How Public Goods can generate

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Central Economics and Mathematics Institute of
Russian Academy of Sciences, Moscow, Russia
How Public Goods can generate
regional structure
simulations on the agent-based model
Valery Makarov
World Congress on Social Simulation 2008, George
Mason University, Fairfax July 14-17, 2008
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The literature, devoted to the structure of
territorial jurisdictions, focused basically on
explanation of causes the structures formation.
Public goods play an important role in the
explanations. See, for example, Besley T. and
Coate Stephen (2003). Much less one can find in
the issue of discovering and analyzing
mechanisms, which lead to this or that regional
format.
The method of analysis is an agent - based model
(ABM) and simulation on the model of various ways
of the jurisdictions formation. For simplicity
I take into account only three types of public
goods, which should be assigned to different
levels of government. First, the basic rules are
formulated, where the behavior of agent is
divided by stages.
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Main rules
The first public good related to primary public
services which any citizen should receive in
his/her everyday life. It is maintenance of local
environment, registration, post service and so
on. The costs to provide this public good depend
on number of people to be served, like
this cost1 k1 c1 n2 where k1 is constant
expenses to keep public service functioning, c1
is a cost to provide one unit of a public service
per a person.
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The other public good is associated with public
schools, hospitals, courts, jails, etc. The cost
of provision the public good is dependent on
number of people also. cost2 k2 c2 n1.5 The
third public good is related to the classic
definition of a pure public good, given by P.
Samuelson, (See Samuelson, P. A. (1954)). It
means no dependence on a number of people.
So, cost3 k3
The ABM deals with finite number of agents. Each
agent has the same wealth equal to k. The wealth
is needed to an agent to pay taxes. The taxes are
going to provide the described public goods.
Preferences of agents to have the public goods
are lexicographical type. Namely, an agent is
ready to pay for the first public good. The rest
is paying for the second public good and finally
for the third one.
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First stage
In the stage the agents move chaotically in a
given 2- dimensional space. 1. If two agents meet
(by chance) each other then they form a
group, 2. The group moves randomly with the speed
which is less then individual speed of an agent
to the number of times equal to the quantity of
the groups members. 3. Territory of a location
of a group depends on its size. It is a circle of
radius r(n) anb, where n quantity of members
in the group, a and b are positive numbers. If n
1, then r 0. 4. If an agent and a group
meet, then there are the two outcomes. (a)
The agent joins the group, if cost1(n1) lt
cost1(n) (b) the agents starts to move in
opposite direction otherwise. 5. If two groups
meet, then there are the two outcomes again.
(a) The groups merge into one, if cost1(n1n2)
lt cost1(n1) cost1(n2) (b) the groups start to
move in opposite directions otherwise. Here n1,
n2 is a size of the population in the first and
the second group relatively.
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Second stage
The second stage begins when no one agent is
alone, and no more merge between groups. The
groups continue to move in slow speed
chaotically. 1. If two groups meet, then there
are the two outcomes. (a) The groups form a
region, if cost2(n1n2) lt k(n1n2) - cost1(n1)
- cost1(n2) (the inequality means that the rest
of the total wealth of the two groups is enough
to provide the second public good) (b) the
groups start to move in opposite directions
otherwise. Remark. The condition of the regions
formation assumes that the agents of the
different groups pay different amounts of taxes.
An agent in lager group pays more.
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2. If a group and a region meet, then there are
the two outcomes. (a) The group joins the
region under conditions cost2 (n N) / (n N)
lt cost2 (N) / N cost2 (n N) / (n N) lt k
cost1 (n) / n Here n and N is a size of the
population in the group and in the region
relatively The first inequality means that the
head tax for provision of the second public good
in the extended region is not greater then the
head tax in primary region. The second inequality
says that the amount of wealth in the group is
sufficient to pay the new tax. (b) The group
and the region start to move in opposite
directions otherwise.
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3. If two regions meet each other we have two
options again (a) The two regions merge into
one if cost2 (N1 N2) / (N1 N2) lt cost2
(N1) / N1 cost2 (N1 N2) / (N1 N2) lt cost2
(N2) / N2. Here N1 and N2 is a size of the
population in the region relatively The first
inequality means that the head tax for provision
of the second public good in the extended region
is not greater then the head tax in the first
primary region. The second inequality says the
same about the second region. (b) The regions
start to move in opposite directions otherwise.
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Third stage
The third stage begins when the process of
mergence between regions get to finish. 1. If
two regions meet, then there are the two
outcomes. (a) Its form a country if they have
enough wealth to provide third public good.
Namely, k(N1N2) (cost1(N1) cost2(N1)
cost1(N2) cost2(N2)) gt k3. Here cost1(N1) is
the total cost to provide first public good
across all jurisdictions in the region 1. (b)
The regions start to move in opposite directions.
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2. If a country and a region meet then there are
the two outcomes. (a) The region joins the
country if the head tax for provision of the
third public good in united country is not
greater then in the original country. (b) The
region and the country start to move in opposite
directions. Under the described conditions of the
third public good provision the option (b) does
not occur because it is beneficial to invite a
region to join a country always. 3. If two
countries meet, they merge into one united
country. The third stage finishes when there is
one country or the existing countries cant meet
each other because of some reasons. I discuss the
reasons later.
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Fourth stage
Voting by feet according to Charles Tiebout. See,
Tiebout Ch. (1956). Randomly chosen agent decides
to move or not to move to the neighboring
jurisdiction within a region according to the
following rule. It moves if the head tax in
his/her jurisdiction is greater then in the
neighboring one. The process is repeating as many
times as it changes the distribution of agents
across jurisdictions.
Planning of experiments
The basic idea of simulations is to find the
parameters and initial conditions, which form
regional structure close to an optimal one after
running all the stages. And what are barriers,
constraints, random causes, which create
difficulties to reach the optimal state.
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The Optimization Problem
The natural optimization problem is related to
the minimization of the head taxes, needed to
provide the all three public goods for the whole
population. So, it is necessary to find the size
of a jurisdiction of the low level and the size
of a region. It is clear that the country should
be one, and all jurisdictions and regions should
be one size. It yields immediately from the
condition that all agents are absolutely
identical to each other.
A hierarchical structure is given by the public
goods. If the number of different types of public
goods is greater, the problem optimal number of
hierarchical ties arises and associated with it
the problem of public goods assignments to the
levels comes too.
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The optimization problem related to a
hierarchical structure arises in different fields
and by various reasons. For example, Qian Yingyi
(1994) considers an economic organization that
owns a capital stock and uses a hierarchy to
control the production. The optimal problem is to
find number of tiers in the hierarchy and optimal
quantity of workers is in each tier. The
objective function in his approach is revenue,
generated from production activity. The trade off
is between the two parameters the number of
bureaucrats to control workers and efficiency of
working activity under the control.
In the paper of Jacob B. L., Chen P. M.,
Silverman S. R. and Mudge T. N. (1996) one can
find a survey and different approaches of the
optimal hierarchical problem in the technical
field, like an organization of computer memory,
etc.
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My case based on a single cause of the
hierarchys emergence of jurisdictions in a
state provision of a certain amount of public
goods by minimal cost. In other words minimal
cost means the minimal head tax.
For example the optimal number of people in the
jurisdiction of the low level (a commune or
municipality) one can find easily. Let k is a
fixed cost to maintain the government functioning
and c is a cost to provide public service for one
person. Let the total cost is cost k cn2 ,
as it was mentioned above. Then the head tax is
cost / n, where n is number of inhabitances in
the jurisdiction. From the first order condition
we immediately obtain where m is the optimal
size of a jurisdiction.
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Needless to say, that there are many causes for
jurisdictions creation. For example, in the
paper Zax J. S. (1988) one can find an empirical
analysis of relations between number and types of
jurisdictions and tastes and other
characteristics of population, based on US data.
Now we see a rising interest to operations on
jurisdictions as among theoreticians (see Alesina
Alberto and Spolaore Enrico (1997)) and among
practitioners too. Russian Federation is under
the total reform of local self governance. And at
the same time there is academic and public
discussion about Federal Constitutional structure
of Russia. See for example, ????? ?. (2004).
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In practice there was merger of the two subjects
of Russian Federation (Perm oblast and Comi
national district). On the agenda there are two
or three mergers more. The general problem is a
big uncertainty in the rules of jurisdictions
creations and liquidations.
The process of new states formation,
unifications and so on, is increasing in the
world last decades. But the more or less precise
rules to do that, which are acknowledged by
international community, are absent. A practical
experience is accumulated gradually, and
theoretical people have to make their
contribution as well.
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The agent-based model
We developed the basic agent based model, which
has a number of versions, depending on questions
one wants to ask. The version, named AS-G1, deals
with simple environment a quadrate in two -
dimensional space. Agents move in the quadrate
chaotically, try to communicate with each other
according the following rules, I describe here.
As it was mentioned above, there are four stages
in the evolutionary process.
The first stage. (1) If two agents see each
other, they start to move to face each other and
form the group. (2) The territory, which is under
control of the group, is a in the circle of
radius r(n) anb, where n is a quantity of the
groups members, a and b positive numbers. If
n 1, then r 0.
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(3) A group moves with a speed less then the
speed of agents movement. Greater group - less
the speed. (4) If an agent sees the group, there
are two outcomes. (a) The agent is taken by the
group, if cost1(n1) lt cost1(n) (b) the agent
start to move in opposite direction
otherwise. (5) If the two groups see each other,
again there are two outcomes (a) the groups
merge, if cost1(n1n2) lt cost1(n1)
cost1(n2) (b) the groups stat to move in
opposite direction otherwise. Here n1, n2 the
size of population in the first and the second
group. The first stage is finished, when there
are no more mergence between groups and agents.
The second stage takes place if agents have money
to pay for the second public good.
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The second stage starts with chaotic movement of
all groups, where the speed of a group depends on
its size. (1) When two groups have a meeting,
there are two outcomes (a) the groups form a
region, if cost2(n1n2) lt k(n1n2) cost1(n1)
cost1(n2) (the inequality means, the quantity
of money of the members of the groups are enough
to produce the second public good) (b) the
groups start to move in opposite directions
otherwise. Remark. Under the outcome (?) variants
are possible because of agreement among the
groups about the quantity of money to pay for
public goods production. (2) If a group and a
region meet, there are two outcomes (a) the
group enters the region under the
conditions i. cost2(n N) / (n N) lt
cost2(N) / N ii. cost2(n N) / (n N) lt k
cost1(n) / n.
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(b) The group and the region move in opposite
directions otherwise. (3) If two regions meet one
has two outcomes again (a) The regions merge
into one, if i. cost2(N1 N2) / (N1 N2)
lt cost2(N1) / N1 ii. cost2(N1 N2) / (N1
N2) lt cost2(N2) / N2 Here N1 and N2 are
quantity of population in the first and in the
second regions. The first inequality says that
the payment for the second public good in the
united region is not greater then in the first
region. The second inequality states the same for
the second region. (b) The regions start to
move in opposite directions otherwise. (4) A
group makes a decision to be transformed to a
region after a given period of time is over. The
decision takes place if the budget constraint
fulfills cost2(n) lt k n cost1(n).
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The second stage is over, when no groups and
region for the merger and transformation. It may
happen that some group stays in the previous
position with no the second public good because
lack of money. The third stage is closed to the
second one from the point of view of substance.
It consists of the process of unification of
region into larger formations, let us call its
by countries. The countries exist to provide the
third public good. (1) When two regions meet, we
have two outcomes again (a) Its form a country,
if the agents have enough money, namely k(N1N2)
(cost1(N1) cost2(N1) cost1(N2) cost2(N2))
gt k3. Here cost1(N1) and cost2(N1) are total
costs of the first region for provision of the
two public goods. Analogously, cost1(N2) and
cost2(N2) costs of the second region.
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(b) Otherwise the regions start to move in
opposite directions. (2) Under meeting of a
region and a country there is one outcome. The
region joins the country. (3) Meeting of two
countries comes also to the mergence. The fourth
stage and the last one relates to the individual
behavior of agents. Each agent looks around and
moves to the group which is better to him/her
according to Tiebouts rule, called voting by
feet (see Tiebout Ch. (1956)). Needless to say,
that there are number of variants to fix the
rules of movement between groups, regions and
countries. Particularly, it is interesting to see
consequences of rules to accept an agent to the
group or region.
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Results of simulations
First I show here how looks a distribution of
agents, groups, regions and countries on
different stages.
Stage 1
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Stage 2
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Stage 3
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It is clear, that the pictures and its substance
depend on a number of parameters. Not always one
can predict what kind of dependence is in place.
In the paragraph I show some results of
simulations with different value of distance
vision of agents, and variety in the size of
groups (the low level).
Let v is a distance, any agent can see other
agent or group and i number of a level of the
hierarchy. The table 1 shows the distribution of
agent along the levels under condition that the
total quantity of agents is 5000. The table 2
shows the distributions in dependence on the
total number of agents (N).
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Table 1
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Table 2
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The simulations and simple analysis show strong
dependence on size of groups both parameters.
The number of groups in a region and the number
of regions in a country are more or less stable.
The first parameter (individual vision or
availability of information) is more important,
then the total size of population.
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References
1. Alesina Alberto and Spolaore Enrico (1997), On
the Number and Size of Nations, The Quarterly
Journal of Economics, CXII, 4, November 1997,
pp1027-1056. 2. Besley T. and Coate Stephen
(2003) Centralized versus decentralized
provision of local public goods a political
economy approach Journal of Public Economics, 87
2611-2637 3. Bewley Truman F. (1981) A Critique
of Tiebouts Theory of Local Public
Expenditures. Econometrica, vol. 49, 3, May,
1981. 4. Boerzel T. A. and Hosti M. O. (2002)
Brussels between Bern and Berlin Comparative
Federalism meets the European Union.
Constitutionalism Web-Papers, Con WEB No. 2/2002.
http//les1.man.ac.uk/conweb/ 5. Jacob B. L.,
Chen P. M., Silverman S. R. and Mudge T. N.
(1996) An Analytical Model for Designing Memory
Hierarchies. IEEE Transactions of Computers,
vol. 45, 10, October 1996. 6. Karpov Ju.
Imitating modeling of systems. Introduction in
modeling with AnyLogic 5 SPb. BHV-Petersburg,
2006 (In Russian).
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References
7. McGuckin R. and Dougherty S. (2003)
Restructuring Chinese Enterprises The Effect of
Federalism and Privatization Initiatives on
Business Performance. The Conference Board
Research Report R-1311-02-RR. 8. Samuelson, P. A.
(1954) The Pure Theory of Public Expenditure.
Review of Economics and Statistics, 37, 4. 9.
Tiebout, C. M. (1956) A Pure Theory of Local
Expenditures, Journal of Political Economy, 64,
5, pp. 416-424. 10. Qian Yingyi (1994)
Incentives and Loss of Control in an Optimal
Hierarchy. Review of Economic Studies,
61(3)527-544. 11. Zax J. S. (1988). The
Effects of Jurisdiction Types and Numbers on
Local Public Finance. In Fiscal Federalism
Quantitative Studies. (1988). Edited by Harvey S.
Rosen. The University of Chicago Press. 1988. 12.
XJ Technologies, http//www.xjtek.com.