Forest Resource Depletion, Soil Dynamics, and Agricultural Development in the Tropics

JOURNAL OF ENVIRONMENTAL ECONOMICS AND MANAGEMENT 18,136-154 (1990)

Forest Resource Depletion, Soil Dynamics, and Agricultural Productivity in the Tropics*

SIMEON K. EHUI

International Institute of Tropical Agriculture, Ibadan, Nigeria

AND

THOMAS W. HERTEL AND PAUL V. PRRCKEL

Department of Agricultural Economics, Purdue University, West hfayette, Indiana 47907

Received July 15,1987; revised October 6,1988

A two-sector dynamic model for agriculture and forestry is proposed. Agricultural yields are a function of the rate of deforestation, the forest stock, and purchased inputs. We examine the impact of changes in the social discount rate, net returns to agriculture, and direct marginal benefits of the forest stock benefits on the optimal deforestation path under the assumption of a quadratic agricultural yield function. Finally, steady-state comparative static analysis is conducted. Q 1990 Academic Press, Inc.

I. INTRODUCTION

The rate at which tropical forests are being consumed is a cause of widespread concern. Closed forest cover in the tropics is decreasing by 10 to 20 million hectares (1 to 2%) per year according to the best available estimates [12,21,23,36]. An estimated 40 to 60% of the remaining tropical forests will be lost by the end of the century if current rates of deforestation are maintained [12,25]. The greatest impetus for forest destruction in the tropics is increasing demand for cropland. Pressures from agriculture are fueled by rapid population growth combined with traditional agricultural practices such as shifting cultivation.’ Lack of employment outside agriculture as well as the absence of property rights also contribute to these pressures [13,21,22,34].

Economic planners in developing countries face a dilemma; forests must be cut and cleared to increase production in the near term, but the loss of forests can reduce agricultural productivity in the long run. This is because many of the tropical soils owe their productive qualities to the protective role of the forest. Indeed, forests help accelerate the formation of topsoil, create favorable soil structure, and store nutrients useful to crop production by retarding erosion and silting and by regulating stream flows [12,21,26,27]. When the forest cover is cleared, this equilibrium is destroyed. The physical and chemical properties of the soil undergo a

*The authors acknowledge the helpful comments and suggestions of Professor S. C. Hu and several anonymous reviewers. This is Journal Paper No. J-11,298 of the Purdue Agricultural Experiment Station.

‘Shifting cultivation is typical of traditional agricultural practice in the tropics. It is defined as a system in which temporary clearings are cropped for a shorter period (typically one to three years) then they are allowed to remain fallow. Clearings are obtained by felling and burning the natural forest [26]. This system results from the low content of nutrients in many tropical soils. Since most of the nutrients are in the living plants, these can only be made available when the land is cleared and burned.

136 0095~0696/90 $3.00 Copyright Q 1990 by Academic Press. Inc. All rights of reproduction in any form reserved.

FOREST DEPLETION IN THE TROPICS 137

series of changes leading to nutrient loss, accelerated soil erosion, and declining yields [14,15,17,18,21,26,31]. In fact, most of the remaining tropical forest land cannot sustain continuous farming under current practices [21].

In spite of widespread concern about forest depletion in the tropics [l, 2,4,5,19,21,23,34], little formal analysis of the socially optimal allocation of land between forest and agricultural uses in the tropics is available. This paper develops a theoretical model of this planning problem. Based on this model, qualitative results are derived which highlight the role of certain agronomic parameters in determining the effect of changes in both the social discount rate and the relative returns in agriculture and forestry on the optimal deforestation path and the steady state-forest stock. The role of technological change in determining the steady-state forest stock is also analyzed. As a result, several key testable hypotheses regarding the nature of the aggregate agricultural production function are identified.

II. THE BASIC MODEL

Consider the problem facing an economic planner who must allocate forest land over time between forest and agricultural use. This requires the use of a social welfare function. The model proposed here has as its objective the maximization of the present value of a utility index of aggregate benefit.2 Both forested and deforested lands are considered as sources of future income. Mathematically, the model is represented as3

yy”= ime-“‘(U[n(D,X,F)]} dt

subject to

7r(-) = B(F) + [L - F(t)]

*[P,(t)* z{ Nt), F(O) - fQ)? x(t))

#(t) = -D(t) = 0 if F(t) = 0

F(O) = F,, L = L.

-

(1)

pxw* WI (2)

(3)

(4)

(5)

‘The model proposed here is in line with the utilitarian approach. This has been popular in the theory of economic growth. It has the advantage of ensuring a reasonable distribution of welfare over time. Although the use of a utility function does not affect the theoretical results, it will have an impact in an empirical application of the model. References that discuss the underlying problems and other ap- proaches include [3,6,8,10,33,35].

3A dot above a variable denotes the time rate of change of that variable, viz., k(r) = dF( r)/dt.

138 EHUI, HERTEL, AND PRECKEL

W is a measure of discounted social welfare. 6 is the social discount rate and thus provides an indication of how future utility is discounted. L represents total arable land and F(t) represents the land area covered by forest at time 1 (forest “stock” in hectares).4 Thus [L - F(t)] measures the total land area devoted to agriculture at time t. Z( *) is the aggregate agricultural yield function (in kilograms per hectare) and is assumed to depend upon the current rate of deforestation D(t), the cumulative amount of deforested land [F(O) - F(t)], and current flow of purchased inputs X(t). P’(t) denotes the per kilogram returns to agriculture at time t. Px(t) is the per kilogram cost of purchased inputs. These are assumed to be exogenously determined in the international marketplace. B(P) is a benefit function represent- ing all services provided by the forests. These include forestry, maintenance of global and local climate, and the preservation of natural habitats, to name but a few. It is assumed that cYB/aF > 0, indicating a positive marginal benefit from the forest stock, and a’B/aF’ I 0, indicating an increasing marginal benefit from a unit of forest stock as the total forest land shrinks.

The utility index, U(e), measures society’s satisfaction at time t. U( .) is assumed to depend only on aggregate benefit at time t. It is also assumed to be time invariant, exhibiting positive but diminishing marginal utility. Formally,

(Al) - = u,(Tr) > 0 ar and

a2u(fl) aT2

= U,*(T) < 0 forallO<Ir< co.

Aggregate benefits in this model, n(e), equal the sum of societal benefits from agriculture and forestry. The equation of motion is given by (3); this describes the change in forest “stock” over time.’ Equation (3) also dictates that if over some interval F(t) = 0, then the rate of deforestation must be constrained so that p(t) = 0 over that interval as well (see Arrow and Kurz [3, p. 411 for more details). Equation (4) gives the nonnegativity conditions on the stock of forest land, the net deforestation rate, and the purchased inputs variable. Finally, Eq. (5) defines the initial endowment of forest land and the total arable land.

The central feature of this land allocation problem is the average agricultural yield function. Past research has studied the effect of deforestation on crop yields at the micro-level. Comparative experimental studies of unfertilized monocultures under continuous cropping in areas of shifting cultivation in tropical South America and Africa indicate that ash addition plus the rapid mineralization of organic matter after clearing and burning a forest provide a sharp increase in available nutrients to the first crop planted. Afterward, yields gradually decline. The extent of this decline varies with soil properties and cropping systems. However, in several studies

4Measuring forest “stock” in hectares is an obvious limitation which could be relaxed in the future. However, this should not cause serious problems as the focus of this research is on the dense and moist tropical rain forests which are fairly homogenous. Most trees in these forests exceed 400 years in age.

‘Note from Eq. (3) that replanting of forest is not explicitly addressed in this model. Rather this model focuses on net deforestation.


cumulative yield losses of 50% were reached by the fourth to sixth consecutive planting [26, 27].6

Theoretical results in this section are built around a set of nine assumptions, (A2)-(AlO), about the first- and second-order partial derivatives of the yield function with respect to its three arguments. It should be emphasized that this is an aggregate average yield function. That is, this function gives the average yield over all land in agriculture at any point in time.

642) a%> all(t) ’ O-

(A2) states that current depletion of forests causes current yields to increase. This occurs because of the nutrient content of ash left after the forest is burned [7,11,26,27].

(A31 aw

a [F(o) - qt)] < O* The second assumption about the aggregate yield function is that increases in the total current period agricultural land area cause yields to fall. This is due to the movement of cropping activity onto more marginal land as deforestation proceeds. Another factor contributing to (A3) is the productivity loss due to removal of organic matter and erosion of the soils [l&20,24, 26-28].7

(A4) states that an increase in the rate of deforestation in the current period causes the aggregate average yield effect of deforestation to fall. This effect also reflects the idea that increasing rates of deforestation tend to occur in a more haphazard manner.

W) a2z(-)

ao(t>a(F(o) - t-(t)) < O. ken [l] showed for the case of Tanzania that, in general, deforestation is also associated with

significant declines in chemical and physical properties of the soil, although in some cases clear felling (particularly with burning) led to short-term increases in availability of important plant nutrients. These include increases in available phosphorous and exchangeable bases which are associated with the decomposition or burning of organic matter after clearing. These additional nutrients decrease over time due to the presence in the soils of free oxides of iron and aluminum which capture available phosphorous to form iron and aluminum phosphates.

‘Deforestation and forest conversion to agricultural uses cause immediate and significant changes in air and soil temperature and relative humidity. Lal and Cummings 1181 estimate the increase in maximum soil temperature of the 0 to 1 cm. layer to be on the order of 20 to 25°C. The cleared land is easily compacted, thus accelerating soil erosion. Roose [24] observed in the humid region of the C&e d’lvoire (the country with the highest rate of deforestation in the world) that runoff and erosion increased 50 to 1000 times following deforestation. With soil erosion and high soil temperatures throughout the year, there is rapid decline in soil organic matter that causes deterioration in soil structure and a lowering of the water and nutrient capacity as well as the infiltration rate [20]. La.l[16] reports a linear decline in soil organic matter content with cumulative soil erosion following deforestation.


(A5) postulates that marginal productivity of current period deforestation declines with increases in the stock of land in agricultural production at time t. This reflects the movement of agriculture onto more marginal land.

(A6) We) ax(t) ’ O.

This assumption states that crop yields increase as more purchased inputs (e.g., fertilizer) are employed. Thus, soil fertility can be maintained or increased by applying sufficient amounts of X [16].

W) a”zC) < o

ax(t)” -

(A7) postulates that diminishing returns prevail with respect to purchased inputs,

Assumption (A8) implies that the marginal yield increases due to deforestation at time t are diminished as additional units of purchased inputs are employed. (A8) can be interpreted as implying diminishing returns to nutrients, since the current deforestation effect is assumed to be equivalent to a good dose of fertilizer [26].8

The remaining second-order derivatives of the yield function are assumed to be equal to zero. This facilitates derivation of qualitative results below.

(A9)

(AlO)

J2ZC) o -= aF(t)2 .

i12z( *) dF(t) 8X(t) = O-

III. COMPARATIVE DYNAMICS

Assuming an interior solution, the current value Hamiltonian for the problem described by (l)-(5) is as follows (time subscripts have been omitted in order to simplify notation):

H(D, F, X, #> = U{ a(& F, X>} - $0. (6)

*Six of the assumptions presented here ((A2)-(A4) and (A6)-(A8)) have been verified empirically using aggregate data from the CBte d’Ivoire, the country with the highest rate of deforestation in the world [9]. Results indicate that deforestation in the current period does indeed contribute positively to yields, and that increases in the cumulative amount of deforested land cause yields to fall. This confirms soil scientists’ hypotheses that deforestation is equivalent to a good dose of fertilizer and that yields decline over time because of the productivity loss of the soils due to leaching of organic matter and erosion. Not surprisingly, yields were found to respond positively to fertilizer but at a decreasing rate.


Here $ is the costate variable associated with the equation of motion (3). The necessary conditions which must be satisfied by a solution to this problem are given bY

and

0 = u,= U,*(L - F)*(P,Z,

J, = u, = U,*[P,Z,(L - F)],

sl+b-$= u,= Un*[BF+PAZF(L-

lim e”‘~(t)F(t) = 0. f’cc

- PX)? (7) (8)

F) - PAZ + P,X] (9)

(10)

Equation (7) indicates that at the optimum, purchased inputs are applied up to the point where the value of the marginal utility equals the net return. Equation (8) indicates that at any point in time, the rate of deforestation should be chosen so that the marginal utility of deforestation (U,) is equal to the opportunity cost of the forest stock (#). Here, I/J measures the future benefit foregone by a decision to deforest today. In other words, it is a measure of the marginal cost of clearing the forest land at time t rather than saving it for future generations.

Equation (9) implies that forest stock services should be employed up to the point where the marginal utility of forest capital is equal to the social cost of this capital. The right-hand side of (9) represents the marginal utility of forest stock. It is composed of two parts: the direct marginal contribution of forests (U..B,) and the indirect marginal contribution of the forest stock through its effect on agricultural productivity. The latter has two components. The first, [U,P,Z,( L - F)], captures the benefit of increased forest cover due to enhanced agricultural yields. The second, [UJ - PAZ + P,X)], measures the net cost of not having an additional hectare of land in agriculture. The left-hand side of (9) measures the cost of employing the services of one unit of the forest capital at any point in time. It includes both an interest charge (S#) and a capital gains term ( - 4).

Totally differentiating (8) with respect to time and combining this result with (9) yields an expression for the time rate of change in the rate of deforestation along the optimal path,

U xx

D = - (~XXUDD - Gx) [-W+ {UC D@J,,- ~UXF))], (11)

where (Y = U,,/U,,. Based on (Al)-(AlO) several relevant partial derivatives of the utility function

may be signed (see Appendix A):

u, > 0; &I, 7 ux*, u,, < 0 ; and u,, = 0. (12)

Using (12) and the second-order condition requiring concavity of U in X and D, the sign of D along the optimal path reduces to

(13)


Recall that U, is the marginal utility of deforestation in current periods. A large value of U, indicates a large agricultural yield response from current period deforestation. This, in turn, translates into a higher marginal utility due to increased profit. This is a one-time effect only, and will be termed the “deforestation motive”.

The term (U, - b&D) represents the net marginal contribution of forest area to utility. Forests contribute to utility through two effects which are implicit in (U, - U,,D). They include the direct return to forestry as well as the protection that forests provide to agricultural yields. The marginal utility of forests lingers into perpetuity and has a present value equal to (U, - U’,D)/S. This term can be described as the “conservation motive”.9 When the conservation motive is relatively weak (U, - v&D < SU,), (13) states that the rate of deforestation must be falling over time (D -C 0). lo It is possible (e.g., when U, = U,, = 0) that b(t) will be nonpositive for all f. In this situation, feasibility requires that the rate of deforestation start out high, declining to zero over time. This is represented graphically by (A) in Fig. 1. (B) in Fig. 1 depicts the converse situation whereby the conservation motive predominates such that D(t) > 0 for all 2. This is infeasible in an infinite horizon problem, since eventually the forest stock must be exhausted.

It is quite possible that D(t) will not be monotonic over the planning horizon. (C) depicts the situation where the conservation motive is initially stronger, becomes relatively weaker in later periods. The opposite situation, (D), is infeasible in the infinite horizon problem for the same reason that (B) was ruled out. Both of these cases will thus be ignored in the subsequent discussion, which focuses on the comparative dynamic effects of changes in the social discount rate and net returns to agriculture and forestry on deforestation rates.

The Role of the Discount Rate

Partial differentiation of (11) with respect to the social rate of discount yields

cd(t) U -=

as [ u,LI,,xx- u&l * uD s O- (14)

This result implies that the higher the discount rate the lower the rate of growth of deforestation. It is illustrated, for the two cases of interest, in Fig. 1. When impatience dominates the conservation motive from t = 0, such that deforestation rates start out high and decline over time, the higher discount rate makes the optimal path of D(t) steeper (the dotted line in Fig. l(A)). Thus, current period

‘The concepts of “conservation motive” and “deforestation motive” are not new. Similar concepts were first employed by Vousden [35].

“The fact that the rate of deforestation falls over time when the conservation motive is weaker does not imply that the total rates of deforestation decline over the planning horizon. This simply indicates how much forest land is cleared today relative to tomorrow. When 8UQ > [C+ - u’$] we expect current rates of deforestation to be higher than those in the future (D -C 0). However, when the conservation motive is stronger such that SlJo -X [V, - U,, D], then we expect the rates of deforestation in the future periods to be higher than those in the earlier periods (thus b > 0).


D(t)

1

A D(t)

C 00

FIG. 1. Comparative dynamics in the model. The dotted line in (A) and (C) illustrates the impacts of a higher discount rate and/or increased relative returns to agriculture. (A) Weak conservation motive (b(r) < 0); (B) Strong conservation motive (b(t) b 0) [infeasible]; (C) Relatively strong conservation motive in early periods followed by weak conservation motive in later periods; (D) Weaker conservation motive followed by strong conservation motive in later periods [infeasible].

deforestation is expected to increase, relative to future deforestation rates.” Now, consider (C) where the stronger conservation motive prevails in the early periods. Here the effect of a higher discount rate is to flatten D(t). However, the optimal path for D(t) becomes steeper in later periods as the conservation motive weakens.

The Effect of Relative Returns in Agriculture and Forestry

Define p = (PAZ - P,X)/B, as the ratio of per hectare returns in agriculture and the marginal benefit from a unit of forest stock. Then from (11) we have (see

D

“Note that we are talking about the effect of 6 on the rate of growth of deforestation [dD( t)/dt], not deforestation D(t). Thus a lower time rate of change of deforestation does not necessarily imply that the actual rate of deforestation declines. In fact, we expect the rate of deforestation to increase in earlier periods relative to future rate. The magnitudes of the early period increases, as well as of future period declines, depend critically on the shape of the aggregate yield function.

144 EHUI, I-IEWIEL, AND PRECKEL

Appendix A)

ah(t) UXX -z- = u,,u,, - uzx) ( * [U, - U,,DQ] < 0. (15)

(15) indicates that the higher the returns to agriculture (relative to forestry), the lower the rate of growth of deforestation. Since this effect is qualitatively identical to that of an increase in the discount rate, Fig. 1 can once again be used to illustrate the various cases of interest. As before, these depend on the strength of the conservation motive relative to societal impatience. In general, when returns in agriculture rise relative to the marginal benefit from forest, current-period deforestation is expected to increase relative to future rates, implying that the rate of deforestation will have to be reduced more quickly for a given forest stock. This is shown by the dotted lines in Fig. 1.

IV. STEADY-STATE RESULTS

General Discussion

In steady state the net deforestation rate is necessarily zero. Setting j = i> = 0 in Eq. (ll), it can be shown that there is a steady-state forest stock (E*) which is uniquely defined by

and

f [U,(D*, F*, X*)] = U,(D*, F*, X”), (16)

Zx(D*, p, X*) = P,/p,, (17)

D* = O.‘* (18)

The left-hand side (LHS) of Eq. (16) can be described as the present value of the stream of marginal utility derived from sustainable economic rents.13 The right-hand side (RHS) of (16) is the marginal utility enjoyed from current deforestation. Thus Rq. (16) asserts that in steady state, the marginal utility of further deforestation (U,) must equal the present value of the foregone marginal future benefit ((l/6)&.).

Note that (16) does not a priori constrain F* to be positive. The sign of F* depends on the strength of the conservation motive (U,) relative to the preference for deforestation (SU,), where F = 0. Consider the two possibilities. First is the case involving a relatively weak conservation motive at D = F = 0, resulting in SU,(O,O, X*) 2 U,(O,O, X*). Based on this preference relationship, Fig. 2(A) provides a phase diagram for the optimal allocation of the forest stock in the (D, F)

t2Note that with the nonnegativity constraint D(r) 2 0 (due to the irreversibility of deforestation), the steady-state solution (D* = 0) is a corner solution with respect to this constraint.

13Note that the left-hand-side of (16) can be interpreted as an expression of marginal user cost [29,30], in that it shows the cost (in terms of welfare loss) of using a unit of the forest land today. Thus this is simply a restatement of the necessary condition (8) with $J replaced by (U,/s).


PANEL A

PANEL B

FIG. 2. Phase diagram for the case where F* < 0

space. T?re isoclines consist of the horizontal axis (# = 0) and a downward sloping curve (D = 0) defined by Eqs. (16), (17), and (18), respectively (see Appendix B for more details). The intersection of these two isoclines defines the optimal steady state forest stock, F*. Here F* < 0, therefore the optimal path involves clearing all the forest lands by some finite point in time, T. l4 This is further illustrated in Figure 2B (the phase diagram in ( $I, F) space) which indicates that the optimal path is to clear all the forest lands if at F = 0 the preference for deforestation is not dominated by the conservation motive.

14A mathematical proof of this is provided by Vousden [35, pp. 140-1411 who addresses the general resource depletion problem.


PANEL A

0

t

FIG. 3. Phase diagram for the case where F* > 0.

The second steady-state situation of interest arises when &Y&0,0, X*) < U,(O, 0, X*). Contrary to Fig. 2, this represents the situation where the conservation motive dominates at D = F 7 0. The phase diagram in (0; F) space (Fig. 3A) shows that in this case the D = 0 locus cuts the F = 0 isocline in the positive orthant. Thus the optimal path involves leaving a stock F* of the forest resource unexploited. The phase diagram in (It, F) space demonstrates why this is so (Figure 3B). It is optimal to leave some forest unexploited because at F = 0 the conservation motive continues to dominate the preference for deforestation motive.

It may be shown that, for a given initial forest stock level F(O), there exists an optimal starting point [D*(O), F(O)] which gives rise to the optimal paths in Figures 2A and 3A. Any other starting point will not be optimal 1321. Since these optimal paths exhibit nonpositive values for b(t), (C) in Fig. 1 may be eliminated. This leaves (A) in Fig. 1 as the only one of the four cases which can be optimal.


Comparative Statics

Equation (16) is implicit in F *. In order to actually solve for the steady-state forest stock, it is necessary to assume a parametric form for the forest benefit function, B(F), and for the yield function, Z(e). For convenience we assume that B(F) is linear, so that B, is constant. Based on (A2)-(A8), note that a second-order approximation to the “ true” underlying function is required. We assume a quadratic functional form as given by

z(t) = PO + P,[X(t)l + &I~(t>l + P,ww - F(t)1 + &PRl

+ i&,[X(t)12 + ‘ia22vw1’ + &2Mt)* x(t)1

+ P,,P(t)l *[F(O) - FWI. (1%

Based on (A2)-(A@, the following signs are expected: Is,, &, p2 2 0; &, &i, pz2, pi2, and f12s 5 0. In addition, for nonzero values of D(t), X(t), and [F(O) - F(t)],

we expect (PI + &JX(t)l + P12tWt)l} > 0, {P2 + k2[X(t)l + P22[WfN + p2J F(0) - F(t)} 2 0, and { & + P2JD(t)]} s 0. An additional variable, TR, is added which provides an index of technological change. The associated coefficient (p_,) is expected to b e positive as a result of exogenous, yield-enhancing technological progress.

Limiting Cases (6 = 0 or 6 = ~0)

Consider first the case where the utility of future revenues is not discounted relative to the utility of current revenues. Thus S = 0 so that Eq. (16) becomes U,( D*, F*, X*) = 0, and the optimal deforestation policy results in maximization of the utility of sustainable economic rent. Solving for the steady state forest stock level yields

F*l 6-o = F(O) + [(Is, + &X* + SBllx*2 + PJR*) - &X*1 - BF + A (20)

2P3 2

where px and BF are the price of purchased inputs and the marginal benefit of forests, relative to the price of agricultural output, i.e., px = P,/P, and BF = BF/PA. The parameter A equals L - FO. It denotes the amount of arable land not under forest cover at time t = 0. TR* is a proxy representation of the level of technology expected in steady state. X* is determined by equation (17), and is a function of the price of fertilizer relative to food:”

x* = j3,‘( Px - Sl). (21)

“Note that the ability to solve X using Eq. (17) alone stems from the fact that the interaction term between X and F has been ignored in Eq. (19).

148 EHUI, HERTEL, AND PRJXKEL

Consider first the situation where A = 0 (i.e., when the total arable land equals the initial forest stock). In this case, the question of whether or not deforestation is optimal hinges entirely on the sign of the second term in Eq. (20). The numerator of this term represents the difference (A) between “modified” steady state net agricultural returns [Le., returns evaluated at F = F,] and the marginal benefit of a unit of forest stock (BF), i.e.,

A = [ (& + &X* + $/3,X** + &TR*) - p,X*] - &. (22)

Thus with A = 0, F*j6_0 < F(O), as long as A is positive. This makes sense since, with higher relative returns to agriculture, we would expect more forest land to be converted to agriculture.

Next, consider the case where L > F,, such that there is arable land present at time t = 0. A positive value for A simply raises the value of F*(,=,. This follows from the fact that the net stock effect, as measured by p,, applies to UN arable land. If A > 0 there will be more agricultural land benefiting from forest cover in steady state, and we expect PJ,,,, to increase accordingly. Deforestation, in this case, will be optimal as long as net returns to agriculture (evaluated at F = Fo) exceed the per hectare returns to forestry by more than the value of the net stock effect on arable land at t = 0 (i.e., A > l&A).

The partial derivatives of PI,,, with respect to px and zF are positive:

cYF* X* T=-->O, JP, w,

and

aFr 1 --,=-->O. aB, 283

That is, an increase in the relative profitability of forestry, brought about either by an increase in agricultural costs or by an increase in the marginal benefits of forest stock, leads to an increase m E*J,,,.

Finally, partial differentiation of F*(6_0 with respect to TR* yields

al+-* i-4 -=-<o. 8 TR* 2& (25)

This result indicates that the higher the expected level of technology, the lower the steady-state forest stock. This is because technological progress raises modified returns to agriculture. In other words, improved technology can offset the loss in productivity due to leaching of nutrients and erosion associated with diminished forest cover. Thus, deforestation is pushed further than would otherwise be the case.

Consider now the special case where the present generation adopts an infinite discount rate. Thus, forest lands are cleared so as to maximize current revenues, and future benefits from forest conservation are disregarded. In this case (assuming U, is bounded), Eq. (16) yields U&D*, I;*, X*) = 0. The steady state forest stock level


is given by Equation (26):16

F*l,+, = F(0) + $*(& + &X*). (26) 23

Note that Fr]8,00 is independent of returns in agriculture and the marginal benefits of forests. It is simply equal to the initial stock, as modified by the ratio of the yield “kick” over the parameter associated with the interaction term between D and F. This is not surprising, since under an open access situation (where 6 -+ cc) the objective is to maximize utility (and hence aggregate profit) from deforestation with no regard to the long-term gain from forestry which includes future incomes from forestry activities, preservation of wildlife and habitat, stabilization of microclimate, as well as increased agricultural yields due to the protecting role of forests. Thus, deforestation is “costless” in a dynamic sense and the problem reduces to one of maximizing aggregate yields. Equation (26) also shows that deforestation is always optimal in this case since [(p2 + &2X*)//323] < 0.17

The Case of a Positive, Finite Discount Rate

In general, the expression for steady-state forest stock is

P = F(0) + & [p + &32,A + P21>

23 (27)

where

and P = w-2 + /312x*) + 3333

y = /.J~ + a2/3,Z,A2 - 2S2p,,A(P2 + 812X*) - 4W2,A.

Other parameters are defined as in the previous manner. The sign of p may be positive or negative, depending on the size of the “yield kick” ( p2 + pi2 X*) relative to the stock effect 21&]/S. However, when ~1 is positive, the sufficiency condition requiring concavity of the Hamiltonian is violated.18 Thus, Al. must be negative. That is, the stock effect must dominate the “yield kick” for an interior solution to obtain. With p < 0 and A = 0, deforestation will be optimal as long as the “modified” per hectare returns to agriculture (evaluated at F = F(0)) exceed those in forestry (i.e., A > 0). When A is positive, then F* < F(0) if the modified per hectare returns to agriculture exceed the per hectare returns to forestry by more than the value of the

‘%olving U,( D*, F*, X*) = 0 yields two values of Fjs,,, namely F,* = L and F*Js _m = F(O) + l/&s * (Ps + & X*). However, F*ls+, = L does not provide a maximum value for aggregate profit and may therefore be eliminated as nonoptimal.

“Note also from Fq. (26) that F*(, _ m can take both signs or even be zero. Its sign reduces to the condition

i.e., the open access steady state forest stock is positive, zero, or negative as long as the initial forest stock exceeds, is equal to, or less than the ratio of the “yield kick” over the parameter associated with the interaction term between D and F.

181n steady state (setting 4 = 0), Fq. (9) yields r/, - SU’ = 0. A sufficient condition for this to be a maximum is CJFF - SC/,, c 0 or, after substitution, (& + pi2 X*) 2 2)/3sl/S.


net stock effect on arable land at t = 0 (i.e., A > (& - p)A). (yl/* must be nonnegative.“)

Partial differentiation of (27) with respect to BF, pX and TR* yields

and

aF* G=Y --v* > 0,

aF*

ap, = x*y-‘12 > 0,

aF*

a TR* = &y-i/2 < 0.

(28)

(29)

Equations (28) and (29) imply that higher relative profitability in forestry (brought about by an increase in either the marginal benefits of forest stock or agricultural costs) give rise to a higher steady-state for forest stock. Equation (30) says that the higher the level of technology (assuming it is costless), the lower the steady-state forest stock. This is because the increase in technological progress restricts the profitability of forestry relative to agriculture.

Turning to the discount-rate effect,

aF* 1 -= -[-&+u

86 S2P2,

-1/2(-&~+&36A)] ~0 ifA> g. (31) 23

Thus, the discount-rate effect is generally indeterminate. Condition (31) states that with F* < F(O), the discount rate has a negative impact on F* if the modified per hectare returns to agriculture (evaluated at F = F(0)) exceed the per hectare returns to forestry by some minimum value which depends upon the magnitude of the agronomic parameters.

V. SUMMARY AND CONCLUSIONS

Economic planners in tropical countries are faced with a dilemma. If forest clearing continues at its present rate, most natural forests will disappear, and the land’s long-run productivity may be irreparably damaged; but if forests are pro- tected from clear-cutting, the current rate of growth in agricultural output may diminish. This is problematic for countries where population growth is rapid and agriculture accounts for a large share of GNP. This paper addresses the trade-off between current and future interests, taking into account the interactions between deforestation and agricultural productivity.

In Section II, a two-sector dynamic model for agriculture and forestry is proposed. The model maximizes the integral of the discounted utility of aggregate benefits over an infinite horizon. The agricultural yield function reflects the hypotheses that (1) ash additions and the rapid mineralization of organic matter after clearing and burning provide a sharp increase in available nutrients to the first crop planted after deforestation; (2) afterwards, crop yields decline and the land is

“From (27) note that p < F(O) if y’/’ > -c - 8&A. With p and &, 5 0 and A 2 0, this implies that y’/* 2 0.

FOREST DEPLETION M THE TROPICS 151

frequently abandoned and left to degrade, possibly causing additional downstream damage; and (3) fertility can be maintained by applying purchased inputs. The optimal rate of growth of deforestation is found to depend on two motives: the deforestation motive and a conservation motive. The deforestation motive is a one-time phenomenon which is measured by the marginal utility of deforestation. The conservation motive is measured by the net marginal utility of forest stock and represents the “flow of benefits” generated by the forestland over the infinite horizon. The rate of growth of deforestation is negative if the conservation motive is weaker than the preference for current deforestation. This will always be the case along the optimal path.

Theoretical results in Sections III and IV illustrate the effects of changes in net returns to agriculture and the marginal benefits of forest stock, as well as the discount rate, on the optimal rate of growth of deforestation and on the steady state forest stock. Greater returns to agriculture (relative to the forest stock) will lead to more deforestation early in the process and less deforestation later, implying that the rate of deforestation will have to be reduced more quickly for a given forest stock. The magnitudes of these comparative dynamic effects are shown to be dependent upon the shape of the aggregate agricultural yield function.

Steady-state, comparative static analysis is conducted under the assumption of a quadratic aggregate yield function. When the social discount rate is infinite. the optimal steady-state forest stock is shown to depend only on the initial forest stock and certain agronomic parameters. In general (i.e., when the discount rate is positive and finite), relative returns in agriculture and forestry play an important role. When the marginal benefits of forest stocks increase (relative to those in agriculture), the steady-state forest stock increases. Technological change serves to lower the steady- state forest stock while the effect of changes in the social discount rate is indeterminate. It depends on the relative returns in agriculture and the magnitude of the forest stock effects, as well as certain agronomic parameters.

Little is currently known about the magnitude of the parameters which link deforestation with aggregate agricultural productivity. Using a set of testable hypotheses, based on agronomic research, this paper has demonstrated the role which these parameters play in determining the socially optimal allocation of forest land over time. It has also identified those agronomic interactions of greatest economic significance. Future empirical work should be directed at testing the hypotheses posed in this paper and narrowing the range of uncertainty about the effects of deforestation on agricultural productivity in the tropics.

APPENDIX A: COMPARATIVE DYNAMICS

The results in (15) may be obtained by writing out the complete expressions for the relevant partial derivatives, thereupon applying (Al)-(AlO):


Furthermore, concavity of U in X and D requires that (U&L/b, - Uix) > 0. Finally, note that U, and U,, cannot be signed apriori (although we expect both to be positive):

i&= U,,[(P,(L-F)Z,)(-P,Z+P,X+ P,(L-F)Z,+B,)] +u,[-P,z,+P,(L-F)Z,,] 2 0

U,= &[-PAZ + P,X+ P,(L - F)Z,+ BF] 2 0.

From the above, we have

ab U xx aB, = - (u&J,, - u&)

[U, - U,,,Dq,] 2 0

and

ab U a(P,z- P,X) = (u&);~ u&)

[U, - U,,DT~] _< 0.

Since both net returns have equal but opposite absolute effects, their ratio (p = (PAZ - P,X)/B,) will have the same absolute effect:

ab U' xx 27 = (u*Jj&) - ?I&)

[U, - U,,DT~] I 0.

APPENDIX B

For purposes of developing the phase diagram in Figure 2A and 3A, the relevant system of equations is

I+ -D (3)

U D = - [tJ&DXl u&] [-sUD + (UC Gd>l* 01)

From (3) and setting # = 0, we have

aD aF &_()= 0.

Thus, the curve describing all the points where j = 0 is given by the horizontal axis. From (11) and setting b = 0, we have

aD Um- Y., 3T &o= su,, ' 0,

by (Al)-(AlO), and assuming U,, > 0.


For phase diagrams 2B and 3B the relevant systems of equations are

p:= -D

\c; = s+ - u,.

(3)

(9)

From (3) and setting P = 0, we have

and from (9) and setting 4 = 0, we have

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Documents

Forest Resource Depletion, Soil Dynamics, and Agricultural Development in the Tropics