I'm teaching Benefit-Cost Analysis (BCA) this semester and using a new book, Zerbe and Bellas, A Primer for Benefit-Cost Analysis. I've gotten through Chapter 3 and all is well (except a bit too much handwringing over the inclusion/exclusion of moral sentiments).

One thing, however. Weighted BCA can be used to right wrongs, such as the maldistribution of income. Consider the following equation:

*w*_{1}NB_{1}+*w*_{2}NB_{2}=*w*NB

where NB is net benefits, *w* are weights and i = 1, 2 groups of folk. The number example given in the book sets *w*_{1} = 1 and *w*_{2} = 1.2 (e.g., suppose group 2 is lower income and they are bearing most of the costs). A problem arises when the sum of the weights are greater than the sample size. In this example, if i = 1, 2 are simply 2 people, adding the weights makes it look like there are 2.2 people. The weighted net benefits will be biased due to the biased population.

The good news is there is a simple fix. You can set your weights were you want them and then adjust them so that their sum is equal to the sample size. The "weighted weights" are equal to the raw weight multiplied by the true sample size and divided by the weighted sample size. In the textbook example, multiple 1 and 1.2 by 2 and divided by 2.2. The weighted weights will be *w*_{1} = 0.91 and *w*_{2} = 1.09.