Mapping Process to Pattern in the LandscapeChange of the Amazonian Frontier

Robert Walker

Department of Geography, Michigan State University

Changes in land use and land cover are dynamic processes reflecting a sequence of decisionsmade by individual landmanagers. In developing economies, these decisionsmay be embedded in the evolution of individual households, asis often the case in indigenous areas and agricultural frontiers. One goal of the present article is to address the land-use and land-cover decisions of colonist farmers in the Amazon Basin as a function, in part, of householdcharacteristics. Another goal is to generalize the issue of tropical deforestation into a broader discussion on forestdynamics. The extent of secondary forest in tropical areas has been well documented in South America andAfrica.Agricultural-plot abandonment often occurs in tandem with primary forest clearance and as part of the samedecision-making calculus. Consequently, tropical deforestation and forest succession are not independent processesin the landscape.This article presents a framework that integrates them into amodel of forest dynamics at householdlevel, and in so doing provides an account of the spatial pattern of deforestation that has been observed in theAmazon’s colonization frontiers. Key Words: Amazon, deforestation, land use change.

Changes in land use and cover occur throughout theworld, but the greatest present-day concernfocuses on tropical deforestation, which drives

species to extinction, releases greenhouse gases, andundermines the sustainability of local environments(Stern, Young, and Druckman 1992; Ojima, Galvin, andTurner 1994;Turner et al. 1995, 2001).Nowhere are theseimpactsmore apparent than in theAmazonBasin,where alarge portion of the forest has vanished in the wake ofgovernment initiatives to bring people without land toland without people (Hecht 1985). The alarm continuesto sound, with rates of forest lost varying between 10,000and 20,000 km2 per year (INPE 2000), and with plans bythe Brazilian government to extend its road-buildingefforts throughout the region (de Cassia 1997; Lauranceand Fearnside 1999).

This article presents amodel of the land-use behavior ofthe colonist farmer, an important agent in currentprocesses of Amazonian deforestation. The elaborationof such a model is not without challenges, given thedynamic institutional environment of the forest frontier.Upon initial occupation of land beyond the forest margin,colonists behave in large part according to the dictates ofthe peasant-economy concept (Thorner, Kerblay, andSmith 1966), focusing their attention on subsistenceproduction in the interests of food security. With frontierexpansion and the arrival of markets, economic behaviorchanges as subsistence agriculture gives way to commer-cial production (Binswanger andMcIntire 1987; DeShazoand DeShazo 1995). After all, colonists often settlefrontiers with the expectation that the risks they take to

subdue the wilderness will be paid off by market rewards(Rudel and Horowitz 1993).

In addition to providing a behavioral model of colonistland-management, this article also seeks to generalize theissue of tropical deforestation into a broader discussion,including its spatial articulation and the often-overlookeddynamic involving the regrowth of secondary vegetation.The extent of secondary regrowth has been well docu-mented for South America (Brown and Lugo 1990),and extensive areas of successional vegetation exist inthe Amazon basin, even with continuing deforestation(Moran et al. 1996). Neglected in the attempt to accountfor this phenomenon is that field abandonmentmay occurin tandem with primary forest clearance and as part of thesame decision-making calculus. In addition to capturingthe transition from subsistence to market-oriented pro-duction, the behavioral model of the colonist alsoaccounts for the simultaneous occurrence of forest lossand regeneration. Moreover, once embedded in a two-dimensional landscape according to the empirical geom-etry of settlement-planning, the model generates spatialoutcomes and patterns of forest fragmentation.

Tropical deforestation and land-cover change ingeneral are complex processes involving a multiplicity ofagents. In the Amazon basin, colonist farmers, corporateranchers, loggers, and gold-miners have all contributed toforest loss, as have government bureaucrats far removedfrom the forest frontier (Smith et al. 1995). Such agentsmay operate with an interdependent logic, a recurrenttheme in discussions on tropical deforestation (Geist andLambin 2001). In an early account addressing the global

Annals of the Association of American Geographers, 93(2), 2003, pp. 376–398r 2003 by Association of American GeographersPublished by Blackwell Publishing, 350 Main Street, Malden, MA 02148, and 9600 Garsington Road, Oxford, OX4 2DQ, U.K.

situation, Myers (1980) describes a process of invasiveforest mobility, in which farmers follow loggers into newlyopened forest. There, as they prepare the land for farming,they finish what the loggers started by taking down thetrees that remain in the wake of selective logging (Leslie1980; Myers 1980; Schmithusen 1980; Office of Technol-ogy Assessment 1984; Walker 1987; Repetto and Gillis1988;Walker and Smith 1993; Kummer and Turner 1994;Brookfield, Potter, and Byron 1995). For the Amazoniancase,Wood (1983) andOzorio deAlmeida (1992) point toanother connected process whereby land consolidation bywealthy ranchers induces migration on the part ofsmallholders, who must pick up and leave for unclaimedlands in primary and old-growth forest. Beyond thesevarious agent interactions, social forces acting in aggregateplay significant roles in processes of land-cover change.The political economy of development in the BrazilianAmazon also offers a prime example of how state policy,interacting with compelling social need, created dramaticpopulation inflows to a forest region (Hecht 1985).

This article takes as given that aggregate social forcesbring people and capital to unsettled regions. As such, itdoes not address the true complexity of land-cover changeprocesses. Nor does it seek to present a broad-scale,statistical account of the deforestation processes presentlyat work in the Amazon basin (e.g., Pfaff 1999). Rather, itseeks to comprehend land use and land cover at the levelof an individual agent—namely, the colonist household.Thus, the forest-dynamics model presented does notprovide a global picture of the forces affecting the tropicalforest biome. Instead, it focuses on the development ofindividual properties of smallholders in theAmazon basin,an endogenous process of change that takes place in thewake of the broad-scale factors affecting migration andcapital flows to the region. Reviews of the global contextand general theories of deforestation can be found inKaimowitz and Angelsen (1998), Geist and Lambin(2001), and Lambin et al. (2001).

The article is organized into two basic parts. It begins byconsidering whatmay be referred to as the classical versionof shifting cultivation, the system of some indigenouspeoples in which rotations of secondary vegetation andfarm plots take place over an infinite time horizon. Thismotivates the colonist model that follows and illustrateshow forest dynamics—including both deforestation andregrowth—can be introduced into a behavioral frame-work. To accomplish this, it is necessary to conceptualizeboth an initial process of land clearance and the sub-sequent implementationof a rotational systemas functionsof natural-resource productivity and labor costs.

After providing a model for shifting cultivation, thearticle presents a model extension for colonist farmers.

This differs from the shifting-cultivation formulation intwo fundamental ways. First, an institutional dynamic isassumed, with market opportunities initially absent andthen present in the evolution of a farming system. Second,the period during which rotation occurs is of finiteduration, in which case, secondary vegetation occurs onlyas a transient feature of a property’s land cover. The crea-tion of a farm is thereby resolvable into two phases: theinitial rotational phase; and the second, which is market-oriented, with permanent agriculture, ranching, or somemix of the two. Such a framework provides an extensionof earlier agent-based efforts that have largely focusedon shifting cultivation, and is able to characterize theexpansion of commercial agriculture into forested areas,much more important as a deforestation driver thanshifting cultivation (Geist and Lambin 2001). The modelis stated in such a way that empirical land-cover changeprocesses, such as the sequence of deforestation actionstaken in creating farmland, are identifiable in the theoreticalstructure.

The secondpart of the article presents a framework thatintegrates twomodel applications. The first is a farm-levelapplication implementing the agent-basedmodels to showhow variables such as size of family workforce, discountrates, and prices affect the deforestation associated withindividual households. Here, numerical methods are usedwith field data to produce land-cover-change magnitudesconsistentwith the empirical setting, an approach that hasbeen previously deployed for similar purposes (Dale et al.1994; Beaumont and Walker 1996; Angelsen 1999;Albers and Goldbach 2000; Evans et al. 2001).1 Thesecond application takes theoretical results from theagent-based framework to represent the process of land-scape evolution. Geographic information system softwareis used to translate individual colonist dynamics into aregional-scale visualization of the ‘‘fishbone’’ pattern ofdeforestation observed in the colonization frontiers ofthe Brazilian Amazon. Thus, in a two-step sequence, thearticle scales up from the decentralized processes ofindividual households to the pattern they make in theaggregate landscape (Brondizio et al. 2002).

Other agent-based models have been advanced todepict tropical deforestation in the Amazon basin (e.g.,Dale et al. 1994; Walker et al. 1997). The present modelseeks to extend earlier formulations by (1) providing abehavioral foundation for the process of landscapechange, (2) linking this behavior to household attributesin the presentation of a spatial model, and (3) reconcilingthe existence of extensive secondary vegetation in theregion with the presence of large populations of low-income farmers. This is accomplished by extending thehousehold-economy model first elaborated by Chayanov

Landscape Change of the Amazonian Frontier 377

(Singh, Squire, and Strauss 1986) to include shifts in ahousehold’s production base. Turner andBrush (1987, 33)have suggested that individual farm households can showhybrid behavior, with subsistence (cf. consumption) andmarket-oriented production (cf. commodity) occurring asopportunities arise (see also Turner and Ali 1996). Thearticle adapts this fundamental insight into a modelconcept that can be deployed to describe land-cover-change processes.

Colonists and Shifting Cultivation

As has been suggested, colonist farming at the house-hold level undergoes an evolution from subsistence-oriented agriculture, in which much of the farm output isconsumed, to commercial activities generating producefor both home consumption and market. For the Ama-zonia case, this typically involves a parallel evolution inthe household’s production base, from shifting cultivationto a ranch, a perennials plantation, or some combination.It is farm production, of course, that provides the mainproximate cause of land-cover change in rural areas,serving as the nexus between objectives of the householdand the use of land (Turner, Meyer, and Skole 1994;Walker and Homma 1996). Consequently, the key tounderstanding land-cover change at the farm level lies inunderstanding how land is utilized for production.

It is important to note at the outset that shiftingcultivators can and do produce for the market. Never-theless, to facilitate the exposition, it shall be assumedherethat they are subsistence farmers producing mainly forhomeconsumption.Colonists begin as shifting cultivators,but ultimately switch both their production system andtheir decision-making calculus to include commercialproduction based on profit maximization. Thus, colonistsare assumed to possess a temporal duality (Turner andBrush 1987) in their relationship to the market, whichruns parallel to their changing production mode.

The exposition of the colonist model focuses primarilyon production, and in particular on the transition fromshifting cultivation to permanent agriculture. Conse-quently, the motivation for the model begins with a briefaccount of shifting cultivation, which, it is argued, haslittle bearing on the colonist systemobserved inAmazoniatoday, despite some claims to the contrary. This leads to anaccount of recent formal modeling efforts directed atshifting cultivation, and a demonstration of their inabilityto describe colonist agriculture.As shall become apparent,consideration of the consumption side of the householdeconomy—in particular, the household’s desire to achieveutility ‘‘maximization’’—provides an important key to

reformulating a statement that reflects the empiricalsetting.

Research on shifting cultivation has beenmotivated, inpart, by concerns about its efficiency (Spencer 1966;Ruthenberg 1971; Watters 1971; Peters and Neuensch-wander 1988)2 and by interest in the role it plays inindigenous agriculture (Posey 1984; Denevan and Padoch1987; Padoch 1987; Unruh 1988, 1990; Irvine 1989).3

Researchers have considered links between the use offallow and fallow ages (Denevan and Treacy 1987; UnruhandAlcorn 1987; Unruh and Paitan 1987; Balee andGely1989), andhave attempted to explain attributes of shiftingcultivation systems (e.g., numbers of fields, fallow lengths)in light of household structure (Salick 1989; Salick andLundberg 1990; Scatena et al. 1996; Coomes, Grimard,and Burt 2000). The use of shifting cultivation, commonin the historic record, remains globally distributed, despitethe advent ofmodern technologies.Although the rotationof land parcels is locally dynamic, the system itself presentsa static, aggregate landscape, resolvable into age cohorts ofsecondary vegetation and cleared lands of active agricul-ture (Walker 1999).

Agronomists, anthropologists, and agricultural econo-mists have suggested that smallholders in forest frontiersin the Amazon basin engage in shifting cultivation, to bedistinguished from ranching as a distinct system, practicedby different agents altogether (Nair 1987, 1991; Serraoand Homma 1993; Faminow 1998). Serrao and Homma(1993) claim that 400,000 shifting cultivators are active intheBrazilianAmazon, and their farm-systemclassification(Serrao and Homma 1993, 298–99) suggests that thesecultivators are subsistence-oriented households growingfood crops, with no involvement in perennial plantationsor ranching. Beckerman (1987) does note the recentappearance of market-oriented bush fallow, but focusesdescriptive attention on the swidden systems of indige-nous peoples.

Describing the Colonist System

Any suggestion that most small-scale farmers in theBrazilian Amazon are subsistence-oriented shifting culti-vators is inconsistent with a great deal of empiricalobservation. Indeed, a narrative appears to be emergingin this regard, encompassing both the ‘‘peasant pioneercycle’’ (Pichon 1997) and processes of farm evolutionlinked to domestic cycles (CAT 1992; DeShazo andDeShazo 1995; Walker and Homma 1996; McCrackenet al. 1999; Perz 2001). The narrative account begins witha newly formed family’s arrival on a piece of forested landbeyond the extensive margin of agriculture. Given littleinitial capital and low levels of experience, the family first

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implements a system of shifting cultivation, producingannual crops such as rice, corn, and beans. This continuesas long as the children are too young to work and theinternal dependency of the household remains high.

With time, however, the family changes. Childrenenter the household workforce, and the parents acquireexperience, gaining confidence in their ability to farm.Such a demographic setting is ripe for investment and theadoption of a system with greater economic potential,such as a ranch, a perennials plantation, or some com-bination of the two. Should the institutional environmentalso change with the arrival of a transportation system andmarkets—the expectation that provides the impetus tocolonization, given the value it brings to land—theagronomic system evolves in parallel with the develop-ment of the household, moving toward an emphasis oncommercial crops. If the children move off-farm as theyform their own families, the process may reverse itself inthe short run, as fields are abandoned with the loss ofhousehold labor.

Anecdotal evidence is accumulating on the land-coverchanges associated with the farm formation process justdescribed. Typically, they involve an initial phase of forestclearing, with deforestation taking place on an annualor semiannual basis in a series of discrete episodes, ordeforestation events. Depending on the household’s finan-cial circumstances, perennials (cocoa, pepper, coffee) andpasture grasses are planted sooner or later, while annualsproduction is mainly continuous throughout. After aboutfive to ten years, deforestation has run its course—at leastfor the first occupants of a property—and the house-hold can focus on agricultural activity (Dale et al. 1993;Browder 1996; Walker and Homma 1996; Walker et al.1997; Vosti, Witcover, and Carpentier 1998a, 1998b;Brondizio et al. 2002).

Figure 1 shows the spatial aspect of farm-propertyevolution for a site in the Brazilian frontier. Moving fromleft to right, from bottom to top, and from upper to lowertier, each panel represents a different time point for the lot.The particular process indicated covers a twelve-yearperiod that is still unfolding. The lot in question wasdelimited by the Brazilian colonization institute, INCRA,which, in the 1970s, provided 100-hectare parcels formigrant families (400m� 2,500m). A salient featureof the land-cover evolution is the continuous advance ofannuals production, from the front of the lot (bottomof each panel) towards the back. This occurs with theclearing of primary forest. Annuals production gives way,in turn, to both pasture and perennial plantings, whichfollow in the wake of deforestation.4

The distribution of farms subject to such evolutionaryprocesses is an empirical question, although survey work

suggests they are widespread (Dale et al. 1993; Browder1994, 1996; Whitcover and Vosti 1996; Pichon 1997;Walker et al. 1997; Fujisaka andWhite 1998;McCrackenet al. 1999). Tables 1 and 2 present data on land cover andhousehold characteristics for surveys of colonist farmers inthe Amazon basin, including Brazil and Ecuador. Theland-cover data (Table 1) establish that, in cross-section,colonist households have diversified farming systemsdominated by pasture. Most of the sites show a similarpattern, with land allocated evenly between perennialsand annuals but with the majority of it in pasture.The magnitudes deforested are also comparable, rangingbetween 18 and 55 hectares.

These data are striking given thewide geographic rangethey represent, and the considerable variations in rainfall,soils, and institutional environment. Actual productioncomponents show strong similarities for three sites in theBrazilian portion of the basin (Table 2A). Area of land inannuals (rice, corn, and beans) is mostly in rice (exceptingthe Acre site), with a sharp drop-off to comparableamounts in the semiannual, cassava. Herd sizes are alsovery close.5 The similarities in the production systemscarry over to attributes of the households (Table 2B).Family size, age of household head, and degree ofdependency (number of children and elderly individualsdivided by total family size) are similar. Only the averageperiod of residence varies, reflecting the older colonizationprocess in the state of Para in the eastern Amazon.

The significance of these agents in the demographyof the region presents a second empirical question.The number provided by Serrao and Homma (1993)—400,000—does not distinguish between identifiable

Figure 1. Property dynamic.

Landscape Change of the Amazonian Frontier 379

groups thatwouldmeet their farming-systemdefinition, inparticular, colonist farmers, indigenous peoples, and long-standing residents of the region who often engage in formsof extraction and fishing. In fact, the 1991 Braziliancensus (S. G. Perz, personal conversation, 2000) indicatesthat the overall population of individuals practicing low-intensity agriculture in the region may be higher thanSerrao andHomma (1993) suggest. In particular, there are1.5 million households deriving income from small-scaleagricultural activity. Assuming, conservatively, an averageof 2.5 individuals per household, this yields a total

population of 3.75 million, which may be compared tothe region’s 150,000 indigenous residents and the ap-proximately 600,000 forest ‘‘extractors’’ and fisherman. Ifthe nonindigenous, small-scale farmers are mostly colo-nists, as seems likely, then such households clearlydominate the rural population of the Brazilian Amazon.Some have argued that such households account for alarge share of deforestation in the region (Walker, Moran,and Anselin 2000), and recent work on the TransamazonHighway (Walker, Wood, et al. 2002) in Para indicatesthat colonist households deforest, on average, at least onehectare annually (between 1986 and 1997). With 1.5million active households, the associated deforestationwould be 15,000 km2 per year, assuming a one-hectareclearing per year per family. This appears to represent avery large proportion of the total amount occurringthroughout the basin.

In sum, research addressing the classical system ofshifting cultivation does not appear to be relevant to theAmazonian colonist. In the colonist system, the shifting offields and the generation of secondary regrowth representa transitory phase in the creation of a farm or ranch.Describing the land-cover processes associated with thissystem is critical to understanding an important compo-nent of Amazonian forest dynamics.

The Shifting Cultivation Problem

Models of Shifting Cultivation

Boserup’s (1965) account of shifting cultivationcontains an implicit concept linking human drivers to

Table 1. Percentage of Deforested Land in Farming System Components

1 2 3 4 5 6 7 8 9 10

Perennials 17 9 30 15 33 10 28 14 3 13Annuals 9 9 22 15 11 10 9 14 16 14Pasture 74 81 48 70 56 80 62 71 81 74Hectares deforested 47 54 23 33 18 55 23 33 29 40

1 Brazil, Rondonia, Ouro Preto; 1991, n5 91; Good soils.a

2 Brazil, Rondonia, Ouro Preto; 1991, n5 91; Moderate soils.a

3 Brazil, Rondonia, Ouro Preto; 1991, n5 91; Restricted soils.a

4 Brazil, Rondonia, Ouro Preto; 1991, n5 91; Unsuitable soils.a

5 Ecuador, Napo and Sucumbios; 1990, n5 412; Full sample.b

6 Ecuador, Napo and Sucumbios; 1990;4100-ha properties.b

7 Ecuador, Napo and Sucumbios; 1990; Accessible properties.b

8 Brazil, Rondonia, Theobroma; 1994, n5 155 (with Acre); Full sample.c

9 Brazil, Acre site; 1994, n5 155 (with Rondonia); Full sample.c

10 Brazil, Para, Uruara; 1996, n5 261; Full sample.d

a Interpreting Figure 6 in Dale et al. (1993).b From Pichon (1997).c FromWhitcover and Vosti (1996), counting ‘‘fallow’’ as pasture.d Unpublished data from National Science Foundation project ‘‘Tenure Security and Resource Use in the Amazon,’’ Chuck Wood, principal investigator.

Table 2A. Farm System Characteristics

Rondonia Acre Para

Rice (ha) 3.36 2.18 3.06Corn (ha) 2.24 2.24 2.40Beans (ha) 1.67 1.67 0.68Cassava (ha) 0.52 0.52 0.62Cattle 22 22 24Property value (Reais/ha) 579 272 276

Table 2B. Household Characteristics

Rondonia Acre Para

Household size 5.38 5.38 5.5Age, household head 47.57 47.57 47.18Adult males 1.95 1.60 1.91Degree of dependency 0.30 0.39 0.37Average period of residence 6.30 7.86 11.46

Note: The Rondonia and Acre data are from Whitcover and Vosti (1996).

The Para data are derived fromNational Science Foundation project ‘‘Tenure

Security and Resource Use in the Amazon,’’ Chuck Wood, principal

investigator.

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land-cover change. In particular, population growthand theneed for increased food production cause a shift in landcover at the regional scale, from old-growth forest underlong rotations toyounger formsof secondary forest (Boserup1965). More recently, geographers and economists havedeveloped formal models of shifting cultivation that gobeyond this demographic explanation. As a rule, thesemodels are either aggregate in nature and consider regionallandscapes emanating from a market center (e.g., Lopezand Niklitschek 1991; Jones and O’Neill 1993a, 1993b;Angelsen 1994), or focus on individual land-managingagents (Barrett 1991; Dvorak 1992; Krautkraemer 1994;Walker 1999; Albers and Goldbach 2000). The modelingframework advanced in this article focuses on individualagents and therefore falls within this latter category.

Most shifting-cultivation models have been statedindependently of the theory of household production,with its emphasis on consumption, production (by familyworkers), and the so-called labor-leisure trade-off (Chaya-nov [1925] 1966; Ellis 1993). Household productiontheory and the concept of the peasant economy considerthe manner in which families produce and consume inorder toachieve themostutility they can (Turner andBrush1987; Ellis 1993). Formulations range from the seminalwork by Chayanov (Thorner, Kerblay, and Smith 1966), inwhich labor markets are absent and the consumption andproduction decisions of households are nonseparable(Mellor 1963; Nakajima 1969), to more modern accountsaddressing smallholder behavior in the presence of well-developed markets for labor, capital, and consumptiongoods (Barnum and Squire 1979; Singh, Squire, andStrauss 1986).

Shifting-cultivation models generally restrict theirfocus to the production side of the household economy,presumably given the task at hand, which is to describe acomplex decision-making process involving optimal crop-ping cycles and periods of fallow.Although the importanceof household welfare (and consumption) is generally

recognized (e.g., Dvorak 1992; Krautkraemer 1994), theframeworks implemented illuminate only technical as-pects of the shifting cultivation system, such as the besttime at which to abandon a farm plot to processes offertility restoration. The models mainly assume profitmaximization and do not provide explicit solutions for thehousehold’s demand for food and land.

One consequence of these various assumptions andrestrictions is that the modeled world of the familypracticing shifting cultivation is often at variance with itsempirical reality (Table 3). Amajor problem in this regardis the existence of production intervals, during whichfarming ceases as the soil regains fertility. An assumptionmade to ensure family survival involves consumption‘‘smoothing’’ during fallow periods, when families tap theirbank deposits to purchase food and other items (AlbersandGoldbach 2000). So long as they can save their profitsduring the cropping cycle, they need not starve. Unfortu-nately, a very large number of shifting cultivators occupyland beyond the extensive margin of agriculture, withlittle or no access to financial institutions or to marketsfor capital, labor, and farm produce (Walker 1999). Thefocus on production technology alsomeans that consump-tion demands—clearly linked to household size andstructure—do not get translated into magnitudes of landuse, critical in any attempt to represent land-cover change.

One solution to the problemof yearly consumption is toconsider a multiplot system in the context of actualhousehold demands and land productivity, an approachtaken byWalker (1999). This leads to the statement of anoptimization problem that unites the household economymodel with shifting cultivation technology. That is, thehousehold is taken to maximize:

X1

t¼0

btUðl; SÞ ð1Þ

where U is a household utility function, b is a discountfactor, l is leisure, and S is subsistence. The model in

Table 3. Shifting Cultivation Models

Behavior Objective FunctionProductionIntervals

LandCreation

SystemSwitching

Dvorak (1992, 811) Utilitya maximization Labor cost minimization No No NoBarrett (1991, 178) Profit maximization Present value of net profit Yes Yesc NoKrautkraemer (1994, 410) Net benefitb maximization Present value of net benefit Yes No NoWalker (1999, 399) Utility maximization Discounted utility No Yes NoAlbers and Goldbach (2000, 271) Profit maximization Net present value of production Yes Yesc NoaDvorak (1992, 811) states that theobjective canbe to‘‘minimize clearing andweeding labor subject to productionof subsistence, or tomaximizeutility of leisure and

consumption.’’ Nevertheless, the actual objective function stated is the minimization of labor costs, subject to a fixed level of subsistence.bKrautkraemer (1994, 410) calls attention to the labor-leisure tradeoff in nonmarket settings and the maximization of utility. However, his solution focuses on net

benefits, and labor poses no explicit constraint on the solution.c Land is created in the sense that shifting cultivation can be initialized from primary forest. Actual land magnitudes are not determined.

Landscape Change of the Amazonian Frontier 381

Walker (1999) provides an explicit solution, giving plotsizes and age of secondary vegetation utilized for amulticrop system as a function of household attributes.Unfortunately, the shifting-cultivation component of themodel suffers from two shortcomings that compromise itsdirect application to thepresent situation. First, it doesnotallow for land creation whereby primary forest is takendown tomakeway for agricultural activity. Second, systemswitches do not occur, particularly the change fromsubsistence production to commercial ventures observedin colonization frontiers (CAT 1992; DeShazo andDeShazo 1995; Walker and Homma 1996; McCrackenet al. 1999; Perz 2001).

The approach this article advances represents anattempt to overcome the shortcomings of previous effortsthat limit their utility in representing land-cover-changeprocesses in settings such as the Amazon frontier. To thisend, the article develops two models representing differ-ent specifications of the general problem represented byfunction (1). The first is for the traditional shiftingcultivation system with long-run cycling between swid-dens and fallow, and the second is for the colonist farmerwho starts out as a shifting cultivator but ultimatelyimplements a commercial system. Both are useful inproviding a full description of the empirical reality.

As has been argued, the colonist farmer is one of thedominant agents of land-cover change in agriculturalfrontiers, especially the Amazon basin. The colonistsystem represents a hybrid of subsistence-oriented activ-ities and production for the market (Turner and Brush1987; Turner and Ali 1996). The particular form ofhybridization to be considered is that occurring throughtime, in parallel to the life-cycle of the household andthe frontier’s advance, which carries with it marketsand new possibilities for economic behavior (Binswangerand McIntire 1987). The modeling challenge istwofold—namely, (1) to demonstrate how a shiftingcultivation ‘‘technology’’ can be incorporated into thehousehold-economy framework, and (2) to reflect farmevolution from nonmarket production to commercialactivity. The operational goal is to develop a formulationthat yields variables describing the process as it is known tooccur in the real world in a series of discrete deforestationevents.

The Constraints of Nature

Before elaborating the model, it is important toconsider the nature of the ‘‘technology’’ of productionunder shifting cultivation. The primary factors definingthis technology are the productivity of the natural-resource base and the labor required to use it. Depending

on intrinsic soil properties and the biomass of standingvegetation, a single slash-and-burn operationmay providerepeatedly high yields. Alternatively, production can beshort-lived due to rapid fertility decline, requiring rapidshifts to new areas. In addition to the soil’s nutrient loadand water-retention capability, the strength of succes-sional response from surrounding areas is important.Weedinvasions can overcome a family’s ability to cultivate foodcrops,making it necessary to quickly abandon plots for thenext production period. Thus, the household’s objectivefunction represented by summation (1) is constrained byenvironmental factors affecting resource productivity andlabor requirements.

In formally modeling the impact of the ‘‘constraints ofnature’’ on household behavior, Barrett (1991), Dvorak(1992), Angelsen (1994), and Walker (1999) elaboratethe concept of a natural production function relating cropoutput to the age of vegetative regrowth slashed andburned in preparing a field. Presumably, output increaseswith age, given that nutrient uptake is continuous in time.Thus, use of older secondary vegetation (and primaryforest) yieldsmore output than shrubby regrowth, becausea greater stock of nutrients is released to the soil withthe slash-and-burn operation. Mathematical functionsof the relationship between soil fertility and age ofregrowth remain theoretical constructs, although ecolo-gical research links biomass accumulation to successionalstage, particularly in the new-world tropics. Table 4 givesdata on forest biomass as a function of age for both second-ary regrowth and mature forest (Uhl 1987; Saldarriagaet al. 1988; Brown and Lugo 1990; Lucas et al. 1993;Salomao, Nepstad, and Vieira 1996; Vieira et al. 1996).These data show that biomass accumulation is increasingin time, and that a substantial period must elapse beforemature forest levels are reached. If nutrient stock is a fixedcoefficient of biomass, then nutrient regeneration and, byimplication, potential crop output are functions of biomassage, not unlike the renewable-resource functions used tomodel forests and fisheries (Clark 1976, 1985).

A second environmental factor affecting householddecisionmaking involves the labor costs required for use ofsecondary vegetation. In particular, the unit area laborcosts associatedwith land clearance andweeding also varyas a function of age of regrowth (Dvorak 1992; Angelsen1994). The stylized fact is thatmature forest requiresmorelabor to prepare than secondary regrowth, which, in turn,requires more maintenance weeding (Pingali andBinswanger 1988). Table 5 presents data for sites inthe Amazon basin, although with little age variation forthe vegetation categories. They show consistently highlabor costs for the preparation of fields frommature forest,and costly weeding operations in the secondary regrowth.

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Mature forest may be more costly overall when the land-clearance requirements measured by the Brazilian Agri-cultural Research Center (EMBRAPA) are assumed(column 8), together with the estimated labor neededfor weeding at specific sites.

Modeling Forest Clearance and Land Creation

To model the empirical circumstances of the colonistfarmer, function (1) must be optimized subject to theproductivity of the resource base, the labor costs of usingit, and the availability of production factors, particularlyfamily labor.6 The case of shifting cultivation is consideredfirst in order to introduce the process of land creation andto formulate the problem in terms of household structure(utility, labor supply). The framework is then modified byadding a switch to commercial activities, yielding the so-

called colonistmodel.Criticisms have beenmade of the useof optimization frameworks to explain behavior in high-risk environments, such as might be encountered in atropical frontier (Lipton 1968). Walker, Perz, Caldas, andTeixeira Silva (2002) implement a risk-minimizationmodel in analyzing farm-system choices of colonists inthe BrazilianAmazon, andHomma and colleagues (1996)have considered the role of risk in explaining the shift fromextractive rainforest activities (Brazil-nut extraction) topasture-based systems. Nevertheless, the frameworkadopted in the article assumes optimization behavior inorder to maintain a focus on land-use dynamics. Manyshifting-cultivation models have been stated in a similarfashion (e.g., Barrett 1991; Dvorak 1992; Krautkraemer1994; Walker 1999; Albers and Goldbach 2000).

In order to facilitate model development, it is assumedthat plots are used for only one year. This assumption

Table 4. Above-Ground Forest Biomass in Select Sites, Tons per Hectare

Location 5 years 10 years 20 years 40 years 70 yearsa Mature

Regional estimatesb 50 95 150 160 165Venezuelac 33.85Venezuelad 58 110 150 255Para, Brazile 13.1 43.9 80.5 105 265.7aAn average age over plots between 60 and 80 years old.bModified from Brown and Lugo (1990) and Lucas et al. (1993).cUhl (1987).dSaldarriaga et al. (1988).eSalomao, Nepstad, and Vieira (1996); Vieira et al. (1996).

Table 5. Labor Costs by Operation, Person-Days per Hectare, Land Preparation

Mature Forest Secondary Regrowth

1 2 3 4 5 6 7 8 1 2 3 4a 4a

Undercutting (Broca) 6.7 * * 1–17b 15 * 12.0 * 12.0 5.5Tree-felling (Derrubada) 11.3 * * 13.0 * 0.0 * 11.5Debris-stacking (Coivara) 6.7 * * * 0.0 *

Burning (Queimada) 1.0 * * * 1.0 *P25.7 25.7 33.1 25 41–42 13.0 19.4

Weeding (Capinas) 12 18.2 24 31.6

Total 37.7 45.9 433 425 441–42 37 51

Notes: These values are for low-intensity, manual practices.

1 State of Para, Brazilian Amazon (CAT 1992).

2 Bolivia (CAT 1992).

3 State of Para, Brazilian Amazon (A. Toniolo and R. Costa personal conversation, 1999).

4 State of Para, Brazilian Amazon (Scatena et al. 1996).

5 State of Para, Brazilian Amazon (Moran 1981).

6 State of Para, Brazilian Amazon (Wagley 1953).

7 State of Para, Brazilian Amazon (Texeira Mendes 1997).

8 Brazil (EMBRAPA 1995; EMBRAPA/CPATU 2000). Values given only for deforestation and burning actions. No weeding.aThe undercutting value is for young secondary growth (2–4 years). The tree-felling value is for secondary growth in excess of 10 years.bValues vary as a function of available equipment. A chainsaw reduces the one-hectare undercutting operation to one day.

Landscape Change of the Amazonian Frontier 383

has the analytically useful effect of equating the age atwhich secondary vegetation is slashed to the number ofdeforestation actions undertaken by the farmer—the so-called deforestation events. Very short cycles—one or twoyears in length—are typical of the region (Walker et al.1997), with its poor soils and humid climate, conditionsthat promote rapid declines in soil fertility and short-livedcropping periods (Barrett 1991).

In this article, deforestation is the clearing of matureforest and not of secondary regrowth. With a single-yearcrop cycle, deforestation on some plot of land starts at timeperiod 0 and continues in successive years until the initialplot, having been abandoned to regrowth in year 1, iscleared, farmed, andabandonedagain.Thus, deforestationtakes place at the same time as regeneration during theinitial phase of land creation, which consists of a sequenceof deforestation events. Beyond this point, deforestationceases and farming occurs on the basis of rotation over thefields created by the deforestation events.

The Shifting-Cultivation Model

The statement of the shifting-cultivation model beginswith the specification of household productive capacityand the labor requirements of the farm activities. Thisspecification, in turn, is germane to the colonist house-hold, given the importance of family labor and the lowlevels of technology utilized by both shifting cultivatorsand colonists. Thus, let household labor endowment be L,and consider two sets of choices, indexed l and r, reflectingthe land-clearance and rotational phases of agriculture,respectively. Let li and wi be the leisure and workassociated with phase i, iA(1,r), where l represents theland creation phase and r the rotational phase. Then, theconstraints imposed by the labor endowment (L) availableto the household are

w1 þ l1 ¼ L and wr þ lr ¼ L

Note that the endowment of household labor remainsconstant (Holden, Pagiola, and Angelsen 1998), anassumption that may be relaxed. With a household-welfare function, U, defined on leisure and food, s, as inWalker (1999), the forest-dynamics problem may now bestated as maximizing, subject to the constraint on labor:

Ws ¼ maxl1;lr;s1;sr

Xa�1

t¼0

btUðl1; s1Þ þX1

t¼a

btUðlr; srÞ ð2Þ

This problem can be solved by choosing the values ofleisure and subsistence that optimize welfare. However, itboth simplifies and facilitates the representation of land-cover change to manipulate the problem to producecontrol variables directly reflective of the change process

itself, which consists of a number of forest-clearanceepisodes of some magnitude and subsequent rotationsthrough the plots so created. In particular, work require-ments are governed by age of the secondary vegetationutilized, a, and the amount of land cleared for each plot, r,or the deforestation eventmagnitude (Dvorak 1992, 810).Hence, let w15 rg(N) andwr5 rg(a), where the unit areacost function, g, is an increasing function in a and N is‘‘age’’ of mature forest (N4a). In addition, total, single-period food production, s, is given as s15 rf(N) for matureforest and sr5 rf(a) for secondary forest, where f is anatural production function, also increasing in a. Substi-tution of these functions into (2) yields a reformulatedproblem solvable in two variables that describe the land-cover change process, namely a and r.7

maxa;r

ð1� baÞU ðL;N;rÞ þ baU ðL;a;rÞ ð3Þ

Recall that under the assumption that plots are used foronly one year, the age at which secondary vegetation isslashed, a, is equivalent to the number of deforestationevents observed on the property. Hence, such a reformu-lation has the effect of transforming both the number andsize of deforestation events into endogenous variables, thevalues of which are functions of household attributes andsite conditions such as size of family workforce, discountrates, and productivity of the resource base.

The solution for a, interpreted as a rotation time,mimics the classical forestry problem, stated by Faust-mann, that identifies an optimal rotation period for cyclesof tree-planting andharvesting (Hirshleifer 1970). Barrett(1991) has pointed out that the shifting cultivator’sproblem is similar to the forester’s, andWalker (1999) hasdirectly applied the principles of Faustmann—as devel-oped by Mitra and Wan (1985)—to shifting cultivation.The cycling of secondary vegetation, with its potential forfood production, is analogous in behavioral terms to thecycling of trees with their market potential.8 The presentformulation differs by allowing for an initial period of landclearance and by introducing the effects of family and sitecharacteristics on the optimal rotation period.

Dvorak (1992, 812) points out that shifting cultivationcan involve the simultaneous management of plots atvarious states of a cropping cycle (as opposed to the useand abandonment of a single plot every year). In sucha situation, the relative labor requirements for weedingand clearing become important. The allocation of timebetween such activities can be incorporated into thepresent framework by allowing for a cropping cycle thatinvolves a choice variable for the number of consecutiveyears that a plot is used before abandonment (seeAppendix A).

Walker384

The Colonist Farmer

The formulation presentedup to this point reflects long-term shifting cultivation in which plots are rotated over aninfinite time horizon with no change in crop types orresponses to altered economic conditions. As has beenargued, shifting cultivation often represents a premarketphase in the evolution of a family’s farming system towardcash-oriented agriculture, as with the development ofcattle-ranching by small producers in Amazonia. In such asituation, initial deforestation occurs as in the precedingcase, followedby rotations through thedeforested land anduse of regrowth in a final pass to create the pastures, whichare then stocked.A similar dynamicunfolds in the creationof perennials plantations, or diversified systems withmixesof pasture and perennials. The objective of the family maythus be represented by a model with periods of deforesta-tion, shifting cultivation, and market-oriented activitybased on a ranch or perennials plantation.

Market involvement can be regarded as a choice forwhich the contemporaneous alternative is subsistenceproduction (Vance and Geoghegan forthcoming). Here,however, entry into the market is taken to be a stage inthe development of a farming system, as portrayed by thenarrative of the colonist farmer. Of course, if the familyexpects to someday engage in market activity, even as itstarts out with a subsistence-oriented system, the shift tomarket activity is not unexpected, but planned. Thus, thecolonist of the model statement is taken to be anindividual with strong expectations and foresight, notsomeone who reacts myopically to unexpected changes incircumstance.

As before, the goal is to optimize welfare. Now,however, a change in household structure is introduced.Let this occur at t5 y, involving both additional house-hold labor and new preferences as represented by theutility function. As children enter the household work-force, the early labor endowment, Le, changes to themature endowment, Lm. Evolving preferences, in turn, arecaptured by the new household utility function, Uy. Ageneral form of the maximization problem as in equation(2) cannot be explicitly stated, since the relative magni-tude of a and y are not known a priori. Thus, consider thecase in which the optimal value of a is known and is lessthan y (e.g., a5 3 and y5 10). The colonist household’soverall utility may then be written as

Wc ¼Xa�1

t¼0

btUðl1; s1Þ þX2a�1

t¼a

btUðlr; srÞ þXy�1

t¼2a

btUðlr; srÞ

þX1

t¼y

btUyðlm; qmÞ ð4Þ

Two points can be made about this summation thatdistinguish it from the shifting cultivator’s problem inequation (2). First, the production of subsistence goods,with labor endowment Le, is as before in the land creationand rotational problem, but at t5 y the productive basis ofthe household economy shifts as labor supply changes toLm. The household moves from a technology dependenton ‘‘nature’’ to a neoclassical apparatus, with a productionfunction and markets for labor and an output, q (Barnumand Squire 1979; Singh, Squire, and Strauss 1986; Ellis1993). Hence, the shifting cultivation phase is transient.This is observed in the fourth term of the summation,where the marketable good, also consumed, is apparent inthe utility function. The second distinction resides in thenew utility function, Uy, which represents new prefer-ences associated with leisure and the marketed good.

Only one complete rotational cycle is given in equation(4), although there is no reason to reject the possibility ofmultiple cycles, which could be explicitly incorporatedinto the model. Nevertheless, multiple cycles wouldprobably extend the transient phase of shifting cultivationbeyond what is actually observed among colonists in theAmazon basin. In addition, direct conversions of tropicalforest to perennials and pasture occur among smallproducers without an intervening period of shiftingcultivation (e.g., Scatena et al. 1996; Walker et al.1997). The goal, however, is to represent the empiricalcase as described and observed—namely, the transient useof secondary regrowth as a phase of farm development.

It is important to note that the ‘‘arrival’’ of markets forgoods and labor is taken to be exogenous to the model,which only represents the behavior of the farmer basedon his or her beliefs. The farmer may have a preciseexpectation about when development will finally occur, aswith the construction of an all-weather highway (CAT1992). This would tend to fix the value of a. Alternatively,the farmer might only know that some degree of land‘‘creation’’ is necessary prior to commercial activity, givenscale economies, and sets out clearing land hoping that‘‘things work out.’’ In this later case, a remains a choicevariable and the farmer’s expectation is weaker, only thatthe opportunity formarket participationwill be in place bythe time he or she is ready. The institutional environmentmay change more slowly than anticipated, or fail todevelop altogether. What matters for the land-covermodel, however, is the farmer’s expectation andderivativebehavior.

Just as with the pure shifting cultivator, the colonist’sproblem involves the maximization of utility over amultiperiod time horizon, as dictated by function (1). Theprocess of maximization is different, however. In particu-lar, the shifting cultivator’s problem is comparable to

Landscape Change of the Amazonian Frontier 385

that of the subsistence household first considered byChayanov, in which utility maximization and the produc-tion decision are nonseparable due to the absence ofmarkets. The household consumes exactly what itproduces and balances the gain from production againstthe immediate loss of leisure. What guides the decision isthe household’s disposition toward labor and associateddrudgery, largely a function of the relative balance ofconsumers to workers in the family.

The colonist starts out this way, but anticipates that bythe time he or she is ready for exchanges of labor andproducts in the market, institutional change will havetransformed opportunities for economic behavior. Now,the colonist firstmaximizes the value of production.Then,with the income so earned functioning as a ‘‘budget’’constraint, the household consumes according to itspreferences in order to maximize overall well-being, orutility. Technical details of the maximization of thecolonist problem are given in Appendices B and C.

This section has demonstrated how the statement of anoptimization problem, founded on received theories of theagricultural behavior of households, can be transformedinto adescriptionof a land-cover changeprocess—namely,the clearing of forest and the implementation of perma-nent agriculture, via shifting cultivation. The followingsection translates this conceptual apparatus into a notionof forest dynamics, gives output from a simulation usingactual field values, and shows how land-cover-changeprocesses at lot level can be scaled up to the landscape.

Forest Dynamics and Spatial Pattern

Forest Dynamics and Locational Considerations

The forest dynamics of both shifting cultivation andcolonist systems can be depicted relying on assumedsolution values for the key endogenous variables, a and r.Let the number of deforestation events be four and thedeforestation event magnitude be 10 hectares (a5 4;r5 10).Then, land creation occurs in both systems duringthe initial four years, with 10 hectares of deforestation ineach year, stabilizing at 40 hectares cleared at the start ofyear 3. Figure 2 depicts this process, which is identi-cal for both systems. Deforestation processes such asthis hypothetical one have been empirically observed andso depicted (Homma et al. 1993; Vosti, Witcover,and Carpentier 1998a, 1998b).

The secondary regrowth graph is different for the twosystems, however, as shown in the lower panel of Figure 2.In shifting cultivation, secondary regrowth builds to30 hectares and remains so thereafter, given that threegarden areas are continuously in fallow (Walker 1999). For

a farming system with conversion to permanent planta-tions or pasture, secondary regrowth areas expand initially,but they are then converted to agricultural use.

An Approach to Spatial Simulation

Actual values of a and r assumed in the construction ofFigure 2 can be obtained through numerical methods(Powell 1978; SAS 1995; Miller 2000; see Appendix C.)Because they are ‘‘control’’ variables for the maximizationproblem confronting the colonist, they can be calculated,once the problemhas been specified and parameterized, toprovide a description of the forest loss process. After landclearance has run its course, the magnitude of deforesta-tion is the product of the amount of land clearedwith eachdeforestation event (r) and the number of these events

Figure 2. (A) Deforestation process at lot level. (B) Land insecondary regrowth.

Walker386

(a). Long-run deforestation occurs under shifting cultiva-tion in spite of the fact that succession leads to regrowth.For theAmazon basin, such regrowth is typically relativelyyoung and not at all suggestive of primary forest cover.

Figures 3 and 4 give results of numerical applications tothe case of a colonist farmer developing a small ranch,under various assumptions about household structure andeconomic conditions (AppendixC). Figure 3describes thecombined impacts of differing discount rates (representedby the beta values) and magnitudes of household labor. Ingeneral, increases in the household endowment of laborlead to increases in the amount of deforestation, acompletely intuitive result. Regarding the impact ofdiscount rates, high beta values (or low discount rates)tend to increase the amount of deforestation. This resultmay seem counterintuitive at first, given the receivedwisdom thathigh discount rates promote excessive rates ofresource exploitation (Clark 1976). In the present case, itwould appear that reduced discounting accentuates theimportance of future ranching values, with associateddemands for land (cf. Angelsen 1999).

Figure 4 brings out the relationship between wages,commodity prices, and deforestation for varying degrees ofdiscounting. The price situation is represented by therelative price for labor, which is actually the ratio of themonetary wage and beef prices (see Appendix C). Hence,an increase in this relative price can represent anincreasing wage rate or a decreasing price for the soldgood. The data were generated assuming a labor endow-ment to the family of 6,000hours.As can be seen, the levelof deforestation diminishes with increases in the relativeprice of labor, presumably due to the relative increase inthe cost of agricultural production, as well as the appeal ofworking off-farm (Angelsen 1995).

The price values range between 0.15 and 3.0, with 0.3taken as representative of current conditions (AppendixC).

Hence, the order of magnitude difference between 0.3and 3 reflects wage variation between the frontier settingand what might be expected in a so-called developedcountry, assuming the commodity price remains constant.In this regard, the figure shows that increasing labor-market remunerationmay not havemuch of an impact oncolonists with higher discount rates, as the associatedgraph (beta5 0.70) shows little slope over the rangepresented. Higher expected wage rates tend to lowerdeforestation for colonists who place a high value onfuture welfare, probably because labor-market rewards arenot available until later in the farm-creation process. Itseems likely that such individuals would be poorlyrepresented in the population at large. Alternatively, withwages constant, decreasing beef prices (increasing relativewage rate) reduce the amount of deforestation, marginallyfor those with high discount rates, but appreciably as therate goes down.

In general, the magnitudes of deforestation calculatedare consistent with empirical observation. As discussed,colonists along the Transamazon Highway in Para re-ceived 100-hectare properties from the government. Todate, individual clearings on holdings that have not beenconsolidated into large enterprises show continued pres-ence of primary forest, meaning that deforestation has notexceeded 100 hectares since initial colonization, whichbegan in the 1970s. Although Brazilian law has set variouslimits on the permitted amount of deforestation forindividual holdings (e.g., 50 percent), many clear forestwith little concern for legal enforcement, given theremoteness of their locations. Nevertheless, the modelresults suggest that colonist families are not likely to clearmuch land given household constraints, at least in the firstgeneration of land occupiers.

An additional conceptual step enables a translation ofthese deforestation behaviors into spatial outcomes. InFigure 3. Deforestation and family labor.

Figure 4. Deforestation and wage rate.

Landscape Change of the Amazonian Frontier 387

particular, if household characteristics are taken to berandomly distributed in a colonization space defined by adevelopment highway, a set of exogenous settlementroads, and surveyed properties, then a and r are alsorandom variables given they are functions of householdcharacteristics. If the colonization space is placed in GISsoftware, realizations of a and r can be distributedaccordingly by an ARC-INFO aml routine. Real-timedepictionof thedeforestationprocess at regional scalemaythen be visualized, as deforestation events occur onindividual lots, expanding each clearing with a property-specific, deforestation-event magnitude.

Figure 5 presents output from such a cartographicmodel (Tomlin 1990) after ten steps in the simulationprocess. Individual lots are occupied in sequence, reflect-ing a process and a rate of colonization, so deforestationstarts sooner on properties closer to the developmenthighway than those farther away. The gray rectanglesaggregate the deforestation for individual holdings to theleft and right of five settlement roads running north andsouth from a development highway such as the Trans-amazon (BR-230), which runs east and west in thediagram. The template of the figure is in rough agreementwith the settlement geometry laid down by INCRA, withsettlement roads spaced 5 kilometers apart and propertiesof 100 hectares shaped in rectangles of 400m� 2500m.The figure ignores the disposition of lots along the mainaxis of the TransamazonHighway, which lie at right anglesto those on the roads behind them.

Note that the settlement roads and developmenthighway constitute a so-called fishbone. Hence, thecartographic model yields a fishbone pattern of deforesta-

tion, which is the spatial aggregate of the many individualclearing activities. Because the as and rs are distributedrandomly, such a simulation cannot be expected toreproduce an accurate account of deforestation at thelot scale. Nevertheless, the overall pattern of fragmenta-tion is consistent with the vaguely pyramidal formobserved along settlement roads in the eastern sector ofthe Amazon basin (Walker, Moran, and Anselin 2000).9

Conclusions

This article has presented amodel that unifies the workof Chayanov with household production theory. In sodoing, it advances a microscale explanation of deforesta-tion that explicitly integrates demographic phenomenaand market conditions. In addition, it provides an empir-ical descriptionof theway that deforestation actually takesplace on smallholder properties in the settlement frontiersof Amazonia. In particular, deforestation is a response tothe farm-creation process, which involves a sequenceof phases moving from shifting cultivation to some formof permanent agriculture. Unlikemost pre-existing effortsat modeling the system of shifting cultivation, criticalvariables in the present formulation describing theland-conversion process are endogenous to householdstructure, and the farming family enjoys direct yearlyconsumption of farm production in the absence offinancial institutions.

The model was solved numerically, yielding values ofdeforestation magnitudes at the lot level, and randomiza-tion of the key land-cover variables (a and r) enabled thetranslation of lot-level processes into an aggregatedpattern of fragmentation, thereby producing the fishbonepattern of deforestation observed widely throughout theAmazon basin. Error assessment was not conducted onthese spatial results (Pontius 2000), given that thecartographic component of the framework was presentedonly in prototype. Nevertheless, refinement of assumedparameter values could lead to a spatial representation ofreal-timedeforestationdynamics in theBrazilianAmazon,at least that part associated with colonization. Such anextension awaits further development.

The deforestation results generated by the variousparameter settings varied between about 20 and 90hectares. This is consistent with magnitudes observed inat least one colonization frontier in the Eastern Amazon.Farming systems of smallholders along the TransamazonHighway in Para show between 29 and 69 hectares ofdeforestation (Walker, Perz, et al. 2002, 194). The loweramount of deforestation is observed among subsistencefarmers (n5 75), while perennial systems with cattle

Figure 5. Stochastic simulation of colonization ‘‘fishbone.’’ Devel-opment highway (e.g., Transamazon) runs east and west; fivesettlement roads run north and south. Individual lots are theelongate rectangles organized in stacks between the settlement roads.Randomized deforestation on individual lots is given as gray areawithin the lots. Results are for a ten-step stochastic simulation usingARC-INFO aml in prototype colonization space.

Walker388

(n5 11) show an average of 69 hectares of landdeforested, despite Brazilian law regarding maximumallowable land clearance. The farmers in the sample alsopossess attributes comparable to the idealized colonist ofthis article. Although the data are cross-sectional and donot capture developmental dynamics, some of the house-holds consume all their production and use only familylabor, while others are profit-oriented, with little relianceon the family workforce or home production for food. Alittle over a third use no off-farm labor and receive littleor no income from off-farm activities (Walker, Perz, et al.2002).

The Transamazon data (Perz andWalker 2002;Walker,Perz, et al. 2002) cannot establish a temporal link betweenthe subsistence and market-oriented systems, but thetheory presented would suggest that the subsistencehouseholds will one day become market-oriented, ahypothesis that could be addressed by future research.Be this as it may, subsistence farmers in the Transamazonsample have been on site for 8.5 years, less than the periodof residence for those associated with the perennials withcattle system,which is 11.3 years on average (Walker, Perz,et al. 2002, 189). On the surface, this would seemconsistent with the theoretical framework, which suggeststhat market involvement follows subsistence-orientedproduction.

One shortcoming of the current formulation is theconceptualization of the lifecycle of the household andfamily labor. In particular, a single generational process isassumed,with a fixed amount of family labor, at least in thesimulations. Nevertheless, it appears that generationaldynamics come into play on some properties, as childrenage and start their own families on the holdings of theirparents (Perz and Walker 2002). Brondizio and colleages(2002) identify pulses of deforestation on individualproperties suggestive of such a phenomena, and Perz andWalker (2002) show that land-cover dynamics of second-ary vegetation are affected by generational extension onindividual holdings. CAT (1992), Walker and Homma(1996), McCracken and colleagues (1999), and Perz(2001) argue that the domestic cycle of the individualcolonist household is critical in explaining switches fromsubsistence farming to perennials and ranching. Never-theless, the time it takes to implement a commerciallyoriented system, as predicted by the model, appears to beless than ten years, which would seem to weaken such‘‘first-generation’’ effects.10 Of probably greater impor-tance to the long-run disposition of a property’s land coveris the generational phenomena as described, for there islittle reason to presume clearing will not start up againif children remain on the farm and begin families oftheir own.

It is important to note that the present model is limitedto a single type of agent—namely, the colonist farmer. Assuch, the picture presented is a partial one at best,especially with respect to the overall mechanism of land-cover change in the Amazon basin, which involves amultiplicity of agents. Besides the colonist, at least twoadditional agents are critical to understandingAmazoniandeforestation: large ranchers, or fazendeiros, and loggers.

Much has been written about the role of large ranchesin Amazonian deforestation, and in particular about howgovernment incentives lured capital to the forest frontiers(e.g., Hecht 1985). There can be little doubt that despitetheir relatively few numbers, given the skewness ofland distribution throughout Brazil (Simmons 2002),large ranches have had a substantial impact on land-cover change at basin scale. The actual amount remainsan empirical question, although it seems clear that therelative importance of small- versus largeholders inthe landscape dynamic at the regional scale is a functionof time and place (Walker et al. 2000). Be that as it may,fully understanding Amazonian deforestation will requirean understanding of fazendeiro behavior and better insightinto process linkage between small- and largeholder landoccupation (Wood 1983; Ozorio de Almeida 1992; Rudel2002).

Another critical linkage to comprehend is that betweenloggers and colonists. While colonists account for‘‘infilling’’ of the landscape with agricultural activityonce theyoccupy their parcels, they themselvesmaynot ex-tend the frontier through road-building activities. Afterthe development highways are in place, the expansion ofthe system is often the outcome of an interaction betweenloggers and farmers, both in Amazonia and elsewhere inthe tropical forest biome (Leslie 1980; Myers 1980;Schmithusen 1980; Office of Technology Assessment1984; Ross 1984; Walker 1987; Repetto and Gillis 1988;Walker and Smith 1993; Kummer and Turner 1994;Brookfield 1995).11

Nevertheless, along development highways such as theTransamazon, roads may be extended after initial colon-ization due to the demand for land arising from latearrivals, and without much logger involvement. In such asituation, the fishbone pictured in Figure 5 (and assumedin the cartographic model) is not entirely exogenous tocolonization, and an appreciable amount of the highlyregular network pattern may be mainly attributable tosmallholder actions. In at least one part of the Transama-zon Highway in the eastern sector of the Amazon basin,the extension of settlement roads beyond the initial spur of6 kilometers (Simmons 2002) occurred when newcomersdemanded that government expand the existing roadnetwork and demarcate new lots for settlement. The

Landscape Change of the Amazonian Frontier 389

government did so, and extended the network to about20 km on both north and south sides of the highway,preserving the geometry of the original plan allowing for100-hectare lots, and a 5-km spacing between the roads.Be this as it may, a general behavioral model of landscapeevolution for forest frontiers awaits empirical descriptionof interactions between colonists and loggers, and anaccount of the spatial decision-making of the road-builders, whoever they are.

Although rural out-migration has been pronouncedin the Brazilian Amazon in recent years (Perz 2000),the colonists who remain in place continue to change thecover of the land they occupy (Walker,Wood, et al. 2002),and there has been little pause in rates of deforestation(INPE 2000). Indeed, colonist-driven landscape changecan be expected to intensify, given plans by the Braziliangovernment to continue with its road-building andinfrastructure improvements (de Cassia 1997; Lauranceand Fearnside 1999;Nepstad et al. 2001). Thus, there willbe a continued need to understand colonist behavior andthe landscapes they shape in their struggle for survival.

Appendices

Appendix A

Let the length of cropping cycle be y, and following(Dvorak 1992, 811),12 consider a rotational system inwhich age of fallow can be freely chosen. This amounts todefining a systembased only on the secondpart of function(2), or:

X1

t¼0

btU ðlr; srÞ ðA1Þ

where the rotation subscripted as r begins in the firstperiod. With a cropping cycle of y years (41), theoptimization problem is to maximize, through yearlychoices of leisure and subsistence, the objective:

maxl0...;ly�1;s0...sy�1

X1

t¼0

Xy�1

i¼0

btyþiU ðli; siÞ ðA2Þ

subject to

wi þ li ¼ L ðA3Þ

w0 ¼ rfg½ða� 1Þy� þ hð0Þg ðA4Þ

wi ¼ rhðiÞ ðA5Þ

si ¼ rf ½ða� 1Þy; i� ðA6Þ

where a weeding function, h, has been added to theconstraint set.13 Note that, as in the previous problem,

leisure and subsistence are ultimately functions of othervariables: namely, fallow length (a), size of plot (r), andcropping cycle (y). Consequently, solutions for thesethree variables yield the optimizing choices of leisureand subsistence. As indicated, the empirical setting issuch that cropping cycles are very short, and taken as 1.Consequently, sufficient conditions for this case (themodel assumption) are established in the following claimby specifying functions for weeding, regrowth, and utility.

Claim:For fixedplot size r, conditions exist underwhichsome single-year cropping cycle is preferred to an arbitrarymultiyear cropping cycle.

Proof: The claim may be established by demonstratingthat, for some multiyear cycle, there exists a single-yearsystem generating greater utility than the multiyearsystem in each year of the cycle. To simplify the exposition,assume h(i)5 0 for i5 0, and let the number of plotsin the single-year cropping cycle be b. Also, let h(i)4g(b)and f(b� 1)4f[(a� 1)y, i] for iZ1. The practicalsignificance of the latter two inequalities is that weedingcosts per unit area always exceed the cost of clearing a newplot under single-year use (with b plots) after the first year,and that fertility decline in the cropping cycle quicklyreduces productivity below the single-year case, also afterthe first year. Assuming these conditions, the single-yearsystemproduces greater utility for some arbitrary plot size rin years beyond the initial year, since there is more leisureand food available.

To establish the claim, it is then only necessary toidentify conditions that yield higher utility for the single-year system in the initial year. Thismay be accomplished byspecification of the utility function. For fixed r, subsistenceproduction may be higher in the multiyear system in the

Figure A1.

Walker390

first year due to older fallow (if (a� 1)y4 b); assume thatthis is so. Because older vegetation is used, however, morework is required, given site preparation costs (slash andburn), allowing for less leisure. Hence, whether the initial-year utility under multicropping is greater than the single-year system depends on the nature of the utility function.Figure A1 demonstrates that a utility function existsyielding higher utility for the single-year system. A similargraph can be depicted for the case in which b4(a� 1)y.

Appendix B

The problem posed differs from previous statements ofthe household economy framework (Thorner, Kerblay,and Smith 1966; Singh, Squire, and Strauss 1986) bycombining nonseparable and separable components. Toshed light on solutions, assume no change in householdlabor endowment, as in the shifting cultivator case. For theseparable part of the problem (e.g., the fourth term ofsummation [4]), single-period optimization has been wellstudied and involves an initial profit maximization,following by utility maximization, or a recursive solution(Singh, Squire, and Strauss 1986, 7). The interest here isto demonstrate that overall optimization of the combinedproblem requires maximization of the separable problemfor all following periods. This result is necessary for thecomputational algorithm implemented in Appendix C. Inaddition, it unifies the Chayanovian and market-basedmodel.

Household Profit Maximization. If labor endowmentremains fixed, and if new preferences are activated as soonas the switch from subsistence to market production takesplace—as in the simulation—the colonist objectivefunction can be stated with two terms, including an initialterm for the utility of land creation and shifting cultivationphases (G(r,a)), and a second term for the utilityassociated with consumption based on commercialproduction (H(r, a, l, q)). Thus, G represents thediscounted welfare stream for both the land-creationand the rotational phases, and H is that for farm or ranchimplementation that comes afterwards. The colonistproblem is to maximize the sum (Wc5G(r, a) 1 H(r, a,l, q)) subject to the constraints on labor and production.Refer to this as the unconditional problem. Alternatively,define a conditional problem as that of maximizingH(r, a, l, q) subject to q5 q(A, w) for fixed values of rand a (A5 ra). Here, q is a production functionfor the agricultural output, dependent on land (A) andlabor (w).14

Given values for r and a, and hence A, the conditionalproblem is that of optimizing the discounted stream of

utilities associated with consumption after commercialactivities begin. Since a unique pair of lm and qm optimizessingle-period utility given appropriate assumptions on theutility function (Singh, Squire, and Strauss 1986), thissame pair optimizes the discounted stream of utilities, inwhich case a solution exists. Given the presence of marketopportunities for labor and output, utilitymaximization—and, hence, the solution to the conditional problem—forsome fixed amount of land A and wage rate v is recursiveand depends on the solution to themaximization of profits(Singh, Squire, and Strauss 1986), or

max q� v w� hðdÞq

subject to q ¼ qðra;wÞðA7Þ

with A fixed (Figure B1) and q as numeraire. Here, h(d)represents the presence of transportation costs modeledwith an ‘‘iceberg technology’’ as a function of distance ofthe production site to somemarket or transshipment point(Samuelson 1952; Nerlove and Sadka 1991; Fujita,Krugman, and Venables 1999).15 Assume v is sufficientlylarge that wrL, the household labor endowment (FigureB1). The profit maximization solution defines a ‘‘budgetconstraint’’ for the optimization of household welfare,some q5 y1 vw5 y1 v(L� l), where y5 q0 � vw0, and(q0,w0) solves the profit maximization problem (see FigureB1).16 Hence, for arbitrary values of r and a (and A), wehave unique utility maximizing values of lm and qm, under

Figure B1. The production function may be interpreted asqt5 q� h(d)q5 [1� h(d)]q. Profit, in turn is qt� vw, where q isnumeraire. Wages may be a function of distance (Angelsen 1994).The income, or ‘‘budget constraint,’’ is q1vl5 y1vL, where (L�w)is leisure.

Landscape Change of the Amazonian Frontier 391

appropriate assumptions on the utility and productionfunctions, and H is maximized conditional on values for rand a. Note that lm and qmmay be regarded as functions ofr and a, as are the choices for labor and subsistence in theland-creation and shifting-cultivation phases. Conse-quently, the empirical process is identifiable in the overallmodel structure as before, with deforestation events andmagnitudes. Note that the household hires nonfamilylabor if wmow0; otherwise, family members work off-farmin addition to pursuing their agricultural activities.

The relationship between the unconditional and theconditional problems is stated in the following claimregardingnecessity. In particular, some set of values (r*, a*,lm* , qm*) solves the unconditional problem only if lm* (andqm*) solve the conditional problem, given (r*, a*). In otherwords, global optimization of household utility over allperiods occurs only if household profits (and utility) aremaximized in the third phase. Proof: The unconditionalproblem is to maximize G(r, a)1H(r, a, l, q) through thechoices of r, a, l, and q. Recalling the definition of Wc, letthere exist a vector, (r*, a*, lm* , qm*), such thatWc(r*, a*, lm* ,qm*)ZWc(r, a, lm, qm) for all feasible vectors, (r, a, lm, qm).Assume further that this vector does not solve theconditional problem. Hence, there exists some (lm**, qm**),distinct from (lm* , qm*), such that H(r*, a*, lm**, qm**)ZH(r*, a*, lm* , qm*). Consequently, Wc(r*, a*, lm**, qm**)ZWc(r*, a*, lm* , qm*), which contradicts the hypothesis,and optimization in the commercial period, dependent onprofit maximization, is necessary to the global solution.

Appendix C

The algorithm works as follows. At iteration k, let thevector of solution variables be Xk (e.g., Xk5 [ak, rk]),which implies knowledge of the gradient vector atthat point, 5 f(Xk). Let the estimated inverse of theHessian matrix, also known, be Bk. The Xk11 vector isthen Xk115Xk�Bk5 f(Xk), which can be used todevelop an equation for obtaining Bk11. Note first thatwith Xk11, 5 f(Xk11) is given. Then Xk11�Xk

5Bk11[5 f(Xk� 1)� 5 f(Xk)] defines Bk11, and thealgorithm proceeds to the next iteration.

To deal with the integer problem, the value of a isconstrained to a set of integers (Mitra andWan 1985) andsolved for the deforestation event magnitude, r, searchingfor theoptimal choice of a and rby reference to thevalueofthe objective function. It is possible that the solutionvalues represent localmaxima, since a global assessment ofsolution values is not undertaken and second-orderconditions are not assessed. However, the optimala values are either 3 or 4, which is consistent with

the observed values of age of secondary vegetation in thestudy area.

The natural production function, f(a), is specified totrack the functions of biomass in time suggested byTable 4(Brown and Lugo 1990; Lucas et al. 1993). The functionalforms are normalized in nutritional units. An asymptoticvalue of grams-protein per hectare is assumed for primaryforest calculated as inWalker (1999), yielding 27 kg/haperyear of protein production from rice and beans, grown inseparate fields in fixed proportion (Moran 1981). The costfunction, g(a), is taken as a monotone increasing functionin a, using the data fromCAT (1992), with a y ‘‘intercept’’of 220 hours (E37 days) indicating the time requirementfor age-insensitive operations (e.g., harvest, bagging, etc.).An asymptote of 450 (E75 days) is adopted, reflectingthe labor involved in working with mature forest. Hence,the age-dependent time costs range from 0 to 230 hours,or up to 38 eight worker-days per hectare. This numbermay be on the low side given data fromEMBRAPA.Cobb-Douglas functions are used for both utility and production(Varian 1993), as are experimental values for exponents.A minimum extent of pasture is taken to be necessary forherd viability. This is set at 20 hectares, which supportsbetween 10 and 20 animals, given regional stockingdensities (Fearnside 1986; Mattos and Uhl 1994). Theproduction function is scaled by observing land produc-tivity for ranches in the region, which yield about 0.50 kgper hectare per year for self-reproducing herds (Mattosand Uhl 1994).

The wage rate for labor power traded against beefproduction is given as follows. First, the minimum wagerate in 1996 was 136 reais per month, yielding about 0.81Real per hour, assuming a forty-hour work week. Fieldinterviews in 1996 indicated a typical dailywage (diario) tobe 10 reais, a number consistent with the minimumwage.Hence, the agricultural wage rate is about U.S.$0.50 perhour. Given an average ‘‘meat on the hoof’’ price of $0.65per kg prior to 1994 (Mattos and Uhl 1994), the relativeprice of labor in a two-commodity (beef and labor power)universe, with beef as a numeraire, is between 0.30 and0.40 kg per hour in the late 1990s, assuming beef pricesheld their value against labor after the devaluation of theReal in 1997.

Appendix D

The sample mean for the distribution of the age of sec-ondary vegetation utilized by colonists (number ofdeforestation events, by the model) is 3.26 years, and itsstandard deviation is 2.52 (unpublished data for theTransamazon Highway; n5 261). The deforestationevent magnitude was taken as a uniform random variable

Walker392

distributed between 2 and 10 hectares (see Homma et al.1993; Dale et al. 1994). The time lag for process initiationwas assumed to be one year for contiguous lots.17

Acknowledgments

This research was mostly supported by the NationalAeronautics and SpaceAdministration under the project,‘‘Pattern to Process: Research and Applications forUnderstanding Multiple Interactions and Feedbacks onLandCoverChange’’ (NAG5-9232), andby theNationalScience Foundation’s Geography and Regional ScienceProgram, under the project ‘‘Tenure Security and Re-sourceUse in theAmazon’’ (SBR-95-11965). I owe a long-standing debt of gratitude, however, to Ariel Lugo of theU.S. Forest Service, who funded my first work in Brazil,and to Adilson Serrao of EMBRAPA/CPATU (BrazilianAgricultural Research Center/Centro de Pesquisa Agro-florestal daAmazoniaOriental)who through the years hasprovidedmewith a great deal of institutional support. I amalso grateful to my Brazilian colleagues, who have beenpatient with me in the field and taught me many things,including Arnaldo Jose de Conto, Alfredo Kingo OyamaHomma,MarcellusCaldas, LuizGuilhermeTeixeira Silva,Pedro Mourao de Oliveira, and Eugenio Arima. Con-versations with Stephen Perz, CharlesWood, andCynthiaSimmons have also proved extremely useful, and I amindebted to them as well. Scott Drzyzga made a criticalcontribution to the article by developing the cartographicmodel used to illustrate the spatial articulation of colonistland use. Although the manuscript was greatly improvedby a number of careful anonymous reviewers, I remainresponsible for any outstanding errors.

Notes

1. The ‘‘new economic geography’’ calls for the use of‘‘computer-assisted thinking’’ and numerical techniques assupplements to analytical results (Fujita, Krugman, andVenables 1999, 8). As Veldkamp and Lambin (2001) note,modeling allows us to conduct experiments to test ourunderstanding of land-use change processes, and to describethem quantitatively.

2. For an opposing view, see Dove (1986) who notes that‘‘inefficient’’ use of land is likely to generate high laborproductivity, very desirable to those who do the work—namely, the farmers.

3. In our usage, ‘‘shifting cultivation’’ (or ‘‘swidden agriculture’’)is a term of broad definition, including sedentary farmersrotating a fixed number of fields (the Amazonian colonistcase) and nomadic groups that shift their residential basewith soil depletion. A ‘‘swidden’’ is Old English for ‘‘burnedclearing.’’ See, for example, Ruthenberg (1971), Peters andNeuenschwander (1988), and Unruh (1988).

4. It is important to note that the term ‘‘pasture,’’ as used byAmazon colonists, does not reflect our common under-standing. In particular, colonists often broadcast grass seedson deforested land, then abandon it until such time as theycan afford to begin stocking with animals, at which point theyreturn and clear the land again. Thus, any regrowth seededwith grass is referred to as ‘‘pasture,’’ despite its ecologicalcharacteristics. Figure 1, therefore, shows a simultaneousprocess of deforestation (loss of mature forest) and regenera-tion of secondary regrowth that is finally converted topermanent use. It also demonstrates the change in therelative importance of the landscape components, whichare mainly annuals in the beginning before shifting to apreponderance of pasture and perennials.

5. Values may appear identical in the Rondonia and Acre sites.They were reported as such because difference of means testscould not reject the null hypothesis of equal values.

6. Colonists often have some access to capital and labormarkets. I abstract from the household economy approachof Singh, Squire, and Strauss (1986) to focus on the land-cover dynamic, and mainly assume a Chayanovian, subsis-tence environment (Nakajima1969).Nevertheless, in a largesample of colonist farmers (n5 261) on the TransamazonHighway in the eastern sector of theAmazon, only 49 percentused off-farm labor in 1996 (Walker, Perz, et al. 2002).Chayanov (Thorner, Kerblay, and Smith 1966) developed hisfamily-labor theory in a setting in which 90 percent of thehouseholds were detached from labor markets.

7. The denominator resulting from the geometric seriesexpressions has been eliminated, given that it is constantand does not affect the solution. Note that the U functionsremain the same, although in equation (3) the arguments arechanged. Both labor and subsistence in equation (2) arefunctions of these new arguments.

8. The present model advances the formulation in Walker(1999)byallowing for an initial cycle of landclearance, criticalto the description of tropical deforestation. Note that thesolution value, a, can be used to characterize a system in termsof measures provided by Ruthenberg (1980) and Boserup(1981), given that agriculture occurs for only one year on anactive plot.Ruthenberg’sRvalue is 100/a, essentially the samemeasure as Boserup’s (1981, 19) ‘‘frequency of cropping.’’

9. The settlement roads in Rondonia, called linhas, are muchmore passable than those in the east, at least on the 300-kmstretch between Altamira and Ruropolis, one of the primarysettlement frontiers in Para. This may explain the broadrectangular pattern observed in the western sector linhas,which stretch for 40 to 60 kilometers between Porto Velhoand Ouro Preto.

10. With a cropping cycle of 1 year and a solution value of a5 4,market entry occurs in the ninth year of operations, given 4years of land-clearance activities and another 4 years usingsecondary vegetation. Nine years is consistent with thelength of residency of the ‘‘high-value’’ colonist systemsreported by Walker, Perz, Caldas, and Teixeira Silva (2002)for the eastern Amazon. Such systems are presumably moremarket-oriented than those of neighboring subsistence farm-ers. Figure 1 suggests an almost complete transition tomarket-oriented production (with perennials and pasture)after twelve years.

11. Fieldwork in the summer of 2000 in both the eastern andwestern sectors of the basin revealed the presence of dense

Landscape Change of the Amazonian Frontier 393

family networks on the individual settlement roads. Distantparcels were being claimed by parents for children who werebecoming adults.One of the leading loggers and private road-builders on the Transamazon Highway in the frontier area ofPara was originally a colonist himself.

12. Dvorak (1992) does not provide an explicit solution to thisproblem. In particular, themodel that is actually solved is for asingle plot in a single period (equations 8–9), despite themultiplot context of the formulation (equations 1–8). Themultiplot decision is only alluded to verbally (Dvorak 1992,812). Thus, the present formulation complements Dvorak’smodel by providing an explicit multiperiod solution, withdiscounting, a labor/leisure tradeoff, and an explicit repre-sentation of the allocation decision across clearing andweeding activities. Walker (1999) has provided an explicitsolution to the multiplot problem, focused not on labor costsbut on complementary subsistence requirements.

13. The weeding function is similar to the vegetation-clearingfunction. It is assumed that a fixed amount of weeding isnecessary to have production, as a function of the year in thecropping cycle.

14. Note that the production function, q, is taken to be freelyavailable to the farmer. However, it is costly to build a herd orto start a plantation of perennials. The present formulationabstracts from such an investment problem to focus on landcreation. However, it could be assumed that seed capitalbecomes available to the farmer from the government, or thatthe farmer arrives on the frontier with an endowment.Perhaps the bestway to view the problem is that q is ameasurenet of loan repayment.

15. Avon Thunian ‘‘accessibility’’ effect has been modeled usingtransport costs only for the produced good. For the case ofcattle, some suggest that they canwalk themselves tomarket,thereby reducing transport costs to the opportunity cost ofthe labor that accompanies them. Nevertheless, ranchers doincur considerable expense through trucking, and the carrretais a vehicle specifically designed for this task. It is important topoint out that inputs to ranching may impose accessibilitycosts, thereby exerting a von Thunian market (or centralplace) effect. In particular, veterinarians and extensionagents may be limited in the area they can reasonably coverwith regularity. See Liu (1999) for a von Thunian interpreta-tion of labor costs in Chinese agriculture. Angelsen (1994)also assumes a distance dependent wage rate.

16. By the first-order conditions of profit maximization, thepartial derivative of the production function in labor mustequal the wage rate, and by the inverse-function theorem(Rudin 1976), we have w0 5 h(ra, v) and therefore q0 5 q(ra,v). Hence, the constraint for utility optimization may bewritten as q5 y(r,a)1v(L � lm), where y(r,a)5 q0 �w0.

17. An analysis of ETM1 satellite images for 1999 shows thatsettlement roads opened along the Transamazon Highway inPara at the rate of 0.77 km per year beginning in the early1970s. If the time steps are interpreted as one year, theextension of the system north and south is underestimated,given the 400-m frontage of individual properties.

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Correspondence: Department of Geography, Michigan State University, East Lansing, MI 48824-1115, e-mail: rwalker@pilot.msu.edu.

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