Ownership Consolidation and Product Characteristics:
A Study of the U.S. Daily Newspaper Market
Online Appendix
Ying Fan
Department of Economics, University of Michigan
C Invertibility of the Penetration Function
In this appendix, I show that the invertibility result in BLP can be extended to a multiple
discrete choice model. I only show the extension for a model where the number of products that an
individual can buy is limited to at most two. The result can be easily extended to a model in which
consumers can choose up to ¯n ≤ J products, where J is the total number of products available in
a choice set.
Penetration Function
Let Φ (·) represent the distribution function of the random term ς
i
. The penetration function
in Section 2.1 is given by
s
j
(δ, x; σ, κ) =
Z
Ψ
(1)
j
(δ, x, ς
i
; σ) dΦ (ς
i
)
+
X
j
0
6=j
Z
Ψ
(2)
j,j
0
(δ, x, ς
i
; σ, κ) − Ψ
(3)
j
(δ, x, ς
i
; σ, κ)
dΦ (ς
i
) ,
where
Ψ
(1)
j
(δ, x, ς
i
; σ) =
exp (δ
j
+ ϑ
ij
)
1 +
P
J
h=1
exp (δ
h
+ ϑ
ih
)
,
is the probability that newspaper j is chosen as the first newspaper (ϑ
ij
is the deviation of household
i’s utility from the mean utility), and the probability that newspaper j is chosen as the second
newspaper when j
0
is the first best is given by the difference between the followings:
Ψ
(2)
j,j
0
(δ, x, ς
i
; σ, κ) =
exp (δ
j
+ ϑ
ij
)
exp (κ) +
P
h6=j
0
exp (δ
h
+ ϑ
ih
)
,
Ψ
(3)
j
(δ, x, ς
i
; σ, κ) =
exp (δ
j
+ ϑ
ij
)
exp (κ) +
P
J
h=1
exp (δ
h
+ ϑ
ih
)
.
36
Invertibility
Since all statements in this section are true for any given (x, σ, κ), these arguments in
s
j
are
omitted for expositional simplicity.
The proof of the invertibility result is slightly different from that in BLP. BLP define a function
F : R
J
→ R
J
pointwise as F
j
(δ) = δ
j
+ ln s
j
− ln
s
j
(δ) and show that F is a contraction when an
upper bound on the value taken by F is imposed. For a single discrete choice model, the value of
δ
j
that solves
P
J
h=1
s
h
=
P
J
h=1
s
h
(δ) when δ
j
0
= −∞ for ∀j
0
6= j is the upper bound of the jth
dimension of a fixed point of F . In a multiple discrete choice model, however, this value does not
exist when
P
J
h=1
s
h
is larger than 1.
24
I first prove the existence and uniqueness of the solution to
s
j
(δ, x; σ, κ) = s
j
for all j directly
without using the function F . I then verify that all conditions in BLP hold so that F is indeed a
contraction mapping – when an upper bound is imposed.
The following inequalities, which will be proven at the end of this section, are useful in the
proof:
∂
s
j
/∂δ
j
<
s
j
(C.4)
∂
s
j
/∂δ
j
> 0 (C.5)
∂
s
j
/∂δ
h
< 0 when h 6= j (C.6)
X
J
h=1
(∂
s
j
/∂δ
h
) > 0 (C.7)
Inequalities (C.5), (C.6) and (C.7) imply that the Jacobian of
s
has a dominant diagonal.
Therefore, there is a unique solution to the equation system of
s
j
(δ) = s
j
for all j.
25
I now prove that all conditions in the theorem in BLP hold.
Condition (1): Inequalities (C.4) and (C.6) imply that
∂F
j
(δ) /∂δ
j
= 1 − (∂
s
j
/∂δ
j
) /
s
j
> 0
∂F
j
(δ) /∂δ
h
= − (∂
s
j
/∂δ
h
) /
s
j
> 0 when h 6= j.
Also, inequality (C.7) implies that
X
J
h=1
∂F
j
(δ) /∂δ
h
= 1 −
X
J
h=1
(∂
s
j
/∂δ
h
) /
s
j
< 1.
24
In a single discrete choice model,
P
J
h=1
s
h
< 1, while in a multiple discrete choice model, the sum of market
penetration for all products
P
J
h=1
s
h
can be larger than 1. But the supremum of
P
J
h=1
s
h
(δ) is 1 when δ
j
0
= −∞
for ∀j
0
6= j.
25
See McKenzie, Lionel (1959), “Matrices with dominant diagonals and economic theory.” In Mathematical methods
in the social sciences (Kenneth Joseph Arrow, Samuel Karlin, and Patrick Suppes, eds.), 47-62, Stanford University
Press.
37
Condition (2): Given the monotonicity of F in all dimensions of δ, a lower bound of function
F is δ = min
j
lim
δ→−∞
J
F
j
(δ)
.
Condition (3): I have already shown that the equation system of
s
j
(δ) = s
j
has a unique
solution. This implies that the mapping F has a unique fixed point. Denote the fixed point by δ
∗
.
Then, F
j
(δ
∗
) = δ
∗
j
for all j. Note that F
j
(δ
∗
+ ∆) −
δ
∗
j
+ ∆
= ln s
j
− ln
s
j
(δ
∗
+ ∆) is strictly
decreasing in ∆ as implied by inequality (C.7). Therefore, F
j
(δ
∗
+ ∆) <
δ
∗
j
+ ∆
for any ∆ > 0.
Define
¯
δ
j
= δ
∗
j
+ ∆. Then, F
j
¯
δ
<
¯
δ
j
for any j. By inequality (C.6), F
j
(δ) < δ
j
for any δ such
that δ
j
=
¯
δ
j
and δ
j
0
≤
¯
δ
j
0
for all j
0
.
I now show inequalities (C.4) to (C.7). Three observations are important:
0 < Ψ
(1)
j
, Ψ
(2)
j,j
0
, Ψ
(3)
j
< 1; Ψ
(2)
j,j
0
> Ψ
(3)
j
; Ψ
(1)
j
> Ψ
(3)
j
.
Inequalities (C.4) and (C.6) follow directly from the three observations:
∂
s
j
/∂δ
j
=
Z
Ψ
(1)
j
1 − Ψ
(1)
j
dΦ (ς) +
X
j
0
6=j
Z
h
Ψ
(2)
j,j
0
1 − Ψ
(2)
j,j
0
− Ψ
(3)
j
1 − Ψ
(3)
j
i
dΦ (ς)
<
Z
Ψ
(1)
j
dΦ (ς) +
X
j
0
6=j
Z
Ψ
(2)
j,j
0
− Ψ
(3)
j
dΦ (ς) =
s
j
,
∂
s
j
/∂δ
h
= −
Z
Ψ
(1)
j
Ψ
(1)
h
dΦ (ς) +
Z
Ψ
(3)
j
Ψ
(3)
h
dΦ (ς) +
X
j
0
6=j,h
Z
−Ψ
(2)
j,j
0
Ψ
(2)
h,j
0
+ Ψ
(3)
j
Ψ
(3)
h
dΦ (ς)
<
X
j
0
6=j,h
Z
−Ψ
(2)
j,j
0
Ψ
(2)
h,j
0
+ Ψ
(3)
j
Ψ
(3)
h
dΦ (ς) < 0 when h 6= j.
To show inequality (C.7), note that
P
J
h=1
∂
s
j
(δ)
∂δ
h
=
∂
s
j
(δ+∆)
∂∆
|
∆=0
, and
∂
s
j
(δ + ∆)
∂∆
|
∆=0
=
Z
Ψ
(1)
j
2
1
e
δ
j
+ϑ
ij
dΦ (ς) +
X
j
0
6=j,0
Z
Ψ
(2)
j,j
0
2
−
Ψ
(3)
j
2
e
κ
e
δ
j
+ϑ
ij
dΦ (ς) > 0.
Combining inequalities (C.6) and (C.7) yields inequality (C.5).
38
D Instrumental Variables
The “excluded” instrumental variables that are assumed uncorrelated with the demand error
ξ
jct
:
• BLP-style instrument (IV
1
)
– the number of competitors
• cost shifters (IV
2
, IV
3
, IV
4
)
– frequency of publication
– households in the home county of newspaper j (It is correlated with Q
jt
in the average
cost of printing and delivery in (10).)
– households in the home counties of other papers that are close-by and are of the same
owner as newspaper j (It is correlated with Q
jt
in (10).)
• demographics in counties of newspaper j’s market (excluding county c) (IV
5
, IV
6
)
– weighted average education and median age (weighted by households in a county)
• demographics in the counties of newspaper j’s competitors (excluding counties in j’s market)
(IV
7
, IV
8
, IV
9
, IV
10
)
– weighted average education, median income, median age and urbanization (weighted by
households in a county)
The table below reports the results of the first-stage regression. Standard errors are clustered
by newspaper. The estimates are largely significant.
39
First-stage Regression Result
newshole
a
opinion reporter local news ratio variety price
coef. s.e. coef. s.e. coef. s.e. coef. s.e. coef. s.e. coef. s.e.
Included IV
log(households in the market) 0.169
∗∗
0.035 0.449
∗∗
0.264 -0.145 1.185 0.331
∗∗
0.053 0.129
∗∗
0.028 12.804
∗∗
2.653
morning edition 0.052
∗∗
0.026 0.654
∗∗
0.313 0.311 0.996 -0.070 0.070 -0.092
∗∗
0.039 2.701 2.525
local dummy -0.008 0.027 -0.041 0.217 0.236 0.863 0.007 0.038 -0.002 0.020 1.044 2.539
county distance (1000km) -0.074 0.366 -0.339 3.319 8.000 12.139 1.373
∗∗
0.539 0.341
∗∗
0.205 103.969
∗∗
43.441
education -0.074
∗∗
0.032 -1.520
∗∗
0.371 -11.998
∗∗
1.418 -0.436
∗∗
0.068 0.227
∗∗
0.038 15.894
∗∗
2.170
median income ($10000) 0.215
∗
0.154 1.343
∗
1.020 1.953 3.742 0.210 0.334 0.163 0.234 38.557
∗∗
14.035
median age -0.005
∗∗
0.002 -0.013 0.013 -0.021 0.049 0.004 0.004 0.003 0.002 0.028 0.160
urbanization 0.076 0.293 2.071 1.713 9.786
∗
7.018 0.077 0.560 0.161 0.362 82.087
∗∗
25.110
time -0.006 0.005 -0.003 0.041 -0.020 0.180 -0.005 0.010 0.002 0.006 -0.422 0.440
Excluded IV
IV
1
0.034
∗∗
0.018 0.235
∗
0.150 1.084
∗∗
0.457 -0.029
∗∗
0.008 -0.012
∗∗
0.004 -1.871
∗∗
0.716
IV
2
3.383
∗∗
0.356 -5.718
∗
3.745 -0.295 14.841 4.867
∗∗
0.895 2.679
∗∗
0.535 229.767
∗∗
35.724
IV
3
3.176
∗∗
0.563 33.451
∗∗
3.265 171.795
∗∗
17.737 -1.461
∗∗
0.536 -0.536
∗∗
0.229 -55.646
∗∗
22.626
IV
4
0.107 0.478 4.334 6.596 16.415 26.662 -0.507 0.482 -0.314 0.334 -16.786 39.376
IV
5
0.724
∗∗
0.328 7.210
∗∗
2.914 29.476
∗∗
10.128 0.089 0.438 -0.466
∗∗
0.156 71.055
∗∗
26.044
IV
6
-0.002 0.001 -0.019
∗
0.012 -0.081
∗∗
0.044 0.002 0.003 0.004
∗∗
0.001 -0.082 0.114
IV
7
-0.659
∗∗
0.368 -7.136
∗∗
3.306 5.685 11.343 0.737 0.732 -0.349 0.399 -41.265 36.571
IV
8
0.241 0.356 5.502
∗∗
3.251 13.255
∗
10.183 -0.384 0.701 0.474
∗
0.340 41.246
∗
27.609
IV
9
-0.007
∗∗
0.003 -0.031
∗
0.023 -0.121
∗
0.083 -0.005 0.005 -0.003
∗
0.002 0.416
∗∗
0.235
IV
10
-0.091 0.120 -1.079 1.071 -13.159
∗∗
3.920 0.284 0.244 0.002 0.096 -16.563
∗
11.301
F test
F(19, 943) 144.10 44.91 46.14 32.67 11.02 48.06
F test of excluded IV
F(10, 943) 16.64 26.72 28.82 9.19 6.92 11.43
** indicates 95% level of significance. * indicates 90% level of significance.
a
The news hole is n
jt
− a
r
jt
, q
jt
, H
jt
; ˆη,
ˆ
λ
, where ˆη and
ˆ
λ are the estimated advertising demand parameters
reported in Table 4.
40