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 the situation usually changes to decreasing returns to scale because it becomes increasingly di¢cult 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 classi...cation under which parameter constellations a saddle point equilibrium is optimal, under which parameter constellations it is optimal to close down by choosing zero investment and when history dependent equilibria occur. In the last scenario we distinguish between di¤erent types of the unstable equilibrium, which can each have their own implication for the ...rm’s investment policy.