MODELLING OF J-CURVE AND S-CURVE USING DIFFERENTIAL EQUATIONS: STUDIES IN ECONOMICS, ENTREPRENEURSHIP AND FINANCE

Abstract

J-Curve phenomenon, which shows how a system responds to an external influence, is observed in many areas like economics, financial investments, healthcare, etc.
We propose a mathematical formulation and framework for the J-Curve in terms of the Riccati differential equation and its associated Laguerre equation. The solutions describing the J-Curve are set up as polynomials similar to the Laguerre polynomials.
We give explicit functional forms for the system characteristics if it has to manifest J-Curve behavior and provide physical interpretations of the various terms in the Riccati equation to help understand the characteristics of any system manifesting a J-Curve behavior. We also set up criteria for any curve to be mathematically validated as a J-Curve.
The Riccati differential equation is used to describe the S-Curve, which describes the cumulative sales growth or population dynamics. Thus, the Riccati equation is shown to unify the mathematical basis of S-Curve and J-Curve.
We analyze five case studies for J-Curve behavior under the defined mathematical framework – a) four parameters of Indian economy are studied from 1960-2020 to validate J-Curve phenomenon post economic liberalization in 1991, b) Internal Rate of Returns for venture investments are proven to exhibit J-Curve, c) long term investments in stock markets are shown to follow J-Curve, d) The GDPs of some countries/regions post the 2007-08 financial crisis are analyzed for J-Curve, 5) GDP of Croatia is shown to exhibit J-Curve post-independence, as well as post global financial crisis.
An interesting property of the Riccati differential equation is also shown to explain the pharmacokinetic absorption of medicines in the body.
This is the first time 1) a mathematical formalism is set up to define the J-Curve phenomenon, 2) an explicit differential equation is defined for the J-Curve, 3) the functional forms of the system’s inertia, environmental damping, as well as the external influence acting on it are given, 4) explicit equation (polynomials with alternating coefficients) which shows the J-Curve behavior is described, 5) the mathematical conditions any curve has to satisfy if it has to qualify as a J-Curve are highlighted, 6) the S-Curve and J-Curve are both shown to be special cases of the generic nonlinear 1st order Riccati differential equation, 7) functional form of external influence on a system to manifest J-Curve behavior is explicitly discussed in the context of pharmacokinetics, and the functional form of medicine absorption in the body is presented.