EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 6 4 6 –6 57

a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n

ANALYSIS

Can sustainable consumption be learned?

A model of cultural evolution

Guido Buenstorf, Christian Cordes⁎

Max Planck Institute of Economics, Evolutionary Economics Group, Kahlaische Str. 10, 07745 Jena, Germany

AR TIC LE I N FO ABS TR ACT

Article history: This paper shows how sustainable consumption patterns can spread within a population via
Received 22 August 2007 processes of social learning even though a strong individual learning bias may favor
Received in revised form environmentally harmful products. We present a model depicting how the biased
11 January 2008 transmission of different behaviors via individual and social learning influences agents'
Accepted 11 January 2008 consumption behavior. The underlying learning biases can be traced back to evolved cognitive
Available online 10 March 2008 dispositions. Challenging the vision of a permanent transition toward sustainability, we argue
that “green” consumption patterns are not self-reinforcing and cannot be “locked in”
permanently.
Keywords: © 2008 Elsevier B.V. All rights reserved.
Consumer behavior
Cultural evolution
Learning
Sustainability
Evolutionary economics

JEL classification:
D11; D83; Q01; C61

1. Introduction Bergh, submitted for publication). To approach a sustainable

state, consumers will have to reduce their level of consump-
Contemporary economies pose severe and potentially disas- tion and/or modify the kinds of goods they consume.
trous challenges to the natural environment. Ecological Substantial prior work has focused on the aggregate level of
economics has developed in reaction to these challenges. Its individual consumption activity (e.g., Heiskanen and Pantzar,
guiding vision is that of (ecological) sustainability: an aggregate 1997; Røpke, 1999). However, it is unclear how reductions in the
pattern of economic activities that can be upheld over time overall level of individual consumption can be induced. In the
without compromising the environment's capacity to support face of apparently insatiable human desires, a reduction based
the needs of future generations (WCED, 1987; Sartorius, 2006). on voluntary restraint is unlikely to happen, particularly as long
As all economic activity is ultimately aimed at satisfying the as productivity increases and technological innovation con-
needs of consumers, consumer behavior has an essential role tinually open up new consumption opportunities. Increasing
to play in any transition toward a more sustainable economy incomes and innovation also limit the power of incentive-based
(e.g., Arrow et al., 2004; Brennan, 2006; Wagner, 2006; van den instruments to reduce overall consumption, because monetary

⁎ Corresponding author. Tel.: +49 3641/686832; fax: +49 3641/686868.

E-mail address: cordes@econ.mpg.de (C. Cordes).

0921-8009/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolecon.2008.01.028
 EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 64 6 –6 57 647

punishments such as eco-taxes would have to reach extremely Section 6 presents some considerations regarding environ-
high levels to remain effective. Finally, engineering a reduction mental policy making. Section 7 concludes the paper.
in overall consumption by forced restraint obviously faces
significant legitimacy issues.
The present paper thus re-focuses the attention on the 2. Consumption as a learning process
composition of consumption activities, discussing whether
and how consumer learning can shift consumption toward Recent theorizing in evolutionary economics studies consu-
activities that are more sustainable in terms of the required mer behavior as shaped by complex processes of individual
inputs of resources and outputs of harmful emissions. For this and social learning. These learning processes originate from
purpose, we model the cultural transmission of consumer innate physiological needs and capacities of the human
behavior via processes of individual and social learning. The cognitive apparatus, including the capacity for culture (see
model incorporates insights from evolutionary and behavioral Witt, 2001; Ruprecht, 2005; Cordes, 2004). In the following, we
economics, anthropology, and psychology. As will be shown, will refer to this theory as the learning theory of consumption
cultural transmission is biased; people tend to acquire some (LTC). The predictions derived from it resonate with related
behavioral variants more easily than others. Moreover, this work in anthropology (e.g., Henrich and McElreath, 2003;
process of cultural transmission is influenced and constrained Corning, 2005).
by humans' evolved psychology that shapes what we learn, LTC remains within the utilitarian tradition of economics.
how we think, and whom we imitate. It suggests that acts of consumption are motivated by their
Consumer learning is critical for the diffusion of sustainable capacity to fulfill human wants. In part, wants are innate and
consumption patterns. We indicate conditions under which universally shared and correspond to basic physiological
consumption of environmentally benign goods can spread needs such as the intake of air, water and food, sleep, and
within a consumer population via processes of social learning the maintenance of the body temperature, but also more
even though an individual learning bias favors an environ- psychic needs such as entertainment and social recognition.
mentally harmful substitute. However, our model also shows Their ability to satisfy innate wants thus provides the most
that “green” consumption patterns are not self-reinforcing and basic hedonistic motivation of consumption activities.
cannot be “locked in” permanently. This finding suggests that In addition to the set of innate wants, agents acquire further
demand-oriented policy measures are limited in their poten- wants through processes of associative learning (conditioning
tial to help achieve a more sustainable economy. through reinforcement). This want learning may operate in an
By building on insights from psychology and anthropology entirely automatic way without conscious control (Witt, 2001).
on how human behavior is conditioned by evolved needs and It takes place when initially neutral activities are repeatedly
learning capacities, our approach contributes to the “natural- performed simultaneously with activities that satisfy pre-
ist” line of research in economics that emphasizes the existing (and currently non-satiated) wants. Over time, an
present-day behavioral implications of human phylogeny association is formed between the initially neutral activity and
(Witt, 2003; Cordes, 2007). We abandon the assumption that the rewarding experience from satisfying the innate want. A
agents rationally choose utility-maximizing items from a new want for the neutral activity and the related “inputs” (i.e.,
given set of alternatives. Instead, we assume that people goods and services) is thus developed. Depending on their
must learn which consumption goods meet their preferences individual learning history shaped by the specificities of their
and that an important force in learning is social observation. exposure to stimuli and interaction with other agents,
Agents are aware of only a fraction of the available informa- associative want learning enables humans to acquire highly
tion. Limits to human rationality in the face of a complex idiosyncratic chains of learned wants. Under conditions of
world induce individuals to adopt culturally transmitted affluence, the link between a specific acquired want and the
behaviors, frequently without independent evaluation of original innate wants is often quite indirect.
their outcomes. Therefore, imitating or learning from others Human learning capacities obviously go beyond simple
is one of the most important means by which humans finesse associative learning based on conditioning (Cordes, 2004),
the bounds of rationality (Boyd and Richerson, 1993; Richerson which is in similar forms also observable in animals. Consumer
and Boyd, 2001).1 behavior is also informed by consciously controlled learning
The remainder of the paper proceeds as follows. In Section processes that lead to the acquisition of explicit knowledge on
2, we sketch the evolutionary approach of analyzing consumer the characteristics of goods as well as on how specific activities
behavior as the outcome of individual and social learning. and goods contribute to the satisfaction of wants. Importantly,
Section 3 discusses some implications of this approach for a this kind of learning does not require reinforcing feedback
potential transition to sustainable consumption. In Section 4, provided by the senses; abstract comprehension of goods
we develop a model depicting how the biased transmission of characteristics and their effects can be sufficient.
different kinds of behavior via individual and social learning LTC suggests that the associative and the consciously
processes influence agents' consumption activities. Implica- controlled levels of consumer learning systematically interact
tions of the model results for learning dynamics and “green” (Witt, 2001). Human attention to stimuli is selective, and
consumption behavior are the subject matter of Section 5. available information is more likely to be consciously pro-
cessed if it relates to activities for which a want already exists.
1
Rational choice is a weak process relative to cultural transmis- In turn, exposure to information may induce changes in
sion in the construction of behavioral repertoires (for a similar consumption activities. These initiate new processes of
argument see Eshel et al., 1998). associative learning, thus furthering the evolution of the
648 EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 6 4 6 –6 57

agent's want structure by reinforcing existing wants and locally favored cultural variants especially if the environment
possibly also adding new ones. changes slowly and the information available to an individual
The acquisition of consumption knowledge is shaped by is poor (Boyd and Richerson, 1985, p. 216ff; Boyd and Richerson,
the information an agent is exposed to. Information avail- 1989). In the face of a complex world, boundedly rational
ability is only to a minor extent within the control of the individuals often adopt culturally transmitted behaviors with-
individual, for example, when she engages in deliberate out independent evaluation of their outcomes (Richerson and
search processes to find some information (e.g., when Boyd, 2001). Therefore, the limitations of human cognitive
pondering purchases of expensive durables). The majority of resources are a fundamental factor underlying cultural evolu-
consumption-related information is provided by her social tion, including the dissemination of consumption knowledge.
environment; it is the consequence of other agents' activities These powerful effects of the social environment on
and communicative acts. Cultural conventions also define consumer learning limit the variability of individual consumer
what kinds of activities and communications are acceptable at behavior (see Leibenstein, 1950). They do not imply, however,
any given time and location. Transmitted culture thus that changes cannot take place at the social level. Changes are
accounts for the bulk of variation in human consumption most likely with regard to the supply of information, for
behavior. Through their agenda-setting effect on the kind of example, via cultural role models. Over sufficient time
information that an agent is exposed to, communication in horizons and/or with sufficiently strong individual re-evalua-
social groups and mass media exert a substantial influence on tions (e.g., because of exogenous shocks such as natural
the acquisition of consumption knowledge. disasters), also the social rewards for particular consumption
The provision of role models is a powerful way in which activities are variable.
social groups and mass media influence individual behavior.
Through observing role models and adopting their successful
behavior, agents can learn from others even in the absence of 3. Implications of consumer learning for
direct verbal communication (Bandura, 1986, ch. 2). Anthro- sustainable consumption
pological evidence likewise indicates that the adoption of
cultural traits is conditioned by the observable attributes The evolutionary dynamics of consumer learning suggested
of individuals exhibiting the trait (Richerson and Boyd, 2005, by LTC have important ecological implications. In what
p. 69; Harrington, 1999). In human phylogeny, selection follows, we focus on activities that involve the consumption
favored social learners who were able to evaluate potential of specific goods and services. These are represented by a set
models and copy the most successful among them, thereby of characteristics (Lancaster, 1966), some of which determine
saving the costs of individual learning (Rogers, 1983; Boyd and their environmental impact.
Richerson, 1985; Henrich and Gil-White, 2001; Labov, 2001). In this context, it is important to note that the majority of
Hence, in model-based learning there is a predisposition to environmentally relevant characteristics are of a non-sen-
imitate successful or prestigious individuals. In general, a sory nature; except for special and extreme cases, agents
model-based bias results if social learners use the value of a cannot physically experience these characteristics with their
second trait that characterizes a model (e.g. prestige) to senses. Some classes of pollutants (such as radioactivity) are
determine the attractiveness of that individual as a model not perceptible for the human senses. In other cases,
for the primary trait (e.g. a consumption behavior). environmental impacts of consumption acts may be too
The social environment's impact on individual consumer ambiguous or indirect to be perceived. Hence most envir-
behavior is further increased through the important motivat- onmentally relevant characteristics of goods and services
ing role of social recognition. As was noted above, social cannot trigger reinforcing sensory feedback. They are there-
recognition is one of the innate wants identified by LTC. It is by fore irrelevant for the satisfaction of innate wants in con-
necessity dependent on how the social environment sanc- sumption activities. The level of CO2 emitted by a heating
tions specific individual acts of consumption. The want for system does not directly affect the consumer's sensory
social recognition combines with what anthropologists refer experience of the pleasant temperature it helps provide.
to as the conformist bias (Aronson et al., 2002, ch. 8; Cialdini and Because of this non-sensory nature, prior learning is required
Goldstein, 2004; Kameda and Diasuke, 2002; Henrich, 2004; for environmentally relevant characteristics to motivate the
Richerson and Boyd, 2005, p. 120ff). Due to the conformist bias, adoption of a good.
agents pick the cultural variant that is used and accepted by Let us first consider associative want learning, which is
the majority of models in a population (Henrich and Boyd, typically not under the agent's conscious control. As outlined
1998), whereas they discriminate against traits that are rare in above, humans are able to form complex chains of wants
the local population.2 through associative learning starting from innate wants. In
Conformist transmission belongs to the class of frequency- this way, wants for the consumption of “green” products can
dependent biases. It increases the likelihood of adoption of be formed. This requires that a good with favorable environ-
mental characteristics has additional characteristics that are
both sensorily perceptible and can be associated with an
2
A counteracting “snob effect” in consumption (Leibenstein,
innate or previously acquired want, or that the good is
1950) may motivate people to refrain from buying a good because
consumed jointly with other satisfying goods and activities.
others are consuming it. Almost by definition, this can only be a
minority behavior. We therefore restrict our analysis to the role Repeated consumption of the good may then establish a new
model and conformity biases, which seem to be more relevant to acquired want that becomes increasingly less dependent on
understand consumption patterns in interacting populations. the initially experienced sensory stimulus.
 EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 64 6 –6 57 649

Yet, as the environmentally relevant characteristics them- However, explicit consumption knowledge is not immune
selves cannot cause the rewarding sensory experience, the to incompatible new information. When the thrust of avail-
associative learning process has to rely on characteristics that able – socially transmitted – information becomes adverse to
are unrelated to the good's environmental impact. While the currently consumed good, an agent may be induced to
allowing for the adoption of sustainable consumer behavior, it change her assessment of it and to abandon its consumption
does not provide a channel for learning sustainable consump- in favor of a substitute. Information on the environmental
tion per se. Environmentally benign consumption behavior has impact of a good or service may no longer be invoked in the
to be motivated by other, non-environmental dimensions of agent's consumption choices if relevant role models, social
the involved goods and services. Once adopted, it can always groups, or the media cease to pay attention to this aspect. Over
be challenged by alternative behaviors that are more attractive time, the environmental dimension of consumption knowl-
in terms of these non-environmental characteristics. In other edge will then lose its importance in informing the agent's
words, there is a hedonistic bias in consumer behavior: consumption choices, particularly when at the same time
environmentally superior goods are easily crowded out by more detailed information about other aspects of the good and
alternatives offering equally (or more) rewarding sensory its substitutes becomes available.
experiences. For instance, the availability of fast and inexpen- As in the case of associative want learning, this suggests
sive air travel in Europe has successfully challenged the more that explicit consumption knowledge is not able to perma-
sustainable, but often less convenient, use of railways. nently lock-in “green” behavior. Consumers may learn how to
In contrast to associative want learning, the consciously behave in a sustainable way when making concrete choices,
controlled acquisition of explicit consumption knowledge about and this knowledge is likely to be retained at least over
differences in the environmental impact of close substitutes intermediate time horizons. Its importance in consumer
may directly enhance the subjective attractiveness of the behavior may nonetheless be eliminated by newly acquired
“green” product for an individual consumer.3 For this to happen, knowledge about other product characteristics or by changed
the individual both needs to be aware of the differences, and she priorities of the social environment. Again, we therefore
has to be convinced that benefits derive from them. These conclude that even if sustainable consumption patterns are
benefits may be related to the consumer's own wants, for adopted, individual behavior is not locked into these beha-
example, because the “green” product promises additional viors, but a future transition away from the sustainable
health benefits. Alternatively, individual consumer behavior behaviors cannot be ruled out.
may even be shaped by knowledge of favorable environmental Summing up, our discussion indicates that individual
benefits that yield no immediate benefits to the consumer learning processes are limited in their capacity to induce
herself, but only to others (or the unspecific “environment”). robust sustainable consumption activities. In contrast to its
Individual acquisition of explicit knowledge about envir- powerful role as a motivator of “hedonistic” consumer
onmentally relevant goods characteristics is highly dependent behavior more generally, associative want learning is inhib-
on the agent's social environment. Over the past decades, ited by the largely non-sensory character of environmentally
“green” interest groups have been successful in putting relevant product characteristics. The acquisition of explicit
environmental issues on the public agenda. This availability consumption knowledge is constrained by the information
of information has enabled consumers to learn about the available to the agent. Moreover, individual consumer beha-
environmental impacts of their own activities, in particular if vior is strongly shaped by the cultural transmission of
they were members of groups that discussed environmental information and social norms. Accordingly, in the remainder
issues. In addition, numerous agents have been willing to act of this paper, we focus on the cultural dimension of consumer
as role models for sustainable consumption, and the environ- learning. In particular, we model the learning of environmen-
mental impact of their activities has become a relevant aspect tally benign consumer behavior through social role models
of social recognition. All these developments favored the and conformist transmission of cultural variants as well as the
cultural transmission of environmentally relevant consump- potentially conflicting dynamics of individual and social
tion knowledge and the adoption of corresponding behavior. learning processes.4

4. A model of evolving consumer behavior

To understand how consumption activities evolve in a population of interacting agents, we need to account for the processes
that increase the frequency of some behavioral variants and reduce that of others. A complex interplay of such processes will
constantly affect any population's consumption “culture”. This section translates the above considerations into a stylized
mathematical model (see, as points of origin, Cavalli-Sforza and Feldman, 1981; Boyd and Richerson, 1985). To keep the model

3
Implicitly, the following discussion assumes that explicit consumption knowledge is effective as a motivator of consumer behavior.
We thus abstract from the well-known problem that agents' activities may be inconsistent with their own knowledge and beliefs.
4
Social effects on individual consumption have also been the subject of other formal approaches: Janssen and Jager (2001), for example,
account for socialization effects in a simulation-based social network analysis of market dynamics, while Windrum and Birchenhall
(1998) simulate the learning processes within a population of consumers and between consumers and producers. For an overview of
different methods applied to environmental economics see van den Bergh et al. (2000).
650 EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 6 4 6 –6 57

traceable, we aggregate all consumer learning into two basic processes, individual associative learning of new wants on the one
hand and social learning of explicit consumption knowledge via cultural role models and conformist transmission on the other.
We assume that consumer learning is ubiquitous and that the processes discussed in the previous sections operate as biases
influencing what agents learn.
The model analyzes the cultural transmission of knowledge related to a specific consumption activity for which two
alternative variants a and b exist. These variants are assumed to differ in their environmental impact. Let a represent the
“sustainable consumption behavior” and b the “hedonistic consumption behavior”. The state of the population of consumers is
determined by the frequency of agents with the variant a, labeled p. Accordingly, the frequency of the b behavior is given by 1 − p.
Now, the task is to find a recursion equation in discrete time that allows us to predict the frequency of p in the next stage of the
cultural transmission process given its frequency in the present stage. The general model is of the form

ptþ1 ¼ pt þ cultural evolutionary forcesðbiasesÞ:

Specifically, we will focus on three cultural evolutionary forces that bias transmission: the hedonistic, role model, and
conformity biases introduced above. All can be traced back to evolved cognitive dispositions. We first derive partial recursions
addressing these biases individually and then analyze the resulting aggregate learning dynamics.
The discussion in the previous section has shown that associative want learning is of limited power to induce sustainable
consumption activities, particularly if “hedonistic” substitutes exist that are more harmful to the environment but have more
attractive characteristics otherwise. In this case, a direct learning bias favoring the “hedonistic” variant results from this
superiority.5 We reflect this bias by assuming that the probability of new adoption differs between the alternative behavioral
variants (see Boyd and Richerson, 1980). Imagine that consumers encounter and experiment with alternative ways of behavior
and then, based on their experience, switch to one of these behaviors with a certain probability. Specifically, we assume that each
a agent has a μab chance of learning to favor the hedonistically attractive variant and each b consumer has a μba chance of learning
to behave in a more sustainable way, where μab ≫ μba, i.e., the sensory pleasures connected to the hedonistic variant strongly favor
its adoption.
The evolution of behavior in a group of consumers is reflected by changes in the frequency of the alternative variants over
time. These are derived from a recursion equation that determines p in the next time step, p′, given the value of p in this period. A
fraction μab of a consumers switch to consumption behavior b, and so are subtracted each learning step. On the other hand, a
fraction μba of b consumers learns to appreciate the sustainable consumption behavior, and so are added to p. Hence, the partial
recursion for the individual learning phase is

pV ¼ p þ ð1  pÞAba  pAab : ð1Þ

Consumers are normally not capable of sensorily experiencing the environmental impact of their consumption. The diffusion
of environmentally benign behaviors between agents is therefore mainly subject to consciously controlled learning, which in turn
is strongly affected by cultural transmission processes. Reflecting humans' evolved psychology, the social learning underlying the
cultural transmission processes is biased; people tend to socially acquire some cultural variants rather than others. The model
reflects the two biases of social learning that where discussed above, the role model bias and the conformity bias.
To allow the relevance of different models (e.g., individuals in different social roles) to differ, we assign different weights to
them. We do this by specifying the probability that a particular set of role models makes an individual to acquire the behavioral
variant a. The importance of the ith model, Ai, in this transmission process is defined by


ai 1F 12 D
Ai ¼
: ð2Þ
P
n
aj 1F 12 D
j¼1

P
Consequently, the actual weight of the ith model depends on (1) her basic weight αi ( i ai ¼ 1), which may depend on an
individual's social role, charisma, or prestige, and (2) the commonness of her behavioral variant in the set of models. The
frequency-dependent component of the models' impact is expressed by the conformity bias parameter D. We assume 0 b D ≤ 1, i.e.,
cultural transmission creates a force increasing the frequency of the more common variant in the group. The parameter D will
take effect when we compose sets of cultural role models below. If D = 0, no conformity bias is present, while if D = 1, its influence is
maximized. Moreover, the weights given by (2) sum up to 1 and are normalized by the denominator so that Ai gives the weight of
the ith model relative to the other models encountered by the individual.6
We assume that one of the cultural role models, M1, always shows behavior a, i.e., this agent or medium is exclusively
exhibiting the environmentally benign consumption behavior. This could relate, for example, to the media or environmental

5
This process can be unconscious. See for an example of such a bias Cordes (2005b).
6
As was pointed out by an anonymous reviewer, even though there is substantial empirical evidence for conformity effects, the exact
form and strength of the conformity bias cannot be unequivocally deduced from social psychology. We accordingly allow the conformity
bias to vary over the range of 0 b D ≤ 1.
 EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 64 6 –6 57 651

interest groups willing to become “agents of collective action” (Witt, 1992). As we focus on the potentially conflicting effects of
(hedonic) individual and social learning regarding the adoption of sustainable behavior, investigating into the effects of such
dedicated cultural role models introduces an important aspect of social learning. The other role models, M2 and M3, may show
either behavior.
Assuming that M1 is a dedicated role model consistently advocating sustainability gives rise to further asymmetry in the
model, which counteracts the asymmetry introduced by the hedonistic bias in individual learning. No similarly dedicated “anti-
sustainability” role model is introduced because this would only be defined in a negative way, while positive role models that
consistently imply non-sustainable behavior are hard to motivate. Even the most unabashed hedonistic lifestyles may include
individual consumption activities that happen to be relatively sustainable — think for example of food connoisseurs who
purchase regionally grown vegetables because of their taste rather than their environmental benefits. Note, moreover, that the
weight of the sustainable role model, M1, is allowed to vary in the following, and that it may reach zero in the limit (the
significance of the assumption on M1 is further discussed in Section 5 below).
With these assumptions, we yield a cultural transmission table (see Table 1) showing the probability of agents acquiring
behavior a or b given a particular set of cultural role models (M1, M2, M3) with different weights (see, for a similar approach, Boyd
and Richerson, 1985, p. 209).
For example, let P(a|aab) denote the conditional probability that a consumer acquires variant a given exposure to models with
variants a, a, and b. Then, from the table, the frequency of a after transmission, p″, given that it was p′ before transmission, is

pW ¼ p V2 þ p Vð1  pVÞfPðajaabÞ þ PðajabaÞg þ ð1  p VÞ2 fPðajabbÞg: ð3Þ

This term computes the frequency of each different set of social models (M1, M2, M3), multiplies this by the probability that a
particular set of social models results in an individual acquiring a particular variant, and then sums over all possible sets of social
models. Note that M1 shows behavior a with probability 1. The parameter D always favors the cultural variant that is more
common among the set of role models. Half of the conformity effect is assigned to each of the “majority models” in a set. As is
shown in the Appendix, inserting the conditional probabilities from Table 1, we can rewrite this equation as follows:

pW ¼ p V2 þ ðp V  1Þ2 ða1 þ Da1 ða1  1ÞÞ þ pVðpV  1Þða2 þ a3  2 þ Dða2 ða2  1Þ þ a3 ða3  1ÞÞÞ: ð4Þ

To keep the model traceable, we assume that the two social models M2 and M3 have the same weights in the transmission
process, i.e., we assume α2 = α3. Furthermore, α1 is then given by 1 − 2α3. Therefore, Eq. (4)) simplifies to

pW ¼ 1 þ 2a3 ðpV  1Þð1 þ D  2Da3 þ DpVð3a3  2ÞÞ: ð4aÞ

The complete recursion for p, obtained by substituting (1) into (4a) is expressed as

pW ¼ 1 þ 2a3 ðp þ Aba ð1  pÞ  pAab  1Þð1 þ D  2Da3 þ Dð3a3  2Þðp þ Aba ð1  pÞ  pAab ÞÞ: ð5Þ

For the purpose of further simplification, we assume a positive value for μab while setting μba = 0 in the following. This implies
that individual learning exclusively favors behavioral variant b. As regards the potentially conflicting social and individual
learning dynamics of the model, this is the theoretically most interesting case. We thus obtain

pW ¼ 1 þ 2a3 ð1 þ pðAab  1ÞÞðDð2a3  1 þ pð3a3  2ÞðAab  1ÞÞ  1Þ: ð5aÞ

This recursion models the change of p in the population over one individual and one social learning step. By setting the
parameters of the system, we can analyze its long run behavior by conceptually iterating Eq. (5a) recursively for many generations.

Table 1 – The probability of agents acquiring consumption behavior a or b given a particular set of models (M1, M2, M3) that
have different intrinsic weights (α1, α2, α3) and frequency-dependent cultural transmission (bias parameter D)
Behavioral variant of Probability that agent acquires behavioral variant

M1 M2 M3 a b

a a a 1 0



a a b ða1 þ a2 Þ 1 þ 12 D a3 1  12 D
1 þ 2 Dð1  2a3 Þ
1
1 þ 2 Dð1  2a3 Þ
1



a b a ða1 þ a3 Þ 1 þ 12 D a2 1  12 D
1 þ 2 Dð1  2a2 Þ
1
1 þ 2 Dð1  2a2 Þ
1



a b b a1 1  12 D ða3 þ a2 Þ 1 þ 12 D
1 þ 2 Dð1  2a1 Þ
1
1 þ 2 Dð1  2a1 Þ
1
652 EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 6 4 6 –6 57

Ap̂
Fig. 1 – Values of Aa3
for D = 0.5, 0 < α3 <0.5, and 0 ≤ μab < 1.

Moreover, we can now calculate the equilibrium frequency of the cultural variant a in the population. At equilibrium the
population does not change so p″ − p = 0. We subtract p from both sides of (5a). One can determine the equilibrium of the set of
coupled recursions implied by (5a) by solving for p̂ denoting the equilibrium frequency of the sustainable consumption behavior
a:7

1
p̂ ¼  ð2a3 ðDð5a3  3Þ  1ÞðAab  1Þ ð6Þ
4a3 ð3a3  2ÞDðAab  1Þ2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
    ffi
þ 4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1ÞðDa3 þ 1Þ þ D2 a3 ða3  1Þ2 þ2ða3 ða3  4Þ þ 2ÞD þ a3 Aab þ 1  1:

Although it is not easy to interpret this term, we can use it to gain insights into the learning dynamics underlying the model.
First, the equilibrium share of agents that stick to the environmentally benign consumption behavior, p̂, is always decreasing in
A p̂
α3.8 This can be seen from Aa 3
, the derivative of (6) with respect to the weight of the cultural role models M2 and M3 (α2 = α3) (see
A p̂
Appendix). Fig. 1 above plots the values of Aa 3
for different values of α3 (0 b α3 b 0.5) and μab (0 ≤ μab b 1) for a given conformity bias of
D = 0.5. The decrease in p̂ is strongest when the variance in the distribution of a and b behaviors in the population is high, i.e.,
when there are many pairings of models holding different behaviors. This is indicated by the “deep valley” on the right hand side
of Fig. 1. Given the fact that M1 is always showing behavior a, the number of b-types among the other models crucially depends on
the hedonistic learning bias μab favoring behavior b. For low values of α3 and μab, the rates of change of p̂ when the parameters
change, i.e., the differences between the single equilibrium states, are low.
Due to the fact that α1 = 1 −α2 −α3 and α2 =α3, a rising weight of M2 and M3 implies a lower influence of M1 in cultural transmission.
This enables the b behavior to spread within the population and gain a higher share via b-types among M2 and M3 introduced by the
hedonistic learning bias μab. On the other hand, low values for α2 and α3 entail a great influence of role model M1, giving rise to a higher
share of behavior a in the group.
The derivative of (6) with respect to D, AADp̂ (see Appendix), shows that p̂ increases when μab is low and then decreases rapidly when
μab exceeds a certain threshold. The higher the conformity bias D, the steeper is this decline. Fig. 2 gives the values of AADp̂ for different
values of D (0b D ≤ 1) and μab (0≤μab b 1) assuming that the weights of the models M2 and M3 are α2 =α3 = 0.3. If the hedonistic learning
bias μab is high, then the conformity bias D favors the spread of consumption behavior b. This is because in this case an individual
encounters many model constellations that are composed of more b than a models. On the other hand, given a small hedonistic bias,
the conformist transmission increases the frequency of the sustainable behavior a in the population because the model M1 is always
a in addition to some a models among the models M2 and M3, which results in many pairings with more a models.
Fig. 3 below plots the derivative of (6) with respect to μab, AAA p̂ab (see Appendix), for different values of μab (0 ≤ μab b 1) and α3
(0 b α3 b 0.5) assuming D = 0.5. The term AAA p̂ab again reflects the effects of a high variance among the role models in combination with
an increasing influence of the cultural role models M2 and M3 (see the “deep valley” in the upper left corner of Fig. 3). The
equilibrium frequency of a, given by p̂, is decreasing in μab (for 0bα2, α3 b 0.5). High values of μab imply that most role models M2 and M3
show the hedonistic behavior and entail a low variance among role models and therefore a small number of agents that potentially
switch between behaviors. As a consequence, the differences between equilibrium states are also small.

7
There is a second solution for p̂ that yields negative values and is therefore ignored in this context. The equilibria denoted by p̂ and
given by Eq. (6) are stable (see Appendix).
8 A p̂
With the only exception being the case when μab = 0. Then Aa 3
¼ 0 for all α3.
 EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 64 6 –6 57 653

5. Learning dynamics and “green” consumption

Some interesting insights concerning agents' consumption

behavior derive from the model developed in the preceding
section. First, Eq. (6) shows that a prominent “green” role model
with a sufficiently large weight in the cultural transmission
process can offset even a strong hedonistic learning bias and
maintain a high level of sustainable consumption behavior
within a population. To see this, Fig. 4 visualizes the values for p̂
depending on α3 and μab given a conformity bias D = 0.5. If the
influence of the “green” model M1 decreases, while μab
introduces new b-type consumers, the level of sustainable
consumption behavior in the population decreases.
However, the presence of an influential, consistently “green”
role model is not required to attain a positive share of sustainable
behavior in the population provided the hedonistic learning bias Ap̂
Fig. 2 – Values of AD for α3 = 0.3, 0 < D ≤ 1, and 0 ≤ μab < 1.
is not too pronounced. For low values of μab, the p̂value may be
significant even if α3 approaches 0.5, which implies that M1 has
more easily. This implies that “green” consumption patterns cannot be
very little influence on the behavior of agents in the population.
“locked in” permanently. A novel – hedonistically more attractive –
Even with such a small influence of M1, individual agents are still
behavior may overcome the social learning forces they are based upon.
induced to switch to the sustainable behavior when encounter-
ing two role models that behave in a sustainable way.9 These Proposition 2 captures the difficulties of maintaining environ-
switches provide a force in the model that counteracts the mentally superior consumption activities that are hedonistically
hedonistic learning bias. On the other hand, facing a large disadvantaged, for example, because of their relative inconve-
hedonistic bias μab, a high frequency of a can only be maintained nience (think of the advantages of power tools over hand-driven
if the influence of M1 in cultural transmission is high. alternatives, or motor traffic versus bicycles or walking). Driving
fast cars may be hedonistically rewarding for many people for it
Proposition 1. Social learning can overcome strong direct biases in potentially meets some basic wants:10 it provides an effortless,
cultural transmission. As a consequence, social models exhibiting convenient, and prestigious means of mobility.11
environmentally benign behaviors can establish a “green” consump- This “non-lock-in” property of the model also holds for the
tion regime notwithstanding a strong hedonistic bias in individual reverse situation. Taking the situation depicted in Fig. 6 as a
learning that favors harmful consumption behavior. starting point, consider an increase in the weight of M1, α1, for
example, the media that propagate the environmentally benign
The properties of the learning dynamics leading to the
behavior a. Then the frequency of a can reach a high share in the
equilibrium values of p, p̂, can be further studied by iterating
population even if the hedonistic bias μab has a high value (0.3) (see
the complete recursion for p, given by Eq. (5a), for many
Fig. 7). Thus, social influences can give rise to a spread of environ-
learning steps. We see that, starting from an all-hedonistic
mentally benign behavior in instances where an agent's individual
population, behavior a may spread to virtually all members of
learning would induce her to adopt the hedonistic alternative.
the population even if the influence of the “green” cultural role
A final implication derived from the equilibrium values of p̂
model M1 is very small (i.e., if the weight of α3 is close to 0.5),
and their determinants concerns the role of the conformity
provided the hedonistic bias, μab, is near zero (Fig. 5). This is
bias D as an inertial force.
due to the fact that there are also a-types among the models
M2 and M3 that, together with the conformity bias D, cause the Proposition 3. A conformity bias stabilizes an established “green”
spreading of the “green” behavior a within the population. consumption regime if the majority of role models in the set of cultural
However, ceteris paribus, the frequency of a decreases models show environmentally benign behaviors. On the other hand, if
steeply as soon as the hedonistic learning bias increases, the environmentally harmful consumption behavior is more common
introducing some more b-variants to the population (Fig. 6 among the role models, this bias hinders the spreading of sustainable
with μab = 0.3). This shows how the (new) availability of a consumption patterns.
hedonistically attractive alternative variant may challenge an
When agents observe only the hedonistically motivated
established “green” consumption regime.
behavior among their peers, it is more difficult to disseminate
Proposition 2. The more hedonistically attractive a certain behavior is, the environmentally benign behavior in this population via a
the stronger is the direct learning bias favoring this behavior. Ceteris 10
For one of these underlying wants – the want for ease – see
paribus, this cultural variant then disseminates within a population
Cordes (2005a).
11
What is more, it offers many opportunities for associative learning
9
Even if M1's influence in cultural transmission approaches of new wants. For these reasons, hedonistically less attractive
zero, she remains in the set of role models and influences the alternative means of transportation, such as trains or trams, suffer
adoption probability of the behavioral variants, which are given in from a potential relative disadvantage as to their adoption probability.
Table 1, via the conformity bias that does not differentiate Explicit knowledge regarding their relative environmental friendli-
between the weights of the models. ness may however act against this preference (see also Jackson, 2002).
654 EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 6 4 6 –6 57

Fig. 3 – Values of AAAp̂ab for D = 0.5, 0 ≤ μab < 1, and 0 < α3 < 0.5.

“green” role model. If, for example, the “green” consumption Fig. 5 – The diffusion of behavior a given that α3 = 0.4975,
behavior is propagated by the media (model M1) while – due to μab = 0.01, D = 0.5, and p = 0 in the beginning.
the hedonistic learning bias – most M2 and M3 cultural role
models (e.g. in an individual's peer group) exhibit the unsustain- taken into account in environmental policy making. The nature
able behavior b, the conformity bias favors this environmentally of human learning is too flexible, and the effect of aggregate
harmful cultural variant. On the other hand, the conformity bias environmental problems on individual consumer behavior is too
creates a bigger “plateau” in Fig. 4 that comprises high values of indirect, for learning to uniformly favor sustainability.
p̂, as can be seen from Fig. 2 where this bias “delays” a decrease A “green” consumption regime that has been established in a
of p̂ for increasing values of μab. In this case, most pairings of population does not “lock in”; it is not immune to changes in the
cultural models comprise more environmentally benign a-types strengths of the alternative biases. If the weight of the “mis-
than b-types and thus the conformity bias impedes the sionary” role model M1 decreases in cultural transmission, the
spreading of the harmful behavior b. hedonistic variant increases its share in a population of
consumers, i.e., a lower p̂ is reached (Fig. 4). This lowering of a
model's weight could be due to new information challenging her
6. Implications for environmental policy making
credibility and reputation. Alternatively, demographic changes
Patterns of consumption are shaped by cultural evolution based may reduce the prestige of some role models as new generations
on social learning within populations, implying that individual of consumers enter the market. Hence, continual efforts at
learning alone is insufficient to explain the attained consump- providing convincing role models are required to maintain the
tion regimes. However, none of the learning dynamics discussed prevalence of sustainable consumption activities in a population.
here can permanently fix a high share of environmentally Through information campaigns or even their own behavior,
benign consumer choices. Accordingly, even if more environ- policy makers may aim at providing such “green” role models
mentally benign consumer choices have been observed for some characterized by a high weight in cultural transmission.
time, this does not rule out that consumers start to opt for less In addition, new substitutes with characteristics that are
“green” substitutes in the future — an important aspect to be better suited to provide satisfying hedonistic sensory

Fig. 6 – The decrease of behavior a given that α3 = 0.4975,

Fig. 4 – Equilibrium values for p̂ (D = 0.5). μab = 0.3, D = 0.5, and p = 1 in the beginning.
 EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 64 6 –6 57 655

processes to make them more compatible with sustainable devel-

opment. In this view, the indirect effects of consumer behavior on
the supply side of markets can be trusted to achieve some of the
required changes in economic activities (see Brennan, 2006). As a
consequence, it is expected that less drastic policy interventions
into the supply side of markets may be required to attain given
goals, and that the acceptability of these interventions can be
enhanced.12 Moreover, since technological change is often
cumulative in nature (Dosi, 1982; Nelson and Winter, 1982), even
a temporary impact on consumer behavior may have permanent
effects because of the innovations it induces. Our analysis does
not contradict any of these conjectures. However, even in the
stylized model we presented above, diffusion of sustainable
patterns of consumption depended on the complex interplay of
several forces and could not be established once and for all. This
suggests that for consumer-oriented policy measures to be
successful, a thorough understanding of the cultural learning
Fig. 7 – The diffusion of behavior a given that α1 = 0.4, μab = 0.3,
dynamics underlying evolving consumption patterns is required.
D = 0.5, and p = 0 in the beginning.

experiences constitute a permanent hazard to “green” con- 7. Conclusions

sumer activities. The discussion has shown that μab critically
affects the level of sustainable behavior attained in the Evidence from psychology, evolutionary anthropology, and
population, particularly if the influence of the “green” role cognitive science suggests that the evolved human psychology
model M1 is small. Where explicit “green” consumption shapes what we appreciate, learn, and propagate. Our cognitive
knowledge has been acquired, it may be challenged by dispositions influence the kinds of cultural variants, for example,
contradictory new information, or its relevance for consump- different consumption behaviors, that spread and persist within a
tion may decay as new information increasingly relates to population. These findings help substantiate assumptions on the
aspects of the good (and its substitutes) other than its evolution of consumption behavior and thus contribute to a
environmental impact. theory of consumption. This paper has shown how environmen-
In principle, the appearance of more attractive hedonistic tally benign consumption patterns can disseminate among
variants is beneficial to the consumer, so trying to prevent them agents via processes of cultural learning even if strong hedonistic
or restricting the flow of information about these goods and biases favor environmentally harmful ways of consumption. A
services seems a policy of questionable legitimacy. However, the stylized model of cultural transmission has been proposed that
availability of new substitutes may justify policy efforts aimed indicates conditions under which a “green” consumption regime
at enhancing the performance of the sustainable product can be maintained in the course of an evolutionary process.
variants, such as targeted procurement, to limit and possibly A key conclusion is that in spite of inertial tendencies due
reduce the hedonistic “distance” between alternatives. Sys- to a conformist bias, consumer learning does not “lock-in”
tematic provision of environmentally relevant information, consumer behavior such that once agents begin to adopt more
such as mandatory information about the resource and energy sustainable patterns of consumption, they necessarily remain
intensity of products (eco-labeling), may also be justified on on this “path”. Hedonistically attractive new variants may be
these grounds. Finally, monetary policy instruments, such as able to crowd out the established “green” behavior. This result
eco-taxes, may be helpful in limiting the disadvantages of runs counter to the widespread notion of a “transition” toward
environmentally superior goods and services. sustainability. We consider it an important caveat against a
Social influences operating through the conformity bias have too one-sided and optimistic outlook on the future. In
a stabilizing effect on individual consumption. But, as indicated particular, it entails the necessity to safeguard improvements
by the model, conformity is a double-edged sword: while it helps in the sustainability of economic activities, which calls for
to stabilize adopted sustainable behaviors, it also makes their ongoing and sometimes even increased policy attention,
initial adoption more difficult. Moreover, once a population of rather than taking for granted what has once been achieved.
interacting agents overcomes the conformity bias because a
“critical mass” (Witt, 1997) of agents shifts in their evaluation of
consumption acts, the information “on the agenda”, or the
Acknowledgements
prestige of certain role models, new patterns of consumption
may be adopted in a rather sudden way. This effect is captured
We are grateful for critical comments and discussions on
by the steep “slope” in Fig. 2. It does not only complicate the
earlier drafts form two anonymous referees.
management of consumer behavior, but may also increase the
environmental hazards, as there is less time to regulate 12
Consumer-oriented policies nonetheless cannot substitute for
activities and/or offset their effects.
the establishment of an adequate institutional and regulatory
It is often suggested that, provided there is sufficient consumer framework as well as the internalization of external effects
demand, producers will voluntarily modify their products and through property rights and pricing.
656 EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 6 4 6 –6 57

Appendix

• Let P(a|aab) denote the conditional probability that an individual acquires the variant a given that she is exposed to models with
variants a, a, b. Then from Table 1


ða1 þ a2 Þ 1 þ 12 D
PðajaabÞ ¼ : ðA1Þ
1 þ 12 Dð1  2a3 Þ

Given the assumed rage of D (0 b D ≤ 1), we can ignore terms of order D2 without altering the qualitative properties of the model,
then, after some simplifications,

PðajaabÞc1  a3 þ Da3  Da32 : ðA2Þ

An analogous derivation yields the expression for P(a|aba). The sum of these two terms is

PðajaabÞ þ PðajabaÞ ¼ 2 þ a2 ðD  1Þ  Da22 þ a3 ðD  Da3  1Þ: ðA3Þ

Due to the symmetry of the model, P(a|abb) = 1 − P(a|baa). Thus

PðabbÞ ¼ a1 þ Da1 ða1  1Þ: ðA4Þ

The combination of these conditional probabilities yields Eq. (4) in the text.

• Proving the stability of the equilibrium at p̂ (Eq. (6)): the value of the derivative of Eq. (5a) with respect to p when p = p̂ is
0 0 1 1
5a3 þ 2ð3a3  2ÞðAab  1Þ  3
B Br ð ð 5a  3 ÞD  1 Þð A  1 Þþ
2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi C C
ApW B B 3 3

ab
   C C
j p̂ ¼ 2a3 ðAab  1ÞBD B
B B 4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1Þða3 D þ 1Þ þ ða3  1Þ a3 D þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1  1 C 2 2
C  1C:
C
Ap @ @ A A
4ð3a3  2ÞDðAab  1Þ2 a3
ðA5Þ
This value is less than 1 and greater than − 1 for 0 ≤ μab b 1, 0 b α3 b 0.5, and 0 b D ≤ 1. Thus, the equilibrium is stable.

• The derivatives of p̂ (Eq. (6)) with respect to α3, D, and μab are:

A p̂ 1
¼
Aa3 4ð2  3a3 Þ2 a23 DðAab  1Þ2
0 1
2a3 ð3a3  2ÞðAab  1Þ
B 0 5Da þ Dð5a  3Þþ 1 C
B 3 3 C
B B 4D2 ðA  1Þa3  6ðD  1ÞDðA  1Þa2 þ 2ðDðDðA  1Þ  8A þ 3Þ þ A  1Þa þ Dð4A  1Þ þ 1 C C
BB ab ab ab ab ab 3 ab C C
B @ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 3
   ffi  1 Aþ C
B C
B 4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1ÞðDa3 þ 1Þ þ ða3  1Þ D2 a3 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1 2 C ðA6Þ
B C
B 0 1 C
B 2a ðð 5a  3 ÞD  1 Þ ð A  1 Þþ C
B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 3

ab
   ffi C
B 3a3 @ Aþ C
B 4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1ÞðDa3 þ 1Þ þ ða3  1Þ D a3 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1  1 2 2 C
B C
B 0 1 C
B 2a ð ð5a  3 ÞD  1 Þ ð A  1 Þþ C
B r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 3

ab
   ffi C
@ ð3a  2Þ@ 2 2
AA
3
4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1ÞðDa3 þ 1Þ þ ða3  1Þ D a3 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1  1

1  2a3 ðAab  1Þ
0 1
Dþ
r a3 ðð2a3  3ÞD þ 2Þ  2ða3 þ ðða3  4Þa3 þ 2ÞDÞAab þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
@     A
4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1Þða3 D þ 1Þ þ ða3  1Þ2 a3 D2 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1  2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 1
A p̂ 4a3 ðAab 1Þðð1a3 Þðða3 1ÞDþ1Þða3 Dþ1Þþðða3 1Þ2 a3 D2 þ2ðða3 4Þa3 þ2ÞDþa3 ÞAab Þþ1
¼ ðA7Þ
AD 4a3 ð3a3  2ÞD2 ðAab  1Þ2

A p̂ 1
AAab ¼
0 2a3 ð3a3  2ÞDðAab  1Þ3 1
2a3 ðð5a3  3ÞD  1ÞðAab  1Þ  a3
0 1
B       C
BB 5D þ a3 D 2Dða3  1Þ2 4a3 þ 11  2 þ 2 ða3  1Þ2 a3 D2 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1 CC
BB C C ðA8Þ
B @ð5a3  3ÞD þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
    ffi  1A C
B C
B 4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1Þða3 D þ 1Þ þ ða3  1Þ a3 D2 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1 2 C:
B C
B C
B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
    ffi C
@ 2
A
ðAab  1Þ þ 4a3 ðAab  1Þ ð1  a3 Þðða3  1ÞD þ 1Þða3 D þ 1Þ þ ða3  1Þ a3 D2 þ 2ðða3  4Þa3 þ 2ÞD þ a3 Aab þ 1  1
 EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 64 6 –6 57 657

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