Keywords: Land use and land cover change; Urban sprawl; von Thunen; Florida Everglades; Economic development
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1. Introduction
Human encroachment of natural areas stemsfrom the demand for both residential space and
agricultural production. These demands havebeen placed in spatial context by bid-rent models,originating with von Thunen, that provide a con-ceptual framework for econometric treatments ofland use and land cover change (e.g. Chomitz andGray, 1996; Walker and Solecki, 1999). The bid-rent formulation in both agricultural and urban
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modes hypothesizes a boundary to human landuse called the urban or agricultural fringe dis-tance, for cities and agricultural hinterlands, re-spectively. This distance can be parameterized inimportant system variables, such as transporta-tion costs and population, to produce functionaldescriptions (see Wheaton, 1974; Fujita, 1989;Nerlove and Sadka, 1991; Jones and O’Neill1993a,b). If natural land lies beyond the fringe,then theoretical results linking fringe to systemattributes provide models of natural areas en-croachment by agriculture or urban uses (Walkeret al., 1997).
Bid-rent formulations are limited in describingencroachment processes such as deforestation,desertification, and wetlands conversion for anumber of reasons. Generally, they focus on long-run equilibria and make strong homogeneity as-sumptions about institutions and resource quality(Bockstael, 1996). In addition, they mainly focuson either urban or agricultural land use, and donot consider their interactions. In urban models(e.g. Fujita, 1989) agricultural rent is constant andprovides solution values for the extent of the city,in which case the city is bounded but the hinter-land is not. Agricultural models set limits to theextent of farming, but collapse the city into adimensionless point (e.g. Nerlove and Sadka,1991; Fujita et al., 1999). Such models are limitedas regional descriptors of land use since cities andneighboring farmlands both possess finite arealextent. Moreover, relationships between the urbanand agricultural sector often underlie processes ofland use and land cover change, and both citiesand farming encroach directly on natural areas(Walker et al., 1997; Walker and Solecki, 1999).
The goal of the present paper is to overcomethe focus of bid-rent models on a single form ofland use. To this end, I combine a von Thunenmodel (Isard, 1956; Fujita et al., 1999) with theurban model of Alonso (1964), as adapted byFujita (1989), in order to address the case ofchanging regimes of natural areas encroachment.The model resulting from this effort is conceptu-ally related to that of Muth (1961), which allowsfor land allocation between two profit-maximizingindustries generating industry-specific bid-rentfunctions. Muth’s formulation focuses on agricul-
tural conversion (i.e. urban sprawl), however,while the model presented in this paper provides amore general description of land cover change,including both urban sprawl and natural areasencroachment, it also considers functional link-ages between city and hinterland, which undergostructural changes in the process of development.In so doing, the model provides a description ofland cover change processes arising in the wake ofeconomic development.
The paper is organized as follows. First I con-sider land use change from the perspective ofMuth, who offered an explicit discussion of urbanand agricultural dynamics. I then provide a dis-cussion and empirical description of natural areasencroachment for the case of South Florida,which has been subject to dramatic land coverdynamics over the past century. The empiricalaccount is based on an application of geographicinformation systems to digital cartographic dataspanning the entire period. I follow this with themodel description, and a statement of theoreticalconditions consistent with the observed processesof change in the South Florida region, both in thelocal economy and land cover. I conclude thepaper with a discussion of model limitations andextensions, and a perspective on policyimplications.
2. Land use change
2.1. Theoretical considerations
Theoretical models for both the residential andagricultural cases have derived explicit functionalrelationships between the extent of land use, orfringe distance, and model attributes such as pop-ulation size and commuting costs. Generally, theyconsider only agricultural or residential land use,with the exception of Muth (1961), who describesthe movement of so-called city limits, or the urbanfringe, in relation to an agricultural hinterlandlying beyond the city. In the Muth model, residen-tial and agricultural industries generate rent, andland is dedicated to the industry providing thehigher value at any given location. Under anappropriate set of assumptions, residential land
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use is found close to the market center, and givesway to farming beyond the urban fringe. Of rele-vance to the present application, the urban fringedistance is endogenous to changes in the system’sexogenous variables. For example, urban land useexpands at the expense of agriculture as demand forhousing grows. Muth analyzes relationships be-tween urban fringe distance and attributes of theeconomic system in place, including wage rates andrelative demand characteristics for housing andagricultural produce.
The Muth approach possesses two shortcomingsfrom the perspective of the present application.First, the extent of agriculture is unbounded (Muth,1961, p. 13), and we cannot observe the advance ofeither urban or agricultural land use into unutilizedlands. This compromises the framework’s appli-cability to the case of natural areas conversion. Thepresent model allows for two fringe distancesassociated with urban land use and farming, andtherefore accounts for the empirical observationthat both farms and urban space encroach onnatural areas. Such encroachments have been re-ported for large metropolitan areas in the US, bothin New Jersey (Walker and Solecki, 1999) andSouth Florida (Walker et al., 1997).
A second shortcoming stems from the conceptu-alization of linkages between city and hinterland.The urban sector is modeled as a residential indus-try maximizing profits, and individuals who may ormay not demand locally-grown produce populatethe city. In the present formulation, I attempt tomake the nature of the structural relationshipbetween hinterland and city more consistent withhistorical observation. To this end, I address boththe changing nature of the relationship and themanner in which agricultural supply functions as aninput to regional production. From a developmen-tal perspective, demand-side linkages internal to aregion are likely to be insignificant in relation to thesupply-side interactions arising, for example, fromcity-based processing of agricultural staples forexport to national and even international markets.1
2.2. The South Florida case
Before presenting the model in formal terms, Iconsider an empirical case that demonstrates ele-ments of the theoretical context just outlined.Natural areas in South Florida, and in particularthe Florida Everglades, have attracted interna-tional attention for their ecological uniqueness andfragility. They have also endured rapid encroach-ments by both residential and agricultural landuses, and are under continuing threat due topopulation growth in the region. At the presenttime, both the Federal Government and State ofFlorida are attempting to undo environmentaldamages wrought by one hundred years of landcover change that has reduced the Evergladesecosystem to about 50% of its original extent, andvirtually eliminated the pine forest that originallycovered the so-called Gold Coast (Broward, Dade,and Palm Beach counties). A restoration effort isunderway to recover the lost hydrologic function ofthe surface waters of the Everglades by reconvert-ing agricultural lands into natural areas.
The initial private-sector stimulus to economicdevelopment in South Florida was provided bytourism with the arrival of Flagler’s railroad at theturn of the century. Prior to this, few people livedin the region, and the economy was based largelyon animal products exploitation.2 By 1900, twohotels had been built in Palm Beach, with a capacitynear 1600 rooms, and Miami possessed a sizeableoperation with 450 rooms.3 In 1909, Miami aloneenjoyed 125 000 tourist visits, ten times
2 Key West, of course, was the third largest city in Floridaat the time with nearly 20 000 inhabitants.
3 The hotel service workforce may be estimated at about1700, and over $1 000 000 per year was generated in revenue.In addition, 4000 construction workers were involved in Fla-gler’s railroad venture through the Florida Keys after Miamihad been reached in 1896. See Derr (1989) for workforceestimates. Revenue estimate assumes visitors purchased one ofFlagler’s 5-week packages for $350, including round-trip coachfrom a northern origin. About 100 private rail cars made theround trip from Jacksonville to Palm Beach each season at acharge of $342 (Derr, 1989).
1 In adaptations of von Thunen, both Nerlove and Sadka(1991) and Fujita et al. (1999) consider the autarchic case, witha trade equilibrium between the city and its hinterland.
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the resident population of about 12 500 (Derr,1989).
State involvement ultimately proved crucial tothe region’s development, and began in earnest atthis time with Governor Broward’s efforts to con-vert the Everglades into a farming region. Agri-cultural expansion followed the successfulcompletion of drainage infrastructure, and by1913 over 360 km of canals had penetrated theEverglades (Derr, 1989), taking tomatoes, beans,and other crops by steamboat to Fort Lauderdaleand Fort Myers for local consumption and ship-ment north (Blake, 1980). By 1920, 23 000 farmerscultivated 138 km2, numbers growing to 92 000inhabitants and 186 km2 by 1927 (Blake, 1980).Between 1907 and the early 1930s, the main fea-tures of the current drainage system had beenimprinted on the sawgrass everglades (Light andDineen, 1994). 708 kilometers of canals were inplace, connecting the inhabited parts of the penin-sula to the southwest and southeast.
Anecdotal evidence suggests the agriculturaland urban economies of South Florida initiallypossessed a certain degree of functional depen-dency. Urban residents added value to the exportof pineapples and oranges to northern markets(through food processing and transportation), asupply-side linkage between farming and the ur-ban sector. In addition, much of the region’sproduce probably served the seasonal touristtrade. Flagler’s initial vision for developing theregion involved shipping tourists south and farmproducts north by rail (Derr, 1989). By way ofcontrast, South Florida presently possesses a ser-vice economy with over 99% of its gross product(about $82 billion in 1990) accounted for by non-agricultural activities (Walker et al., 1998). Al-though farming is a basic sector (Snyder andDavidson, 1994; Mulkey and Clouser, 1988) deliv-ering products to national and even internationalmarkets, the most important crop, sugarcane, hadsales approaching $.5 billion in 1990, a very smallpercentage of the region’s gross product (Alvarezet al., 1994).
The ascension of sugarcane as the key regionalcrop is emblematic of a process of decouplingbetween the agricultural and urban economiesthough the course of the century. Grown in the
region since 1925 (Coale, 1994), the Jones–Costi-gan Act of 1934 and subsequent legislation re-stricted production, so that by 1950 only 153 km2
were committed to cane (Snyder and Davidson,1994). This changed dramatically in the wake ofthe Cuban revolution, with the elimination of theCuban sugar quota in 1960 (Alvarez and Polopo-lus, 1988). South Florida now produces a largeshare of the nation’s sugar, and by 1990 occupied1782 km2 of Everglades land (Coale, 1994), overten times its expanse in 1950. Prices are set innational markets and by federal policy.
The timing of changes in linkages between agri-culture and the urban economy is difficult to trackover the past century given data limitations in theUS Census, which begins regular reporting in theregion in 1930. Vegetable crops were grown in the1930s for the northern winter market (Snyder andDavidson, 1994), when 2.7 percent of the Dadecounty workforce labored in food processing andsteam railroads (US Bureau of the Census, 1930).Price instability for vegetables at this time hasbeen attributed to a highly competitive nationalindustry (Snyder and Davidson, 1994), but mayalso have reflected yearly variation in tourist vis-its. Indeed, one of the noted attractions to hotelsin Florida in the early years was ‘good food’grown on local farms (Derr, 1989). Nor can it beforgotten that transportation remained difficultand inefficient until well into the twentieth cen-tury. The railroad did not arrive in the evergladesagricultural areas until 1925, where steamboatstransported produce until 1921 (Blake, 1980). By1990, the percentage of Dade county workersengaged in food processing and trucking was neg-ligible, at 0.8% (US Bureau of the Census, 1990).
Table 1 presents region-scale dynamics of landcover change occurring in South Florida over thispast century. These data were generated througha GIS application involving remotely sensed dataand maps of the region for the years 1900, 1953,1973, 1988, and 1995, covering the Gold Coastregion of Broward, Dade, and Palm Beach Coun-ties. These five time points allow for the definitionof four conversion episodes, 1900–53, 1953–73,and 1973–88, and 1988–95, which may be used totrack the dynamics of land cover change.
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The table shows that in the first and secondepisodes, agricultural conversion of natural areasdominates urban encroachments, in excess of 70%for both periods. Subsequently, the urban compo-nent increases appreciably. Between 1973 and1988, about 57% of natural areas conversion isaccounted for by urban expansion; this dips to50% in the following period.4 The table alsoshows other land cover dynamics, including agri-
Fig. 1. The linear city system.
cultural conversions and restoration of naturalareas. Loss of agriculture to urban land use (i.e.urban sprawl) is initially minimal relative to natu-ral areas loss, but grows important in subsequentperiods. The abandonment of settled areas, bothagricultural and urban, becomes relatively signifi-cant by the final conversion episode, reflecting,perhaps, current efforts to improve hydrologicfunction by restoring land to natural conditions.
The change in the relative magnitudes of agri-cultural as compared to urban encroachments ofnatural land is consistent with the developmentprocess I seek to model. In particular, in earlyphases with functional coupling, natural areas lossshould be primarily agricultural in nature. But asfarming becomes less important to the regionaleconomy, the stage is set for direct losses ofnatural areas with expansion of the city.
3. Model statement
Consider a region embedded in a national econ-omy where individuals enjoy utility level, u, andearn income, Y, from an unspecified productionprocess. The utility level, u, is fixed, presumablyby a national economy, in which case the city isopen to population movements (Fujita, 1989).The region is a one-dimensional, linear space (As-ami et al., 1990) consisting of a port city and anagricultural hinterland; distances are measuredfrom the port, which functions as the city center.Fig. 1 gives a stylized representation of the spatialsystem, consistent with the overall distribution ofland use in South Florida.
The region is assumed to undergo a develop-ment process as follows. Initially, it provides anexport, q, to the national economy, produced byurban workers who add value to an agriculturalproduct. The export may be taken as a goodstemming from conventional food processing ac-tivities in which workers add value to a locally
Table 1Land cover changes by period (Broward, Dade, and PalmBeach Counties), total area=13737 km2a
1900–53 1953–73 1973–88 1988–95
1681518Natural to 2340 455agriculture
872576 166Natural to 597urban
0 228Agriculture to 583 183urban
0Agriculture to 215 306 133natural
0Urban to 19 122105natural
a Regional data may be obtained for 1900, 1938, 1953, 1973,1975, 1986, and 1995, the years in our archive. For 1938, itappears that agricultural lands include those zoned for agricul-tural use but not yet developed, which inflates the amount ofalready converted land in that year. Our data for 1975 (USGS)is from a different source than the series, 1900, 1953, and 1973,all produced by the Center for Wetlands at the University ofFlorida. The Center data are constructed from aerial photog-raphy except for 1900, which is an ecological ‘reconstruction.’The Center data were put in digital form in a 1 km2 grid.Comparison of 1973 and 1975 suggests some differences in theclassification system used by the US Geological Survey (1990)and the Center for Wetlands, University of Florida. Calculatedland cover changes between 1973 and 1975 would thereforeprobably reflect artifacts of classification differences; in anyevent, a twoyear time period is too short to have muchanalytical value. Consequently, our analysis is based on 1900,1953, 1973, 1988, and 1995. The 1988 and 1995 data are fromthe South Florida Water Management District and are basedon both aerial photography and Landsat TM images. Conver-sion magnitudes were developed with the overlay function ofARC-INFO, based on GIS-generated aggregations of land-cover to agricultural, urban, and natural land.
4 Of course, an initial conversion to farming and then tourban land use may have occurred rapidly within the periodcovered by the conversion episode. Unfortunately, the data donot allow a year by year account.
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grown staple. Alternatively, it may be interpretedas a tourism service, produced by urban workersin conjunction with produce from the region. Theexport economy is initially under the control of aregional monopsonist, who sets price for localproduction but is a price taker in both labor andexport markets. I call this initial developmentalstage a coupled economy (Walker et al., 1997). Instage two of the development process, the linkagebetween urban workers and farming is decoupledas the regional monopsony breaks down and agri-cultural goods are sold directly to national mar-kets. Muth (1961) distinguishes between coupledand decoupled regional economies, the latter ofwhich he refers to as the national food case. ForMuth, linkages between city and hinterland arisefrom the consumption demand for food, and de-coupling takes place when the elasticity of thedemand for food is low given export to a nationaleconomy.
To facilitate the exposition, I also assume tech-nological change in agriculture. In stage one, theproduction factors are land and labor, which ismade available to agriculture on the basis of aresidential equilibrium between urban and hinter-land locations. In stage two, inputs (includingcapital) substitute for labor, and are used in fixedproportion with land. Thus, the stage two econ-omy is modernized, and workers live only in thecity. The two-factor assumption on agriculturalproduction is consistent with Nerlove and Sadka(1991) and Fujita et al. (1999), who assume atechnology based on land and labor. Productionin Fujita et al. (1999) occurs with fixed propor-tions between the factors.
Following the constructive approach advancedby Muth (1961), I assume the stylized fact thaturban land use generally gives way to agriculturemoving out from a city center when both landuses are present.5 As in Muth (1961), this isaccomplished through assumptions on appropri-ate system parameters. The region has a marketcenter and land that can be used for either hous-ing or agriculture. In turn, workers can work at
the center of the city or in the agricultural hinter-land. City workers pay a commuting cost, T(r),for expenses associated with transport to andfrom their residence located at distance, r, fromthe city center. They also pay rent. Farm workerslive and work on the same parcel of land and donot pay commuting costs. In addition, their wagerate is reduced by the amount of rent paid by citydwellers at the urban fringe. This ‘implicit’ rentaccrues to landowners in the agricultural hinter-land, who also receive rent associated with loca-tional advantages in agricultural production.Land use is determined by which form of theparcel bid pays higher rent.
The model is now set out in two steps. First, Iconsider the bid-rents associated with residentialuse of land and agriculture. Then, I consider thedevelopmental framework and implications formodel structure. Although the specification foragriculture changes with the development process,the form of the residential function remains thesame. I follow the presentation with several nu-merical applications to illustrate the main pointsof the conceptual development.
3.1. Bid rents in the city and agriculture
Following Fujita (1989), residential rent paidby urban dwellers, �, is the maximum bid forresidential use of a land parcel, and may bedescribed as a function of the national utilitylevel, u, and distance from the market center, r, or�(u, r). This bid-rent, in turn, may be defined byoptimizing a utility function expressing prefer-ences for a composite good (z) and lot size (s). Ifthe utility function is Cobb–Douglas, as shall beassumed in the sequel, or U=z�s� where �+�=1, and if commuting costs are thr, where th reflectsexpense per unit distance, bid-rent may be derivedas (Fujita, 1989, p. 322):
�(u,r)=��/��(Y− thr)1/�e−u/�
The bid-lot size, s, may also be derived as:
s(u,r)=�−�/�(Y− thr)−�/�eu/�.
Define rmax as the maximum extent of the cityin the absence of agriculture. This is given bysetting �(u, r)=0 and solving for r, or rmax=Y/th.
5 The regional system possesses either agricultural land use,urban land use, or both. If both, agriculture lies beyond thecity, and urban land use does not reappear in the hinterland.
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For agriculture, I consider two functionsreflecting the changes in agricultural technologyassociated with the development stages. Rent isgenerated, of course, by the differential in rev-enues and costs per unit area (Fujita et al., 1999),and for stage one of the development process wehave:
R1(r)= (1/s*)[cp1A(r)−YA]
where R1(r) is agricultural rent at distance r, c isoutput per worker, p1
A(r) is price of the agricul-tural good at distance r, YA is the agriculturalwage rate, and 1/s* is agricultural workforce den-sity, given by the reciprocal of the bid-lot sizeobserved at the urban fringe, r*. The functionalform of price in both development stages is takento reflect a so-called iceberg technology in thetransportation sector for agriculture (Nerlove andSadka, 1991; Fujita et al., 1999), or:
piA(r)=pi
Ae−ar i=1,2
where piA is price of the good at the city center (in
stage one or two) and a accounts for its trans-portation costs from the farm (located at r) to themarket. Such price functions eliminate the need torepresent a transportation sector.
The stage one framework presents theoreticalcomplications owing to urban–agricultural link-ages and the dependency of farm production onlabor, which is spatially determined. The presentformulation differs from Fujita et al. (1999) withrespect to both land productivity and the func-tional relationships between agriculture and thecity.6 Fujita et al. (1999) consider a trade relation-ship between city and hinterland, based on aclosed economy (see also Nerlove and Sadka,1991). Wages and prices are determined endoge-nously, and there is no export to national mar-kets. The productivity of land is fixed, andagricultural labor is used in constant proportionto land under any set of equilibrium conditions.In the present model land productivity is a func-tion of the amount of labor available on a given
parcel (c/s*).7 Although constant for a given equi-librium, changes in the extent of the city alsochange the agricultural labor per unit area, inwhich case output also varies. If the urban fringeadvances at the expense of farming, the availabil-ity of farm labor declines, as does the productivityof the land.
To maintain labor force equilibrium betweenthe two sectors, workers must everywhere enjoythe same utility level. This is accomplished bysetting the agricultural wage equal to the dispos-able income of city residents at the urban fringe,r*.8 Here, commuting costs are thr* and theamount of rent paid by individual workers iss*�(u, r*). What remains of the wage rate, Y,after incurring these expenses is spent on com-posite good, z. Thus, the agricultural wage rate iswritten as YA=Y− thr*−s*�(u, r*), which al-lows workers to consume the same amount of z astheir counterparts on the urban fringe. Since theylive on the land they work, and workforce densityis set at 1/s*, the amount of land individual farmworkers consume through residential occupation isalso the same as their urban counterparts on thefringe, s*. Assuming identical utility functionsacross sectors, the level of utility achieved byworkers in farming is therefore u.9
For stage two agricultural rent, we have:
R2(r)=dp2A(r)−pII
where d is the fixed output per unit land (givenfixed intensity of input application), pI is the inputprice, and I is the amount of input application perunit land. In this construction, the non-land inputis fixed and invariant to distance from the citycenter.
7 Strictly speaking, land is not an input to production insuch a formulation, in that c is a constant. One interpretationis that agriculture functions as a cottage industry, whereby alandlord hires workers to produce an agricultural commodity,and provides the material, land, necessary to do so.
8 I follow Fujita et al. (1999) in my tenurial assumption onagricultural land. A land owner employs agricultural workersand recovers a rent.
9 An implicit assumption is that there are no negative exter-nalities associated with residential occupation of agriculturalland, a possibly strong assumption.
6 Fujita et al. (1999) and Nerlove and Sadka (1991) do notconsider urban land use, and the city is collapsed into adimensionless point.
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3.2. Constructing the sectoral linkages
To this point, I have considered the bid rents inisolation, but they must be linked in order todescribe the stylized development processes ofinterest in the present context. I proceed initiallywith stage one, since this presents a number oftheoretical issues. In particular, agricultural priceis endogenous to the production decision of theregional monopsonist. Once selected, this pricethen sets up the land use regimes by establishingcity limits and the extent of agriculture. In stagetwo, demand for the commodity is taken to beexogenous and perfectly elastic, which fixes priceand essentially collapses the model into two inde-pendent bid rent functions.
Following Fujita (1989), z is numeraire andthere are prices for land, labor, and agriculturalproduce. Also, given the open city assumption(Fujita, 1989), urban bid-rents at given values of rare fixed. I begin by assuming that commutingcosts are large relative to those for transport ofthe agricutural commodity, or th/Y�a. This as-sumption structures the city and its hinterland ina manner consistent with the stylized facts of theirspatial organization.10 The model statement isadvanced in two steps. I initially find a function,p c
A(r), giving the city center farm price that makesresidential and farming rents the same at distancer. Thus, p c
A(r) is actually the same as p1A(0) for
some arbitrary fringe distance. The functionalform is necessary because p1
A(0) is a choice vari-able for the regional monopsonist during phaseone. Consider the equation:
�(u,r)=R1(r) (1)
where R1(r)= (1/s(r))[cp1A(r)−YA(r)]. Eq. (1)
may be solved to yield:
p cA(r)= [(Y− thr)ear]/c. (2)
Thus, for any given urban fringe, r=r*, p cA(r*)
is the city center price that would guarantee iden-tical urban and agricultural rents at r*. Eq. (2)
may be differentiated as dp cA(r)/dr= [a(Y−
thr)ear− thear]/c ; this is negative if and only if :
a(Y− thr)� th,which is true by the assumption a� th/Y. Thus,agriculture moves closer to the market with in-creasing price for the farm good at the city center.This price may be bounded from above so thaturban land use vanishes at higher prices (seeMuth, 1961; p. 11). In particular, at the citycenter the solution to Eq. (2) yields p c
A=Y/c�pmax.
Now, fix some arbitrary value of r, or r*, andconsider the derivatives in r of � and R at dis-tance r*. These are given as d�/dr= − th/s* anddR/dr= −acp c
Ae−ar*/s*. Under the assumptiona� th/Y, we have acpmax� th, which implies thatacp c
Ae−ar� th for all r, and therefore �d�/dr ���dR/dr �. Thus, the agricultural rent function inter-sects the urban bid-rent function from below atr*; moreover, for r�r*, it is also the case that�d�/dr �� �dR/dr �. This does not guarantee thedominance of agricultural land use beyond theurban fringe, which is constructed as follows. Theextent of agricultural land, ra, may be solved bysetting R1(r)=0 for some r* as ra= ln{cp c
A(r*)/�(Y− thr*)}/a, given that YA=Y− thr*−s*�,s*�=�(Y− thr*), and �+�=1. By Eq. (2), wecan substitute for p c
A(r*), obtaining:
ra=r*+ [ ln (1/�)]/a (3)
Note that the extent of agricultural land use,ra−r*= [ln(1/�)]/a, does not vary in price for thefarm good. Hence, the dominance of agriculturebeyond r* is guaranteed by the condition, [ln(1/�)]/a�rmax=Y/th. When r*=0, p c
A is at its max-imum, and agricultural rents are everywheregreater than urban rents, which go to zero atrmax=Y/th. Although spatial extent remains thesame, agricultural supply changes with r*, sinceland productivity is a function of workforce den-sity. In particular, supply increases with price of
11 The value of rmax at 15 is consistent with the numericallyderived curves in Fig. 3, with Y=12 and th= .8, yielding amaximum possible extent of the city as 12/0.8=15. If weassume, as is also the case in Fig. 3, that �=0.4 and a=0.05,then [ln(1/0.4)]/0.05=18.3�15=12/0.8, and the conditionfor agricultural dominance is upheld.
10 This assumption functions analytically in a way similar toMuth’s assumption on the relative values of transportationcosts and technological parameters for residential and farmingindustries (Muth, 1961; p. 11).
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Fig. 2. City and agricultural rents.
the commodity, as production moves closer to thecity center and more labor is devoted to farming.
These various results and observations are illus-trated in Fig. 2, showing one bid-rent curve forresidential housing, and two for agriculture underdifferent price regimes. Potential city rents go tozero at rmax, although both agricultural rents arepositive at this value.11 Note that the agriculturalfringe distance for high prices is less than for lowones. This is accounted for by the result in Eq.(3), which fixes the extent of agriculture beyondthe urban fringe. This result, in turn, is an artifactof the iceberg technology assumption that steep-ens the rent curve in price, and of the specificationof the agricultural wage. In particular, the agricul-tural wage is high when farming is close to thecity center and the price of the farm good is high.
3.3. The regional economy
The structure of the regional economy is givenby the production relations between the city andagriculture. In this regard, stage two is straight-forward, with agricultural demand taken to beperfectly elastic and fixed land productivity. Stageone, however, involves the decisions of a monop-sonist using farm goods and urban labor to pro-duce a regional export. Let the productionfunction for the export good be Q(L,A), where L
is the urban labor force and A is the supplyfunction for agriculture.12 These may be ascer-tained in explicit form by integration:
L=� r*(pc
A)
0
dr/s and A=� ra
r*(p cA)
(c/s*) dr
yielding L(p cA)=��{Y1/�− [Y− thr*(p c
A)]1/�} andA=c/s*(ra−r*), where �=� (�/�)e(−1/�)/th. As isevident, both are functions of the city center pricefor the agricultural good, whose supply is givendirectly as c/s*[ln(1/�)]/a, given that ra=r*+[ln(1/�)]/a. The L and A functions may be differ-entiated in p c
A as dL/dp cA=� [Y− thr*(p c
A)](1−�)/�
dr*/dp cA�0; and dA/dp c
A= −c/(s*)2[ln(1/�)]/a(ds*/dp c
A)�0, since bid-lot size increases withdistance from the market center.13 Hence, A ismonotone increasing in p c
A as is expected forsupply functions. Moreover, since the city con-tracts with increases in city center farm price, L ismonotone decreasing in the price, given the fixednature of the residential density curve under an
12 All agricultural produce in the present formulation is usedfor the export good, and may be regarded as the export base.It is not food grown for subsistence purposes. By way ofcontrast, Fujita et al. (1999) present a food model with en-dogenous consumption of the entire produce of the hinterland.
13 Since pcA increases as distance from the market center
decreases, the derivative, ds*/dpcA, is negative given the rela-
tionship between s and distance.
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Fig. 3. City expansion with technological change. Note: production technology is Cobb–Douglas and export price, pE, is 100.Commodity intensive production is given by q=L0.2A0.8, and urban intensive, by q=L0.8A0.2. Model parameters are: Y=12;th=0.8; a=0.05; u=1; �=0.4, c=1 and �=0.6.
open city system (Fujita, 1989). The regionalmonopsonist chooses p c
A to maximize:
pEQ(L,A)−YL−p cAA (4)
where pE is the price of the export and Y is wagerate, as before. Note that tourism may function asan export sector, with Q serving as the provisionof the tourism service.
Rather than develop a set of theoretical results,I pursue a numerical strategy to demonstrate solu-tions to the monopsonist’s problem, and implica-tions for land use and land cover change. Thismay be accomplished by identifying the r* thatoptimizes Eq. (4) using the relationship betweenp c
A and r* to calculate a profit function in r*. Fig.3 shows results for two settings on the technologyof regional production. Technology that is inten-sive in the agricultural commodity yields a profitfunction maximized with an urban fringe at rci
and, by implication, an agricultural fringe at rci+
[ln(1/�)]/a. With growing emphasis on valueadded in the urban sector, the optimizing citymoves to rui extending the agricultural hinterlandthrough urban sprawl by rui−rci. The shift inimportance of the value-added process in effectcreates employment in the city, requiring addi-tional residential space. Similar treatments can beaccomplished by varying other system parameterssuch as commuting costs. Fig. 4, for example,shows the effect of a reduction in th on the urban(and agricultural) fringe for the same price of thefarm good at the city center.14
To this point, my focus has been on phase oneof the development process due to the theoreticalissues involved in linking the urban and agricul-tural sectors. In fact, as alluded to, the phase twoeconomy presents a trivial problem when price ofthe agricultural good is perfectly elastic, as as-sumed here, and when workers are only urbanbased.15 In such a situation, it is unnecessary to
14 The partial equilibrium effect of commuting cost reduc-tions on fringe distances may be established by totally differen-tiating (2) assuming fixed price, and observing for somearbitrary urban fringe, r, that dr/dth=r/[a(Y− thr)− th]�0,given the condition that a(Y− thr)� th, which is true byassumption.
15 Clearly, modern agriculture uses labor inputs. Neverthe-less, theoretical statements often rely on a two-factor produc-tion function. Stage two agriculture in the present formulationreflects the technological changes that have occurred in agri-culture since the early part of the 19th century.
R. Walker / Ecological Economics 37 (2001) 357–369 367
Fig. 4. Reduced commuting cost.
establish residential equilibrium between the cityand hinterland, and rents remain fixed even asagricultural supply diminishes with advance of thecity. The implication is that with urban growththe agricultural fringe remains stationary, andonce r*=ra, any expansion of the city will lead todirect urban encroachments of natural areas andthe disappearance of agriculture. Fig. 5 illustratesthis case. As residential rents grow, the maximumpotential extent of the city exceeds ra, and passesdirectly into undeveloped land. City expansionmay result from changes in the wage rate (as inFig. 5), in the costs of commuting, in preferencesfor lot size, and in population (due to externalshifts in relative utility levels).
4. Discussion
Although the model is formal in nature, I haveattempted to reflect links between regional eco-nomic development and processes of land coverchange occurring in a particular region, the GoldCoast of South Florida. This is inherently prob-lematic given limitations in the available data, andI have only suggested that the model I present isconsistent with historical observation, scanty as itis. I hypothesize, however, that the model struc-ture possesses a degree of generality given (1) theimportance of agricultural staples in early stages
of regional development; and (2) the historic de-emphasis of agriculture in the aggregate economy,and the emergence of agricultural regions at na-tional level with the passage of time.
Several shortcomings are apparent in thepresent formulation as regards South Florida.Important among these is the existence of yetanother form of rent arising from the amenitiesand services provided by the region’s natural ar-eas. These are typically non-market values ex-pressed as public goods, and in which bothFederal and State government have invested con-siderable resources through the creation of parksand other conservation areas. Such areas would
Fig. 5. Decoupled urban encroachment.
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Fig. 6. Protected area boundary.
5. Conclusions
Despite limitations, the model provides a formaldescription of three land cover change processes,namely, the conversion of natural areas to eitherurban or agricultural land use, the loss of agricul-ture to suburbanization (i.e. urban sprawl), and therestoration of natural areas from prior human use,either urban or agriculture. Thus, the model ad-vances the seminal contribution by Muth, andbegins to bring to the surface the implicit promiseof the von Thunen/Alonso framework in modelingland cover change. Although highly stylized, itcaptures a potentially important aspect of landcover change dynamics resulting from economicdevelopment and the interplay of urban and agri-cultural processes.
Extensions and adaptations of the present modelcould prove useful since theoretical efforts on thesubject have been few and far between. This is aserious shortcoming in the policy arena, given thatpolicy formulation requires a firm basis on behav-ioral understanding. With the growing recognitionof the importance of land use and land coverdynamics to the environment, the need for theoret-ical insight and predictive modeling is likely to growas well. Fortunately, a well-developed theory ofland use already exists in regional and urbaneconomics, and can be deployed as in this paper totell at least part of the story about environmentalchange.
Acknowledgements
I would like to thank Tony Smith for mathemat-ical critique and very helpful interpretative com-ments. I would also like to thank two anonymousreviewers for reactions to an early draft of themanuscript. The research was supported by the USMan and the Biosphere Program (MAB), USDepartment of State, and by the US NationalAcademy of Sciences. I would especially like toacknowledge the support of Roger Soles, executivedirector of the US MAB program, and of MarkHarwell and John Long, chair and co-chair of thatprogram’s Human Dominated Systems Direc-torate. I would also like to acknowledge Judy Bale
represent a zone of effectively infinite rent, and begiven as a border to the expansion of human-dom-inated systems as shown in Fig. 6. Although limitsto expansion were not really relevant to the dynam-ics of land cover change during the first half of thecentury, they are of great importance at the presenttime. In a related vein, the model does not addressstate intervention in either the drainage or currentrestoration phase, which has been key to land coverchange processes in the region.
Another potential shortcoming is the assumptionof phase two regarding the market situation for theprime agricultural export, namely sugar. SouthFlorida supplies a sizeable fraction of the nationaldemand for sugar, in which case encroachments ofagriculture by urban expansion could raise prices.In theoretical terms, this would alter the rentfunction for agriculture, and city growth wouldcontinue pushing farmland into natural areas as inphase one. The data in Table 1 do show continuedexpansion of agriculture even in the later conver-sion episodes. This could reflect (1) differentialrequirements for land suitability by agriculture andurban land use; (2) the release of pent-up demandfor land by the sugar industry in the wake of theembargo on Cuba; (3) the price situation justdescribed; or (4) some other as of yet undescribedphenomenon.16 Further empirical work will berequired to resolve this issue.
16 Coupling between city and hinterland need not be strictlyrelated to food consumption products. Sod production ispresently an important form of agricultural land use in SouthFlorida, serving both local markets and other parts of the state(Bottcher and Izuno, 1994).
R. Walker / Ecological Economics 37 (2001) 357–369 369
and Michael Greene of the National Academy ofSciences. Discussions with my colleague and co-investigator, Bill Solecki, provided me with muchinsight into processes of environmental change inSouth Florida and elsewhere around the world.As always, Steve Hodge of the Florida Resourcesand Environmental Analysis Center (FREAC) atFlorida State University gave superb technicalsupport.
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