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Spatial Structure of a Metropolitan Area with an Agricultural Hinterland
Journal of Urban Economics 48, 307᎐320 Ž2000. doi:10.1006rjuec.1999.2168, available online at http:rrwww.idealibrary.com on
Spatial Structure of a Metropolitan Area with an Agricultural HinterlandU Fumio Takuma and Komei Sasaki † Graduate School of Information Sciences, SKK Building, Katahira 2, Aoba-ku, Sendai 980-8577, Japan E-mail: [email protected] Received September 25, 1998; revised December 6, 1999 A dual-economy metropolitan model is evolved by incorporating both urban and rural sectors to explain the real growth factors of the urban area. The competitive relations between urban and agricultural sectors in both labor and land markets are modeled. Such a dual-economy urban model is also indispensable for evaluating the micro effects of industrial policies Že.g., the policy for liberalization of agricultural products. on the urban spatial structure. Comparative static analysis is performed to evaluate the effects of lowered transport cost, heightened production efficiency, lowered world price of agricultural product, and increased metropolitan population. 䊚 2000 Academic Press Key Words: metropolitan area; dual economy; urban population; rural population; agricultural hinterland.
1. INTRODUCTION In most developed countries, urban areas have expanded by drawing labor, land, and capital from the agricultural sector. On the other hand, the growth of urban areas has increased the demand for agricultural products and has led to the diffusion of new technology into the rural sector, thereby exerting positive effects on the rural area. In this sense, the growth of urban areas depends on the behavior of the agricultural sector. Taking Japan as an example of a developed country, the share of the agricultural sector in total employment has declined sharply and almost monotonically for the last 50 years, from 52 to 5%, while employment has almost doubled. This implies that labor has shifted continuously from the * The original version of this paper was presented at the 1998 annual meeting of the Applied Regional Science Conference. We thank Asao Ando, Suminori Tokunaga, Masahisa Fujita, and the editor and two anonymous referees of this journal for their useful comments. This research was supported in part by Grant-in-Aid for Scientific Research ŽC., which is gratefully acknowledged. † To whom correspondence should be addressed. 307 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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agricultural to nonagricultural sectors. Total farmland in Japan has decreased by 15% over the last 35 years, and a large portion of the converted agricultural land has been used for urban activities such as residences, industrial buildings, roads, and railways. Only a small portion of the converted farmland has been used for non-urban activities such as forestation and mining.1 In conventional monocentric urban models, it is implicitly assumed that a central business district ŽCBD. produces the urban product and exports it to the outside while importing composite goods which will be consumed by urban residents. In such models, the markets for urban goods or composite goods are not explicitly considered. It is also assumed in the conventional models that the land for urban use can be rented from absentee landlords at the opportunity cost to the extent required: there is no competition in the land market between urban and agricultural uses.2 Thus, the conventional monocentric model cannot help in explaining the competitive relationship between urban and rural areas or the interdependence between activities in the urban area and its hinterland. This paper sets out to develop a metropolitan model where both the urban area and its hinterland are incorporated, and the markets of both urban and rural sectors can be considered.3 Migration between urban and rural areas is taken into consideration, and the competitive relation between the two areas in the labor market is modeled. Also, competition in the land market between the urban and rural sectors is incorporated. Within the framework of such a model, these points are analyzed: how land use for urban and agricultural production is determined, and how industrial composition is determined within a metropolitan area.4
Mills and Hamilton w1, pp. 62᎐65x argue that MSAs in the U.S. are labor markets in the sense that most people in a MSA also live there, and vice versa. According to Table 3.5 in their book, the composition ratio of employment in the agricultural sector in MSAs has been lowered while, in particular, that of tertiary industry has been increased. This suggests that a steady shift in labor force from the agricultural to the urban sector has been occurring within MSAs. 2 Most monocentric city models so far have focused on residential location and land rent structure in the residential district. Some models, however, assume that industries demand land Že.g., Sasaki and Kaiyama w7x and Ross and Yinger w4x.. 3 Sasaki w6x intended to analyze the relation between city formation and the activities of the agricultural sector, but the spatial aspect is neglected in his model. 4 The model of Nerlove and Sadka w2x is the only real urban model of a dual economy, but it makes some restrictive and peculiar assumptions. It assumes that land is not used for residences and that a city is completely closed in that equilibrium in the markets of goods, labor, and land is achieved without any trade with other regions. The present model relaxes these assumptions and treats the transportation sector explicitly, which was not introduced in the model of Nerlove and Sadka. 1
METROPOLITAN AREA WITH HINTERLAND
309
Some advanced countries, including Japan, are faced with the problem of liberalization of world agricultural markets. The liberalization effect Že.g., lowered price of domestic agricultural products. on the national economy is sometimes analyzed by a macro economic model ŽYazawa w9x.. However, it is noted that a dual-economy urban model is indispensable for investigating liberalization’s micro effect on land use and rent structure within a metropolitan area Že.g., to determine how much agricultural land will be transformed to urban use; how many workers will shift from agricultural to urban sectors; and how much land rent in the urban residential area will be increased.. In Sect. 2, the basic model is presented and Sect. 3 discusses market equilibrium in the model. A comparative static analysis is performed in Sect. 4, and Sect. 5 states some concluding remarks. 2. THE MODEL 2.1. The General Setting As shown in Fig. 1, a linear metropolitan area is assumed, the inner part of which is used for urban residence, and the outer part for agriculture. There is a ‘‘point’’ CBD in the center of the metropolitan area, where manufacturing goods and transportation services are produced. All the residents in the urban area commute to a CBD and work for either the manufacturing or the transportation sector. It might seem odd that transportation service is produced at the CBD, and the workers in that sector must commute there. We assume that the establishments of the transportation sector are located at the CBD, and every worker is required to insert a ‘‘time card’’ there, and then they will drive trucks or cars. Agricultural output is transported to the CBD after the portion of self-consumption by farmers is deducted. That is, an urban resident incurs commuting costs while a rural resident bears the transport costs of agricultural products. It is assumed that an urban resident purchases the manufacturing goods and agricultural products at the CBD when he or she commutes there, and a rural resident purchases the manufacturing goods
FIG. 1. Spatial structure of a metropolitan area.
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at the CBD when he or she transports the agricultural output there, so that no additional transport cost is incurred by either resident. There are 2 = N people living in this metropolitan area, which is a linear city symmetrical about the CBD, so only the right-hand side of the figure is treated in the subsequent analysis. A firm in the manufacturing sector has a linear production function where labor is the only variable input. The size of a firm is indeterminate, and the aggregated production function is assumed to be represented by F M s a1 N M ,
Ž 1.
where F M , N M , and a1 are, respectively, the total output and employment in the manufacturing industry and the production efficiency coefficient of the manufacturing industry. The profit maximization condition in a competitive market is: a1 p1 y w s 0, Ž 2. where p1 and w are the market price of manufacturing goods and the wage rate, respectively. It is supposed that the number of vehicles Žor the physical capacity of vehicles., such as trucks and cars used in transportation sector, is fixed and only labor is a variable input. In a manner similar to the manufacturing sector, the aggregated production function of the transportation sector is represented as F T s a3 N T , Ž 3. and, at market equilibrium, it holds that a3 p 3 y w s 0,
Ž 4.
where F T, N T, a3 , and p 3 are, respectively, the total supply, employment, the production efficiency coefficient of the transportation sector, and the price of transportation services.5 It is assumed that the metropolitan government rents all the land in the metropolitan area from absentee landlords at the opportunity cost Že.g., equal to the rent of grassland., rents it to urban residents and farmers at the market rent, and then redistributes the net land revenue equally among the people in the metropolitan area. That is, per capita redistribution of the differential land rent is defined as ks 5
H0x a f Ž r Ž x . y r o . dx N
,
Ž 5.
For instance, transportation service is measured by the capacity in unit of floor area of transport Že.g., truck and car-train . per unit distance. In this case, p 3 is the price, for instance, when one car-train with a floor area of 1 m2 is used for transporting passengers or commodities over 1 km.
311
METROPOLITAN AREA WITH HINTERLAND
where r Ž x . is the market rent at the location x miles distant from the CBD, r o is the opportunity cost of land, and x a f is the metropolitan fringe. All the people in the metropolitan area are assumed to have identical preferences and to derive utility from the consumption of manufacturing goods, z1 , agricultural products, z 2 , and residential lot size, q. The utility function is specified in a Cobb-Douglas form, that is: U Ž z1i , z 2i , q i . s Ž z1i .
␣

Ž z 2i . Ž q i .
␥
,
␣ q  q ␥ s 1, for i s u, a, Ž 6 .
where superscripts u and a denote workers of the urban and agricultural sectors, respectively. To simplify the theoretical analysis, the residential lot size is assumed to be the same regardless of residents’ location or occupation, and is given exogenously. That is, q u Ž x . s q a Ž x . s q.
Ž 7.
2.2. Beha¨ ior of an Urban Resident An urban resident works for either the manufacturing or the transportation sector, and obtains the wage income, w. The commuting cost of a worker residing at x is d 0 p 3 x, in which d 0 is the amount of transportation service required for commuting a unit distance.6 Then, this resident’s disposable income is w q k y d 0 p 3 x, which is spent on manufacturing goods, agricultural products, and residence Ži.e., w q k y d 0 p 3 x s p1 z1u Ž x . q p 2 z 2u Ž x . q r Ž x . q .. The metropolitan government rents the land at each location to the resident offering the highest bid-rent. At equilibrium, the utility level of a resident is the same regardless of the location or his occupation, since people have identical preferences, and neither cost nor friction are incurred by moving residential locations or changing occupation. Therefore, the market rent at equilibrium coincides with the bid rent associated with the utility level attained in equilibrium. That is, r Ž x . s r u Ž x ; w, k, ¨ . s max u u
w q k y d 0 p 3 x y p1 z1u y p 2 z 2u
z1 , z 2
subject to U Ž z1u , z 2u , q . s ¨ ,
q
,
Ž 8.
in which r x; w, k, x . is the bid-rent of an urban resident at x, and ¨ the utility level attained at equilibrium. The necessary condition of the problem in Ž8. is Uz 1u p1 s , Ž 9. Uz 2u p2 uŽ
6
Referring to footnote 5, for instance, a unit capacity of car-train is used for commuting 1 km: that is, d 0 s 1.0.
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TAKUMA AND SASAKI
and the optimal solutions are: z1u Ž x . s z 2u
Ž x. s
␣ p2
ž / ž /
rŽ ␣q  .
␣ p2
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
 p1
y ␣rŽ ␣q  .
.
.
Ž 10 .
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
 p1
Ž
.
Ž
.
Ž 11 .
r u Ž x ; w, k , ¨ . s
w q k y d 0 p 3 x y Ž ␣ q  . r␣ Ž ␣r .  r
Ž ␣q  .
p1␣ rŽ ␣ q  . p 2 rŽ ␣ q  . Ž q . y ␥ r
Ž ␣q  .
Ž ¨ . 1r
Ž ␣q  .
q
Ž 12 . The comparative static results on the bid-rent problem in Ž8. are summarized in Table 1. 2.3. Beha¨ ior of a Farmer The production function of each farm is represented in the form 2
f a Ž q2a Ž x . . s a2 Ž q2a Ž x . . ,
0 - 2 - 1,
Ž 13 .
where q2a Ž x . is the farmland area at location x. To simplify the theoretical analysis below, q2a Ž x . is assumed to be the exogenously given value q2 for every location. Also, it is supposed that each farmer’s production level is more than his or her self-consumption. That is, for any farmer, it holds that g a Ž x . s f a Ž q2 . y z 2a Ž x . ) 0,
Ž 14 .
and the excess output, g a Ž x ., is shipped to the market at the CBD. Disposable income of a farmer at x is the sum of the net sales of excess output Ž p 2 y d 2 p 3 x . g a Ž x ., and the distributed differential land rent k, where d 2 is the transportation service required for transporting unit TABLE 1 Comparative Static Results of the Bid-Rent Problem for an Urban Resident Increase in:
w
k
¨
p1
p2
p3
q
d0
x
Effect on z1u z 2u ru
0 0 q
0 0 q
q q y
y q y
q y y
0 0 y
y y ?
0 0 y
0 0 y
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METROPOLITAN AREA WITH HINTERLAND
agricultural products per unit distance.7 This income is expended on manufacturing goods and land for both farming and residence Ži.e., Ž p 2 y d 2 p 3 x . g a Ž x . q k s p1 z1a Ž x . q r Ž x .Ž q q q2 ... Since the utility level of a farmer in equilibrium is equal to ¨ , regardless of his or her location, the market land rent in the rural area is equal to the following bid-rent r Ž x . s r a Ž x ; k , ¨ . s max a a
Ž p 2 y d 2 p 3 x . Ž f a Ž q2 . y z 2a . q k y p1 z1a q q q2
z1 , z 2
subject to U Ž z1a , z 2a , q . s ¨ , and f a Ž q2 . s a2 Ž q2 .
2
.
,
Ž 15 .
The first order condition of Ž15. is Uz 1a
s
Uz 2a
p1 p2
,
Ž 16 .
and the optimal solutions are: z1a
z 2a
Ž x. s Ž x. s
ž ž
␣ P2 Ž x .  p1 ␣ P2 Ž x .  p1
rŽ ␣q  .
/ /
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
.
Ž
.
Ž 17 .
y ␣rŽ ␣q  .
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
.
Ž
.
Ž 18 .
r aŽ x; k, ¨ . s
P2 Ž x . f a qky Ž ␣ q  . r␣ Ž ␣r .  r
Ž ␣q  .
p1␣ rŽ ␣ q  . P2 Ž x .  r
Ž ␣q  .
Ž q . y␥ r
Ž ␣q  .
Ž ¨ . 1r
Ž ␣q  .
q q q2
Ž 19 . in which P2 Ž x . s p 2 y d 2 p 3 x.8 It is noted that consumption of the two goods, z1a Ž x . and z 2a Ž x ., by a rural resident depends on a specific location, while they are the same regardless of location in the urban area Žsee Eqs. Ž10. through Ž12... Comparative static analysis is performed for the problem Ž15., and the results are summarized in Table 2. For an urban resident, a change in the transportation service price, p 3 , does not affect the consumption of manufacturing goods or agricultural products ŽTable 1.. In contrast, in Table 2, 7 Referring to footnote 5, for instance, one-third of the unit capacity of a car-train is used for transporting a unit agricultural product over 1 km: that is, d 2 s 1r3. 8 The consumption price of agricultural product to a farmer is not p 2 but Ž p 2 y d 2 p 3 x . which is the forgone income from selling his output at the CBD market. Thus, consumption price depends on the transportation cost.
314
TAKUMA AND SASAKI TABLE 2 Comparative Static Results of the Bid-Rent Problem for a Farmer Increase in:
k
¨
p1
p2
p3
q
q2
d2
x
a2
Effect on z1a z 2a ra
0 0 q
q q y
y q y
q y q
y q y
y y ?
0 0 ?
y q y
y q y
0 0 q
an increased price of transportation services will generate substitution from manufacturing goods to agricultural products. This is because the price of self-consumed output, z 2a Ž x ., is Ž p 2 y d 2 p 3 x ., which decreases as a result of an increase in p 3 . Another difference from Table 1 is that an increase in the price of agricultural products lowers the bid-rent of urban residents while it heightens the bid-rent of rural residents, since it increases the farmer’s disposable income. 3. MARKET EQUILIBRIUM In this model, it is assumed that an urban residential district is formed in the inner area and the agricultural sector is located in the outer area. ŽThis configuration is ensured if d 0rd 2 G qf a Ž q2 .rŽ q q q2 ., which implies that the slope of the bid-rent curve of an urban resident is steeper than that of a farmer at the urban-rural boundary..9 Letting x u f and x a f denote, respectively, the urban᎐rural boundary and the metropolitan fringe, and expressing the metropolitan net exports to other regions of manufacturing goods and agricultural products, respectively, by Z1f and Z2f , the equilibrium conditions of the metropolitan system are represented by the following equations
FMŽ NM . s
Hx
xaf
f a Ž q2 .
uf
q q q2
H0
xu f
dx s
z1u Ž x . q
H0
xu f
dx q
z 2u Ž x . q
Hx
xaf
z1a Ž x .
uf
q q q2
dx q
Hx
dx q Z1f ,
xaf
z 2a Ž x .
uf
q q q2
dx q Z2f ,
Ž 20 . Ž 21 .
9 Applying the envelope theorem to the problem in Ž8. and Ž15., the slopes of the bid-rent curves of an urban resident and a farmer are obtained, respectively, as yd 0 p 3rq and yd 2 p 3 Ž f a y z 2a .rŽ q q q2 .. Thus, for an urban residential district to be formed in the inner area it must hold that d 0 p 3rq ) d 2 p 3 Ž f a y z 2a .rŽ q q q2 . at the location x u f , where the two bid-rent curves intersect. Obviously, this is ensured if d 0 rd 2 ) qf a Ž q2 .rŽ q q q2 ..
315
METROPOLITAN AREA WITH HINTERLAND
FT Ž NT . s
H0
xu f
q
H0 H0
ks
1 N
½H
0
xu f
xu f
1 q
d0 x
xu f
dx q
1 q
Hx
dx q
Hx
xaf
d 2 x Ž f a Ž q2 . y z 2a Ž x . .
uf
q q q2
dx s N M q N T s N u , xaf
1
uf
q q q2
dx s N u q N a s N,
dx,
Ž 22 . Ž 23 . Ž 24 .
a1 p1 s w,
Ž 25 .
a3 p 3 s w,
Ž 26 . xaf
5
Ž r u Ž x ; w, k, ¨ . y r o . dx q H Ž r a Ž x ; k , ¨ . y r o . dx , xuf
Ž 27 . r u Ž x u f ; w, k, ¨ . s r a Ž x u f ; k, ¨ . ,
Ž 28 .
r aŽ xaf ; k, ¨ . s r o,
Ž 29 .
Ž p1 Z1f q p2 Z2f . y H
xaf
r o dx s 0.
Ž 30 .
0
Equations Ž20. through Ž22. are equilibrium conditions in the markets of manufacturing goods, agricultural products, and transportation service, respectively. Relation Ž23. shows the labor market equilibrium in the urban sector. Equation Ž24. is the population constraint; Ž25. and Ž26. are profit maximization conditions in the urban sectors, and Ž27. defines the redistribution of differential land rent. Equation Ž28. states that, at the urban᎐rural boundary, the bid-rent of an urban resident is equal to that of a farmer. Equation Ž29. states that the market land rent at the metropolitan fringe is equal to the opportunity cost of land. The last relation, Ž30., is derived using Eqs. Ž22. through Ž24. and Ž27.. This represents the current balance of trade with other regions, showing that the total net export value of traded goods is equal to the total Žnet. imported value of land service Žsince the metropolitan government imports land service at the opportunity cost r o .. The system consists of 11 equations and includes 10 unknown variables p 3 , w, k, ¨ , N M , N T , x u f , x a f , Z1f , Z2f 4 . By Walras’s law, one of those equations is omitted.10 p1 and p 2 are exogenous variables, and p1 is set to unity as a numeraire. It is noted that the wage rate, w, and the price ´ of transportation service, p 3 , are affected only by a change in the produc10
In the subsequent analysis, Eq. Ž20. is omitted.
316
TAKUMA AND SASAKI
tion technology in manufacturing industry, a1. In this sense, w and p 3 are almost exogenous variables. 4. COMPARATIVE STATIC ANALYSIS The impact of a change in various parameters is investigated, conforming to the method in Sasaki and Kaiyama w7x: Eqs. Ž20. through Ž30. are totally differentiated, and then the equations are solved for the endogenous variables.11 The results of comparative statics are summarized in Table 3, where the urban population, N u, is defined as the sum of N M and N T , while the rural population, N a, is calculated as Ž N y N u .. In many cases, the direction of impact on the endogenous variables can be unambiguously determined. For each case, the results of comparative statics are explained. The effect of lowered transport cost. Suppose the transport cost is lowered by a decrease in the necessary transport service, d 0 Žor d 2 ., and note that a decrease in d 0 Žor d 2 . implies not a uniform improvement in the transport system but a partial improvement in the system only for transporting passengers Žor agricultural products.. When the technology of transporting passengers is improved, so that d 0 is lowered, the welfare of residents, ¨ , as expected, increases. The urban area represented by x u f expands, since more labor is shifted from the agricultural sector to manufacturing. The rural area shrinks to the extent that the total metropolitan area denoted by x a f is reduced.12 As a result, the output of the manufacturing industry, and its exports, Z1f , increase because employment, N M , in that sector increases. The output and export Ž Z2f . of the agricultural sector decrease because employment, N a, in that sector decreases. When the technology for transporting agricultural products is improved, so that d 2 is lowered, the utility level of metropolitan residents naturally increases. Contrary to the instance of decreased d 0 , labor shifts from the urban to the agricultural sector. Thus, the output and the export of the agricultural sector increases. Interestingly, however, it is ambiguous whether or not output of the manufacturing sector decreases. This is because there is a possibility that some of the labor released from the transportation sector as a result of technological progress will be shifted to the manufacturing sector, whereby the output of manufactured goods will increase. The net export of manufactured goods is necessarily decreased, while that of agricultural products is necessarily increased. 11
The details of the system for the comparative statics are available on request. The effects of lowered commuting cost on the utility level of residents and the urban area in this analysis coincide with those in the literature so far Že.g., Sasaki w5x, Wheaton w8x, Pines and Sadka w3x.. 12
0 0 q 0
Increase in d0 0 d2 0 a1 q a2 0
0 xa f q y y q q ?
0 0 xu f
q
N Effect on: Increase in d0 y d2 q a1 q a2 y y
p2
p2 N
0
w
p3
Effect on:
?
y
?
¡ ~ ¢ ?
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 .
y q ? ?
?
q
q
¡ ~ ¢ ?
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 .
q y ? ?
Z2f
Z1f
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 . y
q
0
y
¡ ~ ¢
y y q q
¨
?
?
? ? ? ?
k
?
y
?
¡ ~ ¢ ?
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 .
y ? ? ?
NM
TABLE 3 Comparative Static Results of the Market Equilibrium
?
?
? ? ? ?
NT
q
y
y q q y
Nu
?
q
q y y q
Na METROPOLITAN AREA WITH HINTERLAND
317
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TAKUMA AND SASAKI
The effect of heightened production efficiency. When production efficiency, a1 , in the manufacturing industry is heightened, the wage rate is increased Žand thus, the price of transport service is increased., whereby labor shifts from the agricultural to the urban sector. The urban area, x u f , is necessarily extended but the metropolitan area measured by x a f decreases. It is noted that the effect on employment in the manufacturing industry is ambiguous: the number of workers, N M , in that sector can be decreased if the production efficiency, a1 , is heightened to the extent that output increases greatly. As a consequence of such adjustments, the utility level, ¨ , increases. An increase in the production efficiency, a2 , in the agricultural sector heightens the welfare level of residents as well. It will cause labor to shift from the urban sector to the agricultural sector, and thereby the urban area shrinks and the metropolitan area expands. The effect on the wage rate is neutral when a2 is heightened. The effects on the net exports of both goods Ž Z1f and Z2f . are ambiguous. The effect of lowered world price of agricultural product. It is hypothesized that p 2 is lowered as the liberalization of agricultural product proceeds. The effect on the utility level, ¨ , is complex: utility increases when the metropolitan area imports Žin net. agricultural product Ži.e., when Z2f - 0., and decreases when it exports Žin net. agricultural product Ži.e., when Z2f ) 0.. An interpretation is that when the consumption of agricultural product exceeds output in the metropolitan area, so that agricultural product is imported Žin net., the lowered price of that product favorably affects the welfare of residents. On the other hand, if the metropolitan area produces more agricultural product than it consumes, such that the excess output is exported to other regions, the lowered price of agricultural product has a negative effect on the utility level by decreasing income level. Regardless of the value of Z2f , when the price of agricultural product, p 2 , is lowered, labor shifts from rural to urban sectors, whereby agricultural output is decreased. As a result, the urban area, x u f , expands, and the metropolitan area, x a f , shrinks. The effect on production and export levels of the manufacturing good is ambiguous. The effect of increased metropolitan population. When N increases, the total employment, N u, in urban sector increases and thereby the urban area, x u f , increases. Thus, the average commuting distance increases, whereby the utility level of metropolitan residents is lowered. It is noted that the metropolitan area, x a f , is not necessarily expanded: if a large amount of labor shifts from rural to urban sectors in response to the
METROPOLITAN AREA WITH HINTERLAND
319
increased demand for manufacturing good and transportation service, then the metropolitan area possibly shrinks.13 In most situations, it is impossible to separate the effect on employment in the manufacturing and transportation sectors; that is, the effect on N M and N T. It is, however, unambiguously demonstrated that a decrease in commuting cost, d 0 , increases the employment, N M , in the manufacturing sector. Land rent is one of the indexes most closely representing the metropolitan spatial structure. However, the effect of a parameter change on the rent is ambiguous in the theoretical analysis, and thus the effect on the per capita redistribution of the differential land rent, k, is indeterminate. Some numerical analyses are needed to capture the effects on land rent. 5. CONCLUDING REMARKS A dual-economy metropolitan model was developed by incorporating both urban and rural sectors for explaining appropriately the real growth factors of the urban area. The competitive relations between urban and agricultural sectors in both labor and land markets have been modeled. Such a dual-economy model is also indispensable for evaluating the micro effects of industrial policies Že.g., the policy for liberalization of agricultural prices. on urban spatial structure. Comparative static analysis was performed for evaluating the effects of lowered transport cost, heightened production efficiency, lowered world price of agricultural product, and increased metropolitan population. In many cases, the direction of impact on the endogenous variables was unambiguously determined. However, the effect of a parameter change on the land rent is indeterminate and needs to be captured by numerical analysis.
13 The effect of an increase in the production efficiency Ž a1 or a2 . on population distribution between sectors makes a sharp contrast between the model of Nerlove and Sadka w2x and ours. In their analysis, the effect is neutral whether a1 or a2 is increased. On the other hand, our results show that an increase in the efficiency of the urban production sector induces labor to shift from the agricultural sector to the urban sector, and this increases urban population. The heightened efficiency of the agricultural sector increases the number of farmers and decreases the urban population. Another interesting contrast between the two models is that the lowered transport cost of agricultural products reduces the urban population in our model, while it increases the urban population in the model of Nerlove and Sadka.
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REFERENCES 1. E. S. Mills and B. W. Hamilton, ‘‘Urban Economics,’’ Fifth ed., HarperCollins College Publishers, New York Ž1994.. 2. M. L. Nerlove and E. Sadka, Von Thunen’s model of the dual economy, Journal of Economics: Zeitschrift Fur Nationalokonomie, 54Ž2., 97᎐123 Ž1991.. 3. D. Pines and E. Sadka, Comparative statics analysis of a fully closed city, Journal of Urban Economics, 20, 1᎐20 Ž1986.. 4. S. Ross and J. Yinger, Comparative statics analysis of open urban models with a full labor market and suburban employment, Regional Science and Urban Economics, 25, 575᎐605 Ž1995.. 5. K. Sasaki, A comparative static analysis of urban structure in the setting of endogenous income, Journal of Urban Economics, 22, 53᎐72 Ž1987.. 6. K. Sasaki, Agricultural sector in a Tiebout model, in ‘‘Exploring Sustainability,’’ Regional Planning Research Group Ž1996.. 7. K. Sasaki and M. Kaiyama, The effects of urban transportation costs on urban spatial structure with endogenous wage income, Regional Science and Urban Economics, 20, 223᎐243 Ž1990.. 8. W. C. Wheaton, A comparative static analysis of urban spatial structure, Journal of Economic Theory, 9, 223᎐237 Ž1974.. 9. N. Yazawa, A macroeconomic analysis of the land use for the evaluation of agricultural liberalization and rationalization policies, Discussion Papers in Socio-Economic Information Science No. 8, Graduate School of Information Sciences, Tohoku University Ž1997..
Spatial Structure of a Metropolitan Area with an Agricultural HinterlandU Fumio Takuma and Komei Sasaki † Graduate School of Information Sciences, SKK Building, Katahira 2, Aoba-ku, Sendai 980-8577, Japan E-mail: [email protected] Received September 25, 1998; revised December 6, 1999 A dual-economy metropolitan model is evolved by incorporating both urban and rural sectors to explain the real growth factors of the urban area. The competitive relations between urban and agricultural sectors in both labor and land markets are modeled. Such a dual-economy urban model is also indispensable for evaluating the micro effects of industrial policies Že.g., the policy for liberalization of agricultural products. on the urban spatial structure. Comparative static analysis is performed to evaluate the effects of lowered transport cost, heightened production efficiency, lowered world price of agricultural product, and increased metropolitan population. 䊚 2000 Academic Press Key Words: metropolitan area; dual economy; urban population; rural population; agricultural hinterland.
1. INTRODUCTION In most developed countries, urban areas have expanded by drawing labor, land, and capital from the agricultural sector. On the other hand, the growth of urban areas has increased the demand for agricultural products and has led to the diffusion of new technology into the rural sector, thereby exerting positive effects on the rural area. In this sense, the growth of urban areas depends on the behavior of the agricultural sector. Taking Japan as an example of a developed country, the share of the agricultural sector in total employment has declined sharply and almost monotonically for the last 50 years, from 52 to 5%, while employment has almost doubled. This implies that labor has shifted continuously from the * The original version of this paper was presented at the 1998 annual meeting of the Applied Regional Science Conference. We thank Asao Ando, Suminori Tokunaga, Masahisa Fujita, and the editor and two anonymous referees of this journal for their useful comments. This research was supported in part by Grant-in-Aid for Scientific Research ŽC., which is gratefully acknowledged. † To whom correspondence should be addressed. 307 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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agricultural to nonagricultural sectors. Total farmland in Japan has decreased by 15% over the last 35 years, and a large portion of the converted agricultural land has been used for urban activities such as residences, industrial buildings, roads, and railways. Only a small portion of the converted farmland has been used for non-urban activities such as forestation and mining.1 In conventional monocentric urban models, it is implicitly assumed that a central business district ŽCBD. produces the urban product and exports it to the outside while importing composite goods which will be consumed by urban residents. In such models, the markets for urban goods or composite goods are not explicitly considered. It is also assumed in the conventional models that the land for urban use can be rented from absentee landlords at the opportunity cost to the extent required: there is no competition in the land market between urban and agricultural uses.2 Thus, the conventional monocentric model cannot help in explaining the competitive relationship between urban and rural areas or the interdependence between activities in the urban area and its hinterland. This paper sets out to develop a metropolitan model where both the urban area and its hinterland are incorporated, and the markets of both urban and rural sectors can be considered.3 Migration between urban and rural areas is taken into consideration, and the competitive relation between the two areas in the labor market is modeled. Also, competition in the land market between the urban and rural sectors is incorporated. Within the framework of such a model, these points are analyzed: how land use for urban and agricultural production is determined, and how industrial composition is determined within a metropolitan area.4
Mills and Hamilton w1, pp. 62᎐65x argue that MSAs in the U.S. are labor markets in the sense that most people in a MSA also live there, and vice versa. According to Table 3.5 in their book, the composition ratio of employment in the agricultural sector in MSAs has been lowered while, in particular, that of tertiary industry has been increased. This suggests that a steady shift in labor force from the agricultural to the urban sector has been occurring within MSAs. 2 Most monocentric city models so far have focused on residential location and land rent structure in the residential district. Some models, however, assume that industries demand land Že.g., Sasaki and Kaiyama w7x and Ross and Yinger w4x.. 3 Sasaki w6x intended to analyze the relation between city formation and the activities of the agricultural sector, but the spatial aspect is neglected in his model. 4 The model of Nerlove and Sadka w2x is the only real urban model of a dual economy, but it makes some restrictive and peculiar assumptions. It assumes that land is not used for residences and that a city is completely closed in that equilibrium in the markets of goods, labor, and land is achieved without any trade with other regions. The present model relaxes these assumptions and treats the transportation sector explicitly, which was not introduced in the model of Nerlove and Sadka. 1
METROPOLITAN AREA WITH HINTERLAND
309
Some advanced countries, including Japan, are faced with the problem of liberalization of world agricultural markets. The liberalization effect Že.g., lowered price of domestic agricultural products. on the national economy is sometimes analyzed by a macro economic model ŽYazawa w9x.. However, it is noted that a dual-economy urban model is indispensable for investigating liberalization’s micro effect on land use and rent structure within a metropolitan area Že.g., to determine how much agricultural land will be transformed to urban use; how many workers will shift from agricultural to urban sectors; and how much land rent in the urban residential area will be increased.. In Sect. 2, the basic model is presented and Sect. 3 discusses market equilibrium in the model. A comparative static analysis is performed in Sect. 4, and Sect. 5 states some concluding remarks. 2. THE MODEL 2.1. The General Setting As shown in Fig. 1, a linear metropolitan area is assumed, the inner part of which is used for urban residence, and the outer part for agriculture. There is a ‘‘point’’ CBD in the center of the metropolitan area, where manufacturing goods and transportation services are produced. All the residents in the urban area commute to a CBD and work for either the manufacturing or the transportation sector. It might seem odd that transportation service is produced at the CBD, and the workers in that sector must commute there. We assume that the establishments of the transportation sector are located at the CBD, and every worker is required to insert a ‘‘time card’’ there, and then they will drive trucks or cars. Agricultural output is transported to the CBD after the portion of self-consumption by farmers is deducted. That is, an urban resident incurs commuting costs while a rural resident bears the transport costs of agricultural products. It is assumed that an urban resident purchases the manufacturing goods and agricultural products at the CBD when he or she commutes there, and a rural resident purchases the manufacturing goods
FIG. 1. Spatial structure of a metropolitan area.
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at the CBD when he or she transports the agricultural output there, so that no additional transport cost is incurred by either resident. There are 2 = N people living in this metropolitan area, which is a linear city symmetrical about the CBD, so only the right-hand side of the figure is treated in the subsequent analysis. A firm in the manufacturing sector has a linear production function where labor is the only variable input. The size of a firm is indeterminate, and the aggregated production function is assumed to be represented by F M s a1 N M ,
Ž 1.
where F M , N M , and a1 are, respectively, the total output and employment in the manufacturing industry and the production efficiency coefficient of the manufacturing industry. The profit maximization condition in a competitive market is: a1 p1 y w s 0, Ž 2. where p1 and w are the market price of manufacturing goods and the wage rate, respectively. It is supposed that the number of vehicles Žor the physical capacity of vehicles., such as trucks and cars used in transportation sector, is fixed and only labor is a variable input. In a manner similar to the manufacturing sector, the aggregated production function of the transportation sector is represented as F T s a3 N T , Ž 3. and, at market equilibrium, it holds that a3 p 3 y w s 0,
Ž 4.
where F T, N T, a3 , and p 3 are, respectively, the total supply, employment, the production efficiency coefficient of the transportation sector, and the price of transportation services.5 It is assumed that the metropolitan government rents all the land in the metropolitan area from absentee landlords at the opportunity cost Že.g., equal to the rent of grassland., rents it to urban residents and farmers at the market rent, and then redistributes the net land revenue equally among the people in the metropolitan area. That is, per capita redistribution of the differential land rent is defined as ks 5
H0x a f Ž r Ž x . y r o . dx N
,
Ž 5.
For instance, transportation service is measured by the capacity in unit of floor area of transport Že.g., truck and car-train . per unit distance. In this case, p 3 is the price, for instance, when one car-train with a floor area of 1 m2 is used for transporting passengers or commodities over 1 km.
311
METROPOLITAN AREA WITH HINTERLAND
where r Ž x . is the market rent at the location x miles distant from the CBD, r o is the opportunity cost of land, and x a f is the metropolitan fringe. All the people in the metropolitan area are assumed to have identical preferences and to derive utility from the consumption of manufacturing goods, z1 , agricultural products, z 2 , and residential lot size, q. The utility function is specified in a Cobb-Douglas form, that is: U Ž z1i , z 2i , q i . s Ž z1i .
␣

Ž z 2i . Ž q i .
␥
,
␣ q  q ␥ s 1, for i s u, a, Ž 6 .
where superscripts u and a denote workers of the urban and agricultural sectors, respectively. To simplify the theoretical analysis, the residential lot size is assumed to be the same regardless of residents’ location or occupation, and is given exogenously. That is, q u Ž x . s q a Ž x . s q.
Ž 7.
2.2. Beha¨ ior of an Urban Resident An urban resident works for either the manufacturing or the transportation sector, and obtains the wage income, w. The commuting cost of a worker residing at x is d 0 p 3 x, in which d 0 is the amount of transportation service required for commuting a unit distance.6 Then, this resident’s disposable income is w q k y d 0 p 3 x, which is spent on manufacturing goods, agricultural products, and residence Ži.e., w q k y d 0 p 3 x s p1 z1u Ž x . q p 2 z 2u Ž x . q r Ž x . q .. The metropolitan government rents the land at each location to the resident offering the highest bid-rent. At equilibrium, the utility level of a resident is the same regardless of the location or his occupation, since people have identical preferences, and neither cost nor friction are incurred by moving residential locations or changing occupation. Therefore, the market rent at equilibrium coincides with the bid rent associated with the utility level attained in equilibrium. That is, r Ž x . s r u Ž x ; w, k, ¨ . s max u u
w q k y d 0 p 3 x y p1 z1u y p 2 z 2u
z1 , z 2
subject to U Ž z1u , z 2u , q . s ¨ ,
q
,
Ž 8.
in which r x; w, k, x . is the bid-rent of an urban resident at x, and ¨ the utility level attained at equilibrium. The necessary condition of the problem in Ž8. is Uz 1u p1 s , Ž 9. Uz 2u p2 uŽ
6
Referring to footnote 5, for instance, a unit capacity of car-train is used for commuting 1 km: that is, d 0 s 1.0.
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TAKUMA AND SASAKI
and the optimal solutions are: z1u Ž x . s z 2u
Ž x. s
␣ p2
ž / ž /
rŽ ␣q  .
␣ p2
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
 p1
y ␣rŽ ␣q  .
.
.
Ž 10 .
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
 p1
Ž
.
Ž
.
Ž 11 .
r u Ž x ; w, k , ¨ . s
w q k y d 0 p 3 x y Ž ␣ q  . r␣ Ž ␣r .  r
Ž ␣q  .
p1␣ rŽ ␣ q  . p 2 rŽ ␣ q  . Ž q . y ␥ r
Ž ␣q  .
Ž ¨ . 1r
Ž ␣q  .
q
Ž 12 . The comparative static results on the bid-rent problem in Ž8. are summarized in Table 1. 2.3. Beha¨ ior of a Farmer The production function of each farm is represented in the form 2
f a Ž q2a Ž x . . s a2 Ž q2a Ž x . . ,
0 - 2 - 1,
Ž 13 .
where q2a Ž x . is the farmland area at location x. To simplify the theoretical analysis below, q2a Ž x . is assumed to be the exogenously given value q2 for every location. Also, it is supposed that each farmer’s production level is more than his or her self-consumption. That is, for any farmer, it holds that g a Ž x . s f a Ž q2 . y z 2a Ž x . ) 0,
Ž 14 .
and the excess output, g a Ž x ., is shipped to the market at the CBD. Disposable income of a farmer at x is the sum of the net sales of excess output Ž p 2 y d 2 p 3 x . g a Ž x ., and the distributed differential land rent k, where d 2 is the transportation service required for transporting unit TABLE 1 Comparative Static Results of the Bid-Rent Problem for an Urban Resident Increase in:
w
k
¨
p1
p2
p3
q
d0
x
Effect on z1u z 2u ru
0 0 q
0 0 q
q q y
y q y
q y y
0 0 y
y y ?
0 0 y
0 0 y
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METROPOLITAN AREA WITH HINTERLAND
agricultural products per unit distance.7 This income is expended on manufacturing goods and land for both farming and residence Ži.e., Ž p 2 y d 2 p 3 x . g a Ž x . q k s p1 z1a Ž x . q r Ž x .Ž q q q2 ... Since the utility level of a farmer in equilibrium is equal to ¨ , regardless of his or her location, the market land rent in the rural area is equal to the following bid-rent r Ž x . s r a Ž x ; k , ¨ . s max a a
Ž p 2 y d 2 p 3 x . Ž f a Ž q2 . y z 2a . q k y p1 z1a q q q2
z1 , z 2
subject to U Ž z1a , z 2a , q . s ¨ , and f a Ž q2 . s a2 Ž q2 .
2
.
,
Ž 15 .
The first order condition of Ž15. is Uz 1a
s
Uz 2a
p1 p2
,
Ž 16 .
and the optimal solutions are: z1a
z 2a
Ž x. s Ž x. s
ž ž
␣ P2 Ž x .  p1 ␣ P2 Ž x .  p1
rŽ ␣q  .
/ /
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
.
Ž
.
Ž 17 .
y ␣rŽ ␣q  .
Ž q . y ␥r ␣q  Ž ¨ . 1r ␣q  , Ž
.
Ž
.
Ž 18 .
r aŽ x; k, ¨ . s
P2 Ž x . f a qky Ž ␣ q  . r␣ Ž ␣r .  r
Ž ␣q  .
p1␣ rŽ ␣ q  . P2 Ž x .  r
Ž ␣q  .
Ž q . y␥ r
Ž ␣q  .
Ž ¨ . 1r
Ž ␣q  .
q q q2
Ž 19 . in which P2 Ž x . s p 2 y d 2 p 3 x.8 It is noted that consumption of the two goods, z1a Ž x . and z 2a Ž x ., by a rural resident depends on a specific location, while they are the same regardless of location in the urban area Žsee Eqs. Ž10. through Ž12... Comparative static analysis is performed for the problem Ž15., and the results are summarized in Table 2. For an urban resident, a change in the transportation service price, p 3 , does not affect the consumption of manufacturing goods or agricultural products ŽTable 1.. In contrast, in Table 2, 7 Referring to footnote 5, for instance, one-third of the unit capacity of a car-train is used for transporting a unit agricultural product over 1 km: that is, d 2 s 1r3. 8 The consumption price of agricultural product to a farmer is not p 2 but Ž p 2 y d 2 p 3 x . which is the forgone income from selling his output at the CBD market. Thus, consumption price depends on the transportation cost.
314
TAKUMA AND SASAKI TABLE 2 Comparative Static Results of the Bid-Rent Problem for a Farmer Increase in:
k
¨
p1
p2
p3
q
q2
d2
x
a2
Effect on z1a z 2a ra
0 0 q
q q y
y q y
q y q
y q y
y y ?
0 0 ?
y q y
y q y
0 0 q
an increased price of transportation services will generate substitution from manufacturing goods to agricultural products. This is because the price of self-consumed output, z 2a Ž x ., is Ž p 2 y d 2 p 3 x ., which decreases as a result of an increase in p 3 . Another difference from Table 1 is that an increase in the price of agricultural products lowers the bid-rent of urban residents while it heightens the bid-rent of rural residents, since it increases the farmer’s disposable income. 3. MARKET EQUILIBRIUM In this model, it is assumed that an urban residential district is formed in the inner area and the agricultural sector is located in the outer area. ŽThis configuration is ensured if d 0rd 2 G qf a Ž q2 .rŽ q q q2 ., which implies that the slope of the bid-rent curve of an urban resident is steeper than that of a farmer at the urban-rural boundary..9 Letting x u f and x a f denote, respectively, the urban᎐rural boundary and the metropolitan fringe, and expressing the metropolitan net exports to other regions of manufacturing goods and agricultural products, respectively, by Z1f and Z2f , the equilibrium conditions of the metropolitan system are represented by the following equations
FMŽ NM . s
Hx
xaf
f a Ž q2 .
uf
q q q2
H0
xu f
dx s
z1u Ž x . q
H0
xu f
dx q
z 2u Ž x . q
Hx
xaf
z1a Ž x .
uf
q q q2
dx q
Hx
dx q Z1f ,
xaf
z 2a Ž x .
uf
q q q2
dx q Z2f ,
Ž 20 . Ž 21 .
9 Applying the envelope theorem to the problem in Ž8. and Ž15., the slopes of the bid-rent curves of an urban resident and a farmer are obtained, respectively, as yd 0 p 3rq and yd 2 p 3 Ž f a y z 2a .rŽ q q q2 .. Thus, for an urban residential district to be formed in the inner area it must hold that d 0 p 3rq ) d 2 p 3 Ž f a y z 2a .rŽ q q q2 . at the location x u f , where the two bid-rent curves intersect. Obviously, this is ensured if d 0 rd 2 ) qf a Ž q2 .rŽ q q q2 ..
315
METROPOLITAN AREA WITH HINTERLAND
FT Ž NT . s
H0
xu f
q
H0 H0
ks
1 N
½H
0
xu f
xu f
1 q
d0 x
xu f
dx q
1 q
Hx
dx q
Hx
xaf
d 2 x Ž f a Ž q2 . y z 2a Ž x . .
uf
q q q2
dx s N M q N T s N u , xaf
1
uf
q q q2
dx s N u q N a s N,
dx,
Ž 22 . Ž 23 . Ž 24 .
a1 p1 s w,
Ž 25 .
a3 p 3 s w,
Ž 26 . xaf
5
Ž r u Ž x ; w, k, ¨ . y r o . dx q H Ž r a Ž x ; k , ¨ . y r o . dx , xuf
Ž 27 . r u Ž x u f ; w, k, ¨ . s r a Ž x u f ; k, ¨ . ,
Ž 28 .
r aŽ xaf ; k, ¨ . s r o,
Ž 29 .
Ž p1 Z1f q p2 Z2f . y H
xaf
r o dx s 0.
Ž 30 .
0
Equations Ž20. through Ž22. are equilibrium conditions in the markets of manufacturing goods, agricultural products, and transportation service, respectively. Relation Ž23. shows the labor market equilibrium in the urban sector. Equation Ž24. is the population constraint; Ž25. and Ž26. are profit maximization conditions in the urban sectors, and Ž27. defines the redistribution of differential land rent. Equation Ž28. states that, at the urban᎐rural boundary, the bid-rent of an urban resident is equal to that of a farmer. Equation Ž29. states that the market land rent at the metropolitan fringe is equal to the opportunity cost of land. The last relation, Ž30., is derived using Eqs. Ž22. through Ž24. and Ž27.. This represents the current balance of trade with other regions, showing that the total net export value of traded goods is equal to the total Žnet. imported value of land service Žsince the metropolitan government imports land service at the opportunity cost r o .. The system consists of 11 equations and includes 10 unknown variables p 3 , w, k, ¨ , N M , N T , x u f , x a f , Z1f , Z2f 4 . By Walras’s law, one of those equations is omitted.10 p1 and p 2 are exogenous variables, and p1 is set to unity as a numeraire. It is noted that the wage rate, w, and the price ´ of transportation service, p 3 , are affected only by a change in the produc10
In the subsequent analysis, Eq. Ž20. is omitted.
316
TAKUMA AND SASAKI
tion technology in manufacturing industry, a1. In this sense, w and p 3 are almost exogenous variables. 4. COMPARATIVE STATIC ANALYSIS The impact of a change in various parameters is investigated, conforming to the method in Sasaki and Kaiyama w7x: Eqs. Ž20. through Ž30. are totally differentiated, and then the equations are solved for the endogenous variables.11 The results of comparative statics are summarized in Table 3, where the urban population, N u, is defined as the sum of N M and N T , while the rural population, N a, is calculated as Ž N y N u .. In many cases, the direction of impact on the endogenous variables can be unambiguously determined. For each case, the results of comparative statics are explained. The effect of lowered transport cost. Suppose the transport cost is lowered by a decrease in the necessary transport service, d 0 Žor d 2 ., and note that a decrease in d 0 Žor d 2 . implies not a uniform improvement in the transport system but a partial improvement in the system only for transporting passengers Žor agricultural products.. When the technology of transporting passengers is improved, so that d 0 is lowered, the welfare of residents, ¨ , as expected, increases. The urban area represented by x u f expands, since more labor is shifted from the agricultural sector to manufacturing. The rural area shrinks to the extent that the total metropolitan area denoted by x a f is reduced.12 As a result, the output of the manufacturing industry, and its exports, Z1f , increase because employment, N M , in that sector increases. The output and export Ž Z2f . of the agricultural sector decrease because employment, N a, in that sector decreases. When the technology for transporting agricultural products is improved, so that d 2 is lowered, the utility level of metropolitan residents naturally increases. Contrary to the instance of decreased d 0 , labor shifts from the urban to the agricultural sector. Thus, the output and the export of the agricultural sector increases. Interestingly, however, it is ambiguous whether or not output of the manufacturing sector decreases. This is because there is a possibility that some of the labor released from the transportation sector as a result of technological progress will be shifted to the manufacturing sector, whereby the output of manufactured goods will increase. The net export of manufactured goods is necessarily decreased, while that of agricultural products is necessarily increased. 11
The details of the system for the comparative statics are available on request. The effects of lowered commuting cost on the utility level of residents and the urban area in this analysis coincide with those in the literature so far Že.g., Sasaki w5x, Wheaton w8x, Pines and Sadka w3x.. 12
0 0 q 0
Increase in d0 0 d2 0 a1 q a2 0
0 xa f q y y q q ?
0 0 xu f
q
N Effect on: Increase in d0 y d2 q a1 q a2 y y
p2
p2 N
0
w
p3
Effect on:
?
y
?
¡ ~ ¢ ?
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 .
y q ? ?
?
q
q
¡ ~ ¢ ?
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 .
q y ? ?
Z2f
Z1f
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 . y
q
0
y
¡ ~ ¢
y y q q
¨
?
?
? ? ? ?
k
?
y
?
¡ ~ ¢ ?
Ž if Z2f - 0 . Ž if Z2f s 0 . Ž if Z2f ) 0 .
y ? ? ?
NM
TABLE 3 Comparative Static Results of the Market Equilibrium
?
?
? ? ? ?
NT
q
y
y q q y
Nu
?
q
q y y q
Na METROPOLITAN AREA WITH HINTERLAND
317
318
TAKUMA AND SASAKI
The effect of heightened production efficiency. When production efficiency, a1 , in the manufacturing industry is heightened, the wage rate is increased Žand thus, the price of transport service is increased., whereby labor shifts from the agricultural to the urban sector. The urban area, x u f , is necessarily extended but the metropolitan area measured by x a f decreases. It is noted that the effect on employment in the manufacturing industry is ambiguous: the number of workers, N M , in that sector can be decreased if the production efficiency, a1 , is heightened to the extent that output increases greatly. As a consequence of such adjustments, the utility level, ¨ , increases. An increase in the production efficiency, a2 , in the agricultural sector heightens the welfare level of residents as well. It will cause labor to shift from the urban sector to the agricultural sector, and thereby the urban area shrinks and the metropolitan area expands. The effect on the wage rate is neutral when a2 is heightened. The effects on the net exports of both goods Ž Z1f and Z2f . are ambiguous. The effect of lowered world price of agricultural product. It is hypothesized that p 2 is lowered as the liberalization of agricultural product proceeds. The effect on the utility level, ¨ , is complex: utility increases when the metropolitan area imports Žin net. agricultural product Ži.e., when Z2f - 0., and decreases when it exports Žin net. agricultural product Ži.e., when Z2f ) 0.. An interpretation is that when the consumption of agricultural product exceeds output in the metropolitan area, so that agricultural product is imported Žin net., the lowered price of that product favorably affects the welfare of residents. On the other hand, if the metropolitan area produces more agricultural product than it consumes, such that the excess output is exported to other regions, the lowered price of agricultural product has a negative effect on the utility level by decreasing income level. Regardless of the value of Z2f , when the price of agricultural product, p 2 , is lowered, labor shifts from rural to urban sectors, whereby agricultural output is decreased. As a result, the urban area, x u f , expands, and the metropolitan area, x a f , shrinks. The effect on production and export levels of the manufacturing good is ambiguous. The effect of increased metropolitan population. When N increases, the total employment, N u, in urban sector increases and thereby the urban area, x u f , increases. Thus, the average commuting distance increases, whereby the utility level of metropolitan residents is lowered. It is noted that the metropolitan area, x a f , is not necessarily expanded: if a large amount of labor shifts from rural to urban sectors in response to the
METROPOLITAN AREA WITH HINTERLAND
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increased demand for manufacturing good and transportation service, then the metropolitan area possibly shrinks.13 In most situations, it is impossible to separate the effect on employment in the manufacturing and transportation sectors; that is, the effect on N M and N T. It is, however, unambiguously demonstrated that a decrease in commuting cost, d 0 , increases the employment, N M , in the manufacturing sector. Land rent is one of the indexes most closely representing the metropolitan spatial structure. However, the effect of a parameter change on the rent is ambiguous in the theoretical analysis, and thus the effect on the per capita redistribution of the differential land rent, k, is indeterminate. Some numerical analyses are needed to capture the effects on land rent. 5. CONCLUDING REMARKS A dual-economy metropolitan model was developed by incorporating both urban and rural sectors for explaining appropriately the real growth factors of the urban area. The competitive relations between urban and agricultural sectors in both labor and land markets have been modeled. Such a dual-economy model is also indispensable for evaluating the micro effects of industrial policies Že.g., the policy for liberalization of agricultural prices. on urban spatial structure. Comparative static analysis was performed for evaluating the effects of lowered transport cost, heightened production efficiency, lowered world price of agricultural product, and increased metropolitan population. In many cases, the direction of impact on the endogenous variables was unambiguously determined. However, the effect of a parameter change on the land rent is indeterminate and needs to be captured by numerical analysis.
13 The effect of an increase in the production efficiency Ž a1 or a2 . on population distribution between sectors makes a sharp contrast between the model of Nerlove and Sadka w2x and ours. In their analysis, the effect is neutral whether a1 or a2 is increased. On the other hand, our results show that an increase in the efficiency of the urban production sector induces labor to shift from the agricultural sector to the urban sector, and this increases urban population. The heightened efficiency of the agricultural sector increases the number of farmers and decreases the urban population. Another interesting contrast between the two models is that the lowered transport cost of agricultural products reduces the urban population in our model, while it increases the urban population in the model of Nerlove and Sadka.
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