Mapping Process to Pattern in the Landscape
Change 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 decisions made by individual land
managers. In developing economies, these decisions may be embedded in the evolution of individual households, as
is often the case in indigenous areas and agricultural frontiers. One goal of the present article is to address the landuse and land-cover decisions of colonist farmers in the Amazon Basin as a function, in part, of household
characteristics. Another goal is to generalize the issue of tropical deforestation into a broader discussion on forest
dynamics. The extent of secondary forest in tropical areas has been well documented in South America and Africa.
Agricultural-plot abandonment often occurs in tandem with primary forest clearance and as part of the same
decision-making calculus. Consequently, tropical deforestation and forest succession are not independent processes
in the landscape. This article presents a framework that integrates them into a model of forest dynamics at household
level, and in so doing provides an account of the spatial pattern of deforestation that has been observed in the
Amazon’s colonization frontiers. Key Words: Amazon, deforestation, land use change.
C
hanges in land use and cover occur throughout the
world, but the greatest present-day concern
focuses on tropical deforestation, which drives
species to extinction, releases greenhouse gases, and
undermines the sustainability of local environments
(Stern, Young, and Druckman 1992; Ojima, Galvin, and
Turner 1994; Turner et al. 1995, 2001). Nowhere are these
impacts more apparent than in the Amazon Basin, where a
large portion of the forest has vanished in the wake of
government initiatives to bring people without land to
land without people (Hecht 1985). The alarm continues
to sound, with rates of forest lost varying between 10,000
and 20,000 km2 per year (INPE 2000), and with plans by
the Brazilian government to extend its road-building
efforts throughout the region (de Cassia 1997; Laurance
and Fearnside 1999).
This article presents a model of the land-use behavior of
the colonist farmer, an important agent in current
processes of Amazonian deforestation. The elaboration
of such a model is not without challenges, given the
dynamic institutional environment of the forest frontier.
Upon initial occupation of land beyond the forest margin,
colonists behave in large part according to the dictates of
the peasant-economy concept (Thorner, Kerblay, and
Smith 1966), focusing their attention on subsistence
production in the interests of food security. With frontier
expansion and the arrival of markets, economic behavior
changes as subsistence agriculture gives way to commercial production (Binswanger and McIntire 1987; DeShazo
and DeShazo 1995). After all, colonists often settle
frontiers 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 colonist
land-management, this article also seeks to generalize the
issue of tropical deforestation into a broader discussion,
including its spatial articulation and the often-overlooked
dynamic involving the regrowth of secondary vegetation.
The extent of secondary regrowth has been well documented for South America (Brown and Lugo 1990),
and extensive areas of successional vegetation exist in
the Amazon basin, even with continuing deforestation
(Moran et al. 1996). Neglected in the attempt to account
for this phenomenon is that field abandonment may occur
in tandem with primary forest clearance and as part of the
same decision-making calculus. In addition to capturing
the transition from subsistence to market-oriented production, the behavioral model of the colonist also
accounts for the simultaneous occurrence of forest loss
and regeneration. Moreover, once embedded in a twodimensional landscape according to the empirical geometry of settlement-planning, the model generates spatial
outcomes and patterns of forest fragmentation.
Tropical deforestation and land-cover change in
general are complex processes involving a multiplicity of
agents. In the Amazon basin, colonist farmers, corporate
ranchers, loggers, and gold-miners have all contributed to
forest loss, as have government bureaucrats far removed
from the forest frontier (Smith et al. 1995). Such agents
may operate with an interdependent logic, a recurrent
theme in discussions on tropical deforestation (Geist and
Lambin 2001). In an early account addressing the global
Annals of the Association of American Geographers, 93(2), 2003, pp. 376–398
r 2003 by Association of American Geographers
Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, and 9600 Garsington Road, Oxford, OX4 2DQ, U.K.
Landscape Change of the Amazonian Frontier
situation, Myers (1980) describes a process of invasive
forest mobility, in which farmers follow loggers into newly
opened forest. There, as they prepare the land for farming,
they finish what the loggers started by taking down the
trees that remain in the wake of selective logging (Leslie
1980; Myers 1980; Schmithusen 1980; Office of Technology Assessment 1984; Walker 1987; Repetto and Gillis
1988; Walker and Smith 1993; Kummer and Turner 1994;
Brookfield, Potter, and Byron 1995). For the Amazonian
case, Wood (1983) and Ozório de Almeida (1992) point to
another connected process whereby land consolidation by
wealthy ranchers induces migration on the part of
smallholders, who must pick up and leave for unclaimed
lands in primary and old-growth forest. Beyond these
various agent interactions, social forces acting in aggregate
play significant roles in processes of land-cover change.
The political economy of development in the Brazilian
Amazon also offers a prime example of how state policy,
interacting with compelling social need, created dramatic
population inflows to a forest region (Hecht 1985).
This article takes as given that aggregate social forces
bring people and capital to unsettled regions. As such, it
does not address the true complexity of land-cover change
processes. Nor does it seek to present a broad-scale,
statistical account of the deforestation processes presently
at work in the Amazon basin (e.g., Pfaff 1999). Rather, it
seeks to comprehend land use and land cover at the level
of an individual agent—namely, the colonist household.
Thus, the forest-dynamics model presented does not
provide a global picture of the forces affecting the tropical
forest biome. Instead, it focuses on the development of
individual properties of smallholders in the Amazon basin,
an endogenous process of change that takes place in the
wake of the broad-scale factors affecting migration and
capital flows to the region. Reviews of the global context
and general theories of deforestation can be found in
Kaimowitz and Angelsen (1998), Geist and Lambin
(2001), and Lambin et al. (2001).
The article is organized into two basic parts. It begins by
considering what may be referred to as the classical version
of shifting cultivation, the system of some indigenous
peoples in which rotations of secondary vegetation and
farm plots take place over an infinite time horizon. This
motivates the colonist model that follows and illustrates
how forest dynamics—including both deforestation and
regrowth—can be introduced into a behavioral framework. To accomplish this, it is necessary to conceptualize
both an initial process of land clearance and the subsequent implementation of a rotational system as functions
of natural-resource productivity and labor costs.
After providing a model for shifting cultivation, the
article presents a model extension for colonist farmers.
377
This differs from the shifting-cultivation formulation in
two fundamental ways. First, an institutional dynamic is
assumed, with market opportunities initially absent and
then present in the evolution of a farming system. Second,
the period during which rotation occurs is of finite
duration, in which case, secondary vegetation occurs only
as a transient feature of a property’s land cover. The creation of a farm is thereby resolvable into two phases: the
initial rotational phase; and the second, which is marketoriented, with permanent agriculture, ranching, or some
mix of the two. Such a framework provides an extension
of earlier agent-based efforts that have largely focused
on shifting cultivation, and is able to characterize the
expansion of commercial agriculture into forested areas,
much more important as a deforestation driver than
shifting cultivation (Geist and Lambin 2001). The model
is stated in such a way that empirical land-cover change
processes, such as the sequence of deforestation actions
taken in creating farmland, are identifiable in the theoretical
structure.
The second part of the article presents a framework that
integrates two model applications. The first is a farm-level
application implementing the agent-based models to show
how variables such as size of family workforce, discount
rates, and prices affect the deforestation associated with
individual households. Here, numerical methods are used
with field data to produce land-cover-change magnitudes
consistent with the empirical setting, an approach that has
been previously deployed for similar purposes (Dale et al.
1994; Beaumont and Walker 1996; Angelsen 1999;
Albers and Goldbach 2000; Evans et al. 2001).1 The
second application takes theoretical results from the
agent-based framework to represent the process of landscape evolution. Geographic information system software
is used to translate individual colonist dynamics into a
regional-scale visualization of the ‘‘fishbone’’ pattern of
deforestation observed in the colonization frontiers of
the Brazilian Amazon. Thus, in a two-step sequence, the
article scales up from the decentralized processes of
individual households to the pattern they make in the
aggregate landscape (Brondizio et al. 2002).
Other agent-based models have been advanced to
depict tropical deforestation in the Amazon basin (e.g.,
Dale et al. 1994; Walker et al. 1997). The present model
seeks to extend earlier formulations by (1) providing a
behavioral foundation for the process of landscape
change, (2) linking this behavior to household attributes
in the presentation of a spatial model, and (3) reconciling
the existence of extensive secondary vegetation in the
region with the presence of large populations of lowincome farmers. This is accomplished by extending the
household-economy model first elaborated by Chayanov
378
Walker
(Singh, Squire, and Strauss 1986) to include shifts in a
household’s production base. Turner and Brush (1987, 33)
have suggested that individual farm households can show
hybrid behavior, with subsistence (cf. consumption) and
market-oriented production (cf. commodity) occurring as
opportunities arise (see also Turner and Ali 1996). The
article adapts this fundamental insight into a model
concept that can be deployed to describe land-coverchange processes.
Colonists and Shifting Cultivation
As has been suggested, colonist farming at the household level undergoes an evolution from subsistenceoriented agriculture, in which much of the farm output is
consumed, to commercial activities generating produce
for both home consumption and market. For the Amazonia case, this typically involves a parallel evolution in
the household’s production base, from shifting cultivation
to a ranch, a perennials plantation, or some combination.
It is farm production, of course, that provides the main
proximate cause of land-cover change in rural areas,
serving as the nexus between objectives of the household
and the use of land (Turner, Meyer, and Skole 1994;
Walker and Homma 1996). Consequently, the key to
understanding land-cover change at the farm level lies in
understanding how land is utilized for production.
It is important to note at the outset that shifting
cultivators can and do produce for the market. Nevertheless, to facilitate the exposition, it shall be assumed here
that they are subsistence farmers producing mainly for
home consumption. Colonists begin as shifting cultivators,
but ultimately switch both their production system and
their decision-making calculus to include commercial
production based on profit maximization. Thus, colonists
are assumed to possess a temporal duality (Turner and
Brush 1987) in their relationship to the market, which
runs parallel to their changing production mode.
The exposition of the colonist model focuses primarily
on production, and in particular on the transition from
shifting cultivation to permanent agriculture. Consequently, the motivation for the model begins with a brief
account of shifting cultivation, which, it is argued, has
little bearing on the colonist system observed in Amazonia
today, despite some claims to the contrary. This leads to an
account of recent formal modeling efforts directed at
shifting cultivation, and a demonstration of their inability
to describe colonist agriculture. As shall become apparent,
consideration of the consumption side of the household
economy—in particular, the household’s desire to achieve
utility ‘‘maximization’’—provides an important key to
reformulating a statement that reflects the empirical
setting.
Research on shifting cultivation has been motivated, in
part, by concerns about its efficiency (Spencer 1966;
Ruthenberg 1971; Watters 1971; Peters and Neuenschwander 1988)2 and by interest in the role it plays in
indigenous agriculture (Posey 1984; Denevan and Padoch
1987; Padoch 1987; Unruh 1988, 1990; Irvine 1989).3
Researchers have considered links between the use of
fallow and fallow ages (Denevan and Treacy 1987; Unruh
and Alcorn 1987; Unruh and Paitán 1987; Balée and Gély
1989), and have attempted to explain attributes of shifting
cultivation systems (e.g., numbers of fields, fallow lengths)
in light of household structure (Salick 1989; Salick and
Lundberg 1990; Scatena et al. 1996; Coomes, Grimard,
and Burt 2000). The use of shifting cultivation, common
in the historic record, remains globally distributed, despite
the advent of modern technologies. Although the rotation
of land parcels is locally dynamic, the system itself presents
a static, aggregate landscape, resolvable into age cohorts of
secondary vegetation and cleared lands of active agriculture (Walker 1999).
Agronomists, anthropologists, and agricultural economists have suggested that smallholders in forest frontiers
in the Amazon basin engage in shifting cultivation, to be
distinguished from ranching as a distinct system, practiced
by different agents altogether (Nair 1987, 1991; Serrão
and Homma 1993; Faminow 1998). Serrão and Homma
(1993) claim that 400,000 shifting cultivators are active in
the Brazilian Amazon, and their farm-system classification
(Serrão and Homma 1993, 298–99) suggests that these
cultivators are subsistence-oriented households growing
food crops, with no involvement in perennial plantations
or ranching. Beckerman (1987) does note the recent
appearance of market-oriented bush fallow, but focuses
descriptive attention on the swidden systems of indigenous peoples.
Describing the Colonist System
Any suggestion that most small-scale farmers in the
Brazilian Amazon are subsistence-oriented shifting cultivators is inconsistent with a great deal of empirical
observation. Indeed, a narrative appears to be emerging
in this regard, encompassing both the ‘‘peasant pioneer
cycle’’ (Pichón 1997) and processes of farm evolution
linked to domestic cycles (CAT 1992; DeShazo and
DeShazo 1995; Walker and Homma 1996; McCracken
et al. 1999; Perz 2001). The narrative account begins with
a newly formed family’s arrival on a piece of forested land
beyond the extensive margin of agriculture. Given little
initial capital and low levels of experience, the family first
Landscape Change of the Amazonian Frontier
implements a system of shifting cultivation, producing
annual crops such as rice, corn, and beans. This continues
as long as the children are too young to work and the
internal dependency of the household remains high.
With time, however, the family changes. Children
enter the household workforce, and the parents acquire
experience, gaining confidence in their ability to farm.
Such a demographic setting is ripe for investment and the
adoption of a system with greater economic potential,
such as a ranch, a perennials plantation, or some combination of the two. Should the institutional environment
also change with the arrival of a transportation system and
markets—the expectation that provides the impetus to
colonization, given the value it brings to land—the
agronomic system evolves in parallel with the development of the household, moving toward an emphasis on
commercial crops. If the children move off-farm as they
form their own families, the process may reverse itself in
the short run, as fields are abandoned with the loss of
household labor.
Anecdotal evidence is accumulating on the land-cover
changes associated with the farm formation process just
described. Typically, they involve an initial phase of forest
clearing, with deforestation taking place on an annual
or semiannual basis in a series of discrete episodes, or
deforestation events. Depending on the household’s financial circumstances, perennials (cocoa, pepper, coffee) and
pasture grasses are planted sooner or later, while annuals
production is mainly continuous throughout. After about
five to ten years, deforestation has run its course—at least
for the first occupants of a property—and the household 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-property
evolution for a site in the Brazilian frontier. Moving from
left to right, from bottom to top, and from upper to lower
tier, each panel represents a different time point for the lot.
The particular process indicated covers a twelve-year
period that is still unfolding. The lot in question was
delimited by the Brazilian colonization institute, INCRA,
which, in the 1970s, provided 100-hectare parcels for
migrant families (400 m 2,500 m). A salient feature
of the land-cover evolution is the continuous advance of
annuals production, from the front of the lot (bottom
of each panel) towards the back. This occurs with the
clearing of primary forest. Annuals production gives way,
in turn, to both pasture and perennial plantings, which
follow in the wake of deforestation.4
The distribution of farms subject to such evolutionary
processes is an empirical question, although survey work
379
Figure 1. Property dynamic.
suggests they are widespread (Dale et al. 1993; Browder
1994, 1996; Whitcover and Vosti 1996; Pichón 1997;
Walker et al. 1997; Fujisaka and White 1998; McCracken
et al. 1999). Tables 1 and 2 present data on land cover and
household characteristics for surveys of colonist farmers in
the Amazon basin, including Brazil and Ecuador. The
land-cover data (Table 1) establish that, in cross-section,
colonist households have diversified farming systems
dominated by pasture. Most of the sites show a similar
pattern, with land allocated evenly between perennials
and annuals but with the majority of it in pasture.
The magnitudes deforested are also comparable, ranging
between 18 and 55 hectares.
These data are striking given the wide geographic range
they represent, and the considerable variations in rainfall,
soils, and institutional environment. Actual production
components show strong similarities for three sites in the
Brazilian portion of the basin (Table 2A). Area of land in
annuals (rice, corn, and beans) is mostly in rice (excepting
the Acre site), with a sharp drop-off to comparable
amounts in the semiannual, cassava. Herd sizes are also
very close.5 The similarities in the production systems
carry over to attributes of the households (Table 2B).
Family size, age of household head, and degree of
dependency (number of children and elderly individuals
divided by total family size) are similar. Only the average
period of residence varies, reflecting the older colonization
process in the state of Pará in the eastern Amazon.
The significance of these agents in the demography
of the region presents a second empirical question.
The number provided by Serrão and Homma (1993)
—400,000—does not distinguish between identifiable
380
Walker
Table 1. Percentage of Deforested Land in Farming System Components
Perennials
Annuals
Pasture
Hectares deforested
1
2
3
4
5
6
7
8
9
10
17
9
74
47
9
9
81
54
30
22
48
23
15
15
70
33
33
11
56
18
10
10
80
55
28
9
62
23
14
14
71
33
3
16
81
29
13
14
74
40
1 Brazil, Rondônia, Ouro Preto; 1991, n 5 91; Good soils.a
2 Brazil, Rondônia, Ouro Preto; 1991, n 5 91; Moderate soils.a
3 Brazil, Rondônia, Ouro Preto; 1991, n 5 91; Restricted soils.a
4 Brazil, Rondônia, Ouro Preto; 1991, n 5 91; Unsuitable soils.a
5 Ecuador, Napo and Sucumbios; 1990, n 5 412; Full sample.b
6 Ecuador, Napo and Sucumbios; 1990; 4100-ha properties.b
7 Ecuador, Napo and Sucumbios; 1990; Accessible properties.b
8 Brazil, Rondônia, Theobroma; 1994, n 5 155 (with Acre); Full sample.c
9 Brazil, Acre site; 1994, n 5 155 (with Rondônia); Full sample.c
10 Brazil, Pará, Uruará; 1996, n 5 261; Full sample.d
a
Interpreting Figure 6 in Dale et al. (1993).
b
From Pichón (1997).
c
From Whitcover 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.
groups that would meet their farming-system definition, in
particular, colonist farmers, indigenous peoples, and longstanding residents of the region who often engage in forms
of extraction and fishing. In fact, the 1991 Brazilian
census (S. G. Perz, personal conversation, 2000) indicates
that the overall population of individuals practicing lowintensity agriculture in the region may be higher than
Serrão and Homma (1993) suggest. In particular, there are
1.5 million households deriving income from small-scale
agricultural activity. Assuming, conservatively, an average
of 2.5 individuals per household, this yields a total
Table 2A. Farm System Characteristics
Rice (ha)
Corn (ha)
Beans (ha)
Cassava (ha)
Cattle
Property value (Reais/ha)
Rondônia
Acre
Pará
3.36
2.24
1.67
0.52
22
579
2.18
2.24
1.67
0.52
22
272
3.06
2.40
0.68
0.62
24
276
Table 2B. Household Characteristics
Household size
Age, household head
Adult males
Degree of dependency
Average period of residence
Rondônia
Acre
Pará
5.38
47.57
1.95
0.30
6.30
5.38
47.57
1.60
0.39
7.86
5.5
47.18
1.91
0.37
11.46
Note: The Rondônia and Acre data are from Whitcover and Vosti (1996).
The Pará data are derived from National Science Foundation project ‘‘Tenure
Security and Resource Use in the Amazon,’’ Chuck Wood, principal
investigator.
population of 3.75 million, which may be compared to
the region’s 150,000 indigenous residents and the approximately 600,000 forest ‘‘extractors’’ and fisherman. If
the nonindigenous, small-scale farmers are mostly colonists, as seems likely, then such households clearly
dominate the rural population of the Brazilian Amazon.
Some have argued that such households account for a
large share of deforestation in the region (Walker, Moran,
and Anselin 2000), and recent work on the Transamazon
Highway (Walker, Wood, et al. 2002) in Pará indicates
that colonist households deforest, on average, at least one
hectare annually (between 1986 and 1997). With 1.5
million active households, the associated deforestation
would be 15,000 km2 per year, assuming a one-hectare
clearing per year per family. This appears to represent a
very large proportion of the total amount occurring
throughout the basin.
In sum, research addressing the classical system of
shifting cultivation does not appear to be relevant to the
Amazonian colonist. In the colonist system, the shifting of
fields and the generation of secondary regrowth represent
a transitory phase in the creation of a farm or ranch.
Describing the land-cover processes associated with this
system is critical to understanding an important component of Amazonian forest dynamics.
The Shifting Cultivation Problem
Models of Shifting Cultivation
Boserup’s (1965) account of shifting cultivation
contains an implicit concept linking human drivers to
Landscape Change of the Amazonian Frontier
land-cover change. In particular, population growth and the
need for increased food production cause a shift in land
cover at the regional scale, from old-growth forest under
long rotations to younger forms of secondary forest (Boserup
1965). More recently, geographers and economists have
developed formal models of shifting cultivation that go
beyond this demographic explanation. As a rule, these
models are either aggregate in nature and consider regional
landscapes emanating from a market center (e.g., López
and Niklitschek 1991; Jones and O’Neill 1993a, 1993b;
Angelsen 1994), or focus on individual land-managing
agents (Barrett 1991; Dvorak 1992; Krautkraemer 1994;
Walker 1999; Albers and Goldbach 2000). The modeling
framework advanced in this article focuses on individual
agents and therefore falls within this latter category.
Most shifting-cultivation models have been stated
independently of the theory of household production,
with its emphasis on consumption, production (by family
workers), and the so-called labor-leisure trade-off (Chayanov [1925] 1966; Ellis 1993). Household production
theory and the concept of the peasant economy consider
the manner in which families produce and consume in
order to achieve the most utility they can (Turner and Brush
1987; Ellis 1993). Formulations range from the seminal
work by Chayanov (Thorner, Kerblay, and Smith 1966), in
which labor markets are absent and the consumption and
production decisions of households are nonseparable
(Mellor 1963; Nakajima 1969), to more modern accounts
addressing smallholder behavior in the presence of welldeveloped markets for labor, capital, and consumption
goods (Barnum and Squire 1979; Singh, Squire, and
Strauss 1986).
Shifting-cultivation models generally restrict their
focus to the production side of the household economy,
presumably given the task at hand, which is to describe a
complex decision-making process involving optimal cropping cycles and periods of fallow. Although the importance
of household welfare (and consumption) is generally
381
recognized (e.g., Dvorak 1992; Krautkraemer 1994), the
frameworks implemented illuminate only technical aspects of the shifting cultivation system, such as the best
time at which to abandon a farm plot to processes of
fertility restoration. The models mainly assume profit
maximization and do not provide explicit solutions for the
household’s demand for food and land.
One consequence of these various assumptions and
restrictions is that the modeled world of the family
practicing shifting cultivation is often at variance with its
empirical reality (Table 3). A major problem in this regard
is the existence of production intervals, during which
farming ceases as the soil regains fertility. An assumption
made to ensure family survival involves consumption
‘‘smoothing’’ during fallow periods, when families tap their
bank deposits to purchase food and other items (Albers
and Goldbach 2000). So long as they can save their profits
during the cropping cycle, they need not starve. Unfortunately, a very large number of shifting cultivators occupy
land beyond the extensive margin of agriculture, with
little or no access to financial institutions or to markets
for capital, labor, and farm produce (Walker 1999). The
focus on production technology also means that consumption demands—clearly linked to household size and
structure—do not get translated into magnitudes of land
use, critical in any attempt to represent land-cover change.
One solution to the problem of yearly consumption is to
consider a multiplot system in the context of actual
household demands and land productivity, an approach
taken by Walker (1999). This leads to the statement of an
optimization problem that unites the household economy
model with shifting cultivation technology. That is, the
household is taken to maximize:
1
X
bt Uðl; SÞ
ð1Þ
t¼0
where U is a household utility function, b is a discount
factor, l is leisure, and S is subsistence. The model in
Table 3. Shifting Cultivation Models
Behavior
Dvorak (1992, 811)
Barrett (1991, 178)
Krautkraemer (1994, 410)
Walker (1999, 399)
Albers and Goldbach (2000, 271)
a
Utilitya maximization
Profit maximization
Net benefitb maximization
Utility maximization
Profit maximization
Objective Function
Labor cost minimization
Present value of net profit
Present value of net benefit
Discounted utility
Net present value of production
Production
Intervals
No
Yes
Yes
No
Yes
Land
Creation
No
Yesc
No
Yes
Yesc
System
Switching
No
No
No
No
No
Dvorak (1992, 811) states that the objective can be to‘‘minimize clearing and weeding labor subject to production of subsistence, or to maximize utility of leisure and
consumption.’’ Nevertheless, the actual objective function stated is the minimization of labor costs, subject to a fixed level of subsistence.
b
Krautkraemer (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.
382
Walker
Walker (1999) provides an explicit solution, giving plot
sizes and age of secondary vegetation utilized for a
multicrop system as a function of household attributes.
Unfortunately, the shifting-cultivation component of the
model suffers from two shortcomings that compromise its
direct application to the present situation. First, it does not
allow for land creation whereby primary forest is taken
down to make way for agricultural activity. Second, system
switches do not occur, particularly the change from
subsistence production to commercial ventures observed
in colonization frontiers (CAT 1992; DeShazo and
DeShazo 1995; Walker and Homma 1996; McCracken
et al. 1999; Perz 2001).
The approach this article advances represents an
attempt to overcome the shortcomings of previous efforts
that limit their utility in representing land-cover-change
processes in settings such as the Amazon frontier. To this
end, the article develops two models representing different specifications of the general problem represented by
function (1). The first is for the traditional shifting
cultivation system with long-run cycling between swiddens and fallow, and the second is for the colonist farmer
who starts out as a shifting cultivator but ultimately
implements a commercial system. Both are useful in
providing a full description of the empirical reality.
As has been argued, the colonist farmer is one of the
dominant agents of land-cover change in agricultural
frontiers, especially the Amazon basin. The colonist
system represents a hybrid of subsistence-oriented activities and production for the market (Turner and Brush
1987; Turner and Ali 1996). The particular form of
hybridization to be considered is that occurring through
time, in parallel to the life-cycle of the household and
the frontier’s advance, which carries with it markets
and new possibilities for economic behavior (Binswanger
and McIntire 1987). The modeling challenge is
twofold—namely, (1) to demonstrate how a shifting
cultivation ‘‘technology’’ can be incorporated into the
household-economy framework, and (2) to reflect farm
evolution from nonmarket production to commercial
activity. The operational goal is to develop a formulation
that yields variables describing the process as it is known to
occur in the real world in a series of discrete deforestation
events.
The Constraints of Nature
Before elaborating the model, it is important to
consider the nature of the ‘‘technology’’ of production
under shifting cultivation. The primary factors defining
this technology are the productivity of the naturalresource base and the labor required to use it. Depending
on intrinsic soil properties and the biomass of standing
vegetation, a single slash-and-burn operation may provide
repeatedly high yields. Alternatively, production can be
short-lived due to rapid fertility decline, requiring rapid
shifts to new areas. In addition to the soil’s nutrient load
and water-retention capability, the strength of successional response from surrounding areas is important. Weed
invasions can overcome a family’s ability to cultivate food
crops, making it necessary to quickly abandon plots for the
next production period. Thus, the household’s objective
function represented by summation (1) is constrained by
environmental factors affecting resource productivity and
labor requirements.
In formally modeling the impact of the ‘‘constraints of
nature’’ on household behavior, Barrett (1991), Dvorak
(1992), Angelsen (1994), and Walker (1999) elaborate
the concept of a natural production function relating crop
output to the age of vegetative regrowth slashed and
burned in preparing a field. Presumably, output increases
with age, given that nutrient uptake is continuous in time.
Thus, use of older secondary vegetation (and primary
forest) yields more output than shrubby regrowth, because
a greater stock of nutrients is released to the soil with
the slash-and-burn operation. Mathematical functions
of the relationship between soil fertility and age of
regrowth remain theoretical constructs, although ecological research links biomass accumulation to successional
stage, particularly in the new-world tropics. Table 4 gives
data on forest biomass as a function of age for both secondary regrowth and mature forest (Uhl 1987; Saldarriaga
et al. 1988; Brown and Lugo 1990; Lucas et al. 1993;
Salomão, Nepstad, and Vieira 1996; Vieira et al. 1996).
These data show that biomass accumulation is increasing
in time, and that a substantial period must elapse before
mature forest levels are reached. If nutrient stock is a fixed
coefficient of biomass, then nutrient regeneration and, by
implication, potential crop output are functions of biomass
age, not unlike the renewable-resource functions used to
model forests and fisheries (Clark 1976, 1985).
A second environmental factor affecting household
decision making involves the labor costs required for use of
secondary vegetation. In particular, the unit area labor
costs associated with land clearance and weeding also vary
as a function of age of regrowth (Dvorak 1992; Angelsen
1994). The stylized fact is that mature forest requires more
labor to prepare than secondary regrowth, which, in turn,
requires more maintenance weeding (Pingali and
Binswanger 1988). Table 5 presents data for sites in
the Amazon basin, although with little age variation for
the vegetation categories. They show consistently high
labor costs for the preparation of fields from mature forest,
and costly weeding operations in the secondary regrowth.
Landscape Change of the Amazonian Frontier
383
Table 4. Above-Ground Forest Biomass in Select Sites, Tons per Hectare
Location
Regional estimates
Venezuelac
Venezuelad
Pará, Brazile
b
5 years
10 years
20 years
40 years
70 yearsa
50
33.85
95
150
160
165
110
105
150
58
43.9
13.1
80.5
Mature
255
265.7
a
An average age over plots between 60 and 80 years old.
Modified from Brown and Lugo (1990) and Lucas et al. (1993).
c
Uhl (1987).
d
Saldarriaga et al. (1988).
e
Salomão, Nepstad, and Vieira (1996); Vieira et al. (1996).
b
called colonist model. Criticisms have been made of the use
of optimization frameworks to explain behavior in highrisk environments, such as might be encountered in a
tropical frontier (Lipton 1968). Walker, Perz, Caldas, and
Teixeira Silva (2002) implement a risk-minimization
model in analyzing farm-system choices of colonists in
the Brazilian Amazon, and Homma and colleagues (1996)
have considered the role of risk in explaining the shift from
extractive rainforest activities (Brazil-nut extraction) to
pasture-based systems. Nevertheless, the framework
adopted in the article assumes optimization behavior in
order to maintain a focus on land-use dynamics. Many
shifting-cultivation models have been stated in a similar
fashion (e.g., Barrett 1991; Dvorak 1992; Krautkraemer
1994; Walker 1999; Albers and Goldbach 2000).
In order to facilitate model development, it is assumed
that plots are used for only one year. This assumption
Mature forest may be more costly overall when the landclearance requirements measured by the Brazilian Agricultural Research Center (EMBRAPA) are assumed
(column 8), together with the estimated labor needed
for weeding at specific sites.
Modeling Forest Clearance and Land Creation
To model the empirical circumstances of the colonist
farmer, function (1) must be optimized subject to the
productivity of the resource base, the labor costs of using
it, and the availability of production factors, particularly
family labor.6 The case of shifting cultivation is considered
first in order to introduce the process of land creation and
to formulate the problem in terms of household structure
(utility, labor supply). The framework is then modified by
adding a switch to commercial activities, yielding the so-
Table 5. Labor Costs by Operation, Person-Days per Hectare, Land Preparation
Mature Forest
1
2
3
*
*
*
*
*
*
*
*
Weeding (Capinas)
6.7
11.3
6.7
1.0
25.7
12
25.7
18.2
Total
37.7
45.9
Undercutting (Broca)
Tree-felling (Derrubada)
Debris-stacking (Coivara)
Burning
(Queimada)
P
33.1
433
4
13.0
Secondary Regrowth
5
6
7
1–17b
15
*
*
*
*
8
25
41–42
425
441–42
1
2
3
4a
12.0
0.0
0.0
1.0
13.0
24
*
*
*
*
12.0
5.5
19.4
31.6
37
51
Notes: These values are for low-intensity, manual practices.
1 State of Pará, Brazilian Amazon (CAT 1992).
2 Bolivia (CAT 1992).
3 State of Pará, Brazilian Amazon (A. Toniolo and R. Costa personal conversation, 1999).
4 State of Pará, Brazilian Amazon (Scatena et al. 1996).
5 State of Pará, Brazilian Amazon (Moran 1981).
6 State of Pará, Brazilian Amazon (Wagley 1953).
7 State of Pará, Brazilian Amazon (Texeira Mendes 1997).
8 Brazil (EMBRAPA 1995; EMBRAPA/CPATU 2000). Values given only for deforestation and burning actions. No weeding.
a
The undercutting value is for young secondary growth (2–4 years). The tree-felling value is for secondary growth in excess of 10 years.
b
Values vary as a function of available equipment. A chainsaw reduces the one-hectare undercutting operation to one day.
4a
11.5
384
Walker
has the analytically useful effect of equating the age at
which secondary vegetation is slashed to the number of
deforestation actions undertaken by the farmer—the socalled deforestation events. Very short cycles—one or two
years in length—are typical of the region (Walker et al.
1997), with its poor soils and humid climate, conditions
that promote rapid declines in soil fertility and short-lived
cropping periods (Barrett 1991).
In this article, deforestation is the clearing of mature
forest and not of secondary regrowth. With a single-year
crop cycle, deforestation on some plot of land starts at time
period 0 and continues in successive years until the initial
plot, having been abandoned to regrowth in year 1, is
cleared, farmed, and abandoned again. Thus, deforestation
takes place at the same time as regeneration during the
initial phase of land creation, which consists of a sequence
of deforestation events. Beyond this point, deforestation
ceases and farming occurs on the basis of rotation over the
fields created by the deforestation events.
The Shifting-Cultivation Model
The statement of the shifting-cultivation model begins
with the specification of household productive capacity
and the labor requirements of the farm activities. This
specification, in turn, is germane to the colonist household, given the importance of family labor and the low
levels of technology utilized by both shifting cultivators
and colonists. Thus, let household labor endowment be L,
and consider two sets of choices, indexed l and r, reflecting
the land-clearance and rotational phases of agriculture,
respectively. Let li and wi be the leisure and work
associated with phase i, iA(1,r), where l represents the
land creation phase and r the rotational phase. Then, the
constraints imposed by the labor endowment (L) available
to the household are
w1 þ l1 ¼ L and wr þ lr ¼ L
Note that the endowment of household labor remains
constant (Holden, Pagiola, and Angelsen 1998), an
assumption that may be relaxed. With a householdwelfare function, U, defined on leisure and food, s, as in
Walker (1999), the forest-dynamics problem may now be
stated as maximizing, subject to the constraint on labor:
Ws ¼ max
l1 ;lr ;s1 ;sr
a1
X
t¼0
bt Uðl1 ; s1 Þ þ
1
X
bt Uðlr ; sr Þ
ð2Þ
t¼a
This problem can be solved by choosing the values of
leisure and subsistence that optimize welfare. However, it
both simplifies and facilitates the representation of landcover change to manipulate the problem to produce
control variables directly reflective of the change process
itself, which consists of a number of forest-clearance
episodes of some magnitude and subsequent rotations
through the plots so created. In particular, work requirements are governed by age of the secondary vegetation
utilized, a, and the amount of land cleared for each plot, r,
or the deforestation event magnitude (Dvorak 1992, 810).
Hence, let w1 5 rg(N) and wr 5 rg(a), where the unit area
cost function, g, is an increasing function in a and N is
‘‘age’’ of mature forest (N4a). In addition, total, singleperiod food production, s, is given as s1 5 rf(N) for mature
forest and sr 5 rf(a) for secondary forest, where f is a
natural production function, also increasing in a. Substitution of these functions into (2) yields a reformulated
problem solvable in two variables that describe the landcover change process, namely a and r.7
max ð1 ba Þ U ðL; N;rÞ þ ba U ðL;a;rÞ
a;r
ð3Þ
Recall that under the assumption that plots are used for
only one year, the age at which secondary vegetation is
slashed, a, is equivalent to the number of deforestation
events observed on the property. Hence, such a reformulation has the effect of transforming both the number and
size of deforestation events into endogenous variables, the
values of which are functions of household attributes and
site conditions such as size of family workforce, discount
rates, and productivity of the resource base.
The solution for a, interpreted as a rotation time,
mimics the classical forestry problem, stated by Faustmann, that identifies an optimal rotation period for cycles
of tree-planting and harvesting (Hirshleifer 1970). Barrett
(1991) has pointed out that the shifting cultivator’s
problem is similar to the forester’s, and Walker (1999) has
directly applied the principles of Faustmann—as developed by Mitra and Wan (1985)—to shifting cultivation.
The cycling of secondary vegetation, with its potential for
food production, is analogous in behavioral terms to the
cycling of trees with their market potential.8 The present
formulation differs by allowing for an initial period of land
clearance and by introducing the effects of family and site
characteristics on the optimal rotation period.
Dvorak (1992, 812) points out that shifting cultivation
can involve the simultaneous management of plots at
various states of a cropping cycle (as opposed to the use
and abandonment of a single plot every year). In such
a situation, the relative labor requirements for weeding
and clearing become important. The allocation of time
between such activities can be incorporated into the
present framework by allowing for a cropping cycle that
involves a choice variable for the number of consecutive
years that a plot is used before abandonment (see
Appendix A).
Landscape Change of the Amazonian Frontier
The Colonist Farmer
The formulation presented up to this point reflects longterm shifting cultivation in which plots are rotated over an
infinite time horizon with no change in crop types or
responses to altered economic conditions. As has been
argued, shifting cultivation often represents a premarket
phase in the evolution of a family’s farming system toward
cash-oriented agriculture, as with the development of
cattle-ranching by small producers in Amazonia. In such a
situation, initial deforestation occurs as in the preceding
case, followed by rotations through the deforested land and
use of regrowth in a final pass to create the pastures, which
are then stocked. A similar dynamic unfolds in the creation
of perennials plantations, or diversified systems with mixes
of pasture and perennials. The objective of the family may
thus be represented by a model with periods of deforestation, shifting cultivation, and market-oriented activity
based on a ranch or perennials plantation.
Market involvement can be regarded as a choice for
which the contemporaneous alternative is subsistence
production (Vance and Geoghegan forthcoming). Here,
however, entry into the market is taken to be a stage in
the development of a farming system, as portrayed by the
narrative of the colonist farmer. Of course, if the family
expects to someday engage in market activity, even as it
starts out with a subsistence-oriented system, the shift to
market activity is not unexpected, but planned. Thus, the
colonist of the model statement is taken to be an
individual with strong expectations and foresight, not
someone who reacts myopically to unexpected changes in
circumstance.
As before, the goal is to optimize welfare. Now,
however, a change in household structure is introduced.
Let this occur at t 5 y, involving both additional household labor and new preferences as represented by the
utility function. As children enter the household workforce, the early labor endowment, Le, changes to the
mature endowment, Lm. Evolving preferences, in turn, are
captured by the new household utility function, Uy. A
general form of the maximization problem as in equation
(2) cannot be explicitly stated, since the relative magnitude of a and y are not known a priori. Thus, consider the
case in which the optimal value of a is known and is less
than y (e.g., a 5 3 and y 5 10). The colonist household’s
overall utility may then be written as
Wc ¼
a1
X
bt Uðl1 ; s1 Þ þ
t¼0
þ
1
X
t¼y
2a1
X
t¼a
bt Uy ðlm ; qm Þ
bt Uðlr ; sr Þ þ
y1
X
bt Uðlr ; sr Þ
t¼2a
ð4Þ
385
Two points can be made about this summation that
distinguish it from the shifting cultivator’s problem in
equation (2). First, the production of subsistence goods,
with labor endowment Le, is as before in the land creation
and rotational problem, but at t 5 y the productive basis of
the household economy shifts as labor supply changes to
Lm. The household moves from a technology dependent
on ‘‘nature’’ to a neoclassical apparatus, with a production
function and markets for labor and an output, q (Barnum
and Squire 1979; Singh, Squire, and Strauss 1986; Ellis
1993). 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 in
the utility function. The second distinction resides in the
new utility function, Uy , which represents new preferences 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 of
multiple cycles, which could be explicitly incorporated
into the model. Nevertheless, multiple cycles would
probably extend the transient phase of shifting cultivation
beyond what is actually observed among colonists in the
Amazon basin. In addition, direct conversions of tropical
forest to perennials and pasture occur among small
producers without an intervening period of shifting
cultivation (e.g., Scatena et al. 1996; Walker et al.
1997). The goal, however, is to represent the empirical
case as described and observed—namely, the transient use
of secondary regrowth as a phase of farm development.
It is important to note that the ‘‘arrival’’ of markets for
goods and labor is taken to be exogenous to the model,
which only represents the behavior of the farmer based
on his or her beliefs. The farmer may have a precise
expectation about when development will finally occur, as
with the construction of an all-weather highway (CAT
1992). 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, given
scale economies, and sets out clearing land hoping that
‘‘things work out.’’ In this later case, a remains a choice
variable and the farmer’s expectation is weaker, only that
the opportunity for market participation will be in place by
the time he or she is ready. The institutional environment
may change more slowly than anticipated, or fail to
develop altogether. What matters for the land-cover
model, however, is the farmer’s expectation and derivative
behavior.
Just as with the pure shifting cultivator, the colonist’s
problem involves the maximization of utility over a
multiperiod time horizon, as dictated by function (1). The
process of maximization is different, however. In particular, the shifting cultivator’s problem is comparable to
386
Walker
that of the subsistence household first considered by
Chayanov, in which utility maximization and the production decision are nonseparable due to the absence of
markets. The household consumes exactly what it
produces and balances the gain from production against
the immediate loss of leisure. What guides the decision is
the household’s disposition toward labor and associated
drudgery, largely a function of the relative balance of
consumers to workers in the family.
The colonist starts out this way, but anticipates that by
the time he or she is ready for exchanges of labor and
products in the market, institutional change will have
transformed opportunities for economic behavior. Now,
the colonist first maximizes the value of production. Then,
with the income so earned functioning as a ‘‘budget’’
constraint, the household consumes according to its
preferences in order to maximize overall well-being, or
utility. Technical details of the maximization of the
colonist problem are given in Appendices B and C.
This section has demonstrated how the statement of an
optimization problem, founded on received theories of the
agricultural behavior of households, can be transformed
into a description of a land-cover change process—namely,
the clearing of forest and the implementation of permanent agriculture, via shifting cultivation. The following
section translates this conceptual apparatus into a notion
of forest dynamics, gives output from a simulation using
actual field values, and shows how land-cover-change
processes 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 and
colonist systems can be depicted relying on assumed
solution values for the key endogenous variables, a and r.
Let the number of deforestation events be four and the
deforestation event magnitude be 10 hectares (a 5 4;
r 5 10). Then, land creation occurs in both systems during
the initial four years, with 10 hectares of deforestation in
each year, stabilizing at 40 hectares cleared at the start of
year 3. Figure 2 depicts this process, which is identical for both systems. Deforestation processes such as
this hypothetical one have been empirically observed and
so depicted (Homma et al. 1993; Vosti, Witcover,
and Carpentier 1998a, 1998b).
The secondary regrowth graph is different for the two
systems, however, as shown in the lower panel of Figure 2.
In shifting cultivation, secondary regrowth builds to
30 hectares and remains so thereafter, given that three
garden areas are continuously in fallow (Walker 1999). For
Figure 2. (A) Deforestation process at lot level. (B) Land in
secondary regrowth.
a farming system with conversion to permanent plantations 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 of
Figure 2 can be obtained through numerical methods
(Powell 1978; SAS 1995; Miller 2000; see Appendix C.)
Because they are ‘‘control’’ variables for the maximization
problem confronting the colonist, they can be calculated,
once the problem has been specified and parameterized, to
provide a description of the forest loss process. After land
clearance has run its course, the magnitude of deforestation is the product of the amount of land cleared with each
deforestation event (r) and the number of these events
Landscape Change of the Amazonian Frontier
(a). Long-run deforestation occurs under shifting cultivation in spite of the fact that succession leads to regrowth.
For the Amazon basin, such regrowth is typically relatively
young and not at all suggestive of primary forest cover.
Figures 3 and 4 give results of numerical applications to
the case of a colonist farmer developing a small ranch,
under various assumptions about household structure and
economic conditions (Appendix C). Figure 3 describes the
combined impacts of differing discount rates (represented
by the beta values) and magnitudes of household labor. In
general, increases in the household endowment of labor
lead to increases in the amount of deforestation, a
completely intuitive result. Regarding the impact of
discount rates, high beta values (or low discount rates)
tend to increase the amount of deforestation. This result
may seem counterintuitive at first, given the received
wisdom that high discount rates promote excessive rates of
resource exploitation (Clark 1976). In the present case, it
would appear that reduced discounting accentuates the
importance of future ranching values, with associated
demands for land (cf. Angelsen 1999).
Figure 4 brings out the relationship between wages,
commodity prices, and deforestation for varying degrees of
discounting. The price situation is represented by the
relative price for labor, which is actually the ratio of the
monetary wage and beef prices (see Appendix C). Hence,
an increase in this relative price can represent an
increasing wage rate or a decreasing price for the sold
good. The data were generated assuming a labor endowment to the family of 6,000 hours. As can be seen, the level
of deforestation diminishes with increases in the relative
price of labor, presumably due to the relative increase in
the cost of agricultural production, as well as the appeal of
working off-farm (Angelsen 1995).
The price values range between 0.15 and 3.0, with 0.3
taken as representative of current conditions (Appendix C).
Figure 3. Deforestation and family labor.
387
Figure 4. Deforestation and wage rate.
Hence, the order of magnitude difference between 0.3
and 3 reflects wage variation between the frontier setting
and what might be expected in a so-called developed
country, assuming the commodity price remains constant.
In this regard, the figure shows that increasing labormarket remuneration may not have much of an impact on
colonists with higher discount rates, as the associated
graph (beta 5 0.70) shows little slope over the range
presented. Higher expected wage rates tend to lower
deforestation for colonists who place a high value on
future welfare, probably because labor-market rewards are
not available until later in the farm-creation process. It
seems likely that such individuals would be poorly
represented in the population at large. Alternatively, with
wages constant, decreasing beef prices (increasing relative
wage rate) reduce the amount of deforestation, marginally
for those with high discount rates, but appreciably as the
rate goes down.
In general, the magnitudes of deforestation calculated
are consistent with empirical observation. As discussed,
colonists along the Transamazon Highway in Pará received 100-hectare properties from the government. To
date, individual clearings on holdings that have not been
consolidated into large enterprises show continued presence of primary forest, meaning that deforestation has not
exceeded 100 hectares since initial colonization, which
began in the 1970s. Although Brazilian law has set various
limits on the permitted amount of deforestation for
individual holdings (e.g., 50 percent), many clear forest
with little concern for legal enforcement, given the
remoteness of their locations. Nevertheless, the model
results suggest that colonist families are not likely to clear
much land given household constraints, at least in the first
generation of land occupiers.
An additional conceptual step enables a translation of
these deforestation behaviors into spatial outcomes. In
388
Walker
particular, if household characteristics are taken to be
randomly distributed in a colonization space defined by a
development highway, a set of exogenous settlement
roads, and surveyed properties, then a and r are also
random variables given they are functions of household
characteristics. If the colonization space is placed in GIS
software, realizations of a and r can be distributed
accordingly by an ARC-INFO aml routine. Real-time
depiction of the deforestation process at regional scale may
then be visualized, as deforestation events occur on
individual lots, expanding each clearing with a propertyspecific, deforestation-event magnitude.
Figure 5 presents output from such a cartographic
model (Tomlin 1990) after ten steps in the simulation
process. Individual lots are occupied in sequence, reflecting a process and a rate of colonization, so deforestation
starts sooner on properties closer to the development
highway than those farther away. The gray rectangles
aggregate the deforestation for individual holdings to the
left and right of five settlement roads running north and
south from a development highway such as the Transamazon (BR-230), which runs east and west in the
diagram. The template of the figure is in rough agreement
with the settlement geometry laid down by INCRA, with
settlement roads spaced 5 kilometers apart and properties
of 100 hectares shaped in rectangles of 400 m 2500 m.
The figure ignores the disposition of lots along the main
axis of the Transamazon Highway, which lie at right angles
to those on the roads behind them.
Note that the settlement roads and development
highway constitute a so-called fishbone. Hence, the
cartographic model yields a fishbone pattern of deforesta-
Figure 5. Stochastic simulation of colonization ‘‘fishbone.’’ Development highway (e.g., Transamazon) runs east and west; five
settlement roads run north and south. Individual lots are the
elongate rectangles organized in stacks between the settlement roads.
Randomized deforestation on individual lots is given as gray area
within the lots. Results are for a ten-step stochastic simulation using
ARC-INFO aml in prototype colonization space.
tion, which is the spatial aggregate of the many individual
clearing activities. Because the as and rs are distributed
randomly, such a simulation cannot be expected to
reproduce an accurate account of deforestation at the
lot scale. Nevertheless, the overall pattern of fragmentation is consistent with the vaguely pyramidal form
observed along settlement roads in the eastern sector of
the Amazon basin (Walker, Moran, and Anselin 2000).9
Conclusions
This article has presented a model that unifies the work
of Chayanov with household production theory. In so
doing, it advances a microscale explanation of deforestation that explicitly integrates demographic phenomena
and market conditions. In addition, it provides an empirical description of the way that deforestation actually takes
place on smallholder properties in the settlement frontiers
of Amazonia. In particular, deforestation is a response to
the farm-creation process, which involves a sequence
of phases moving from shifting cultivation to some form
of permanent agriculture. Unlike most pre-existing efforts
at modeling the system of shifting cultivation, critical
variables in the present formulation describing the
land-conversion process are endogenous to household
structure, and the farming family enjoys direct yearly
consumption of farm production in the absence of
financial institutions.
The model was solved numerically, yielding values of
deforestation magnitudes at the lot level, and randomization of the key land-cover variables (a and r) enabled the
translation of lot-level processes into an aggregated
pattern of fragmentation, thereby producing the fishbone
pattern of deforestation observed widely throughout the
Amazon basin. Error assessment was not conducted on
these spatial results (Pontius 2000), given that the
cartographic component of the framework was presented
only in prototype. Nevertheless, refinement of assumed
parameter values could lead to a spatial representation of
real-time deforestation dynamics in the Brazilian Amazon,
at least that part associated with colonization. Such an
extension awaits further development.
The deforestation results generated by the various
parameter settings varied between about 20 and 90
hectares. This is consistent with magnitudes observed in
at least one colonization frontier in the Eastern Amazon.
Farming systems of smallholders along the Transamazon
Highway in Pará show between 29 and 69 hectares of
deforestation (Walker, Perz, et al. 2002, 194). The lower
amount of deforestation is observed among subsistence
farmers (n 5 75), while perennial systems with cattle
Landscape Change of the Amazonian Frontier
(n 5 11) show an average of 69 hectares of land
deforested, despite Brazilian law regarding maximum
allowable land clearance. The farmers in the sample also
possess attributes comparable to the idealized colonist of
this article. Although the data are cross-sectional and do
not capture developmental dynamics, some of the households consume all their production and use only family
labor, while others are profit-oriented, with little reliance
on the family workforce or home production for food. A
little over a third use no off-farm labor and receive little
or no income from off-farm activities (Walker, Perz, et al.
2002).
The Transamazon data (Perz and Walker 2002; Walker,
Perz, et al. 2002) cannot establish a temporal link between
the subsistence and market-oriented systems, but the
theory presented would suggest that the subsistence
households will one day become market-oriented, a
hypothesis that could be addressed by future research.
Be this as it may, subsistence farmers in the Transamazon
sample have been on site for 8.5 years, less than the period
of residence for those associated with the perennials with
cattle system, which is 11.3 years on average (Walker, Perz,
et al. 2002, 189). On the surface, this would seem
consistent with the theoretical framework, which suggests
that market involvement follows subsistence-oriented
production.
One shortcoming of the current formulation is the
conceptualization of the lifecycle of the household and
family labor. In particular, a single generational process is
assumed, with a fixed amount of family labor, at least in the
simulations. Nevertheless, it appears that generational
dynamics come into play on some properties, as children
age and start their own families on the holdings of their
parents (Perz and Walker 2002). Brondizio and colleages
(2002) identify pulses of deforestation on individual
properties suggestive of such a phenomena, and Perz and
Walker (2002) show that land-cover dynamics of secondary vegetation are affected by generational extension on
individual holdings. CAT (1992), Walker and Homma
(1996), McCracken and colleagues (1999), and Perz
(2001) argue that the domestic cycle of the individual
colonist household is critical in explaining switches from
subsistence farming to perennials and ranching. Nevertheless, the time it takes to implement a commercially
oriented system, as predicted by the model, appears to be
less than ten years, which would seem to weaken such
‘‘first-generation’’ effects.10 Of probably greater importance to the long-run disposition of a property’s land cover
is the generational phenomena as described, for there is
little reason to presume clearing will not start up again
if children remain on the farm and begin families of
their own.
389
It is important to note that the present model is limited
to a single type of agent—namely, the colonist farmer. As
such, the picture presented is a partial one at best,
especially with respect to the overall mechanism of landcover change in the Amazon basin, which involves a
multiplicity of agents. Besides the colonist, at least two
additional agents are critical to understanding Amazonian
deforestation: large ranchers, or fazendeiros, and loggers.
Much has been written about the role of large ranches
in Amazonian deforestation, and in particular about how
government incentives lured capital to the forest frontiers
(e.g., Hecht 1985). There can be little doubt that despite
their relatively few numbers, given the skewness of
land distribution throughout Brazil (Simmons 2002),
large ranches have had a substantial impact on landcover change at basin scale. The actual amount remains
an empirical question, although it seems clear that the
relative importance of small- versus largeholders in
the landscape dynamic at the regional scale is a function
of time and place (Walker et al. 2000). Be that as it may,
fully understanding Amazonian deforestation will require
an understanding of fazendeiro behavior and better insight
into process linkage between small- and largeholder land
occupation (Wood 1983; Ozório de Almeida 1992; Rudel
2002).
Another critical linkage to comprehend is that between
loggers and colonists. While colonists account for
‘‘infilling’’ of the landscape with agricultural activity
once they occupy their parcels, they themselves may not extend the frontier through road-building activities. After
the development highways are in place, the expansion of
the system is often the outcome of an interaction between
loggers and farmers, both in Amazonia and elsewhere in
the tropical forest biome (Leslie 1980; Myers 1980;
Schmithusen 1980; Office of Technology Assessment
1984; 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 the
Transamazon, roads may be extended after initial colonization due to the demand for land arising from late
arrivals, and without much logger involvement. In such a
situation, the fishbone pictured in Figure 5 (and assumed
in the cartographic model) is not entirely exogenous to
colonization, and an appreciable amount of the highly
regular network pattern may be mainly attributable to
smallholder actions. In at least one part of the Transamazon Highway in the eastern sector of the Amazon basin,
the extension of settlement roads beyond the initial spur of
6 kilometers (Simmons 2002) occurred when newcomers
demanded that government expand the existing road
network and demarcate new lots for settlement. The
390
Walker
government did so, and extended the network to about
20 km on both north and south sides of the highway,
preserving the geometry of the original plan allowing for
100-hectare lots, and a 5-km spacing between the roads.
Be this as it may, a general behavioral model of landscape
evolution for forest frontiers awaits empirical description
of interactions between colonists and loggers, and an
account of the spatial decision-making of the roadbuilders, whoever they are.
Although rural out-migration has been pronounced
in the Brazilian Amazon in recent years (Perz 2000),
the colonists who remain in place continue to change the
cover 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 change
can be expected to intensify, given plans by the Brazilian
government to continue with its road-building and
infrastructure improvements (de Cassia 1997; Laurance
and Fearnside 1999; Nepstad et al. 2001). Thus, there will
be a continued need to understand colonist behavior and
the 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 in
which age of fallow can be freely chosen. This amounts to
defining a system based only on the second part of function
(2), or:
1
X
bt U ðlr ; sr Þ
ðA1Þ
leisure and subsistence are ultimately functions of other
variables: namely, fallow length (a), size of plot (r), and
cropping cycle (y). Consequently, solutions for these
three variables yield the optimizing choices of leisure
and subsistence. As indicated, the empirical setting is
such that cropping cycles are very short, and taken as 1.
Consequently, sufficient conditions for this case (the
model assumption) are established in the following claim
by specifying functions for weeding, regrowth, and utility.
Claim: For fixed plot size r, conditions exist under which
some single-year cropping cycle is preferred to an arbitrary
multiyear cropping cycle.
Proof: The claim may be established by demonstrating
that, for some multiyear cycle, there exists a single-year
system generating greater utility than the multiyear
system in each year of the cycle. To simplify the exposition,
assume h(i) 5 0 for i 5 0, and let the number of plots
in 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 practical
significance of the latter two inequalities is that weeding
costs per unit area always exceed the cost of clearing a new
plot under single-year use (with b plots) after the first year,
and that fertility decline in the cropping cycle quickly
reduces productivity below the single-year case, also after
the first year. Assuming these conditions, the single-year
system produces greater utility for some arbitrary plot size r
in years beyond the initial year, since there is more leisure
and food available.
To establish the claim, it is then only necessary to
identify conditions that yield higher utility for the singleyear system in the initial year. This may be accomplished by
specification of the utility function. For fixed r, subsistence
production may be higher in the multiyear system in the
t¼0
where the rotation subscripted as r begins in the first
period. With a cropping cycle of y years (41), the
optimization problem is to maximize, through yearly
choices of leisure and subsistence, the objective:
max
1 X
y1
X
l0 ...;ly1 ;s0 ...sy1
btyþi U ðli ; si Þ
ðA2Þ
t¼0 i¼0
subject to
wi þ li ¼ L
ðA3Þ
w0 ¼ rfg½ða 1Þy þ hð0Þg
ðA4Þ
wi ¼ rhðiÞ
ðA5Þ
si ¼ rf½ða 1Þy; i
ðA6Þ
where a weeding function, h, has been added to the
constraint set.13 Note that, as in the previous problem,
Figure A1.
Landscape Change of the Amazonian Frontier
first year due to older fallow (if (a 1)y 4 b); assume that
this is so. Because older vegetation is used, however, more
work is required, given site preparation costs (slash and
burn), allowing for less leisure. Hence, whether the initialyear utility under multicropping is greater than the singleyear system depends on the nature of the utility function.
Figure A1 demonstrates that a utility function exists
yielding higher utility for the single-year system. A similar
graph can be depicted for the case in which b4(a 1)y.
Appendix B
The problem posed differs from previous statements of
the household economy framework (Thorner, Kerblay,
and Smith 1966; Singh, Squire, and Strauss 1986) by
combining nonseparable and separable components. To
shed light on solutions, assume no change in household
labor endowment, as in the shifting cultivator case. For the
separable part of the problem (e.g., the fourth term of
summation [4]), single-period optimization has been well
studied and involves an initial profit maximization,
following by utility maximization, or a recursive solution
(Singh, Squire, and Strauss 1986, 7). The interest here is
to demonstrate that overall optimization of the combined
problem requires maximization of the separable problem
for all following periods. This result is necessary for the
computational algorithm implemented in Appendix C. In
addition, it unifies the Chayanovian and market-based
model.
Household Profit Maximization. If labor endowment
remains fixed, and if new preferences are activated as soon
as the switch from subsistence to market production takes
place—as in the simulation—the colonist objective
function can be stated with two terms, including an initial
term for the utility of land creation and shifting cultivation
phases (G(r,a)), and a second term for the utility
associated with consumption based on commercial
production (H(r, a, l, q)). Thus, G represents the
discounted welfare stream for both the land-creation
and the rotational phases, and H is that for farm or ranch
implementation that comes afterwards. The colonist
problem is to maximize the sum (Wc 5 G(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 maximizing
H(r, a, l, q) subject to q 5 q(A, w) for fixed values of r
and a (A 5 ra). Here, q is a production function
for the agricultural output, dependent on land (A) and
labor (w).14
Given values for r and a, and hence A, the conditional
problem is that of optimizing the discounted stream of
391
utilities associated with consumption after commercial
activities begin. Since a unique pair of lm and qm optimizes
single-period utility given appropriate assumptions on the
utility function (Singh, Squire, and Strauss 1986), this
same pair optimizes the discounted stream of utilities, in
which case a solution exists. Given the presence of market
opportunities for labor and output, utility maximization—
and, hence, the solution to the conditional problem—for
some fixed amount of land A and wage rate v is recursive
and depends on the solution to the maximization of profits
(Singh, Squire, and Strauss 1986), or
max q v w hðdÞq
ðA7Þ
subject to q ¼ qðra; wÞ
with A fixed (Figure B1) and q as numeraire. Here, h(d)
represents the presence of transportation costs modeled
with an ‘‘iceberg technology’’ as a function of distance of
the production site to some market or transshipment point
(Samuelson 1952; Nerlove and Sadka 1991; Fujita,
Krugman, and Venables 1999).15 Assume v is sufficiently
large that w r L, the household labor endowment (Figure
B1). The profit maximization solution defines a ‘‘budget
constraint’’ for the optimization of household welfare,
some q 5 y 1 vw 5 y 1 v(L l), where y 5 q 0 vw 0, and
(q 0 ,w 0 ) solves the profit maximization problem (see Figure
B1).16 Hence, for arbitrary values of r and a (and A), we
have unique utility maximizing values of lm and qm, under
Figure B1. The production function may be interpreted as
qt 5 q h(d)q 5 [1 h(d)]q. Profit, in turn is qt vw, where q is
numeraire. Wages may be a function of distance (Angelsen 1994).
The income, or ‘‘budget constraint,’’ is q1vl 5 y1vL, where (L w)
is leisure.
392
Walker
appropriate assumptions on the utility and production
functions, and H is maximized conditional on values for r
and a. Note that lm and qm may be regarded as functions of
r and a, as are the choices for labor and subsistence in the
land-creation and shifting-cultivation phases. Consequently, the empirical process is identifiable in the overall
model structure as before, with deforestation events and
magnitudes. Note that the household hires nonfamily
labor if wmow 0 ; otherwise, family members work off-farm
in addition to pursuing their agricultural activities.
The relationship between the unconditional and the
conditional problems is stated in the following claim
regarding necessity. In particular, some set of values (r*, a*,
l*m, q*m) solves the unconditional problem only if l*m (and
* ) solve the conditional problem, given (r*, a*). In other
qm
words, global optimization of household utility over all
periods occurs only if household profits (and utility) are
maximized in the third phase. Proof: The unconditional
problem is to maximize G(r, a)1H(r, a, l, q) through the
choices of r, a, l, and q. Recalling the definition of Wc, let
* , qm
* ), such that Wc(r*, a*, lm
*,
there exist a vector, (r*, a*, lm
* )ZWc(r, a, lm, qm) for all feasible vectors, (r, a, lm, qm).
qm
Assume further that this vector does not solve the
conditional problem. Hence, there exists some (l**
m , q**
m ),
*
*
*
*
**
**
distinct from (lm, qm), such that H(r , a , lm , qm )Z
* , qm
* ). Consequently, Wc(r*, a*, lm
**, qm
**)
H(r*, a*, lm
* , qm
* ), which contradicts the hypothesis,
ZWc(r*, a*, lm
and optimization in the commercial period, dependent on
profit maximization, is necessary to the global solution.
Appendix C
The algorithm works as follows. At iteration k, let the
vector of solution variables be Xk (e.g., Xk 5 [ak, rk]),
which implies knowledge of the gradient vector at
that point, 5 f(Xk). Let the estimated inverse of the
Hessian matrix, also known, be Bk. The Xk11 vector is
then Xk11 5 Xk Bk 5 f(Xk), which can be used to
develop an equation for obtaining Bk11. Note first that
with Xk11, 5 f(Xk11) is given. Then Xk11 Xk
5 Bk11[ 5 f(Xk 1) 5 f(Xk)] defines Bk11, and the
algorithm proceeds to the next iteration.
To deal with the integer problem, the value of a is
constrained to a set of integers (Mitra and Wan 1985) and
solved for the deforestation event magnitude, r, searching
for the optimal choice of a and r by reference to the value of
the objective function. It is possible that the solution
values represent local maxima, since a global assessment of
solution values is not undertaken and second-order
conditions are not assessed. However, the optimal
a values are either 3 or 4, which is consistent with
the observed values of age of secondary vegetation in the
study area.
The natural production function, f(a), is specified to
track the functions of biomass in time suggested by Table 4
(Brown and Lugo 1990; Lucas et al. 1993). The functional
forms are normalized in nutritional units. An asymptotic
value of grams-protein per hectare is assumed for primary
forest calculated as in Walker (1999), yielding 27 kg/ha per
year of protein production from rice and beans, grown in
separate fields in fixed proportion (Moran 1981). The cost
function, g(a), is taken as a monotone increasing function
in a, using the data from CAT (1992), with a y ‘‘intercept’’
of 220 hours (E37 days) indicating the time requirement
for age-insensitive operations (e.g., harvest, bagging, etc.).
An asymptote of 450 (E75 days) is adopted, reflecting
the 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 number
may be on the low side given data from EMBRAPA. CobbDouglas 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 for
herd viability. This is set at 20 hectares, which supports
between 10 and 20 animals, given regional stocking
densities (Fearnside 1986; Mattos and Uhl 1994). The
production function is scaled by observing land productivity for ranches in the region, which yield about 0.50 kg
per hectare per year for self-reproducing herds (Mattos
and Uhl 1994).
The wage rate for labor power traded against beef
production is given as follows. First, the minimum wage
rate in 1996 was 136 reais per month, yielding about 0.81
Real per hour, assuming a forty-hour work week. Field
interviews in 1996 indicated a typical daily wage (diario) to
be 10 reais, a number consistent with the minimum wage.
Hence, the agricultural wage rate is about U.S.$0.50 per
hour. Given an average ‘‘meat on the hoof’’ price of $0.65
per kg prior to 1994 (Mattos and Uhl 1994), the relative
price of labor in a two-commodity (beef and labor power)
universe, with beef as a numeraire, is between 0.30 and
0.40 kg per hour in the late 1990s, assuming beef prices
held their value against labor after the devaluation of the
Real in 1997.
Appendix D
The sample mean for the distribution of the age of secondary vegetation utilized by colonists (number of
deforestation events, by the model) is 3.26 years, and its
standard deviation is 2.52 (unpublished data for the
Transamazon Highway; n 5 261). The deforestation
event magnitude was taken as a uniform random variable
Landscape Change of the Amazonian Frontier
distributed between 2 and 10 hectares (see Homma et al.
1993; Dale et al. 1994). The time lag for process initiation
was assumed to be one year for contiguous lots.17
Acknowledgments
This research was mostly supported by the National
Aeronautics and Space Administration under the project,
‘‘Pattern to Process: Research and Applications for
Understanding Multiple Interactions and Feedbacks on
Land Cover Change’’ (NAG 5- 9232), and by the National
Science Foundation’s Geography and Regional Science
Program, under the project ‘‘Tenure Security and Resource Use in the Amazon’’ (SBR-95-11965). I owe a longstanding debt of gratitude, however, to Ariel Lugo of the
U.S. Forest Service, who funded my first work in Brazil,
and to Adilson Serrão of EMBRAPA/CPATU (Brazilian
Agricultural Research Center/Centro de Pesquisa Agroflorestal da Amazônia Oriental) who through the years has
provided me with a great deal of institutional support. I am
also grateful to my Brazilian colleagues, who have been
patient with me in the field and taught me many things,
including Arnaldo José de Conto, Alfredo Kingo Oyama
Homma, Marcellus Caldas, Luiz Guilherme Teixeira Silva,
Pedro Mourão de Oliveira, and Eugenio Arima. Conversations with Stephen Perz, Charles Wood, and Cynthia
Simmons have also proved extremely useful, and I am
indebted to them as well. Scott Drzyzga made a critical
contribution to the article by developing the cartographic
model used to illustrate the spatial articulation of colonist
land use. Although the manuscript was greatly improved
by a number of careful anonymous reviewers, I remain
responsible for any outstanding errors.
Notes
1. The ‘‘new economic geography’’ calls for the use of
‘‘computer-assisted thinking’’ and numerical techniques as
supplements to analytical results (Fujita, Krugman, and
Venables 1999, 8). As Veldkamp and Lambin (2001) note,
modeling allows us to conduct experiments to test our
understanding of land-use change processes, and to describe
them quantitatively.
2. For an opposing view, see Dove (1986) who notes that
‘‘inefficient’’ use of land is likely to generate high labor
productivity, 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 farmers
rotating a fixed number of fields (the Amazonian colonist
case) and nomadic groups that shift their residential base
with soil depletion. A ‘‘swidden’’ is Old English for ‘‘burned
clearing.’’ See, for example, Ruthenberg (1971), Peters and
Neuenschwander (1988), and Unruh (1988).
393
4. It is important to note that the term ‘‘pasture,’’ as used by
Amazon colonists, does not reflect our common understanding. In particular, colonists often broadcast grass seeds
on deforested land, then abandon it until such time as they
can afford to begin stocking with animals, at which point they
return and clear the land again. Thus, any regrowth seeded
with grass is referred to as ‘‘pasture,’’ despite its ecological
characteristics. Figure 1, therefore, shows a simultaneous
process of deforestation (loss of mature forest) and regeneration of secondary regrowth that is finally converted to
permanent use. It also demonstrates the change in the
relative importance of the landscape components, which
are mainly annuals in the beginning before shifting to a
preponderance of pasture and perennials.
5. Values may appear identical in the Rondônia and Acre sites.
They were reported as such because difference of means tests
could not reject the null hypothesis of equal values.
6. Colonists often have some access to capital and labor
markets. I abstract from the household economy approach
of Singh, Squire, and Strauss (1986) to focus on the landcover dynamic, and mainly assume a Chayanovian, subsistence environment (Nakajima 1969). Nevertheless, in a large
sample of colonist farmers (n 5 261) on the Transamazon
Highway in the eastern sector of the Amazon, only 49 percent
used off-farm labor in 1996 (Walker, Perz, et al. 2002).
Chayanov (Thorner, Kerblay, and Smith 1966) developed his
family-labor theory in a setting in which 90 percent of the
households were detached from labor markets.
7. The denominator resulting from the geometric series
expressions has been eliminated, given that it is constant
and does not affect the solution. Note that the U functions
remain the same, although in equation (3) the arguments are
changed. Both labor and subsistence in equation (2) are
functions of these new arguments.
8. The present model advances the formulation in Walker
(1999) by allowing for an initial cycle of land clearance, critical
to the description of tropical deforestation. Note that the
solution value, a, can be used to characterize a system in terms
of measures provided by Ruthenberg (1980) and Boserup
(1981), given that agriculture occurs for only one year on an
active plot. Ruthenberg’s R value is 100/a, essentially the same
measure as Boserup’s (1981, 19) ‘‘frequency of cropping.’’
9. The settlement roads in Rondônia, called linhas, are much
more passable than those in the east, at least on the 300-km
stretch between Altamira and Ruropolis, one of the primary
settlement frontiers in Pará. This may explain the broad
rectangular pattern observed in the western sector linhas,
which stretch for 40 to 60 kilometers between Porto Velho
and Ouro Preto.
10. With a cropping cycle of 1 year and a solution value of a 5 4,
market entry occurs in the ninth year of operations, given 4
years of land-clearance activities and another 4 years using
secondary vegetation. Nine years is consistent with the
length of residency of the ‘‘high-value’’ colonist systems
reported by Walker, Perz, Caldas, and Teixeira Silva (2002)
for the eastern Amazon. Such systems are presumably more
market-oriented than those of neighboring subsistence farmers. Figure 1 suggests an almost complete transition to
market-oriented production (with perennials and pasture)
after twelve years.
11. Fieldwork in the summer of 2000 in both the eastern and
western sectors of the basin revealed the presence of dense
394
12.
13.
14.
15.
16.
17.
Walker
family networks on the individual settlement roads. Distant
parcels were being claimed by parents for children who were
becoming adults. One of the leading loggers and private roadbuilders on the Transamazon Highway in the frontier area of
Pará was originally a colonist himself.
Dvorak (1992) does not provide an explicit solution to this
problem. In particular, the model that is actually solved is for a
single plot in a single period (equations 8–9), despite the
multiplot context of the formulation (equations 1–8). The
multiplot decision is only alluded to verbally (Dvorak 1992,
812). Thus, the present formulation complements Dvorak’s
model by providing an explicit multiperiod solution, with
discounting, a labor/leisure tradeoff, and an explicit representation of the allocation decision across clearing and
weeding activities. Walker (1999) has provided an explicit
solution to the multiplot problem, focused not on labor costs
but on complementary subsistence requirements.
The weeding function is similar to the vegetation-clearing
function. It is assumed that a fixed amount of weeding is
necessary to have production, as a function of the year in the
cropping cycle.
Note that the production function, q, is taken to be freely
available to the farmer. However, it is costly to build a herd or
to start a plantation of perennials. The present formulation
abstracts from such an investment problem to focus on land
creation. However, it could be assumed that seed capital
becomes available to the farmer from the government, or that
the farmer arrives on the frontier with an endowment.
Perhaps the best way to view the problem is that q is a measure
net of loan repayment.
A von Thunian ‘‘accessibility’’ effect has been modeled using
transport costs only for the produced good. For the case of
cattle, some suggest that they can walk themselves to market,
thereby reducing transport costs to the opportunity cost of
the labor that accompanies them. Nevertheless, ranchers do
incur considerable expense through trucking, and the carrreta
is a vehicle specifically designed for this task. It is important to
point out that inputs to ranching may impose accessibility
costs, thereby exerting a von Thunian market (or central
place) effect. In particular, veterinarians and extension
agents may be limited in the area they can reasonably cover
with regularity. See Liu (1999) for a von Thunian interpretation of labor costs in Chinese agriculture. Angelsen (1994)
also assumes a distance dependent wage rate.
By the first-order conditions of profit maximization, the
partial derivative of the production function in labor must
equal the wage rate, and by the inverse-function theorem
(Rudin 1976), we have w 0 5 h(ra, v) and therefore q 0 5 q(ra,
v). Hence, the constraint for utility optimization may be
written as q 5 y(r,a)1v(L lm), where y(r,a) 5 q 0 w 0.
An analysis of ETM1 satellite images for 1999 shows that
settlement roads opened along the Transamazon Highway in
Pará at the rate of 0.77 km per year beginning in the early
1970s. If the time steps are interpreted as one year, the
extension of the system north and south is underestimated,
given the 400-m frontage of individual properties.
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