Your assumption is that non-carbon energy will be more expensive. That may be true in the short term, considering capital costs of replacing some of our infrastructure, but it is likely to be cheaper in the medium-longer term, as most renewable energy sources have zero ongoing fuel costs and often lower maintenance too. Consider that most cost comparisons of solar, wind, wave, geothermal etc vs existing coal, gas, nuclear etc compare amortised capital + ongoing costs of renewable energy plants vs ongoing costs only of the legacy plants (as capital costs are paid off already) - and they are still remarkably close. When you compare apples to apples (total lifetime costs for both, or ongoing costs only) then renewables already look much cheaper, in the majority of cases.
Consider also the projected (financial and human) costs of climate change, which will certainly impact poorer populations the most, as they can't afford to mitigate those costs. Drought, floods, famine, weather extremes, water shortages, rising sea levels, increased risk of tropical diseases - these are all discussed in the IPCC AR5 WGII Impacts section, and while the wealthier populations can move farmlands, irrigate, vaccinate, build levees, pay for disaster insurance etc, much of the world's poor will be SOL. Numerous studies (can cite if you like) have shown that the short-term costs of avoiding further climate change are far outweighed by the longer-term costs of adapting to the consequences instead.