For a change of total capacity x, which a ects all c ≥ ci we obtain:
dπi(x, qEQ dx ci)
Z θ θ c i E Q , i
P Q E Q , θ ¢ + P q
x Ã 1 +
d Q dx ci) E Q i
c i !
k(ci) = 0
(ii) Second order conditions: The cross derivatives with respect to di erent technologies equal zero,
d 2 π i ( x , q c 1 , c 2 ) dx
) M c 0 , c 0 0 )
=0 for (c1 < c2 < c0 < c00) and
d 2 π i ( x , q d x c 0 , c 0 0 ) d x M c i ) )
The second derivatives with respect to the same technologies can be shown to be neg-
d 2 π i ( x , d(x q M c 0 , c 0 0 ) )
< 0 and
d2πi(x,q d(x c i ) ) M 2
< 0, if the following conditions are satisfied:
( a ) D e m a n d i s l i n e a r , i . e . P q q = 0 ,
f00(θ) ≤ 0 whenever
x 0 0 j
(c) ≤ 0 ∀c, j =6
f 0 ( θ ) > 0 ,
Proof see appendix 8.
Also for the case of strategic interaction we observe first of all that again the second order conditions have a very special and simple form: all cross derivatives equal to zero. Again (for given investment decisions X i(c)) the profitability of substituting investment in technology c0 by investment in technology c00 is solely determined by the investment level xi(c) forc ∈ [c0, c00] but not by the investment decision in other technologies c < c0 or c > c00.
Verification of second order condition thus reduces to checking for negative second
derivatives with respect to the same technologies, i.e.
d 2 π i ( x , M q
c 0 , c 0 0 ) )
< 0 and
d(x d 2 π i ( x , q c i ) ) M 2
The computations involved are relatively burdensome, and we restrict to the case of lin- ear demand in order to maintain tractability of the problem (It seems however that there are no major obstacles when extending the present analysis of second order conditions to the nonlinear case). Furthermore in order to ensure concavity of the problem two further assumptions are required: first the density of uncertainty should not increase too steeply (condition (b)) and second the investment functions chosen by all rivals should become flatter and flatter as the capacity bound xj is approached (condition (c)).
The first order conditions can be interpreted similar to the case of welfare or profit
maximization as analyzed in sections 3 and 4. For the interior solution, firms face the trade off of substituting investment in technology c0 by investment in technology c00. Again this decision is driven by the mass above the critical demand realization versus the difference i n i n v e s t m e n t c o s t , i . e . 1 F ( θ c E Q , i ) + k c ( c ) . U n d e r S t r a t e g i c i n t e r a c t i o n , h o w e v e r also take into account the impact of their investment decision on the rivals spot market , fi r m s