Paper_Hierarchical levels in a state. Two ways of formation.


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NES 20th Anniversary Conference, Dec 13-16, 2012
Article "Hierarchical levels in a state. Two ways of formation" presented by Valery Makarov at the NES 20th Anniversary Conference.
Author: Valery Makarov

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Paper_Hierarchical levels in a state. Two ways of formation.

  1. 1. Hierarchical levels in a state. Two ways of formation. Makarov Valery L.Any state has a hierarchical structure coming from the necessity to have suitablecontrol and some other reasons. One can formulate optimal problems related to thehierarchical structure. Among its: What is optimal number of the levels? What isoptimal distribution of power and responsibilities between levels? How manyitems should be under control of a level?In the paper I deal with two different approaches to answer the given questions.Both approaches come from real practice of various countries. The first approach,named "top - down", is used, for example by the USSR, modern Russia and China.Top authorities in the countries have decided, how many provinces, municipalitiesshould be, what types of public goods must be provided on each level, and so on.The second approach, named "bottom - up", comes from the real experience of aformation and evolution of federal states like Germany or USA.So, the "top - down" direction means that a central organ like social plannercalculates optimal structure. In the paper one finds formulation of somemathematical optimization problems and a way, how to solve its. There are alsosome comparisons of artificial solutions with what we have in reality.On the contrary the "bottom - up" direction in the paper is represented bycomputer experiments, based on the agent - based model. The agent - based modelruns artificial population of individuals, which interact with each others, try tomaximize some individual cost function.1. Social planner problem.The optimization problem related to a hierarchical structure arises in differentfields and by various reasons. For example, Ian Yingyi (1994) considers aneconomic organization that owns a capital stock and uses a hierarchy to control theproduction. The optimal problem is to find number of tiers in the hierarchy andoptimal quantity of workers in each tier. The objective function in his approach isrevenue, generated from production activity. The trade off is between the twoparameters: the number of bureaucrats to control workers and efficiency ofworking 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 problemin the technical field, like an organization of computer memory, etc.
  2. 2. In this paper I concentrated on a single cause of the hierarchy’s emergence ofjurisdictions in a state: provision of a certain amount of public goods by minimalcost. In other words minimal cost means the minimal head tax.Needless to say, that there are other causes for jurisdiction’s creation. For example,in the paper Zax J. S. (1988) one can find an empirical analysis of relationsbetween number and types of jurisdictions and tastes and other characteristics ofpopulation, based on US data.Now we see a rising interest to operations on jurisdictions as among theoreticians(see Alesina Alberto and Spolaore Enrico (1997), Weber Sh…) and amongpractitioners too. Russian Federation is under the sizable reform of local selfgovernance. And at the same time there is academic and public discussion aboutFederal Constitutional structure of Russia. See for example, Юрьев М (2004)….In practice there was merger of the some subjects of Russian Federation (Forexample, Perm oblast and COMI national district). The general problem is a biguncertainty in the rules of jurisdictions’ creations and liquidations. The process ofnew states’ formation, unifications and so on, is increasing in the world lastdecades. But the more or less precise rules to do that, which are acknowledged byinternational community, are absent. A practical experience is accumulatedgradually, and theoretical people have to make their contribution as well.So the social planner problem arises as a demand of a central government, whichis responsible to provide the reforms.2. The ModelThere is a country with N citizens, who have identical tastes. The problem is toprovide a public service to each of them in a quantity equal to everybody byminimal cost.Notations:с - expenses of a government to provide one unit of a public service per a person;q - number of a hierarchical level (tier) for a given government; q = 0,1, 2, …; Here 0 is the number for a central government.kq - costs for keeping functioning of the government on hierarchical level q, undercondition that the level is lowest;nq - quantity of governments under subordination of the level’s q government;fq - total costs for the provision of a public service for the whole population pluscosts to keep all governments functioning;Objective function – total costs.
  3. 3. Remember that it is convenient to formulate problems in term of costs then interms of utility function.Now basic assumption:c * n2 - costs for provision of a unit of public service to n people;One needs to explain, why c * n2. The classic definition of a pure public good,given by P. Samuelson, (See Samuelson, P. A. (1954)) means no dependence on anumber of people. This is correct if we take a certain (concrete) public good. Butthinking about generalized public good, which is rather public service, one hasstrong dependence on the number of people being served. Bewley Truman F.(1981) discussed the difference between public goods and services in the contextof costs’ dependence on a number of people. The other point is why n2. When weare talking about public services at large including police, taxation, registration,etc, it is natural to take into consideration a “distance” of a person from a (local)government. In the literature, see, for example, Alesina Alberto and SpolaoreEnrico (1997), one can find different definitions of the distance. I assume here thatthe cost is proportional to the information links among people being served to keepthe quality of service. Roughly speaking the number of information links among npeople is equal to n2. There are calculations based on empirical data for variouspublic services showing certain functional form of dependence on quantity ofpeople being served. The social service based on a social security network isclosed to the quadratic dependence.In the application 1 there are curves, coming from real statistical data of variousregions.kq * ln(nq) - costs to keep government of the level q functioning, under conditionthat the government controls nq governments of lower level;It is natural to assume that the cost depends on the number of governments undersubordination in a decreasing return to scale. The logarithmic function used for thatis just as an example.Then if q = 0, then no tree, no sub-government under subordination. So one hastotal costs to provide one unit of public good and costs to keep the governmentfunctioning asf0 = k0 + c * N2 .Here the first term is costs of government’s functioning and the second one is coststo provide public service for the whole population.
  4. 4. If q >= 1, then to calculate total costs is a little bit more difficult. It is easy to dounder the assumption that all governments of a given level control the samenumber of governments. The number nq indicates exactly the condition. Thenumber does not depend on particular copy of the level’s q government. Namely,f1 = k0 * ln(n0)+ n0 * (k1 + c * (N/n0)2) = k0 * ln(n0)+ n0 * k1 + c * N2 / n0.Under q = 2 total costs are:f2 = k0 * ln(n0)+ n0 * k1 * ln(n1) + n0 * n1 * k2 + c * N2 / n0 * n1.Going along the induction one obtains the total costs for arbitrary number of levelsq:fq = k0 * ln(n0)+ n0 * k1 * ln(n1) + n0 * n1 * k2 * ln(n2) +…+ n0 * n1 * …* nq-2 *kq-1 * ln(nq-1) + n0 * n1 * …* nq-1 * kq + c * N2 / n0 * n1 * …* nq-1.Remember that if the hierarchical tree looks less symmetric, the total costsformula is more complicated.3. The Optimization Problems.Based on the model and the given objective function we can formulate at leastthree immediate optimization problems.(a) Optimal number of government’s levels (tiers).The problem consists of finding the q*, which gives minimal total costs forprovision of public service in quantity 1 for any person. By other words: q* = arg Min(fq)Here Min is taken over q. But it is clear that functions fq depend on the otherparameters participating in the definition of the function, that is on N, с, kq, nq .Hence the number q* depends on the named parameters.(b) Optimal quantity of inhabitances in a country.What is more effective from the point of view of total costs to provide publicgoods? To be in large Federal State or to create smaller state (probably federal onetoo). Much depends on relation between the numbers kq. The population has tocompare the total costs (and hence amount of taxes) under staying in the initialFederation or secession in a certain stake. Namely, one has to compare {Min(fq)/n},
  5. 5. where n runs from 1 to N. The N can be equal to infinity. Min is taken over n andq.Here Min(fq)/n is a head tax in the case where the size of population is equal to n.The country has hierarchical (or federal, if you like) structure, if q*>0.(c) Optimal size of a country with a fixed number of tiers.Let us suppose that q is given. Then the optimal size of population n*(q) is going tobe dependent on the given q. The problem makes sense in some practical issues aswe see below.4. Numerical calculations.I made numerical calculations with artificial numbers, which have about the sameorder as real ones. First, the (a) optimization problem.I used an algorithm of direct перебора. So, numbers N, c, k0, k1, k2 , k3 , k4 , k5 , k6 aregiven and keep the same during the algorithm worked. Its are:с = 0,0001k0 = 10 k1 = 1,6k2 = 1 k3 = 0,5 k4 = 0,25k5 = 0,12k6 = 0,12The algorithm starts with numbers n0, n1 , n2 , n3 , n4 , n5 , which are chosen close tomy understanding of an optimal solution.In the algorithm the only variable is changing. Namely, N is equal consequently to100, 316.228, 10000, 100000, 1 million, 10 millions, 100 millions, 1 billion, 10billions.Here the number 316.228 is the solution of the following optimization problem:Find N*, which provides minimum for the formula: f0 / N =( k0 + c * N2) / N .So, 316.228 is the optimal quantity of population under condition, that a centralgovernment provides public services directly. And it is the global minimum.The calculations with (a) Optimization problem.
  6. 6. If the population of a country is equal to 100, then no local government is needed.The cost per a person (head tax) to provide a public service is equal to 0.11.If the population is equal to 316.288, then the optimal structure changes. Namely,best way is to have two local governments, where 158 inhabitances are located ineach jurisdiction. The head tax is 0.0597. To keep previous structure we have headtax = 0.0632.If the population of a country is equal to 10000, then the optimal solution givestwo levels of a local government. The first level has 8 governments and the secondone – 12. It means that the central government controls 8 governments, and eachgovernment of the first level controls 12 governments of the lowest level. The totalquantity of government of the type is equal to 8*12 = 96, each serves to 104people. The jurisdictions of low level contain 1250 people. The head tax = 0.0245.Under the population of 100 thousands one more level occurs. The first level has 9governments, second – 13, and the third one – 13 as well. A lowest jurisdictioncontains 66 people, head tax = 0.01485, the quantity of local governments onbottom is 9*13*13 = 1521.Under the population of one million people we have additional level, e. g. totally 4levels plus central one. A lowest jurisdiction contains 59 people, head tax =0.0134, the quantity of local governments on bottom is 16940.Under the population of 10 millions people one more level is added, 5 +1 in total.A lowest jurisdiction contains 54 people, head tax = 0.0109, the quantity of localgovernments on bottom is 186340.Under the population of 100 millions people we have the same number of levels.The distribution of local governments along levels is 8, 18, 18, 18, 50. A lowestjurisdiction contains 43 people, head tax = 0.00912, the quantity of localgovernments on bottom is 2 332 800.Under the population of one billion people the general picture is about the same.Finally, under the population of 10 billions people one more level is needed. Thedistribution of local governments along levels is 10, 29, 29, 30, 30, 68. A lowestjurisdiction contains 20 people, head tax = 0.0041, the quantity of localgovernments on bottom is 514 692 000.The summary of the calculations one can see in the table^Quantity of 100 316 10 th- 100 th- 1 mln. 10 100 1 10population nd. nd. mln. mln. bln bln
  7. 7. . .Quantity of 0 2 8 9 7 7 8 13governments, 1 levelQuantity of 0 0 12 13 11 11 18 31governments, 2 levelQuantity of 0 0 0 13 11 11 18 31governments, 3 levelQuantity of 0 0 0 0 20 11 18 31governments, 4 levelQuantity of 0 0 0 0 0 20 50 65governments, 5 levelQuantity of 0 0 0 0 0 0 0 0governments, 6 levelN. of people 100 158 104 66 59 54 43 40in lowestjurisdictionHead tax 0,1 0,059 0,024 0,0148 0,013 0,0109 0,00912 1 7 5 5 4N. of 1 2 96 1521 16940 18634 233280government 0 0s on thelowest levelTotal N. of 1 3 97 1522 16941 18634all types of 1government.The two conclusions immediately follow from the calculations: 1. Greater population – local government must be closer to people. 2. Under given cost to maintain local governments the head tax is less in large countries. Now I come back to the second approach, named "bottom - up", as it was mentioned above.
  8. 8. 5. The agent-based model of a countrys population.A countrys population is located in a square. Any agent has the same quantity ofmoney k. The agents problem is to spend the money for getting maximal quantityof public goods. The preference of an agent is lexicographical. It means that he/shedirects money to a first public good, then to a second one and so on.The cost to provide the first public good iscost1 = k1 + c1 * n2 ,where n – number of agents in a group, k1 - constant expenditure to keep thegroups functioning, c1 - cost per a person.Accordingly the formulas for the second and the third public goods are:cost2 = k2 + c2 * n1.5cost3 = k3The running of the model is suitable to consider by phases.The first phase. The agents move in two- dimensional space chaotically. 1. If two agents see each other, they move to each other and form a group. The place, where the group is located, depends on its size. This is the circle of the radius r(n) = a*nb, where n – quantity of group members, a and b - positive numbers. If n = 1, then r = 0. 2. A group moves as well, but the speed is less then for individual agent. 3. If an agent sees a group, then there are two outcomes. (a) The agent is accepted by the group, if cost1(n+1) <= cost1(n); (b) the agent starts to move in opposite direction. 4. If two groups see each other, then again there are two outcomes: (a) the groups merge, cost1(n1+n2) <= cost1(n1) + cost1(n2); (b) the groups start to move in opposite direction, on contrary. Here n1, n2 – quantity of population in the first and the second group respectively.The first phase is over, when all groups are formed. The second phase takes placeif agents have money to pay for the second public good.The second phase starts with chaotic movements of all groups. The groups movewith different speeds depending on their size. 1. If two groups meet each other, then there are the two cases: (a) groups are inequality means, that a budget of the agents is enough for production of the
  9. 9. second public good); (b) on contrary, the groups start to move in opposite directions. Remark. Under realization of the outcome (а) variants are possible in dependence of some agreement between the agents of the groups to pay for the production of the second public good. 2. If a group and a region meets, there are two outcomes: (a) The group enters to the region under conditions: i. cost2(n+ N) / (n + N) <= cost2(N) / N; ii. cost2(n + N) / (n + N) <= k - cost1(n) / n. Here n and N are a quantity of population in the group and the region consequently. The first inequality means that the payment for the second public good in extended region is not greater then in the original region. The second inequality states, that the budget of the group is sufficient for the production of the public good. (b) The group and the region start to move in opposite directions on contrary. 3. Under meeting of two regions there are two outcomes again. (a) The two 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 is a size of population in the first and the second region respectively. The first inequality states that the payment for the second good in the united region is not greater then in the first region. The second inequality says the same about the second region. (b) In opposite case the regions start to move from each other. 4. A group makes decision to be transform into a region after some fixed time, if it has enough money. Namely, cost2(n) <= k * n – cost1(n).The second phase is finished when there are no groups for merging. As a result wehave a number regions, each of its consists of one or several groups. Still there aregroups which are not transformed to regions, if its have no money. The populationof the groups has no possibility to use the second public good.The third phase is closed to the second one by the substance. It is a process of theregions merging into more wide groups. Let call its countries for convenience.The mergers aim is to produce the third type of public good. 1. Under meeting of the two regions there are the outcomes: (a) The country is formed if the regions have enough money to produce the third public good. Namely k*(N1+N2) – (cost1(N1) + cost2(N1) + cost1(N2) + cost2(N2)) >= k3. 2. Here cost1(N1) and cost2(N1) are expenses of the first region for production of the first and the second public good respectively. Analogously
  10. 10. cost1(N2) и cost2(N2) – the expenses of the second region. (b) On contrary the regions move in opposite directions. 3. Under meeting of a region and a country there is one outcome. The region joins the country because both have a benefit in providing the third public good. 4. Under meeting of two countries merge is happened also.The forth phase is related again to individual behavior of the agents. When thehierarchical system of groups, regions, countries is established, the agents start tolook at neighboring communities for "voting by feet", according to Tiebouttheory. See, Tiebout Ch. (1956).So on the fourth phase all groups do not move, their location is stable. The agentsfind best position for themselves.There are number of numerical calculations on the described agent based model.Of course these are just some exercises based on artificial figures. But we findparameters which give a hierarchical structure close to the optimal structure,presented in the first part of the paper.In the Power Point Presentation followed the text one can find some of thecalculations.AppendixEvidence from different countries.The information about Chinese hierarchical system is borrowed from McGuckin R.and Dougherty S. (2003)Federal level:Three tiers: 1. Central government 1 2. Provincial regions 31
  11. 11. 3. Prefectures 331Local level:More three tiers: 4. Counties 2109 5. Townships 44800 6. Villages 737400Average number of inhabitances in townships & villages is about 1500.Russian Federation.Formally according to the Constitution of Russian Federation there are three levels: 1. Federal government 1 2. Subjects of Federation 87 3. Municipalities 25000In fact Russia has five (six) levels: 1. Federal government 1 2. Federal districts 7 3. Subjects of Federation 87 4. Municipal districts 5. Townships (городские поселения) 6. Villages (сельские поселения)The average number of inhabitances in townships & villages will be about 5000.USABorrowed from Zax J. S. (1988), based on 1985 data. States – 51; Counties – 3130;municipalities (including special districts and school districts) more then 82000.So, the qualitative picture of reality does not contradict to the exercise withartificial figures.Literature.
  12. 12. Alesina Alberto and Spolaore Enrico (1997), On the Number and Size of Nations,The Quarterly Journal of Economics, CXII, #4, November 1997, pp1027-1056.Besley T. and Coate Stephen (2003) “Centralized versus decentralized provisionof local public goods; a political economy approach” Journal of Public Economics,87 2611-2637Bewley Truman F. (1981) “A Critique of Tiebout’s Theory of Local PublicExpenditures”. Econometrica, vol. 49, #3, May, 1981.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. B. L., Chen P. M., Silverman S. R. and Mudge T. N. (1996) “An AnalyticalModel for Designing Memory Hierarchies”. IEEE Transactions of Computers, vol.45, # 10, October 1996.McGuckin R. and Dougherty S. (2003) “Restructuring Chinese Enterprises: TheEffect of Federalism and Privatization Initiatives on Business Performance”. TheConference Board Research Report R-1311-02-RR.Samuelson, P. A. (1954) The Pure Theory of Public Expenditure. Review ofEconomics and Statistics, 37, 4.Qian Yingyi (1994) “Incentives and Loss of Control in an Optimal Hierarchy”.Review of Economic Studies, 61(3):527-544.Zax J. S. (1988). “The Effects of Jurisdiction Types and Numbers on Local PublicFinance”. In: Fiscal Federalism: Quantitative Studies. (1988). Edited by Harvey S.Rosen. The University of Chicago Press. 1988.Муниципальная власть №1 (2004).Иноземцев В. Л. (2004) «Специфические особенности европейскойсоциальной модели». Журнал Института Европы РАН Современная Европа,№1. Стр. 87-99.Казаков А. И. (2004) Российский этнический федерализм: угроза целостностистраны? ж. Федерализм, №1.Евсеенко Т. Солопова Н. (2004) Конфедерация как форма государственногоустройства.ж. Федерализм, №1.
  13. 13. «Численность населения Российской Федерации по городам, поселкамгородского типа и районам на 1 января 2004г.» (2004). Федеральная службагосударственной статистики, Москва 2004г.Юрьев М (2004), «Реформа территориального устроения». Ж. Главная тема№1, Ноябрь 2004г.