Eeckhout

Eeckout (AER04)
命題 1 (都市規模の切断分布とZipf法則)
都市規模∼対数正規分布
⇓
都市規模下限値
都市数
} Zipf係数推定値
(対数線形回帰による)
1
Source: Census Bureau, 2000.
FIGURE 12. ENTIRE SIZE DISTRIBUTION AGAINST RANK
AND LINEAR REGRESSION LINES FOR DIFFERENT
TRUNCATION POINTS
2
of cities
cities of
heterogen
the size s
space.
Define
Time is d
a set of l
Each city
Si,t, and t
S ϭ ¥I S
and can p
productiv
logical ad
of motion
city exp
shock ␴i,t
44
THE AMERICAN ECONOMIC REVIEW
DECEMBER 200
TABLE 3—PARETO COEFFICIENT REGRESSIONS
Truncation point
Estimates
Sគ
City
Kˆ (s.e.)
aˆ (s.e.) (GI s.e.)
R2
135
155,554
Chattanooga (city), TN
19,383
5,000
6,592
Attalla (city), AL
12,500
1,378
Fullerton (city), NE
25,000
42
1.354
(0.011) (0.165)
1.314
(0.002) (0.042)
1.125
(0.002) (0.023)
0.863
(0.002) (0.011)
0.534
(0.001) (0.005)
0.99
2,000
21.099
(0.144)
20.648
(0.017)
18.588
(0.019)
15.944
(0.014)
13.029
(0.010)
Nគ
Lyndhurst (CDP), NJ
Paoli (town), CO
0.997
0.985
0.961
0.860
Notes: Dependent variable: Rank (ln). s.e. standard error; GI s.e. Gabaix-Ioannides (2003)
corrected standard error (aˆ(2/N)1/ 2).
Source: Census Bureau, 2000.
※ 都市定義は通常用いられる都市圏とは大きく異なり、
of cities and allow for competition betwee
cities of different sizes. The space in whic
むしろ、行政区に近い。
heterogeneous cities are considered is therefor
the size space rather than a given geographic
後述の経済モデルで導入される都市外部効果の及ぶ範囲としては、
space.
都市圏が妥当。
3
Define an economy with local externalities C
1434
THE AMERICAN ECONOMIC REVIEW
2000年の「都市」と「都市圏」
TABLE 1—TEN LARGEST CITIES
Rank
City
IN THE
UNITED STATES
Population S
B. The Size Distr
SNY /S
Over the entire size distrib
1
New York, NY
8,008,278
1.000
city has a population of 1,338
2
Los Angeles, CA
3,694,820
2.167
empirical density function on
3
Chicago, IL
2,896,016
2.753
mic (ln) scale, together with t
4
Houston, TX
1,953,631
4.099
normal density for the em
5
Philadelphia, PA
1,517,550
5.277
mean and variance. Figure 3
6
Phoenix, AZ
1,321,045
6.062
7
San Diego, CA
1,223,400
6.546
tive density function. The sa
8
Dallas, TX
1,188,580
6.738
standard error in brackets) is
9
San Antonio, TX
1,144,646
6.996
and the standard deviation i
10
Detroit, MI GIBRAT’S LAW
951,270
EECKHOUT:
FOR (ALL) 8.419
CITIES
1435
theoretical density function
Note: SNY /S denotes the ratio of population size relative to
size distribution is normal in
TABLE
2—TEN LARGEST METROPOLITAN AREAS IN THE UNITED S␾
TATES
New York.
(␮ˆ , ␴ˆ ):
VOL. 94 NO. 5
Rank
1
2
3
4
5
6
7
8
9
10
Source: Census Bureau, 2000.
MA
New York-Northern
New
Jersey-Long
Island, NY-NJ-CT-PA
A substantial
portion
of research
into the size
Los Angeles-Riverside-Orange
County,
CA
distribution of the U.S. population has been
Chicago-Gary-Kenosha, IL-IN-WI
19
done
using
the
MA
as the unit of measureWashington-Baltimore, DC-MD-VA-WV
ment (see, for example,
San Francisco-Oakland-San
Jose, CA Krugman, 1996;
Gabaix, 1999; IoannidesCity,
andPA-NJ-DE-MD
Henry G. OverPhiladelphia-Wilmington-Atlantic
Boston-Worcester-Lawrence,
man, 2003). An MAMA-NH-ME-CT
typically covers one (or
Detroit-Ann
Arbor-Flint,
MI The largest metropolitan
several)
large cities.
Dallas-Fort Worth, TX
area is New York-Northern New Jersey-Long
Houston-Galveston-Brazoria, TX
Island, including the cities of New Haven, Conand Trenton,
New Jersey,
Note: SNY /S necticut,
denotes theNewark
ratio of population
to New and
York.
4 size relative
Population S SNY /S
1
21,199,865
(1)
␾1.000
͑ ␮ˆ , ␴ˆ ͒ ϭ
eϪ
16,373,645 1.295
␴ˆ 2 ␲
9,157,540 2.315
7,608,070 2.787
A Kolmogorov-Smirnov
(
7,039,362
3.012
ness of fit
of the empirical
6,188,463
3.426
5,819,100
3.643
against the
lognormal with s
5,456,428
7.28 and 3.885
sample standard de
5,221,801 4.060
generates the KS test statistic
4,669,571 4.540
ͱ
the corresponding p-value obt
This is supporting evidence i
US (2007年):930都市
(a) US
17
Kesten:
930都市
(b) Kesten
4
Rank size
Gabaix-Ibragimov
Upper average
16
2
15
0
14
-2
13
12
-4
11
-6
10
9
-8
-1
0
1
2
3
4
5
6
7 -1
0
1
2
3
4
5
6
7
※ 対数正規分布より、Kesten過程の定常分布に近い。
5
iid成長過程
都市規模・成長率の分布:
成長率
都市規模(2000年時)
ガウス平滑化関数
h:バンド幅
6
規模sの都市の人口成長率(ノンパラメトリック推定値):
都市iの正規化成長率(実現値):
平準化バンド幅
カーネル平滑化関数 (Epanechnikovカーネル):
7
95%信頼区間
ブートストラップ
(500ランダムサンプリング w/ replacement)
※ 規模以外の性質(e.g., 地域)によって成長率が異なる場合にも、iid成
長率を棄却できない。
8
規模sの都市の人口成長率分散(ノンパラメトリック推定値):
外れ値を除外した推定値
9
モデル
都市 ID: i 2 {1, . . . , I}
総都市規模: S =
都市 i のTFP:
X
I
Si,t
第 t 期の都市 i 規模
Ai,t = Ai,t
1 (1
+ i,t )
| {z }
(外生的)生産性ショック
2(0,1) i.e., ショック:十分小
労働の限界生産物:
yi,t = Ai,t a+ (Si,t )
a0+ (Si,t ) > 0 :都市人口規模の正外部効果
10
実質的労働力: Li,t = a (Si,t ) li,t
| {z }
都市 i の総労働投入/労働者
1 li,t )
(余暇: 2[0,1]
都市費用(都市人口規模の負外部効果)
a0 (Si,t ) < 0
↵
1 ↵
効用関数: u(ci,t , hi,t , li,t ) = ci,t hi,t (1 li,t )
| {z }
財消費
余暇
土地消費
(各都市の土地賦存量=H)
11
立地均衡: σi,tの決定 → 立地の決定(ゼロ移動費用)
u⇤ (Si,t ) = u⇤ (Sj,t ) = U
Ai,t ⇤(Si,t ) = Aj,t ⇤(Sj,t )
都市規模の純効果: ⇤(Si,t ) = a+ (Si,t )a (Si,t )Si,t
Λが負のべき関数の場合 (i.e., 一極集中しない場合):
e.g.,
}
}
12
/↵
✓
a+ (Si,t ) = Si,t
a (Si,t ) = Si,t
}
⇤(Si,t ) =
No-black hole条件: ✓ <
Si,t
/↵
(✓,
> 0)
+ /↵ (i.e., Λは減少関数)
K
=
=
1
⇤ (Ai,t )
⇤
K
=
⇤ 1 (Ai,t
1
=
⇤ 1 (1 +
|
{z
✓
Si,t
⌘1+✏i,t
1 (A
K
1 (1 +
i,t
1
1 (1 +
1) ⇤
i,t )
}
·Si,t
13
※
i,t ))
i,t )
∵べき関数の性質
1
⇤
1
(1 +
NBH条件
i,t )
2 (0, 1)
仮定:十分小
Si,t
Si,t 1 = ✏i,t Si,t
T
X
Si,t Si,t
t=1
Si,t
1
T
X
=
1
⇡
ln Si,t = ln Si,t
1
✏i,t
t=1
Z Si,T
Si,0
dSi
= ln Si,T
Si
ln Si,0
+ ✏i,t
ln Si,T = ln Si,0 + ✏i,0 + · · · + ✏i,t
中心極限定理
SiT:漸近的に対数正規分布
命題3(Eeckhout, 04)
TFPのランダム成長
Λ:減少べき関数
⇒
{
}
都市規模の成長過程:Gibrat法則に従う
i.e., 都市規模分布 :漸近的に対数正規分布
14
※ Census Designated Places (CDP)と
15