Japanese municipalities, 1970 present

Japanese municipalities, 1970 present 3000 2500 Number of municipalities 2000 1500 1000 500 1980 1990 2000 2010 Year

m M q m N m θ m q m c(x m ) c(x m ) X m X m c(n m ) m τ m Y m = i m y i i m T m (q m,τ m ) q m c(x m )=τ m Y m + T m. u i (q m,τ m,θ m )=β 0 ((1 τ m )y i )+β 1 (q m β 3 )+β 2 l(i, θ m )+ɛ m, β 3 l(i, θ) i θ i ɛ m ɛ X m

q m τ m τm u i (q m,τ m,θ m )=β 0 (1 τ m )+β 1 (q m β 3 )+β 2 l(i, θ m )+α i + ɛ m, α i = β 0 (y i ) θ m θm u mm (T m ) m T m u mm (T m )=β 0 (1 τ m)+β 1 (q m β 3 )+β 2 l m (θ m)+α m + ɛ m α m = 1 N m i m α i l m (θ) = l(i, θ) τ q θ 1 N m i m τm =1 β 0 Y m + T m β 3 c(x m ) β 0 + β 1 Y m qm = β 1 Y m + T m β 3 c(x m ) + β 3 β 0 + β 1 c(x m ) θm = l m (θ). θ S M S S m

π Π i S m S u i (τ S,q S,θ S )=β 0 (1 τ S )+β 1 (q S β 3 )+β 2 l(i, θ S )+α i + ɛ S. T S S τ S q S θ S (q,τ,θ ) qs τ S θ S S m S m S u ms = β 0 (1 τ S)+β 1 (q S β 3 )+β 2 l m (θ S)+α m + ɛ S. S m S S m m Y m α m S S

m S u ms u mm = β 0 ( (1 τs) (1 τm)) + β 1 ( (qs β 3 ) (qm β 3 )) + β 2 (l m (θs) l m (θm)) + ɛ S ɛ m. m m S Y S+T S β 3 c(x S ) Y m +T m β 3 c(x m ) Y m Y S > Y S+T S β 3 c(x S ) c(x S ) > Y m+t m β 3 c(x m ) c(x m ) T S T m m T b W (T )= w m u mm (T m ) b T m m M m M {m} u mm u m{m} θm l m θs θ m m

w m m m τ m q m θ m T m = β 3 c(x m ) Y m + w m(β 0 + β 1 ). b y T m = β 3 c(x m ) ay m, a =1 w yb (β 0+β 1 )

M = {1, 2, 3} {1, 2, 3} 1 {1} 1 {1, 3} 1 {1, 2}, {1, 2, 3} 2 {2} 2 {1, 2} 2 {2, 3}, {1, 2, 3} 3 {3} 3 {2, 3} 3 {1, 3}. {{1, 2}, {3}} {{1, 2}, {3}} {2, 3} Π Π Π {{1, 2}, 3} {2, 3} {1, 3} Π π / Π π π π Π π

π Π S S π, m S u ms >u ms {m}, {m } π, S = {m, m }, u mm >u ms u m m >u m S S π π l(i, θ m )

Prefectures of Japan Shizuoka µ Decimal Degrees 0 0.5 1 2 3 4 τy m τ τ τ ɛ l α i

Decimal Degrees 0 0.05 0.1 0.2 1995 borders 4 Shizuoka Prefecture c T m = ( c(x m ).75 τy m, 0). c

Decimal Degrees 0 0.05 0.1 0.2 1995 borders 2006 borders 4 Shizuoka Prefecture X m τy m c c 0 c 0 c c 0 c 0 c c c c

w m c c 0 T 0 w c 0 X m c c c c c c ψ c c 0 = c

12 Cost per capita H 100,000L 10 8 6 4 2 c c c é 0 1000 10 4 10 5 10 6 Population c 0 (N m )=(1+ H 0 (N m )) c H 0 (N m )=ψh(n m ). ψ ψ H0 H H = β 4 H0 ˆβ 4 c(n m )=(ψ 0 + ψ 1 H(N m )) c ψ 0 =1 β 1 β 3 ψ ψ β 4 β 4 =0 c β 4 =1

~ c (blue), and without hypothesized spurious adjustments (pink) 12 10 8 6 4 2 10^3 10^4 10^5 10^6 Population S T S = T S + S T S = ( c(x S ).75 τy S, 0). T S S S 0 S 1 S c 0

β 3

Cost per capita ( 100,000) 10 5 10^3 10^4 10^5 10^6 Population 15 Cost per capita H 100,000L 10 5 0 1000 10 4 10 5 10 6 Population

0.0 0.1 Fraction of c ~ 0.2 0.3 0.4 10^3 10^4 10^5 10^6 Population 0.0-0.1 Fraction of c -0.2-0.3-0.4 1000 10 4 10 5 10 6 Population

0.2 Fraction of c ~ 0.1 0.0 10^3 10^4 10^5 10^6 Population 0.20 0.15 Fraction of c 0.10 0.05 0.00 1000 10 4 10 5 10 6 Population

0.8 Fraction participating in mergers 0.6 0.4 0.2 0.3 0.2 0.1 0.0 "Stick" incentive (fraction of c ~ ) 0.8 Fraction participating in mergers 0.6 0.4 0.2 10^3 10^4 10^5 10^6 Population

0.8 Fraction participating in mergers 0.6 0.4 0.2 0.5 1 2 5 SFR per capita 0.8 Fraction participating in mergers 0.6 0.4 0.2 0.1 0.2 0.5 1 2 SFR (fraction of SFN)

ˆβ S β m S S S β ɛ β β 0 β 1 β 2 β 3 β 4 c

ɛ S = {m 1,m 2,...,m 14,m 15 } S = {m 1,m 2,...,m 14 } ɛ S ɛ S ɛ S N(0, 1) S ɛ S ω i N(0, 1) i m N m ω m = 1 N m s 2 m = N m ω i i=1 1 N m 1 N m i=1 (ω i ω m ) 2, N m S ω S s 2 S m S ω M s M ω M M m ω m ɛ m = f(x m ) s 2 m, S f(x) > 0 S = A B f(x S ) >f(x A ) f(x S ) >f(x B )

ɛ S S ω M s M ɛ( ω M,s M ) N f(x S )= S 1 ɛ 2 S N(0, 1) ( ) 100 10 10 13 S

β 0 h( ω, s X) h h (π X, β) = ω M,s M h( ω M,s M X) π Π (ɛ( ω M,s M ) X, β) Π ω M s M π h h π Π h ( ω, s) ɛ( ω, s) h β m S u ms (β) =v ms (β)+ɛ S

h( ω 0 M,s 0 M X) h (π 0 X, β 0 ) ω 0 M s0 M ω M s M π 0 Π (ɛ( ω 0 M,s0 M ) β0 ) π 0 π 0 ( ω M 0,s0 M ) g 1 (π, β X) =E ωm,s M [h( ω M,s M X)] h (π X, β) = h( X) h (π X, β). h( X) h( ω M,s M X) ω M s M g 1 β 0 E π [g 1 (π, β 0 X)] = h( X) E π [h (π X, β 0 )] h( X) E ωm,s M [h( ω M,s M X)] = 0 ( ω M,s M,π) π ( ω M,s M ) ( ω M,s M ) Π π Π ω M s M π Π (ɛ( ω M,s M ) X, β 0 ) ω M s M π h( ω M,s M ) h (π X, β 0 ) ( ω M,s M,π) h ω s h h ˆβ h =0 β

g 1 h h h h h( ω M,s M X) = m,m N m N m ( ω m ω m ) 2 + Nm + N m m M N m 1 (s 2 m 1) 2 2 h h h h ɛ ɛ S µ Q h ω ˆβ ˆβ ω s 2

Q g 2 (Q, β X) =Q µ Q(β X) E[g 2 ] 0 Q 99 µ 99 Q g 3 (Q, β X) =Q µ Q (β X) E Q g 3 (Q 99,β 0 X) =µ 99 Q µ 99 Q (β 0 X) 0 π Π τ

τ m = (τ m(β)+ε m, τ), τ m γ =0 τ m = (τ m(β)+γx mk + ε m, τ), g 4 (β,x) γ γ =0 g 4 γ

c τy (β) =[ḡ 1 (β)] 2 +[ḡ 1 (β)] 2 +[ḡ 1 (β)] 2 + [ḡ 2 (β)] 2 +[ḡ 2 (β)] 2 +[ḡ 2 (β)] 2 + [ḡ 3 (β)] 2 +[ḡ 4 (β)] 2 ḡ [x] = (x, 0) ˆβ (β) ˆβ 2 ˆβ 3 c ḡ 4

v ms m S β 0 219.56 118.18 (145., 454.) (39., 348.) β 1 5.04 2.80 (2.6, 12.1) (0.0, 9.8) β 2 0.22 0.23 ( 0.32, 0.04) ( 0.39, 0.0) β 3 1.04 1.06 (0.95, 1.12) (0.97, 1.25) β 4 0.49 0.51 (0.38, 0.60) (0.29, 0.66) N (a, b) τ ˆβ4 H β 4 =0 H 0 β 4 =1

N m

Nm = β 0 ( yn m β 3 c(n m ) )+β 1 ( yn m β 3 c(n m ) )+β 2 l (N m ) N m yn m c(n m ) l (N m ) N m 2 T 0 w c c 0 c 0 c(n m x) x 0.20N m S S S = S S S

ŵ m c 0 c w c ŵ ˆβ 4 c T 0 β 3 =1 β 4 =1 ˆβ 3 ˆβ 4 T 1

Π ɛ

ɛ ɛ m m T = {T m,t m,t S } S T m T m ɛ T m T m ɛ S ɛ m ɛ S ɛ m v u ms (T S )=v ms (T S )+ɛ S u mm ɛ

T (T m (T m + T m (T m T m)+(t m Tm)+(T m T m ) TS T S e +(1 η) wm e b = ({S} T ) 1 ({S} T ) T S )+η w m b Tm )+(T S T S ) =1+ ({S} T )(1 ({S} T )) (β 0 + β 1 ) ({S} T ) T T m T m ({S} T ) v m T m T m ({S} T ) vm w m e b w m b e = η = ({S} T ) v m ({S} T ) v S β 0 + β 1 1 ({S} T ) m v S e = E[u ms u mm u m S = u m m,u ms u mm,t] m m e (TS T S ) (Tm Tm) (Tm T m ) TS Tm Tm TS TS T m Tm Tm Tm

TS Tm Tm ({S} T )(1 ({S} T )) ({S} T ) v S ɛ ɛ T 2 ɛ m = ɛ m =0 ɛ S (0,σ) c(n m )=c 0 + c 1 N m 2

({S} T )(1 ({S} T )) ({S} T ) v S = σ N m 2Tm TS T m Tm T S T S c(n m ) c(n m ) T m Tm β 0 β 1 β 2 β 3 =1 β 4 =1 c 0 = 1.3 c 1 = 130, 000 y b/w T m = T m (1 + T S 2T m 2Tm T ) 1 S TS =2T m TS = T S Tm = Tm Tm τm T m

ɛ T 0 I S S = {m, m } I S = (T 0 m T 1 m)+(t 0 m T 1 m )+(T 1 S T 0 S) T 0 m + T 0 m T 0 S e e I e e I [0, 1] I S = {m, m } m m I N m N m I ˆβ ŵ T N m N m

1.0 10^6 0.8 10^5 0.6 Population 10^4 0.4 0.2 10^3 0.0 10^3 10^4 10^5 10^6 Population

6 4 Density 2 0 0.2 0.4 0.6 0.8 1.0 1.2 Incentive T ɛ ɛ T I T

ˆβ T T I I

m M m {S M m S} m π(m) m π π m π π(m) m π (m) π S π S m S, π m π < Π Π Π (Π,<) π, π Π π π

π / Π, π Π π<π π S π π S π π S π S π π π π π π π π π π π S π S π π S π π S π π \ π = S π S π S = Q Q π S S π S π π S π S π π S π π \ π = S S = Q Q π Q Q Q Q π Π π π π π {π 1,...,π n } π π 1... π n π π 1 S π 2 S π 3 π 1 π 2 π 3 {π 3 } π 1 π 3

Π = {π π π π } (Π, ) Π π / Π {π 1,...,π n } Π π π 1 π n π n Π π n = π l l<n Π Π \ Π = Π Π Π \ Π Π Π (Π, ) Π \ Π Π \ Π Π =Π c(x m ) 24 c(x m )= X mk c k (1 + Hk (X m )). k=1 Π Π {π Π π Π,π π }

X mk c k H k k k m 160 2 X m Hk H k (X m )= j J1 H j k (X mj)+ 1 X mk c k j J 2 Hj k (X mj). J 1 J 2 k J 2 X mk c k c(x m ) 24 c(x m )= X mk c k (1 + H k (X m )) + ζ m k=1 H k (X m )= j J1 H j k (X m). J 1 X m ζ

10^3 10^4 10^5 10^6 10^7 10^3 10^4 10^5 10^6 10^7 10^5.5 10^5.0 10^4.5 planning taxcollection registration other Estimated Salary Expense ( per capita) 10^4.0 10^5.5 hygiene firefighting elderly otherland agriculture othereducation commerce welfare 10^5.5 10^5.0 10^4.5 10^4.0 10^5.0 10^4.5 10^4.0 10^3 10^4 10^5 10^6 10^7 10^3 10^4 10^5 10^6 10^7 Population 2

q m qm f =5 qm r =3 c(x) H 24 c(x m )= X mk c k (1 + H k (X m )) + ζ m k=1 H k = ψh k 24 c(x m )= X mk c k (1 + H k (X m )) + ζ m k=1 c k H k H k c ζ m β 4 < 1

c k H k b β 4 τ

H 1 H 0 c 1 c 0 c 0 c 1 c 0 H H k c ψ b c 0 c 1 H c 1 k X mk H k c = c β 3 =1

c 0 c 1 m S T m T S T m T S m S β 3

c k k c c 1 TS 1 T 0 S c 0