How Public Goods can generate

1. 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

2. The literature, devoted to the structure of territorial jurisdictions, focused basically on explanation of causes the structure’s 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 jurisdiction’s 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.

3. 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.

4. 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.

5. 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 group’s members. 3.Territory of a location of a group depends on its size. It is a circle of radius r(n) = a*nb, 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(n+1) <= 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(n1+n2) <= 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.

6. 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(n1+n2) <= k*(n1+n2) - 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 region’s formation assumes that the agents of the different groups pay different amounts of taxes. An agent in lager group pays more.

7. 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) <= cost2 (N) / N; cost2 (n + N) / (n + N) <= 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.

8. 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) <= cost2 (N1) / N1; cost2 (N1 + N2) / (N1 + N2) <= 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.

9. 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*(N1+N2) – (cost1(N1) + cost2(N1) + cost1(N2) + cost2(N2)) >= 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.

10. 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 can’t meet each other because of some reasons. I discuss the reasons later.

11. 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.

12. 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.

13. 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.

14. My case based on a single cause of the hierarchy’s 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 + c*n2 , 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.

15. Needless to say, that there are many causes for jurisdiction’s 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, Юрьев М. (2004).

16. 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.

17. 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) = a*nb, where n is a quantity of the group’s members, a and b – positive numbers. If n = 1, then r = 0.

18. (3) A group moves with a speed less then the speed of agent’s 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(n+1) <= 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(n1+n2) <= 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.

19. 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(n1+n2) <= k*(n1+n2) – 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 (а) 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) <= cost2(N) / N; ii. cost2(n + N) / (n + N) <= k – cost1(n) / n.

20. (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) <= cost2(N1) / N1; ii. cost2(N1 + N2) / (N1 + N2) <= 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) <= k * n – cost1(n).

21. 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 it’s 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*(N1+N2) – (cost1(N1) + cost2(N1) + cost1(N2) + cost2(N2)) >= k3. Here cost1(N1) andcost2(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.

22. (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 Tiebout‘s 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.

23. Results of simulations First I show here how looks a distribution of agents, groups, regions and countries on different stages. Stage #1

24. Stage #2

25. Stage #3

26. 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).

27. Table 1

28. Table 2

29. 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.

30. 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 Tiebout’s 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).

31. 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.