Accessible Requires Authentication Published by De Gruyter April 8, 2021

Media Capture and Bias in the Market for News

Abhra Roy

Abstract

We analyze a model of media bias under government capture and a free press. The government wants citizens to invest in a project. Citizens gain from investing only if the state of the economy is good. The state is unobserved. The media firm receives a noisy signal about the actual state and makes a report about whether or not the state of the economy is good. Citizens read the report and decide whether or not to invest. In this context, we show that media bias under government capture may be smaller (greater) than that under free press if the cost of investment is sufficiently high (low) provided that the signal noise is below a certain threshold. Finally, we show that the difference between the bias under government capture and free press diverges (converges) when the cost of investment is sufficiently high (low) in response to a reduction in noise.

JEL Classification: L82; L29; L33; D73

Corresponding author: Abhra Roy, Department of Economics, Finance and Quantitative Analysis, Kennesaw State University, 560 Parliament Garden Way, Kennesaw, GA30144, USA, E-mail:

Appendix

Lemma 1:

In equilibrium under free press, it is always optimal for the media firm to follow an editorial policy such that the signal s = g is reported truthfully.

Proof:

Consider the following editorial policies:

Editorial policy A:

Pr [r=g|s=b]=σP, Pr [rP=b|s=b]=(1σP), Pr [rP=g|s=g]=1, Pr [rP=b|s=g]=0 and

Editorial Policy B:

Pr [rP=g|s=b]=0, Pr [rP=b|s=b]=1, Pr [rP=g|s=g]=1σˆP, Pr [rP=b|s=g]=σˆP.

It is easy to see that the good signal is reported truthfully under editorial policy A and the bad signal is reported truthfully under editorial policy B. We will show that the media firm will always prefer editorial policy A over B. Now consider editorial policy B. Since citizens only gain from investing when the actual state is good, they calculate the likelihood of the good state following any report using Bayes’ theorem. Therefore, the likelihood of a good state given report b is given by:

(20)Pr[S=G|rP=b]=Pr[rP=b|S=G]Pr[S=G]Pr[rP=b],=θ{(1q)+qσˆP}θ{(1q)+qσˆP}+(1θ){q+(1q)σˆP}.

Let XˆbP be the minimum return required to justify investment following a report rP = b under editorial policy B. As a consequence, we must have:

(21)XˆbP=c{1+(1θθ)q+(1q)σˆP(1q)+qσˆP}.

It is easy to check that XˆbP<Xb=c{1+(1θθ)q1q}. Similarly, the likelihood of the good state given a report r = g under editorial policy B is given by the following:

(22)Pr[S=G|rP=g]=Pr[rP=g|S=G]Pr[S=G]Pr[rP=g],=θqθq+(1θ)(1q).

As before, let XˆgP be the minimum return to justify investment under editorial policy B which is given by:

(23)XˆgP=c{1+(1θθ)1qq}.

It is straightforward to check that XˆgP<Xg=c{1+(1θθ)(1q)+qσPq+(1q)σP} for σP. As before, consider the intervals (XˆgP,X) and (X,XˆbP) under editorial policy B. In the interval (XˆgP,X), citizens do not invest based on the prior but invest in the project if the media firm issues a report rP = g. In the interval (X,XˆbP), citizens always invest based on the prior but do not invest if the media firm reports rP = b. Therefore, the net benefit, NgP, of subscribing to the media outlet in the interval (XˆgP,X) is given by the following:

(24)NgP=Pr[rP=g]Pr[S=G|r=g](xc)Pr[rP=g]Pr[S=B|rP=g]c,=θq(1σˆP)(Xc)(1θ)c(1q)(1σˆP).

Recall that the cost of following news is captured by d. As a result, the minimum return required from the investment to justify subscribing to the news is given by:

(25)Xˆ˜gP=c{1+(1θθ)1qq+dθq(1σˆP)}.

Similarly, the minimum return to justify subscribing to news in the interval (X,XˆbP) is given by:

(26)Xˆ˜bP=c{1+(1θθ)q+(1q)σˆP(1q)+qσˆP}dθ((1q)+qσˆP).

Since anyone with an expected return less than XˆgP does not invest in the project, they do not have any need to subscribe to the media outlet. Similarly, any citizen with a return above XˆbP always invests in the project and do not subscribe to the news. Therefore, the expected number of subscribers under editorial policy B are citizens whose expected return lies in the interval Xˆ˜bPXˆ˜gP, which, after a little algebra yields:

(27)Sˆ˜P={c2m(1θθ)(2q1)d2mθ(1σˆP)}1q((1q)+qσˆP).

Whereas the expected number of subscribers under editorial policy A, given that the magnitude of the bias is the same under both editorial policies, is given by the following:

(28)SˆP={c2m(1θθ)(2q1)d2mθ(1σˆP)}1(1q)(q+(1q)σˆP).

Since q(12,1), we must have 1q((1q)+qσˆP)<1(1q)(q+(1q)σˆP) . Therefore, SˆP>Sˆ˜P. Further, since σˆP is not the optimal choice of bias under editorial policy A, the expected number of subscribers, in equilibrium, must be higher. Given the profit function is monotonic in the number of subscribers as in Equation (1), this must imply that the profit under editorial policy A are higher than that under B. Hence proved. The result of lemma 1 can be shown to remain unaltered even if we include subscription prices▪[10]

Lemma 2:

For a sufficiently largem, the equilibrium level of bias under free pressσP*(0,1).

Proof:

Note that the payoff of the function of the media firm, from Equation (1), is given by:

πm=aSˆP(σP)+b(σP)SˆPαR0.

Differentiating πm with respect to σP yields Equation (13) which is as follows:

πmσP=aSˆP(σp)+b(σP)SˆPαR0+αb(σP)SˆPα1SˆP(σP)R0.

where SˆP(σP)=c2m(1θθ)(2q1)(q+(1q)σP)2+d(1q)2θm(q+(1q)σP)2<0. Now consider the following:

(29)φ=SˆPα1(σP)R0[b(σP)SˆP(σP)+αb(σP)SˆP(σP)].

Evaluating φ|σP=0 we get

(30)φ|σP=0=SˆPα1(0)R0{θ(1q)(1c2m)c2m(1q)2+α(2q1)qd(1q)(1α)2mq}.

Note that for a sufficiently large m, φ|σP=0>0. Therefore, for a sufficiently large R0, πmσP|σP=0>0. Differentiating the first order condition with respect to σP and some algebra yields:

(31)2πmσP2=aSˆP(σP)α(1α)b(σP)SˆPα2(σP)SˆP(σP)2R0.

Clearly, 2πmσP2<0 for a sufficiently large R0. Since S˜P(1)=0, we must have σP(0,1)

Lemma 3:

The equilibrium level of media bias under capture,σGov[0,1).

Proof:

The objective function of the government from Equation (17), after some algebra, turns out to be:

VGov=γ4θmn(σGov)ν(σGov)h(σGov)TGov.

After some algebra, we can show that VGov is strictly concave in σGov. Further, since S(1) = 0, we must have σGov[0,1)

Lemma 4:

For a sufficiently largem, the equilibrium level of bias under free press with subscription price,σP*(0,1).

Proof:

Let p be the subscription price for the news outlet. Therefore, the payoff of the media firm is captured by the following equation:

(32)πm=(p+a)SˆP+b(σ)SˆPαR0 .

where SˆP=ν(σP)h(σP)2θm and ν(σP)=2θm(q+(1q)σP)cn(σP)p. The media firm first determines what price to charge in equilibrium. Maximizing the objective function with respect to p yields:

(33)πmp=(p+a)2θm(q+(1q)σP)+SˆPαb(σ)SˆPα1(σP)R04θ2m2(q+(1q)σP)2.

which can be rewritten as g(σP)h(σP)(p+a)αb(σP)SPα1R0=0. This yields p*=p(σP). With a little bit of algebra, we can show that p(σP)<0. Plugging p=p(σP) in the objective function πm we now have:

πm=(p(σP)+a)SˆP+b(σP)SˆPαR0 .

The rest of the theorem is similar to the proof of lemma 2. We can show that for a sufficiently large mR0, πmσP|σP=0>0. Further, citizens do not subscribe to the news when σP=1 since the news is completely uninformative. Therefore, πm|σP=1=0. As a result, we must have σP(0,1) (since πm is continuous in σm). Further, we can show that πm is strictly concave if p(σP)0 (sufficient but not neccessary) which implies that σP is unique □

Proposition 1:

c(0,c0)such that

  1. (a)

    Media bias under capture is greater (lesser) than under free press if the cost of investment is below (above) a thresholdc, i.e.,c<c,σGov*>σP*andc(c,c0),σGov*<σP*wherec0=inf[c|SˆP=0]for everymandθ(0,1).

  2. (b)

    Media bias under capture decreases with the cost of investment, i.e.,dσGov*dc<0

Proof (a):

Note that from Equation (17) we have:

VGov=φ2θmn(σGov)ν(σGov)h(σGov)TGov.

Differentiating Equation (17) with respect to σGov and evaluating it at σGov=σP yields:

(34)VGovσGov|σGov=σP=φ2θmλcγc2(q+(1q)σP).

where

(35)λc={2θm(q+(1q)σP)cn(σP)d}n(σP).

and

(36)γc=n(σP){c(1θ)(2q1)(q+(1q)σP)d(1q)(q+(1q)σP)}.

Now consider λc, for every m,d, c0 such that λc=0(λcγc)|c=c0<0 since c(1θ)(2q1)d(1σP)>0. Since γc is increasing in c, c<c0 such that λcγc=0. Therefore, c<c, λcγc>0 and c(c,c0), λcγc<0. Since VGov is concave in σGov, λcγc>0σGov>σP and λcγc<0σGov<σP

Proof (b):

Differentiating Equation (17) with respect to σGov yields:

(37)D(σGov)=n(σGov)ν(σGov)h(σGov)+n(σGov)[ν(σGov)h(σGov)+ν(σGov)h(σGov)],=0

It is easy to check that VGov is strictly concave with respect to σGov. Differentiating Equation (37) with respect to c using the implicit function rule gives us the following:

(38)Dcdc+DσGovdσGov=0.

Since VGov is strictly concave in σGov, DσGov<0. Further, with a little algebra we can show that Dc < 0 which yields our desired result □

Proposition 2:

The difference between equilibrium media bias under capture and under free press increases with q, i.e.,{σGovσP}q>0.

Proof:

First consider free media. Recall that, under free press, the first order condition for profit maximization with respect to σP from Equation (13) yields:

πPσP=a2θm{ν(σP)h(σP)+ν(σP)h(σP)}+b(σP)SˆPαR0+b(σP)αSˆPα1R0,=aSˆP(σP)+b(σP)SˆPαR0+b(σP)αSˆPα1R0,=F(σP(q),q).

Using the implicit function rule, we get dσPdq=FqFσP<0 where Fq is as follows:

(39)Fq=aSˆPq(σP)+bq(σP)SˆPα(σP)R0+αb(σP)SˆPα1(σP)R0+αbq(σP)SˆPα1(σP)SˆP(σP)R0α(1α)b(σP)SˆPα2(σP)SˆP(σP)SˆPq(σP)R0+αb(σP)SˆPα1(σP)SˆPq(σP)R0.

After some algebra, we can show that Fq < 0 provided that R0 is sufficiently large. Therefore, σPq<0. Now consider the first order conditions for maximizing VGov under capture which is given by Equation (40):

(40)D(σGov)=n(σGov)ν(σGov)h(σGov)+n(σGov)[ν(σGov)h(σGov)+ν(σGov)h(σGov)],=0

Dividing both sides of Equation (40) by n(σGov)ν(σGov)h(σGov) we get:

(41)D(σGov)=n(σGov)n(σGov)+ν(σGov)ν(σGov)+h(σGov)h(σGov),=0

Using the implicit function theorem on Equation (40) we get

(42)DσGovdσGov+Dqdq=0.

which yields dσGovdq=DqDσGov. Since VGov is strictly concave with respect to σGov, DσGov>0. Therefore, the sign of dσGovdq depends on the sign of Dq which is given by:

(43)Dq=ddq[n(σGov)n(σGov)+ν(σGov)ν(σGov)+h(σGov)h(σGov)].

It is easy to check that ddq(n(σGov)n(σGov)), ddq(h(σGov)h(σGov)) and ddq(ν(σGov)ν(σGov)) are positive. Therefore σGovσ increases with q

Proposition 3:

c(0,c0)such that welfare under capture is smaller (greater) than that under free press if the cost of investment is smaller (greater) than the thresholdc, i.e.,c<c, WGov < WPandc(c,c0), WGov > WP.

Proof:

From Equations (18) and (19) and after some algebra, we get

(44)WGovWP=θ4m(q+(1q)σGov)(2mXgGov)2(q+(1q)σP)(2mXgP)2.

From Equation (15), we have:

(45)SˆG=Pr[XgGovX2m]=Pr[X2m]Pr[XXgGov]=1XgGov2m.

Further note that SˆGov=v(σGov)h(σGov)2θm from Equation (16). As a result we must have 2mXgGov=v(σGov)h(σGov)θ. After a little algebra, we see that Equation (44) reduces to:

(46)WGovWP=14θmν(σGov)2h(σGov)θ4m(q+(1q)σP)(2mXgP)2.

Differentiating WGovWP with respect to σGov* yields:

(47)(WGovWP)σGov=14θmν(σGov)[ν(σGov)h(σGov)+2ν(σGov)h(σGov)].

Since ν(σGov)<0 and h(σGov)<0, we must have (WGovWP)σGov<0. Note that WσGov=σPGov=WP and since (WGovWP)σGov<0, using Proposition 1, we obtain our result  □

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Received: 2020-07-17
Accepted: 2021-02-25
Published Online: 2021-04-08

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