School quality and the education production function

The education production function $y = f(x_1, x_2, \dots, x_k)$ relates school, family and peer inputs, such as class size, teacher qualifications and parental time, to a measure of student output $y$, such as a test score or later earnings. Its marginal products tell a school system which inputs are worth paying for. Because inputs are not randomly assigned, credible estimates come from natural and randomized experiments, which show that school quality raises achievement and earnings, most of all for disadvantaged students.

The intuition

A school works like a firm: it combines inputs to produce skill. A school board that wants the most achievement for its budget should spend so that the last dollar produces the same gain everywhere,

$$\frac{MP_1}{p_1} = \frac{MP_2}{p_2} = \dots = \frac{MP_k}{p_k},$$

where $MP_j$ is the marginal product of input $j$ and $p_j$ its price, as in a firm’s cost minimization. Hiring more teachers to cut class size is worth it only if the marginal product of a smaller class, per dollar, beats that of better-paid teachers, books or tutoring. Applying the rule requires knowing the marginal products, and that is the hard part.

Why inputs are not randomly assigned

Comparing pupils who received different inputs does not reveal those marginal products, because inputs go to different kinds of pupils:

• families choose where to live, and so which school their children attend;
• districts with richer residents spend more per pupil;
• schools put struggling pupils into smaller remedial classes.

Each of these correlates an input with pupils’ backgrounds, which also affect achievement, and the resulting bias can run in either direction. Remedial classes make small classes look harmful; rich districts make spending look powerful.

Score advantage from family background (points)

12.0

True effect of one more pupil

−0.40 points

Slope of the naive comparison, A to B

0.46 points

Open in Graph Lab

Figure 1. Within each group of pupils one more pupil per class lowers the average score by 0.4 points, but school B serves better-off families and has larger classes than school A. The line through the two schools slopes upward, at 0.46 points per pupil: comparing schools mixes class size with family background.

The sign reversal in Figure 1 is not hypothetical. In Israeli elementary schools in the early 1990s, the raw correlation between class size and test scores was positive, because larger classes were found in schools with better-off pupils.¹ Surveys of studies that relate school inputs to outcomes without dealing with this problem, notably Hanushek’s, find mixed and often weak effects.²

School quality and the return to schooling

Card and Krueger asked whether the quality of the schools a man attended changes what each year of schooling is worth. Using 1980 Census earnings of men born between 1920 and 1949, they related each man’s return to schooling to the quality of schools in his state of birth, measured by the pupil-teacher ratio, the length of the school term and teachers’ relative pay. Men educated in states with better schools earned a higher return: a fall in the pupil-teacher ratio from 30 to 25 was associated with a return 0.4 percentage points higher. The relationship was similar for Black and white men, which the authors read as causal, since school quality for Black students in this period improved largely through political and court pressure. Once school quality was held fixed, parents’ income and education had no effect on state returns.³

Quality here raises the slope of the wage-schooling relationship, not just its level: with $\ln w = a + (b + \delta)s$, a better school adds $\delta$ to every year. Later work challenged how robust the estimates are.⁴

A natural experiment: Maimonides’ rule

In the period Angrist and Lavy studied, Israeli public schools split each grade into classes of at most 40 pupils, following a rule set out by the twelfth-century scholar Maimonides. A grade of 40 pupils is taught as one class of 40; a grade of 41 is split into two classes averaging 20.5; a grade of 81 into three of 27.¹ With enrolment $e$ and a cap of 40, average class size is

$$\text{class size} = \frac{e}{\lceil e/40 \rceil},$$

where $\lceil e/40 \rceil$ is the number of classes, $e/40$ rounded up.

Enrolment in the grade

41

Maximum class size

40

Number of classes

2

Average class size

20.5

Average class size with one fewer pupil

40.0

Open in Graph Lab

Figure 2. Average class size under a cap of 40 pupils. Class size rises one for one with enrolment to 40, then drops to 20.5 at 41 and to 27 at 81, so schools with nearly identical enrolment have very different class sizes.

Schools just below and just above each multiple of 40 are nearly the same size and plausibly similar in other respects, yet their class sizes differ by nearly 20 pupils. Comparing them is a regression discontinuity design: any jump in test scores at the cut-off can be attributed to class size. Angrist and Lavy found that smaller classes raised reading and mathematics scores of fifth graders substantially and the reading scores of fourth graders by less, with no clear effect for third graders. The gains were largest for pupils from disadvantaged backgrounds.¹

A randomized experiment: Project STAR

Tennessee’s Project STAR assigned about 11,600 kindergarten pupils and their teachers at random, starting in 1985–86, to small classes (13 to 17 pupils), regular classes (22 to 25) or regular classes with a teacher’s aide. Random assignment makes class size independent of family background. Pupils in small classes scored about 4 percentile points higher in their first year, and the advantage grew by about 1 percentile point a year after that. Aides had little effect, and gains were larger for minority pupils and those eligible for free lunch.⁵ Followed into adulthood, pupils from small classes were more likely to attend college; the effect on earnings at age 27 was not statistically significant, but it was imprecisely estimated.⁶

School quality and racial inequality: the Rosenwald schools

In the early twentieth century, public schools for Black children in the rural American South were severely underfunded. From 1914 to 1931 the Rosenwald Rural Schools Initiative, begun by Booker T. Washington of the Tuskegee Institute and the Chicago businessman Julius Rosenwald, helped build nearly 5,000 schools for rural Black children. The Rosenwald fund paid only when local communities raised matching money and local authorities cooperated; local Black communities and state and county governments supplied most of the funding. The schools followed modern designs with proper lighting, ventilation and sanitation, were fully equipped with books and desks, and came with minimum teacher salaries and teacher training. By the end of the program, about 36 percent of rural Black school-age children in the South could have attended one.⁷

Aaronson and Mazumder find large effects:⁷

• rural Black children with access completed over a year more schooling, enough to explain close to 40 percent of the Black-white convergence in schooling in the South for cohorts born between 1910 and 1925;
• a Rosenwald school in the county raised rural Black literacy by 9.3 percentage points, with no effect on rural white children;
• access raised cognitive test scores, measured in Second World War enlistment records, and migration to the North;
• about two thirds of the extra schooling came from children who would have attended school anyway, so quality, not only access, mattered;
• gains were largest in the most disadvantaged counties, and the program earned an internal real rate of return of about 7 to 9 percent on its full cost.

Because grants required local contributions, counties that got schools might have valued education more and improved anyway. The authors find that Black school attendance in 1910, and its change since 1900, were similar in counties with and without schools, and that the spread of the schools was unrelated to earlier Black socioeconomic conditions.

Key idea

Inputs are not randomly assigned, so raw comparisons can even get the sign wrong. Natural and randomized experiments show that school quality raises achievement and earnings, with the largest gains where schooling was worst.

Worked example

Maimonides’ rule caps classes at 40 pupils.

1. Predicted class sizes. A grade of 40 forms one class of 40. A grade of 44 needs $\lceil 44/40 \rceil = 2$ classes of 22. A grade of 121 needs 4 classes, averaging $121/4 = 30.25$.

2. Two groups of schools. Schools enrolling 36 to 40 pupils in a grade have one class each, averaging $(36 + 37 + 38 + 39 + 40)/5 = 38$ pupils. Schools enrolling 41 to 45 have two classes each, averaging $(20.5 + 21 + 21.5 + 22 + 22.5)/5 = 21.5$ pupils.

3. Read off the effect. Suppose average reading scores are 72.0 in the first group and 75.3 in the second. The groups are alike apart from class size, so the effect of one more pupil is $(75.3 - 72.0)/(21.5 - 38) = 3.3/(-16.5) = -0.2$ points.

4. Use it. Cutting class size from 31 to 25 would then raise reading scores by $6 \times 0.2 = 1.2$ points. Comparing all large and small schools instead would mix class size with differences in their pupils.

Exercise: Card and Krueger’s estimate in earnings

A fall in the pupil-teacher ratio from 30 to 25 raises the return to each year of schooling by 0.4 percentage points. For a worker with 12 years of schooling, log earnings rise by $12 \times 0.004 = 0.048$, so earnings are about 4.8 percent higher (exactly $e^{0.048} - 1 = 4.9$ percent). The gain applies to every year of schooling, not only to one.

Common mistakes

Traps on exams

• Reading a raw correlation as an effect. Inputs follow pupils’ backgrounds, so the bias can even reverse the sign.
• Confusing level and slope. Card and Krueger’s result is about the return per year of schooling, not a one-off earnings gain.
• Calling Maimonides’ rule random assignment. Only schools near the cut-offs are comparable; schools with 40 and 120 pupils differ in many ways besides class size.
• Ignoring the setting. Effects found where classes are large, as in Israel, need not carry over to schools whose classes are already small.

Check yourself

1.

What is the education production function?

1The relationship between inputs such as class size, teacher qualifications and parental time and a measure of student output such as test scores or later earnings2The relationship between a worker's years of schooling and lifetime earnings3The rule that sets class size from the number of pupils enrolled4The cost to a school board of producing one more year of schooling

Check answer

2.

In both groups of pupils, one more pupil per class lowers the average score by 0.4 points. School A serves lower-income families and has classes of 22; school B serves higher-income families and has classes of 36. A line through the two schools slopes upward. Why?

Score advantage from family background (points)

12.0

True effect of one more pupil

−0.40 points

Slope of the naive comparison, A to B

0.46 points

1The comparison mixes the effect of class size with the effect of family background, because the school with bigger classes also serves better-off pupils2Bigger classes raise scores for higher-income pupils3The true effect of class size is positive once enough schools are compared4School A's scores are measured with more error because it has fewer pupils

Check answer

3.

Which finding comes from Aaronson and Mazumder's study of the Rosenwald schools?

1Rural Black children with access to a Rosenwald school completed over a year more schooling, enough to explain close to 40 percent of Black-white convergence in schooling for Southern cohorts born 1910 to 19252The schools raised enrolment but had no effect on literacy or test scores3The schools mostly benefited white children in the same counties4The gains were largest in counties where Black school attendance was already high

Check answer

Practise more questions on this topic

Sources

1. Angrist, J. D. and V. Lavy (1999). Using Maimonides’ rule to estimate the effect of class size on scholastic achievement. Quarterly Journal of Economics, 114(2), 533 to 575. NBER working paper 5888. ↩ ↩² ↩³

2. Hanushek, E. A. (1986). The economics of schooling: Production and efficiency in public schools. Journal of Economic Literature, 24(3), 1141 to 1177. ↩

3. Card, D. and A. B. Krueger (1992). Does school quality matter? Returns to education and the characteristics of public schools in the United States. Journal of Political Economy, 100(1), 1 to 40. NBER working paper 3358. ↩

4. Heckman, J., A. Layne-Farrar and P. Todd (1996). Human capital pricing equations with an application to estimating the effect of schooling quality on earnings. Review of Economics and Statistics, 78(4), 562 to 610. ↩

5. Krueger, A. B. (1999). Experimental estimates of education production functions. Quarterly Journal of Economics, 114(2), 497 to 532. NBER working paper 6051. ↩

6. Chetty, R., J. N. Friedman, N. Hilger, E. Saez, D. W. Schanzenbach and D. Yagan (2011). How does your kindergarten classroom affect your earnings? Evidence from Project STAR. Quarterly Journal of Economics, 126(4), 1593 to 1660. NBER working paper 16381. ↩

7. Aaronson, D. and B. Mazumder (2011). The impact of Rosenwald schools on Black achievement. Journal of Political Economy, 119(5), 821 to 888. Figures here are from the revised working paper (Federal Reserve Bank of Chicago WP 2009-26, September 2011). ↩ ↩²