given interval of marginal cost of production [c0, c00] we show concavity. Furthermore, most importantly, all cross derivatives with respect to changing investment at different level of production cost (i.e. the two disjunct intervals [c0, c00] and [c1, c2]) equal zero. This implies that optimality of increasing or reducing investment for a given interval of marginal cost of production [c0, c00] is independent of changes in investment made at any other level of marginal cost of production, i.e. [c1, c2].
T h i s i s i l l u s t r a t e d i n fi g u r e 1 0 . W h e n e v e r c u m u l a t i v e i n v e s t m e n t x D E V b i
(c) is in the
Figure 10: Deviation from the symmetric equilibrium candidate is not profitable.
r e g i o n b e l o w x ∗ E Q , i
(c) for some c ∈ [c0, c00], then firm i can increase it’s profits by increasing
investment in c ∈ [c0, c00]. Since cross derivatives equal zero this does not interfere with changing investment for other levels of marginal cost of production. We can thus conclude
that firm i will
c a n d i d a t e x ∗ E Q
always reduce it’s profits when deviating from the symmetric This proves existence of the symmetric equilibrium.
Part III) Show that an asymmetric equilibrium cannot exist: We finally show that an asymmetric equilibrium of the investment market game cannot