Environmental endogenous discounting and multiple equilibria: A comment

ABSTRACT

The purpose of this paper is to supplement the study of Dioikitopoulos et al. (2020), which introduces an endogenous time preference and presents a model for the comprehensive investigation of economic growth and environmental issues. In particular, this paper explores some aspects of the numerical analysis in the original paper and presents important numerical examples regarding multiplicity and local stability of equilibria.

KEYWORDS:

• Endogenous time preference
• environmental quality
• multiple equilibria
• numerical analysis

JEL CLASSIFICATION:

• E70
• O44
• Q56

Acknowledgments

The author would like to thank the Editor-in-Chief (Mark P. Taylor) and the anonymous referee for helpful comments and suggestions. The author also gratefully acknowledges the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number JP21K01507) and the Japan Center for Economic Research for financial support. Any errors are my own responsibility.

Disclosure statement

No potential conflict of interest was reported by the author.

Availability of data and material

The data used can be provided upon request.

Code availability

The code used can be provided upon request.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

¹ From here on, a hat attached to a variable indicates that it is a steady-state value.

² $r(\hat{z})$ is the equilibrium interest rate, which can be expressed as $r(\hat{z})=Aa(1-\tau ){\hat{z}}^{a-1}-\delta$.

³ ${\rho }_N=\frac{\mathrm{\partial }\rho (N)}{\mathrm{\partial }N}=-\gamma \le 0$ can be confirmed. Therefore, we find that a property of decreasing marginal impatience is assumed with respect to environmental quality.

⁴ In the supplementary appendices, a stability analysis is also performed, and in the process, the Jacobian related to the model’s dynamic system is presented in detail. The results of the analysis using the Jacobian are correct, but the notation contains typographical errors. The corrected version of the Jacobian is presented in Appendix D of the present paper.

⁵ In this numerical example, the endogenous time preference function is specified as $\rho (N)=\bar{\rho }-\gamma N$. The positivity condition for the growth rate in the steady-state equilibrium is $\hat{z}>{\left(\frac{\delta }{Ab\tau }\right)}^{1/a}$.

⁶ The MATLAB code for this calculation is available from the author upon reasonable request.

⁷ The combination of the eigenvalues is similar to the superior equilibrium case for the two equilibria in Table 2.

Additional information

Funding

Grant-in-Aid for Scientific Research (C) (JSPS KAKENHI Grant Number JP21K01507) from the Japan Society for the Promotion of Science and the Japan Center for Economic Research.