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Where θdenotes the critical demand realization where firms start to become capacity constrained and cis the corresponding technology where firms start to be capacity con- strained. Denote by θcap(c) the locus of all points satisfying (ii) and remember that the

l o c u s o f a l l p o i n t s s a t i s f y i n g ( i ) i s d e n o t e d b y θ c F B

. In the following we show that these two

lines intersect exactly once whenever firms are active on the market, i.e. (θ, c) exists and is uniquely defined by the above system:

existence:

If we have

is solved for some θcap

R c ( θ c ) > c , w h e r e a s c o n d i t i o n ( i ) i s s o l v e d b y θ cdF (θ) > k(c), then indeed the capacity condition (ii) c F B = c. Thus

θcap

( c ) l i e s s t r i c t l y a b o v e θ c F B

at c. If on the other hand we consider some sufficiently

high cH , condition (i) will be solved at θ. Since we assume however k(c) > 0 for all

c (see assumption 2 (i)) condition (ii) can only be satisfied for θcap

(cH ) < θ (if not,

the LHS of (ii) would drop to zero). Thus θcap

( c ) l i e s s t r i c t l y b e l o w θ c F B

at cH , which

proofs that at least one solution to the above system of two equations must exist.

  • Uniqueness: Deriving the slope of θcap(c) we obtain by implicit function theorem:

Whenever θcap

θ c a p 0 ( c ) = ( c ) i n t e r s e c t s θ c F B 1 , w e o b t a i n : θ c a p kc(c) F (θcap (c)) 0 ( c ) =

1

kc(c F(

) )

= 1. Since all interior

solutions satisfying

condition (i)

exhibit

positive slope (i.e.

θ F B

(c)

  • >

    1 we observe

that whenever the locus where condition (ii) is satisfied intersects the locus where (i)

is

satisfied,

then

θcap

(c)

intersects

necessarily

from

above

(i.e.

at

(θ, c),

the

locus

θcap

( c ) i s l e s s s t e e p t h a n t h e l o c u s θ c F B

), which proofs uniqueness.

Proof lemma 4

The argument with respect to the cross derivatives is analogous to the proof of lemma 2 in appendix 8. The second derivatives with respect to the same technologies are obtained 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 dx c F B ( c 0 , c 0 0 )

dc < 0.

(23)

This expression is negative since

dx

c M

c 0 , c 0 0 )

  • >

    0, ( compare expression (10)). For the second

derivative with respect to capacity choice we obtain:

d 2 π i ( x , q F B d (x(ci)) 2

)

=

Z c F B

P q

X ¢ + P q q

X¢ X dF (θ) < 0.

(24)

35

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