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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.

equilibrium

Part III) Show that an asymmetric equilibrium cannot exist: We finally show that an asymmetric equilibrium of the investment market game cannot

46

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