X hits on this document





11 / 48

Computation of the second derivatives is relegated to appendix 8. A crucial property of the way the problem of investment in the optimal technology mix is presented throughout this article lies in the simplicity of the second order conditions, namely the cross derivatives with respect to different technologies always equal to zero. This is due to the fact, that the profitability of changing investment xi(c) for some technologies c [c0, c00] is the same for all different investment functions which coincide for the technologies c [c0, c00] under consideration but exhibit completely different values at other technologies. Optimality of investment xi(c0) at some technology c0 can thus be determined independently from the shape of the remaining function.

Let us now briefly provide some economic intuition for the first order conditions as given i n l e m m a 2 . F i r s t n o t i c e t h a t v a r i a t i o n o f i n v e s t m e n t x i ( c ) f o r c [ c 0 , c 0 0 ] a s i l l u s t r a t e d i n fi g u r e 3 c a n b e i n t e r p r e t e d a s s u b s t i t u t i n g t e c h n o l o g y c 0 0 b y t e c h n o l o g y c 0 ( i n t h e c a s e o f

d x ( c 0 , c 0 0 )

> 0 , t h e r e v e r s e f o r d x ( c 0 , c 0 0 )

  • >


The impact of such variation will reduce expected production cost on the one hand, it will increase however the investment cost of firm i. More specifically this implies that firms when choosing their optimal technology–mix face the following tradeoff: The extra revenues from such substitution in the case of perfect competition are given by the savings

in production cost (c00

c0), which occur with probability (1

F ( θ c F B

)) and are given by

R c 0 0 c 0 ( 1

F ( θ c F B

))dc. The extra expenditures of such variation on the other hand are given

by the extra investment cost

R c 0 0 c 0

kc(c)dc = k(c00)

k(c0) < 0.

Thus whenever 1

F ( θ c F B

) + kc(c) > 0 firms want to increase investment in the more

efficient technology c0 by substitution with the less efficient technology c00.

For the converse,

1 F ( θ in c00. c F B

)+kc(c) < 0 firms want to invest less in the more efficient technology c0 by reducing

We now denote by cthat technology where firms just start to invest and denote by θ= cthat demand realization where firms start to produce for given investment. 16 Investment

below ccannot be profitable, this lower bound is characterized by 1

F (c) + kc(c) = 0,

see lemma 3 (i). For c > c, the optimal technology mix then satisfies 1

F ( θ c F B

) + kc(c) = 0 for all

technologies c [c, c]. Notice however that this interior first order condition is only a relative statement, establishing the optimal mix of technology choice. It still leaves the question of absolute capacity choice, untouched. This is tackled in expression (7) of lemma 2, the first order condition for a capacity change. It pins down the trade off a firm faces at

16i.e. X(c)=0, for c cand firms do not produce for θ θfor given investment.


Document info
Document views212
Page views212
Page last viewedMon Jan 23 13:25:15 UTC 2017