Journal of Environmental Economics and Management 46 (2003) 86–105
Options, uncertainty and sunk costs:
an empirical analysis of land use change
Todd Schatzki
LECG, 350 Massachusetts Avenue, Suite 300, Cambridge, MA 02139, USA
Received 13 June 2000; revised 15 December 2001
Abstract
In a real-option model of land conversion incorporating return uncertainty and sunk costs, optimal
conversion thresholds are significantly higher than those from expected net present value models not
accounting for these factors. Empirical tests of conversion from agriculture to forest suggest that
landowners value the option to convert when making conversion decisions. Higher uncertainty in returns to
all potential uses and lower correlation between shocks to agricultural and forest returns decreases the
likelihood of conversion. Estimates indicate a significant impact on conversion decisions, with
approximations of option values ranging from 7% to 81% of the expected value of the land asset.
r 2003 Elsevier Science (USA). All rights reserved.
Keywords: Real options; Land use; Uncertainty
1. Introduction
Existing empirical analyses of land use conversion typically assume deterministic decisionmaking based on the expected net present value (ENPV) of returns to alternative land uses. These
models, however, may be inadequate to either explain observed frictions in land use conversion or
provide a reliable basis for projecting the effectiveness of future policies. Developing better land
use decision-making models becomes increasingly important as public policies increase reliance on
use land conversion programs to achieve environmental policy goals, such as habitat preservation
or carbon sequestration from tree planting.
This paper examines the effects of uncertainty and sunk costs on conversion decisions. Realoption investment models show that when returns are uncertain (i.e., exhibit some random walk
component) and costs are sunk that the returns required to induce investment are significantly
higher than in models based on an ENPV approach [11]. Land conversion represents an
E-mail address: [email protected].
0095-0696/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.
doi:10.1016/S0095-0696(02)00030-X
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
87
investment to switch production from one use to another, where conversion is optimal when the
relative returns between uses rises above a conversion threshold.1 Under the ENPV approach,
landowners convert when the present value of the discounted stream of returns from the
alternative use exceeds that of the current use, net of conversion costs. The real options approach,
however, considers the fact that sunk costs cannot be recouped and conversion decisions can be
delayed. When landowners can delay conversion, more information about future returns is gained
(specifically, expectations about returns are updated) before making the irreversible investment.
As a result, the option to convert land uses is valuable. Optimal conversion thresholds from real
options models thus reflect both the expected relative returns to alternative land uses and the
value of the conversion options.
The real option model provides a potential explanation of the friction in land use decisions
observed in previous studies. Conversion decisions in these studies generally appear relatively
unresponsive to changes in the underlying returns to alternate land uses, particularly between
agriculture and forested uses. Stavins and Jaffe [30], for example, estimate that only 32–69% of
land owners in the Mississippi Delta convert land when it is optimal, where the underlying
decision-making is based on an ENPV decision-making model. Their findings suggest the presence
of factors not accounted for in either their decision-making model or their estimation of economic
returns. Other researchers have similarly noted these land conversion frictions based on findings
of low elasticities of land use between forest and agriculture [20,22].2
Friction in land conversion is also evident in the performance of public land conversion
programs. The Conservation Reserve Program (CRP), for example, has had limited success
promoting cropland conversion to more permanent uses, including forests. The CRP pays rental
fees and a portion of conversion costs to landowners converting land to more conserving uses,
such as grassland, forest, wetlands, or other habitat.3 This program has had difficulty achieving
initial program targets for forest enrollment despite mid-stream changes in eligibility requirements
giving preferential treatment to forest conversions [8,19]. At present, conversions to forests
comprise only 7% of enrollments, which is only 40% of the minimum program enrollment goals
for forests [19]. In comparison, about 85% of land enrolled in CRP has been converted to
grasslands. Other US programs, such as the Agricultural Conservation Program or Forestry
Incentives Program, have experienced similar difficulties promoting more permanent conversions,
particularly to forestland [8].
A number of other hypotheses have been offered for these frictions, including: non-pecuniary
returns to landowners [21]; liquidity constraints; linkages between human capital and land; spatial
1
Cappozza and Li [5], Geltner, Riddiough, and Stojanovic [13], and others use the real-option approach to model
development of agricultural land for housing, while Albers [1] examines tropical deforestation as an irreversible process.
Schatzki [27] develops a theoretical model of conversion between uses appropriate for forest-to-agricultural conversions
in which (1) profits to both uses are uncertain (rather than only the developed use), and (2) conversion can be reversed,
although the costs of conversion are sunk.
2
There is other suggestive evidence. Cubbage and Haynes [7], in their survey of timber supply, report that forest
stumpage elasticities for land in the southern US are about 0.47 for the forest industry and 0.345 for non-industrial
owners. Such estimates include management responses, such as forest thinnings and more active replanting, which are
likely to be more elastic in comparison to land use responses.
3
Farmers actually submit bids, although, over the period studied, rental rates were almost uniformly at the county
bid cap since all bids were accepted.
88
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
land constraints; and transaction costs [27,29]. Relatively little empirical analysis, however, has
examined these factors. Land use hysteresis resulting from the effect of options on decision-making
may explain these observed frictions. Land use hysteresis occurs when land converted as a result of a
shift in relative returns does not convert back when relative returns revert to original values. The
asymmetric effect of options on conversion between land uses may cause such hysteresis.
Characteristics of land conversion decisions suggest that options should be an important factor
in decision-making. First, conversion decisions can be delayed without sacrificing the option to
convert. Second, the initial conversion costs (including tree planting, ‘‘sunk forest rents’’,4 and
other habitat restoration measures) are sunk. Third, the cost of converting back to agriculture
(e.g., removal of stumps) may be large. Note that for some types of land use conversions, these
costs may be much lower, thus preserving the value of the option to convert back to the original
land use. Forests represent a relatively permanent conversion, with high conversion costs; in
contrast, grasslands have low conversion costs, since grassland can be easily plowed under and
put back into production, and a temporary shift to grassland also has long-run soil productivity
benefits. Differences in the cost of converting back to agriculture may explain the observed
preference of grasslands to forests in the CRP program. Since landowners can shift land back into
production after 10-year CRP contracts expire, low conversion costs increase the value of the
option to convert back into agriculture.
To examine whether option values affect landowner decision-making, empirical tests are
performed to determine the effect of uncertainty in agriculture and forest returns, and the
correlation between changes in these returns on land conversion decisions. The empirical results on a
sample of agricultural plots in the state of Georgia show that: (1) the conversion threshold increases
with greater uncertainty in the returns to either agriculture or forests; and (2) the conversion
threshold decreases with greater correlation between changes in returns to agriculture and forests.
These results are consistent with the effect of options on decision-making. Further, approximations
of the option value suggest that they are an important factor in actual decision-making.
These results are also consistent with other empirical research examining the effect of option values
on decision-making. Quigg [24] measures the difference between the value of land when vacant and
developed, finding that the uncertainty level implicit in this difference is consistent with an option
model where the vacant land is valued as an option to develop. Holland et al. [15] find that changes
in the level of uncertainty have a negative effect on the aggregate rate of new residential, retail, and
commercial construction. Leahy and Whited [18] and Eberly [12] use historical measures of
uncertainty in a manner similar to this paper to examine other irreversible investments.5
4
The rents to forestland are sunk until the time of harvest. Since growth in stand value is non-linear, optimal rents
cannot be extracted if harvest occurs before the optimal rotation period. Early harvest reduces the average rental rate
compared to this optimal level. Consequently, early harvest results in an implicit opportunity cost that varies over the
stand’s lifetime.
5
Leahy and Whited [18] examine both idiosyncratic and systematic risk to distinguish the effects of uncertainty due to
irreversibility and risk aversion (i.e. through the Capital Assets Pricing Model). They find that investment levels of
individual firms are consistent with the presence of option values but not risk aversion. Eberly [12] tests the effect of
uncertainty upon individual’s car purchases, finding that uncertainty affects the difference between the trigger level at
which new investments occur and the target level of durables as a portion of wealth. An alternative approach tests the
underlying decision-making with structural estimation of a dynamic-stochastic decision model. A number of dynamic
decisions have been studied with this approach, including capital replacement [26] and forest harvest [23].
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
89
The paper proceeds as follows. Section 2 develops the model of conversion between forests and
agriculture under uncertain returns with sunk costs of conversion. Section 3 develops testable
hypotheses from this model. Section 4 develops an econometric model of conversion from
agriculture to forests. Section 5 discusses model results, and Section 6 provides conclusions.
2. A model of land conversion under uncertainty with sunk costs
To motivate the empirical work, a theoretical model of land conversion under uncertainty with
sunk costs is developed. A risk neutral landowner faces the choice of continuing production in the
current land use, or converting it to another productive use.6 For present purposes, possible land
uses are restricted to agriculture and forest.7 To maximize returns from the land, the owner of
agricultural land chooses the maximum of (1) the expected one-period returns in agriculture plus
the expected value of land in agriculture, or (2) the expected value of bare forest land net of
conversion costs, i.e.:
f
a
; ert E½V1;tþ1
Cta g;
maxVta ¼ maxfE½Rat þ ert E½Vtþ1
ð1Þ
a
where E½ is the expected value, Rat is the annual returns to agriculture, Vtþ1
is the value of land in
f
is the value of land in forests with stand age t; Cta is the one-time costs of
agriculture, Vt;tþ1
conversion from agriculture to forests, r is the discount rate, and t denotes the period. Similarly,
the land owner whose land is in forest use must consider three options: convert the land to
agriculture, harvest the land (but keep the land in forests), or allow the forest to continue to grow:
f
f
f
a
¼ maxfFt;t þ E½Rat þ ert E½Vtþ1
Ctf ; Ft;t þ ert E½V0;tþ1
; ert E½Vtþ1;tþ1
g;
maxVt;t
ð2Þ
where Ft;t is the value of the standing forest of age t; and Ctf is the one-time cost of conversion
from forest to agriculture. The value of land in forests depends upon the current stand age, t;
because the value of the current stand increases with age as the stand grows older and increases in
volume.8 Conversion costs in either direction are sunk costs.9
6
Assumptions made to simplify the conversion problem include constant returns to scale, no constraints on credit or
landowner’s ability to liquidate their land assets, no linkages between the farmer’s human capital and the land asset,
and conversion costs that are linear in acreage.
7
The model can be extended to include multiple land uses, although analytical solutions are not feasible.
8
In contrast to agriculture, where revenues are received annually, revenues from forestry are received after a multiyear rotation period. This delay is the result of non-linear forest growth which necessitates multi-year rotation periods
to allow the forest stand to grow to an optimal economic size [17]. As noted in footnote 4, forest rents are ‘‘sunk’’ until
harvest, with early harvest resulting in a diminishment in rents. Schatzki [27] provides further details on this problem.
The model presented here abstracts from the constraint imposed by sunk rents by implicitly nesting the optimal harvest
decision within the land use decision. This representation is reasonable for present purposes because the timing of the
forest harvest decision is independent of current forest prices when prices follow geometric Brownian motion [6, 25].
Under these conditions, the expected future price will be the same as the current price, so delaying incurs additional
opportunity costs from delayed harvest, without any expected increase in harvest revenues.
9
The portion of reforestation costs that might be retrieved is limited, since the actual biomass of replanted trees is
small and the value of young trees is significantly lower.
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
90
It follows that the landowner will convert cropland to forest when
f
a
oert E½V1;tþ1
Cta :
E½Rat þ ert E½Vtþ1
ð3Þ
Similarly, the landowner will convert forestland to agriculture when
f
f
a
Ctf 4maxfFt;t þ ert E½V0;tþ1
; ert E½Vtþ1;tþ1
g:
Ft;t þ E½Rat þ ert E½Vtþ1
ð4Þ
Further analytic simplification of the problem requires a specification of the process by which
returns to different uses change over time. Only under particular sets of assumptions do recursive
problems of this type lead to closed-form analytic solutions. Recent attention has focussed upon a
class of models that assume (1) returns follow a random walk, or Brownian motion process, and
(2) the conversion decision is continuous rather than discrete [5,10,11,13]. Under these conditions,
the decision to convert from agriculture to forests can be summarized by a simple decision rule
which states that conversion is optimal when the relative return of forest to agriculture, Rt ¼
Rft =Rat ; rises above a conversion threshold that is independent of the current return from either
use. Here, Rft is the annualized return to forests (or the rental rate) based on the current level of
timber prices and growth rates. The conversion rule is then
convert if
Rt 4RF ðsat ; sft ; mat ; mft ; rt ; r; C a ; gf ðtÞÞ;
ð5Þ
where RF is a function of the discount rate, conversion costs, parameters of the diffusion process
of returns, ðsa;t ; st;f ; ma;t ; mf ;t ; rt Þ; and forest growth rates, gf ðtÞ: The s’s are the variance
parameters reflecting uncertainty, m’s are drift parameters reflecting deterministic trends, and r is
the correlation between shocks to agriculture and forest returns. Similarly, the decision to convert
from forest to agriculture can be summarized by a similar decision rule:
convert if
Rt oRA ðsat ; sft ; mat ; mft ; rt ; r; C a ; gt ðtÞ; tÞ:
ð6Þ
When prices or returns follow a random walk, the optimal conversion thresholds, RA and RF ;
developed under these conditions differ significantly from those developed using ENPV models.
The higher threshold is the result of the option to delay conversion, which must be relinquished
upon conversion.
Fig. 1 illustrates the implications of option values for conversion thresholds. This figure shows
conversion thresholds for the option model—RA and RF —and the ENPV model—R̃A and R̃F
when forest growth is constant. The assumption of constant growth eliminates the constraint of
conversion caused by non-linear growth, which greatly simplifies the numerical derivations and
maintains much of the intuition.10 The option model illustrated here assumes that returns follow a
10
This growth assumption is made at the cost of some realism, since forests could be harvested annually with
constant growth. The need to harvest forests after multi-year rotations, and the resulting sunk rents (see footnote 4),
gives rise to an important constraint on conversion from forest to agriculture. Consequently, with non-linear growth,
conversion thresholds from agriculture to forest (forest to agriculture) would be higher (lower) than those shown in
Fig. 1. In addition, conversion thresholds from forest to agriculture would vary with stand age. Conversion thresholds
(i.e., RA oRft =Rat ) decrease when growth in stand value is at its highest rate, which is typically in the middle years of a
stand’s cycle. Note that growth in stand value is the result of both incremental stand growth, and the increased
marketability of stands with larger volume trees. The derivation of the analytic model with constant forest growth rates
is described in [27] and [28].
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
91
1.4
B
RF
Current Relative Returns - R = Rf / Ra
1.3
1.2
~
RF
1.1
A
1
~
RA
0.9
0.8
RA
0.7
C
0.6
0.5
0
0.5
1
1.5
2
2.5
3
~
Conversion Costs - C a
Fig. 1. Examples of hysteresis: conversion thresholds for the deterministic and reversible option models.
geometric Brownian motion, i.e., dRa =Ra ¼ ma dt þ sa dza and dRf =Rf ¼ mf dt þ sf dzf where dz
is the increment of a standard Wiener process, such that EðdzÞ ¼ 0; Eðdz2 Þ ¼ dt; and Eðdzf dza Þ ¼
r: The parameters of the diffusion process are based on state-wide estimates of the diffusion
parameter for Georgia across multiple crops and forest products: sa ¼ 0:09; sf ¼ 0:05; ma ¼
0:009; mf ¼ 0:015; r ¼ 0:2; and r ¼ 0:04: The ENPV model assumes the threshold R̃i ¼ 1 þ rC i ;
where iAfA; F g: This deterministic model does not consider the option to delay, which may be
valuable even under completely deterministic conditions. The graph illustrates how the conversion
thresholds increase significantly when conversion decisions incorporate uncertainty and
sunk costs. Conversion costs are measured in terms of annual returns to agriculture, so that
C̃a ¼ C a =Rat :
The gap between thresholds for conversion to agriculture and forests may lead to land use
hysteresis, where land converted in response to an increase in relative returns does not convert
back when returns shift back to original values. While there is also such a gap with the ENPV
model due to conversion costs, the gap between RA and RF in Fig. 1 is significantly larger. The
potential for dampened price response can be seen by examining the conversion decisions of a
landowner facing changing returns. When returns rise from A to B, land is optimally converted
from agriculture to forests. A shift in returns back to their original level at A, however, is not
sufficient to induce conversion back to agriculture. Instead, returns must fall back below the
threshold RA (i.e., from B to C) before the land will convert back to agriculture. Such wide swings
in prices may only occur over long periods of time, suggesting that reversal of land use change
may be difficult or slow.
92
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
3. Testable implications of the real-option conversion model
There are a number of testable implications of the model summarized in (5) and (6). Since the
empirical analysis examines agricultural to forest conversion, the discussion focuses only on this
type of conversion.11 One implication is that the conversion thresholds are increasing in the level
of uncertainty (sa and sf ) of either land use (i.e., dRF =dsf 40 and dRF =dsa 40). Numerical
simulations indicate that when returns follow geometric Brownian motion, this hypothesis holds
under all conditions for sf and under most all conditions for sa :12 An increase in sa or sf leads to
an increase in the uncertainty of relative returns. When uncertainty is greater, delaying provides
more information about future returns, since the random shock is more likely to be larger.
Greater uncertainty thus increases the value of the option to delay and, as a result, the conversion
threshold.
Another implication arises from the relationship between returns to agriculture and forests.
As the correlation between the changes in agricultural and forest returns increases, the uncertainty
in relative returns declines. When the correlation is positive, a positive (negative) change in one
use’s returns will likely be accompanied by a positive (negative) change in the alternate use’s
returns. Delay under these circumstances provides less information about changes in
future relative returns, thus reducing the option value. Consequently, the conversion threshold
is decreasing in the correlation between changes in agriculture and forest returns (i.e.,
dRF =dro0).
These hypotheses are developed using a theoretical model that assumes that the diffusion
process for returns follows geometric Brownian motion. In this model, the variance parameter is
assumed to be constant. In the empirical model, however, the variance parameter is allowed to
evolve over time. The results of numerous theoretical models with stochastic variance parameters
indicate that it is reasonable to extend these hypotheses to a situation with stochastic variance
parameters [14,16]. These models also show that option values are increasing in the variance
parameter regardless of whether the variance parameter is constant or stochastic.
4. Econometric model of land conversion
To test these hypotheses, land conversion from agriculture to forests in the state of Georgia is
examined over the period 1982–1992. Land use data comes from the National Resources
Inventory (NRI), which is a random, stratified panel of plot-level observations for the years 1982,
1987 and 1992. The analysis considers the change in land use from agriculture to forest over two
periods: 1982–1987, and 1987–1992. About 9% of plots originally in cropland were converted to
forests over the ten-year survey period. In contrast, only 0.35% of plots in forest convert to
agriculture. Due to the limited conversion from forests to agriculture, the analysis only considers
conversion of cropland to forests.
11
Another implication that will not be tested is that the expected stream of returns in forests will greatly exceed the
expected stream of returns in agriculture at the point of conversion. Data limitations, particularly due to the short time
series of observations, make estimation of a structural model that could test such a hypothesis unwarranted.
12
Simulations show that RF may be decreasing in sa for combinations of low sa and very high r.
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
93
Table 1
Summary of variables
Agricult. to forest conversion
sa
oa
la
sf
of
lf
Cor(forest, agric. revenues)
ma
mf
Current agricultural profit
Current forest profit (rent)
Land quality (LCC)a
Irrigation
Conservation practice
CRP eligibility
Population density (p/sq ml)
% of county in pasture
a
Mean
Standard deviation
Minimum
Maximum
0.086
0.197
0.221
0.159
0.078
0.073
0.080
0.426
0.051
0.095
68.174
53.136
2.275
0.186
0.291
0.175
56.419
0.077
0.221
0.100
0.134
0.070
0.022
0.026
0.022
0.187
0.058
0.065
71.983
9.288
0.946
0.389
0.454
0.380
81.351
0.034
0
0.051
0.040
0.039
0.034
0.018
0.039
0.202
0.329
0.247
28.53
14.40
1.00
0
0
0
5.56
0.046
1
0.724
0.818
0.483
0.155
0.131
0.139
0.844
0.081
0.019
268.38
79.20
7.00
1
1
1
1838.08
0.255
In regressions. LCC is inverted (LCC ¼ 7 LCC) so that higher values represent better land quality.
Some further restrictions are placed on the sample of observations. First, agricultural plots are
excluded if they convert to urban use or pasture, removing 46 observations that convert to urban
use and 294 that convert to pasture. Second, NRI observations are included only if they are
cropped in one of the five major crops (soybeans, corn, wheat, cotton, peanuts) or are idled.13
These restrictions reduce the sample about 11%. None of these restrictions change the basic
conclusions presented. The sample is pooled over the two time periods (i.e., 1982–87 and 1987–92)
for a total of 5414 observations. Table 1 summarizes the variables used in the analysis.
From (5), the landowner converts plot i from agriculture to forests when current relative returns
to production rise above a threshold, RFit :
convert if
Rit 4RFit ðsat ; sft ; mat ; mft ; r; r; CÞ:
The probability of conversion is then
Rfit
F
PrðconversionÞ ¼ Pr
4Rit :
Rait
13
ð7Þ
ð8Þ
Idled land is included since it is likely tied to wheat and corn production through Acreage Reduction Plans (ARPs)
that are required in order to receive subsidies. The NRI included land in other crops, such as horticulture (e.g., berries
and nuts), tobacco, hay or vegetables. These crops are affected by a different set of economic factors than the majority
of cropland. For example, horticulture has long-term, fixed investment characteristics that lead to differences from the
model used. Tobacco cropping is significantly affected by regulated quotas, while hay production may be tied to cattle
operations.
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T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
After taking the log of each side of the inequality, the probability of conversion can be written:
PrðconversionÞ ¼ Prðln Rfit ln Rait 4ln RFit Þ:
ð9Þ
The model is estimated by specifying Rfit ; Rait ; and RFit : Current returns to forests are specified
c
as Rfit ¼ gf R% fit1 ; where j denotes the county plot i is in, and R% fit is the product of a plot specific
productivity measure from the NRI, fi ; and a county-level price index, pjt [31]. Over this period,
the CRP has a substantial influence on landowner conversion decisions by increasing the return to
forest ownership.14 The effect of CRP is identified by whether a plot is eligible for enrollment in
the CRP program. The eligibility variable, EL, is based on the same data and criteria used by the
Farm Service Agency (FSA) when making initial eligibility determinations.15 The addition of
rental payments is specified as a proportional increase in the returns to forests, resulting in the
c
following specification for forest returns: Rfj ¼ gf ec2 EL R% fit1 :
Plot-level returns to agriculture, Rait ; are a function of county-level average returns, R% ajt ; and
intra-county land quality and management characteristics (irrigation and use of conservation
practices). The following functional form is used for the implicit plot-level returns to cropland:
!b2
b1 LCCi
Rait ¼ ga R% ajt
ey1 ðCPi CPj Þ ey2 ðIRRi IRRj Þ eit ;
ð10Þ
LCC j
where a bar above a variable represents the variable’s county-level mean. LCC is the Land
Capability Classification, a composite index reflecting multiple factors including soil type, land
features, and other factors that pose limitations to agriculture production [32]. IRR identifies if
the plot is irrigated. CP identifies if conservation practices are used. The most common
conservation practices are terraces, contour plowing, and conservation tillage. Factors affecting
the return to agriculture production observed by the landowner, but not the econometrician, are
specified as eit :
The conversion threshold, RFit ; is specified as a function of the trend and variance of returns to
agriculture and forestry, and the correlation between changes to these returns. Variables are
measured at the county-level using historical data on agricultural revenues, and net returns (rents)
in forests.16 Variation in uncertainty across counties can be expected due to differences in
cropping patterns, transportation costs [3], weather conditions, and other factors. Farm-level
shocks may also affect conversion decisions but are not captured in the data.
14
Georgia has the largest state CRP enrollment in forests (about 646,000 acres as of 1996) and almost all CRP
enrollment in Georgia (91%) is in forests rather than other conserving uses such as grasslands [19].
15
While enrollment for CRP started in 1986, farmers who had already converted land prior to the inception of the
program were eligible for incentives so long as the land had been cropped for three of the previous 5 years. Thus farmers
who had cropped in 1983, but then converted to forests, were still eligible for incentives in 1986. Under certain
circumstances, ineligible land may receive CRP funding and some initial ineligibility designations may be changed, so
the variable may underestimate the full effect of CRP incentives on actual conversion decisions. There is no reason to
believe, however, that eligibility is correlated with the variance of returns or correlation of changes in returns to
different land uses, so estimates of these parameters of interest should be unbiased.
16
Similar parameter estimates and significance levels are obtained when agricultural returns are used in place of
revenues.
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
95
Uncertainty is measured assuming returns follow geometric Brownian motion (i.e.,
P
2
1=2
; where Rjk is the average annual returns
E½Rt ¼ Rt1 ), so that sjt ¼ ðð t6
k¼t ðRjk Rj;k1 Þ Þ=6Þ
17
per acre in county j for agriculture or forests. Estimates assume a stochastic uncertainty
parameter, where the variance is based on returns from the most recent 6-year period. Trends in
profits to agriculture and forests are calculated from least squares estimates over the previous
seven years also using annual average returns per acre in each county.18 Correlations between
agricultural revenues and forest rents are estimated using all observations between 1974 and 1992,
i.e., rjt ¼ covðDRfjt ; DRajt Þ=varðDRfjt ÞvarðDRajt Þ: Interest rates and conversion costs are not
included in the regressions since there is no observable cross-sectional variation in these variables.
Since there is no closed-form solution to the conversion threshold, RFit ; the following flexible nonlinear functional form, which uses a power function for uncertainty variables, is used to model the
conversion threshold: R̂Fit ¼ xskajt1 skfjt2 ek3 majt þk4 mfjt þk5 rjt : This specification does not impose a particular
decision-making structure on the conversion threshold, allowing it to accommodate different
behavioral responses to the uncertainty and correlation measures.
While the model restricts conversion to only forested uses, other conversion options
may exist that potentially affect the likelihood of converting to forests. When a landowner has
more than one alternative land use option, the conversion threshold to each use increases
above the threshold level absent competing alternatives. This increase is due to a ‘‘rational
indecision’’ that makes the choice of the optimal alternative difficult when current returns to these
alternatives are similar [13].19 Consequently, controls are added to the conversion threshold for
residential housing and pasture, resulting in the following specification for the conversion
threshold
RFit ¼ R̂Fit PDkjt6 ek7 PASjt ;
ð11Þ
where PD; the population density of county j; measures the importance of the option to
convert to urban uses, and PAS, the percentage of the county land use in pasture, measures the
importance of the option to convert to pasture uses. To arrive at the final specification, (10), (11)
and Rfit are substituted into (9). The resulting limited dependent variable regression
models whether a plot of land in agricultural production converts to forests or remains in
agriculture.
17
All uncertainty and correlation measures are calculated after first detrending and normalizing returns and revenues
so that deviations represent percentage rather than absolute changes in returns, consistent with geometric rather than
standard Brownian motion. Normalization thus insures that uncertainty measures are not biased toward counties with
higher absolute levels of returns.
18
The data limits the complexity of the forecast mechanism for the variance and trend. The results presented,
however, are robust to expanded measures of the variance and trend encompassing data over longer time periods. Using
measures based on (1) the entire time series (1972–1992) and (2) the 12 years of data surrounding the conversion
decision does not affect the general results and conclusions. Results and conclusions are similarly unaffected for the
alternative uncertainty measures presented in subsequent sections. Future research might examine the implications of
alternative, more complex forecast mechanisms on the effect of uncertainty on conversion.
19
At an extreme, if two new alternatives have equal returns, the hurdle rates are infinitely large, since the landowner
will always wait to see which alternative becomes more valuable.
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
96
5. Results
Results of the basic specification are presented in column (I) of Table 2. The results are
consistent with both of the earlier hypotheses, suggesting that option values are a factor in
landowner decision-making. Increased uncertainty in returns to either land use reduces the
likelihood of conversion, with coefficient estimates significant at the 1% level for uncertainty in
agricultural revenues and at the 5% level for uncertainty in forest returns. Conversion thresholds
are higher for plots with greater uncertainty in the either forest or agricultural returns, consistent
with the conclusion that option values affect decision-making. The effect of the correlation
between returns is also consistent with the conclusion that option values affect conversion
decisions. Increased correlation between changes in agricultural revenues and forest returns
increases the likelihood of conversion, with a coefficient estimate significant at the 5% level. The
results are consistent with the real-option conversion model since increases in the correlation
between changes in returns decreases conversion thresholds, thus increasing the likelihood of
conversion.
Column (IV) of Table 2 also provides results using the following alternative specification for the
conversion threshold
R̂Fit ¼ xek1 sajt þk2 sfjt þk3 majt þk4 mfjt þk5 rjt :
ð12Þ
This specification is somewhat less flexible than (11) since the threshold is necessarily convex with
respect to sa and sf : These results also support the conclusion that option values affect land use
decision-making. Coefficients on sa ; sf ; and r all have the same sign as regressions using (11),
with coefficients on sa and sf significant at the 1% level and the coefficients on the r significant at
the 10% level. The magnitude and significance the coefficient estimates for other variables change
relatively little.
The coefficient estimates for other variables are generally consistent with expectations about the
effect of different variables on conversion decisions. Agricultural land is more likely to convert to
forests when: (1) the returns (and trend in returns) to agriculture are low, (2) the returns (and
trend in returns) to forests are high, (3) land is not irrigated, (4) conservation practices are not
used, (4) land is eligible for the CRP, (5) intra-county land quality is low,20 (6) population density
is low, and (7) the proportion of land in pasture is low. Coefficient estimates across all
specifications are statistically significant at, at least, the 5% level, except for the trends in
agricultural and forest returns, which are statistically insignificant.
5.1. Alternative measures of uncertainty
The hypotheses from Section 3 are developed under the assumption that returns follow
geometric Brownian motion, consistent with most theoretical models assessing option values.
Actual returns, however, may not follow a pure random walk. Although some components of
shocks may be permanent (e.g., some demand shifts and structural supply shifts), some
20
This result extends previous studies by showing that micro-level land quality, reflected in intra-county land quality
variation, positively affect land returns in addition to aggregate measures land quality [20,22].
k1
k2
k3
k4
k5
b1
C1
y2
y1
C2
b2
k6
k7
Uncertainty in agricultural revenues
Uncertainty in forest returns
Trend in agricultural revenues
Trend in forest returns
Correlation between returns/revenues
Current revenues in agriculture
Current returns in forests
Irrigation
Conservation practices
CRP eligibility
LCC (land quality)
Population density
% of county land in pasture
0.586
(0.141)
0.763
(0.327)
1.602
(1.269)
0.710
(1.398)
0.674
(0.324)
0.451
(0.077)
0.507
(0.266)
0.708
(0.181)
0.922
(0.154)
0.329
(0.136)
1.062
(0.163)
0.192
(0.083)
0.270
(0.110)
6.061
(1.513)
1403.50
1.074
(0.145)
0.410
(0.185)
1.794
(1.278)
0.245
(1.144)
0.634
(0.324)
0.483
(0.075)
0.467
(0.265)
0.725
(0.185)
0.901
(0.151)
0.346
(0.137)
1.047
(0.164)
0.202
(0.084)
0.317
(0.115)
5.796
(1.366)
1390.9
0.954
(0.133)
0.457
(0.232)
1.387
(1.257)
0.078
(1.231)
0.583
(0.322)
0.398
(0.072)
0.506
(0.267)
0.723
(0.184)
0.895
(0.150)
0.323
(0.137)
1.052
(0.163)
0.192
(0.083)
0.274
(0.115)
6.297
(1.404)
1393.17
l
o
s
(III)
(I)
(II)
Power specification
Note: Standard errors in parentheses.
Significant at the 1% level; significant at the 5% level; significant at the 10% level.
Log likelihood
Intercept
Coeff.
Variable description
2.754
(0.703)
13.326
(4.745)
1.158
(1.242)
1.012
(1.395)
0.624
(0.320)
0.522
(0.085)
0.479
(0.261)
0.701
(0.180)
0.931
(0.155)
0.333
(0.136)
1.067
(0.164)
0.195
(0.084)
0.276
(0.109)
0.983
(1.304)
1403.54
s
5.801
(0.896)
6.676
(2.662)
0.962
(1.250)
0.041
(1.133)
0.571
(0.321)
0.605
(0.085)
0.433
(0.259)
0.709
(0.181)
0.909
(0.152)
0.349
(0.137)
1.055
(0.165)
0.198
(0.083)
0.336
(0.112)
0.535
(1.335)
1394.18
o
(IV)
(V)
Exponential specification
Table 2
Conversion from agricultural lands to forests: tests of alternative specifications and measures of uncertainty and correlation
6.725
(1.059)
8.335
(3.289)
1.250
(1.242)
0.228
(1.237)
0.520
(0.318)
0.484
(0.077)
0.499
(0.263)
0.708
(0.182)
0.911
(0.152)
0.333
(0.137)
1.056
(0.163)
0.185
(0.082)
0.295
(0.113)
1.166
(1.332)
1395.12
l
(VI)
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
97
98
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
components may not be permanent (e.g., weather-related shocks).21 Augmented Dickey–Fuller
tests do not reject a unit root for forest returns in all 159 counties, and reject a unit root in 4 of 159
counties for agricultural revenues. Although these results are generally consistent with a random
walk, the time series of returns and revenues is too short for these tests to be conclusive.22 To
address the concern that returns may not follow a pure random walk, this section evaluates the
option value hypotheses under the assumption that returns have some degree of mean reversion.
General conclusions on the effect of uncertainty on decision making still hold when returns follow
a diffusion processes with mean-reversion, assuming only a limited degree of mean reversion is
present [9,11]. As the degree of mean reversion increases, however, forest asset values may become
increasing in forest uncertainty since owners can opt to harvest when forest prices are high relative
to the long-run mean price [4].23 The larger the variance, the greater likelihood that prices will be
high relative to mean values. Thus, as mean-reversion increases, optimal conversion thresholds
are decreasing in the uncertainty of forest returns, providing a counter hypothesis for the effect of
forest uncertainty on conversion decisions.
Table 2 provides the results of regressions using two new measures of uncertainty reflecting
different forms of mean-reversion. One measure assumes a moving average, where E½Rtþ1 ¼
Pt6
P
s
2
1=2
:
d N
s¼0 ð1 dÞ Rts and d ¼ 0:5 is assumed. Uncertainty is thus ojt ¼ ðð
k¼t ðRjk E½Rjk Þ Þ=6Þ
Another measure assumes mean-reversion through an Ohrenstein–Uhlenbeck process, where dR ¼
%
ZðR RÞR
dt þ R dt and R% is a fixed value. As Z increases, the degree of mean-reversion increases.
P
2
1=2
The resulting measure of uncertainty is ljt ¼ Zðð t6
; where the fixed value R% is
k¼t ðRjk Rj Þ Þ=6Þ
assumed to be the county mean of returns and Z ¼ 0:25 is assumed.
Columns (II) and (V) of Table 2 present results using o to measure uncertainty in agricultural
revenues and forest returns, while columns (III) and (VI) present results using l to measure
uncertainty. The coefficients on these new uncertainty measures are negative and statistically
significant at the 1% level for agricultural uncertainty and the 5% level for forest uncertainty,
consistent with the earlier results. These results further confirm the effect of option values on land
use decisions. These results also suggest that any potential increase in forest values due to the time
harvest decisions with up-turns in market prices is outweighed by the effect of options.
The coefficient on the correlation remains positive in all specifications, although the significance
of the coefficient estimate falls when alternative uncertainty variables are used. The coefficient on
the correlation remains significant at the 10% level in all runs except when the exponential
conversion threshold (13) and l are used (column VI). The lower significance of r across the
specifications, particularly in comparison to uncertainty variables, is not entirely surprising. There
21
Changes in returns will depend on shocks to prices, costs and yields. Each of these shocks may have permanent and
non-permanent components, with the change in returns reflecting a combination of these components. Shocks to prices,
for example, may reflect weather-related events that may not be completely permanent. Weather-related shocks may
also, however, affect yields, resulting in partially offsetting effects. The net effect on returns may, as a result, not be a
pure random walk, but a diffusion process with some mean-reversion.
22
Data on agricultural returns for Georgia are available for 1972–1995, while forest returns are available for 1973–
1995. Longer time series are available, but only on a national level. Augmented Dickey–Fuller tests are performed with
an AR(2) process for each county’s forest returns and agricultural revenues. With only 23 or 24 observations, however,
this is not a powerful test, so it is difficult to reach any empirical conclusions.
23
The author thanks an anonymous referee for raising this point. In [4], returns have a stationary distribution, with
no random walk component.
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
99
may be large differences in the availability of information on the volatility or uncertainty of
returns compared to the correlation between shocks to returns to different uses. While
information on the volatility of agricultural and forest returns is available through various
services and publications that report yield and price information, there is significantly less
information on the relationship of between returns to alternate land uses. Limited access to
information would reduce the opportunity for landowners to respond to these market signals.
5.2. Estimates of the size of the option to convert
The results indicate that increased uncertainty has a statistically significant effect on land
conversion decisions, resulting from the effect of option values on decision-making. In this
section, the magnitude of this effect is examined. A direct measurement of the option values
cannot be made since the parameters in (9) are not fully identified by the estimation procedure.
The approach used relies instead on substituting estimated coefficients into the first-order
conditions of (9) and solving the resulting differential equation.
The first-order conditions are derived by starting with the optimal conversion conditions.
Conversion is optimal when the returns to forest net of conversion costs are greater than the
returns to agriculture plus the net option value. The net option value is the difference between the
value of the option to convert to forests ðOVf Þ; which is surrendered upon conversion, and the
value of the option to convert back to agriculture ðOVa Þ; which is obtained upon conversion.
Conversion is thus optimal when
Rf
Ra
C4
þ OV ;
r mf
r ma
ð13Þ
where OV is the net option value, OV ¼ OVf OVa : At the point where conversion is just
optimal, (13) becomes an equality. This is the so-called value matching conditions common to real
options problems [11].
After rearranging terms and dividing by Ra ; which normalizes the conversion threshold, the
relationship between the relative returns at the point of conversion and the option value is
r mf
OV ðs; m; r; r; CÞ þ C
Rt ¼
;
ð14Þ
þ ðr mf Þ
Ra
r ma
where OV is written explicitly as a function of the diffusion parameters of the returns to each use
and other variables, and Rt ¼ Rf =Ra : Since Rt ¼ RF at the point of conversion, the RHS of (14) is
equal to RF : Taking the log of each side and then taking the first order conditions with respect to
sa leads to the following equation:
@ ln RF
1
@OV
¼ Ra
:
@sa
ðrm Þ þ OV þ C @sa
ð15Þ
a
This equation assumes that sa is independent of interest rates, conversion costs, and other terms in
the diffusion process of returns. Rearranging terms, the first-order condition for the option value
100
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
with respect to sa is
@OV
Ra
@ ln RF
¼
þ OV þ C
:
@sa
r ma
@sa
ð16Þ
A value for the last term on the RHS can be estimated using the econometric results. Taking the
log of (11) and then taking the derivative with respect to sa produces the following:
@ ln RF @ðln x þ k1 ln sa þ k2 ln sf þ k3 ma þ k4 mf Þ
¼
@sa
@sa
k1
¼ :
ð17Þ
sa
After substituting (17) into (16), the marginal change in the option value for a change in sa can be
written as
@OV
Ra
k1
¼
þ OV þ C
:
ð18Þ
@sa
r ma
sa
A similar equation can be developed for the exponential specification:
@OV
Ra
¼
þ OV þ C k1 :
@sa
r ma
ð19Þ
Both (18) and (19) can also provide estimates of the marginal effects of sf on option values by
substituting sf for sa and using the appropriate estimated coefficient. Solutions for OV can be
developed by solving the partial differential equations in (18) and (19). These solutions can then
be used to estimate the change in option value from an increase either sa or sf : This approach will
not provide an estimate of the total option value, since only uncertainty in one variable is
considered. To solve the differential equations, the boundary condition OV ðsa ¼
0; sf ; m; r; r; CÞ ¼ 0 is assumed. This assumption provides a lower bound estimate, since option
values will be positive when sa ¼ 0 and sf 40: For (19), the assumption is OV ðsa ¼
0:001; sf ; m; r; r; CÞ ¼ 0; since @OV =@sa is undefined at sa ¼ 0: Eqs. (18) and (19) are solved
using the Runge–Kutta method [2].24
Fig. 2 provides estimates of the option values for changes in the three uncertainty measures
using (18), while Fig. 3 provides estimates using (19). These figures assume a 5% landowner
discount rate and ma ¼ 0:009: Comparison of the figures shows the effect of specification choice on
option value estimates. Option value estimates are typically higher for the power specification,
and higher for almost all values presented in Figs. 2 and 3. Above low levels of uncertainty (i.e.,
above the asymptote in Fig. 3), option value estimates are relatively linear in uncertainty for both
specifications. The option value curve for the exponential specification, however, is significantly
steeper than curve from the power specification (above the initial asymptote). Consequently,
option value estimates using the exponential specification are more sensitive to the level of
24
This Runge–Kutta method provides numerical approximations for the value of a variable, y, where y is the solution
to a differential equation of the form dy=dx ¼ f ðy; xÞ: Solutions for OV are developed by iteratively solving the
equation OV kþ1 ¼ OV k þ Ds=6ðLk;1 þ 2Lk;2 þ 2Lk;3 þ Lk;4 Þ; where Lk;1 ¼ f ðsk ; OV k Þ; Lk;2 ¼ f ðsk þ Ds=2; OV k þ
Ds=2Lk;1 Þ; Lk;3 ¼ f ðsk þ Ds=2; OV k þ Ds=2Lk;2 Þ; and Lk;4 ¼ f ðsk þ Ds; OV k þ DsLk;3 Þ:
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
101
1000
900
ωa
Option Value ($)
800
700
600
σa
500
400
300
200
λa
100
0
0.000
0.050
0.100
0.150
Uncertainty (σa, ωa, λa)
0.200
0.250
Fig. 2. Change in option value for changes in agricultural uncertainty: exponential specification based on Eq. (19).
1000
900
ωa
Option Value ($)
800
700
600
500
λa
400
σa
300
200
100
0
0.000
0.050
0.100
0.150
Uncertainty (σa, ωa, λa)
0.200
0.250
Fig. 3. Change in option value for changes in agricultural uncertainty: power specification based on Eq. (18).
uncertainty. Fig. 4 provides the 5% and 95% confidence intervals for the option value curves,
using the conversion threshold in (19) and sa as a measure of uncertainty.
Table 3 provides estimates of the option value measured at the sample mean and one standard
deviation above and below the mean for each of the uncertainty variables. The results assume
discount rates of 5% and 10%, representing different possible landowner discount rates. At the
sample mean and 5% discount rate, option value estimates range from $339 to $1096 for sa -0;
while estimates range from $145 to $967 for sf -0: Consistent with Figs. 3 and 4, results indicate
that option value estimates using the exponential specification are generally more sensitive to the
level of uncertainty than those based on the power specification. A one standard deviation
reduction in variance decreases option values by less than 15% for 5 of 6 measurements with the
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T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
800
Option Value ($)
700
600
500
400
300
200
100
0
0.000
0.050
0.100
0.150
0.200
0.250
Uncertainty (σa)
Fig. 4. Confidence intervals (5% and 95%) for change in option value: Exponential specification, s-measure of
uncertainty for agriculture.
power specification and by more than 25% for 4 of 6 measurements with the exponential
specification.
Table 3 also provides the ratio of option values to classical measurements of the land asset
value, measured as the expected stream of returns in agriculture. With a 5% real discount rate,
option values range from 11% to 81% of expected returns, while at a 10% real discount rate,
option values range from 7% to 67% of expected returns. These results suggest that option values
decrease in size relative to expected returns as the discount rate increases. This result is consistent
with the general theory of options. As discount rates increase, delaying conversion has increasing
opportunity costs, as current gains are forgone. Thus, the value of the option to delay conversion
declines.
The results suggest that not only does landowner behavior appear to be consistent with valuing
the option to convert, but that the magnitude of these option values is significant. The estimates
reported here should be used with some caution since they are based on only a partial elimination
of uncertainty, the use of elasticities outside of the observed range of uncertainty, and may reflect
potentially unidentified factors. Nonetheless, the results suggest that option values do play a
significant role in landowner decisions to convert land from agriculture to forests.
6. Conclusion
The real-option model indicates that landowners should optimally delay conversion when
facing sunk conversion costs and return uncertainty. The results of this paper suggest that actual
land owner decision-making incorporates these option values into land conversion decisions, and
that the magnitude of these options is potentially large. These results provide an explanation for
observed friction in the land conversion process, particularly in conversions to more ‘‘permanent’’
uses.
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
103
Table 3
Estimates of the value of the option to convert
Estimated option value (r ¼ 0:05)
One S.D. below Mean
Ratio of option value to expected returns
One S.D. above r ¼ 0:05
r ¼ 0:10
Power specification
Uncertainty in:
Agricultural revenues sa
oa
la
Forest returns
sf
of
lf
349.1
964.8
295.4
0.46
0.81
0.25
0.45
0.67
0.21
787.9
484.6
153.8
0.56
0.34
0.11
0.46
0.28
0.09
644.8
586.1
206.7
738.7 774.6
1076.3 1140.0
348.9 475.3
0.38
0.55
0.18
0.19
0.33
0.11
788.2
402.3
169.4
967.4 1100.5
576.0 716.6
228.4 248.8
0.50
0.30
0.12
0.29
0.18
0.07
712.6
414.1
133.6
623.3 740.6
1096.4 1114.7
338.7 370.9
755.2
455.4
144.7
Exponential specification
Uncertainty in:
Agricultural revenues sa
oa
la
Forest returns
sf
of
lf
Note: Option values calculated by numerically solving (18) and (19), as described in text. Option value estimates are
provided for one standard deviation above and below the sample mean for each variance measure.
These results have potentially significant policy implications, with lessons for the design of
programs promoting shifts in behavior and development of estimates of program performance.
These lessons are relevant to environmental programs as well as programs in other policy areas
(e.g., social policies). Program design should consider the effect of sunk costs to participation,
participation alternatives, and uncertainty over outcomes on participation decisions. Options
values will predispose participants toward alternatives with lower initial sunk costs and greater
flexibility (e.g., lower conversion costs out of the alternative). In addition, uncertainty over future
program availability may encourage enrollment, since delaying under these circumstances may
cost the landowner the opportunity to take advantage of the incentive. Thus, short-lived or onetime initiatives may more effectively promote participation than permanent programs that allow
participants to delay joining.
Incorporating option values into policy forecasts may also provide more realistic assessments of
policy performance. For example, estimates of supply curves for environmental benefits from land
conversion based on ENPV decision-making may overstate the price-responsiveness of
landowners, leading to overly optimistic projections of costs or participation. Supply curve
estimates reflecting option values can be developed using micro-engineering approaches with
104
T. Schatzki / Journal of Environmental Economics and Management 46 (2003) 86–105
assumed real-option decision-making models, or using econometrically estimated models that
incorporate uncertainty into conversion decision-making. Future research and program
evaluation should consider whether option values are likely to have an important role in
decision-making, and ensure that models accurately consider these effects if they are potentially
important.
Acknowledgement
The author thanks the Crump Foundation for funding for initial stages of this research. The
author also thanks Robert Stavins, William Hogan, Chris Avery, Andrew Plantinga, Roger
Claasen, J.R. DeShazo, Karen Fisher-Vanden, Doug Staiger, and two anonymous referees for
comments at various stages of this research.
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