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technologies yields:

πi(x, q) =

Z c i c i

Ã"

Z

c i

[P (Q, θ)

c] dF (θ)

k(c)

#

dxi(c)

dc

!

dc

This expression can be transformed by applying the rule of integration by parts:

πi(x, q) =

Z c i c i

Ã

d dc

"

Z

c i

[P (Q, θ)

c] dF (θ)

k ( c ) # x i ( c )

!

dc +

Z

c i

i

([P (Q, θ)

ci] xi) dF (θ)

k(ci)xi

(2)

The first summand of (2) yields profits of interior investment xi < xi(ci) and the second summand yields profits of investment at the capacity bound xi = xi(ci).

We now analyze the impact of a variation of the investment function xi(c) on firm i’s total profits. We have to take into account two different types of such variations as illustrated in figure 3.

Figure 3: Changes of Investment xi(c), interior case and boundary case.

F i r s t w e c o n s i d e r i n t e r i o r c a s e s , w h e r e i n v e s t m e n t i s c h a n g e d b y s o m e a m o u n t d x ( c 0 , c 0 0 )

for all technologies c [c0, c00], with c0 < c00 ci. Such variation is denoted by

d π i ( · dx ) c 0 , c 0 0 )

and is

given by expression (3) of lemma 1. Second we consider variations of the overall capacity xi by the amount dx(ci), changing investment for all technologies above the highest technology

c ci. Such variation is denoted by

d π i ( · dx ) c i )

and is given by expression (4) of lemma 1.

When computing those first derivatives it is important to notice that the spot market equilibrium Q(x, θ) depends on firms investment choices. Thus both the critical demand

8

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