The locus of the optimal capacity choice condition is illustrated in figure 9. The candidate equilibrium has to satisfy differential equation (37) and pass through the intersection of the

l o c u s x ∗ 0 ( c ) w i t h t h e l o c u s x ∗ C A P

(c) (point ”B” in figure 9) as boundary condition.

In order to show uniqueness of the symmetric candidate equilibrium it remains to show t h a t t h e b o u n d a r y c o n d i t i o n i s i n d e e d u n i q u e , i . e . t h a t t h e i n t e r s e c t i o n o f t h e l o c u s x ∗ 0 ( c )

w i t h t h e l o c u s x ∗ C A P

(c) is unique.

# In order to do so we observe that the integrand of

e x p r e s s i o n ( 3 8 ) c a n b e r e w r i t t e n i n t e r m s o f 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 use of expression (17): c E Q

, by making

P ( n x , θ ) + P q x

c=θ

(B(nx)

P q ( n x ) x + c ) = θ

θ c E Q

The pair (x^{∗}, c^{∗}) which has to lie both on the locus x_{0}(c) and the locus x_{CAP }thus be characterized by the following two conditions:

(c) can

a)

b)

F (θ^{∗}) Z_{∗ }θ

(k_{c}(c^{∗}) + 1) = 0

θ^{∗}dF (θ) = k(c^{∗})

## Where x^{∗ }=

n

x : P ( x , θ ∗ ) + P q ( x ) x

c ∗ = 0 o , θ ∗ d e n o t e s 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

where firms start to become capacity constrained and c^{∗ }is the corresponding technology where firms start to be capacity constrained. Existence and uniqueness of (θ^{∗}, c^{∗}) have already been established in lemma 3. We can conclude that theorem 3 characterizes a unique symmetric equilibrium candidate.

# Part II) Show that deviation from the candidate equilibrium is not pro table

We now show that deviation of firm i = 1, . . . , n is not profitable if all other firms

s t i c k t o t h e c a n d i d a t e e q u i l i b r i u m x ∗ E Q , i

. By construction of the symmetric candidate

equilibrium the first order conditions of firm i both for an interior change (expression (18) of lemma 5) and for a change of total capacity are satisfied (expression (19) of lemma

5 ) . M o r e o v e r w h e n e v e r fi r m i i s c a p a c i t y c o n s t r a i n e d ( i . e . x 0 E Q , i

= 0 , x ∗ E Q

(c) = x^{∗ }for all

c ≥ c^{∗}, ”above point S in figure 9”) then the first order condition for an interior change (expression (18) of lemma 5) is negative, which implies that firm i indeed finds it optimal to be capacity constrained for all c ≥ c^{∗ }(Remember, this is how the unique symmetric equilibrium candidate has been determined in Part I) of the current proof).

In lemma 5 (ii) we furthermore show sufficiency of the first order conditions of firm i. That is, when considering the second derivative with respect to changing investment for a

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