Download an evolutionary new economic geography model

JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000, pp. 671–695
AN EVOLUTIONARY NEW ECONOMIC GEOGRAPHY MODEL*
Wei Fan
Department of Economics, University of Michigan, Ann Arbor, MI 48109, U.S.A. E-mail:
[email protected]
Frederick Treyz and George Treyz
Regional Economic Models, Inc., 306 Lincoln Ave, Amherst, MA 01002. E-mail:
[email protected] and [email protected]
ABSTRACT. In this paper we present a general new economic geography model with
multiple industries and regions, full labor and capital mobility, land use in production and
consumption, and a dynamic adjustment process in which consumers maximize utility and
firms respond to nonzero profits. All industries use intermediate inputs as well as land,
labor, and capital. Systems of cities form endogenously within this framework, including
asymmetrical urban hierarchies and cities of different sizes and industry compositions.
Each urban area has a bid-rent gradient and zones with land uses and densities as in the
von Thünen model. The equilibrium depends not only on initial conditions but also on
speeds of adjustment. The model is a prototype for empirical implementation, as illustrated
with a simulation of the effects of transportation cost reductions.
1.
INTRODUCTION
The emerging area of study known as the new economic geography has
greatly enriched spatial economic analysis as evidenced by the burgeoning
literature in regional, urban, and international economics (see Krugman, 1998a;
Fujita, Krugman, and Venables 1999 for recent reviews). Models developed in
this field have led to fresh insights on a wide range of issues such as specialization in a system of cities (Abdel-Rahman, 1996), industry location decisions
across countries (Venables, 1996), the effects of congestion (Brakman et al.,
1996), and preindustrial agglomeration (Duranton, 1998). However, researchers
recognize the need for a more general spatial economic model. Fujita and
Krugman (1995) unify von Thünen and Chamberlin models in a way that they
suggest will “lead towards the development of a general equilibrium model of
urban systems.” Krugman (1998b) calls for development of a “computable
*We are grateful to two anonymous referees for their insightful comments and Dr. Omar
El-Gayar’s help in many respects. Wei Fan is also grateful to Gordon Hanson, Alan Deardorff, and
Michelle White for their constructive discussions, and for financial support from Regional Economic
Models, Inc.
Received November 1998; revised May 1999 and November 1999; accepted March 2000.
© Blackwell Publishers 2000.
Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA and 108 Cowley Road, Oxford, OX4 1JF, UK.
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
geographical equilibrium” model that can answer questions about the effects of
technological shocks and policy changes on the economy.
In this paper we present a new economic geography model with endogenous
land use, labor mobility, interindustry purchases, and N-locations in one- or
two-dimensional space. This general structure produces stylized metropolitan
forms with rich structures, including an endogenously generated bid-rent and
density gradient, and concentric zones specializing in different industries.
Additionally, the model presented in this paper has much in common with
regional computable general equilibrium models. This suggests uses of the
model to evaluate regional economic issues, such as the effect of environmental
regulations, economic development initiatives, and productivity changes on
regional economic development. As an example, simulation results presented in
this paper illustrate the effect of transportation cost changes on the spatial
economic structure.
The model that we present is not the first that moves towards the goal of a
generalized new economic geography model. For example, Puga (1999) presents
a model that unifies different strands of work relating to international trade
theory. However, his model is not applicable to regional or urban analysis
because it does not include land explicitly. A generalized regional economic
model is presented in Fujita, Krugman, and Mori (1999) and Fujita, Krugman,
and Venables (1999) but the formulation makes constraining assumptions such
as the existence of an unlimited amount of land in the economy and the absence
of capital stock. The model presented here relaxes some of the most restrictive
assumptions of the majority of new economic geography models. In the formulation of this paper we assume the following: workers are mobile between sectors
and regions; land is explicit in consumption and production for all sectors,
differentiated inputs are used in production; and geographic space is discrete
so that numerical solution methods may be applied easily in either one- or
two-dimensional space. As in many other models cited in this paper the agglomerative forces in our model are the price effects (both the consumers’ price and
producers’ price) and the wage effect. However, unlike models based on immobile
production sectors, the spreading force in this model comes solely from demand
for a limited supply of land as it is in Helpman (1998). Helpman’s model has
only two regions and a single differentiated industry. In addition to land and
labor we also have capital as a production factor (Baldwin, 1999a). Despite some
common features with existing models, our model is an initiatory model with
endogenous land and capital use and labor mobility in a formulation with
N-locations and interindustry transactions.
In Section 2 we formally present the model. In Section 3 we describe the
evolutionary solution method used to solve for the geographical economic
equilibrium. We present results of some simulations in Section 4. In particular,
the economy in this model organizes itself into cities with different sizes and
different industrial structures depending on the initial population and capital
stock distributions and adjustment speeds. Simulations show the change in the
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FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL
673
regional and urban structure caused by reductions in transportation costs. In
Section 5 we summarize the paper and discuss possible model extensions.
2.
THE MODEL
In this model there are N regions (i = 1,2, . . . , N). The population in the
economy is L. The population, land area, and capital stock in region i are denoted
by Li, Si, and Ki, respectively. Identical individuals each provide one unit of labor
and have the same preferences. All consumers consume land and M commodities
produced in the economy with an identical Cobb-Douglas utility function
M
∏c
ui = siθ0
(1)
θm
mi
m=1
where si is the amount of land and cmi is the amount of composite commodity m
consumed by a representative consumer in region i. Throughout the paper
subscripts i and j are reserved for regions and m is reserved for industries or
commodities. A subscript combination mi stands for industry m in region i or is
simply called regional industry mi. All m commodities are differentiated. In each
region the composite commodity m is composed of different varieties in the
Dixit-Stiglitz (1977) form
F
= G∑ c
GH
nm
cm
σ m −1
σm
mi, v
v= 1
σm
σ m −1
I
JJ
K
where nm is the total number of commodity m varieties produced in all N regions,
cmi,v is the amount of variety v of commodity m consumed by a representative
region i consumer, and σm is the elasticity of substitution between two commodity m varieties. With a Cobb-Douglas utility function a representative consumer
spends a constant budget share on land and each of the M commodities. The
budget shares for land and the M commodities are θ0 and θm, m = 1, 2, . . . , M,
M
respectively, and θ 0 +
∑θ
m
= 1.
m= 1
All commodities are produced with increasing returns-to-scale technology
using capital, land, labor, and intermediate inputs. A profit-maximizing firm
produces only one variety of commodity because of the benefits of increasing
returns. Therefore, the same index v is used to represent a firm or a variety. The
production function for a representative firm in each industry is assumed to be
in the following Cobb-Douglass production function with increasing returns to
scale
(2)
φ mk
φ ml
φ ms
M
d k i d s i dl i ∏ d z
mi, v
mi, v
m′ mi, v
mi, v
m′= 1
© Blackwell Publishers 2000.
i
φ mm′
= α m + zmi, v
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
where zmi,v is the amount of variety v produced by firm v in regional industry
mi; φmk, φms, φml are the capital, land, and labor shares in industry m,respectively;
and φmm′ is the share of intermediate input m′ in industry m. The fixed cost is
αm. Finally, kmi,v, smi,v, and lmi,v are the amount of capital, land, and labor used
in the production, and zmm′i,v is the composite intermediate input produced by
industry m′ used by firm v in regional industry mi. To simplify the solution
method, intermediate inputs are assumed to be of the same form as the
composite consumption commodities. The production functions for all industries
are in the same form but they can differ in input shares, fixed costs, and marginal
costs. However, all regions share the same production technology.
Industries can also differ in transportation costs. In order to set up a
tractable model, transportation costs are assumed to take the iceberg form. That
is, for each unit of commodity m goods shipped from the port of origin region i
for a distance of dij, only (1 + γmdij)–1 unit arrives at the port of destination region
j (γm > 0). Consequently, for industry m the c.i.f. price will be higher than its f.o.b.
price by a factor of (1 + γmdij), that is
Pmij = Pmii (1 + γmdij)
(3)
where dij is the distance between region i and region j. Pmii is the f.o.b. price of
commodity m in region i. As is well known, under the assumption that there are
a large number of firms in each industry, all firms face the same elasticity of
substitution (Krugman, 1980). Assuming that all firms practice mill pricing,
each firm will then charge the same f.o.b. price to consumers from all regions. It
can be shown that with the production function given in Equation (2) and the
CES utility function, the f.o.b. price charged by profit-maximizing firms in
regional industry mi is
Pmii =
(4)
σm
Ω mi
σm − 1
where Ωmi is the marginal cost function of a representative cost-minimizing firm
in regional industry mi. Furthermore, it can be shown that
M
(5)
Ω mi = rki ψ mki + rsi ψ msi + wi ψ mli +
∑P
m′ i ψ mm′ i
m′= 1
where rki, rsi, and wi are the capital rental rate, land rental rate, and wage rate
in region i, respectively. Pmi is the price index of commodity m defined as
F
= G∑ n
GH
N
(6)
Pmi
j =1
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1− σ m
mj Pmji
I
JJ
K
1
1− σ m
FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL
675
where Pmji is the c.i.f. price in given in Equation (3). The terms ψmki, ψmsi, ψmli,
ψmm′i are the standard factor requirement functions for capital, land, and labor,
and the intermediate input requirement functions for regional industry mi,
respectively. They are functions of the three factor prices and the M intermediate
input prices, for example
(7)
ψ mki
LF r I F φ I OP
= MG J G
MNH r K H φ JK QP
si
mk
ki
ms
φ ms
LMF w I F φ
MNGH r JK GH φ
φ ml
i
mk
ki
ml
I OP LMF P I F φ
JK PQ ∏ MNGH r JK GH φ
M
m′= 1
m′i
mk
ki
mm′
I OP
JK PQ
φ mm′
The profit made by a representative firm in industry m can be shown as
(8)
π mi = Ω mi
Fz
GH σ
mi,v
−1
m
− αm
I
JK
The zero-profit output level for a representative firm in industry mi is
b
g
0
zmi
= αm σm − 1
and the equilibrium number of firms after entry in industry mi is
(9)
0
nmi
=
F L
GH α σ
mi
m
I bψ g
JK
−1
mli
m
Consumers, who are also the owners of the firms, have three income sources:
labor income, rental income, and profits. For simplicity, land and capital in all
regions are assumed to be owned equally by all consumers in the economy, and
each consumer provides one unit of labor. Under these two assumptions, per
capita income in region i can be expressed as
(10)
yi = wi + π i +
1
L
N
∑ dr
sj S j
+ rkj K j
i
j =1
The first term on the right-hand side of Equation (10) is the labor income or the
wage rate w. The second term is per capita profit defined as follows
πi =
1
Li
M
∑n
mi π mi
m= 1
The last term is rental income, which is simply the total rental spending in the
economy divided by the total number of people in the economy. It can be shown
that the total spending on capital and land are
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
L
O
Fφ I
∑ MMNGH φ JK w L PPQ
M
rki K i =
mk
i
ml
m=1
L
O
Fφ I
∑ MMNGH φ JK w L PPQ
M
rsi Si = θ 0 yi Li +
mi
ms
i
mi
ml
m=1
Notice that there is both consumption spending and production spending on
land. One can solve per capita income from Equation (10) and express it as
r
r r r
y = A −1 w + π + ∆
e
j
where
LM1 − L θ
MM LL
θ
−
A=M
L
MM LM
MN − L θ
1
1
0
0
1
0
L2
θ0
L
L
1 − 2 θ0
L
M
L2
θ0
−
L
−
LN
θ0
L
L
...
− N θ0
L
...
M
LN
θ0
... 1−
L
′
−
...
OP
PP
PP
PP
Q
r
y = y1 , y2 ,..., y N
b
g
r
′
w = bw , w ,..., w g
r
′
∆ = b ∆, ∆,..., ∆ g
r
′
π = b π , π ,..., π g
LF φ + φ I
O
1
w L P
∆ = ∑ MG
J
∑
L
MNH φ K
PQ
1
2
1
2
N
N
M
N
mk
ms
j
ml
m= 1
j
j =1
Finally, B and H are defined as the factor share matrix and intermediate
input share matrices, respectively
LM
MN
φ 1k
B = φ2k
φ3k
φ 1s
φ2s
φ 3s
OP
PQ
φ 1l
φ 2l ,
φ 3l
and
LM
MN
φ 11 φ 12
H = φ 21 φ 22
φ 31 φ 32
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φ 13
φ 23
φ 33
OP
PQ
FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL
677
Intermediate inputs are produced with the three production factors, so B* is
defined as the effective factor share matrix. One can show that
B* = (I – H)–1B
The effective factor income shares in the economy are
M
Θf =
(11)
∑θ
*
m B mf
f = k, l
m= 1
and
M
Θs = θ +
(12)
∑θ
*
m B ms
m=1
Finally, Region 1’s labor is chosen as the numeraire, that is, Region 1’s wage
rate is always normalized to one.
3.
THE DYNAMIC ALGORITHM
Fujita and Mori (1997) formally introduced the evolutionary solution
method from complexity theory into the field of economic geography to numerically solve this type of model, which is otherwise very difficult to solve analytically. We adopt their basic methodology to solve the model proposed in Section 2.
Essentially, the economy starts with a given initial population distribution and
the autarkic equilibrium industrial composition. Then, the economy evolves by
itself at each period based on a set of prescribed laws of motion. The evolution
process continues until the economy reaches a full equilibrium in which all
markets clear and utility levels are the same in all regions. The laws of motion
describe how the economy adjusts to its equilibrium from an off-equilibrium
position.
In this model, there are (M + 3) markets in each region: M commodity
markets plus the capital market, the land market, and the labor market. The
algorithm assumes that the land rental rate adjusts fast enough to clear the
land market in each region in every period. For the capital market, it is assumed
that in each period the capital stock is fixed in each region and the capital rental
rate adjusts instantaneously so that the capital market is cleared in each region.
However, it takes time for capital to flow from one region to the other. Therefore,
the capital rental rate can be different among different regions during the
adjustment process. Wage rates adjust based on labor market conditions. The
number of firms in each regional industry adjusts according to its profit level.
Wage rates adjust according to the demand for labor at zero-profit production
level. The output level produced by a firm is determined by total expenditure on
the regional industry’s output and the number of firms in the regional industry.
Price levels evolve by iteration according to Equations (3) to (6). Labor moves
across industries based on the difference between the actual employment level
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
and the hypothetical zero-profits employment level. Finally, the population in
each region adjusts through migration to equalize the utility level in all regions.
Specifically, the algorithm starts with initial conditions including population distribution in space {Li, t = 0}, capital stock distribution in space {Ki, t = 0},
labor allocation among industries in each region {Lmi,t = 0}, capital stock allocation
among industries in each region {Kmi, t = 0}, and initial wage rates {wmi, t = 0}. The
capital rental rate and land rental rates are set to clear the capital market and
land market, respectively, in each region
rki =
rsi =
1
Ki
M
Fφ I
∑ GH φ JK w L
mk
i
m=1
LM
MN
i
ml
M
F I
JK
φ ms
1
θ 0 yi Li +
wi Li
Si
φ ml
m=1
∑ GH
OP
PQ
The initial factor and intermediate requirement functions ψmki, ψmsi, ψmli,
and ψmm′i are first calculated as if there were no intermediate inputs. Then in
each period, commodity prices are calculated using Equations (3) to (6) with
previous cost values in Equation (5). Equations represented by Equation (7)
update the factor and intermediate input requirement functions and unit cost
functions with the newly calculated commodity prices. The initial number of
firms in each regional industry is set to the zero-profit number of firms given in
Equation (9).
The spending by a region on each of the M commodities produced in a
particular region is
Emij =
nmiω mi µ mij
ESmj
N
∑n
mkω mk µ mkj
k= 1
where Emij is the spending by consumers and producers in region j on commodity m produced in region i. On the right-hand side, ESmj is total spending in
region j on commodity m including both consumption spending and production
spending shown as
M
ESmj = θ m y j L j +
Fφ I
∑ GH φ JK w L
m′= 1
The ωs and µs are defined as follows
ω mi = Ω 1mi− σ m
and
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m′m
m′l
j
j
FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL
d
µ mij = 1 + γ m dij
i
679
1− σ m
In dollar value, the demand for all commodity-m varieties produced in region i
is
N
EDmi =
∑E
mij
j
The demand for a regional industry’s output is transformed into the hypothetical zero-profit demand for labor by firms in this regional industry, that is
D
Lmi
=
φ ml EDmi
wi
and
M
LiD =
∑L
D
mi
m= 1
is the total zero-profit demand for labor in region i. The total zero-profit demand
is the demand for labor in a regional industry for a given demand for the regional
industry’s output if profits for all firms were eliminated by competition.
The difference between the zero-profit demand and the actual supply of
labor reflects tensions on the labor market. The wage rates adjust according to
the zero-profit demand for labor in excess of supply with a damped adjustment
factor, that is1
wi,t + 1 = wi,t + λ w LiD,t − Li,t
e
j
where λw is the speed of adjustment for the wage rate.
The indirect utility level can be calculated using the income, the rental rate,
the commodity price indices and the utility function in Equation (1), as follows
(13)
Vi = yi rsi− θ 0
M
∏P
−θm
mi
m= 1
The migration decision is based on the indirect utility level for the representative consumer in each region. If the utility level in a region is lower than
the average utility level in the economy, residents in that region will move to
regions with relatively higher utility levels. For regions with above-average
1
The way zero-profit demand is calculated and the way adjustment process for wage rates is
modeled assume no forward-looking behavior in labor contract negotiation. Baldwin (1999b) shows
that inclusion of forward-looking expectation leads to absolutely no change in the main results of
the standard core-periphery model.
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
utility level the amount of in-migration into the region is greater with higher
utility level. Our model simply assumes that the number of migrants to region
i in period t, migi, t, that an above-average region receives is proportional to the
deviation of its utility level from the average. Formally
Li,t + 1 − Li,t = migi,t
R|λ dV − V i L
S|λ ρ ∑ dV − V i L
T
L
i,t
t
i,t
L i,t
t
k, t
k,t
if Vi,t ≤ Vt
if Vi,t > Vt
Vk, t < Vt
where λL is the migration rate, Vt is the population average of indirect utility,
and ρi,t is the proportion of migrants an above-average region i receives. These
are defined by
1
Vt =
L
N
∑V
i,t Li,t
i=1
and
ρ i,t =
Vi,t − Vt
∑ dV
k,t
− Vt
i
Vk, t > Vt
Capital stock moves based on the capital rental rate differences in the N
regions. In this model, it follows a similar dynamic process to the movement of
people. That is, capital flows into regions with an above-average capital rental
rate and out of regions with a below-average capital rental rate. The capital
migration rate is λk.
In each region and each period labor moves, with friction, from industries
with a low zero-profit demand for labor to industries with a high zero-profit
demand. If there are any new in-migrants, they are allocated proportionally to
each industry’s zero-profit demand for labor because they have already left their
old job by out-migration, or in equation
F
GH
Lmi,t + 1 = Lmi,t + λ m Li,t
D
Lmi
,t
LiD,t
I
JK
− Lmi,t + migi,t
D
Lmi
LiD,t
where λm is the intersector migration rate.
Finally, the output level of a representative firm in regional industry mi is
calculated from the material balance equation
zmi,v =
© Blackwell Publishers 2000.
EDmi
nmi Pmii
FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL
681
The profit level can then be calculated from Equation (8). At the end of each
period the number of firms in each regional industry changes due to the entry
and exit decision as follows
nmi,t = nmi,t–1 + λnπmi,t–1
(14)
where λn is the entry and exit rate.
The key to the evolutionary method is that the industrial composition in
each region is determined endogenously. A regional industry may disappear and
new regional industries may be born as the economic system evolves. A regional
industry disappears when the number of firms in that regional industry goes
below one following the dynamics specified by Equation (13). A new industry
will be born if a tentative firm in the new industry has enough demand for its
output to make a positive profit. Specifically, if in each period the number of
firms in a regional industry goes down to less than one then the regional industry
disappears—both the number of firms and amount of labor in the regional
industry are set to zero. On the other hand, for each missing regional industry
the profitability of establishing a firm in that industry is determined by calculating the profit level given that there is one firm in the regional industry. If a
new firm will be profitable it will then come into existence (a new firm appears);
otherwise that regional industry does not form.
With the existence of multiple equilibria the economy’s off-equilibrium
behavior matters a great deal to the final equilibrium. The path-dependence
phenomena in economics is now well known not only in economic geography, but
also in other areas as well (for example, Arthur, 1994). Yet, the dynamic process
specified in this paper is still more for convenience than for realism. In reality
an off-equilibrium economy is characterized by unemployment, temporary
profit, inventory, involuntary saving, and so forth. The laws of motion described
in this paper are an abstraction of the more complicated adjustment process of
the real economy.
4.
SIMULATION RESULTS
In this section we present results for different model specifications to show
the endogenous generation of a spatial economy with urban agglomerations,
density and rent gradients within metropolitan areas, and complex systems of
cities. An eight-region and a fifty-region “racetrack” model illustrate land uses
and rents, as well as change that results from lower transportation costs.
Table 1 presents the input-output relationships used throughout the paper
unless otherwise specified. Agriculture, manufacturing, and service industries
all use intermediate inputs and three factors: capital, land, and labor. Factor use
reflects the land intensity of agriculture, capital intensity of manufacturing, and
labor intensity of services. Interindustry relationships reflect the use of manufactures in agriculture and the reflexive dependence of manufacturing and
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
TABLE 1: Factor and Intermediate Input Share in Production
Agriculture
Manufacture
Services
0.01
0.10
0.04
0.10
0.60
0.15
0.05
0.15
0.10
0.15
0.10
0.45
0.02
0.10
0.20
0.10
0.08
0.50
A-Product
M-Product
S-Product
Capital
Land
Labor
TABLE 2: Consumption Share
Land
A-Product
M-Product
S-Product
0.20
0.10
0.30
0.40
TABLE 3: Technological Parameters
Agriculture
Manufacturing
Service
1.00
1.00
0.20
6.00
5.00
1.00
0.05
3.00
3.00
1.00
0.50
3.00
Fixed Cost
Marginal Cost
Transportation Cost
Elasticity of Substitution
services.2 Table 2 shows consumption shares in land, agricultural and manufactured products, and services. Table 3 shows technological parameters and
transportation costs. These reflect a high fixed cost and low transportation cost
for manufactured goods, a low fixed cost for agricultural products, and very high
transportation costs for services. The elasticity of substitution for production
reflects a relatively homogenous agricultural product and differentiated manufactured goods and services.
An Eight-Region Model
A single-city economy. In this section we present results for an eight-region
racetrack economy where the regions are uniformly distributed on a circle and
each has the same land area. All goods must be transported along the circumference. The distance between two consecutive regions is one unit of distance.
Therefore, there is no first-nature locational advantage (Krugman, 1993). The
initial condition is specified such that Region 1 has 93 percent of the population
and the other 7 regions each have 1 percent of the population. All regions have
the same wage rates and allocate their labor according to their autarkic demand
calculated with the effective labor demand as
2
We also ran simulations using more real-world coefficients, such as those in Sonis, Hewings,
and Gazel (1995). However, in the experiment the agglomerative force dominated the outcome which
located all nonagricultural economic activity in a single location.
© Blackwell Publishers 2000.
FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL
Lmi =
683
θ m (B*) ml
Li
Θl
The adjustment speeds are λL = 0.1, λw = 0.01, λK = 0.002, λn = 0.1, and
λm = 0.25. These speeds remain the same throughout this paper unless otherwise specified.
The population distribution in the economic geographic equilibrium is
shown in Figure 1. The figure shows the final population distribution and the
final labor allocation in each regional industry. It shows that Region 1 develops
into service centers, and Regions 3 and 8 specialize in the manufacturing
industry. Regions 4, 5, and 6 are occupied by agriculture. Region 2 is a mixture
of services and manufacturing and Region 7 is a mixture with mostly manufacturing and agriculture. Regions 1, 2, 3, 7, and 8 could be seen as cities whereas
Regions 4, 5, and 6 form a rural area. The spatial distributions of factor prices
are shown in Figure 2. As is expected, capital rental rates are the same in all
regions and land rental rates are the highest in the service centers and lowest
in the rural regions. Moreover, the disparity in wage rates is much smaller than
that in land rental rates. This result is consistent with the large variance in
rents and the relatively smaller variance in wages observed among U.S. cities.
However, the results show higher wage rates in rural regions than in the cities.
With labor mobility, at equilibrium, whether workers in the cities receive
higher wages depends on whether the disutility of congestion outweighs the
benefit of lower price index from the variety effect. It is possible that the benefit
of living in a city is sufficiently high that people are willing to accept a lower
wage rate for the privilege of city life. In other words, farmers may be highly
paid in order to compensate them for living in remote areas with a low variety
of goods and services.
Smaller wage rate differences are an outcome of the indirect utility function
in Equation (13). The effective land income share and labor income share in the
economy for the input-output coefficients given in Table 1 can be calculated using
Equations (11) and (12). The effective wage share in income is
Θl
0.5108
=
≈ 0.8
Θ l + θ0 + Θ s
0.5108 + 0.1271
b
g
A 1 percent increase in the wage rate increases income and indirect utility by
about 0.8 percent. On the other hand, a 1 percent increase in the rental rate
results in a Θ0 percent (0.2 percent) decrease in indirect utility. Therefore, other
things being equal, a 1 percent difference in regional wage rates must be
compensated by a 4 percent difference in the rental rate for the indirect utility
to be equal. This explains why the variance in equilibrium land rental rates is
much greater than the variance in equilibrium wage rates.
Path dependence. It is well known that with increasing returns, economic
geographic equilibrium depends not only on the preference and industry
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
FIGURE 1: The Economic Geographic Equilibrium.
(The populations are in thousands)
structures in the economy but also on the evolution path leading to the equilibrium. Krugman (1995) is one of several to show the dependence of the final
equilibrium on the initial configuration. In this section we reaffirm these earlier
results within the context of a generalized new economic geography model.
Moreover, the result shows the dependence of the final equilibrium choice on
another aspect of the evolution path—the adjustment speed, a phenomenon that
has not been observed in previous work in the field.
© Blackwell Publishers 2000.
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685
FIGURE 2: Factor Prices in Economic Geographic Equilibrium.
The eight-region model is used to generate 1,000 outcomes starting with
1,000 different random initial population and capital stock distributions. A total
of 27 different equilibria are found in the 1,000 outcomes.3 Figure 3 shows
3
Two distributions are considered as the same if they differ by a translation or reflection
operation because of the symmetry of the system. For example, for a distribution of ABCDEFGH a
translation operation to the right by one position changes the distribution to BCDEFGHA. A
reflection operation around B changes the original distribution to CBAHGFED
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
population distributions of the ten most frequent equilibria. The differences in
total population distribution between some of the equilibria (e.g., between
equilibria #1 and #2, between equilibria #5 and #7) are so small that they are
virtually invisible on the bar charts. However, they represent distinct equilibria
associated with different industrial composition.
FIGURE 3: Effect of Initial Population Distribution on Geographic
Equilibrium. (The populations are in thousands)
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687
Table 4 shows the 27 equilibria that resulted in an experiment of 1,000
outcomes starting from random initial conditions. The table shows the industry
mix for each outcome along with the probability (frequency) of its occurrence.
A, M, and S stands for a region specializing in agriculture, manufacturing, or
service, respectively. AM stands for a region with a mix of agriculture and
manufacturing, MS for a region with a mix of manufacturing and services, and
so on. One can see that equilibria #1 and #2 are the most likely outcomes with
a combined probability of over 64.5 percent. Although the difference in the
population distributions is very small, the industry distributions are clearly
different. There are two regions specializing in agriculture in equilibrium #1,
but there are three of them in equilibrium #2. In all 27 outcomes there is no
region with a mix of agriculture and service. Services and agriculture do not mix
because both the forward and the backward linkages are weak. If a region
chooses to produce in two industries it is more cost-efficient to produce either
in agriculture and manufacturing if the land rental rate is low, or in manufacturing and service if the land rental rate is high. The probability of the
TABLE 4: The Equilibrium Industrial Structure
Equilibrium
Configuration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Probability
of Occurence
Industrial Structure
A
M
A
M
A
MS
A
S
S
AM
MS
S
AM
S
M
AM
M
S
S
AM
M
AM
S
S
A
MS
A
A
AM
A
A
MS
S
A
AM
M
S
S
AM
AM
MS
A
S
A
M
M
S
S
AM
AM
AM
M
S
AM
© Blackwell Publishers 2000.
AM
A
A
A
S
M
A
A
A
MS
M
A
S
M
A
M
MS
A
MS
AM
AM
S
A
AM
AM
AM
S
M
A
M
A
M
A
MS
A
A
AM
A
AM
M
A
A
AM
S
A
S
MS
A
AM
AM
MS
S
A
AM
S
A
S
M
S
A
S
A
A
A
A
MS
A
A
AM
MS
AM
AM
M
AM
A
A
S
AM
MS
A
AM
MS
M
MS
MS
AM
A
M
AM
MS
M
S
S
AM
AM
S
S
AM
S
A
S
M
A
AM
S
A
AM
S
M
MS
S
S
A
MS
S
S
S
S
AM
AM
S
A
M
M
S
MS
A
M
MS
MS
MS
AM
M
S
MS
AM
S
M
MS
A
M
MS
M
M
AM
AM
AM
MS
M
S
A
AM
AM
AM
A
S
S
AM
AM
S
M
M
0.335
0.310
0.073
0.053
0.046
0.034
0.027
0.027
0.025
0.017
0.010
0.007
0.006
0.006
0.005
0.003
0.003
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
0.001
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formation of a two-city equilibrium with at least two pure service regions
separated by nonservice regions is very small. From Table 4, this probability can
be seen to be 5.6 percent. In fact, among the top-ten list, only the tenth is a
two-city equilibrium.
As far as we know, there has been no study on the effect of adjustment speed
on equilibrium choice. This is not surprising. For most of the models developed
so far there are two adjustment processes at most: the wage-rate adjustment
and migration. Migration is assumed to occur after wage rates adjust to clear
the markets. Under these assumptions, changing the adjustment speeds is
equivalent to changing the unit of time in two separated processes. Therefore,
there is no effect on the equilibrium the economy eventually reaches. In this
model, all adjustment processes occur in the same time so a change in relative
adjustment speed can have real effect on the economy. For example, a change in
capital migration rate may result in a higher per capita capital stock in some
regions, and a lower per capita capital stock in others.
Figure 4 illustrates this property of the model with two population migration rates. For the eight-region model starting with 93 percent of the population
in Region 1, a migration rate λL of 0.1 produces a configuration with two large
equally populated regions. One of Region 1’s neighbors ends up in exactly the
same position as Region 1, although they started with different initial conditions.
On the other hand, the outcome of a lower migration rate of 0.001 is a different
equilibrium with the first region remaining as the biggest center for the
economy. It is interesting to notice that the reflection symmetry about Region
1 of the initial distribution is broken in the first equilibrium, but is maintained
in the second equilibrium. The dependence on the adjustment speed reemphasizes the importance of the off-equilibrium behavior in economies characterized
by increasing returns.
Urban systems. Elaborate metropolitan areas emerge within a regional
system as more regions are added to the model. This is illustrated with a
fifty-region economy. Figure 5 shows the equilibrium distributions of total
population, labor forces employed in each of the three industries, capital stock,
capital rent, land rent, and wage rate in each region. Characteristics of the von
Thünen isolated state begin to emerge, including patterns of population density,
industry composition, and land rents that are found in a spatial economy with
urban, suburban, and rural regions.
Figure 5 illustrates a system of cities and rural areas in which there are
multiple metropolitan centers, each with increasing population densities and
land rents moving from the hinterland to the urban core. There are five separate
cities with the largest one centered on Region 37. The two medium-sized cities
centered on Regions 19 and 50 are about the same size. The two smallest cities
are located at Regions 10 and 26, respectively. There are five small towns
scattered in rural regions specializing in manufacturing at Regions 6, 29, 32, 42,
and 44. Within each metropolitan center, the service industry is located at the
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FIGURE 4: Effect of Migration Rate on Geographic Equilibrium.
(The populations are in thousands)
center and is surrounded by the manufacturing industry. This is qualitatively
similar to observed patterns of urban development in which a central business
district is occupied by banking, insurance, and other service firms, and is ringed
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FIGURE 5: The Economic Geographic Equilibrium.
(The populations are in thousands)
by zones of manufacturing and agriculture. There are 12 pure-service regions,
14 pure-manufacturing regions, 12 pure-agriculture regions, 10 regions with a
mix of agriculture and manufacturing, and 2 regions with a mix of manufacturing and service. Not surprisingly, capital stock is heavily concentrated in the
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metropolitan centers. The land rental rate is the highest in Region 37, the center
of the largest city, with a value of 68.30. The lowest rental rate is in Region 6
with a value of 14.26. The highest land rental rate is about 4.8 times as high as
the lowest land rental rate. Again, the difference in wage rates is very small,
and the wage rates in the rural region are slightly higher than the wage rates
in the metropolitan centers. The highest wage rate is in Region 31, a pureagriculture region, with a value of 1.105, while the lowest wage rate of 0.997 is
in Region 18, a suburban region of one of the medium-sized cities with a mix of
manufacturing and service. The wage difference is only about 10 percent.
The effect of reductions in transport costs on urban systems. The model
presented may be used to show the equilibrium effects of various exogenous
shocks. Figure 6 shows the equilibrium distributions when the transportation
costs for all three industries are reduced to 30 percent of the original costs after
the economy reaches the equilibrium presented in Figure 5. One can see that
reduction in transportation costs changes the equilibrium configuration significantly. The most substantial change is that population becomes more concentrated around a large metropolitan center in Regions 21 and 22. The agriculture
production that existed in Regions 23 and 24 is squeezed out from the large
metropolitan center. The largest metropolitan center that used to be in Region
37 shrinks to the second largest metropolitan center. The other four metropolitan
centers are all significantly smaller than previously. Furthermore, the five small
towns are all gone. A dramatic change also occurs with the land rental rate. The
distribution for the land rental rate peaks in Region 21 and again in Region 36.
The land rental curve is flattened in the sense that the highest rental rate (55.7
in Region 21) is only about 3.2 times higher than the lowest one (17.6 in Region 6)
compared to 4.8 times before the reduction. Population densities also become
more evenly distributed. The largest population decreases from 63.8 thousand
to 49.7 thousand, while the lowest population increases from less than 1.7 thousand to 1.9 thousand. The more uniform metropolitan population densities and
rents are not surprising in the context of the underlying role of transportation
costs in the determination of rent and density gradients in urban economic
theory (e.g., Fujita, 1989). In a regional context, the simulation is consistent with
the two-region result of Kilkenney (1998), which shows that very low transportation costs lead to the development of rural areas.
If transportation costs are reduced to one-tenth of the original level, rather
than 30 percent, the new equilibrium configuration comes very close to a
monocentric core-periphery structure. The new equilibrium shown in Figure 7
has only one large metropolitan center in Region 25. The land rental rate
decreases almost monotonously as one moves away from the center, except for
a couple of small deviations at Region 9 and 37. Agriculture is all located in the
periphery. The major reason for this is that the agricultural sector uses the
largest proportionate land and transportation inputs. Thus, as transportation
costs decrease it pays farmers to move to the periphery as they increase the
isolation of the city centered at the two ends so that it ultimately disappears.
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FIGURE 6: Equilibrium with Reduced Transportation Cost.
(Populations are in thousands)
The other cities then combine with services (the heaviest user of its own and
nonfarm intermediate inputs and the smallest user of land) and next with
manufacturing that uses both land and the intermediate inputs surrounding it.
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FIGURE 7: Equilibrium with Further Reduced Transport Cost.
(Populations are in thousands)
5.
CONCLUSION
This paper presents an N-region, N-industry, three-factor evolutionary
economic geography model with monopolistic competition. The distinguishing
features include: (1) multiple industries and regions, (2) complete mobility of
labor between industries and regions, (3) land use in both consumption and
production, (4) intermediate inputs used in all industries, (5) concurrent adjustment processes including firm location changes based on nonnegative profits,
and (6) discrete formulation for easy numerical implementation. The prototype
model presented in this paper can generate urban systems with cities of
different sizes and industry compositions. The simulation results show that the
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economic-geographic equilibrium reached by the economy depends not only on
initial conditions, but also on the speeds of adjustment.
In contrast to the models presented in Fujita, Krugman, and Venables
(1999), which separately include land rental markets, intermediate inputs, and
urban hierarchies, this paper brings together these separate lines of inquiry into
a single, unified model. The model generates an economy with multiple urban
centers, each of which has a von Thünen-type land use pattern and rent gradient.
This outcome is due to all industries and households using land and being able
to substitute between land, labor, and capital. The result is that households and
each of the industries concentrate in separate zones and compete with each other
in the market for land and in all other markets. The model structure produces
complex, asymmetrical equilibrium outcomes. Most new economic geography
models produce only two spatial configurations (economic activity is either
concentrated in one location or is equally dispersed); only Kilkenney (1998) and
Helpman (1998) have obtained asymmetrical results. Furthermore, we show
that although many equilibrium configurations are possible, some are more
likely to occur.
A simulation based on transportation cost reductions illustrates how the
model can be used to evaluate the effects of external shocks on the economy. The
simulation shows changes in the land use by industry, population densities, and
the number and location of cities. In particular, as transportation costs decrease
industries with a high use of intermediate inputs and relatively low land
intensity tend to agglomerate. On the other hand, industries with fewer backward linkages and higher land use are more dispersed. These particular results
depend on the model parameters.
The basic structure may be extended to achieve its potential as a practically
useful model. For example, it can be expanded to incorporate less than instantaneous speeds of adjustment in land and capital markets, and to allocate
workers across sectors based on wage mechanisms during the transitory period.
Heterogeneous labor within and among skill or industry categories can be
incorporated into the model. To fully capture income and product relationships
the model can be extended to account for government activity and for the savings
and capital formation process.
The model also serves as a prototype for an empirical new economic
geography model. Although initial conditions can be observed in a real economy,
estimates of parameters are also required to develop a model that can be used
for realistic policy analysis. Key parameters include the speed of response with
which economic actors respond to market signals and elasticities. Such a model
can then put the field of economic geography on a solid foundation that provides
insights about policy interventions, as called for by Fujita, Krugman, and
Venables (1999).
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© Blackwell Publishers 2000.