School District Consolidation: Market Concentration and the Scale-Efficiency
Tradeoff*
Timothy J. Gronberga, Dennis W. Jansena, Mustafa U. Karakaplanb, Lori L. Taylorc,
a
Department of Economics, Texas A&M University, College Station, TX 77843
b
Department of Finance and Economics, Texas State University, San Marcos, TX, 78666
c
Bush School of Government & Public Service, Texas A&M University, College Station, TX
77843
* The authors thank the Education Research Center at Texas A&M for research support on this
project. All views expressed are those of the authors alone. Gronberg and Jansen thank the
Private Enterprise Research Center at Texas A&M for research support. The conclusions of this
research do not necessarily reflect the opinion or official position of the Texas Education Agency
(TEA), the Texas Higher Education Coordinating Board (THECB), or the State of Texas.
Abstract
Consolidation is often proposed as a strategy for increasing school district quality without
increasing educational funding. If consolidation reduces competition in the local school market
and reduces efficiency, any savings from exploiting economies of scale may be lost to increased
inefficiency. We use a stochastic cost function to estimate investigate these effects for districts in
Texas. We find important economies of scale, but we also find that increased market
concentration leads to increased cost inefficiency. Finally, we illustrate the practical importance
of these two potentially offsetting factors in a simulation that considers consolidating Texas
school districts to county-level districts. We find that failure to consider the effect on
competition can lead to substantial overestimates of the benefits of consolidation.
KEY WORDS: Economics of Education, Scale, Competition, Efficiency, Stochastic Cost
Function
2
1. Introduction
School district consolidation to take advantage of scale economies is often proposed as a
strategy for increasing the quality of education services without increasing funding. An
unintended consequence of consolidation is increased concentration and reduced competition in
the local school market. If lessened competition increases inefficiency, the cost savings from
exploiting economies of scale may be reduced, perhaps substantially.
Most studies of school district consolidation focus on scale economies. In this paper we
emphasize the competitive effect as an important link to understanding the net effect of
consolidation. We use a stochastic cost function frontier to estimate the direct and indirect effects
of school district consolidation on school district costs in Texas. We find large economies of
scale from consolidating very small districts. We also find that increases in market concentration
leads to increases in inefficiency, suggesting that existing estimates overstate the potential
benefits from district consolidation. We illustrate the quantitative importance of this scale –
efficiency tradeoff by simulating a systemic consolidation of school districts in Texas and find
that the increased inefficiency significantly counteracts any savings due to scale economies.
2. The Literature on Economies of Scale and the Effects of Competition
Many researchers have examined the relationship between school district size and the
cost. Andrews, Duncombe and Yinger (2002) surveyed the literature on educational cost
functions and concluded that the cost-minimizing scale for districts is in the 2,000 - 5,000 pupil
range, with costs sharply higher for very small districts with fewer than 300 pupils.
Some researchers specifically examined scale effects of school district consolidation, and
nearly all found substantial potential gains. Duncombe, Miner and Ruggiero (1995) find large
cost savings from consolidating districts with fewer than 500 students, as do Duncombe and
Yinger (2007). Dodson and Garrett (2004) simulate the savings from consolidating Arkansas
3
districts and find per student cost savings range from 19–54 percent. Zimmer, DeBoer and Hirth
(2009) also find large potential gains from consolidation.
Researchers have also examined the relationship between competition and student
performance. Belfield and Levin (2002) and Taylor (2000) survey the earlier literature and find
that most studies report a positive correlation between competition and student performance.
Hoxby (2003) reports that the introduction of charter schools – charter competition -- increased
student performance in math and reading at traditional public schools. Sass (2006) finds evidence
that competition from charter schools increased student performance in Florida; Booker et al.
(2008) find similar effects in Texas. On the other hand, Bettinger (2005) and Bifulco and Ladd
(2006) find no effect of charter competition on student performance, while Ni (2009) finds that
competition from charter schools had a negative effect on student achievement in Michigan.
A few studies have focused on the relationship between competition and relative
efficiency of public school districts. Husted and Kenny (1997) estimate efficiency frontiers for
the educational system in each state and find that inefficiency is higher in states with a low
number of districts per capita. Grosskopf et al. (1999, 2001) find district inefficiency in Texas is
substantially higher in metropolitan areas with less competition for enrollment (both public and
private). Millimet and Collier (2008) estimate efficiency scores for Illinois school districts and
find school districts become more efficient as other districts in the county become more efficient,
a pattern they attribute to competition. Kang and Greene (2002) estimate production efficiency
frontiers for New York school districts, and conclude efficiency is lower in more concentrated
counties. In contrast, Duncombe, Miner and Ruggiero (1997) find cost efficiency for New York
districts is lower where private school enrollment is higher.
3. A Model of Cost and Consolidation
4
We use a stochastic frontier approach to estimate the direct and indirect effects of school
district consolidation on district spending. This approach is well-suited for analyzing public and
non-profit supplier behavior. The possibility of cost inefficiency looms larger in the absence of
profit-maximizing incentives, and incentives for cost efficiency are weakened when competition
is lessened. The stochastic cost frontier lets the data reveal the degree of cost inefficiency. The
advantages and challenges of applying the stochastic frontier methods to estimate school cost
functions are discussed in a recent paper by Gronberg, Jansen and Taylor, 2011.
The standard stochastic frontier model specifies
𝐸 = 𝐶(𝑍 | 𝛽) ∙ exp(𝑣 + 𝑢),
(1)
where 𝐸 is observed spending, 𝐶(𝑍 | 𝛽) is the cost frontier, 𝑍 = {𝑤1 , … , 𝑤𝑘 ; 𝑧1 , … , 𝑧𝑚 ; 𝑦} is a
vector including variables affecting the cost frontier, 𝑤𝑙 are input prices, 𝑧𝑗 are quasi-fixed
inputs, 𝑦 is outcome(s), 𝛽 is the cost parameter vector, 𝑣 is a two-sided error term representing
exogenous shocks and 𝑢 is a one-sided error term that captures cost inefficiency. This cost
frontier is the neo-classical cost function, the object of discovery. Inefficiency increases cost
above minimum cost, so 𝑢𝑖 ≥ 0. Cost efficiency is defined as 𝐶𝐸𝑖 = exp(−𝑢𝑖 ) ≤ 1.
Educational outcomes include a quantity dimension, student enrollment, and a quality
dimension, gains in human capital. Building on the structure of Andrews, Duncombe and Yinger
(2002), we let 𝑁 denote district enrollment and 𝑆 student achievement. We follow the common
convention in the education literature and estimate a per-pupil cost model:
𝐸∗ ≡
𝐸 𝐶( 𝑤1 , … , 𝑤𝑘 ; 𝑧1 , … , 𝑧𝑚 ; 𝑆, 𝑁 | 𝛽) ∙ exp(𝑣 + 𝑢)
=
𝑁
𝑁
(2)
Taking natural logarithms of equation (2) gives
ln 𝐸 ∗ = ln 𝐶(∙) − ln 𝑁 + 𝑣 + 𝑢
(3)
5
Economies of scale could be measured with respect to quantity, 𝑁, or with respect to
quality, 𝑆.1 The most common measure, and the one relevant to consolidation, is enrollment.
Andrews, Duncombe and Yinger (2002) label this ‘economies of size.’ We define economies of
size as the enrollment elasticity of per pupil expenditures (𝜂 = 𝜕ln𝐸 ∗ /𝜕ln𝑁), holding constant
student achievement 𝑆, input prices 𝑤, quasi-fixed inputs 𝑧, and cost inefficiency. From (3):
𝜂 = 𝜃 − 1,
(4)
where 𝜃 = 𝜕ln𝐶/𝜕ln𝑁 is the enrollment elasticity of total cost. Economies of size exist if 𝜂 < 0.
The competitiveness of the education market facing a school district is one factor that can
influence cost efficiency in a district, as competition serves to discipline the tendency of not-forprofit decision-makers to engage in excessive spending. The competition hypothesis predicts a
negative relationship between district competitiveness and district cost inefficiency.
The stochastic cost frontier framework accommodates testing of the competition
hypothesis via the one-sided error term, 𝑢. We specify that
𝑢 = 𝑢(𝑥, 𝛿), 𝑤𝑖𝑡ℎ 𝑢 ≥ 0
(5)
where 𝑥 is a vector of environmental cost efficiency factors, including a measure of competition,
and 𝛿 is a parameter vector. We substitute (5) into the per pupil expenditure equation (3) to get
ln 𝐸 ∗ = ln 𝐶(∙) − ln 𝑁 + 𝑣 + 𝑢(𝑥, 𝛿)
(6)
We use this model to examine the effects of a school district consolidation on per pupil
expenditures. A change in N also impacts efficiency, so equation (4) is modified to include this
effect. Let 𝑥1 denote our competition measure. We differentiate (6) with respect to ln𝑁 to get
𝜂 = (𝜃 − 1) + (
𝜕𝑢
𝜕𝑥1
)∙(
)∙𝑁
𝜕𝑥1
𝜕𝑁
(7)
1
Duncombe and Yinger (1993) identify three scale economy measures. Besides 𝑁 and 𝑆, they discuss economies of
scale in the production of school activities, 𝐺, an intermediate public input in the production of student achievement.
6
This decomposes the response of per pupil spending from a consolidation-induced change in N
into a cost economy effect (𝜃 − 1) and a competitive efficiency effect (𝜕𝑢⁄𝜕𝑥1 ) ∙ (𝜕𝑥1 ⁄𝜕𝑁) ∙
𝑁. The competitive efficiency hypothesis implies (𝜕𝑢⁄𝜕𝑥1 ) < 0 and (𝜕𝑥1 ⁄𝜕𝑁) < 0, so when
(𝜃 − 1) < 0 per pupil cost savings from consolidation are dampened by increased inefficiency.
4. Specification of the Econometric Model
Our baseline model is a modified translog frontier specification, which nests the CobbDouglas function form and specifications which assume full efficiency.2 The dependent variable
is operating expenditures per pupil (𝐸 ∗ ). Right-hand-side variables include output variables
(enrollment, 𝑁 = 𝑞1 , and quality measures 𝑞𝑖 ), input prices denoted 𝑤𝑙 , and environmental
factors denoted 𝑧𝑗 . All variables not already expressed as percentages are in natural logarithms.
School district size varies greatly in Texas so we include a cubic term for enrollment.3
Our model for district expenditures per pupil is:
𝑛1
𝑛2
𝑛3
𝑛1 𝑛1
ln 𝐸 ∗ = 𝛼0 + ∑ 𝛼1𝑖 𝑞𝑖 + ∑ 𝛼2𝑖 𝑤𝑖 + ∑ 𝛼3𝑖 𝑧𝑖 + 0.5 ∑ ∑ 𝛼4𝑖𝑗 𝑞𝑖 𝑞𝑗
𝑖=1
𝑖=1
𝑖=1
𝑖=1 𝑗=1
𝑛3 𝑛3
𝑛2 𝑛2
𝑛1 𝑛2
+ 0.5 ∑ ∑ 𝛼5𝑖𝑗 𝑤𝑖 𝑤𝑗 + 0.5 ∑ ∑ 𝛼6𝑖𝑗 𝑧𝑖 𝑧𝑗 + ∑ ∑ 𝛼7𝑖𝑗 𝑞𝑖 𝑤𝑗
𝑖=1 𝑗=1
𝑛1 𝑛3
𝑖=1 𝑗=1
(8)
𝑖=1 𝑗=1
𝑛2 𝑛3
+ ∑ ∑ 𝛼8𝑖𝑗 𝑞𝑖 𝑧𝑗 + ∑ ∑ 𝛼9𝑖𝑗 𝑤𝑖 𝑧𝑗 + 𝛼10 ∙ 𝑞13 + 𝑣 + 𝑢
𝑖=1 𝑗=1
𝑖=1 𝑗=1
We impose the usual symmetry restrictions, 𝛼𝑎𝑖𝑗 = 𝛼𝑎𝑗𝑖 , for 𝑎 = 4, 5 𝑎𝑛𝑑 6.
4.1. Modeling the One-Sided Error
2
The model is a modified translog in that variables measured in percentages are not logged (e.g. the share of low
income students). This avoids a technical issue, as some of these variables can have a value of zero.
3
Gronberg, Jansen and Taylor (2011) also use this specification. Other researchers have dealt with this issue by
excluding the largest Texas districts from analysis (e.g., Imazeki and Reschovsky, 2004).
7
Our analysis is predicated on the assumption that competition affects school district
efficiency, here the one-sided error, but not the educational production technology itself.
Simar, Lovell and Eeckaut (1994) suggest modeling u as:
𝑢 = 𝑢(𝑥𝑖 , 𝛿) = ℎ(𝑥𝑖 , 𝛿) ∙ 𝑢∗
(9)
where the 𝑥𝑖 are the exogenous variables affecting cost efficiency, 𝑢∗ is a nonnegative random
variable that is not a function of the 𝑥𝑖 and ℎ(𝑥𝑖 , 𝛿) is a nonnegative function of 𝑥𝑖 . Wang and
Schmidt (2002) say this specification exhibits the ‘scaling property’ because 𝑥𝑖 affects the scale
of u through ℎ(𝑥𝑖 , 𝛿), but does not affect the shape of 𝑢 defined by the basic distribution, 𝑢∗ .
Intuitively, the scaling property decomposes 𝑢(𝑥𝑖 , 𝛿) into two logically and economically
independent terms. The basic random term 𝑢∗ can be viewed as the district’s idiosyncratic cost
efficiency which reflects such random elements as administrator talent. Then the term ℎ(𝑥𝑖 , 𝛿)
depends on market factors, e.g. the degree of competition, that scale up the idiosyncratic term.
Fried, Lovell and Schmidt (2008) discuss candidates for the scaling function ℎ(𝑥, 𝛿), and
there are choices for the basic distribution 𝑢∗ .4 We follow Wang and Schmidt (2002), with:
𝑢 ≡ 𝒩 + [ 0 , exp(𝛿0 + 𝛿 ∙ 𝑥𝑖′ ) ]
(10)
Thus, 𝑢∗ has a half-normal distribution and the variance of 𝑢, 𝜎𝑢2 , is defined as exp(𝛿0 + 𝛿 ∙ 𝑥𝑖′ ).
This specification has the scaling property -- the mean and standard deviation of 𝑢 change as 𝑥𝑖
changes, but the shape of the distribution remains half-normal.5
4.2. Modeling the Two-Sided Error
We model the variance of ν to account for heterogeneity based on district size:
1
𝜎𝑣2 = exp (𝛾0 + 𝛾1 ∙ )
𝑁
2
where 𝑣𝑖 ~ 𝑖𝑖𝑑 𝒩(0, 𝜎𝑣 ).
4
5
(11)
Suitable basic distribution specifications include half normal, exponential or truncated normal distributions.
An example of such a specification is presented in Caudill, Ford and Gropper (1995).
8
4.3. Estimating Cost Efficiency
With our specification above, the log-likelihood function can be written as
𝑁
1
2
𝜀𝑖 𝜆
𝜀𝑖2
ln𝐿 = ∑ { ln ( ) − ln𝜎𝑆 + lnΦ ( ) − 2 }
2
𝜋
𝜎𝑆
2𝜎𝑆
(12)
𝑖=1
where 𝜎𝑆 = √𝜎𝑢2 + 𝜎𝑣2 , 𝜆 = 𝜎𝑢 ⁄𝜎𝑣 , 𝜀𝑖 = 𝑢𝑖 + 𝑣𝑖 , and Φ(∙) is the cumulative distribution
function of the standard normal distribution.
Jondrow et al. (1982) suggests estimating ui based on the conditional distribution of 𝑢
given 𝜀. That is,
𝑢̂𝑗𝑒 = E(𝑢𝑖 | 𝜀𝑖 ) = 𝜇∗𝑖 + 𝜎∗ {
𝜙(−𝜇∗𝑖 ⁄𝜎∗ )
}
1 − Φ(− 𝜇∗𝑖 ⁄𝜎∗ )
(13)
where 𝜇∗𝑖 = (𝜀𝑖 𝜎𝑢2 )⁄(𝜎𝑢2 + 𝜎𝑣2 ) , 𝜎∗ = 𝜎𝑢 𝜎𝑣 /√𝜎𝑢2 + 𝜎𝑣2 , and 𝜙(∙) is the standard normal density
function. This estimate of the one-sided error can then be used to generate an estimate of cost
efficiency by substituting this point estimate of 𝑢𝑖 into exp{−𝑢𝑖 }.
Cost efficiency can also be estimated directly. Battese and Coelli (1988) suggest:
𝐶𝐸𝑖 ≡ E{exp(−𝑢𝑖 ) |𝜀𝑖 } = {
1 − Φ(𝜎∗ − 𝜇∗𝑖 /𝜎∗ )
1
} ∙ exp (−𝜇∗𝑖 + 𝜎∗2 )
1 − Φ(−𝜇∗𝑖 /𝜎∗ )
2
(14)
where 𝜎∗ = 𝜎𝑢 𝜎𝑣 /√𝜎𝑢2 + 𝜎𝑣2 , and 𝜇∗𝑖 = (𝜀𝑖 𝜎𝑢2 )/𝜎𝑢2 + 𝜎𝑣2 .
5. Data
The data for our analysis come from administrative files and public records of the Texas
Education Agency. The unit of analysis is the school district, and the analysis includes all 967
traditional public districts with complete data that served grades K-12 during the 2010-11 school
year.6 Table 1 provides descriptive statistics on the variables used in this analysis.
6
Open enrollment charter schools are excluded because they may have a different education technology.
9
The dependent variable is the log of current operating expenditures per pupil excluding
food and student transportation expenditures.7 Transportation expenditures and food
expenditures are excluded as they are unlikely to be explained by the same factors that explain
student performance, and thus would add unnecessary noise to the analysis. We also exclude
community service, debt service, facility acquisition and construction, and intergovernmental
payments. Remaining average per pupil expenditure in Texas districts during our sample period
are $9,354, with a low of $5,706 and high of $26,554.
As noted above, our independent variables include a quantity dimension of output
(enrollment) and two quality dimensions of student performance. The first is the change in
passing rate on the Texas Assessment of Knowledge and Skills (TAKS). The TAKS is
administered annually in grades 3-11 in mathematics and reading/language arts, and passing
rates are a key element of a district’s evaluation under both the Texas and federal school
accountability systems. We used administrative data on individual student scores to calculate the
percentage of students in each district who passed the TAKS in 2011 and compared it to the
percentage of those same students who passed in 2010, thereby calculating the change in passing
rate for a matched cohort of students. Our second indicator of quality is the percentage of
students completing an advanced course in high school. This addresses the common criticisms
of standardized tests, that they are too easy or the passing standards too low.
The independent variables in our analysis include a price indicator, the wages of teachers.
We use a hedonic wage index of beginning teacher salaries modeled after Gronberg et al (2011)8.
7
The expenditures data are adjusted to account for school districts that are fiscal agents for other school districts.
Fiscal agents collect funds from member districts in a shared service agreement, and make purchases or pay salaries
with those shared funds on behalf of member districts. Thus spending of fiscal agents is artificially inflated while
spending by member districts is artificially suppressed. We use TEA data from F-33 files indicating the amount
fiscal agents spent on behalf of member districts to allocate spending to member districts on a proportional basis.
8
The hedonic wage model was estimated using administrative data on individual salaries for the six years from
2005-06 through 2010-11, The index was generated by predicting the full-time-equivalent monthly salary for a first-
10
Salaries of administrators and other professional staff are too highly correlated with teacher
salaries to include them both.
Ideally we would include direct measures of local prices for instructional equipment and
classroom materials. Unfortunately, such data are not available. Since prices for other
instructional materials are largely set in a national market, any district-level variation in price
should largely be a function of transportation costs. We include a measure of geographic
isolation, the linear distance to the nearest major metropolitan area, as a proxy for these prices.
Finally, the model includes indicators for environmental factors that influence the cost of
education. We include the percentages of students in each district classified as economically
disadvantaged, limited English proficient (LEP), Special Education, or high school students.
The empirical measure of competition is central to our analysis. We measure
competitiveness using Herfindahl-Hirschman indices (HHI) of market concentration based upon
student enrollment. This approach to measuring public school competition is common in the
literature (e.g. Hoxby (2000), Grosskopf et al. (2001), Holmes, Desimone and Rupp (2006),
Booker et al. (2008)), with labor markets used to define educational markets Thus, the HHI for
an educational market is the sum of squared enrollment shares for all of the public and private
school districts in the metropolitan area, micropolitan area or county.9
As Figure 1 illustrates, the delivery of education services is highly concentrated in Texas.
The U.S. Department of Justice Antitrust Division labels markets with an HHI between 0.10 and
0.18 as moderately concentrated, and markets with an HHI over 0.18 as concentrated. By these
standards, 193 of the 198 Texas education markets in our sample were at least moderately
concentrated in 2010-2011. Districts with low HHI measure are usually in large metropolitan
year teacher who had average characteristics with respect to gender and ethnicity, educational attainment, teaching
assignment, certification status, and coaching status. Details available upon request.
9
Data on enrollment and location of traditional public and charter schools come from the Texas Public School
directory, and of private schools come from the NCES Private School Universe Survey and the TEA.
11
areas with competing public school districts, charters, and private schools. Districts that are
highly concentrated by the HHI measure are typically in smaller urban areas or rural areas.
6. Regression Results
Table 2 presents modified marginal effects from our estimates of the model in equation
(8). Because of interaction terms, marginal effects depend not only on the estimated model
parameters but also on the values of the explanatory variables. We report the marginal effects of
each variable evaluated at the mean of all other explanatory variables.10 For example, the
marginal effect of enrollment, which is η from equation (4), is:
𝑛2
𝑛3
2
(𝜃̂
− 1) = 𝛼̂11 + ∑ 𝛼̂71𝑗 𝑤𝑗 + ∑ 𝛼̂81𝑗 𝑧𝑗 + 2𝛼̂411 𝑞1 + 3𝛼̂10 𝑞1
𝑗=1
(15)
𝑗=1
We present estimates for three alternative specifications: The first excludes any measure
of competition from the one-sided error. The second has HHI enter linearly in the one-sided
error, and the third specification has HHI entering as a quadratic form.
Results in Table 2 show that our estimated cost function is generally consistent with prior
expectations. It costs significantly more to produce higher output quality, whether increases in
passing rates or in the percent of students taking advanced courses. Districts paying higher wages
to teachers have higher costs, as do districts in geographically remote areas. Districts with
student populations that require more resource-intensive instructional technologies also have
higher costs. For example, the mean marginal effect of economically disadvantaged students is
0.18. A district with average characteristics would see a 0.18 percent increase in cost from a one
percentage point increase in the share of economically disadvantaged students.
10
The full set of estimated coefficients is available from the authors.
12
Adding HHI to the model (the column titled Linear HHI) improves the fit substantially.
Results indicate that more concentrated education markets (higher HHI) are more cost
inefficient. The third column indicates that adding the quadratic in HHI does not significantly
improve the log likelihood ratio and does not much change the estimated marginal effects.
Figure 2 plots estimates of cost inefficiency for Texas school districts. Average efficiency
is just under 90%. The distribution is substantially skewed and reveals substantial inefficiency.
The marginal impact of HHI on the one sided error calculated from 𝑢̂𝑗𝑒 is 0.04, so an
increase of 0.1 in the HHI (equivalent to the change from a market with ten equally-sized school
districts to a market with five equally-sized school districts) results in an increase in 𝑢̂𝑗𝑒 by
0.004.11 The impact on efficiency is larger, involving the exponential of the one-sided error.
Table 3 presents coefficient estimates and standard errors from the linear HHI model for
the enrollment variable and the interactions of enrollment with all other variables. This illustrates
the nonlinear relation between cost and enrollment—the linear and quadratic enrollment terms
are statistically significant at the 1 percent level, and the cubic enrollment term is statistically
significant at the 5 percent level. The interaction between enrollment and the percentage of
economically disadvantaged students is positive and statistically significant suggesting that
economies of scale are smaller for school districts with a large fraction of economically
disadvantaged students. The statistically significant and negative coefficient on the interaction
between enrollment and distance and the interaction between enrollment and teacher salaries
suggest that economies of scale are also smaller where input prices are lower.
6.1. Endogeneity in the Stochastic Cost Frontier Model
11
The appendix (available upon request) describes how this marginal effect is calculated.
13
The literature identifies potential sources of endogeneity in this analysis. As discussed in
Gronberg et al. (2011) there are reasons to expect that the student outcomes are endogenous.
Hoxby (2000) discusses reasons to be concerned that the level of competition is endogenous.
Neither of these potential sources of endogeneity is easily handled. Standard instrumental
variables (IV) approaches cannot be directly employed in stochastic frontier analyses, and the
literature has yet to develop a strategy for addressing endogeneity with respect to the scaling
factors in the one-sided error of a stochastic frontier model.12 Therefore we approach the issue
of endogeneity as a robustness check on our baseline specifications. We estimate first-stage
relationships for the potentially endogenous variables as if the analysis were two-stage least
squares, and demonstrate that our chosen instruments are well correlated with the potentially
endogenous variables. We then use predicted values from the first stage as regressors in the
second-stage, stochastic frontier model.
6.1.1. Identifying Potential Instruments
Because the problems associated with weak instruments are well documented (e.g.
Angrist and Pischke 2009; Bound, Jaeger and Baker 1995; or Murray 2006), identifying strong
instruments is a necessary precondition for any instrumental variables analysis. Furthermore,
given the translog specification, we must find instruments not only for competition and the two
measures of school quality but also for the relevant interaction terms.
Hoxby’s influential analysis suggests that topology can be a source of reliable
instruments for school district competition (Hoxby, 2000). We follow Hoxby (2000) and
Rothstein (2007) and use the U.S. Geological Survey (USGS) Geographic Names Information
System (GNIS) data to create two categories of streams based on the length of the streams,
12
Guan et al. (2009) present a two step method to handle endogenous frontier regressors in a panel setting. Their
approach does not generalize to models where the one-sided error is a function of specified regressors.
14
defining streams longer than 3.5 miles as large, and others as small.13 We use the number of
large streams and small streams in a county as exogenous sources of variation.
Indicators of the local demand for education are commonly used as instruments for
school quality14 Possible determinants of the local demand for education include the share of
district population over age 65, the share of adult district population with at least a bachelor’s
degree, the share of owner-occupied housing stock, and the distance to the nearest institution of
higher education.15 We use these variables as instruments for our two measures of school quality.
To construct instruments for the interaction terms, we interacted all of the potential instruments
with each exogenous variable.
Heteroskedasticity-robust identification and instrument relevance statistics from the first
stage regression of the IV estimation are presented in Table 4. As illustrated, the proposed
instruments are correlated with the potentially endogenous regressors, but none of the first stage
regressions satisfy the common rule of thumb that the first-stage F-test for the excluded
instruments exceeds 10. We conclude that the instruments commonly suggested in the literature
weakly identify educational quality and market concentration in Texas.16
More importantly, as indicated in Table 5, replacing the potentially endogenous variables
(and their interaction terms) in the stochastic frontier model has little qualitative impact on our
analysis. The main difference between the results in Table 2 for the column labeled ‘Linear HHI’
and the results in Table 5 for the column labeled ‘Pseudo-IV’ are in the estimated marginal effect
of the variables ‘Change in TAKS Passing Rate’ and ‘Percent Taking Advanced Courses.’ For
We use the “alternative version of GNIS data” mentioned by Rothstein which includes coordinates of two points
for each stream in each county: one point is the origin where that stream starts traversing the county, and the other
point is the destination where that stream ends traversing that county. The length between these two points is
calculated by using the haversine formula.
14
See Imazeki and Reschovsky 2006, Gronberg, Jansen and Taylor (2011) or Baker (2012).
15
The data come from Census 2000 voter demographics.
16
Gronberg, Jansen and Taylor (2011) also reached this conclusion.
13
15
these two variables, the estimated marginal effect is substantially higher when we use the
Pseudo-IV model. For the Change in TAKS Passing Rate, the marginal effect rises from 0.392
to 0.831, and for the Percent Taking Advanced Courses the marginal effect rises from 0.096 to
0.364. The estimated marginal effects of school district size are largely unaffected, and the
effects of competition on efficiency are increased, making our analysis appear conservative.
6.2. Additional Robustness Checks
We estimate several alternative specifications that are reported in Table 5. Houston ISD
and Dallas ISD are the two biggest school districts in Texas by a factor of two, and are located in
the two most competitive educational markets. The first column of results in Table 5 reports
estimates from a sample that excludes these two districts. The estimated marginal effects are not
much changed, indicating our findings are not sensitive to inclusion of Houston and Dallas ISDs.
Enrollment density—the number of students per square mile in a school district—may be
an important confounding factor. There are reasons to believe that low density school districts
are less efficient. Low-density districts may have higher administrative costs due to geographical
dispersion and monitoring costs. Low density school districts also tend to be located in low
competition environments. To explore this possibility we exclude districts with fewer than 0.42
students per square mile (i.e. districts below the 5th percentile of density). The results are
presented in Table 5. Excluding low density districts does not appreciably change our findings.
A final robustness check uses a different measure of market competition. In place of HHI
we use the 4-firm concentration index, CR4. The pairwise correlation between CR4 and HHI is
0.8. The mean of CR4 is 78% in our sample, with a minimum of 37% and a maximum of 100%,
also suggesting that education markets are highly concentrated.
Results using CR4 are reported in the CR4 column of Table 5. The estimated marginal
effects are consistent with those from our model using HHI, and the marginal effects for the one
16
sided error variance are understandably different but indicate a positive impact of increased
concentration on inefficiency.
7. Simulations
We find that increases in market concentration increase expenditures by increasing
inefficiency. We also find that increases in school district size can reduce the cost of education
via economies of scale. School district consolidation is a double-edged sword.
To explore this trade-off we trace out the estimated relationship between expenditures,
scale and competition. The expenditure per pupil for a given district can be predicted by adding
an estimate of 𝑢 and 𝑣 to the estimate of cost per pupil. We estimate ν with its expected value, 0.
Then expected expenditure per pupil are
∗ ′
̂ 𝑖∗ = E( ln𝐸𝑖∗ | 𝑞𝑖 , 𝑤𝑖 , 𝑧𝑖 , 𝑥𝑖 ) = 𝛼̂0 + 𝑍𝑖𝑗
ln𝐸
𝛼̂𝑗 + 𝑢̂𝑖
(16)
where 𝑍𝑗∗ is a vector of variables in the translog specification and 𝛼̂0 and 𝛼̂𝑗 are the maximum
likelihood estimates of 𝛼0 and 𝛼𝑗 , respectively.
We use Jondrow et al.’s (1982) method for estimating 𝑢̂𝑖 conditional on 𝜀𝑖 to estimate the
one sided error in our sample. For a hypothetical consolidated district we cannot generate this
estimate, as we have no estimate of the hypothetical district’s error term to decompose into oneand two-sided errors. We adopt the following strategy. We assume 𝑢 ≡ 𝒩 + [ 0 , 𝜎𝑢2 ] and define
𝜎𝑢2 = exp(𝛿0 + 𝛿 ∙ 𝑥𝑖′ ), with 𝑢 a half-normal distribution. The basic half-normal distribution of 𝑢
is not a function of 𝑥, and the mean of any variable with a half-normal distribution can be written
as √2𝜎 2 ⁄√𝜋 where 𝜎 2 is the variance of the normal distribution determining the half-normal.
Hence we define 𝑢̂ as17
17
If 𝑢 is assumed to have the scaling property with a different distribution, say truncated normal, 𝑢̂ can be defined
as exp(𝛿 ∙ 𝑥 ′ ) ∙ 𝜇 ∗ where 𝜇 ∗ is the mean of the truncated normal distribution, which is equal to 𝜇 +
𝑎−𝜇
𝑏−𝜇
𝜎
𝜎
𝜙( 𝜎 )−𝜙( 𝜎 )
𝑏−𝜇
𝑎−𝜇 𝜎
Φ(
)−Φ(
)
17
√2𝜎𝑢2
√2
∙ √exp(𝛿0 + 𝛿 ∙ 𝑥 ′ )
√𝜋
√𝜋
Taking the derivative of equation (17) with respect to 𝑥 yields the marginal effect on cost
𝑢̂ = E(𝑢 | 𝑥) =
=
(17)
inefficiency of a hypothetical district:
𝜕𝑢̂
√exp(𝛿0 + 𝛿 ∙ 𝑥 ′ ) ∙ √2 𝛿𝑗
=
∙
𝜕𝑥𝑗
2
√𝜋
(18)
7.1. Iso-Expenditure Curves
We present iso-expenditure curves summarizing the trade-off between competition and
scale in Figure 1. These iso-expenditure curves hold all cost factors except for enrollment and
HHI constant at sample mean values. Each iso-expenditure curve presents all possible pairs of 𝑁
and HHI that would result in the same level of expenditure when other variables affecting
expenditure are held fixed.
As the figure illustrates, a small district operating in a perfectly competitive market can
achieve the same expenditure per student as a monopoly district with many more students, all
other things being equal. In Figure 1, the vertical axis ranges from 0 (monopoly) to 1 (perfect
competition). There are scale economies for enrollments between 47 and 47,124 students – in
this region, expenditures per student decline with increases in enrollment. For enrollments from
47,124 to 203,294 students, expenditures per pupil increase with increases in district enrollment.
The cubic term would eventually have expenditures per pupil decline again as enrollment grows
much beyond 200,000 students, but this is outside our sample range.
In Figure 1 the maximum expenditure per pupil would be $17,059 and occur at point O
where enrollment is 47 and competition is 0. This is essentially the origin. The minimum
where 𝜇 and 𝜎 are the mean and standard deviation of the normal distribution determining the truncated normal
distribution, and 𝑎 and 𝑏 are the truncation minimum and maximum boundaries.
18
expenditure per pupil is at point A where enrollment is 47,124 and competition is 1. At this
point, per pupil expenditure would be $7,385. We present a series of iso-expenditure curves for
various expenditure levels between $7,385 and $17,059. One of these curves is E2 with
expenditure per pupil of $9,418, the level of per pupil expenditure predicted for a district with
2,000 students if the district is a monopoly. Curve E3, with expenditure per pupil of $8,119,
illustrates the level of per-pupil expenditure predicted for a district with 2,000 students if the
district is perfectly competitive. For a district with 2,000 students, the 16 percent difference in
expenditure between E2 and E3 is fully attributable to the difference in efficiency between
competition and monopoly.
The competitive effect and the scale effect work in the opposite directions over the range
of 47 to 48,077 students. This range includes 98 percent of the school districts in Texas. For
these districts, there is a trade-off between economies of scale and economies of competition,
and the gains from consolidation are reduced by the increased inefficiency. As district size
increases beyond 48,077 students, the model predicts that consolidation would lead to increased
costs due to diseconomies of scale and increases in inefficiency.
7.2. Consolidation Simulation
We illustrate the magnitude of the offsetting scale and competition effects by simulating
the impact of a policy action that consolidates all districts in Texas at the county level. This is
not a completely abstract exercise. Other states such as Florida currently have a county school
district system. In January 2011, former Texas State Representative Fred Brown, R-Bryan,
proposed a bill (Texas House Bill 106) to reduce the number of public school districts in Texas
by consolidating all traditional public school districts into county-wide school districts.
Our simulation analysis is designed to illustrate the potential importance of considering
the market structure effect of consolidation. It is not intended to be a policy analysis of an actual
19
county district reform in which spending might be constrained by funding formula
considerations. That is, an actual consolidation might result in lower funding of schools due to
the state funding formula, and that lower funding could result in lower student achievement. In
our consolidation simulation we calculate costs and expenditures for the consolidated district
with given student achievement levels, to illustrate the potential cost savings or dissavings due to
economies of scale and due to decreased competition and resulting increased inefficiency.
We compare the predicted total expenditures in a county pre-consolidation with predicted
total expenditure post-consolidation.18 We predict expenditure per pupil using equation (16) with
our estimate of 𝑢̂ and the coefficient estimates from our stochastic frontier estimation presented
in the linear HHI model.19 Our pre-consolidation expenditure calculation uses observed districtlevel values for q, w, z and x. We use state averages to impute any missing values so that we can
include most traditional public school districts in this simulation.20
21
We predict expenditures after consolidation assuming the total enrollment and
demographic composition of a county is unchanged by the consolidation. Basically we are
assuming that parents will not move in response to a consolidation. We recalculate the HHI
measures of counties after consolidation assuming that there would be no changes to the private
or charter school enrollments. Finally, we assume the characteristics of the new county-level
districts are a pupil-weighted average of the characteristics of the constituent, pre-consolidation
districts. For example, we assume that a county’s change in passing rates after consolidation
18
We also compared actual pre-consolidation total expenditures of districts within a county to predicted postconsolidation expenditures of the county. This did not change our qualitative findings. Quantitatively, it led to a
greater increase in total spending for the state after consolidation.
19
We also used equation (16) with 𝑢̂𝑗𝑒 to calculate the sum of the predicted log expenditures of districts in a county
before consolidation, and used equation (16) with our estimate 𝑢̂ to calculate predicted log expenditures in a county
post consolidation. In this case more districts incur losses after consolidation and the total losses are greater.
20
Replacing the missing values with county averages instead of state averages is an alternative approach. It leads to
relatively small quantitative differences, and the winners and losers after consolidation do not change.
21
We excluded Loving County from the simulation as their school system was consolidated into Wink County's ISD
in 1972 due low levels of enrollment. Bowie County is excluded from the simulation because Texarkana MSA
encompasses Miller County in Arkansas, which complicates the calculation of our competition measure.
20
would be equal to the enrollment weighted average of the change in passing rates of the districts
in that county. Summary statistics before and after consolidation are presented in Table 6.
Consolidation results in much larger districts on average. The maximum district size increases
dramatically, from 203,294 to 784,992. The HHI measure also increases dramatically, from an
average of 0.327 pre-consolidation to 0.844 post consolidation.22
Our simulation indicates that consolidation would cause expenditures to decrease in 109
counties and increase in 99 counties. There are 44 counties that are already county-districts
before consolidation and hence are unaffected.
Expenditures are predicted to rise in the largest counties in Texas. Bexar County, for
example, is one of the 8 counties in Greater San Antonio—one of the largest metropolitan areas
in the U.S. In 2011, there were 39 public school districts in Greater San Antonio and 15 of them
were in Bexar County. Before consolidation, the HHI in San Antonio is 0.098. After
consolidation, the HHI increases to 0.522. At the same time, consolidation changes the size of
the districts—before consolidation the average school district enrollment in Bexar County is
20,885 students with the smallest district enrolling 985 students. After consolidation the
enrollment in the new Bexar County district is 313,278 students. Expenditures rise by $984 per
pupil, partly because the new school district enrollment exceeds the (local) minimum-cost size,
and partly because the reduction in competition increases inefficiency.
The influence of the competitive effect is most evident in the smaller counties on the
fringe of large metropolitan areas. In Bastrop County, there are currently four districts with an
average enrollment of 3,727 students. Consolidation increases the HHI from 0.115 to 0.311, and
the post-consolidation district size is 14,907 students. There are significant savings due to the
22
Most metropolitan areas include more than one county, so county-level consolidation does not eliminate
competition.
21
increase in size, as these districts are all in the economies of scale region of the cost frontier.
Nonetheless, these savings are offset by the change in the competitive environment and Bastrop
County ends up with a predicted increase in expenditures after consolidation. Bastrop County
provides an example where decreased competition more than offsets economies of scale
Table 7 summarizes the results of our simulation. Before consolidation, Texas’s total cost
is $34.75 billion, while total expenditures are $38.27 billion because inefficiency has
expenditures $3.52 billion above costs. After consolidation Texas’s total cost is estimated to be
$34.32 billion, a decline in costs of $0.43 billion due to net savings from scale economies. At
the same time the consolidation decreases competition and increases inefficiency, and postconsolidation expenditures due to inefficiency are $5.48 billion, an increase of $1.96 billion.
Total expenditures after consolidation are then $39.80 billion, a net increase of $1.53 billion in
expenditures due to the consolidation.
If we conduct the above simulation in the No HHI model from table 2 we find significant
cost savings from consolidation. In that exercise expenditures only increase in six large counties
-- Travis, El Paso, Harris, Tarrant, Dallas, and Bexar. For these six counties, consolidation
creates mega-districts and there are diseconomies of scale. Statewide, the reduction in
expenditures after consolidation is about $376.3 million. In other words, failing to consider the
competitive effect leads to a substantial overestimate of the benefits of consolidation
8. Conclusions
We use a stochastic cost frontier to provide evidence that competitiveness within the
public school market is linked with efficiency. Our model suggests that many districts operate
where there are economies of scale, and a natural suggestion in this case is consolidation. We
also find, however, that consolidation leads to less competition and increased inefficiency.
22
More specifically, we find that the district cost function exhibits significant economies of
size, with per pupil costs falling with district size up to 47,000 students. This range of declining
per pupil costs is much greater than that found in previous studies. We also find that a less
competitive market structure is associated with greater cost inefficiency and thus with higher
expenditures per pupil, ceteris paribus. For a district large enough to be considered scale efficient
in the literature, we predict that expenditures would be 16 percent higher under monopoly than
under perfect competition, all other things being equal.
Policy makers face a tradeoff between scale and efficiency when considering district size.
We provide a simulation exercise of a consolidation of Texas districts to the county level. Our
simulation results suggest that including the effect of changes in the competitive structure of the
market may be quite important—the difference in the predicted change in aggregate expenditures
from an analysis that ignores the competitive structure effect and an analysis that considers the
competitive structure effect is almost two billion dollars. Ignoring the potential efficiency
implications could lead to a seriously misleading assessment of the benefits of any consolidation
or deconsolidation proposal.
Finally, while our results are about the importance of considering market conditions and
the impact of market conditions on efficiency when looking at public schools and possible public
school consolidation, the general idea is more widely applicable. Researchers have found
inefficiency based on production, cost, or profit functions in many areas including banking,
health care, agriculture, and manufacturing. Banking mergers or acquisitions, for instance, may
allow firms to take advantage of economies of scale or scope, but also will lead to changes in
market conditions that may lead to changes in firm efficiency. Any such changes in efficiency
can alter the net benefits of mergers or acquisitions, and hence the possible impact of this
phenomenon is an important consideration.
23
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27
Figure 1: The Distribution of School District Competition (HHI) in Texas
Note: The HHI (Herfindahl-Hirschman Indices) reported here are based on student enrollment in
all public and private schools.
28
Figure 2: The Distribution of Cost Efficiency among Texas School Districts
Note: Cost efficiency estimated by the Battese and Coelli formula. The mean estimated value is
0.895, with a minimum of 0.540 and maximum of 0.985.
29
Figure 3: The Estimated Iso-Expenditure Curves
30
Table 1: Descriptive Statistics
Variable
Mean
Standard
Deviation
Minimum
Maximum
Operational expenditure per pupil
9,354
2,337
5,706
26,554
Enrollment
4,913
13,550
47
203,294
Change in TAKS passing rate
-0.002
0.037
-0.175
0.273
Percent taking advanced courses
0.245
0.094
0
0.856
Percent low income students
0.581
0.181
0
1
Percent limited English proficiency
0.081
0.093
0
0.655
Percent Special Education students
0.097
0.025
0.03
0.219
Percent high school students
0.286
0.04
0.053
0.519
Distance from major metro areas
107.5
81.4
5
365.3
Hedonic predicted salary index
1.449
0.115
1.242
1.714
HHI
0.325
0.242
0.053
1
Number of observations = 967
31
Table 2: Mean Marginal Effects from Main Specifications
Dependent variable:
log(Expenditure per pupil)
Enrollment (log)
Change in TAKS passing rate
Percent taking advanced
courses
Percent low income students
Percent limited English
proficiency
Percent Special Education
students
Percent high school students
Distance from major metro
areas (log)
Hedonic predicted salary index
(log)
Dependent variable: log(𝜎𝑢2 )
No HHI
Linear HHI
-0.098 ***
(0.005)
0.375 ***
(0.142)
0.115 **
(0.048)
0.175 ***
(0.030)
0.085
(0.079)
0.623 ***
(0.180)
0.206
(0.131)
0.049 ***
(0.007)
0.484 ***
(0.077)
-0.090 ***
(0.005)
0.392 ***
(0.137)
0.096 **
(0.046)
0.184 ***
(0.028)
0.052
(0.077)
0.693 ***
(0.172)
0.245 *
(0.126)
0.036 ***
(0.007)
0.460 ***
(0.075)
Quadratic
HHI
-0.090 ***
(0.005)
0.399 ***
(0.137)
0.096 **
(0.046)
0.184 ***
(0.028)
0.050
(0.077)
0.683 ***
(0.172)
0.241 *
(0.126)
0.035 ***
(0.007)
0.477 ***
(0.077)
-4.094 ***
(0.134)
2.127 ***
(0.271)
-4.644 ***
(0.146)
2.449 ***
(0.366)
-4.789 ***
(0.186)
HHI
Constant
Dependent variable: log(𝜎𝑣2 )
31.660 ***
28.012 ***
27.385 ***
(2.751)
(2.885)
(2.947)
-9.880 ***
-9.568 ***
-9.482 ***
Constant
(0.463)
(0.476)
(0.484)
Observations
967
967
967
Log Likelihood
761.8
792.1
793.0
Notes: Standard errors are in parentheses. Asterisks indicate significance at the
1% (***), 5% (**) and 10% (*) levels. The coefficient estimate of HHI squared
variable in the Quadratic HHI column is 1.062 and its standard error is 0.811. A
likelihood ratio test would reject adding the HHI squared variable in the
specification.
1 / log(Enrollment)
32
Table 3: Coefficient Estimates for the Enrollment Variable
and Its Interaction Terms in the Linear HHI Model
-0.395 ***
(0.138)
0.048 ***
× Enrollment (log)
(0.016)
-0.110
× Change in TAKS passing rate
(0.125)
-0.020
× Percent taking advanced courses
(0.037)
0.070 ***
× Percent low income students
(0.024)
-0.025
× Percent limited English proficiency
(0.050)
0.136
× Percent Special Education students
(0.152)
-0.264 **
× Percent high school students
(0.118)
-0.021 ***
× Distance from major metro areas (log)
(0.005)
-0.155 **
× Hedonic predicted salary index (log)
(0.070)
-0.001 **
Enrollment (log), cubed
(0.001)
Notes: Standard errors are in parentheses. Asterisks indicate
significance at the 1% (***), 5% (**) and 10% (*) levels.
Enrollment (log)
33
Table 4: Identification and Instrument Relevance Statistics from the First Stage
Regression of the Pseudo-IV Estimation
Shea’s
F Statistic p-value of
Variable
Partial R2
2
(57, 873) F Statistic
Partial R
Change in TAKS passing rate
0.0750
0.1450
1.65
0.002
× Enrollment
0.0570
0.1108
1.64
0.002
× Change in TAKS passing rate
0.0846
0.2085
1.06
0.354
× Percent taking advanced courses
0.0538
0.1284
1.46
0.017
× Percent low income students
0.0977
0.1556
1.55
0.007
× Percent limited English proficiency
0.0733
0.1300
1.03
0.424
× Percent Special Education students
0.0593
0.1937
1.52
0.009
× Percent high school students
0.0682
0.1496
1.62
0.003
× Distance from major metro. areas
0.0544
0.1567
1.60
0.004
× Hedonic predicted salary
0.0553
0.1296
1.63
0.003
Percent taking advanced courses
0.0766
0.1204
3.31
0.000
× Enrollment
0.0767
0.1391
3.47
0.000
× Percent taking advanced courses
0.0562
0.1417
3.67
0.000
× Percent low income students
0.0848
0.0918
1.91
0.000
× Percent limited English proficiency
0.0682
0.1416
1.33
0.057
× Percent Special Education students
0.0646
0.1109
3.50
0.000
× Percent high school students
0.0955
0.1208
3.32
0.000
× Distance from major metro. areas
0.0792
0.1106
2.93
0.000
× Hedonic predicted salary
0.0647
0.1404
3.53
0.000
H
0.1711
0.2829
5.53
0.000
Number of Observations = 967
Number of Regressors = 57
Number of Instruments = 94
Number of Excluded Instruments = 57
Note: First stage test statistics are heteroskedasticity-robust.
34
Table 5: Mean Marginal Effects from Alternative Specifications
Dependent variable:
log(Expenditure per pupil)
Enrollment (log)
Change in TAKS passing rate
Percent taking advanced courses
Percent low income students
Percent limited English
proficiency
Percent Special Education
students
Percent high school students
Distance from major metro areas
(log)
Hedonic predicted salary index
(log)
Dependent variable: log(𝜎𝑢2 )
HHI
Pseudo-IV
Houston &
Dallas
Excluded
-0.084 ***
(0.008)
2.170 ***
(0.745)
0.376 **
(0.184)
0.173 ***
(0.037)
0.049
(0.081)
0.737 ***
(0.185)
0.157
(0.138)
0.029 ***
(0.008)
0.468 ***
(0.083)
-0.090 ***
(0.005)
0.391 ***
(0.136)
0.095 **
(0.046)
0.185 ***
(0.028)
0.057
(0.077)
0.694 ***
(0.172)
0.257 **
(0.127)
0.035 ***
(0.007)
0.459 ***
(0.075)
Low
Density
Districts
Excluded
-0.086 ***
(0.005)
0.392 ***
(0.144)
0.083 *
(0.046)
0.199 ***
(0.028)
0.029
(0.076)
0.709 ***
(0.175)
0.173
(0.129)
0.029 ***
(0.007)
0.392 ***
(0.076)
3.208 ***
(0.383)
2.112 ***
(0.270)
1.741 ***
(0.297)
CR4
Constant
-4.993 ***
(0.175)
-4.627 ***
(0.145)
-4.593 ***
(0.148)
CR4
-0.090 ***
(0.005)
0.424 ***
(0.139)
0.099 **
(0.047)
0.179 ***
(0.029)
0.059
(0.078)
0.641 ***
(0.175)
0.242 *
(0.127)
0.044 ***
(0.007)
0.580 ***
(0.079)
2.146 ***
(0.321)
-5.583 ***
(0.280)
Dependent variable: log(𝜎𝑣2 )
29.061 *** 28.151 *** 27.822 *** 26.967 ***
(3.489)
(2.907)
(3.277)
(3.072)
-9.947 ***
-9.603 ***
-9.523 ***
-9.418 ***
Constant
(0.577)
(0.479)
(0.526)
(0.502)
Observations
967
965
918
967
Log Likelihood
839.4
789.1
793.4
783.2
Notes: Standard errors are in parentheses. Asterisks indicate significance at the 1% (***), 5%
(**) and 10% (*) levels.
1 / log(Enrollment)
35
Table 6: Descriptive Statistics Before and After Consolidation
Mean
Standard
Deviation
Minimum
Maximum
Enrollment
4,696
13,276
20
203,294
Change in TAKS passing rate
-0.002
0.041
-0.257
0.273
Percent taking advanced courses
0.245
0.092
0
0.856
Percent low income students
0.578
0.185
0
1
Percent limited English proficiency
0.08
0.094
0
0.655
Percent Special Education students
0.096
0.026
0.00
0.219
Percent high school students
0.273
0.071
0
0.519
Distance from major metro areas
107.7
81.3
5
365.3
Hedonic predicted salary index
1.446
0.114
1.242
1.714
HHI
0.327
0.24
0.053
1
Enrollment
18,877
67,283
80
784,992
Change in TAKS passing rate
-0.004
0.022
-0.074
0.103
Percent taking advanced courses
0.249
0.065
0.034
0.5
Percent low income students
0.599
0.129
0.224
0.944
Percent limited English proficiency
0.085
0.08
0
0.604
Percent Special Education students
0.096
0.019
0.04
0.157
Percent high school students
0.28
0.03
0
0.413
Distance from major metro areas
132.7
85.4
8.9
363.7
Hedonic predicted salary index
1.433
0.103
1.259
1.714
HHI
0.844
0.232
0.292
1
Before Consolidation, n=1,013
After Consolidation, n=252
36
Table 7: The Results of the Consolidation Simulation
Before Consolidation
After Consolidation
State’s predicted total expenditure
$38,269,743,104
$39,802,163,200
State’s predicted total cost
$34,748,907,520
$34,318,032,896
State’s predicted total inefficiency
$3,520,835,584
$5,484,130,304
𝑢̂ at average H
0.111
0.192
Mean of the distribution of 𝑢̂
0.115
0.197
37
Appendix A (not intended for publication)
In order to analyze the marginal effect of each variable that explains the variance of cost
inefficiency on the cost inefficiency, we take the derivative of 𝑢̂𝑗𝑒 with respect to that 𝑥𝑗 variable.
In this appendix, we presents how we calculate 𝜕𝑢̂𝑗𝑒 ⁄𝜕𝑥𝑗 . First, let’s rewrite equation (13)
𝑢̂𝑗𝑒 = E(𝑢𝑖 | 𝜀𝑖 ) = 𝜇∗𝑖 + 𝜎∗ {
𝜙(−𝜇∗𝑖 ⁄𝜎∗ )
}
1 − Φ(− 𝜇∗𝑖 ⁄𝜎∗ )
(A1)
where 𝜇∗𝑖 = (𝜀𝑖 𝜎𝑢2 )⁄(𝜎𝑢2 + 𝜎𝑣2 ) , 𝜎∗ = 𝜎𝑢 𝜎𝑣 /√𝜎𝑢2 + 𝜎𝑣2 , 𝜎𝑢2 = exp(𝛿0 + 𝛿 ∙ 𝑥𝑖′ ), 𝜙(∙) is the
standard normal density function, and Φ(∙) is the cumulative distribution function of the standard
normal distribution. Now, let Γ equal to the derivative of 𝜇∗𝑖 with respect to 𝑥𝑗 , that is
Γ=
𝜕𝜇∗𝑖 𝛿𝑗 ∙ 𝜀𝑖 ∙ 𝜎𝑢2 ∙ 𝜎𝑣2
=
(𝜎𝑢2 + 𝜎𝑣2 )2
𝜕𝑥𝑗
(A2)
where 𝛿𝑗 is the coefficient of 𝑥𝑗 . Moreover, let Λ equal to the derivative of 𝜎∗ with respect to 𝑥𝑗 ,
that is
𝛿𝑗 ∙ 𝜎𝑢 ∙ 𝜎𝑣3
𝜕𝜎∗
Λ=
=
𝜕𝑥𝑗 2 ∙ (𝜎𝑢2 + 𝜎𝑣2 )3/2
(A3)
Finally, let Ω equal to the derivative of 𝜙(−𝜇∗𝑖 ⁄𝜎∗ )⁄[1 − Φ(− 𝜇∗𝑖 ⁄𝜎∗ )], that is
𝜙(−𝜇∗𝑖 ⁄𝜎∗ )
}
1 − Φ(− 𝜇∗𝑖 ⁄𝜎∗ )
Ω=
𝜕𝑥𝑗
𝜙(−𝜇∗𝑖 ⁄𝜎∗ ) ∙ (−Γ 𝜎∗ + Λ 𝜇∗𝑖 ) ∙ [𝜇∗𝑖 − 𝜇∗𝑖 ∙ Φ(− 𝜇∗𝑖 ⁄𝜎∗ ) + 𝜎∗ ∙ 𝜙(−𝜇∗𝑖 ⁄𝜎∗ )]
=
𝜎∗3 ∙ (1 − Φ(− 𝜇∗𝑖 ⁄𝜎∗ ))2
𝜕{
(A4)
So we write the derivative of 𝑢̂𝑗𝑒 with respect to that 𝑥𝑗 as
𝜕𝑢̂𝑗𝑒
𝜙(−𝜇∗𝑖 ⁄𝜎∗ )
=Γ+Λ∙{
} + 𝜎∗ ∙ Ω
𝜕𝑥𝑗
1 − Φ(− 𝜇∗𝑖 ⁄𝜎∗ )
(A5)
38