Optimal longevity of a dynasty

Abstract

Standard optimal growth models implicitly impose a ``perpetual existence'' constraint, which can ethically justify infinite misery in stagnant economies. This paper investigates the optimal longevity of a dynasty within a Critical-Level Utilitarian (CLU) framework. By treating the planning horizon as an endogenous choice variable, we establish a structural isomorphism between static population ethics and dynamic growth theory. Our analysis derives closed-form solutions for optimal consumption and longevity in a roundabout production economy. We show that under low productivity, a finite horizon is structurally optimal to avoid the creation of lives not worth living. This result suggests that the termination of a dynasty can be interpreted not as a failure of sustainability, but as an {altruistic termination} to prevent intergenerational suffering. We also highlight an ethical asymmetry: while a finite horizon is optimal for declining economies, growing economies under intergenerational equity demand the ultimate sacrifice from the current generation.

Figures (7)

• Figure 1: Left: Utility function of an individual. The consumption level for neutrality (or well-being subsistence) is denoted by $\nu$. Marginal utility and average utility coincide at $c=\omega$. Right: Contribution to the critical-level utilitarian population value, where $\alpha$ denotes the critical level. At $c=\kappa$, the utility reaches the critical level. In all cases, the domain of permissible consumption levels $c$ for existence is $c>\nu$.
• Figure 2: Left: The solid lines represent $g[N]$, as defined in (\ref{['fngn']}), for cases I--IV with clear correspondences, i.e., the steepest slope corresponds to case I, while case IV corresponds to the scenario with a negative slope. The dashed line represents $f[N]$, which is also defined in (\ref{['fngn']}). Right: The plots show population value functions $\mathcal{V}[N]$ for cases I--IV with obvious correspondences.
• Figure 3: Population value function (top), seleted trajectories of undiscounted contribution (middle) and capital intensity (bottom), for AK setting cases I (left), III (center), and IV (right). The trajectories are selected for $N=200, 400, 600$.
• Figure 4: Population value function (top), selected trajectories of undiscounted contribution (middle) and capital intensity (bottom), for ZD setting cases V (left), VI (center), and VII (right). The trajectories are selected for $N=200, 400, 600$. Note that the population value function of the top-left panel is increasing indefinitely. The parameters corresponding to each of the cases are given in Table \ref{['tab_parameters']}.
• Figure 5: Lorenz curves for $N=200$ (solid line), $N=400$ (dotted line), and $N=600$ (dashed line). The top row panels (from left to right) correspond to AK settings I, III, and IV. The second row panels (from left to right) correspond to ZD settings V, VI, and VII.

Theorems & Definitions (6)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• Proposition 1
• proof