the capacity bound, where the right hand side is just marginal welfare at the spot market from additional investment in technology c while the left hand side is marginal cost of investment.The overall capacity choice X and the corresponding boundary technology c^{∗ }consequently have to satisfy both the interior first order condition (6) and the boundary condition (7). In lemma 3 we now characterize the set of active technologies [c^{∗}, c^{∗}]:

Lemma 3 Active Technologies) If investment in some technologies is profitable, i.e. R (θ c) dF (θ) > k(c) for some c ≥ 0, then firms will invest in the technologies c ∈ [c^{∗}, c^{∗}]. c

(i) Lower Bound c^{∗}): The lowest technology for which investment is profitable c^{∗ }is

F ( c ∗ ) + k c ( c ∗ ) = 0 . T h e d e m a n d r e a l i z a t i o n f o r w h i c uniquely characterized by: 1 h fi r m s s t a r t t o p r o d u c e i s g i v e n b y θ ∗ = c ∗ .

(ii) Upper Bound c^{∗}): The highest technology for which investment is profitable c^{∗}, and the demand realization when firms are capacity constrained θ^{∗ }are uniquely char- acterized by:

(a)

(b)

1 Z_{∗ }

F (θ^{∗}) + k_{c}(c^{∗}) = 0

θ

θ^{∗}dF (θ) = k(c^{∗})

(8)

Proof See appendix 8 I n t h e o r e m 1 w e p u t a l l t h e s e r e s u l t s t o g e t h e r c h a r a c t e r i z i n g I n d u s t r y i n v e s t m e n t X which obtains under perfect competition: ∗ F B

(c)

# Theorem 1 Competitive Solution) Under perfect competition

the

industry

invest-

ment

function

X ∗ F B

(c)

is

unique

and

characterized

as

follows:

X ∗ F B

(c) =

ï£± ï£´ ï£´ ï£² ï£´ ï£´ ï£³

0 (X, c) : X:

{ P ( X , θ c F B n P ³ X , θ ∗ ´ ) = c} = c ∗ o

c < c^{∗ }c^{∗ }≤ c ≤ c^{∗ }c^{∗ }< c.

(9)

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

for the interior solution solves 1

F ( θ c F B

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

U n d e r S y m m e t r y e a c h fi r m c h o o s e s x ∗ F B

(c) =

1 n

X ∗ F B

(c).

For an illustration of industry investment obtained under perfect competition compare figure 4.

The entire analysis was conducted under the hypothesis, that all technologies c ∈ [c^{∗}, c^{∗}]

are part of a solution, i.e.

t h a t t h e s o l u t i o n X ∗ F B

(c) is strictly monotonic in c.

We obtain

12