Transitioning to a low-carbon economy will require significant investment to transform energy systems, alter the built environment and adapt infrastructure. A strategy to finance this investment is needed if the limit of a 2°C increase in global mean temperatures is to be respected. Also, high-income countries have pledged to pay the "agreed full incremental costs" of climate-change mitigation by developing countries, which are not necessarily the same as incremental investment costs. Building on simulations using Integrated Assessment Models and historical evidence, this paper explores some of the issues posed by this dual financing challenge. We discuss the "fiscal self-reliance" of the energy sector, finding that carbon pricing would generate sufficient fiscal revenues within each region to finance total investment in energy supply. Even when allowing for trade in emission permits, regional carbon fiscal revenues should still suffice to cover both their own investment in energy supply and permit purchases from abroad. We show that incremental energy-supply investment (and saving) needs are well within the range of past variation of aggregate investment, and argue that the challenge is rather to ensure that revenues from carbon pricing and other sources are complemented by investment in the appropriate sectors. But fairness and equity are likely to warrant transfers from advanced industrial countries to developing nations.