Dynamic Savings with Adaptive Adjustment in the Solow–Swan Model: Bistability and Escape from a Poverty Trap

We extend the Solow–Swan growth model by replacing the constant savings rate with a savings rate that adjusts gradually toward a capital-dependent target of Hill form. The savings rate becomes a state variable, turning the model into a two-dimensional dynamical system. For a sufficiently savings steep target, the system admits a poverty-trap regime with three equilibria: the origin, a saddle, and a stable high-capital equilibrium. We characterize the stability of every positive equilibrium by the elasticity of the savings target; in the bistable phase portrait, the saddle’s stable manifold provides the natural separatrix between the two observed attraction regions. We then study escape by a temporary public-investment flow. A direct geometric argument identifies an upper forward-invariant region from which the uncontrolled system converges to the high-capital equilibrium. If the investment flow satisfies a sufficient condition, the policy drives capital above a buffer level in finite time; savings then catches up through adaptive adjustment, after which the stimulus can be removed permanently. The model therefore converts the one-dimensional threshold of a static savings specification into a two-dimensional basin problem, where both the level of capital and the speed of savings adjustment are necessary for escape.