The monotonic path and its value loss when an optimal path is non-monotonic

Under standard economic assumptions, the optimal paths in optimal growth models can be non-monotonic and, at times, extremely complex. In contrast, real-world policies are typically based on the assumption of a monotonic progression towards objectives. To address this discrepancy, this study investigates the characteristics and value loss associated with an alternative monotonic path when the optimal path is non-monotonic in discrete-time, one-state variable optimal growth models. We assume that the planner selects the best path from a class of monotonic paths (i.e., either monotonically increasing or decreasing paths). We show that if the optimal path is increasing (or decreasing), the corresponding monotonic path will also be increasing (or decreasing). Monotonic paths generically encounter time inconsistency when reaching their steady states. If the monotonic path is revised at this point, the transition from increasing to decreasing, or vice versa, in the monotonic path occurs in tandem with a similar transition in the associated optimal path. Distinct features of the monotonic paths compared to the optimal paths include time inconsistency and the finite time to reach the steady state. Moreover, the monotonic path with revision exhibits differences in the local stability of the common interior steady state compared to the optimal policy. Regarding value loss, in three models demonstrating chaotic optimal paths, the study finds that the upper bounds of the value loss ratios incurred by adopting monotonic paths without revision range from 10-5 to 10-13 relative to the optimal value function. We argue the potential generality of this marginal value loss. Furthermore, we discuss several implications of these findings, including a possible rationale for why complex solutions to optimization problems can describe human behavior that is not universally optimal.