Lecture 2D: Rank-Size Rule

Transcription

1 Econ 460 Urban Economics Lecure 2D: Rank-Size Rule Insrucor: Hiroki Waanabe Summer Hiroki Waanabe 1 / 56 1 Rank-Size Rule 2 Eeckhou 3 Now We Know 2012 Hiroki Waanabe 2 / 56 1 Rank-Size Rule US Ciies a a Glance Zipf s Law & Pareo Disribuion Overall Disribuion & Lognormal Disribuion 2 Eeckhou 3 Now We Know 2012 Hiroki Waanabe 3 / 56

2 US Ciies a a Glance 2012 Hiroki Waanabe 4 / 56 US Ciies a a Glance Quesion 1.1 (Ciies and Their Size) Wha is he mos ypical American ciy? Does ha represen American ciies? Some ciies are large and some are small. Wha disribuion do hey follow? A bell shape? 2012 Hiroki Waanabe 5 / 56 US Ciies a a Glance 2012 Hiroki Waanabe 6 / 56

3 US Ciies a a Glance Cerainly no a bell shape. Fac 1.2 (Rank-Size Rule) Large ciies are known o closely follow he Pareo disribuion Hiroki Waanabe 7 / 56 Zipf s Law & Pareo Disribuion Definiion 1.3 (Pareo Disribuion) A random variable s is said o follow he Pareo disribuion if he cumulaive disribuion funcion is F(s) = 1 s L for s s s L 0 for s < s L, where s L (> 0) is he lower suppor (i.e., he size of he smalles ciy in he daa se). Remark The parameer is called a Pareo index or degree exponen Hiroki Waanabe 8 / 56 Zipf s Law & Pareo Disribuion 5 Probabiliy Densiy Funcion a=0.1 a=0.5 a=1.0 a=2.5 a= s (wih lower suppor s L =1) 2012 Hiroki Waanabe 9 / 56

4 Zipf s Law & Pareo Disribuion 1 Cumulaive Densiy Funcion a=0.1 a=0.5 a=1.0 a=2.5 a= s (wih lower suppor s L =1) 2012 Hiroki Waanabe 10 / 56 Zipf s Law & Pareo Disribuion The Pareo disribuion is heavy ailed. So are he ciy-size disribuions Hiroki Waanabe 11 / 56 Zipf s Law & Pareo Disribuion US Census 2000 daa for upper 300 ciies (Census-defined places, no MSA) Hiroki Waanabe 12 / 56

5 Zipf s Law & Pareo Disribuion Census 2000 Pareo Disribuion Frequency Ciy Size x Hiroki Waanabe 13 / 56 Zipf s Law & Pareo Disribuion Census 2000 Pareo Disribuion Frequency Ciy Size (in Log Scale) 2012 Hiroki Waanabe 14 / 56 Zipf s Law & Pareo Disribuion For empirical implemenaion, urban economiss usually ake logarihms. For s s L, F(s) = 1 s L s 1 F(s) = s L s log(1 F(s)) = log(s L ) log(s) Hiroki Waanabe 15 / 56

6 Zipf s Law & Pareo Disribuion 1 F(s) roughly measures he normalized rank of he ciy. rank 1 Empirical CDF is given by 1 oal # of ciies. rank Normalized rank is ( 1 F(s)). oal # of ciies Hence, log(r) = c log(s r ), (1) where r marks he rank of he ciy of size s r and c is a consan (called Zipf s Law) Hiroki Waanabe 16 / 56 Zipf s Law & Pareo Disribuion 1 CDF(s) a=0.1 a=0.5 a=1.0 a=2.5 a= s (wih lower suppor s L =1) 2012 Hiroki Waanabe 17 / 56 Zipf s Law & Pareo Disribuion How s he mach? log(r) = } {{ }} {{ } log(s r ), R 2 =.9877 (!?!!) (107.4) ( 22.54) Numbers in parenheses denoe -values 1. 1 For he null is = Hiroki Waanabe 18 / 56

7 Zipf s Law & Pareo Disribuion 10 2 Census 2000 (Places) OLS Esimaion Size in Log Scale 2012 Hiroki Waanabe 19 / 56 Zipf s Law & Pareo Disribuion Was ha jus US in 2000? Does he rank-size rule hold in 1 ime series? 2 cross-secion? 2012 Hiroki Waanabe 20 / 56 Zipf s Law & Pareo Disribuion MSA-level (US Census) Year inercep () Pareo index () R (162.6).8545 (19.31) (161.1).8492 (19.99) (149.7).8395 (19.97) Hiroki Waanabe 21 / 56

8 Zipf s Law & Pareo Disribuion Rank Size Plo 10 2 US MSA 1990 US MSA 2000 US MSA Census 1990 OLS Esimaion Ciy Size in Log Scale Size in Log Scale Census 2000 OLS Esimaion Census 2010 OLS Esimaion Size in Log Scale Size in Log Scale 2012 Hiroki Waanabe 22 / 56 Figure 1: Zipf s Law & Pareo Disribuion Counry Yr # inercep () Pareo index () R 2 Canada (58.31).8196 (10.33).9861 China (95.80) (-17.85).9758 France (33.26) (-7.182).9576 Germany (200.7).9625 (5.687).9910 Korea (57.65).8697 (6.582).9602 Taiwan (125.1).9744 (2.453) Daa provided couresy of Kwok Tong Soo Hiroki Waanabe 23 / 56 Zipf s Law & Pareo Disribuion Rank Size Plo Rank Size Plo Belgium 84 Belgium 88 Belgium 93 Belgium 00 Canada 02 Canada 07 Canada Ciy Size in Log Scale 10 6 Ciy Size in Log Scale Rank Size Plo Rank Size Plo 10 2 China 90 China Ehiopia 07 Ehiopia Ciy Size in Log Scale Ciy Size in Log Scale 2012 Hiroki Waanabe 24 / 56

9 Zipf s Law & Pareo Disribuion Rank Size Plo Rank Size Plo France 82 France 92 France 99 France 06 France Germany 95 Germany 01 Germany 05 Germany Ciy Size in Log Scale 10 2 Rank Size Plo Ciy Size in Log Scale Taiwan 81 Taiwan 91 Taiwan 01 Taiwan Ciy Size in Log Scale 2012 Hiroki Waanabe 25 / 56 Overall Disribuion & Lognormal Disribuion The daase we had is runcaed. Wha does he overall ciy-size disribuion look like? Lognormal disribuion describes he overall ciy-size disribuion well. If log(s) follows he normal disribuion, hen s follows he lognormal disribuion Hiroki Waanabe 26 / 56 Overall Disribuion & Lognormal Disribuion 1000 Census 2K Lognormal 800 Frequency Ciy Size (in Log Scale) 2012 Hiroki Waanabe 27 / 56

10 1 Rank-Size Rule 2 Eeckhou Landscape Producion Consumer Marke Clearance Gibra s Law Zipf s Law 3 Now We Know 2012 Hiroki Waanabe 28 / 56 Landscape Eeckhou s model ([Eec04]). Ciies are indexed by = 1,,. Ciy size is S R +, where = 0, 1, 2, denoes ime. Naional populaion is S = =1 S. Ciy s produciviy is represened by A. Law of moion (recurrence equaion): A = A 1 (1 + σ ), where σ is IID, E =1 σ = 0 and (1 + σ ) > 0. Is his an AR(1) process? 2012 Hiroki Waanabe 29 / 56 Landscape 0.5 σ 1 σ 2 Technological Shock σ i 0 σ 3 σ 4 σ Time 2012 Hiroki Waanabe 30 / 56

11 Landscape 7 Produciviy Parameer A i A 1 A 2 A 3 A 4 A Time 2012 Hiroki Waanabe 31 / 56 Producion A firm employs one worker, who exhibis CRS echnology. A worker produces A +(S ) unis of consumpion goods. + (S ) measures he posiive exernaliy, wih + (S ) > 0. A ypical firm in ciy solves max 0 L 1 π (L ) = 1 A +(S )L L, where L is effecive labor. The labor marke is compeiive and profis are compeed away in equilibrium: A +(S ) =. (2) 2012 Hiroki Waanabe 32 / 56 Producion Quesion 2.1 (Prediced Ciy-Size Disribuion) So far, we have a CRS echnology and one cenripeal force. Wha kind of a disribuion do we expec o see in equilibrium? Does ha explain he acual ciy-size disribuion? 2012 Hiroki Waanabe 33 / 56

12 Producion S spli concenrae NY LA NY LA S/2 S/2 S 0 A 1 +(S/2) A 2 +(S/2) A 1 +(S) Hiroki Waanabe 34 / 56 Producion Each worker/residen is endowed wih one uni of ime. Liz splis her ime in 1 hours of labor 2 1 hours of leisure Commuing in NY is coslier han commuing in Monana. Effecive labor is defined by L := (S ), where 0 < ( ) 1 and ( ) < Hiroki Waanabe 35 / 56 Consumer Wage is paid for he effecive labor L raher han he labor iself. i.e., he firm does no pay for he ime los in commuing. Zero-profi condiion (2) implies A +(S ) =. hours of labor earns: L = A +(S )L = A +(S ) (S ) Hence Liz s income is A +(S ) (S ) Hiroki Waanabe 36 / 56

13 Consumer The amoun of residenial land in ciy is H. An individual consumes h unis of land a he price of p Hiroki Waanabe 37 / 56 Consumer Quesion 2.2 (Uiliy Maximizaion Problem) Liz solves max c,h, (c, h, ) = c subjec o α h c + p h (S ) β 1 γ for each, where α, β, γ 0 and α + β + γ = 1. The budge consrain can also be wrien as c + p h + (1 ) (S ) 1 (S ) Hiroki Waanabe 38 / 56 Consumer Which one should be endogenous? Ciy size or uiliy level? A he beginning of he period Liz observes all σ, Assume cosless relocaion. S is endogenous. hen chooses he ciy o maximize he uiliy level. Everyone achieves he same uiliy in equilibrium. If no, wha would happen? 2012 Hiroki Waanabe 39 / 56

14 Consumer How do we solve Quesion 2.2? Fac 2.3 (Workaround for Cobb-Douglas UMP) max 1, 2, 3 ( 1, 2, 3 ) = α 1 β 2 γ 3 subjec o m = p p p 3 3, (α, β, γ 0, α + β + γ = 1) α, β, γ measure he expendiure share of 1, 2, 3, i.e., p 1 1 = αm, p 2 2 = βm, p 3 3 = γm Hiroki Waanabe 40 / 56 Consumer Then he soluion o Quesion 2.2 is: c = α (S ) h = β (S ) p 1 = γ (S ) (S ) (3) 2012 Hiroki Waanabe 41 / 56 Marke Clearance Household H/S i c Landlord Firm 2012 Hiroki Waanabe 42 / 56

15 h = H S Marke Clearance Marke clearance: Commodiy A firm produces c and sells for $1 for each. Already incorporaed in (3). Labor A firm employs L and pays = A +(S ) per hour. CRS implies = A +(S ). Subsiue his wih (3). Land A landlord supplies H o mee he S demand h and charges p per uni. Subsiue his wih (3) Hiroki Waanabe 43 / 56 Marke Clearance The marke clearing condiions imply: c = αa +(S ) (S ) = α + β (4) p = β S H A +(S ) (S ). = A +(S ) Hiroki Waanabe 44 / 56 Marke Clearance Comparaive saics on = A +(S ): high A implies high. Does his mean ha all he people move o he ciy wih he bes echnology? No. Noe ha high A also implies high p. High ren evens ou he high wage Hiroki Waanabe 45 / 56

16 Marke Clearance Wage level and ren will be differen among he ciies in equilibrium. How do we find he equilibrium size disribuion S? In equilibrium, Liz and Kenneh should achieve he same uiliy level U, regardless of heir ciy of residence. Plug (4) back in he uiliy funcion o obain (c, h, ) = β αa +(S ) (S ) α H γ γ. (This is called an indirec uiliy funcion). S 2012 Hiroki Waanabe 46 / 56 Gibra s Law Since (c, h, ) = (cj, hj, j ) = U for all ciies and j, αa +(S ) (S ) α β H S γ γ = αa j +(S j ) (S j α β ) H γ γ A +(S ) (S ) S β α S j = A j +(S j ) (S j ) S j β α A (S )Θ = A j (Sj )Θ, where (S )Θ := + (S ) (S ) S β α Hiroki Waanabe 47 / 56 Gibra s Law Proposiion 2.4 (Ciy Size and Technology) A ciy wih beer echnological shock σ will have a larger size S if posiive exernaliy is no oo large. Proof. See Eeckhou [Eec04] Hiroki Waanabe 48 / 56

17 Gibra s Law How does S evolve over ime? S = K A 1/Θ 1/Θ = K A 1 (1 + σ ) = S 1 (1 + σ ) 1/Θ =: S 1 (1 + ε ), (5) where (1 + ε ) := (1 + σ ) 1/Θ Hiroki Waanabe 49 / 56 Gibra s Law (5) leads o Gibra s law: S S 1 = ε. S Hiroki Waanabe 50 / 56 Zipf s Law (5) also leads o Zipf s law: log S = log S 1 + log(1 + ε 1 ) = log S 2 + log(1 + ε 2 ) + log(1 + ε 1 ). = log S 0 + τ=0 (1 + ε τ ) log S 0 + τ=0 ε τ. By he cenral limi heorem, he sum of ε aysmpoically follows he normal disribuion. Then S follows he lognormal disribuion Hiroki Waanabe 51 / 56

18 1 Rank-Size Rule 2 Eeckhou 3 Now We Know 2012 Hiroki Waanabe 52 / 56 Rank-size rule Fa-ail disribuion Eeckhou s model 2012 Hiroki Waanabe 53 / 56 References [Eec04] Jan Eeckhou. Gibra s law for (all) ciies. American Economic Review, Hiroki Waanabe 54 / 56

19 Map du Jour No, I didn forge hem Hiroki Waanabe 55 / 56 Index A, see produciviy +, see posiive exernaliy AR(1), 29 cenral limi heorem, 51 ciy size, 29 Cobb-Douglas uiliy funcion, 40 degree exponen, 8 Eeckhou, 29 effecive labor, 32, 35 equilibrium uiliy level, 46 Gibra s law, 50 heavy ail, 11 L, see effecive labor, see labor labor, 35 law of moion, 29 lognormal disribuion, 26, 51 naional populaion, 29 Pareo disribuion, 8 Pareo index, 8 posiive exernaliy, 32, 48 produciviy, 29 S, see naional populaion S, see ciy size U, see equilibrium uiliy level uiliy maximizaion problem, 38, see wage wage, 36 Zipf s law, 16, Hiroki Waanabe 56 / 56