We turn now from marginal revenue to costs. We model costs as depending on the
endogenous variables and on a vector of cost shocks, ω
t
, with one shock for each decision.
As an example somewhat in the spirit of
Fan (2013), we could model costs as
C ¼q
t
mc a
t
, y
t
, θ
q
ðÞ+ ω
q
½+ F
a
a
t
, θ
a
ðÞ+ ω
a
a
t
½+ F
y
y
t
, θ
y
ðÞ+ ω
y
y
t
½: (3.24)
The first term is quantity times marginal cost, where marginal cost varies with advertise-
ments and quality as well as a parameter vector and a cost shock. The second term in
brackets is the fixed cost of selling and producing advertisements, also depending on a
parameter and a cost shock. The third term is the fixed cost of quality. Looking forward
to a relatively straightforward IV or GMM estimation method, a key feature of
(3.24) is
that the incremental cost of each endogenous variable is linear in its associated cost unob-
servable. To capture this, we might assume the more general cost function:
C ¼
Cq
t
, a
t
, y
t
, x
t
, θðÞ+ ω
q
q
t
+ ω
a
a
t
+ ω
y
y
t
, (3.25)
where the specification of
Cq
t
, a
t
, y
t
, w
t
, θðÞwould depend on the details of the applica-
tion and the data.
Given the profit function defined by
(3.20)–(3.22) and (3.25), the three first-order
conditions (for subscription price, ad price, and quality respectively) then resemble
traditional “multi-product” firm pricing first-order conditions:
q
t
+ p
t
@
C
@q
t
+ ω
q
t
dq
t
dp
t
+ r
t
@
C
@a
t
+ ω
a
t
da
t
dp
t
¼0, (3.26)
a
t
+ r
t
@
C
@a
t
+ ω
a
t
da
t
dr
t
+ p
t
@
C
@q
t
+ ω
q
t
dq
t
dr
t
¼0, and (3.27)
p
t
@
C
@q
t
+ ω
q
t
@D
q
@y
t
+ r
t
@
C
@a
t
+ ω
a
t
@D
a
@q
t
@D
q
@y
t
@
C
@y
t
+ ω
y
t
¼0: (3.28)
For example, in Equation
(3.26) a one-unit increase in subscription price increases profits
by (i) subscription quantity q
t
plus (ii) the price-marginal cost margin on subscriptions
times the total change in q
t
induced by the price change plus (iii) the advertising margin
times the total change in a
t
induced by the change in p
t
. Again, recall that the total deriv-
atives in the first-order conditions capture both the direct and indirect effects of price
changes, via a use of the implicit function theorem.
Note that in a Nash price (and quality) setting equilibrium, the oligopoly first-order
conditions take the same form as
(3.26)–(3.28), except that the derivatives of the demands
with respect to prices and qualities depend on the derivatives of differentiated products
demand, holding other firm’s prices and qualities fixed. Multi-product oligopoly first-
order conditions are also easy to derive.
Another possibility, following
Fan (2013), would be to model quality as being chosen
prior to prices. In the oligopoly case, this will lead to a different final equilibrium
112 Handbook of Media Economics
prediction. Note that the two pricing first-order conditions implicitly define equilibrium
prices as a function of quality, y
t
. Again, following Fan, the derivative of prices with
respect to quality, say dp
t
/dy
t
and dr
t
/dy
t
, can be found via the implicit function theo-
rem applied to the pricing first-order conditions (
3.26 and 3.27). This in turn allows Fan
to make use of first-order conditions for newspaper quality that look forward to the later
equilibrium in prices, without computing the “second-stage” pricing equilibrium.
To complete an empirical specification,
Rosse (1970) assumes that incremental costs
@
C=@q
t
,@
C=@a
t
,@
C=@y
t
ðÞ
are all linear in data and parameters. The three first-order
conditions (
3.26–3.28), plus linear subscriber and advertiser demand equations, then
make up the five linear equations that
Rosse (1970) takes to the newspaper data.
7
Iden-
tification then follows from classic linear simultaneous equations arguments.
Fan (2013)
uses the oligopoly versions of these same equations, modified for a first-stage quality deci-
sion together with carefully chosen functional forms and a more realistic demand side.
BLP suggest estimating the cost parameters via classic Generalized Method of
Moments (GMM) techniques that are closely related to nonlinear (in the parameters)
IV methods. As usual, we need excluded instruments that are assumed to have zero
covariance with the unobservable ωs. The moment conditions used in estimation are
then the interaction of the implied unobservable ωs with the instruments.
To implement the GMM method, given any guess for the demand and cost param-
eters, it must be possible to solve the first-order conditions (
3.26–3.28) for the cost shocks
ω. This is very close to the supply-side estimation technique in BLP, where the first-order
pricing conditions for multi-product firms have multiple cost errors in each equation and
one must show how to “invert” the multi-product oligopoly first-order conditions to
solve for the cost shocks. Conditional on the solution for the ωs, the GMM econometric
techniques of BLP carry directly over to the case of two-side markets.
Note that the first-order conditions are linear in the cost unobservables (with coef-
ficients that are total demand derivatives). A sufficient condition to solve
(3.26)–(3.28) for
the costs shocks is that the first two equations can be solved for the quantity and ad
demand errors, as the third equation can then be solved for the quality error. Since
the equations are linear in the ωs, this solution is unique as long as the appropriate deter-
minant condition is satisfied; here that condition is
@q
t
@p
t
@a
t
@r
t
6¼
@q
t
@r
t
@a
t
@p
t
:
7
Rosse (1970), in the manner of that day, simplifies the problem by simply “tacking on error terms”
(disturbances) at the end of each first-order condition, but this obscures a discussion of exclusion restric-
tions, which ought to be based on the idea that there are various incremental costs on the right-hand side of
the first-order conditions.
113
Empirical Modeling for Economics of the Media
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