Richer nations need to plow at least $400 billion a year into the developing world by 2050 to help cut greenhouse gases and fight climate change, according to a study that that would require a quadrupling of aid pledged.

The annual sum required may even be as high as $2 trillion, according to the study by academics at two divisions of the London School of Economics. By comparison, wealthier countries including the U.S., Japan and European Union members have promised to deliver an annual $100 billion by 2020.

"The climate finance goals to which high-income countries have committed are rather unambitious," the researchers wrote in the study published Monday. "Transfers never exceed 1 to 2 percent of gross domestic product for high-income country regions in any of the model projections. The cost to high-income countries, while substantial, is not likely to be prohibitive."

Climate finance is a key element of negotiations among more than 190 nations as they try to devise by year-end a new agreement to fight climate change that will come into effect in 2020.

Developing countries argue they need the cash to take expensive measures needed to both cut emissions and fight the effects of rising temperatures that are already being seen. Those include higher sea levels, more frequent droughts and heat waves.

The study’s authors examined the costs of reducing greenhouse gas emissions to a level that would contain warming since the Industrial Revolution to the globally-agreed goal of 2 degrees Celsius (3.6 degrees Fahrenheit). They then devised a formula to distribute the costs "equally" according to a country’s economic output.

The study is by the LSE’s Grantham Research Institute on Climate Change and the Environment and the ESRC Center for Climate Change Economics and Policy, a researcher jointly hosted by LSE and the University of Leeds in northern England.