Lazear model of a private school

The economist Edward P. Lazear defined a theoretical of a (private) profitmaximizing school and was able to derive some interesting conclusions fromthe model. Here we describe the model.

The school has a fixed number, $Z$ of students.The cost of a teacher is $W>0$.Each student is assumed to be not disruptive with a probability $p$. So ifthe class has $n$ students the probability of ”peace in the classroom”is $p^n$. Only when there is peace can any learning take place.The school charges according the class size and all students in the classof size $n$ pay the same amount. Assuming that the value of a unit oflearning is $V>0$ and that there are $m$ classes, the average classsize is $n=Z/m$. The expected revenue then is$Vmn{p}^{Z/m}=VZ{p}^{Z/m}$. And the cost for these $m$ classes is $Wm$.Then the expected profit is $VZ{p}^{Z/m}-Wm$. This is a function to be maximized.

Using this theoretical framework Lazear was able to show:

1. As $p$ increases it is optimal to use fewer classes, so that class size increases.

2. As $W$ increases it is optimal to use fewer classes, so that class size increases.

Lazear briefly considered an extension of the model to the case wherethere are two types of students, having different disruption probabilities.He concluded that if the school wants to maximize revenue it would segregate thestudents, that is, there would be no mixed classes.