.065 0.025 **

Pseudo R-Squared 0.446 0.446

* Significant at the 95% confidence level.


As expected, the mean baseline price coefficient is negative and large enough that price

responsiveness is negative for all ranges of temporal distance in both PCM and RCL. The es-

timation results indicate that both β1i and β2i are nearly zero, indicating that price response

varies little with temporal distance for tickets purchased more than 20 weeks prior to consump-

tion. Thus, temporal distance does not seemingly influence price responsiveness among the 13%

of cruise bookings reserved by “early birds” (defined as those purchasing 20 or more weeks in

advance). Price responsiveness amplifies (in absolute value) over time in the final 20 reservation

weeks, when the remaining 87% of cruise bookings take place. Both PCM and RCL show that

β3i and β4i are statistically significant at the 95% confidence level.

The size of this effect is most easily interpreted in terms of price elasticity. We calculate

price elasticities of product-level market shares for all j, t, and r, (i.e.,∂sjtr/sjtr∂Pjtr/Pjtr

), and summarize

how price elasticity point estimates and 95% confidence intervals change with temporal distance

r. PCM estimates in panel A of Figure 11 show that price elasticity increases (in absolute

value) in the final 20 weeks prior to departure, reaching -1.32 in the final reservation week.

Confidence intervals exclude zero in the final 19 weeks prior to departure. Point estimates cross

the threshold of unitary elasticity five weeks prior to departure. The confidence intervals of the

price elasticity estimates reject the null hypothesis that demand was price elastic until 12 weeks

prior to departure, and price elasticity monotonically increases (in absolute value) in the second

half of the reservation period. In the final two weeks, price elasticities significantly exceed (in

absolute value) unitary elasticity at the 95% level, indicating that demand becomes price elastic

24

Figure 11: Time-varying price elasticities

A. PCM Estimates B. RCL Estimates

-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.20

0.20.40.60.81

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0


-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.20

0.20.40.60.81

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0


Figure 12: Temporal distance main effects


0

1

2

3

4

5

6

7

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1


0

1

2

3

4

5

6

7

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1


shortly before the opportunity to sell more tickets expires.18

At first glance, one might think that the observed price elasticity pattern implies that a

decreasing price path would be optimal. However, in order to understand how demand changes

with temporal distance, the main effect of temporal distance on demand must also be considered.

Figure 12 shows that the main effect of temporal distance on demand increases strongly between

the start and end of the advance sales period, reflecting a preference for purchase flexibility.

However, this influence is non-monotonic, as the main effect falls slightly between 10 and 19

weeks prior to departure.

Figure 13 shows how the empirical distribution of perceived quality estimates (and 95%

confidence intervals) change with temporal distance. The mean utilities increase as r falls until

five weeks prior to departure, showing that the change in price responsiveness was not sufficient

18RCL estimates in panel B of Figure 11 show a nearly identical pattern except that the 95% confidence intervalexcludes zero throughout all reservation periods.

25

Figure 13: Distributions of perceived quality estimates

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1


to offset the general increase in the main effect of temporal distance on perceived quality.19 The

increasing tendency of product quality estimates shows that the firm’s observed pricing policy

is directionally consistent with its demand incentives, despite the changes in price elasticity

corresponding to temporal distance.

4.3.2 Price responsiveness across products

Table 4 presents the effects of itineraries and cabin types on price responsiveness. The PCM

and RCL models show nearly identical patterns. Price responsiveness varies significantly across

products, showing that aggregate demand for shorter itineraries is less responsive to price than

demand for longer cruises, with a correlation of -0.5 between price-itinerary interaction coeffi-

cients and itinerary durations. Demand for Interior and Ocean View cabins is less responsive

to price than demand for Balcony cabins. We view these attribute-specific effects as control

variables and include them to prevent confounds from contaminating the effects of temporal

distance on price sensitivity.

19The perceived quality estimates continue to rise in the periods between 10 and 19 weeks of temporal distance,despite the small decline in main effects of temporal distance, because advertising response parameter estimatesare largest during this range (as shown in Tables 9, 10, 13 and 14).

26

Table 4: Price coefficients by itineraries and cabin types

PCM Myopic RCL


Itineraries - Base: Itinerary 19

Itinerary 23 0.025 0.007 ** 0.025 0.006 **

Itinerary 28 0.064 0.007 ** 0.064 0.007 **

Itinerary 30 -0.139 0.007 ** -0.139 0.007 **

Itinerary 31 0.164 0.007 ** 0.164 0.006 **

Itinerary 34 0.019 0.006 ** 0.019 0.006 **

Itinerary 37 0.230 0.006 ** 0.230 0.006 **

Itinerary 38 0.138 0.006 ** 0.138 0.006 **

Itinerary 39 -0.028 0.005 ** -0.028 0.005 **

Itinerary 43 0.252 0.005 ** 0.252 0.005 **

Itinerary 55 0.018 0.007 ** 0.018 0.007 **

Cabin Types - Base: Balcony

Interior Cabin 0.244 0.005 ** 0.244 0.005 **

Ocean View Cabin 0.128 0.004 ** 0.128 0.004 **



4.3.3 Advertising effects

For ease of interpretation, we present advertising elasticities here and full parameter estimates

in the Online Appendices. Advertising elasticities are calculated as follows:

∂∑

(j,r)∈Rt sjtr/∑

(j,r)∈Rt sjtr

∂at−τ/at−τfor τ th lag of own ads in week t, and

∂∑

(j,r)∈Rt sjtr/∑

(j,r)∈Rt sjtr

∂bt−τ/bt−τfor τ th lag of competitive ads in week t.

Figure 14 presents point estimates and 95% confidence intervals based on PCM and RCL

model estimates. All advertising elasticities are estimated to be positive and statistically sig-

nificant. Mean elasticity estimates for own contemporaneous and lagged ads are around 0.2 –

0.3 in both models, slightly higher than the average short-term sales elasticity of advertising of

.12 (Sethuraman et al. 2011). Competitor advertising is positively associated with focal cruise

line bookings, logically consistent with the review of advertising content that showed advertising

primarily promoted category consumption. The category expansion effect is also directionally

27

consistent with estimates from other industries (e.g. Wosinska 2002, Iizuka and Jin 2005, Shapiro

forthcoming).

Figure 14: Own and Competitive Ad Elasticities


.00

.10

.20

.30

.40

.50

.00

.10

.20

.30

.40

.50

4.3.4 Strategic purchase deferral results in PCM

Figure 15 presents a box-and-whiskers plot of estimated purchase probabilities (from Equation

12) across different temporal distances r.20 Median values all exceed 99%, indicating that deferral

is predicted to occur in less than 1% of purchase occasions. Purchase deferral is predicted most

frequently early in the advance sales period, when negative expected price changes are most

frequent (as was shown in Figure 7).

4.4 Policy experiments: alternate pricing slopes

Interesting questions to consider include, was the firm’s observed price slope across weeks of

temporal distance near optimal? How much would optimal itinerary-specific dynamic price

slopes vary across cruise itineraries?

We undertake these questions with substantial caution, as important caveats apply to the

results. The data do not provide variation in the firm’s observed pricing policy, so there are no

data to identify consumer or competitor reactions to changes in firm pricing policy. Everything

20κ0 = 4.90 and κ1 = .001. κ0 is statistically significant at the 99% confidence level, but κ1 is not significant.

28

Figure 15: Consideration probability estimates across temporal distances

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 7 6 5 4 3

0.99

250.

9926

0.99

270.

9928


that follows must be interpreted as the model’s prediction of an immediate market response in a

change in pricing policy, prior to considering any response in competitor or consumer behavior.

In other words, we undertake a partial-equilibrium analysis with the knowledge that the Lucas

Critique applies to the results.

With these caveats firmly in mind, we are interested in using the empirical model to predict

how cruise itinerary revenues would respond to local changes in the slope with which cruise

itinerary price changes with temporal distance. We do three things to try to limit concerns

about external validity. First, we constrain policy experiments to vary the pricing policy for

only one cruise itinerary at a time. Second, we impose a constraint on all policy experiments

that the average cruise price must remain constant; the only thing that varies is the slope,

i.e. the rate at which prices change with temporal distance. Third, we only consider relatively

small changes in price slope. Overall, cruise prices rose at a weekly rate of 0.6% in the final

25 reservation weeks. We investigated cruise-specific revenue changes accruing to increments of

0.1% in price slope, for all increments between -1% and 2%. We focused on the data periods

of 3 ≤ r ≤ 25, where both price elasticity and total demand steadily increased, in order to

investigate the tradeoffs in revenues. In applying these constraints, we hope to consider fairly

subtle policy experiments, rather than stretching the model far beyond the bounds of the data

used to estimate it.

29

The results indicate that, for six of 11 cruise itineraries, the weekly 0.6% increase in cruise

price is approximately optimal. For one itinerary, the model predicts that a weekly price slope

of -1% would increase revenues. For four cruise itineraries, the model predicts that a price slope

of 2% would increase revenue.

Figure 16 illustrates the predicted changes in revenues for four of the 11 itineraries. The

four itineraries depicted were selected simply to facilitate understanding of the general results.

Itineraries 23 and 55 show internal revenue maximums that are approximately equal to the

observed average weekly change in price (0.6%). Itinerary 33 is predicted to increase revenue

at a higher price slope, whereas itinerary 43 is predicted to generate more revenue with a

downward-sloping pricing policy.

Figure 16: Predicted changes in revenues by price slope

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

3%

-1.0

%-0

.9%

-0.8

%-0

.7%

-0.6

%-0

.5%

-0.4

%-0

.3%

-0.2

%-0

.1%

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

1.0%

1.1%

1.2%

1.3%

1.4%

1.5%

1.6%

1.7%

1.8%

1.9%

2.0%

Rev

enue

cha

nges

Weekly price changes over the reservation window

Itinerary 23

Itinerary 30

Itinerary 43

Itinerary 55

Figure 17 illustrates why the optimal slopes varied across itineraries. For all cruise itineraries,

a larger price slope means lower projected bookings, as it implies higher prices in the periods

when demand is greatest. However, the degree to which sales are predicted to fall varies across

itineraries. The itineraries where faster price slopes are predicted to increase revenues are those

for which sales are predicted to fall the least. Although only four itineraries are pictured, this

explanation applies to all eleven itineraries studied.

We do not want to push these policy experiments too far. Still, we find it interesting that

without imposing any assumptions about optimal pricing behavior in estimation, the model

30

Figure 17: Predicted changes in bookings by price slope

-15%

-10%

-5%

0%

5%

10%

15%

20%

-1.0

%-0

.9%

-0.8

%-0

.7%

-0.6

%-0

.5%

-0.4

%-0

.3%

-0.2

%-0

.1%

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

1.0%

1.1%

1.2%

1.3%

1.4%

1.5%

1.6%

1.7%

1.8%

1.9%

2.0%

Tick

et sa

les

chan

ges

Weekly price changes over the reservation window

Itinerary 23

Itinerary 30

Itinerary 43

Itinerary 55

predicted that the observed price slope is approximately optimal for most itineraries. In four of

the five remaining cases for which the model predicts revenue could be improved, the optimal

price slope is predicted to be larger than what was observed.

These results are driven primarily by the offsetting effects of temporal distance on perceived

quality. Consumers value flexibility, on average, and therefore most consumers prefer to purchase

late in the reservation window. At the same time, conditional on the firm’s observed policy of

increasing prices, demand is most responsive to price late in the advance sales period. The

demand intercept effect dominates the price elasticity effect for most cruises, leading the model

to predict that the observed rate of change was either near optimal or too slow for most cruise

itineraries.

5 Robustness Checks

This section presents evidence that the results of primary interest are robust to alternate mod-

eling assumptions, parameterizations, partitions of the data and alternate explanations.

31

5.1 Alternate model specifications and parameterizations

An alternate model might reasonably posit that consumers do not substitute across ranges of

temporal distance. For example, consumers may focus in on particular departure weeks in such

a way that the PCM and RCL models overstate the the true set of choices considered. Therefore,

we estimate the following log-log demand model as a benchmark for comparison:

Tjtr = βjrpjtr + Cjδ + Ijα+Kt+rω1 +Krω2 +Atr +Btr + εjtr, (17)

where Tjtr is log of the number of tickets sold, and the predictors include price, attributes,

timing variables and advertising, as before. The time-varying price coefficient is specified as

βjr = β0+CjβC+Ijβ

I+f (r|β), identical to Equation (5) expect for the unobserved heterogeneity

specification. For comparison, we also estimate the following multinomial logit (MNL) model:

logSjtr − logS0t = βjrpjtr + Cjδ + Ijα+Kt+rω1 +Krω2 +Atr +Btr + εjtr, (18)

where Sjtr is the observed market share for a combination of j and r booked in week t, and S0t

is the outside share.

We further investigate the following questions: i) whether the piecewise specification with 10

week brackets affects the main empirical results, ii) whether inclusion of the final three reserva-

tion weeks (which featured a sharp drop in sales) affects the results, and iii) whether exclusion

of potentially endogenous advertising variables affects the results. Therefore, we estimate both

Equations (17) and (18) with the following specifications and data samples:

• M1. f (r|β) including 40 week-specific temporal distance indicators, estimated using the

full sample

• M2. f (r|β) including 5-week ranges of temporal distance, estimated using the full sample

• M3. f (r|β) including 10-week ranges of temporal distance, estimated using the full sample

• M4. f (r|β) including 40 week-specific temporal distance indicators, dropping data from

the final three weeks of temporal distance

32

• M5. f (r|β) including 5-week ranges of temporal distance, dropping data from the final

three weeks of temporal distance


three weeks of temporal distance

• M7. f (r|β) including 40 week-specific temporal distance indicators, estimated using the

full sample, without advertising covariates

• M8. f (r|β) including 5-week ranges of temporal distance, estimated using the full sample,

without advertising covariates

• M9. f (r|β) including 10-week ranges of temporal distance, estimated using the full sam-

ple, without advertising covariates

• M10. f (r|β) including 40 week-specific temporal distance indicators, dropping data from

the final three weeks of temporal distance, without advertising covariates


three weeks of temporal distance, without advertising covariates


three weeks of temporal distance, without advertising covariates

Figure 18: Time-varying price elasticities in log-log models

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1


M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

33

Figure 19: Time-varying price coefficients in multinomial logit models

-1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1


M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

Figure 18 presents point estimates for f (r|β) in all 12 models, showing that price elasticities

are amplifying over time regardless of model specification, parameterization, exclusion of anoma-

lous reservation weeks, or exclusion/inclusion of advertising covariates. Figure 19 presents point

estimates for f (r|β) in multinomial logit models, showing that price coefficients are increasing

(in absolute value) with temporal distance in the second half of the reservation window and

do not change much with temporal distance in the first half. The reason the multinomial logit

models show a kink at about 20 weeks prior to cruise departure, whereas the log-log models do

not, is because of the inflection point in weekly ticket sales observed at r = 20 in Figure 2. This

sales acceleration causes the demand quantity (Tjtr) to increase more sharply over time than

the log odds ratio (logSjtr− logS0t) before the 20-week mark. In the second half of the advance

sales period, Tjtr and logSjtr − logS0t change at nearly identical rates.

5.2 Partitioned regressions

Another potential question is whether price elasticity patterns might be explained by different

types of consumers arriving to purchase in different times of the year. To investigate, we partition

the data by reservation week and estimate log-log demand models within each partition:

Tjtr = θ0t + θ1trpjtr +Krθ2t + Cjθ3t + Ijθ4t + εjtr, (19)

34

where θ0t is a partition-specific intercept and εjtr is a mean-zero error term. θ1tr is a partition-

specific parameter that estimates price elasticity for each value of temporal distance r within

each reservation week partition, and θ2t, θ3t, and θ4t are fixed-effects parameter vectors estimated

separately by partition.

Figure 20: Time-varying price elasticities by reservation weeks

-6

-5

-4

-3

-2

-1

0

1

2

3

4

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1


t=Week 1 Week 2 Week 3 Week 4 Week 5

Week 6 Week 7 Week 8 Week 9 Week 10

Figure 20 presents point estimates of θ1tr for the final 20 weeks of temporal distance, limited

to the first ten partitions for visual interpretability. Of course, the estimates are quite noisy,

since each is based on a 1/52 subset of data. Still, amplifying price elasticities are estimated

within each partition. It appears that cohort differences are not likely to explain the results of

primary interest.

Another reasonable alternate model is that some consumers fix a departure date prior to

shopping for a cruise. This might be the case if some consumers are more schedule-constrained

than others, for example, if they would only cruise during summer or a major holiday week

when schools are out of session. Or, some consumers’ purchases might be driven by concerns

that their desired cruise during a high-demand period may sell out or become significantly more

35

expensive, suggesting differences in risk tolerance between consumers. If so, then the results of

primary interest might be driven by some form of latent consumer heterogeneity.

To investigate, we first categorized departure weeks into three seasons, according to revenue:

Christmas (the highest revenue at $9 million), high-season weeks with revenue of $5-8 million,

and low-season weeks (below $5 million). We estimated a multinomial logit model identical

to M6 in the previous subsection, where f (r|β) interacts with three seasonality dummies. As

presented in Figure 21, the time-varying price coefficients increase over time in the final 20

weeks of temporal distance, suggesting that the main result does not change much between

peak, high-season and low-season periods.21

Figure 21: Time-varying price coefficients across different seasons

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3


Xmas (MNL) Peak (MNL) Off Peak (MNL)

Another way to investigate differences in consumer types is to partition the data by cabin

type or itinerary, since sales patterns differ across weeks of temporal distance by observable

cruise product attributes. To investigate, we estimated a multinomial logit model identical

to M6, where f (r|β) interacts with both itinerary and cabin-type fixed effects. As shown in

Figure 22, although the level of price elasticity varies across product attributes, the relationship

between price elasticity and temporal distance is fairly consistent across cruise itineraries and

cabin types.

21Similar results are observed in log-log demand models (not reported).

36

Figure 22: Time-varying price coefficients across different itineraries and cabins

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3


Baseline Itin 23 Itin 28 Itin 30 Itin 31 Itin 34 Itin 37

Itin 38 Itin 39 Itin 43 Itin 55 Cabin 2 Cabin 3

5.3 Alternate explanation: sellout-driven substitutions

Figures 3 and 4 show that more expensive itineraries and cabins have higher purchase shares

early in the advance sales window, whereas less expensive tickets have higher shares late in

the reservation period. Could the estimated relationship between temporal distance and price

elasticity be driven by sellouts of expensive itineraries early in the advance sales period? If yes,

then the main result may simply be an artifact of consumer switching to lower-priced alternatives

later in the reservation period as more expensive alternatives gradually become less available.

To explore this scenario, Table 5 presents sellout frequencies of the three highest-selling

itineraries. Non-pictured cruise itineraries show similar patterns. Cruises very rarely sell out

before the final two weeks of temporal distance, suggesting that sellout-driven substitution is

unlikely to cause the observed price elasticity pattern (especially in the range of 10-19 weeks of

temporal distance).

Figure 23 summarizes average capacity utilization (i.e., the cumulative number of tickets

sold / the total number of tickets available) of these three itineraries over the advance sales

window. Mean capacity utilization is actually higher in the week of departure for the cheaper

cruise (Itinerary 43) than the more expensive cruises. In addition, capacity utilization of the

more expensive cruises (23 and 55) increases almost linearly over time, whereas that of the

37

Table 5: Frequency of sellouts out of top three itineraries

5 weeks 4 weeks 3 weeks 2 weeks 1 week Total # ofahead ahead ahead ahead ahead cruise products

Itinerary 23 0.00% 0.00% 0.00% 3.47% 17.36% 144($665 on average)Itinerary 43 0.00% 0.00% 0.00% 0.00% 10.20% 98($256 on average)Itinerary 55 0.00% 1.35% 1.35% 1.35% 12.16% 74($663 on average)

cheaper cruise (43) increases sharply at the last minute. This pattern suggests that consumers

buying more expensive options are less constrained by capacity in the final few weeks of temporal

distance.

Figure 23: Mean capacity utilization over time of top three itineraries

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0





A similar tendency is observed across cabin types. Figure 24 shows that mean final-week

capacity utilization is higher for the lowest priced cabin (93% for Interior) than for higher priced

cabins (80% for Ocean View and 86% for Balcony). This again suggests that buyers of high-price

cabins are less constrained by capacity than low-price cabin buyers.

We note that we lack data on cabin quality other than cabin type. It is plausible that cabins

sold earlier in the advance sales period have different unobserved attributes from cabins sold

later, such as proximity to the deck or engine room. Therefore, unobserved differences in cabin

quality may contribute to explaining the paper’s primary result.

To further check whether the main result is driven by substitution among cabin types, we

38

Figure 24: Temporal distance and mean capacity utilization by cabin type

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0


Interior

Oceanview

Balcony

partitioned the data by cabin type and estimated three separate multinomial logit models within

each partition. The models incorporated interactions between price and each possible value of

temporal distance. Figure 25 graphs the interactions, showing that the pattern of increasing

price sensitivity is observed within each partition of the data.

Figure 25: MNL Price coefficients estimated within cabin type partitions

-1.60

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

20 18 16 14 12 10 8 6 4 2


Interior

Oceanview

Balcony

5.4 Alternate explanation: competitor prices (evidence from 2015)

It is possible that competitive cruise prices covaried with the focal firm’s cruise prices. If

competitors’ prices changed with temporal distance, the apparent relationship between temporal

distance and price elasticity may be an artifact of changes in price competition corresponding to

39

Figure 26: Temporal distance and average prices in 2015

600

800

1,000

1,200

1,400

1,600

1,800

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0


Focal Cruise Line

Competitor A

Competitor B

Competitor C

ranges of temporal distance. Unfortunately, despite our best efforts, we were not able to obtain

competitive cruise prices corresponding to the 2004 booking data.22

To investigate correlations among competitors’ prices, we purchased newer data from a cruise

price tracking firm. The data report weekly Florida cruise ticket prices for the focal cruise line

and its three most similar competitors from June–December 2015, by itinerary and cabin type.23

One pattern that may lead to the observed price elasticity finding is convergence of compet-

itive prices late in the advance purchase window. If cruise lines differentiate their prices early

in the advance sales window then gradually converge to a more competitive price range late

in the booking period, aggregate demand may show an increasing sensitivity to price due to

increasing competition among competing cruises. Figure 26 presents average prices of the four

main competitors over the advance sales window. It is clear that the four cruise lines showed no

regular pattern in price competition across the final 40 weeks of temporal distance.

Another possibility is that competitive prices may be falling with temporal distance, whereas

the focal firm’s prices rise with temporal distance. Then it would appear that the outside option

was becoming more attractive, increasing the apparent price sensitivity of aggregate demand

22We were told by pricing managers at the focal cruise line that they did not systematically incorporate com-petitors’ prices in their pricing algorithm in 2004. We were unable to find any source of historical price data.

23Despite our best efforts, we were not able to find any source of corresponding booking data for 2015 pricingdata. Therefore, we must assume that correlations between competitors’ prices and temporal distance were nottoo dissimilar between 2004 and 2015.

40

Figure 27: Intertemporal log price variation across temporal distance

7.4

7.6

7.8

8.0

8.2

8.4

8.6

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2


Focal Cruise Line

Competitor A

Competitor B

Competitor C

late in the advance sales period. To isolate how competitors’ prices vary with temporal distance

from other drivers of price variation, we run a separate log-price regression for each competitor

l2015 (∈ L2015) observed in 2015 on the indicator variables in Equation (1):

pl2015jtr = C l2015j χl20151 + I l2015j χl20152 +K l2015t+r ψ

l20151 +K l2015

r ψl20152 + ξl2015jtr . (20)

Figure 27 presents point estimates of temporal distance fixed effects, ψl20152 , to compare price

trends across competitors. Competitors A and C show increasing tendencies overall, whereas

competitor B shows no monotonic pattern. Therefore, competitors’ pricing patterns do not

suggest that increasing utility of the outside option can explain the observed relationship between

temporal distance and price elasticity.

Although the competitive data did not come from the same data period as our demand

observations in 2004, they showed that the focal cruise line instituted similarly sized price

increases over the final 14 weeks of temporal distance in both 2004 and 2015.

5.5 Alternate explanation: strategic purchase delay

It also seems unlikely that the amplifying price elasticity pattern was driven by strategic purchase

delay, for several reasons. First, mean prices rose 20% during the final 25 weeks of temporal

distance, suggesting that the optimal dynamic strategy in most cases was to buy on the spot.

41

Airfares also tend to rise as temporal distance falls, further disincentivizing purchase delay. The

PCM model did not estimate a high incidence of strategic purchase deferral, even in cases where

cruise price is expected to fall in the subsequent reservation week.

In fact, firm pricing managers explicitly discussed with us the dynamic game between the

firm and its consumers as a strong disincentive to last-minute discounts, even on cruises with

substantial capacity remaining. They said explicitly that they did not want to train their con-

sumers to expect last minute deals. Therefore, we view the estimated relationship as consistent

with a model in which the most knowledgeable, experienced and committed consumers arrive

early and buy early. Less committed cruisers arrive late and pay more (if they purchase at all).

The series of robustness checks indicates that the estimated relationship between price elas-

ticity and temporal distance is remarkably robust to model specifications, parameterizations,

data subsamples and alternate explanations.

6 Discussion and Conclusion

The main result of this paper is that the airline-based conventional wisdom does not always

apply: aggregate demand can become more responsive to price as the advance sales period

proceeds. As far as we know, this is the first field evidence of a negative causal relationship

between price elasticity and temporal distance in any advance sales industry. However, this

absolute increase in price elasticity does not necessarily imply that the optimal price path declines

with temporal distance, as the main effect of temporal distance on demand was estimated to

more than offset the change in price sensitivity during most of the advance sales period. Policy

experiments showed that, subject to some important caveats and constraints, observed price

slopes were approximately optimal for more than the half of the cruise itineraries.

It is important to be aware that the estimated relationship between price elasticity and

temporal distance – indeed the entire dataset and the entire analysis – conditions on the firm’s

observed pricing policy and consumer response to the observed pricing policy. There is no

evidence that either of those two factors varied during the sample, so we cannot say how the

42

findings would differ under an alternate pricing policy.24 Therefore, the available data cannot

disentangle the underlying mechanism, i.e. how much of the aggregate effect occurs “naturally”

among cruise purchasers and how much of it was induced by the firm’s pricing policy. We see

this as a compelling topic for future research.

It would also be interesting to better understand consumer search for cruises at the micro

level. For example, one could investigate consumer-level online search data for cruise attributes

and prices. This might be possible using data from a cruise line that sells online and requires

shoppers to log in prior to searching for fares and attributes, or data from a panel of internet

users’ visits to web sites.

A third interesting topic for related research would be examining dynamic interactions be-

tween firm pricing policy and consumer demand. If the firm moves to monetize its inelastic

consumers early in the reservation window, or if the firm removes the disincentives to buy late

by flattening the price path, how many consumers would change their shopping behavior, and

how quickly? Such dynamic consumer reactions to meaningful changes in firms’ dynamic pricing

strategies have rarely been explored empirically, but in this context the strategic interactions

between sellers and buyers could be fascinating.

We hope this paper will stimulate further empirical study of advance sales industries; such

work would complement the already substantial theoretical literature on the topic. The re-

lationship between temporal distance and consumer preferences is critical to optimal pricing,

so understanding how this relationship may vary across advance sales industries and dynamic

pricing policies is needed to improve allocative efficiency. Further empirical studies in various

industrial and competitive contexts are needed to better understand how much of the existing

knowledge relies on general, as opposed to industry-specific, conduct and behavior. We are

confident that further research will refine our understanding and test theories about how firms

and consumers behave in dynamic pricing games.

24We are not aware of any papers that estimate consumers reactions to changes in firms’ advance sales policies.The nearest paper on the topic is Lazarev (2013), who uses data from monopoly airline markets to estimate astructural model of optimal advance pricing. The model is then used to evaluate a policy experiment in whichconsumers can mitigate the discriminatory effects of intertemporal pricing by offering tickets on a resale market.However, purchasers’ optimal behavior is predicted based on theory-driven assumptions, as it is not identifiablein the available data.

43

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Online Appendix A: Instrumental variables

This appendix provides evidence that crude oil prices predict cruise prices. We first fit a log-price

regression on all exogenous cruise attribute and timing variables,

pjtr = Cjφ1 + Ijφ2 +Kt+rη1 +Krη2 + εjtr, (21)

then estimate a log-price regression on exogenous variables, crude oil price Zt, and interactions

between Zt and exogenous variables as below:

pjtr =Cjφ1 + Ijφ2 +Kt+rη1 +Krη2 (22)

+ Ztν1 + (Cj · Zt) ν2 + (Ij · Zt) ν3 + (Kr · Zt) ν4 + εjtr.

Model fit without the instruments is 0.852. Including the instruments increases the model

fit to 0.861. The F-statistic comparing the two models is 23.99, strongly rejecting the null

hypothesis that the instruments do not predict log price, and much higher than 10, the empirical

threshold to gauge first-stage instrument strength suggested by Stock et al. (2002). Tables 6 and

7 present parameter estimates for crude oil prices (ν1, ..., ν4 in Equation 22). Most interactions

are statistically and economically significant. The interactions between crude oil price and

itinerary dummies are sensible, with a correlation of 0.65 between itinerary duration and effect

of the interaction on cruise price (i.e., ticket prices for longer cruises are more strongly correlated

with crude oil prices, because such cruises travel longer distances and burn more crude oil).

Overall, these empirical results suggest that crude oil prices are strong predictors of cruise

ticket prices.

Online Appendix B: Standard deviation parameter estimates of random coef-

ficients in demand model

Table 8 presents standard deviation parameter estimates of random coefficients in both PCM

and RCL model. Although they are not statistically significantly different from zero, the point

estimates indicate that the heterogeneity distribution is fairly narrow in both models.

47

Table 6: Interaction Parameters with Product Fixed Effects

Point Est. Std. Err.

Zt 0.193 0.099

Itinerary 23 ·Zt 0.323 0.059 **

Itinerary 28 ·Zt 0.398 0.090 **

Itinerary 30 ·Zt 0.343 0.071 **

Itinerary 31 ·Zt 0.638 0.071 **

Itinerary 34 ·Zt 0.252 0.067 **

Itinerary 37 ·Zt 0.143 0.073

Itinerary 38 ·Zt -0.179 0.087 *

Itinerary 39 ·Zt 0.308 0.064 **

Itinerary 43 ·Zt -0.045 0.061

Itinerary 55 ·Zt 0.444 0.067 **

Interior Cabin ·Zt 0.199 0.044 **

OceanView Cabin ·Zt 0.085 0.044



Online Appendix C: More PCM demand parameter estimates

This online appendix reports other PCM parameter estimates that are not included in the main

body. Tables 9 and 10 present own and competitive advertising parameters, respectively. Ad-

vertising effects are directionally increasing as temporal distance decreases, but the interactions

with temporal distance are not statistically significant. Table 11 presents the intercept and fixed

effects of itineraries and cabin types from Equation (3). The baseline utilities of itineraries 23,

28, 30, and 55 are significantly higher than that of itinerary 19, and those of Interior and Ocean

View cabins are lower than that of Balcony cabin after controlling for prices and advertising.

Table 12 presents fixed effect estimates of departure weeks (Kt+r) in Equation (3). They control

for baseline demand over different seasons of departure.t

Online Appendix D: More RCL demand parameter estimates

This online appendix reports RCL estimates for demand parameters that are not reported in

the manuscript. Most of the estimates are substantially identical to the estimates in PCM, so

we omit interpretation for brevity.

48

Table 7: Interaction Parameters with Temporal Distance Fixed Effects


40 Weeks Ahead·Zt 0.111 0.217 20 Weeks Ahead·Zt -1.257 0.144 **


38 Weeks Ahead·Zt -0.660 0.427 18 Weeks Ahead·Zt -1.094 0.123 **




34 Weeks Ahead·Zt -0.290 0.344 14 Weeks Ahead·Zt -0.898 0.114 **


32 Weeks Ahead·Zt -0.520 0.238 * 12 Weeks Ahead·Zt -0.733 0.104 **

31 Weeks Ahead·Zt -0.529 0.212 * 11 Weeks Ahead·Zt -0.632 0.100 **

30 Weeks Ahead·Zt -0.733 0.199 ** 10 Weeks Ahead·Zt -0.496 0.095 **




26 Weeks Ahead·Zt -1.100 0.192 ** 6 Weeks Ahead·Zt -0.195 0.089 *

25 Weeks Ahead·Zt -1.480 0.196 ** 5 Weeks Ahead·Zt -0.112 0.088

24 Weeks Ahead·Zt -1.403 0.189 ** 4 Weeks Ahead·Zt -0.115 0.087

23 Weeks Ahead·Zt -1.144 0.178 ** 3 Weeks Ahead·Zt 0.050 0.087

22 Weeks Ahead·Zt -1.345 0.167 ** 2 Weeks Ahead·Zt 0.351 0.088 **

21 Weeks Ahead·Zt -1.319 0.156 ** 1 Week Ahead·Zt 0.525 0.088 **



Table 8: Standard Deviation Estimates of Random Coefficients

PCM Myopic RCL

Std. Dev. (Std. Err.) Std. Dev. (Std. Err.)

Baseline β0i 0.043 (6.653) 0.043 (6.714)

Time-varying Slope β1i (r ≥ 30) 0.001 (2.349) 0.001 (2.684)

Time-varying Slope β2i (20 ≤ r < 30) 0.001 (0.672) 0.001 (0.947)

Time-varying Slope β3i (10 ≤ r < 20) 0.002 (0.282) 0.001 (0.414)

Time-varying Slope β4i (r < 10) 0.000 (1.348) 0.000 (1.348)

Interior Cabin 0.209 (4.728) 0.209 (5.287)

Ocean View Cabin 0.206 (5.144) 0.206 (6.059)

49

Table 9: Own Ad Coefficients by Temporal Distance from Consumption in PCM


Baseline γ0t -0.030 0.006 **

Time-varying Slope γ1t (r ≥ 30) 0.008 0.030

Time-varying Slope γ2t (20 ≤ r < 30) 0.001 0.038


Time-varying Slope γ4t (r < 10) 0.008 0.024

Baseline γ0,t−1 -0.004 0.006

Time-varying Slope γ1,t−1 (r ≥ 30) -0.007 0.034

Time-varying Slope γ2,t−1 (20 ≤ r < 30) -0.011 0.048

Time-varying Slope γ3,t−1 (10 ≤ r < 20) 0.007 0.043

Time-varying Slope γ4,t−1 (r < 10) 0.002 0.035

Baseline γ0,t−2 -0.042 0.007 **

Time-varying Slope γ1,t−2 (r ≥ 30) 0.006 0.019





Table 10: Competitive Ad Coefficients by Temporal Distance from Consumption in PCM


Baseline λ0t -0.008 0.005

Time-varying Slope λ1t (r ≥ 30) -0.001 0.030

Time-varying Slope λ2t (20 ≤ r < 30) -0.012 0.030

Time-varying Slope λ3t (10 ≤ r < 20) 0.002 0.047

Time-varying Slope λ4t (r < 10) -0.009 0.027

Baseline λ0,t−1 0.043 0.007 **

Time-varying Slope λ1,t−1 (r ≥ 30) 0.000 0.032

Time-varying Slope λ2,t−1 (20 ≤ r < 30) -0.012 0.026

Time-varying Slope λ3,t−1 (10 ≤ r < 20) 0.019 0.041

Time-varying Slope λ4,t−1 (r < 10) -0.001 0.034

Baseline λ0,t−2 -0.026 0.006 **

Time-varying Slope λ1,t−2 (r ≥ 30) -0.001 0.034





50

Table 11: Fixed Effects Estimates for Itineraries and Cabin Types in PCM


Intercept -7.111 0.008 **

Itinerary 23 0.173 0.002 **

Itinerary 28 -0.178 0.002 **

Itinerary 30 1.080 0.001 **

Itinerary 31 -0.681 0.002 **

Itinerary 34 -0.403 0.002 **

Itinerary 37 -0.752 0.002 **

Itinerary 38 -0.359 0.001 **

Itinerary 39 0.057 0.002 **

Itinerary 43 -1.252 0.004 **

Itinerary 55 0.426 0.002 **




51

Table 12: Fixed Effects for Cruise Departure Weeks in PCM


Week 1 0.337 0.039 ** Week 26 0.103 0.037 **

Week 2 0.191 0.056 ** Week 27 0.190 0.043 **

Week 3 0.330 0.045 ** Week 28 0.062 0.039

Week 4 0.343 0.039 ** Week 29 0.167 0.044 **

Week 5 0.442 0.040 ** Week 30 0.002 0.041

Week 6 0.393 0.035 ** Week 31 0.046 0.048

Week 7 0.566 0.030 ** Week 32 -0.117 0.044 **

Week 8 0.455 0.027 ** Week 33 -0.049 0.048

Week 9 0.573 0.029 ** Week 34 -0.312 0.052 **

Week 10 0.361 0.032 ** Week 35 -0.266 0.060 **

Week 11 0.459 0.030 ** Week 36 -0.234 0.056 **

Week 12 0.248 0.030 ** Week 37 -0.289 0.066 **

Week 13 0.442 0.036 ** Week 38 -0.314 0.059 **

Week 14 0.226 0.031 ** Week 39 -0.197 0.072 **

Week 15 0.329 0.037 ** Week 40 -0.218 0.055 **

Week 16 0.240 0.030 ** Week 41 -0.214 0.062 **

Week 17 0.399 0.030 ** Week 42 -0.250 0.075 **

Week 18 0.250 0.030 ** Week 43 -0.164 0.083 *

Week 19 0.398 0.031 ** Week 44 -0.197 0.056 **

Week 20 0.251 0.033 ** Week 45 -0.140 0.065 *

Week 21 0.305 0.036 ** Week 46 -0.157 0.051 **

Week 22 0.163 0.035 ** Week 47 -0.092 0.062

Week 23 0.270 0.040 ** Week 48 -0.285 0.051 **

Week 24 0.081 0.038 * Week 49 -0.028 0.058

Week 25 0.233 0.043 ** Week 50 0.019 0.040



52

Table 13: Own ad coefficients by temporal distance from consumption in RCL


Baseline γ0t -0.030 0.005 **

Time-varying Slope γ1t (r ≥ 30) 0.008 0.030



Time-varying Slope γ4t (r < 10) 0.008 0.024

Baseline γ0,t−1 -0.004 0.006

Time-varying Slope γ1,t−1 (r ≥ 30) -0.007 0.034

Time-varying Slope γ2,t−1 (20 ≤ r < 30) -0.011 0.048



Baseline γ0,t−2 -0.042 0.007 **

Time-varying Slope γ1,t−2 (r ≥ 30) 0.006 0.019





Table 14: Competitive ad coefficients by temporal distance from consumption in RCL


Baseline λ0t -0.008 0.005

Time-varying Slope λ1t (r ≥ 30) -0.001 0.030

Time-varying Slope λ2t (20 ≤ r < 30) -0.012 0.030

Time-varying Slope λ3t (10 ≤ r < 20) 0.002 0.047

Time-varying Slope λ4t (r < 10) -0.009 0.027

Baseline λ0,t−1 0.043 0.007 **

Time-varying Slope λ1,t−1 (r ≥ 30) 0.000 0.031




Baseline λ0,t−2 -0.026 0.006 **

Time-varying Slope λ1,t−2 (r ≥ 30) -0.001 0.034





53

Table 15: Fixed effects estimates for itineraries and Cabin Types in RCL


Intercept -7.111 0.008 **

Itinerary 23 0.173 0.002 **

Itinerary 28 -0.178 0.002 **

Itinerary 30 1.080 0.001 **

Itinerary 31 -0.681 0.002 **

Itinerary 34 -0.403 0.002 **

Itinerary 37 -0.752 0.002 **

Itinerary 38 -0.359 0.001 **

Itinerary 39 0.057 0.002 **

Itinerary 43 -1.252 0.004 **

Itinerary 55 0.426 0.002 **




54

Table 16: Fixed effects for cruise departure weeks in RCL


Week 1 0.337 0.038 ** Week 26 0.103 0.037 **

Week 2 0.191 0.055 ** Week 27 0.190 0.043 **

Week 3 0.330 0.045 ** Week 28 0.062 0.039

Week 4 0.343 0.038 ** Week 29 0.167 0.044 **

Week 5 0.442 0.040 ** Week 30 0.002 0.041

Week 6 0.393 0.035 ** Week 31 0.046 0.048

Week 7 0.566 0.030 ** Week 32 -0.117 0.044 **

Week 8 0.455 0.027 ** Week 33 -0.049 0.048

Week 9 0.573 0.029 ** Week 34 -0.312 0.052 **

Week 10 0.361 0.032 ** Week 35 -0.266 0.059 **

Week 11 0.459 0.030 ** Week 36 -0.234 0.056 **

Week 12 0.248 0.030 ** Week 37 -0.289 0.065 **

Week 13 0.442 0.036 ** Week 38 -0.314 0.058 **

Week 14 0.226 0.031 ** Week 39 -0.197 0.072 **

Week 15 0.329 0.037 ** Week 40 -0.218 0.054 **

Week 16 0.240 0.030 ** Week 41 -0.214 0.061 **

Week 17 0.399 0.030 ** Week 42 -0.250 0.074 **

Week 18 0.250 0.030 ** Week 43 -0.164 0.082 *

Week 19 0.398 0.030 ** Week 44 -0.197 0.055 **

Week 20 0.251 0.033 ** Week 45 -0.140 0.065 *

Week 21 0.305 0.036 ** Week 46 -0.157 0.050 **

Week 22 0.163 0.035 ** Week 47 -0.092 0.061

Week 23 0.270 0.040 ** Week 48 -0.285 0.051 **

Week 24 0.081 0.038 * Week 49 -0.028 0.057

Week 25 0.233 0.042 ** Week 50 0.019 0.040



55

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