Optimal hierarchy

in a Federal State

Makarov Valery (CEMI, Moscow)

A Federal State emerges by variety of reasons. Some of the reasons come from a general problem of control in complex multidimensional systems.

See, for example: Qian Yingyi (1994) “Incentives and Loss of Control

in an Optimal Hierarchy”. Review of Economic Studies, 61(3):527-544.

The report considers an economic organization that owns a capital stock an 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 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.

There is sizable literature devoted to optimal design of technical devices, like memory for computers and others.

See, for example: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.

Historically a Federal State appeared in a process of unification of relatively small states to be more powerful (militarily, politically economically) together.

For example, there are Germany, Canada, USA. The opposite top – down process takes place too. It based on keeping multinational state sustainable.

A fundamental problem for the necessity of federal state’s structure arises in relation with efficiency of provision of local and national public goods.

See 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.

In the paperZax 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.one can find analysis of impact of variations in tastes of population on the number of types of jurisdictions.

In the report I formulate and solve relatively simple optimization problem where the only factor for Federal State’s existence matters. Namely, it is size of population.

Notations:N - total number of citizens in a country;

с - costs of a government to provide one unit of a public good (public service) per a person;q - number of a hierarchical level (a tier) for a given government; q = 0,1, 2, …;kq - costs for keeping functioning of the government on hierarchical

level q, under condition that the level is lowest;nq - quantity of governments under subordination of the level’s q

government; fq - total costs for provision of a public good (quantity is equal to

one) for the whole population plus costs to keep all governments functioning;

Objective function – total costs

c*n2 - costs for provision of a unit of public good (service) for n people;

One needs to explain why the cost is c*n2.

The classic definition of a pure public good given by P. Samuelson, (See Samuelson, P. A. (1954)) means no dependence on a number of people.

This is correct if we take a certain (concrete) public good. But thinking about generalized public good, which is rather public service, one has strong dependence on the number of people being served. Bewley Truman F. (1981) discussed the difference between public goods and services in the context of costs’ dependence on a number of people.

The other point is why n2.

When we are 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 Spolaore Enrico (1997), one can find different definitions of the distance.

I assume here that the cost is proportional to information links among people being served to keep the quality of service. Roughly speaking the number of information links among n people is equal to

n2.

Then if q = 0, one has total costs to provide one unit of public good and costs to keep the government functioning as

f0 = k0 + c*N2.

Here the first term is costs of government’s functioning (central one) and the second one is costs to provide public good for the whole population.

kq*ln(nq) - costs to keep government of the level q functioning,

under condition that the government controls nq governments of

lower level;

It is natural to assume that the cost depends on the number of governments under subordination in a decreasing return to scale. The logarithmic function used for that is just as an example.

If q >= 1 to calculate total costs is a little bit more difficult.

It is easy to do under assumption that all governments of a given level control the same number of governments.

The number nq indicate exactly that condition.

The number 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 levels q:

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

The problem consists of finding the q*, which provides

minimal total costs for provision of public good in quantity 1. By other words:

q* = arg Min(fq)

Here Min is taken over q. But it is clear that functions fq

depend on the other parameters participating in the definition of the function, that is on N , с , kq , nq . Hence the number q* depends

on the named parameters.

Optimal number of government’s levels (tiers)

Optimal quantity of inhabitances in a countryWhat is more effective from the point of view of total costs to

provide public goods? To be in large Federal State or to create smaller state

(probably federal one too). Much depends on relation between the numbers kq. The

population has to compare the total costs (and hence amount of taxes) under staying in the initial Federation or secession in a certain stake.

Namely, one has to compare

{Min(fq)/n},

where n runs from 1 to N. The N can be equal to infinity.

Min is taken over n and q. Here Min(fq)/n is a head tax in the case

of the size of population is equal to n. The country has federal

structure if q*>0.

Optimal size of a country with a fixed number of ties

Let us suppose that q is given. Then optimal size of population n*(q) is going to be dependent on the given q.

The problem makes sense in some practical issues as we see below.

100

316

10 000

100 000

1 000 000

10 000 000

100 000 000

1 000 000 000

10 000 000 000

one level

two levels

three levels

four levels

five levels

Total number of citizens in a country

Optimal number of government’s levels (tiers)

100 316 104 105 106 107 108 109 1010$

$

$ $ $

$

$

$

$

0,060

0,11

0,024

0,015

0,015

0,012

0,010

0,009

9

0,008

5

Head

tax

Total number of citizens in a country

Optimal quantity of inhabitances in a country

100 316 104 105 106 107 108 109 1010

Total number of citizens in a country

Number of inhabitances in a municipality

100

158

104

66 72

60

5247

41

World government is efficient

Greater population – local government closer to people

How Chinese Jurisdictions are defined

Local levelMore three tiers:

4. Counties 2109 3705. Townships 448006. Villages 737400

Federal levelThree tiers:

1. Central government 12. Provincial regions 31 10 – 113. Prefectures 331 6 – 7

Average number of inhabitances in townships & villages is 1500

Russian FederationFormally according to the Constitution of Russian Federation there

are three levels:Federal government 1Subjects of Federation 89 128Municipalities 11429 12600

In fact Russia has or will have soon five (six) levels:Federal government 1Federal districts 7 12Subjects of Federation 89 (87)Municipal districtsTownshipsVillages

The reform of the local governance considers creating 24000-30000 townships and villages in total. The townships and villages are at the same

level but with a little bit different status. Then the average number of inhabitances in townships & villages will be about 5000.

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. 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.

5. 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.

6. Samuelson, P. A. (1954) The Pure Theory of Public Expenditure. Review of Economics and Statistics, 37, 4.

7. Qian Yingyi (1994) “Incentives and Loss of Control in an Optimal Hierarchy”. Review of Economic Studies, 61(3):527-544.

8. 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.