Intra-school competition

Applied Economics, 2006, 38, 1641–1647

Intra-school competition and student achievement
Melvin V. Borland, Roy M. Howsen* and Michelle W. Trawick
Department of Economics, Western Kentucky University, Bowling Green,
KY 42101, USA

Within the economics of education literature, numerous studies have investigated the relationship between educational market competition and educational achievement. Educational market competition has been defined as either the availability of vouchers within a community or the number of schools or school districts within the relevant market structure.
While these studies have shown that increases in inter-district competition result in increased student achievement, no studies, to our knowledge, have yet investigated the effect of intra-school competition on student achievement. Within this study, a measure of intra-school competition is developed and the findings indicate that increased intra-school competition leads to increased student achievement.

I. Introduction
Within the economics of education literature numerous articles exist, such as those by Borland and
Howsen (1992), Epple and Romano (1998), Hoxby
(2000b), Marlow (2001), and Hanushek and Rivkin
(2003), concerned with the relationship between the degree of educational market competition and various educational input and output measures,
i.e., student achievement, school-student composition, school curriculum, school employment, and teacher quality. In these studies, educational market competition is typically defined in terms of the availability of vouchers within a community, or the number of schools or school districts within the relevant educational market area. The implication is that increases in such availability and number provide greater choice for parents of students and increased competition across schools. While these studies have shown that increases in inter-district competition result in increased student achievement, no studies, to our knowledge, have yet investigated the effect of intra-school competition on student achievement.

In a general reference to school effects, i.e., teacher quality, class size, and expenditure per pupil,
Hanushek (1986) indicates that the evidence on the influence of such effects on student achievement is ambiguous and controversial. Further, he suggests that school effects are, for the most part, statistically insignificant in terms of affecting student achievement or, at the least, confounding. Hanushek argues that schools do matter primarily in terms of teacher behaviour – behaviour, however, that has defied detailed description. Thus, the question exists of whether or not it is possible to construct a set of variables that accounts for differences in teacher behaviour. In a later study, Hanushek et al. (2002) find that teacher behaviour resulting from teacher experience is an important determinant of student achievement. He finds that fourth and fifth grade students who have teachers with two years of experience or less do not perform as well academically as students who have teachers with more than two years of experience.
The study reported in this paper adds to the body of knowledge concerned with the relationship

*Corresponding author. E-mail: [e-mail]
Applied Economics ISSN 0003–6846 print/ISSN 1466–4283 online ß 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00036840500426991

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1642 between teacher behaviour and student achievement by including a measure of the degree of intra-school teacher competition as an attribute of teacher behaviour. To the extent that teacher behaviour matters in student achievement, the inclusion of this measure is expected to provide additional explanation of variations in student achievement.

II. Intra-school Competition
If Hanushek is correct in arguing that teacher behaviour matters, then measures of variables to capture such behaviour would assist educational officials in their attempt to increase student achievement. In addition to previously considered teacher associated variables, i.e. teacher salary, teacher rank, and teacher experience, a variable that could provide even further explanation, at least, for part of the teacher effects on student achievement is the degree of teacher competition faced within a school. Teachers who are assigned a particular class within a school and who face increased competition with a greater number of alternative classes of the same grade, i.e. greater intra-school teacher competition, are expected to behave differently in terms of the combination of variables that they would choose in their objective function than teachers who face less competition within their school. The hypothesis relevant to this study is that increased intra-school competition causes teachers to behave in a manner that is more consistent with the objective of maximizing student academic achievement.
One aspect of increased intra-school teacher competition is that teachers are pressed to employ inputs that are more efficient in the production of a given level of output. For example, teachers would be expected to select that combination of inputs, i.e. lecture, homework, computer usage, that he/she believes would maximize student achievement. This behaviour results because teachers have concerns about their class performances relative to other classes both from principal and parental points of view.
An alternative explanation for the intra-school teacher competition effect on student achievement could be couched in terms of cooperation, even if unintended, among teachers within the same school, also known as teacher peer effects. This peer effect may take the form of interaction and learning from others on such matters as teaching technique, student motivation, and the handling of disciplinary problems. This potential increased interaction and learning among teachers may be expected to result in

M. V. Borland et al. increased student achievement. In schools where fewer classes of third graders exist, for instance, this interaction and learning among teachers would be lessened. Thus, as the degree of intra-school teacher competition increases, resulting from the increase in the number of classes within a school, an increase in student achievement may be expected because of the increased potential interaction and learning among a larger number of teachers.
The measure of intra-school teacher competition that is incorporated in this paper is the intra-school
Herfindahl index. Specifically, let csij denote third grade enrollment in a particular class i within school j and t3sj the total third grade enrollment in school j.
Let sij be the enrollment share of class i within school j, given by sij ¼ csij/t3sj. Squaring sij and then summing this value across a particular school generates the intra-school competition variable, i.e. the intra-school Herfindahl index, for school j and is given by istcj. If only one class exists within a school, then the intra-school Herfindahl index is equal to 1.
As the degree of competition increases within the school, the intra-school Herfindahl index approaches
0. Thus, the larger the number of classes within a school, ceteris paribus, the lower the intra-school
Herfindahl index. Employing the intra-school
Herfindahl index as a measure of intra-school teacher competition provides an opportunity to advance the body of knowledge with respect to factors that account for differences in teacher behaviour – behaviour that matters in terms of affecting student achievement. Data for this study originated with the Kentucky
Department of Education, KDE. The KDE regularly collects two separate data sets, one, that provides teacher data concerned with variables such as name of school employed, grade taught, rank, experience, and salary, and another, that provides individual student data on such variables as name of school attended, grade level, teacher, gender, race, age, innate ability, and educational achievement by academic subject. The combination of these two data sets, i.e. a merger of these two data sets for all third grade classes throughout the state of Kentucky for the academic year ending in 1990, is the basis of the data employed in this study. The resulting merged data set contains slightly over 30 000 individual student observations.

III. Model of Student Achievement
The statistical model employed in this study is consistent with a model used by Borland, Howsen

Intra-school competition and student achievement

1643

Table 1. Variable definitions
Student achievement (sa)
Intra-school competition (istc)
Class size (cs)
Class size squared (cs2)
Teacher salary (ts)
Inter-district competition (h)
Innate ability (ia)
Teacher rank (trk)
Teacher experience (te)
Variation in ability (sdia)
Union status (u)
Attendance rate (a)
College (c)
Total in school (t3s)
Total in district (t3d)
Mean of ability (miac)
Student gender (sg)
Percentage black (pb)
Percentage Catholic (pc)
Population (pop)
Income (y)
Tax effort (tax)
Administration cost (adm)
Teacher race (tr)

An individual 3rd grade student’s normal curve equivalent over all score for maths, reading, and spelling.
Sum of squares of class size shares within a particular school.
Actual number of students for which test scores exist in a particular class at the end of the associated academic year
The square of class size
The salary of the teacher for a particular class in dollars
Sum of squares of third grade enrollment shares of districts or independent schools within a county
Cognitive skills index for each student within a class. The test of cognitive skills is an assessment of academic aptitude that includes verbal, nonverbal, and memory skills.
Level of education for each teacher within a particular class
The number of years of teaching for a teacher of a particular class
Standard deviation of innate ability within a class
Dummy variable equal to one if the union bargains for the local teachers, and equal to zero otherwise
Aggregate day’s attendance divided by the aggregate day’s membership
Percentage of students within a school district who go on to college
Number of third grade students within a school
Number of third grade students within a school district
Mean innate ability by class
Dummy variable equal one if the student is male, and equal to zero otherwise
Percentage of the county population that is black
Percentage of the county population that is Catholic
Number of individuals within a county
Median family income within a county
Local revenue per child in daily attendance divided by the assessed property value per child in daily attendance
Administration cost divided by total expenditures within the school district
Dummy variable equal one if the teacher is white, and equal to zero otherwise

and Trawick (2005). Support for this specification can be found in Hanushek et al. (2002), Hoxby
(2000a), and Borland and Howsen (2000). The statistical model used in this paper is specifically indicated below: sa ¼ fðistc, cs, cs2, ts, h, ia, trk, te, sdia, u, a, c, t3s, t3dÞ ð1Þ cs ¼ gðia, sa, ts, t3s, t3d, sdia, miac, sgÞ

ð2Þ

h ¼ kðsa, pb, pc, pop, c, y, csÞ

ð3Þ

ts ¼ iðsa, h, te, cs, tax, u, adm, y, trÞ

ð4Þ

within which sa is a measure of individual student achievement, istc is the measure of intra-school teacher competition (i.e. an intra-school Herfindahl index), cs is class size, cs2 is class size squared, ts is teacher salary, h is a measure of inter-district competition (i.e. an inter-district Herfindahl index), ia is individual student innate ability, trk is the teacher’s rank, te is teacher experience, sdia is the standard deviation of innate ability by class, u represents the presence or absence of a union, a

represents student attendance rate, c represents the percentage of students within the school district who go to college, t3s is the number of third grade students in a school, t3d is the number of third grade students in the school district, miac is the mean of innate ability by class, sg is a dummy variable which represents student’s gender, pb is the proportion of the population that is black, pc is the percentage of the population that is catholic, pop is the population of the community, y represents a measure of income for the community, tax represents tax effort of the community, adm represents school administrative expenditures, and tr represents teacher’s race.
Table 1 provides detailed descriptions of variables within the four-equation system. Table 2 provides corresponding mean and standard deviation values for variables in Equation 1.
The focus of this study is on the empirical results of the student academic achievement equation and, in particular, the impact of teacher effects on student achievement. Thus, a discussion of the variables will be limited to those variables in the student achievement equation, i.e.
Equation 1 of the system.

M. V. Borland et al.

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Table 2. Descriptive statistics, n ^ 30, 731
Mean
Student achievement (sa)
Intra-school competition (istc)
Class size (cs)
Teacher salary (ts)
Inter-district competition (h)
Innate ability (ia)
Teacher rank (tr)
Teacher experience (te)
Variation in ability (sdia)
Union status (u)
Attendance rate (a)
College (c)
Total in school (t3s)
Total in district (t3d)

Std. dev.

52.604
0.405
20.956
24967.790
0.720
105.361
1.993
13.476
14.124
0.083
94.937
50.409
69.634
356.613

20.263
0.249
3.154
3923.090
0.259
15.253
0.687
8.413
2.375
0.276
0.859
11.291
37.431
236.683

The sign on the coefficient for the intra-school teacher competition variable, istc, is of particular interest. If increased competition and/or interaction and learning among teachers contribute to higher student achievement, one expects the sign of the coefficient on istc to be negative. Recall that an increased value of the Herfindahl index suggests decreased competition and/or interaction and learning. Thus, a decrease in the number of classes and therefore teachers within a school implies a decrease in competition and/or interaction and learning among the school’s third grade teachers, ceteris paribus, and lower academic test scores for students.
The anticipated signs of the coefficients on the teacher salary, teacher rank, and teacher experience variables are positive. Consistent with past studies, increases in teacher salary, teacher rank, and teacher experience are expected to lead to higher student achievement. Previous studies report mixed results with respect to the effect of class size on student performance.
Rainey (2004) finds that class size has a negative effect on student performance while Marlow’s (2001) results indicate that reducing class size does not lead to higher student performance. Borland, Howsen and
Trawick’s (2005) results indicate that the coefficients for class size and class size squared should be positive and negative, respectively. As class size increases from small to large in enrolment, students within the class are expected to simultaneously experience both positive and negative effects on their level of achievement. There is the expected positive peer effect on student achievement in the sense that students learn from one another because of an increase in associated student skills as the number of students within a class increases. To the extent that students compete with one another, there is an

additional positive effect on student achievement as class size increases. As class size increases, there is, as well, an expected negative effect on student achievement in the sense that a teacher’s contribution specific to an individual student is reduced with less time spent per child. Given that these aforementioned effects stand in direct contrast to one another, the anticipated signs on the coefficients for class size and class size squared are positive and negative, respectively. Thus, the relationship between class size and achievement is expected to be not only nonlinear, but nonmonotonic as well.
The anticipated sign on the coefficient for the interdistrict competition variable, i.e. h, is negative.
Administrators of a school district who face competition from another school district within the same county are more likely to behave in a manner that maximizes student academic achievement because of the potential loss of students and corresponding financial aid than school administrators who do not face such competition and the corresponding loss of financial aid. As the Herfindahl index increases, i.e. as inter-district competition is reduced, measures of student achievement are expected to decrease.
The coefficients on the student innate ability, student attendance, and percentage of children within a school district who go on to college variables are expected to display a positive sign. Within a student achievement equation, the inclusion of a student innate ability variable in the model specification is of importance because it avoids the potential for model misspecification. For a discussion of the importance of the inclusion of an innate ability variable, see Hanushek (1996).
In an attempt to measure the influence of student peers on achievement, the standard deviation of the innate ability variable within a class in included in the student achievement equation. The influence of student peer characteristics on student achievement is uncertain. Zimmer and Toma (2000) report positive influences of peers on individual student achievement, Vandenberghe (2002) finds no systematic evidence regarding grouping policies on student performance, and Hanushek (1972, 1992) finds no peer effects on student achievement. Similarly, studies of teacher union effects on student achievement have produced mixed results, see Hoxby (1996). Thus, the teacher union variable may display a positive or negative sign with respect to its impact on student achievement. The expected signs associated with the number of third grade students within a school and the number of third grade students within a school district are ambiguous. To the extent that these two variables proxy for school size and school district size, past

Intra-school competition and student achievement studies provide conflicting results. For instance,
Caldas (1993) and Lamdin (1995) find that the effect of school size on student achievement is uncertain. Similarly, Driscoll et al. (2003) find that the district size effect on student achievement is unclear. Sawkins (2002) provides evidence that there exists a u-shaped relationship between student achievement and school size.
The model described above results in a fourequation system that satisfies the necessary and sufficient rank condition required for identification.
The endogeneity of the class size squared variable does not necessitate a separate equation within the system. See Greene (2000, pp. 675, 676). However, the class size squared term in the primary equation requires a first-stage estimation procedure using a
Taylor approximation to the reduced form equations in the system as suggested by Greene (2000, pp. 689,
690) and Kelejian (1971). Second-stage estimates were obtained in the usual way by using the predicted value from these first stage equations.

IV. Estimation Results
Table 3 reports the estimated regression coefficients for each independent variable for Equation 1, only.
The first number represents the parameter estimate

Table 3. Second state achievement equation

regression

results:

student

Estimate (Std. error)
Intercept
Intra-school competition (istc)
Class size (cs)
Class size squared (cs2)
Teacher salary (ts)
Inter-district competition (h)
Innate ability (ia)
Teacher rank (tr)
Teacher experience (te)
Variation in ability (sdia)
Union status (u)
Attendance rate (a)
College (c)
Total in school (t3s)
Total in district (t3d)

À77.8145
À1.3812a
2.36011a
À0.0506a
À0.0001
À0.7416b
1.0223a
À0.6099
0.0396b
À0.2103c
À0.6468d
0.0429
À0.0022
À0.0201c
À0.0011c

(9.5665)
(0.5017)
(0.4946)
(0.0125)
(0.0001)
(0.3562)
(0.0049)
(0.2065)
(0.0179)
(0.0324)
(0.2879)
(0.0902)
(0.0076)
(0.0034)
(0.0003)

Adjusted R2

0.5967

n

30730 a Notes: Statististically at the 1% level for 1-tailed test. b Statististically at the 5% level for 1-tailed test. c Statististically at the 1% level for 2-tailed test. d Statististically at the 5% level for 2-tailed test.

1645 and the number directly below represents the corresponding standard error.
The coefficient of the intra-school competition variable, istc, is negative and statistically significant at the 1% level for a one-tailed test. As expected, this result suggests that greater levels of competition and/ or interaction and learning among teachers in the school result in increased student achievement, even after controlling for class size. To the extent that past studies have omitted a measure of intra-school teacher competition from their model specification, omitted variable bias may exist. Thus, attributing increases in student achievement to decreases in class size in such models may overstate the effect of class size when, in fact, at least part of this increase in student achievement resulting from smaller classes should be more appropriately attributed to the competition and/or interaction and learning among an increased number of teachers within a school associated with class size reduction.
The teacher experience variable is positive and significant in terms of impacting student achievement. This finding is in agreement with the study by
Hanushek and Rivkin (2003). The remaining teacher effect variables, teacher rank and teacher salary, are insignificant in terms of influencing student achievement. Although this finding is in opposition to the
Jacques and Brorsen (2002) study, it is not surprising; see Hanushek (2002).
The class size and class size squared variables were positive and negative, respectively, and each statistically significant. The optimal class size is estimated to be 23.3601 students, a result not inconsistent with
Borland, Howsen and Trawick (forthcoming). This result suggests, of course, that efforts to decrease class size will not necessarily increase student achievements. The innate ability variable’s coefficient is positive and significant. This suggests that empirical studies of student achievement should include a measure of student cognitive skills as an explanatory variable; see
Borland and Howsen (1996). Additionally, the estimated coefficient of the standard deviation of innate ability within a class is negative and significant. This result suggests that as the variation in student ability increases, student performance decreases. One implication of this finding is that students placed in homogeneous classes with respect to innate ability tend to perform better than students placed in heterogeneous classes.
The coefficient for the inter-district Herfindahl index is negative and significant at the 5% level for a one-tail test. This finding indicates that student achievement increases whenever students attend a school within a particular district that faces increased

M. V. Borland et al.

1646 competition from other school districts. This finding is supported by Marlow (1997), Hoxby (2000a) and
Borland and Howsen (2000), who all suggest that increased market competition leads to higher student academic achievement. The coefficients on the total number of third grade students within a school and within the district are negative and significant. Thus, efforts to consolidate schools may lead to lower student performance.
The coefficient of the unionization variable is negative and significant, thereby suggesting that the existence of a collective bargaining unit within the school is associated with less student achievement.
The attendance rate at the school and the proportion of the school’s students going on to college are not statistically important predictors of student achievement in this model.

V. Conclusions
This study of student achievement contributes to the existing literature by including a teacher behaviour variable through the level of competition and/or interaction and learning among teachers resulting from having additional classes within a specific grade.
For example, suppose that a specific school has two third grade classes. If each teacher is held accountable for their students’ performance, then one would expect that each teacher would adjust behaviour to be more student achievement enhancing than if the teacher was the only teacher in the third grade.
In such an environment, teachers realize that their relative accomplishments, as measured by their students’ performances, matter. Likewise, cooperation among teachers, i.e. teacher peer effects, could play a significant role in influencing student achievement. The statistically significant and negative coefficient of the intra-school competition variable supports this view. The results regarding class size suggest that reducing class sizes may not necessarily be achievement enhancing. However, efforts by school administrators and the public to do so may inadvertently be promoting more competition and consequently increases in student achievement.
Future research studies should be undertaken in an effort to separate out the influence of establishing additional classes within a school in order to determine if these additional classes, i.e. with a lower intra-school Herfindahl index value, lead to higher student achievement because of additional competition or from increased interaction and learning from others among teachers. The data set

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