What Do Cost Functions Tell Us About the Cost of an Adequate Education?

Abstract

Econometric cost functions have begun to appear in education adequacy cases with greater frequency. Cost functions are superficially attractive because they give the impression of objectivity, holding out the promise of scientifically estimating the cost of achieving specified levels of performance from actual data on spending. By contrast, the opinions of educators form the basis of the most common approach to estimating the cost of adequacy, the professional judgment method. The problem is that education cost functions do not in fact tell us the cost of achieving any specified level of performance. Instead, they provide estimates of average spending for districts of given characteristics and current performance. It is a huge and unwarranted stretch to go from this interpretation of regression results to the claim that they provide estimates of the minimum cost of achieving current performance levels, and it is even more problematic to extrapolate the cost of achieving at higher levels. In this article we review the cost-function technique and provide evidence that draws into question the usefulness of the cost-function approach for estimating the cost of an adequate education.

This article was a conference paper for From Equity to Adequacy to Choice: Perspectives on School Finance and School Finance Litigation, Show-Me Institute and the Truman School of Public Affairs of the University of Missouri, October 30, 2007.

We acknowledge the support of the Missouri Show-Me Institute. The usual disclaimers apply.

Notes

^{1 1}The alternative methods are discussed in Ladd, Chalk, and Hansen (1999). Particular attention to the use of cost functions can be found in Gronberg et al. (2004), Duncombe (2006), and Baker (2006a). Critiques can be found in Hanushek (2006, 2007).

²To an economist, this is a doubly redundant phrase, as “cost” implies efficiency, which in turn implies minimum spending necessary to achieve a given outcome. Because this usage may not be universal, we use this phrase for clarity and emphasis.

³See Baker (2006b) and Duncombe (2007). Baker was retained by the main group of plaintiff districts, the Committee for Educational Equality, and Duncombe was retained separately by the City of St. Louis. For the defense, Costrell was retained by the Attorney General of Missouri and Hanushek by the Defendant Intervenors (Shock, Sinquefield, and Smith).

⁴In addition to some of the studies previously cited, a partial list would also include Duncombe and Yinger (1997, 2000, 2005, 2007), Imazeki (2008), Imazeki and Reschovsky (2004a, 2004b), and Reschovsky and Imazeki (2003). Imazeki and Reschovsky, in their various publications about costs in Texas, alternately used an efficiency index derived from a data envelope analysis, including a Herfindahl index, or ignore the issue.

⁵ Figure 2, Figure 3, Figure 4, Figure 5 are based on the Duncombe data and analyses. The Baker data and analyses are very similar, and the corresponding figures are available upon request. Both studies pool data across several years, although these diagrams depict only 1 year.

⁶Similarly, there is a very wide horizontal range: At any given spending level, performance varies widely.

⁷In fact, if these data suggest any relationship at all, it is U-shaped, rather than linear, which means a negative relationship between performance and spending over the lower ranges of performance and a positive relationship over the higher ranges. The linear relationship has an R ² of only 4%, the portion of the variation in spending accounted for by variations in performance. A quadratic relationship, depicting the U-shaped curve, provides a much better fit, with an R ² of 30%.

⁸In the interest of simplicity, the text omits a number of technical details. For example, these equations are often estimated in logarithmic form for the dependent variable and some independent variables. Also, typically the estimation uses instrumental variables for the performance variable (and perhaps others, such as teacher salaries), as is discussed in a later section.

⁹To be sure, it is uncontroversial that higher FRL is associated with lower district performance, but the statistical evidence that extra spending systematically raises performance over the observed range is highly controversial. Student-level data from Missouri indicates no relationship between spending and performance of African American FRL students (Podgursky, Smith, & Springer, 2007, Figure 10).

¹⁰Because of the specific functional form, the estimate varies modestly depending on the percentage of students that are Black. The estimate just given is for the average district in the state, whereas for St. Louis, the figure is 85% (Baker, 2006b). The estimate in Duncombe (2007) also implies a substantial premium, but because of the way that race entered the equation (interacted with FRL) the interpretation is less straightforward.

¹¹The collective bargaining environment is a textbook case of the violation of the competitive price-taking assumption for inputs, as the impersonal forces of the market are replaced by relative bargaining power.

¹²Some practitioners (including Baker, 2006b) use regional cost indexes instead of teacher salaries. This avoids some of the difficulties previously discussed but may only weakly reflect prices faced by districts.

¹³Duncombe's equation includes performance (instrumented), teacher salaries (instrumented), % FRL, % FRL × % black, % SPED, indicator for K-12 district, a set of indicators for district size, property values, district income, state aid relative to income, % college educated, % age 65 or older, % housing units owner occupied, median housing price relative to average property values, and a series of year indicators.

¹⁴This variation could be the result of inadequate controls for true differences across districts. For example, the percentage of FRL students is likely to be a poor measure of the variation in resources that students receive at home across districts, especially across relatively high-poverty districts. Yet these coarse measures are often the only measures available to researchers or to those designing and implementing school funding formulas. However, as previous analyses of achievement show, even with exceptional measures of district characteristics, much of the variation in achievement for districts with the same spending is likely to remain.

¹⁵Other methods have also been used, which attempt to identify statistically the points at or near the bottom of figures comparable to Figure 3. These methods, stochastic frontier analysis and data envelopment methods, have been used by Duncombe and others in earlier publications (see, e.g., Duncombe, Ruggiero, & Yinger, 1996; Grosskopf, Hayes, Taylor, & Weber, 1997). Recent work, however, including that presented in court, focuses on the method discussed in the text.

¹⁶Cost function analysts acknowledge that they are only estimating “average efficiency,” a term that would seem to modify the definition of cost. However, they continue to state that the estimated cost figures represent what is “necessary” or “required” to achieve any given result, which effectively restores the original definition. Figure 4 and Figure 5 use the “required” terminology, from Duncombe (2007).

¹⁷ Duncombe (2007) identifies this as the Missouri School Improvement Program standard for St. Louis in 2008. This target happened to be near the state average in 2005, of 25.6.

¹⁸The fact that the process starts from a logically flawed base can still be seen in Figure 5 by examining the large number of “deficit” dots to the right of the vertical line. These are districts that are found to spend less than “required” to meet the standard that they are already meeting.

¹⁹ Duncombe (2006) and Baker (2006a) have argued that the upward tilt in diagrams such as this in Kansas (Duncombe) and other states (Baker) provide some evidence in support of the approach's statistical validity (albeit a “fairly weak validity test” in Duncombe's view). However, as our step-by-step derivation shows, the tilt simply reflects the estimated sign of Å₁. The point is that any positive estimate of Å₁, even if it is highly problematic (for reasons such as those discussed in the next section), will necessarily generate a positive tilt in a diagram such as Figure 5. Thus, a positive tilt is of no independent value in assessing the validity of the cost-function estimates.

²⁰When “teacher salaries” is used as an input price control (as in Duncombe, 2007), it is also treated as a troublesome explanator and instrumented.

²¹The Missouri cost functions suffered from both problems discussed in this paragraph, although the point was somewhat moot because the instruments chosen were invalid.

²²It should be pointed out that Figure 6 and Figure 7 are not necessarily representative of all student outcome measures. If one took a different grade to look at these relationships or looked at reading instead of math, most alternatives actually give insignificant relationships between spending and achievement, and frequently they have the wrong sign. This might be expected, as the regressions are drawing lines through these clouds of points with little shape to the points that allow estimating such a relationship. A few districts performing at a slightly different point in the cloud can change the slope of the relationship.