In low-income, capital-scarce economies that face financial and fiscal constraints, managing revenues from newly found natural resources can be a daunting challenge. The policy debate is how to scale up public investment to meet huge needs in infrastructure without generating a higher public deficit, and avoid the Dutch disease. This paper uses an open economy dynamic stochastic general equilibrium model that is compatible with low-income economies and calibrated on Ugandan's data to tackle this problem. The paper explores macroeconomic dynamics under three stylized fiscal policy approaches for managing resource windfalls: investing all in public capital, saving all in a sovereign wealth fund, and a sustainable-investing approach that proposes a constant share of resource revenues to finance public investment and the rest to be saved. The analysis finds that a gradual scaling-up of public investment yields the best outcome, as it minimizes macroeconomic volatility. The analysis then investigates the optimal oil share to use for public investment; the criterion minimizes a loss function that accounts for households' welfare and macroeconomic stability in an environment featuring oil price volatility. The findings show that, depending on the policy maker's preference for stability, 55 to 85 percent of oil windfalls should be invested.