Sustainability with endogenous discounting

• Author: Ngo Van Long,John M. Hartwick
• Date: 01 June 2020
• Published date: 01 June 2020
• DOI: http://doi.org/10.1111/ijet.12237

doi: 10.1111/ijet.12237

Sustainability with endogenous discounting

John M. Hartwick*and Ngo Van Long†

We construct a dynamic competitivemodelwithastockofhuman‐made capital and several stocks

of natural resources and ask under what conditions consumption will be constant if infinitesimal

households with heterogeneous preferences and endowments discount their utility at an endogenous

rate that depends on some macroeconomic variables. We show that for consumption to be constant,

this function must be the marginal product of capital function. We demonstrate that Hartwick’srule

holds in a modified form that takes account of natural growth of resource stocks.

Key words constant consumption, endogenous discount rate, Hartwick’s rule, natural re-

sources, sustainability

JEL classification D63, O44, Q01, Q20, Q33

Accepted 25 August 2017

1 Introduction

The concern for sustainable development has led to several formulations of the concept of sus-

tainability (e.g. Solow 1974; Pezzey 1992; Asheim 2010; Mitra, Asheim, Buchholz, and Withagen

2013; Cairns and Martinet 2014; Fleurbaey 2015). The simplest sustainability concept is that of

constant consumption. The standard textbook model of the infinitely lived consumer assumes that

individuals discount the future at a constant rate, which generically rules out constant con-

sumption. There is an alternative stream of literature that assumes endogenous discounting

(Uzawa 1968; Epstein 1987; Obstfeld 1990; Pittel 2002, ch. 5; Ayong le Kama and Schubert 2007;

Yanase 2011). For example, the discount rate that applies at time

t

may depend on the consumer’s

wealth, or consumption level, or the quality of the environment at that time.

In this paper, we suppose that infinitesimal households use an instantaneous discount rate that

depends on some macroeconomic variable. We show that for constant consumption to hold, that

macroeconomic variable must be the marginal product of the aggregate capital stock. We demonstrate

that under these circumstances, Hartwick’s rule (that along the constant consumption path, resource

rents must be invested in human‐made capital) holds in a modified form that takes account of natural

growth of resource stocks.

2 The basic model

There is a continuum of infinitely lived individuals (or households), indexed by θ, where θranges

from

0

to

1

. Their utility functions may differ from each other, and are denoted by

θ(( ))

θ

u

ct,

,

International Journal of Economic Theory 16 (2020) 216–221 © 2019 IAET216

*

Department of Economics, Queen’s University, Kingston, Ontario, Canada.

†

Department of Economics, McGill University, Montreal, Canada. Email: ngo.long@mcgill.ca