The structure of cities

The structure of cities G Université Paris-Dauphine et University of British Columbia Paris, July 6, 2009 (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 1 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen But there are also negative externalities, which create centrifugal forces: The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: The two processes will yield di erent structures (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen workers live close to ther jobs But there are also negative externalities, which create centrifugal forces: The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: The two processes will yield di erent structures (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen workers live close to ther jobs convenience stores and entertainment industries But there are also negative externalities, which create centrifugal forces: The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: The two processes will yield di erent structures (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen workers live close to ther jobs convenience stores and entertainment industries But there are also negative externalities, which create centrifugal forces: congestion The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: The two processes will yield di erent structures (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen workers live close to ther jobs convenience stores and entertainment industries But there are also negative externalities, which create centrifugal forces: congestion polluting industries The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: The two processes will yield di erent structures (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen workers live close to ther jobs convenience stores and entertainment industries But there are also negative externalities, which create centrifugal forces: congestion polluting industries The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: throught the market The two processes will yield di erent structures G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The economic determinants of city structure Why do we have cities in the rst place? Because of positive externalities, which create centripetal forces: craftsmen learn from other craftsmen workers live close to ther jobs convenience stores and entertainment industries But there are also negative externalities, which create centrifugal forces: congestion polluting industries The city comes to existence because the former are stronger than the latter. But then these forces determine the structure of the city. There are two possible ways to allot land: throught the market by a benevolent planner The two processes will yield di erent structures G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 2 / 20

The market (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 3 / 20

The market for land the shape of the city is prescribed as some Ω R d Some additional features (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

The market for land the shape of the city is prescribed as some Ω R d workers move freely into the city and choose the place where they work and the place where they live Some additional features (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

The market for land the shape of the city is prescribed as some Ω R d workers move freely into the city and choose the place where they work and the place where they live rms move freely into the city and choose the place where they localize Some additional features (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

The market for land the shape of the city is prescribed as some Ω R d workers move freely into the city and choose the place where they work and the place where they live rms move freely into the city and choose the place where they localize there are two competing uses for land: residence and business Some additional features (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

The market for land the shape of the city is prescribed as some Ω R d workers move freely into the city and choose the place where they work and the place where they live rms move freely into the city and choose the place where they localize there are two competing uses for land: residence and business the land belongs to absentee landlords, who rent it to the highest bidder Some additional features (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

The market for land the shape of the city is prescribed as some Ω R d workers move freely into the city and choose the place where they work and the place where they live rms move freely into the city and choose the place where they localize there are two competing uses for land: residence and business the land belongs to absentee landlords, who rent it to the highest bidder Some additional features there is a single good in the economy, which all rms produce and all residents consume (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

The market for land the shape of the city is prescribed as some Ω R d workers move freely into the city and choose the place where they work and the place where they live rms move freely into the city and choose the place where they localize there are two competing uses for land: residence and business the land belongs to absentee landlords, who rent it to the highest bidder Some additional features there is a single good in the economy, which all rms produce and all residents consume rents and surplus production leave the city G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 4 / 20

Equilibrium conditions: residents All inhabitants are identical. Their utility is U(c, S), where c is consumption and S the area they rent. U satis es the usual conditions. All people who live at x pay the same rent Q (x) and have the same take-home pay ϕ (x) (salary net of transportation costs), so they all solve the same problem: max fu (c, S) j c + Q (x) S ϕg c,s The optimal value ū must be the same for all x: ū := max fu (c, S) j c + Q (x) S ϕ (x)g c,s ū is given exogeneously (income earned outside the city). This determines N (ϕ), relative density of residents, and Q (ϕ) G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 5 / 20

Equilibrium conditions: rms All rms are identical and have constant returns to scale.. The production per unit of land at y is f (z, n (y)), where n (y) is the relative density of jobs. The productivity z = z (y) is given by: Z z(y) = g( ρ(y 0, y)ν(y 0 )dy 0 ) where ρ 0, g is increasing and ν (y) is the absolute density of jobs. All rms located at y pay the same rent q (y) and the same salary ψ (y), and make the same pro t f (z, n) ψn q (y), which is zero because of perfect competition: 0 = max ff (z, n) ψ (y) n q (y)g n0 This gives n (ϕ), relative density of jobs G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 6 / 20

Transportation costs Workers who live at x and work at y, where they pick up a salary ψ (y), have a revenue ϕ (x) = ψ (y) c (x, y) with c 0 and c (x, x) = 0. It may be convex or concave - the rst case is typical of the structure of cities, the second of interregional trade. A transport plan is a map x! P x, where P x is a probability on Ω. We understand P x (y) as the proportion of the residents of x who work at y. This becomes a transport map τ : Ω! Ω if P x = δ τ(x ) is a Dirac mass, ie if all those who live at x work at the same place τ (x). This is the case when the cost is convex. G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 7 / 20

Equilibrum: de nition We start from two densities µ (x) (residents) et ν (y) (jobs) and two functions ϕ (x) (take-home pay at x) et ψ (y) (salary at y) such that: Z µ = Z ν ϕ (x) = max fψ (y) c (x, y)g y ψ (y) = min fϕ (x) + c (x, y)g x (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 8 / 20

Equilibrum: de nition We start from two densities µ (x) (residents) et ν (y) (jobs) and two functions ϕ (x) (take-home pay at x) et ψ (y) (salary at y) such that: Z µ = Z ν ϕ (x) = max fψ (y) c (x, y)g y ψ (y) = min fϕ (x) + c (x, y)g x The associated transportation plan must map µ on ν and conversely (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 8 / 20

Equilibrum: de nition We start from two densities µ (x) (residents) et ν (y) (jobs) and two functions ϕ (x) (take-home pay at x) et ψ (y) (salary at y) such that: Z µ = Z ν ϕ (x) = max fψ (y) c (x, y)g y ψ (y) = min fϕ (x) + c (x, y)g x The associated transportation plan must map µ on ν and conversely We deduce the productivity z (y), and the relative densities N = N (ϕ (x)) and ν = ν (z (y), ψ (y)) as well as the rents Q = Q (ϕ (x)) and q = q (z (y), ψ (y)) for residences and rms (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 8 / 20

Equilibrium: de nition In view of q (y) and Q (x), the landlords allot the land. If θ (x) is the proportion of land for industrial use, we must have: 8 < 0 si q(z, ψ) Q(ϕ) < 0 θ = 0 θ (x) 1 si q(z, ψ) Q(ϕ) = 0 : 1 si q(z, ψ) Q(ϕ) > 0 G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 9 / 20

Equilibrium: de nition In view of q (y) and Q (x), the landlords allot the land. If θ (x) is the proportion of land for industrial use, we must have: 8 < 0 si q(z, ψ) Q(ϕ) < 0 θ = 0 θ (x) 1 si q(z, ψ) Q(ϕ) = 0 : 1 si q(z, ψ) Q(ϕ) > 0 We deduce new distributions µ and ν by: µ (x) = (1 θ(z, ψ, ϕ))n (ϕ (x)) ν(y) = θ(z, ψ, ϕ)n(z, ψ) and there is a unique pair ( ϕ, ψ) such that R µ = R ν G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 9 / 20

Equilibrium: de nition In view of q (y) and Q (x), the landlords allot the land. If θ (x) is the proportion of land for industrial use, we must have: 8 < 0 si q(z, ψ) Q(ϕ) < 0 θ = 0 θ (x) 1 si q(z, ψ) Q(ϕ) = 0 : 1 si q(z, ψ) Q(ϕ) > 0 We deduce new distributions µ and ν by: µ (x) = (1 θ(z, ψ, ϕ))n (ϕ (x)) ν(y) = θ(z, ψ, ϕ)n(z, ψ) and there is a unique pair ( ϕ, ψ) such that R µ = R ν We have an equilibrium if (µ, ν, ϕ, ψ) = ( µ, ν, ϕ, ψ) G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 9 / 20

Existence Theorem There is always an equilibrium It can be checked that people move: there is no equilibrium with all residents living where they work. R. E. Lucas, Jr. and E. Rossi-Hansberg, On the Internal Structure of Cities, Econometrica, vol. 70 (2002), pp. 1445-1476. (Treats the case of a circular city with radial transport and iceberg costs) G. Carlier and I. Ekeland, "Equilibrium structure of an bidimensional assymmetric city". Nonlinear Analysis TWA, vol 8 (2007): 725-748. (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 10 / 20

The planner (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 11 / 20

The benevolent president. The president has a large empty space at his disposal, and assigns locations to rms and workers in order to maximize the social optimum The number of workers a given rm employs is exogeneously prescribed (say 1) w (x) density of residence at x f (x) density of jobs at x Z Z w (x) dx = f (x) dx = 1 R 2 R 2 (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 12 / 20

The benevolent president. The president has a large empty space at his disposal, and assigns locations to rms and workers in order to maximize the social optimum The number of workers a given rm employs is exogeneously prescribed (say 1) Number of rms and workers prescribed exogeneously w (x) density of residence at x f (x) density of jobs at x Z Z w (x) dx = f (x) dx = 1 R 2 R 2 (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 12 / 20

The benevolent president. The president has a large empty space at his disposal, and assigns locations to rms and workers in order to maximize the social optimum The number of workers a given rm employs is exogeneously prescribed (say 1) Number of rms and workers prescribed exogeneously Shape of the city is not prescribed (one- or two-dimensional) w (x) density of residence at x f (x) density of jobs at x Z Z w (x) dx = f (x) dx = 1 R 2 R 2 G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 12 / 20

Externalities There are several externalities, some positive, some negative: Congestion: 1 2 γ w w 2 (x) and 1 2 γ f f 2 (x) Transport: c (x, y) R R k Firm- rm (production) C 1 2 kx Residence-residence (utility) R R 2 kx yk 2 w (x) w (y) dxdy C 2 θ 2 yk 2 f (x) f (y) dxdy Firme-residence (utility) : C 3 ρ R R 2 kx yk 2 w (x) f (y) dxdy We can have ρ > 0 (convenience) or ρ < 0 (pollution). G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 13 / 20

The planner s problem It consists of minimizing J (f, w) := γ R w 2 w 2 dx + γ R R R f 2 f 2 dy + W c (f, w) + k 2 2 kx yk 2 f (x) f ( R R 2 kx yk 2 w (x) w (y) dxdy + ρ R R 2 kx yk 2 w (x) f (y) dxd + θ 2 over all probability densities (w, f ) and all maps T which transport w to f. Here W c is the Wasserstein distance. Existence obtains provided centifupetal forces dominate centrifugal ones γ f > 0, γ w > 0, k > 0, θ > 0 k + ρ > 0, θ + ρ > 0, ρ > 1 2 (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 14 / 20

The quadratic case In the case of quadratic cost, there is an explicit solution: T (x) = λx w (x) = 1 γ w " f (y) = 1 γ f C w θ + ρ + 1 λ) 2 C f k + ρ + 1 λ 1 ) 2 x 2 +! y 2 # + (1) (2) (3) So the optimal city is circular, the optimal transport map is radially linear, and the distributions w and f are concentric and bell-shaped. Firms are more concentrated than residences if ϱ + k > 0. The size of the city increases with γ and decreases with θ, ρ + k, and c In the case of superquadratic costs, c (x, y) = 1 p kx ykp the symmetry will break if congestion costs are relatively small. (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 15 / 20

The two-sector city C 1 : = γ Z w (w 1 + w 2 ) 2 + γ Z f (f 1 + f 2 ) 2 (congestion) 2 2 C 2 : = tw c (f 1, w 1 ) + tw c (f 2, w 2 ) (transportation) 2 Z Z k i C 3 : = jx yj 2 f i (x)f i (y)dxdy (worker/worker externalities) i=1 2 C 4 : = θ Z Z jx yj 2 (w 1 (x) + w 2 (x))(w 1 (y) + w 2 (y))dxdy ( rms/ rms 4 C 5 : = 2 ρ i i=1 2 Z Z jx yj 2 f i (x)(w 1 (y) + w 2 (y))dxdy ( rms/workers) G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 16 / 20

Frame Title Existence obtains under similar conditions (centripetal parameters should dominate centrifugal ones), and we have a structural result: Theorem Assume that k 1 + ρ 1 6= k 2 + ρ 2. Then there is segregation: f 1 f 2 = w 1 w 2 = 0 (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 17 / 20

How the planner harnesses market forces (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 18 / 20

Implementation To implement the social optimum, the president can impose two land taxes: B (y) for business use and R (x) for land use. Firms are perfectly competitive, and choose their location so as to maximize pro t, ie production minus salary minus business tax. Workers and competitive and choose choose their residence so as to maximize their take-home pay, net of transportation cost and residential tax. We nd (in the quadratic, one-sector case, γ f = 0) B (y) = C 4 + ρ + k y 2 2 ρ + k R (x) = C 5 t t + ρ + k + θ + ρ x 2 2 s (x) = C 6 (ρ + k) x 2 where s (x) = C 6 (ρ + k) x 2 is the salary at x. Note the counter-intuitive fact that B (x) is an increasing function of the distance to the center G (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 19 / 20

Guillaume Carlier, Ivar Ekeland and Jean-Charles Rochet, "Tax and the city", in preparation (Université Paris-Dauphine The structure et University of cities of British Columbia) Paris, July 6, 2009 20 / 20