I am finding in California grassland productivity is about 95% based on what species are present plus soil nutrients. The climate, land use, land cover all come in as a very distant influence.  When you say "grassland productivity" I assume that you mean potential productivity, as if it was left ungrazed for a year and you measured the biomass produced per area?

For example in the arid Great Basin, the most productive grass was the six foot tall Great Basin Wild Rye, that also produced the richest soils.  However, when you graze that species out, and you lower the soil nutrient levels below what those seedlings need to survive (see http://www.ecoseeds.com/greatbasin.html) then you have dropped the potential productivity to maybe 1/10th or 1/100th the potential if you still had the Great Basin Wild Rye still covering the land.

Great answer. This awoke another question: "How to partition the environmentan and vegetation (and changes) effects on grassland productivity (carbon fluxes)?"

I want to add another wild card to this question and issue--PSEUDOMONAS HOST PLANTS?

All native grassland around the world, probably contained Pseudomonas host plants, that helped create the rainfall for those areas.  But if the grazing animals eradicate all of the Pseudomonas host plants, did the climate change on its own, or were the cattle and sheep mostly responsible by eliminating the plants that produced the rainfall of the area?  Which came first, the climate change or the elimination of the plants that produce the rainfall? 

You can see the difference in still having the Pseudomonas plant around in the mountains above Salalah Oman, or a few miles away where they have been grazed out, at http://www.ecoseeds.com/cool.html