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# 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.