Are human influences on climate really small is the title of a post by Steven Koonin on Climate Etc. The magnitude of the radiant gas forcing, CO2 and equivalent "greenhouse" gases, can be relative to your perspective. That really isn't the way it should be. Increasing one force on a stable system should be fairly easy to figure out.
For a "global" impact though you can pick a number of frames of reference. Surface air temperature aka lower troposphere temperature, as I have mentioned is the worst possible choice of frames. From the "surface" you have radiant, latent and convection energy flows that has different impacts on the "surface" temperature. You can use an "effective" surfaces energy and temperature, for example 395 Wm-2 radiant, 90 Wm-2 latent plus 25 Wm-2 convective would produce an effective surface energy of 510 Wm-2 which would be roughly and effective temperature of 308 K degrees. With a Top of the Atmosphere energy value of 240 Wm-2 which would have an effective temperature of 255 K degrees. So the combined "greenhouse" effect from the "surface" would be 510-240=270 Wm-2 per 308-255=53 K degrees. That produces a 5.1Wm-2/K forcing ratio. There is of course some uncertainty involve so the "standard" of 5.35 Wm-2/K is close enough.
So if you are going to determine "small" for a doubling of CO2 which happens to use the "standard" dT=5.35ln(CO2f/CO2i), you should stick with an apples to apples comparison. If you neglect latent and convective heat flux you would get 395-240=155 Wm-2 per 289-255=34 K degrees producing a 4.56 Wm-2/K ratio.
With a doubling of CO2 expected to add 3.7 Wm-2 of "forcing", everything else remaining equal, you would have a 3.7/270 yields 1.3% to 3.7/155 yields 2.4% impact on "surface" energy/temperature provided temperature and energy have a linear relationship. They don't have a linear relationship of course, but for small changes you have small error with that assumption.
A simpler "guestimate" is to use the estimated forcing versus the estimated "force" or Down Welling Longwave Radiation (DWLR) estimated at about 340 Wm-2. That gives you a touch over 1% impact, all things remaining equal, and the DWLR value should include all feedbacks for your small change and hopefully small estimation error. I prefer that simplification since DWLR energy is roughly equal to the average ocean energy at an average temperature of ~4C degrees. That produces an all things remaining equal estimate of almost one C degrees per CO2 equivalent doubling or 3.7 Wm-2 of anthropogentic forcing.
So if you pick the worse thermodynamic frame of reference and assume away the reasons that it is the worst frame of reference you get a bigger impact than if you pick a simpler frame of reference. Imagine that?
Following Koonin's post the "let's" play thermodynamics games begin. The simple minded warmists pick the "global" surface temperature sans complications, the coldists pick their cherry, but if you consider dT~5.35ln(CO2f/CO2i) you should pick effective surface temperature and energy which includes latent and convective heat loss estimates.
The you could blow that off and use the "subsurface" temperatures which have less latent and convective flux to worry about. I have been saying this for some time, but I noticed some others are moving into a similar train of thought.
Issac Held has a post on "Addicted to Global Mean Surface Temperature" and Roy Spencer an anti-skydragon post or soil temperatures. Neither quite gets to the real simplicity of "sub-surface" energy which is the actual best possible reference for a change in atmospheric forcing. It will be interesting though to see just how complex and convoluted the extreme factions of the debate will go to preserve their personal ideology.