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Appendix

Proof of lemma 1

Part (i): Since only technologies c [c0, c00] 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 c0 and c00. This yields:

i(x, q) d x ( c 0 , c 0 0 )

=

Z c 0 0 c 0

à 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 simplified36

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 )

xi(c)

=

c i

#

dc

.

Part (ii): Notice that changing total capacity at ci leaves the profits from technologies c < ci 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 xi, yielding expression (4):

i(x, q) d x ( c i )

=

Z c i i ·

P (Q(θ), θ)

ci +

dP (Q(θ), θ) d x ( c i )

x i ¸ d F ( θ )

k(ci)

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

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