# Department of mathematical illusion

Matt explores the phenomenon of decreasing returns to scale:

Via Andrew Sullivan, Eric dePlace notes that "You save more fuel switching from a 15 to 18 mpg car than switching from a 50 to 100 mpg car." And so you do. A 15 MPG car would require 1,000 gallons of gas to drive 15,000 miles while an 18MPG car could get it done in just 833 gallons. That saves 167 gallons of gasoline. By contrast, since a 50 MPG only uses 300 gallons to go 15,000 miles, upgrading to 100 MPG can't save that much gas -- the super-efficient car uses 150 gallons.

One moral of the story is that the MPG statistic is probably misleading a lot of people who aren't quantitatively sophisticated. In policy terms, meanwhile, the upshot is simply that it makes more sense to focus on raising the efficiency of the least-efficient vehicles than creating new super-cars. Of course, the genius of pricing carbon through a tax or through auctioned emissions permits is, once again, that is spares people the burden of trying to do all the math in our heads and just lets price signals automatically find the most economical way of reaching the targets.

This is a common mistake in assessing ratios. I've just finished a piece on the impending retirement of the Baby Boomers, and one of the things that's hard to explain is the accelerating nature of the problem. Since 1945, the year the Baby Boomers were born, the number of workers per retiree has fallen from 40 to about 3. The drop from 40 workers per retiree to, say, 10 workers *sounds* very large. But for the workers, it's a smaller change than the decline from 3 to 2. Just keeping retiree income steady (as a percentage of GDP) over the next 20 years will require every worker to increase their contributions to same by 50%.