Ability Tracking or Comprehensive Schooling?

A Theory on Peer Effects in Competitive and

Non-Competitive Cultures

Kathrin Thiemann∗

February 10, 2016

Abstract

We develop a model of student decision making that shows that it depends on the

culture of competitiveness in a country or region whether it is optimal to choose a

school design with ability tracking or comprehensive schooling. Students with differ-

ent cultural background differ in their concern for relative position in the classroom,

which is modeled by reference dependent preferences. We contrast competitive cul-

tures, where students compare their performance to the best performance in class,

and non-competitive cultures where the reference point is the average performance.

Taking into account students with heterogeneous abilities, we show that the average

performance in competitive cultures is maximized under comprehensive schooling

and in non-competitive cultures under ability tracking. Segregation of abilities in

non-competitive cultures, however, leads to a higher dispersion of performances.

This means high ability students perform higher and low ability students lower

under ability tracking than under comprehensive schooling.

JEL-Code: I28, J24, D83

Keywords: Loss Aversion, Reference Dependence, Ability Tracking, Peer Effects,

Culture, Competitiveness

∗Department of Economics, University of Hamburg, von-Melle-Park 5, 20146-Hamburg, Germany.Phone: +49 40 42838 6331. Email: kathrin.thiemann@uni-hamburg.de.

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1 Introduction

Motivation Learning behavior of students differs to a huge extent with respect to their

cultural background. These differences in learning styles are unconsciously adopted by

students due to cultural differences in cognitive patterns and modes of thinking. In

economic research that strives to determine optimal school systems and teaching practices,

cultural differences in learning behavior should thus play a crucial role. However, culture

as a determinant for outcomes in education has received little attention in economic

research so far.

In a classroom many aspects that are influenced by culture play a role, such as self-

control of students, student interactions or student-teacher relationships. In this paper

we are concentrating on educational peer effects, that is the question of how the perfor-

mance of classmates influences the individual student’s performance. We assume that

this influence works through the channel of social comparison, which varies in its extent

and nature from culture to culture. For instance, for students from competitive and per-

formance oriented cultures comparison to higher performing peers is essential for their

self-perception and drives them to try to be more successful than their classmates. Dif-

ferences in the quality of peer effects between cultures may thus stem from differences in

competitiveness. Evidence for this link of cultural background and competitive behavior

has been found by Gneezy et al. (2009). They conduct an experiment in a Maasai tribe

in Tanzania and a Khasi tribe in India, showing that Khazi women are selecting into a

competitive environment much more often than Maasai women. The reverse is true for

men from the two tribes.1

The constellation of classmates, in particular whether they are of high or low ability,

accordingly influences the individual student’s effort choice. Therefore a crucial question

that schools and governments face is whether students should be grouped according to

their ability or whether students of all abilities should be educated together. This contro-

versy between ability tracking (also referred to as streaming, phasing or ability grouping)

and comprehensive schooling manifests itself already in the fact that some countries seg-

regate students into different schools as early as age 10 (e.g. Austria, Germany, Hungary,

Slovak Republic) whereas others (e.g. Canada, Japan, Norway, Sweden, United Kingdom,

United States) keep a comprehensive school system till the end of secondary education.

However, also in countries with an entirely comprehensive school structure, placement

into ability tracks within schools is frequently practiced. Here the arguments for or

1Competitive behavior here means that individuals choose an incentive scheme that does not only paythem based on their performance but promises higher returns if they outperform other individuals in agiven task.

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against ability tracking stay essentially the same. The main trade-off is perceived to be

between equity and efficiency. Especially the inequality increasing effect of tracking is

well-documented in the literature (e.g. Hanushek and Woessmann (2006)). Whether the

effects of ability tracking differ systematically with different cultures of competitiveness

has to the best of our knowledge not been investigated so far.

The concept of cultural differences in the domain of competitiveness has been consid-

ered in cross-country value surveys such as the GLOBE study (House et al., 2004) or the

Hofstede dimensions (Hofstede et al. (2010)). For instance they show that South East

Asian countries, Western European countries and the USA are more performance oriented

and competitive than countries from Eastern Europe, the Middle East or Africa. Apply-

ing his cultural dimensions to teaching and learning behavior at school Hofstede (1986)2

describes a competitive or as he terms it ”masculine” culture as one where students are

overly ambitious, where the best student is the norm and where failures are seen as dis-

asters. In non-competitive or ”feminine” cultures students practice mutual solidarity, the

average student is the norm and failures in school are relatively minor incidents. These

categorizations of student behavior in competitive and non-competitive cultures serve as

a foundation of our theoretic modeling.

Framework We develop a model of student decision making that allows us to distinguish

between competitive and non-competitive cultures. With this we show that it depends on

the culture of competitiveness whether a school design with ability tracking or comprehen-

sive schooling is to be preferred. The student’s effort choice including social comparison

is modeled by reference dependent preferences as in Tversky and Kahneman (1979). We

contrast two extremes, namely competitive cultures, where the students’ reference point

is the best performance in class, and non-competitive cultures where students compare

their performance to the average performance. Students that perform below their refer-

ence point suffer from a high utility loss due to loss aversion. To avoid this loss of utility

students are strongly motivated to reach their reference performance. Due to their high

reference point students in competitive cultures are more affected from loss aversion than

students in non-competitive cultures. Accordingly we find a kind of following behavior

of students with respect to their reference point. This means the higher loss aversion the

harder students work in order to reach their reference point. Assuming that a certain cul-

ture of competitiveness in a country is given, the policy maker is left with the decision on

an appropriate school system. Taking into account students with heterogeneous abilities,

2See also Oettingen (1995) for a more detailed description of learning and teaching behavior in differentcultures.

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we compare a comprehensive school design, where all abilities are together in one class

with an ability tracked school design, where high abilities are sorted into a high track and

low abilities respectively into a low track. We determine average performance and the

variance of performances under the two systems as decision criteria.

Results We show that in a competitive culture a comprehensive school system provides

a higher average performance and a lower variance of performances. A comprehensive

school system is thus always to be preferred in a competitive culture. The intuition is

that also low ability students have well performing students to look up to, which is highly

motivating. In the case of a non-competitive culture ability tracking provides a higher av-

erage performance. However, we face a trade-off between maximizing performances and

minimizing inequality, since a segregation of abilities here leads to a higher dispersion

of performances. This means that high ability students perform higher and low ability

students perform lower under ability tracking than under comprehensive schooling. The

gain from high ability students’ performances, however, is higher than the loss from low

ability students’ performances.

The remainder of this paper is organized as follows: Section 2 provides an overview of

the related literature. Section 3 introduces the general model and characterizes optimal

performance. In section 4 we analyze a competitive culture, where the best student’s per-

formance is the reference point, and in section 5 turn to a non-competitive culture, where

it is the average student’s performance. Section 6 discusses the results and introduces

some extensions. Section 7 concludes.

2 Related Literature

Most closely related to our work are theories on peer effects that consider peers inter-

actions as a determinant on the decision between comprehensive schooling and ability

tracking. There are furthermore empirical studies that try to determine what system is

to be preferred. To the best of our knowledge there are no studies that incorporate the

influence of culture into research on peer effects.

An overview of the literature on peer effects is provided by Epple and Romano (2011).

Usually peer effects are tackled by incorporating mean ability in class in the education

production function. In this linear-in-means model mean ability in class is assumed to

have a positive linear influence on the individual student’s performance. In the presence

of linear peer effects the overall sum of students’ performances is still equally high when

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students of all abilities are together in one class or in classes grouped by ability. Thus,

there are no efficiency gains from ability tracking compared to comprehensive schooling.

However, inequality would increase under ability tracking, since high ability students

would gain from the high mean ability in the high track and low abilities would suffer

from the low mean ability in the low track. Differences in efficiency between ability

tracking and comprehensive schooling, can be found in the presence of non-linear peer

effects. For instance in an early paper Arnott and Rowse (1987) attempt to find a rationale

for the optimal school system by maximizing a welfare function in which welfare increases

in the sum of all students’ final skills, but decreases with inequality. Mean ability in class

here enters a Cobb-Douglas production function of students’ skills, representing the peer

effect. However, no clear cut recommendation on the optimal school design can be made,

since results depend sensitively on the exponents in the production function.

More recent work by Benabou (1996) suggests that the peer effect (average ability)

that enters the educational production function can be measured by a CES (constant

elasticity of substitution) index. In the case that the elasticity of substitution is smaller

than 1, different abilities in the classroom are substitutes, meaning that heterogeneity of

students is a source of gain. As Argys et al. (1996) have shown, comprehensive schooling

then leads to efficiency gains compared to tracking. This is because low ability students

gain substantially more from comprehensive schooling than high ability students lose

in a comprehensive school class compared to ability tracking. The opposite is true if

the elasticity of substitution is bigger than 1, that is when heterogeneous abilities are

complements. Here heterogeneity of students is a source of loss. The gains of high

ability students under tracking outweigh the losses of the low ability students under

tracking. Thus ability tracking yields efficiency gains here. There are studies surveying

students’ behavior suggesting that abilities rather work as complements (see Foster and

Frijters, 2010), however, this literature says little about what determines the elasticity of

substitution. One contribution of our theory is that we argue that this could be due to

culture. In addition our contribution to the existing theoretical literature is that we do

not only consider peer effects being driven by the average performance but also by the

best performance.

Experimental and field studies on peer effects document that peers do have an influence

on the individual student’s performance (e.g. Falk and Ichino, 2006; Kandel and Lazear,

1992; Sacerdote, 2001; Zimmerman, 2003). It also is investigated how the composition of

one’s peers, that is their ethnicity, income and family background, adds to this influence

(e.g. Hoxby, 2000; Ammermueller and Pischke, 2009). More interesting for our research

are investigations into how the ability distribution in class influences peer effects. Lavy

5

et al. (2012) explore whether the exceptionally low- and high-achieving peers as opposed

to average performing classmates, drive the impact of peers performance on students’

achievements. They mainly find evidence for extremely low ability students having a

negative impact on their peers achievement. Hoxby and Weingarth (2005) have proposed

several models of peer effects to describe different peer behaviors. Among them is the

”Shining Light” model, where ”a single student with sterling outcomes can inspire all

others to raise their achievement” and the ”Bad Apple” model, where ”the presence of

a single student with poor outcomes spoils the outcomes of many other students” Hoxby

and Weingarth (2005, p.6). Evidence is not found for either of these models, nor for the

popular linear-in-means model.

There is a strand of literature that tries to empirically determine whether ability track-

ing or comprehensive schooling is to be preferred. Most of this literature does not contrast

systems in different countries, but analyzes the differences between ability grouping and

comprehensive teaching in one school or country. There are roughly two categories of

studies, those that try to estimate the effect of ability tracking on mean achievement and

those that look at the effects on the distribution. Effects on mean achievement of ability

grouping are usually low and non-significant (a review is presented in Slavin, 1990). In

terms of distribution studies usually find that tracking harms low ability students but ben-

efits high ability students (e.g. Argys et al., 1996; Hoffer, 1992). Still, there are also some

studies that have exploited the regional variation across countries in terms of tracking

and relate this to the data from international student achievement tests (e.g. Hanushek

and Woessmann, 2006; Brunello and Checchi, 2007). Predictions of these studies include

that ability tracking does not have a significant impact on students’ performance, but it

does increase educational inequality. This coincides with theory on linear peer effects.

Among the related research is also the growing literature on loss aversion, based on

original work on prospect theory by Tversky and Kahneman (1979) and Tversky and

Kahneman (1991), henceforth referred to as KT (1979) and KT (1991). Loss aversion

means that, compared to a reference point, ”losses loom larger than gains” (KT, 1979,

p.279). In terms of numbers it is often suggested that individuals dislike losses twice

as much as they like equally high gains (e.g. Johnson et al., 2006). For example a

loss of one Dollar starting from a reference point of zero Dollars hurts utility twice as

much as a gain of one Dollar benefits utility. The phenomenon has been used to explain

outcomes in diverse fields of research. Closest to our research is the literature referred

to as ”Catching up with the Joneses” (e.g. Abel, 1990; Gali, 1994). Originally used in

the context of asset pricing this literature assumes that individuals do not only get utility

from their absolute level of income or consumption, but also from relative comparison

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to some social reference group. Thus, mean income or mean consumption of neighbors,

peers or colleagues is incorporated into prospect theory as a reference point. Clark and

Oswald (1998) develop a micro-economic model of behavior, when individuals care about

relative position, that is they exhibit loss aversion compared to mean action in society.

When their peers start to act in a certain way, individuals find that their own marginal

utility from acting that way increases because of loss aversion. The model thus predicts

following and herding behavior, a phenomenon that we also observe in our model. In

an educational setting the concept of loss aversion has been used by Levitt et al. (2012),

who conduct experiments on students and find that incentives framed as losses motivate

more than incentives framed as gains. To the best of our knowledge the concept of loss

aversion has not been used in an educational setting with respect to the performance of

classmates.

3 The Model

Consider a population of students, where each student simultaneously chooses its optimal

performance level pi.3 The utility maximization problem including reference dependent

preferences according to KT (1979) is given by:

Maxpi

ui = (1 − s)pi + s ∗ v(pi − ri) − c(pi, ai) (1)

Utility depends on a linear combination of a direct private component of utility and a

comparison oriented component given by the value function v(⋅). The reference point

ri is the performance student i is comparing his own performance to. The s, with 0 <s ≤ 1, is a weight of private utility versus comparison oriented utility. As s → 0 the

standard non-behavioral model holds, where preferences are only self-interested. As s→ 1

relative position becomes more important, thus s can be interpreted as the degree of

social comparison. Costs are given by c(pi, ai) = p2i

2ai. They are convex in performance and

linearly decreasing in ability ai. Students are heterogeneous in ability, thus high ability

students face lower costs than low ability students. Specifying the value function we look

at the simple linear case:

v(pi − ri) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

(pi − ri) if pi > riλ(pi − ri) if pi < ri

(2)

3This is equivalent to an effort-choice model, when assuming that effort linearly translates into per-formance without any noise.

7

Students compare their performance to a reference point ri. They get a positive utility

as high as the difference between their own performance and the reference point if they

perform higher than their reference point. Likewise they suffer from a loss in utility if

they perform lower. Losses in utility are higher than the simple difference between ri and

pi, because of loss aversion. This is captured by λ, the coefficient of loss aversion, with

λ > 1. Students thus dislike losses more than they like equally high gains. Overall utility

is linearly increasing in performance, but the student has a higher marginal utility in the

pi < r case. There is thus a kink of the value function at the reference point.

How does culture enter our model? Assume, first of all, that all students in one

classroom have the same culture. Thus culture only varies from country to country.

We assume that culture is transmitted from one generation to the next by parents and

peers, but also through institutions. That is teachers, curricula and performance feedback

rules at schools also shape the preferences of students. The aspect of culture we are

looking at here is the tendency to competitive behavior. The model above allows us to

integrate concepts of competitiveness at three points consistent with Hofstede’s (1986)

interpretation of competitiveness: The first indicator is the reference point ri, which is

assumed to be higher in more competitive cultures. That is students compare their own

performance to a higher reference performance in class. Reference point setting is also

influenced by teachers. Oettingen describes that teachers in competitive countries ”single

out high-achieving students as the ideal” Oettingen (1995, p.156) and highlight their

academic successes in front of the class. Secondly, λ the coefficient of loss aversion, can

be interpreted as an indicator of competitiveness. According to Hofstede for students

in competitive cultures ”failure in school is a severe blow to his/her self-image” and

in non-competitive cultures ”failure in school is a relatively minor accident” Hofstede

(1986, p.315). In economic terms this translates into students from competitive cultures

exhibiting higher loss aversion. The third indicator is s, the degree of social comparison.

Since s expresses the relative importance of social comparison, a high s also represents

competitive preferences. The higher the s the more important are the reference point

and the coefficient of loss aversion. This indicator s can thus be seen as a multiplier of

the other two indicators. In general social comparison is also facilitated by institutions

in more competitive cultures. For instance regular and frequent performance feedback is

provided and test rankings are made public to the class (Oettingen, 1995). This reinforces

the high s of students’ preferences in competitive cultures.

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3.1 Characterizing Optimal Performance

Marginal benefits are different for the two cases of the value function. Assume that the

reference point ri does not depend on pi:

pi < ri ∶ MB = (1 − s) + sλ ≡ µ

pi > ri ∶ MB = (1 − s) + s = 1

For ease of notation we substitute (1 − s + sλ) ≡ µ > 1, for the rest of the analysis, since

the expression captures the joint effects of s and λ. Note that for the highest possible

s = 1, µ is equal to the coefficient of loss aversion λ. Marginal costs are MC = piai

, thus

linearly increasing in performance and steeper for lower abilities. The following graph

illustrates marginal benefits and marginal costs for a given reference point ri and abilities

a1 < a2 < a3:

1

MB,MC

MB

µ

ri

a1

a2

a3

performance

Figure 1: Marginal Benefits and Marginal Costs for Different Levels of Ability, witha1 < a2 < a3

Marginal benefits equal marginal costs at p∗i = µai below the reference point and at

p∗i = ai above the reference point. The sufficient condition for a local maximum is fulfilled

in both cases, since ∂2u∂p2i= − 1

ai< 0. In the case of figure 1 students with low ability, like

a1, reach their utility maximum below the reference point and high ability types, like

a3, optimally perform above the reference point. In the case of ability type a2 optimal

performance is found at the corner solution p∗i = ri. Observe that for ability type a2

marginal utility from performing is increasing in the pi < ri case (marginal benefits are

9

bigger than marginal costs), but decreasing in the pi > ri case (marginal costs are bigger

than marginal benefits). Thus there must exist a maximum at p∗i = ri. All ability types

for which the interior solution pi = µai is bigger than ri and those for which the interior

solution pi = ai is smaller than ri have their utility maximum at this corner solution.

Summarizing we can fully characterize optimal performance:

Lemma 1.

(i) For a given reference point ri and a given ability ai the optimal performance of

student i is given by:

pi(ri, ai) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

µai if ai < riµ case 1

ri if riµ ≤ ai < ri case 2

ai if ai ≥ ri case 3

(3)

(ii) Students’ performances are monotonously increasing in ability: ∂pi(ai,ri)∂ai

≥ 0.

Since optimal performance depends on the reference point, which is given by other stu-

dents performances, it can be termed best response function. In both interior solutions the

reference point does not influence the level of performances, however, it does determine in

which case the student performs. With a growing ri the thresholds riµ and ri that distin-

guish the cases are increasing, making a lower case performance for a given ai more likely.

In both interior solutions optimal performance is linearly increasing in ability ai. More-

over, for a given reference point ri ability determines in which case the student performs,

with a higher ability resulting in a higher case. We can thus conclude that performance is

monotonously increasing in ability (see Lemma 1.ii). The student with the highest ability

is thus also the student with the highest performance. This can also be observed in figure

1. In terms of µ performance in case 1 is linearly increasing in this parameter. That

means the more competitive the student the higher his performance below the reference

point. This is because of the higher marginal benefit from performance due to higher loss

aversion. Students above the reference point are not affected by this motivational force.

Performances here do not depend on competitiveness. Furthermore, from figure 1 it can

be seen that the higher µ the higher the range for a corner solution. Thus, the more

competitive the student the more likely that he reaches the reference point.

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3.2 The Reference Point

We already mentioned that in a competitive culture students are assumed to compare

their performance to a higher reference performance than students in non-competitive

cultures. According to Hofstede (1986) a competitive culture is one where the best student

in class is the norm, whereas a non-competitive culture is one, where students compare

their performance to the average student. These are to be thought of as extremes on

two opposite ends of possible levels of competitiveness. In line with Hofstede (1986)

we contrast the two extreme reference points: the average performance among the other

students (non-competitive reference point) and the best performance among the other

students (competitive reference point). In case of the average performance we take the

maximizing student itself out of the average, because this ensures that student i itself

cannot influence the reference point by choosing his optimal performance. In case of

the best performance the supplement ”among the other students” ensures that the best

student has someone to compare to, namely the second best student in class.

We assume that all students in one culture do not only have the same reference point,

but also the same loss aversion λ and the same degree of social comparison s, with all

indicators being higher in a competitive culture. To answer the question what school

system (comprehensive schooling vs. ability tracking) a non-competitive culture and a

competitive culture should have, we contrast the two regimes in each culture separately.

We consider two criteria to determine which system is to be preferred: the average perfor-

mance and the variance of performances under the two regimes. Usually a system would

be preferred if it provides a higher average performance, but a lower variance, assuming

inequality aversion. These goals might, however, be conflicting if there is a trade-off be-

tween increasing average performance and decreasing the variance of performances. We

start by analyzing a competitive culture and then turn to a non-competitive culture.

4 Competitive Culture

In a competitive culture the reference point is given by the best student among the other

students in class: ri = max (pj). This performance is different under a comprehensive

or ability tracked system. Consider a continuum of students, who are located uniformly

along the ability segment ai ∈ (0, a]. To find Nash equilibrium performances under a

comprehensive school regime we analyze the case of students of all ability types ai ∈ (0, a]in one class, each with the best response function from (3). In the next subsection we

look for Nash equilibria in two separate classes of a system with ability tracking. Here low

ability students ai ∈ (0, a2] are gathered in a low track and high ability students ai ∈ ( a2 , a]

11

are gathered in a high track.

4.1 Comprehensive Schooling

Proposition 1. For a given ability ai ∈ (0, a], in a competitive culture under comprehen-

sive schooling with µ = 1 − s + λs > 1, Nash equilibrium performances are given by:

p∗i (ai) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

µ ai if ai < p∗aµ

p∗a if ai ≥ p∗aµ

, for ai ≠ a (4)

p∗a ∈ [a, µa) (5)

Proof We first prove by contradiction that the highest performing student in class is

the student with the highest ability, thus student a. We then derive the equilibrium

performance of the best student in class. With this as the reference point of all other

students we get their best response functions:

(i) Suppose the best performing student in class is a student with a lower ability than

a, say a − ε with ε > 0. Since this student, by definition of the reference point, is

performing above the reference point, his performance must be pa−ε = a − ε. Now

look at the best response function of student a:

pa(pa−ε) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

µa if µa < a − ε

a − ε if µa ≥ a − ε > a

a if a ≥ a − ε

Student a’s best response is to perform a. This is, however, bigger than a − ε and

student a − ε would not be the best student in class anymore. Thus, it cannot be

an equilibrium that the highest performing student in class is not the student with

the highest ability.

(ii) Student a is thus the best performing student in class. Since he performs above or

equal to the reference point, his performance is either a or equal to the performance

of the second best student. The second best student must be a student with lower

ability than a, say a − ε, where ε > 0. Figure 2 shows the best response functions of

the best and second best student:

12

pa−ε

pa

aa − ε µaµ(a − ε)

a

a − ε

µ(a − ε)

µa BRa

BRa−ε

Figure 2: Best Response Functions of the Best and Second Best Student

Mutual best response is found at p∗a = p∗a−ε = p∗, with p∗ ∈ [a, µ(a − ε)]. The second

best student must be the student with the second highest ability (see Lemma 1.(ii)).

In a continuous distribution the second best student must be the student for which

ε is infinitely small, so that the class of possible equilibria becomes p ∈ [a, µa). The

higher µ the more possible equilibria.

(iii) All other students have the best performance as a reference point: ri = p∗a ∈ [a, µa).They will never perform above the reference point, that is in case 3. Substituting p∗ainto the best response function as in (3) yields equilibrium performances for students

ai ≠ a as in Proposition 1. ∎

Equilibrium performances as stated in Proposition 1 are increasing in µ as long as the

student is performing below the reference point. For a high enough µ the student even-

tually switches to performance at the reference point ri = p∗a, namely once µ ≥ p∗aai

. For

µ→∞ performances converge to a symmetric equilibrium with every student performing

at p∗ai . Figure 3 illustrates equilibrium performances depending on µ for a discrete choice

of students from the ability distribution. The figure is an example, where the best student

performs at the lower bound of possible equilibria, namely at pa = a. This performance is

highlighted in black. The other students perform at p∗i = µai until µ ≥ aai

and henceforth

they also perform at p∗i = a. Other possible equilibria look like the below figure, except

that the performance of student a is higher and it takes a higher µ for the other students

to reach the reference point.

13

1 2 3 4 5 6 7 8 9 10

0

a

a2

µ

perf

orm

ance

Figure 3: Competitive Culture, Comprehensive Schooling

It can be seen that higher competitiveness µ drags performances up to the best perfor-

mance in class. This is what Clark and Oswald (1998) describe as following behavior. The

implication is that performances in more competitive and in particular more loss averse

cultures are not as dispersed and generally higher than in less competitive cultures. This

is because higher loss aversion can be translated into a higher motivation of students to

reach the reference point. In the given case of the best student being the reference point,

all students except the best student itself are affected from this higher return to perfor-

mance. Remember that µ captures the joint effect of λ and s. As a benchmark empirical

estimates (e.g. Johnson et al. (2006)) mostly conclude that monetary losses are about

twice as painful as gains, suggesting a λ = 2. In terms of cultural differences Arkes et al.

(2010) find a loss aversion of about 1.65 for relatively non-competitive Asians and 1.9 for

relatively competitive US Americans. Even for the highest possible s = 1 we would thus

not expect µ to be much bigger than 2 in a competitive culture. From figure 3 it can be

seen that for µ’s of this size performances of relatively high ability students have already

converged to the best student’s performance, whereas lower ability students’ performances

are still rather low and relatively unaffected by loss aversion. Overall convergence towards

the best student’s performance is thus rather unrealistic.

4.2 Ability Tracking

Under ability tracking low ability students ai ∈ (0, a2] are taught together in a low track

and high ability students ai ∈ ( a2 , a] are taught together in a high track. Suppose that

14

ability is fully observable and students are correctly assigned to tracks according to their

ability. The following Nash equilibrium performances are derived:

Proposition 2. For a given ability ai ∈ (0, a], in a competitive culture under ability

tracking with µ = 1 − s + λs > 1, Nash equilibrium performances are given by:

High track: ai ∈ ( a2 , a)

p∗i (ai) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

µ ai if ai < p∗aµ

p∗a if ai ≥ p∗aµ

(6)

p∗a ∈ [a, µa) (7)

Low track: ai ∈ (0, a2)

p∗i (ai) =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

µ ai if ai <p∗a

2

µ

p∗a2

if ai ≥p∗a

2

µ

(8)

p∗a2

∈ [ a2, µa

2) (9)

Proof To find Nash equilibria in the tracks we again determine first, how the best student

in the respective track performs and then derive the best response functions of the other

students:

(i) High track: The best student a is performing at p∗a ∈ [a, µa) for the same reasons as

stated in the proof for Proposition 1. This is the reference point of all other students

in the high track: ri = p∗a, for ai ∈ ( a2 , a). Substituting this in the best response

function (3) yields high track equilibrium performances.

(ii) Low track: The best student here is student a2 . Again it can be shown as in in

the proof for Proposition 1 that he performs at p∗a2

∈ [ a2 , µ a2) in equilibrium. This is

the reference point for all other students in the low track: ri = p∗a2

, for ai ∈ (0, a2).

Substituting this into the best response function (3) yields low track equilibrium

performances. ∎

Equilibrium performances in the high track are the same as under comprehensive school-

ing. For these high ability students, thus, nothing changes when switching from an ability

tracked system to a comprehensive school system. A symmetric equilibrium, with all

15

students in the high track performing at p∗i ∈ [a, µa), is reached once the lowest ability

student in class a2 crosses the

p∗aµ threshold. For p∗a = a this is when µ ≥ 2 and bigger µ’s

are needed the bigger p∗a. This indicates that performances can converge to a symmetric

equilibrium at a high level already for a low level of µ. In the low track on the other

hand the reference point and thus the threshold in the case condition are now different

from the situation under comprehensive schooling. In order to change from the first case

to performing at the reference point, µ must now surpassp∗a

2

2ai, which is much lower than

under comprehensive schooling. Remember that µ can be interpreted as a form of moti-

vation for the students. Thus switching from the first case to the second in which they

lose this motivation is detrimental for performance. With the reference point being so low

in the low track low ability students lose motivation. A symmetric equilibrium in the low

track with every student performing at p∗i ∈ [ a2 , µ

a2) is reached only when µ →∞. Figure

4 illustrates equilibrium performances depending on µ for a discrete choice of students.

The figure is an example, where the best student of the high track performs at pa = a and

the best student of the low track at p∗a2

= a2 .

1 2 3 4 5 6 7 8 9 10

0

a

a2

µ

perf

orm

ance

Figure 4: Competitive Culture, Ability Tracking

Compared to the comprehensive school case it can be seen that low ability types per-

formances lose under ability tracking. Under comprehensive schooling the high reference

point drags up high ability types as well as low ability types. Under ability tracking,

however, the reference point in the low track is lowered to the performance of student a2 ,

thus constraining the motivational power of loss aversion for the low ability types. For

16

the high ability types the situation under ability tracking or comprehensive schooling is

the same.

4.3 Comprehensive Schooling vs. Ability Tracking

Formally comparing the average performances, pC and pT , and the variances of perfor-

mances, σC and σT , under the two regimes in competitve cultures leads to Proposition 3:

Proposition 3.

1. In a competitive culture the average performance under comprehensive schooling is

strictly higher than under ability tracking: pC > pT for any µ > 1.

2. The variance of performances is strictly higher under ability tracking than under

comprehensive schooling: σT > σC for any µ > 1.

The formal proof of Proposition 3 is provided in the appendix. Ability tracking is thus

never the better option in a culture, where students take the best student as a reference

point. The intuition behind this is that in comprehensive schools all students are affected

from the motivating force of the high reference point. In an ability tracked system this

high motivating force is restricted to the high ability students. Low ability students’ per-

formances lose, because they do not have a high achieving peer to look up to. Since these

low ability students do not converge to high performance levels under ability tracking,

the variance of performances is higher under a segregating regime.

5 Non-Competitive Culture

5.1 Comprehensive Schooling

The non-competitive reference point is average performance among the other students in

class. Since we are again considering an ability distribution ai ∈ (0, a] with a continuum of

students, the reference point in a comprehensive school is given by: ri = 1a ∫

a

0 pj(aj)daj =1a ∫

a

0 pi(ai)dai = pC . We do not have to take into account that student i is not considered

in the average, since he has measure 0 in the continuum. The reference point is thus the

same for every student, namely average performance in the comprehensive school class pC .

Proposition 4. For a given ability ai ∈ (0, a], in a non-competitive culture under com-

prehensive schooling with µ = 1 − s + λs > 1, Nash equilibrium performances are given by:

17

p∗i (ai) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

µ ai if ai < a√µ+µ

a√µ√

µ+1 if a√µ+µ ≤ ai <

a√µ√

µ+1

ai if ai ≥ a√µ√

µ+1

(10)

Proof

(i) We first determine average performance in a comprehensive school class pC , which

is given by:

pC = 1

a

⎛⎝∫

pC

µ

0µaidai + ∫

pC

pC

µ

pCdai + ∫a

pCaidai

⎞⎠= a

2µ + (µ − 1)(pC)2

2aµ

Solving for pC yields:

pC =aõ

õ + 1

(ii) Average performance is the reference point of all students: ri = pC . Substituting

this into the best response function (3) gives equilibrium performances as in Propo-

sition 4.

(iii) Beyond these Nash equilibria there could be symmetric equilibria, where all students

perform in the second case. The average performance (reference point) then equals

individual equilibrium performance. These equilibria exist for all reference points

that fulfil ai ≤ ri < µai for every student. For the highest and the lowest ability

student these intervals only overlap in p∗i (ai) = p∗ ∈ [a, µa) where a is the lowest

ability in the ability distribution. Since a → 0 in my model, these equilibria do not

exist. ∎

Taking a closer look at the thresholds that distinguish the cases in equilibrium performance

given in (10), it can be shown that pC = a√µ√

µ+1 increases in µ and pC

µ = a√µ+µ decreases in µ.

Take a look at the following graph to see what this means for equilibrium performances:

18

1 2 3 4 5 6

0

a

a2

µ

case 1

case 2

case 3

pC

µ

pC

perf

orm

ance

Figure 5: Case Thresholds for Students of Abilities (0, a] under Comprehensive Schooling

From the graph it can be seen in which case the students perform depending on µ. We

find the case of an Equal Division Equilibrium for µ → 1. In this type of equilibrium all

high ability students ai ∈ ( a2 , a] perform in the third case, that is above the reference point,

and all low ability students ai ∈ (0, a2] perform in the first case below the reference point.

The bigger µ, the more students, starting with students close to the reference point,

are switching to a performance at the average. This is because case 1 performance is

linearly increasing in µ such that low ability students can surpass the average. However,

also the average performance increases in µ since it encompasses case 1 performances.

The fewer students perform in the first case, the flatter is average performance in µ.

Because of this growing reference point also more and more students from case 3 fall

below this threshold. These students then also perform at the average. As µ → ∞ we

reach a symmetric equilibrium where all students perform at the reference point in case 2.

Since this reference point is approaching a as µ → ∞, we again observe a convergence

towards the highest possible performance at a as we have already seen in a competitive

culture. Figure 6 illustrates performances for some representative students depending on

µ. Average performance is highlighted in black:

19

1 2 3 4 5 6 7 8 91 10

0

a

a2

µ

perf

orm

ance

Figure 6: Non-Competitive Culture, Comprehensive Schooling

It can be seen that with a growing µ there is a convergence towards the average perfor-

mance, which is again converging towards the performance of the best student in class,

student a. This confirms the result from the analysis of the competitive culture, that

we observe generally higher and not as dispersed performances as µ increases. In a non-

competitive culture, that we consider here, we assume that µ is rather low, that is well

below 2. For these values performances under comprehensive schooling are still rather

diverging. Generally only low ability students benefit from a reference point at the aver-

age. For µ’s below 2 most students with a higher ability than a2 are not affected by loss

aversion and thus not motivated to perform higher. We now compare this to equilibrium

performances under ability tracking.

5.2 Ability Tracking

Equilibrium performances under ability tracking with students ai ∈ (0, a2] in a low track

and high ability students ai ∈ ( a2 , a] in a high track are given in Proposition 5:

Proposition 5. For a given ability ai ∈ (0, a], in a non-competitive culture under ability

tracking with µ = 1 − s + λs > 1, Nash equilibrium performances are given by:

20

Low track: ai ∈ (0, a2]

p∗i (ai) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

µ ai if ai < a2(µ+√µ)

aõ

2(√µ+1) if a2(µ+√µ) ≤ ai <

aõ

2(õ+1)

ai if ai ≥ a√µ

2(õ+1)

(11)

High track: ai ∈ ( a2 , a]

p∗i (ai) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

µ ai if ai <a(√µ+2)2(µ+√µ)

a(µ+2√µ)

2(√µ+1) ifa(√µ+2)2(µ+√µ) ≤ ai <

a(µ+2√µ)

2(õ+1)

ai if ai ≥a(µ+2

õ)

2(õ+1)

if 1 < µ < 2 (12)

p∗i = p∗ ∈ [a, µa2

) if µ ≥ 2 (13)

Proof

(i) In the low track average performance pL is given by:

pL = 2

a

⎛⎝∫

pL

µ

0µai dai + ∫

pL

pL

µ

pL dai + ∫a2

pLai dai

⎞⎠= a

2µ + 4(µ − 1)(pL)2

4aµ

Solving for pL yields:

pL =aõ

2õ + 2

This is the reference point of all students in the low track: ri = pL. Substituting

this reference point into the best response function (3) yields low track equilibrium

performances as in Proposition 5.

(ii) In the high track average performance pH is given by:

pH = 2

a

⎛⎝∫

pH

µ

a2

µai dai + ∫pH

pH

µ

p dai + ∫a

pHai dai

⎞⎠

= (µ − 1)(pH)2

aµ− aµ

4+ a

21

Solving for pH yields:

pH =⎧⎪⎪⎪⎨⎪⎪⎪⎩

a(µ+2√µ)

2(√µ+1) if 1 < µ < 2

a if µ ≥ 2

This is the reference point of all students in class ri = pH . Substituting this reference

point into the best response function (3) yields high track equilibrium performances

as in Proposition 5.

(iii) Symmetric equilibria in the high track exist for all reference points that fulfill ai ≤ri < µai for every student. For the highest (a) and the lowest ability student ( a2)

these intervals overlap in p∗i (ai) = p∗ ∈ [a, µa2 ) if µ ≥ 2. ∎

Behavior in the tracks is similar to that in a comprehensive school class, that is per-

formances converge to the respective average performance in the tracks. Look at the

following graph to see the case thresholds pH and pH

µ in the high track and pL and pL

µ

respectively for the low track:

1 2 3 4 5 6

0

a

3a4

a2

a4

pH

pH

µ

pL

pL

µ

µ

case 2

case 2

case 1

case 1

case 3

case 3

perf

orm

ance

Figure 7: Case Thresholds for Students of Abilities (0, a] under Ability Tracking

The graph shows in which case of their best response function the students are performing

depending on µ. In the high track we find an equal division equilibrium for µ → 1, such

that every student with higher ability than 3a4 performs in case 3, that is at p∗i = ai, and

every student with ability below 3a4 performs in case 1 at p∗i = µai. For the low track we

find the same pattern, only that students are divided into a group below and above a4 .

22

With µ increasing, more and more students in the high and low track perform in case

2, that is at the average performance of the respective track. Since average performance

is equal to the upper threshold, we can observe that average performance in both tracks

is converging towards the highest ability, a in the high track and a2 respectively in the

low track. The striking difference to comprehensive schooling is that in the high track

a symmetric equilibrium with every student performing in case 2 is reached already for

µ ≥ 2. In the low track this is only reached as µ →∞. Figure 8 illustrates representative

equilibrium performances as functions of µ:

1 2 3 4 5 6 7 8 9 10

0

a

a2

µ

perf

orm

ance

Figure 8: Non-Competitive Culture, Ability Tracking

On the whole performances again converge to the best student in class the higher µ. Com-

pared to comprehensive schooling, however, high ability students are converging faster.

This is because once only high ability students are gathered in a class, the reference point

is much higher. Students ( a2 ,

3a4], now find themselves below the average. This motivates

them strongly to perform higher and reach the reference point. For low ability students

the reverse is true. With only low ability types in a class the average performance is quite

low. Students ( a4 ,

a2] are now above the average and are thus not motivated at all, unlike

the situation under comprehensive schooling. Especially these relatively high ability types

in the low track are the losers in an ability tracked system. Even for very high µ’s per-

formances in the low track do not exceed a performance of p = a2 . Under comprehensive

schooling, however, the performances of these low ability types were raised much higher,

for an extreme µ even to the best student a. Altogether figure 8 suggests that low ability

students lose compared to comprehensive schooling, whereas high ability students gain in

23

terms of higher performance. Whether the average performance, however, is higher under

ability tracking or comprehensive schooling is formally analyzed in the next section.

5.3 Comprehensive Schooling vs. Ability Tracking

Formally comparing the average performances, pC and pT , and the variances of perfor-

mances σC and σT , under the two regimes in non-competitive cultures leads to Proposi-

tion 6.

Proposition 6.

1. The average performance in a non-competitive culture is strictly higher under ability

tracking than under comprehensive schooling (pT > pC) if 1 < µ < 4.

2. The variance of performances in a non-competitive culture is strictly higher under

ability tracking than under comprehensive schooling: σT > σC for any µ > 1.

A formal proof of Proposition 6 is given in the appendix. Proposition 6.1. states that

ability tracking yields a strictly higher average performance in a non-competitive culture

for 1 < µ < 4. Since for the maximum s = 1 a µ = 4 implies a coefficient of loss aversion

of λ = 4, we see that this range covers most realistic s − λ− combinations. The result

that ability tracking outperforms comprehensive schooling when the reference point is the

average performance is therefore almost independent of loss aversion and the degree of

social comparison. The differences between a competitive and non-competitive culture

mainly stem from the difference in reference points.

Figure 8 suggests that ability tracking yields a higher average performance because

high ability students benefit more than low ability students lose compared to a comprehen-

sive school system. To understand that this must be the case consider the main difference

between the two school systems. Basically the policy-maker has to decide whether he

wants to boost the performance of rather high ability students ( a2 ,

3a4] by putting them

below the average performance in an ability tracked system. Or, he can boost performance

of lower ability students ( a4 ,

a2] that are performing below the average in a comprehensive

school. Motivating high ability students, however, is more beneficial than motivating low

ability students, since their performance increases more steeply in µ due to their lower

costs. Thus average performance must be higher under ability tracking. However, we face

a trade-off between maximizing performances and minimizing inequality, since a segrega-

tion of abilities here leads to a higher variance of performances. This result mirrors past

empirical research (e.g. Argys et al. (1996), Hoffer (1992)) that already argues that ability

24

tracking benefits good students but harms low ability students. A clear recommendation

on the optimal school system cannot be given, but depends on the political objective. If

the political claim was not to leave any child behind, then ability tracking should not be

chosen.

We have now compared ability tracking and comprehensive schooling both in a competi-

tive culture (best student benchmark) and in a non-competitive culture (average student

benchmark). Table 1 summarizes the results so far:

Table 1: Summary of Results

Average Performance Variance

pC > pT σT > σC

Non-Competitive Culture µ > 4 µ > 1

Competitive Culture µ > 1 µ > 1

The results show that in competitive cultures comprehensive schooling is the better regime

since it provides a higher average performance and a lower variance. In a non-competitive

culture we face a trade-off between the two objectives. One important result, that is also

consistent with existing empirical studies (e.g. Hanushek and Woessmann (2006)), is that

ability tracking always increases the variance and thus the inequality among students.

6 Discussion

6.1 Non-linear Value Function

The simplified linear value function used in our model might be subject to criticism.

Compared to the original value function from Tversky and Kahneman (1979) it does not

model diminishing sensitivity. That means whereas our linear value function assumes

that all students below the reference point face equal marginal utility from performance,

marginal utility is differently high in a non-linear function depending on how close to the

reference point a student’s performance is. The value function from the original Tversky

and Kahneman (1979) model is convex below the reference point and concave above:

v(pi) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

(pi − r)α if pi ≥ r

−λ(r − pi)β if pi < r,with v′′(pi) < 0 for pi > 0 and v′′(pi) > 0 for pi < 0

25

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3value

performance

Figure 9: Value Function with Diminishing Sensitivity and r = 0

Diminishing sensitivity as one part of prospect theory has not played a major role in eco-

nomic research. Barberis (2012) summarizes that diminishing sensitivity, in contrast to

loss aversion and reference dependence, is much less important and has played an unclear

role. In our school environment a value function like this would imply that students just

below the reference point are highly motivated to reach the performance of the reference

student, whereas students far away are less motivated. In our performance context this

assumption does not seem to be unrealistic. For instance, Gill and Prowse (2012) show in

a real effort experiment on disappointment aversion that competing students are discour-

aged by a high effort choice of their opponent. That is, they exert less effort the less likely

it seems to them that they can reach the performance of their counterpart. Even though

diminishing sensitivity is not modelled with our linear value function, it is taken account

of this effect by the convex cost function that depends negatively on students’ ability.

This evokes that students with lower ability are less motivated to reach the reference

point since a marginal increase in performance is much more costly to them compared to

high ability students. Remember that equilibrium performance below the reference point

is given by pi = µai, and thus increases in ability, reflecting the different motivation of

different ability types. Despite the intuitive importance of diminishing sensitivity, using

the above value function would lead to unrealistic results. Together with the convex costs

in my model, there would not be any interior solutions in the convex part of the value

function. There would, however, be a corner solution, where students do not perform at

all. In the case of the best student benchmark there would only be extreme equilibrium

performances at the reference point or at zero performance. This would also hold for linear

costs. These scenarios do not seem to reflect peer effects realistically. Another possible

criticism is that symmetric equilibria at the reference point do only exist in our model,

because of the kink in the linear value function at the reference point. In the case of the

26

above non-linear value function there would not be the corner solution at the reference

point, since the function is continuous here. However, there would still be a convergence

towards the reference point with increasing competitiveness, since the function becomes

steeper around the reference point with increasing λ and increasing s.

6.2 Multi-Track System

So far we have only tackled the alternatives of a comprehensive school class versus a two

track system. What happens if we institute more than two tracks? In the case of a

competitive culture with the best student as a reference point, the introduction of tracks

is never beneficial. Introducing even more than two tracks further worsens the situation.

Average performance decreases in the number of tracks and the variance of performances

increases in the number of tracks. In a non-competitive culture the picture is not as clear.

Three equally sized tracks yield a higher average performance than a two track system for

1 < µ < 1.63. Four tracks yield a higher average performance than a three track system

only for 1 < µ < 1.38. Thus it depends on the exact preferences of the students whether

more than two tracks still increase average performance. The institution of more than

four tracks will not be beneficial. Also, the variance of performances increases further

the more tracks we introduce. The trade-off between increasing average performance and

minimizing inequality thus aggravates.

6.3 Tracks of Unequal Size

So far we have only taken into account tracks of equal size. Can it be beneficial to in-

troduce tracks in different sizes? As an example consider Germany, where the school

system is split into three schools for different ability types. The existing system has fa-

vored high inequality of performances, with a small proportion of students forming an

elite that is exceptionally well-performing in international comparisons, but a large pro-

portion of students from the lower end of the distribution that is performing particularly

poorly (Baumert et al., 2001). Voices have become loud that call for an abolition of the

stratification in favor of a comprehensive school system. However, many do not want to

give up the ”Gymnasium”, the academic track in the school system, since they are afraid

that low performing students will slow down high performers. There are suggestions to

merge the two lower school types, while keeping the ”Gymnasium”. There would thus

be a smaller high track to optimally incentivize high ability students, while putting the

majority of students into a lower track to avoid losing very low ability students. With

our model we can analyze the possible effects of the merging. Assuming that Germany

27

is a non-competitive culture, take a look at figures 10 and 11 to see the impacts of such

a policy. Compared to a three track system the suggested system will decrease average

performance.4 The targets aimed at, however, are reached. For high performing students

nothing changes, still the variance of performances decreases. The losers are average abil-

ity students that have been in the middle track under the three track system. Low ability

students benefit from the merging of the two lower tracks, but this does not compensate

the loss in performance of the average ability types.

1 1.5 2 3 4 5 6 7 8

0

a

2a3

a3

µ

perf

orm

ance

Figure 10: Three Equally Sized Tracks

1 1.5 2 3 4 5 6 7 8

0

a

2a3

a3

µ

perf

orm

ance

Figure 11: Merging Low and Middle Track

6.4 Participation Constraint

Our model, especially in the case of a competitive culture, rests on the assumption that

all students are motivated by loss aversion to reach the performance of the best student

in class. The higher the reference point the higher is average performance in class in our

model. Loss aversion with respect to a very high reference point, however, also means

that students suffer from a high loss of utility. Some might argue that students would

rather opt out of competition in order to avoid this loss of utility. For instance research

has shown that especially in competitive cultures the high degree of social comparison

with respect to the best students in class, can harm the self-image of low performing

students. Oettingen (1995) compares self-efficacy beliefs of students in West and East

Berlin. Atmosphere at Eastern Berlin schools is described as very competitive, with high

degrees of public evaluation of students and social comparison. Investigations show that

especially low ability students have a significantly lower self-efficacy belief in East Berlin

than low ability students in West Berlin. Among high ability students the difference is not

significant. Moreover, East Berlin pupils showed higher correlations between their efficacy

4Formally comparing the average performance of the two systems shows that a three track systemyields a strictly higher average performance as long as 1 < µ < 4

28

beliefs and course grades (Oettingen, 1995, pp.162). The constant comparison to high

achieving classmates undermines the motivation of low performing students, they have

lower aspirations and give up more readily in the face of difficulties. To take account of

these effects, we can introduce a opt-out-option for students in our model. We introduce a

participation constraint, that states that students choose not to participate in competition

as soon as their utility becomes negative. That is pi = 0 if ui < 0. Rearranging ui < 0 we

get another case in our best response function:

pi(ri, ai) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 if ai < 2λsri(1−s+λs)2

(1 − s + λs)ai if 2λsri(1−s+λs)2 ≤ ai < ri

1−s+λs

ri if ri1−s+λs ≤ ai < ri

ai if ai ≥ ri

Note that we resubstituted µ = 1 − s + λs. Concentrating on a competitive culture, we

compare a comprehensive school class with reference point ri = a to a system with ability

tracking with reference points ri = a in the high track and ri = a2 in the low track.

Figures 12 and 13 show equilibrium performances under the two regimes for a fixed level

of loss aversion at λ = 2 depending on s:

0 0.1 0.2 0.3 0.4 0.5

0

a

a2

s

perf

orm

ance

Figure 12: Comprehensive Schooling withParticipation Constraint

0 0.1 0.2 0.3 0.4 0.5

0

a

a2

s

perf

orm

ance

Figure 13: Ability Tracking with Participa-tion Constraint

From the graphs it can be seen that the participation constraint is rather strict. Low

ability students already opt out at very low levels of the degree of social comparison s

of below 0.1. Formally comparing average performance under the two regimes leads to

the result that ability tracking yields a strictly higher average performance. This is the

contrary to the result without participation constraint. The mechanism for this becomes

29

clear from the graph. Under comprehensive schooling low ability students suffer a huge

utility loss since the difference between their performance and the reference point is very

high. They are easily demotivated and opt out already for very low levels of s. Under

ability tracking low ability students are not that far away from their reference point and

are thus not as easily demotivated. Following this line of argumentation it would be

beneficial to institute even more than two tracks. In reality we do not observe students

that do not perform at all to such an extent. How realistic this extension is remains to

be tested.

7 Conclusion

In this paper we integrated aspects of culture into a model of reference-dependent pref-

erences to show that differences in cultural competitiveness of students matter for the

decision on optimal school design. We have shown that a comprehensive school design is

to be preferred in a competitive culture, whereas ability tracking yields a higher average

performance in a non-competitive culture. The difference in outcomes mainly stems from

the difference in reference points chosen in the two extreme cultures. A high reference

point like the best performance in class motivates all students in class, even the lowest

ability types, which makes a comprehensive school system preferable. A low reference

point like the average student’s performance leaves high ability students unambitious.

Tracking according to ability here yields the opportunity to also stimulate high ability

types. At the same time low ability students’ motivation is undermined by the low ref-

erence point in a low track. Here the choice between ability tracking and comprehensive

schooling thus induces a trade-off between equity and efficiency.

These results show that the cultural background of students does matter for the de-

cision on institutional design. This fact is often neglected in educational research. The

role of culture should in particular also play a major role when considering international

student achievement tests that strive to identify best practices to be instituted across

countries. Copying school systems from successful countries is not a good strategy when

the culture of the country under consideration is substantially different.

The main factor that drives performances in our theory is loss aversion. Students

are solely motivated because they strive to avoid a loss of utility due to a high differ-

ence between own performance and reference performance. Thus, the higher loss aversion

the more do performances converge to the best performance in class. As a policy maker

whose only aim is to increase performances, this mechanism can be used. Independently

of whether a tracked or comprehensive system is in place, the schooling context should be

30

designed to reinforce competitive preferences. For instance, regular and frequent perfor-

mance feedback and rankings should be provided to facilitate social comparisons. And,

teachers should highlight the best students’ achievements in order to induce high refer-

ence points. According to our theory this increases performances, however, also note that

students’ utility or well-being is decreasing the further away the reference point and the

fiercer competition. In the worst case students might choose not to engage in competition.

That is they are demotivated by reference points that are unrealistic to achieve, which

leads them to exert even less or no effort. High reference points and an emphasis on social

comparison may thus be destructive. In our section on a possible participation constraint

we take account of this effect.

There is much more research to be done in the area of culture’s influence on students’

learning behavior. Direct possible extensions to our model include the analysis of other

possible (endogenous) reference points or the inclusion of uncertainty. In a next step we

will test our model empirically with field data and (or) in lab experiments.

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Appendix

A Proof of Proposition 3

(i) Suppose there is a β, with 1 ≤ β < µ, so that equilibrium performance of the best

student p∗a ∈ [a, µa) can be characterized by p∗a = βa in a comprehensive school class

as well as in a high track and p∗a2

= βa2 in the low track. Average performances

under comprehensive schooling pC and ability tracking pT are given by the following

integrals over the ability distribution ai ∈ (0, a]:

pC = 1

a(∫

βaµ

0µaidai + ∫

a

βaµ

(βa)dai) = aβ(2µ − β)2µ

pT =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1a (∫

βa2µ

0 µaidai + ∫a2βa2µ

βa2 dai + ∫

βaµ

a2

µaidai + ∫aβaµ(βa)dai) if µ < 2β

1a (∫

βa2µ

0 µaidai + ∫a2βa2µ

βa2 dai + ∫

aa2(βa)dai) if µ ≥ 2β

=⎧⎪⎪⎪⎨⎪⎪⎪⎩

a(10βµ−5β2−µ2)8µ if µ < 2β

aβ(6µ−β)8µ if µ ≥ 2β

In the case of tracking we need to distinguish two cases since a2 > βa

µ for µ > 2β.

This is the case when a symmetric equilibrium is reached in the high track. Average

performance under comprehensive schooling is strictly higher if pC > pT :

1st case: 1 < µ < 2β

aβ(2µ − β)2µ

> a (10βµ − 5β2 − µ2)8µ

⇔ (µ − β)2 > 0, holds for any 1 < µ < 2β.

2nd case: µ ≥ 2β

aβ(2µ − β)2µ

> aβ(6µ − β)8µ

⇔ µ > 3

2β,holds for anyµ ≥ 2β.

Thus pC > pT holds for any µ > a+βa , which is given in any equilibrium. QED

(ii) The variances under comprehensive schooling σC and ability tracking σT are given

34

by:

σC =1

a(∫

βaµ

0(µai − p

C)2dai + ∫

a

βaµ

(βa − pC)2dai) =

a2β3(4µ − 3β)

12µ2

σT =1

a(∫

βa2µ

0(µai − p

T )2dai + ∫

a2

βa2µ

(βa

2− pT)

2

dai + ∫

βaµ

a2

(µai − pT )

2dai + ∫

a

βaµ

(βa − pT )2dai)

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a2(−75β4+156β3µ−114β2µ2+60βµ3−11µ4)192µ2 if µ < 2β

a2β2(−3β2+20βµ+12µ2)192µ2 if µ ≥ 2β

The variance under ability tracking is strictly higher if σT > σC holds for any µ > 1:

1st case: 1 < µ < 2β

a2 (−75β4 + 156β3µ − 114β2µ2 + 60βµ3 − 11µ4)

192µ2>a2β3(4µ − 3β)

12µ2⇔ µ <

27β

11, holds for any µ < 2β.

2nd case: µ ≥ 2β

a2β2 (−3β2 + 20βµ + 12µ2)

192µ2>a2β3(4µ − 3β)

12µ2⇔ µ > 0 , holds for anyµ ≥ 2β. QED ∎

B Proof of Proposition 6

(i) Average Performance: We have already derived average performances pC , pL and pH

when determining the reference points under comprehensive schooling and ability

tracking. To determine average performance under ability tracking suppose that

there is a γ, with 0 ≤ γ < µ2 , such that symmetric equilibria in the high track can be

denoted by p∗ = γa.

pT = 0.5pL + 0.5pH =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0.5aõ

2(√µ+1) + 0.5a(µ+2

õ)

2(√µ+1) if 1 < µ < 2

0.5aõ

2(√µ+1) + 0.5γa if µ ≥ 2=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a(µ+3√µ)

4(√µ+1) if 1 < µ < 2

a(√µ(1+2γ)+2γ)4(√µ+1) if µ ≥ 2

Average performance under ability tracking is strictly higher than under compre-

hensive schooling if pT > pC is fulfilled:

1st case: 1 < µ < 2

a (µ + 3√µ)

4(õ + 1) >aõ

õ + 1

⇔ µ > √µ, holds for any 1 < µ < 2.

35

2nd case: µ ≥ 2

a (√µ(1 + 2γ) + 2γ)4(√µ + 1) >

aõ

õ + 1

⇔ 2γ > (3 − 2γ)√µ,

holds for any (1 < γ < 3

2∧ µ < 4γ2

4γ2 − 12γ + 9) ∨ γ ≥ 3

2.

For a γ = 1 this is true for 1 < µ < 4. QED

(ii) Variances under comprehensive schooling σC and ability tracking σT are given by:

σC = 1

a(∫

pµ

0(µai − pC)

2dai + ∫

p

pµ

(pC − pC)2dai + ∫

a

p(ai − pC)

2dai)

σT =1

a

⎛⎝∫

pL

µ

0(µai − pT )

2dai + ∫

pL

pL

µ

(pL − pT )2dai + ∫

a2

pL(ai − pT )

2dai

+∫pH

µ

a2

(µai − pT )2dai + ∫

pH

pH

µ

(pH − pT )2dai + ∫

a

pH(ai − pT )

2dai

⎞⎠

Substituting pC and pT and rearranging yields if the symmetric equilibrium in the

high track is at p∗ = a:

σC = a2

3 (õ + 1)2

σT =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

a2(0.125µ3/2−0.0416667µ3+0.3125µ2−0.4375µ+0.375)(√µ+1)2 if 1 < µ < 2

a2(0.0625µ+0.25√µ+0.291667)

(√µ+1)2 if µ ≥ 2

The variance under tracking is strictly higher than under comprehensive schooling

if σT > σC . This must also hold for any symmetric equilibrium in the high track

p∗ > a, since this implies an even higher variance under tracking.

1st case: 1 < µ < 2

a2 (0.125µ3/2 − 0.0416µ3 + 0.3125µ2 − 0.4375µ + 0.375)(√µ + 1)2 > a2

3 (õ + 1)2

⇔ 1 > µ3 − 7.5µ2 + 10.5µ − 3µ3/2, holds for any 1 < µ < 2.

36

2nd case: µ ≥ 2

a2 (0.0625µ + 0.25√µ + 0.2916)

(õ + 1)2 > a2

3 (√µ + 1)2 ⇔ µ + 4√µ > 2

3, holds for any µ ≥ 2.

QED ∎

37