On Sustainability and Survivability in the Matchbox Two-Sector Model: A Complete Characterization of Optimal Extinction

TL;DR

This paper fully characterizes the conditions under which an economy in a two-sector growth model will go extinct, revealing how discount factors and sector capital intensities influence long-term outcomes.

Contribution

It provides a complete analysis of optimal extinction, including thresholds for extinction, the role of sector capital intensity, and bifurcation points, extending previous models.

Findings

Extinction occurs when discount factor is below a critical threshold.

Staggered extinction depends on sector capital intensity.

Multiple bifurcation thresholds influence extinction timing.

Abstract

We provide a complete characterization of optimal extinction in a two-sector model of economic growth through three results, surprising in both their simplicity and intricacy. (i) When the discount factor is below a threshold identified by the well-known $\delta$-normality condition for the existence of a stationary optimal stock, the economy's capital becomes extinct in the long run. (ii) This extinction may be staggered if and only if the investment-good sector is capital intensive. (iii) We uncover a sequence of thresholds of the discount factor, identified by a family of rational functions, that represent bifurcations for optimal postponements on the path to extinction. We also report various special cases of the model having to do with unsustainable technologies and equal capital intensities that showcase long-term optimal growth, all of topical interest and all neglected in the antecedent literature.