On the Allocative Efficiency of Small School Districts
by
Lori L. Taylor
Bush School of Government & Public Service and
Department of Economics, Texas A&M University
1098 Allen Building, 4220 TAMU
College Station, TX 77843
Shawna Grosskopf
Department of Economics
Oregon State University
Corvallis, OR 97331-3612
and
Kathy J. Hayes
Department of Economics
Southern Methodist University
Dallas, TX 75275
February 2016
On the Allocative Efficiency of Small School Districts
Abstract
A flood of new data on student performance has revealed a profound disconnect between
educational outlays and student performance, and has triggered a scramble to implement policy
changes that will increase the technical and allocative efficiency of school districts. Small school
districts are particularly sensitive to this renewed focus on efficiency because indivisibilities in
the educational process make it more difficult for small districts to avoid allocative
inefficiencies.
The policy presumption is that small school districts are administratively top heavy.
However, it could just as easily be the case that indivisibilities at the classroom level lead small
districts to overuse teachers rather than administrators. Therefore, the nature of the allocative
inefficiency of small districts is an open and interesting question.
This analysis uses administrative data on students, staff and spending to estimate the
technical and allocative efficiency of Texas public schools during the 2013-14 school year.
Following Grosskopf et al. (2001), we use an input distance function to model school production
and generate measures of technical and allocative inefficiencies. The efficiency measures are
bootstrapped to address measurement error in the input prices and value-added outputs. Our
results suggest that small districts are no more likely to overuse administrators than to overuse
teachers, and that there is no relationship between school district size and the degree of either
technical or allocative inefficiency once other likely determinants of inefficiency are taken into
account. The results are robust to alternative definitions of labor input and the inclusion or
exclusion of very large school districts.
Keywords:
school district efficiency, school finance, administrative overhead
1
Introduction
The No Child Left Behind Act of 2001 (NCLB) had an enormous impact on the U.S.
education system. NCLB required all states to develop and administer student exams in math,
reading, science and any other subjects the state deems appropriate. Importantly, states were
required to publish the results at the state, district and school levels—a feature that has not
changed despite the recent passage of the Every Student Succeeds Act of 2015.1 At all levels, the
states must provide separate information on the performance of low-income students, minority
students, special education students and students with limited English proficiency.
The flood of new data on student performance has revealed a profound disconnect
between educational outlays and student performance, and has led parents and taxpayers to agitate
for substantial improvements in school district effectiveness. Meanwhile, NCLB established
severe financial penalties for states and school districts if they fail to make progress toward
student proficiency. Facing voter pressure on the one hand, federal pressure on the other, and
budgets that are still recovering from recent fiscal crises, state and local policymakers throughout
the nation are scrambling to implement policy changes that will increase the technical and
allocative efficiency of school districts.
Small school districts are particularly sensitive to this renewed focus on efficiency.
Conventional wisdom holds that indivisibilities in the educational process make it more difficult
for small districts to avoid allocative inefficiencies. After all, even the smallest school district
must have a superintendent, the usual complement of central administrators and a sixth grade
teacher—even if the resulting sixth grade class is inefficiently small. However, little is known
about the nature of those inefficiencies.
1
Science exams became part of the report cards required by NCLB starting with the 2007-08 school year.
2
The policy presumption is that small school districts are administratively top heavy. For
example, the State of Texas holds school districts accountable for their “administrative cost ratio”
(which is defined as the ratio of spending on instructional leadership and general administration,
divided by spending on instruction, instructional resources, curriculum and staff development and
guidance counseling). Districts with more than 10,000 students receive the highest rating on this
indicator if their administrative cost ratio is less than 0.086; districts with fewer than 500 students
receive the same rating if their administrative cost ratio is less than 0.24, or nearly three times
larger. There is little recognition of the fact that indivisibilities at the classroom level could lead
small districts to overuse teachers rather than administrators.
In this paper, we use Texas data on students, staff and spending to explore two interrelated questions. First, are small school districts systematically less technically and allocatively
efficient than other school districts? And second, what is the pattern of that inefficiency? Are
small districts more likely to overuse administrators or more likely to overuse teachers? Our
results suggest that small districts are no more likely to be allocatively inefficient than their larger
brethren. Furthermore, when small districts are allocatively inefficient they are no more likely to
overuse administrators than to overuse teachers. We can detect no support for the policy
presumption that small school districts are systematically top heavy. These results are robust to
alternative definitions of labor input and the inclusion or exclusion of very large school districts.
Texas Public Schools
The Texas public school system is particularly well suited to an analysis of the relative
efficiency of small school districts. There are more than 1,000 traditional public school districts in
Texas serving nearly 5 million students each year. The state’s long-established accountability
system provides annual, student level data on student performance, making it possible to generate
3
detailed measures of school district output. The personnel and financial accounting systems
support reliable estimates of educational inputs and input prices.
More importantly, Texas has a wealth of small districts. Although some districts in the
state (such as the Houston or Dallas Independent School Districts) are among the largest in the
country, most Texas school districts are small. The median Texas school district has fewer than
1,000 students in fall enrollment, and 85 percent of Texas school districts have fewer than 5,000
students. This wealth of small districts facilitates an exploration of the relative efficiency of small
districts.
Measuring School District Efficiency
The literature suggests several alternative strategies for measuring school district
efficiency.2 Typically, researchers model the relationship between school district inputs and
school district outputs using either a production function, a cost function or a distance function.3
Those relationships are then estimated, and efficiency scores extracted, using data envelopment
analysis (e.g. Duncombe, Miner and Ruggiero 1997 and 1995, Ruggiero 2007 or Grosskopf et al.
1999), stochastic frontier analysis (e.g. Gronberg et al. 2005), or corrected least squares analysis
(e.g. Grosskopf et al. 2001, Schwartz et al. 2005 or Schwartz and Zabel 2005).4
Following Grosskopf et al. (2001), we use an input distance function to model school
district production and generate measures of school district inefficiency. The input distance
function is similar to the cost function but requires data on input quantities rather than input
2
For more on school efficiency analyses, see Johnes (2004) or Taylor (2010b).
3
Distance functions are representations of the production technology that allow for multiple outputs as well as
multiple inputs (Fӓre and Primont 1995).
4
As discussed in Taylor(2010b), analyses using panel data and indicator variables for each school or district (as in
Schwartz et al. 2005 or Schwartz and Zabel 2005) can be thought of as a special case of C-OLS.
4
prices. Like a cost function, it can accommodate multiple educational outputs but, unlike a cost
function, it does not require the assumption that decision-makers are trying to minimize costs.5
The distance function can also be used to generate separate measures of technical and allocative
efficiency. As such, the input distance function can be a useful tool for analyzing both dimensions
of school district efficiency.
As discussed in the technical appendix, we use the input distance function (D(y,x)) to
estimate the degree of technical and allocative efficiency for each school district. Those efficiency
measures can be most easily understood by examining figure 1. Assume that schools use two
variable inputs to produce an array of educational outputs (y). In figure 1, the isoquant L(y)
represents all of the technically efficient combinations of x1 and x2 that can produce y. A school
district producing output y at point K is using too much of both inputs. It could reduce both inputs
proportionally and still produce output level y. We measure the technical efficiency of each
school district s (τs ) as OK’/OK.6 If τs =1, the school district is technically efficient. If τs < 1 then
the school district is technically inefficient. The smaller the level of τs, the greater the technical
inefficiency. For example, technical efficiency of 0.8 would imply that the school district could
use 80 percent of its current inputs (i.e. reduce both inputs proportionally by 20 percent) yet still
produce the same output level y.
While most of the literature on school district efficiency focuses on technical or cost
efficiency, the input distance function also allows us to estimate allocative efficiency, or the
extent to which the school district is using the wrong mix of inputs. The slope of w*w* in figure 2
(i.e. the slope of the isoquant L(y) at the technically efficient point K’) represents the relative
5
While the cost function assumes cost minimizing behavior, inefficiency can be allowed for in the cost function using
techniques outlined by Schmidt and Sickles (1984).
6
This is Farrell technical efficiency, which is the reciprocal of the input distance function, i.e. τs=1/D(x,y).
5
marginal product, or equivalently the relative shadow prices, of the two inputs for observation K.7
The school district’s budget constraint goes through K’ but has the same slope as w’w’ in the
figure. A school district at K’ is allocatively inefficient (choosing the wrong mix of inputs) when
the slope of w*w* in figure 2 does not equal the slope of the w’w’. We define κs as the degree to
which the shadow relative prices match the observed relative prices for school district s.
κs w*1 /w*2
w*1 /w'1
w'1 /w'2
w*2 /w'2
(1)
If κs =1 then the school district is allocatively efficient. When κs > 1 then the marginal
product per dollar paid for input 1 (w1*/w1') is greater than the marginal product per dollar for
input 2 (w2*/w2') and input 1 is underutilized relative to input 2. When κs < 1, then the marginal
product per dollar for input 1 is less than the marginal product per dollar for input 2, and input 1 is
over utilized. In figure 2, κs < 1 and input 1 is over utilized relative to input 2. Reallocation to K*
would reduce variable costs.
Data and Model Specification
Analysis using the input distance function requires information on the quantity of school
district outputs, the quantity of school district inputs and the relative prices of those inputs.
Therefore, the initial steps of our analysis involve estimating outputs, inputs and input prices. We
discuss each in turn.
As a general rule, the data for this analysis come from the administrative files of the Texas
Education Agency (TEA). We focus on traditional public school districts during the 2013-14
school year, which is the most recent year for which complete data are available. We focus on
traditional public school districts because charter schools (which are a deregulated form of public
7
The partial derivatives of the input distance function with respect to the input quantities provide estimates of those
shadow prices. See Grosskopf et al. (2001).
6
school district) may have systematic differences in their educational technologies. We base our
measures of school district output on student performance on standardized tests, so only
traditional public districts that served students in all of the tested grades are included in the
analysis. Complete data were available for 930 traditional public school districts fitting this
description.
Output Quantities
As part of its school accountability system, Texas administers annual examinations in
reading and mathematics to every student in tested subjects and grades. We measure educational
output using a normalized gain score indicator of student performance on the Texas Assessment
of Knowledge and Skills (TAKS) and the state’s End of Course (EOC) exams. Each year, Texas
administers TAKS exams in grades 3-8 and EOC exams at the high school level.8 Although we
recognize that schools produce unmeasured outcomes that may be uncorrelated with math and
reading test scores, and that standardized tests may not measure the acquisition of important
higher-order skills such as problem solving, these are the performance measures for which
districts are held accountable by the state, and the most common measures of school district
output in the literature (e.g. Gronberg, Jansen and Taylor 2011a and 2011b, or Imazeki and
Reschovsky 2004). Therefore, we rely on them here.
As an approach to dealing with mean reversion, we normalize the annual gain scores (Y)
in each subject as in Reback (2008). In this normalization, we use test scores for student (i), grade
(g), and time or year (t), denoted as Sigt. (Subject-matter subscripts are suppressed.) All students
statewide are included in this calculation. We measure each student’s performance relative to
others with the same past score in the subject as:
8
Tests in other subjects such as science and history are also administered, but not in every grade level.
7
Yigt 
Sigt  E ( Sigt | Si ,g 1,t 1 )
[ E ( S | Si ,g 1,t 1 )  E ( Sigt | Si , g 1,t 1 )2 ]0.5
2
igt
(3)
In calculating Yigt for math, we calculate the average test score in math at time t, grade g, for
students scoring Si,g-1,t-1 in math at time t-1, grade g-1. Thus, for example, we consider all sixthgrade students with a given fifth-grade score in math, and calculate the expected score on the
sixth-grade test as the average math score at time t for all sixth-grade students with that common
lagged score. Our variable Yigt measures individual deviations from the expected score, adjusted
for the variance. This is a type of z-score. Transforming individual TAKS and EOC scores into zscores allows us to aggregate across different grade levels despite the differences in the content of
the various tests. For ease of interpretation, we further transform the z-scores into normal curve
equivalent (NCE) scores. NCE scores are a monotonic transformation of z-scores that are
commonly used in the education literature.9 Thus, our two measures of educational output are the
school district average NCE gain scores in reading and mathematics.
Input Quantities
As in Grosskopf et al. (2001, 1997), labor is the only variable input in our analysis. We
focus on two types of labor—instructional and non-instructional personnel. Our measure of
instructional labor input is a weighted sum of the full-time-equivalent number of teachers and
teacher aides employed in the school district, where the weights are the relative predicted wages
for the two groups.10 (See the section on input prices for a description of the process for
generating wage predictions.) Thus, if salary models predict that teacher aides are paid half the
9 The NCE score equals 50+21.063*z-score.
10
This approach treats teachers as homogeneous, which is a common assumption in the literature on school
efficiency. (See Taylor 2010b.) Contracted instructional personnel are treated as teachers for this analysis. As a
robustness check, we also estimate an alternative specification in which teacher salary is used as an indicator of
teacher quality.
8
salary of teachers in a school district, then each teacher aide in that school district is counted as
one half of a teacher. Our measure of non-instructional labor input is a similarly weighted sum of
the full-time-equivalent numbers of administrative, support and auxiliary personnel.11
Ideally, we would like to have a direct measure of school district capital as well as school
district labor. Unfortunately, such data are unavailable. However, Gronberg, Jansen and Taylor
(2011b) find that school district expenditures on maintenance are positively correlated with
school district capital stocks, suggesting that they can be used as a reasonable proxy. Therefore,
we measure capital stocks as a five-year average of expenditures on maintenance. We treat this
measure of school district capital stocks as a quasi-fixed input in our analysis.
Am important feature of the Texas public school system is its geography. School districts
in metropolitan areas tend to be geographically compact while school districts in rural counties
can encompass thousands of square miles. Compactness is an asset to the education process, as it
allows school districts to have more discretion about the size and location of campuses.
Therefore, we treat geographic compactness (which we define as the inverse of the number of
square miles in the jurisdiction) as a quasi-fixed input.
Finally, we consider the students themselves as inputs into the educational process that are
beyond the control of school districts—at least in the short term. Therefore, we include as quasifixed inputs the number of students in fall enrollment, the percentage of students who are not low
income, the percentage who are not special education students, the percentage of students who are
not limited English proficient and the percentage of students who are in high school.12
11
Support personnel include supervisors, counselors, librarians, nurses, physicians and special service personnel.
Auxiliary personnel are nonprofessional school personnel such as clerical, cafeteria and maintenance workers.
12
For reasons of parsimony, we did not include enrollment counts for each grade level. We did differentiate between
high school students and other students because previous analyses of Texas suggest that the cost of educating high
school students is different from the cost of educating non-high school students (Gronberg et al, 2004).
9
Input prices
Estimating allocative efficiency involves comparing shadow relative prices with observed
relative prices. Therefore, it is crucial to have reliable estimates of the relative price of
instructional and non-instructional labor. Grosskopf et al. (2001) used average wages by labor
market as the prices. However, subsequent research suggests that there can be both considerable
variation in worker characteristics across labor markets, and considerable variation in predicted
wages across school districts within labor markets (e.g. Taylor 2004, 2005, 2010a). Observed
salaries may also include rents (Taylor 2010a; Bruner and Squires 2013).Therefore, we base our
measures of relative wages on five labor cost models, one for each type of school district
employee (teachers, teacher aides, administrators, support staff, and auxiliaries).
The labor cost models are estimated from five-year panels of individual payroll records for
everyone working in a traditional public school district in Texas.13 The five-year panels cover the
period from the 2009-2010 school year through the 2013-2014 school year.
The modeling strategy follows Gronberg et al. (2011a). Full-time-equivalent annual wages
(in logs) are a function of labor market characteristics, worker characteristics and job
characteristics.14 Fixed effects for metropolitan areas and counties (as appropriate) capture local
variations in amenities and the cost of living, while individual fixed effects capture unobservable
differences across workers. Time-varying worker characteristics such as years of experience,
educational attainment, and indicators for a new hire are included in all models. In addition, the
teacher model includes indicators for teaching assignments (e.g. math, science, health and P.E.,
13
Incomplete and inconsistent records (such as records that indicate that full-time equivalent employees were paid
less than the federal minimum wage) were excluded from the analysis.
14
A full-time equivalent teacher, aide or support staffer is presumed to work 187 days, which is the length of a
standard academic year in Texas. A full-time equivalent administrator is presumed to work 226 days (the modal
length administrator contract) and a full-time equivalent auxiliary worker is presumed to work 260 days (the modal
length of employment for auxiliaries).
10
computer, special education, or secondary school) while the administrator and support staff
models include indicators for occupation (principal, assistant principal, nurse, etcetera). To
control for district-specific compensating differentials, all of the models also include student
demographics (the percentages of students who are economically disadvantaged and limited
English proficient) school district size (the log of enrollment and the log of enrollment squared),15
the average distance in miles from the center of the closest metropolitan area, the average distance
in miles from the nearest teacher certifying institution, the county unemployment rate, and the US
Department of Housing and Urban Development’s estimate of fair market rents for the county.
Appendix table A1 presents the labor models.
For each school district, the labor models generate predicted monthly salaries for each of
the five types, holding worker characteristics constant at the statewide mean for that type, but
allowing wages to vary according to the district values for all other variables in the model. We
interpret those salary predictions as the district-specific prices for those types of labor. Thus, the
price of teachers in Dallas Independent School District (DISD) is the annual salary that DISD
would have been expected to pay in 2013-14 to a teacher with state average demographics.
The salary predictions are used to generate district-specific weights for aggregating the
five worker categories into instructional and non-instructional personnel. For example, the
predicted salary for teachers in DISD is $54,359 while the predicted salary for teacher aides is
$21,851. Therefore, when calculating the number of full-time-equivalent instructional personnel
for DISD, teacher aides represent 40 percent of a teacher. Across districts, salaries indicate that
teacher aides represent as much as 70 percent and as little as 24 percent of a teacher. Similar
15
For purposes of the salary analysis, district enrollment is top-coded at 25,000 students.
11
calculations indicate that, on average, support staff represent 65 percent of an administrator while
auxiliaries represent 32 percent of an administrator.
Importantly, the relative price of non-instructional personnel is also used in calculating
allocative efficiency. The relative price of non-instructional personnel is the ratio of predicted
administrator wages to predicted teacher wages. The average administrator is paid 72 percent
more per year than the average teacher so, on average, the relative price of non-instructional
personnel is 1.72. It ranges from a low of 1.30 to a high of 2.24. Table 1 provides descriptive
statistics on this variable as well.
Estimation
The translog cost function has a long history of use in estimating cost functions because of
its flexibility and ability to nest various hypotheses within its structure. In this analysis we
estimate a translog form for the input distance function. All of the independent variables that are
not already in percentage terms are transformed into logs. Then, all of the variables are interacted
with all of the other independent variables under analysis. There are two discretionary inputs
(instructional personnel and non-instructional personnel), seven non-discretionary inputs (capital
stock, the number of high school students, the number of square miles, the percent not low
income, the percent not special education, the percent not limited English proficient, and the
percent high school) and two outputs (mathematics output and reading output). The dependent
variable is the log of the input distance function, which by definition is set equal to zero. See the
technical appendix for more details on the estimation procedure.
One advantage of the translog specification is that by Shephard's lemma the first
derivative of the input distance function with respect to the log of instructional personnel equals
the instructors’ share of payroll expenditures. By estimating the distance function and the share
equation together in a system of simultaneous equations we can improve the efficiency of the
12
estimated parameters. We use the observed input quantities and the predicted salaries of teachers
and administrators (P=w2/w1) in each district to define instructional payroll shares (S1= x1 / (x1 +
Px2 )) for each observation. We then estimate a system of two equations – one for the input
distance function and the other for the share equationBusing restricted least squares.
By definition, the input distance function is bounded from below by one. However, the
predicted values of the log of the distance function are distributed around zero. Therefore, we
follow Grosskopf et al (2001) in adjusting the intercept term by adding the absolute value of the
most negative residual. The scaling yields estimated values of the log of the distance function that
are greater than or equal to zero, and estimated values for the input distance function that are
greater than or equal to one. While all school districts are likely to exhibit at least some
inefficiency relative to the true but unobserved technology, our method assigns one school district
to be technically efficient in the best-practice sense. The literature refers to this method for
estimating efficiency scores as corrected OLS. Jensen (2005) and Ruggiero (1999) find that in a
great many cases, corrected OLS analysis produces more accurate technical efficiency rankings
than does stochastic frontier analysis.
As noted by Grosskopf et al. (2001), inverting the value of the input distance function for
each observation yields our measure of Farrell technical inefficiency, τs. Values of τs range from
zero to one, with a value of one indicating that the school district is technically efficient (in the
sense that the variable inputs cannot be proportionally reduced without reducing current output
levels).
The predicted values from the share equation (together with the variable input quantities
and the ratios of average prices P=w2/w1) provide sufficient information to generate a point
estimate of κ for each school district (κs). If κs > 1 then the wage-deflated marginal product of
instructors is greater than the wage-deflated marginal product of administrative staff for school
13
district s, and the school district is underutilizing teachers. Similarly, if κs < 1 then the wagedeflated marginal product of instructors is less than the wage-deflated marginal product of
administrative staff for school district s and the school district is overutilizing teachers, relative to
administrators. We use the value of κs as our measure of allocative inefficiency: the farther κs is
from one, the greater is the difference between the relative market price and the relative shadow
price and the more allocatively inefficient is the school.
At this point, we face two estimation problems. Our first problem arises because the
value-added outputs and relative prices are known to be measured with error, and measurement
error can bias estimates of school district efficiency (Taylor 2010b). Second, statistical
significance cannot be determined for our efficiency measures because they represent
transformations of the predicted values from the estimated system of equations.
We address these problems by bootstrapping the outputs and prices. As district averages
of individual NCE scores, each output measure has an associated standard error. Each salary
prediction also has an associated standard error. Therefore, we replicate the original data set 1,000
times. For each observation in each replicated data set, mathematics (reading) output per pupil is
set equal to the school’s point estimate for mathematics (reading) output per pupil, plus a random
error drawn from a normal distribution with a standard deviation equal to the standard error of the
school’s point estimate. Similarly, each predicted salary is set equal to the school’s point-estimate
prediction, plus a random error drawn from a normal distribution with a standard deviation equal
to the standard error of the school district’s point estimate for the salary. Because the variable
inputs are weighted averages of the labor types, with the weights based on the salary predictions,
the variable input quantities are recalculated for each replication. We then re-estimate the system
14
of equations 1,000 times – once for each of the 1,000 data sets.16 By bootstrapping the data in this
way, we are also in effect conducting a robustness check on the estimates of the value-added
outputs and relative input prices.
Results
Each of the 1,000 replications generates an estimate of τs and κs for each of the 952 school
districts under analysis. Table 2 presents the distribution of district-specific means for these
efficiency measures.
As the table illustrates, there is a wide range of efficiency in Texas school districts.
Consider first the estimates of technical efficiency. The estimates of technical efficiency are
centered on 64 percent and have a rather straightforward interpretation. Relative to the best
practice in the state, the typical school district in Texas could reduce its labor inputs by 36 percent
without reducing student performance in mathematics and reading. While schools undoubtedly
produce valuable outputs that are not correlated with math and reading scores, and some of the
apparent inefficiency may be attributable to the cost of producing those unmeasured outputs,
these estimates suggest that most Texas school districts are far from the best practice frontier.
Notably, these estimates of technical efficiency are similar to those found in previous
estimates of school district efficiency in Texas. Reschovsky and Imazeki (2004) found that “the
average district in Texas is 59 percent as efficient as the most efficient districts in the state” (p.
41). Gronberg, Jansen and Taylor (2011 and 2010) found that the average Texas district was 89
percent efficient.
16
Tables indicating the distribution of coefficient estimates are available from the authors.
15
Districts where κs = 1 are allocatively efficient, so deviations from one in either direction
indicate allocative inefficiency. On average, relative shadow prices diverge from market prices by
5.6 percent. The divergence exceeds 25 percent in a handful of districts.
To explore the distribution of allocative inefficiency further, we divide the sample into
three groups – school districts that overuse administrators, school districts that overuse instructors
and allocatively efficient school districts. Any district where 950 or more of the 1,000 iterations
indicate that κs > 1 is considered a district that overuses administrative personnel (relative to
instructional personnel). Similarly, any district where 950 or more of the iterations indicate that κs
< 1 overuses instructional personnel, a pattern that could arise if class sizes are smaller than
optimal, for example. All other districts are deemed allocatively efficient because we lack
evidence that they systematically overuse or underuse administrators. They may still be
allocatively inefficient, but we are not able to detect it or to determine the direction of the
inefficiency.
If Texas public schools are systematically top-heavy–as is frequently alleged—then κs
should be greater than one for most school districts. Instead, we find that most school districts in
Texas cannot be characterized as overusing administrators. For 549 of the 952 school districts
under analysis, κs is not systematically different from one. In 216 districts, κs is consistently
below one, indicating that the school district overuses teachers. In the remaining 187 districts, κs
is consistently above one, indicating that the school district overuses administrators. Thus, the
evidence suggests that only 20 percent of the traditional public school districts in Texas
systematically overuse administrators, and 23 percent of the districts systematically underuse
administrators.
Table 3 illustrates the relationship between our efficiency measures and per pupil
spending. Not surprisingly, spending per pupil is highest among schools in the bottom quartile.
16
(The correlation between technical efficiency and per pupil spending is -0.2029.) Somewhat
surprisingly, districts that are allocatively efficient spend significantly more than districts that are
allocatively inefficient. There is no evidence that an over-reliance on administrative personnel in
and of itself leads to unusually high spending by Texas public schools.
There also is no evidence that small districts are inefficiently top heavy. Table 4 presents a
cross-tabulation between our allocative efficiency groups and school district size categories. (The
size categories correspond to those TEA uses in its Financial Integrity Rating System of Texas—
FIRST—to set standards for maximum administrative cost ratios.) As the table illustrates, school
districts in the largest size category (those with at least 10,000 students) are nearly three times as
likely to overuse teachers as to overuse administrators. Districts in the next largest category (those
with enrollments between 5,000 and 9,999) are also somewhat more likely to overuse teachers
(35%) than to overuse administrators (29%). In contrast, there are no systematic differences
among school districts with fewer than 5,000 students; on average, 19% of districts overuse
teachers and 20% overuse administrators.17
Small districts are much more likely than large districts to be categorized as allocatively
efficient. There are two possible explanations for this pattern. First, measurement error is clearly
part of the story. Small districts are more prone than large districts to measurement error in the
outputs, and may also be more susceptible to measurement error in the prices. Measurement error
increases the likelihood that a district will be found to overuse administrators in some of the 1,000
iterations and underuse them in others, leading to a categorization of allocative efficiency (or,
rather, an inability to characterize the nature of the allocative inefficiency).
17
The hypothesis that class size category predicts efficiency group is rejected when districts with 5,000 or more
students are excluded. The probability of a greater chi-squared test statistic is 0.295.
17
Second, small districts might actually be allocatively efficient more often than large
districts. This is an unexpected finding, but consistent when viewed through the lens of the Texas
regulatory environment. In Texas, school districts are required to maintain a maximum class size
of 22:1 in grades K-4. However, small districts can easily obtain waivers if it would be a financial
hardship to reduce the class size to that level. As a result, the regulation tends to only be binding
in larger school districts. If the optimal class size is larger than that required by law, then a
regulation that is more likely to bind large districts could be leading to systematically more
allocative inefficiency among large districts, and the observed greater tendency for large districts
to overuse teachers.
While small districts are less likely to be classified as allocatively inefficient, their
allocative inefficiencies are larger in magnitude than those of large districts. As Table 5
illustrates, we find a significant, negative relationship between our measure of allocative
inefficiency (absolute value of κs ) and school district size. We find no such relationship for
technical efficiency.
Of course, the simple correlation between district size and allocative efficiency might be
spurious. To further explore the relationship between measured efficiency and enrollments, we
consider a few other factors discussed in the literature18 as likely to influence school district
efficiency—enrollment density and the degree of competition in the local education market—as
well as the administrative cost ratio that is used by the TEA as an indicator of allocative
efficiency. We measure enrollment density as the natural log of the number of students per square
mile. We measure competition in the labor market with a Herfindahl index of public school
18
For example, see Taylor (2010b), Haelermans, De Witte, and Blank (2012) or Grosskopf et al. (2001). Another
factor frequently posited to influence school district efficiency is unionization. Because Texas is a right to work state,
union activity is low and there is no information on differences (if any) in the degree of unionization from one Texas
school district to another.
18
enrollments (or equivalently, the sum of squared enrollment shares) in each county or
metropolitan area. The administrative cost ratio is defined as current operating expenditures on
central administration divided by current operating expenditures on instruction.19 As Table 5
illustrates, these variable are also significantly correlated with our measure of allocative
inefficiency.
Table 6 provides results from a regression analysis of the relationship between
enrollments and technical and allocative efficiency, holding constant the administrative cost ratio,
enrollment density and Herfindahl index of enrollment concentration. Here, we exploit the
bootstrapped nature of our data by replicating the regression with each iteration of the dataset and
reporting the means of the 1,000 regression coefficients and their 5th and 95th percentiles.
As the table illustrates, after controlling for other plausible sources of inefficiency we find
no evidence that school district size is associated with lower technical efficiency or higher
allocative inefficiency. The first and third columns present results from models that do not control
for any other possible determinants of inefficiency. In these specifications, it appears that
increases in school size are associated with increases in technical efficiency and decreases in
allocative inefficiency. However, when other possible determinants of inefficiency are included in
the model (columns 2 and 4) any effect of school district size goes away.
As to the other determinants of school district efficiency, we find that school districts in
Texas are more technically and allocatively efficient where the enrollment density is higher.
Increases in the Herfindahl index (which imply increases in market concentration) are associated
with higher technical efficiency but also higher allocative inefficiency. We also find no evidence
that higher administrative cost ratios are associated with higher levels of inefficiency. Instead, we
19
Only operating expenditures from the Foundation School Program (i.e. the state’s school finance formula) are
included.
19
find that districts with a higher administrative cost ratio tend to be no more allocatively inefficient
and somewhat more technically efficient that other districts, ceteris paribus.
The Sensitivity of the Analysis to Outlier Districts
Two Texas school districts – Dallas Independent School District and Houston Independent
School District – stand out from the rest. While the median district under analysis has less than
1,000 students in fall enrollment, Dallas and Houston ISDs each have more than 150,000
students. Each is more than 40% larger than the next largest district, and both were classified as
overusing teachers in the baseline analysis. Readers familiar with Texas might wonder if these
two outlier districts have undue influence on the estimation.
To address this concern, we re-estimated the baseline model excluding Dallas and
Houston. We found that the analysis is not sensitive to the inclusion or exclusion of these very
large districts. The Pearson correlation between the two estimates of technical efficiency is 0.999.
The Pearson correlation between the two estimates of allocative efficiency is also 0.999.
Excluding Dallas and Houston Independent School Districts from the estimation changes the
efficiency classification for a handful of districts (two previously identified as efficient are
reclassified as overusing teachers and three previously identified as inefficient are reclassified as
efficient) but has no substantive effect on the findings of the analysis
The Sensitivity of the Analysis to the Definition of Labor Inputs
Our measure of instructional labor (the number of full-time-equivalent teachers plus a
weighted count of full-time-equivalent teacher aides) treats teachers as homogeneous. Therefore,
a school district with 15 teachers has the same labor input whether they are 15 rookies with a
bachelor’s degree or 15 experienced teachers with a master’s degree. Although the literature
suggests that differences in a teacher’s experience and educational attainment have little
20
correlation with her effectiveness, this approach may understate the labor inputs of schools with
highly qualified (and therefore highly compensated) teachers.
To address this concern, we replicated the analysis using estimates of the effective
quantities of instructional and non-instructional inputs instead of the full-time-equivalent counts.
We defined the effective quantity of instructional labor as the school’s total payroll for teachers
and teacher aides, divided by the predicted teacher wage. We similarly defined the effective
quantity of non-instructional labor as the total non-instructor payroll divided by the predicted
administrator wage. In so doing, we treat compensation as a direct indicator of educator quality,
an approach that is consistent with work by Loeb and Page (2000) indicating that once researchers
control for variations in non-pecuniary labor market conditions there is a positive and statistically
significant relationship between teacher salaries and teacher quality. In this alternative
specification, schools that spend more on instructors are presumed to be using more instructor
input than other schools.
As tables 7, 8 and 9 illustrate, using effective labor quantities has an effect on the details
of the analysis, but not on the conclusions regarding the relationship between enrolment and
efficiency. Where the baseline analysis indicates that the average public school district has a
technical efficiency score of 63.8 percent, the analysis based on effective inputs indicates that the
average public school district has a technical efficiency score of 65.5 percent. The average
deviation from allocative efficiency remains 5.5%.
Using these alternative input quantities, we find that school districts are somewhat less
likely to be identified as efficient (Table 8). However, it is still the case that large districts tend to
overuse teachers, and that small and midsized districts (those with fewer than 5,000 students) are
21
no more likely to overuse administrators than to overuse teachers.20 Of the 779 small and
midsized districts, 162 are found to overuse administrators and 157 are found to overuse teachers.
Table 9 presents our analysis of the determinants of efficiency, using the efficiency
estimates from the alternative specification of input quantities. As the table illustrates, we again
find that school district size has no power to explain allocative inefficiency. Small districts are
no more allocatively inefficient than larger districts, once other possible influences on efficiency
are taken into account. There is some evidence that larger districts are more technically efficient
than smaller ones but the marginal effect is very small. Increasing district size from 5,000
students to 10,000 students is predicted to increase the technical efficiency measure by 0.004.
On the other hand, conclusions regarding the other possible determinants of efficiency are
sensitive to the strategy for measuring inputs. When teachers are considered homogeneous (the
baseline specification) increases in market concentration are associated with increases in technical
efficiency, but when salary is interpreted as an indicator of teacher quality (the alternative
specification) increases in market concentration are associated with decreases in technical
efficiency. This pattern is deserving of further study.
Conclusions
Our analysis of school districts in Texas finds evidence of widespread technical efficiency.
Based on the best practice of traditional public schools in Texas, we estimate that Texas schools
could cut their operating expenses by 36 percent with no decrease in measured student
performance. Therefore, Texas could benefit enormously from policies that encouraged school
districts to use their resources more efficiently.
20 The hypothesis that class size category predicts efficiency group is rejected when districts with 5,000 or more
students are excluded. The probability of a greater chi-squared test statistic is 0.313.
22
On the other hand, we find no evidence that school districts in Texas are top-heavy or that
inducing schools to spend a greater share of their budgets on instruction will lead to increased
efficiency technical or allocative. Most Texas school districts are choosing an efficient mix of
instructional and non-instructional labor, and districts are more likely to be devoting too many
personnel resources to the classroom than too few.
So, is there any validity to using the administrative cost ratio—with or without
adjustments for school district size—as an indicator of school district efficiency? The nonprofit
community clearly thinks so: there are a number of researchers who use some version of an
administrative expense ratio as an indicator of efficiency in models of nonprofit activity (e.g.
Jacobs and Marudas, 2009; Chikoto and Neely, 2013; or Tinkelman and Mankaney, 2007) and the
Better Business Bureau (BBB) recommends that donors avoid nonprofits with administrative
expenditures that exceed 35% of their total expenses (BBB 2015). The educational policy
community also seems to think so: states other than Texas—including Louisiana, Kansas and
Georgia—have experimented with requiring school districts to spend no more than a threshold
percentage of their resources on administration.
Our evidence would suggest the opposite. We find that increases in the administrative cost
ratio are associated with increases, not decreases in the technical efficiency of school districts.
Furthermore, we also find that larger districts—for whom the expectations regarding a lower
administrative cost ratio are more stringent—are more likely to overuse teachers than to overuse
administrators. In other words, school districts in Texas would be more efficient if the
administrative cost ratio were higher, not lower. As such, our analysis casts doubt on the validity
of applying simple rules of thumb in the hopes of improving complex economics systems like
public education.
23
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27
Table 1: Descriptive Statistics
Mean
Standard
Minimum Maximum
Deviation
4.58
27.64
67.02
3.06
37.69
61.74
707.38
3.43 10417.80
701.16
4.79 10171.26
922.95
10.77 12245.45
321.00
2.23
4686.71
11383.53
112.37 178047.50
14113.20
103.00 210716.00
363.73
5.13
4865.72
3.94
26.78
65.31
18.62
0.00
100.00
9.57
28.80
100.00
2.19
79.20
98.10
0.11
1.30
2.24
Math Output (per pupil)
49.53
Reading/Language Arts Output (per pupil)
49.81
Math Output (total)
257.74
Reading/Language Arts Output (total)
257.27
Instructional Personnel
362.81
Non-instructional Personnel
120.69
Capital Index
4364.56
Number of Students
5158.81
Square miles
267.96
Percent high school
43.58
Percent not low income
42.20
Percent not limited English Proficient
91.46
Percent not Special Education
90.93
Relative price of administrators
1.72
Wage Indices
Teachers
$45,093
$4,324
Aides
$17,909
$2,539
Administrators
$77,332
$7,020
Support staff
$49,920
$3,937
Auxiliaries
$24,990
$4,049
Note: There were 932 school districts in the estimation sample.
$35,918
$11,460
$51,926
$36,706
$10,213
$55,055
$27,223
$114,829
$62,365
$54,062
Table 2: Descriptive Statistics for School District Efficiency Measures
Mean
Standard
Minimum Maximum
Deviation
τs
0.638
0.058
0.286
1.000
1.003
0.063
0.770
1.312
κs
| κs-1 |
0.056
0.038
0.009
0.330
Note: Efficiency measures are averages over 1,000 iterations for each of the 952 school districts
in the estimation sample.
28
Table 3: Per Pupil Spending by Efficiency Categories
Mean
Technical Efficiency
Top 25%
Interquartile Range
Bottom 25%
Allocative Efficiency
Efficient
Overuse teachers
Overuse administrators
Standard
Deviation
Minimum Maximum
238
476
238
$9,695
$9,829
$10,875
$1,914
$2,111
$2,720
$6,113
$6,917
$7,824
$18,330
$30,812
$25,302
544
218
190
$10,299
$9,651
$9,830
$2,392
$1,831
$2,347
$6,917
$6,774
$6,113
$30,812
$16,898
$25,302
Table 4: Cross-tabulation of Allocative Efficiency and Enrollment Categories
Overuse
Overuse
Enrollment Category
Teachers
Efficient
Administrators
>=10,000
46 (44%)
43 (41%)
16 (15%)
5,000 to 9,999
24 (35%)
24 (35%)
20 (29%)
1,000 to 4,999
59 (18%)
191 (59%)
74 (23%)
500 to 999
40 (20%)
117 (60%)
39 (20%)
< 500
49 (19%)
169 (65%)
41 (16%)
All Districts
218 (23%)
544 (57%)
190 (20%)
Total
105 (100%)
68 (100%)
324 (100%)
196 (100%)
259 (100%)
952 (100%)
Table 5: Efficiency Correlations
Technical
Efficiency
(τs)
0.0363
0.0372
Allocative
Inefficiency
(| κs-1 |)
-0.1020
-0.2925
Pearson Correlation with Administrative Cost Ratio
Spearman Correlation with Administrative Cost Ratio
0.0448
0.0576
0.2172
0.2448
Pearson Correlation with Land Area in Square Miles
Spearman Correlation with Land Area in Square Miles
0.0144
-0.0066
0.2634
0.3050
Pearson Correlation with Herfindahl Index
Spearman Correlation with Herfindahl Index
0.0287
-0.002
0.3787
0.4024
Pearson Correlation with Enrollment
Spearman Correlation with Enrollment
29
Table 6: Modeling the Inefficiency
Technical Efficiency
Allocative Inefficiency
(τs)
(| κs-1 |)
Model 1
Model2
Model 3
Model 4
Enrollment (log)
0.001
0.0001
-0.004
0.001
(0.0004, 0.002)
(-0.001, 0.001)
(-0.006, -0.003)
(-0.003, 0.004)
Admin. Cost Ratio
0.066
0.063
0.071
0.048
(0.044, 0.088)
(0.041, 0.086)
(0.013, 0.132)
-0.010, 0.108)
Density (log)
0.001
-0.003
(0.001, 0.002)
(-0.006, -0.001)
Herfindahl Index
0.008
0.043
(0.004, 0.013)
(0.031, 0.055)
Constant
0.624
0.625
0.079
0.033
(0.597, 0.653)
(0.598, 0.654)
(0.062, 0.096)
(0.012, 0.054)
Note: Mean coefficients from 1,000 bootstrapped OLS regressions. 5th and 95th percentiles in
parentheses.
Table 7. Descriptive Statistics for School District Inefficiency, Alternative Specifications
Mean
Standard
Deviation
Minimum Maximum
Baseline Analysis
τs
κs
| κs-1 |
Excluding Dallas and Houston ISDs
τs
κs
| κs-1 |
Effective Labor Inputs
τs
κs
| κs-1 |
30
0.638
1.003
0.056
0.058
0.063
0.038
0.286
0.770
0.009
1.000
1.312
0.330
0.636
1.003
0.056
0.058
0.063
0.038
0.287
0.770
0.010
1.000
1.311
0.329
0.655
1.003
0.055
0.066
0.063
0.039
0.393
0.798
0.008
1.000
1.465
0.466
Table 8: Cross-tabulation of Allocative Efficiency and Enrollment Categories, Effective
Labor Inputs
Enrollment
Overuse
Overuse
Category
Teachers
Efficient
Administrators
Total
>=10,000
50 (47%)
40 (38%)
15 (14%)
105 (100%)
5,000 to 9,999
25 (37%)
24 (35%)
19 (28%)
68 (100%)
1,000 to 4,999
61 (19%)
184 (57%)
79 (24%)
324 (100%)
500 to 999
42 (21%)
116 (59%)
38 (19%)
196 (100%)
< 500
54 (21%)
160 (62%)
45 (17%)
259 (100%)
All Districts
232 (24%)
524 (55%)
196 (21%)
952 (100%)
Table 9: Modeling the Inefficiency, Effective Labor Inputs
Technical Efficiency
Allocative Inefficiency
(τs)
(| κs-1 |)
Model 1
Model2
Model 3
Model 4
Enrollment (log)
0.0033
0.006
-0.004
0.002
(0.003, 0.004)
(0.005, 0.006)
(-0.006, -0.003)
(-0.001, 0.004)
Admin. Cost Ratio
0.162
0.170
0.100
0.080
(0.143, 0.185)
(0.142, 0.193)
(0.043, 0.159)
(0.023, 0.138)
Density (log)
-0.003
-0.004
(-0.004, -0.003)
(-0.006, -0.001)
Herfindahl Index
-0.024
0.034
(-0.028, -0.020)
(0.024, 0.046)
Constant
0.612
0.610
0.072
0.030
(0.584, 0.649)
(0.582, 0.647)
(0.056, 0.089)
(0.010, 0.051)
th
th
Note: Mean coefficients from 1,000 bootstrapped OLS regressions. 5 and 95 percentiles in
parentheses.
31
32
Appendix Table
Table A.1: Salary Models for Instructional and Non-instructional Staff
Years of Experience (log)
Years of Experience (log), squared
First year teacher
No degree
MA
PhD
New hire
Multi-campus teaching assignment
Elementary education
Language arts
Math
Science
Social Studies
Health & PE
Foreign language
Fine arts
Teachers
-0.0114***
(0.00114)
0.0203***
(0.000633)
-0.0130***
(0.000743)
-0.000733
(0.00127)
0.0217***
(0.000344)
0.0329***
(0.00263)
-0.00518***
(0.000212)
0.00345***
(0.000404)
-0.000733***
(0.000166)
-0.00113***
(0.000171)
-0.000590***
(0.000197)
-0.00116***
(0.000206)
-0.00103***
(0.000190)
0.00917***
(0.000274)
-0.00512***
(0.000369)
-0.00136***
(0.000247)
33
Aides
0.0109***
(0.00255)
-0.00342***
(0.00106)
Administrators
0.0124
(0.00882)
-0.00283
(0.00264)
-0.0184***
(0.00396)
-0.000402
(0.0145)
-0.0717
(0.0489)
-0.0173***
(0.00112)
-0.0104
(0.0162)
0.0155***
(0.00272)
0.0509***
(0.00678)
-0.00317
(0.00258)
Support Staff
0.0471***
(0.00432)
-0.0113***
(0.00153)
-0.00383
(0.00536)
0.0251***
(0.00280)
0.0919***
(0.0123)
-0.00919***
(0.00180)
Auxiliaries
0.0441***
(0.00264)
-0.0143***
(0.000965)
-0.0180***
(0.00614)
-0.0150
(0.0171)
0.000514
(0.0402)
-0.0559***
(0.00153)
Computer
Vocational/technical
Special education
Self-contained classroom
No grade
Elementary grade
Secondary grade
Pre-kindergarten
Kindergarten
Department head
Support staff
Administration
Percent LEP
Percent Economically Disadvantaged
Enrollment (log)
Enrollment (log), squared
Fair market rent
Unemployment rate
Miles from metro center
-0.000786**
(0.000316)
-0.000524
(0.000513)
-0.000262
(0.000320)
0.000268
(0.000175)
-0.000399**
(0.000171)
-0.00143***
(0.000154)
0.00125***
(0.000163)
-0.000756
(0.000461)
-0.00214***
(0.000248)
0.0134***
(0.00314)
0.00155
(0.00307)
0.0379***
(0.00982)
-0.00618***
(0.00113)
0.00331***
(0.000748)
0.0539***
(0.00311)
-0.00116***
(0.000183)
0.0627***
(0.00243)
0.00119***
(0.000117)
-0.000587***
34
0.157
(0.247)
1.285***
(0.000541)
0.124
(0.0866)
-0.00639
(0.00543)
-0.0117***
(0.00339)
-0.0498*
(0.0287)
0.00366**
(0.00175)
0.0315*
(0.0163)
0.00214***
(0.000563)
-0.00229***
-0.0208
(0.0136)
-0.0427***
(0.00904)
-0.00871
(0.00601)
0.115***
(0.0198)
-0.00495***
(0.00122)
-0.0575**
(0.0291)
-0.00142
(0.000975)
-0.000576*
0.0182**
(0.00758)
-0.0416***
(0.00631)
0.0415***
(0.00611)
0.0173
(0.0135)
-0.000219
(0.000843)
0.00306
(0.0168)
-0.00215***
(0.000725)
-0.000757***
0.0946***
(0.0352)
-0.0509***
(0.00645)
-0.0836**
(0.0333)
0.00643***
(0.00200)
-0.00693
(0.0171)
-0.000573
(0.000781)
-0.000762
Miles from teacher certifying institution
School year 2009-10
School year 2010-11
School year 2011-12
School year 2012-13
(4.30e-05)
4.63e-05
(5.28e-05)
-0.0402***
(0.000672)
-0.0311***
(0.000548)
-0.0359***
(0.000390)
-0.0229***
(0.000229)
(0.000381)
0.000846*
(0.000512)
-0.0884***
(0.00118)
-0.0690***
(0.00139)
-0.0653***
(0.00120)
-0.0391***
(0.000827)
Asst. principal
Principal
Program director
Athletic director
(0.000299)
0.000564
(0.000358)
-0.101***
(0.00279)
-0.0751***
(0.00251)
-0.0679***
(0.00203)
-0.0432***
(0.00149)
-0.207***
(0.00771)
-0.113***
(0.00717)
-0.132***
(0.00746)
-0.132***
(0.0235)
Nurse
Counselor
Educational diagnostician
Teacher facilitator
Constant
10.18***
(0.0177)
9.700***
(0.158)
10.84***
(0.103)
(0.000283)
0.000642**
(0.000271)
-0.0799***
(0.00179)
-0.0607***
(0.00186)
-0.0578***
(0.00150)
-0.0378***
(0.00101)
(0.000764)
0.00251**
(0.00101)
-0.101***
(0.00137)
-0.0728***
(0.00160)
-0.0725***
(0.00133)
-0.0438***
(0.000952)
0.00594
(0.0194)
0.0230***
(0.00585)
0.0141**
(0.00596)
-0.00861**
(0.00377)
10.58***
(0.0719)
10.35***
(0.161)
Observations
1,526,002
299,867
108,716
268,240
Number of individuals
417,475
107,308
31,533
88,334
Note: All models also contain individual fixed effects and fixed effects for CBSA or county. Robust standard errors in parentheses
*** p<0.01, ** p<0.05, * p<0.1 35
878,515
280,055
Technical Appendix:
This technical appendix provides additional detail on the strategies used to estimate the input
distance function and the input share equation.
Input Distance Function
By definition, the input distance function is
,
:
where
: We specify the following equation to identify the parameter estimates of the input distance
function, D(y,x):
(A1)
1  D ( y , x )  exp( )
where D(y,x) is linearly homogeneous in x and v is an error term. Taking the log of (A1) yields:
(A2)
0 =lnD(y,x) + υ.
In this analysis we use a translog form for the distance function. Thus, equation A2
becomes
0
∑
∝
∑
ln
∑ ∑
∑ ∑
ln
∑ ∑
ln
ln
ln
ln
ln
∑ ∑
∑ ∑
ln
ln
36
ln
ln
ln
∑
∑ ∑
ln
ln
ln
(A3)
where xj is the quantity for discretionary inputs (INST and NINST), zr is the quantity for nondiscretionary inputs (CAPINPUT, ENROLL, COMPACTNESS and the exponential value for the
variables that are already in percentage terms—HIGHSCHL, HIGHSES, NOTSPECIAL, and
NOTLEP) and y m is the output quantity (MATH and READING). We impose homogeneity in the
discretionary inputs (β j = 1, β j k = 0, ρ j m = 0, γ j r = 0 where all of the sums are over values of
j) as required by the definition of the input distance function (Grosskopf et al. 2001). In addition
we also impose symmetry (e.g. β j k=β k j).
One advantage of the translog specification is that by Shephard's lemma the first derivative
of (A3) with respect to lnx1 equals the expenditure share for input 1 (S 1 = w 1x 1/(w 1x 1 + w 2x 2)).
By estimating the distance function and the share equation together in a system of simultaneous
equations we can improve the efficiency of the estimated parameters. We use the observed input
quantities and the predicted salaries of teachers and administrators (P=w 2/w1) in each district to
define instructional expenditure shares (S1= x 1 / (x1 + Px 2 )) for each observation. The relative
price of administrators (P) is defined in terms of predicted salaries rather than the observed salaries
because the observed salaries may include rents (Bruner and Squires 2013).
Thus, we estimate the following system of equations:
0
∑ ∑
∑ ∑
∑ ∑
∑
∝
∑ ∑
ln
ln
ln
∑ ∑
ln
ln
∑ ∑
ln
ln
ln
ln
ln
ln
ln
,
37
ln
∑
ln
∑
(A4)
ln
ln
ln
ln
ln
using restricted least squares. Although Equation A4 appears to not be estimable given the
nonvariance of the left hand side of the first equation, such a system can be estimated by first
imposing homogeneity restrictions and then using restricted least squares estimation. See Hayes,
Grosskopf and Hirshberg (1995) for details. Our measure of technical efficiency, τs is the inverse
of the predicted value of the distance function in equation A4, corrected for the minimum
predicted value.
As discussed in Grosskopf et al. (2001), the first derivatives of the input distance function
with respect to input quantities yield (cost-deflated) shadow or support prices of those inputs.
Therefore, the predicted values from the instructional share equation (together with the variable
input quantities and the ratios of average prices P=w2/w1) provide sufficient information to
generate a point estimate of κ for each school district (κs). With some rearrangement, the definition
of κs given in equation 1 becomes
/
/
/
/
∙
/
(A5)
Therefore, given the translog specification,
Ŝ/
Ŝ /
∙
(A6)
where Ŝ is the predicted value from the share equation.
38