Revisiting urban economics in light of data

Transcription

1 Revisitig urba ecoomics i light of data Marc Barthelemy CEA, Istitut de Physique Théorique, Saclay, Frace EHESS, Cetre d Aalyse et de Mathématiques sociales, Paris, Frace marc.barthelemy@cea.fr

2 Outlie Modellig mobility q q Mobility: gravity law The radiatio model Relatio betwee commutig distace ad icome q q q Empirical results Testig the McCall model of job search The closest opportuity model Mobility: statistical properties

3 Motivatios: uderstadig mobility Structure of cities: spatial distributio of resideces ad activity ceters Useful for modellig may practical applicatios: q q q Urba plaig (trasport plaig) epidemic spread

4 Geeral models: gravity ad radiatio

5 The gravity model Area i Distace r Area j P i P j Number of trips betwee i ad j? T ij = K P ip j r Gravity law (Reilly 1929, Zipf 1946)

6 The gravity model Problems with the gravity model - Cogestio? - Theoretical derivatio? A derivatio proposal (Wilso, 1967): umber of ways to costruct a cofiguratio {T ij } = Q T! ij T ij Maximize with X costraits: i T ij = T j, X T ij = T, ij X T ij = T i j X T ij C ij = C ij

7 The gravity model We the obtai T ij / T i T j e C ij But the cost eeds to be give C ij / ( d ij ) T ij e d(i,j) log d(i, j) ) T ij 1/d(i, j)

8 The radiatio model (Simii et al, 2012) Alterative to the Gravity model Distace r Home How to choose a job Office

9 The radiatio model (Simii et al, 2012) Area i Distace r Area j P i P j Each idividual has his threshold Ad looks for the closest job z j such that z j >z i z i A choice z i = max{x 1,X 2,...,X Pi } z j = max{x 1,X 2,...,X Pj } with X p(x) (ad cumulative F)

10 The radiatio model (Simii et al, 2012) The probability of beig emitted at i ad absorbed at j is: Z P (i! j) = dzp Pi (z)p sij (<z)p Pj (>z) where P Pi (z) =P i F (z) P i P sij (<z)=f (z) s ij P Pj (>z)=1 F (z) P j 1 df dz s ij i+ j

11 The radiatio model (Simii et al, 2012) We the have: Z P (i! j) =P i dzf(z) P i a chage of variables the gives (u=f) P (i! j) = P i P j (P i + s ij )(P i + P j + s ij ) 1 df dz F (z)s ij (1 F (z) P j ) s ij i+ j Note: lik with the rak = # idividuals betwee i ad j (ie. such that d(i,w)<d(i,j))

12 Results Compariso with empirical data (US data):

13 Results Compariso with empirical data (US data):

14 Mobility ad socio-ecoomics: Commutig distace ad icome What is the relatio betwee icome ad commutig distace? with G. Carra, I. Mulalic, M. Fosgerau

15 Icome ad commutig distace Distace r Home Icome Y Office Fidig/choosig a job Questios: q Average commutig distace r versus Y q Distributio P(r Y)?

16 Icome ad commutig distace: UK data r Y 0.5

17 Icome ad commutig distace: DK data r Y 0.8

18 Icome ad commutig distace: US data r Y 0.0

19 Distributio of commutig distace (UK) Distributio P (r Y ) r 2 [2.65, 3.0]

20 Distributio of commutig distace (US) Distributio P (r Y ) r 2 [2.8, 3.3]

21 Summary: empirical results Average commutig distace r Y where depeds o the coutry Distributio: broad law where 3 P (r Y ) r

22 Classical model: McCall (1971) Optimal strategy (stoppig problem) q Offers draw from cumulative distributio F(x) q Waitig time cost c q Goal: maximize expected value v(w) for a give offer w i had v(w) =h 1X t=0 t y(t) o er = wi where : ( ( w if accepts offer y(t) = w if acceptsoffer y(t) = c if refuses offer c if refuses

23 Classical model: McCall (1971) Bellma equatio q v(w) = max 1 First term: accepts offer w Z w, c + v(w 0 )df (w 0 ) q Secod term: refuses; average over all possible offers w Solutio of the Bellma equatio: optimal strategy with a reservatio wage - If w< cotiue search - If w> accept offer Probability to accept a offer: p = Z 1 df (w 0 )=1 F ( )

24 Addig space to the McCall model We assume that the offers are distributed uiformly i 2d space with desity. Idividuals are startig at r=0 (home). Each time we ecouter a offer, there is a probability p to accept it. The probability to accept the N th offer is: P (N) =(1 p) N 1 p Probability to be at distace r with N poits (uiform distributio): P (R = r N) = 2 (N 1)! 1 r ( r2 ) N e r 2 R

25 Addig space to the McCall model P(R=r N) ad P(N) combied together lead to: P (r Y )=2 rpe p r2 -> Expoetially decreasig fuctio! Cetered at with fiite width r 1/ p p 1/ p p r e r2 The McCall model is ot i agreemet with data! Optimal strategy assumptio??

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29 The closest opportuity model Radom reservatio wage distributed accordig to the same distributio (cumulative F) as offers Start from home, explore space with icreasig r, accept the first job search such that the offer>reservatio wage Desity of job offers depeds o the icome: higher skills jobs less dese tha lower skills jobs (skills <=> icome) where = 0 Y characterizes the job market i the coutry with: G. Carra, I. Mulalic, M. Fosgerau

30 The closest opportuity model Radom reservatio wage distributed accordig to We the have Z P (r)dr = ( )P (x < ) r2 1 P (x < ) 2 r dr d Probability to have value Probability that there are o iterestig offer i the disk <r Probability that there is at least oe iterestig offer i the rig [r,r+dr]

31 The closest opportuity model The expressio Z P (r)dr = ( )P (x < ) r2 1 P (x < ) 2 r dr d ca be rewritte as P (R = r) = 2 r Z ( )F ( ) r2 log F ( )d which is idepedet from F ad is equal to P (r) = 2 r (1 + r 2 ) 2

32 The closest opportuity model Cosequeces of Average distace: Distributio decays as Data collapse 2 r P (r) = (1 + r 2 ) 2 r(y )= 1 r Y /2 2 0 P (r) 1/r 3 r! r/y /2 P (u) = 2 0 u (1 + 0 u 2 ) 2

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37 Discussio Optimal strategy does ot seem to be realistic for the job search problem Empirical results: broad tail for the commutig distace distributio A simple stochastic model q q Predicts the uiversal broad tail Shows the importace of the relatio betwee the desity of jobs ad skills (ot sigificative for the US, strog for the UK ad very strog for DK).

38 Mobility ad statistical physics: a multilayer view

39 Huma mobility: Levy flight? May small jumps ad some rare log jumps P ( r) 1 r

40 Huma mobility: Levy flight? Empirical studies o the displacemet distributio P ( r) 1 ( r + r 0 ) e r/apple (Trucated) Levy flight? Very empirical: model, mechaism? (Brockma et al, Nature 2006; Gozalez et al, Nature 2008)

41 Huma mobility: empirical results Empirical results o the trip duratio (GPS, 800,000 private cars i Italy) P (t/ t ) Roma Milao Napoli Torio Palermo Geova P (t) e t/hti t/ t Gallotti, Bazzai, Rambaldi, MB, Nature Comms 2016

42 Huma mobility: the further, the faster Average velocity with the trip duratio hvi 'v 0 + at Cars : v 0 ' 17.9km/h a ' 16.7km/h 2 Public trasport : v 0 ' 7.4km/h a ' 12km/h 2 Gallotti, Bazzai, Rambaldi, MB, Nature Comms 2016

43 Huma mobility: a simple model Average trip: two phases (acceleratio ad decceleratio). Gallotti, Bazzai, Rambaldi, MB, Nature Comms 2016

44 Huma mobility: a simple model Acceleratio phase: Radom acceleratio kicks v = v 0 + k(t) v v m = v 0 + k(t m ) v with : P (k) =e k /k! = pt/2 Gallotti, Bazzai, Rambaldi, MB, Nature Comms 2016

45 Huma mobility: a simple model This model predicts: P (v t) e pt/2+ v v 0 v 0 log(pt/2) (1 + v v 0 v 0 ) Determie p ad δv by fittig all P(v t) for all duratios v (from t=5ms to 180ms) p=2 jumps/hour ad δv=40km/h (cosistet with ) Gallotti, Bazzai, Rambaldi, MB, Nature Comms 2016

46 Huma mobility: recap Recap (all parameters determied): P (t) e t/t,p(v t) P ( r) r = vt Z P ( r) = P (v t)p (t) ( r vt)dvdt Which gives for : P ( r) 1 r e C r with δ=1/2 (γ depeds o the parameters) Gallotti et al, Nature Comms 2016

47 Huma mobility: predictios Result: 10 0 Private Cars data Trucated Power Law Predictio 10 2 P ( r ) This is ot a fit! This is ot a Levy walk! r (km) Gallotti et al, Nature Comms 2016

48 Summary ad Perspectives The multilayer view allows: - to describe ad uderstad importat features due to the couplig of layers - to characterize them ad their efficiecy (ew tools eeded). Helpful for comparig systems, ad testig ad fidig specific optimizatio strategies. Simple models allow to uderstad the essetial mechaisms of mobility