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d

d x c 1 , c

2

h

i(x,q dx c FB 0 , c 0 0

)i

= 0, for all [c1, c2] different from [c0, c00]. Formally this is due to the fact

t h a t t h e c r i t i c a l d e m a n d r e a l i z a t i o n θ c F B

for c [c0, c00] as defined by (5) does only depend

on X(c) for c [c0, c00], but not on X(c) for c [c1, c2]. Thus expression (6) is not a function of X(c) for c [c1, c2] and the cross derivative equals zero. The argument is analogous for

d dx c i )

h

i(x,q dx c FB 0 , c 0 0

)i

=0

The second derivatives with respect to the same technologies are given as follows:

c00

d 2 π i ( x , q F B d (x 2 ( c 0 , c 0 0 ) )

)

=

c0

f ( θ c F B

)

θ c F B d x ( c 0 , c 0 0 )

dc < 0.

(21)

This expression is negative since

dx

c FB

c 0 , c 0 0 )

  • >

    0, (compare expression (5)).

And for the second derivative with respect to capacity choice we obtain:

d 2 π i ( x , q d (x(ci)) F B 2

)

=

Z c F B

P q

X, θ¢

dF (θ) < 0.

(22)

N o t i c e : S i n c e t h e i n t e g r a n d i s c o n t i n u o u s a t θ c F B

(equals zero), the derivative with respect

to this lower limit drops out according to Leibnitz rule.

Proof of lemma 3

A c c o r d i n g t o l e m m a 2 , t h e o v e r a l l c a p a c i t y b o u n d X nology cneeds to satisfy the following to conditions: F B

under perfect competition at tech-

(i)

F (θ)

(kc(c) + 1) = 0

(ii)

X F B

:

Z cF B

P ( X F B

, θ)

c= k(c)

W e r e w r i t e t h e i n t e g r a n d o f ( i i ) i n t e r m s o f t h e c r i t i c a l d e m a n d r e a l i z a t i o n θ c F B use of its definition (5) and obtain:

by making

P ( X F B

, θ)

c= θ

B ( X F B

)

c= θ

θ c F B

We obtain expression (8) as given in lemma 3:

(i)

(ii)

F (θ) Zθ

(kc(c) + 1) = 0

θdF (θ) = k(c)

34

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