We develop a dynamic general equilibrium model to study the effect of technological progress on the optimal transition to a renewable energy fueled world economy. Energy is used as an input in production and can come from fossil or renewable sources. Both require the use of capital, which is also needed in the production of final goods. Renewable energy firms can invest in improving the productivity of their capital stock. The actual improvement is subject to spillovers and involves an opportunity cost, which we interpret as the cost from scraping existing capital. These results in underinvestment in the productivity of renewable energy. In the presence of environmental externalities, the optimal allocation can be implemented through a Pigouvian tax on fossil fuel, together with a policy that promotes new renewable technologies. We calibrate our model using world-economy data and characterize the transition toward a low carbon economy. We find that it is optimal for renewables to "start small" and pick up their market penetration only later. In the short run, investment is needed mainly to improve productivity in the renewable energy sector. Later, renewable energy contributes by becoming a "modest" engine of economic growth. It takes approximately 150 years before fossil fuel is phased out entirely, resulting in a 2.8 degree Celsius temperature increase.