Stochastic continuous time growth models that allow for closed form solutions

• Published in: Journal of economics (Vienna, Austria) Vol. 124; no. 3; pp. 213 - 241
• Main Authors: Menoncin, Francesco, Nembrini, Stefano
• Format: Journal Article
• Language: English
• Published: Vienna Springer 01.07.2018; Springer Vienna; Springer Nature B.V
• Subjects: Consumption; Economic models; Economic theory; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Economics and Finance; Expected utility; Game Theory; Growth models; Macroeconomics/Monetary Economics//Financial Economics; Microeconomics; Public Finance; Risk; Risk aversion; Risk factors; Simulation; Social and Behav. Sciences; Technology; CRRA preferences; O4; Closed-form solution; Dynamic stochastic general equilibrium models; HARA preferences; Autonomous consumption; E2
• ISSN: 0931-8658; 1617-7134
• DOI: 10.1007/s00712-017-0567-z

We find a closed form solution that maximises the expected utility of an agent's inter-temporal consumption subject to a stochastic technology, which is a linear combination of AK and Cobb-Douglas technologies. Additionally, we consider two cases of agent preferences: (i) Constant Relative Risk Aversion (CRRA) preferences, which treat optimal consumption as a linear function of capital, and (ii) Hyperbolic Absolute Risk Aversion (HARA) preferences, which treat optimal consumption as an affine function of capital. By establishing a minimum (subsistence) level of consumption in the HARA model, we are able to create a framework that more accurately represents real-world circumstances than previous studies have done. Furthermore, for both the CRRA and HARA cases we show the suitable, consistent stochastic differential equation which describes the capital dynamics. Finally, we perform a numerical simulation based on the CRRA case and calibrate US data for the HARA case.