Forest Conservation in Costa Rica
when Nonuse Benefits are
Uncertain but Rising
Erwin Bulte, Daan P. van Soest, G. Cornelis van Kooten,
and Robert A. Schipper
Stochastic dynamic programming is used to investigate optimal holding of primary tropical forest in
humid Costa Rica when future nonuse benefits of forest conservation are uncertain and increasing.
The quasi-option value of maintaining primary forests is included as a component of investment
in natural capital. Although the impact of uncertainty on conservation incentives is substantial,
our results indicate that a rising trend in future benefits and compensation by the international
community for beneficial spillovers are more important factors in determining optimal holdings of
forest stocks. Without compensatory payments, however, further deforestation may be warranted.
Key words: investment under uncertainty, tropical deforestation, nonuse forest benefits.
Forest conservation is an investment in natural capital that yields biodiversity, carbon sink
and tourism benefits, as well as net revenues
from the sustainable harvests of commercial timber and nontimber forest products.
The opportunity costs of forest conservation
are the foregone net returns to agriculture.
Under ideal conditions where governments
do not use primary forests as an outlet for
urban population pressure or as a source of
short-term revenue (van Kooten, Sedjo, and
Bulte), the objective of land use policy should
be to maximize social welfare by balancing
the marginal benefits and costs of forest conservation. Although there appears to be consensus that tropical deforestation is socially
excessive (Pearce, Putz, and Vanclay), there
is little empirical evidence to support or
refute this supposition. Existing empirical
work is often based on total or average values, rather than marginal ones (Ehui and
Hertel; Ruitenbeek). Further, most studies employ a deterministic framework that
Erwin H. Bulte is an associate professor in the Department of
Economics, Tilburg University, The Netherlands. Daan P. van
Soest is an associate professor in the Department of Economics,
Tilburg University, The Netherlands. G. Cornelis van Kooten is
a professor in the Department of Applied Economics and Statistics, University of Nevada and in the Agricultural Economics
and Rural Policy Group, Wageningen, The Netherlands. Robert
A. Schipper is an associate professor, Development Economics
Group, Wageningen University, The Netherlands.
Subject to the usual qualifier, the authors wish to thank two
anonymous referees and the journal editor for helpful comments
and suggestions.
assumes current and future benefits and costs
of forest conservation are somehow known
(Bulte, Joenje, and Jansen). In reality, surprisingly little is known about the future environmental benefits of forest conservation.
Major potential benefits of sustainable
forestry are related to carbon uptake and
storage and preservation of biodiversity, but
future values of these benefits are uncertain. Similarly, whereas there is evidence that
forest conservation yields substantial (transboundary) nonuse benefits, little is known
about the future demand for these natural
amenities. Such demand depends on future
developments regarding income growth and
the availability of substitutes, which are
inherently uncertain.
For the Atlantic Zone of Costa Rica, Bulte,
Joenje and Jansen conclude that there is currently too much forest area: “The government of Costa Rica may have set aside more
of its tropical forests than is socially optimal. Therefore, developing part of the natural heritage in the future that is currently
protected should receive serious consideration from policy makers” (p. 503). However,
they disregard uncertainty about future forest
benefits. In this article, we extend their analysis by explicitly recognizing uncertainty about
forest conservation benefits and that development implies that (some of) these benefits are lost forever. We model uncertainty as
Amer. J. Agr. Econ. 84(1) (February 2002): 150–160
Copyright 2002 American Agricultural Economics Association
Bulte et al.
geometric Brownian motion, and provide a
numerical application of quasi-option value
to the management of tropical forests. We
also integrate data from a detailed multidisciplinary model of agricultural activities into
a stochastic dynamic optimization model, and
provide a much-improved approximation of
the opportunity costs of forest conservation (benefits of agriculture foregone). Our
estimates reject Bulte, Joenje and Jansen’s
conclusion that further deforestation is economically justifiable.
The objectives of the article are, first, to
compute the (socially) optimal forest stock
in humid Costa Rica using detailed data
on shadow prices of agricultural land. To
establish whether current deforestation is
socially excessive, we compare the current
forest stock to two benchmarks: (1) the
domestically optimal stock when transboundary spillover benefits of conservation are
ignored, and (2) the globally optimal stock
that takes such benefits into account. Second,
we explore the impact of time trends and
uncertainty about forest conservation benefits on the incentive to conserve forests. We
quantify quasi-option value and assess its relative importance for forest conservation in
Costa Rica.
A Stochastic Model of Forest Conversion
In this section, we develop a stochastic model
of forest conservation and conversion that
is based on Conrad. Compared to net agricultural revenues, many benefits of forest
conservation are uncertain, especially those
related to environmental functions such as
watershed protection, carbon flux and biodiversity (but see the concluding section).
Let B denote the annual per-hectare benefits of sustainable forestry, which vary with
the “perspective” taken (Pearce, Putz, and
Vanclay). From a global perspective, these
benefits equal the sum of carbon, tourism
and biodiversity (and other nonuse) benefits, plus the benefits of sustainable harvesting
of timber and nontimber forest products.
A national analysis is narrower, typically
excluding carbon and nonuse benefits. We
assume that B is constant across the forest
(which assumes that the study area is homogeneous and small compared to total tropical forest area in the region), but that future
benefits are uncertain.
Forest Conservation in Costa Rica
151
Uncertainty about the benefits of forest
protection adds considerable complexity to
the issue of forest conservation, particularly
since conversion to agriculture constitutes
in some respects an irreversible event. The
combination of uncertainty and irreversibility implies that there is a premium associated
with caution and delay of land conversion.
One component of forest conservation that is
overlooked in deterministic analyses is quasioption value (QOV), which measures the
benefit from delaying development in the current period as more becomes known about
future benefits and costs in the next period
(Arrow and Fisher; Graham-Tomasi; Albers,
Fisher, and Hanemann). As the prospect of
receiving better information in the future
improves, the incentive to remain flexible
and take advantage of this information also
increases. That is, the expansion of choice by
keeping alternative options open and delaying development of primary forests (e.g.,
delaying loss of endangered species) represents a welfare gain to society.1
We assume that the only alternative use of
forestland is agriculture, and that the size of
so that A
=
the total land area is fixed at A,
At + F t, where At and F t, respectively, denote land in agriculture and forestry
at time t. We also assume that “new information” about the uncertain benefits of forest conservation becomes available at various
times, so that forest benefits B may be modeled as geometric Brownian motion (GBM):2
(1)
dB = B dt + B dz
where and are trend
and uncertainty
√
parameters, and dz = dt with ∼ N 0 1
(or dz is the increment of a Wiener process). Forest benefits are assumed to change
1
Quasi-option value is a different concept than option value
(OV), which is also related to uncertainty. OV is the added
amount a risk averse person would pay for some amenity, over
and above its current value in consumption, to maintain the
option of having that amenity available for the future, given that
the future availability of the amenity is uncertain. While OV
assumes uncertainty in supply, it derives from risk aversion on
the part of demanders. QOV assumes uncertain benefits, but is
derived under risk neutrality: as the prospect of receiving better information in the future improves, the incentive to remain
flexible and take advantage of this information also increases
(Graham-Tomasi, p. 595). QOV measures the benefit of information and remaining flexible by avoiding irreversibility.
2
Alternatively, we may assume that forest benefits are
described by a mean reverting process (Dixit and Pindyck).
However, it is not obvious what the long-run trend should be
for nonmarketed forest benefits, and furthermore the analysis
in Appendix B supports the assumption of GBM. By assuming
GBM rather than mean reversion, we may overestimate QOV:
strong mean reversion implies that there is less uncertainty about
future values of B (see below).
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February 2002
Amer. J. Agr. Econ.
over time at rate . The expected change is
EdB = B dt (where E is the expectations
operator), with variance 2 B 2 dt (Dixit and
Pindyck, pp. 70–71). Parameter can be positive (e.g., reflecting an increasingly important carbon sink function as atmospheric CO2
concentration rises), but it may also be negative (say, due to improvements in combinatorial chemistry that lead to a reduced need
for primary genetic material).
Total benefits of forest conservation, Mt,
are assumed to be proportional to forest size:
− At]. The
MB F = BtF t = Bt [A
ecological interlinkages between forest conservation and agricultural productivity are
ignored (see Ehui and Hertel). The opportunity costs of forest conversion are the foregone net benefits derived from the alternative
land use. It is possible to calculate the shadow
price of agricultural land, PA , as a function
of the area under cultivation, A. The shadow
price traces out the net marginal return to
land as forest is converted to agriculture, and
is specified as: PA A = A− , where and
are positive parameters. The parameter measures the net return to the “best” hectare
in the study region, and is determined by soil
quality, market conditions, and distance to
market. The net marginal return to agriculture declines as more forestland is converted
to agriculture, mainly because such land is of
lower quality and/or further away from markets. Total agricultural benefits, N A, are
given by the area under the marginal returns
function:
A
A
(2) N A =
PA a da =
a− da
a=0
a=0
=
A1− 1 − Land either remains forested or is converted
to agriculture. Then the region’s annual flow
of net benefits is simply the sum of agricultural and forest benefits:
(3)
At Bt
− At
At1− + BtA
=
1−
The social planner aims to maximize the
discounted flow of net benefits, taking into
account the uncertainty of lost benefits from
forest conservation. We model the decision
to deforest as a “threshold control model”
(Dixit and Pindyck, pp. 359–67). The underlying model differs somewhat from those of
Albers, Fisher and Hanemann, and Conrad,
because forest conversion is incremental
rather than all-or-nothing. Whereas Conrad
focused on a single conservation project and
determined the threshold level of forest benefits that justified continued conservation of
a given forest, we determine the optimal forest stock (i.e., an interior solution). Unlike
Albers, Fisher and Hanemann, who restrict
their analysis to three periods (but also consider agroforestry as an intermediate land
use option), we consider the optimal land
use problem for a risk-neutral central planner with an infinite time horizon.
As derived in Appendix A, the critical
value of forest benefits, B C , or the threshold
curve linking the stock of agricultural land
and the threshold value for the stochastic
state variable B, is given by:
r − A− − c
(4) B C A =
−1
r
where c denotes the forest conversion cost,3
and is the usual root of the fundamental quadratic that involves the discount rate
and the drift and volatility coefficients of the
stochastic process (1). A necessary condition
for conversion to be optimal is that r > ,
which means that the trend in forest benefits should not exceed the discount rate (see
Appendix A). As is clear from equation (4),
the critical value of forest benefits, B C , is a
decreasing function of the agricultural land
area, because of decreasing marginal returns
in agriculture.
The term / − 1 takes on values
between 0 and 1, and basically measures the
importance of uncertainty—it can be considered as a type of risk mark down. The
same holds for the critical value function
if forest benefits are assumed to be certain. The critical value under certainty is calculated by letting approach zero. Then
/ − 1 → 1 and (4) reduces to the standard
expected net present value (NPV) criterion:
B ∗ /r − = /rA− − c, where B ∗ /r −
are the discounted forest benefits foregone, with B ∗ the threshold NPV for the
deterministic case (ignoring QOV). As uncertainty rises, the critical level of forest benefits falls, warranting additional deforestation,
which explains why there is a mark down
rather than mark up.
3
Marginal land conversion costs c may be negative if there is
a positive one-time net benefit from logging the site that exceeds
the costs of preparing the harvested site for crop production.
Bulte et al.
Forest Conservation in Costa Rica
153
N
10
7
2
5
8
1
4
12
0
10
9
20 Kilometers
3
6
11
Figure 1. Humid Costa Rica Atlantic zone: Twelve subregions
The Atlantic Zone of Costa Rica
We apply our model to the humid Atlantic
zone of Costa Rica for which detailed data
are available on the benefits of agricultural
development (Schipper et al.) and limited
data on the benefits of forest conservation
(Bulte, Joenje, and Jansen). We compute the
forest benefits threshold function, B C A,
compare it to actual benefits, and then use
it to compute optimal forest stocks in the
region.
Benefits of Agricultural Conversion
Foregone net agricultural revenues are the
most important opportunity cost of sustainable forest exploitation. As agricultural area
expands, crop production encroaches on land
with increasingly lower net marginal benefits
(either because higher quality land is used
first or because of increasing transportation
costs) and farmers cultivate crops with lower
(financial) returns. To model the decreasing marginal benefits of deforestation, we
employ a detailed linear programming (LP)
model developed during more than a decade
of multidisciplinary fieldwork in the study
region (Schipper et al.). The objective in
the LP model is to maximize the discounted
value of social welfare, defined as the sum
of producers’ and consumers’ surpluses from
agricultural production, subject to biophysical and economic constraints. For a given
agricultural area, the LP model enables one
to compute the shadow prices for agricultural
land at the margin.
In the LP model, the Atlantic zone of
Costa Rica is divided into 12 subregions
(figure 1). There are three types of land
quality or soils (fertile & well drained, fertile & poorly drained, and fertile), nine crop
and five pasture activities, and a variety of
farm management practices. Overall there
are 1,352 possible crop systems and 1,756 pasture systems in the model. The agricultural
land use model for Costa Rica captures interactions via the labor market (expanding of
labor-intensive crops drives up the wage rate)
and has downward-sloping demand functions
for agricultural crops where applicable. Additional details are found in Schipper et al. It
is worth emphasizing that the LP model has
thus far been used to determine the optimal
allocation of crops, and not to compute the
extensive margin of agriculture.
An important assumption of the LP model
is that agriculture is sustainable, so that productivity does not decline over time. On that
basis, the LP model generates the annuity
shadow prices of land. We simulate the LP
model for an area of 320,000 ha, applying a
step size of 1,000 ha.4 Using the output from
the LP simulations as the input for regression
4
To ensure convergence, we omit 25,000 ha of the best farmland. Agricultural benefits as a function of land in production are
defined by PA = A . When > 1, convergence may not occur
(see equation 2), but when the very best soils are deleted from
the sample, the condition < 1 is satisfied. Forest conservation
can never compete with agriculture on the best soils in any event.
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February 2002
Amer. J. Agr. Econ.
Table 1. Annual Benefits of Primary Forests, Atlantic Zone, Costa Rica
$ ha−1 year−1
Item
Production Function
Sustainable timber harvests
Sustainable extraction of
nontimber forest products
Sustainable ecotourism
Regulatory Function
Carbon uptake and storagea
Habitat Function
Existence value
Biological prospecting
(incl. Pharmaceutical
value)
$75
50
20
5
$105
105
$20
10
10
a Assumes
100 tonnes of carbon stored per ha, $15 per tonne C, and a
7% discount rate.
analysis, we estimate the following marginal
function for agriculture (in 1998 US$):
(5)
lnPA = 1576 − 089 lnA
378 −253
where the t-values are provided in parentheses and the adjusted R2 is 0.69.5
Benefits of Forest Conservation
Forests play an important role in the production of timber and nontimber forest products,
ecological services and wildlife habitat. Estimates of the monetary values of these services are available from Bulte, Joenje and
Jansen, who indicate that these benefits may
be considerable for the study region—$200
per hectare in annual benefits of natural forest conservation (table 1).6 These are global
values and include transboundary benefits for
which actual compensation may or may not
be forthcoming.
There is little information about the uncertainty of the benefits in table 1; little is known
about the trend and uncertainty parameters and . No time series data on shadow prices
of carbon and species are available, nor are
there reliable estimates of future nonuse values. Only time series data on tourism benefits are available. In Appendix B, we show
5
Inclusion of a quadratic term increases R2 to 0.94, but this
results in a marginal function that no longer guarantees convergence of the optimal control model.
6
Our estimate of forest benefits of $200 per ha per year is
much lower than that of Costanza et al. (1997), who provide a
very high estimate of the value tropical forest protection that
exceeds ten times that provided here.
that tourism benefits can be modeled as geometric Brownian motion, and that the parameters associated with this dynamic process
are roughly = 005 and = 010 (see also
Conrad). It is heroic to assume that these
parameter values apply to the sum of all forest benefits over time, but in the absence of
further data we employ these parameter values for a base case scenario, using sensitivity analysis to encompass a wider range of
parameter values.
Determining Optimal Forest Stocks
Substituting (5) in (4) and using the information on the benefits of forest conservation provides an expression for the critical
value of forest benefits as a function of agricultural land. Assuming a discount rate of
7%, a particular threshold curve, B C A, is
depicted in figure 2. For convenience, we also
depict the threshold curve for the deterministic (NPV) case, B ∗ A. This curve is located
entirely above the stochastic threshold curve,
and hence uncertainty matters when deforestation is modeled as an irreversible process:
the criterion warranting additional deforestation is more stringent when uncertainty is
taken into account.
The dashed line in figure 2 represents the
barrier function; below this line additional
deforestation is warranted when uncertainty
is taken into account. To the northeast of
this line, forest benefits (as measured on
the vertical axis) are such that conservation
of the remaining forest area is optimal. As
soon as forest benefits drop below the critical value in (4), just enough additional deforestation occurs to move horizontally back
to the curve. The vertical distance between
the deterministic critical NPV and that under
uncertainty can be interpreted as the quasioption value of retaining natural forests.
The Costa Rican government is concerned
only with domestic benefits of forest conservation, and it ignores spillover benefits
for the most part. The reason is that, at
the present time, there is no comprehensive
scheme of international transfers that compensates Costa Rica for the carbon sink or
biodiversity benefits of its forests, although
some compensatory payments admittedly do
take place. Specifically, assume that the government takes only the production function of natural forests into account, and
not the regulatory function and existence
values, so forests are valued annually at
Bulte et al.
Forest Conservation in Costa Rica
155
1.2
Critical Forest
Benefits ($000s)
1
0.8
Inertia Region
0.6
Certainty
0.4
Uncertainty
0.2
Region of immediate deforestation
0
0
40
80
120
160
200
240
280
320
Agricultural Land ('000s ha)
Figure 2. Critical values of forest benefits under certainty = 0 and uncertainty = 0125,
with = 0025, r = 007, = 6990062, = 0887, c = 0
$75/ha (table 1). Using the parameter values from the analysis of tourism in Costa
Rica as a reasonable first approximation for
the stochastic model ( = 005 and = 01;
see Appendix B), we find that the optimal
forest stock in the Atlantic Coast region is
220,000 ha (table 2). This compares to an
extant forest stock of about 80,000–100,000
ha. Consistent with many assertions in the literature, it appears that tropical deforestation
is indeed excessive. Only for low trend and
uncertainty values, additional deforestation is
warranted.
The model allows us to analyze some spatial dimensions of deforestation. We find that
some forest in the lowlands (in the core
of the study region) should be cleared and
that forest conservation should occur at the
fringes (e.g., in the mountainous and swamp
areas). This is consistent with actual deforestation patterns in the region. The shaded
areas in figure 1 indicate remaining forest
areas. The domestically optimal forest stock
is arrived at by keeping regions 5 and 7–12
completely forested; regions 2 and 6 with
Table 2. Domestically Optimal Forest
Stocks, Various Parameter Values with
= 320000 ha
A
= 0000
= 0025
= 0050
= 0075
= 0100
= 0125
= 000
= 0025
= 005
0
0
0
0
15000
36000
37000
42000
53000
68000
83000
99000
211000
212000
213000
216000
220000
225000
more than 80% forest cover; regions 1 and 4
about 70% forest cover; and region 3 some
60% forest cover. There are issues of forest fragmentation and quality, as mountainous sites are likely to support different forest
systems than those in the lowlands. Nonetheless, without better data on forest conservation amenities, we have no basis for assigning
different values for forest preservation to different forest types.
There are many reasons why too much forest may have been cleared for agriculture
(see Brown and Pearce; van Kooten, Sedjo
and Bulte), and one of these is apparent from
table 2. If governments ignore the trend in
nonuse benefits and quasi-option value, the
perceived optimal forest stock is substantially
smaller than the current stock or the truly
optimal stock. Indeed, complete deforestation
may appear optimal when a deterministic
approach is adopted that ignores uncertainty
and trend in nonuse benefits. The results
are highly sensitive to assumptions about the
trend and uncertainty parameters, so we are
reluctant to draw any firm conclusions about
whether the current stock is too large or
too small. Both are possible. If the trend
is smaller ( = 0025, with = 01), then it
appears that current forest stocks are about
right. If is below 0.025 and/or is below
0.1, then the optimal forest stock would be
below the current stock even when QOV is
taken into account.
The results from table 2 indicate that outcomes are more sensitive to assumptions with
respect to the trend parameter than to
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February 2002
Amer. J. Agr. Econ.
Table 3. Globally Optimal Forest Stocks,
= 320000 ha
Various Parameter Values, A
= 0000
= 0025
= 0050
= 0075
= 0100
= 0125
= 000
= 0025
= 005
176000
186000
195000
203000
210000
217000
218000
219000
223000
228000
233000
238000
275000
276000
276000
277000
279000
280000
assumptions about the uncertainty parameter . This is consistent with Albers, Fisher
and Hanemann, who report that QOV only
constitutes 1.6% of total economic value for
the benchmark case in their Thailand study.7
If this result is confirmed by other studies,
much simpler models that capture only trend
but ignore uncertainty may be sufficient for
policy analysis. Estimating the magnitude of
these parameters may yet be an important
direction for future research.
Now assume that the government of Costa
Rica takes transboundary spillovers into
account, perhaps because of compensation
by the international community. The forest is
now valued at $200/ha per year (table 1). The
optimal forest stocks for this case and the
same parameter combinations as per table 2
are displayed in table 3. The results unambiguously indicate that socially optimal forest
stocks are greater than current stocks.
Discussion and Conclusions
Because the benefits of forest conservation
are uncertain, and deforestation is to a certain extent irreversible, deterministic cost–
benefit analyses produce biased outcomes
because they overlook quasi-option value as
a legitimate component of the return to
investments in natural capital. In this study,
therefore, the benefits of investments in natural capital were modeled as a stochastic, nonstationary process, but with the addition of
an upward trend. Although uncertainty about
future forest benefits has a significant positive
impact on optimal forest stocks, our analysis
7
However, Albers, Fisher and Hanemann mention that,
despite its small size, QOV value creates a “dramatic difference in the optimal land use pattern.” Conrad (p. 101) also
reports that the critical amenity value justifying preservation of
the Headwaters Forest was “particularly sensitive to changes in”
the discount rate and trend parameter.
indicates that the trend in nonuse values is likely more important than uncertainty
per se.8 When uncertainty and trend are
taken into account, it appears that too much
primary tropical forest has been converted
to agriculture in the humid region of Costa
Rica, supporting arguments that more of
the natural environment should be preserved
than indicated by current valuations (Barrett;
Porter; Fisher, Krutilla, and Cicchetti).
The results presented here also differ considerably from those of Bulte, Joenje and
Jansen, who found that extant forest stocks
in the same region of Costa Rica were excessive, even from a global perspective. These
authors used five observations and a linear
specification to estimate the shadow opportunity cost of agriculture, and they disregarded
uncertainty and the upward trend in nonuse
benefits.
The rate at which sustainable forest benefits increase and the actual payment of compensation by countries benefiting from Costa
Rica’s efforts to protect tropical forests are
more important than quasi-option value. For
low values of the trend parameter, additional deforestation may be warranted from
a domestic perspective; for high values this
conclusion does not hold. In contrast, if compensatory payments are made (say for carbon
sink functions), the optimal level of forest
stocks is unambiguously higher than current
stocks (table 3). This suggests that international compensation for externalities from
conservation of natural capital need to be
taken seriously, and institutions and incentives need to be developed to bring them
about.
To provide some perspective, we calculate
the domestic benefits of land use in agriculture and forestry for both the optimal and
extant forest stocks (assuming = 005 and
= 010). The domestic optimum forest stock
is 220,000 ha, and the current stock is about
100,000 ha (table 2), whereas the globally
optimal stock is some 279,000 ha (table 3).
By integrating the annual profit flow (equation (3)), we calculate the discounted value of
future net benefits (assuming instantaneous
adjustment) from the perspective of Costa
Rica and the world. The results are presented in table 4. There is no difference in
the domestic and world value of agricultural
8
The assumption of GBM implies that we may even overestimate true QOV if the true dynamics are, for example, a mean
reverting process.
Bulte et al.
Table 4.
Forest Conservation in Costa Rica
157
Domestic and Global Net Discounted Revenues by Sector (106 1998 US$)
Current
(100,000 ha)
Sector
Agriculture
Forestry
Total
Domestically Optimal
(220,000 ha)
Globally Optimal
(279,000 ha)
(i)
Costa Rica
(ii)
World
(iii)
Costa Rica
(iv)
World
(v)
Costa Rica
(vi)
World
3548
375
3923
3548
1000
4548
3245
825
4070
3245
2200
5445
2934
1046
3980
2934
2790
5724
revenues as there are no spillover benefits.
The difference between the domestic benefits for the current versus the domestically
optimal forest stocks (columns i versus iii)
may be interpreted as the “cost” of not having appropriate land use institutions in place.
From table 4, lack of appropriate institutions
results in a domestic loss of $147 million
in net present value terms. The global loss
of excessive deforestation amounts to about
$1.176 billion (columns vi versus ii).
If it were possible to travel back to the
time before modern agriculture began in the
study area, what transfers would have been
needed to induce Costa Rica to choose the
globally optimal stock of primary forest?
From table 4 (columns iii and v), the minimum once-and-for-all transfer that should
have been required to compensate Costa
Rica to adopt the globally optimal level of
forest protection is US$90 million, or US$6.3
million annually (at r = 7%). The international community’s maximum willingness
to pay for the forest size to be increased
from the domestic to the global optimum is
$279 million ($19.5 million annually; compare
columns iv and vi). However, this assumes
that appropriate instruments or incentives
are in place domestically to ensure that
landowners take into account spillover benefits within the country. If all externalities
of forest conservation are to be compensated by the international community, an
annual transfer of US$1.74 billion (=$125/
ha × 279,000 ha) is required.
These results clearly suggest that choosing the appropriate terms of reference for
implementing the Clean Development Mechanism, say, has important distributional consequences (Karp and Liu). They also indicate that substantial amounts are at stake,
highlighting why it is virtually impossible
to draft international agreements that are
acceptable to all participants. So far, patchwork policies, such as project-level funds
made available through the Global Environment Facility, have been preferred, perhaps
because of their relatively modest costs.
The total outstanding debt of Costa Rica
amounted to about $4.2 billion in 1999
(World Bank), so a “total” transfer of some
$1.7 billion to internalize all spillover benefits constitutes some 40% of the total debt,
a sizable amount. In comparison, the Global
Environment Facility approved $8 million for
Costa Rica in 2000 to provide financial incentives for the conservation of primary forests
and encouragement of sustainable management in secondary forests (World Bank).
Globally, about $6.0 billion annually is currently spent on nature conservation worldwide (James, Gaston, and Balmford). This
further indicates that substantial increments
in international transfers are needed when
external effects are to be fully internalized.
Some caveats remain. First, future benefits of agricultural development may also be
uncertain and subject to trends (see Porter;
Zinkhan), and this will affect the magnitude
of the markup term associated with uncertainty and irreversibility. Theoretically, this
could reverse some of the results. Because we
believe that uncertainty about preservation
benefits is more important than uncertainty
about development benefits, we focused
exclusively on the former, but extending the
model to allow for multiple sources of uncertainty is clearly desirable.
Second, we may underestimate the true
opportunity costs of conservation, because, as
a result of data availability, the set of alternative land use options was restricted to those
that were considered sustainable. It has been
observed, however, that many landowners in
the region prefer to mine their soils and then
abandon the land. Such behavior may be
rational, because for positive discount rates
unsustainable production may yield a higher
net present value than sustainable agriculture
(see also McConnell). Underestimating the
158
February 2002
true opportunity costs of forest conversion
implies overestimating the true optimal forest stock, although future generations are left
with cut forests and depleted soils. Further,
the current study does not incorporate possible positive feedback effects of forest conservation on agricultural production. Indeed,
it is assumed that successive marginal increments to the agricultural land base may be
regarded as independent little projects, which
is at odds with empirical evidence reported
by Ehui and Hertel. We may underestimate
the true optimal forest stock for this reason.
These considerations need to be taken into
account in future modeling efforts.
Finally, while we assume that forest conversion is “irreversible,” it is clear that this
does not hold for some of the forest functions discussed above, as they may be equally
well served by secondary forests. Notably, for
mitigating climate change, if storage in wood
products is taken into account (not currently
permitted under Kyoto rules), it may not
matter whether primary or secondary forests
sequester carbon; yet, the carbon uptake and
sink functions account for a significant proportion of the benefits of forest conservation.
For this reason, we may overestimate the
importance of QOV, and the benefits of tropical forest conservation. This further suggests
that time trends are more important determinants of optimal forest stocks than uncertainty considerations.
[Received July 2000;
accepted March 2001.]
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Appendix A
Solving the Stochastic Model
The social planner maximizes the discounted value
of land use, W , by choosing the optimal amount
of land to convert from forest to agriculture in
each period, U t. Deforestation is warranted
when forest benefits drop below a certain threshold (boundary) value, which is a decreasing function of the amount of land already in agricul = At + F t, with F t the
ture, At, where A
area in forestry. Benefits of forest conservation are
denoted B.
Consider a short interval dt. Given (stochastic)
changes in per-hectare forest benefits over time,
dB, forest benefits may fall below the threshold
value. In that case, a new optimal land area A
can
be calculated, with optimal deforestation U ∗ equal
to A
− A. Suppressing time notation, W A B
equals the sum of benefits over the period dt and
the maximized discounted value of land use thereafter. Leaving open the possibility that in the next
period forest benefits change dB, so that agricultural land should be adjusted to A
, W equals:
(A1)
W A B
= max
A
− A dt
A1− + BA
1−
1
!EW A
B + dB
1 + r dt
− cA − A" +
Here r is the continuous-time discount rate and
A
maximizes the RHS of this expression (which
may or may not exceed A). Equation (A1) gives
the maximized net discounted benefits, taking
into account that the planner can adjust agricultural area in response to changes in forest conservation benefits (but only upward because of
irreversibility). Alternatively, this equation can be
written as:
(A2) rW AB = max
A
159
A1− +BA−A
1−
−cA
−A+
1
E dW AB
dt
where dt1 E dW A B is the expected gains of
holding onto forestland. Land can be viewed as
an asset that should pay a normal return per unit
of time. The annual dividend from holding onto
the land is equal to the sum of the period’s net
land benefits and the expected capital gains (the
RHS of A2), which should be equal to the normal
return on the value of the asset, rW (the LHS of
A2). Maximizing (A1) with respect to additional
land under cultivation A
yields EWA A
B +
dB = c, where WA = #W /#A. As dt → 0, the
optimization decision tends to EWA A
B =
c. If the expected discounted marginal value
of agricultural land exceeds marginal conversion
costs, EWA A B > c, agricultural land will be
expanded U ∗ = A
− A to the point where equality is restored. If EWA A B ≤ c, ex post too
much deforestation has taken place and the planner will choose not to expand agricultural area further U ∗ = 0.
Suppose that forest benefits are such that the
optimal policy rule is not to deforest. Using A
= A
in equation (A1), the value of all land is given by
− A dt
A1− + BA
(A3) W A B =
1−
+
1
!EW A B + dB"
1 + r dt
Applying Ito’s lemma, the expected value of the
change in forest benefits is
(A4) EW A B + dB
1
= E W A B + WB dB + WBB dB2
2
1
= W A B + BWB dt + 2 B 2 WBB dt
2
Substituting (A4) in (A3) and letting the higher
order terms of dt go to zero yields
− A + BWB A B
A1− + BA
(A5)
1−
1
+ 2 B 2 WBB − rW A B = 0
2
To solve the homogeneous part of differential
equation (A5), representing the value of having
the option to deforest, try W h = ZB . This yields:
1 2 2
1 2
−
− −r = 0
(A6)
2
2
with roots 1 2 = 21 − 2 ± 21 − 2 2 + 2r2 , where
1 (2 is the positive (negative) root.
160
February 2002
Amer. J. Agr. Econ.
The homogeneous solution is of the form
W h A B = Z1 AB 1 + Z2 AB 2 . As B increases, the value of the option to develop land
decreases because the probability that B falls
below critical level B C · decreases. Therefore,
Z1 A = 0. The particular solution, representing the expected value of never deforesting, is
A−A
+ r1−
A1− , so that the value
W p A B = Br−
of the total land area equals:
(A7) W A B = Z2 AB 2 +
A1−
r1 − − A
BA
+
r −
Note that r > should hold to let the discounted
stream of forest benefits converge.
The constant of integration, Z2 A, and the critical value of per-hectare forest benefits that trigger additional forest clearing, B C , can now be calculated using two conditions that should hold at
the boundary. First, the value-matching condition
states that the expected marginal benefits of clearing should equal the marginal clearing costs:
(A8) B C 2
BC
#Z2 A −
+ A −
= c
#A
r
r −
Second, the smooth-pasting condition requires
that the marginal benefits of land clearing and its
marginal costs meet tangentially at the threshold
(Dixit and Pindyck, pp. 130–32, 364). In mathe#c
A
matical terms, this implies #W
= #B
. Since c is a
#B
constant, the smooth-pasting condition boils down
toWAB = 0. Differentiating (A7) with respect to
B C gives:
1
#Z A C −1
(A9) 2 2
B −
= 0
#A
r −
Combining (A8) and (A9) yields equation (4) in
the text.
Appendix B
Testing that Tourism Exhibits GBM
Using data on the number of visitors to Costa Rica
for the period 1975–95, we test whether tourist
benefits follow GBM.9 Denoting tourist numbers
by R, we first test if lnR follows ordinary Brownian motion. Next, the appropriate parameter values can be calculated (e.g., see Conrad). Following Stewart (pp. 199–203), we estimate unrestricted and restricted regression equations and
9
The data were gathered from the UN Statistical Yearbooks
(various issues) and from the World Tourism Organization’s
Yearbook of Tourism Statistics (various issues). We have also
examined data on tourist revenues, finding similar results. The
datasets and the regression results are available from the authors
on request.
Table B1. Estimation Results for Restricted
and Unrestricted Models (t-values in parentheses)
Dependent Variable:
Change in log
Tourist Numbers
Coefficient
Unrestricted
&
Restricted
−13711
−1790
00008
1828
−0122
9414
0528
2361
0374
0120
'
(−1
)
R2
SSR
0022
1032
0492
2516
0250
0144
use an F -test to determine whether tourism numbers exhibit GBM. Denoting r = ln R, the unrestricted regression equation is:
(B1)
rt − rt−1 = & + <t + * − 1rt−1
+ )rt−1 − rt−2 + t The null hypothesis is that r follows Brownian
motion, or H0 , <= 0, * = 1. The restricted model
is:
(B2) rt − rt−1 = + )rt−1 − rt−2 + -t From the regression results in table B1, the F statistic is calculated to be 1.7. The Dicky–Fuller
critical F -value at the 5% level (for T is lower
than 25) is 7.24, so the null hypothesis of GBM
motion cannot be rejected (Stewart, p. 203).
Having established that changes in tourist numbers exhibit GBM, we proceed by estimating the
relevant trend and uncertainty parameters. The
first derivative of ln R is:
(B3) d ln R =
1
1
dR2 dR −
R
2R2
Substituting dR = R dt + R dz in (B.3) and
applying Ito’s Lemma, gives:
(B4) d ln R = dt + dz − 21 2 dt
= − 21 2 dt + dz
= a dt + s dz
The trend and uncertainty parameters are determined using the relationship between a, s, and
as presented in (B4). Estimates of a and s are
simply equal to the mean and standard deviation
of the d ln R series, and are found to be equal to
0.049 and 0.095, respectively. Hence, = 0054 and
= 0098.