Markets, Social Networks, and Endogenous Preferences1
Michal Kvasniˇcka2
Abstract. This paper generalizes the Bell’s model (“Locally interdependent preferences
in a general equilibrium environment,” JEBO, 2002), and models an interaction
between a market, endogenous preferences, and a general social network. Contrary
to Bell’s results, 1) the system need not to converge, 2) the agents’ preferences need
not to be polarized, 3) the agents’ preferences need not to adjust in the proportion to
the availability (only the more abundant good can be consumed in one type of equilibrium),
and 4) the agents with the same preferences need not to be clustered.
Keywords: endogenous preferences, market, social network, agent-based simulation
JEL classification: D83, R13, D51, D85, C69
AMS classification: 68U20, 05C82, 91D30, 91B24, 91B42, 91B52, 91B69
1 Introduction
In economic theory, agents’ preferences are usually taken as exogenous and fixed. While there are many appealing
reasons for this approach, it seems more than likely that preferences of real-world people are endogenous, change
over time, and are influenced by their social interactions. There is a growing body of literature on endogenous
preferences and impact of social networks on economic behavior. One of the studies in the preference adaptation
in the context of social networks is A. M. Bell’s seminal paper “Locally interdependent preferences in a general
equilibrium environment” [1]. In this paper, Bell explored the evolution of agents’ preferences by means of an
agent-based simulation. In her basic model, the prices of two goods were set in a centralized market. The agents’
preferences were endogenous: the agents increased their taste for the good that was consumed more in their local
neighborhood. In equilibrium, each agent consumed only one of the goods, the agents with the same preferences
were geographically clustered, and fewer agents specialized in consumption of the scarcer good.
Bell simulated her model only for one size and one type of social network (the grid). Therefore it is not known
whether her findings generalize for other sizes and types of networks. It is also not known whether her findings
were caused by the centralized market, the social network, or their interaction. The purpose of this study is to
extend Bell’s model to find answers to these two questions. To do it, the basic version of Bell’s model (the model
with a given endowment and no production) will be extended and simulated for various sizes and types of networks
and their parametrization. It will be explored how much the characteristics of the resulting equilibrium depend on
the characteristics of the social network. One special case discussed is disconnected agents. This will help us to
understand what part of the Bell’s findings are caused by the social network, and what are caused by the centralized
market. It will be shown that many of Bell’s results have to be modified.
2 Model
The used model is a straightforward generalization of the Bell’s model of “exchange economy”, see [1]. The only
change is that the implicit grid network was replaced with the explicit description of a general social network. In
general, the agents consume two kinds of goods (e.g. black and white t-shirts) which are comparable, i.e. their
total consumption has a natural meaning (how many t-shirts an agent consumes). In every period, all agents get
an initial endowment of each good. Then they trade these goods with each other at the centralized market at the
market clearing price. The agents’ preferences evolve over the time: each agent increases her preference for the
good that has been recently more popular (i.e. more consumed) in her neighborhood. Her neighborhood consists
of the agent herself and the agents she has relationship with. The set of all relationships in the population forms a
social network.
1This paper has been created as a part of project of specific research no. MUNI/A/0796/2011 at the Masaryk University.
2Masaryk University, Faculty of Economics, Department of Economics, Lipov´a 41a, Brno, 602 00, the Czech Republic,
michal.kvasnicka@econ.muni.cz
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More formally, there are N agents indexed i = 1, . . . , N. In every period, each agent gets the same endowment:
e1 units of good 1 and e2 units of good 2 (e1 ≤ e2). Agent i then demands xi1 unit of good 1 and xi2 units of good 2
to maximize her one-period Cobb-Douglas utility function subject to the constraint given by her endowment, i.e.
max
xi1,xi2
xait
i1 x1−ait
i2 s.t. p1xi1 + p2xi2 = p1e1 + p2e2, (1)
where p1 and p2 are the prices of good 1 and good 2 respectively, and ait is agent i’s relative preference for good 1 at
time t. The initial value of the preference parameter ai0 is drawn independently for each agent from the continuous
uniform distribution U(0, 1).
Agent i’s demand for the two goods is then
xi1(p1, p2) = ait e1 +
p2
p1
e2 , xi2(p1, p2) = (1 − ait)
p1
p2
e1 +
p2
p1
e2 . (2)
Since the total endowment is given, the market clearing relative price is
p∗
1
p∗
2
=
e2 ∀ j ajt
e1 ∀ j(1 − ajt)
(3)
The social network is represented by an undirected graph G, in which agents are vertices and their relationships
are edges (connections). Two agents i and j have a relationship if they are connected with an edge; we then write
i ∼ j ∈ G. We define the agent i’s neighborhood n(i) = {j : j = i ∨ j ∼ i ∈ G}, i.e. as the set of indices of all agents
who are connected to agent i and the index of the agent i herself. The social network is for each simulation created
randomly by a given algorithm; it is fixed within the simulation.
After observing the consumption in her neighborhood, each agent adjusts her preferences in such a way that
she increases the preference for the good that is consumed more in her neighborhood. Specifically, agent i sets the
future value of her preference parameter ai,t+1 at
ai,t+1 = ait + r
j∈n(i) xj1
j∈n(i) xj1 + j∈n(i) xj2
− 0.5 (4)
where the adjustment parameter r ∈ (0, 1) regulates the speed of the preference adjustment.
The evolution of agents’ preferences and consumption is simulated in the agent-based computational fashion
(for introduction to it, see e.g. [3]). First, the model is initialized: a given social network consisting of N agents
is created (see [5] for the network-creation algorithms), and each agent is assigned a random initial preference ai,0
drawn from U(0, 1). The simulation than proceeds in steps repeated until the model converges (i.e. the agents’
preference parameters change no more), or the maximal amount of steps is reached. In each step, 1) the market
clearing relative price p∗
1/p∗
2 is calculated for the agents’ current preferences (equation 3), 2) each agent’s
equilibrium consumption is calculated (equation 2), and 3) the agents’ preferences are adjusted (equation 4).
3 Results of simulations
The model has been simulated for N = 12, 25, 100, 156, 506, 992, 1980, 2500 agents and the following standard
types of network (see Figure 1 for the intuition or [5] for a detailed description of the networks): grid on torus,
star, ring, tree, small world, power network, complete network and disconnected agents (for the complete network,
the maximal N was 992 because the used simulation software was not able to carry more connections). In the ring
network and initial small world network, each agent could have a connection with 2, 4, 6, or 8 closest agents.
In the tree network, each agent could have 2 to 5 outgoing connections. In the power network, each agent was
created with 1 to 5 new connections. The rewire probability in the small world network was 0.05, 0.1, 0.25, 0.5.
The endowment of good 1 was e1 = 1, 5, 10, 20, . . . , 90, 95, 99, 100, while the endowment of good 2 was e2 = 100.
The adjustment constant r was set to 0.5 as in [1]. The model has been simulated thirty times for each feasible
combination of parameters. The maximal amount of simulation steps was set to 10 000. The total number of the
simulation runs was 123 900. The model was simulated in NetLogo 5.0 [4]. The simulation results have been
analyzed in R [2].
Since the full parametric space of the simulation is huge and non-rectangular, only the most salient stylized
facts will be presented here. Specifically, the possible outcomes of the simulations, the role of the market and the
interactions in the social network, clusters, and the determinants of the relative price will be discussed.
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(a) complete (b) star (c) ring (d) grid (e) tree (f) small world (g) power
Figure 1 Examples of the network structures. The figures are taken from [5]. The grid is depicted as von Neumann
neighborhood only for simplicity; Moore neighborhood was used in simulations.
3.1 Possible simulation outcomes
Bell [1] discusses only two kinds of simulation outcomes, both very similar. In both, the system converges, each
good is consumed by some agents, and fewer agents consume the scarcer good than the more abundant one. The
difference between the outcomes lies in the agents’ equilibrium preferences. In the first case, each agent specializes
in consumption of only one kind of good (i.e. ait ∈ {0, 1}), while in the second case, at least some agents consume
both goods (i.e. have ait ∈ (0, 1)). Bell claims that the second outcome is unstable, and hence cannot occur in a
simulation [1, p. 321]. Indeed, all Bell’s simulation outcomes were of the first kind (she calls it “polarized”).
My more general simulations produced a richer set of outcomes. There were four states: 1) the simulation did
not converged (0.205 % of runs), 2) it converged but some agents remained non-polarized (0.220 % of runs), 3) the
simulation converged, all agents were polarized but consumed only the more abundant good (13.835 % of runs),
and 4) the simulation converged, all agents were polarized and both kinds of good were consumed by some agents
which is the Bell’s only outcome (85.739 % of runs). Since the above stated frequencies are somewhat artificial, it
is instructive to decompose them.
Let us start with the state 1, the non convergence. This only occurred when the endowment of the two goods
was precisely the same (i.e. e1 = e2), the network was either complete or star, and the number of agents was
relatively large. In particular, the simulation of complete network always converged with N = 12 and in 93.333 %
of runs with N = 25; otherwise it did not converge. The simulation of star network always converged with
N = 12, . . . , 100; then the frequency of the converged runs decreased with the number of agents: 93.333 % of
runs with N = 256 converged, 70 % of runs with N = 506, 40 % of runs with N = 992, 33.333 % of runs with
N = 1980, and 10 % of runs with N = 2500. It is interesting that these two types of network are in a sense
opposite to each other and the reasons why they do not converge are opposite too. In the complete network, all
agents are symmetric. If the network does not converge, it is because each agent interacts directly with all other
agents and there is no “locality”. Thus any change of an agent’s preferences forces all other agents to change their
preferences, which makes the next adjustment necessary, and so on. In theory, there can be both many polarized
and non-polarized equilibria (e.g. one half agents prefer good 1 and the other half prefer good 2, or each agent’s
preference is 0.5 etc.) but all these equilibria are very fragile. In the star network, one central agent is connected
to all other (branch) agents, while each branch agent is connected only to the central agent. The branch agents tend
to be polarized in this case—roughly one half of them prefers good 1 and the other half good 2. It is the central
agent who wavers between the two and thus precludes the convergence.
The state 2, converged but not-polarized, occurred most often when the endowment of the two kinds of good
was the same too—96.703 % of runs in this state occurred when e1 = e2. However, it occurred with other
proportions of the endowment too, even though only rarely. Most often this state occurred when e1 = e2, the
network was either complete or star, and number of agents was small. All runs in the complete network with
N = 12 and 6.667 % of runs with N = 25 ended in this state. (No other run on the complete network converged,
see above.) As for the star network, all runs with N = 12, . . . , 100, 93.333 % of runs with N = 256, 70 % of runs
with N = 506, 40 % of runs with N = 992, 33.333 % of runs with N = 1980 and 10 % of runs with N = 2500
ended in this state. Beside these, two other networks (power and small world) tended to end up in this state. With
few agents and many connections, these networks are close to the complete network. Usually, the power network
ended in this state when e1 = e2 and N = 12. The frequency of this state raised with the number of connections
from 3.333 % with 2 connections to 46.667 % with 5 connections. The power network with 5 connections ended
in this state once also with N = 25 and once with N = 256 and e1 = 0.8e2. The small world network ended
in this state most often with N = 12 and e1 = e2; however, it ended in this state several times even with other
configuration, e.g. with N = 1980 and e1 = 0.5e2. Beside these, only tree network ended in this state once with
N = 25 and e1 = 0.5e2. In general, the results for this state suggest that even though the non-polarized state is
fragile, it can arise from the simulation, especially if the number of agents is small. It is also possible, that the runs
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denoted as “non-convergent” might have converged if the simulation was given more time.
The state 3, the converged polarized state where only the more abundant good 2 is consumed, occurred with
any network including the disconnected one. However, with disconnected network, this state occurred only when
the amounts of the two goods were extremely asymmetric (e1 = 0.01e2 or e1 = 0.05e2) and the number of agents
was small (N = 12, . . . , 100 when e1 = 0.01e2 and N = 12 in the other case). With increase in the endowment
of good 1 or the number of agents, the frequency of state 3 decreases and soon vanished with the disconnected
network. The situation was dramatically different with complete network. With it, the state 3 occurred always
when e1 < e2 no matter what was the number of agents. The behavior of other networks was between these two
extremes. In general, the increase in the endowment of the scarcer good e1 or in the number of agents decreased
the probability that all agents consumed only good 2. However, individual networks somewhat differed in their
propensity to this state. Paradoxically, grid network used by Bell was relatively more prone to end up in this
state than most other networks, with exception of complete and star networks and small world network with the
comparable number of connections. Surprisingly, the closest to the behavior of complete network was again star
network. The overall probability that the state 3 occurs can be modeled with the logistic regression. The results
are summarized in Table 1. The presence of a social network increases the probability that only the more abundant
good 2 is consumed (all network dummy variables coefficients are positive). The rise in the number of agents or
in the amount of the scarcer good decreases the probability, while the rise of the average number of connections
between agents increases it.
Estimate Std. Error z value Pr(>|z|)
(Intercept) −1.5718 0.1233 −12.75 0.0000
grid network 3.3075 0.1363 24.26 0.0000
power network 2.3564 0.1263 18.66 0.0000
small-world network 2.4546 0.1241 19.78 0.0000
ring network 2.6996 0.1269 21.28 0.0000
star network 4.5173 0.1359 33.23 0.0000
tree network 1.3319 0.1290 10.33 0.0000
complete network 10.4102 0.1689 61.62 0.0000
number of agents −0.0013 0.0000 −65.32 0.0000
endowment e1 −0.0796 0.0007 −108.29 0.0000
average number of connections 0.0018 0.0002 7.13 0.0000
log likelihood: −25089.08, McFadden’s pseudo R2
: 0.496, correctly predicted: 92.65 %
Table 1 Binary logit model. Dependent variable is one if the state is state 3—converged, polarized, and only the
more abundant good is consumed; otherwise it is zero. Network dummy variables are in contrast to disconnected
agents. The average number of connections in the tree network is only approximate. Only data from the converged
runs were used.
3.2 Role of market and of social network
Bell claims that “the number of agents consuming a good in the steady state is proportional to the availability of
the goods...” [1, p. 311]. If this was true, the relative price p1/p2 would be close to unity with no regard to the
proportion of the endowments as long as e1 > 0 and e2 > 0. Bell does not attempt to judge what part of this result is
caused by the centralized market and what part is caused by the interactions on the social network. She only claims
that “the price acts as a negative feedback mechanism that limits consumptions of scarce ... goods and encourages
consumption of plentiful ... goods” [1, p. 311]. Similarly, the “bandwagon” effect of the preference adjustment
in the social network creates a positive feedback mechanism: if one good is more abundant than the other one,
the agents can see it more often, and learn to like it more. (The previous section shows that the bandwagon effect
can be so strong that all agents learn to consume only the abundant good, and throw away the scarce one, which
contradicts the Bell’s claim stated above.)
Disconnected agents were included among the networks to allow to disentangle the impact of the two types
of feedback: disconnected “network” includes only the market negative feedback; the rest networks include both
the negative feedback and the bandwagon positive feedback. Their impacts can be seen either in the relative
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price p1/p2 or in the relative consumption, i.e. the average consumption of the agents who consume only good 1
divided by the average consumption of the agents who consume only good 2. However, the later is not necessary
since it is the reciprocal value of the former. It is because all agents have the same endowment e1 and e2, and hence
the same income p1e2 + p2e2. (The relative price p1/p2 = 0 means that no agent consumes the scarcer good 1, and
hence the relative consumptions is not defined in this case. For this reason, the relative price p1/p2 will be used in
the analysis.)
It is the market negative feedback (together with the preference-adjustment algorithm) what brings the system
into an equilibrium (disconnected “network” always converges). In the equilibrium, usually both goods are
consumed (the state 3 occurs least often with disconnected agents). And it is this force that presses the relative
price p1/p2 to unity. The left part of Table 2 shows that the scarcer good 1 is usually more expensive than the
more abundant good 2 (p1/p2 ≥ 0) when agents are disconnected but as e1/e2 and the number of agents raises,
the market negative feedback is able to push the relative price p1/p2 to unity. The consumers of the scarce good 1
have in average lower consumption that the consumers of the abundant good 2 in this case.
The presence of a social network adds the bandwagon effect which destabilizes the network (and possibly creates
the clusters, see below). First, the bandwagon effect can preclude the convergence and increases the probability
that the non-polarized state 2 occurs when the system converges (see above). Moreover, it raises the demand for the
more abundant good and thus increases its price. The right part of Table 2 shows the example of the relative price
for the grid network: it is almost always below unity, which means that the bandwagon positive feedback usually
more than offsets the stabilizing impact of the market negative feedback. In the limit case, the scarcer good 1 is
not consumed at all and its relative price p1/p2 = 0. If the scarce good 1 is consumed at all, its consumers have
higher consumption than the consumers of the abundant good 2 in this case.
disconnected grid
N / e1 1 5 10 20 95 99 100 1 5 10 20 95 99 100
12 1.82 1.60 1.57 1.56 1.16 1.14 1.15 0.00 0.00 0.00 0.00 1.05 1.01 1.00
25 0.98 1.52 1.66 1.60 1.15 1.16 1.15 0.00 0.00 0.00 0.00 0.98 1.06 1.05
49 1.40 1.64 1.85 1.62 1.05 1.04 1.03 0.00 0.00 0.00 0.48 0.96 0.99 1.02
100 1.67 1.70 1.71 1.55 1.03 1.01 1.00 0.00 0.00 0.46 0.57 0.97 1.00 1.01
256 1.40 1.64 1.74 1.53 1.00 1.00 1.00 0.00 0.48 0.58 0.61 0.95 0.97 0.97
506 1.34 1.77 1.75 1.58 1.01 1.00 1.00 0.00 0.59 0.59 0.62 0.96 0.98 0.99
992 1.29 1.74 1.73 1.55 1.01 1.00 1.00 0.00 0.59 0.62 0.63 0.96 1.01 1.01
1980 1.33 1.77 1.73 1.56 1.01 1.00 1.00 0.00 0.60 0.61 0.66 0.94 0.97 0.97
2500 1.36 1.79 1.75 1.56 1.01 1.00 1.00 0.00 0.59 0.60 0.66 0.95 0.99 0.99
Table 2 Average relative price p1/p2 for selected endowments e1 (columns), number of agents N (rows), and
networks (panels). Averaging can bias the relative price slightly upward. Only data from states 3 and 4 were used
in the computation.
3.3 Clusters
Bell reported that the agents with the same preference were clustered together in the polarized equilibria. This
is indeed what happens in the polarized equilibria in the networks in which each agent has many connections
(e.g. grid network). Each agent can retain her preference only if the number of the agents with the same preference
is in her neighborhood higher than the number of agents with the opposite preference, i.e. if the agents are clustered.
This, however, is not necessary in networks where some agents have only one connection (e.g. in tree network).
There might survive also agents who have preference different from their surroundings. The sufficient condition
is that 1) the dissenting agent i has only one connection, prefers the cheaper scare good 1 (p1/p2 < 1), and hence
has higher consumption than the consumers of the more abundant good 2, and 2) agent j in the neighborhood n(i)
of agent i has more than one connection and she and most other agents in her neighborhood n( j) consume the
expensive abundant good 2, and hence have lower consumption than agent i. In this case, there can survive
consumers of the scarce good 1 outside clusters; there even need not to be any clusters at all. This also explains
relatively higher resistance of this type of networks against the state 3.
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3.4 Quantitative determinants of relative price
The type and size of the social network do not affect only the qualitative, but also the quantitative characteristics
of the equilibrium. Table 3 shows the determinants of the relative price p1/p2. The presence of the social network
lowers the relative price. The rise in the number of agents and the endowment e1 of the scarce good 1 brings the
relative price to unity (notice that the corresponding parameters for the social network and disconnected agents
have the opposite signs). Obviously, the used type of network strongly affects the equilibrium relative price.
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.63 0.03 54.28 0.00
grid network -1.37 0.03 -45.23 0.00
power network -1.44 0.03 -47.89 0.00
small-world network -1.35 0.03 -44.84 0.00
ring network -1.27 0.03 -42.01 0.00
star network -1.67 0.03 -54.71 0.00
tree network -1.24 0.03 -41.30 0.00
complete network -1.98 0.03 -64.68 0.00
number of agents -0.00 0.00 -2.02 0.04
endowment e1 -0.01 0.00 -18.85 0.00
average number of connections -0.00 0.00 -5.04 0.00
number of agents × social network 0.00 0.00 6.01 0.00
endowment e1 × social network 0.01 0.00 41.00 0.00
s: 0.2422, R2
: 0.6145, adjusted R2
: 0.6144, F: 1.6380 on 12 and 123360 DF, p < 2.2e − 16
Table 3 Linear regression model. Dependent variable is the relative price p1/p2. The network dummy variables
and their summary “social network” are reported in contrast to disconnected agents. The robust standard errors of
the parameter estimates are reported. The average number of connections in the tree network is only approximate.
4 Conclusions
The simulation results show that Bell’s conclusions have to be somewhat modified. There are more possible
outcomes than she expected: the system need not to converge, the agents’ preferences need not to be polarized,
and the number of agents consuming a good need not to be “proportional to the availability”—the agents can
consume only the abundant good and throw away the scarce one. The agents need not to be clustered. In general,
the convergence to an equilibrium, the type of the resulting equilibrium, and the equilibrium relative price are quite
sensitive to the type and size of the used network because it is the bandwagon effect what destabilizes the system.
This means that a careful specification of the social network might be crucial for modeling real-world markets.
References
[1] Bell, A. M.: Locally interdependent preferences in a general equilibrium environment, Journal of Economic
Behavior & Organization 47 (2002), 309–333.
[2] R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for
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[3] Tesfatsion, L.: Agent-Based Computational Economics: A Constructive Approach to Economic Theory. In:
Handbook of Computational Economics, vol. 2, (L. Tesfatsion and K. L. Judd, eds.), Elsevier, 2006, 831–880.
[4] Wilensky, U. (1999). NetLogo. Center for Connected Learning and Computer-Based Modeling, Northwestern
University, Evanston, IL, http://ccl.northwestern.edu/netlogo/.
[5] Wilhite, A.: Economic activity on fixed networks. In: Handbook of Computational Economics, vol. 2, (L. Tesfatsion
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