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- ふさこ とみもと
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1 2007 3
2 i
3 A 85 A A A B 90 B B ii
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5 [22] % 36.5% % 49.8% 9.1% 36.8% 3.9% 70% 10km 20km 30km 40km 43.7%, 44.5%, 54.2% 60.2% 60.9%
6 Kaneko [8] 2 2 3
7 Waldman [20] 4
8 1.3 Böhm-Bawerk [17] 1 Böhm-Bawerk 3 20 Neumann-Morgenstern [18] Böhm-Bawerk Shapley-Shubik [16] Böhm-Bawerk Shapley-Shubik [16] Shapley-Shubik [16] 5
9 Kaneko [7] Shapley-Shubik Kaneko [7] Shapley-Shubik Kaneko [7] Kaneko [8] Kaneko [8] Kaneko [8] Kaneko [8] Kaneko-Ito-Osawa [9] [9] Ricardo [15] Ricardo [15] 3 Gerber [5] Kaneko [8] 6
10 Ricardo Thünen [19] Ricardo [15] Thünen Thünen [19] Ricardo Thünen 20 Thünen [19] Alonso [1] Thünen Thünen central business district: CBD CBD Alonso [1] Muth [14] Mills [12] Fujita [3] Fujita-Krugman-Venables [4] 4 Braid [2] Braid [2] CBD 7
11 Böhm-Bawerk [17] Kaneko [8] Ricardo [15] Kaneko-Ito-Osawa [9] 4 Ito [6]
12 1.1: 9
13 CBD k = 1,..., T k 1 1 p 1,..., p T k Gk I Gk f T r f r = r 1,..., r f Ricardo [15] JR
14 k I Gk, r k k ÎĜk ˆr k 3.2 ÎĜk I Gk ˆr k r k ÎĜ1 I G1 ÎĜ2 I G2 ÎĜf 1 I Gf ÎĜk I Gk ˆr 1 r 1,, ˆr f 1 r f ÎĜ1 I G1 ÎĜ2 I G2 ÎĜf 1 I Gf h k + gc
15 h k k gc c 1.3 c k h k ĥk r = r 1,..., r f 4.2 k h k ĥk k r = r 1,..., r f r = r 1,..., r f 4 k k r k ˆr k k k 1,..., 1 r 1 r k 1 r k r k+1 r f ˆr 1 ˆr k 1 ˆr k ˆr k+1 ˆr f 1.2: k I Gk ÎĜk k 1.2 k k r k ˆr k 1.2 k 4 t > k k 12
16 k 1,..., 1 k k k 1,..., r 1 r k 1 r k r k+1 r T ˆr 1 ˆr k 1 ˆr k ˆr k+1 ˆr T 1.3: k 4.5 k k k k r 1,..., r T T k=1 d p k d r k
17 1.4 p k d k p k d k d 1.4 r k d p k d r k 2 U t, s, c = αt + βs + γ c t s c α β γ 1.5 α β γ JR p k d p k d p k 2 k = 1,... T T k=1 d R a = p k d r k d p k d p k 2 T k=1 p k k 1 w k wk d=1 p k d w k k 1.6 R a r 1,..., r T
18
19 JR M, N M M = {1,..., m}, N N = {1,..., n } i M j N T 2 CBD 16
20 A B A B 2.5 j N Ω := { e 0, e 1,,e T } R + { e 0, e 1,,e T } T e 0 e k k = 1,..., T R + e k k 1 k 1 e k e 0 Ω i x i, c I i > 0 i M k p k, x i = e k I i p k i M A-1 A-2 u i : Ω R A-1 i M, u i x i, c c A-2 i M k = 1,..., T u i e 0, I i > ui e k, 0 A-1 A A-1 A-2 M, N A-1 A-2 B
21 j C j 0 = 0 < C j 1 C j : Z + R + C j y j B j N k k = 1,..., T y j Z + C j y j + 1 C j y j C j y j + 2 C j y j + 1 Z + C j y j j y j C j y j y j y j C j 0 = 0 < C j 1 1 C j y j + 1 C j y j y j + 1 j 1 B 2 1 k 1 j k N = {1,..., n } N = {1,..., T } N k k k = 1,... T j N N N = N 1 N T N k {C j } j Nk k 2 18
22 j N k y j C k : Z + R + { C k y k = min C j Nk jy j : } y j Nk j = y k and y j Z + for all j N k 2.1 k N = {1,..., T } k N k 2.1 M, N p R T + x { e 0, e 1,,e T } m y Z T + p, x, y 2.1 p, x, y i M I i px i 0 I i px i 0 x i { e 0, e 1,,e T } u i x i, I i px i u i x i, I i px i k = 1,..., T p k y k C k y k p k y k C ky k i M x i = T k=1 y ke k. px i = T k=1 p kx ik x ik x i k T T e k y k p x { e 0, e 1,,e T } m y Z T + p, x, y p M, N Shapley-Shubic[16] u i 1 Kaneko [7] Kaneko-Yamamoto [10] M, N 19
23 2.2 M, N p, x, y M, N p p p p M, N Shapley-Shubic[16] Kaneko [8] Miyake [13] 2.3 M, N p. A.1 p p p M, N 3 I 1 I 2 I m
24 A-1 A-2 u i : Ω R C i, i M u i, = u i, CBD JR 2.5 CBD C C E 2.2 u i i u D u x i, c > u x i, c δ > 0 u x i, c = u x i, c + δ E u x i, c = u x i, c c < c δ > 0 u x i, c + δ > u x i, c + δ D E 2 x i, c x i, c x i x i c < c δ x i D δ > 0 21
25 ux i, c δ < ux i, c δ F 1,..., T F u e 1, 0 > u e 2, 0 > > u e T, 0 2.3a F 0 u e T, 0 k = 1,..., T F 1 T B D E F c 0 u e 1, c > u e 2, c > > u e T, c 2.3b A-1 F u e t, c u e t, 0 > u e t+1, 0 D δ > 0 u e t, 0 = u e t+1, δ E u e t, c u e t+1, δ + c A-1 u e t+1, δ + c > u e t+1, c u e t, c u e t+1, δ + c > u e t+1, c u e t, c > u e t+1, c F 2.3b CBD M, N p = p 1,..., p T M, N p, x, y M, N 22
26 p, x, y 1 i M x i = e k k < k p k > p k 2 x i = e k, x i = e k I i > I i k k. k < k i M x i = e k p k p k A-1 2.3b u e k, I i p k > u e k, I i p k u e k, I i p k i xi = e k 2 I i > I i k > k p k > p k i u e k, I i p k u e k, I i p k D δ 0 u e k, I i p k = u e k, I i p k + δ p k > p k δ 0 I i p k < I i p k I i p k + δ I i I i = δ > 0 E u e k, I i p k + δ > u e k, I i p k + δ + δ u e k, I i p k + δ = u e k, I i p k + I i I i = u e k, I i p k u e k, I i p k + δ + δ = u e k, I i p k + δ + I i I i = u e k, I i p k + δ u e k, I i p k > u e k, I i p k + δ u e k, I i p k A-1 i x i = e k k k 0 p k > p k G G f p, x, y f T k = 1,..., f y k > 0, k = f + 1,..., T y k = 0 Ricrdo G
27 1 p 1 > p 2 > > p f 1 > p f I 1 I m Gk k = 1,..., T k Gk = y t k = 1,..., T. 2.5 t=1 G k f Gk k k > f Gk 1 Gk k Gk I Gk k 2.1 Gk I Gk u e f 1, I Gf 1 r f 1 = u e f, I Gf 1 r f u e f 2, I Gf 2 r f 2 = u e f 1, I Gf 2 r f 1. u e 1, I G1 r 1 = u e 2, I G1 r
28 2.1: Gk k = 1,..., f r 1,..., r f 2.6 k 1 e k k e k+1 Gk k = 1,..., f f 1 Gf 1 f 1 1 r f 1 Gf 1 1 Gf 1 f 1 r f 1 Gf 1 f 1 1 f 2 Gf 2,..., G1 1 r f 1 r f 1 r f 1 Gf 1 f 1 f 1 1 r f f Gf 1 r f 1 r f f f 1 r f 1 Gf
29 r f 2 r f 3,..., r 1 Gf 3,..., G1 2.6 u e 1, 0 < u e f, I Gf 1 rf rf 2.6 r 1,..., r f 1, r f = r f r 1 > > r f 1 > r f = r f. A r 1 > > r f 1 > r f = r f 1.3 Ricardo r 1 > > r f 1 > r f = r f r f r 1,..., r f 1, r f r f r f r 1,..., r f 3 r = r 1,..., r f p = p 1,..., p T 26
30 2.8 p, x, y 1 2 r f = p f r 1,..., r f = p f p 1,..., p f 1 k = 1,..., f 1 I Gk = I Gk+1. 2 k = 1,..., f 1 p k < C k y k + 1 C k y k.. A p f p, x, y C k y k C k y k 1 p k C k y k + 1 C k y k 2 p k C k y k + 1 C k y k < k = 1,..., f 1 y k k = 1,..., f 1 y k + 1 p k B 2.1 y k A.3 Kaneko-Ito-Osawa [9] 27
31 B M, N 2.3 CBD JR 400km CBD 5m 2 25m 2 25m 2 45m 2 45m 2 65m 2 2.2: JR 28
32 2.1: m k h k w k = m 2, 25 45m 2, 45 65m 2 15, 35, 55m 2 T = 4 3 = k, h k w k u : Ω = { e 0, e 1,,e 12} R + R U u : Ω R + R U U t, s, c = 2.1t + 3.2s + 80 c
33 t 18, 28, s 15, 35, 55 c U 2.3 u t, s k e k u U u U t + 3.2s t = 18 s = 55 m 2 h 1 = = h 1 12 h k k s = 55 4 t = 18 s = 35 k = 0, 1,..., 12 u e k, c = h k + 80 c, 2.8 k 1 h k = 2.1t + 3.2s h 0 A i k u e k, I i p k p k I i 2.7 U s, t, I i p ks,t = 2.1t + 3.2s + 80 c,
34 p ks,t I i kt, s t, s A-1 A-2 C-F 80 c E 2.1 w k Yahoo! a k y k y k w k C k y k = L y k w k a k k 1 a k < p k k = 1,..., f L I k w k a k p k w k w k + 1 w k w k 2.1 f k=1 w k = 6388 I 1,..., I m m = I 1 = 850 I 6388 = r = r 1,..., r a k k 31
35 2.2: m k r k p k r f=12 = p k Yahoo! 1 p 1 = 1 w d=1 p 1d p 1d k = 1,..., 12 r k p k 2.6 M, N M, N M, N M, N 1 M, N 5 r f 5 32
36 2.3: 33
37 CBD M, N M, N 34
38 3 3.1 M, N M, N r = r 1,..., r f 2.6 ˆM, ˆN ˆM, ˆN M, N M, N A-1 F M, N ˆM, ˆN C0 1,..., T u, 35
39 C1 M = {1,..., m} ˆM = {1,..., ˆm} C2 I 1,..., I m Î1,..., Î ˆm C3 C k Ĉk k = 1,..., T C4 f ˆf r f ˆr ˆf C5 I Gk ÎĜk C5 M, N m ˆM, ˆN ˆm < m k ˆM, ˆN k M, N I Gk ÎĜk I Gk 3.1 r ˆr M, N ˆM, ˆN k 1 k minf, ˆf 1 ÎĜk I Gk ˆr k r k ˆr k r k ˆr k+1 r k , >, <, = 3.1 k ÎĜk I Gk ˆr k r k k k ÎĜk ˆr k I Gk r k ˆr k ˆr k+1 r k r k k I Gk r k ˆM, ˆN k k + 1 r k r k+1 36
40 ˆM, ˆN r k r k+1 k k k 3.2 E k u e k, I Gk r k = u e k+1, I Gk r k ˆr k r k ˆr k+1 r k+1 ÎĜk I Gk ˆr k r k ÎĜk I Gk > ˆr k r k ˆr k r k > ˆr k+1 r k+1 ÎĜk I Gk > ˆr k r k δ = ÎĜk I Gk ˆr k r k > 0 M, N u e k, I Gk r k = u e k+1, I Gk r k+1 E u e k, I Gk r k + δ > u e k+1, I Gk r k+1 + δ u e k, ÎĜk ˆr k = u e k+1, ÎĜk ˆr k+1 > u e k+1, I Gk r k+1 + δ ˆM, ˆN A-1 ÎĜk ˆr k+1 > I Gk r k+1 +δ = ÎĜk r k+1 ˆr k r k ˆr k r k > ˆr k+1 r k+1 ÎĜk I Gk ˆr k r k ˆr k r k ˆr k+1 r k+1 ÎĜk I Gk ˆr k r k ÎĜk ˆr k I Gk r k δ = I Gk r k ÎĜk ˆr k 0 ˆM, ˆN u e k, ÎGk ˆr k = u e k+1, ÎGk ˆr k+1 E u e k, ÎGk ˆr k + δ u e k+1, ÎGk ˆr k+1 + δ u e k, I Gk r k = u e k+1, I Gk r k+1 u e k+1, ÎGk ˆr k+1 + δ M, N A-1 I Gk 37
41 r k+1 ÎGk ˆr k+1 + δ = ˆr k+1 + ˆr k + I Gk r k ˆrk r k ˆr k+1 r k M, N M, N ˆM, ˆN r = r 1,..., r f ˆr = ˆr 1,..., ˆr ˆf f ˆf Gk k f Ĝk k ˆf f ˆf 3.2 ÎĜ1 I G1 ÎĜ2 I G2 ÎĜf 1 I Gf 1, 3.3 ÎĜ1 I G1 ÎĜ2 I G2 ÎĜf 1 I Gf M, N k = 1,..., f 1 1,..., f 1 ÎĜ1 I G1, ÎĜ2 I G2,, ÎĜf 1 I Gf 1 38
42 k {ÎĜ1 I G1,, ÎĜl1 I Gl 1 }, {ÎĜl1+1 I Gl 1 +1,, ÎĜl2 I Gl 2 },. {ÎĜlk 1+1 I Gl k 1+1,, ÎĜlk I Gl k }, 3.5 l 1, l 2..., l k t = 1,..., k 1 ÎĜlt 1 +1 I Gl t 1 +1 < < ÎĜl t I Gl t ÎĜl t +1 I Gl t +1 ÎĜl t+1 I Gl t+1 ÎĜlt 1 +1 I Gl t 1 +1 ÎĜl t I Gl t ÎĜl t +1 I Gl t +1 < < ÎĜl t+1 I Gl t < < ÎĜ1 I G1 ÎĜ2 I G2 1 ÎĜ1 I G1 <... < ÎĜl 1 I Gl 1 2 ÎĜl 1 +1 I Gl 1 +1 ÎĜl 2 I Gl 2 ÎĜ1 I G1, ÎĜ2 I G2,, ÎĜf 1 I Gf k k
43 3.2 r ˆr M, N ˆM, ˆN k 1 k f a ˆr k+1 r k+1 < ˆr k r k ˆr k r k < ˆr k 1 r k 1 < < ˆr 1 r 1 ; b ˆr k r k < ˆr k+1 r k+1 ˆr k+1 r k+1 < < ˆr f r f a ˆr k r k > ˆr k+1 r k+1 ˆr k+1 r k+1 > > ˆr f r f ; b ˆr k+1 r k+1 > ˆr k r k ˆr k r k > ˆr k 1 r k 1 > > ˆr 1 r a b 3.3 a k k + 1 ˆr k+1 r k+1 < ˆr k r k < b ˆr k+1 r k+1 > ˆr k r k > a l k l f 1 ˆr l r l > ˆr l+1 r l+1 ˆr l+1 r l+1 > ˆr l+2 r l+2 ˆr l r l > ˆr l+1 r l ÎĜl+1 I Gl+1 ÎĜl I Gl > ˆr l r l > ˆr l+1 r l+1 ÎĜl+1 I Gl+1 > ˆr l+1 r l ˆr l+1 r l+1 > ˆr l+2 r l+2 2 b l 2 l k 1 ˆr l +1 r l +1 > ˆr l r l ˆr l r l > ˆr l 1 r l 1 ˆr l +1 r l +1 > ˆr l r l ÎĜl 1 I Gl 1 ÎĜl I Gl < ˆr l r l ÎĜl 1 I Gl 1 < ˆr l r l δ = ˆr l r l ÎĜl 1 I Gl 1 > 0 M, N u e l 1, I Gl 1 r l 1 = u e l, I Gl 1 r l 40
44 I Gl 1 r l 1 < I Gl 1 r l δ > 0 E u e l 1, I Gl 1 r l 1 δ < u e l, I Gl 1 r l δ = u e l, ÎĜl 1 ˆr l ˆM, ˆN u u e l 1, I Gl 1 r l 1 δ < u e l 1, ÎĜl 1 ˆr l 1 e l 1, ÎĜl 1 ˆr l 1 A-1 I Gl 1 r l 1 δ < ÎĜl 1 ˆr l 1 ˆr l 1 r l 1 < ˆr l r l r ˆr M, N ˆM, ˆN. 1 : 3.3 k 1 k 2 1 k 1 k 2 f a ˆr 1 r 1 > > ˆr k1 r k1 ; b ˆr k1 r k1 = = ˆr k2 r k2 ; c ˆr k2 r k2 < < ˆr f r f. 2 : 3.4 k 1 k 2 1 k 1 k 2 f a ˆr 1 r 1 < < ˆr k1 r k1 ; b ˆr k1 r k1 = = ˆr k2 r k2 ; c ˆr k2 r k2 > > ˆr f r f. 2.5 r ˆr ˆr k r k k = 1,..., f k 1 k 2 k 1 = k 2 = k 1 = k 2 = f b 41
45 . 2 1 ˆr 1 r 1 ˆr 2 r 2 k 1 = 1 ˆr 1 r 1 < < ˆr k r k k k 1 ˆr k1 r k1 = = ˆr k r k k k 2 k 2 k 2 = k 1 k 2 = f k 1 k 2 a b k 2 ˆr k2 r k2 > ˆr k2 +1 r k2 +1 ˆr k2 r k2 < r k2 +1 r k a c b r k2 r k2 > r k2 1 r k2 1 > > r 1 r 1 k 1 k JR M, N ˆM, ˆN ˆr M, N ˆM, ˆN r ˆr M, N k = 1, 2, 3, 4, 5 25 r f=12 = % ˆr f=12 = U t, s, c = 2.1t + 3.2s + 80 c ˆM, ˆN k = 1,..., f = 12 2 M, N 12 k=1 w k = = 125 ˆM, ˆN 12 k=1 ŵk = = 6263 ŵ 1 = 244, ŵ 2 = 242, ŵ 3 = 235, ŵ 4 = 736, ŵ 5 = 159 k = 6,..., 12 ŵ k = w k M, N ˆM, ˆN 3 42
46 3.1: 1,000 I Gk ÎĜk ÎĜk I Gk r k ˆr k ˆr k r k ÎĜk I Gk ÎĜ1 I G1 < ÎĜ2 I G2 < < ÎĜ5 I G ÎĜ6 I G6 > ÎĜ7 I G7 > > ÎĜ12 I G k = 2 ˆr 3 r 3 = > ˆr k=2 r k=2 = b ˆr k=2 r k=2 = > ˆr 1 r 1 43
47 ˆr 1 r 1 = 4.98 k = 3 ˆr k=3 r k=3 = > ˆr 4 r 4 = a ˆr 4 r 4 = 4.99 > ˆr 5 r 5 ˆr 5 r 5 = l 1 = 5,..., l 2 = f = ÎĜk I Gk ˆr k r k k = 1,..., k = 3 k = k 1 k 2 k 1 = k 2 = 3 k 1 = k 2 = M, N M, N ˆM, ˆN
48 3.1:
49 2 M, N 46
50 M, N M, N ˆM, ˆN A-1 F M, N ˆM, ˆN 47
51 u : Ω R u e k, c = h k + gc h k k = 1,..., T k h 1 > h 2 > > h T g : R + R gc c 4.1 e k, c h k gc g c lim c + gc = + h 0 + gi m > h 1 + g0 1 h 0 e A-1 F h k k h k h k ĥk h k ĥk 4.1 h k gc M, N ˆM, ˆN M, N k M, N h 8 = 10.2 ĥ8 = 24.0 ˆM, ˆN M, N 8 1 h 0 + gi m > h 1 + g0 A-2 h 1 > > h T I 1 I m g i M k = 1,..., T h 0 + gi i > h k + g0 48
52 4.1: 8 m k h k w k ˆM, ˆN 8 CS1 ˆf = f ˆr ˆf = r f CS2 t f 1 ÎĜt = I Gt CS3 ĥk > h k t k t f 1 ĥt = h t CS1 f r f CS2 t f 1 CS3 k k h k ĥk ĥ k 1 = h k 1 > ĥk > h k 49
53 4.1: k 4.1 CS1 CS2 CS3 i t = k + 1,..., f 1 ˆr t = r t, ii ˆr k > r k, iii t = 1,..., k 1 ˆr t r t, iv t = 1,..., k 2 0 ˆr t r t ˆr t+1 r t+1. I Gk 1 > I Gk iii iv M, N ˆM, ˆN r ˆr i k t = k + 1,..., f ˆr t = r t 1 ii k iii k t = 1,..., k iv k 1 t = 1,..., k 2 r t ˆr t > 0 50
54 4.1 k t = 1,..., k i ˆM, ˆN k ˆM, ˆN Gk 1 k k 1 ˆr k 1 k 1 r k 1 ˆr k 1 k k 1 k 1 E k 4.2 k CS2 CS3 CS2 CS3 CS2 ÎĜk > I Gk t k t f 1 ÎĜt = I Gt CS3 t f 1 ĥt = h t 4.1 k k I Gk ÎĜk k 51
55 4.2: k t f 1 h t f CS1 CS2 CS3 i t = k + 1,..., f 1 ˆr t = r t, ii ˆr k > r k, iii t = 1,..., k 1 ˆr t > r t, iv t = 1,..., k 2 0 < ˆr t r t < ˆr t+1 r t k ii k 1,..., k iii iv 4.2 ˆM, ˆN 52
56 t = 1,..., k 1 ˆr t > r t iv t = 1,..., k 1 k 1 t = 1,..., k 2 ˆr t r t > k r k u e k, I Gk r k = u e k+1, I Gk r k+1 r k Gk k r k k + 1 r k+1 ˆM, ˆN k + 1,..., f k k k M, N Gk e k, I Gk r k e k+1, I Gk r k+1 ˆM, ˆN ˆr k Ĝk k k + 1 ˆM, ˆN k Ĝk ÎĜk = I Gk ˆr k r k ÎĜk = I Gk ˆr k+1 = r k+1 k + 1 ˆM, ˆN Gk ˆr k > r k k 1 Gk M, N e k, I Gk r k ˆM, ˆN ê k, I Gk ˆr k ê k 53
57 ˆr k Gk 1 Gk 2 E Gk 1 M, N e k, I Gk 1 r k ˆM, ˆN ê k, I Gk 1 ˆr k Gk 1 ˆM, ˆN M, N Gk 1 e k, I Gk 1 r k e k 1, I Gk 1 r k 1 Gk 1 ˆM, ˆN ê k, I Gk 1 ˆr k M, N e k 1, I Gk 1 r k 1 k 1 Gk 1 ê k, I Gk 1 ˆr k e k 1, I Gk 1 ˆr k 1 Gk 1 ˆr k 1 < r k 1 1,..., k 2 k k M, N Gk e k, I Gk r k e k+1, I Gk r k+1 ˆM, ˆN Ĝk Gk k k +1 Ĝk e k, ÎĜk r k e k+1, ÎĜk r k+1 ˆM, ˆN e k, ÎĜk ˆr k e k+1, I Gk r k+1 e Ĝk k, ÎĜk r k e k, ÎĜk ˆr k k ÎĜk ˆr k ÎĜk r k ˆr k > r k k 1 Gk 1 e k, ÎĜk 1 ˆr k ˆr k Gk 1 ÎĜk 1 = I Gk 1 ˆM, ˆN e k 1, ÎĜk 1 ˆr k 1 e k, ÎĜk 1 ˆr k Gk 1 M, N ˆr k 1 > r k 1 2 I Gk 1 > I Gk 54
58 C0 M, N ˆM, ˆN CS1 M, N ˆM, ˆN h t + g I Gt r t = ht+1 + g I Gt r t+1 t = 1,..., f 1; 4.2 ĥ t + gîĝt ˆr t = ĥt+1 + gîĝt ˆr t+1 t = 1,..., f CS3 t k t f 1 ĥt = h t CS2 t f 1 ÎĜt = I Gt t k t f 1 g I Gt ˆr t g IGt r t = g IGt ˆr t+1 g IGt r t i t = f CS1 ˆr f = r f g I Gf 1 ˆr f 1 g IGf 1 r f 1 = g IGf 1 ˆr f g IGf 1 r f = 0. g I Gf 1 ˆr f 1 = g IGf 1 r f 1 g IGf 1 ˆr f 1 = I Gf 1 r f 1 ˆr f 1 = r f 1 t = f 2,..., k + 1 t = k + 1,..., f 1 ˆr t = r t ii t = k CS3 ĥ k+1 = h k+1 CS2 ÎĜk = I Gk i ˆr k+1 = r k+1 ĥ k + gi Gk ˆr k = h k + g I Gk r k 4.5 CS3 ĥk > h k gi Gk ˆr k < g I Gk r k g I Gk ˆr k < I Gk r k ˆr k > r k iii ˆr k 1 r k 1 δ = I Gk 1 I Gk 0 g 55
59 4.5 ĥk + gi Gk ˆr k + δ h k + g I Gk r k + δ δ ĥ k + gi Gk 1 ˆr k h k + g I Gk 1 r k 4.6 t = k 1 ˆM, ˆN 4.3 t = k 1 M, N ĥ k 1 + gi Gk 1 ˆr k 1 h k 1 + g I Gk 1 r k CS3 ĥk 1 = h k 1 gi Gk 1 ˆr k 1 g I Gk 1 r k 1 g IGk 1 ˆr k 1 I Gk 1 r k 1 ˆr k 1 r k g I Gk 2 ˆr k 2 g IGk 2 r k 2 = g IGk 2 ˆr k 1 g IGk 2 r k 1 0 g ˆr k 2 r k 2 t = 1,..., k 1 ˆr t r t iv CS2 iii ÎĜ1 I G1 = 0 ˆr 1 r ˆr 1 r 1 ˆr 2 r 2 CS2 ÎĜ2 I G2 = 0 ˆr 2 r ˆr 2 r 2 ˆr 3 r 3 t = 1,..., k 2 0 ˆr t r t ˆr t+1 r t+1 iii δ = I Gk 1 I Gk iii g 4.2 ii 4.2 i iii iv 4.1 ii ii M, N 4.2 t = k h k + g I Gk r k = hk+1 + g I Gk r k
60 CS2 δ = ÎĜk I Gk > 0 r k > r k+1 I Gk r k < I Gk r k+1 g 4.8 h k + gi Gk r k + δ > h k+1 + gi Gk r k+1 + δ δ h k + gîĝk r k > h k+1 + gîĝk r k CS3 ĥk = h k ĥk+1 = h k ĥk + gîĝk r k ĥk+1 + gîĝk r k+1 = ĥk+1 + gîĝk ˆr k+1 i r k+1 = ˆr k+1 ĥ k + gîĝk r k > ĥk+1 + gîĝk ˆr k+1 = ĥk + gîĝk ˆr k 4.10 ˆM, ˆN 4.3 gîĝk r k > gîĝk ˆr k g ÎĜk r k > ÎĜk ˆr k ˆr k > r k CS h 10 = 10.8 ĥ10 = h 7 = 25.9 ĥ 7 = h 5 = 64.7 ĥ5 =
61 4.2: 10, 7, 5 m k h k w k M, N h = h 1, h 2,..., h 12 h 10 ĥ10 h 2 h = h 1,..., ĥ10, h 11, h h 7 ĥ7 h 3 h 3. h 5 ĥ5 h 4 ĥ 4.3 h h h ĥ 4.1 r ˆr h 10 ĥ M, N r 10 r 1,..., r h 7 ĥ ,..., 6 1 M, N r 1,..., r
62 4.3: 4.1 r 7 h 5 ĥ5 M, N r = r 1,..., r 12 1, 2, 3, 4, 6, U t, s, c = 2.1t + 3.2s + 80 c , 7, = 12.6 M, N r = r 1,..., r 12 1, 2, 4, 5, 6, 8, 9, 10 59
63 4.4: l 1 h t CS3 k h t t = k, k + 1,..., k + l 1 l 1 CS3 4.4 CS3 r f I Gt CS1 CS2 CS3 t = k, k + 1,..., k + l 1 l 1 ĥt > h t t t t f 1 ĥt = h t 4.4 k ˆr k > r k ˆr k r k k k 1 ˆr k 1 r k 1 t = k 2,..., iii k k ˆr k 1 < r k 1 60
64 4.3 CS1 CS2 CS3 ˆr t > r t, t = k,, k + l 1, 1 l l ˆr k+l r k+l ˆr k 1 < r k l l k + l k l = 2 C0 M, N ˆM, ˆN CS1 M, N ˆM, ˆN h t + g I Gt r t = ht+1 + g I Gt r t+1 t = 1,..., f 1; 4.11 ĥ t + gîĝt ˆr t = ĥt+1 + gîĝt ˆr t+1 t = 1,..., f t = k CS2 ÎĜk 1 = I Gk 1 CS3 ĥk 1 = h k g I Gk 1 ˆr k 1 g IGk 1 r k 1 = { g } 4.13 I Gk 1 ˆr k g IGk 1 r k + ĥ k h k t = k CS2 ÎĜk = I Gk ĥ k h k = { g } I Gk ˆr k+1 g IGk r k+1 { g } I Gk ˆr k g IGk r k + ĥk+1 h k+1 t = k + 1 ĥ k+1 h k+1 = { g } I Gk+1 ˆr k+2 g IGk+1 r k+2 { g } I Gk+1 ˆr k+1 g IGk+1 r k+1 + ĥk+2 h k+2 61
65 4.13 ĥk h k ĥk+1 h k+1 2 g I Gk 1 ˆr k 1 g IGk 1 r k 1 = { g } { } I Gk 1 ˆr k g IGk 1 r k g IGk ˆr k g IGk r k + { g } { } I Gk ˆr k+1 g IGk r k+1 g IGk+1 ˆr k+1 g IGk+1 r k+1 +g I Gk+1 ˆr k+2 g IGk+1 r k+2 + ĥ k+2 h k ˆr k > r k ˆr k+1 > r k+1 g I Gk 1 I Gk I Gk { g IGk 1 ˆr k g IGk 1 r k } { g IGk ˆr k g IGk r k } 0 { g IGk ˆr k+1 g IGk r k+1 } { g IGk+1 ˆr k+1 g IGk+1 r k+1 } 0 ˆr k+2 r k+2 g g I Gk+1 ˆr k+2 g IGk+1 r k+2 0 CS3 ĥk+2 h k+2 > g I Gk 1 ˆr k 1 g IGk 1 r k 1 > 0 g ˆrk 1 < r k 1 l g I Gk 1 ˆr k 1 g IGk 1 r k 1 l 1 { } { } = g IGk+s 1 ˆr k+s g IGk+s 1 r k+s g IGk+s ˆr k+s g IGk+s r k+s s=0 + g I Gk+l 1 ˆr k+l g IGk+l 1 r k+l + ĥ k+l h k+l 4.15 CS3 ˆr k+l r k+l ĥk+l h k+l > g ˆr k 1 < r k
66 4.4 CS1 CS2 CS3 i t = k + l,..., f 1 ˆr t = r t, ii ˆr k+l 1 > r k+l 1, iii t = 1,..., k 1 ˆr t r t, iv t = 1,..., k 2 0 ˆr t r t ˆr t+1 r t+1.. i ii iv 4.1 i ii iv iv r k 1 r k g I Gk 1 ˆr k 1 g IGk 1 r k 1 = { g } I Gk 1 ˆr k g IGk 1 r k + ĥ k h k ĥk > h k ˆr k r k g I Gk 1 ˆr k 1 g IGk 1 r k 1 > 0 ˆr k 1 < r k 1 ˆr k > r k k + 1 t k + l 1 t ˆr t r t 4.3 ˆr k 1 < r k 1 k + 1 t k + l 1 t ˆr t > r t g I Gk 1 ˆr k 1 g IGk 1 r k 1 l 1 { } { } g IGk+s 1 ˆr k+s g IGk+s 1 r k+s g IGk+s ˆr k+s g IGk+s r k+s s=0 + g I Gk+l 1 ˆr k+l g IGk+l 1 r k+l + ĥ k+l h k+l i ˆr k+l = r k+l g I Gk+l 1 ˆr k+l g IGk+l 1 r k+l = 0 ĥ k+l h k+l = 0 g I Gk 1 ˆr k 1 g IGk 1 r k 1 > 0 ˆr k 1 < r k 1 63
67 4.3: 1,000 m k r k ˆr k h 8 = 10.2 ĥ8 = 24.0 CS3 CS1 CS ˆr 8 = 83.0 > r 8 = ii 4.3 iii 7 1 t = 1,..., 6 ˆr t r t ˆr t+1 r t+1 ˆr 1 r 1 = > ˆr 2 r 2 = > ˆr 3 r 3 = 1.154, ˆr 3 r 3 = > ˆr 4 r 4 = > ˆr 5 r 5 = 1.170, ˆr 5 r 5 = > ˆr 6 r 6 = > ˆr 7 r 7 = iv
68 4.5 r t r t ˆr t r t 1, : r
69
70 5 5.1 M, N Yahoo! M, N M, N 2.5 U t, s, c = 2.1t + 3.2s + 80 c 2.7 M, N 3 4 M, N 2 M, N
71 U t, s, c = 2.1t + 3.2s + 80 c 2.7 U t, s, c = αt + βs + γ c 5.1 α β γ 5.1 t s c 5.1 r f=12 r 1,..., r 11, r 12 r f=12 α β γ r 12 r 1,..., r 11, r 12 p k d 12 k=1 d p k d r k p k d k d p k d d = 1 d = Yahoo! 5.2 p k d r k d p k d r k 2 k = 1,..., 12 α β γ r 12 α β γ r 12 r 1,..., r 11, r
72 5.1: 2.5 U t, s, c = 2.1t + 3.2s + 80 c r 12 = min r k 12 k=1 p k d r k d 5.3 r 1,..., r 12 = p 1,..., p 12 p k k p k = w k d=1 p k d 12 k=1 p k d p k 2 d 12 k=1 p k d r k d p k d p k 2 k = 1,..., r 1,..., r 12 d 69
73 5.1: Kaneko-Ito-Osawa[9] m k r k p k R a R a = 12 k=1 d 12 k=1 p k d r k p k d p k 2 d 2.5 U t, s, c = 2.1t + 3.2s + 80 c r 12 = 49.0 R a = = % R a 5.1 Kaneko- Ito-Osawa[9] m 2 85m =
74 U t, s, c = 2.2t + 3.1s + 80 c r 20 = 49.6 R a R a = 20 k=1 d 20 k=1 d p k d r k 2 = p k d p k R a 5.2 U t, s, c = 2.1t + 3.2s + 80 c r 12 = 49.0 R a R a U t, s, c = αt+βs+γ c 5.1 t s t s R a B [21] 71
75 Yahoo! r = αs β r s m 2 5 U t, s, c = αt + βs + γ c m t t = 7, 15, lns-lnr r = αs β lnr = α + β lns 5.6 r 1 β = dr/r ds/s
76 5.2: t 7 lnr = lns lnr = lns lnr = lns lnr = lns β Yahoo! % , 20, 30, 40, 50, 60 m k r k w k t 73
77 5.3: m : m : m : m : k r k w k 74
78 5.2: 7 t 5.3: 15 t 75
79 5.4: 21 t 5.5: 35 t 76
80 , 20.0,..., 60.0 m 2 lns-lnr 9 15 m M, N % m m m 2 77
81
82 Ricardo 79
83
84
85 2 C CBD JR 2 C 1 U t, s, c = 2.1t + 3.2s + 80 c 5 M, N 5 C 1 Kleiber-Kotz[11] 82
86 83
87 84
88 A A p, x, y k = 1,..., T p k k p k p k, x k, y k T p 1, x 1, y 1,, p T, x T, y T A.1 k, l = 1,..., T p k k > pl k pl l > pk l x k i = ek x l i = el i M A.1 i M x k i = ek x l i = el p k k = pl k pl l = pk l A.1. p k p l p k k pl k pl l pk l i x k i = ek x l i = el p k, x k, y k i u i e k, I i p k k u ie l, I i p k l pl l pk l A-1 u i e l, I i p k l u ie l, I i p l l u i e k, I i p k k u ie l, I i p l l A.1 p l, x l, y l i u i e l, I i p l l u i e k, I i p l k A.1 u i e k, I i p k k u ie l, I i p l l u ie k, I i p l k A.2 p k k > pl k A-1 u ie k, I i p l k > u ie k, I i p k k A.2 u i e k, I i p k k > u ie k, I i p k k p k k = pl k pl l > pk l A-1 u ie l, I i p k l > u ie l, I i p l l 85
89 p l, x l, y l i p l l = pk l A.1 T p 1, x 1, y 1,, p T, x T, y T k, l = 1,..., T p k k = pl k pl l = pk l pk, x k, y k = p l, x l, y l T p 1, x 1, y 1,, p T, x T, y T p k 2 k T p 1 p k k = p1 k p1 1 = pk 1 p1, x 1, y 1 p k, x k, y k k = 2,..., T k = 1 l = 1,..., T p k 3 k T p 2 p k k = p2 k p 2 2 = pk 2 p2, x 2, y 2 p k, x k, y k k = 1, 2 l = 1,..., T k = 1 l = 1,..., T p 1, x 1, y 1 = p 2, x 2, y 2 p 1, x 1, y 1 p 2, x 2, y 2 k = 1, 2 l = 1, 2,..., T p 1, x 1, y 1,, p T, x T, y T A.1 p 1, x 1, y 1,, p T, x T, y T i M x k i = ek p, x, y k = 1,..., T p k = pk k, i M x i = e k k x k i = ek, e 0 k N k = 1,..., T y k = yk k. 86
90 p, x, y 1 p, x, y A.1 i k x k i = e k x, y k x k, y k p, x, y i x i = e 0 x i = e k p k x i pk, x k, y k l k l u i x i, I i p x i = u i x k i, I i p k x k i ui e l, I i p k e l p l l pk l A-1 u i e l, I i p k e l u i e l, I i p l e l = u i e l, I i p e l l k l u i x i, I i p x i u i e l, I i p e l k p k pk, x k, y k A F u e f, I Gf 1 rf D δ > 0 u e f, I Gf 1 rf = u e f 1, δ > u e 1, 0 > > u e f 1, 0 A-1 δ I Gf 1 δ = r f 1 r f A.3 = r f 2.6 1, A-1, F u e f 1, I Gf 2 r f 1 u e f 1, I Gf 1 r f 1 = u e f, I Gf 1 rf > u e 1, 0 > > u e f 2, 0 D δ > 0 u e f 1, I Gf 2 r f 1 = u e f 2, δ A.4 A-1 δ I Gf 2 δ = r f 2 r f = r f r 1,..., r f 1, r f = r f F u e f 1, 0 > u e f, 0 D b > 0 u e f 1, 0 = u e f, b E u e f, I Gf 1 rf = u e f 1, I Gf 1 r f 1 > u e f, I Gf 1 r f 1 + b A.5 87
91 A.3 A.5 A-1 I Gf 1 r f > I Gf 1 r f 1 + b > I Gf 1 r f 1 r f 1 > r f b > 0 r 1 > > r f 1 > r f = r f A A.2 p, x, y k = 1,..., f 1 x i = e k I i = I Gk i M,x i = e k+1 I i = I Gk+1 i M. i i i M I i = I Gk, x i = e l l > k, I i = I Gk, x i = e l l < k x i = e t t k i M I i > I i = I Gk Gk x i = e t t k i M I i < I i = I Gk Gk 2.8. A.2 k = 1,..., f 1 x i = e k I i = I Gk i,x i = e k+1 I i = I Gk+1 i u e k, I Gk p k u e k+1, I Gk p k+1 u e k+1, I Gk+1 p k+1 u e k, I Gk+1 p k I Gk = I Gk+1 2 u e k, I Gk p k = u e k+1, I Gk p k k = 1,..., t 1 u e k, I Gk p k = u e k+1, I Gk p k+1 88
92 u e t, I Gt p t > u e t+1, I Gt p t+1 t 1 t f 1 p t, p t 1,..., p 1 p t, p t 1,..., p 1 p k < C k y k + 1 C k y k, A.6 u e k, I Gk p k = u e k+1, I Gk p k+1 k = 1,..., t 1, u e t, I Gt p t > u e t+1 A.7, I Gt p t+1 A.2 1 k f 1 k x i = e k I i = I Gk i M e k, I i p k A-2 u e k, I i p k u e 0, I i > u e k, 0 I i p k > 0 p t, p t 1,..., p 1 A.6 A.7 p p k = p k k t p k k > t A.8 p x, y p, x, y x, y p A.8 A.6 i x i = e k k > t i p p x i = e k k > t i x i = e k k t i E k < k k > k u e k, I i p k u e k +1, I i p k +1 u e k, I i p k u e k +1, I i p k +1 x i = e k k t 89
93 B B.1 B.1 2 1, 2 3 u x, c = h k + c k = 1, 2, 3 h 1 = 10, h 2 = 7, h 3=f = 0 I 1 = 100, I 2 = 52 p 1 = 51, p 2 = u e 1, I 1 p 1 = = 17 > u e 2, I 1 p 2 = = 15.66, u e 2, I 2 p 2 = = > u e 1, I 2 p 1 = = 11. B.1 B.2 2 1, u e 1, I 1 p 1 = u e 2, I 1 p 2 B.1 B.2 Ut, s, c t s 5.1 Ut, s, c = α 1 t + α 2 + β 1 s + β 2 + γ c B.3 α 1 α 2 β 1 β 2 γ α 1 t + α
94 5.2 U t, s, c = αt + βs + γ c 5.1 c B.3 t s B B.3 α 2 β R a Kaneko-Ito-Osawa[9] 5.1 B.3 R a α 1 α 2 β 1 β 2 γ B.4 Ut, s, c = 400t s c B.4 B.4 B.3 α 1 = 400 α 2 = β 1 = 1800 β 2 = γ = 80 B.4 α 2 β 2 R a B.4 R a R a = = r 12 p 12 = 51.5 R a B.4 α Ut, s, c = 400t s c R a R a = = α R a B.1 B.1 α 2 R a B.1 R a 2.5 U t, s, c = 2.1t + 3.2s + 80 c R a = B.1 t s B
95 B.1: α 2 R a α 2 β 12 2 k=1 d p k d r k 2 R a r k p k d 5.2 α 2 β 2 B.2 α 2 B β B.2 α 2 β 2 R a B.3 α 2 β 2 α 1 t + α 2 β 1 s + β 2 t s R a B.4 400t Ut, s, c = 1.9t s c R a = = Ut, s, c = 1.9t + 3.2s + 80 c 92
96 B.2: β 2 R a α 2 β 12 2 k=1 d p k d r k 2 R a R a = = R a t s 93
97 [1] Alonso, William. Location and Land Use. Harvard University Press, Cambridge, MA, [2] Braid, Ralph M. The short-run comparative statics of a rental housing market. Journal of Urban Economics, Vol. 10, pp , [3] Fujita, Masahisa. Urban Economic Theory: Land Use and City Size. Cambridge University Press, New York, [4] Fujita, Masahisa, Krugman, Paul, and Venables, Anthony. J. The Spatial Economy : Cities, Regions, and International Trades. MIT Press, Cambridge, MA, [5] Gerber, Robert I. Existence and description of housing market equilibrium. Regional Science and Urban Economics, pp , [6] Ito, Tamon. Effects of quality changes in rental housing markets with indivisibilities. Regional Science and Urban Economics. to appear. [7] Kaneko, Mamoru. The central assignment game and the assignment markets. Journal of Mathematical Economics, Vol. 11, pp , [8] Kaneko, Mamoru. Housing market with indivisibilities. Journal of Urban Economics, Vol. 13, pp , [9] Kaneko, Mamoru, Ito, Tamon, and Osawa, Yuichi. Duality in comparative statics in rental housing markets with indivisibilities. Journal of Urban Economics, Vol. 59, pp ,
98 [10] Kaneko, Mamoru and Yamamoto, Yoshitsugu. The existence and computation of competitive equilibria in markets with an indivisible commodity. Theory, Vol. 38, pp , Journal of Economic [11] Kleiber, Christian and Kotz, Samuel. Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley, New York, [12] Mills, Edwin S. Studies in the Structure of the Urban Economy. Johns Hopkins University Press, Baltimore, [13] Miyake, Mitsunobu. Comparative statics of assignment markets with general utilities. Journal of Mathematical Economics, Vol. 23, pp , [14] Muth, Richard F. Cities and Housing. University of Chicago Press, Chicago, [15] Ricardo, David. The Principles of Political Economy and Taxation. J. M. Dent and Sons, London, : original. [16] Shapley, Lloyd S. and Shubik, Martin J. Assignment game I: the core. International Journal of Game Theory, Vol. 1, pp , [17] von Böhm-Bawerk, Eugen. Positive Theory of Capital. Books for Libraries Press, New York, : original, translated by William A. Smart. [18] von Neumann, John and Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton University Press, Princeton, [19] von Thünen, Johann Heinrich. Die Isolierte Staat in Beziehung auf Landwirtshaft und Nationalökonomie. Pergamon Press, New York, : original, translated by Carla M. Wartenberg, edited with an introduction by Peter Hall. [20] Waldman, Michael. Durable goods theory for real world markets. Journal of Economic Perspectives, Vol. 17, pp , [21].., [22]
131 はじめに エコノミア 第 64 巻第 1 号 (2013 年 5 月 ), 頁 [Economia Vol. 64 No.1(May 2013),pp ]
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