Economics Letters 118 (2013) 213–215
Contents lists available at SciVerse ScienceDirect
Economics Letters
journal homepage: www.elsevier.com/locate/ecolet
Rational students and resit exams
Peter Kooreman
Department of Economics, Tilburg University, P.O.B. 90153, 5000 LE Tilburg, The Netherlands
article
info
Article history:
Received 29 June 2012
Received in revised form
24 September 2012
Accepted 12 October 2012
Available online 24 October 2012
JEL classification:
D01
I21
abstract
Resit exams – extra opportunities to do an exam in the same academic year – are widely prevalent in
European higher education, but uncommon in the US. I present a simple theoretical model to compare
rational student behavior in the case of only one exam opportunity versus the case of two exam
opportunities. Numerical examples for a wide range of plausible parameter values show that a second
exam opportunity increases the ultimate passing probability only slightly, but strongly reduces the
average total student effort.
© 2012 Elsevier B.V. All rights reserved.
Keywords:
Frequency of examinations
Education production function
Rationality
1. Introduction
Resit exams – extra opportunities for all students to do an exam
in the same academic year – are widely prevalent in European
higher education, but uncommon in the US. This note presents
a simple theoretical model to compare student behavior in the
case of only one exam opportunity versus the case of two exam
opportunities.
Numerical examples with plausible parameter values show
that a second exam opportunity increases the ultimate passing
probability only slightly, but dramatically reduces the expected
total student effort. In some cases students who are lucky to pass
at the first attempt spend less than one fifth of the time they would
have spent with only one exam opportunity.
While the present paper is related to the educational literature
on optimal testing procedures and setting educational standards
– see e.g. Costrell (1994) and Lazaer (2006) – the economics of
resit exams does not seem to have received any attention in the
literature.1
2. The model
Consider the simplest possible model that captures the key
issues of interest. A risk-neutral rational student needs to decide
E-mail address: [email protected].
1 In an empirical study, De Paola and Scoppa (2011) compare the case of two
exams that each cover different parts of the course material versus one exam
covering the whole course material. In the present note the course material is the
same for both exam opportunities.
0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.econlet.2012.10.015
how much time to spend on preparing for an exam with a
binary outcome, pass or fail. The student wishes to maximize the
probability of passing, but faces a constant opportunity cost w per
unit of time. The student’s utility function might be specified as
U = Y + RUP . Here, Y = w(Z − t ) is income, with Z denoting
the total time available, and t the time spent preparing for the
exam. R = 1 if the student ultimately passes and 0 otherwise.
UP is the utility of passing. UP can be normalized to 1 without
loss of generality through redefining the time unit in which w is
expressed.
2.1. One exam opportunity
First consider the case with only one exam opportunity. The
relationship between the probability of passing, p, and the amount
of time spent on exam preparation (or simply ‘‘effort’’), t, is
specified as

p(t ) = max 0,
α + βt
1 + α + βt

(1)
with β > 0. Specification (1) captures a number of key features
and will allow for analytic solutions of the optimization problem.
The passing probability is increasing in effort, with decreasing
returns. The parameter α can be interpreted as an indicator of
exam difficulty. If α > 0, it is possible to pass without exerting
any effort, as in the case of multiple-choice exams. If α < 0, it is
impossible to pass without spending at least a minimum amount
of time tmin = −α/β ; see Fig. 1 (dotted line: α > 0; solid line:
α < 0). The case α < 0 can also be interpreted as a case with
214
P. Kooreman / Economics Letters 118 (2013) 213–215
As in (3), the passing probability is
1
1
p̃2 = 1 − w 2 β − 2 .
(5)
Note that if the student fails at the first attempt, the total time spent
preparing for both exams, T = t1 + t2 , is exactly the same as the
total time spent when there is only one exam, in the absence of
depreciation (i.e. δ = 0).
The optimal time investment for the first attempt follows from
maximizing
p1 (t1 ) + (1 − p1 (t1 ))p̃2 − w t1
− w(T − (1 − δ)t1 )(1 − p1 (t1 ))
− 21
(6)
− 12
with respect to t1 , where T = β w −αβ −β . The solution
is given by the positive root of a quadratic equation in t1 .
For δ = 0 (no depreciation), the expressions for the optimal
time investment at the first attempt, the passing probability at the
first attempt, and the ultimate passing probability are given by

1
−1 −3
−1
−1

t̃1 = 2 2 w 4 β 4 − αβ − β
1
1
− 12
−
p̃1 = 1 − 2 w 4 β 4

1
3
3

P̃ = 1 − 2− 2 w 4 β − 4 .
Fig. 1. The relationship between time spent and passing probability.
fixed costs (monetary or mental) of doing an exam. The parameter
β reflects the ability of the student, with β being larger for a bright
student than for a less-talented student.2
The optimal effort follows from maximizing p(t ) − w t. The
solution is given by
1
1
t̃ = β − 2 w − 2 − αβ −1 − β −1 ,
(2)
with implied passing probability
1
1
p̃ = p(t̃ ) = 1 − w 2 β − 2 .
−1
(7)
For δ = 1 (full depreciation), the expressions are

−1 −3
−1
−1

t̃1 = w 4 β 4 − αβ − β
1
1
p̃1 = 1 − w 4 β − 4

3
3

P̃ = 1 − w 4 β − 4 .
(8)
The analysis shows that a second exam increases the ultimate
passing probability (P̃ > p̃). Students who pass at the first attempt
spend less time preparing as when there is only one exam opportunity.
(3)
I assume that w/β < 1 and is sufficiently small to ensure
positive effort and passing probability. Higher ability leads to a
higher passing rate, but the effect of ability on effort is ambiguous.
High opportunity costs lead to lower effort and a lower passing
probability. Note that in this model an easy exam (high α ) lowers
student effort to such an extent that the passing probability
remains unaffected. Hence, for a group of homogeneous students,
passing rates are uninformative about exam difficulty.
2.2. Two exam opportunities
If there is a second unconditional exam opportunity, the ex ante
ultimate passing probability is P = p1 + (1 − p1 )p2 , where p1
and p2 are the passing probabilities at the first and second attempt,
respectively.
Optimal effort follows from backward induction. Suppose that
the student fails the first time. Then he/she is back in the situation
described in the previous subsection, with the only difference
being that a time investment equal to (1 − δ)t1 has already been
done. δ is a depreciation factor, and it is likely to be larger the
longer the time span between the two exam opportunities. Then
it follows from (2) that the optimal time spent preparing for the
second opportunity, conditional on t̃1 , is given by
1
−1
1
t̃2 = −(1 − δ)t̃1 + β − 2 w − 2 − αβ −1 − β −1 .
(4)
2 It could be argued that both α and β reflect both student ability and exam
difficulty. For example, if an exam is poorly designed, β can be close to 0 even for a
bright student. In this note I will not elaborate on this.
2.3. A numerical illustration
In the model, students are characterized by four parameters, α ,
β , δ , and w. For each parameter, I consider two different plausible
numerical values and calculate the outcomes of the model for all
16 combinations. The results are presented in Table 1.3
The results show that, for a wide range of parameter values,
a second exam opportunity increases the ultimate passing
probability only slightly (P /p), but dramatically reduces the
expected student effort (T /t, with T = p1 t1 + (1 − p1 )T ), in some
cases by almost two thirds. Students who are lucky to pass at the
first attempt spend only a fraction of the time they would have
spent with only one exam opportunity (t1 /t), in some cases less
than one fifth.
3. Discussion
The results presented in this note suggest that unconditionally
offering students more than one exam opportunity makes little
sense. The main effect is that it provides them with a potential
windfall gain of passing at the first attempt with minimal effort.
Thus the number of resits can strongly affect student effort and
thereby the quality of a degree.
Of course, a number of qualifications should be kept in mind
when interpreting the results. For example, the model assumes
that students have perfect knowledge about the shape of the
relationship between effort and passing probability. In practice,
students sometimes use the first exam opportunity to learn about
3 To interpret numerical values of w note that a student is indifferent between
passing and 1/w time units (hours) of non-exam activities.
P. Kooreman / Economics Letters 118 (2013) 213–215
215
Table 1
Rational time inputs and passing probabilities.
α
β
0
−2
0
−2
0
−2
0
−2
0
−2
0
−2
0
−2
0
−2
w
δ
One opportunity
Two opportunities
t
p
t1
p1
t2
p2
T
P
t1 /t
Comparison
T /t
P /p
0.1
0.1
0.2
0.2
0.001
0.001
0.001
0.001
0
0
0
0
90
110
65.7
75.7
.90
.90
.93
.93
34.7
54.7
21.6
31.6
.78
.78
.81
.81
55.3
55.3
44.1
44.1
.90
.90
.93
.93
47.1
67.1
29.9
39.9
.98
.98
.99
.99
.39
.50
.33
.42
.52
.61
.45
.53
1.09
1.09
1.06
1.06
0.1
0.1
0.2
0.2
0.0005
0.0005
0.0005
0.0005
0
0
0
0
131.4
151.4
95
105
.93
.93
.95
.95
43.2
63.2
26.6
36.6
.81
.81
.84
.84
88.2
88.2
68.4
68.4
.93
.93
.95
.95
59.8
79.8
37.4
47.4
.99
.99
.99
.99
.33
.42
.28
.35
.45
.53
.39
.45
1.06
1.06
1.04
1.04
0.1
0.1
0.2
0.2
0.001
0.001
0.001
0.001
1
1
1
1
90
110
65.7
75.7
.90
.90
.93
.93
21.6
41.6
13.8
23.8
.68
.68
.73
.73
90
110
65.7
75.7
.90
.90
.93
.93
43.2
63.2
27.6
37.6
.97
.97
.98
.98
.24
.38
.21
.31
.48
.57
.42
.50
1.08
1.08
1.06
1.06
0.1
0.1
0.2
0.2
0.0005
0.0005
0.0005
0.0005
1
1
1
1
131.4
151.4
95
105
.93
.93
.95
.95
27.6
47.6
17.4
27.4
.73
.73
.78
.78
131.4
151.4
95
105
.93
.93
.95
.95
55.2
75.2
34.7
44.7
.98
.98
.99
.99
.21
.31
.18
.26
.42
.50
.37
.50
1.06
1.06
1.04
1.04
its nature and difficulty. Also, the model assumes that students
do not derive any intrinsic utility from exam preparation. Highly
motivated and conscientious students might act as if the second
exam opportunity does not exist and use it only as a last resort.
The numerical results also depend on the degree of randomness
in the relationship between effort and passing. A well-designed
exam exhibits a steep increase in the passing probability beyond
a target effort level. Note however that for parameter values closer
to that case (α = −2; β = 0.2) the adverse effects of two exam
opportunities still emerge.
Empirical research, preferably based on field experiments, is
required to obtain more insight into the benefits and costs of resit
exams (including the costs of designing and grading exams for
faculty) and the optimal institutional design of evaluating students.
Acknowledgments
I thank an anonymous reviewer for useful comments and Eric
van Damme, Tobias Klein, Asako Ohinata, and Jan Potters for
helpful discussions.
References
Costrell, Robert M., 1994. A simple model of educational standards. American
Economic Review 84, 956–971.
De Paola, Maria, Scoppa, Vincenzo, 2011. Frequency of examinations and student
achievement in a randomized experiment. Economics of Education Review 30,
1416–1429.
Lazaer, Edward, 2006. Speeding, terrorism, and teaching to the test. Quarterly
Journal of Economics 121, 1029–1061.
Economics Letters 121 (2013) 141–142
Contents lists available at ScienceDirect
Economics Letters
journal homepage: www.elsevier.com/locate/ecolet
Erratum
Corrigendum to ‘‘Rational students and resit exams’’ [Econom. Lett.
118 (1) (2013) 213–215]
Peter Kooreman
Department of Economics, Tilburg University, P.O.B. 90153, 5000 LE Tilburg, The Netherlands
article
info
Article history:
Available online 29 July 2013
The equations in (8) and, as a consequence, some of the
numbers in Table 1 in Kooreman (2013) are incorrect. I apologize
for the error and wish to express my gratitude to Mariam
Arutyunyan, student at the Faculty of Economics, National
Research University Higher School of Economics, Moscow, Russia,
for pointing it out.
The corrected expressions are given by:

−2
−2 1
−1
−1
−1 −3

t̃1 = (2w 2 β 2 − αβ − β ) 2 − αβ − β
1
1
1
p̃1 = 1 − (2w − 2 β 2 − α − 1)− 2

1
1
1
1
1

P̃ = 1 − w 2 β − 2 (2w − 2 β 2 − α − 1)− 2 .
(8*)
Table 1
Rational time inputs and passing probabilities; corrected.
One opportunity
Two opportunities
t
p
t1
p1
t2
p2
T
P
t1 /t
Comparison
T /t
P /p
α
α
α
α
= 0,
= −2,
= 0,
= −2,
β
β
β
β
= 0.1,
= 0.1,
= 0.2,
= 0.2,
w
w
w
w
= 0.001,
= 0.001,
= 0.001,
= 0.001,
δ
δ
δ
δ
=0
=0
=0
=0
90
110
65.7
75.7
.90
.90
.93
.93
34.7
54.7
21.6
31.6
.78
.78
.81
.81
55.3
55.3
44.1
44.1
.90
.90
.93
.93
47.1
67.1
29.9
39.9
.98
.98
.99
.99
.39
.50
.33
.42
.52
.61
.45
.53
1.09
1.09
1.06
1.06
α
α
α
α
= 0,
= −2,
= 0,
= −2,
β
β
β
β
= 0.1,
= 0.1,
= 0.2,
= 0.2,
w
w
w
w
= 0.0005,
= 0.0005,
= 0.0005,
= 0.0005,
δ
δ
δ
δ
=0
=0
=0
=0
131.4
151.4
95
105
.93
.93
.95
.95
43.2
63.2
26.6
36.6
.81
.81
.84
.84
88.2
88.2
68.4
68.4
.93
.93
.95
.95
59.8
79.8
37.4
47.4
.99
.99
.99
.99
.33
.42
.28
.35
.45
.53
.39
.45
1.06
1.06
1.04
1.04
α
α
α
α
= 0,
= −2,
= 0,
= −2,
β
β
β
β
= 0.1,
= 0.1,
= 0.2,
= 0.2,
w
w
w
w
= 0.001,
= 0.001,
= 0.001,
= 0.001,
δ
δ
δ
δ
=1
=1
=1
=1
90
110
65.7
75.7
.90
.90
.93
.93
33.6
55.8
21.1
32.0
.77
.78
.81
.82
90
110
65.7
75.7
.90
.90
.93
.93
46.5
67.6
29.7
40.1
.98
.98
.99
.99
.37
.51
.32
.42
.52
.61
.45
.53
1.09
1.09
1.06
1.06
α
α
α
α
= 0,
= −2,
= 0,
= −2,
β
β
β
β
= 0.1,
= 0.1,
= 0.2,
= 0.2,
w
w
w
w
= 0.0005,
= 0.0005,
= 0.0005,
= 0.0005,
δ
δ
δ
δ
=1
=1
=1
=1
131.4
151.4
95
105
.93
.93
.95
.95
42.2
64.1
26.2
37.0
.81
.82
.84
.84
131.4
151.4
95
105
.93
.93
.95
.95
59.3
80.2
37.2
47.6
.99
.99
.99
.99
.32
.42
.28
.35
.45
.53
.39
.45
1.06
1.06
1.04
1.04
DOI of original article: http://dx.doi.org/10.1016/j.econlet.2012.10.015.
E-mail address: [email protected].
0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.econlet.2013.07.005
142
P. Kooreman / Economics Letters 121 (2013) 141–142
The corrected table is provided in Table 1; the numbers that
deviate from the previous version are indicated in bold.
The key message of the paper does not change. A second
exam opportunity increases the ultimate passing probability only
slightly, but strongly reduces average total student effort.
In some cases students who pass at the first attempt spend
less than one third (not one fifth, as stated in the previous
version) of the time they would have spent with only one exam
opportunity.
References
Kooreman, P., 2013. Rational students and resit exams. Economics Letters 118,
213–215.