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Optimal transportation and the structure of

cities

G. Carlier

, I. Ekeland

March 17, 2003

Abstract

Following the recent analysis of Lucas and Rossi-Hansberg [10], we study the equilibrium structure of cities. By adopting a different (monetary) specification of commuting costs, we are able to prove the existence of equilibrium in more general situations: non-circular cities, or multi-sectorial production. The main mathematical tool in this paper is the use of optimal transportation theory. This theory offers a new framework, both powerful and robust for problems in geographical economics.

Keywords: equilibrium city, optimal transportation, production exter-nalities, zoning.

We are particularly grateful to Robert E. Lucas Jr. and Esteban Rossi-Hansberg for lengthy and helpful discussions. We gratefully acknowledge the hospitality of the Department of Economics at the University of Chicago and the support of Pierre-Andr´e Chiappori.

Universit´e Bordeaux I, MAB, UMR CNRS 5466 and Universit´e Bordeaux IV, GRAPE,

UMR CNRS 5113, carlier@math.u-bordeaux.fr.

Universit´e Paris Dauphine, Institut de Finance and CEREMADE, UMR CNRS 7534,

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1

Introduction

Most models in economic geography take the underlying structure as given, and consider factors of production which are free to move between different locations which are given ex ante (see [6] for a recent survey). In a pathbreak-ing paper [10], Lucas and Rossi-Hansberg gave the first model explainpathbreak-ing the internal structure of a city endogeneously, as an equilibrium solution between different possible uses of land. In their model, there are two possible uses for land, residential and business. The driving force for concentration, and indeed for the existence of cities in the first place, is a positive externality of labor: the more workers are concentrated at a given place, the more pro-ductive each of them becomes. The driving force for dispersion, on the other hand, is the desire of every individual to spread his living quarters, occupying as much residential land as possible.

Both these forces are constrained. As concentration increases, and more and more workers aggregate in smaller and smaller business districts, the rental price in these districts increases so much that it becomes worthwhile for firms to look for cheaper locations. The dispersion effect, on the other hand, is constrained by the size of the city, since its boundaries, in the Lucas and Rossi-Hansberg model, are given a priori. So firms will enter in competition with individuals for land use. It will be up to the landowners to apportion the land between business use and residential use, taking into account, of course, the rents they can obtain from either side.

What about the internal structure of the city? We know how firms want to locate: they want to be together, in order to benefit from the positive externality of labour. We also know where individuals want to dwell: far away from each other, in order to increase their living quarters, and close to their working place, in order to avoid losing time and money in transportation.

Lucas and Rossi-Hansberg were the first ones to work out an equilibrium model for the internal structure of cities. In their model, the transportation costs take the ”iceberg” form which is classical in the economic literature since the pionneering work of von Th¨unen [14] (see chapter 4), and Samuel-son [11]. All individuals carry one unit of labor, and some of it is lost in commuting to work. The result is that, although all workers are identical, they do not all supply the same quantity of labor to the market: those who travel farthest contribute the least. As a consequence, the total supply of labor will depend on the locations of firms, the location of houses, and the actual paths people follow to go to work. In other words, the total supply of labor is not known a priori, but must be worked out with all the other features of equilibrium. This introduces great mathematical complications,

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which Lucas and Rossi-Hansberg have been able to overcome in the case when the shape of the city is a circle, and the structure is assumed to be rotationally symmetric. They have been able to show the existence of an equilibrium for this situation. In subsequent work [13], Rossi-Hansberg has solved the planner’s problem for the same case.

Although there is nothing in the economic argument to prevent their solution to be carried over to non-circular cities, and perhaps to other situ-ations (several goods, externalities of dwelling), the mathematics simply do not seem to be there. The Lucas and Rossi-Hansberg solution makes heavy use of the symmetry assumption, which essentially reduces the situation to one dimension, and despite much work we have found no way to extend it to more general situations.

We therefore propose a new model, which differs from the preceding one in one crucial respect: the form taken by the transportation costs. In our model, they will be monetary. Living at x and working at y will cost an amount c (x, y), which is taken of the wage the individual is earning at y. Thus, all individuals contribute one unit (a ”job”) to the labour market, and the total supply of labour therefore is no longer a feature of the equilibrium: it is just the total number of workers in the city. So the number of people living in the city is precisely equal to the number of jobs on the labour market: this is what mathematicians call a conservation law.

Our model has a desirable economic feature: by changing the function c (x, y), one can vary the cost of transportation relative to labour, which is not possible in the standard ”iceberg” models. In addition, because of the conservation law, it acquires totally new mathematical features, enabling us to use the theory of optimal transportation. As a consequence, we are able to find an equilibrium structure for cities of any shape, and to extend the result to the case where several goods are produced. In addition, our solution has an immediate economic interpretation: it gives us at one stroke, not only the location of firms and houses, but also tells us where people who live at x are going to work, and where people who work at y come from. Indeed, it is another desirable feature of our model that all individuals living at x work at the same place, and that all individuals working at y come from the same place.

Our model is quite robust. The assumptions on the utility functions of consumers and production functions of firms are standard, except for the production externality, which is modelled as in [10]. The cost function is of the form c(x, y) = C (x − y), with C (x) ≥ C (0) = 0, and C convex. Landowners just rent out the land to the highest bidder; in case of equal bids, they are indifferent whom they rent to.

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struc-ture of the city, showing business districts (no housing), residential districts (no firms), and mixed districts, with positive densities both of dwellings and firms. Uniqueness is an open question.

The structure of the paper is as follows. Our main result, Theorem 3, is stated and proved in section 7. The proof is by a limiting argument from a mathematically simpler situation, where landowners apportion land in proportion of the rents offered for residence or production, but never wholly exclude either. In other words, landlords are prevented (by city regulations, say) from renting all their land either to business or to housing. Every neighbourhood must carry some firms and some dwellings: all districts must be mixed, there can be no purely residential or purely business district. This may not be very realistic from the economic point of view, but it enables us to evade the mathematical complexities caused by zero densities.

This ”zoning” model is set out in section 2. In section 3, we describe rational behavior of workers, firms and landowners. In section 4, we define equilibria. In section 5, we express the equilibrium problem as a fixed point problem; the main argument in this section relies on the duality theory for mass transportation problems. In section 6, we prove the existence of a fixed point, and hence of an equilibrium for the zoning model (Theorem 1).

In section 7, we do without the zoning regulations, and we finally prove the existence of an equilibrium for our main model, where landowners simply rent land to the highest bidder. In section 8, we extend this result to the case where there are several sectors of production, and hence different types of firms (Theorem 4). In section 9, we indicate directions for future research.

2

The two models

The city consists of inhabitants, which work and consume, of firms and of land owners all of whom have to fit inside Ω, a given open bounded and connected subset of R2. The city produces a single good which is both consumed and used as num´eraire. Whatever fraction of the production is not consumed within the city is sold abroad.

All the inhabitants are identical. They are equally productive and have the same utility function (c, S) 7→ U (c, S), where c is their consumption of the produced good and S is the amount of residential land they rent. We assume that U is strictly concave, strictly increasing in both arguments and continuously differentiable. All workers are endowed with the same quantity of labor that they supply inelastically to the productive activity.

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There is a commuting cost c (x, y). Inhabitants who work at y and live at x get a wage ψ and bring home ψ − c (x, y) .

All firms in the model have the same technology, with constant returns to scale. The production function per unit of land at a given location is (z, n) 7→ f (z, n), where z is the productivity at this location and n is number of workers per unit of business land. It is assumed that f is continuously differentiable, strictly increasing with respect to both arguments and strictly concave with respect to n.

Given the overall density of jobs (number of jobs per unit of land, irre-spective of its use) ν (x) at each location x of the city, the productivity z at x reflects employment at neighboring locations through the formula:

z(x) = Zν(x) := χ(

Z

ρ(x, y)ν(y)dy), x ∈ Ω (1) where the weighting kernel ρ(., .) is nonnegative and continuous on Ω × Ω (hence uniformly continuous, Ω × Ω being compact) and χ is a continuous increasing bounded function χ : R+ → [z, z] with +∞ > z > z > 0. The fact

that χ is increasing reflects the positive externality of labour. The bounds on χ mean that productivity cannot tend to infinity even if the density of em-ployment is very high (people get in each other’s way), and that productivity is positive even if there is no employment around.

In the terminology of Lucas and Rossi-Hansberg, city land is owned by absentee landlords: agents who play no role in the production or consumption processes. It is assumed that they extract all surplus from workers and firms. In the general case, landlords let out their land to the highest bidder. This case will be treated in section 7. In the zoning case, they apportion land between business use and residential use, taking into accounts the rent offered, but without shutting out the lowest bidder.

3

Rational behaviour

We are looking for an equilibrium, that is, a situation where there is no incentive for inhabitants to change the place they live nor the place they work, and no incentive for firms to move. This definition will be made more precise later on. But already it has two important consequences: at equilibrium, all firms should make the same profit, and all inhabitants should have the same utility level, so that firms have no incentive to move their location and inhabitants no incentive to change their dwelling place. So, at equilibrium, all firms make profit 0, the surplus being appropriated by the landlord, and all consumers have utility level u, exogeneously given (for instance, it could

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be the reservation utility inhabitants could get from moving out of the city). Note that inhabitants could still want to move their working place.

For this section, following closely the analysis of Lucas and Rossi-Hansberg [10], we first describe for given wages, revenues and productivities, the ra-tional behaviour of consumer-workers, firms and landlords. In particular we shall determine rents for business use and for residential use as functions of wages, revenues and productivities. Then we deduce from free mobility of la-bor that optimal commuting choice imposes sharp restrictions on wages and revenues that take the form of conjugacy relations between these quantities.

3.1

Workers

Assume that inhabitants dwelling at x ∈ Ω have an available revenue (take-home pay, net of commuting costs) ϕ. It is is known that their utility level is u . An inhabitant of the city dwelling at x divides his revenue between consuming c and renting land S at respective prices 1 and Q. The available revenue ϕ must be enough to yield utility u. Defining:

V (Q) := min {c + QS | U (c, S) ≥ u} (2) we have ϕ = V (Q) and since V is obviously strictly increasing and continu-ous, one can invert this relation. The rent Q is given by Q = Q(ϕ) = V−1(ϕ), and it is easy to see that ϕ 7→ Q(ϕ) is continuous and strictly increasing.

Inhabitants living at x with revenue ϕ then solve program (2) with Q = Q(ϕ) which yields the optimal consumption level c(ϕ) and residential space S(ϕ). Let N be the number of inhabitants housed at x. Since each one occupies S(ϕ) units of land, we find that the relative density of residents (number of residents per unit of residential land) at x only depends on the revenue ϕ at x and is given by:

N = 1

S(ϕ) =: N (ϕ).

3.2

Firms

Assume that the wage paid at a location y is ψ and the productivity is z, then firms located at y choose the level of employment at y by solving the program:

Π(z, ψ) := max {f (z, n) − ψ · n | n ≥ 0} (3) Let n(z, ψ) the corresponding optimal level of employment. The function Π(z, ψ) is the rent for business use and n(z, ψ) is the relative density of jobs (number of jobs per unit of land used for production) at y if productivity at y is z and wage is ψ.

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3.3

Landlords (zoning case)

Since land owners extract all surplus from workers and firms, given wage ψ, productivity z and revenue ϕ at location x, landlords at location x are offered the rent Π = Π(z, ψ) for business use and the rent Q = Q(ϕ) for residential use, those two rents being determined by the firms’s and workers-consumers’ respective programs. Landlords then determine a fraction θ(Π, Q) of land allocated to business, so that 1 − θ(Π, Q) is the fraction of land for resi-dential use. The fraction θ(Π, Q) is naturally assumed to be nondecreasing with respect to Π, and nonincreasing with respect to Q. For mathematical reasons, we also want θ to be bounded away from 0 and 1 (say θ(., .) takes values in [θ, θ] with 0 < θ < θ < 1) and to be jointly continuous with respect to the rents Π and Q. The economic interpretation would be that there are zoning regulations which ask that no neigbourhood in the city be completely devoid of firms or dwellings: even in mainly business areas there must be some housing and even in mainly residential areas there must be some em-ployment. In the sequel, slightly abusing notations, we shall simply denote θ(Π(z, ψ), Q(ϕ)) by θ(z, ψ, ϕ).

3.4

Free mobility of labor

If we denote by ϕ(x) the available revenue at location x and by ψ(y) the wage paid at location y, functions ϕ and ψ are linked by free mobility of labor. Indeed, people living at x rationally choose to work at a location y as to maximize their revenue ψ(y) − c(x, y). Hence, we have:

ϕ(x) = sup {ψ(y) − c(x, y) | y ∈ Ω} (4) This condition implies that ψ(y) ≤ inf {ϕ(x) + c(x, y) | x ∈ Ω} but if the inequality was strict at y then no one would work at y: residents from every area would bring home more money by working somewhere else. Since we have assumed that there always are firms located at y and hence people who work there, we must have:

ψ(y) = inf {ϕ(x) + c(x, y) | x ∈ Ω} (5) This equality can also be interpreted as saying that firms located at y try to attract residents who would work for the lowest wages.

In the sequel we shall say that a pair of functions (ψ, ϕ) that satisfies (4) and (5) are conjugate to each other. Indeed the previous conjugacy relations are incentive-compatibility conditions which express free mobility of labor.

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In addition, they define a commuting network over the city. Define

s(x) = arg max {ψ(y) − c(x, y) | y ∈ Ω} (6) t (y) = arg min {ϕ(x) + c(x, y) | x ∈ Ω} (7) provided the right-hand sides are well-defined and unique. Then the maps s : Ω → Ω and t : Ω → Ω are inverse to each others, and they tell us where people who live at x work, and where people who work at y live1.

4

Equilibrium

Using the previous section, one is able to derive from wage ψ , revenue ϕ and productivity z at location x, the number of residents and the number of jobs at x. This yields an overall density of residents µ and an overall density ofe jobs eν which are defined as functions of (z, ψ, ϕ) by the formulas:

e

µ(z, ψ, ϕ) : = (1 − θ(z, ψ, ϕ))N (ϕ) (8) e

ν(z, ψ, ϕ) : = θ(z, ψ, ϕ)n(z, ψ) (9) So any pair of conjugate functions (ψ, ϕ) together with a productivity function z determines an overall density of jobs µ and an overall density ofe jobsν. Clearly the total number of inhabitants should equal the total numbere of jobs: again, this is a basic feature of our model which is absent from the one of Lucas and Rossi-Hansberg. This will give us an equilibrium condition, we will express mathematically through the notion of measure-preserving maps. Definition 1 Given a Borel measure m on Ω and a Borel map X : Ω → Ω we will denote by X(m) the image measure, defined for every Borel subset B of Ω by:

X(m)(B) := m(X−1(B)).

We will say that the map X pushes m forward to X(m), or that X(m) is the image of m through X. In what follows, |B| denotes the 2-dimensional Lebesgue measure of a Borel subset B of R2. As usual, Rn++ denotes the set

of vectors in Rn with strictly positive components.

An equilibrium consists of:

1It is not obvious that relations (6) and (7) define s and t as single-valued maps. This

will be proved later on to be the case for the class of transportation costs we consider (see Lemma 2).

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• an overall density of residents µ, and an overall density of employment ν,

• a productivity function z,

• a wage function ψ and a revenue function ϕ,

• a transportation map s , or commuting network, which maps residence locations x to working places s(x),

such that:

• productivity z is obtained from the density of employment ν by formula (1),

• the total labor supply (number of residents) equals the total labor de-mand (number of jobs),

• s pushes µ (x) dx forward to ν (x) dx,

• the transportation map s is consistent with wages: workers living at x rationally choose to work at s(x), given wages ψ defined in the whole city,

• wages and revenues are consistent with free mobility of labor, i.e. are conjugates in the sense of (4), (5)

• densities of employment and of residents are consistent with rational behaviour of workers, firms and landowners, i.e. µ and ν are obtained from (z, ψ, ϕ) by (8) and (9).

This yields the formal definition:

Definition 2 An equilibrium is a collection (µ, ν, z, ψ, ϕ) of positive contin-uous functions on Ω and a pair of measurable maps (s, t): Ω → Ω satisfying:

1. Rµ =Rν, 2. z = Zν,

3. (ψ, ϕ) satisfy conditions (4), and (5), 4. for all x ∈ Ω:

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5. s(µ) = ν and t(ν) = µ,

6. s(t(y)) = y a.e. and t(s(x)) = x a.e., 7. (s, t) satisfy conditions (6), and (7),

Note that, using the fact that (ψ, ϕ) are conjugates by (4), (5), the last requirement in the definition is equivalent to:

ϕ(x) = ψ(s(x)) − c(x, s(x)), (10) ψ(y) = ϕ(t(y)) + c(t(y), y) (11) To prove existence, we introduce the following assumptions:

Assumption 1 Ω is an open bounded and connected subset of R2 such that |∂Ω| = 0, which satisfies an additional regularity condition: there exists δ0 >

0 and ε0 ∈ (0, π) such that for every y ∈ Ω, there is some ε ∈ [0, 2π) such

that Ω − y contains the intersection of the open ball of radius δ0 with a cone

of angle 2ε0:

{y + (t cos(v), t sin(v)), t ∈ [0, δ0], v ∈ [ε − ε0, ε + ε0]} ⊂ Ω.

Assumption 2 The commuting cost is of the form c(x, y) = C(x − y) where C is a strictly convex2 and differentiable function from R2 to R

+ with C (x) ≥

C(0) = 0 for all x.

Assumption 3 The overall density of residents eµ : [z, z] × R2++ → R++,

defined by formula (8), is continuous, and satisfies the relations: [z0 ≤ z, ψ0 ≥ ψ, ϕ0 > ϕ] =⇒µ(ze 0, ψ0, ϕ0) >µ(z, ψ, ϕ)e

∀(z, ψ) ∈ [z, z] × R++, limϕ→+∞µ(z, ψ, ϕ) = +∞.e

Assumption 4 The overall density of jobs ν : [z, z] × Re 2++→ R++, defined

by formula (9), is continuous, and satisfies the monotonicity relation: [z0 ≤ z, ψ0 > ψ, ϕ0 ≥ ϕ] =⇒ν(ze 0, ψ0, ϕ0) <ν(z, ψ, ϕ).e

2A more general assumption, the generalized Spence-Mirrlees condition on the cost

would lead to similar results. We omit it here for the sake of simplicity. See Carlier [3] for some results with this broader class.

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Assumption 5 For every constant m > 0, the function x 7→ν(z, kxk, m) ise not integrable near x = 0.

Some comments are in order. Note that assumptions 4 and 5 imply that for all fixed (z, ϕ) ∈ [z, z] × R++:

lim

ψ→0ν(z, ψ, ϕ) = +∞.e

Assumption 1 is purely technical. It is a kind of uniform cone condition which is satisfied for instance when Ω is convex.

As for the monotonicity assumptions, we recall that θ(z, ψ, ϕ) is nonde-creasing with respect to Π(z, ψ) and noninnonde-creasing with respect to Q(ϕ). Note also that Π(., .) is nondecreasing with respect to z, nonincreasing with respect to ψ and that Q is nondecreasing, which proves that if z0 ≥ z, ψ ≤ ψ0

and ϕ ≤ ϕ then θ(z0, ψ0, ϕ0) ≥ θ(z, ψ, ϕ). Hence assumptions 3, 4 and 5 onνe and µ , actually are conditions on the behavior of N (.) and n(., .).e

As in Lucas and Rossi-Hansberg, consider the Cobb-Douglas case, namely: f (z, n) = zγ0nα0,

U (c, S) = cβ0S1−β0

with γ0 > 0 and (α0, β0) ∈ (0, 1)2. In this example, our assumptions are

satisfied, provided α0 ≥ 1/2. Indeed, explicit computations lead to:

n(z, ψ) =  α0z γ0 ψ 1−α01 , (12) N (ϕ) = ββ0/(1−β0) 0 u 1/(1−β0)ϕβ0/(1−β0) (13)

so that, with the monotonicity properties of θ(., ., .), assumptions 3 and 4 are satisfied. For any constant m > 0, one has:

e

ν(z, kxk, m) ≥ θn(z, kxk) = θ α0z

γ0

kxk 1−α01

so that assumption 5 is satisfied provided 1−α1

0 ≥ 2.

We can now state our first result. The assumptions in section 2 are understood to be satisfied.

Theorem 1 If conditions 1, 2, 3, 4 and 5 are satisfied, there exists at least one equilibrium for the zoning model.

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5

Reformulation as a fixed point problem

5.1

Optimal transportation

A fundamental tool in our approach is the theory of optimal transportation problems, in particular the dual formulation of such problems. The literature related to this subject that started with G. Monge by the end of the 18th century, is now much too vast even to try to give exhaustive references. For what is needed in the sequel and the results recalled in this section we refer to Brenier [1], McCann and Gangbo [2], Rachev and R¨uschendorf [12], Levin [8] and Carlier [3]. Indeed the conjugacy relations (4), (5) together with the mass conservation condition in the definition of equilibrium are tightly connnected with the dual formulation of optimal transportation problems.

Given (µ, ν) two continuous positive functions on Ω with same total mass R µdx = R νdx and a cost function of the form c(x, y) = C(x − y) with C strictly convex and differentiable, consider the following problem:

(Mµ,ν) inf s Z Ω C(x − s(x))µ(x)dx | s(µ) = ν  (14) and the so-called dual problem:

(Dµ,ν) sup ψ,ϕ Z Ω ψν − Z Ω ϕµ | ψ(y) − ϕ(x) ≤ C(x − y), (x, y) ∈ Ω2 

Notice that (Dµ,ν) is a linear program, and that changing ψ to ψ + a and

ϕ to ϕ + a, for any constant a, does not change the value of the criterion. Notice also that the admissible set for this problem:

{(ψ, ϕ) | ψ(y) − ϕ(x) ≤ C(x − y), (x, y) ∈ Ω × Ω} consists of those pairs (ψ, ϕ) such that for all (x, y) ∈ Ω × Ω:

ϕ(x) ≥ sup

y∈Ω

ψ(y) − C(x − y), ψ(y) ≤ inf

x∈Ωϕ(x) + C(x − y)

This actually suggests that if the supremum is attained at (ψ, ϕ) then ψ and ϕ are conjugate to each other, that is, satisfy (4) and (5).

The results from optimal transportation theory that we shall use are summarized in the next result:

Theorem 2 If µ and ν are two continuous positive functions on Ω with same total mass and if the cost function C is strictly convex and differentiable, we have

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1. The supremum in (Dµ,ν) is attained by a pair (ψ, ϕ) of functions

satis-fying (4) and (5),

2. (ψ, ϕ) is the unique solution of (Dµ,ν) up to the addition of the same

constant to both functions,

3. the duality relation inf(Mµ,ν) = sup( Dµ,ν) holds,

4. there exists a Borel map s : Ω → Ω, unique up to a.e. equivalence, such that for all x ∈ Ω

ϕ(x) = ψ(s(x)) − C(x − s(x)) 5. s(µ) = ν and s is the unique solution of (Mµ,ν).

6. s is invertible in the sense that there exists another Borel map t : Ω → Ω such that s ◦ t and t ◦ s coincide a.e. with the identity map. Moreover one has t(ν) = µ and for all y ∈ Ω:

ψ(y) = ϕ(t(y)) + C(t(y) − y).

Actually, more precise characterizations of the solutions can be obtained. If ψ0 and ϕ0 are conjugate to each other and such that for all x:

ϕ0(x) = ψ0(s(x)) − C(x − s(x))

where s is the solution of (Mµ,ν), then necessarily (ψ0, ϕ0) solves (Dµ,ν).

Similarly if s0 pushes forward µ to ν and there exist conjugate functions ψ0 and ϕ0 such that for all x:

ϕ0(x) = ψ0(s0(x)) − C(x − s0(x))

then s0is the solution of (Mµ,ν), and (ψ0, ϕ0) solves (Dµ,ν). Hence the previous

Theorem actually provides a full characterization of optimal solutions to both problems (Dµ,ν) and (Mµ,ν).

5.2

Deducing revenues and salaries from distributions

Theorem 2 is of particular interest in our equilibrium problem since it implies that wages and revenues can be deduced from densities. Indeed, this result from optimal transportation theory implies that given a continuous positive density of residents µ and a continuous positive density of employment ν there

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exists a unique (up to the addition of a constant) pair of wage and revenue functions ψ and ϕ which induce a consistent transportation map. More precisely, given (µ, ν) there exist a unique (up to an additive constant again) pair of wage/revenue (ψ, ϕ) and a unique pair of maps (s, t) which satisfy requirements 3, 5, 6 and 7 of the definition of an equilibrium. Moreover s is the only solution of the optimal transportation (Mµ,ν), and (ψ, ϕ) are

solutions of the linear program (Dµ,ν).

Let us fix µ and ν two positive continuous functions on Ω having the same integral. Define a wage function ψ and a revenue function ϕ as the unique solution of the program (Dµ,ν) which satisfies the normalization condition

infΩψ = 1.

Note that by (4) ϕ ≥ ψ ≥ 1. The regularity of (ψ, ϕ) is given in the next result:

Lemma 1 Let c0 := maxk∇C(x − y)k | (x, y) ∈ Ω × Ω . Then all (ψ, ϕ)

that satisfy the conjugacy relations (4), (5) are Lipschitz continuous with Lipschitz constant at most c0.

Proof: Let (x1, x2) ∈ Ω × Ω and let yn ∈ Ω be such that

ϕ(x1) ≤ ψ(yn) − C(x1− yn) + 1 n Since ϕ(x2) ≥ ψ(yn) − C(x2− yn) we have ϕ(x1) − ϕ(x2) ≤ 1 n + C(x2− yn) − C(x1− yn) ≤ 1 n + c0kx1− x2k Since this inequality holds for all n and using a similar argument, reversing the order of (x1, x2), we get the result for ϕ. The same reasoning shows that

ψ is also c0-Lipschitz.

Note that this enables us to extend ψ and ϕ to Ω.

We already mentioned the fact that a conjugate pair of revenue/wage functions actually defines a commuting network. This assertion is made precise by the following result:

Lemma 2 Let (ψ, ϕ) satisfy the conjugacy relations (4), (5), then for (Lebesgue) almost every x ∈ Ω the set:

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is single-valued. Similarly, for almost every y ∈ Ω the set: arg minϕ(x) + C(x − y) | x ∈ Ω is single-valued.

Proof: It follows from Lemma 1 that the two sets defined in the above statement are nonempty and compact in Ω. Furthermore since ψ is c0

-Lipschitz continuous, Rademacher’s Theorem (see for instance [5]) implies that ψ is differentiable almost everywhere. Let y ∈ Ω be a point where ψ is differentiable and let t(y) be such that ψ(y) = ϕ(t(y)) + C(t(y) − y) since:

t(y) ∈ arg minϕ(x) + C(x − y) | x ∈ Ω and since ψ is differentiable at y one has:

∇ψ(y) = −∇C(t(y) − y)

and since C is smooth and strictly convex, one can rewrite this equation as: t(y) = y + ∇C∗(−∇ψ(y))

where C∗ denotes the Fenchel Transform of C. A similar reasoning for ϕ finally shows proves the desired uniqueness result.

5.3

Deducing new distributions from revenues and salaries

Given densities µ and ν as before, so that they have the same total mass, assume now that we are given the wage function ψ and the revenue function ϕ satisfying the normalization min ψ = 1 as in the previous paragraph. We now aim to prove that there exists a unique constant λ > −1 (so that ψ + λ and ϕ + λ remain positive) such that:

Z Ωe µ(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx = Z Ωe ν(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx. So µ(Ze ν(x), ψ(x) + λ, ϕ(x) + λ) and ν(Ze ν(x), ψ(x) + λ, ϕ(x) + λ) will be a new density of residents and a new density of jobs, still having he same total mass.

Lemma 3 For all ν ∈ C0(Ω, R

++) and all pair of conjugate functions (ψ, ϕ)

with min ψ = 1, the equation : Z Ωe µ(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx = Z Ωe ν(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx

admits a unique root λ ∈ (−1, +∞). This root satisfies λ ∈ [λ1, λ2], where

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Proof: For λ > −1 define: F (λ) := Z Ωe µ(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx G(λ) := Z Ω e ν(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx

Because of assumptions 3 and 4 those two functions are continuous. Using assumption 3, and since ϕ ≥ ψ ≥ 1 on Ω and z ≥ Zν ≥ z, we find that:

F (λ) ≥ eF (λ) := |Ω| ·µ(z, 1 + λ, 1 + λ)e G(λ) ≤ eG(λ) := |Ω| ·eν(z, 1 + λ, 1 + λ)

Note that the functions eF and eG do not depend on ψ, ϕ or Zν, that eF is

increasing and eG decreasing, that both are obviously continuous, and lim

λ→−1( eF − eG)(λ) = −∞, λ→+∞lim ( eF − eG)(λ) = +∞. (15)

Hence the equation eF (λ) = eG (λ) admits a unique root that we denote λ2 ∈ (−1, +∞).

Since minψ = 1, using Lemma 1, the fact that C is bounded on Ω − Ω and the conjugacy relations, we deduce that there exists a constant c1 > 1

(which does not depend on ψ and ϕ) such that for all x ∈ Ω 1 ≤ ψ(x) ≤ ϕ(x) ≤ c1

Using assumption 3, we first get:

F (λ) ≤ F (λ) := |Ω| ·µ(z, ce 1+ λ, c1+ λ)

Note that F is an increasing function of λ. Let y0 ∈ Ω be such that

ψ(y0) = 1 = min ψ. Using Lemma 1, we get:

ψ(y) ≤ 1 + c0ky − y0k, ∀y ∈ Ω

with assumption 4, we obtain then G(λ) ≥

Z

e

ν(z, 1 + λ + c0ky − y0k, c1+ λ)dy

Using assumption 1, we then get: G(λ) ≥ G(λ) := ε0 π Z B(0,δ0) e ν(z, 1 + λ + c0kxk, c1+ λ)dx

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When λ → −1, the integrandeν(z, 1 + λ + c0kxk, c1+ λ) converges

mono-tonically to eν(z, c0kxk, c1− 1). Using assumption 5 and the monotone

con-vergence Theorem we obtain that the (decreasing and continuous function) G tends to +∞ as λ tends to −1. Together with assumption 3, this yields:

lim

λ→−1(F − G)(λ) = −∞, λ→+∞lim (F − G)(λ) = +∞. (16)

Hence the equation F = G admits a unique root in (−1, +∞) that we denote λ1 ∈ (−1, +∞). Since eF − eG ≤ F − G ≤ F − G and those three functions

are increasing continuous, the equation F = G admits a unique solution λ ∈ (−1, +∞) and λ2 ≥ λ ≥ λ1. Since λ2 and λ1 do not depend on ψ, ν and

ϕ, we are done.

Note that with λ determined as in the previous Lemma the two functions ψ + λ and ϕ + λ are strictly positive, of course they still are conjugate, still solve the program (Dµ,ν) and the solution s of the optimal transportation

problem ( Mµ,ν) and its inverse map t still satisfy (10 ) with ψ + λ and ϕ + λ

instead of ψ and ϕ.

Besides, we have obtained the two new densities: e

µ(Zν(x), ψ(x) + λ, ϕ(x) + λ)

e

ν(Zν(x), ψ(x) + λ, ϕ(x) + λ).

By construction those densities have the same integral. Finally, these densities are strictly positive and continuous, for they arise by composition of the two mappings [Zν(.), ψ(.) + λ, ϕ(.) + λ] and [µ(., ., .),e ν(., ., .)], whiche are continuous by assumption.

5.4

The fixed point problem

Let us define: ∆ :=  (µ, ν) ∈ C0(Ω, R++)2, Z Ω µdx = Z Ω νdy 

We sum up the preceding results by showing that the proof of Theorem 1 reduces to a certain fixed-point problem for a certain operator T : ∆ → ∆ which we construct as follows.

Start from (µ, ν) ∈ ∆, a pair of densities. Step 1:

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Define the productivity function z by : z = Zν. It is immediate to check

that z ∈ C0(Ω, [z, z]).

Step 2:

Let (ψ, ϕ) be the only solution of problem (Dµ,ν) such that min ψ = 1.

Denote (ψ, ϕ) := T1(µ, ν). Notice that ψ, ϕ are continuous functions and

that ϕ ≥ ψ ≥ 1. Step 3:

Find the only constant λ such that: Z Ω e µ(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx = Z Ω e ν(Zν(x), ψ(x) + λ, ϕ(x) + λ)dx

and define (ψ + λ, ϕ + λ) := T2(ψ, ϕ). Lemma 3 guarantees that T2 is

well-defined, whereas Lemma 1 insures that T2(ψ, ϕ) is a pair of

Lipschitz-continuous strictly positive functions. Step 4:

Compute the new element (µ0, ν0) of ∆, defined for all x ∈ Ω by: µ0(x) : =µ(Ze ν(x), ψ(x) + λ, ϕ(x) + λ) = eµ(Zν(x), T2(ψ, ϕ)(x)) ν0(x) : =eν(Zν(x), ψ(x) + λ, ϕ(x) + λ) = eν(Zν(x), T2(ψ, ϕ)(x)) and denote by T (µ, ν) := (µ0, ν0) this new pair of densities. It is immediate to check that by construction T (µ, ν) ∈ ∆.

Existence of an equilibrium actually reduces to the existence of fixed points of T . This is the content of the next result:

Proposition 1 A pair (µ, ν) ∈ ∆ is a fixed point of T if and only if (µ, ν, Zν, T2◦

T1(µ, ν), s, t) is an equilibrium, where s denotes the solution of the optimal

transportation problem (Mµ,ν) and t is its inverse map.

Proof: Assume first that (µ, ν) ∈ ∆ is a fixed point of T : (µ, ν) = T (µ, ν). Let (ψ, ϕ) := T2 ◦ T1(µ, ν), s be the solution of (Mµ,ν) and t be its inverse.

Continuity and positivity of ψ and ϕ follow from Lemma 1 and Lemma 3. Requirements 1 and 2 of equilibrium are obviously satisfied, the conju-gacy requirement 3 follow from Theorem 2 and the definition of T1 and T2,

properties 5 and 6 follow from the definition of s and t and property 7 fol-lows from Theorem 2. Finally since (µ, ν) is a fixed point of T and since (ψ, ϕ) := T2◦ T1(µ, ν), we have:

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so that requirement 4 is satisfied.

Conversely assume that (µ, ν, z, ψ, ϕ, s, t) is an equilibrium. Property 1 implies that (µ, ν) ∈ ∆, property 2 implies z = Zν. Properties 3, 5 and

7 together with Theorem 2 imply that s is the solution of (Mµ,ν) and that

(ψ, ϕ) is a solution of the dual problem (Dµ,ν), in particular there is a constant

λ such that (ψ + λ, ϕ + λ) = T2 ◦ T1(µ, ν). Using conditions 1 and 4 of

equilibrium, we derive: Z Ω e µ(Zν(x), ψ(x), ϕ(x))dx = Z Ω e µ(Zν(x), ψ(x), ϕ(x))dx

using the definition of T2 and Lemma 3 we obtain (ψ, ϕ) = T2 ◦ T1(µ, ν).

Using requirement 4 again we finally obtain that (µ, ν) = T (µ, ν).

6

Existence proof

In this section we shall prove that the operator T satisfies the assumptions of Schauder’s fixed point Theorem so that it admits at least one fixed point.

6.1

Continuity properties

Lemma 4 Defining the constant c0 as in Lemma 1, we have:

1. There exist two constants m1 and m2 with m2 > m1 > 0 such that, if

(µ, ν) ∈ ∆ and (ψ, ϕ) = T2 ◦ T1(µ, ν), then ψ and ϕ are c0 -Lipschitz

and:

m1 ≤ ψ ≤ m2, m1 ≤ ϕ ≤ m2

2. There exist two constants α and β with β > α > 0 such that: T (∆) ⊂ ∆α,β := {(µ, ν) ∈ ∆ | α ≤ µ ≤ β, α ≤ ν ≤ β}.

Proof: Assertion 1 follows immediately from Lemmas 1, 3 and the conju-gacy relations.

As for assertion 2, we have:

T (µ, ν) = (µ(Ze ν, T2◦ T1(µ, ν)),ν(Ze ν, T2◦ T1(µ, ν)))

Using assertion 1 and assumptions 3 and 4, we immediately obtain assertion 2 with:

α := min(µ(z, me 1, m1),eν(z, m2, m2)), β := max(eµ(z, m2, m2),ν(z, me 1, m1)) this implies in particular that T (∆) ⊂ ∆α,β.

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Note that Lemma 4 enables us to consider only the set ∆α,β, so that the

densities we will deal with remain uniformly bounded away from 0 and infin-ity (this is crucial of course if we want to use problems (Dµ,ν) and (Mµ,ν)).

Lemma 5 The mapping Z: Z : C

0(Ω, [α, β]) −→ C0(Ω, [z, z])

ν 7−→ Zν

is continuous where C0(Ω, [α, β]) and C0(Ω, [z, z]) are endowed with the sup

norm.

Proof: The fact that Z maps C0(Ω, [α, β]) into C0(Ω, [z, z]) is

straightfor-ward. First note that for ν and ν0 in C0(Ω, [α, β]) we have:

max x∈Ω Z Ω ρ(x, y)|ν(y) − ν0(y)|dy ≤ Ckν − ν0k∞ (17) where C = max x∈Ω Z Ω ρ(x, y)dy

Since χ is uniformly continuous on [0, βC], for all ε > 0 there exists δ > 0 such that |χ(t) − χ(t0)| ≤ ε for all (t, t0) ∈ [0, βC] such that |t − t0| ≤ δ. This proves that if kν − ν0k∞ ≤ δ/C then kZν− Zν0k ≤ ε.

Lemma 6 T1 is a continuous mapping from ∆α,β into C0(Ω, R+)×C0(Ω, R+),

those two spaces being endowed with the sup norm.

Proof: Let a sequence (µn, νn) in ∆α,β converge uniformly to a limit (µ, ν).

Let (ψn, ϕn) := T1(µn, νn). Using Lemma 1, the fact that (ψn, ϕn) is

uni-formly bounded and Ascoli’s Theorem, we may assume that some subse-quence again labeled (ψn, ϕn) converges uniformly to a limit, say (ψ, ϕ).

Let us first check that ψ satisfies our normalization selection assumption min ψ = 1. Obviously minψ ≥ 1. Now there exists xn ∈ Ω such that

ψn(xn) = 1 and taking subsequence if necessary we may assume that xn

converges to some x ∈ Ω, but since

|ψn(xn)−ψ(x)| ≤ |ψn(xn)−ψn(x)|+|ψn(x)−ψ(x)| ≤ C0kxn−xk+kψn−ψk∞

we get min ψ = 1.

Let us prove now that (ψ, ϕ) = T1(µ, ν),. To that end it is enough to

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constraints of (Dµ,ν). Let (ψ0, ϕ0) be any admissible pair for problem (Dµ,ν)

namely. ψ0(y) − ϕ0(x) ≤ C(x − y) for all x, y. By the definition of (ψn, ϕn)

we have: Z Ω ψnνn− Z Ω ϕnµn ≥ Z Ω ψ0νn− Z Ω ϕ0µn

passing to the limit and since (ψ0, ϕ0) is arbitrary we get Z Ω ψν − Z Ω ϕµ = sup(Dµ,ν)

and since T1(µ, ν) is the only solution of (Dµ,ν) satisfying our normalization

condition we get (ψ, ϕ) = T1(µ, ν).

The previous argument actually proves that T1(µ, ν) is the only cluster

point of the relatively compact sequence (ψn, ϕn) so that the whole sequence

converges to T1(µ, ν), which proves the desired result.

Lemma 7 T2◦T1 is a continuous mapping from ∆α,β to C0(Ω, R+)×C0(Ω, R+)

those two sets being endowed with the sup norm.

Proof: Let a sequence (µn, νn) in ∆α,β converge uniformly to a limit (µ, ν).

Using Lemma 6, we know that (ψn, ϕn) := T1(µn, νn) converges uniformly to

(ψ, ϕ) := T1(µ, ν). Let λn∈ R be such that T2◦ T1(µn, νn) = (ψn+ λn, ϕn+

λn). By the definition of T2, we have:

Z Ωe µ(Zνn, ψn+ λn, ϕn+ λn) = Z Ωe ν(Zνn, ψn+ λn, ϕn+ λn) (18)

and by Lemma 3, the sequence λnis bounded, taking values in [λ1, λ2], up to

a subsequence again denoted λn, we may then assume that λn converges to

some λ ∈ [λ1, λ2]. Since (ψn, ϕn) converges uniformly to (ψ, ϕ) and since Zνn

converges uniformly to Zν by Lemma 5, and since both integrands in (18)

are uniformly bounded, passing to the limit in (18) yields: Z Ωe µ(Zν, ψ + λ, ϕ + λ) = Z Ωe ν(Zν, ψ + λ, ϕ + λ)

Using Lemma 3, this means precisely that T2◦ T1(µ, ν) = (ψ + λ, ϕ + λ).

This actually proves that T2◦T1(µ, ν) is the only cluster point of the relatively

compact sequence T2 ◦ T1(µn, νn) = (ψn+ λn, ϕn+ λn), so that the whole

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Lemma 8 T is a continuous mapping from ∆α,β into itself endowed with the

sup norm.

Proof: This result follows from Lemmas 5-6-7 and the well-known fact that the Nemytskii operator T3:

T3 :  C0(Ω, [z, z]) × C0(Ω, [m1, m2]) 2 −→ C0 (Ω, R)2 (z(.), ψ(.), ϕ(.)) 7→ (µ(z(.), ψ(.), ϕ(.)),e eν(z(.), ψ(.), ϕ(.)))

is continuous for the uniform topology sinceµ(., ., .) ande ν(., ., .) are uniformlye continuous on the compact set [z, z] × [m1, m2] × [m1, m2].

6.2

Compactness

Lemma 9 The set Z C0(Ω, [α, β]) is relatively compact in C0(Ω, [z, z]) en-dowed with the sup norm.

Proof: Define first the linear operator K by: K : C

0(Ω, [α, β]) −→ C0(Ω, R)

ν(.) 7−→ Kν(.) :=

R

Ωρ(., y)ν(y)dy

Note first that K is continuous and maps C0(Ω, [α, β]) into C0(Ω, [0, βC]) (see Lemma 5). Let us prove now that the set K C0(Ω, [α, β]) is uniformly

equicontinuous. Let ε > 0 and let δ > 0 be such that:

sup{|ρ(x, y) − ρ(x0, y)| | (x, x0, y) ∈ Ω3, |x − x0| ≤ δ} ≤ ε β|Ω|

then for all ν ∈ C0(Ω, [α, β]) and all (x, x0) ∈ Ω2 such that |x − x0| ≤ δ, we have |Kν(x) − Kν(x0)| ≤ ε. Hence K C0(Ω, [α, β]) is uniformly

equicontin-uous.

Since K C0(Ω, [α, β]) is bounded and uniformly equicontinuous, by As-coli’s Theorem it is relatively compact in C0(Ω, [0, βC]). The desired result

follows from composition with the continuous function χ.

Lemma 10 T (∆α,β) is relatively compact in ∆α,β endowed with the sup

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Proof: By Lemma 4, T2◦T1(∆α,β) is relatively compact in C0(Ω, [m1, m2])

2 , and by Lemma 9, Z C0(Ω, [α, β]) is relatively compact in C0(Ω, [z, z]). Now

since the map T3 defined in the proof of Lemma 8 is continuous, the set

T (∆α,β) is relatively compact since by construction, it is included in the

im-age of the relatively compact set Z C0(Ω, [α, β]) × T

2 ◦ T1(∆α,β) by T3.

Combining those results we are finally able to prove Theorem 1. Indeed, the continuous operator T maps the closed convex subset ∆α,β of C0(Ω, R)

into itself and its image is relatively compact for the uniform norm. By Schauder’s fixed point Theorem, T admits at least one fixed point (µ, ν) ∈ ∆α,β. Using Proposition 1, this finally proves that there exists at least one

equilibrium.

7

The general model

Along the lines of Lucas and Rossi-Hansberg [10], we now consider the gen-eral situation where the behaviour of landlords is not restricted by zoning regulations: they simply rent their land to the highest bidder. This will lead to a more complicated structure, where the city is divided into business districts, residential districts, and mixed areas. In other words, the overall density of jobs or residences can vanish in certain areas, which will cause mathematical difficulties.

Rational behavior of workers given their revenue ϕ is exactly the same as described in subsection 3.1, this yields the two functions N (ϕ) and Q(ϕ) as previously. Similarly, rational behavior of firms given wages ψ and pro-ductivity z is the same as described in subsection 3.2; this leads to the two functions n(z, ψ) and Π(z, ψ) as previously. Rational behaviour of landlords now is different, and subsection 3.3 must then be replaced by the following:

7.1

Landlords (general case)

At location x ∈ Ω, given rents for business and residential use Π(z(x), ψ(x)) and Q(ϕ(x)), landowners determine a fraction θ(x) ∈ [0, 1] devoted to busi-ness use that satisfies:

Π(z(x), ψ(x)) > Q(ϕ(x)) ⇒ θ(x) = 1 (19) Π(z(x), ψ(x)) < Q(ϕ(x)) ⇒ θ(x) = 0 (20) Note that, if Π = Q, landlords are indifferent between allocating land for

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residential or for business use. The values of θ on the set {x | Π(z(x), ψ(x) = Q(ϕ(x))} will be determined by the equilibrium conditions.

In this framework, the overall density of employment ν is given by ν(x) = θ(x)n(z(x), ψ(x))

and the overall density of residents µ by µ(x) = (1 − θ(x))N (ϕ(x)). Recall that the support of µ, denoted supp(µ), is the closure of the set where µ (x) > 0; similarly for ν.

7.2

Equilibrium

Now, free mobility of labor implies conjugacy relations between wages and revenues that are slightly different in the zoning case since there is only employment on supp(ν) and residents on supp(µ). Those relations therefore take the form:

ϕ(x) = sup {ψ(y) − c(x, y) | y ∈ supp(ν)} (21)

ψ(y) = inf {ϕ(x) + c(x, y) | x ∈ supp(µ)} (22) Definition 3 A general equilibrium is a pair of nonnegative Lebesgue inte-grable functions (µ, ν) on Ω, a collection of functions (z, ψ, ϕ, θ) on Ω and a pair of measurable maps (s, t): Ω → Ω satisfying:

1. Rµ =Rν > 0, 2. z = Zν,

3. (ψ, ϕ) satisfy conditions (21), and (22), 4. for almost every x ∈ Ω, θ(x) ∈ [0, 1] and :

µ(x) = (1 − θ(x))N (ϕ(x)), ν(x) = θ(x)n(z(x), ψ(x)), 5. conditions (19) and (20) are satisfied,

6. s(µ) = ν and t(ν) = µ,

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8. (s, t) satisfy for µ-a.e. x and ν-a.e. y:

ϕ(x) = ψ(s(x)) − c(x, s(x)) (23) ψ(y) = ϕ(t(y)) + c(t(y), y) (24) Note that, in contrast with the preceding case, s and t are not defined on all of Ω, but on the supports of µ and ν respectively.

Assumption 6 Both functions N and Q are continuous from R++ to R++,

increasing and

lim

ϕ→+∞N (ϕ) = limϕ→+∞Q(ϕ) = +∞.

Assumption 7 Both functions n and Π from [z, z] × R++ to R++ are

con-tinuous, and satisfy the monotonicity relations:

[z0 ≤ z, ψ0 > ψ] =⇒ n (z0, ψ0) < n (z, ψ) and Π (z0, ψ0) < Π (z, ψ) .

Assumption 8 The function x 7→ n(z, kxk) is not integrable near x = 0. Note that assumptions 7 and 8 imply that for all fixed z ∈ [z, z]:

lim

ψ→0n(z, ψ) = +∞.

Assumption 9 The function Π satisfies: lim

ψ→0Π(z, ψ) = +∞.

The main result of this paper is the following.

Theorem 3 Under assumptions 1, 2, 6, 7, 8, 9, there exists at least one general equilibrium.

The proof of this result involves three steps: in the first one we shall approximate the general problem with a sequence of zoning problems that fullfil the assumptions of Theorem 1, then we shall obtain bounds on this sequence, finally we shall exhibit a general equilibrium by passing to the limit.

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Step 1 : approximation

We approximate the landowner’s behavior as follows. Let k ≥ 2 be an integer. Assume that if the rent for business use is Π(z, ψ) and the rent for residential use Q(ϕ) then the landlord determines the fraction θ devoted to business use by solving:

max  θΠ(z, ψ) + (1 − θ)Q(ϕ) − 1 2kθ 2 | 1 k ≤ θ ≤ k − 1 k  (25) This determines a function (z, ψ, ϕ) 7→ θk(z, ψ, ϕ) and two functions:

e

µk(z, ψ, ϕ) : = (1 − θk(z, ψ, ϕ))N (ϕ)

e

νk(z, ψ, ϕ) : = θk(z, ψ, ϕ)n(z, ψ)

which satisfy assumptions 3, 4, 5 of Theorem 1. There exists then, for all k ≥ 2 a zoning equilibrium that we denote by (µk, νk, zk, ψk, ϕk) and (sk, tk) (the

commuting maps). To shorten notations, we will denote by θk(.) the function

θk(zk(.), ψk(.), ϕk(.)) and we note for further reference that θk = 1 − 1/k on

the set {Π(zk, ψk) − Q(ϕk) ≥ 1/k(1 − 1/k)}, and that θk = 1/k on the set

{Π(zk, ψk) − Q(ϕk) ≤ 1/k2}).

Step 2 : a priori bounds

Lemma 11 Both sequences ψk and ϕk are bounded in the sup norm. There

exists a positive constant a such that ψk is bounded from below by a for all

k, and lim inf Z Ω µk = lim inf Z Ω νk > 0

Proof: Assume for instance that ϕk is not bounded. Then, using the fact

that the functions ϕk are uniformly Lipschitz and that ψk and ϕk are

conju-gates, we find that (up to a subsequence) both ϕk and ψkconverge uniformly

to +∞. In this case Q(ϕk) converges uniformly to +∞ and Π(zk, ψk) is

uni-formly bounded (by assumption 7). In particular for k large enough θk ≡ 1/k.

Since n(zk, ψk) is uniformly bounded (by assumption 7 again) we have

µk = (1 − 1 k)N (ϕk) → +∞ νk = 1 kn(zk, ψk) → 0

a contradiction with the mass balance assumption R µk =R νk. This proves

then that ψk and ϕk are uniformly bounded. In particular this implies that

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Let us prove the second assertion. Assume on the contrary that δk :=

inf ψk goes to 0 as k goes to infinity. As already noted ψk extends by uniform

continuity to Ω; let yk ∈ Ω be such ψk(yk) = δk. Since for all y ∈ Ω we have

ψk(yk) ≤ δk+ c0ky − ykk, we have, by assumption 7,

n(zk(y), ψk(y)) ≥ n(z, δk+ c0ky − ykk)

and similarly, using assumption 7 again, we get

Π(zk(y), ψk(y)) ≥ Π(z, δk+ c0ky − ykk)

Using assumptions 7 and 9, the fact that Q(ϕk) is bounded and the fact

that δk tends to 0, we deduce that there exists k0 and r > 0 such that

if y ∈ Ω ∩ B(yk, r), and if k ≥ k0 then θk(y) = (1 − 1/k). Hence using

assumption 1, we get for k ≥ k0:

Z Ω νk ≥ (1 − 1/k) ε0 π Z B(0,min(δ0,r)) n(z, δk+ C0kyk)dy

Now using assumption 8, and the monotone convergence Theorem, we deduce that lim k Z Ω νk= +∞

By the mass balance condition, the sequence µkcannot be bounded, and this

is the desired contradiction.

The last assertion of the lemma is an easy consequence of the previous ones and assumptions 6 and 7.

The previous Lemma implies in particular that N (ϕk) and n(zk, ψk) are

uniformly bounded.

Step 3 : convergence and existence of an equilibrium.

By the previous Lemma and Ascoli’s Theorem, and extracting a subse-quence if necessary, we may assume that (ψk, ϕk) converges uniformly to some

pair (ψ, ϕ) (with ϕ ≥ ψ ≥ a > 0). Since, θk ∈ [0, 1], N (ϕk) and n(zk, ψk) are

uniformly bounded, we may assume that µk and νk converge weakly in L2 to

some limits µ and ν. The previous Lemma implies that R µ = R ν > 0. It is also immediate to check that zk converges pointwise to z := Zν and that ψ

and ϕ are conjugate to each other in the sense that they satisfy (4) and (5). The proof of Lemma 5 carries over if we replace strong convergence by weak convergence, since the maps involved are all linear. It follows that (ψ, ϕ) is a solution of (Dµ,ν).

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Since µ (x) dx and ν (y) dy are absolutely continuous with respect to the Lebesgue measure, the mass transportation problem (Mµ,ν) admits a

solu-tion (see for instance [7], [2], [8] or [3]). Note that it does not follow directly from Theorem 2 since µ and ν are not continuous and may vanish on some zones of Ω. Moreover one has for µ -a.e. x:

s(x) ∈ argmax{ψ(y) − c(x, y) | y ∈ supp(ν)} and ϕ(x) = ψ(s(x)) − c(x, s(x)). Since s(x) ∈ supp(ν) this yields

ϕ(x) = max{ψ(y) − c(x, y) | y ∈ supp(ν)}

for µ a.e. x and since ϕ is continuous this implies relation (21). Similarly, there exists a map t from supp(ν) to supp(µ) such that t(ν) = µ, and ψ(y) = ϕ(t(y)) + c(t(y), y) ν-a.e. so that (22) is satisfied. Moreover, t(s(x)) = x, µ-a.e. and s(t(y)) = y, ν-a.e..

Since θk∈ [0, 1] (taking a subsequence if necessary) we may assume that

θkconverges in the weak ∗ topology of L∞to some limit θ0, with 0 ≤ θ0 ≤ 1.

Lemma 12 If Π(z(x), ψ(x)) > Q(ϕ(x)) then θ0(x) = 1, µ(x) = 0 and

ν(x) = n(z(x), ψ(x)). Similarly, if Π(z(x), ψ(x)) < Q(ϕ(x)) then θ0(x) = 0,

µ(x) = N (ϕ(x)) and ν(x) = 0. Proof: Define:

Ω+:= {x ∈ Ω | Π(z(x), ψ(x)) > Q(ϕ(x))}

and assume that x ∈ Ω+. For k large enough Π(zk(x), ψk(x)) − Q(ϕk(x)) ≥

1/k − 1/k2 so that θ

k(x) = 1 − 1/k for k large enough. In particular θk

converges to 1 a.e. in Ω+ and in L1(Ω+). Since θk converges to θ0 for

the weak ∗ topology of L∞(Ω+) we have θ0 ≡ 1 on Ω+. Similarly on Ω+,

νk = θkn(zk, ψk) converges a.e. and in L1(Ω+) to n(z, ψ), hence ν(x) =

n(z(x), ψ(x)) a.e. on Ω+. Finally, on Ω+, µk= (1 − θk)N (ϕk) converges a.e.

and in L1(Ω

+) to 0, hence µ(x) = 0 a.e. on Ω+. The proof of the other

assertion is similar.

Lemma 13 The following relation is satisfied for (Lebesgue) a.e. x ∈ Ω: µ(x)n(z(x), ψ(x)) + ν(x)N (ϕ(x)) = n(z(x), ψ(x))N (ϕ(x)) (26)

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Proof: To shorten notations let us set Nk := N (ϕk), nk := n(zk, ψk).

Obviously one has

µk(x)nk(x) + νk(x)Nk(x)) = nk(x)Nk(ϕ(x)), x ∈ Ω (27)

We recall that Nk converges uniformy to N (ϕ) and that nk converges

pointwise and in L2 (by the dominated convergence Theorem) to n(z, ψ),

hence the rightmost member of (27) converges pointwise and in L2to n(z, ψ)N (ϕ).

We also recall that µkand νkconverge weakly in L2 to µ and ν. Obviously

νk· Nk converges weakly in L2 to ν · N (ϕ). Let us prove that µk· nkconverges

weakly in L2 to µ · n(z, ψ). Take g ∈ L2, setting n := n(z, ψ) we have:

Z Ω g(µk· nk− µ · n) = Z Ω g · nk(µk− µ) + Z Ω g · µ(nk− n)

The second integral goes to 0 as k goes to ∞ because µ ∈ L∞ and nk− n

converges to 0 in L2. To prove that the first one also goes to 0 it is enough

to prove that g · nk converges strongly in L2 to g · n. Since nk is uniformly

bounded g2(n

k− n)2 is bounded by κg2 ∈ L1 for some constant κ > 0, and

since g(nk − n) converges a.e. to 0 we deduce from Lebesgue’s dominated

convergence Theorem that g · nk converges strongly in L2 to g · n.

So µk· nk converges weakly in L2 to µ · n(z, ψ). Taking weak L2 limits of

both members of equation (27) yields the desired result.

Note first that µ ≤ N (ϕ) and ν ≤ n(z, ψ). Second, let us remark that N (ϕ) and n(z, ψ) are bounded from below by a strictly positive constant. Dividing (26) by N (ϕ) · n(z, ψ), we get:

ν

n(z, ψ) = 1 − µ

N (ϕ) ∈ [0, 1] define then for all x ∈ Ω

θ(x) := ν(x)

n(z(x), ψ(x)) = 1 −

µ(x) N (ϕ(x))

by construction, requirement 4 of the definition of an equilibrium is satisfied. It remains to prove that (19) and (20) are satisfied. If x ∈ Ω+, Lemma 12

implies that µ = 0, and ν = n(z(x), ψ(x)) so that θ(x) = 1 which proves (19), the proof of (20) being the same.

Finally the collection (µ, ν, z, ψ, ϕ, θ, s, t) satisfy all the requirements of the definition of a general equilibrium.

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8

The multi-sector extension

In this section, we state and prove an equibrium theorem in a multi-sector setting. For mathematical simplicity, we revert to the situation where the behaviour of landlords is constrained by zoning regulations. The general case should be derived by a limiting argument, as above.

The rational behaviour of workers is as described in section 3.1. Given an available revenue ϕ, we derive a rent Q(ϕ) and a relative density of residents N (ϕ).

The rational behaviour of firms is as described in section 3.2, but now the firms are divided in N sectors of activity i = 1, ..., N . Workers are equally productive in each sector and therefore are offered the same wage by firms from different sectors in each location of the city. If a firm of sector i is located at a place where the wage paid is ψ and its productivity is zi, the

level of rents Πi(zi, ψ) for sector i and the level of employment ni(zi, ψ) for

sector i are determined by the maximization program: max

ni≥0

{fi(zi, ni) − ψ · ni} (28)

The rent Πi(zi, ψ) is the value of this program and ni(zi, ψ) is the

correspond-ing optimal level of employment.

Landowners determine a fraction θi of land allocated for business use in

sector i (so that 1 −PN

i=1θi is the fraction used for residences) by solving

the program: max ( N X i=1 (θiΠi(zi, ψ) − γi(θi)) + (1 − X i θi)Q(ϕ) | θi ≥ 0, X i θi ∈ [θ, θ] ) (29) The functions γi are strictly convex increasing cost functions and the

constants θ, θ are as in subsection 3.3. The solution of this program de-termines the functions θi(z1, . . . , zN, ψ, ϕ), and finally we obtain densities of

employment in each sector i, νei and density of residents eµ as functions of (z1, . . . , zN, ψ, ϕ) by the formulas: e µ(z1, . . . , zN, ψ, ϕ) : = 1 − X i θi(z1, . . . , zN, ψ, ϕ) ! N (ϕ) (30) e νi(z1, . . . , zN, ψ, ϕ) : = θi(z1, . . . , zN, ψ, ϕ)ni(zi, ψ). (31)

Given densities of jobs in all the sectors, νi, productivity of sector i, zi

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We assume that the productiviy of sector i at location x is given by: zi(x) = Zνi1,...,νN(x) = χi  ( Z Ω ρij(x, y)νj(y)dy)j=1,..,N  . (32) Where the weighting kernels ρij(., .) are nonnegative continuous on Ω × Ω

and each function χi is continuous χi : R+→ [zi, zi] with +∞ > zi > zi > 0.

Finally free mobility of labour implies that wages and revenues are con-jugate in the sense of (4), (5) and that conservation conditions are satisfied between µ and the total density of jobs ν :=P

iνi. We introduce the formal

definition:

Definition 4 A zoning multi-sector equilibrium is a collection (µ, νi, zi, ψ, ϕ),

i = 1, . . . , N , of positive continuous functions on Ω and a pair of measurable maps (s, t): Ω → Ω satisfying: 1. Rµ =PN i=1 R Ωνi, 2. zi = Zνi1,...,νN, i = 1, . . . , N ,

3. (ψ, ϕ) satisfy conditions (4), and (5), 4. for all x ∈ Ω: µ(x) =eµ(z1(x), ..., zN(x), ψ(x), ϕ(x)) , νi(x) =νei(z1(x), ..., zN(x), ψ(x), ϕ(x)), i = 1, . . . , N, 5. s(µ) =PN i=1νi and t( PN i=1νi) = µ,

6. s(t(y)) = y a.e. and t(s(x)) = x a.e., 7. (s, t) satisfy conditions (6), and (7).

To prove existence, we need to adapt assumptions 3, 4 and 5 in section 2:

Assumption 10 The overall density of residents µ : [ze 1, z1] × ... × [zN, zN] ×

R2++ → R++, defined by formula (30), is continuous, and satisfies the

rela-tions: [zi0 ≤ zi ∀i, ψ0 ≥ ψ, ϕ0 > ϕ] =⇒eµ(z 0 1, . . . , z 0 N, ψ 0 , ϕ0) > µ(ze 1, . . . , zN, ψ, ϕ) and ∀(z1, . . . , zN, ψ) ∈ [z1, z1] × ... × [zN, zN] × R++ lim ϕ→+∞µ(ze 1, . . . , zN, ψ, ϕ) = +∞.

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Assumption 11 The overall density of jobs in sector i: e

νi : [z1, z1] × ... × [zN, zN] × R2++ → R++,

defined by formula (31), is continuous, and satisfies the monotonicity rela-tion: [zi0 ≤ zi ∀i, ψ0 > ψ, ϕ0 ≥ ϕ] =⇒νei(z 0 1, ...., z 0 N, ψ 0 , ϕ0) < νei(z1, ..., zN, ψ, ϕ).

Assumption 12 There exists i ∈ {1, . . . , N } such that for every constant m > 0, the function x 7→ νei(z1, ..., zN, kxk, m) is not integrable near x = 0.

Theorem 4 Under assumptions 1, 2, 10, 11 and 12, there exists a zoning multi-sector equilibrium.

We sketch the proof by directly translating the equilibrium into a fixed point problem. Let us define first:

∆N := ( (µ, ν1, . . . , νN) ∈ C0(Ω, R++) N +1 | Z Ω µdx = n X i=1 Z Ω νidx )

and let (µ, ν1, . . . , νN) ∈ ∆N be the densities of residents and of employment

in each sector. Step 1:

Compute the total density of employment ν := P

iνi and compute the

productivity function of each sector zi by formula (32). Denote by z the

vector (z1, . . . , zN).

Step 2:

Compute a pair of wage/revenues that is consistent with µ and ν by selecting a solution (ψ, ϕ) of problem (Dµ,ν).

Step 3:

As in Lemma 3, find a constant λ such that Z Ωe µ(z, ψ + λ, ϕ + λ) =X i Z Ωe νi(z, ψ + λ, ϕ + λ) Step 4:

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Compute new densities (µ0, ν10, . . . , νN0 ) by setting:

µ0 =µ(z, ψ + λ, ϕ + λ), νe i0 =eνi(z, ψ + λ, ϕ + λ)

and let us denote (µ0, ν0) =: T (µ, ν).

The assumptions then guarantee that T is well-defined and admits at least one fixed point.

Note that an approximating procedure, similar to the one we described in section 7, would enable us to prove the existence of a general (non-zoning) multi-sector equilibrium.

9

Conclusion

In this paper, we have proved the existence of spatial equilibria under robust assumptions. The mathematical results of optimal transportation theory have played a crucial role, and they have turned out to have very natural economic interpretations, which have not previously been described in the literature. We have no doubt that they could be applied to many other situations in geographical economics than the internal structure of cities.

A particular mention should be made of the structure of transportation costs. Here we have assumed them to be convex, which is natural in that context: indeed, if costs were concave, there would be gains in spreading out, and it would be difficult to justify the existence of the city in the first place. Concave costs, or more precisely cost functions of the type C(kx − yk), where C is concave and C (t) ≥ C (0) = 0, lead to very different situations. For instance, in that case it is no longer true that, in equilibrium, all those who dwell at x go to work at the same place y: typically, some of them would stay at x to work, while others would find work elsewhere (all of them, of course, bringing home the same revenue). We plan to study such situations in a forthcoming paper.

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Références