Figure 2: Investment decision xi(c) of firm i.
In order to determine total profits of firm i associated to some investment choice xi(c), we first determine profits associated to some partial investment dxi(c0) as illustrated in figure 2. That is we determine profits generated by technology c0, the amount invested in this technology is given by dxi(c0). Observe that such investment will only yield positive r e v e n u e s f o r d e m a n d r e a l i z a t i o n s θ > θ c 0 i . E x p e c t e d r e v e n u e s g e n e r a t e d b y d x ( c 0 ) a r e g i v by the expected markup in all those demand realizations where indeed production is at or above marginal cost c0 times the amount dx(c0) invested: e n
[P (Q(x, θ), θ)
c0] dF (θ) ∗ dxi(c0)
On the other hand, the cost of investment in technologies dxi(c0) is given by the (constant) marginal cost k(c0) of investment times the amount times the amount dx(c0) invested:
k(c0) ∗ dxi(c0)
In the context of electricity generation, the above analysis corresponds to determining expected profits generated by a small power plant of size dxi(c0) which produces at marginal cost c0 and comes at a cost of investment given by k(c0).
In order to obtain total profits associated to the entire investment-choice xi(c), we need to sum up for all technologies, where investment took place. Suppose firm i did choose to invest into technologies c ∈ [ci, ci], with 0 ≤ ci < ci, then integration over all those