# Appendix

## Proof of lemma 1

Part (i): Since only technologies c ∈ [c^{0}, c^{00}] are affected by that change, we just have to apply the product rule of differentiation to the integrand of the second summand of (2)

evaluated within the limits c^{0 }and c^{00}. This yields:

dπ_{i}(x, q) d x ( c 0 , c 0 0 )

=

Z c 0 0 c 0

Ã d d c d x ( c 0 , c dc d " d 0 0 ) Z

c i

c i [P (Q(θ), θ) ^{"}Z [P (Q(θ), θ) c] dF (θ) c] dF (θ) k(c)

#

k ( c ) # x i ( c )

!

dc

## This can be further simplified^{36 }

and yields:

Z c 0 0 c 0

d

dc

"

Z

c i

## [P (Q(θ), θ)

c] dF (θ)

k(c)

#

d

dc

"

Z c i ·

d P ( Q ( θ ) , θ ) d x ( c 0 , c 0 0 )

¸

d F ( θ ) # x i ( c ) d c

N o t i c e t h a t t h e l o w e r l i m i t s o f i n t e g r a t i o n θ c i o f b o t h s u m m a n d s c l e a r l y d o d e p e n d o n however only the integrand of the first summand is function of c through the expression ” c”. We obtain expression (3) by Leibnitz rule after some rearranging: c ,

Z c 0 0 c 0

"

1

F ( θ c i ) + k c ( c ) +

d θ dc c i

f ( θ c i )

µ

P (Q(θ), θ)

c+

dP (Q(θ), θ) d x ( c 0 , c 0 0 )

x_{i}(c)

¶

=

c i

#

dc

.

Part (ii): Notice that changing total capacity at c_{i }leaves the profits from technologies c < c_{i }unaffected. The First derivative with respect to changing total capacity is thus just given by differentiation of the first summand of (2) with respect to total capacity x_{i}, yielding expression (4):

dπ_{i}(x, q) d x ( c i )

=

Z c i i ·

P (Q(θ), θ)

c_{i }+

dP (Q(θ), θ) d x ( c i )

x i ¸ d F ( θ )

k(c_{i})

## Proof of lemma 2

The second order conditions for the case of perfect competition can easily be verified: First we observe that the cross partial derivative with respect to different technologies

3 6 M a y b e m e n t i o n t h e c o n t i n u i t y o f p r o fi t s i s s u e , s o d i e r e n t i a t i o n o f t h e l i m i t s θ c i w r t d x d o e s n o t p l a role for the first derivatives. a y

33