Abstract
We examine the investment rule that must be satisfied by an efficient and egalitarian path in a discrete-time version of the Dasgupta–Heal–Solow model of capital accumulation and resource depletion. In the discrete-time model, competitive valuation of net investments in terms of early and late pricing differs. We redefine Hartwick’s rule to require zero value of net investments at a valuation rule intermediate between these two. Using this definition, we show that along an efficient and egalitarian path, Hartwick’s rule is followed in all time periods. We thereby establish the converse of Hartwick’s result in discrete time, and we do so under weaker assumptions than those in the existing literature on how output varies as a function of capital and resource use. Our redefinition of Hartwick’s rule follows naturally if discrete time is viewed as providing information at discrete points in time of an underlying continuous-time process.
Highlights
Hartwick’s rule for sustainability prescribes reinvesting resource rents, keeping the value of net investments equal to zero
This paper shows how the converse of Hartwick’s result obtains in the model of capital accumulation and resource depletion under weaker assumptions than those imposed by Dasgupta and Mitra (1983)
Even though the competitive prices along such an efficient and egalitarian path do not provide the precise price ratio for which the value of net investments is zero, we show that such prices at the discrete points in time yield a narrow interval for the price ratio
Summary
Hartwick’s rule for sustainability prescribes reinvesting resource rents, keeping the value of net investments equal to zero. That gives zero value of net investments when going from one point in time to the The proof of this result is based on the following observation made by Malinvaud (1953) in discrete time and developed in the context of efficient and egalitarian paths in continuous time by Mitra (2002): Vectors of initial stocks that are sufficient to maintain well-being at or above a given level c form a convex set S(c). By weakening the assumptions on the production function, we establish a significantly stronger result With their additional assumption, namely that resource use has a non-vanishing functional share of output as resource use approaches zero, Dasgupta and Mitra (1983, Theorem 1) show that if it is feasible to keep consumption bounded away from zero starting with given initial stocks of capital and resource, these stocks can give rise to an efficient and egalitarian path. All proofs—except the demonstration of our main result—are contained in the “Appendix”
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