California's quality-blind layoffs law harms teachers and students

Los Angeles Times article "California's quality-blind layoffs law harms teachers and students" (March 24th, 2010) while making a valid point, misunderstands optimal hiring practices. It suggests schools, rather than retaining teachers who have been employed the longest, should only keep the best teachers.

After all, if layoffs are unavoidable, you would think that it would be in the interest of everyone to keep the best teachers and cut those who are least effective.

Corrections agrees that length of tenure is suboptimal. However, only hiring the best is similarly suboptimal. While the problem of optimal hiring given a fixed budget and discretely heterogeneous population can take on the dimensions of a Knapsack Problem requiring a dynamic programming solution, Corrections suggests that the essence of the problem is quite simple: hire the best teachers per dollar, which may or may not be the best teachers.

Imagine there are six types of teachers, as displayed graphically below (click to enlarge). Teachers quality is denoted by number of boxes--teacher price and quality units per dollar are also displayed numerically. Then, let us say that we have $1260 dollars. We could hire 105 Type I Teachers, 140 Type II, 180 Type III, 252 Type IV, 420 Type V, or 1260 Type VI (we could also hire a combination of the teachers--our point is robust to allowing this). We would then get 630, 700, 728, 756, 840, or 1260 units of quality by hiring each type. We can see that we are best hiring the worst teachers, because they are the cheapest teachers per unit of quality. While the Times is correct that firing based on tenure is poor decision making, it is not correct that schools should hire the best teachers if they want the highest quality education possible.