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