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X M

which gives rise to those critical demand realizations, we observe different investment

behavior, firms will invest less than under perfect competition.

For the boundary case on the other hand, the first order condition pins down the trade off firms faces at the capacity bound. The left hand side is marginal revenues at the spot market from additional investment in technology c0 while the right hand side is marginal cost of investment.

The following theorem 2 provides a full characterization of firms’ investment decision X M ( c ) u n d e r j o i n t p r o fi t m a x i m i z a t i o n .

Theorem 2 Monopoly Solution) Under profit maximization the industry investment

f u n c t i o n X M ( c ) i s (X, c) : X u n i q u e a n d c h a r a c t e r i z e d a s f o l l o w s : X M ( c ) = 0 : { P ( X , θ c F B ) + P q ( X ) X = c } n P ³ X , θ + P q ´ X¢ X = c

o

c < ccc cc< c.

(14)

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 (c [c, c]) is identically to

c h o o the First best case characterized by 1 s e s x M ( c ) = 1 n X M ( c ) .

F ( θ c F B

) + kc(c) = 0. Under symmetry each firm

For an illustration compare figure 4.

Remark 1 In

the

peak

load

pricing

literature

(see

for

example

Crew

and

Kleindorfer

(1986), pp. 77–79), a so called “second best solution” is proposed, where welfare is max- imized under a profit constraint. The tightness of the profit constraint can be expressed through a shadow price denoted by 0 and normalization yields λ = 1+ [0, 1]. For λ [0, 1] all solutions lying within the two extreme points of welfare maximum and profit maximum can be obtained (see Crew and Kleindorfer (1986) expression (4.58) on p. 79). As it turns out, within our approach of a continuous technology set all those solutions share

t h e s a m 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 s θ c F B

, which solve 1

F ( θ c F B

) + kc(c) = 0, c [c, c].

F i r m s i n v e s t m e n t x λ ( c ) = 1 n X λ ( c ) d i e r s , h o w e v e r , d e p e n d i n g o n t h e t i g h t n e s s o f t h e p r o fi t

X λ ( c ) = (X, c) : X : constraint, as expressed through the parameter λ. We obtain 0 { P ( X , θ c F B ) + P q ( X ) λ X = c } n P ³ X , θ + P q ´ X¢ λX = c

o

c < ccc cc< c.

(15)

15

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