The discriminatory incentives to bundle in the cable television industry

Abstract

An influential theoretical literature supports a discriminatory explanation for product bundling: it reduces consumer heterogeneity, extracting surplus in a manner similar to second-degree price discrimination. This paper tests this theory and quantifies its importance in the cable television industry. The results provide qualified support for the theory. While bundling of general-interest cable networks is estimated to have no discriminatory effect, bundling an average top-15 special-interest cable network significantly increases the estimated elasticity of cable demand. Calibrating these results to a simple model of bundle demand with normally distributed tastes suggests that such bundling yields a heterogeneity reduction equal to a 4.7% increase in firm profits (and 4.0% reduction in consumers surplus). The results are robust to alternative explanations for bundling.

Appendices

Appendix

Appendix 1: Proofs of propositions

Proof of Proposition 1

Suppose there are n discrete products (components) supplied by a monopolist and consumers differ in their preferences (willingness-to-pay) for each of these products, given by a type vector, v _i  = (v _i1, ..., v _in ). Let each v _ic , c = 1, ..., n, be independent with means μ _c and variances σ _c . Let $x_{in} \equiv \frac{1}{n} \sum_{c=1}^{n} v_{ic}$ be the per-good valuation for consumer i of a bundle of n goods, let μ _n be its mean, and let $\sigma^2_n$ be its variance. Note that μ _n and $\sigma^2_n$ follow the well-known formulas for the mean and variance of an average of (independent) random variables:

$$\mu _{n} = \frac{1}{n}{\sum\limits_{c = 1}^n {\mu _{c} = \overline{{\mu _{c} }} } }$$

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and

$$\sigma ^{2}_{n} = \frac{1}{{n^{2} }}{\sum\limits_{c = 1}^n {\sigma ^{2}_{c} = \frac{1}{n}{\left[ {\frac{1}{n}{\sum\limits_{c = 1}^n {\sigma ^{2}_{c} } }} \right]} = \frac{1}{n}\overline{\sigma } ^{2}_{c} } }$$

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Because the sequences v _ic are uniformly bounded, $\lim_{n \rightarrow \infty} \bar{\mu}_c$ and $\lim_{n \rightarrow \infty} \bar{\sigma}^2_c$ exist. Let $\lim_{n \rightarrow \infty} \bar{\mu}_c = \mu$ and $\lim_{n \rightarrow \infty} \bar{\sigma}^2_c = \sigma^2$. Note this implies lim_{n → ∞} μ _n  = μ and $\lim_{n \rightarrow \infty} \sigma^2_n = \lim_{n \rightarrow \infty} \frac{\sigma^2}{n} = 0$.

Let $q_n(p) \equiv \int_{p}^{\infty} dF(x_{in})$ give the market share of a bundle of size n offered at per-good price p, where F(x _in ) is the CDF of x _in . Note that $q_n(\mu - \epsilon) = \mbox{Prob}(x_{in} > \mu - \epsilon) = \mbox{Prob}(x_{in} - \mu > - \epsilon)$.

Let $\epsilon^{n}(p) \equiv - \frac{\partial q^n(p)}{\partial p} \frac{p}{q_n}$ be the (absolute value of the) elasticity of the per-good demand curve evaluated at per-good price p and let $\tilde{\epsilon}^{n}(\tilde{p}) \equiv -\frac{\partial q^n(\tilde{p})}{\partial p} \frac{\tilde{p}}{q_n(\tilde{p})}$ be the corresponding (aggregate) elasticity of the bundle demand curve evaluate at total price $\tilde{p}$. For a bundle of size n, $\tilde{p} = np$.

By the weak law of large numbers and symmetry, $\mbox{Prob}(x_{in} - \mu < -\epsilon) \leq \frac{1}{2}\frac{\sigma^2}{\epsilon^2 n}$, implying $q_n(\mu - \epsilon) \geq 1 - \frac{1}{2}\frac{\sigma^2}{\epsilon^2 n}$. By a similar argument, $q_n(\mu + \epsilon) \leq \frac{1}{2}\frac{\sigma^2}{\epsilon^2 n}$.

Case I Per-good elasticity. We first prove the proposition for the per-good elasticity, εⁿ.

Let $\omega = \frac{\sigma^2}{\epsilon^2}$ and consider a change in price from μ − ε to μ + ε on the per-good demand for a bundle of size n.

$$\begin{array}{*{20}l} {{\frac{{\partial q_{n} }}{{\partial p}}} \hfill} & {{ \leqslant \frac{1}{2}\frac{\omega }{n} - {\left( {1 - \frac{1}{2}\frac{\omega }{n}} \right)}} \hfill} \\ {{} \hfill} & {{ \leqslant \frac{\omega }{n} - 1} \hfill} \\ \end{array} $$

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Then

$$ \in ^{n} {\left( {\mu - \in } \right)} = - \frac{{\partial q_{n} }}{{\partial p}}\frac{{\mu - \in }}{{q_{n} }} \geqslant - {\left( {\frac{\omega }{n} - 1} \right)}\frac{{\mu - \in }}{{1 - \frac{1}{2}\frac{\omega }{n}}} \geqslant \frac{{n - \omega }}{{n - \frac{\omega }{2}}}{\left( {\mu - \in } \right)}$$

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Differentiating this with respect to the bundle size n yields

$$\label{dpergood} \frac{\partial \epsilon^{n}(\mu - \epsilon)}{\partial n} \geq \frac{\omega}{2 \left[n - \frac{\omega}{2}\right]^2} ~ (\mu - \epsilon) > 0 $$

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Let $\epsilon = \mu - p^*_n$ so that $p^*_n = \mu - \epsilon$.⁴⁶ Then increasing bundle size makes the per-good bundle demand curve more elastic when evaluated at the profit-maximizing price for a bundle of size n.

Case II Aggregate bundle elasticity. When considering the impact of increases in n on the aggregate bundle elasticity, one has to accommodate that a given change in the per-good bundle price has a larger effect on the aggregate price for a larger bundle than for a smaller bundle. While this does not impact the elasticity of the size-n bundle demand curve evaluated at price p _n , it does impact the elasticity of the size-(n + 1) bundle demand curve evaluated at price p _n . In particular,

$$\begin{array}{*{20}l} {{\widetilde{ \in }^{{n + 1}} {\left( {\widetilde{p}_{n} } \right)}} \hfill} & {{ = \in ^{{n + 1}} {\left( {p_{n} } \right)}\frac{n}{{n + 1}}} \hfill} \\ {{} \hfill} & {{ = \in ^{{n + 1}} {\left( {p_{n} } \right)}A{\left( n \right)}} \hfill} \\ {{} \hfill} & {{ \geqslant \frac{{n - \omega A{\left( n \right)}}}{{n - \frac{{\omega A{\left( n \right)}}}{2}}}\widetilde{p}_{n} } \hfill} \\ \end{array} $$

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Under A4, it is easy to show that for the per-good elasticities, $\epsilon^{n+1}(p_n) \geq \epsilon^{n}(p_n)$. For the aggregate size-(n + 1) bundle elasticity, however, we must scale $\epsilon^{n+1}(p_n)$ by A(n) < 1. What impact does this have on the comparison? One can show that $\tilde{\epsilon}^{n+1}(\tilde{p}_n) \geq \tilde{\epsilon}^{n}(\tilde{p}_n)$ whenever the right-hand side of the last inequality in Eq. 8 is greater than the right-hand side of the last inequality in Eq. 6. This holds for all n.⁴⁷

Proof of Proposition 2

Let preferences be as for Proposition 1 above except in allowing for correlation between consumer valuations, v _i  = (v _i1, ..., v _in ). Let ρ _c,d = corr(v _ic ,v _id ). With correlation, the variance of the per-good valuation for a bundle of size n, x _in , may be written as

$$\begin{array}{*{20}l} {\sigma ^{2}_{n} }&{ = \frac{1}{{n^{2} }}{\left( {{\sum\limits_{c = 1}^n {\sigma ^{2}_{c} + 2{\sum\limits_{c = 1}^{n - 1} {{\sum\limits_{d = c + 1}^n {\rho _{{c,d}} \sigma _{c} \sigma _{d} } }} }} }} \right)}} \\ {}&{ = \frac{1}{n}{\left[ {\frac{1}{n}{\left( {{\sum\limits_{c = 1}^n {\sigma ^{2}_{c} + 2{\sum\limits_{c = 1}^{n - 1} {{\sum\limits_{d = c + 1}^n {\rho _{{c,d}} \sigma _{c} \sigma _{d} } }} }} }} \right)}} \right]}} \\ {}&{ = \frac{1}{n}\overline{\sigma } ^{2}_{c} } \\ \end{array} $$

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The primary benefit of bundling is due to heterogeneity reduction as measured by the variance of per-good tastes for the bundle, $\sigma^2_n$. Unlike for Proposition 1 above, once we allow for correlation in tastes, bundle size, n, is not a sufficient statistic for $\sigma^2_n$. In particular, adding a new good to a bundle changes $\sigma^2_n$ by both (1) changing $\bar{\sigma}^2_c$ and (2) increasing n.

Let $\eta = \frac{\omega}{n} = \frac{\sigma^2}{\epsilon^2 n} = \frac{lim_{n \rightarrow \infty} \bar{\sigma}^2_c}{\epsilon^2 n}$. Then we may re-write Eq. 6 above as

$$\begin{array}{*{20}l} {{ \in ^{n} {\left( {\mu - \in } \right)}} \hfill} & {{ = - \frac{{\partial q_{n} }}{{\partial p}}\frac{{\mu - \in }}{{q_{n} }}} \hfill} \\ {{} \hfill} & {{ \geqslant - {\left( {\eta - 1} \right)}\frac{{\mu - \in }}{{1 - \frac{1}{2}\eta }}} \hfill} \\ {{} \hfill} & {{ \geqslant \frac{{1 - \eta }}{{1 - \frac{\eta }{2}}}{\left( {\mu - \in } \right)}} \hfill} \\ \end{array} $$

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Differentiating this with respect to η yields

$$\label{dpergoodeta} \frac{\partial \epsilon^{n}(\mu - \epsilon)}{\partial \eta} = \frac{-1}{2 \left[1 - \frac{\eta}{2}\right]^2} ~ (\mu - \epsilon) < 0 $$

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This is a more general statement of Eq. 7 above.⁴⁸ Reducing the (limiting) variance of the bundle (e.g. by increasing n or reducing σ ²) makes per-good demand for a bundle of size n more elastic.

The result of the proposition follows from Eq. 11. To see this, suppose the bundle had only two goods (i.e. component 2 was the nth good). It is easy to see that $\frac{\partial \sigma^2_n}{\partial \rho_{1,2}} > 0$, i.e. making the correlation between components 1 and 2 more negative reduces $\sigma^2_n$. Since, $\frac{\partial \eta}{\partial \sigma^2} >0$ it follows that $\frac{\partial \epsilon^{n}}{\partial \rho_{\tilde{1},2}} < 0$: making correlations more negative makes the bundle demand curve more elastic. For the case of general n, simply note that the variance of a bundle of size n can be decomposed into the variance of a bundle of size (n − 1), the variance of component n, and twice the covariance between a bundle of size (n − 1) and component n.

$$\begin{array}{*{20}l} {\overline{{\sigma ^{2}_{c} }} }&{ = \frac{1}{n}{\left( {{\sum\limits_{c = 1}^n {\sigma ^{2}_{c} + 2} }{\sum\limits_{c = 1}^{n - 1} {{\sum\limits_{d = c + 1}^n {\rho _{{c,d}} \sigma _{c} \sigma _{d} } }} }} \right)}} \\ {}&{ = \frac{1}{n}{\left[ {{\left( {{\sum\limits_{c = 1}^{n - 1} {\sigma ^{2}_{c} + 2} }{\sum\limits_{c = 1}^{n - 2} {{\sum\limits_{d = c + 1}^{n - 1} {\left. {\rho _{{c,d}} \sigma _{c} \sigma _{d} } \right)} }} }} \right)} + \sigma ^{2}_{n} + 2{\sum\limits_{c = 1}^{n - 1} {\rho _{{c,n}} \sigma _{c} \sigma _{n} } }} \right]}} \\ {}&{ = \frac{1}{n}{\left[ {\sigma ^{2}_{{n\widetilde{ - }1}} + \sigma ^{2}_{n} + 2\rho _{{n\widetilde{ - }1,n}} \sigma _{{n\widetilde{ - }1}} \sigma _{n} } \right]}} \\ \end{array} $$

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Appendix 2: Instruments

In this appendix, we present an analysis of the instruments used for prices and network carriage in the econometric analysis.

Price instruments To assess the power of the price instruments, Table 7 presents results from reduced form regressions of prices on the instruments and exogenous variables.⁴⁹ The results are organized in sets of three columns. For each set of three, the first column reports the point estimates from the regression of the price of basic service, p _b , on the instruments and included exogenous variables. Similarly in the second and third columns for the price of Expanded basic services I and II, p_I and p_II, if offered.

Table 7 First-stage estimation, prices

The first set of three columns report estimates using cost shifters as instruments for cable prices. As these shifters do not vary across services, we interact them with cable service dummy variables to allow their effects to differ by service. Reported are the estimated parameters for these interactions.⁵⁰ Evidence in support of the cost instruments is mixed. While homes passed does not appear to be an important cost shifter in any equation, the remaining variables enter intermittently. Most influential are affiliation (negative and significant in the first and third columns) and MSO subscribers and its square (negative for large values and occasionally significant in the first and third columns). Channel capacity enters as expected only in the second column. That said, p values associated with the hypothesis test of joint insignificance for all parameters are trivially small in all but the expanded I equation.⁵¹ On balance, while supporting their use as instruments, lack of variation across services and an indirect connection to marginal costs suggests the cost shifters may be weak instruments.

The second set of three columns report estimates using prices of cable services of other systems within an MSO as instruments.⁵² The results are quite promising. Other-system prices within an MSO provide strong and significant effects for both basic and Expanded I equations, particularly for prices of the same service. Results for a second expanded service are poor, possibly due to relatively few observations. As expected, p values associated with the hypothesis of joint insignificance are soundly rejected for the basic and Expanded I equations.

Network instruments To assess the power of the network instruments, Table 8 presents a synopsis of reduced form (probit) regressions of network carriage on the instruments and included exogenous variables. As above, the results are organized in sets of three columns. As we must predict the carriage of each of the top-15 cable networks (as well as the sum of other cable networks) on all the exogenous variables and instruments, the number of estimations performed was considerable.⁵³ Rather than report the point estimates of the instruments for each specification, we simply report the p value from the hypothesis test of joint insignificance of the instrument set. As can be seen from the table, the instruments have considerable power, at least for the basic and first expanded basic equation.⁵⁴ Coefficient estimates were as expected—particularly powerful predictors of the carriage of network q on service s was the corresponding likelihood it was carried on service s by other systems within its MSO.

Table 8 First-stage estimation, network carriage