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# Gregor Zoettl† - page 44 / 48

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Figure 9: Derive the symmetric equilibrium candidate, solutions of differential equation (37) and boundary condition given by point ”B”.

t h e l o c u s x 0 ( c ) , fi r m s w i l l n e v e r fi n d i t o p t i m a l t o r e m a i n c a p a c i t y c o n s t r a i n e d b u t w i always choose to slightly increase their investment (notice all optimal trajectories exhibit l l x 0 ( c ) > 0 ) a n d f o r a l l p a i r s ( x , c ) a b o v e t h e l o c u s x 0 ( c ) fi r m s w i l l n o t fi n d i t o p t i m increase investment, but will stop to invest choosing to be capacity constrained (which de facto implies x0(c) = 0). We can conclude that in the candidate equilibrium the pair (x, c) a l t o

w h e r e fi r m s s t a r t t o b e c a p a c i t y c o n s t r a i n e d n e c e s s a r i l y m u s t l i e o n t h e x 0 ( c ) l o c u s .

Finally the candidate solution must not only satisfy the interior optimality condition given by expression (18) of lemma 5, but also the optimality condition for optimal overall capacity choice given by expression (19) of lemma 5. For the symmetric case, expression

(19) yields all pairs (x, c) which satisfy the optimal total capacity choice condition.

We

d e n o t e t h e i r l o c u s b y x C A P

(c) which is given as follows:

x C A P

(c) :

(

(x, c) :

Z c E Q

P ( n x , θ ) + P q x

cdF (θ) = 0

)

(38)

44

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