Optimal longevity of a dynasty

This paper develops a dynamic optimization framework for evaluating the optimal longevity of a dynasty under a critical-level utilitarian social welfare criterion. Unlike standard models that assume an infinite planning horizon and apply discounting to future generations, we endogenize the planning horizon itself and examine the welfare-maximizing number of generations in a roundabout production economy with capital accumulation. The model incorporates a well-being threshold that must be met for generational existence to be considered ethically justified, providing an alternative to traditional utilitarian aggregation. Using finite-horizon dynamic programming, we derive closed-form consumption trajectories and analyze how technological parameters and intertemporal preferences affect the optimal generational span. Our results show that the optimal horizon need not be infinite -- even under zero discounting -- and that intergenerational equity may require limiting dynasty length.