04-04 vs. 1 2 1 980-8579 6-6-06 E-mail: shuhei.yamaguchi.p7@dc.tohoku.ac.jp 2 980-8579 6-6-06 E-mail: akamatsu@plan.civil.tohoku.ac.jp Fujita and Ogawa(1982) Fujita and Ogawa Key Words: agglomeration economy, multiple equilibria, urban subcenter formation, stochastic stability 1. Fujita and Ogawa 1) ( FO ) 2 Fujita and Thisse 2) FO Osawa and Akamatsu 3) FO FO (Sandholm 4) ) FO 5) (NEG) FO 1
FO 2 FO 3 4 FO 5 6 7 8 2. FO (1) 1 K K 1,, K} S S S/K N M (2) i j j W j z U(s = S H, z) s S H > 0 1 W j T ij = S H R i + z (1) T ij R i i i j T ij = t i j (2) t > 0 i j where z ij = arg max z max z ij (3) i,j U(s = S H, z) =W j T ij S H R i (4) i j i j n ij (3) i Π i max i Π i (m) = maxa i (m) S F R i L W i } (5) i S F, L m [m i ] A i (m) ( ) m A i (m) = j D ij m j (6) D ij τ > 0 D ij = exp( τ i j ) (7) (4) FO a) z = W j T ij S H R i if n ij > 0 z W j T ij S H R i if n ij = 0 i, j K (8) z Π i Π = A i S F R i LW i if m i > 0 Π A i S F R i LW i if m i = 0 i K Π (9) 2
b) S = S H n ij + S F m i if R i > 0 j K S S H j K n ij + S F m i if R i = 0 i K (10) R i n ij = Lm i if W j > 0 i K n ij Lm i if W j = 0 i K j K (11) W j c) FO m i = M (12) i K n ij = N (13) 3. FO (1) 1 logit dynamics t m [m i ] i Π i K (m) m M t 1 i j ρ ij 1 FO m exp[θπ j (m e i + e j )] ρ ij (m) = k S exp[θπ k(m e i + e k )] i, j S (14) e i i 1 0 K θ θ = 0 θ t t + 1 m m m i p m m = M ρ ij m = m e i + e j 0 otherwise (15) t m π t m π t t + 1 m π t+1 = P π t P P [p m m ] (2) π = P π π m π m m 1 π θ π m > 0 (3) 2 Π [Π i ] Z Z(m) m j Z(m) m i = Π j Π i i, j S (16) Sandholm 4) (14) 3
πm Z πm = 1 M! H k K (Mm exp(θz(m)) (17) i)! H m M π m = 1 2 πm πm = k K(Mm k )! k K (Mm k)! exp[θ(z(m) Z(m ))] (18) θ 1 lim θ θ log π m πm = lim (Z(m) Z(m )) + 1θ log k K(Mm k )! } θ k K (Mm k)! =Z(m) Z(m ) m, m M (19) Z(m) > Z(m ) π m > π m 1 Z(m) m FO 4. FO FO (1) FO FO 1 FO m [m i ] n [n ij ] 2 max Z(m, n) = m 0,n 0 ZF (m) Z H (n) (20) s.t. n ij S H + m i S F S i K (21) j K Lm j i K n ij j K (22) n ij = N (23) m i = M (24) i K 1 2 Z F (m) = 1 m i D ij m j (25) 2 Z H (n) = n ij T ij (26) KKT FO ( I ) (2) (20) (24) Bendes decomposition 2 m [P] min n ij T ij (27) n 0 s.t. (21) (22) (23) [P] [D] max z,r,w ZH (z, R, W m) z N + (R i S F + W i L)m i S R i (28) i K i K s.t. z W j S H R i T ij i, j K (29) R i 0 i K (30) W j 0 j K (31) [P] [D] m ẐH (m) ẐH (m) ẐH (m) = ẐH (m) max m 0 ẐF (m) 1 m i D ij m j 2 ẐH (m) (32) s.t. (24) (20) (24) m ẐF (m) m i = Π i i S (33) ẐF (m) Π(m) FO ẐF (m) 4
1 5. (1) FO 1 S x [ S/2, S/2] x S 1 m, n m(x), n(x, y) z ij, Π i (m), R i, W j z(x, y), Π(x), R(x), W (x) A i (m) A[m(x)] = D(x, y)m(y)dy x S (34) S D(x, y) D(x, y) exp[ τ x y ] (35) x y 2 x, y i, j i j T (x, y) t x y (36) Z(m(x), n(x, y)) = Z F (m(x)) Z H (n(x, y)) (37) Z F (m(x)) = 1 D(x, y)m(x)m(y)dxdy (38) 2 S S Z H (n(x, y)) = T (x, y)n(x, y)dxdy (39) S S Z F (m(x)),z H (n(x, y)) (20) 1 2 (21),(22) S H n(x, y)dy + S F m(x) 1 x S (40) S n(x, y)dx L m(y) y S (41) S (40),(41) S H N + S F M S (42) N LM (43) 2 1 (N = LM) 2 (S H N + S F M = S) (2) 2 3 ( BA) ( RA) ( IA) m(x) m(x) BA RA IA ( BA/RA ) (e.g., 2 b 0, b 1, b 2 ) BA/RA BA RA 2 BA 4 N BA N 5 ( a) e)) a) 2 (a) BA/RA 5
3 4 e) 2 Z c b) 2 (b) IA BA RA IA RA BA BA/RA b (1) [b 0, b 1 ] T Z (1) c) 1 2 (c) 1 x [ b 0, b 0 ] BA BA/RA b 0 = MSF 2 Z m d) 2 BA 2J (J N) 3 x 0 x = 0 RA RA BA BA/RA b (2J) [b 0, b 1,..., b 2J 1 ] T Z (2J) 3 BA 2J + 1 (J N) 4 x 0 x = 0 BA RA BA BA/RA b (2J+1) [b 0, b 1,..., b 2J ] T Z (2J+1) 6. 2 BA/RA 1 BA/RA 2 1 (1) BA/RA b (i) BA/RA Z (i) (b (i) ) i BA/RA b (i) BA/RA Z(i) Z (i) = max b (i).z (i) (b (i) ) i (44) 6
(2) 1 Step 1 τ, t, M, L, S F, S H Step 2 Z (i) := max.z (i) (b (i) ) ( i) b (i) Step 3 Zc, Zm, Z(i) ( i) Step 4 maxz c, Z m, max i Z (i) }} Step 5 5 7. 2 τ, t τ, t 33 (M, L, S F, S H ) = (100, 1, 1, 1) (1) 5 (τ, t) t t t t 1, t 2 t < t 2 (e.g., t = 0.05 5 ) τ 6 t = 0.05, τ = 0.2 3 τ 6 τ 1 1 τ 1 2 τ 1 τ t < t 2 τ 2 5 1 7
7 1 6) Core-Periphery 7) SISC (2) FO 7 3 t t t < t τ 1 1 t 1 8. FO 3 Osawa and Akamatsu 3) 7 BA/RA 1 FO FO NEG FO 1 2 FO FO FO I 1 4.(1) KKT 2.(4) (20) (24) L(m, n, R, W, z ) = Z(m, n) + i + j W j (L m j i R i (S H n ij + S F m i S) j n ij ) + z ( i s.t. n ij, m i, R i, W j, z 0 n ij N) j (I.1) R i, W j, z 1 n ij n ij = 0 n ij 0, n ij 0 m i m i = 0 m i 0, m i 0 i, j i (I.2) (I.3) 8
R i S S H n ij S F m i = 0 j S S H n ij S F m i 0, R i 0 j i (1) Z F (τ) = m2 m [exp( τs) + τs 1] (τ) 2 ( ) W j n ij Lm j = 0 i n ij Lm j 0, W j 0 i N i n ij = 0 j j (I.4) (I.5) (I.6) (I.4) (10) (I.5) (11) (I.6) (13) n ij = T ij + R i S H W j + z i, j (I.7) (I.2) (8) m i = j D ij m j + R i S F LW i I, i (I.8) Π = 0 (I.3) (9) II (38),(39) BA/RA Z F (τ) Z H (t) Z(τ, t) = Z F (τ) Z H (t) m b = 1/S F x B m(x) = m r = 0 x R (II.1) m m = 1/(S F + LS H ) x I n b = 0 x B n(x) = n r = 1/S H x R (II.2) n m = L/(S F + LS H ) x I n(x, y)dy n(x) B, R, I S BA,RA,IA x (38),(39) Z H (t) = 0 (2) 1 b 0 b 1 b 1 = L 1 + L b 0 + M 2m b Z F (τ) = 2m2 m (τ) 2 τb 0 exp[ τb 0 ] sinh[τb 0 ]} + 2m2 b (τ) 2 τ(b 1 b 0 ) exp[ τb 1 ](sinh[τb 1 ] sinh[τb 0 ])} + 2m b (τ) 2 (m b 2m m ) sinh[τb 0 ](exp[ τb 1 ] exp[ τb 0 ])} y x x x ( b 1, b 1 ) b 1 b 0 y(x) = (x b 1 ) + b 0 x [b 1, S/2] S/2 b 1 b 1 b 0 (x + b 1 ) b 0 x [ S/2, b 1 ] S/2 b 1 Z H (t) = n r (S/2 b 1 )(S/2 b 0 )t (3) 1 Z F (τ) = m2 b (τ) 2 [exp ( b 0τ) + b 0 τ 1] y x b 0 (x b 0 ) x [b 0, S/2] S/2 b 0 y(x) = b 0 (x + b 0 ) x [ S/2, b 0 ] S/2 b 0 x otherwise Z H (t) = n r S/2(S/2 b 0 )t 9
(4) 2J J N [ J 1 Z F (τ) = 2m2 b (τ) 2 τ(b 2i+1 b 2i ) i=0 + sinh[τb 2i ](exp[ τb 2i+1 ] exp[ τb 2i ]) exp[ τb 2i+1 ](sinh[τb 2i+1 ] sinh[τb 2i ]) J 1 2(exp[ τb 2i+1 ] exp[ τb 2i ]) i=1 }] i 1 (sinh[τb 2j+1 ] sinh[τb 2j ]) j=0 } (II.3) 2 (II.3) 4,5 b 2k,2k+1 = nr Lm b (b 2k b 2k 1,2k ) + b 2k b 2k+1,2k+2 = Lm b n r (b 2k+1 b 2k,2k+1 ) + b 2k+1 where k = 0, 1,..., J 1 b 2k,2k+1 b 2k (x b 2k 1,2k ) + b 2k b 2k b 2k 1,2k x [b 2k 1,2k, b 2k ] y(x) = b 2k+1 b 2k,2k+1 (x b 2k+1 ) + b 2k,2k+1 b 2k+1,2k+2 b 2k+1 x [b 2k+1, b 2k+1,2k+2 ] where k = 0, 1,..., J 1 J 1 Z H (t) =tn r [(b 2k b 2k 1,2k )(b 2k,2k+1 b 2k 1,2k ) k=0 + (b 2k+1,2k+2 b 2k+1 )(b 2k+1,2k+2 b 2k,2k+1 )] (5) 2J + 1 J N [ J Z F (τ) = 2m2 b (τ) 2 τ(b 2i b 2i 1 ) i=1 + (sinh[τb 2i 1 ] sinh[τb 0 ])(exp[ τb 2i ] exp[ τb 2i 1 ]) exp[ τb 2i ](sinh[τb 2i ] sinh[τb 2i 1 ]) } sinh[τb 0 ](exp[ τb 2i ] exp[ τb 2i 1 ]) J 2[(exp[ τb 2i ] exp[ τb 2i 1 ]) i=2 } i 1 (sinh[τb 2j ] sinh[τb 2j 1 ]) j=1 + τb 0 exp[ τb 0 ] sinh[τb 0 ] ] (II.4) 3 (II.4) 5,6 b 2k,2k+1 = Lm b (b 2k b 2k 1,2k ) + b 2k b 2k+1,2k+2 = n r nr Lm b (b 2k+1 b 2k,2k+1 ) + b 2k+1 where k = 0, 1,..., J 1 b 0 (x b 0 ) b 0,1 b 0 y(x) = x [b 0, b 0,1 ] b 2k 1,2k b 2k 1 (x b 2k 2,2k 1 ) + b 2k 1 b 2k 1 b 2k 2,2k 1 x [b 2k 2,2k 1, b 2k 1 ] b 2k b 2k 1,2k (x b 2k ) + b 2k 1,2k b 2k,2k+1 b 2k where k = 1, 2,..., J x [b 2k, b 2k+1 ] Z H (t) = tn r [b 0,1 (b 0,1 b 0 ) J + (b 2k 1 b 2k 2,2k 1 )(b 2k 1,2k b 2k 2,2k 1 ) k=1 + (b 2k,2k+1 b 2k )(b 2k,2k+1 b 2k 1,2k )}] 1) Fujita, M., Ogawa, H. : Multiple equilibria and structural transition of non-monocentric urban configurations, Regional science and urban economics, Vol. 12, No.2, pp. 161-196, 1982. 2) Fujita, M. and Thisse, J.-F., Economics of Agglomeration: Cities, Industrial Location, and Globalization (2nd Edition), Cambridge University Press, 2013. 3) Osawa, M.,Akamatsu, T.: Stochastically stablity analysis of a model of endogenous urban subcenter formation, (CD-ROM), Vol.54, 2016. 4) Sandholm, W. H.: Population games and evolutionary dynamics, MIT press, 2010. 5) Reza Sobhaninejad D3() Vol.69, No.1, pp.53-63, 2013. 6) 1 D Vol. 66, No. 4, pp.442-460, 2010. 7) Social Interaction D3( ) Vol.67 No.1 pp.1-20 2011. (2018. 4. 27 ) 10
STOCHASTIC STABILITY ANALYSIS OF A MODEL OF POLYCENTRIC URBAN CONFIGURATIONS: LINEAR CITY VS. CIRCULAR CITY Shuhei YAMAGUCHI and Takashi AKAMATSU 11