but it is a negative externality on the B’s. The subsequent analysis picks up the narrative
with the simplification of neglecting integer constraints in the number of firms, but the
broad story is coherent with that just outlined, and extends it to more firms.
1.3.7 Multiple Stations
To analyze preference externalities in largermarkets with two taste groups and with multiple
stations of each type, we ignore the integer constraint, and first determine the equilibrium
numbers of firms of each type. There are two equations in two unknowns (n
w
and n
b
):
M
W
W
b
+ M
B
B
b
¼F
M
W
W
w
+ M
B
B
w
¼F,
which are the zero profit conditions for b and w stations respectively.
These equations define two entry reaction functions for the two station types in terms
of numbers of each type. We solve for the numbers via a more indirect method. That is,
we write the second equation above in terms of the variables in the first one, which
enables us to solve for the equilibrium market shares of each station. We then use the
share expressions to solve for the equilibrium numbers, and with this information we
can find the equilibrium consumer surplus for each viewer type.
First note that
W
w
¼
W
b
exp2s and
B
w
¼
B
b
exp 2sðÞ(using the odds ratios of the
different types),
30
and thus we can rewrite the two equations to be solved simultaneously as
M
W
W
b
+ M
B
B
b
¼F
M
w
W
b
exp2s + M
B
B
b
exp 2sðÞ¼F:
The solution yields the two choice probabilities as
W
b
¼
F
M
W
1
exp 2sðÞ+1
and
B
b
¼
F
M
B
exp 2sðÞ
exp 2sðÞ+1
(with analogous expressions for the w stations). These expressions
already give us the breakdown of listeners as
B
b
W
b
¼
M
W
M
B
exp 2sðÞ: In equilibrium then,
the ratio of own-side to other-side listening of any given station is proportional to the
ratio of other-side to own-side populations! The fraction of listeners to a b station is
slanted more toward B-types when there are more W’s around because then there are
a lot of w stations so the W’s are much more likely to find what they want among the
w stations, and few will listen to the b station. So, while it may seem that the listenership
is quite segregated by this metric of relative listenership, it is rather that the majority group
has a lot of choices. Conversely, a population with a large B representation will have a
more even listenership for each b station.
30
Hence own type is preferred by the factor exp(2s).
25
Preference Externalities in Media Markets
Now we can find the equilibrium numbers from the conditions
B
b
¼
exps
1+n
b
exps + n
w
exp sðÞ
B
b
¼
exp sðÞ
1+n
w
exps + n
b
exp sðÞ
:
These imply that
1+n
b
exps + n
w
exp s
ðÞ
¼
M
B
F
exps + exp s
ðÞðÞ
1+n
w
exps + n
b
exp sðÞ¼
M
W
F
exps + exp sðÞðÞ,
and from there we can solve out for the equilibrium number of w- stations as
n
w
¼
exps
exp2s +1
+
M
W
F
exp2s
exp2s 1ðÞ
M
B
F
1
exp2s 1ðÞ
:
The equilibrium n
b
expression just transposes M
W
and M
B
.
Before we turn to the precise preference externality that we get from analyzing the equi-
librium welfare of the two groups, there are several take-aways from these numbers. First of
all, clearly the number of stations increases in own-side market presence, and decreases (lin-
early) in other-side market presence. That is, B presence crowds out w stations, ceteris paribus.
This effect was already apparent in the monopoly analysis above. Below we look at the ratio
of the two types of station as a function of the group populations. Second, if market presence
is the same for both groups (M
W
¼M
B
), then the total number of station is
2exps
exp 2sðÞ+1
+
M
F
,
where M is total population. Recalling the single market case had n ¼
M
F
exp sðÞ
, this implies
that there are more stations in a homogeneous market.
31
The reason is that the population’s
tastes as a whole are better matched on average. Of course, there are several caveats to such
conclusions: for example, the analysis has assumed that stations are symmetric. In many
instances stations differ substantially in their profitability and listener base sizes.
Third, the equilibrium ratio of firms is
n
w
n
b
¼
exps 1 exp2sðÞ+ exp 2s +1ðÞ
M
W
F
exp2s
M
B
F
exps 1 exp2sðÞ+ exp 2s +1ðÞ
M
B
F
exp2s
M
W
F
:
To get some traction on how this depends on the total population make-up, if the M’s are
large then this is approximately
M
W
F
exp2s
M
B
F
M
B
F
exp2s
M
W
F
¼
kexp 2s1
exp2sk
, where we have set M
W
¼kM
B
.
31
Comparing numbers, the statement holds if exp sðÞ<
2exp s
exp2s +1
, which is true for s > 0:
26
Handbook of Media Economics
In the relevant range (for positive numbers of each station type), this is an increasing and
convex function of k. Hence, the fraction of w-type stations increases with the fraction
of W-types, and does so at an increasing rate. As with the monopoly analysis earlier, the
b-stations get increasingly crowded out by w-types, which nonetheless attract B-type
listeners as they provide more and more alternatives. The majority tastes increasingly
dominate the market’s offerings. However, this market tyranny is perhaps somewhat
more benign than it might appear because in the model the B’s do benefit from the
increased variety of w stations.
To analyze this effect in more detail, we now turn to the groups’ welfare. From the
Log-Sum formula for consumer surplus in the logit model, the expected welfare of an
arbitrary W-type is ln n
w
exps + n
b
exp sðÞ+1ðÞ. Using the equilibrium values for sta-
tion numbers, this welfare is an increasing function of the expression
exps
exp2s +1
+
M
W
F
exp2s
exp2s 1
M
B
F
1
exp2s 1
exps
+
exps
exp2s +1
+
M
B
F
exp2s
exp2s 1
M
W
F
1
exp2s 1
exp sðÞ:
Clearly per W welfare increases in W market presence, so there are positive own-side
preference externalities. But the other striking feature of the expression is that it is inde-
pendent of M
B
. This says that there are zero preference externalities from the other side.
Given the empirical findings in this regard (none or mildly negative cross effects), this is
quite a compelling benchmark. Here two effects are canceling out. First, a larger B pres-
ence would mean more b stations, which has a beneficial effect on W welfare through
more choice. And indeed the total number of stations is higher. But more B’s also implies
some crowding out of more highly valued w stations, which depresses welfare.
To put this last point in a wider perspective, recall that the model has simplified by
assuming strong symmetry in station valuations, namely that the other-side attractiveness
is s. If instead the other-side valuation were higher, then the preference externality
would be positive: the first effect would dominate.
32
However, if the other-side valua-
tion were lower, then the crowd-out effect would dominate, and the preference exter-
nality would be negative.
Finally, with the welfare analysis in hand, we can look at the equilibrium mix of lis-
teners in the market, which is another empirically measurable statistic that can be tracked
as a function of population composition. The fraction of W’s listening is
n
w
exps + n
b
exp sðÞ
1+n
w
exps + n
b
exp sðÞ
while the B fraction is analogous. We already effectively determined
the behavior of this expression in the welfare analysis. In particular, the numerator is
independent of M
B
and so therefore is the denominator. This means that the fraction
32
To see this, suppose both station types had the same attractiveness, s. Then more of the “other” type is
good, because they are equally valued, and more other-side presence just increases variety.
27
Preference Externalities in Media Markets
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