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NEW ECONOMIC SCHOOL

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# BSP/2007/094 R

2007 " " .. (, ) .. (, ). , . , XXI .

Moscow 2007


.. / # BSP/2007/094 R. - .: , 2007. - 19 . (.)
( , , ..). . , . . , , . . : , ,

Belyakov Anton. On The Dynamics of People's Unions / Working Paper # BSP/2007/094 R. - Moscow, New Economic School, 2007. - 19 p. (Rus.)
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

ISBN ї .., 2007 . ї , 2007 .




1 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 . . . . . . . . . . . . . . . . . . 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 6 7

4
5 5 6 7 8 8 9 10 11 13 13



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. , , , . (, 1976), , , . , , , , . . , , . ( ), . . (Alesina & Spolaore, 1997) , , , . , (Bolton & Roland, 1997). (Tiebout, 1956) , , . (Bewley, 1981). . , . , , . , , , 1867 4


. . , . , , Е , . , Е (, 2007). (, 2007).

1



(. 1) S1 , S2 , W1 , W2 . -

S1 0 x

S2 1

E

. 1: . , , , . Е .

1.1



, . 5


W1 (x) + W2 (x), . W1 (x) + W2 (x) max .
x[0,1]

- (FOC) FOC: (W1 (x) + W2 (1 - x)) = 0, x (1)

- (SOC) SOC: 2 (W1 (x) + W2 (1 - x)) 0. x2 (2)

, (1) (2), .. , .

1.2



, , . 1 2 .
W1 (x) x 1 W2 (1-x) 2 x

-

.

x S1 : (W1 (x) + W2 (1 - x)/2) = 0 ? ? x

x:

(3)

, 2 1. , S1 x : x: (W1 (x)/2 + W2 (1 - x)) = 0 x 6 (4)


, [x, x] , , .

E

0

x

x

1

. 2: 1. x < x. , . , , .

2



, . (1) (2) p1 = W1 (S1 ) , S1 p2 = W1 (S2 ) , S2 (5)

S1 = x S2 = 1 - x. Е FOC: p1 = p2 , SOC: p1 p2 + 0. S1 S2 (6) (7)

p1 p2 () , 1 2 () () . , Е 7


p- p+ , p- > p+ , , , . (6) p- p+ , 1 2 + p p- . 1 2 () , () .

2.1



, t+ > 0, t
-

0.

p+ = p - t+ , p- = p + t- .

p- > p+ , , .

2.2



(Kahneman, D. & A. Tversky (1979)), , , ( ) . 3. , p- p+ . p- , - .

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. 3:

3



, , . , i- Wi = Wi (Si , L) Е Wi > 0, Si Е L (W1 + W2 ) = - < 0. L (8) 2 Wi < 0 Si2

, Е . 1 2,

'$

S

1

L rE

S

2

&%

. 4: , 9


S1 (. 4). , . 2. FOC: p1 = p2 + r - . , , p1 p2 r. (9) p1 p2 + , r2 , r (9)

r , p1 , p2 , . - , .

4



uj = A j S k , j = 1 . . . N,

S - , k - , Aj - j - , N - .

10


:
N k -1 j =1

p = kS

N Sk Aj = k SN

N

Aj = k n u,
j =1

(10)

n = N/S - , u =

uj /N -

. p, n u , (10) . , p - , n - , u - ( ) k - . Aj , , , . . . , , u p .

5



, , , . ? ? (Grossman & Mendoza, 2001, 2004) , , , : , . - , . . 1 : , 2 : (. . 5). , . , 11


ЕЕ Е ЕЕ Е Е -p2 , p2 Еr Е Е ЕЕ 2 rr Е Е rr Е Е rr ЕЕ ЕЕ r Е rЕ ЕЕ r rr 1 rrr 1 rr rr rr r rr r p1 - 3p2 , 0 r p1 - p2 , -p2 r

Е Е Е

p1 - p2 , 0

. 5: . , , Е , . 1 2, p1 2 p1 . , . p2 , . , Е , . , , , , , . , ( 2) p2 + p2 , 1, p1 . , 2p2 > p1 , . , , (p1 p2 ) , , - ( , SPNE). , , .

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6



, , , , . , . , , , , . , , , , . , , , . , . , - . , , , .

7



, , , . , . , , , Ks . , Kd . Ks Kd - . , N

13


. 6: (13) dN = Ks n0 dS - Kd n dS, (11)

n - , n0 - , dS - . N = S n dN = S dn + n dS . (11) , :
n n
1

dn = Ks n0 - (1 + Kd )n

S S1

dS , S

(12)

S1 - , n1 - . (12), : (K n0 - n)/(K n0 - n1 ) = (S/S1 )-(1+Kd ) , K = Ks /(1 + Kd ). , S/S1 = (r/r2 )2 , Е : n = K n0 +(n1 -K n0 )(r1 /r)2+2Kd . (10), Е : p = p2 + r - r , = k T (n1 - K n
-2(1+Kd )

, p2 = K n0 k T . -

2(1+Kd ) 0 )r1

(9) : p2 + r
2(1+Kd )

p0 +

. r

(13)

14


L2 L1 (13) 6 : a) n1 > K n0 b) n1 < K n0 . n1 = K n0 , L1 p2 . , , p2 > p0 .. K k u > k0 u0 . , Ks = 0, K = 0. , , K , .. . a) rA , rB - . , rB , - rA . r1 < rA rA . L2 L1 a), . b) r0 , . . L2 L1 b), . b) (k ) / (u) . a) - . (L1 - L2 ) , a) , b) , b) L1 a). b) ( , ): . a), , , - 15


, .

16



(, ) ( , , ..). , . , . , , , . . , Е , . . . , , . b) . , KS .

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1. x x. , (W1 (x) < 0, W2 (x) < 0 x [0, 1]) W1 (x) W1 (x) W2 (x) W2 (x) 1 W1 (x) = - W2 (x), 2 (3) (4), W1 (x) W2 (x), , W2 (x) < 0 W1 (x) > 0. , x < x. 2. F OC : d(W1 + W2 ) = 0, Wi Wi dSi + dL. Si L 1 W2 (x) = - W1 (x) 2

dWi (Si , L) =

(5) (8), d(W1 + W2 ) = p1 dS1 + p2 dS2 - dL. dS2 = -dS1 dL = dS1 p1 - p2 -
r dS r
1

; -

= 0, (9).

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, . . (2007). " (, ..)", . 43(2), 118- 122. , . . (1979). " " ., , (1001, 3734, 3735). Alesina, A. and E. Spolaore (1997). "On the Number and Size of Nations", Quarterly Journal of Economics Vol. 113, 1027-1056. 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. 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.

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