Документ взят из кэша поисковой машины. Адрес оригинального документа : http://belyakov.imec.msu.ru/papers/Belyakov_eng.pdf Дата изменения: Mon Sep 8 14:32:50 2008 Дата индексирования: Mon Oct 1 19:39:42 2012 Кодировка: Windows-1251

NEW ECONOMIC SCHOOL

Anton Belyakov

On The Dynamics of People's Unions

Working Paper # BSP/2007/094 E

This paper is based on the Master Thesis prepared at NES in 2007 in the framework of the research project "Heterogeneity and Diversity" under the supervision of prof. V.L. Makarov (NES and CEMI) and prof. A.V. Savvateev (NES and CEMI). The research was supported by Ford Foundation, World Bank, John D. and Catherine T. MacArthur Foundation. The author would like to thank his advisors and a consultant for the project prof. Shlomo Weber for their guidance and support, express gratitude to all the participants of XXI NES research conference.

Moscow 2007


Belyakov Anton. On The Dynamics of People's Unions / Working Paper # BSP/2007/094 E. - Moscow, New Economic School, 2007. - 18 p. (Engl.)
A new mechanism governing the dynamics of territory changing between groups of people such as countries is proposed based on trading with approval of both sides under particular voting rule (veto rule, majority rule, etc.). Conquest of the territory is viewed as a special case of trading. One- and two-dimensional cases are considered, where in the latter case the cost of border between countries is proportional to its length. A state equation is obtained based on particular personal utility function. Under migration rules maximal territory expansion and minimal possible territory are evaluated for small group surrounded by a big one. Parallels with statistical physics processes are revealed. Key words: public economics, allocation theory, size of nations

.. / # BSP/2007/094 E. - .: , 2007. - 18 . (.)
( , , ..). . , . . , , . . : , ,

ISBN ї .., 2007 . ї , 2007 .



Introduction
1 Mechanisms of b order moving by trade 1.1 Veto rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ma jority rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Multiple steady equilibria 2.1 Presence of transaction costs . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Behavioral effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Countries on plain 4 Example of state equation 5 War versus trade 6 Mechanisms of migration 7 Limits of the country expansion

4
5 5 6 7 8 8 9 10 11 12 12

Conclusion

16

Appendix

17

References

18

3


Introduction
This work studies the processes of origination and development of people's groups which own a particular territory. Not only countries could be an example of such groups, but also the municipalities of different levels and every group of land owners which members are not allowed to sell its land without permission of the group. The historical examples could be taken from the works by Lev Gumilev (e.g., Gumilev, 1990) where he determined the concept of ethnos as a group of people having common model of behavior allowing them to mark out themselves from the others. Gumilev showed on the historical examples that people in the ethnos not necessarily have the same ethnicity, nationality, religion, language and so on. Gumilev thought that every ethnos has the particular stages of its development which are determined by the number of passionary persons it contain. He states passionarity as a feature of a person's temper to prefer the unusual (for the ordinary peoples) values such as glory, power and so on. In the given study we will try to give an economical basis for these concepts. In the classical paper by Alberto Alesina and Enrico Spolaore (Alesina & Spolaore, 1997) the size and number of nations on the continuum of uniformly distributed individuals is considered particulary as an equilibrium where each individual at the border between two countries can choose which country to join with her land. There have been considered the coalition equilibrium as well as in (Bolton & Roland, 1997). Charles M. Tiebout in his paper (Tiebout, 1956) considered equilibrium where the consumer-voter is fully mobile and will move to one of the fixed number of communities where her preference pattern is best satisfied. The results of this paper was criticized by Bewley (Bewley, 1981). In contrast with the just mentioned papers in the work in hand there is no free will of agent supposed. An individual can leave her country only with permission of other citizens and she can join the other country if its citizens agree with it. The second assumption is that the utilities of all agents are transferable so that individual can pay for permission to leave and to join the country. Thus, countries can trade their territory, a historical example of which is the buying Alaska from Russia by the United States of America in 1867. Stability of the result of the trade depends on the form of the utility functions and on the voting rules in the countries. The war is considered as a mechanism preventing cheating during the trade. The countries which have approximately the same marginal 4


utility of the territory at the border would fight rather than trade as a result of equilibrium in the corresponding game. We also study two regimes of the group's size dynamics, where maximal and minimal territory of the group could be found. In this problem the parallels with processes of bubbling of the overheated liquid could be revealed. This work gives the microeconomic basis for some of author's results in (Belyakov, 2007).

1

Mechanisms of b order moving by trade

Let us consider two countries located on a unit interval (fig. 1) with territories S1 , S2 , and with transferable quazilinear social utility functions W1 , W2 . Territory could be exchanged

S1 0 x

S2 1

E

. 1: Two nations on the interval for money with approval by voting in both contries. Decision could be maid by veto rule when every citizen ought to approve it or by majority rule when more than half of the citizens should vote for it. For simplicity we will further assume positive and monotonically decreasing marginal social utility of the country with respect to its territory.

1.1

Veto rule

When every citizen of the both countries ought to approve the deal, the Pareto efficiency is obtained. Since each person should be paid in order to compensate her disutility and social utility functions are quazilinear the problem of finding an equilibrium border positions is equivalent to the problem of finding extremums of sum social utility functions of both countries W1 (x) + W2 (x). Stable border positions correspond to the maximum points W1 (x) + W2 (x) max .
x[0,1]

5


Here the first order condition (FOC) FOC (stationarity): (W1 (x) + W2 (1 - x)) = 0 x (1)

is the condition of equilibrium, while the second order condition (SOC) SOC (stability): 2 (W1 (x) + W2 (1 - x)) 0 x2 (2)

is the condition of stability to small perturbations of the border position. If both countries have decreasing marginal utilities to their territories, there exists the unique internal solution for the equation (1) and inequality (2), i. e. a stable border position which guaranties them a Pareto optimum.

1.2

Ma jority rule

If approval of only one half of a given country citizens is sufficient for selling a part of its territory, buyer could pay lower price in comparison with the veto rule case. Let us suppose that country 1 pays an additional surplus of all its citizens to the ruling half of country 2 citizens. Such deals can be done while x of the territory S1 satisfies the condition: (W1 (x) + W2 (1 - x)/2) = 0 ? ? x
W1 (x) x 1 W2 (1-x) 2 x

-

. Hence a maximum

x:

(3)

The same trick could be made by country 2 if it byes territory from country 1. So the lowest possible territory S1 is x obtaining from the condition: x: (W1 (x)/2 + W2 (1 - x)) = 0 x (4)

So there could be an interval [x, x] where border is possible but any point of this interval is not an equilibrium. Proposition 1. With assumption of monotonically decreasing marginal utilities of the countries, this interval is not empty and x < x. This result can easily be generalized on the case when a particular percentage of citizens 6


E

0

x

x

1

. 2: Possible positions of the border under ma jority rule make a decision about selling territory. Obviously the lower this percentage is the lower is the minimum possible territory of the country.

2

Multiple steady equilibria

We will give two examples where the steady equilibrium position of the border is not unique. Let us at first rewrite FOC (1) and SOC (2) in terms of marginal welfares p1 = W1 (S1 ) , S1 p2 = W1 (S2 ) , S2 (5)

where S1 = x and S2 = 1 - x. Thus stable equilibrium border should satisfy FOC (stationarity): p1 = p2 , SOC (stability): p1 p2 + 0. S1 S2 (6) (7)

Variables p1 and p2 are the maximum (minimum) prices that countries 1 and 2 could pay (take) for increasing (decreasing) their territory by one unit. If welfare functions are not continuously differentiable it might happen that its left derivation p- is not equal to its right derivation p+ , specifically p- > p+ which is consistent with decreasing marginal utility required for stability. Then equilibrium condition (6) has the following view p- p+ , 1 2 p+ p- . 1 2

Left (right) border of the continuum of equilibria could be obtained from the point where first (second) inequality becomes the equality. 7


2.1

Presence of transaction costs

Suppose that there are marginal costs of the new territory accommodation t+ > 0 and of the territory selling t- 0. Then the left and the right derivations of welfare with respect to territory have the view p+ = p - t+ , p- = p + t- .

Obviously p- > p+ which means that all points in the equilibrium interval are stable.

2.2

Behavioral effect

Prospect Theory by Kahneman and Tversky (Kahneman, D. & A. Tversky (1979)) based on psychological experiments suggests that personal utility function is defined on deviations from the reference point; generally concave for gains, convex for losses; and steeper for losses than for the gains. Example of such a utility function is drawn on the figure 3. From this figure it could be seen that derivation p
-

is lager than p+ . But p

-

. 3: Utility function

is increasing which means that borders of the equilibrium interval sometimes could be unstable. 8


3

Countries on plain

In order to be sure that the unique equilibrium border exists, let us assume in what follows that all countries are governed under the veto rule. Now consider the two-dimensional case where the social welfare of i-th country Wi = Wi (Si , L) depends on its territory Wi > 0, Si and on the length if the border L (W1 + W2 ) = - < 0. L (8) 2 Wi < 0 Si2

The letter implies that the expenses for supporting the border are proportional to its length and paid by both countries in a proportion that doesn't matter. If country 1 is

'$

S

1

L rE

S

2

&%

. 4: Circular country surrounded by infinite country surrounded by country 2, the optimal form of the border at the constant territory S1 is a circle (fig. 4). This form will be approved by all citizens because it minimizes the expenses on the border's support. Proposition 2. Under above conditions the equilibrium border is obtained from FOC (stationarity): p1 = p2 + where r - is the border radius. Thus, the border is in the equilibrium when the price of land for the inner country p
1

, r

(9)

is equal to that for the outer country plus the cost of the border unit divided by the border radius r. 9


Formula (9) has a direct analogy in physics with the condition of bubbling of overheated liquid p1 p2 + , r2

where bubble of radius r could exist if the pressure of the gas in the bubble p1 is not less than pressure outside p2 plus an additional pressure dew to the surface tension which is inversely proportional to the squared radius of the curvature of the bubble surface. The radius is squared because three-dimensional bubble has a two-dimensional border in contrast with our one-dimensional border case of the country on plain.

4

Example of state equation

Let the individual utility function be uj = A j S k , j = 1 . . . N,

where S is the territory of a group, k is the technological parameter of the group, Aj is the individual utility share of a citizen j , N is the number of people in the group. Then, marginal welfare will be
N k -1 j =1

p = kS

Aj = k

N Sk SN

N

Aj = k n u,
j =1

(10)

where n = N/S is the people's concentration, u =

uj /N is the average utility in the

group. Variables p, n and u could be considered as the state variables of the group. There is a direct analogy in physics with the state equation of the ideal gas, where p stands for a pressure, n for a concentration of molecules, u for a temperature (mean energy of molecules) and k - Boltzmann constant. The individual utility share Aj reflects her passionatity, i.e. her preference of the increase of territory to the increase of money. Thus, the higher is passionatity of individuals, the more average utility in the group u is, and the more is the price of its territory. 10


5

War versus trade

Let us now assume that, instead of paying for the territory, a country could spend same money for hiring the army and conquer the territory. Which of the two alternatives - to buy or to conquer the country - would it prefer? The same dilemma has been studied before (e. g. Grossman & Mendoza, 2001, 2004) in the economic theory of empire building using examples of the Roman, Mongol, Ottoman, and Nazi German empires, where three strategies where considered: Uncoerced Annexation, Coerced Annexation, and Attempted Conquest. In order to answer this question, we will use a different game-theoretic model, where a country could cheat, for example take money but not give territory or vice versa. Country 1 has the strategies to buy or to conquer, while country 2 has the strategies to cheat or not to cheat (see figure 5). p1 - p2 , 0

ЕЕ Е -p2 , p2 ЕЕ Е Е Е rr not conquer Е Е Е2 r buy Е Е rr Е ЕЕ r ЕЕ Е cheat rr Е rr Е ЕЕ r 1 rrr 1 rrr rr rr r r conquer r p1 - p2 , -p2 conquer r p1 - 3p2 , 0 r r

Е not cheatЕ Е Е

. 5: Game Let the war process be the same as that of the trade with the only difference that the seller does not get a payment. Country which hires the greatest army wins. The Army could be treated as a third player which can not gain utility from the land itself but only conquers territory for the country which pays the highest price. If country 1 wants to conquer a part of country 2 it pays the army a little bit more than p1 because otherwise country 2 can conquer territory back by paying p1 . Thus, the conquest looks like an auction. Let the territory be traded with the minimum seller price p2 , otherwise the conquest is preferable. A country pays the army only if its welfare surplus allows it to win, otherwise it retreats. Hence, countries never fight, one hires the army and occupy a part of the territory of another country or does not hire the army because another could hire not less 11


army itself. Thus the cheater (country 2) after getting money invests p2 + p2 in the army to protect from country 1 attack with maximum army cost p1 . Hence, the cheating condition is 2p2 > p1 , and it is a Nesh equilibrium in pure strategies in this subgame. The following conclusion is that the countries near the equilibrium (i.e. p1 p2 ) will always fight for the territory instead of trading because the threat of cheating (it is the subgame perfect Nash equilibrium, SPNE). Only the country which values territory at least twice as its neighbor can afford itself buying instead of conquering.

6

Mechanisms of migration

Migration between two countries proceeds also with permission from both countries involved and if the individual utility of migrant increases. With the assumption of decreasing personal utility function on the concentration of people, there are two ways of migration. The first way is when a person sells her land in her country and buys land in the country of which citizen she wants to be. Thus, she freely lives her country because her former compatriots get her land at a little bit cheeper price and then she is accepted by voting in new country because she compensates its citizens their loss of personal space buying land at a little bit higher price. The second way of migration is when a part of the country territory where person is living has been conquered or sold. In the second case person has a choice whether to stay in a new country or to leave in her own country, maybe taking compensation from expanding country or just because she wants to live in her native country. The first case is impossible under veto rule because all people get payment sufficient to compensate their loss in utility due to their territory contraction. Thus we will consider the case of migration due to ma jority rule or conquest.

7

Limits of the country expansion

In this section we will describe the expansion of a circle country surrounded by a big one in which not all citizens are properly compensated for the loss of their land, as it usually is in the case of conquest. In this case some people would prefer to stay on their land and become the citizens of expanding country. For the sake of simplicity let the share of 12


people who prefer to become the citizens of the expanding country be a constant, denote it by Ks . We also suppose that a share Kd of people of the expanding country on the new territory just dies. The coefficients Ks and Kd are exogenous. Thus, the total number N of people in the expanding group will change according to the following law dN = Ks n0 dS - Kd n dS, (11)

where n is the concentration of people in the country, n0 is the concentration of people in the surrounding country, and dS is the territory increment of the expanding country. From the expression for the people number N = S n we also get dN = S dn + n dS . Substituting the latter into (11) and separating variables, we obtain an integral equation:
n n
1

dn = Ks n0 - (1 + Kd )n

S S1

dS , S

(12)

where S1 is the initial square of the country, n1 is the initial concentration of people in the country. Taking the integrals in (12) we get: (K n0 - n)/(K n0 - n1 ) = (S/S1 )-(1+Kd ) , where constant K = Ks /(1 + Kd ). Since the country has a form of a circle S/S1 = (r/r2 )2 , after some transformations we get the expression for the concentration of people in the country depending on the radius of its territory n = K n0 + (n1 - K n0 )(r1 /r)2+2 expanding country on its radius p = p2 + r
-2(1+Kd ) Kd

. Substituting the

latter expression into the equation (10) we get the dependence of a land price inside the , where p2 = K n0 k T - the land price . Substituting the latter expression
2(1+Kd ) 1

at r , and coefficient = k T (n1 - K n0 )r

into (9) we get a condition for country expansion p2 + r
2(1+Kd )

p0 +

. r

(13)

In order to determine the maximal and minimal possible sizes of the country we depict the values of the right L2 and left L1 sides of the inequality (13) on the figure 6 for two cases: a) n1 > K n0 , and b) n1 < K n0 . When n1 = K n0 the curve L1 degenerates into horizontal line at the level p2 . From the both plots it is clearly seen that for existence of the domain of unlimited expansion the inequality p2 > p0 is necessary i. e. K k u > k0 u0 . It is impossible if the share of staying peoples of surrounding country Ks = 0 because it means that K = 0. Hence, for the optimal expansion policy it is necessary to increase K

13


. 6: Left and right sides of the inequality (13)

maximally, i. e. no one of the people from expanding country should die and all people on the occupied territory should join the expanding country. On the plot a) at the point rA the country is in the unstable equilibrium, while at the point rB it is in stable. Because any small negative deviation of the radius from rB leads to the unlimited expansion while, a positive one to the decreasing of the radius to rA . At the initial radius r1 < rA the expansion of the country is limited by the radius rA . If the curves L2 and L1 not intersect on a) the country expands unlimitedly at any initial radius. On the plot b) there is only the unstable equilibrium point r0 which is the minimal possible radius of the country. At the grater radius the country unlimitedly expands. If the hyperbolas L2 and L1 do not intersect on the plot b) the country can not exist at any size. In the regime b) the expansion proceeds only because technical k and energetic u superiority of the country in comparison with its surroundings. While in the regime a) group additionally expands because of its superiority in the concentration of people. Nevertheless if we suppose the positive dependence of the expansion velocity on the price difference at the border L1 - L2 then in the case a) the unlimited expansion proceeds slower than in the case b) at the same initial price difference because in b) the inner price L1 rises while the radius increase in contract with a). There were historical examples of expansion in the quick regime b) when the initial people concentration of expanding group is less then that of the surroundings: conquest of the New World by Europeans, and joining of the Siberia to the Russia. The slow regime a) appears to be typical for long term interacting groups which have the same levels of 14


technical development and passionarity because of the technology and people exchange but due to some reasons one group has higher people concentration.

15


Conclusion
In this work a new mechanism governing the dynamics of territory changing between groups of people such as countries is proposed, based on trading with approval of both sides under particular voting rule (veto rule, ma jority rule,...). The possible reasons for multiple stable equilibria could be transaction costs and different marginal utility for gains an losses. Conquest of the territory is viewed as a special case of trading where a "seller" country does not get a payment for its territory. It was found that only the country which values territory at least twice as its neighbor can afford itself buying instead of conquering. In the two-dimensional case was considered, where the cost of the border between countries is proportional to its length, an influence of the border curvature on the land price has been studied. A state equation of the group was obtained based on particular personal utility function. This equation appeared to be similar to the state equation of ideal gas. Under migration rules the maximal territory expansion and the minimal possible territory are evaluated for a small group surrounded by a big one. Regime b) is similar to the process of bubbling of the overheated liquid. The possible extension of this work is to make endogenous the share KS of the people who stays on their land after its conquest.

16


Appendix
Proof of Proposition 1. Let x x. Then, from monotonic decreasing property of marginal utilities (W1 (x) < 0, W2 (x) < 0 x [0, 1]) we have W1 (x) W1 (x) W2 (x) W2 (x) substituting here 1 W1 (x) = - W2 (x), 2 from (3) and (4) and summing we obtain W1 (x) W2 (x) which is a contradiction since W2 (x) < 0 and W1 (x) > 0. Thus x < x. Proof of Proposition 2. F OC : d(W1 + W2 ) = 0, Wi Wi dSi + dL. Si L 1 W2 (x) = - W1 (x) 2

dWi (Si , L) =

Taking into account (5) and (8) after summing we get d(W1 + W2 ) = p1 dS1 + p2 dS2 - dL. From geometry of the problem we have got expressions dS2 = -dS1 and dL = substitution and dividing by dS1 we get p1 - p2 -
r dS r
1

; after

= 0 and consequently (9).

17


References
Alesina, A. and E. Spolaore (1997). "On the Number and Size of Nations", Quarterly Journal of Economics Vol. 113, 1027-1056. Belyakov, A. O. (2007). "About mathematical description of the people's association development processes (ethnoses, firms staff etc.)", Economics and Mathematical Methods Vol 43(2), 118-122. (in Russian) Bewley, T. F. (1981). "A Critique of Tiebout's Theory of Local Public Expenditures", Econometrica 49(3), 713-740. Bolton, P. and G. Roland (1997). "The Break-up of Nations", Quarterly Journal of Economics Vol. 131, 1057-1090. Grossman, H. I. and J. Mendoza (2001). "Annexation or Conquest? The Economics of Empire Building", NBER Working Paper No. 8109. Grossman, H. I. and J. Mendoza (2001a). "Butter and guns: Complementarity between economic and military competition", Economics of Governance Vol. 2(1), 25-33. Grossman, H. I. and J. Mendoza (2004). "Peace and War in Territorial Disputes", NBER Working Paper No. 10601. Gumilev, L. (1990). "Ethnogenesis and the biosphere", Moscow, Progress, 386 p. (first published in 1979 in Russian) Kahneman, D. and A. Tversky (1979). "Prospect Theory: An Analysis of Decision under Risk", Econometrica Vol. 47(2), 263-292. Tiebout, C. M. (1956). "A Pure Theory of Local Expenditures", The Journal of Political Economy Vol. 64(5), 416-424.

18