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

TL;DR

This paper introduces a fractional calculus approach to economic modeling, generalizing classical production functions to incorporate long-term memory effects using Caputo derivatives.

Contribution

It develops a new class of memory-invariant production functions based on fractional derivatives, extending classical models like Cobb-Douglas.

Findings

Mittag-Leffler functions naturally describe fractional growth trajectories.

The fractional invariants converge to classical Cobb-Douglas as the order approaches one.

New generalized production functions incorporate long-term memory effects.

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. Using this fractional approach, we 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.