An Urban Economic Model over a Continuous Plane with
Spatial Characteristic Vector Field
Yuzuru Miyata*
Graduate School of Architecture and Civil Engineering, Toyohashi University of Technology
1. Introduction
Among others Beckmann (1952) firstly introduced the concept of a two dimensional continuous space into
economics. This great step had unfortunately not shown further expansion in economics. Through several papers
related to Beckmann’s initiation, Beckmann and Puu (1985) at last reached a systematic treatment of the continuous spatial economics. However, consideration of households and firms locations is not necessarily sufficient, resulting in reconsideration from a new urban economics point of view. Miyata (2010) has solved all partial differential equations and suggested the empirical way of this theory. This article extends the author’s previous study
introducing spatial characteristic vector field (which will be mentioned later) in the model, and shows asymmetry
in land use pattern in a two dimensional continuous space.
2. Assumptions of the Model(1)There is a city in a two dimensional continuous space (a plane). The land characteristics are heterogeneous. The
characteristics of each point are denoted by (lc1(x1, x2),···,lcn(x1, x2)) named spatial characteristic vector field(SCVF). For example, one can consider uneven land and/or a gradient. SCVF gives impact only on transport cost.
(2)Transports of commodities and labor are considered in this study. The transportation cost is expressed as vonThünen type. The transport cost of unit volume of commodity at point (x1, x2) is specified as a scalar function over
SCVF denoting it by ε(x1, x2). Regarding workers commuting, transport cost per head and per distance is denotedby ζ(x1,x2).
(3)There are N households and M firms in the city. N and M denote the numbers of households and firms, respectively, and they are sufficiently large. Households and firms are homogeneous, respectively.
(4)Land in the city is owned by absentee landowners, who reside outside the city. There is a unique local government in the city, and it rents all land in the city from the absentee landowners. The local government rents the land
to households and firms at market rent, and then redistributed it equally to households.
(5)The service of capital stock is assumed to be the numerare, since the capital stock is assumed to move in free ofcost in the city leading to unique equilibrium capital return rate being irrespective firms’ locations.
(6)As mentioned later, the parameters in the locational potential function for each firm is sufficiently large. In thiscase a simple von Thünen ring becomes an equilibrium urban configuration.
3. Firms Behavior
The production function of a firm at location x = (x1, x2) is specified as a Cobb-Douglas type of homogeneousdegree of unity, and agglomeration economy is taken into account. The agglomeration economy is represented bylocational potential function, Ω(x), which was introduced by Fujita and Ogawa (1982). The locational potentialfunction is defined as follows:Ω( x ) ≡∫ ∫ µb( y) exp(−ω || x − y ||)dy dy
A

1

(1)

2

where A : city area, b(y) : density of firms at location y, µ : monetary conversion parameter in locational potential, ω :parameter expressing the effects of distance between different points, || x – y || : distance between location x and y-1-Then the production by a firm at location x may be written as follows:q( x ) = Ω( x )q A k d ( x )α k ld ( x )α l mB ( x )α m

(2)where q(x) : output of a firm at location x, kd(x) : input of capital stock at location x, ld(x) : input of labor at location
x, mB(x) : input of land at location x, qA : efficient parameter in the production function, αk, αl, αm : elasticity parameters in the production function (αk + αl + αm = 1 )We assume that each firm is a price taker for commodities and production factors. The firms’ locational equilibrium condition is that the profit in each firm is equalized at every point. Due to the linear homogeneity of degree
one in each firm’s technology, the equilibrium profit in each firm becomes zero. Then the bid rent function in eachfirm is defined as follows:⎡ p ( x )q ( x ) − r ( x )k d ( x ) − w( x )l d ( x ) ⎤αkαlαmg B ( x ) ≡ max ⎢⎥ subject to q( x ) = Ω( x )q A kd ( x ) ld ( x ) mB ( x ) (3)
m
(
)
x
B
⎣
⎦
π B ( x) = 0
with respect to k d ( x ), l d ( x ), and m B ( x )where p(x) : price of commodity at location x, r(x) : capital return rate at location x, w(x) : wage rate at location x,
gB(x) : bid rent by a firm at location x, πB(x) : profit in a firm at location x
Solving this optimization problem, we obtain the conditional demands for labor and capital associated withfirm’s output q(x), the bid max lot size and the bid rent function.
kd ( x) =α k p( x )r ( x)−

1mB ( x ) = ( Ω( x ) q A ) α m p ( x )ld ( x ) =(4)q( x )αkαl
α m −1ααα m ⎡ r ( x ) ⎤ m ⎡ w( x ) ⎤ m⎢⎥⎣ αk ⎦⎢⎥⎣ αl ⎦

α l p( x )w( x )q( x )1q ( x ) (6)

(5)
αk

αl

⎡ α ⎤αm ⎡ α ⎤αmg B ( x ) = α m ( p( x )Ω( x )q A ) α m ⎢ k ⎥ ⎢ l ⎥ (7)⎣ r ( x ) ⎦ ⎣ w( x ) ⎦4. Households
The household utility function at location x is assumed to be expressed as follows:u (c( x ), mH ( x )) ≡ c( x ) βc mH ( x ) βm ( β c + β m = 1)(8)where u(c(x), mH(x)) : household utility function at location x, c(x) : household consumption at location x,
mH(x) : household land input at location x, βc and βm : elasticity parameters in the utility function
Each household endows available working time ls, and capital stock ks, and is assumed to inelastically supply
them to firms obtaining income of w(x) ls+ r(x) ks plus redistributed income from the local government πH. In
household locational equilibrium, the household utility level takes the same value u*, being irrespective of household residential places. Therefore the household bid rent function for land is specified as follows:g H ( x ) ≡ max

w( x )l s + r ( x )k s +π H − p ( x )c( x )m H ( x)subject to u(x) = u*(9)with respect to c( x ) and m H ( x )c( x ) =βcp( x )βc(w( x)l s + r ( x)k s +π H )(10)⎡⎤ βm * 1p( x )mH ( x) = ⎢⎥ u βm⎣ β c ( w( x )l s + r ( x )k s +π H ) ⎦

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(11)

βc

1

⎡ β ⎤ β m ⎡ w( x )l s + r ( x )k s +π H ⎤ β m
g H ( x) = β m ⎢ c ⎥ ⎢⎥u*⎦⎣ p( x ) ⎦ ⎣(12)

Solving the optimization problem (9), one obtains per capita consumption, the bid max lot size and the bid rent
function.
5. Land Market Equilibrium Condition
We consider a monocentric land use patter, since it has been demonstrated by Miyata (2010) that a monocentric
land use patter becomes an equilibrium one if parameters in the locational function are appropriately chosen. The
market rent function is denoted by;g ( x ) ≡ max {g B ( x ), g H ( x ), g A } where gA stands for agricultural land rent exogenously given.

(13)

If the business district is located surrounding the city center, and the residential area exists surrounding the business district, the equilibrium land rent is denoted as follows:g ( x ) = gB ( x ) ≥ gH ( x ) : x ∈ business district (14) g ( x ) = gB ( x ) = gH ( x ) : x ∈ boundary between thebusiness district and the residential area (15) g ( x ) = gH ( x ) ≥ gB ( x ) : x ∈ the residential area (16)g ( x ) = gH ( x ) = gA ( x ) : x ∈ on the city boundary (17)The constraints on the numbers of firms and the households are as follows:AB1
dx1 dx 2
mB ( x)where AB is business district.

(18)

AH

1
dx1 dx 2
mH ( x)where AH is residential area.

(19)

M=

∫∫

N=

∫∫

6. Local Balance Equations for Commodity, Labor, and Capital StockLet us assume the transport technology for commodities as von Thünen technology with a variable ratio ε (x) ≡ε (x1, x2) which depends on location. That is, the transport cost is incurred in the shipped commodities being expressed as ε (x)·q in shipping q unit of commodities in the unit distance. Let φ(x) be a two dimensional vector ofcommodities shipped to location x in unit time and in unit area size. Then the following local balance equation forcommodities is realized (Beckmann and Puu (1985)).

div ϕ ( x ) = q( x )b( x ) − c( x )h( x ) − ε ( x ) || ϕ ( x ) ||(20)

where b(x) : firms’ density at location x ( = 1 / mB(x) ), h (x) : households’ density at location x ( = 1 / mH(x) ),║·║ : norm of a two dimensional vector
7. Global Equilibrium Condition in Commodity and Labor Markets
Integrating the commodity local balance equation (20), and applying Gauss divergence theorem, one can obtain
the global equilibrium condition on commodities as presented in equation (21), since all commodities produced in
the city are consumed by households in the city, and we assume that there is no import and export of commodities.
So the right hand side in equation (21) becomes zero.

∫ ∫ div ϕ ( x )dx dx = ∫ ∫ [q( x)b( x ) − c( x )h( x ) − ε ( x ) || ϕ ( x) ||]dx dx = ∫
A

1

2

1

A

2

∂A

c n ( x ( s ))ds = 0

where ∂A: city boundary, cn(x(s)): net export whose direction is normal to the city boundary

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(21)

8. Commodity Transport Agent (C.T.A.)
Shipment of commodities is assumed to be done by the commodity transport agent (C.T.A.) (Beckmann and
Puu (1985)). The C.T.A. buys p(x)q(x)b(x) of commodities at point x, and sells p(x)c(x)h(x) of commodities tohouseholds. Thus the profit of C.T.A. at x is expressed as follows:

π Tq ( x ) = p( x )c( x )h( x ) − p ( x )q ( x )b( x ) = − p ( x )div ϕ ( x ) − p( x )ε ( x ) || ϕ ( x ) ||(22)

The C.T.A. aims to find the optimal route which maximizes the profit earned over the entire city area. This isdone by applying the variational principle to the integral of the local profit (22) over the entire city area. The optimal condition is finally reduced as equation (23).
p ( x )ε ( x )ϕ ( x)= grad p ( x )|| ϕ ( x ) ||(23)

In equation (23), φ(x) / || φ(x) || stands for the direction along which commodities are shipped, and that direction
coincides with the gradient of commodity price. That is, the direction along which commodities are transported isthe direction where commodity price gets highest. This is famous finding as the so-called gradient law by Beckmann (1952)
9. Commodity Price
The commodity price equation (23) can be transformed into the following non-linear first order partial differential equation.
22⎛ ∂ln p( x ) ⎞ ⎛ ∂ln p( x ) ⎞⎟⎟ = ε ( x ) 2⎟⎟ + ⎜⎜
⎜⎜
∂
∂
x
x
1
2
⎠
⎠ ⎝
⎝

(24)

Equation (24) can mathematically be solved, and its solution is formula (25) when the initial manifold is degenerated to a single point.
p ( x1 , x2 ) = p0 exp ε ( x1 , x2 ) x12 + x22(25)

The initial price p0 is obtained by the price on the city boundary determined by general equilibrium in the entire
city. The formula (25) is called integral conoid showing a cone shaped surface in the three dimensional space aspresented Miyata (2010), if ε(x1, x2) is a constant. Regarding variable ε(x1, x2), its graphical illustration will beshown later.
10. The Optimal Transport of Commodities
In this section, we try to solve the optimal volume of shipment of commodities. Transforming equation (23), thefollowing quasi linear first order partial differential equation is obtained.

~~
x1 ∂ || ϕ ( x ) ||x 2 ∂ || ϕ ( x ) ||+= q ( x )b( x ) − c( x )h( x )− || ϕ ( x ) || (ε ( x ) + div x~/ || x~ ||)~~|| x || ∂x|| x || ∂x
1

(26)

2

where ~
x1 ≡ (∂ε ( x ) / ∂x1 ) || x || +ε ( x ) x1 / || x || , ~x2 ≡ (∂ε ( x ) / ∂x2 ) || x || +ε ( x ) x2 / || x || , x~ ≡ ( ~x1 , ~x2 )The left hand side in equation (26) stands for a directional derivative with respect to the volume of commodity
shipment. Its direction is obtained by normalizing ( ~x1 , ~
x2 ) , and a change in commodity flow volume in the direction where the transport cost gets higher is larger than in other directions.
The first term in the right hand side expresses the supply of commodities at location x, the second term depicts
household consumption and finally the third term implies the depreciation in goods in transporting and general--4-ized diffusion effect. Here generalized diffusion effect means that a change in shipment of commodities getssmaller as the location of commodities transported gets further from the city center. This fact never appears unless
SCVF and land density are taken into account.
Although div x~/ || x~ || can be expanded, it becomes a very complex formula and it is not easy to see its economic implication. However when a change in ε(x) is small as compared with the distance between x and the city
center, div x~/ || x~ || becomes a decreasing function of distance showing a property of diffusion effect.
The right hand side in equation (26) is a scalar. Therefore if ~x1 / || ~x || in the left hand side is large (i.e. thetransport cost is high in the direction of x1), it is natural that ∂ || ϕ ( x ) || / ∂x1 gets smaller. That is, the volume ofcommodities transported becomes smaller in the direction where the transport cost gets higher. This matches oursense in observing usual transport phenomena.
Now solving equation (26) introducing a distance parameter s with the initial value of zero at the city center, we
obtain equation (27).
|| ϕ ( x ( s )) ||=⋅ exp ⎛⎜ −⎝∫s

∫ [q( x(τ ))b( x (τ )) − c( x (τ ))h( x(τ ))]0

div x~ (η ) / || x~ (η ) || dη ⎞⎟exp ⎛⎜ −
τ
⎠
⎝
s

s

∫τ

(27)

ε ( x (η )) dη ⎞⎟ dτ⎠Moreover the vector of commodity flow is denoted by;ϕ ( x ) = (|| ϕ ( x ) || ~x1 / || x~ ||, || ϕ ( x ) || ~x2 / || ~
x ||)

(28)

If SCVF is even, the direction of commodity shipment shows a line. But when SCVF is uneven, the direction
will illustrate a curve.
Now let us explain the economic implication of equation (27). [ · ] in the integral is commodities produced minus household consumption. The second term describes the generalized diffusion effect in which the volume ofcommodities shipped to the point x more decreases as x gets further from the city center. This is the first theoreticalfinding among literature, and it never appears without variable land density and SCVF.
The last term is an exponential integral of the variable transport cost showing that the commodities are depreciated by transport services. Incidentally if ε is a constant, then the last term becomes exp (-ε·(s-τ)) and one can intuitively see its economic implication. The equation (27) can be described in words as follows:
commodity volume shipped to point x = (total supply of commodites – total consumption of commodities)×
generalized diffusion effect – depreciation in commodities by transport services
11. Equilibrium Conditions
Equilibrium conditions in the model are summarized as follows:
Commodity market: || ϕ ( x ( S )) ||= 0 in equation
Labor market: || ψ ( x (0)) ||= 0 (the number of households commuting into the city center = 0)
Capital market: ks( x ) = k d ( x )b( x ) and∫ ∫ ks( x)dx dx = N ⋅ k
1

ALand market: equations (14) to (17)Location conditions: π B ( x ) = 0 : x ∈ business district
The numbers of firms and households:

M=

∫∫

AB

1
dx1 dx 2 (AB : business district) (34)mB ( x)

-5-

2

s

(29)
(30)
(31)

(32)

u ( x ) = u * : x ∈ residential area∫∫1
dx1 dx 2 (AH : residential area) (35)mH ( x)

N=

AH

(33)

From these conditions, bid rents gB(x) and gH(x), bid max lot sizes mB (x) and mH (x) and the equilibrium utility
level are obtained determining the equilibrium urban configuration.
12. Introducing Spatial Characteristic Vector Field
As an example of SCVF, we consider a case where the transport costs increase at certain place in the city. Figure
1 shows a change in heights from the city center to a point (7, 7). The land shape shows a slope from a circle passing a point (3. 3), and the height hits the peak of 4.5 at a point (7, 7). This change is expressed by a part of the normal distribution surface, and then the spatial configuration of commodity price shows a surface presented in Figure 2 hitting its peak at point (5,5).
In the slope area, the isoprice set is calculated as a circle of (x1-5)2+(x2-5)2 = r 2 differing from the isoprice set
from the city center x12 + x22 = r 2. Thus transport routes become a set of curves from points (3, x2). Then commodity supply, household consumption, land use pattern etc. are determined under the general equilibrium framework.
Further considerations are left as a future task.
height

5

4.5

標 4.5
高

4
3.5
3
2.5
2
1.5
1
0.5
0

4
3.5

0

1

2

3

4

都心 from
部からの距離distance
the city center

5

3
2.5
2
1.5
1
0.5
0

10
6

7

8

9

5

10

distance from the city center

Figure 1. Hypothetical Geographical Condition

0
-5
-10
-10

-6.5

-3

0.5

4

7.5

distance from the city center

Figure 2. A Change in Spatial Configuration of Commodity Price
13. Concluding Remarks
This paper has examined the patterns of commodity and labor shipments in a monocentric city by solving thepartial differential equations under the framework initiated by Beckmann (1952) and Beckmann and Puu (1985).
This study extends my previous achievement into a case of variable von Thünen transport coefficient introducing
SCVF. The new findings in this paper are summarized as follows:(1)Transport networks can be internalized by applying the variational principle.
(2)Although only linear roads in a network appear over an even SCVF, curved roads in a network can be internalized over an uneven SCVF.
(3)By introducing uneven SCVF, generalized diffusion effect and generalized congestion effect can be obtained.
(4)Commodity and labor flows irregularly change over an uneven SCVF.
(5)A curved road relates the light changing in a continuously uneven medium. Relationship our approach and thewave equation could be an interesting theme.

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