A Stochastic, Dynamic Model for Optimizing Home Video Release

Franco Berbeglia

∗

, Timothy Derdenger, Sridhar Tayur

Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 15213

September 21, 2021

Abstract

We study how innovations in the home video viewing experience affect the optimal home

video window—the time duration between theater market exit (a random variable) and home

video release—in the movie industry. We specifically analyze the case when there are two home

turn impact home videos of all other technological qualities. We estimate the model parameters

using panel data (for action movies) on the weekly level for home videos and box office. We

conduct a counterfactual analysis in which the home video windows for DVDs and Blu-rays are

jointly optimized. We find that an immediate after-theater release of lower technological quality

home videos (DVDs) combined with a 5-week delay on higher technological quality home videos

(Blu-rays) is optimal. We attribute this result to consumer heterogeneity, where there is greater

substitution between higher technological quality home videos and theaters. Our methods can

also be applied to the “streaming versus theater” trade-off analysis.

1 Introduction

Home video release timing is among the most important decisions for movie distribution. As a

home video experience has improved, from VHS in the 1990s to 4K Ultra HD today. This poses

a question for the movie industry: is it optimal to further decrease the home video window as the

technological quality of home videos increases, or should the window be increased? Furthermore, as

several technologies of home videos are available at the same time, for example DVD and Blu-ray

during the 2010’s, which technological quality home video should be released first, and how far

ahead of the other, in order to maximize studio revenue?

Advertising plays an important role in movie distribution. In general, 80% of the advertising

budget is spent during theatrical release, while the remaining 20% is left for the home video re-

lease. This generates incentives for studios to release their home videos early, as the advertising

expenditure in theaters will have a stronger spillover effect on home video consumption. However,

performance, and the associated run length, are stochastic, adding to the complexity of the setting.

In the past few years, we have seen several efforts from distributors to shrink the home video

window. With the start of the COVID-19 pandemic in December 2019 and the limited market

power of theaters, these efforts expanded. In July 2020 AMC (the world’s largest movie theater

chain) and Universal struck an agreement to release home videos just 17 days after theatrical

release (Watson 2020). Similar agreements were implemented by other studios, with Warner Bros.

announcing it will release all of its 2021 films simultaneously in theaters and on the HBO Max

platform (Schwartzel and Flint 2020). Recently, ViacomCBS launched its own streaming platform,

Paramount+, which will release Paramount and some MGM titles just 45 days after theatrical

release (Kit 2021). These changes were made in the midst of a market dominanted by home

where beliefs about other agents’ actions matter. The benefit of structural models are the powerful

counterfactual analyses they may provide, wherein our case these pertain to the supply side and

the optimization of the home video window (e.g. the revenue impact of delaying or advancing

home video release). Specifically, this paper develops a dynamic structural model for the box office

market and the home video market, consisting of two technological qualities, DVDs and Blu-rays

The model is built around release timing strategies, and it is able to quantify the change in revenue

associated with advancing or delaying home video releases. In order to do this, modeling consumer

forward looking behavior is critical. We model the home video market as an infinite horizon model

for each movie and technological quality, where consumers decide whether to buy the disc, or to

delay their purchase in each period. We model the box office market for each movie as a dynamic

how the industry functions and which decisions studios and theaters make is equally important

to correctly model both the demand and supply sides. For instance, the time a movie remains in

theaters is not a decision the studios make unilaterally, but one that theaters decide, depending

on film performance. Thus, consumers in the box office market do not know the horizon of such

a market, and have to form expectations about its duration in each period. Under our setting,

consumers form dynamic expectations about the time a movie remains in theaters. These expec-

tations depend on movie performance and time since release. Consumers are also forward looking

in the evolution of the movie quality over time, the home video market value, and the home video

window, all of which determine the terminal continuation value of the box office market. A major

technical effort in this paper is the development and estimation of the demand for all the three

This can be seen as a proxy for Full HD and 4K videos today. It is simple to add other home videos as the data

becomes available

the industry practice at the time of data collection. We leave everything constant, while adjusting

for the advertising build-up (as having a shorter home video window increases the advertising

build-up coming from theaters) and the box office revenue signal. We find that it is optimal to

set the DVD and Blu-ray release 2.3 weeks after theatrical exit to achieve a 4.47% increase in

studio revenue

modifying the home video window for DVDs and Blu-rays. This allows the studio to further

exploit market segmentation, as consumers express heterogeneous preferences depending on the

technological quality of the home video player they possess. We find that the optimal strategy

is to release DVDs within a week of theatrical exit, but to delay Blu-rays about 5.15 weeks from

theatrical exit. This strategy increases studio revenue by 5.36% with respect to the data, and again,

shortening the home video window. Thus, the studios can reap all the benefits of the advertising

spillover effect from theaters while posing minimal competition to theaters. The trade-off is more

nuanced for Blu-rays. Shortening their home video window will increase substitution from theaters,

and will further impact the DVD market, as the quality signal coming from box office revenue is

reduced. This result captures consumer heterogeneity in the box office market; Blu-ray player

owners value picture quality above everything else, whereas DVD player owners value timing and

pricing above picture quality. This allows the studios to perform market segmentation strategies

on releases to boost revenue.

The technical contributions of our paper include the modeling and estimation of (a) a finite

horizon, dynamic, discrete choice model, with a horizon length that follows a path dependent

Studio revenue captures all of the home video revenue, and a share of the box office revenue that comes from

imposing a standard theater-distributor contract.

consumer trends are likewise unknown in a path dependent manner, and can have continuation

value after the season.

The remainder of this paper is organized as follows: Section 2 presents the related literature,

Section 3 describes the industry and shows the main attributes of the dataset used, Section 4

presents the structural model including consumer utility specifications and the consumers’ optimal

purchase problem, Section 5 explains the estimation and identification procedure for the model

parameters, Section 6 shows and describes the parameter estimates of the model, and Section 7

presents the counterfactual analysis on the home video windows. We conclude in Section 8.

2 Literature Review

Below we divide the relevant literature into two distinct areas: structural estimation in empirical

operations management and movie related research–including movie release timing.

The use of structural estimation has been growing in empirical operations management research.

Terwiesch et al. (2020) presents a detailed summary of the current papers, presenting the research

demand and cost estimates, whereas Nevo (2000) develops a practitioner’s guide for estimating

random-coefficients logit models. In our case, we base our demand model on the framework of

Gowrisankaran and Rysman (2012), which combines Berry et al. (1995) for modeling the discrete

choice demand and Rust (1987) for the optimal stopping decisions to estimate a dynamic discrete

choice demand model. We also employ the works of Nevo (2000) to obtain the final fixed-effect

estimates, and Derdenger (2014) to connect the box office and home video markets.

Recently, there has been several structural estimation papers in the service management space.

waiting cost and reward for service. After the utility parameters are estimated, the authors conduct

a counterfactual analysis to assess the impact of changes in the service discipline. This presents

several similarities with our paper, where the decision-making for customers is also modeled as

an optimal stopping problem for each market, and, furthermore, we use counterfactual analysis to

assess the impact of changes in the home video window on movie revenue. In a similar fashion, Hu

et al. (2021) study the mechanism of customer retrials in call centers through a dynamic structural

model in order to provide economically feasible solutions to reduce retrials. Guajardo et al. (2016)

use the framework from Nevo (2000) to study the impact of service attributes (warranty length,

after-sales service quality) on consumer demand in the U.S. automobile industry. Allon et al. (2011)

use the framework from Berry et al. (1995) to study the consumer cost of waiting in the fast food

The problem of finding the optimal inter-release times of sequentially released products has been

widely studied in several industries. Moorthy and Png (1992) analyze the optimal timing and

inter-release times. We additionally focus on how the optimal inter-release times change as the

technological quality of home videos increases from DVDs to Blu-rays, while creating heterogeneous

consumer preferences between these products. Another innovation of our model is that consumers

form expectations about the evolution of movie quality during each market, as well as on the

time a movie remains in theaters. We develop a discrete hazard model to asses the probability

distribution of the remaining time in theaters of a movie that is dependent on the time since

seasonality and freshness for DVD; see Mukherjee and Kadiyali (2018). August et al. (2015) and

Calzada and Valletti (2012) analyze, through theoretical models in different settings, the conditions

under which day-and-date, direct-to-video, or delayed home video releases are optimal release timing

strategies for the movie industry.

Finally, there are numerous papers that empirically analyze demand in the movie industry. This

includes Eliashberg and Shugan (1997), Eliashberg et al. (2000), Elberse (2007), Eliashberg et al.

(2014), and Packard et al. (2016), who study the impact of critics’ reviews, star actors, networks

of cast and crew, and opening weekend box office on overall revenue performance. Lehmann and

Weinberg (2000), Elberse and Anand (2007), and Rao et al. (2017) analyze the impact of advertising

on box office revenue. However, these papers only do so in a non-structural reduce form manner.

3 Industry Setting and Data

The motion picture industry comprises three stages: production, distribution and exhibition. The

production stage consists of the development of a motion picture and is a creative process with

important economic implications for the parties involved. The process usually begins with an idea,

her chances of obtaining bank loans and securing favorable distribution and exhibition deals. These

contracts benefit the studios, as they provide a constant inflow of products from successful films.

Many producers face financing issues when they cannot reach a deal with a studio; in such cases

the studio must obtain financing from other sources, which is difficult when no distribution deals

are guaranteed (Vogel 2014).

The distribution stage begins once a movie has completed production, and it includes the

distribution to theaters, home video markets, as well as the marketing activities in each market

where the movie is released. Distributors face a wide range of decisions in this stage, including

distributors—usually referred to as “The Big Six”—include Paramount, Sony, Twentieth Century

Fox, Universal, Walt Disney, and Warner Bros. These studios produce, finance, and distribute their

own movies. At the same time, they also finance and distribute films produced by independent film

makers who are associated with the studio. Ensuring a strong US theatrical box-office gross is very

important for these studios, because it is a performance metric to indicate sales potential in other

distribution channels such as global theatrical, home video, and pay television (Eliashberg et al.

2006). Simultaneously, the growing importance of non-theatrical channels as a source of revenue

is generating incentives for the studios to reduce the time between theatrical and non-theatrical

releases. Figure 1 shows the evolution of the average DVD release windows, (time between theatrical

and DVD releases), for major studios and years. This reduction in non-theatrical windows poses

In the exhibition stage, major studios have limited control. The contractual agreements between

exhibitors and distributors involve a minimum playing time, as well as terms on how the box office

revenue is shared between the parties involved. Beyond these, it is up to each individual theater

to control the total playing time for each movie (beyond that set minimum). Studios generate a

strong buzz prior to and during their theatrical release—combining advertising, word of mouth,

and media attention—which drives demand for the motion picture in other distribution channels

(Eliashberg et al. 2006).

We estimate our model using panel data for box office, DVD, and Blu-ray sales. The data are at

the weekly level for all panels. For each week that a movie is in the box office, our data include

149 movies, across box office, DVD and Blu-ray, with observations between the years 2009 and

images from www.themoviedb.org and applied color theory on them to extract hue information;

the procedure is described in section 5.3.4. We create market shares by dividing the unit sales

by the number of consumers, for the box office, or number of DVD/Blu-ray player owners for the

home videos. We use data from the U.S. Census Bureau to get the number of consumers within

each movie rating per year, and data from the Consumer Electronic Association to get information

on DVD/Blu-ray player ownership. These data are interpolated to the monthly level, assuming

a linear growth rate. We adjust inflation in all revenues to January 2019 dollars using data from

the Bureau of Labor Statistics. To create our final dataset, we remove box office panel weeks

with fewer than 100 theaters and we define the start of the home window as the week in which

a movie is in less than 100 theaters. The final dataset consists of 1, 797 observations across 149

time and across movies. It is important to remark that we don’t explicitly express the dependence

of f

on k since j (movie/medium) already includes that information.

The estimation of the model parameters involves a second stage in which the movie medium

fixed effects are regressed on time fixed movie characteristics, following Nevo (2000). Examples

of movie characteristics involve opening box office revenue, distributor, the home window length,

poster colors and release year.

depends on the technological quality of the home video player owned. This heterogeneity enters

exclusively in the outside option and not in the purchase utility, as we assume that the box office

ticket purchase continuation value is zero for both consumer types

exits the box office market for movie j. If consumer i decides to purchase a box office ticket for

movie j in time period t, she obtains utility given by

) + 

where f

is the flow utility from observable and unobservable box office characteristics, p

price of the box office ticket, and 

is an idiosyncratic shock. The parameter φ is a scaling

parameter that permits the comparison between the home video and box office markets; it serves

as a utility normalization factor, since we forced the error terms for both markets to have the same

variance.

A consumer who does not watch movie j in period t and who owns a home video player of

This can be transformed into a setting in which consumers obtain a flow utility in each period and the discounted

sum of these utilities is equal to the purchase utility in our model

and time since release, number of theaters available, etc. are important to forming expectations

about the last period of the market). Finally, V

(Ω

) represents the value of having a box office

ticket purchase possibility when the state of movie j is Ω

player of technological quality k ∈ {DVD, Blu-ray}. Note that the outside option depends on j

since we are modeling each movie as a monopoly, and the continuation value of delaying the box

office purchase decision is different for each movie.

The box office flow utilities depend on the movie features with the following relation:

+ α

+ ξ

unobservable components of the flow utility that vary both over time, and across movies. Examples

of time-dependent box office movie characteristics include advertising expenditure, average price,

month, age, and performance up to current period. Similarly to the home video market, the

estimation of the model parameters involves a second stage in which fixed effects are regressed

on time invariant movie characteristics. Examples of movie characteristics involve distributor,

production budget, and release year. Note that in (4) and (6) we assume that the consumer

state Ω

. This is analyzed within the consumer’s problem, in section 4.3.2.

We now outline the consumer’s decision process, which incorporates the forward looking behavior

about the evolution of the movie industry states Ω

. We begin by describing the decision process

for the home video market as it is independent of the value functions of the box office market. We

then describe the finite horizon model of the box office market, where the home video value enters

(Ω

) =

Z

max



+ α

+ 

(Ω

(~

)d~

indicates the purchase utility, while the second indicates the expected discounted value of delaying