TY - JOUR AU - Atre, Madhusudan V PY - 2022/04/26 Y2 - 2024/03/28 TI - MODELLING OF J-CURVE AND S-CURVE USING DIFFERENTIAL EQUATIONS: STUDIES IN ECONOMICS, ENTREPRENEURSHIP AND FINANCE JF - Global journal of Business and Integral Security JA - GBIS VL - IS - SE - Thesis DO - UR - http://gbis.ch/index.php/gbis/article/view/72 SP - AB - <p>J-Curve phenomenon, which shows how a system responds to an external influence, is observed in many areas like economics, financial investments, healthcare, etc.<br>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.<br>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.<br>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.<br>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.<br>An interesting property of the Riccati differential equation is also shown to explain the pharmacokinetic absorption of medicines in the body.<br>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.</p> ER -