Optimal hierarchy

1
Optimal hierarchy in a Federal State
Makarov Valery (CEMI, Moscow)
2
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.
3
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.
4
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.
5
A fundamental problem for the necessity of
federal states 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 paper 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. one can find
analysis of impact of variations in tastes of
population on the number of types of
jurisdictions.
6
In the report I formulate and solve relatively
simple optimization problem where the only factor
for Federal States existence matters. Namely, it
is size of population.
Notations N - total number of citizens in a
country ? - 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 levels 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
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Objective function total costs cn2 - costs
for provision of a unit of public good (service)
for n people One needs to explain why the
cost is cn2. 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.
8
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.
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kqln(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.
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12
Optimal number of governments levels (tiers)
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 , ? , kq , nq . Hence the
number q depends on the named parameters.
13
Optimal quantity of inhabitances in a country
What 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 qgt0.
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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.
15
Numerical
calculations
16
Optimal number of governments levels (tiers)
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
Total number of citizens in a country
five levels
17
Optimal quantity of inhabitances in a country
0,11
1
2
3
4
5
0,060
0,024
0,015
0,015
0,012
Head tax
0,010
0,0099
0,0085
Total number of citizens in a country
100
316
104
105
107
108
109
1010
106
18
Number of inhabitances in a municipality
1
2
3
4
5
158
100
104
72
66
60
52
47
41
Total number of citizens in a country
100
316
104
105
107
108
109
1010
106
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1
World government is efficient
2
Greater population local government closer to
people
20
How Chinese Jurisdictions are defined
Federal level Three tiers 1. Central government
1 2. Provincial regions
31 10
11 3. Prefectures
331 6 7
Local level More three tiers 4. Counties
2109
370 5. Townships
44800 6. Villages
737400
China
Average number of inhabitances in townships
villages is 1500
21
Russian Federation
Formally according to the Constitution of
Russian Federation there are three
levels Federal government
1 Subjects of Federation 89
128 Municipalities
11429 12600
In fact Russia has or will have soon five (six)
levels Federal government
1 Federal districts
7 12 Subjects of
Federation 89 (87) Municipal
districts Townships Villages
Russia
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.
22
Literature

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. 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.

23
Literature

1. Samuelson, P. A. (1954) The Pure Theory of Public
   Expenditure. Review of Economics and Statistics,
   37, 4.
2. Qian Yingyi (1994) Incentives and Loss of
   Control in an Optimal Hierarchy. Review of
   Economic Studies, 61(3)527-544.
3. 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.