# The perfectly competitive spot market outcome for fixed industry investment X(c) and fixed realization of uncertainty θ ≥ 0 is given by the well known condition of ”price = marginal cost”, i.e. P (X(c), θ) = c ∀θ. As in the previous section for the general case, we

n o w d e fi n e 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

which makes firms produce at marginal cost

c . W e o b t a i n f o r θ c F B

and the corresponding spot-market output Q^{F }^{B }

(θ):

Q^{F B }θ (θ) c F B

= =

c + B(X(c)) ∀c { ( Q , θ ) : ( Q = X ( c ) , θ = θ c F B

∀c)}

(5)

Having specified the outcomes at the spot market under perfect competition for given investment choice, we can now turn towards solving for equilibrium investment choice at stage one. We already have derived general first order conditions in lemma 1. It now remains to adapt those conditions for the case of perfect competition and to verify their sufficiency, as summarized in lemma 2.

Expressions (6) and (7) of lemma 2 can be directly derived from expressions (3) and (4), since the last summand of it’s integrands equals zero. This is due to the following two

observations: first under perfect competition firms take the market price as given i.e.

dP (·) dx

=

0 a n d s e c o n d , b y d e fi n i t i o n o f θ c F B

a s e s t a b l i s h e d a b o v e i n ( 5 ) , w e h a v e P ( Q ( θ c F B

) , θ c F B

) = c.

# We obtain the following equilibrium conditions under perfect competition:

Lemma 2 Optimality Conditions, Perfect competition) .

(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 :

c^{00 }

dπ_{i}(x, q^{F B }d x ( c 0 , c 0 0 )

)

=

Z

(1

F ( θ c F B

) + k_{c}(c)) dc = 0.

(6)

c^{0 }

# For a change of total capacity x, which a ects all c ≥ c_{i }we obtain:

dπ_{i}(x, q^{F B }d x ( c i )

)

=

Z c F B

P X, θ^{¢ }

c dF (θ)

k(c) = 0.

(7)

(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 ) FB c 0 , c 0 ) 0 )

= 0 ( f o r c 1 < c 2 < c 0 < c 0 0 ) a n d

dx d 2 π i ( x , q c 0 , c 0 0 ) dx FB c i ) )

=0.

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

d^{2}π_{i}(x,q^{FB }d(x c 2 0 , c 0 0 ) ) )

< 0 and

d^{2}π_{i}(x,q d(x c i ) ) FB 2

)

< 0.

10