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# Gregor Zoettl† - page 45 / 48

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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 x0(c) and the locus xCAP thus be characterized by the following two conditions:

(c) can

a)

b)

F (θ) Zθ

(kc(c) + 1) = 0

θdF (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 cis 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) = xfor 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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