90 H. Bonesrønning Economics of Education Review 18 1999 89–105
model for a sample of Norwegian upper secondary school students, and by separating between income and
substitution effects, we find that hard grading 5 the income effect has positive effects on student achieve-
ment. There are several reasons why these results should not be regarded as definitive.
First, an assumption applied by Correa and Gruver — and by us in the empirical work reported above — is
that students passively accept the teachers’ grading. This assumption is restrictive. It seems more? likely that the
students recognize that they can improve perceived achievement by allocating time to influence teacher grad-
ing practices. We augment the Correa and Gruver’s model with rent seeking students, and show theoretically
that when students put pressure on teachers for easy grading, the relationship between grading and student
achievement will be mediated through the students’ time constraints.
The crucial question then is which of the two models outlined above we should repose most confidence in.
This is an empirical question: We find that teachers’ grading is systematically associated with teacher charac-
teristics and discuss whether this finding is consistent with the Correa and Gruver’s model, the alternative
model, or both. It turns out that none of the models can be excluded on the basis of this discussion. Thereafter,
we investigate which theory provides a superior expla- nation in a formal statistical sense. We find no support
for the rent seeking hypothesis, which imply that the conceptualization of grading as a teacher’s instrument
seems appropriate.
Second, it turns out that the effects of teachers’ grad- ing cannot be empirically revealed unless relevant mea-
sures of teacher heterogeneity are included. Montmar- quette and Mahseredjian 1989 present an illustrative
discussion of the problems of separating between teach- ing quality and grading effects when teacher character-
istics correlate poorly with student achievement. Now, teacher heterogeneity represents the achilles heel of edu-
cation production function studies. It is well known that student achievement adjusted for student traits, varies a
lot across teachers, but that purchased teacher character- istics education, experience cannot explain much of the
variation in student achievement. To deal with this prob- lem this paper utilizes the teachers’ own exam results
from the universities as an additional measure of teacher heterogeneity. Fortunately, the teachers’ own exam
results turns out to be an important determinant of teacher effectiveness.
Third, student self-selection might obfuscate the true relationships. After one year, the students in the upper
secondary school are sorted into mathematics, social studies or foreign language by interaction between a
selection criterion which is the probability of getting admission to the university and an underlying distri-
bution of abilities and preferences. An example is that students who respond to hard grading by increasing their
studying time, may find that their probabilities of getting admission to the university increases when the grading
gets harder. Students with preferences for hard grading may consequently sort themselves into the subjects with
the hardest grading. We investigate whether the teachers’ grading practices affect the students’ specialization
choices, and subsequently, we make the appropriate con- trols for selectivity bias while estimating the education
production functions.
The paper is organized as follows. The model is laid out in Section 2. Section 3 provides empirical specifi-
cations. In Section 4 data and results are presented, while concluding remarks are offered in Section 5.
2. The Model
Our basic framework is the teacher-student interaction model presented by Correa and Gruver 1987.
2
This model is within the education production function tra-
dition insofar it has an achievement production function embedded, but it is a much richer approach including
behavior assumptions regarding the two central actors; teachers and students. The major issue is that teachers
and students react to each other’s working strategies. The model highlights the problem that policies attempting to
increase student achievement by increasing the allocation of teacher time to education may have little or even
negative effect on achievement due to reallocation of time on the part of the student. Grading is introduced as
an instrument by which the teacher can reduce input sub- stitutions.
We make two modifications of the basic framework. First, Correa and Gruver apply the assumption that stu-
dents care about perceived achievement, which is defined as the product of real achievement and teacher grading.
Teacher grading then is like a wage increase in a labor supply model. It produces counteracting income and sub-
stitution effects, which are troubling in empirical work. We therefore introduce an alternative teacher grading
model where the income and substitution effects are for- mally separated.
3
Second, Correa and Gruver treat teacher grading as a policy variable by which the teacher can manipulate
students’ studying time. This assumption may be too restrictive. Bishop 1989 has focused on how students
establish peer pressure against studying hard when there
2
The first contribution in the literature discussing incentives on students which we are aware of, is Brown and Saks 1980,
who introduce a model incorporating many of the same elements as Correa and Gruver.
3
The separation between substitution and income effects is suggested by an anonymous referee.
91 H. Bonesrønning Economics of Education Review 18 1999 89–105
is a lack of external standards for judging academic achievement. According to Bishop, this is part of the
explanation for the apathy experienced in U.S. high schools. The institutional setting of the Norwegian upper
secondary school is somewhat different. Admission to the university is decided on the basis of a mix of real
and perceived achievement, which implies that students have to care about both components of achievement. In
this setting it seems likely that students put pressure on teachers for easy grading, i.e. grades set above the real
achievement level. Inspired by Bishop, we therefore aug- ment the Correa and Gruver’s model with rent seeking
students. Student rent seeking may take many forms; we specify it as a time consuming activity. Examples are
that students use some of their time in the classroom being non-cooperative if they receive too hard a grading.
Successful rent seeking activities take grading away from the level preferred by the teacher. The grading level pre-
ferred by the teacher is assumed to be equal to the grad- ing level preferred by the school principal. The rationale
for this assumption is that a principal facing many teach- ers
prefers consistent
grading practices
across classrooms, due to both equity and efficiency consider-
ations. To achieve this goal the principal introduces non-pecuniary costs for the teachers who set grades
that diverge from the principal’s preferred grading.
The Correa and Gruver’s model augmented with a modified relationship between perceived and real
achievement and with rent seeking students is given by the following equations:
v 5 v e,a, v
e
9 0, v
a
9 0,v
e
0 , 0,v
a
0 , 1
u
s
5 u
s
w
s
, o
s
, u
s ws
9 0, u
s os
9 2
w
s
5 G
3
1 g
s
v 2 3 3
G
3
5 G r 2 aGr 2 G
T
, G
r
9 0, G
rr
4 ,
0, G0 5 G
T
, a 0 e 1 o
s
1 r 5 k
s
5 u
T
5 u
T
g
T
v ,o
T
6 a 1 o
T
5 k
T
7 Eq. 1 is an achievement production function where
student achievement v is assumed to be a concave func- tion of a student’s studying time e and a teacher’s teach-
ing time a. Different educational methods and qualitative characteristics of the teacher and the student can be rep-
resented by different functional forms or by different values of the parameters. By teaching quality we under-
stand the stock of teacher characteristics innate abilities, education,
experience, the
stock of
teaching technologies that is positively associated with student
achievement, net of teacher teaching time and net of the teacher’s grading practices.
Eq. 2 says that student utility u
s
is an increasing function of student time to other activities o
s
and per- ceived achievement w
s
. The student derives utility from perceived achievement because it is one of the two pri-
mary determinants of admission to a university and to a field of study. The marginal rate of substitution between
w
s
and o
s
reflects the achievement orientiation of the student.
Eq. 3 portrays the teachers’ grading policy in the Norwegian upper secondary school. In Norway the tea-
cher’s grades as well as the grades from the nationally administred exams are given on a 0–6 scale, where 0 and
1 represent failure and 6 is the best result. In Eq. 3 G
3
is the teacher’s grade given a student who gets 3 on the external exam. G
3
describes the teacher as an easy grader G
3
3 or an hard grader G
3
, 3. g
s
describes the degree to which the teacher’s grades vary with real
achievement as measured by the external exam. Eq. 4 says that G
3
is a weighted sum of the student rent seeking function Gr, and the grading level pre-
ferred by the school principal G
T
. The rent seeking func- tion is a concave function of student rent seeking time
r . The exact functional form of Gr is determined by
teacher and student characteristics. We assume that the teacher sets grades equal to the grades preferred by the
school principal, G
T
, when students allocate no time to rent seeking. a reflects the cost to the teacher of deviat-
ing from the principal’s preferred grading. a 5 0 implies that there are no such costs, while a 5 1 implies that
these costs are prohibitive large. The teacher’s grading practice is thus the result of student pressure for easy
grading, the teacher’s ability to withstand pressure for easy grading, and the teacher’s costs of deviating from
the principal’s preferred grading practices. By teacher strength
we understand the teacher characteristics that shift the rent seeking function.
The assumption underlying Eq. 4 is that students influence the income effect G
3
, but not the substitution effect g
s
. This restrictive model is rich enough to illus- trate the new mechanisms that are introduced by rent
seeking students. In the empirical part of the paper we also investigate whether g
s
is a function of student rent seeking.
Eq. 5 is the student’s time constraint where k
s
is total available time, and the other variables are defined above.
Eqs. 6 and 7 are the teacher’s utility function and time constraint respectively. g
T
is the evaluation of the teacher made by the school principal, and should be
interpreted as the school principal’s instrument to affect the teacher’s teaching time. o
T
is the teacher’s time to other activities. The marginal rate of substitution
between g
T
v and o
T
reflects the academic orientation of the teacher.
Our concern is the relationship between the teacher’s grading and student achievement. To see how the model
portrays this relationship it might be instructive to first
92 H. Bonesrønning Economics of Education Review 18 1999 89–105
consider the case without rent seeking students. That is, the Correa and Gruver’s assumption of exogenous grad-
ing instruments applies. In this case, and under the assumption that the teacher
has set hisher teaching time equal to a 5 a9, the rational utility maximizing student chooses leisure and studying
time according to Eq. 8. u
os
9 u
ws
9 5 g
s
v
e
9 8
Eq. 8 reads that the marginal increase in perceived achievement with respect to studying time is equal to the
ratio of marginal utilities. If the marginal returns to the external exam grades
rises g
s
↑ , and G
3
is adjusted to keep the utility level unchanged, the student’s response will be to allocate
more time to studying and less time to other activities the substitution effect. If the teacher shifts from hard
to easy grading G
3
↑ , the student’s response will be to
allocate less time to studying and more time to other activities in the case when leisure is a normal good, and
more time to studying and less time to other activities in the case when leisure is an inferior good. Unfortunately,
students’ attitude towards leisure may be an important dimension of student heterogeneity. The restrictive
model then does not provide clearcut hypotheses about the relationship between G
3
and student achievement. Next, assuming rent seeking students, the optimality
conditions are: u
os
9 u
ws
5 g
s
v
e
9 5 G9 r1 2 a
9 The equality to the right says that in optimum an extra
hour should yield the same increment in perceived achievement no matter if it is spent on studying or
rent seeking. If the teacher rises the marginal returns to the external
exam grades g
s
↑ , the student’s response will be to use
more of hisher time for studying relative to rent seeking. Thus we expect as above that g
s
is positively correlated with real student achievement as measured by the exter-
nal grades. If the teacher’s costs of deviating from the principal’s preferred grading increases a↑, the marginal
returns to rent seeking decrease and the student will real- locate hisher time towards studying time and time to
other activities. It turns out that students with “high” elasticities of substitution between achievement and time
to other activities in their utility functions respond to an increase in a by allocating more time to studying. An
increase in a is analogous to increased teacher strength. Appendix B provides a formal mathematical analysis for
the special case of CES utility and achievement pro- duction functions.
One additional observation from the model is that low achieving students face the largest opportunities for rent
seeking. To see this, note that neither w
s
nor v can be larger than 6. Assuming for simplicity that g
s
5 1, we
have from Eq. 3 w
s
5 G
3
1 v 2 3 6
10 A high achieving student experiences v-values close
to 6, while a low achieving student experiences v-values far below 6. Eq. 10 then implies that the opportunity
set for G
3
is much larger for low achievers than for high achievers.
This does not necessarily imply that low achieving students use more time for rent seeking than high achiev-
ing students. The crucial question of course is whether the low achieving students also face the strongest incen-
tives for rent seeking. The answer is that they might do, if they face smaller marginal returns to studying time
than high achieving students.
So far we have focused on the grading issues. Some comments on the other elements of student-teacher inter-
action are warrented. From the first order conditions of the student’s maxim-
ation problem we can define the student’s reaction func- tions
e 5 f
a9,g
s
,G
T
,b
s
,b
v
,b
g
11 r
5 g a9,g
s
,G
T
,b
s
,b
v
,b
g
12 where a9 is the teaching time set by the teacher, g
s
is the degree to which a teacher’s grades vary with achieve-
ment, G
T
is the grading practices preferred by the school principal and b
i
, i 5 s,v,g, represent the characteristics of the student’s utility function, the achievement production
function and the rent seeking function respectively. Some of the comparative static issues are discussed in
Appendix B.
From the first order conditions of the teacher’s maxim- ation problem we can define the teacher’s reaction func-
tion a
5 h e9,g
T
,b
v
,b
uT
13 where b
uT
represents the characteristics of the teacher’s utility function and the other variables are explained
above. It is clear that the teaching time assumed by the stud-
ent a9 may differ from the equilibrium teaching time a
, and no adjustment mechanisms are modelled. Cor- rea and Gruver pay much attention to the game theoreti-
cal equilibrium of teacher-student interaction. We realize that the problems of statistically analysing the model
becomes much more complex when the game theoretical equilibrium is considered, but do not elaborate upon
these issues here.
We have discussed how grading may affect studying time once a subject is chosen. We close this section by
making some comments about how the grading practices might affect the students’ subject choices. After one year
93 H. Bonesrønning Economics of Education Review 18 1999 89–105
the students in the Norwegian upper secondary school have to choose which of the specialization subjects —
mathematics, social studies or foreign language — they will study for the last two years in the upper secondary
school. It seems likely that the demand for specialization subjects is determined by the expected pecuniary and
non-pecuniary returns in interaction with abilities, tastes and some other factors. The teachers’ grading practices
may affect the expected returns, for instance by affecting the probabilities of getting admission to the universities.
When the teachers practice hard grading, some students may find that their probability of getting admission to
higher education decreases, while other students may find that their admission probability increases. That is,
the students may sort themselves across subjects depen- dent on their own abilities and preferences. An example
is that the students who choose mathematics given hard grading, may be those students who respond to hard
grading by allocating more time to studying. Thus, even if there exists no relationship between grading and stud-
ent achievement for all students, such relationships may be revealed for mathematics due to sorting.
3. Empirical Specifications
