Two Modes of Scheduling in a Simple Economic Agent-Based Model
Sarah Wolf
1,2
, Steffen F
¨
urst
1,2
, Sophie Knell
2
, Wiebke Lass
2
, Daniel Lincke
1,2
, Antoine Mandel
3
,
Jonas Teitge
2
and Carlo Jaeger
1
1
Global Climate Forum, Berlin, Germany
2
Potsdam Institute for Climate Impact Research, Potsdam, Germany
3
Centre d’
´
Economie de la Sorbonne, Universit
´
e 1 Panth
´
eon-Sorbonne, Paris, France
Keywords:
Economic Agent-based Models, Scheduling, Climate Policy, Win-win Strategies.
Abstract:
Agent-based models (ABMs), and with them simulation, are gaining importance in economics. As they allow
to study coordination problems in a dynamic setting, they can be helpful tools for identifying win-win strate-
gies for climate policy. This paper argues that strongly simplified models can support a better understanding
of economic ABMs. We present work in progress on an example case: while in economic systems in the real
world many actions and interactions by various agents take place in parallel, often ABMs use sequential com-
putation. With a simple economic agent-based model of firms that trade and produce goods, we explore and
discuss two alternative modes of scheduling: the timetable model, where all agents complete one step after the
other, and the heliotropic model, where one agent after the other completes steps. We find that the timetable
model is better suited for working with data from national statistics, while the heliotropic model dispenses
with random shuffling that is often introduced to guarantee symmetric expectations for agents. The latter can
be used in a completely deterministic fashion, providing a baseline case for studying the system’s dynamics.
1 INTRODUCTION
Simulation plays an ever more important role in eco-
nomics as agent-based models (ABMs) are used more
frequently. These represent the economy as a com-
plex system in which macro-features emerge from
the interaction of many heterogeneous agents. Imple-
menting a system of agents on a computer and observ-
ing simulation runs to study the system’s behaviour
poses the question whether some of the observations
owe to computational features of the implementation
rather than being characteristic of the system under
study. For example, in real-world economic systems,
many actions take place in parallel, while on the com-
puter, parallel actions are often represented by se-
quential steps. Some observations might occur due
to the sequencing chosen by the modeller
1
.
Various platforms for agent-based modelling
(such as Swarm, Repast, MASON, Netlogo, etc.) pro-
vide tools for representing time and scheduling ac-
1
Parallel computation may provide ways to avoid this
problem, but parallelisation is beyond the scope of this
short paper: interdependence between the agents makes the
model used difficult to parallelise. Also, we aim at a simple
model, while parallel computation raises complexity.
tions by different agents. ABMs may implement a
simple sequence of agents all conducting the same ac-
tion one after the other, or complex message passing
systems between agents that trigger actions in event-
driven simulation. In some cases, randomness needs
to be introduced to to guarantee symmetric expecta-
tions for agents. For example, in a representation of
trade, the first firms buying goods might find full in-
ventories of all others, while the last ones may find
inventories rather empty. To avoid such a bias, which
would be an artefact of computing sequentially, the
order in which agents act is often determined by ran-
dom shuffling. This means that randomness is intro-
duced for computational reasons
2
.
Most works related to ABMs provide little detail
on how simulations are executed and rather focus on
describing agents and their environment, as stated by
Mathieu and Secq (2012), who find that the represen-
tation of time and scheduling in the simulator used, as
well as sequential or parallel execution of actions can
have crucial impacts on simulation results.
The present paper focuses on the case of sequen-
2
Other sources of randomness, such as random muta-
tions to represent innovation, may be essential to the model,
but are not of interest here.
303
Wolf S., Fürst S., Knell S., Lass W., Lincke D., Mandel A., Teitge J. and Jaeger C..
Two Modes of Scheduling in a Simple Economic Agent-Based Model.
DOI: 10.5220/0004032203030308
In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2012),
pages 303-308
ISBN: 978-989-8565-20-4
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
tial computations with a fixed schedule
3
. We use a
simplistic economic ABM of firms that trade and pro-
duce goods in order to explore and discuss two al-
ternative modes of scheduling: the timetable model,
where all agents complete one step after the other, and
the heliotropic model, where one agent after the other
completes steps. In particular, our work in progress
focuses on the questions how much randomness is
necessary, whether and how the model behaviour be-
comes mathematically tractable, and how model in-
put and output relates to economic data from national
statistics. We argue that a better understanding of
these questions, and of economic ABMs in general, is
relevant to climate policy making. Therefore, the cli-
mate policy background is sketched in Section 2 be-
fore Section 3 briefly describes the simplistic model.
First results are presented in Section 4 and discussed
in Section 5, before Section 6 concludes.
2 CLIMATE POLICY AND ABMs
The motivation for our work on economic ABMs
stems from the climate change context. Climate
policy analysis and recommendations are generally
based on standard economic modelling; the most fre-
quently used models are computable general equilib-
rium models (Capros et al., 1999, for example) or op-
timal growth models (Nordhaus, 2008; Stern, 2007).
The general set-up is to concentrate on a a busi-
ness as usual (BAU) growth path as the single sta-
ble equilibrium path of the system, that is optimal
in the short run, and compare this with a situation
where measures for the mitigation of greenhouse gas
emissions are taken. Mitigation is framed as a wel-
fare trade-off between present and future generations:
greenhouse gas emissions are an external effect of the
present upon future generations, so that for these the
path is not optimal. The current generation uses the
atmosphere (as also a few previous generations did) as
a “waste dump” for emissions without considering the
future negative effects this will due to climate change
resulting from the emissions. Introducing a price on
emissions, this externality can be internalized, that is,
eliminated. However, on the BAU growth path this
implies costs in the short run, usually expressed in
terms of a reduction in GDP, because the current gen-
eration will have to pay for using the atmosphere, that
before it simply used “for free”.
This widely accepted welfare trade-off argument
has coined a narrative of mitigation as a problem of
3
Here fixed is meant as in opposition to event-driven.
The schedule may still involve randomness in the order in
which agents act.
burden sharing (Jaeger et al., 2012). While the miti-
gation costs are legitimated by the benefits of avoided
climate change and its impacts, these benefits lie in
a rather far away future, so that on shorter planning
horizons, such as the election periods of politicians,
the costs seem much more relevant. In this setting, in-
ternational negotiations have made little progress to-
wards significant world-wide reductions of emissions.
There is, however, in this argument a fundamen-
tal assumption that is problematic: the existence of a
unique stable equilibrium growth path of the system
is warranted neither by economic theory nor, much
more importantly, by real-world observations as sum-
marized in empirical data. For example, Ormerod
et al. (2009) find that the US, the UK, and the Ger-
man economic system from time to time switch from
a steady to a weak pattern. Also, the comparison of
countries that at a certain point in time were in similar
situations but now differ in economic growth, such as
Poland and Hungary at the moment, deserves the con-
sideration of different growth paths.
Likewise for economic theory: the Arrow-Debreu
framework of general equilibrium shows the existence
of equilibria in an abstract setting a growth path is
determined by prices that, for each time-step under
consideration, balance supply and demand. However,
equilibria need not be unique nor stable. Assump-
tions made to guarantee uniqueness and stability in-
clude that of a single representative agent, discussed
for example by Kirman (1992).
Hence, the narrative of climate change mitigation
as a problem of burden sharing is not the only story
to be told. Jaeger et al. (2012) suggest to “reframe
the problem of climate change, from zero-sum game
to win-win solutions”, i.e., mitigation measures which
are beneficial for the economy. Win-win strategies for
climate policy can be identified when widening the
focus – from concentrating on a single equilibrium to
considering several possible equilibria (Jaeger et al.,
2010; Shi and Zhang, 2011). A low carbon econ-
omy needs a good deal of restructuring as compared
with the current economic situation, for example from
fossil to renewable energy. Such a structural change,
from the perspective of economic theory, involves a
shift to another equilibrium growth path. Win-win op-
portunities arise when the new path is in some sense
“better” than the current one. In this case, micro-costs
which occur due to mitigation measures, such as in-
creased energy prices due to emission trading, can be
more than compensated by macro-benefits, such as
higher growth and less unemployment, that the new
growth path entails (Jaeger et al., 2010).
Equilibrium selection is a coordination problem
(Jaeger, 2012): as for the case of conventions (such
SIMULTECH2012-2ndInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
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304
as driving on the left or driving on the right) there are
several viable alternatives – which one is in place de-
pends on interactions of many agents and on institu-
tions evolving between them. In order to shift an eco-
nomic system to a different equilibrium, agents need
to re-coordinate to another alternative. Agent-based
models are a promising tool for studying such prob-
lems as they allow to consider coordination problems
between many agents in a dynamic setting. However,
few ABMs have been applied in the climate change
context, and even less to study climate economics (see
Balbi and Giupponi, 2009, for a review).
Macro-economic ABMs, of particular interest in
the climate policy context, are often rather complex
models with a huge amount of state variables (Gintis,
2007; Dawid et al., 2011; Mandel et al., 2009). Study-
ing equilibrium selection as a coordination problem
between many agents requires much more complex
models than the standard representative agent models.
However, while the simulations of a complex ABM
allow to observe the modelled system’s dynamics, this
does not necessarily mean that one understands these
dynamics. In a simulation, iterations compute one
new state after the other from a given initial state of
all agents and their environment. Hence, an ABM
could be described as a dynamical system given by a
state space together with a transition function. How-
ever, models are too complex to write these down ex-
plicitly, and it is not even clear whether the dynami-
cal system should be deterministic (Epstein, 2006) or
a Markov process, that is, probabilistic (Tesfatsion,
2006; Gintis, 2007). For the model’s implementation,
randomness can be discussed away with the argument
that the computer generates pseudo random numbers.
Many ABMs have random elements that are essen-
tial to the modelled system, but others might be intro-
duced for implementational/computational reasons.
Macro-economic agent-based modelling for (cli-
mate) policy does not always start from simple mod-
els that are then further elaborated. Therefore, as a
second track besides complex economic ABMs, we
advocate strongly simplified economic ABMs to gain
a better understanding of their properties as dynam-
ical systems and related methodological issues. In
the following, we present a preliminary example in
this spirit. While this work is further away from real-
world applications such as climate policy analysis, it
aims at small improvements of our understanding of
economic ABMs which in turn may be very help-
ful in that context. In particular, the interdependence
between agents, that makes scheduling an issue here,
is an essential aspect also in more closely climate
change related problems.
3 THE SIMPLE ABM
The model used here is based on the Lagom model
family of economic ABMs (Mandel et al., 2009; Wolf
et al., 2012) but it radically simplifies the economic
system considered: 100 firms, grouped into 5 sectors,
“trade”
4
and produce the good of their sector. Pro-
duction requires intermediary inputs, the production
structure is based on an input output table.
Firms’ parameters are a desired production d
0
, input coefficients for circulating capital γ
[0, 1]
5
, and an inventory depreciation rate
5
r [0, 1].
It suffices to consider their inventory I
0
as state
variables in this simplistic model. Economic input
data (total production, an input-output table, and the
inventory depreciation rate) is used to initialize the
model.
3.1 Activities: Trade and Production
For each sector s = 1, . . . , 5, firms try to “buy” the
inputs for their desired production d. Being my-
opic, they only see a few firms; the number of firms
observed, and whether these are fixed or randomly
drawn firms, are model input parameters. In the case
of fixed suppliers, a firm “sees” the firms listed just
before itself in the model’s firm list. The set of ob-
served firms from sector s is denoted O
s
. Firms first
find out the available supply of inputs among the firms
they observe for each sector. This is a
s
=
iO
s
I
i
for
sector s. The quantity to be produced is determined
as q = min
d, min
sS
a
s
γ
s
and the required amounts
of inputs are then given by q · γ
s
. The firm then buys
q · γ
s
for each sector s, and the sellers subtract the re-
spective amounts from their inventories.
6
Production corresponds to a simple change of
state variables for a firm: the produced quantity q
is added to the inventory. In particular, production
changes state variables for the active firm only, while
4
The model uses a caricature of trade, without any pay-
ments being made. Rather than of “buying goods”, one
might speak of “obtaining presents”.
5
This rate can be seen as a work-around for the miss-
ing consumption in the simplistic model: inventory is de-
creased, which it would also be if the good was sold to
consumers. As money does not play a role here, this ap-
proximation is good enough for our purposes.
6
This is a typical case of interaction in an economic
ABM that is easy to describe in words and rather easy to
implement, but not so easy to describe as a mathematical
dynamical system. State variables of “passive” seller firms
change, and the amount q that is bought may depend on the
inventory of some firm in O
s
that again may depend on how
much other firms have previously bought from this firm.
TwoModesofSchedulinginaSimpleEconomicAgent-BasedModel
305
trade makes changes to the state variables of other
firms as well.
3.2 Two Modes of Scheduling
Time evolves discretely. In each period, each firm car-
ries out each activity once. Within periods, we con-
sider two modes of scheduling.
In the timetable model, two steps constitute a pe-
riod: trade and production. In each step, all firms
carry out the corresponding activity. The real-world
time interval that a period represents may be chosen
by the model user by calibrating the input data accord-
ingly. In each period, goods are produced using only
goods from previous periods as inputs. This means
that the aggregate production of all firms in one period
corresponds to the production amount in the time span
represented by this period. Therefore, data from na-
tional statistics can easily be used in this model, both
to calibrate parameters in such a way as to represent a
given real-world economic system at a given moment
in time, and to validate the model output by compari-
son with national statistics data. The trade step, how-
ever, has a “first come first serve”-element, therefore,
the order in which firms trade is important.
In the heliotropic model, firms enter activity
phases one after the other. Within an activity phase,
a firm first trades and then produces. Here, all agents
are “in the same situation”: each firm finds some other
firms that have just produced goods with a full inven-
tory, while others may have sold most of their inven-
tory since they last produced. That is, in this model
no bias arises from the order in which agents act, or in
other words, firms have symmetric expectations even
though they always act in the same order. However,
production is not simultaneous here. A period can-
not easily be mapped to a time interval in the real
world because the production from one period is not
clearly separated from that of another one: firms can
use goods that others have already produced in the
same period as inputs for production. This compli-
cates the link with national statistics data.
4 SIMULATIONS
Input parameters have been calibrated in order to
identify the range of parameter values where the
model is susceptible to changes in scheduling. This
means considering an economic system in which the
production inputs are scarce, so that trading first or
last in the timetable has an effect. The inputs used are
toy data, for example, all entries in the input-output
table are equal to 1 for simplicity. With a production
value of 5.1 for all sectors, input coefficients are high
compared with the desired production and initial in-
ventories. Also, the inventory depreciation rate is set
rather high, so that inventories stay low compared to
the demand of production inputs in later periods.
4.1 Shuffling in the Timetable Model
Figure 1 shows that shuffling of firms at the begin-
ning of the trade step is indeed necessary to avoid a
bias that favours firms who trade first: reduced pro-
ductions occur only for the last firms in the list when
no shuffling takes place, while, when firms are shuf-
fled at the beginning of each trade step, the reduced
production amounts are scattered over all firms.
Figure 1: Timetable model with (above) and without (be-
low) shuffling before trade. The model output shown is the
produced quantity of firms, columns are sectors, cells show
firms. All parameters (including random seed) are the same
in the two example runs. The desired production is at the
upper border of the firms’ cells, the downward spikes show
when a firm produces less than desired.
Without scarcity, the order in which firms trade
would not have mattered here. However, it might
not always be as obvious that shuffling is required to
provide symmetric expectations for agents. In more
complex models, higher prices or less skilled workers
might be examples of how agents acting last are disad-
vantaged. Such biases between agents are artificially
introduced by sequential operations on the computer
that represent parallel actions in the real world.
4.2 Scarcity in the Heliotropic Model
The heliotropic model can – without a need to shuffle
firms – produce output that resembles the one seen in
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306
Fig. 1 with shuffling. However, with the same input
data, one sees less scarcity effects, and the inventory
depreciation rate needs to be raised to obtain a similar
picture as above. This was to be expected because
firms trading late in a period may buy goods from
firms that have already produced in that period, but
it underlines the difficulty of mapping output to na-
tional statistics data. Produced goods may enter back
into the production scheme in the same period when
they were produced, meaning the separation between
production inputs and produced goods is lost.
With fixed suppliers, the heliotropic model can be
used to construct a completely deterministic baseline
case in which randomness is eliminated. Consider-
ing this model as a dynamical system, it is a truly
deterministic system. Writing down state space and
transition function is still a lengthy undertaking, and
therefore not done in this short paper. However, it be-
comes clearer how the system functions: depending
on the choice of data for the initialisation one can cre-
ate an economy in equilibrium, where stocks remain
the same or grow, or an economy in decline that will
in the end have to crash. For simplicity, the inventory
depreciation rate can be “switched off”, that is, set to
0. With an input of 1 unit from each sector, the econ-
omy stagnates when the production is set to 5. For
production values below this, the economy declines,
and production goes down to 0 (except for round-off
errors), and this the more quickly, the lower the ini-
tial production value is set, of course. In fact, it suf-
fices to have one sector with a production that is not
sustainable to create an economy in decline when all
goods are needed as inputs. With an initial production
value greater than 5, firms always succeed in produc-
ing their desired production, and inventories simply
increase. The heliotropic model can thus be a starting
point for adding bits and pieces of complexity (e.g.,
an evolving, instead of a fixed desired production), for
observing, and hopefully understanding, its effects.
5 DISCUSSION
The results presented are of course no deep modelling
results. Nevertheless, this simple model highlights
some interesting points concerning economic ABMs
that use sequential computation. In particular, both
modes of scheduling come with advantages and dis-
advantages.
In the timetable model, shuffling introduces ran-
domness that is not actually essential to the mod-
elled system, but required in order to provide sym-
metric expectations for agents, owing to the sequen-
tial implementation of the model. Randomness in
ABMs implies that the interpretation of model out-
put necessitates many model runs and statistical eval-
uations of these to learn about the distributions of
states. Looking at single trajectories, one might see
some effects of improbable events having taken place
in just this run. While in ergodic systems the proba-
bility distributions over states converge regardless of
the initial state chosen, one does not necessarily know
when the system arrives “close enough” to the limit
and while distributions converge, this does not mean
one only sees “average” trajectories. The heliotropic
model eliminates this randomness, allowing for a sim-
pler starting point for analysing the ABM. In the ef-
forts that are being made to create economic ABMs
which are mathematically tractable, this is a step for-
ward. In fact, the deterministic dynamical system that
arises when using fixed matching rules for trade in
this model shows behaviour that depends on the input
data used in a clearly understandable way.
At the same time, efforts are being made to create
economic ABMs which are empirically satisfactory
(Fagiolo et al., 2006; Boero and Squazzoni, 2005, for
example). Here, the timetable model shows an advan-
tage. The “separation” of periods, meaning that all
goods produced in a given period can be used as pro-
duction inputs at the earliest in the next period, allows
to consider the aggregate production of all firms in a
period as corresponding to the real-world production
of the time span that this period represents. Greater
ease of mapping data from national statistics to input
data, and vice versa mapping model output to data
from national statistics facilitates the empirical vali-
dation of the model. This is an important advantage
also in the climate policy context, where empirically
relevant models are badly needed to “compete” with
single equilibrium models. As most data are in a for-
mat fit for standard modeling approaches, ABMs that
can be fed and validated with these data formats have
the edge over those that cannot.
6 CONCLUSIONS AND
OUTLOOK
This paper presented work in progress on the compar-
ison of two scheduling modes for a simple economic
ABM. We find that the output is similar. However, the
timetable model, where all firms complete one step af-
ter the other, is closer to data from national statistics,
facilitating their use as input and the interpretation
of aggregates in the model output. The heliotropic
model, where one firm after the other completes all
steps, can do without shuffling that in the other model
is necessary to avoid favouring agents who act first.
TwoModesofSchedulinginaSimpleEconomicAgent-BasedModel
307
Thus, the heliotropic model eliminates randomness –
artificial, because required due to sequential compu-
tations – producing a simpler model.
For the task of helping to generate theoretical in-
sights, it can be helpful to start from simple economic
ABMs and add complexity in a step-by-step manner.
In fact, this is the aim of the models used here: too
simple to study real-world economic systems, they
may allow a better understanding of issues that can
be studied already at this simple stage. Indirectly, a
better understanding of economic ABMs as dynami-
cal systems then may contribute to policy analysis of
real-world problems – as for example the questions of
multiple equilibria and win-win strategies for climate
policy sketched above.
Further work on simple economic ABMs, in par-
ticular on a completely deterministic version of the
heliotropic model as a benchmark and on these mod-
els as dynamical systems in the mathematical sense,
seems worthwhile.
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