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investment choice, we can now turn towards solving for optimal investment choice at stage one. Again relying on the general first order conditions derived in lemma 1, we can directly deduce the first order conditions under profit maximization as summarized in lemma 4.

Expression (12) is directly obtained from (3) since it’s last integrand equals zero for the case of profit maximization. This is due to the following two observations: first

d P ( Q dx M c 0 , c 0 0 ) ,

)

x i ( c ) = P q ( · )

d Q dx M ( ) c 0 , c 0 0 x i ( c ) = P q X ( c ) , a n d s e c o n d b y t h e d e fi n i t i o n o f θ c M

as estab-

l i s h e d i n ( 1 0 ) , w e h a v e P ( Q ( θ c M ) , θ c M )

c + P q ( Q ( θ c M

))X(x)

=

0.

We

obtain

the

following

optimality

conditions

under

profit

maximization:

Lemma 4 Optimality Conditions, Profit Maximization) .

(i) First order conditions: F o r a n i n t e r i o r c h a n g e a e c t i n g c [ c 0 , c 0 0 ] w h e r e ( c 0 < c 0 0 c i ) w e o b t a i n :

c00

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

=

Z

(1

F ( θ c M ) + k c ( c ) ) d c = 0 .

(12)

c0

For a change of total capacity x, which a ects all c ci we obtain:

i(x, qM ) d x ( c i )

=

Z ci M

P

X , θ ¢ + P q

X¢ X

c dF (θ)

k(c) = 0.

(13)

(ii) Second order conditions: The cross derivatives with respect to di erent technologies equal zero,

i.e.

dx

d 2 π i ( x , q c 1 , c 2 ) dx

) M c 0 , c 0 0 )

=0 for (c1 < c2 < c0 < c00) and

d 2 π i ( x , q d x c 0 , c 0 0 ) d x M c i ) )

=0.

The second derivatives with respect to the same technologies are always negative, i.e.

d 2 π i ( x , M q

d(x

c 0 , c 0 0 ) )

)

2

< 0 and

d d(x 2 π i ( x , q c i ) ) M 2

)

< 0.

Computation of the second order conditions is relegated to appendix 8.

Those first order conditions look very similar to the case of perfect competition as

obtained in the last section. Most notably the locus of m a k i n g fi r m s p r o d u c e a t s o m e l e v e l o f m a r g i n a l c o s t c , θ c M c o m p e t i t i o n s i n c e i t i s a l s o g i v e n a s t h e s o l u t i o n t o 1 the critical demand realizations, is identical to the case of perfect )+kc(c) = 0, in the following we F ( θ C m

w i l l t h u s k e e p t h e n o t a t i o n θ c F B

.

Thus under profit maximization firms will choose to invest

into the same active set in lemma 3 for the case

of of

technologies c [c, c] which has already been characterized perfect competition. When backing out Industry investment

14

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