A number of eco-friendly turf management sources say something along the lines of
Increasing grass height only 1/8 inch adds about 300 square feet more energy-collecting leaf surface for each 1,000 square feet of lawn. So, it has a greater ability to produce its own food and less need for fertilizer.1
If you are currently mowing at less than 3” every 1/8” of grass blade added increases leaf surface area by 30%. - from the Perfect Earth Project Toxin-Free Lawn and Landscape Manual available here.
Does this have to do with the fact that the first couple of inches are non- or low-photosynthesizing crowns or stems? Wouldn't having greater leaf area up high lead to shading out a grass plant's own or neighbor's leaves?