Product-Line Decisions and Rapid Turnover in Movie Markets

Abstract

We present empirical evidence on a firm’s decision to eliminate a product from a multi-product line, in a market with rapid product turnover, using weekly micro-level data from a major metropolitan movie market. While film offerings are broadly similar across the market, the timing for removing a specific film varies at the theater level. Theaters are more likely to end a film’s run when the nearest competitor is showing the same film and is owned by the same chain; a film is more likely to survive when the nearest theater is showing the film, but is under different ownership. Theater- and movie-specific factors affect a film’s run length, including the film’s within-theater relative revenues, number of screens, and film rating.

Appendices

Appendix 1

In the accelerated failure time (AFT) model, the survival function for an individual with covariates X is $S\left( {t{|}{\varvec{X}}} \right) = S_{0} \left( {\exp \left\{ { - {{\varvec X}^{\prime}{\varvec \beta} }} \right\}t} \right).$ This survival function is related to the linear regression equation

$${\text{ln}}t = {{\varvec X}^{\prime}{\varvec \beta} } + z.$$

(4)

The error term, z, represents the baseline distribution of ${\text{ln}}t$ when the covariates equal zero.

Following Rodríguez (2010), define $T_{0} = \exp \left\{ z \right\}$ for this baseline case. The probability that the baseline individual survives beyond time t is

$$S_{0} \left( t \right) = \Pr \left\{ {T_{0} > {\text{t}}} \right\} = { }\Pr \{ \exp \left\{ {\text{z}} \right\} > {\text{t}}\} = {\text{Pr}}\{ z > {\text{ln}}t\} .$$

(5)

Exponentiating both sides of EQ-4 implies that T is distributed according to

$$\exp \left\{ {{{\varvec X}^{\prime}{\varvec \beta} } + z} \right\} = \exp \left\{ {{{\varvec X}^{\prime}{\varvec \beta} }} \right\}\exp \left\{ z \right\} = T_{0} {\text{exp}}\left\{ {{{\varvec X}^{\prime}{\varvec \beta} }} \right\}.$$

(6)

Thus, the covariates have this multiplicative effect on the baseline survival time. It follows from EQ-6 that the probability of survival beyond time t, given covariates X, is

$$S\left( {t{|}{\varvec{X}}} \right) = \Pr \{ T > t|{\varvec{X}}\} = \Pr \left\{ {T_{0} \exp \left\{ {{{\varvec X}^{\prime}{\varvec \beta} }} \right\} > t} \right\},$$

(7)

$$S\left( {t{|}{\varvec{X}}} \right) = \Pr \left\{ {T_{0} > \exp \left\{ { - {{\varvec X}^{\prime}{\varvec \beta} }} \right\}t} \right\}.$$

(8)

EQ-8, with EQ-5, implies that $S\left( {t{|}{\varvec{X}}} \right) = S_{0} \left( {\exp \left\{ { - {{\varvec X}^{\prime}{\varvec \beta} }} \right\}t} \right)$, which is the AFT specification.⁵⁹

Appendix 2

As was shown in Appendix 1, the estimated parameters in the AFT model measure the multiplicative effect of covariates on baseline survival time. Exponentiating an estimated coefficient gives a time ratio. This ratio compares the survival time that is associated with a one-unit change in the covariate, to the survival time prior to the change, as follows.⁶⁰

Consider the linear regression equation, EQ-4, in Appendix 1. Exponentiating both sides gives the following survival time, for subject j, with a given set of covariates ${\varvec{X}} = \left( {X_{1} \ldots X_{k} } \right)$:

$$t_{j} = {\text{exp}}\left( {\beta _{1} X_{1} + \cdots + \beta _{k} X_{k} } \right)T_{j} .$$

(9)

In the case of a one-unit increase in $X_{1}$, for example, the new survival time would be

$$t_{j}^{*} = {\text{exp}}\left( {\beta_{1} \left( {X_{1} + 1} \right) + \cdots + \beta_{k} X_{k} } \right)T_{j} .$$

(10)

The time ratio measures $t_{j}^{*}$ relative to $t_{j}$ as follows:

$$\frac{{t_{j}^{*} }}{{t_{j} }} = \left[ {\exp \left( {\beta_{1} \left( {X_{1} + 1} \right) + \cdots + \beta_{k} X_{k} } \right)T_{j} } \right]/[\exp \left( {\beta_{1} X_{1} + \ldots + \beta_{k} X_{k} } \right)T_{j} ] = \exp \left( {\beta_{1} } \right).$$

(11)

Thus exponentiating the coefficient estimate from the AFT regression gives the factor by which survival time increases for a one-unit increase in the covariate. A time ratio that is greater than one implies that the covariate lengthens expected survival time; a time ratio that is less than one indicates that the covariate shortens the survival time, which implies that an earlier exit is more likely.