Sustainability with endogenous discounting when utility depends on consumption and amenities

We study the issue of sustainability using a model with a stock of man-made capital and a stock of exhaustible natural resource that provides a flow of amenity services as well as an input for the production of a consumption good. We ask under what conditions the utility flow will be a constant if infinitesimal households discount their utility using an endogenous utility discount rate that depends some macroeconomic variables. Our main result is that for the utility flow to be constant, the utility discount function must be the marginal product of capital function adjusted for the growth rate of aggregate consumption weighted by the elasticity of the consumer's marginal rate of substitution between the final good and the amenity services. We demonstrate that Hartwick's Rule holds but the Hotelling Rule must be modified. We also provide an explicit analytical example to confirm the general result.