USER GENERATED CONTENT AND BIAS IN NEWS MEDIA:
WEB APPENDIX
Proof of Lemma 1: In order to support segmentation the expression inside the radicals in (7) and (8) (the
discriminants of the quadratic equations) have to be positive. Note that (8) can be rewritten as:
)̂
(
(
= √(
)
)
(
)(
)(
)
.
Therefore, for segmentation we need:
)̂
(
(
)
.
(WA.1)
Also note that when segmentation exists, ̂
satisfied when
(
) √(
. Using (14) in (WA.1), the last inequality is
)
(
) √(
)
(
) √(
)
̂
(
) √(
)
̂
(
and
) √(
) √(
)
)
. Notice that
for
. Thus,
. The restrictions for Newspaper 1 are similarly found. Thus,
̂
̂
(
We designate by
(
̂
̂
.
) the added utility that a reader having beliefs b derives from the
print over the online edition of Newspaper 2 given that the online segment comprises of readers in the
interval [ ̂
̂
] (note that at the symmetric equilibrium
(
.) Then, from (3) and (4):
̂
(
(
(̂
(̂
̂
)
(
̂
)
)
(
)
.Then, using (12) and
(̂
̂
Note that for
,
)= ( ̂
(
̂
(
[(
. We define ( ̂
))]
) by
)
̂
̂
and
)
)
(
) as the function obtained by multiplying
(
in
)
( ) is a concave function of ̂
̂
(
), we obtain:
)(
̂
))
(̂
)
exists, we
investigate whether H( ) changes its sign from positive to negative over the interval ( ̂
yields:
)
and
(̂
)
.
, implying that it can change sign from
positive to negative at most once. To guarantee that a root to the equation
Specifically, whether ( ̂
̂
. Evaluating the function ( ) at ̂
̂
).
and ̂
(̂
)
(̂
)( (
(
(
)
)
)
Requiring that ( ̂
(
) √(
)
))
, and
(WA.2)
.
)( (
(
(WA.3)
(̂
and
)
(
)
yields:
) √(
)
(
))
.
if and only if
.
(̂
Note that the function H( ) is concave because
and ̂
(
̂
(̂
)
)
(̂
. It obtains its maximum value in the range [0,
) √(
)
(
)
)
))
̂
. Because
)
. It follows, therefore, that the root of
. In Figure WA. 1 we depict how H( ) changes as a function of ̂
𝐻( )
̂
𝑏
𝑏̂ 𝐸 𝐸
(
𝛼) √(
𝛼)
𝑎
when
as well. The threshold reader who is indifferent between
the print and the online editions satisfies the equation ( ̂
the last equation is bigger than
(
] at
. It is easy to show that ̂
(̂
, it follows that
̂
(
)
𝑎(
𝑎)
𝑏
𝑏
(
𝛼)
√(
𝛼)
𝛼
8𝛼(
𝛼)
𝑏
𝐸𝐸
𝑏̂ 𝐸 𝐸
.
Figure WA. 1: Implicit Determination of the Value of ̂
𝑏
𝑏̂ 𝐸 𝐸
.
Proof of Proposition 1: In what follows we will first show how (14) is derived. Then, we will use this
equation to demonstrate the results in this proposition.
From (9) and (10) optimizing with respect to the subscription prices
,
and
yields the following first order conditions:
((
̂
) ((
)
(
))
) ((
)
(
)))
̂
(
(
̂
(
̂
((
) ((
)
) ((
)
̂
(
(
(
̂
̂
(
)
Substituting (5) and
))
,
(WA.4)
(
)
)
,
(WA.5)
)
))
(
))
(
(
(̂
According to (7)
̂
(
̂
))
,
(WA.6)
)(
)
)
.
(WA.7)
)
. Using this relation and summing (WA.4) and (WA.5), we get:
.
(WA.8)
(
(
derived from (5) into (WA.8) we obtain:
)
)
)(
.
(WA.9)
Using a similar approach, the subscription fee for the print edition of Newspaper 2 can be derived as:
(
)
)(
Solving (WA.9) and (WA.10) for
(
(
)
̂
(
(̂
)
)
(
)
̂
(
) ̂
)
(
))
, and substituting
from (7), we obtain:
(̂
(
(WA.10)
we get the expressions in (11). Rewriting (WA.4) as
)
(
(
and
.
)((
)
̂
(
)
)
.
)
(WA.11)
Using a similar approach for Newspaper 2 we find:
(
)
(
)
(
(
)
̂
)((
)
̂
(
)
)
.
(WA.12)
Using the Envelope Theorem when optimizing with respect to
and
we get:
̂
,
(WA.13)
.
(WA.14)
̂
̂
̂
To illustrate the derivation of the equilibrium location choices, we focus on the optimization of
Newspaper 2. From (10) and (WA.14):
̂
(
((
(
)
))
Using (12), and
̂
(
)
(
)
(
)
(
(
)
̂
)
((
(
(
)(
,
(
(
)
)
(WA.15)
,
)
), and
)
))
(
) from (5), (8), and (11)
in (WA.15) yields:
(
)
̂
( (
̂
) (
(
)
At the symmetric equilibrium,
(WA.16) and solve for
( (̂
(
Finally,
)
)=
(̂
(
(
)
(
̂
(
)
(̂
)
1
(WA.16)
and
with
) ).
)
)
(
̂
)
when segmentation exists according to Lemma 1.
̂
(
)
̂
(
)
̂
(
̂
)
(
(
̂
)
)
( ̂
(
) )
when segmentation is possible.
(̂
̂
)
when
(̂
)
, given
( ) changes its sign from positive to negative at this point in order to support segmentation.
Note that in a similar fashion the first order condition with respect to
(
)
in
(WA.17)
,
Proof of Proposition 2: From the proof of Lemma 1 we know
1
))
required for segmentation implies
since
that
)
therefore, we can replace
(
)
)
since ̂
(
)(
to get (14), which can be rewritten as:
Notice that ̂
. Further, (
)
(
( (
̂
) (
̂
)
(
can be derived as:
(
)(
)
)).
̂
Using the Implicit Function Theorem we can write:
{
Notice that
{
(̂
̂
}
)
}
(̂
{
(̂
(
{
)
} since
(
(
(̂
̂
)
)
̂
) )
}
)
or .
, since
̂
(
̂
, and
when
(̂
)
̂
)
̂
>0,
when segmentation exists.
̂
̂
(
(̂
) , which implies:
)
)
(( ̂
)
(̂
̂
. Thus, from (14):
(̂
Again from (14),
(̂
)
and
(
since according to Lemma 1,
̂
)
)
̂
)
((
(̂
̂
)̂
(
)
segmentation is supported. That is,
(
)
(̂
(̂
̂
)). Substituting for
)
)
(̂
and
)
into this expression,
we obtain:
(̂
(
(̂
)
)̂
( (
)
(̂
(
(
)
)
)̂
)̂
(
Notice that the denominator in (WA.18) is negative since
)̂
(
(
)
(
according to Lemma 1. Thus,
(
̂
)
)
(̂
̂
)
(
)̂
. In the numerator, (
)
)
(
)
(
)
).
(̂
(WA.18)
)
since ̂
.
Proof of Proposition 3: Using a similar approach, as in Section 3.1 of the paper, the expressions for the
and ̂
threshold consumers
as functions of the decisions made by the newspapers in the
three stages of the game are:
=
(
)
(
̂
(
(
)
(
)
)
√(
)
)
,
(
)
(
(WA.19)
(
(
)
)
.
)
(WA.20)
The payoff functions of the newspapers are:
(
(
)
̂
)
(
(
),
)
(WA.21)
(̂
)
(
).
(WA.22)
From (WA.21) and (WA.22) optimizing with respect to
and
, yields
expressions similar to those derived when both newspapers extend their product lines. Specifically,
similar expressions to (11) and (WA.12), as follows:
(
(
)
(
)
)
(
(
)(
(
)(
(
(
)
(
)
(
),
)
̂
), and
̂
)
)((
(WA.23)
(
(WA.24)
)
)
.
)
(WA.25)
Using the Envelope Theorem when optimizing (WA.21) and (WA.22), with respect to
and
we get:
(
)(
)
,
(WA.26)
̂
(WA.27)
̂
̂
((
)
(
(
))
)(
)
Substituting (WA.23) and (WA.24) in (WA.19) yields:
(
)
(
)
.
(WA.28)
).
(WA.29)
From (WA.24):
(
)
(
Substituting (WA.23), (WA.28) and (WA.29) into (WA.26) we obtain:
.
(WA.30)
According to (WA.19)
=
. Substituting in (WA.27) the last relation, (WA.24),
(
(WA.25), the fact that from (WA.19)
(
equation in
(
)
)
(
)
, and from (WA.23)
(
and
)
(
) yields a quadratic
as follows:
)
((
(
)
8
(̂
)
(
)
))
.
(WA.31)
Solving (WA.31) for
and choosing the root to ensure that
in (16). Substituting (16) in (WA.30), we obtain
follows that
̂
(
when segmentation is supported.
, we will first show that ̂
In order to demonstrate that
solution for ̂
as given
as expressed in (15). From these solutions, it
, since ̂
, and
, we obtain
̂
̂
. Note that the
can be obtained implicitly as in the proof of Lemma 1 as follows. We designate by
) the added utility that a reader having beliefs b derives from the print over the online
edition of Newspaper 2, given that the online segment comprises of readers in the interval [ ̂
].
Then, from (3) and (4):
̂
(
)
̂
(
[(
)
(
)
̂
(
(
)
))]
.
We define
(̂
̂
using (WA.25) and
(̂
)
(
in
( ̂
̂
(
)
( (̂
Notice that
(
(̂
)(
)
(
)
)
√
(
( ) from the proof of Lemma 1,
the function defined in (WA.32) is negative when evaluated at ̂
, it follows that ̂
(
√
(̂
) by
(
)
.Then,
))
(WA.32)
as in (16).
)
Hence, using the definition of function
̂
̂
), we obtain:
is expressed in terms of ̂
where
(̂
) as the function obtained by multiplying
̂
As a result, from (16)
)
(
)
)
(̂
(
)
(̂
)
)
(̂
)
). Since
and it should be equal to zero at
.
Finally, substituting (15), (16), (WA.23) and (WA.24) into (WA.19) yields:
(
√
(̂
)
(
)
)
.
(WA.33)
Proof of Proposition 4: For ease of exposition we drop the superscript E,E in all the variables. When
, the cutoff points ̂ are still given as in (6). However, because readers have no
Newspaper i chooses
ability to affect
via UGC, they consider
(
̂
The expression for
)
exogenous. Specifically,
.
(WA.34)
remains as in (5) and the objectives of the firms are still given in (9) and (10).
Optimizing with respect to
[
(
[
(
and
in Stage 3 yields for
)
(
)
(
)
)
Substituting the expressions for
(
a solution identical to (11) and for
)(
(
)],
)(
)].
(WA.35)
back into the expression for ̂ in (WA.34), yields that
and
̂
segmentation is feasible at the symmetric equilibrium, specifically at
(
)
(
)(
(
)
)
(
This implies from (WA.35) that:
)
(
](
{[
)
)(
).
(WA.36)
(
)
̂
)
(
and
)
and
yields:
(
)
̂
},
(WA.37)
,
(WA.38)
is given in (5) and ̂ in (WA.34). Evaluating (WA.37) and (WA.38) at the symmetric
where
equilibrium where
expressions for
{
,
and
,
,
)
[(
(
̂ , while using the equilibrium
)(
(
(
̂
from (11) and (WA.35) yields:
)
(
)]
)
(
) [
(
(
)
)(
(
)
},
(WA.39)
)
) (
)
)
(
Assuming an interior equilibrium with
(
if:
. In the second stage, each newspaper chooses
. For Newspaper 1, differentiating (9) with respect to
(
:
(
)
(
)
].
, and since (
implies that
)
),
(
)
(
)(
)
(
)
.
(WA.40)
Substituting (WA.40) into (WA.39) yields that:
[
(
)
]
(
)
.
(WA.41)
(
To ensure segmentation, the lower bound on
(WA.40) implies that
of
and
)
from (WA.36) should hold, which combined with
. Using the last inequality in (WA.41) implies that
for all values
. Hence, Newspaper 1 will choose the lowest bias consistent with
that
, implying
and no segmentation arises. A similar argument holds also for Newspaper 2.
Proof of Proposition 5: Substituting (15), (16), (WA.23), and (WA.33) into (WA.21) yields:
(
)
(̂
√
(
(
)
)
)
(WA.42)
Evaluating (11) and (12) at the symmetric equilibrium (when
̂
̂
̂
and
), we obtain:
–
(
,
)
(WA.43)
(
̂
(
(
)
)
)((
)̂
(
)
).
(WA.44)
Substituting (WA.43), (WA.44) and (14) into (9) we get:
(
̂
(
(
)
(̂
)(
From (WA.45),
(
(̂
)
̂
(
̂
)
̂
(
)
)
)
(WA.45)
, when segmentation exists. Because the
as well.
Since from the proof of Proposition 3, ̂
(
)
since
)
firms are symmetric
(
)
(̂
)(
(
)
)
(̂
)
(
̂
̂
)
, it follows that:
(
)
)
√
(
(̂
)
(
)
)
(
(
(
))
)
.(WA.46)
The second term subtracted inside the parenthesis of (WA.46) is:
(
̂
)(
(̂
)
(
(̂
)
)
(
)
̂
(
)
)
(
√
(̂
)
(
)
)
(WA.47)
(WA.47) decreases with ̂
(
)
(
√(
))
. Therefore at ̂
which is less than
, it obtains its maximum value of
when
possible according to Lemma 1. Thus,
Note that
(
̂
)
, which holds when segmentation is
.
(
)
(̂
)
(
)
̂
(
)
(
(̂
)
)
(
Substituting (15), (16), (WA.24), and (WA.33) into
(
(̂
√
(
)
(
)
(̂
√
) (
(
)
(
)
(
)
)
)
(
(
).
) implies
)
)
(
̂
, since
)
, when segmentation is supported.
Proof of Corollary 1: Notice that when
,
Therefore, in our formulation, choosing
is equivalent to not expanding the product line. From Proposition 5 we know that when one
newspaper does not expand, the other chooses to expand. Hence, the outcome
cannot
correspond to an equilibrium. We also know from Proposition 5 that when one newspaper expands, the
other chooses to expand as well. Therefore, in equilibrium each newspaper chooses a positive
.
Weights of Reporting Strategy: We start by deriving the optimal weights for the case {NE, NE}. Let the
reporting strategy of newspaper be specified as:
(
)
.
Calculating the expected utility of a reader of type
(
̅–
)
The expression for
(
(
[
when subscribing to , we obtain:
) ]
(WA.48)
can be derived by solving the equation
)
, where
for . We obtain:
[ (
)
((
)
(
) )]
.
The objective functions of the newspapers can be expressed as:
(
)
(
, and
Each newspaper chooses
and
)
.
in Stage 2 and
subsequently in Stage 3 (assuming that in Stage 1
both newspapers choose NE). Optimizing for prices we can obtain the Nash Equilibrium prices of the
third stage as follows:
[
(
)
],
[
(
)
].
Substituting the Nash equilibrium prices back into the objective functions, we can obtain the second stage
payoff functions when
̂
and
are selected.
[
(
)
] ,
̂
(
[
)
] .
Optimizing the second stage payoff functions, we obtain2:
and
(
)
.
Hence, the optimal weights are as stated in the main text. Note that the optimal
magnitude of the coefficient of
ensures that the
in the expected utility of each reader given in (WA.48) is minimized.
Because we are unable to obtain close form solutions for the subgames {E, E} and {NE, E}, we
cannot use the same approach for deriving
as we do for the subgame {NE, NE}. Instead, we will
provide an intuitive explanation for why we believe the weights are likely to remain the same in the other
subgames as well.
To develop this intuition, we consider first a simplified Hotelling model of horizontal product
differentiation with quadratic transportation costs combined with an additional attribute of vertical
differentiation that we designate by
the consumer’s type
(
̅
, and
)
.
when buying from firm is:
( )
,
where ( ) obtains its maximum value where
( )
, irrespective of the value of . Formulating the
environment as a two stage game, where in the first stage each firm chooses
chooses its price
and attribute
is distributed along the horizontal dimension uniformly on the interval
The net utility of a consumer of type
( )
for firm . Each firm chooses its location
and
and in the second it
, it is easy to show that the Nash Equilibrium is characterized by
and
( )
, where
. Hence, the vertical attribute is chosen so as to maximize the expected
utility of the consumer, and the horizontal attribute is chosen as in the {NE, NE} subgame.
Next we argue that in our environment as well there are two dimensions of differentiation
between the newspapers. The location
is the choice along the horizontal dimension, and the weight
is the choice along the vertical dimension. The expected utility of an individual when buying the print
version of newspaper as a function of the location, weights, and price chosen by the newspaper is:
(
̅–
)
[
(
) ]
,
and when buying the online version it is:
2
The expression for ̂ can be written as: ̂
[ (
)
)
((
(
)
(
) )]
and
(
(
)
(
)] where
. Instead of optimizing ̂ with respect to
calculations are simpler by optimizing it with respect to
optimizing ̂ .
)[
and
and
directly, the
. A similar simplification can also be used when
(
̅–
)
(
[
) ]
The expected utility expressions can be written, therefore, as:
̅
where
(
( )
)
̅
, and
is a quadratic function of the product
( )=0 or
( ) where
the function
second stage and
and
(
(
(
)
)
( )
,
), and the value of the coefficient
. In our three stage model,
and
maximizes
are chosen in the
are chosen in the third stage. Based upon expected utility expressions, the
profit functions of the newspapers will depend on the locations only via expressions of the form
. The dependence on weights will include also an additional term that depends on
and
independent of
as implied by the term ( ) in the expected utility of the reader.
When the newspaper optimizes its profits in the second stage, the optimization with respect to the
coefficient
product
includes two terms. The first term is optimization of expressions that include the product
) as derived from the function ( ) and the second term is optimization of
(or
expressions that include
independent of
or
optimization of the first term with respect to
to
, as derived from the function ( ). However, the
is equivalent to the optimization of this term with respect
, implying that the derivative of this term with respect to
The optimal value of
(
is selected optimally.
will be determined, therefore, from the optimization of the second term, in which
in the function ( ). Hence, the optimal
appears independent of
implying that
vanishes when
)
will still maximize this function,
irrespective of the subgame considered. Intuitively, given that
is an attribute
of vertical product differentiation, each newspaper is forced to choose its value in a way that maximizes
the reader’s expected utility in order not to lose market share to the competing newspaper.
Expansion of Readership Facilitated by Online Editions: In this section we demonstrate that the
profits of each newspaper may rise with the introduction of an online edition, if in the absence of offering
such editions the market of readers is not fully covered. In Figure WA.2 we depict this possibility.
Withdraw
from the
Market
Buy print
edition of
Newspaper 1
̃
Buy print
edition of
Newspaper 2
Withdraw
from the
market
̃
Figure WA.2: Market Less Than Fully Covered When Only the Print Version is Offered
Less than full coverage implies that the expected utility of readers with very extreme political
opinions is negative when newspapers offer only print editions, namely
̃ , and
for ̃
for
̃ and
̃ . Readers located at ̃ and ̃ are just indifferent between
buying the print edition of newspapers 1 and 2, respectively, and withdrawing from the market. Solving
for ̃ and ̃ yields:
̃
(
√(
)
[
)
̅
(
)
(
)
] ,
(WA.52)
̃
(
√(
)
[
)
̅
(
)
(
)
] .
Note that the expressions in the brackets included in the radicals of (WA.52) are positive, since
the expected utility of a reader located at b=0 is positive according to Figure WA.2. Hence, ̃
and ̃
(
)
. The objectives of the two newspapers are:
(
)
̃ )(
(
)
Optimizing with respect to
,
, and
̃( ̃
(
) (̃
̃
̃
(̃
;
)
(
)
)
)
.
(WA.53)
yields at the symmetric equilibrium when
,
̃ that:
)
(
)(
and
)̃
(
,
(WA.54)
where the subscript less in (WA.54) indicates that the market is less than fully covered. It is possible to
find an upper bound on the equilibrium profits in (WA.54). Specifically,
.
(WA.55)
Now, assume that extending the product mix by introducing the online edition allows the
newspapers to cover the entire market. Specifically, the expected utility of readers located at
is strictly positive at the {E, E} equilibrium. Hence, [
(
)
and pays the fee
]
when reader
and
is exposed to the bias
. From (WA.45), it is possible to derive a lower bound on the expected
profits of each newspaper for the region of
values that support segmentation, as specified in Lemma 1.
Specifically,
(
)
(WA.56)
A comparison of (WA.55) with (WA.56) implies that the expansion of the readership that is
facilitated by the extension of the product mix will definitely increase the profits of each newspaper
provided that
8 , namely that readers are not overly concerned about inaccurate reporting. Note
that this condition does not contradict the requirement for less than full coverage in the absence of
segmentation. A necessary condition for the latter is that
values for the ratio
. Hence, there is a nonempty interval of
that is consistent with the result that introducing the online edition may increase the
profits of the newspapers. This increase in profits is different from the result reported in Proposition 4,
when the market was fully covered even in the absence of segmentation.
An Additional Dimension of Heterogeneity among Readers: In this section we extend our model by
allowing for a second dimension of heterogeneity among readers with respect to their preference for the
print versus the online editions unrelated to political opinions. This preference may be related, for
instance, to the age of the reader, with younger readers usually preferring the online edition, and older
readers being more comfortable with the traditional, print edition. Specifically, we assume that the
expected utility of a young reader increases by
and that of an older reader decreases by
when choosing the online edition. Modifying (3) we obtain:
={
̅
[(
̅
) ]
[(
) ]
With the above modified expected utility, the threshold reader who is indifferent between the
online and print editions of a given newspaper is different for the young and old populations. Specifically,
̂
̂
and ̂
̂
(
̂
)
(
̂
(
)
√(
)
(
. For Newspaper 2, for instance, adjusting (8) yields:
)
(
)
(
√(
)
)
(
(
)
(
)
(
,
)
(
)
(
)
)
.
)
We assume that the populations of young and old readers are still each uniformly distributed on
the interval [-
] according to their political opinions, and that these distributions are independent of
age. The proportions of young and old in the general population of readers are (1-q) and q, respectively.
We derive Proposition WA.1 from this modified model.
Proposition WA.1: When there exists an additional dimension of reader heterogeneity, unrelated to
political opinions, each newspaper chooses to intensify the bias of its print edition and the expected size
of the online segment declines. Moreover, when the variability in the population that is unrelated to
political opinions increases ( i.e., when (
) and ( ̂
̂
) are bigger), the polarization of
the newspapers becomes more significant. Polarization remains, however, more moderate than in an
environment with only print editions.
Proof: We drop the superscript E,E to simplify the notation, and write the objective of Newspaper 2 as:
{( ( ̂
(
)
)(
)
{(( ̂
(
)(
)
̂
)(
(
̂
))}
)(
))}.
A similar expression can be derived for the objective of Newspaper 1. Using an approach similar to that
leading to the first order condition (WA.16), yields that the optimization with respect to
in the second
stage can be expressed as:
|
(
[
(
where [ ̂ ]
[ ̂ ]) (
(
) (
)
)
̂
( [ ̂ ]) )
[ (
)( ̂
̂
)( ̂
(
(
)̂
(
) (
)̂
]
)
)
]
,
(WA.57)
. The first term of (WA.57) coincides with the first order
condition (WA.16) that was derived when politics was the only differentiating attribute among readers,
with the only difference being that [ ̂ ] replaces ̂
in (WA.16). The second term is positive, and
measures the extent of heterogeneity between the young and old populations. This second term is bigger
when the variance due to the different ages in the general population is bigger (the product (
when the difference ( ̂
and
̂
) is more significant (as implied by the different values of
.) Evaluating (WA.57) at the point when [ ̂ ]
2 has to increase
beyond
intensifies, and since
̂
implies that
[ ̂ ])
[(
)
(
(
[ ̂ ])
(
(
̂
the equilibrium
)
, and
[
.
)( ̂
(
. As a result, to satisfy (WA.57)
is still smaller than
̂
)
because the term inside the brackets is positive and bigger than
(̂
, hence Newspaper
in order to satisfy the first order condition (WA.57). Hence, bias
and [ ̂ ] move in the same direction, [ ̂ ]
In addition note that
)) and
(
̂
)( ̂
)
) (
(
)
)̂
)
]
], given that [ ̂ ]
,
, implying that at
. In addition, the expected equilibrium profits are still smaller
when both editions are offered given that the newspapers would not choose segmentation according to
politics if they had full control over the attributes of both editions.
The result reported in Proposition WA.1 is consistent with that reported in Proposition 4.
According to Proposition 4, if newspapers could fully control the bias of their online editions, they would
choose it to be identical to the bias of their print editions. Once some of this control is transferred to
readers via UGC, the political segmentation of readers leads to intensified bias of the online version and
reduced bias of the print version. However, if there is additional heterogeneity among readers that is
unrelated to politics, newspapers can move closer to the outcome they would choose if they had full
control over the characteristics of the online editions. Specifically, while the bias of the print version
remains smaller than
, it moves closer to this value. As well, the segment of consumers who choose
the more biased online editions shrinks. Moreover, as the variability in the population that is unrelated to
political opinions increases, the equilibrium moves closer to that described in Proposition 4, when
newspapers have the exclusive right to choose the online bias.
Product Proliferation in Models of Horizontal Differentiation: In this section, we consider a
traditional Hotelling model with quadratic transportation costs to investigate whether firms find it optimal
to proliferate their product mixes at the equilibrium. We consider a Hotelling model where consumers’
preferences are uniformly distributed over the support
the transportation cost
where
, as in our paper. Each consumer incurs
is the distance between the location of the consumer and the location
of the brand the consumer chooses. There are two firms with each offering potentially two brands. The
brands of Firm 2 are located to the right of the brands offered by Firm 1. The locations of Firm 2’s brands
are designated by
Hence,
and
and
with
and those of Firm 1 by
and
with
.
designate more extreme versions on the product line offered by Firm 2 and 1,
respectively. If the equilibrium is characterized by product proliferation by both firms, the segmentation
of the consumers is as depicted in Figure 1.
As in our model, we designate by
Firm 2, and similarly
and
and
the prices of the two brands (
and
designate the prices chosen by Firm 1. We formulate the game as a two
stage model. In the first stage, each firm chooses the locations of its two brands
and
second stage, after observing the locations of all four brands, each firm chooses its prices
As in our derivations, we designate by
and in the
and
.
the consumer who is indifferent between brands
and by ̂ , the consumer who is indifferent between
and
) chosen by
and
. We obtain the expressions for
these threshold consumers as:
̂
,
(
)
(
)
, ̂
(
.
)
(WA.58)
With product proliferation in the second stage the firms choose prices to maximize their profits as
follows:
[(
[( ̂
where ̂ and ̂ , and
locations as follows:
̂ )
)
(̂
(
) ],
̂ ) ]
(WA.59)
are given by (WA. 58). These optimizations yield prices in terms of the
[ (
(
)
(
)(
)
(
],
)(
[ (
),
(
)
),
)
].
(WA.60)
If an interior equilibrium exists, in the first stage Firm 2 optimizes with respect to
(
)
̂
(WA.61)
By the Envelope Theorem derivatives with respect to
symmetric equilibrium with
with
as follows:
̂
)
(
and
and
and
vanish for Firm . Considering the
, it is easy to show that there is no equilibrium
that satisfies the system of equations (WA. 61). Hence, there is no symmetric equilibrium
with product proliferation. Considering the case that
and optimizing locations in the absence of
product proliferation, yields a solution we report in the paper (In Proposition 4), namely that
. However, this solution implies that firms choose locations outside the support
optimization with respect to
, namely the
is unconstrained (similar to the assumption made in Zhang (2011)). Next,
we consider whether an equilibrium with product proliferation would arise if the locations chosen by the
firms are constrained to be inside the interval, namely if locations have to satisfy
and
similarly for Firm 1.
Constrained Optimization: Given the extreme locations for
and
chosen in the unconstrained
and –
optimization, we consider the possible existence of an equilibrium with
. Evaluating the prices and the derivatives
̂
and
such an equilibrium, we obtain the following expression for
(
)(
)
=
(
from (WA.60) and (WA.58) at
in (WA.61):
(
) (
)
.
)
(WA.62)
|
Evaluating the right hand side of (WA.62) at
yields
there is an interior solution
that satisfies
at such an equilibrium (when
) yields:
(
)
(
)
[ (
)
(
and at
yields
. Moreover, evaluating
)]
. Hence,
from (WA.61)
for all
. Hence, choosing
as the biggest value for
in the feasible region is, indeed,
optimal.
To ensure that product proliferation will indeed arise at the constrained optimization we have to
compare Firm 2’s profits when choosing a single product at
, where
with its profits when
solves (WA.62). The profits of Firm 2 when choosing one variant
is
Its profit when choosing two variants can be calculated as:
(
)
(
.
)
(WA.63)
It can be verified that the right hand side of (WA.63) is smaller than
for all
. Hence,
proliferation does not arise in the constrained environment as well.
Are There Incentives to Block Off the Online Version in Stages 2 and 3?
In this section we investigate whether each newspaper has an incentive to eliminate the online segment by
unilaterally deviating from the equilibrium behavior of {E, E} regime. In Stage 2 the newspaper can do it
by modifying its location
and/or weight
, and in Stage 3 it can do it by raising
to eliminate
demand for the online version.
Is Deviation in Stage 2 Optimal?: We first derive the expected profits of the deviating newspaper by
assuming that it can eliminate demand for the online version (i.e., ̂
̂
for Newspaper 2), while still keeping the coefficient
contemplates the deviation and chooses the new location
location of
(
for Newspaper 1 and
)
. Assuming that Newspaper 1
, while Newspaper 2 sticks to the equilibrium
, we can use the derivations of the {NE, E} regime to obtain from (WA.30) that:
.
The price that the deviating newspaper will charge in the third stage can be derived from (WA.23) as:
(
)
(
)(
(
)
)
And the expression for the newspaper’s market share
(
)
(
)
can be obtained from (WA.19). (WA.23),
and (WA.24) as follows:
.
Hence, the profits of the deviating newspaper can be derived as:
(
)
(
) .
In Tables WA.1-WA.3 we calculate this profit for various values of the parameters and demonstrate that it
is always lower than the equilibrium profits
. This derivation implies that eliminating the demand for
the online version in the second stage reduces the profits of the newspaper even when it does not have to
use different values of the coefficient
. As discussed in the section “Weights of the Reporting Strategy”
above, we believe that using a value other than (
)
is suboptimal. Therefore, profits are likely to decline
even further if in order to eliminate this demand the newspaper has to modify the value of the coefficient
. Hence, the newspaper is not likely to have an incentive to eliminate the online version in the second
stage.
Is Deviation in Stage 3 Optimal?: In this stage the newspaper can eliminate the demand for the online
version by simply charging a high enough
deviation and chooses prices
and
. Assuming, once again, that Newspaper 1 contemplates the
, we can use the derivation of the third stage equilibrium in the
{NE, E} regime to argue that the price it will charge for the print edition upon deviation
the same as at the equilibrium,
will remain
. Because in Stage 3, locations have already been determined at
, it follows from (WA.23) that:
(
)
which is equal to
from
(11). Moreover, from (WA.19) it follows that its market share will remain at 50% as it is at the
equilibrium. Recall that at the equilibrium with segmentation the markup derived from the online version
exceeds the markup from the print version (i.e., (
)
(
)). Hence, deviation to block off
the demand for the online version cannot be profitable, given that the newspaper loses the higher markup
from online readers while maintaining the same markup from the print readers and the same market share
as at the equilibrium.
Table WA. 1: Changes in Model Variables with Respect to
0.10
̂
0.800
1.473
)
1.563
1.491
-1.514
1.457
-0.010
3.931
3.928
4.00
3.887
4.041
0.11
0.883
1.482
1.585
1.494
-1.510
1.471
-0.006
3.953
3.951
4.00
3.923
4.027
0.12
0.943
1.490
1.607
1.497
-1.505
1.484
-0.004
3.974
3.972
4.00
3.957
4.015
0.13
0.990
1.498
1.627
1.499
-1.501
1.497
-0.001
3.995
3.994
4.00
3.991
4.003
(
Note: In all instances,
,
,
= 0.80,
,
, and conditions given in Lemma 1 hold.
Table WA. 2: Changes in Model Variables with Respect to
0.80
̂
0.800
1.473
)
1.563
1.491
-1.514
1.457
-0.010
3.931
3.928
4.00
3.887
4.041
0.81
0.846
1.479
1.571
1.493
-1.511
1.466
-0.007
3.945
3.943
4.00
3.911
4.032
0.82
0.887
1.484
1.578
1.495
-1.508
1.475
-0.006
3.958
3.957
4.00
3.933
4.024
0.83
0.924
1.489
1.585
1.496
-1.506
1.483
-0.004
3.971
3.970
4.00
3.954
4.016
(
Note: In all instances,
,
,
,
, and conditions given in Lemma 1 hold.
Table WA. 3: Changes in Model Variables with Respect to
1.333
̂
0.800
1.473
)
1.563
1.491
-1.514
1.457
-0.010
3.931
3.928
4.00
3.887
4.041
1.423
0.857
1.480
1.573
1.493
-1.511
1.468
-0.007
4.213
3.946
4.27
4.179
4.299
1.513
0.901
1.486
1.581
1.495
-1.507
1.478
-0.005
4.496
3.962
4.54
4.470
4.561
1.603
0.937
1.491
1.588
1.497
-1.505
1.485
-0.003
4.780
3.975
4.81
4.763
4.825
(
Note: : In all instances,
,
,
= 0.80,
, and conditions given in Lemma 1 hold.