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