Characterizing Optimality in Dynamic Settings: A Monotonicity-based Approach

We develop a novel analytical method for studying optimal paths in dynamic optimization problems under general monotonicity conditions. The method centers on a locator function -- a simple object constructed directly from the model's primitives -- whose roots identify interior steady states and whose slope determines their local stability. Under strict concavity of the payoff function, the locator function also characterizes basins of attraction, yielding a complete description of qualitative dynamics. Without concavity, it can still deliver sharp results: if the function is single crossing from above, its root identifies a globally stable steady state; if the locator function is inverted-U-shaped with two interior roots (a typical case), only the higher root can be a locally stable interior steady state. The locator function further enables comparative statics of steady states with respect to parameters through direct analysis of its derivatives. These results are obtained without solving the full dynamic program. We illustrate the approach using a generalized neoclassical growth model, a rational (un)fitness model, and a learning-by-doing economy.