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# Proof of Lemma 5

## Part (i): First order conditions

Throughout section 5 we already have shown how expressions (18) and (19) are derived from the general first order conditions stated in lemma 1.

In appendix 8 we have derived expression (27), which is given by:

d Q dx E Q i c 0 , c 0 0 )

=

³ Pj=6 i R j ´ ( 1 ) C i q q ( C i q q P q ) ( 1 + P j = 1 R j ) A f t e r m u l t i p l i c a t i o n w e o n b t a i n : and expression (32) is given by:d

c EQ,i

dc

=

( C i q q

P q ) ( 1 + P n j = 1

C i q q

Rj ) .

c EQ,i

i EQ dQ

dc

0 00 dx(c ,c )

= ( 1)

X

j R=

X

j =6 i

j =6 i

( P q

( · ) + P q q x j ( c ) ) x 0 j ( c E Q j

1

P q x 0 j ( c E Q j

(θ))

(θ))

T h i s i s d u e t o t h e f o l l o w i n g t w o o b s e r v a t i o n s : R j i s d e fi n e d i n ( 2 6 ) b y R j : = P q P q q q j C j q q P q S . i n c e C j q ( q ) i s t h e i n v e r s e o f i n v e s t m e n t x j ( c ) o f fi r m j , f o r a n y q 0 = x j ( c 0 ) t h e f o l l o w i n g w 1 e l l k n o w n r e l a t i o n s h i p i s s a t i s fi e d : C j q q ( q 0 ) = x 0 j ( c 0 ) .

## We thus obtain slightly rewriting (18) for the interior first order condition:

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

)

=

c00

## Z

c0

Ã

1

F ( θ c E Q , i

) + k c ( c ) + f ( θ c E Q , i

)

Ã jX=6 i

R j !

( P q ) x i ( c )

!

dc

(34)

Part (ii): Second order conditions In order to verify second order conditions we first analyze the cross partial deriva- tives with respect to different technologies and observe that they equal to zero:

d

d x c 1 , c

2

h

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

)i

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

for c [c0, c00] as defined by (16) and the ) does only depend on xi(c) for c [c0, c00], but f a c t 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 E Q r e s u l t i n g S p o t m a r k e t e q u i l i b r i u m Q E Q ( θ c E Q

not on xi(c) for c [c1, c2], where [c0, c00] an [c1, c2] are arbitrary non-overlapping intervals. Thus expression (18) 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 EQ 0 , c 0 0

)i

=0

We now focus on the second derivatives with respect to the same technologies, first we

f o c u s o n

t h e i n t e r i o r c a s e , i . e w e n e e

dt

o s h o w

d 2 π i ( x , q

### EQ)

d(x

c 2 0 , c 0 0 ) )

< 0 . I n o r d e r t o p r o o f c o n c a v i t y

o f t h e i n t e r i o r s o l u t i o n , w e d i ff e r e n t i a t e e x p r e s s i o n ( 3 4 ) w i t h r e s p e c t t o d x ( c 0 c 0 0 ) :

d π 2 i ( x , q E Q d (x 2 ( c 0 , c 0 0 ) ) )

=

c00

## Z

c0

d

d x ( c 0 , c 0 0 )

Ã

1

F ( θ c E Q , i

) + k c ( c ) + f ( θ c E Q , i

)

Ã jX=6 i

R j !

( P q ) x i ( c )

!

dc

40

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