The paper offers a complete analysis of the welfare-maximising capital investment and resource depletion policies in the Dasgupta-Heal-Solow-Stiglitz (DHSS) model with capital depreciation and any returns to scale. We establish a general existence result and show that an optimal admissible policy may not exist if the output elasticity of the resource equals 1. We characterise the optimal policies by applying an appropriate version of the Pontryagin maximum principle for infinite-horizon optimal control problems. We conclude the paper with an economic interpretation and a discussion of the welfare-maximising policies.