The Agglomeration-Differentiation Tradeoff in ... - Yale University

2.1.3. The Profit Function
Conforming to the multiplicative specifications so far, the observed component of the cost
multiplier, ˆ C fl (Equation 5), is specified as:
B
ˆ ⎛ ⎞
Cfl( xl) = exp⎜∑
γ fbxxbl ⎟
(14)
⎝ b=
1 ⎠
where, bl x are the observed cost shifters at distance band b around location l and γ fbx are format
and band specific cost parameters.
Substituting the expressions for revenue and cost into our profit specification, Equation
(3), then taking the log transformation, and after making some trivial sign reversals, we have a
equation for the transformed profit function that is very similar to equation (1):
( v pr ) ( Cˆ
)
( ) ( ) ( )
16
( )
r c m
π = ln π = ln + ln + ω − ln + ω + ξ + ε
ifl ifl fl fl l fl l il
2.4 Equilibrium Choice Probabilities:
Recall that the idiosyncratic cost shock, ε il , is known to rivals only in distribution. Due
to such incomplete information about rivals’ profits, a firm cannot exactly predict rivals’ discrete
actions but it can have rational expectations about rivals’ strategies. Hence, for a given set of
pr r c
vectors of price, revenue and cost shocks across all locations ( ω , ω , ω ), firm i can form
rational expectations about the number of firms that will enter the market, N m , and the location
and format choices of the (N m m m m m
-1) entering rivals, P P1 , P2 ,... PF
(15)
= ⎡
⎣
⎤
⎦
. That is, corresponding
to each format f (f’) firms we will have a vector of lm conditional choice probabilities (CCPs),
{ 1, 2,...,
}
P = p p p
m
f f f flm
m ( Pf ' { pf '1, pf '2 ,..., pf
'l
} ) m
= . For instance, fj
p ( f ' j)
p is a CCP of a f-