Quality choice and advertising regulation in broadcasting markets

Abstract

We consider the role of the endogenous choice of platform quality in a broadcasting duopoly market where competing media platforms also choose their levels of advertising. We compare the equilibrium levels of quality, advertising and welfare under private and mixed duopoly competition. We show that the welfare comparison between the private and mixed duopoly regimes depends crucially on the interplay between the net direct effect of advertising on welfare and the degree of substitutability between platforms. We also consider the effects on quality and welfare of recent policies that tend to eliminate advertising as a source of financing for publicly-owned platforms.

Appendix

1.1 The model with an endogenous tax on advertising revenues

This subsection incorporates an initial stage in which the government sets a tax on the revenue of the rival private platform to finance the publicly-owned platform. Thus, the revenue obtained by the private platform is considered to consist of its advertising revenue after tax, and the revenue obtained by the publicly-owned platform to consist of the tax revenue collected from the private platform. So profits are given, respectively, by

$$\begin{aligned} \pi _{1}=\tau \gamma a_{2}x_{2}-\frac{v_{1}^{2}}{2}; \quad \pi _{2}=\left( 1-\tau \right) \gamma a_{2}x_{2}-\frac{v_{2}^{2}}{2}, \end{aligned}$$

where $\tau$ represents the direct tax on the revenue of the private platform. By substituting the demand function (2) in the definition of profits and taking into account that $a_{1}=0$, the following can be obtained:

$$\begin{aligned} \pi _{1}=\tau \gamma a_{2}\frac{v_{2}-v_{1}+t-\delta a_{2}}{2t}-\frac{ v_{1}^{2}}{2}; \quad \pi _{2}=\left( 1-\tau \right) \gamma a_{2}\frac{ v_{2}-v_{1}+t-\delta a_{2}}{2t}-\frac{v_{2}^{2}}{2}.\nonumber \\ \end{aligned}$$

(24)

From maximizing the profit of the private platform, the level of advertising by private platform 2 can be obtained, which is (19). By substituting it in the profit functions (24) the market shares and profits are obtained:

$$\begin{aligned} x_{1}&= \frac{v_{1}-v_{2}+3t}{4t}; \; \pi _{1}=\tau \frac{k\left( v_{2}-v_{1}+t\right) ^{2}}{8t}-\frac{v_{1}^{2}}{2}; \\ x_{2}&= \frac{v_{2}-v_{1}+t}{4t}; \; \pi _{2}=\left( 1-\tau \right) \frac{k\left( v_{2}-v_{1}+t\right) ^{2}}{8t}-\frac{v_{2}^{2}}{2}. \end{aligned}$$

Now consider the quality choice by platforms, so the publicly-owned platform maximizes social welfare and the private one maximizes its profit. Taking into account that the social welfare ($W$) at the first stage is now given by (21), from the first order conditions in the first stage of the game, the reaction function of each platform can be obtained:

$$\begin{aligned} v_{1}\left( v_{2}\right) =\frac{\left( 2k-7\right) t+\left( 2k+1\right) v_{2} }{2k-8t+1}; \; v_{2}\left( v_{1}\right) =\frac{\left( 1-\tau \right) \left( t-v_{1}\right) k}{4t+k\left( \tau -1\right) }. \end{aligned}$$

From the intersection of the quality reaction function of the platforms the NE levels of advertising, market shares and profits at the second stage of this game are obtained:

$$\begin{aligned} v_{1}\left( \tau \right)&= \frac{2k\left( 1-\tau \right) +\left( 2k-7\right) t }{2k\left( 2-\tau \right) +1-8t}; \; v_{2}\left( \tau \right) =\frac{2k\left( \tau -1\right) \left( t-1\right) }{4k-8t-2k\tau +1}; \; a_{1}\left( \tau \right) =0; \; a_{2}\left( \tau \right) =\frac{-4t\left( t-1\right) }{\left( 1+2k\left( 2-\tau \right) -8t\right) \delta }.\nonumber \\ x_{1}\left( \tau \right)&= -\frac{6t-4k+2k\tau +1}{4k-8t-2k\tau +1};\; x_{2}\left( \tau \right) =\frac{-2\left( t-1\right) }{4k-8t-2k\tau +1}; \;\pi _{2}\left( \tau \right) =\frac{2k\left( 1-\tau \right) \left( t-1\right) ^{2}\left( 4t-k+k\tau \right) }{\left( 4k-8t-2k\tau +1\right) ^{2}}.\nonumber \\ \pi _{1}\left( \tau \right)&= -\frac{4k^{2}t^{2}-8k^{2}t\tau +8k^{2}t+4k^{2}\tau ^{2}-8k^{2}\tau +4k^{2}-16kt^{3}\tau +32kt^{2}\tau -28kt^{2}+12kt\tau -28kt+49t^{2}}{2\left( 4k-8t-2k\tau +1\right) ^{2}}\nonumber \\ W\left( \tau \right)&= V-\frac{\left( t-1\right) \left( \left( 4t\tau ^{2}-8t\tau +8t-8\tau +8\right) k^{2}-16kt^{2}+32kt\tau -44kt-4k\tau +4k+56t^{2}-7t\right) }{2\left( 4k-8t-2k\tau +1\right) ^{2}}\nonumber \\ \end{aligned}$$

(25)

By maximizing the welfare function (25) with respect to the tax, the optimal level of the tax is found to be zero.¹⁵

$$\begin{aligned} \frac{\partial W(\tau )}{\partial \tau }=-2k\left( 2k\tau +1\right) \left( t-1\right) ^{2}\frac{8t-2k-1}{\left( 8t-2k\left( 2-\tau \right) -1\right) ^{3}}<0 \end{aligned}$$

1.2 Proof of Proposition 1

As explained in the main text, substituting (17) in ( 15) yields the SPE level of welfare in the mixed duopoly, which is given by

$$\begin{aligned} W^{M}(k,t)=-\frac{\Omega \left( k,t\right) }{2\left( 2k-1\right) ^{2}\left( 2t+8k^{2}t-8kt+k^{2}-4k^{3}\right) ^{2}} \end{aligned}$$

where

$$\begin{aligned} \Omega \left( k,t\right) \equiv \left( \begin{array}{c} 8k^{8}\left( 98t-\left( 30t+4\right) k+k^{2}+108t^{2}+3\right) \\ -2k^{6}\left( 2k+\left( 460k-249\right) t+\left( 1608k-2390\right) t^{2}+16t^{3}\left( 26k-113\right) \right) \\ +k^{4}t\left( 11-124k+\left( 1561-3680k\right) t+\left( 6480-6560k\right) t^{2}\right) \\ -2t^{2}\left( -10t-653k^{2}t+1890k^{3}t+124kt-16k^{2}+174k^{3}\right) \end{array} \right) . \end{aligned}$$

Recall that the welfare level at the SPE of the zero duopoly is given by $W^{Z}(k,t)=V+\frac{\left( 1-t\right) \left( 8k^{2}t+8k^{2}-16kt^{2}-44kt+4k+56t^{2}-7t\right) }{2\left( 4k-8t+1\right) ^{2}}$.

Therefore the welfare comparison between the two alternative duopolies is given by the sign of the function $F(k,t)=W^{M}(k,t)-W^{Z}(k,t).$ The implicit equation $F(k,t)=0$ is represented in Fig. 3. From this figure it is easy to obtain the region $Z$ where $F(k,t)=W^{M}(k,t)-W^{Z}(k,t)<0$, and region $M$, where $F(k,t)=W^{M}(k,t)-W^{Z}(k,t)>0$. Due to the complexity of function $W^{M}(k,t)$, Fig. 3 is obtained by using the mathematical tools available in Scientific Word 5.5. The technical details needed to obtain Fig. 3 are available upon request.

1.3 Proof of Proposition 2

The welfare comparison between the mixed and private duopolies is given by the function $G(k,t)=W^{M}(k,t)-W^{P}(k,t)$. The implicit equation $G(k,t)=0$ is represented in Fig. 4. From this figure the regions $M_{1}$ and $M_{2}$ are obtained where $G(k,t)=W^{M}(k,t)-W^{P}(k,t)>0$, and the region $P$ where $G(k,t)=W^{M}(k,t)-W^{P}(k,t)<0$. As in the proof of Proposition 1, the technical details needed to obtain Fig. 4 are available upon request.

1.4 Proof of Proposition 3

As in the proofs of Propositions 1 and 2, the regions plotted in Fig. 5 are obtained by comparing the functions $W^{Z}(k,t)$, $W^{M}(k,t)$, and $W^{P}(k,t)$. By choosing the appropriate intercepts between these three functions the relevant frontiers plotted in Fig. 5 are obtained. As in the proofs of the two previous propositions, the technical details of this result are available upon request.

1.5 Parameter restrictions in the private duopoly

From (4) the Second Order Conditions (SOC) with respect to $a_{i}$ imply $\frac{\partial ^{2}\pi _{i}(a_{i},a_{j})}{\partial a_{i}^{2}}=-\gamma \frac{\delta }{t}<0$, which is consistent with Assumption 1. Similarly, according to (7), the SOC with respect to $v_{i}$ imply $\frac{\partial ^{2}\pi _{i}(v_{i},v_{j})}{\partial v_{i}^{2}}= \frac{1}{9}\frac{k-9t}{t}<0$, which is also consistent with Assumption 1. Finally, from (9) it is clear that the SPE levels of $a_{i}$, $v_{i}$, and $x_{i}$ are positive.

1.6 Parameter restrictions in the mixed duopoly

Note, first, that the SOC of platform 2 with respect to $a_{2}$ is the same as in the private duopoly. From (11) the SOC with respect to $a_{1}$ implies $\frac{\partial ^{2}W}{\partial a_{1}^{2}}=-\frac{1}{2} \delta \frac{2\gamma -\delta }{t}=-\frac{1}{2}\delta ^{2}\frac{2k-1}{t} <0\leftrightarrow k>0.5$, which is clearly ensured by Assumption 1.

Now, consider the SOC for a SPE with respect to quality levels:

From $\pi _{2}$ and $W$ in (15), the SOC with respect to qualities implies $\frac{\partial ^{2}W}{\partial v_{1}^{2}}=\frac{1}{2} \frac{2t-4kt+k^{2}}{t\left( 2k-1\right) }<0$ and $\frac{\partial ^{2}\pi _{2} }{\partial v_{2}^{2}}=\frac{-t-4k^{2}t+4kt+k^{3}}{t\left( 2k-1\right) ^{2}} <0$.

The two inequalities above are satisfied under Assumption 1. The proof is shown by plotting the two above expressions and checking that they are negative for the values of $t$ and $k$ consistent with Assumption 1. The details of this result are available upon request.

Now consider the equilibrium levels for the endogenous variables at expression (17). The proof that all these variables are non-negative under Assumption 1 is shown by plotting each variable as a function of $t$ and $k$ and checking that its value is non-negative for the relevant set of parameter values. Again, the technical details of this proof are available upon request.

1.7 Parameter restrictions in the zero duopoly

Note that in this case the SOC with respect to platform 2’s advertising is the same as in the mixed duopoly. Therefore this condition satisfies Assumption 1. (Recall that $a_{1}^{Z}=0$, which means that there is no need to consider the SOC with respect to this variable).

Now consider the SOC with respect to quality levels. From (20) and (21) the following is obtained:

$$\begin{aligned} \frac{\partial ^{2}\pi _{2}}{\partial v_{2}^{2}}=\frac{1}{4}\frac{k-4t}{t}<0 \;\text { and }\;\frac{\partial ^{2}W}{\partial v_{1}^{2}}=\frac{1}{8}\frac{ 2k-8t+1}{t}<0. \end{aligned}$$

The two above inequalities are clearly satisfied under Assumption 1.

Finally, to prove that the SPE levels of the endogenous variables at (23) are positive under Assumption 1, plot each variable as a function of $k$ and $t$ and check that all the functions are non-negative under the restriction imposed by Assumption 1. Again, the technical details of the proof are available upon request.