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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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