Public policy makers are often pressured to reduce class size in public schools in order to raise student achievement.
A: One way to model the production process for student achievement is to view the “teacher/student” ratio as the input. For purposes of this problem, let t be defined as the number of teachers per 1000 students; i.e. t = 20 means there are 20 teachers per 1,000 students. Class size in a school of 1000 students is then equal to 1000/t.
(a)Most education scholars believe that the increase in student achievement from reducing class size is high when class size is high but diminishes as class size falls. Illustrate how this translates into a production frontier with t on the horizontal axis and average student achievement a on the vertical.
(b) Consider a school with 1,000 students. If the annual salary of a teacher is given by w, what is the cost of raising the input t by 1—i.e. what is the cost per unit of the input t?
(c) Suppose a is the average score on a standardized test by students in the school, and suppose that the voting public is willing to pay p for each unit increase in a. Illustrate the “production plan” that the local school board will choose if it behaves analogously to a profit maximizing firm.
(d) What happens to class size if teacher salaries increase?
(e) How would your graph change if the voting public’s willingness to pay per unit of a decreases as a increases?
(f) Now suppose that you are analyzing two separate communities that fund their equally sized schools from tax contributions by voters in each school district. They face the same production technology, but the willingness to pay for marginal improvements in a is lower in community 1 than in community 2 at every production plan. How do the isoprofit maps differ for the two communities?
(g) Illustrate how this will result in different choices of class size in the two communities.
(h) Suppose that the citizens in each of the two communities described above were identical in every way except that those in community 1 have a different average income level than those in community 2. Can you hypothesize which of the two communities has greater average income?
(i) Higher level governments often subsidize local government contributions to public education, particularly for poorer communities. What changes in your picture of a community’s optimal class size setting when such subsidies are introduced?
B: Suppose the production technology for average student achievement is given by a = 100t 0.75, and suppose again that we are dealing with a school that has 1000 students.
(a) Let w denote the annual teacher salary in thousands of dollars and let p denote the community’s marginal willingness to pay for an increase in student achievement. Calculate the “profit maximizing” class size.
(b) What is the optimal class size when w = 60 and p = 2?
(c) What happens to class size as teacher salaries change?
(d) What happens to class size as the community’s marginal willingness to pay for student achievement changes?
(e) What would change if the state government subsidizes the local contribution to school spending?
(f) Now suppose that the community’s marginal willingness to pay for additional student achievement is a function of the achievement level. In particular, suppose that p (a) = Baβ−1 where β ≤ 1. For what values of β and B is the problem identical to the one you just solved?
(g) Solve for the optimal t given the marginal willingness to pay of p (a). What is the optimal class size when B = 3 and β= 0.95 (assuming again that w = 60.
(h) Under the parameter values just specified, does class size respond to changes in teacher salaries as it did before?