Optimal investments with convex–concave revenue: a focus-node distinction

This paper considers a capital accumulation model in which revenue is a convex–concave function of the capital stock. While for certain capital values increasing returns to scale are reasonable, usually this property does not hold in general. In particular for large capital stock values it becomes increasingly difficult and thus expensive to produce more and more because of limitations of resources or infrastructure, lack of trained personnel in the region, etc. We give a complete classification under which parameter constellations a saddle point equilibrium is optimal, when to close down by choosing zero investment and when history dependent equilibria occur. In the last scenario we distinguish between different types of the unstable equilibrium, which can each have their own implication for the firm's investment policy.