## Abstract

We offer a model of media as a multisided platform, providing entertainment and news to viewers, commercial opportunities to advertisers, and political influence to politicians, thanks to the presence of influenceable voters among the media audience. We characterize a political economic equilibrium, determining simultaneously media choices and politicians’ electoral positions. We show that as the value of political influence increases, the media transitions from catering to commercial advertisers to selling political influence, resulting in policy choices that hurt influenceable voters.

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## Notes

- 1.
We do not specify details of the trade between viewers and advertisers as it is not the focus of the current paper; see for instance Anderson and Coate (2005). The assumption that advertisers appropriate all the gains form trade is immaterial. Otherwise, the value of watching the media for viewers has to be amended to include not only an entertainment but also a commercial motive.

- 2.
In practice, media can both slant the content of news and provide explicit or implicit political propaganda. In terms of the model, what is important in either case is that it yields electoral benefits for politicians and a nuisance for viewers.

- 3.
The actual format of the auction is not important–since the value of the political space for politicians is common knowledge, and is the same for both politicians, a first prize auction, a second price auction, or simply a posted price would lead to the same results. See Fang (2002) and Bikhchandani et al. (2013).

- 4.
Our assumption that media’s entertainment value is larger for poorer consumers is consistent with evidence that the number of hours spent watching TV is negatively correlated with income; see, e.g., Dooe (2013), who uses US data from the General Social Survey (NORC 2014). Similarly, TV watching in Mexico is concentrated in the lower socioeconomic levels, with the most watched programs being news and soap operas (IBOPE AGB 2009).

- 5.
We use

*D*to denote the differential operator. - 6.
For the Figure, we let

*F*be a uniform distribution on \([\underline{\upomega }_1,\overline{\upomega }_1] = [1,2]\), let \({\upomega }_2=1\), and let*G*be a standard uniform distribution. We also let the income parameter and the nuisance parameter of the viewers’ payoffs be given by \(\uplambda = \upgamma =1/2\), and the political influence of media be given by \(\updelta =4\). We can calculate \(\overline{a}_c=1/3\) and \(\overline{a}_e=1/4\). - 7.
For simplicity, we have viewers and advertisers becoming active whenever they are indifferent. This is without loss of generality, since on the equilibrium path the sets of indifferent viewers and advertisers are zero measure.

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## Acknowledgements

We thank David Levine, Tom Palfrey, Andrea Mattozzi, Stephane Wolton, two anonymous referees, and several seminar audiences for comments and suggestions.

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## Appendix

### Appendix

###
*Proof of Theorem 3.1*

To characterize equilibrium behavior, we proceed backwards from the last stages of the game. Citizens can anticipate a consumption level equal to their initial endowment of good 1, that is \({\upomega }_1\), regardless of whether they are active or inactive, since all the gains from trade are appropriated by advertisers. Given their expected consumption levels, at the voting stage the ideal levels of the public good for citizens are given by \(X({\upomega }_1)\). Given their Euclidian preferences over levels of the public good, in any political economic equilibrium uninfluenced citizens at the voting stage vote for the politician whose proposal is closer to their ideal public good level, splitting their votes exactly in case of indifference, while influenced viewers vote for the winner of the political auction.

At the electoral competition stage, the bid paid by the winner of the political auction is already a sunk cost, so that both politicians seek to maximize their vote shares. Moreover, since influenced voters’ behavior is predetermined by the result of the political auction, both politicians seek to maximize their votes among uninfluenced voters. It is simple to check that in equilibrium both politicians offer policies that are medians of the ideal levels of the public good among uninfluenced citizens. This is because, if any politician expects to obtain less than half the votes, then the politician can adopt the policy choice of the other politician and obtain half the votes. In particular, if the median is unique, politicians offer the same policy.

At the watching stage, viewers’ optimal behavior and correct expectations about their consumption levels imply that in equilibrium, for any previous history, they watch the media if and only^{Footnote 7} if \(1 - \uplambda {\upomega }_1 - \upgamma a \ge 0\), or equivalently, if and only if

The equilibrium fraction of active viewers, then, for any history at the watching stage is given by

At the ad market stage, advertisers’ optimal behavior and correct expectations about decisions of other advertisers and future decisions of viewers imply that in equilibrium, for any choice of \(a_e\) and *p* by the firm, an advertiser becomes active if

or equivalently if

Thus, the equilibrium fraction of active advertisers is given by the solution to

For \(a_c >0\), we can rewrite this expression as

Note that for every \(0 \le a_e < (1-\uplambda \underline{\upomega }_1)/\upgamma\) and every \(0<p \le {\upomega }_2 F((1 - \upgamma a_e)/\uplambda )\), there is a unique solution \(a_c \in [0,(1-\uplambda \underline{\upomega }_1)/\upgamma -a_e)\) to Eq. (11); that is the level of commercial advertising in the unique equilibrium of the subgame following the firm’s decision \((p,a_e)\). If the firm sets \(p=0\), all advertisers become active in the ensuing subgame, since they are at least indifferent between buying an ad or not. If the firm sets instead \(p \ge {\upomega }_2 F((1 - \upgamma a_e)/\uplambda )\), the level of commercial advertising in the ensuing subgame is zero.

At the ad market stage as well, equilibrium behavior in the auction and correct expectations about decisions of advertisers and future decisions of viewers imply that politicians bid

(This is the value of winning the political auction, when politicians anticipate correctly that regardless of who wins the political auction, they will split equally the votes of non influenced citizens.)

It is easy to see that \(p=0\) cannot be revenue-maximizing. Similarly, choosing any price \(p > {\upomega }_2 F((1 - \upgamma a_e))/\uplambda )\) is revenue equivalent to setting \(p ={\upomega }_2 F((1 - \upgamma a_e))/\uplambda )\). Thus, we can write the problem of the firm as choosing both political and commercial advertising under the constraint

Since increasing \(a_e\) is detrimental for the firm for \(a_e \ge 1/\updelta\), we can further restrict our attention to

Given the choices of advertisers and viewers along the equilibrium path in the subgame following a firm’s choice of \(a_e\) and *p*, it is tedious but straightforward to verify that the distribution of ideal levels of the public good for non influenced voters is given by \(H(a_c,a_e)\) if \(p=P(a_c,a_e)\).

Using Eqs. (11) and (12) and the objective of the firm, the profit maximization problem for the firm, then, can be written as

subject to

Since the expression for the objective of the firm in (M) is continuous and differentiable, and the choice set for \(a_c\) and \(a_e\) is compact, a solution for the problem of the firm exists and moreover it satisfies the usual first order conditions.

To show that the solution to problem M is unique, observe that, by assumption B, \({\upomega }_2 \uppi (a_c) +\updelta r a_e\) is a concave function of \(a_c\) and \(a_e\), which in turn implies that it is also a log-concave function of \(a_c\) and \(a_e\). Similarly, since *F* is log-concave, \(F((1 - \upgamma (a_c + a_e))/\uplambda )\) is a log-concave function of \(a_c\) and \(a_e\). Since the product of log-concave functions is log-concave, it follows that the objective function in problem M is log-concave as well, and therefore has a unique maximum. To check that \(a_c + a_e \le (1 -\uplambda \underline{\upomega }_1)/\upgamma\) is never binding, note that if the inequality is not strict, the value of the objective function is zero, but the firm can make positive profits by setting \(a_e\) and \(a_c\) close enough to zero, since by assumption \(\underline{\upomega }_1 < \uplambda\). Similarly, \(a_c \le 1\) is never binding, since the value of the objective function in M can be increased by reducing \(a_c\) and increasing \(a_e\) pari passu whenever \(a_c=1\), given that \(\uppi '(1)<0\) but \(\updelta r >0\). Thus, the problem of the firm can be formulated as in part (ii) of the theorem, and it has a unique solution. The remainder of the equilibrium path can be obtained retracing our steps. In particular, substituting \(P(a^*_c,a^*_e)\) for *p* in the ideal points of voters we obtain that the distribution of ideal points is given by \(H(x|a^*_c,a^*_e)\), as described by part (iii) of the theorem. By construction, the equilibrium path is unique, as required by part (i) of the theorem. \(\square\)

###
*Proof of Proposition 4.3*

For any pair \(x',x'' \in {\mathfrak {R}}\) such that \(x'<x''\), let

and for any \(x' \in {\mathfrak {R}}\) let

where \(a_c(r)\) and \(a_e(r)\) are the equilibrium choices of commercial and political advertisement as a function of *r*. Intuitively, \(m(r|x',x'')\) is the measure of the set of uninfluenced voters with ideal points in the interval \((x',x'']\), given that the level of political rents is *r*. Similarly, \(m(r|-\infty ,x'')\) and \(m(r|x',+\infty )\) are the measure of the sets of uninfluenced voters with ideal points respectively weakly below and strictly above \(x'\).

Let *x*(*r*) denote the median of \(H(x|a_c(r),a_e(r))\), that is the equilibrium policy choice as a function of political rents. It is easy to see that

and moreover, \(x>x(r)\) if and only if

Now suppose that for the initial value of political rents \(r'\) we have \(a_c(r')>0\), \(a_e(r')>0\), and \(x(r')< X((1-\upgamma a(r'))/\updelta )\); that is, in the initial situation the median voter is an inactive viewer. If \(r''\) is larger than but close enough to \(r'\), it must be that \(x(r'')<X((1-\upgamma a(r''))/\updelta )\). Thus,

because all citizens with ideal points below \(x(r')\) are inactive viewers, whose ideal points are unaffected by changes in advertising. However,

where the inequality in the second line follows from Proposition 4.2. Hence,

so that \(x(r')>x(r'')\).

Suppose instead that for the initial value of political rents \(r'\) we have \(a_c(r')>0\), \(a_e(r')>0\), and \(x(r') > X((1-\upgamma a(r'))/\updelta )\); that is, in the initial situation the median uninfluenced voter is an active viewer. Note that this implies \(a_e(r')< 1/\updelta\); otherwise all the active viewers would be influenced. If \(r''>r'\) is larger than but close enough to \(r'\), it must be that \(x(r'')>X((1-\upgamma a(r''))/\updelta )\). We can calculate

because all voters with ideal points above \(x(r')\) are active viewers, and the fraction of active viewers who are uninfluenced decreases from \(1-\updelta a_e(r')\) to \(1-\updelta a_e(r'')\). From Proposition 4.1 (i) we have \(a(r'')>a(r')\) implying \(X((1-\upgamma a(r''))/\updelta )) > X((1-\upgamma a(r'))/\updelta ))\). Thus,

Hence,

so that \(x(r')>x(r'')\). \(\square\)

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Castañeda, A., Martinelli, C. Politics, entertainment and business: a multisided model of media.
*Public Choice* **174, **239–256 (2018). https://doi.org/10.1007/s11127-017-0496-y

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### Keywords

- Media effects
- Three-sided platform
- Mass media