analyses in a relatively short time-frame. In the forest, this means being able to project long-term
production levels for a given management regime, founded on empirical data over only a few years.
Our study utilized this approach to calculate the commercial value of enrichment planting with
data collected over three years. Such enrichment planting consists of planting and managing com-
mercially valuable tree species inside the existing forest. The study was carried out in the rainforest
at the biological research station of the Universi- dad Nacional Auto´noma de Me´xico in Los Tuxt-
las Veracruz, Mexico, about 30 km from the city of San Andre´s Tuxtla for more information see
Ibarra-Manrı´quez et al., 1997. The employed tree species, Pouteria sapota Jacquin H.E. Moore
and Stearn ‘mamey’; Sapotaceae, is native to this forest but occurs naturally only in low densi-
ties of about one tree per hectare. Its sweet fruits of 200 to over 600 g are found in local markets in
southern Mexico, as well as in Mexico City’s supermarkets. The native range of P. sapota is
probably southern Mexico to Nicaragua Pen- nington, 1990, but it has been introduced in
Florida, the Caribbean, South America, and Southeast Asia Morton, 1987; Oyen, 1991.
The seedlings of P. sapota have their maximum growth rate in the forest understory. In an ecolog-
ical analysis, the optimal canopy openness during the first 2 years after germination and transplant-
ing was 60 Ricker, 1998. Furthermore, existing adult trees can produce high fruit yields both
inside and outside the forest environment. These ecological characteristics make the species suitable
for forest enrichment.
2. Methods and results
The commercial value of a planted Pouteria sapota seedling was calculated as its expected net
present value NPV, a standard concept in eco- nomics and business administration:
NPV =
i = 1 MA
[F
i
S
i
P − Ce
− ri
] − K. 1
In this equation, i is the age as a discrete integer number the harvest is once a year. Planting
occurs at i = 0, and MA is the maximum age of production. The variable F
i
is the expected annual fruit yield of the tree at age i F
i
= 0 in the first
years, while S
i
is the expected survivorship at age i as a proportion of the initial cohort size S
i
ranges between 0 and 1. The economic parame- ters are the per-unit market price P, the per-unit
cost C for harvesting, transportation and com- mercialization, the discount rate r, and the ex-
pected present value K of all planting and management costs of the tree. Eq. 1 calculates
the expected net return of the annual fruit yield in a given year F
i
S
i
P – C, and discounts it by multiplying with e
– ri
to take the time delay for future harvests into account from an investor’s
perspective. Then it sums up the present values of all net returns, and subtracts the present value of
the management costs K. The separation of per- unit costs at harvest C and management costs K is
convenient because C must be paid per kilogram fruit at the time of harvest, while K must be paid
per tree or per hectare at times that are indepen- dent from the harvesting times. Consequently, it
would be difficult to convert K into a component of C.
Eq. 1 models the expected or average com- mercial value per tree of one cohort, planted at
present, and without any replanting to replace dead trees. Prices and costs are assumed to remain
constant over time. Fruit prices P were observed in the regional market of San Andre´s Tuxtla,
approximately every two weeks between March 1995 and April 1997. The average price was
Mex4.52kg [US1 = Mex9 – 10 ‘pesos’ in May 1999; n = 28 times the fruits were found in the
market]. The per-unit cost of harvesting, trans- portation, and marketing costs C was estimated
to be Mex1.62kg. Of this amount, the compo- nent for harvesting was Mex0.61 37.7 after
measuring an average harvesting time of 3.7 min kg with two persons and using a labor cost of
Mex30.00person for 8 h. The component for transportation
to San
Andre´s Tuxtla
was Mex0.56kg
34.6, when
using a
public pickup-truck service with a cost of Mex28.00 for
one person plus Mex28.00 for 100 kg freight. Finally, the component for comercialization was
estimated to be Mex0.45kg 27.8, assuming
that 12 h of labor on the market would sell 100 kg of fruits.
The discount rate r reflects the value of time for an investor e.g. a year is worth a 5 increase of
invested money. Empirical analysis shows that a 5 discount rate is accurate in the long-run, i.e.
considering decades in forestry Berck, 1979; Ricker and Daly, 1998. Finally, K sums up the
present values of all planting, management and equipment costs. Costs in later years are discounted
in the same way as fruit harvests in later years. K was estimated to be Mex30.00tree. This includes
Mex2.23seedling 7.4 nursery costs, Mex4.77 seedling 15.9 site preparation and planting
costs, Mex6.36tree 21.2 present value of the watering costs over the first three years, and
Mex13.27tree 44.2 present value of weeding costs [present values were calculated by multiplying
costs with e
– ri
; for more details see Ricker 1998]. These components for K sum up to Mex26.63tree
88.7, which, however, was rounded up to Mex30.00 to leave a margin for uncontemplated
costs. Watering was considered to become unneces- sary after 3 years, once the trees have established
themselves with sufficient roots and within a hu- midity-conserving forest environment. Weeding
was considered to be necessary every 3 months for the first 3 years, and once a year thereafter.
To determine annual fruit yield F
i
and sur- vivorship S
i
, we inventoried 100 naturally occur- ring P. sapota trees of varying trunk diameter over
3 years 1995 – 1997. The 100 trees were found in an area of about 5 × 5 km, both inside the forest
and in the open. Of these, we were able to measure the trunk circumference increments of 93 trees and
the annual fruit yield of 79 trees in at least 1 of the 3 years. Two trees were cut down in the study
period one of which formed part of the 93 trees, while six trees died or disappeared and in the latter
case were assumed to have died naturally. In a number of cases, tree owners did not give their
permission to measure fruit yield.
2
.
1
. A new method to determine tree ages and project yield
Studies of forest ecology and management depend heavily on knowing the ages of trees,
for example, to know at what age timber trees reach their harvestable size. In temperate zones,
annual growth rings in the trunk wood can be counted, but in the winterless lowland tropics no
reliable annual growth rings are formed Jacoby, 1989. We have developed a new method for
determining expected tree ages from annual incre- ment data, based on the Bertalanffy – Richards –
Chapman BRC growth model Pienaar and Turnbull, 1973; Zeide, 1993. The method can be
applied with data collected over only one year, though several years are preferable here 3 years;
see point 1 below.
The basic functional form of the BRC model for diameter growth is:
D = D
Max
1 − e
– aAge b
. 2
In Eq. 2, D is the trunk diameter, and D
Max
, a, and b are regression coefficients. D
Max
is the max- imum possible diameter reached at infinity age
here in cm; note that it is also a regression coefficient and not a measured value. The coeffi-
cient a can be interpreted as a growth rate parame- ter per year, and b as a shape parameter without
a unit.
Following, an equivalent functional form of the BRC model is developed that relates trunk
diameter and relative growth, without includ- ing age as a variable. First, Eq. 2 is solved for
Age:
Age = ln [1 − DD
Max 1b
] – a. 3
Next, an equation for the instantaneous relative growth dDdAgeD is derived from Eq. 2 by
taking the derivative with respect to Age and dividing by D:
RG = dDdAgeD =
[abD
Max
1 − e
– aAge b – 1
e
aAge
][D
Max
1 − e
– aAge b
] =
abe
aAge
− 1.
Then, in this equation of relative growth, the variable Age is replaced with Eq. 3. Finally,
the logarithm is taken to spread the data points and to avoid a nonlinear regression that would
unrealistically predict negative values of relative growth:
ln [RG] = ln {ab[11 − DD
Max 1b
− 1]}. 4
The empirical relative growth RG was calculated for each tree as the annual diameter increment
divided by the diameter D. Trunk diameters were determined from their circumferences circ,
measured at a marked height with a measurement tape D = circp. Circumference increments were
measured at the beginning and the end of a year. The final measurements were not taken exactly
365 days after the initial ones. Therefore, the increment measurements were slightly inter- or
extrapolated linearly to 365 days [annual incre- ment = 365measured
incrementnumber of
days]. Furthermore, in such data sets from tape measurements of irregular trunks with buttresses,
the measurement error can be large. Conse- quently, there are unrealistic outliers. Here, di-
ameter increments appearing to be \ 6.0 cm and 5
0 cm were eliminated. In this way, 13 4.6 of 282 original data points were eliminated.
The SYSTAT 5.03 software was used to adjust the parameters D
Max
, a, and b of Eq. 4 with nonlinear regression and ordinary least squares.
The result is shown in Fig. 1 R
2
= 0.61, n = 269
increments for 93 trees and 3 years. The software also estimates standard errors. Having determined
the parameters a, b, and D
Max
, the estimated age of each tree can be calculated from its diameter
D with Eq. 3. The largest tree was estimated to be 191 years old diameter at breast height = 131
cm. This age was set as the maximum age MA for fruit production. However, there was no con-
tribution of yields after 150 years to the seedling’s net present value; from a commercial viewpoint it
was too far in the future.
Fruit production was measured by actually har- vesting all fruits in the period from May to July
of each year over three years, and weighing them on a market scale. The annual fruit yield was
graphed over the estimated tree ages, and the BRC model employed a second time with nonlin-
ear regression to determine an average growth curve for the expected annual fruit yield per tree
F:
F = F
Max
1 − e
– cAge d
. 5
Similar to the regression coefficients in Eq. 2, F
Max
is the maximum possible annual fruit yield reached at infinity age here in kg; again not a
measured value, c represents a growth rate parameter per year, and b a shape parameter
without a unit. Eq. 5 represents a continuous function therefore the notation F rather than F
i
. However, it may be applied only with ages that
are 1 year apart in order to predict the annual fruit yields as discrete events.
The result of adjusting Eq. 5 to the data with nonlinear regression is shown in Fig. 2 R
2
= 0.33,
n = 225 harvests for 79 trees. Note that the BRC- model captures well the typical pattern of the
development of annual fruit yield, taking an aver- age path and remaining conservative without be-
ing unduly influenced by outliers.
A few precautionary notes are in place for applying this new method to estimate ages and to
project yields: 1. The new method for determining tree ages
relies on the assumption that the years of increment measurements are representative of
other years. In particular, climate fluctuations are of concern, as they can cause lower or
higher increments in some years. In this study, the average increments over 3 years were at
least relatively consistent, 1.1 cm in 1995, 0.8 cm in 1996, and 1.0 cm in 1997 for the 81
trees.
Fig. 1. Nonlinear regression adjusted D
Max
, a, and b in Eq. 4 for 93 trees of Pouteria sapota, measured over 3 years in Los
Tuxtlas R
2
= 0.61, n = 269. The diameter is at breast height.
Fig. 2. Measured and expected annual fruit yield of Pouteria sapota at Los Tuxtlas. F
Max
, c, and d of Eq. 5 were adjusted with nonlinear regression and using the average age estimates
for the 79 trees measured over 3 years R
2
= 0.33, n = 225.
Note that the continuous function of expected yield represents a discrete process, i.e. yields occur only once a year.
the sample size improves the estimate, each error term has to be uncorrelated with its
corresponding X-variable, here D and Age, respectively Maddala, 1992, chapter 9.1. We
calculated Pearson correlation coefficients for each of the three cases, none of which was
statistically significant [r
u4 − u5
= 0.02 n = 221,
r
u4-D
= – 0.02
n = 269, r
u5-Age
= 0.04
n = 225]. Thus, there is no evidence that these
assumptions were violated. 4. Finally, the three regression coefficients of the
BRC model are correlated with each other in Eqs. 4 and 5. The SYSTAT software calcu-
lated automatically these correlations, the re- sult
being r
D
Max
-a
= – 0.44,
r
D
Max
-b
= 0.70,
r
a-b
= 0.11 n = 269; and r
F
Max
-c
= – 0.72, r
F
Max
-
d = – 0.69, r
c-d
= 0.99 n = 225. Highly corre-
lated parameters
may cause
misleading confidence intervals when these are con-
structed on the assumption that each parame- ter individually presents a normal distribution
independently from the other two parameters. Here, the confidence intervals constructed un-
der this assumption in Table 1 lead to negative c and d, an impossibility in the BRC model.
Truncating c and d to zero leads to extreme situations that are of interest, but that do not
reflect 95 confidence limits: With c = 0, there is no fruit yield at all, and with d = 0, from
year 1 to the last year the fruit yield is the maximum one F
Max
.
2
.
2
. Predicting tree sur6i6orship Turning to survivorship S, the simplest model
that has been employed is the negative exponen- tial function Lieberman et al., 1985; Condit et al.,
1995:
S = e
– mAge
. 6
In this model, the expected annual mortality rate m of a cohort is assumed to be constant through-
out the tree’s life. Though this model is not accurate for very early and very late tree ages
when mortality is high, it is useful for the type of data employed here. One has to assume that at
i = 1 year, the seedling stock is well-established in the field, thus having passed already through the
2. A statistical problem arises from measuring the same trees repeatedly over years because
these measurements are generally expected to be correlated. A tree that grows well in one
year tends to grow well in other years. Here, the Pearson correlation coefficients for loga-
rithmic relative growth of the trunk diameters between years 1995 – 97 were all statistically
highly significant [r
95 – 96
= 0.78 n = 92, r
95 –
97 = 0.67 n = 84, r
96 – 97
= 0.82 n = 85]. To a
lesser extent, significant correlations existed also for fruit yields of the same trees between
years [r
94 − 95
= 0.50
n = 72, r
94 − 96
= 0.45
n = 71, r
95 − 96
= 0.31
n = 74]. While
parameter estimates remain unaffected by these correlations, variance estimators may be
biased, and confidence intervals may be inac- curate Sullivan and Reynolds, 1976; West,
1995. However, here the key objective was to estimate the regression parameters, and this
problem is only of secondary concern.
3. Carrying out the two nonlinear regressions for Eqs. 4 and 5 independently assumes that
their error terms i.e. residuals u are uncorre- lated Borders, 1989. Furthermore, to get a
consistent regression estimate i.e. increasing
M .
Ricker et
al .
Ecological Economics
31 1999
439 –
448
Parameter Original
Confidence Change of
New NPV in Change of
Ratio of Explanation
Changed NPV
parameter changes
parameter value parameter value
Mex interval
value a year
–1
0.0023363 Growth rate
Lower 95 –11.0
25.19 –51.0
4.7 0.0026237
0.0029111 Upper 95
+ 11.0
81.61 +
58.8 5.4
parameter for age determina-
tion 1.0590
0.9479 Lower 95
–10.5 143.13
+ 178.5
–17.0 Growth form
b parameter for
1.1701 Upper 95
+ 10.5
7.21 –86.0
–8.2 age determina-
tion Lower 95
–19.6 D
Max
cm 7.63
–85.2 –4.4
Maximum pre- 351.19
282.47 419.90
Upper 95 +
19.6 108.01
+ 110.1
5.6 dicted average
diameter Truncated
–100.0 c
–30.00 Growth rate
–158.4 1.6
0.093059 0.190115
Upper 95 +
104.3 410.12
+ 697.9
6.7 parameter for
annual fruit yield
87.318 Truncated
–100.0 2453.81
+ 4673.9
–46.7 Growth form
d year
–1
488.586 Upper 95
+ 459.5
–7.58 –114.7
parameter for –0.2
annual fruit yield
49.093 Lower 95
F
Max
kg –20.9
34.34 –33.2
1.6 Maximum pre-
62.102 75.112
Upper 95 +
20.9 61.05
+ 18.8
0.9 dicted average
annual fruit yield
No –50.0
120.34 +
134.1 –2.7
m Constant an-
0.02 0.01
No +
100.0 –4.10
–108.0 0.04
–1.1 nual mortality
rate 32.03
–37.7 P Mexkg
2.5 Per-unit price
4.52 3.83
Lower 95 –15.3
70.76 +
37.7 2.5
+ 15.3
Upper 95 5.21
Per-unit cost –61.7
79.46 +
54.6 –0.9
1.62 0.62
C Mexkg No
23.33 –54.6
–0.9 +
61.7 No
2.62 –40.0
258.22 +
402.4 –10.1
r Annual discount
0.05 0.03
No +
40.0 –4.10
–108.0 –2.7
No rate
0.07 K Mextree
30.00 10.00
No –66.7
71.40 +
38.9 –0.6
Present value 50.00
No +
66.7 31.40
–38.9 –0.6
per tree of all management
costs
a
NPV = net present value, calculated with Eq. 1; US1 = Mex9–10 in May 1999; ‘ change of parameter value’: 100 = original parameter value; ‘ change of NPV’: 100 = Mex51.40, the result with the original parameter values; ‘Ratio of changes’ = elasticity of NPV to parameter = change of NPV change of
parameter value.
initial phase of relatively high mortality. An early constantly low mortality can also be supported by
management. Late ages are of no concern because the discounting factor e
– ri
results in very low or no contributions of old trees to the net present
value. The critical parameter to be adjusted in Eq. 6
for a specific site and species is the constant annual mortality rate m. Over the three study
years, 6 of 100 individuals of a wide range of diameters i.e. ages died naturally at different
times 4 in the first year, 0 in the second, and 2 in the third. Therefore, m was estimated to be 0.02
for a planted cohort [ln [1 + 4100 + 096 + 2 963]. Mortality data on adult trees is difficult to
obtain in rainforest species with low densities because large survey areas are required to analyze
several populations of, say, 100 trees. However, our estimated annual mortality rate of 2 is
consistent with results from other studies on rain- forest trees involving larger tree propulations as
well as varying site and climate conditions Lieberman et al., 1985; Condit et al., 1995.
2
.
3
. Sensiti6ity analysis To analyze which parameters are critical in
determining the net present value, a sensitivity analysis for each model parameter is provided in
Table 1. The first column ‘Parameter’ shows the 11 input parameters, and the second column ‘Ex-
planation’ a short explanation about their mean- ing. The third column ‘Original parameter value’
presents the original input parameter values, re- sulting together in an average estimate for the net
present value of Mex51.40. Next, in column 4 ‘Changed parameter value’ an upper and lower
bound is chosen. When possible, 95 confidence intervals based on a normal distribution were
chosen to get these bounds column 5, ‘Confi- dence interval’. In two cases for c and d, the
lower confidence limits were unrealistically less than zero, and were truncated to be zero see
point 4 in Section 2.1. For parameters without standard errors m, C, r, K, ranges were chosen
so as to be realistically possible.
For each upper or lower parameter, the whole net present value calculation was carried out,
keeping all other parameters at their original value. In the case of a, b and D
Max
, lower and upper ages for each of these three parameters
were re-calculated for the fruit yield data of the 79 trees i.e. a total of six possible ages for each tree.
Subsequently, the second nonlinear regression with Eq. 5 was carried out for each of the six
cases, and the corresponding net present value was calculated each time with the new F
Max
, c, and d combined.
The sixth column of Table 1 ‘ change of parameter value’ calculates the percent deviation
of changed parameters from their original value. The seventh column ‘new NPV’ gives the new
net present value, and the eighth column ‘ change of NPV’ the percent change relative to
the original net present value of Mex51.40 per seedling. Finally, the last column ‘Ratio of
changes’ divides the ‘ change of NPV’ by the ‘ change of parameter value’. This ratio indi-
cates the sensitivity of the net present value to the parameter, i.e. the average percent change of the
net present value resulting from a 1 parameter change. In economic terms, this ratio would be
called the elasticity of the net present value to the parameter.
As an example, take the parameter a in Table 1. In nonlinear regression with Eq. 4, the parame-
ter a was calculated to be 0.0026237 ‘Original parameter value’. The 95 confidence limits of a,
assuming here
a normal
distribution, are
0.0023363 and 0.0029111. The column ‘ change of parameter value’ indicates that 0.0023363 is
11.0 lower than the original value of 0.0026237 [0.0026237 − 0.00233631000.0026237].
Using a = 0.0023363 instead of 0.0026237, with all other
parameters remaining at their original values, the net present value is Mex25.19 ‘New NPV’. The
column ‘ change of NPV’ shows that Mex25.19 is 51.0 lower than the original net present value
of Mex51.40 [51.40 − 25.1910051.40]. Fi- nally, the last column ‘Ratio of changes’ shows
that the elasticity of the net present value to the parameter a is 4.7. Each one percent-decrease of
the value of a decreases the net present value on average by 4.7 note that calculations were car-
ried out with more than one decimal so that 4.7 rather than − 51.0 – 11.0 = 4.6 resulted.
One may be concerned about several parameters varying at the same time. However, given multiple
and independent factors influencing the parameter values, one can expect that when one parameter is
above its
predicted average
value, another
parameter is likely to be below its predicted aver- age value. In this way, the effect of stochastically
varying parameters on the net present value is expected to be buffered to some extent. Neverthe-
less, the determination of the distribution and confidence limits for the net present value will be
an interesting future research topic.
3. Discussion
