Caputo-Type Memory Invariants: A Fractional Generalization of the Cobb-Douglas Production Function

Abstract

Standard dynamical systems approaches to economic modeling, such as those deriving the Cobb-Douglas and CES production functions from exponential growth trajectories, typically rely on integer-order differential equations. While effective, these models assume that economic output depends solely on the instantaneous state of capital and labor, effectively ignoring the long-term ``memory effects'' inherent in policy, infrastructure, and technological adoption. This paper extends the exponential framework by introducing the Caputo fractional derivative into the underlying dynamical systems governing factor inputs. By replacing standard growth rates with fractional-order counterparts of order $0 < \alpha \le 1$, we model economic trajectories where the rate of change is a non-local function of the system's entire history. We demonstrate that the Mittag-Leffler function emerges as the natural growth solution in this context, providing a nested generalization of the classical exponential model. Unlike exogenous fractional frameworks, we use this fractional approach to derive a new class of time-independent invariants that serve as generalized production functions. We show that as the fractional order approaches unity, these forms converge exactly to the classical Cobb-Douglas function.