THE SLASH AND BURN AGRICULTURE

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					          EEE WORKING PAPERS SERIES - N. 14




   THE SLASH AND BURN AGRICULTURE


                        Hugo Loza
Ecological and Environmental Economics Program – ICTP, Italy
 Universidad Mayor, Real y Pontificia San Francisco Xavier de
  Chuquisaca, Centro de Postgrado e Investigación, Bolivia




       This document was prepared on January, 2004.
                               The slash and burn agriculture
                                               Hugo Loza1
                    Universidad Mayor, Real y Pontificia San Francisco Xavier de Chuquisaca
                                     Centro de Postgrado e Investigación
                                    Calle Urriolagoitia 155, Sucre, Bolivia

                                               29 January 2004

Abstract

   We use dynamical equations to represent and analyse the slash and burn agriculture. The slash
and burn agriculture is a production system that is broadly diffused among the peasants who
inhabit the low lands of Bolivia. The study of this production system is important because it has
an unfavorable impact on the farm economy as well as on the environment. Nevertheless, it may
represent only the first stage towards sustainable forms of production. We find that, under certain
conditions, the farm can reach a stable equilibrium. The equilibrium can be reached following
three different kinds of paths, corresponding to three local forms of convergence for the
economy. We associate a slow evolution of the farm with an equilibrium, corresponding to two
real eigen-values in the matrix of the linearized dynamic system; a moderate evolution with
another equilibrium, corresponding to one eigen-value; and an evolution with fluctuations to a
third equilibrium, associated with two complexes and conjugated eigen-values.

Keywords: Developing countries; Agriculture of colonization; Dynamic models

JEL classification: Q1

1. Introduction

  The lack of means of production among some peasants, namely Bolivians of the low lands
settling in the virgin forest, forces them to adopt the only production technology within the reach
of their possibilities. The fast fall of the natural fertility of soil, for the repeated cultivation in the
same parcel, is the cause of this itinerant agriculture. The cost of clearing new parcels, as well as
the burnt of vegetable matter have an unfavorable impact on production costs and the
environment. The slash and burn agriculture constitutes, however, in the historical experience of
the colonization of new lands, only the first stage in the transit towards a diversified agriculture.

  Coming to the zone of colonization, the peasants receive, from the State, an endowment of
woodland that they prepare and sow with the help of manual tools. They usually apply a labour
intensive technology to the culture of rice. The agricultural works comprehend two types of tasks:

1
 Hugo Loza
Tel. + fax (591) 3 343 72 51
Casilla 5079, Santa Cruz
hloza@cotas.com.bo
the deforestation of the plot and the cultivation of rice properly speaking. By deforestation we
mean cutting down the trees and burning the stubbles. A part of the ground is then enabled for
sowing, the cultural works follows and, finally, the rice is harvested. The experience of
colonization shows that the culture of rice without rotation, on the same plot, rapidly exhausts the
natural fertility of soil, and, as a consequence, the yields fall. Because of the failure to control the
multiplication of brush, competing with the principal culture for the soil nutrients, the peasant
prepares a new plot of woodland, abandoning the previous parcel as fallow.

  In this paper, we present a model, which explicitly formulates behavioural assumptions and
considers the specific technical and economic characteristics of the slash and burn agriculture.
We present the main results of the stability analysis, emphasizing the dynamic properties of the
equilibrium. Subsequently, we address the issue of sustainability of the slash and burn
agriculture, in terms of exogenous parameters like market prices and wages. Finally, we illustrate
the possible evolution of a farm in three different scenarios, corresponding to three local forms of
convergence. We associate a slow evolution of the farm with an equilibrium, corresponding to
two real eigen-values in the matrix of the linearized dynamic system; a moderate evolution with
another equilibrium, corresponding to one eigen-value; and an evolution with fluctuations to a
third equilibrium, associated with two complexes and conjugated eigen-values.

2. The model

 We represent the productive process of the slash and burn agriculture by means of the following
dynamic model2.

    The first equation describes the evolution of fertile land surface available for production:

            •
            F = a1H − a2 F                                                                    (1)

  According to (1), the change in this area F ensues from the balance between the new added
parcels H and the exhausted plots put in fallow. We suppose that the areas put in fallow are
proportional to the entire cultivated area, where 0 < a 2 < 1 represents the constant rate of
formation of fallow.

  The second and third equations represent the production function for new parcels. The available
technology does not allow any replacement between factors. It is described by means of a
Leontief production function with constant coefficients, where the new cleared parcels are
proportional to labour L1 and capital (tools) K1 .

                   L K 
            H = min 1 , 1 
                    b1 b2 



2
    Capital letters stand for variables, and small letters stand for positive parameters.

                                                               2
             L1
       H=                                                                                 (2)
             b1

             K1
       H=                                                                                 (3)
             b2

  The fourth equation shows the growth rate of labour devoted to land clearing, according to a
linear decreasing function of the cultivable surface, where c1 represents the growth rate of the
clearing labour in the absence of fertile soil available for production. The threshold parameter c2
could be interpreted as the expected farm size in terms of fertile soil area.

        •
       L1 = c1 (c2 − F ) L1                                                               (4)

 The fifth and sixth equations represent the production function for rice Y, where the farmer uses
two substitutable production factors, the work L2 and the cultivable area F, as well as a third
complementary factor K2 . A Cobb-Douglas function drives the substitution between labour and
cultivable area, where 0 < e < 1 represents the production elasticity with regard to L2 and
0 < 1 − e < 1 , with regard to F.

                             K 
       Y = min d1 Le F 1− e , 2 
                    2
                             d2 

       Y = d1 Le F 1− e
               2                                                                          (5)

            K2
       Y=                                                                                 (6)
            d2

  The last equation (7), establishes the budget balance of the farm, equalling income to expenses
(remuneration of production factors labour L1 + L2 , evaluated at the wage rate w, and the two
forms of capital K1 and K 2 , whose price is normalized to one).

        pY = w ( L1 + L2 ) + K1 + K 2                                                     (7)

3. Analysis of stability

  The qualitative behaviour of the solutions of this model can be put in evidence by means of the
analysis of the variables trajectories and speeds in the space of phases. It is possible to simplify
this analysis by reducing the size of the system trough elimination of an equal number of
variables and equations. Using this procedure we discard the variables in equations (1) and (4)
others that F and L1 . Thus, we obtain the following system of differential equations:


                                                 3
         •
                a1
        F=         L1 − a2 F
                b1
                                                                                                  (8)
        •
        L1 = c1 (c2 − F ) L1

It is a non-linear system. To identify the stationary points, we write the following equivalences:

         •
                                 a2b1
        F = 0 ⇔ L1 =                  F                                                           (9)
                                  a1

        •
        L1 = 0 ⇔ F = c2 ; L1 = 0                                                                  (10)

Thus, we have two equilibrium points:

             F = L1 = 0                                                                           (11)
and
                                 a 2 b1c2
             F = c2 ,     L1 =                                                                    (12)
                                    a1


 To analyse the stability of these equilibrium, it is convenient to refer first, to the phase diagram
presented in Figure 3.1. We restrict the representation to the first quadrant, since only positive
variable values have economic relevancy. In this diagram, the states of the system where the
                                            •
cultivable area is stationary ( F = 0) , correspond to the points on the straight line that departs
from the origin, corresponding to the equation L1 = (a 2 b1 / a1 ) F . Over this straight line
                                                •
( L1 > (a 2 b1 / a1 ) F ) , F increases ( F > 0) , whereas below it ( L1 < (a 2 b1 / a1 ) F ) , it diminishes
 •
( F < 0) . As for the states of the system where the labour assigned to the deforestation L1 is
                •
stationary ( L1 = 0) , these are represented by the axis of the abscissas ( L1 = 0 ) and by the vertical
straight line F = c 2 . To the left of this straight line ( F < c 2 ) , and for L1 > 0 , labour L1 increases
 •                                                               •
( L1 > 0) , whereas to the right hand ( F > c 2 ) , diminishes ( L1 < 0) .

  The two equilibrium states of the system are at the intersection of the straight line of equation
 L1 = (a 2 b1 / a 1 ) F with the axis of abscissas L1 = 0 , and with the vertical straight line of equation
F = c 2 . The evolution of the system in a neighbourhood of these equilibria depends on the zone
where the trajectory lies (Figure 3.1). The system follows the direction of arrows, first, towards
North-East; then, towards South-East; then, towards South-West and finally, towards North-
West. Hereby, the system moves away from the origin, which is, therefore, an unstable
equilibrium. On the contrary, as we will demonstrate later on, the second equilibrium (12) is

                                                     4
locally stable and the system moves, directly or in spiral, towards this point. In this case, labour
assigned to deforestation is not null, the farmer depends on a precarious technology, and is caught
in a scheme of simple reproduction, which does not offer him any progress or improvement.



                                      L1
                                             &
                                             F>0
                                           &
                                           L1 > 0



                                                       &
                                                       F<0
                                                     &1 < 0
                                                     L

                                                              F


                                   Figure 3.1. Phase diagram for system (8)

  We now demonstrate, considering the linear part of the system (8), that the origin corresponds
to a saddle point and, as a consequence, is an unstable equilibrium; whereas the second point is
an asymptotically stable equilibrium. Indeed, the matrix of the linear form associated with the
system (8) can be written as follows:

                            a1      
             −a 2
        A=                  b1
                                                                                         (13)
                                    
           − c1 L1    c1 ( c 2 − F )

The value of the determinant of this matrix at the point of coordinates (0, 0) is:

        DetA = − a 2 c1 c2 < 0 ;                                                          (14)

so, indeed, the origin is an unstable equilibrium. On the other hand, if we calculate the
determinant and the trace of the same matrix at the alternative equilibrium point, we obtain the
following expressions:

        DetA = a 2 c1 c 2 > 0
                                                                                          (15)
        TrA = − a 2 < 0 ;

allowing us to affirm that this equilibrium point is asymptotically stable.



                                                      5
4. The parameters in the stable equilibrium

  We now focus on analysing the impact of changes in parameter values for the economy of the
farm. We shall analyse these changes in the case of stable equilibrium and, to highlight the role
that each of the eleven parameters plays in the model, we introduce the taxonomy of Table 3.1.

  We can distinguish, on one hand, technical - agronomic parameters, corresponding to
characteristics of the production, such as the rate of fallow formation a 2 , the unitary coefficient
of labour for parcel deforestation b1 and the product - elasticity of work e in the rice production
function. On the other hand, there is a second group of technical – economic parameters,
including the unitary expenditure in deforestation tools b2 , the unitary expenditure in capital
inputs and labour for rice production d 2 . The values for these two groups of parameters have
been estimated as averages from a random sample of farms. The price of rice p and the wage rate
w, belong to a third group of economic parameters, whose levels the market determines. Other
parameters assure the compatibility of units of measurement, for example a 1 and d 1 ; the first one
compares the fertility of the soil, for one hectare of new deforested plot, and the second appears
in the rice production function.

                                           Table 3.1
                                  Taxonomy of model parameters

                   Parameter Description                       Type

                   a2        formation of fallow               technical - agronomic
                   b1        labour for deforestation          id.
                   e         labour elasticity of production   id.
                   b2        expenses for deforestation        technical - economic
                   d2        inputs for production             id.
                   p         market rice price                 economic
                   w         market wage rate                  id.
                   a1        compatibility                     dimension
                   d1        id.                               id.
                   c1        velocity of development           decision
                   c2        farm size                         id.

 Finally, we have a scale parameter c2 , corresponding to the farm size, and c1 which is a control
parameter, representing the velocity of evolution of the variables in the system. This parameter
will turn out to be critical for the determination of the eigen-values of the dynamic system (8).

 In the stable equilibrium, the surface of fertile soil equals the scale parameter of the farm:

        F q = c2                                                                            (20)

                                                  6
  The labour preparing the production plots increases with the increase of the farm size. The
labour for deforestation purposes increases also, in the equilibrium, with the rate of formation of
fallow and with the loss of efficiency of the labourers working on deforestation tasks.

         q      a 2 b1 c2
        L1 =                                                                               (21)
                   a1

  As the deforested area is proportional to the labour invested in the corresponding activity, it is
also proportional to the farm size and to the rate of loss of the natural soil fertility:

                 a 2 c2
        Hq =                                                                               (22)
                  a1

  There is an additional parameter to those appearing in the previous equations. It represents the
payment amortization for capital tools services. It depends on factors such as the exchange rate or
tariffs, as well as on the evolution of the industrial technology:

                a 2 b2 c2
        K1q =                                                                              (23)
                    a1

 The following equation identifies a feasibility condition for the slash and burn agriculture,
showing the restrictions on parameter values that allow the existence of positive solutions for
production variables such as labour for cultural works.

          1     ac
               ( 2 2 ( wb1 + b2 ) + wL2 ) = d1c1− e Le
                                               2     2                                     (24)
        p − d 2 a1

                                       63
                                       54
                                       45
                                       36
                                       27
                                       18
                                        9
                                        0
                                            0   16   32    48   64   80   96
                                                          Labour



                    Figure 3.2. Feasibility condition for the slash and burn agriculture




                                                           7
  The expression on the left hand side of (24) represents a straight line, where labour is the
variable, whereas the right hand side identifies a decreasing yield production function. Only the
first intersection of these curves is interesting from an economic point of view (Figure 3.2). This
solution tends to vanish with a falling market price for rice. Indeed, a price reduction displaces
the point of intersection of the straight line with the vertical axis and increases his slope.

  On the other hand, an increasing wage produces similar effects. In the absence of a solution,
however, an alternative interpretation is possible. As the slash and burn agriculture is an activity
of subsistence, we can not discard the possibility of a peasant accepting, at least temporarily, a
remuneration for his labour under the market wage level, to make the agricultural production
feasible.

 Though the market parameters are most susceptible to variations, we must not discard the
possibility of changes in technical - economic parameters. Thus, an increase in the price of
deforestation tools b2 shifts the intersection point of the straight line with the vertical axis in
Figure 3.2, jeopardizing again the feasibility of production. Increases in the price of seeds, or
other inputs, in d 2 generates similar effects.

  Finally, we observe that the slash and burn agriculture, as it happens in reality and as it is
represented in the model, is a surviving activity whose limits are established by the size of the
peasant family. Indeed, we observe that an increase of the scale parameter c2 displaces the
intersection point in Figure 3.2, menacing the feasibility of the farm’s production. This shift is
not compensated by a decrease of the slope, which is kept positive.

5. The evolution of the farm

 In this section we first study the determination of the last parameter, corresponding to the
development velocity of the farm c1 . Subsequently, suggested parameter values to be adopted in
some simulation experiments are presented. We then show how to solve the model and finally we
discuss some simulation results.

      The growth velocity of the economy

  To establish an interval of values for the parameter representing the velocity of development for
the farm c1 we refer to the formula that allows the calculation of the eigen values associated with
the matrix of the linearized dynamical subsystem of the model:

            1
       λ=               2
              (−a 2 ± a 2 − 4a 2 c1 c2 )                                                  (25)
            2

  In (15) it was demonstrated that the model exhibits an asymptotically stable equilibrium, but the
trajectories of the farm’s economy depend on the velocity of growth. Indeed, if we are interested
in the values of c1 that cancel out the value of the discriminant in (25), we have:



                                                 8
          2
        a 2 − 4a 2 c1 c2 = 0                                                                (26)

which is equivalent to the expression:

        c1 = a 2 / 4c 2                                                                     (27)

Therefore, when c1 takes the value given by (27), the characteristic polynomial has only one root
and the farm moves at a moderate velocity; whereas when this parameter is such that
 0 < c1 < (1 / 4 )(a 2 / c2 ) there are two real eigen values and the farm moves at a slower velocity,
as will be shown later in the simulations. When c1 > (1 / 4)(a 2 / c 2 ) two complexes conjugated
eigen values appear, implying trajectories that are wrapped in spiral around the stable
equilibrium. In this case, the economy is initially exposed to fluctuations, the stronger the far
away is the value of c1 from the threshold (27). These fluctuations weaken in time, as the
trajectories come closer to the equilibrium point.

  From a decisional point of view, it is likely that a peasant may discard the values of c1 resulting
in movements with fluctuations. Indeed, excluding exceptional situations, such as the illness of
some family members, or the arrival of some member of the widespread family, allowing to
increase (or diminish) substantially the availability of labour, the peasant would choose the
annual cultivated area on the basis of available resources. On the other hand, it is also unlikely
that a peasant would choose a slow development, where a more rapid evolution is possible.
Therefore, the threshold (27) represents a sort of ideal value, allowing a smooth and steady
development for the farm.

      The parameters of the model

  The technical - agronomic, technical - economic, and purely economic parameters, as well as
the parameter of scale c2 have been estimated from survey data collected in 1991 from a random
sample of farms settled to the north of Santa Cruz de la Sierra.




                                                  9
                                                   Table 2
                                      Identification of the parameters

                             Parameter Value                Unit of
                                                          measurement

                                 a1          1   [1]
                                 a2        0.9   [1]
                                 b1         30   days/hectare, [JO/HA]
                                 b2     15.93    bolivianos/hectare, [BS/HA]
                                 c1    0.0575    1/hectare, time unity, [1/HA UT]
                                 c1    0.0058    [1/HA UT]
                                 c1    0.5754    [1/HA UT]
                                 c2        3.9   [HA]
                                 d1       4.48   [TN/HA (HA JO ) ]
                                                                  e


                             d2          80.3    bolivianos/tonelada, [BS/TN]
                             e           0.35    [1]
                             p           264     [BS/TN]
                             w             10    [BS/JO]

 To estimate the product-elasticity of work e, we used historical information emerged during the
so called crisis of fallow.

  For the development velocity of the farm c1 , we already argued for the selection of the repeated
eigen value in the matrix of the linearized system. To illustrate the implications on growth, we
chose two values slightly below and above the threshold value. Finally, initial values of fertility
and labour are established so as to represent the situation of the farmer when he arrives on the
colonization zone.

      Solving the equations

  We solve the differential system (8) by the method of Runge Kutta, after the identification of
the initial values of fertility and labour. We then proceed to evaluate other variables in the model,
for twenty years. The following equations allow the calculation of the cleared area and tool
expenditures:

                   L1 (t )
        H (t ) =                                                                           (28)
                    b1

        K 1 ( t ) = b2 H ( t )                                                             (29)




                                                     10
  From (5), (6) and (7) we establish for each year in the agricultural calendar, the condition of
feasibility for the slash and burn agriculture:

          1
               (( wL1 (t ) + K 1 (t )) + wL2 (t )) = d1Le (t ) F 1− e (t )
                                                        2                                     (30)
        p − d2

We solve this implicit equation, by applying the rules described in the following iterative process:

        L2 (r ) → g1 ( L2 (r )) → g 2 ( L2 (r + 1)) =
                          −
        g1 ( L2 (r )) → g 2 1 g 2 ( L2 (r + 1)) =
        L2 (r + 1) → g1 ( L2 (r + 1))                                                         (31)

where g 1 ( L2 (r )) represents the linear function of the left member in (30) and g 2 ( L2 (r )) the one
in the right member; while r represents the process step.

 Values for expenditures in tools can be obtained by means of:

                       d2
        K 2 (t ) =          ( w( L1 (t ) + L2 (t )) + K1 (t ))                                (32)
                     p − d2

 Finally, the production path is given by:

                   K 2 (t )
        Y (t ) =                                                                              (33)
                    d2

      Simulation results

 The figures 3.3, 3.4 and 3.5 show the phase space for three different values of the velocity of
convergence for the economy c1 , corresponding to a moderate motion, a slow motion, and an
evolution with fluctuations.




                                                            11
                        11

                                                                   5
                        9
               Labour




                                                          Labour
                        7
                                                                   4
                        5


                        3
                                                                   3
                             1     2        3   4
                                                                       1     1.5       2

                                    Fertility                              Fertility



                             Figures 3.3 and 3.4. Phases diagram corresponding to
                                         a moderate and a slow motion

  According to the theory of dynamical systems, when the value of c1 corresponds to a single
eigen value, the path converges towards equilibrium, following, locally, a straight line (Figure
3.3). The changes of fertility and labour towards their equilibrium values happen at a moderate
velocity, in comparison with the other two motions.

  On the other hand, if we choose the value of c1 in the open interval between the origin and the
threshold value, the orbit moves toward the equilibrium point along a parable. The motion should
stop on the equilibrium point if we should be disposed to permit the evolution during a time
superior to twenty years.

  By choosing a value of c1 above the threshold value, corresponding to the repeated eigen value,
the orbits move toward the equilibrium point following a spiral path. The rapid motion of the
variables towards the equilibrium causes damping fluctuations, whose amplitude increases with
the value of c1 .




                                                     12
                                           18

                                           15

                                           12




                                  Labour
                                           9

                                           6

                                           3
                                                1   2        3      4   5

                                                        Fertility



                      Figure 3.5. Phases diagram for a fast motion economy


  We established the initial conditions for soil fertility and labour on the base of average values
reported at San Julián South, where the settlements are recent. At the beginning of the
colonization process, the initial farm endowments are limited, so they correspond to a point in the
phase space near the origin. In this case, the initial loss of fertility does not compromise the
convergence towards the stable equilibrium.

  On the basis of the distinction between moderate, slow and fast fluctuating growth, we present
here below an analysis of the farm evolution, on a temporary horizon of twenty years. The fast
motion farm shows better results in terms of resources and production levels. Indeed, the fast
motion farm mobilizes a 146% more labour and capital than the slow motion farm and 25% more
of these resources than the moderate motion one. These differences are widened if wages and
input costs are annualised at a higher interest rate.




                                                        13
                                            Table 3.3
                           Evolution of the slash and burn agriculture
                            (moderate motion: one real single root)

       Year Fertility    Labour     Area Expenses Labour Expenses Product
              F(t)         L1(t)     H(t)  K1(t)    L2(t)     K2(t)   Y(t)
             (HA)     (10 journeys) (HA) (10 BS) (journeys) (100 BS) (tons)

       0      1.27      3.81
       1      1.43      4.41            1.47    2.3         1.05         2.08       2.59
       2      1.64      5.06            1.69    2.7         1.22         2.38       2.97
       3      1.86      5.72            1.91    3.0         1.36         2.70       3.36
       4      2.1       6.39            2.13    3.4         1.48         3.01       3.75
       5      2.35      7.04            2.35    3.7         1.59         3.31       4.12
       6      2.58      7.65            2.55    4.1         1.68         3.59       4.48
       7      2.8       8.19            2.73    4.4         1.76         3.85       4.79
       8      3.01      8.68            2.89    4.6         1.81         4.07       5.07
       9      3.19      9.08            3.03    4.8         1.85         4.26       5.31
       10     3.34      9.42            3.14    5.0         1.88         4.42       5.50
       11     3.47      9.69            3.23    5.1         1.90         4.55       5.66
       12     3.57      9.90            3.30    5.3         1.91         4.64       5.78
       13     3.66      10.07           3.36    5.3         1.92         4.72       5.88
       14     3.72      10.19           3.40    5.4         1.92         4.78       5.95
       15     3.77      10.28           3.43    5.5         1.92         4.82       6.00
       16     3.81      10.35           3.45    5.5         1.92         4.85       6.04
       17     3.83      10.39           3.46    5.5         1.92         4.87       6.06
       18     3.85      10.43           3.48    5.5         1.92         4.88       6.08
       19     3.87      10.45           3.48    5.5         1.92         4.90       6.10
       20     3.88      10.47           3.49    5.6         1.92         4.90       6.11

  Table 3.3, Figure 3.6 and 3.7, show the results, in terms of the seven variables characterizing
the farm, for a moderate motion economy. The initial conditions are the same as for the two other
types of motion.

  As already mentioned, the slash and burn system includes two stages. During the first one, the
colonist clears a virgin parcel, with the intention to make available the natural soil fertility for
production. This can be seen in the variables appearing in the first four columns (after that
referred to the reference year) of Table 3.3. During the second stage, the peasant sows the plot
previously cleared, executes the cultural tasks, and finally harvests the rice. This can be seen in
the last three columns of Table 3.3, showing variables such as labour, expenditures and
production level.




                                                14
                12
                         Labour                                        6           Product
                10
                 8
                                         Tools                         4
                 6
                                                                                      Expenses
                 4                               Fertility
                                                                       2
                 2
                                  Area                                                  Labour
                 0                                                     0
                     0     4        8      12      16        20            0   4        8      12   16   20
                                        Year                                                Year




                     Figures 3.6 and 3.7. Evolution of the slash and burn agriculture
                                    (moderate motion: one real root)

  The harmonious evolution of the system and, especially, the steady growth of the natural soil
fertility, keep almost constant the amount of labour needed for the weed control, showing that is
possible to find a technological equilibrium that avoids the virulence of one of the main causes of
fallow crisis. Historically, an equilibrium of this type, balancing new contributions with
permanent losses of fertility, was empirically discovered, by a costly method of trial and error, by
the peasants themselves, and it is now a knowledge resource for the new generations of migrant
colonists.

  Whereas in the moderate motion case the economy reaches an equilibrium point toward the
twentieth year, in the slow motion case this equilibrium is not reached before the end of the
simulation period. For example, the production of rice in the twentieth year is below the second
year level of the moderate case. In these circumstances, the economy exhibits difficulties in
overcoming a stagnant situation. The primary reason is the slow reaction of the farmer against the
loss of natural fertility.




                                                                  15
                                           Table 3.4
                          Evolution of the slash and burn agriculture
                                 (slow motion: two real roots)

Year Fertility    Labour     Area Expenses Labour Expenses Product
       F(t)         L1(t)     H(t)  K1(t)    L2(t)     K2(t)   Y(t)
      (HA)     (10 journeys) (HA) (10 BS) (journeys) (100 BS) (tons)

0    1.27               3.81
1    1.36               3.87                   1.29     2.1           0.78                1.81         2.26
2    1.41               3.92                   1.31     2.1           0.76                1.84         2.29
3    1.45               3.98                   1.33     2.1           0.76                1.87         2.32
4    1.47               4.04                   1.35     2.1           0.76                1.89         2.36
5    1.49               4.09                   1.36     2.2           0.77                1.92         2.39
6    1.52               4.15                   1.38     2.2           0.78                1.94         2.42
7    1.54               4.21                   1.40     2.2           0.79                1.97         2.45
8    1.56               4.26                   1.42     2.3           0.80                2.00         2.49
9    1.58               4.32                   1.44     2.3           0.81                2.02         2.52
10   1.60               4.38                   1.46     2.3           0.82                2.05         2.56
11   1.62               4.44                   1.48     2.4           0.83                2.08         2.59
12   1.65               4.49                   1.50     2.4           0.84                2.11         2.62
13   1.67               4.55                   1.52     2.4           0.85                2.13         2.66
14   1.69               4.61                   1.54     2.4           0.86                2.16         2.69
15   1.71               4.67                   1.56     2.5           0.88                2.19         2.73
16   1.73               4.73                   1.58     2.5           0.89                2.22         2.76
17   1.75               4.79                   1.60     2.5           0.90                2.24         2.79
18   1.78               4.85                   1.62     2.6           0.91                2.27         2.83
19   1.80               4.91                   1.64     2.6           0.92                2.30         2.86
20   1.82               4.97                   1.66     2.6           0.93                2.33         2.90

       6                                                      3
                                                                           Product
               Labour

       4                                                      2
                          Tools                                               Expenses

       2                               Fertility              1


                          Cleared area                                             Labour
       0                                                      0
           0       4       8      12      16       20             0    4       8      12     16   20

                               Year                                                Year




           Figures 3.8 and 3.9. Evolution of the slash and burn agriculture
                             (slow motion: two real roots)



                                                        16
  The slow motion economy is on the borderline of viability. If a shock further reduces the
reaction velocity of the farmer, the farm may not survive.

                                          Table 3.5
                         Evolution of the slash and burn agriculture
                              (fast motion: two complex roots)

      Year Fertility    Labour     Area Expenses Labour Expenses Product
             F(t)         L1(t)     H(t)  K1(t)    L2(t)     K2(t)   Y(t)
            (HA)     (10 journeys) (HA) (10 BS) (journeys) (100 BS) (tons)

      0     1.27      3.81
      1     2.48      13.63          4.54   7.2         11.19          6.76   8.42
      2     4.62      15.63          5.21   8.3         4.51           7.39   9.21
      3     4.51      9.77           3.26   5.2         1.17           4.55   5.67
      4     3.77      8.64           2.88   4.6         1.15           4.03   5.02
      5     3.60      10.08          3.36   5.4         1.98           4.73   5.89
      6     3.85      11.20          3.73   5.9         2.38           5.26   6.55
      7     4.01      10.86          3.62   5.8         2.00           5.09   6.34
      8     3.95      10.33          3.44   5.5         1.78           4.83   6.02
      9     3.87      10.31          3.44   5.5         1.84           4.83   6.01
      10    3.87      10.51          3.50   5.6         1.95           4.93   6.13
      11    3.90      10.58          3.53   5.6         1.96           4.96   6.17
      12    3.91      10.52          3.51   5.6         1.91           4.93   6.14
      13    3.90      10.48          3.49   5.6         1.90           4.91   6.11
      14    3.90      10.49          3.50   5.6         1.91           4.91   6.12
      15    3.90      10.51          3.50   5.6         1.92           4.92   6.13
      16    3.90      10.51          3.50   5.6         1.92           4.92   6.13
      17    3.90      10.50          3.50   5.6         1.92           4.92   6.13
      18    3.90      10.50          3.50   5.6         1.91           4.92   6.13
      19    3.90      10.50          3.50   5.6         1.92           4.92   6.13
      20    3.90      10.50          3.50   5.6         1.92           4.92   6.13




                                             17
                                                               12
               16
                                                               10
                             Labour
               12                                              8                    Product
                                                               6
                 8
                                Tools
                                                               4
                 4                                                                      Expenses
                                         Fertility             2
                               Area                                                 Labour
                 0                                             0
                     0   4     8    12     16        20             0   4   8      12    16    20
                                Year
                                                                            Year



                 Figures 3.10 and 3.11. Evolution of the slash and burn agriculture
                                 (fast motion: two complex roots)

  Table 3.5 presents results for the fast and fluctuating motion. Here, the orbit reaches the
equilibrium point in the tenth year of activity. However, the rapid convergence is obtained
through fluctuations as high as the value of c1 is far from the threshold value; that is, when the
farmer’s reaction is overestimated.

  Under exceptional circumstances, such as unexpected additional contributions of labour in the
family endowment, the emergence of fluctuating paths, such as those displayed in Figures 3.10
and 3.11, is quite likely. Otherwise, the moderate motion case seems to be the most likely
scenario.

6. Conclusions

   The slash and burn agriculture is practiced since remote times in the basin of the Amazon
River, but this practice has recently raised concerns about its effects on deforestation and
biodiversity. Considering that the natural fertility of the soil is recovered after seven years from
the last harvest, we presented a model showing that, under certain conditions, it is possible to
reach a sustainable, self-reproductive equilibrium in time. The model results appear to be
sufficiently realistic.

7. Bibliographical references

Campen, R., Castillo, A., Loza, H., Méndez, M. 1981. Apuntes sobre colonización. Centro de
estudios y proyectos. La Paz, Bolivia.

CIPCA, CORDECRUZ, SACOA. 1992. Diagnóstico socioeconómico de las colonias
Antofagasta, Huaytú y San Julián. Vol. 2, Economía. Unidad de Planificación y Proyectos,
Corporación regional de desarrollo. Santa Cruz, Bolivia.

Lévi-Strauss, C., 1964, Mythologiques. Le cru et le cuit, Librairie Plon, Paris


                                                          18
Loza, H. 1999. Dinámica económica de la agricultura de colonización. Thèse de doctorat
présentée à la Faculté de sciences économiques et sociales de l’Université de Genève. Centre
universitaire d’étude des problèmes de l’énergie.

Maxwell, S. 1979. Colonos marginalizados al norte de Santa Cruz. Avenidas de escape a la
crisis del barbecho. Documento de trabajo No 4; Centro de investigación agrícola tropical.
Santa Cruz, Bolivia.

Moll, M. 1981. Problemas ecológicos en el desarrollo agrícola de Santa Cruz, Bolivia.
Direktion fuer Entwicklungszusammenarbeit und Humanitaere Hilfe. Bern, Schweiz.

Sanchez, P. 1992, Tropical Region Soils Management, in Sustainable Agriculture and the
Environment: Perspectives on growth and constraints, édité par Vernon W. Ruttan, Westview
Press, Boulder-San Francisco-Oxford.

Sioli, H., 1985, The effects of deforestation in Amazonia, The Geographical Journal, vol. 151,
London




                                             19
                    The slash and burn agriculture revisited
                                               Hugo Loza1
                Universidad Mayor, Real y Pontificia San Francisco Xavier de Chuquisaca
                                 Centro de Postgrado e Investigación
                                Calle Urriolagoitia 155, Sucre, Bolivia

                                               11 February 2004




1. Modifications to the model

  We refer to The Slash and Burn Agriculture paper. We maintain the equations (1) to
(6), without modifications. The new equation (7) is written as follows:

        pY = w( L1 + L2 ) + K 1 + K 2 + S ;                                               (7)

where the well-known variables maintain their old meaning; while S represents the
surplus or the benefit of the farm.

    We add to the model the following equation:

        L2 = c3 [( p − d 2 )eY − wL2 ] ;
        &                                                                                 (8)

where all the variables and parameters have been previously defined, except c3 which
represents the adjusting speed of the working time assigned to the control of weeds L2 .
We are, by this way, supposing that the farmer is willing to hire additional labour for
the control of weeds, as far as its contribution to the product is superior to the
corresponding cost.

  Considering the possibility of a positive surplus or benefit S in (7), and posing a new
behaviour rule regarding the labour assigned to weed’s control in (8), we have a system
of eight equations and the same number of endogenous variables.

  We can deduce relationship (8), reasoning as follows. We define the income I, and the
cost C, as functions of the labour assigned to the weed’s control as follows:

        I ( L2 ) = pd1 Le F 1−e
                        2                                                                 (9)

       C ( L2 ) = w( L1 + L2 ) + K 1 + K 2 ;                                              (10)

which we can write as follows:

1
 Hugo Loza
Phone + fax (591) 3 343 72 51
Casilla 5079, Santa Cruz
hloza@cotas.com.bo
      C ( L2 ) = w( L1 + L2 ) + K1 + d 2 d1 Le F 1−e
                                             2                                     (11)

If we are interested to the marginal income, to the marginal cost, and to their difference,
we obtain the following expressions:

              dI
      I′ =       (L2 ) = pd1eLe2−1 F 1−e                                           (12)
             dL2

             dC
      C′ =       ( L2 ) = w + d1d 2 eLe −1 F 1−e
                                      2                                            (13)
             dL2

      I ′ − C ′ = d1eLe−1 F 1−e ( p − d 2 ) − w
                      2                                                            (14)

In this way, we can think that the rate of growth of the working time assigned to the
weed’s control, shall be positive, as far as this difference (between the income and the
cost), shall be in the margin, positive, just as we write in the following equation:

                [                            ]
      L2 = c3 ( p − d 2 )ed1 Le−1 F 1−e − w L2 ;
      &
                              2                                                    (15)

from which we obtain easily the equation (8), observing that the first term of the
expression between brackets, represent an expression for the product, when carrying out
the corresponding multiplication.

2. Analysis of stability

  Reducing the size of the system we obtain the following differential system of three
equations and three endogenous variables.

      & a
      F = 1 L1 − a 2 F                                                             (16)
         b1

      L1 = c1 (c2 − F )L1
      &                                                                            (17)

                [                            ]
      L2 = c3 ( p − d 2 )ed1 Le−1 F 1−e − w L2
      &
                              2                                                    (18)

From the following equivalences we identify the stationary points of the system:

      &           a b
      F = 0 ⇔ L1 = 2 1 F                                                           (19)
                   a1

      &
      L1 = 0        ⇔   F = c 2 ; L1 = 0                                           (20)

                                                   1

      &                              w         e −1
      L2 = 0        ⇔    L2 =                  F;        L2 = 0                  (21)
                               ( p − d 2 )d1e 


                                                       2
As a result there are the two equilibrium positions that follow:

      F = L1 = L2 = 0                                                                         (22)

                                                                1
                            a 2 b1c 2                 w         e−1
      F = c2 ,       L1 =             ,   L2 =                  c2                          (23)
                               a1               ( p − d 2 )d1e 

Changes of L2 , not having any impact on F or on L1 , as we notice examining the
system (16), (17) and (18), the analysis of stability presented in The Slash and Burn
Agriculture paper, remain valid.

3. The variables at the equilibrium

 In the stable equilibrium, the variables are written as follows:

      F = c2                                                                                  (24)

             a 2 b1c2
      L1 =                                                                                    (25)
                a1

             a2 c2
      H=                                                                                      (26)
              a1

             a 2 b2 c 2
      K1 =                                                                                    (27)
                 a1

                                  1
                  w         e−1
      L2 =                  c2                                                              (28)
            ( p − d 2 )d1e 

                                      e
                    w         e−1
      Y = d1                  c2                                                            (29)
              ( p − d 2 )d1e 

                                          e
                          w         e −1
      K 2 = d1 d 2                  c2                                                      (30)
                    ( p − d 2 )d1e 

                                              e                         1
                               w         e −1         w         e −1 a 2 b1c 2       b 
      S = ( p − d 2 )d1                  c2 − w                 c2 −              w + 2  (31)
                                                                                     
                         ( p − d 2 )d1e         ( p − d 2 )d1e          a1           b1 
                                                                                             




                                                        3
4. Simulation results

  On Table 1, we compare the behaviour of the variables on the stable equilibrium, in
the two versions of the model. On the second and on the third columns of this table, we
show the results corresponding to the values of the parameters just as they appear in the
reference paper. We are surprised by the significant increment of the labour assigned to
the weed’s control (1.9 at 19.9), and the very important increment of the production
level (6.1 at 30.9), when the colonist discovers the possibility of the surplus or benefit
maximization modifying the labour assignment. Although this is the expected result, the
magnitude of the increment of the production level seems out of any reasonable
consideration. Checking then, the estimate of the labour elasticity e as well as the
estimate of the parameter of compatibility in the same production function d1 we find
out that values of these parameters significantly lower are compatible with the rare
available observations, provided that the fall of the biological fertility of the soil is not
as marked as we had initially supposed.

  On the fourth and fifth columns of this table we present the results that we judge more
coherent with the disposable observations. Indeed, we see that in the case of the first
model, the equilibrium value of the production level doesn't suffer a remarkable
modification passing from 6.13 to 6 tons; as for the modified model we obtain a value
equal to 13.33 tons. Although this value means an important increment of the
production level (6 at 13.33), it represents a yield of 3.4 tons for hectare of fertile soil;
which is not out of the reach of the slash and burn technology.

                                          Table 1
                         The variables on the stable equilibrium.
                         Results for the two versions of the model

                 Variables 1ª version 2ª version 1ª version 2ª version
                           e = 0.348, d1 = 4.478 e = 0.094, d1 =3.593

                  F (HA)         3.9        3.9         3.9         3.9
                  H (HA)         3.5        3.5         3.5         3.5
                  L1 (JO)       105         105        105          105
                  K1 (BS)        56         56          56           56
                  L2 (JO)       1.92       19.86       0.00         2.30
                  K2 (BS)       492        2480        482         1071
                  Y (TN)        6.13       30.88       6.00        13.33
                  S (BS)          0        2582          0         1114

5. Conclusions

  Although there is not agreement on this point, it is probable that the colonist doesn't
have the means to reason in terms of the marginal income and cost when he is trying to
improve his production surpluses. He is however, in position to compare the increments
of the production level before supplementary labour assignments to the weed’s control,
discovering empirically, the rule that we have formulated as equation (8). In this way,
we are able to model a procedure that had already become evident when observing in
Figure 3.2 and in the corresponding relationship (24) of the reference paper, the
existence of a surplus then, virtual and now real, of production.


                                             4

				
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