By explicitly taking into account the demand linkages between media markets and markets for advertisements, we found that the market outcome is likely to differ substantially from what is predicted by standard oligopoly theory. When products are strongly differentiated, equilibrium beliefs are symmetric. This yields a symmetric duopoly outcome. When products are closer substitutes, asymmetric equilibria and natural monopolies are equally plausible outcomes. Hence, there is some theoretical justification for concerns about excessive concentration in media markets, especially in markets where media goods are of low intrinsic quality and where consumers are not bound by subscription contracts. From a policy point of view, we may
conclude that regulations which reduce price competition among media firms may increase the scope for monopoly.
An interesting extension would be to allow for different levels of intrinsic product quality.
In such a framework, it might be possible to study the incentives to invest in quality enhancing technologies. This could potentially have a strong impact on the process that determines market structure. From an empirical perspective, it would be interesting to examine the relation between product quality, for instance measured in terms of the amount of editorial material, and market structure. A further aspect would be to test whether media markets where subscription contracts are common tend to be less concentrated.
References
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F)(x) ' &8 1&r
Proof of Proposition 1: Strict concavity, g(0) > 0 and g(xmax) = 0 implies that there exists exactly one point x*0[0, x ) such that g(x ) = x and consequently g(g(x )) = x .max * * * *
Suppose x is an asymmetric equilibrium point. Then g(x) = 2x where 2x is a point such that g(2x) = x. Thus 2x is also an equilibrium point since, by assumption, g(g(2x)) = g(x) = 2x. Moreover, note that g(x) > x for x < x . Thus if x < x and g(x) < x then x < g(x) < g(g(x)). Hence, such an* * * x cannot be an equilibrium. Consequently, if x < x and g(g(x)) = x then 2x = g(x) > x .* *
To simplify notation let µ = (2-c )/c and 8 = (4-c )(1-c )/(bc) where 8 $ 0 and µ $ 1, since2 2 2 c0[0,1]. Then . If g(g(x)) can intersect the 45E line from below only once, there exist at most three equilibria. Since g(g(0)) > 0, such intersections must lie on the increasing segment between two peaks (the interval [argmax g(x), max{x* g(x) = argmax g(x)}]) where g´(g(x)) = µ-(8/r)g(x)1/r-1 < 0 and g´(x) = µ-(8/r)x < 0. To prove that there is at1/r-1 most one intersection from below, it suffices to show that the slope of the equilibrium condition g(g(x)) - x = 0, which we denote F(x) = g´(g(x)) g´(x)-1, is strictly quasi-concave on the interval.
Thus, FO(x) is negative for r $ 1/2 since g´(g(x)) and g´(x) < 0. To show quasi-concavity also for r < 1/2, we evaluate FO(x) at points where FN(x) = 0. Using FN(x) = 0 for substitution in the first and third terms and simplifying, (A2) can be rewritten as follows:
8(1&r) r4g(x) x
µ&8
rx1/r&1 r(1&2r) x1/r&1%g(x)1/r&1 % (1%r)8g(x)1/r&1x1/r&1 .
x1/r ' a%bx This expression is strictly negative for r < 1/2 and consequently F is strictly quasi concave on the interval.
Finally, there exists at most one natural monopoly equilibrium {g(x ) = 0 , g(0) = x }NM NM since there is only one x such that g(x ) = 0. (For existence see Proposition 2). ~NM NM
Proof of Proposition 2: The proof proceeds in three steps. First, we characterize c and c and* **
show that if b > 0 then (i) a symmetric equilibrium exists for c0[0, 1]. Second, we prove that: (ii) an asymmetric equilibrium exists for c0[c , c ], and (iii) a natural monopoly equilibrium exists* **
for c0[c , 1]. Third, it is shown that only a symmetric equilibrium exists if b < 0. The same* reasoning applies when the media good is public, but then the equilibrium is conditional on p =Mi p = 0. The expressions corresponding to the public good case are indicated by a PG index.Mj
Step I: Let c (r) be the set of {c, r} for which 1) the equilibrium is symmetric and 2)**
g(g(x)) is tangent to the 45E line. In a symmetric equilibrium, expressions (5) and (1) must hold for x = x = x , which yields;i j
These expressions have unique interior solutions for c0[0, 1] which establishes (i). The expressions in (A3) combined with Mg(g(x))/Mx = 1, implies that c (r) is implicitly defined by:**
[ra]r ' (rc)r a(2&c2&c) Similarly, let c (r) be the set of {c, r} such that the smallest equilibrium quantity is equal to zero.* Thus, c (r) must satisfy 0 = g(g(0)) which, in turn, means that c (r) is implicitly defined by:* *
The right-hand sides of (A4) and (A5) strictly decrease in c. Hence, c (r) and c (r) are unique.* **
We now show that 0 # c (r) # c (r) # 1. First, note that* **
and and that c (0) = c (0) = 1. Furthermore,** ** PG
For c (r) = c (r) in the private goods case the right-hand sides of (A4) and (A5) must be equal.* **
This implies that which is satisfied only by r = 1. The corre-sponding PG condition, is also satisfied only by r = 1. Hence, c* # c**
for public, as well as private, goods. Finally, note that c and c are monotonously decreasing in* **
b. In the limit as b 6 0 , c = c = 1 and as b 6 +4, c = 0. Hence, 0 # c # c # 1. * ** * * **
Step II: Asymmetric equilibria cannot exist unless g(g(x)) intersects the 45E line more than once. The limiting case occurs for c (r) when g(g(x)) is tangent to the 45E line at x, where x lies**
on the upward sloping segment between two peaks of g(g(x)) (see proof of Proposition 1). In this interval g(g(x)) is first strictly convex and then strictly concave. Moreover, for c = c (r),**
M g(g(x))/MxMc < 0 and M g(g(x))/Mx = 0. Hence, an increase in c will cause g(g(x)) to cut the 45E2 2 2 line from above (one fixed point), while a reduction will cause it to cut from below (three fixed points). Since c (r) is unique, this result is independent of the magnitude of the change in c.**
Hence, there are three fixed points for c < c and one for c > c .** **
x1/r' (a%bx)/2.
By construction, c (r) partitions the c r- space [0,1] so that the small firm has a strictly* 2 positive market share if and only if c > c . Thus, an asymmetric equilibrium with strictly positive* market shares for both firms exists for c0[c , c ], which establishes claim (ii). * **
We proceed to claim (iii). Let firm 2 be the small firm and assume that q = q = 0. We--M2 --A2 will show that these beliefs are fulfilled in equilibrium for c $ c (r). *
We begin by examining the public good case. Suppose q = q = 0. The demand facing--M2 --A2 firm 1, xPGm, is then the solution to This follows from expression (1) for c = q =A2 0. The threshold advertising level that makes q = 0 rational for consumers isM2 This follows from expression (1) for q = q = 0. It can be shown that xM2 A2 PGm > x if and only if^PG c > cPG*. Hence, if c < cPG* there exists no equilibrium with zero quantity expectations. We can also conclude that in the public good case x = xNM PGm.
Now consider the private good case. Suppose q = q--M2 ~ = 0. Firm 1, being anA2 unconstrained monopolist, chooses x , which is the solution tom This follows from expression (5), letting c = q = 0. The threshold advertising level that makes q = q = 0A2 M2 A2 rational for consumers is This follows from expression (3) for q = q = 0.M2 A2 It can be shown that x > x if and only if c > c where c is defined implicitly by m ^ *** ***
We can verify that c (0) = c (0) and that * *** It can also be shown that c (r) $ c (r) for r0(0, 1]. Thus, for c $ c , zero quantity expectations for the*** * ***
small firm are supported by unconstrained monopoly behavior by the large firm, i.e., x = x .NM m
x1/r ' bx(2&c2)
[r(2&c2%c)%c2%c&2]1&r and b[ra]r
a(1&cPG) ' 1
[r(cPG%1)&(1&cPG)]1&r
For c0[c , c ] the natural monopolist’s behavior is constrained by the presence of firm* ***
2. By rewriting expression (3) we obtain the following condition determining the duopoly level of advertising, x , given q = q = 0: d --A2 --M2
By construction, x = x for c = c and x = x for c = c . Since x decreases monotonically in c,^ d * ^ m *** ^ we have that x # x # x when c0[c , c ]. Now, what expectations regarding firm 1 are consistentm ^ d * ***
with q = q = 0? If c0[c , c ] and q = q = 0 then firm 1 optimally sets p so that x = x.--A2 --M2 * *** --A2 --M2 M1 ^ If x > x, firm 1 can raise p (towards the monopoly level) without threat of competition, while^ M1 if x < x (whereby firm 2 faces a positive demand), firm 1 would want to lower p (towards the^ M1 duopoly level). Thus, x = x = x and q = q = 0 is a rational expectations equilibrium forNM ^ --A2 --M2 c0[c , c ]. If c < c , then x > x which is inconsistent with existence of a natural monopoly* *** * ^ d equilibrium.
Step III: Finally we show that there exists a unique symmetric equilibrium for b < 0. If b < 0 then no c0[0, 1] satisfies (A4) and (A5). Since c (0) and c (1) decrease in b, it follows that* * c , c > 1 for b < 0. Thus, only symmetric equilibria exist. The equations determining symmetric* **
equilibria (A3) yield unique positive equilibrium quantities also for b < 0 and r > 1. ~
Proof of Proposition 3: The expressions in (A4) can be rewritten as
respectively. No {r, c}0[0,1] satisfies these equalities simultaneously. Hence, c (r) in the private2 **
good case cannot intersect the corresponding function in the public good case and, by example, we know that c (r) is smaller in the latter case. The same argument applies for c (r). ~** *
the large firm in an (non-monopolistic) asymmetric equilibrium and dotted lines the small firm.