Back-of-the-envelope Calculations: Global Warming

In the introductory chapter of  The Demon Haunted World: Science as a candle in the Dark Carl Sagan writes:

It was from [G. P.] Kuiper that I first got a feeling for what is called a back-of--the-envelope calculation: A possible explanation to a problem occurs to you, you pull out an old envelope, appeal to your knowledge of fundamental physics, scribble a few approximate equations on the envelope, substitute in likely numerical values, and see if your answer comes anywhere near explaining your problem. If not, you look for a different explanation, It cut through nonsense like a knife through butter.

Given the controversies that are currently swirling around Climate change (the President of the United States has called it a "hoax"), it would be useful for the non-scientists to have a back-of-the-envelope calculation to "cut through nonsense". There is a simple equation, called the Kaya Identity that would be very useful to understand.

I first became aware of the Kaya Identity on one of the first IPCC reports. The Kaya Identity proves a back-of-the-envelope way to calculate global temperature increase, T, from population growth. The current IPPC report, AR15, doesn't mention it in the introduction but it should. AR15 claims that it is already too late to limit anthropogenic (human caused) climate change. It is  dire warning but how can the non-scientists check the assertion. Let's work through he Kaya identity slowly and see what the calculations tell us.

The first step in the causal chain is the effect of population growth, N,  on economic production, Q. The Kaya equation is Q = qN where q is output per capita. In the Kaya framework, q is called an intensive variable while N and Q are extensive variables.