Accelerators in macroeconomics: Comparison of discrete and continuous approaches

TL;DR

This paper demonstrates that standard discrete-time macroeconomic models do not accurately replicate continuous-time models and proposes an exact discrete approach using finite differences to ensure consistent economic behavior predictions.

Contribution

It introduces a self-consistent discrete-time modeling method based on exact finite differences, aligning discrete models with their continuous counterparts.

Findings

Standard discrete models differ from continuous models in solutions and predictions.

Exact finite difference approach aligns discrete and continuous models.

Using Harrod-Domar model, both models produce identical solutions.

Abstract

We prove that the standard discrete-time accelerator equation cannot be considered as an exact discrete analog of the continuous-time accelerator equation. This leads to fact that the standard discrete-time macroeconomic models cannot be considered as exact discretization of the corresponding continuous-time models. As a result, the equations of the continuous and standard discrete models have different solutions and can predict the different behavior of the economy. In this paper, we propose a self-consistent discrete-time description of the economic accelerators that is based on the exact finite differences. For discrete-time approach, the model equations with exact differences have the same solutions as the corresponding continuous-time models and these discrete and continuous models describe the same behavior of the economy. Using the Harrod-Domar growth model as an example, we show that equations of the continuous-time model and the suggested exact discrete model have the same solutions and these models predict the same behavior of the economy.