## Where θ^{∗ }denotes the critical demand realization where firms start to become capacity constrained and c^{∗ }is the corresponding technology where firms start to be capacity con- strained. Denote by θ^{cap}(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 θ^{cap }

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 c^{∗}dF (θ) > k(c^{∗}), then indeed the capacity condition (ii) c ∗ F B = c^{∗}. Thus

θ^{cap }

( 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 c^{H }, 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 θ^{cap }

(c^{H }) < θ (if not,

the LHS of (ii) would drop to zero). Thus θ^{cap }

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

at c^{H }, which

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

•

Uniqueness: Deriving the slope of θ

^{cap}(c) we obtain by implicit function theorem:

Whenever θ^{cap }

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 k_{c}(c) F (θ^{cap }(c)) 0 ( c ∗ ) =

1

k_{c}(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

θ^{cap }

(c)

intersects

necessarily

from

above

(i.e.

at

(θ^{∗}, c^{∗}),

the

locus

θ^{cap }

( 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:

c^{00 }

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

)

=

## Zµ

c^{0 }

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