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WORKING PAPERS IN ECONOMICS & ECONOMETRICS
ADOPTING NEW TECHNOLOGIES IN THE CLASSROOM
Pedro Gomis Porqueras
Australian National University Research School of Economics
Jose A. Rodrigues-Neto Australian National University Research School of Economics
[email protected] School of Economics
JEL codes: D1, I1, J22, J24, O33
Working Paper No: 528 ISBN: 086831 528 1
September 2010
Adopting New Technologies in the Classroom∗
Pedro Gomis Porqueras†
Australian National University
Research School of Economics
Jose A. Rodrigues-Neto‡
Australian National University
Research School of Economics
Abstract
This paper explores the incentives that students and instructors face when a newtechnology that grants access to online class materials is introduced. We examine theconsequences for attendance and for the composition of live lectures. We also analyzehow various sources of heterogeneity in students’ characteristics, learning styles, andtechnologies affect individual incentives to attend lectures when different degrees ofaccess to online resources are available. In particular, we consider heterogeneity in theoutside options of students and the effectiveness of different online materials. We obtainsome testable implications that may guide empirical researchers towards estimationstrategies that better capture how granting access to online class materials impactsattendance and class composition.
Keywords: absenteeism, online policies.JEL classification: D1, I1, J22, J24, O33.
∗We would like to thank Hugo Mialon, Martin Richardson, Matthew Ryan, as well as seminar participantsat the Australian National University and Auckland University for helpful comments and suggestions. Wewould also like to thank Merrilyn Larusson for English editing. All remaining errors are our own responsi-bility.†Research School of Economics, H.W. Arndt Building 25A, Australian National University, Canberra -
ACT - 0200, Australia.‡Research School of Economics, H.W. Arndt Building 25A, Australian National University, Canberra -
ACT - 0200, Australia.
1 Introduction
During pre-historic times, education mostly occurred via observation and imitation. Tradi-
tions, beliefs, values, practices, and local knowledge were passed orally from person to person
for generations. The young learned informally from their parents, extended family, and kin.
At later stages in their lives, individuals received more structured and formal instruction.1
This educational system forced students to be in direct contact with instructors. Moreover,
could only be reviewed it had been committed to memory.
The educational landscape changed drastically after 3500 BC because new technologies
allowed humans to create writing systems.2 These new developments greatly changed the
way knowledge was transmitted from instructors to students. Written systems expanded the
methods for acquiring knowledge because students no longer had to be in direct contact with
an instructor and because teaching materials could be more easily reviewed.
A similar ”educational revolution” is now possible thanks to the development of digital
technologies.3 Access to online class materials gives students greater flexibility. Compared
with traditional live lectures, online materials have the advantage of allowing students to
choose the time, location, and method of study. These new technologies can have important
consequences for student attendance, class composition, and academic outcomes.
Coincident with the introduction of online technologies at universities, student behavior
has also drastically changed. Contemporary students attend fewer classes and spend less
time studying than their predecessors. Babcock and Marks (2010) find that full-time stu-
dents in 1961 allocated 40 hours per week to class and studying; in 2003, students invested
about 27 hours in academic study. In addition, more full-time enrolled students have some
form of paid employment during the semester, and the amount time devoted to work and
leisure has risen among US college students from 1961 to 2003. At Australian universi-
ties, McInnis and Hartley (2002) find that students worked 14.7 hours in the most recent
week they were employed.4 These employment and attendance patterns vary by discipline,
emphasizing the importance of student, course, and labor market heterogeneity. For in-
1See Adeyemi and Adeyinka (2002) and Akinnaso (1998) for more on this issue.2For instance, the original Mesopotamian writing system was derived from a method of keeping accounts,
and by the end of the 4th millennium BC this had evolved into a process of recording numbers by pressinga triangular-shaped stylus into soft clay. See Fischer (2004) for more on the history of writing.
3This practice is becoming more common around the globe, with many institutions implementing orconsidering automated recording of lectures. See Bell, Cockburn, McKenzie, and Vargo (2001) for more onthis issue.
4The authors find that 38% of respondents work 16 hours or more per week and 18% work 21 hours ormore. Respondents with 15 and less contact hours work an average of 15.3 hours, but once course contacthours exceed 15 hours per week, paid work drops to 14.7 hours for those with 16-20 course contact hoursand to 13.2 hours for respondents with 21 or more contact hours per week.
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stance, McInnis and Hartley (2002) find that average hours worked per week by students in
15.3 for Arts/Humanities/Social Science, 15.3 for Education, 14.4 for Science, and 16.4 for
Commerce/Business/Administration.
Results from past studies on classroom attendance and student labor supply suggest that
absenteeism might be consistent with utility-maximizing behavior. The issue becomes one of
time allocation, as students choose between competing academic and non-academic activities.
Economic theory can offer useful insights into how students respond to the introduction of
online materials; their strategic behavior may give rise to some unintended consequences.
Analyzing these consequences is one of our objectives.
This paper explores whether adoption of online technology is consistent with recent facts
on employment and attendance. In particular, we study how several economic factors (het-
erogeneity in the student population, online technologies, and individual learning styles)
affect student incentives to attend class. We show that these incentives depend on the mag-
nitude of the students’ outside options, which, for some students, may be related to the state
of the economy. We consider only situations where the students are already enrolled at a
traditional university, where live lectures are offered. In other words, the incentive problems
of students that prefer to enroll just in online universities and enrollement decisions are
beyond the scope of this paper.
Not all technologies available to the lecturer have equal impact on attendance. Not
only is it important to decide whether to allow access to online materials, but it is also
crucial to determine the type of digital technologies to use because they induce different
incentives for students to attend class. The lecturer may maximize attendance by choosing
online delivery systems that complement, but are poor substitutes for, live lectures. As the
usefulness of online class materials increases for absentee students, attendance diminishes
because individuals will substitute class participation with online materials. On the other
hand, attendance is relatively better when online materials are more useful for students who
attend class because the materials complement learning at home. In other words, if online
materials are relatively good substitutes, and relatively poor complements, for class, then
attendance decreases with the degree of access students have to online materials.
We evaluate the effect on class composition of changing the level of access to online mate-
rials using several different hypotheses regarding the joint distribution of students’ learning
and their outside options. If there are two groups of students, one with relatively strong
ability to substitute classes with online materials and another with relatively poor substi-
tution ability, then the deviation of class composition from the enrolment proportions is an
increasing function of students’ access to online materials. By providing more access, the
lecturer distorts class composition. We also consider the possibility of correlation in the
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distributions of outside options and how well technologies substitute class, with the goal of
evaluating attendance and class composition.
Finally, we note that mandatory online policies are unlikely to be universally beneficial
across campus, given that the demands and usefulness of technology can differ substantially
across disciplines. It is possible that a given technology is a good complement (or substitute)
to classes in one subject, while in another subject the same technology is a poor complement
(or substitute) to lectures. The exact same technology can impact distinct academic areas
differently. Thus, this paper highlights the importance of measuring the effectiveness of
online technologies on learning outcomes, an aspect not fully explored by the empirical
literature. Moreover, a one-size-fits-all policy across campus is unlikely to be effective, given
that the lecturer cannot control students’ outside options and different groups of students
face different attendance thresholds.
The next section reviews the empirical literature examining class attendance. Section 3
presents a benchmark model that analyzes students’ incentives to attend class. Section 4
examines the implications of offering technologies with different degrees of substitutability,
relative to attending live lectures. Section 5 examines the consequences of having student
populations that differ in both their degree of substitutability and their outside options.
Section 6 concludes the paper.
2 Literature Review
People receive information in three ways, sometimes referred to as modalities: visual –sights,
pictures, diagrams, and symbols; auditory – sounds and words; and kinesthetic –taste, touch,
and smell. An extensive body of research has established that most people learn most
effectively with one of the three modalities and tend to miss or ignore information presented
in either of the other two; see for instance, Barbe and Milone (1981), Dunn and Dunn (1978),
or Waldheim (1987), among others. Once information is received, complex mental processes
convert it into knowledge which can be grouped into: active experimentation and reflective
observation.5 Thus, how much a given student learns also depends on how compatible his
or her learning style is with the instructor’s teaching style.6 Thus, information processing
varies by student, which in turn is influenced by a lecturer’s teaching style. In this paper, we
study the impact of learning and style heterogeneity on class attendance and composition by
5Active experimentation involves doing something in the external world with the information –discussingit or explaining it or testing it in some way. Reflective observation involves examining and manipulating theinformation introspectively. See Kolb (1984) for more on these processes.
6Instructors develop a teaching style based on their beliefs about what constitutes good teaching, theirpersonal preferences and abilities, and the norms of their particular discipline.
3
considering technologies that give students access to online materials outside the classroom.
The education literature has identified several benefits of attending live lectures. The
importance of student attendance is not limited to its value as a good predictor of academic
performance.7 Attendance is also a key component in the education experience as students
learn to manage their differences, work in teams, and share and observe the experience of
other students. By observing arguments of others, students develop and polish their critical
thinking skills. Moreover, sharing experiences helps university students establish long-lasting
friendships and construct valuable professional networks. Depending on the subject and
teaching style, kinesthetic learners may most likely benefit from live lectures.
The literature has identified various underlying determinants for absenteeism. For in-
stance, Romer (1993) finds that absenteeism is lower for courses with a significant math-
ematical component, for core courses, and for courses with a higher perceived quality of
instruction. Cohn and Johnson (2006) find that factors that influence a student’s deci-
sion to attend class include a student’s race, GPA (higher GPA implies better attendance),
SAT (higher SAT implies lower attendance), college experience (freshmen appear to attend
more classes than juniors and seniors), and residence (state residents have better atten-
dance records). Other determinants are students’ social activities. For instance, Longhurst
(1999) finds that 46% of sampled students admitted missing classes for social, recreational,
and leisure activities. Similarly, Marburger (2001) finds that absenteeism was significantly
higher on Fridays.
In keeping with the findings of Cohn and Johnson (2006), Wagner et al., (2004) and
McCoy and Smyth (2004) emphasize the importance of examining labor market conditions,
given that high levels of part-time work are important in explaining student absenteeism.
Similarly, McInnis and Hartley (2002) examine the extent to which full-time undergraduate
students combine full-time enrolment with substantial hours of paid work at Australian
universities.
Finally, there is an increasing number of papers in the education literature that examine
the link between the introduction of digital technologies and class attendance. For instance,
Massingham and Herrington (2006) report that the main reason noted by students for not
attending lectures is the availability of materials online. Similarly, Chang (2007) finds that
55.1% of academic staff in an Australian university reported that student attendance had
decreased as a result of recording lectures. Moreover, when asked why they do not attend
lectures, 68.3% of students surveyed reported that they learned just as effectively using
7For instance, Romer (1993) finds that the difference in performance between a student who attendsregularly and one who attends sporadically is approximately a full letter grade. In the same spirit, Durdenand Ellis (1995) find that absenteeism is strongly associated with poor academic performance.
4
recordings as attending the corresponding lecture in person.
These empirical findings highlight the importance of considering different sources of het-
erogeneity when examining the role of technology in the classroom, class attendance, and
class composition. These findings motivate our work.
3 Model Setup
Assume that students have limited time and obtain utility both from learning in a class
in which they are already enrolled and from an alternative activity. By introducing online
access to class materials, students are able to free up some time by not attending the live
lecture. These technologies can be a substitute as well as a complement to attendance to live
lectures. In this environment, we examine how a student’s choices to attend or skip lectures
changes when there is stochastic access to online class materials. Afterwards, we analyze
how the introduction of these new technologies affects class composition.
3.1 Homogeneous Students
As a benchmark, we consider an environment where there is a single class/lecture in the
course. To consider the strategic interactions we analyze a simultaneous two player game.
Player 1 is the student who decides to attend or skip class and player 2 is the lecturer. Player
2 decides the probability of granting access to online materials, which we denote by p. This
situation captures the possibility that the lecturer grants access to online materials that cover
just a fraction of the material conveyed in the live lecture. This probability also reflects the
possibility that technical difficulties prevent the recording or accessing of online materials.
The online materials this paper considers are power point slides, audio, and video recordings,
as well as multimedia resources offered by some textbooks. Since there is evidence linking
attendance and academic performance, it is important to consider the potential implications
of choosing the probability p for students.
Let L > 0 be a measure of how much a student learns in the absence of online materials
(in this case p = 0). If the student goes to class and accesses online materials, she learns
L+βL for some constant β > 0; where β denotes the degree of complementarity ; that is, the
additional proportion of knowledge acquired by accessing online materials after attending
class. The larger β, the more helpful online materials are to students who attended class.
When the student does not attend class, she learns zero (relative to attending the live
lecture) in the absence of online materials, and αL if she can access online materials; where
α ≥ 0 denotes the degree of substitutability of this new technology relative to not attending
the live lecture. This can be thought of as the share of the knowledge conveyed in the lecture
5
that students can learn by just accessing online materials. The larger α, the more helpful
online materials are to the student who missed class. Notice that we do not restrict α ≤ 1.
Thus, we allow the possibility that, by replaying lectures, students who missed class learn
even more than those who attend live lectures and do not review online materials.
Different combinations of online materials can have quite distinct β and α. In what
follows, we consider a situation where the lecturer does not have control over the set of
online materials. The lecturer only can decide whether to provide access or not. The choice
of the actual β and α is beyond the scope of this current paper.
A student has choices about how to use her time; thus, not attending class offers the
payoff of engaging in an alternative activity, which we denote by W > 0. To simplify our
discussion, we henceforth restrict the outside option just to the utility gains from working.
This is consistent with McInnis and Hartley’s (2002) Australian survey which finds that 78%
of full-time enrolled students have some paid employment and 72.5% have paid employment
during semester (75.7% of females and 68.6% of males). Similarly in the UK, Callender and
Kemp (2000) find that over 60% of students had been employed at some time during the
academic year.
In order to capture strategic behavior we consider a normal form game between a repre-
sentative student who takes a course, and the lecturer who teaches it. Implicit in the payoffs
of the game, we assume that the lecturer only cares about the student learning outcomes
and the student cares about her learning outcomes as well as other non-academic uses of her
time. The payoffs of pure action profiles are described below:
Online Access (p) No Access (1− p)Attend class L+ βL, L+ βL L, LSkip class W + αL, αL W , 0
Having no access to online materials corresponds to p = 0. The other extreme, p =
1, represents a situation where all lectures are available online, which we can think of as
a mandatory online policy. Notice that the payoffs exclude any additional benefit that
working brings to the return on learning. This is consistent with Long and Hayden’s findings
(2001) that the great majority of the jobs held by full-time students are not directly or even
indirectly related to the course of study.
Suppose that access to online class materials is granted with probability p ∈ [0, 1]. Then,
the payoff for the student of playing action ”skip class” is given by:8
U1(Skip class, access with prob p)] = W + pαL. (1)
8If we allow the student to play a mixed action, then she will mix if and only if L+βL = W +αL, whichis a razor’s edge condition. In the next section, when we let L and W be continously distributed, this willbe an event of measure zero. Hence, this case will not change the analysis.
6
On the other hand, the payoff for the student playing ”attend class” is:
U1(Attend class, access with prob p)] = L+ pβL. (2)
Assume that the provision of online materials is costless to the lecturer. Thus, the possible
payoffs for the lecturer are:
U2(Skip class, access with prob p)] = pαL,
and
U2(Attend class, access with prob p)] = L+ pβL.
Since β > 0 and α > 0, a dominant strategy for the lecturer is to grant full access to
online material (p = 1). Knowing this, the student compares the payoffs L+βL and αL+W
to decide if she attends class or not.
Proposition 1 Consider the previous normal form game between the student and the lec-
turer.
(a) Suppose that technologies and outside options for the student are such that:
W < L+ p(β − α)L. (3)
Then, the normal form game has a unique Nash equilibrium in which the student attends
class and the lecturer grants full access to online material; p = 1.
(b) Suppose that technologies and outside options for the student are such that:
W > L+ p(β − α)L. (4)
Then, the normal form game has a unique Nash equilibrium in which the student skips class
and the lecturer grants full access to online material; p = 1.
The proof of this proposition and all subsequent results can be found in the appendix.
Proposition 1 reveals that the incentives to go to class depend on the magnitude of the
student’s outside options, which may be related to the state of the economy, reflecting the
opportunity cost of students. Thus, labor market conditions impact attendance once online
policies are in place.9
Proposition 1 also highlights the role of the difference between the degree of comple-
mentarity and the degree of substitutability of online technologies, β − α, when analyzing
9This is relevant because students’ outside options are not under the lecturer’s control, and can easilyvary from institution to institution or across time.
7
attendance. For instance, the difference between the degrees of substitutability and com-
plementarity of audio digital recordings in mathematics is probably quite different to the
analogous difference when one audio records a history or law class. The exact same technol-
ogy impacts distinct disciplines in different ways.
Different disciplines or even institutions using very different technologies (with distinct
absolute levels for α and β) may provide the same incentives for their students. What
really matters is the relative online benefits, value of β − α, and the outside options of the
student. Hence, it is important to classify online class materials in terms of their degrees of
substitutability and complementarity to the live lectures.10
Corollary 1 (a) Suppose that the degree of complementarity is larger than the degree of
substitutability; β > α. If W < L, then the student prefers to attend class regardless of the
probability p, and the specific values of the degrees of substitutability and complementarity of
the underlying technology.
(b) Suppose that the degree of complementarity is smaller than the degree of substitutabil-
ity; β < α. If W > L, then the student prefers to skip class regardless of the probability p,
and the specific values of the degrees of substitutability and complementarity of the underlying
technology.
Proposition 1 and Corollary 1 emphasize the importance of classifying online technologies
according to α and β. Moreover, it is crucial to estimate the relative returns, aspects not
addressed in the empirical literature.
In the next subsections we examine the roles of various sources of heterogeneity that have
less clear implications for attendance. This can provide further insights on the relationship
between attendance and technology adoption in the classroom.
3.2 Heterogeneity
It is well documented that students have different ex post learning returns from attending
class, which could be captured by different L. This reflects the fact that not all students
absorb the exact same proportion of class materials. For instance, one student may be able
to learn all the content explained in a given live lecture while a different student may learn
10In practice, this is relevant because lecturers and universities have some control over the degree ofsubstitutability and complementarity of the different online materials. For instance, a lecturer can recordlectures using mp3 formats that allow students to download lectures and listen to them when they want.On the other hand, the lecturer can provide recording materials that cannot be downloaded and have a timewindow where students can access those materials. These different options change the relative degree ofsubstitutability and complementarity of online materials, thus the incentives to attend class change. Theseaspects have not been fully explored by the education literature when examining the impact of online policies.
8
only half of it. Moreover, for some students the content of a particular course may be very
important for their future careers, while for some other group the material is relatively less
relevant.11
Another source of heterogeneity is the different outside options that students have, which
may come from different job skills or from different wealth levels; that is, the same salary may
provide more utility to poorer students.12 It also reflects the fact that local students have
better connections to local business or that some group of students enjoy more favorable labor
market conditions.13 This heterogeneity assumption is consistent with the findings of Goldin
(1990), who documents the wage gap between males and females in the US. Similarly, Trejo
(1997) establishes that Mexican-American men earn substantially lower wages on average
than their white counterparts.
3.2.1 Setup
In order to incorporate these sources of heterogeneity into the benchmark model, consider
jointly and continuously distributed random variables L and W , defined on [0,+∞) ×[0,+∞), with joint cumulative distribution FL,W . Let fL,W denote the joint density as-
sociated with the joint distribution FL,W . Let FL and FW denote, respectively, the marginal
distributions with respect to L and W .
Let fL and fW denote, respectively, the marginal densities with respect to L and W .
Assume that the marginal densities are such that fL(l) > 0, ∀l ∈ [0,+∞); and fW (w) > 0,
∀w ∈ [0,+∞).
3.2.2 Attendance
Inspired by part (a) of Proposition 1, attendance is defined as follows:
Definition 1 For every p ∈ [0, 1], attendance to the live lecture, denoted A(p;α, β), is:
A(p;α, β) =
∫ +∞
l=0
∫ l+(β−α)lp
w=0
fL,W (l, w) dw dl. (5)
From now on, we always assume that the following restriction on the technological pa-
rameters β and α holds:
β − α > −1. (6)
11This may reflect the fact that students take the easiest possible course just to complete the minimumnumber of non-major credits.
12We refer to Cohn and Johnson (2006), Wagner et al. (2004), and McCoy and Smyth (2004), amongothers, for more on this issue.
13In many countries, immigration regulations limit employment opportunities for foreign students.
9
Inequality (6) guarantees that A(p;α, β) ≥ 0, regardless of the value p ∈ [0, 1], and
restricts the set of online technologies. Equation (6) is mathematically equivalent to:
1 + (β − α)p > 0, ∀p ∈ [0, 1], (7)
which, itself, is equivalent to:
l + (β − α)lp > 0, ∀l > 0.
This means that, at least in the extreme case of zero as an outside option (W = 0), condition
(3) holds for every p ∈ [0, 1]. In other words, every student with realization L = l > 0 and
outside option W = 0 prefers to attend class regardless of the online policies.
Depending on the joint distribution FL,W , the degrees of complementarity and substi-
tutability, attendance can respond quite differently to alternative online policies. The next
result examines how changes in the online probability impact attendance.
Proposition 2 Suppose that condition (6) holds.
(a) If the degree of complementarity is smaller than the degree of substitutability, β < α,
then attendance decreases with the probability of granting online access to class materials.
Formally, function p 7→ A(p;α, β) is decreasing. Hence, attendance is minimized under full
access to online materials (p = 1), and maximized if p = 0.
(b) When the degree of complementarity is larger than the degree of substitutability, β > α,
then attendance increases with the probability of granting online access to class materials, p.
Formally, function p 7→ A(p;α, β) is increasing. Hence, attendance is maximized under full
access to online materials (p = 1), and minimized if p = 0.
(c) If α = β, then attendance is independent of the probability of granting online access
to class materials, p. Function p 7→ A(p;α, β) is constant.
The previous result does not depend on the specific distribution of students’ outside
options. Given the expression in equation (5), which shows that A(p;α, β) varies continuously
with α and β, part (c) of Proposition 2 reveals that when the difference β − α is relatively
small, changes in the online policy (p) do not impact attendance very much. Extreme actions
by the lecturer choosing p = 0 or p = 1 may lead to very similar levels of attendance.
As we can see, not all technologies available to the lecturer have equal impact on atten-
dance. It is not only important to decide whether to allow access or not, but it is also crucial
to determine the type of digital technologies available because the difference between the
degrees of complementarity and substitutability induce different incentives to students.
10
3.2.3 Gap in Attendance
In order to further explore the consequences of technology on attendance consider the fol-
lowing definition.
Definition 2 Fix the technology parameters α and β. The gap in attendance, denoted
G(p;α, β), due to access to online materials with probability p, is equal to the difference
between attendance when the probability of granting access to online materials is zero and
when it is equal to p ∈ [0, 1]. Mathematically:
G(p;α, β) = A(0;α, β)− A(p;α, β). (8)
The gap depends on the underlying technology parameters (β and α). As mentioned
earlier, the technology parameters α and β are functions of the lecturer’s delivery system.
The gap also depends on the online policy (p) and on the joint distribution of outside options
and students’ skills, FL,W . This joint distribution is not under the lecturer’s control.
If p = 0, clearly the gap is zero. If α > β and p > 0, the gap G(p;α, β) is positive,
regardless of the joint distribution of L and W . Mathematically:
G(p;α, β) =
∫ +∞
l=0
∫ l
w=l−(α−β)lp
fL,W (l, w) dw dl > 0. (9)
However, if the benefit of granting access to online materials is greater for those students
who were present than to those who were not; that is, β > α, and p > 0, then the gap
G(p;α, β) is negative, regardless of the joint distribution of L and W . Formally, if β > α
and p > 0, then:
G(p;α, β) = −∫ +∞
l=0
∫ l+(β−α)lp
w=l
fL,W (l, w) dw dl < 0. (10)
The next result examines how changes in the online probability impact the attendance
gap.
Proposition 3 Suppose that condition (6) holds. Then, regardless of the joint distribution
of L and W :
(a) If the degree of complementarity is smaller than the degree of substitutability, β < α,
then the gap G(p;α, β) is positive when p > 0, and increasing with the probability of granting
online access to class materials, p. In this case, the gap in attendance is maximized under
full access (p = 1), and minimized if p = 0.
(b) If the degree of complementarity is larger than the degree of substitutability, β > α,
then the gap G(p;α, β) is negative when p > 0, and increasing with the probability of granting
11
online access to class materials, p. In this case, the gap in attendance is minimized under
full access (p = 1), and maximized when p = 0.
(c) For every given probability p > 0, the gap in attendance G(p;α, β) is an increasing
function of the degree of substitutability (α), and a decreasing function of the degree of
complementarity (β). This is independent of the sign of β − α.
Parts (a) and (b) of Proposition 3 show that an arbitrarily small change in the difference
between the degrees of complementarity and substitutability may be sufficient to induce a
major change in the policy that leads to a minimum gap. Therefore, it suffices to have a
change in the technological parameters that alter the sign of the difference β−α. Of course,
when β − α is relatively close to zero, the gap G(p;α, β) is also relatively close to zero,
regardless of p.
Part (c) of Proposition 3 highlights the importance of choosing different online delivery
systems that complement live lectures but are poor substitutes for them. This consideration
is more relevant when p is large. As the usefulness of online class materials increases for
absent students, the smaller attendance becomes. On the other hand, attendance is larger
when the usefulness of access to online materials increases for attending students; that is, a
larger the degree of complementarity (β).
4 The Role of Different Degrees of Substitutability
In this section we explore the consequences of allowing students to differ in their degree of
substitutability when designing online policies.
4.1 The Basics
Based on the empirical findings of the education literature, we now allow students to have
different degrees of substitutability.14 For instance, kinesthetic learners may benefit mostly
from attending live lectures, while auditory learners may benefit from repeatedly listening
to audio recorded classes. Finally, visual learners may benefit the most from replaying video
materials.
For expositional purposes consider just two degrees of substitutability, denoted by α+
and α− where α+ > α− > 0, and a unique degree of complementarity β > 0. Let 0 < θ < 1
be the fraction of students that have a degree of substitutability equal to α+. The analysis
that follows is isomorphic to the case of constant α and varying β.
14See Chang (2007) for more on this issue.
12
In what follows we consider an environment where: (1) the joint distribution of the
random variables L and W is independent of the group to which a student belongs; and (2)
technology is such that inequality (6) still holds for all values of α. Under these conditions,
for every p ∈ [0, 1] and every l > 0, it is always the case that l + lpβ − lpα+ > 0 and
l + lpβ − lpα− > 0. This assumption is similar to (6), and implies that in the following
definitions A+ ≥ 0 and A− ≥ 0.
Definition 3 For each p ∈ [0, 1] and each α ∈ {α+, α−}, attendances in the α+ and α−
groups, denoted A+(p;α, β) and A−(p;α, β), respectively, are given by:
A+(p;α, β) = θ
∫ +∞
l=0
∫ l+(β−α+)lp
w=0
fL,W (l, w) dw dl.
A−(p;α, β) = (1− θ)∫ +∞
l=0
∫ l+(β−α−)lp
w=0
fL,W (l, w) dw dl.
Given these definitions, attendance is equal to A = A+ + A−. Since p ≥ 0, l ≥ 0, and
α+ > α− > β, then l + (β − α−)lp ≥ l + (β − α+)lp, with equality holding if and only if
p = 0. This implies the following:∫ +∞
l=0
∫ l+(β−α−)lp
w=0
fL,W (l, w) dw dl ≥∫ +∞
l=0
∫ l+(β−α+)lp
w=0
fL,W (l, w) dw dl. (11)
Inequality (11) implies that the proportion of students in the α− group who attend
lectures is at least as large as the proportion of students in the α+ group who attend lectures;
that is, absenteeism is relatively larger in the α+ group. Mathematically:
A−1− θ
≥ A+
θ.
Claim 1 If p > 0, then attendance is decreasing with the proportion of students having
degree of substitutability equal to α+, and increasing with the proportion of students having
degree of substitutability α−. Mathematically:
dA
dθ< 0, and
dA
d(1− θ)> 0.
When the benefit to absent students of using a technology varies across the entire student
body, access to online materials changes the composition of students attending class. This
should be taken in account when designing online policies and deciding the type of technology
available to students.
13
4.2 Students Attending Lectures
In this section, we explore how introducing new technologies affects the attendance of stu-
dents belonging to different groups. If a student’s group membership is correlated with other
characteristics such as major, gender, age, nationality, religion, or social status, then the ex-
istence of online material disproportionally impacts the incentives for these groups to attend
class.15 This observation is consistent with the findings of McInnis and Hartley (2002) and
Cohn and Johnson (2006), among others.16
The composition of students attending a live lecture changes as the probability of access
to online materials varies. Without any access, p = 0, the Lebesgue measures of members
of groups α+ or α− are, respectively, θ and 1 − θ. However, if granting access to online
class materials is stochastic, p > 0, then the Lebesgue measures of members of the α+ group
decreases from θ to the fraction A+/A, which is smaller than θ. Formally:
A+
A=
(1 +
(1− θ)θ
∫ +∞l=0
∫ l+(β−α−)lp
w=0fL,W (l, w)dw dl∫ +∞
l=0
∫ l+(β−α+)lp
w=0fL,W (l, w) dw dl
)−1
.
To measure the distortion on class composition arising from the stochastic online policy,
consider the following definition.
Definition 4 The deviation (from the population proportions) in class composition is defined
as D = θ − A+/A.
Claim 2 describes how different online policies affect the attendance of the two different
groups of students.
Claim 2 If p > 0, then the deviation in class composition is positive; i.e., D > 0. If p = 0,
then D = 0.
The following result establishes a general rule that is valid for all joint distributions of L
and W .
Proposition 4 Suppose that the degree of complementarity lies in between the two possible
values for the degree of substitutability; α− ≤ β ≤ α+. Then, the deviation from the popu-
lation proportions in class composition is an increasing function of the online probability p.
Granting full access to online materials (p = 1) maximizes the deviation D.
15Some students could use this fact to argue that the new technology is discriminatory.16McInnis and Hartley (2002) report that compared with all other students, those studying Com-
merce/Business/Administration had significantly fewer course contact hours and spent fewer days on campus.It is not surprising then to find that these students had the highest average weekly hours of work (16.4 hours).
14
To sharpen our predictions, we now examine how the deviation in class composition
varies with the probability of granting online access when we impose a condition on the joint
distribution for the random vector (L,W ). Inequality (3) is equivalent to:
W
L< 1 + (β − α)p.
Hence, we can express the deviation D as depending on the distribution of random variable
Y = W/L. The next result explores this observation.
Proposition 5 Assume that one of the following two alternative hypotheses hold:
(1) The degree of complementarity is smaller than the smallest degree of substitutability
(β < α−) and the cumulative distribution FW/L of the ratio W/L is continuous and log-
concave; that is, y 7→ ln(FW/L(y)) is a concave function.
(2) The degree of complementarity is larger than the greatest degree of substitutability
(β > α+) and the cumulative distribution FW/L of the ratio W/L is continuous and log-
convex; that is, y 7→ ln(FW/L(y)) is a convex function.
In both cases, the deviation in class composition is an increasing function of the online
probability p. Granting full access to online materials (p = 1) maximizes the deviation D.
Corollary 2 Let L0 > 0 and W0 > 0. Suppose that the degree of complementarity is no
larger that the greatest degree of substitutability; that is, β ≤ α+. If the random variables
L and W are independent and uniformly distributed on the intervals [0, L0] and [0,W0],
respectively, then the deviation from the population proportions in class composition is an
increasing function of the online probability p. Granting full access to online materials (p =
1) maximizes the deviation D.
Introducing an online policy negatively affects attendance to live lectures. We now ex-
plore the robustness of this result as we relax the assumptions that students’ have a constant
outside option and payoff from learning. Given that in labor economics the wage offer distri-
bution has been modeled as following a log concave distribution, we explore the consequences
of this assumption.17
Corollary 3 Suppose that the degree of complementarity is no larger than the greatest degree
of substitutability; that is, β ≤ α+. Suppose that L is constant and the distribution function
FW (w) is continuous and log-concave; that is, w 7→ ln(FW (w)) is a concave function. Then,
the deviation D in class composition is an increasing function of the probability p. Full access
to online materials maximizes this deviation.17One of the earliest economic applications of log-concavity is in the job search literature, where, under the
assumption that the wage offer distribution is log-concave, one is able to assign many comparative dynamicsderivatives. See Burdett and Ondrich (1985) and Flinn and Heckman (1983).
15
The implications of this result are more likely to hold at universities located in large
urban areas and for students taking an introductory or required course. This is the case as
the log concave wage offer distribution is more likely to hold in such environments and the
return on learning tends to be flatter for such courses; hence the assumption of constant L.
The next result explores a situation where students face a constant outside option W
and the returns to their learning, L, is continuously distributed.
Corollary 4 Suppose that the degree of complementarity is no larger that the greatest degree
of substitutability; that is, β ≤ α+. Suppose that the outside option W is constant for all
students, and the distribution of learning FL(l) is continuous and has a non decreasing hazard
rate; that is, l 7→ fL(l)/(1−FL(l)) is a weakly increasing function. Then, the deviation from
the population proportions in class composition is an increasing function of the probability
p. Full access to online materials maximizes the deviation D. Choosing p = 0 minimizes
deviation D.
The implications of Corollary 4 are more likely to hold at universities located in small
urban or rural centers which offer limited job opportunities, thus offering a fairly constant
outside option. The non-constant return on learning is more likely to hold for students taking
a course that is relevant to their major and future careers.
The results in this section highlight the importance of measuring different α and β for
different subgroups of the population by stratifying the data according to important observ-
ables such as age, major and nationality, for example.
5 Correlated Types
This section explores the consequences of having sub-populations of students that differ in
both their degree of substitutability and in their outside options. This last assumption tries
to capture gender/race based wage gaps in the job market or the fact that foreign students
typically do not enjoy the same outside options as domestic students. This is consistent with
the findings of McInnis, James, and Hartley (2000). These authors report that commencing
undergraduate students born in Australia and who speak English at home are more likely to
be studying and working, than studying only. Students born overseas or who live in a home
where a language other than English is spoken are more likely to be studying only, rather
than working.18
18Moreover, these authors report that around half of international full fee-paying students worked 6-10hours per week, 17% worked 11-15 hours and 14% worked 16-20 hours. Under current government regulations,international students in Australia are allowed to work up to 20 hours per week.
16
5.1 Heterogeneous Degree of Substitutability and OutsideOptions
Assume that all students have the same value for the learning L > 0 and the same degree of
complementarity β ≥ 0. Suppose that the distribution of outside options is finite, with two
possible values, w+ > w−. We refer to a student with W = w+ as ”high outside option”, and
W = w− as ”low outside option”. Assume that the distribution of degrees of substitutability
has only two possible values, α+ > α− > 0. We refer to a student with α = α+ as ”high
degree of substitutability”, and α = α− as ”low degree of substitutability”. Assume that
w− > L > 0 and β < α−. Apart from these restrictions, the values L, β, α+, α−, w+ and w−
are arbitrary. Hence, the distributions of outside options and degree of substitutability may
be correlated.
Suppose that the population of students is divided into four groups of students which we
denote as (ij); where i ∈ {+,−} denotes degree of substitutability and j ∈ {+,−} represents
the student’s outside options. For instance, type +− represents a student with degree of
substitutability α+ and outside option w−, while type ++ represents a student with degree
of substitutability α+ and outside option w+. In order to analyze how incentives to attend
class vary across the different groups, consider the following definitions.
Definition 5 Define the following threshold probabilities:
p++ =L− w+
L(α+ − β), p+− =
L− w−L(α+ − β)
, p−+ =L− w+
L(α− − β), and p−− =
L− w−L(α− − β)
.
Simple algebra proves that:
p−− > max{p+−, p−+} > p++.
Now, we explore how different probabilities regarding access to online class materials
affect class composition. First, consider the case where probability p is such that p++ < p <
min{p+−, p−+}. In this case, only students of type ++ do not attend lecture. To see why
this is the case, note that p++ < p is equivalent to:
w+ > L+ p(β − α+)L.
According to Proposition 1, type ++ students do not attend lectures. On the other hand,
type +− students attend lectures because p < p+− is equivalent to:
w− < L+ p(β − α+)L,
which, again, follows from Proposition 1. Similarly, p < p−+ (respectively p < p−−) implies
that type −+ (−−) students go to class.
17
Now, consider a situation where probability p satisfies:
max{p+−, p−+} < p < p−−.
In this case, only students of type −− attend lecture.19
As we can see, the groups considered here have very different attendance thresholds when
considering whether to go to live lectures or not.
5.2 Full Attendance: Providing Incentives to all Students
In general, in order to motivate all students to attend class, it is necessary and sufficient to
provide incentives to type ++ students because these students have the highest opportunity
cost and the highest degree of substitutability.
Claim 3 Type ++ students attend lectures if and only if all students attend class. This is
true if and only if p < p++, or equivalently:
α+ <L− w+ + pβL
pL⇔ w+ < L+ p(β − α+)L ⇔ L >
w+
1 + p(β − α+). (12)
Under a policy granting full access to online materials (p = 1), students of type ++
attend class if and only if:
L >w+
1 + β − α+
.
The benefit from going to class has to overcome not only the opportunity cost w+, but it
also has to be larger than the product of the outside option and the factor (1 + β − α+)−1,
which is greater than 1.
Full attendance is more likely to be achieved as the maximum values for the degree of
substitutability, α+, and outside options, w+, decrease, or as the benefit of the lecture, L,
or the degree of complementarity, β, increase, or, as the probability p decreases.
5.3 Class Composition
In order to simplify exposition, suppose for the remaining of this Section that p+− = p−+.
Let θ++ > 0, θ−+ > 0, θ+− > 0, θ−− > 0 be constants such that:
θ++ + θ−+ + θ+− + θ−− = 1.
Consider the following table that defines the measure of the student population in each
group:w+ w−
α+ θ++ θ+−α− θ−+ θ−−
19The proof requires, again, Proposition 1, and is analogous to the previous case.
18
Given this structure, the next remark is a direct application of well-known results in statistics.
Remark 1 There is positive (respectively, negative) correlation between the random vari-
ables α and w if and only if θ++θ−− > θ+−θ−+ (respect., θ++θ−− < θ+−θ−+). These random
variables are independent if and only if θ++θ−− = θ+−θ−+.
There are two alternative ways of measuring deviation in class composition. Under full
attendance, the proportion of students with high degree of substitutability is θ++ + θ+−.
With online polices, this value may decrease. Let Aα(p) denote the proportion of students
with high degree of substitutability present in class when online materials are available with
probability p.
Definition 6 The α-deviation in composition, denoted Dα, is:
Dα(p) = θ++ + θ+− − Aα(p).
Since Aα(p) is weakly decreasing, then Dα(p) is weakly increasing in p. Similarly, under
full attendance, the proportion of students with high outside option is θ++ + θ−+. With
online policies, this value may decrease. Let Aw(p) denote the proportion of students with
high outside option present in class when online materials are available with probability p.
Definition 7 The w-deviation in composition, denoted Dw, is:
Dw(p) = θ++ + θ−+ − Aw(p).
Since Aw(p) is weakly decreasing, then Dw(p) is weakly increasing in p. Given these
definitions we can establish some results regarding the implications of online policies for the
different types of deviations.
Claim 4 Under full attendance, both α-deviation in composition and the w-deviation in
composition are zero. In other words, if p < p++, there is no deviation in class composition;
neither α-deviation, nor w-deviation.
The next result describes deviations from the population proportions when the probability
p increases to some value in the interval p++ < p < p+− = p−+.
Claim 5 Suppose that p++ < p < p+− = p−+. Then:
(a) The proportion of students with degree of substitutability α = α+ in lecture and the
α-deviation in composition are, respectively:
Aα(p) =θ+−
θ−+ + θ+− + θ−−, and Dα(p) = θ++ + θ+− −
θ+−
θ−+ + θ+− + θ−−.
19
The α-deviation in composition increases with θ++ and θ−−.
(b) The proportion of students with outside option w = w+ in class and the w-deviation
in composition are, respectively:
Aw(p) =θ−+
θ−+ + θ+− + θ−−, and Dw(p) = θ++ + θ−+ −
θ−+
θ−+ + θ+− + θ−−.
The w-deviation in composition increases with θ++ and θ−−.
The next result assumes that the probability p increases even more relative to the previous
case.
Claim 6 Suppose that p+− = p−+ < p < p−−. Then:
(a) The proportion of students with degree of substitutability α = α+ in lecture and the
α-deviation in composition are, respectively:
Aα(p) = 0, and Dα(p) = θ++ + θ+−.
The α-deviation in composition increases with θ++ and θ+−.
(b) The proportion of students with outside option w = w+ in class and the w-deviation
in composition are, respectively:
Aw(p) = 0, and Dw(p) = θ++ + θ−+.
The w-deviation in composition increases with θ++ and θ−+.
Full Attendance - +
+ -
- -
Who Goes to Class?
- -Empty Class
=
Partial Bias Maximum Bias Undefined BiasUnbiased
Figure 1: Attendance thresholds.
The previous claims highlight the fact that the technology parameter α, and the outside
option of students (w) induce different incentives for students to attend class. These obser-
vations are even more relevant when there is heterogeneity among the student population.
Claims 4, 5 and 6 provide some guidance on how to disentangle the effects of technology
and students’ outside options in class composition by focusing on different student groups.
Figure 1 summarizes the effect of different online policies according to the different incentive
for attendance thresholds.
20
6 Concluding Remarks
The recent emergence of web-based lecture technologies has increased the use of online class
materials, but few studies have considered how different online policies impact teaching and
learning. This paper analyzes how granting online access to class materials changes the
incentives for students to attend live lectures, which has consequences for class composition.
These aspects have not previously been addressed by the literature.
A student’s incentive to attend class depends on the magnitude of his/her outside options.
Not all technologies available to the lecturer have equal impact on attendance. Moreover,
this paper highlights the importance of choosing online delivery systems that complement
live lectures, but are poor substitutes for them, unless the course is taught only online.
Given that a lecturer cannot control the outside options of students, and that different
student groups have different attendance thresholds, a one-size-fits-all policy across campus
is unlikely to be effective. Moreover, degrees of substitutability and complementarity (α and
β) differ across disciplines. For instance, the difference between the degrees of substitutability
and complementarity of audio digital recordings in mathematics is probably quite different to
the analogous difference in a audio-recorded history or law class. The exact same technology
impacts distinct disciplines in different ways.
Online polices have significant consequences for the composition of student groups attend-
ing live lectures. Why should we care about class composition? If class content is influenced
by students who attend live lectures, online polices may modify the content of courses. In
particular, if the most skilled students are more likely to belong to the group of students
with higher degrees of substitutability and better outside options, then these students may
not attend classes that provide online access to lecture materials. Given that this group of
students typically asks the majority of clarifying questions, their absence could reduce learn-
ing opportunities for the entire class. If group membership is correlated to other observable
characteristics such as gender, age, nationality, religion, or social status, then providing ac-
cess to online materials will impact the attendance of these distinct groups differently. These
findings are also relevant for teaching evaluations, which will be biased towards a particu-
lar group when (as they often are) they based on data from students who are present in
class. Thus, a lecturer’s choice of technology changes the pool of students participating in
teaching evaluations.20 Further, if academic career development partly depends on teaching
ratings, a lecturer could be motivated to tailor his/her delivery method to the preferences of
a particular group of students.
20In some universities, students who do not go to class may evaluate teaching via an online system. Buteven if all students are able to provide evaluations, those who attend class may have a completely differentopinion about the course than those who only study via online materials.
21
Our results based on the strategic interactions of students and lecturers are consistent
with employment and attendance stylized facts. The framework presented in this paper can
provide some guidance for designing empirical strategies that better capture the effect of
online technologies on class attendance and composition. In particular, this work highlights
the need to classify the different technologies in terms of returns to learning among the
different groups of students attending class. These returns might vary by class size, discipline,
and specific curricula; that is, elective versus mandatory courses.
A Appendix: Proofs
Proof. of Proposition 1:
Granting full access to online materials is a dominant strategy for the lecturer. Attending
lecture is optimal to the student if and only if:
E[U(Attend class | access with prob p)] ≥ EU(skip class | access with prob p).
Substituting (2) and (1) in this inequality and simplifying:
L ≥ W + pαL.
Simple algebra reveals that this inequality, W ≤ L + Lpβ − Lpα and W/L − 1 ≤ (β − α)p
are all mathematically equivalent. This completes the proof.
Proof. of Corollary 1:
(a) Equation (3) is equivalent to:
W
L− 1 < (β − α)p. (13)
If W < L, then the left-hand side of equation (13) is negative. Since the right-hand side
of (13) is non negative (because β > α and p ≥ 0), inequality (13) holds. By part (a) of
Proposition 1, the student prefers to attend class.
(b) Similarly, if W > L and β < α, then W > L > L+ p(β − α)L. Hence, condition (4)
holds. By part (b) of Proposition 1, the result holds. This concludes the proof.
Proof. of Proposition 2:
Since fL(l) > 0, ∀l ∈ [0,+∞); and fW (w) > 0, ∀w ∈ [0,+∞), then attendance A(p;α, β)
- given by equation (5) - is the integral of the joint density fL,W (l, w) ≥ 0, a function that is
positive on some relevant region of the L×W plane. The sign of this integral depends only
on the relative order of the integration limits. Three cases may occur:
(a) Case β < α. The function p 7→ l + (β − α)lp is decreasing. This implies that the
function p 7→ A(p;α, β) is also decreasing.
22
(b) Case β > α. The function p 7→ l + (β − α)lp is increasing, which implies that the
function p 7→ A(p;α, β) is also increasing.
(c) Case β = α. Attendance is a constant function because when α = β, the function
p 7→ l + (β − α)lp is constant. This concludes the proof.
Proof. of Proposition 3:
The gap G(p;α, β) is equal to the integral of the joint density fL,W (l, w) ≥ 0, a function
that is positive on some relevant region of the L×W plane. The sign of this integral depends
only on the relative order of the integration limits. When β = α, then G(p;α, β) = 0. Other
two possibilities may occur.
Case (a): β < α. Equation (9) reveals that the gap is positive when p > 0. Moreover,
l− (α−β)lp ≤ l, and equality holds if and only if p = 0. Hence, the function p 7→ G(p;α, β)
is increasing. Clearly, G(p;α, β) is maximized when p = 1, and minimized when p = 0.
Case (b) β > α. Equation (10) reveals that the gap is negative when p > 0. Moreover,
l+ (β−α)lp ≥ l, and equality holds if and only if p = 0. Hence, the function p 7→ G(p;α, β)
is decreasing. Clearly, G(p;α, β) is minimized when p = 1, and maximized when p = 0.
Part (c): Since p > 0, then equation (9) reveals that the region that we integrate
fL,W (l, w) ≥ 0 increases in α, and decreases in β. Thus, an increase in α leads to a larger
gap, and an increase in β leads to a smaller gap. This concludes the proof.
Proof. of Claim 1:
Attendance is given by:
A =
∫ +∞
l=0
∫ l+(β−α−)lp
w=0
fL,W (l, w) dw dl − θ∫ +∞
l=0
∫ l+(β−α−)lp
w=l+(β−α+)lp
fL,W (l, w) dw dl, (14)
or
A =
∫ +∞
l=0
∫ l+(β−α+)lp
w=0
fL,W (l, w) dw dl+ (1− θ)∫ +∞
l=0
∫ l+(β−α−)lp
w=l+(β−α+)lp
fL,W (l, w) dw dl. (15)
Condition (11) is equivalent to:∫ +∞
l=0
∫ l+(β−α−)lp
w=l+(β−α+)lp
fL,W (l, w) dw dl ≥ 0. (16)
Since by hypothesis p > 0, then inequalities (11) and (16) hold strictly. Equation (14)
shows that attendance is an affine and decreasing - because of inequality (16) - function of θ.
Equation (15) is obtained by subtracting and adding to the right-hand side of equation(14)
the term: ∫ +∞
l=0
∫ l+(β−α−)lp
w=l+(β−α+)lp
fL,W (l, w) dw dl,
23
which is equal to:∫ +∞
l=0
∫ l+(β−α−)lp
w=0
fL,W (l, w) dw dl −∫ +∞
l=0
∫ l+(β−α+)lp
w=0
fL,W (l, w) dw dl.
From (15) and inequality (16), it is clear that attendance is increasing in 1−θ. This concludes
the proof.
Proof. of Claim 2:
A direct calculation reveals that D > 0 if and only if:∫ +∞l=0
∫ l+lpβ−lpα−w=0
fL,W (l, w) dw dl∫ +∞l=0
∫ l+lpβ−lpα+
w=0fL,W (l, w) dw dl
> 1.
This inequality is equivalent to (11), which itself is a direct consequence of the hypothesis
α+ > α−. This concludes the proof.
Proof. of Proposition 4:
For every p ∈ [0, 1] and every α > 0, let h(p, α) be defined as follows:
h(p, α) =
∫ +∞
l=0
∫ l+lpβ−lpα
w=0
fL,W (l, w)dwdl.
Consider the following function:
p 7→ h(p, α−)
h(p, α+). (17)
This function is increasing (respectively, constant, or decreasing) if and only if the deviation
D from the population proportions in class composition is increasing (respectively, constant,
or decreasing) in p. This is true because D is the composition of two decreasing functions
of h(p, α−)/h(p, α+); that is:
D = θ − A+
A= θ −
(1 +
(1− θ)θ
h(p, α−)
h(p, α+)
)−1
.
The function defined on equation (17) is increasing if its derivative is positive, which mean
that:
h(p, α+)∂h(p, α−)
∂p> h(p, α−)
∂h(p, α+)
∂p. (18)
Apply Leibniz integral rule to compute the derivative of h(p, α) with respect to p:
∂h(p, α)
∂p= (β − α)
∫ +∞
l=0
l.fL,W (l, l + lpβ − lpα)dl.
Thus, the function defined by (17) is increasing if:
h(p, α+)(β−α−)
∫ +∞
l=0
l.fL,W (l, l+ lpβ− lpα−)dl> h(p, α−)(β−α+)
∫ +∞
l=0
l.fL,W (l, l+ lpβ− lpα+)dl.
(19)
24
If α− ≤ β ≤ α+, then inequality (19) is true as its left-hand side becomes positive (or
zero when β = α−) and it right-hand side becomes negative (or zero when β = α+). This
concludes the proof.
Proof. of Proposition 5:
Let FW/L denote the cumulative distribution of the ratio W/L, and fW/L the correspond-
ing density function. Observe that:
h(p, α) = FW/L(1 + (β − α)p) ,
and taking the derivative with respect to p:
∂h(p, α)
∂p= (β − α)fW/L(1 + (β − α)p) .
Case (1): β < α− and log concave FW/L. When β < α− (which implies that β < α+),
inequality (18) is equivalent to:(α− − βα+ − β
)fW/L(1 + (β − α−)p)
FW/L(1 + (β − α−)p)<fW/L(1 + (β − α+)p)
FW/L(1 + (β − α+)p). (20)
The first factor on the left-hand side is positive and smaller than 1 because β < α− < α+.
The hypothesis that FW/L is log concave means that the function y 7→ fW/L(y)/FW/L(y) is
weakly decreasing. Since 1 + (β − α−)p > 1 + (β − α+)p, then inequality (20) holds. This
implies that inequality (18) also holds, proving the result for this case.
Case (2): β > α+ and log convex FW/L. When β > α+ (which implies that β > α−),
inequality (18) is equivalent to:(β − α−β − α+
)fW/L(1 + (β − α−)p)
FW/L(1 + (β − α−)p)>fW/L(1 + (β − α+)p)
FW/L(1 + (β − α+)p). (21)
The first factor on the left-hand side is positive and greater than 1 because β > α+ > α−.
The hypothesis that FW/L is log convex means that the function y 7→ fW/L(y)/FW/L(y) is
weakly increasing. Since 1 + (β − α−)p > 1 + (β − α+)p, then inequality (21) holds. This
implies that inequality (18) also holds, proving the result for this case as well.
Proof. of Corollary 2:
If α− ≤ β ≤ α+, then the result follows from Proposition 4. If β < α−, then the result
comes from Proposition 5 because the distribution of W/L is log concave. In both cases, the
deviation in the class composition is an increasing function of the probability p, and making
p = 1 maximizes this deviation. This completes the proof.
Proof. of Corollary 3:
If α− ≤ β ≤ α+, then the result follows from Proposition 4. Suppose that β < α− and L
is constant and W is continuously distributed. Then, since L is constant, the log concavity
25
of W is equivalent to the log concavity of the ratio W/L. Thus, the result comes from
Proposition 5. Hence, the following function is decreasing:
p 7→ A+
A=
(1 +
1− θθ
FW (L+ Lpβ − Lpα−)
FW (L+ Lpβ − Lpα+)
)−1
.
Thus, the function p 7→ D is increasing. This completes the proof.
Proof. of Corollary 4:
If α− ≤ β ≤ α+, then the result is a direct consequence of Proposition 4. Suppose that
β < α−, and W = w is always constant, say W = w, for some fixed w ∈ R, and L is
continuously distributed. Then, A = A+ + A−, where:
A+ = θ
[1− FL
(w
1 + pβ − pα+
)],
A− = (1− θ)[1− FL
(w
1 + pβ − pα−
)].
Hence:
A+
A=
1 +
(1− θθ
)1− FL(
W1+pβ−pα−
)1− FL
(W
1+pβ−pα+
)−1
.
To see how A+/A changes with p, we need to analyze the monotonicity of the following
function:
p 7→1− FL
(W
1+pβ−pα−
)1− FL
(W
1+pβ−pα+
) . (22)
This function is increasing if and only if:
R(α−)
R(α+)(β − α−) > (β − α+)
(1 + pβ − pα−1 + pβ − pα+
)2
, (23)
where, for every α > 0, R(α) is defined as:
R(α) =fL
(W
1+pβ−pα
)1− FL
(W
1+pβ−pα
) .
Since β < α− < α+, then equation (23) becomes:
R(α−)
R(α+)<
(α+ − β)
(α− − β)
(1 + pβ − pα−1 + pβ − pα+
)2
.
By hypothesis, α 7→ R(α) is weakly increasing. Hence, R(α−)/R(α+) ≤ 1. Since α− < α+
and β < α−, then some algebra reveals that:
1 <(α+ − β)
(α− − β)
(1 + pβ − pα−1 + pβ − pα+
)2
.
26
Thus:R(α−)
R(α+)≤ 1 <
(α+ − β)
(α− − β)
(1 + pβ − pα−1 + pβ − pα+
)2
.
The function defined by (22) is increasing. Hence, the function p 7→ A+/A is decreasing.
Therefore, the function p 7→ D is increasing. This completes the proof.
Proof. of Claim 3:
Direct calculations show that all inequalities in (12) are mathematically equivalent to
p < p++. Since p++ = min{p−−, p+−, p−+, p++} and p < p++, then p < p+−, p < p−+, and
p < p−−. Thus, by Proposition 1, all types will go to class. Conversely, if all students go
to class, then, by Proposition 1, p < min{p−−, p+−, p−+, p++} = p++. This completes the
proof.
Proof. of Claim 4:
If p < p++, then Claim 3 establishes that all students go to class. Under full attendance,
Aα(p) = θ++ +θ+−. By definition, the α-deviation in composition is equal to zero. Similarly,
under full attendance, Aw(p) = θ++ + θ−+. By definition, the w-deviation in composition is
equal to zero. This completes the proof.
Proof. of Claim 5:
If p++ < p < p+− = p−+, by Proposition 1, only type ++ students do not attend class.
Proof of part (a). A measure of θ−+ + θ+− + θ−− go to lecture. Out of these students, a
measure θ+− has α = α+. Hence
Aα(p) =θ+−
θ−+ + θ+− + θ−−.
Using this and the definition Dα(p) = θ++ + θ+− − Aα(p), we obtain:
Dα(p) = θ++ + θ+− −θ+−
θ−+ + θ+− + θ−−.
A direct computation reveals that:
dDα(p)
dθ++
> 0 anddDα(p)
dθ−−> 0.
This completes the proof of part (a).
Proof of part (b). A measure of θ−+ + θ+− + θ−− students go to lecture. Out of these
students, a measure θ−+ has w = w+. Hence:
Aw(p) =θ−+
θ−+ + θ+− + θ−−.
Using this and the definition Dw(p) = θ++ + θ−+ − Aw(p), we obtain:
Dw(p) = Dw(p) = θ++ + θ−+ −θ−+
θ−+ + θ+− + θ−−.
27
A direct computation reveals that:
dDw(p)
dθ++
> 0 anddDw(p)
dθ−−> 0.
This completes the argument of part (b), and concludes the proof.
Proof. of Claim 6:
Since p+− = p−+ < p < p−−, by Proposition 1, only students of type −− attend lecture.
Proof of part (a). Since only type −− students attend lectures, Aα(p) = 0, and this
implies that Dα(p) = θ++ +θ+−. Clearly, Dα(p) = θ++ +θ+− increases in both θ++ and θ+−.
This finishes the proof of this part.
Proof of part (b). Since only type −− students attend lectures, Aw(p) = 0, and this
implies that Dw(p) = θ++ + θ−+. Clearly, Dw(p) is increasing in both θ++ and θ−+. This
finishes the argument for this part, and completes the proof.
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