Mathematical Appendix
Ramsey Pricing
PROOF OF THEOREM 1:
I maximize social welfare V subject to π > K . The Lagrangian is
V + κ(π − K )
the associated first-order conditions are that for each I
J NJ = 0
(1 + κ) P I − C I − cN J − κµI + bJ + κ bf
or
I
I
P − C − cN
or letting λ ≡
J
−
κ
1+κ
I
µ +
κ
1
J
J NJ = 0
b +
bf
1+κ
1+κ
κ
1+κ
J NJ
P I = C I + cN J + λµI − [1 − λ]bJ + λbf
Then solve for λ by substituting into the desired profit condition π = K :
i
h
f
A−b
B N A N B = K + bA + bB − c N A N B
λ µA N A + µB N B + bA + bB − bf
I now turn to the two alternative forms of the Ramsey problem mentioned in the text.
PROPOSITION 8: Interior Ramsey prices maximizing user surplus subject to the constraint
that the firm makes a rate of return r on her variable costs must solve
(A 1)
J + [1 − λ]bJ + λµI
P I = (1 + r ) C I + cN J − N J λbf
|
{z
}
|
{z
}
cost-plus
two-sided classical Ramsey pricing (Theorem 1
subsidy (in cost-plus terms) required for Pigouvian prices
z
}|
{
A B
− [1 + r ] c N N
where λ ≡
A A
B B
f
A−b
B NANB
N µ + N µ + bA + bA − bf
|
{z
}
bA
+ bB
demoninator from Theorem 1
1
2
THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2009
PROOF:
The Lagrangian is
V − (1 − κ)π − r κ cN A N B + C A N A + C B N B
with corresponding first-order conditions
J NJ = 0
κ P I − κ(1 + r )(C I + cN J ) + (1 − κ)µI + bJ − [1 − κ]bf
or letting λ ≡ 1 − κ1
J + [1 − λ]bJ
P I = (1 + r ) C I + cN J + λµI − N J λbf
Plugging into the rate-of-return constraint π = rC yields
h
i f
A−b
B
α µA N A + µB N B + bA + bB − bf
= bA + bB − [1 + r ]c N A N B
which yields the desired solution.
Thus user Ramsey pricing with a required rate of return over variable costs is a natural combination of classical cost-plus regulation and the classical two-sided Ramsey pricing of Theorem
1. Prices mark-up costs by the required rate of return, incorporating two-sided internalization
of cross-benefits and market power in the weighted-average fashion of classical Ramsey pricing
above. Now that there is no required profit threshold, the weighting is given the amount necessary to offset the subsidy needed to achieve cost-plus Pigouvian pricing while maintaining the
required rate of return. As noted earlier, if r = 0 and the platform must simply break even, the
solution is identical to the break-even classical OT pricing.
PROPOSITION 9: Interior Ramsey prices maximizing social surplus subject to the constraint
that the firm makes a rate of return r on her variable costs must solve
(A 2)
J + λµI
P I = (1 + λr )(C I + cN J ) − N J λbJ + [1 − λ]bf
|
{z
}
|
{z
}
cost-plus
two-sided classical Ramsey pricing (Theorem 1)
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
VOL. VOL NO. ISSUE
target profits
3
Pigouvian subsidy
}|
}| {
z
z
{ A B
A A
B B
A B
A
B
r C N + C N + cN N + b + b − c N N
where λ ≡ f
A−b
B + 2r c N A N B
rC A + µA N A + rC B + µB N B + bA + bB − bf
|
{z
}
(local) profit gain from moving towards rate-of-return maximization
PROOF:
The Lagrangian is now
h
i
V + κ π − r C A N A + C B N B + cN A N B
giving first-order conditions
I N J − (1 + κ[1 + r ]) cN J
(1 + κ)P I − (1 + κ[1 + r ]) C I − κµI + bI + κ bf
κ
1+κ
this takes the form in the text. Plugging this back into P A − [1 + r ]C A N A +
P B − [1 + r ]C B N B − (1 + r )cN A N B = 0 yields the definition for λ.
Letting λ ≡
The social Ramsey pricing problem with a required rate of return over variable costs is a slight
modification of user-optimal Ramsey pricing in Proposition 9.
Generalization
PROOF OF THEOREM 2:
I compute VJI . By definition
R
V I (N)) = θ I ∈2I (N) u I N; θ I f I θ I dθ I
R
= θ I :u I N;θ I ≥P I (N) u I N; θ I f I θ I dθ I
fI is the boundary of 2I , by the Leibnitz Integral Rule
Because 2
R
VJI = θ I ∈2I u IJ N; θ I f I θ I dθ I +
R
I
I − P I [N] u I N; θ I f I θ I dθ I =
g
I u J N; θ
J
θ I ∈2
u IJ
(A 3)
N
J
+P
I
Z
g
I
θ I ∈2
h
i
u IJ N; θ I − PJI [N] f I θ I dθ I
By the implicit function theorem for J 6= I and letting g
N1I ≡
PJI
g
NI
=− J
g
NI
R
=
|{z}
1 The Leibnitz Integral Rule
I
∂g
NI
∂ PI
N; θ I f I θ I dθ I
R
I θ I dθ I
g
I (N) f
θ I ∈2
g
I (N) u J
θ I ∈2
4
THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2009
Thus term second term in A 3 vanishes for I 6= J . On other other hand for I = J by the implicit
function theorem
R
I N; θ I f I θ I dθ I − 1
g
I −1
u
N
g
I ∈2
I
J
θ
(N)
=
PII = J
R
I θ I dθ I
g
g
I (N) f
NI
θ I ∈2
1
fI ) is P I . Thus the
So the second term of A 3 is simply P I (as the gross utility of all users in 2
first-order condition that marginal social benefit equal marginal cost becomes
X
VJI = CI ⇐⇒
J
P I = CI −
X
J
uJ
I N
J
which is the Pigouvian condition reported in the text.
PROOF OF THEOREM 3:
Here I begin by computing marginal revenues. For J 6= I :
∂ P I (N) N I
I
=
= PJI N I =
RJ
∂NJ
while
where I ≡
R
u IJ N; θ I f I θ I dθ I
I NJ
N J = uf
R
J
I θ I dθ I
f
g
I
I
θ ∈2 (N)
g
I (N)
θ I ∈2
I
I NI − P
RII = P I + PII N I = P I + uf
I
I
∂g
N I PI
∂ PI g
NI
is the price elasticity of participation (holding fixed anticipated participa-
tion). As usual, let µI ≡
PI
I
be the market power, that is the difference between price and
classical marginal revenue. Then equating marginal revenue to marginal cost requires
P I − µI +
Xf
J
uJ
I N = CI
J
which can easily be re-arranged to yield either of the formulae in the theorem.
Comparative Statics
I begin by showing that ρ I = −
µI
2
N I ∂ π2
∂NI
and that local strict concavity of profits is equivalent
to the local version of 2SC, while violation of local weak 2SC implies local non-concavity. Then
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
VOL. VOL NO. ISSUE
5
taking these facts as given, I establish Theorem 4.
I
d
P
ρI ≡
I
dC Thus ρ I > (≥)0 ⇐⇒
∂ 2π
∂N
A2
∂2π 2
2
∂NI
∂ 2π
∂N
B2
NJ
I
d
N
= P1I
I
dC =−
NJ
µI
NI
∂2π
2
∂NI
< (≤)0 and
> (≥)
µA µB
∂ 2π 2
⇐⇒
> (≥)ρ A ρ B χ 2
∂ N A∂ N B
NANB
which shows that strict local 2SC implies local concavity (by the classical Hessian test) and weak
2SC is necessary for local concavity. Based on these, Theorem 4 is easy to establish.
Two-sided contraction
PROOF OF THEOREM 4:
The forward direction of the proof follows directly from above: global concavity of profits
(which is equivalent to the negative definiteness of profits, which is 2SC) in N A , N B is inde
pendent of C A , C B by inspection. For the converse, suppose that 2SC were violated at some
D
E
g
A, g
pair of participation rates N
N B . Let
fI ≡ P I − µI + bf
J N J − cN J
C
D
E
D
E
g
g
fB has a solution at
A, g
A, C
be evaluated at N
N B . Then clearly equation (2) given costs C
D
E
g
A, g
N
N B . However because 2SC is violated here, this must be a local minimum or saddle-point
of the firm’s profits. Therefore it cannot be the optimal participation pair.
In the Armstrong case, from equation (12) χ = bA + bB so 2SC becomes
2
µA µB
A B
A
B
>
ρ
ρ
b
+
b
NANB
In the special case of linear demand that is typically analyzed,
NI
µI
NI
=
PI, N J
1
φI
where φ I is the (con
Q I 2g
P I −P I −bI N J
2g
PI
stant) slope of demand. If we parameterize demand by
=
f
I
I
so that Q demand at price 0 and P is the optimal price when there is no two-sidedness and
6
THE AMERICAN ECONOMIC REVIEW
C I = 0 then
1
σI
=
QI
2g
PI
SEPTEMBER 2009
so the condition simplifies, after a bit of algebra, to
bA + bB Q A Q B
q
<8
g
Ag
QA QB P
PB
4
which is that the surplus generated by two-sidedness at zero price should not be more than 8
times the geometric mean of optimal profits on the two sides if there were no two-sidedness, as
optimal profits for linear demand with this parameterization are
QI g
PI
2 .
Pigouvian second-order conditions
Now I derive the second-order conditions for the Pigouvian problem. The first derivative of
social welfare V with respect to participation on side I is
P I − C I − cN J + bJ N J
Taking the derivative with respect to N I yields
P1I
+ bJ
2N
J
∂
µI
=− I +
N
µI
− I +
N
Z
∞
R∞ R∞
−∞ P J (N J ,N I )−bJ N I
bJ f J (B J , bJ )d B J dbJ
∂NI
=
J bJ f I P J [N J , N I ] − bJ N I , bJ dbJ
bJ − bf
−∞
NJ
µI
=− I + J
N
µ
2
J
J 2 − bf
bg
µI
NJ
=− I + J
N
µ
2
J
J 2 − bf
bg
Thus the own-second derivative condition for concavity is that on both sides I
µA µB
> σfI
NANB
where σfI is the variance of interaction benefits among marginal users on side I. The cross partial
is then
P2A
BNB
∂
∂ bf
A−c+
−c+
= bf
B
∂N
µB
A−c+
bf
R∞
−∞ b
B fI
R∞ R∞
−∞ P B (N B ,N A )−bB N A
bB f B (B B , bB )d B B dbB
∂NB
P B [N B , N A ] − bB N A , bB dbB
NB
f
A+b
B −c
= bf
=
VOL. VOL NO. ISSUE
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
7
So the positive determinant condition requires
2
f
A+b
B −c
bf
A B fB
> A Bg
N N σA
N N σ
NANB
−
1
−
1
µA µB
µA µB
µA µB
(A 4)
If imposed globally, just as with 2SC, these ensure sufficiency of first-order conditions. By the
same argument as in Section III for 2SC, these conditions are weak in the sense of Theorem 4. In
the RT2003 case, heterogeneity is unidimensional so all marginal users have the same preferences
f
A =σ
B = 0. Therefore inequality A 4 becomes
and σf
2
m A m B > pA + pB − c
I as there are only interaction values. By the first-order condition for the Pigouvian
as p I = bf
problem in the RT2003 case p A + p B − c = −s A = −S B so this becomes
m A m B > s A · s B = ρ A · ρ B m A m A ⇐⇒ 1 > ρ A · ρ B
which is my social second-order condition from E. Glen Weyl (2009). In the Armstrong case
2
µA µB
A
B
>
b
+
b
NANB
In the linear case, when ρ I =
1
2
this is exactly the private 2SC, except four times stronger.
Positive comparative statics
PROOF OF THEOREM 5:
Letting R I N I , N J ≡ P I N I , N J N I be revenue on side I the cross partial is
χ=
A
R12
+
B
R12
∂ P A + P1A N A
∂ P B + P1B N B
∂ M RA ∂ M RB
−c =
+
−c =
+
−c =
∂NB
∂NA
∂NB
∂NA
B
µA
µe
f
f
b
b
A A
B B
f
f
f
ANA +b
B NB −c = b
B +b
A +b
B −c− e
A +b
P2A + P12
N + P2B + P12
−
N − c = bf
1
1
NB NA
where M R I is the classical marginal revenue on side I. Thus there are substitutes (complements)
8
THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2009
f
J thus
I
I
I
J
if e
bN A N B > (<)µe
b . But by equation (7), µ = P − C + b − c N
fB − C B N B = π + e
bN A N B
BA − C A N A + B
µ = 2e
bN A N B + g
Clearly µ, π > 0 so dividing through there are substitutes (complements) if
e
µe
bN A N B
< (>) b
µ
π +e
bN A N B
⇐⇒ 1 −
e
µe
µe
bN A N B
π
> (<)1 − b ⇐⇒
> (<) B
A
B
A
B
e
e
µ
µ
π + bN N
π + bN N
If µ e
B < 0 then we clearly have substitutes. Otherwise this is equivalent to
e
µe + µ e
µe
π +e
bN A N B
bN A N B
B
< (>) b
⇐⇒
< (>) b
π
µe
π
µe
B
B
Normative comparative statics
PROOF OF THEOREM 6:
By implicit differentiation of equation (1)
J
N I ρ I χ ddCN J
dNI
=−
dC J
µI
So
dNI
dC J
dNJ
dC J
=−
N I ρI χ
µI
I
J
dV I
I dN
I dN
I
I
f
I −b
I
=
V
+
V
∝
−N
ρ
χ
+
b
1
2
dC J
dC J
dC J
where the constant of proportionality is positive by concavity of profits. Clearly the effect of
an exogenous increase in side J participation, driven by cost, is the opposite of the effect of an
increase in cost (in sign, which is all I am concerned with).
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
VOL. VOL NO. ISSUE
9
The Scale-Income Model
In the SI model biI = β I BiI . I therefore index a user’s type by her membership benefit BiI . A
user i on side I will participate if
BiI + β I BiI N J > P I
that is if BiI >
tive.
PI
1+β I N J
Intuitively, ν I
dimension; if
νI
=
νI N J P I
I
β
if
νI
βI
is positive and if BiI <
νI N J P I
I
β
if
νI
βI
is nega-
(roughly) represents the fraction of heterogeneity that is along the interaction
is negative, then as prices rise interaction values tend to fall. Dividing by β I
converts this into membership benefits. It may be either the large or small scale users that participate and they may have either positive or negative membership benefits. In what follows, I derive
everything from first principles, as in unidimensional models my assumption of smoothness of
f I fails. All the results would follow identically from directly specializing my more general
results, as they did in the RT2003 and Armstrong cases; however, for expositional purposes it
is useful to show how direct analysis of a unidimensional model leads to the same results as
specializing general results from the RT2006 model.
I
νI N J P I
N I (P I , N J ) = 1 − F I
if βν I > 0 and N I = F I if
βI
νI
βI
< 0 where F I is
the c.d.f.
of membership values. Let D I (B I ) ≡ 1 − F I or F I as appropriate. Then N1I =
I I0 I
2
0
0
I
− βν I f I , where f I = F I , and N2I = − βν I ν β IP f I . By the definition, ν I = − ν I ,
yielding
N2I
νI
=
!
P I νI
βI
2
βI
νI =ν P I
β I
I
Letting P I (N I , N J ) be the inverse of N I with respect to its first
by the inverse and
argument,
µI
N I βI I
I
I
I =
implicit function theorems we have P1 = − N I where µ ≡ f I ν I and P2 = ν I P I = bf
−1
I . Surplus on side I is
β I DI
N I = β I bf
νI
J ! Z
N
βI
ν I [N J ]
βI
·∞ νI N J P I
βI
[
]
Using the change of variables φ I =
BI βI
νI
BI + βI BI N J − P I
f I (B I )d B I
this becomes
Z
νI ∞ VI = I φI − P I f I
β PI
νI φI
βI
!
dφ I =
Z
∞
PI
DI
νI φI
βI
!
dφ I
10
THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2009
by the standard integration by parts for surplus. Therefore V1I = µI and
∞
Z
V2I
=−
∞
Z
PI
0
νI φI
νI
βI
βI
PI
!
f I dφ I − ν I P I N I =
νI ν I φ I I f I dφ I − ν I P I N I = ν I V I
β cI ρc
cI is the
I where µ
By E. Glen Weyl and Michal Fabinger’s (2009)’s Theorem 3, V I = N I µ
1
I is an average of
inverse hazard rate of D I with respect to φ at φ = P I and ρc
over
d
I
1− ∂ µ I
∂N
1
P1I
cI = µI
φ > P I . Therefore to establish the formulae given in the text, I must show that, in fact, µ
I = ρI .
and ρc
NI
N I β I c
I
µ = − I
= I I = µI
∂D f ν I
∂φ I I
φ =P
I to an average of ρ I , let me calculate ρ I .
For the equivalence of ρc
ρI = −
µI
NI
∂2π
I2
∂N
=−
µI
N I P1I − µI1
I
= ρc
Following a similar logic
χ = P2A + ν B N B P1B + ν B P B − µI2 − c = ν A P A + ν B P B − c − ν B N B
µI2
∂
=
N I β I 1
I
f ( N I ) |ν I [ N J ]|
∂NJ
=
NI
I 1
β ∂ β I + N J fI
∂NJ
I
= (ν )
N I β I fI
µB
− µI2
NB
= ν I µI
therefore
χ=
i
X h
ν I P I − µI − c
I =A,B
Because the natural analogs of the naı̈ve simplifications from above are now apparent, I revert to
directly using these, but direct derivations are available on request.
VOL. VOL NO. ISSUE
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
11
The sign of equilibrium network effects from side I to side J are now determined by

ν J µJ ρ J + ρ J 

X h
ν K P K − µK
i
− c
K =A,B
As stated in the text, if the second term is positive (as in the newspaper case) and the first term also
is (as on the advertiser side, because ν B > 0) then clearly equilibrium externalities from readers
to advertisers are positive. On the reader’s side the second term is positive, but the first is negative
as ν A < 0. The equilibrium network effect will tend to be positive when complementarity is large
relative to reader-side market power and when pass-through is decreasing, so that inframarginal
harms are small relative to the pass-through of marginal cost changes.
The own-side cost-welfare effect on side I


− µA µB + 


X h
ν K P K − µK
i
− c ρ J ν I µI ρ I N A N B  ∝
K =A,B


− µJ + ν I ρ J ρ I N A N B 

X h
ν K P K − µK
i
− c
K =A,B
For advertisers this is clearly negative as both terms are. For readers this is again ambiguous:
while they lose from the higher prices the gain from resulting reduction in the Spence distortion
and therefore advertising. Thus it is not clear that readers will be worse off as the newspaper
business becomes less competitive, because on both sides of the market reduced competition is
likely to lead to higher advertising rates and this may improve reader welfare.
Applications
In this appendix I describe formally the GASH model of duopoly, derive equilibrium conditions for it and use these to state formally and prove the results I referred to in Subsection VI.C.
Armstrong only considers a linear demand, product differentiated model where all users must
choose one platform in the spirit of Harold Hotelling (1929), making it impossible to compare
monopoly to competition as under monopoly the platform would charge infinite prices.
Demand
Users on each side I have a usage benefit bI which depends only on which side of the market
they are on, not which platform they use nor their identity. users have a membership benefit for
12
THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2009
each platform 1 and 2, BiI ,1 and BiI ,2 respectively, drawn from a symmetric density function
g I : R 2 → R. Given a vector, consisting of a pair of side I prices and a pair of participation
levels on side J , P I ,i , P I ,−i , N J ,i , N J ,−i , we can write the number of side I agents that
choose to participate on platform i. Let us call this N I ,i , where
N I ,i (P I ,i , P I ,−i , N J ,i , N J ,−i ) =
Z ∞
Z BI,i +bI N J ,i −P I,i +P I,−i −bI N J ,−i
P I,i −bI N J ,i
g I (B I ,i , B I ,−i )d B I ,−i d B I ,i .
−∞
Let R I ,i ≡ bI N J ,i − P I ,i be the payoff an agent on side I receives from joining platform i,
net of membership benefits. We can then express the number of side I agents that participate of
platform i as N I ,i = N I ,i (P I ,i , N J ,i , R I ,−i ), where
N I ,i (P I ,i , N J ,i , R I ,−i ) =
Z ∞
P I,i −bI N J ,i
Z BI,i +bI N J ,i −P I,i −R I,−i
g I (B I ,i , B I ,−i )d B I ,−i d B I ,i .
−∞
Note that as N I ,i is strictly decreasing in it’s first argument, it can be inverted, yielding a well
defined inverse demand P I ,i N I ,i , N J ,i , R I ,−i .
Duopoly equilibrium
Supposing that both firm 1 and firm 2 offer insulating tariffs on both sides of the market; that
is, P I ,i P J ,i = bI N J ,i + h I ,i for some chosen h I ,i as the insulating tariff for the Armstrong
model is just full insurance. We can the consider each firm i to choose participation levels on
each side, taking as given R A,−i and R B,−i . Thus, in equilibrium, each firm maximizes, with
respect to N I ,i , I = A, B,
(A 5)
[P A,i (N A,i , N B,i , R A,−i ) − C A ]N A,i + [P B,i (N B,i , N A,i , R B,−i ) − C B ]N B,i
The first-order condition for maximization of (A 5), with respect to N I ,i , is
(A 6)
P I ,i − C I + P1I ,i N I ,i + P2J ,i N J ,i = 0
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
VOL. VOL NO. ISSUE
Note that, by the inverse function theorem, P1I ,i =
rem, P2I ,i = −
N1I ,i
Z
N2I,i
N1J ,i
=−
−∞
|
Z
, and that by the implicit function theo-
. Calculating N1I ,i , we find
−R I,−i
−
1
N1I,i
13
g I (P I ,i (N J ,i ) − bI N j,i , B I ,−i )d B I ,−i
{z
}
market expansion
∞
P I,i (N J ,i )−bI N J ,i
|
g I (B I ,i , B I ,i + bI N J ,i − P I ,i (N J ,i ) − R I ,i )d B I ,i .
{z
}
cannibalization
Calculating N2I ,i , we find
N2I ,i =
Z
−R I,−i
bI g I (P I ,i (N J ,i ) − bI N j,i , B I ,−i )d B I ,−i
−∞
∞
Z
+
P I,i (N J ,i )−bI N J ,i
bI g I (B I ,i , B I ,i + bI N J ,i − P I ,i (N J ,i ) − R I ,i )d B I ,i
= −bI N1I ,i
Thus, (A 6) can be rewritten
(A 7)
P I ,i − C I + bJ N J ,i =
N I ,i
−N1I ,i
≡ µIo ,i
where µIo ,i is firm i’s own-price market power on side I. Equation (A 7) can be thought of
as governing firm i’s best-response to firm −i’s choices of R I ,−i . Thus the conditions for a
symmetric-across-firms (SAF) equilibrium in insulting tariffs, what I call an symmetric insulated
equilibrium (SIE), it must be the case that for I = A, B and i = 1, 2
(A 8)
P I − C I + bJ
NJ
= µIo
2
where P I is the SIE price on side I, N J is the total market participation on side J and µIo is
the SAF own-price market power on side I.
14
THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2009
Monopoly
Suppose firms i and −i can act, via merger or some other means, in such a way so as to
maximize joint profits. Then, assuming the platforms choose a SAF strategy, they solve
(
max 2
Xh
N I,m
P
I ,m
(N
I ,m
,N
J ,m
)−C
I
i
N
I ,m
)
I
where P I ,m (N I ,m , N J ,m ) is the inverse, with respect to the first argument, of
N I ,m P I ,m , N J ,m , the number of side I agents who join any one of the two symmetric platforms
N I ,m (P I ,m (N J ,m ), N J ,m ) =
Z
Z BI,m
∞
P I,m (N J ,m )−bI N J ,m
g I (B I ,m , B I ,−m )d B I ,−m d B I ,m
−∞
The first-order condition of the platforms is
P I − C I + bJ
NJ
N I ,m
≡ µI
=
2
−N1I ,m
where µI is the joint market power and
N1I ,m = −
Note that µI N I , N J
P I,m −bI N J ,m
Z
g I (P I ,m − bI N J ,m , B I ,−m )d B I −m
−∞
> µIo N I , N J for any pair of participation rates N I , N J as the
denominator of µI does not include the (strictly positive) cannibalization term of µIo .
Merger effects
PROPOSITION 10: Assume 2SC for the Armstrong model applied to the monopoly optimum
2
A B
in this context: ρ I > 0 for both I and NµA µN B > ρ A ρ B bA + bB . Then participation and
welfare are higher on both sides of the market under duopoly than under monopoly.
PROOF:
Rewrite condition (A8) as
MI
P I − C I + bJ N J = µI − µI 1 − oI
M
!
APPENDIX TO A PRICE THEORY OF MULTI-SIDED PLATFORMS
VOL. VOL NO. ISSUE
Let G I N I , N J = 1 −
?
?
MoI
MI
15
?
?
and let the value of G I at SIE participation rates N A , N B
be G I . Note that G I , G I > 0 as noted above. Inthe monopoly problem costs are given by
C I − t GI
for t ∈ [0, 1]. Clearly for t = 0 the solution is the same as the SAF monopoly optimum
(which is unique by the assumed second-order conditions). For t = 1, the allocation must be
the same as at the SIE equilibrium. But by my reasoning in Section III.A, participation rates on
the two sides of the markets are complements, so a fall in either cost raises both participation
rates. Thus an increase in t must raise both participation rates. Therefore participation on both
sides of the market is higher under the SIE equilibrium than under the SAF monopoly optimum.
Furthermore in the Armstrong model welfare on each side of the market depends (directly) only
on participation on that side as there is no Spence distortion, so both groups of users must be
better off. Finally, because participation rates are below their optimal level for any amount of
market power on both sides, the resulting increase in participation on both sides must be social
welfare enhancing. Thus a merger without efficiencies from a SIE equilibrium is harmful to
participation, both groups of users’ welfare and overall social welfare.
This result contrasts significantly with the RT2003 model, where competition can easily harm
one or even both groups of users. Of course the result depends on a particular assumption about
tariffs, namely that firms use insulating tariffs. In a previous version of this paper I proved that
this result was significantly more robust and holds in a very wide range of reasonable cases with
non-insulating tariffs. Those results, omitted for the sake of brevity and clarity, are available on
request.
REFERENCES
Hotelling, Harold. 1929. “Stability in Competition.” Economic Journal, 39(153): 41–57.
Weyl, E. Glen. 2009. “Monopoly, Ramsey and Lindahl in Rochet and Tirole (2003).” Economics
Letters, 103(2): 99–100.
Weyl, E. Glen, and Michal Fabinger. 2009. “Pass-Through as an Economic Tool.”
http://www.fas.harvard.edu/∼weyl/research.htm.