Movie Industry Demand and Theater Availability

Abstract

Consumers can only choose to see a movie if it is available in theaters. Explicitly taking into account movie theater availability, we estimate a structural model of movie demand with the use of U.S. movie data from 1995 to 2017. Estimation results indicate that the impact of theater availability on movie demand is both statistically and economically significant. We also find that movie budget predictions based on the model that incorporates theater availability is more consistent with the data, while the model that ignores theater availability on average over-predict production budgets.

Appendices

Appendix 1

In this appendix, we provide the necessary steps in deriving Eqs. (4) and (5) in the main text. Let $v_{jt}=\theta _j-\lambda (t-r_j)+\xi _{jt}+\delta \ln (m_{kt})$, where $m_{jt}$ is the proxy for theater availability, and $\theta _{j}$ is movie j’s perceived quality specified in Eq. (2) of the main text. Consumer i’s taste shock in week t, $\varsigma _{it}+(1-\sigma )\varepsilon _{ijt}$, follows an extreme value distribution. In addition, the outside option is always considered by consumers, so $v_{0t}=-\tau _t$. Using the standard nested logit specification (McFadden 1978), the probability of consumers choosing movie j in week t is:

$$\begin{aligned} s_{jt}=\frac{\exp (v_{jt}/(1-\sigma )) \cdot [\sum _{k\in J_t}\cdot \exp (v_{kt}/(1-\sigma ))]^{-\sigma }}{[\exp (-\tau _t/(1-\sigma ))]^{(1-\sigma )}+[\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]^{(1-\sigma )}}. \end{aligned}$$

In this specification, the outside option is in a different nest and can have a different substitutability from all the movies. The fraction of population that would go to theaters in given week is:

$$\begin{aligned} 1-s_{0t}&=\sum _{k\in J_t}s_{kt} = \frac{\sum _{k\in J_t}\exp (v_{kt}/(1-\sigma )) \cdot [\sum _{k\in J_t}\cdot \exp (v_{kt}/(1-\sigma ))]^{-\sigma }}{[\exp (-\tau _t/(1-\sigma ))]^{(1-\sigma )}+[\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]^{(1-\sigma )}}\\&=\frac{[\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]^{(1-\sigma )}}{[\exp (-\tau _t/(1-\sigma ))]^{(1-\sigma )}+[\sum _{k\in J_t}\exp (v_{kt}/(1-\sigma ))]^{(1-\sigma )}}. \end{aligned}$$

Correspondingly, the proportion of population choosing the outside option is:

$$\begin{aligned} s_{0t}&=\frac{[\exp (-\tau _t/(1-\sigma ))]^{(1-\sigma )}}{[\exp (-\tau _t/(1-\sigma ))]^{(1-\sigma )}+[\sum _{k\in J_t}\exp (v_{kt}/(1-\sigma ))]^{(1-\sigma )}}\\&=\frac{[\exp (\tau _t/(1-\sigma ))]^{(\sigma -1)}\cdot [\sum _{k\in J_t}\exp (v_{kt}/(1-\sigma ))]^{\sigma }}{[\exp (\tau _t/(1-\sigma ))]^{(\sigma -1)}\cdot [\sum _{k\in J_t}\exp (v_{kt}/(1-\sigma ))]^{\sigma }+\sum _{k\in J_t}\exp (v_{kt}/(1-\sigma ))}\\&=\frac{[\exp (\tau _t/(1-\sigma ))]^{\sigma }[\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]^{\sigma }}{[\exp (\tau _t/(1-\sigma ))]^{\sigma }[\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]^{\sigma }+\exp (\tau _t/(1-\sigma ))\cdot [\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]}\\&=\frac{D_t^{\sigma }}{D_t^{\sigma }+D_t}, \end{aligned}$$

where

$$\begin{aligned} D_t=\exp (\tau _t/(1-\sigma ))\cdot \sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))=\sum _{k\in J_t}\exp ((v_{kt}+\tau _t)/(1-\sigma )). \end{aligned}$$

The within-industry market share of movie j in week t is

$$\begin{aligned} \frac{s_{jt}}{1-s_{0t}}&=\frac{\exp (v_{jt}/(1-\sigma )) \cdot [\sum _{k\in J_t}\cdot \exp (v_{kt}/(1-\sigma ))]^{-\sigma }}{[\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))]^{(1-\sigma )}}\\&=\frac{\exp (v_{jt}/(1-\sigma )) }{\sum _{k\in J_t} \exp (v_{kt}/(1-\sigma ))}\\&=\frac{\exp (\tau _t/(1-\sigma ))\exp (v_{jt}/(1-\sigma ))}{\exp (\tau _t/(1-\sigma ))\sum _{k\in J_t} \exp ((v_{kt})/(1-\sigma ))}\\&=\frac{\exp ((v_{jt}+\tau _t)/(1-\sigma ))}{D_t}. \end{aligned}$$

Furthermore,

$$\begin{aligned} \frac{s_{jt}}{s_{0t}}&=\frac{\exp (v_{jt}/(1-\sigma )) \cdot [\sum _{k\in J_t}\cdot \exp (v_{kt}/(1-\sigma ))]^{-\sigma }}{[\exp (-\tau _t/(1-\sigma ))]^{(1-\sigma )}}\\&= \frac{\exp (v_{jt}/(1-\sigma )) \cdot \exp (\tau _t/(1-\sigma )) }{[\exp (\tau _t/(1-\sigma ))]^{\sigma }[\sum _{k\in J_t}\cdot \exp (v_{kt}/(1-\sigma ))]^{\sigma }}\\&=\frac{\exp ((v_{jt}+\tau _t)/(1-\sigma ))}{D_t^{\sigma }}=\frac{\exp ((v_{jt}+\tau _t)/(1-\sigma ))}{(\exp ((v_{jt}+\tau _t)/(1-\sigma )))^{\sigma }}\\&\quad \left( \frac{\exp ((v_{jt}+\tau _t)/(1-\sigma ))}{D_t}\right) ^{\sigma }\\&=(\exp ((v_{jt}+\tau _t)/(1-\sigma )))^{1-\sigma }\cdot \left( \frac{s_{jt}}{1-s_{0t}}\right) ^{\sigma }. \end{aligned}$$

Taking a log-transformation, we have

$$\begin{aligned} \ln (s_{jt})-\ln (s_{0t})&= (1-\sigma )\ln \left( \exp ((v_{jt}+\tau _t)/(1-\sigma )))\right) \cdot \sigma \ln \left( \frac{s_{jt}}{1-s_{0t}}\right) \\&=\delta \ln (m_{jt})+\theta _j-\lambda (t-r_j)+\tau _t+\sigma \ln \left( \frac{s_{jt}}{1-s_{0t}}\right) +\xi _{jt}\\&=\alpha +\beta \ln (B_j) + \mu _g + \psi _y -\lambda (t-r_j)+\tau _t+\delta \ln (m_{jt})\\&+\sigma \ln \left( \frac{s_{jt}}{1-s_{0t}}\right) +\epsilon _j +\xi _{jt}. \end{aligned}$$

Also from the derivation of $s_{0t}$, we have $1-s_{0t}=\frac{D_t}{D_t^{\sigma }+D_t}$. It follows that

$$\begin{aligned} s_{jt}&=\frac{s_{jt}}{1-s_{0t}}\cdot (1-s_{0t})=\frac{\exp ((v_{jt}+\tau _t)/(1-\sigma ))}{D_t}\cdot \frac{D_t}{D_t^{\sigma }+D_t}\\&=\frac{\exp \left( \frac{\theta _j-\lambda (t-r_j)+\tau _t+\delta \cdot \ln (m_{jt})+\xi _{jt}}{1-\sigma }\right) }{D_t^{\sigma }+D_t} \end{aligned}$$

Appendix 2

In this appendix, we show the detailed derivation of Eq. (8) in the main text. Let $\chi _{jt}=\exp \left( \frac{\theta _j-\lambda (t-r_j)+\tau _t+\delta \cdot \ln (m_{jt})+\xi _{jt}}{1-\sigma }\right)$, then from the “Appendix 1”, we know that $s_{jt}=\chi _{jt} / (D_t^{\sigma }+D_t)$ and $\frac{d \chi _{jt}}{d \theta _j}=\frac{\chi _{jt}}{1-\sigma }$. In addition, we know that $D_t=\sum _k \chi _{kt}$, where $\frac{\partial D_t}{\partial \chi _{jt}}=1$.

Therefore,

$$\begin{aligned} \frac{d s_{jt}}{d \theta _j}= & {} \frac{d \chi _{jt}}{d \theta _j}(D_t^{\sigma }+D_t)^{-1} - \chi _{jt}(D_t^{\sigma }+D_t)^{-2} (\sigma D_t^{\sigma -1}+ 1)\frac{d \chi _{jt}}{d \theta _j}\\= & {} \frac{d \chi _{jt}}{d \theta _j}(D_t^{\sigma }+D_t)^{-1} \left[ 1 - \chi _{jt}(D_t^{\sigma }+D_t)^{-1}(\sigma D_t^{\sigma -1}+ 1)\right] \\= & {} \frac{s_{jt}}{1-\sigma } [1-s_{jt}(\sigma D_t^{\sigma -1}+ 1)]. \end{aligned}$$

Notice that $s_{0t}=\frac{D_t^{\sigma }}{D^{\sigma }_t+D_t}$, this means that $D_t=\left( \frac{s_{0t}}{1-s_{0t}}\right) ^{\frac{1}{\sigma -1}}$, so the above becomes

$$\begin{aligned} \frac{d s_{jt}}{d \theta _j} = \frac{s_{jt}}{1-\sigma } \left[ 1-s_{jt}\left( \sigma \frac{s_{0t}}{1-s_{0t}}+ 1\right) \right] . \end{aligned}$$

From Eq. (2) of the main text, we also know that $\frac{\partial \theta _{jt}}{\partial B_{j}}=\beta \frac{1}{B_{j}}$.

Assuming studio profit $(R(B_j)+q_j)\sum _t p_{jt} M_t W(t-r_j) s_{jt} -B_j$, where $R(B_j)$ is the worldwide to domestic box-office ratio and $q_j$ captures the additional revenue source outside of theaters. Then the first order condition is

$$\begin{aligned}&\beta \frac{1}{B_{j}} \cdot (R(B_j)+q_j) \sum _t p_{jt} M_t W(t-r_j) \frac{s_{jt}}{1-\sigma } \left[ 1-s_{jt}\left( \sigma \frac{s_{0t}}{1-s_{0t}}+ 1\right) \right] \\&\quad + \beta _R\frac{1}{B_{j}}\sum _t p_{jt} M_t W(t-r_j) s_{jt} =1. \end{aligned}$$

We rearrange the above to get Eq. (8) in the manuscript.

Appendix 3

The international box-office ratio is a function of production budget $B_j$, a movie’s genre, and release year. We specify the regression model as the follows:

$$\begin{aligned} R_{j}=\alpha _R+\beta _R \ln (B_j) + {\tilde{\mu }}_g + {\tilde{\psi }}_y +{\tilde{\epsilon }}_j \end{aligned}$$

To control for the potential endogeneity, we use the total production budgets of the movie producer in the previous year as a instrument variable. The regression results are presented in the Table 7.

Table 7 International box-office ratio regression result