Key Words : social network, core-periphery model, traffic congestion, spatial interaction, social externality

Transkript

1 ( ) E-mal: ohra@stu.kobe-u.ac.jp 2 ( ) E-mal: ota@opal.kobe-u.ac.jp Helsley & Zenou (2014) 2 Key Words : socal network, core-perphery model, traffc congeston, spatal nteracton, socal externalty 1. 1) 2) 3) face-to-face communcaton 5),6) Ballester et al. 17) Helsley & Zenou 18) Ballester et al. Helsley & Zenou

2 2. 2 (1) N = {1,..., n} (n 2) g G = [g j ] g j (, j N) j (f agents and j are drectly connected) g j = 0 (otherwse) g j = 1 g j = 1 g = 0 G 0 (0, 1) g j = g j = 1 2, j E g g = N, E 1 g G g 11 g 12 g G = g 21 g 22 g 23 = (1) g 31 g 32 g C Center P Perphery = 1,..., n C x 0 (f agent locates n C) x = (2) 1 (f agent locates n P) n x = {x 1,..., x n } C n c P n p = n n c x C C P 2 1 C : g = {1, 2, 3}, {12, 13} lnk1 lnk2 C P x =0 x =1 2: (2) = 1,..., n x x = 0, 1 U U (v, v, g, h ) = z + u (v, v, g) + f (h ) (3) z u (v, v, g) f (h ) v v {v 1,..., v 1, v +1,..., v n } h u u (v, v, g) = αv 1 2 v2 + θ g j v v j (4) α, θ (4) u 2 U / v 2 < 0 3 θ, j (g j = 1) j v 742

3 j v j 2 U /( v v j ) = θ > 0 (for g j = 1) 3 (5) y = z + [ p + t(x, v) ] v + R(x )h (5) (5) p, t(x, v) p α > p > 0 x v t 1, t 2 ( 0) 1, 2 t 1 v j, t 2 x x j v j (6) t(x, v) t(x, v) = t R(x ) x R(x ) ( N) g v j ( j = 1,..., n, j ) U z v 1. (4) 3 2. (5) 2 13) Jovanovc & Rob 14) Berlant & Fujta 15) ) Helsley & Zenou Helsley & Zenou (3) Nash (3), (5) v j ( j = 1,..., n, j ) v = α p t t 1 v t 2 x v + θ g j v j (7) x = x 2 ( = 1,, n) (7) v Nash A 1 n X k x k (k = 1,, n) n (7) [(1 + t 1 )I + t 2 X]v = (α p)1 t 1 Av t 2 X AXv + θgv (8) 1 1 I (8) v Nash v [I θg + t 1 (I + A) + t 2 X(I + AX)] 1 I. v = [I θg + t 1 (I + A) + t 2 X(I + AX)] 1 (α p)1 (9) (9) v v g Katz-Bonacch II. ( ) 1 U = y t 1 + t 2 x v 2 R(x )h + f (h ) (10) 2 Helsley & Zenou (4) n = e. Katz-Bonacch 1, 2, 3 C 4, 5 P 743

4 (a) SN (b) SN (c) SN 4 3: SN 2 SN 4 n=5, # of lnks= 4 1: SN 2 SN 3 SN 4 v U v U v U v W W =n W = [U + R(x )h ] (11) =1 h = 1 R(x ) = 1 f (1) = 0 α = 2.5, θ = 0.25, y = 6, p = 0.5, t 1 = 0.05, t 2 = v SN 2 2 5, 3 4; SN 3 3 4; SN 4 2,3 4,5 5 () C ().e. g j = 1 j C () P () () j SN 3 SN 4 () SN 1 () SN 1 2 () 1 3. (1) v U (11) =n max W = max [U + R(x )h ] v ( N) v ( N) =1 =n { = max y [p + t(x, v)]v v ( N) =1 +αv 1 2 v2 + θ 1 α p 2t 1 v j 2t 2 x x j v j v +2θ g j v v j + f (h ) } g j v j = 0 (12) v O = α p 2t 1 v j 2t 2 x x j v j + 2θ g j v j (13) (7) v O v v O = v t 1 v j t 2 x x j v j + θ, j, j g j v j (14) Nash v 744

5 (2) 1, 2 d O 1 = t 1 v j, d O 2 = t 2x, j, j x j v j (15) = θ g j v j (16) s O U = y pv t(x, v)v (d O 1 + do 2 )v R(x )h +(α + s O )v 1 2 v2 + θ g j v v j + f (h ) = y p + t 1 v j + +t 2 x x j v j +, j, j +αv 1 2 v2 + 2θ v j x j v j v R(x )h g j v v j + f (h ) (17) (17) 2. v O = [I 2θG + 2t 1 A + 2t 2 X AX] 1 (α p)1 U O (17) 1 ( ) 1 U O = y t 1 + t 2 x v O2 R(x )h + f (h ) (18) (10) 1 Nash v 1, 2 d O 1 = t 1 n v j, d O 2 = t 2 x n x j v j s O = θ n g j v j (18) =n [ W O = U O + R(x )h + d O 1 v + d O 2 x ] v s O vo =1 =n [ ( ) 1 = y t 1 + t 2 x v O2 + f (h ) =1 (3) + t 1 v j v + t 2 x x j v j v θ g j v j (19) d O 1, do 2 so s O 1.e. 1 1, 2 c 1, c 2 x max U = y pv v t 1 v j + c 1 v t 2 x j v j + c 2 x v R(x )h +αv 1 2 v2 + θ g j v v j + f (h ) (20) (20) c 1, c 2 > 0 1, 2 1 c 1, c 2 c 1, c 2 < 0 1 v = α p c 1 c 2 x t t 1 v t 2 x v + θ g j v j (21) v = 1,..., n ˆv ˆv = 1,..., n 745

6 2: SN 2 SN 3 SN 4 ˆv Û ˆv Û ˆv Û C (c 1 ) (-0.82) (-1.10) (-1.25) C (c 2 ) (0.64) (0.85) (0.68) Ŵ : SN 2 SN 3 SN 4 v O U O v O U O v O U O T T S W O ˆv (21) v = [I θg + t 1 (I + A) + t 2 X(I + AX)] 1 [(α p c 1 )I c 2 X ] 1 (22) ˆv ( ) 1 Û = y t 1 + t 2 x ˆv 2 + f (h ) R(x )h (23) c 1, c 2 =n ] } max Ŵ = max {[Û + R(x )h + c1 v + c 2 x v c 1,c 2 c 1,c 2 =1 (24) 3 4 1, 2 C 1 = c 1 v, C 2 = c 2 x v =1 c 1, c 2 =1 (4) (24) Ŵ c 1, c T 1, T 2 S 1, 2 T 1 = d O 1 vo, T 2 = d O 2 x v O, S = =1 =1 =1 s O vo W O (19) =n [ ] W O = U O + R(x )h + T1 + T 2 S (25) =1 SN 3, 4 v SN SN v 1 < v 2 v O 1 > vo 2 2 SN 2 SN 4 c 1 c

7 I. Helsley & Zenou 18) [I θg] 1 θ ρ(g) < 1 (26) ρ(g) G I θg + t 1 (I + A) + t 2 X(I + AX) (26) II. Bonacch 20) Bonacch 5) 17) n G k k G k G k g [k] j 0 g j k( 1) G 0 = I (I ) θ > 0 g θ M M(g, θ) = [I θg] 1 = + k=0 θ k G k θ M(g, θ) m j (g, θ) = + k=0 θ k g [k] j g j k θ k + k=0 θ k G k G < r = lm nf k θk 1/k = 1 θ r G G. G ρ(g) = max ι λ ι λ ι G θρ(g) < 1 M 1 (Bonacch ) g n G θ M(g, θ) = [I θg] 1 g θ Bonacch b(g, θ) b(g, θ) = [I θg] 1 1 (27) Bonacch M b (g, θ) = m j = + k=0 θ k g [k] j (28) 1) ) Duranton, G. and Puga, D. : Mcro-foundatons of Urban Agglomeraton Economes, In: Henderson, J. V. and Thsse, J. -F. (Ed.), Handbook of Regonal and Urban Economcs, Vol. 4, North Holland, pp , ) Storper, M. and Venables, A.J.: Buzz: face-to-face contact and urban economy, Journal of Economc Geography, Vol.4, pp , ) Beckmann, M. J. : Spatal equlbrum n the dspersed cty, In: Papageorgou, G. J. (Ed.), Mathematcal Land Use Theory, Lexngton, MA: Lexngton Books, pp , ) Jackson, M. O. : Socal and Economc Networks, Prnceton: Prnceton Unversty Press,

8 6) ) No. 12, pp.1-13, ) Marshall, A. : Prncples of Economcs, London: Macmllan, ) Fujta, M. and Thsse, J. F. : Economcs of Agglomeraton Ctes, Industral Locaton, and Regonal Growth, Cambrdge: Cambrdge Unversty Press, ) Beckmann D, Vol.66, No.2, pp , ) Mossay, P. and Pcard, P.: On spatal equlbra n a socal nteracton model, Journal of Economc Theory, Vol. 146, Issue 6, pp , ) Helsley, R.W. and Strange, W.C. : Urban nteractons and spatal structure, Journal of Economc Geography, pp. 1-20, ) No. 590/IV-39, pp.11-22, ) Jovanovc, B. and Rob, R. : The Growth and Dffuson of Knowledge, Revew of Economc Studes, Vol. 56, pp , ) Berlant, M. and Fujta, M. : Knowledge Creaton as a Square Dance on the Hlbert Cube, Internatonal Economc Revew, Vol. 49, No. 4, pp , ) pp , ) Ballester, C., Calvò-Armengol, A. and Zenou, Y. : Who s who n networks. Wanted: The key player, Econometrca, Vol. 74, pp , ) Helsley, R. W. and Zenou, Y. : Socal Networks and Interactons n Ctes, Journal of Economc Theory, Vol.z 150, pp , ) Cabrales, A., Calvò-Armengol, A. and Zenou, Y. : Socal Interactons and Spllovers, Games and Economc Behavor, Vol. 72, pp , ) Bonacch, P. : Power and Centralty: A Famly of Measures, Amercan Journal of Socology, Vol. 92, No. 5, pp , ) Corchòn, L. C. and Mas-Colell, A. : On the stablty of best reply and gradent systems wth applcatons to mperfectly compettve models, Economcs Letters, Vol. 51, pp , POLICY ANALYSIS OF SOCIAL NETWORKS AND SPATIAL INTERACTIONS Yuk OHIRA and Toshmor OTAZAWA In ths paper we extend Helsley & Zenou (2014) s socal nteracton model and consder traffc congestons. Focusng on negatve congeston externaltes and postve communcaton externaltes, we defne the frst-best congeston tax and subsdy on nteractons. Snce the frst-best tax and subsdy polcy could be dffcult to mplement because of dffcultes n collecton of nformaton about socal networks, we propose a second-best polcy n whch a sngle value of tax/subsdy on each lnk s appled. Numercal examples are then undertaken for nvestgatng the nterrelatonshps between socal networks and publc polces. 748