Deforestation due to uncertainty in land and capital prices: The case of cattle
ranching in the humid tropics of Costa Rica
P.C. Roebeling1, E. Hendrix2 and R. Ruben1
1)
2)
Development Economics Group, Department of Social Sciences, Wageningen University
Operations Research & Logistics Group, Department of Social Sciences, Wageningen University
Information regarding corresponding author:
Name
P.C. Roebeling
Title
Ph.D. researcher
Organization / Affiliation
Wageningen University / Development Economics Group
Mailing address
P.C. Roebeling
Development Economics Group
Department of Social Sciences
Hollandseweg 1
6706 KN Wageningen
The Netherlands
Telephone
+31-317-482972
Fax
+31-317-484037
E-mail
Peter.Roebeling@ALG.OE.WAU.NL
Paper title
Deforestation due to uncertainty in land and capital prices:
The case of cattle ranching in the humid tropics of Costa Rica
JEL Codes
Q15 / Q18
Abstract
This paper examines the effect of land speculation on investment decisions by cattle ranchers in
Latin America, and the subsequent consequences for deforestation under varying levels of land
price uncertainty and interest rate subsidy. Based on the Neoclassical investment theory, a
stochastic reversible investment model with adjustment costs is developed in which land prices
are modeled as a geometric Brownian motion. Analytical results indicate that: i) the optimal rate
of investment is dependent on the price of land at a specific moment in time, ii) the value of the
farm increases with the variance in land prices due to speculative returns from land, and iii) the
maximum farm size attained over time will be larger, the larger the variance in land prices. For
the numerical example of an average cattle rancher in the Atlantic Zone of Costa Rica, it is
shown that not correcting the development in land prices for fluctuations in interest rates leads to
overestimated resource use levels and overestimated speculative returns from land. Policy
simulations show that lower fluctuations in land prices lead to a reduction in these speculative
returns, as well as a reduction in the maximum farm size attained over time. This effect is further
reinforced in combination with lower interest rate subsidies. Consequently, it can be concluded
that variability in land prices alone is a sufficient condition for land speculation and subsequent
inflated levels of deforestation of agrarian frontier areas by cattle ranchers in the Atlantic Zone of
Costa Rica, and that this deforestation is further promoted by subsidized livestock credit.
Key words: dynamic optimization, land speculation, deforestation, cattle ranching
1
1
Introduction
This paper examines the effect of land speculation on investment decisions by cattle ranchers in
Latin America, and the subsequent consequences for deforestation under varying levels of land
price uncertainty and interest rate subsidy. A stochastic reversible investment model with
adjustment costs for cattle ranchers is developed, in which uncertainty in land prices is modeled
as a geometric Brownian motion. In the numerical example, the development in land prices is
corrected for fluctuations in interest rates.
Over the period 1980 to 1990, global deforestation in the tropics amounted up to 15.4*106
ha yr-1 or about 0.8% of the tropical forest area (Hecht, 1992; Faminow, 1998). Almost 50% of
global deforestation took place in Latin America, where pasture for cattle ranching has been the
principal replacement for cleared primary forest (van Hijfte, 1989; Hecht, 1992; Kaimowitz,
1996; Smith et al, 1997). Forest clearing and subsequent conversion to pasture for cattle ranching
has important environmental effects, like the loss of bio-diversity, resource degradation (Bouman
et al, 1998b; Kaimowitz, 1996), and elevated levels of CO2, N2O and NO emission (Hecht,
1992; Veldkamp, 1993, Plant & Bouman, 1999).
Cattle ranching in Latin America is mostly pasture-based, oriented towards beef
production for the market, and takes place on large land holdings (Seré & Jarvis, 1992; Squires
& Vera, 1992). Consequently, cattle ranching plays a predominant role in terms of land use.
Over the last century the total pasture area about tripled, and by 1985 the total pasture area
accounted for 75% of the total agricultural area (Faminow, 1998). The expansion of the livestock
sector in Latin America is generally explained by (Hecht, 1992; Kaimowitz, 1996; Faminow,
1998): 1) the characteristics of livestock production, 2) the increased demand for beef and dairy
products, 3) government policies, and 4) land speculation.
Land speculation by cattle ranchers is often considered the principal cause of deforestation
in Latin America, especially in combination with interest rate subsidies that were widely
provided to cattle ranchers in the 1980’s and 1990’s (van Hijfte, 1989; Hecht, 1992; Kaimowitz,
1996; Smith et al, 1997; Faminow, 1998). Cattle ranching is viewed as the easiest vehicle to
securing the land while waiting for land prices to rise, while at the same time it also provides
economic returns in the form of beef production (Hecht, 1992; Faminow, 1998). Consequently, it
is argued, cattle ranchers hold more pasture land for beef production than what would be optimal
from a productive point of view (Kaimowitz, 1996; Jansen et al, 1997; Smith et al, 1997).
Proof for the hypothesis that land speculation causes deforestation, is relatively limited and
contradicting. Fearnside (1990), Diegues (1992) and Kaimowitz (1996) claim that real land
prices have risen considerably over the last couple of decades due to infrastructure development,
favorable livestock product prices during the 1960’s and 1970’s, population growth and
urbanization, while Kaimowitz also claims that deforested lands sell at a higher price than
forested lands. Faminow (1998), however, states that the empirical evidence is limited, and
shows that annual rates of returns to land speculation in the Brazilian Amazon exceed the
opportunity cost of capital to a limited extent and in a small number of cases.
Based on the Neoclassical investment theory (Eisner & Strotz, 1963; Gould, 1968;
Treadway, 1969; Mussa, 1977; Abel, 1990), a stochastic reversible investment model with
adjustment costs is developed in which land prices are modeled as a geometric Brownian motion.
The model is solved analytically and a numerical example of the model will be given for cattle
ranchers in the Atlantic Zone of Costa Rica. Model base run results are generated on the basis of
1995 data, while correcting the development in land prices for fluctuations in interest rates.
Model simulations are performed for varying levels of uncertainty in land prices and varying
levels of interest rate subsidy.
The paper is structured as follows. Section 2 provides a discussion on the development in
real land prices and interest rates in Costa Rica over the last couple of decades, and is followed
2
by an empirical determination of the stochastic process underlying the development in land
prices as well as an empirical determination of the magnitude of interest rate subsidy targeted
towards the livestock sector. In Section 3, the deterministic and stochastic reversible investment
model with adjustment costs for cattle ranchers is developed. Land prices are modeled as a
geometric Brownian motion in the stochastic version of the model. Section 4 provides a
numerical application of the model with uncertainty in land prices to the case of Costa Rica. Base
run results in which the development in land prices is corrected for fluctuations in interest rates,
are compared to base run results in which the development in land prices is not corrected for
fluctuations in interest rates, as well as the to actual situation in 1986 and 1996. Subsequently, a
comparative static sensitivity analysis with respect to uncertainty in land prices and interest rate
subsidies is performed, and results are analyzed. Finally, Section 5 offers concluding remarks
and observations.
2
Land prices and interest rates in Costa Rica
The first Subsection starts with an overview of the debate on the development of land prices in
Latin American agrarian frontier areas, and is followed by the empirical determination of the
stochastic process underlying the development in real land prices in the Atlantic Zone of Costa
Rica over the period 1965 to 1991. The development in land prices is modeled as a geometric
Brownian motion, and parameter values are used in the numerical application of the model in
Section 4. The second Subsection gives an overview of the development in real interest rates in
Costa Rica over the period 1982 to 1997, while giving special attention to interest rates subsidies
that are targeted towards specific sectors in the economy. The magnitude of interest rate subsidy
that is targeted towards the livestock sector, is used in the numerical application of the model in
Section 4. The last Subsection provides a discussion on the relation between land prices and
interest rates in Costa Rica over the period 1984 to 1991. The development in land prices is
corrected for fluctuations in real interest rates, and parameter values are used in the numerical
application of the model in Section 4.
2.1
Land prices
The development in land prices in Latin American agrarian frontier areas, is an important issue
of debate in literature. Kaimowitz (1996) argues that real land prices in Central America have
risen considerably over the last couple of decades, for a number of reasons. First, real land prices
have risen at a rate greater or equal to the opportunity cost of capital due to infrastructure
development, favorable livestock product prices during the 1960’s and 1970’s, as well as
population growth and urbanization. Second, he states that deforested lands usually sell at a
much higher price than forested lands. Faminow (1998), however, states that real land prices
have not risen at all. He shows that annual rates of returns to land speculation in the Brazilian
Amazon are greater than the opportunity cost of capital in only 32% of the cases and even
negative in 34% of the cases. For some years, however, he found annual rates of return that
highly exceeded the opportunity costs of capital.
The estimation of trend and standard deviation in land prices requires time series data of
land prices over the largest period possible. However, there do not exist published data on land
transactions and land prices in Costa Rica. Annual land price data over the period 1965 to 1991
are obtained from the Institute of Agricultural Development (IDA), an institute that was
concerned with the registration, distribution and sale of land and that maintained a database of
land transactions over the earlier mentioned period. Real land prices are calculated using the
1995 consumer price index (World Bank, 1999).
Figure 1 shows real annual land prices together with the change in real land prices over
the period 1965 to 1991. Real land prices vary considerably over time, while there is no visually
3
evident general trend in land prices. There are, however, two distinct periods in which real land
prices have risen considerably. As of 1976 land prices are elevated to a higher plane. This is the
period in which the government started to play an active role in the protection of natural
resources through the establishment of natural parks and a more strict control on deforestation in
agrarian frontier forest areas (Salas, et al., 1983). Over the period 1986 to 1988 land prices have
risen temporarily, due to the following causes. First, this is the period in which the “FODEA”
law was passed, which cancelled certain debts, provided longer pay-back periods, and lowered
interest rates on past debts provided to, especially, cattle ranchers (Hijfte, 1989). Second, this is
the period in which a major highway is constructed that connects the capital (San José) with the
major export harbor (Limón) and that passes right through the middle of the Atlantic Zone.
Figure 1. Real land prices in the Atlantic Zone of Costa Rica: levels and percentage changes
over the period 1965 – 1991.
Source: Instituto de Desarrollo Agrícola (IDA).
Data regarding percentage changes in real land prices dpI,t / pI,t are important for the stochastic
model developed in Section 3, which is a continuous time model of a competitive farm that buys
land at an exogenously given land price pI,t. It is assumed that land prices evolve according to a
geometric Brownian motion,1 which is given by
dpI ,t / pI ,t  dt  dz
(1)
where  is the instantaneous drift,  is the instantaneous standard deviation, where dz is a Wiener
process with mean zero and unit variance, and where  – ½2 is the expected rate of growth in
land prices (Dixit & Pindyck, 1994). This specification is used to determine the constants  and
, while using the data presented in Figure 1.
Over the period 1965 to 1991, the Chow breakpoint test indicates three distinct periods of
changes in real land prices at the 5% level of significance. Parameter values for these three
periods are given in Table 1. During the period 1965 to 1975, the expected rate of growth in land
prices was negative, about – 8.1% yr-1. The period 1976 to 1979 can be considered as a transition
period, in which the rate of growth in land prices was no less than 50.3% yr-1. Finally, as of 1980
the rate of growth in land prices was about 0.8% yr-1. For the numerical example provided in
Section 4, we will use the parameter values of  and  for this latter period.
Table 1. Drift and standard deviation in real land prices over the period 1965 – 1991.1
Coefficients:
Drift 
Standard deviation 
Number of observations
Notes:
1
1
1965 – 1975
Period
1976 – 1979
1980 – 1991
-0.045
0.277
10
0.745
0.695
4
0.131
0.496
12
Dependent variable is the percentage change in land price.
Geometric Brownian motion is generally used to model economic and financial variables (Dixit and Pindyck,
1994). Given the occurrence of a jump in real land prices over the period 1976 – 1979, it might have been
more realistic to model land prices as a Poisson process. The available data on real land prices, however, do
not permit an accurate estimation of the mean arrival time and size of the jump in real land prices.
4
2.2
Interest rates
The development in real interest rates in Costa Rica over the period 1982 to 1997, is shown in
Figure 2. Large fluctuations in real interest rates can be observed, while real interest rates were
even negative at the beginning of the 1980’s. Economic growth in Costa Rica, which was mainly
agricultural driven, slowed at the end of the 1970’s due to a decrease in the growth of agricultural
production (Nieuwenhuyse et al., 2000). This resulted in an economic crisis at the beginning of
the 1980’s, with subsequent high levels of inflation (up to 90% in 1982) and currency
depreciation (up to 130% in 1981, as compared to the US$). Consequently, structural adjustment
programs were developed which aimed at the lowering of inflation, the balancing of fiscal and
external accounts, the lowering of trade barriers, as well as the reform of the financial and state
sector (Nieuwenhuyse et al., 2000).
Figure 2. Real interest rates in Costa Rica over the period 1982 – 1997.
Source: World Bank development indicators (1999).
The gradual implementation of structural adjustment programs lead to a relative stabilization of
real interest rates, with a geometric mean of about 6.9% per year since 1984. Fluctuations in real
interest rates remained, however, relatively large. From 1993 to 1995, for example, the real
interest rate was about 8% points above the average since 1984, leading to a reduction in
investment combined with a contraction in consumption. This recession was (at least partially)
caused by a government bail-out of depositors associated to a major bank that collapsed in 1994
(Nieuwenhuyse et al., 2000).
Political support for the livestock industry in Latin America in the 1970’s has been
reflected in the terms and availability of livestock credit (Kaimowitz, 1996; Faminow, 1998).
Livestock credit provided in this period was heavily subsidized, it went mostly to cattle ranchers,
and it was allocated to a relatively small group of cattle ranchers. Figure 3 shows the percentage
difference in real interest rates charged on livestock and agricultural credit, as compared to the
national average credit interest rate over the period 1984 to 1996.2 These data confirm the belief
that real interest rates for livestock credit were subsidized in the past. Especially before 1992,
interest rates on livestock and, to a minor extent, agricultural credit were well below the national
average. Interest rates charged on livestock and agricultural credit were, respectively, 0.7% and
0.3% below the national average, with a peak in 1985 when the livestock and agricultural interest
rate were, respectively, 2.7% and 1.4% below the national average. Comparison of the livestock
and agricultural interest rates shows that the interest rates charged on livestock credit was about
0.4% below the agricultural interest rate, with a peak of 1.3% difference in 1985 and 1995.
Figure 3. Percentage difference in real interest rates for livestock and agricultural credit
compared to the national average over the period 1984 – 1996.
Source: Banco Nacional de Costa Rica (BNCR).
Over the decade before the economic crisis at the beginning of the 1980’s, livestock credit
subsidies were even larger than those reported in Figure 3. Kaimowitz (1996), who refers to an
unpublished report by Aguilar & Solís (1988), shows that real interest rates for livestock credit
2
The national average real interest rate includes the agricultural, livestock, construction, housing, and other
sectors.
5
were even negative between 1970 and 1983. After the economic crisis at the beginning of the
1980’s, the gradual implementation of structural adjustment programs lead to a reduction in the
deviation of livestock and agricultural interest rates from the national average (see Figure 3). Put
differently, public livestock credit has become less available and less subsidized since the second
halve of the 1980’s.
Subsidized livestock credit promoted cattle ranching in a number of ways (Kaimowitz,
1996). First, credit helped cattle ranchers to overcome credit constraints which otherwise would
have limited pasture and herd expansion. Second, credit subsidies made cattle ranching a
relatively attractive investment option compared to alternative investment options. Finally, credit
facilitated the transfer of cattle between ranchers. As a consequence, credit subsidies lead to an
allocation of scarce capital away from investments with higher returns, thereby reducing the
national income (Faminow, 1998).
The role of subsidized livestock credit in the conversion of forest to pasture should,
however, not be overstated. In his study on the impact of livestock credit on deforestation in
Panama, Ledec (1992) shows that less than 10% of deforestation could be attributed to livestock
credit, that most of this deforestation took place in small forested areas outside the agricultural
frontier, and that banks preferred to provide credit to large established cattle ranchers rather than
to relatively small colonist cattle ranchers. Hence, livestock credit subsidies explain only part of
the expansion of cattle ranching and subsequent deforestation in Latin America.
2.3
Relation between land prices and interest rates
Real land prices as well as real interest rates fluctuate strongly over time, as can be observed
from Figure 1 and 2, respectively. According to the macroeconomic IS-LM theory, an increase in
prices leads to a decrease in the interest rate, and vice versa (Branson, 1989). When real prices
increase, real money supply decreases, an excess demand in the money market is created, and,
for the money market to clear, the interest rate must increase given a certain income level. In this
section the development in real land prices is corrected for fluctuations in real interest rates
In Section 3, a stochastic reversible investment model with adjustment costs is developed
in which land prices are modeled as a geometric Brownian motion. Given that land prices
fluctuate over time while all other prices are held constant, the development in real land prices
needs to be adjusted for fluctuations in interest rates. If not, fluctuations in land prices and,
subsequent, speculative returns from land will be overestimated.
The development of real land prices and livestock interest rates in Costa Rica over the
period 1984 to 1991 (the longest period for which land prices and livestock interest rates were
found for the case of Costa Rica) are shown in Figure 4. It is clear from the figure that high land
prices are accompanied with low interest rates, and vice versa. Over this same period,
fluctuations in beef and, especially, input prices were relatively low and mostly determined on
the world market,3 while labor costs for cattle and pasture maintenance are relatively low
compared to other costs given the labor extensive character of cattle ranching (Kaimowitz,
1996). Consequently, it is sufficient to correct the development in real land prices for fluctuations
in real interest rates.
Figure 4. Real land prices and livestock interest rates in Costa Rica over the period 1984 – 1991
Source: Real land prices from the Instituto de Desarrollo Agrícola (IDA), and real interest rates from the Banco
Nacional de Costa Rica (BNCR).
3
Trend and standard deviation in real beef prices are –1.3% and 15.2% per year, respectively, over the period
1980 to 1998.
6
In order to determine the development in real land prices while correcting for fluctuations in real
interest rates, fixed capital costs (i.e. interest payments on the value of land) are calculated for the
concerned period. Percentage changes in fixed capital costs are calculated, and drift and standard
deviation in fixed capital costs are determined ( = 0.124 and  = 0.476), equivalent to the
calculation of drift and standard deviation in real land prices in Section 2.1. For the numerical
example given in Section 4, we will use the parameter values for drift and standard deviation in
fixed capital costs, as these reflect the drift and standard deviation in real land prices in case
interest rates are held constant.
3
Model specification
3.1
The deterministic model
Based on the dynamic Neoclassical investment theory with adjustment costs (Eisner & Strotz,
1963; Gould, 1968; Treadway, 1969; Mussa, 1977; Abel, 1990), a model for cattle ranchers in
Latin America is developed. In the model it is assumed that: 1) the production function employs
one variable and one fixed factor of production, produces one single output, is twice
differentiable and reflects constant returns to scale, 2) the variable input can be adjusted
costlessly and instantaneously, 3) the fixed capital input can be adjusted and is subject to an
investment cost function that is convex in the rate of investment, 4) the fixed capital input is
subject to depreciation, 5) all factor, output and capital markets function perfectly, and 6) there is
perfect certainty concerning the present and future.
The cattle rancher uses cattle St and land At to produce beef according to a production
function Q(St, At) that is quadratic in the stocking rate S/A, while assuming constant returns to
scale (c.r.s.). The rancher undertakes gross investment It, while facing convex increasing
investment costs C(It). The annual net income stream tis
 ( S t , At , I t )  pQ ( S t , At )  p S S t  p A At  C ( I t )
with
Q(S t , At )  aSt  bSt2 At1

(3)
where n/(n-1) and n  {2, 4, 6, …}
C ( I t )  p I ,t I t  c3 I t
(2)
(4)
and where p represents the beef price, pS is the price of cattle and cattle maintenance, pA is the
price of pasture maintenance, pI is the price of land, c3 is the adjustment cost scale parameter, and
where  is the adjustment cost elasticity.4 The farm maximizes the present value of net income
streams over time, subject to the equation of motion for land. The optimization problem is
formulated as follows
Maximize





V  At   max  s e rs ds
I s ,Ss 

t


(5)
subject to
A t  I t  At
(equation of motion for At)
and
A0 > 0 and I0 = 0
At  0 and St  0
(initial conditions)
4
Abel & Ebely (1997) show that the adjustment cost function is real-valued and convex for negative and
positive It in case  = n/(n-1), with n an even positive integer.
7
where r is the time discount rate, and where a dot over a variable denotes the derivative of that
variable with respect to time t. The equation of motion for land is determined by gross
investments It in land as well as depreciation  of land. The value function in Eq. (5) satisfies the
Hamilton-Jacobi-Bellman equation
rV  At   max pQ( S t , At )  p S S t  p A At  C ( I t )  ( I t  At )V A 
(6)
I t , St
where VA is the derivative of V(At) with respect to A. This optimality condition requires that, over
an infinitely small time interval, the instantaneous net income equals the required return rV(At).
Using Eq. (3) and (4), maximization of Eq. (6) with respect to I and S, respectively, yields the
first order conditions (f.o.c.’s)
1 /(  1)
 1
V A  p I  c3 I t

pS  p a  2bSt At1
or

or
 V  pI 

I t   A
c

 3

( p S  pa) At
St 
2 pb
(7)
(8)
Eq. (7) states that the optimal rate of investment is such that marginal valuation of land VA is
equal to the marginal cost of investment in land, i.e. the sum of purchase and adjustment costs.
Eq. (8) states that the farm employs cattle up to the point where marginal costs equal marginal
returns. Substitution of It and St into Eq. (6) yields
rV  At   (h  V A ) At  V A  p I 
where h 
 /(  1)

(9)
2 p S pa  p 2 a 2  p S2  4 p A pb
4 pb
1 /(  1)

   1

 c3  

 c3  1

 c3  
 /(  1)
Note that VA is the marginal valuation or shadow value of installed land, in correspondence with
Tobin’s marginal q (Tobin, 1969). To find an analytical solution of Eq. (9), we take V(At) to be a
linear function of the land stock At, thus
V  At   qAt  C
(10)
where C and q = VA are constants to be determined. Substitution of Eq. (10) into Eq. (9) yields
rqAt  rC  (h  q) At  q  p I 
 /(  1)

(11)
which holds for all values of At. Consequently, the term multiplying At on the lhs must equal the
term multiplying At on the rhs, and, similarly, the term not involving At on the lhs must equal the
term not involving At on the rhs. This yields, while solving for q and C, respectively
q
2 p S pa  p 2 a 2  p S2  4 p A pb
h

r 
4 pb(r   )
(12)
8
C  r 1 q  p I 
 /(  1)
  r 1 q  p I 
 /(  1) 
1 /(  1)

 1 c  
 3 

 c3  1

c

 3 
 /(  1)



(13)
Consequently, taking V(At) to be a linear function of At leads to a solution of Eq. (9). This means
that q is independent of the land stock At, in line with the assumption that the farm faces a
constant returns to scale production function and exogenous prices.5 Substitution of q and C back
into Eq. (10) and (7), respectively, yields the fundamental value of the farm V(At) and the
optimal rate of investment It, which are given by
V  At   qAt  r 1 q  p I 
I t*
 q  pI
 
 c3 



1 /(  1)
 /(  1) 
1 /(  1)

 1 c  
 3 

 c3  1

c

 3 
 /(  1)



(14)
 2 p pa  p 2 a 2  p S2  4 p A pb  4 p I pb(r   ) 

  S

4 pb(r   )c3 


1 /(  1)
(15)
As assumed in Eq. (10), it is also clear from Eq. (14) that the fundamental value of the farm is a
linear function of the land stock At, given that q is independent of the land stock. The value of the
farm will increase for positive and negative differences between the shadow price of land q and
the purchase price of land pI, as land can be bought for future productive returns and sold for
immediate sales returns.6 Eq. (15) states that the optimal rate of investment is an increasing
function of q, and independent of A and t. When the shadow price of land q is greater (smaller)
than the purchase price of land pI, the farm buys (sells) land and gross investment is positive
(negative).7
Using the equation of motion, the land stock path At is obtained by solving the differential
equation At / t  At  I t . It can be verified through direct substitution that the specific solution
to this nonhomogeneous linear differential equation is given by
I 
I

At   A0  t  e t  t



(16)
where A0 is the initial land stock. Eq. (16) shows that the steady state land stock A* equals It/,
and that the land stock grows (falls) in an exponential fashion towards this equilibrium level
given the initial land stock below (above) this equilibrium.
3.2
The stochastic model with uncertainty in land prices
Based on the deterministic model presented in the previous Subsection, a stochastic reversible
investment model for cattle ranchers is developed in which uncertainty in land prices is modelled
as a geometric Brownian motion. Consequently, the description of the model will be brief.
Again, cattle St and land At are used to produce beef according to the c.r.s. production function
Q(St, At), gross investment It is subject to convex investment costs C(It, pI,t), and the annual net
income stream t is given by
 (St , At , I t )  pQ(St , At )  pS St  p A At  C(I t , pI ,t )
5
6
7
(17)
Abel (1983) shows that q equals the present value of marginal revenue products of land.
Note that  / (– 1) is even for n  {2, 4, 6, …}, so that q  p I 
 /(  1)
> 0 for all q – pI.
Note that 1 / (– 1) is odd for n  {2, 4, 6, …}, so that I > 0 for q > pI and I < 0 for q < pI.
9
Q(St , At )  aSt  bSt2 At1
with
C ( I t , p I ,t )  p I ,t I t  c3 I t
(18)
where n/(n-1) and n  {2, 4, 6, …}
(19)
where p is the beef price, pS is the price of cattle and cattle maintenance, pA is the price of pasture
maintenance, pI,t is the price of land at a specific moment in time t, c3 is the adjustment cost scale
parameter, and  is the adjustment cost elasticity. The farm maximizes the present value of net
income streams over time, subject to the equation of motion for land and the land price process.
The optimization problem becomes





V At , p I ,t   max Et   s e rs ds
I s ,Ss


t


Maximize
(20)
subject to
A t  I t  At
dpI ,t / pI ,t  dt  dzt
(equation of motion for At)
(land price process pI,t)
and
A0 > 0 and I0 = 0
At  0 and St  0
(initial conditions)
where r is the time discount rate. The equation of motion for land is determined by gross
investments It and depreciation . The land price is modeled as a geometric Brownian motion,
where  is the instantaneous drift parameter,  is the instantaneous standard deviation, and dzt is
an increment to the standard Wiener process with mean zero and unit variance. The value
function in Eq. (20) satisfies the Bellman equation

rV  At , p I ,t   max pQ( S t , At )  p S S t  p A At  C ( I t , p I ,t )  Et {dV } / dt
I t , St

(21)
This optimality condition requires that, over an infinitely small time interval, the instantaneous
net income plus the expected capital gain Et{dV}/dt equals the required return rV(At, pI,t). The
expected capital gain Et{dV} is calculated using Ito’s lemma and the constraints in Eq. (20) that
describe the evolution of At and pI,t, so that
Et {dV} / dt  ( I t  At )VA  pI ,tV pI  12  2 pI2,tV pI pI
(22)
Substitution of Eq. (22) into Eq. (21) yields the basic condition for the stochastic optimal control
problem in Eq. (20)


rV At , p I ,t   max pQ( S t , At )  p S S t  p A At  C ( I t , p I ,t )  ( I t  At )V A  p I ,tV pI  12  2 p I2,tV pI pI (23)
I t , St
Using Eq. (18) and (19), maximization of Eq. (23) with respect to I and S yields the f.o.c.’s
1 /(  1)
V A  p I ,t  c3 I t 1

pS  p a  2bSt At1
or

or
 V A  p I ,t 

I t  
 c3  
( p  pa) At
St  S
2 pb
(24)
(25)
10
which are similar to the first order conditions in the deterministic case, though now the rate of
investment It is a stochastic process. Substitution of Eq. (24) and Eq. (25) into Eq. (23) yields
rV At , p I ,t   hAt  VA  p I ,t 
 /(  1)
where h 
  AtVA  p I ,tV pI  12  2 p I2,tV pI pI
(26)
2 p S pa  p 2 a 2  p S2  4 p A pb
4 pb
1 /(  1)

   1

 c3  

 c3  1

 c3  
 /(  1)
To find an analytical solution of Eq. (26), we take V(At) to be a linear function of the land stock
At. So
V At , pI ,t   qAt  g
with
q  q( pI ,t ) and g  g ( p I ,t )
(27)
where q and g are functions to be determined. Substitution of Eq. (27) into Eq. (26) yields
rqAt  rg  hAt  q  p I ,t 
 /(  1)
  At q
(28)
 p I ,t q pI At  p I ,t g pI  12  2 p I2,t q pI pI At  12  2 p I2,t g pI pI
which should hold for all values of At. Consequently, the term multiplying At on the lhs must
equal the sum of terms multiplying At on the rhs, and, similarly, the term not involving At on the
lhs must equal the sum of terms not involving At on the rhs. This yields
q pI pI 
2  2
2(r   )  2
q pI 
q
p I ,t
p I2,t
1
2

 2 2 q  p I ,t 
2 2
2r 2
g pI 
g

p I ,t
p I2,t
p I2,t
 /(  1)
g pI pI 

p 1b 1 2  2 p S pa  p 2 a 2  p S2  4 p A pb
p I2,t

(29)
(30)
It can be verified that the particular solution to Eq. (29) is given by
q
2 p S pa  p 2 a 2  p S2  4 p A pb
4 pb(r   )
(31)
This expression for q is identical to the q in the deterministic setting, and does not involve any of
the parameters of the adjustment cost function. Abel & Eberly (1997) note that q is the present
value of expected marginal revenue products, which is exogenous for a competitive firm with
constant returns to scale. The q is independent of the specification of the adjustment cost
function, as the path of marginal revenue products is independent of the farm’s investment.
Similarly, it can be verified that the particular solution to Eq. (30), for  = 2, is given by
g ( p I ,t ) 
1
q
q2
2
p

p

I
,
t
I
,
t
2c3 (r   )
4c3 r
4c3 r  2   2


(32)
which equals the present value of the expected rents accruing to the adjustment technology, as
represented by the adjustment cost function (Abel and Eberly, 1997).
11
The fundamental value of the farm V(At, pI,t) is obtained through substitution of q and
g(pI,t) into Eq. (27), which yields
V ( At , p I ,t )  q( p I ,t ) At  g ( p I ,t )  qAt 
1
q
q2
2
p

p

I
,
t
I
,
t
2c3 (r   )
4c3 r
4c3 r  2   2


(33)
The value function is comprised of two additive parts (Abel and Eberly, 1997). The first part,
q(pI,t)At, represents the value of existing land, and equals the expected present value of the
returns to the existing land stock. The second part, g(pI,t), represents the value of the adjustment
technology, and equals the present value of the expected rents accruing to the adjustment
technology as represented by the investment cost function C(It, pI,t) in Eq. (19).
Looking at uncertainty in land prices  while assuming  = 0, we see that the value of the
farm is higher for 0 < 2 < r and lower for 2 > r, as compared to the deterministic case (i.e. the
case with constant land prices). In both cases, expected land prices decrease in a negative
exponential fashion towards zero over time. For 0 < 2 < r , the value of the farm increases with
the variance in land prices as the costs of adjusting the land stock will be below the expected
discounted returns obtained from this adjustment. For 2 > r, the value of the farm has the
asymptotic value V ( At , p I ,t )  qAt  qp I ,t / 2c3 r  q 2 / 4c3 r as  2   , which is p I2,t / 4c3 r below
the value of the farm in the deterministic case.
Looking at the trend in land prices , while assuming  = 0, it can be verified that the
value of the farm is higher for  < 0, 0 <  < ½ r and  > r, and lower for ½ r <  < r, as
compared to the deterministic case (i.e. the case with constant land prices). For  < 0, expected
land prices decrease in a negative exponential fashion towards zero over time. The value of the
farm has the asymptotic value V ( At , p I ,t )  qAt  q 2 / 4c3 r as    , which is
p I2,t / 4c3 r  qp I ,t / 2c3 r below the value of the farm in the deterministic case. As land prices fall
more rapidly over time, investment costs will decrease, land is purchased for beef production,
and, consequently, the value of the farm will increase. Similarly, for  > r, expected land prices
increase in an exponential fashion towards infinity over time. The value of the farm has the
asymptotic value V ( At , p I ,t )  qAt  q 2 / 4c3 r as    , which is p I2,t / 4c3 r  qp I ,t / 2c3 r below
the value of the farm in the deterministic case. As land prices increase more rapidly over time,
investment costs will increase, available land is sold at the current (increasing) land price pI,t,
and, consequently, the value of the farm will increase. Finally, for 0 <  < r the behavior of the
value function is complex, due to asymptotic values of the value function for  = ½ r and  = r.
Similarly, the rate of investment It is obtained through substitution of q (which is equal to
VA) into Eq. (24), so that
I ( p I ,t ) 
q  p I ,t
2c3

2 p S pa  p 2 a 2  p S2  4 p A pb  4 p I ,t pb(r   )
8 pb(r   )c3
(34)
which only differs from the rate of investment in the deterministic setting, in the sense that the
rate of investment is dependent on the price of land pI,t at a specific moment in time t.
Consequently, the rate of investment fluctuates over time according to the land price process
depicted in Eq. (20), and gross investment I(pI,t) is positive (negative) when the shadow price of
land q is greater (smaller) than the purchase price of land pI,t.
Finally, looking at the expected land price E(pI,t) we know that the land price process in
Eq. (20) has the following solution (Dixit & Pindyck, 1996)
12
E( p I ,t )  p I ,0 e (  
2
/ 2) dt
(35)
where pI,0 is the land price in t = 0, and where  – 2/2 is the expected rate of growth in land
prices. In case  – 2/2 < 0, land prices are expected to decrease in a negative exponential
fashion towards zero over time. The rate of investment has the asymptotic level I ( pI ,t )  q / 2c3
as pI ,t  0 , and the corresponding asymptotic land stock equals At   q /( 2c3 ) . In case  –
2/2 > 0, land prices are expected to increase in an exponential fashion towards infinity over
time. The rate of investment is decreasing as p I ,t   , and the corresponding land stock first
increases, reaches a maximum in case A0 < Amax, and then decreases until the farm is closed.8
4
Model results
This Section provides a numerical application of the model with uncertainty in land prices to the
case of Costa Rica, for the reference year 1995. Base run situations are generated for the case in
which the development in land prices is either or not corrected for fluctuations in interest rates.
The base run in which the development in land prices is corrected for fluctuations in interest rates
(corrected base run, where  = 0.124 and  = 0.476) is generated, and compared to the base run
in which the development in land prices is not corrected for fluctuations in interest rates
(uncorrected base run, where  = 0.131 and  = 0.496), as well as the to actual situation in 1986
and 1996. Subsequently, a comparative static sensitivity analysis with respect to uncertainty in
land prices and interest rate subsidies is performed on the basis of the corrected base run, and
results are analyzed.
Base run results are obtained for the year 1995. The price of beef (p = 1.41 US$ kg-1),
cattle and cattle maintenance (pS = 213.2 US$ AU-1), and pasture maintenance (pA = 55.0 US$
ha-1) are obtained from the PASTOR system, which is calibrated for the Atlantic Zone of Costa
Rica for the year 1995 (Bouman et al., 1998a). The base price of land with road access, poor soil
drainage and no further facilities for an average beef cattle farm in the Atlantic Zone (pI = 630.0
US$ ha-1) is obtained from the Dirección de Tributación Directa of the Ministry of Housing.
Trend and standard deviation in real land prices over the period 1980 – 1991 are determined in
Section 2.1 ( = 0.131 and  = 0.496), and trend and standard deviation in real land prices over
the period 1984 – 1991 that are corrected for fluctuations in real interest rates are determined in
Section 2.3 ( = 0.124 and  = 0.476). The constant rate of land depreciation is set at  = 0.10, in
line with Faminow (1998) who states that the carrying capacity of pasture land falls rapidly in
case it is used for cattle ranching. In about 7-10 years after pasture establishment the field is
abandoned or recuperated. Quadratic adjustment costs related to investment in land are
determined on the basis of secondary data regarding labor supply, costs and requirements,
thereby assuming that 25% of all beef cattle farms invest simultaneously (c3 = 3.432 and  = 2).
The time discount rate (r = 6.9% yr-1) is obtained from the World Development Indicators
(World Bank, 1999), and is calculated as the geometric mean of the real interest rates over the
period 1984 to 1997. The official 1995 exchange rate (179.73 Colones US$-1) is, also, obtained
from the World Development Indicators (World Bank, 1999). Production function parameter
estimates for pasture based beef production on poorly drained soil types in the Atlantic Zone of
Costa Rica, are determined on the basis of technical input-output coefficients generated by the
PASTOR system (a = 354.3 and b = - 70.9).
8
The rate of investment will be positive until the point in time t where the shadow price of land q equals the
purchase price of land pI,t.
13
4.1
Base run results
In the previous Section it is shown that the rate of investment in the stochastic setting fluctuates
over time according to the land price process, the latter being modeled as a geometric Brownian
motion. As such, this finding does not provide information on the effect of uncertainty in land
prices on investment decisions by cattle ranchers in Costa Rica. Cattle ranchers do, however,
anticipate on the expected development in land prices over time. In Section 3 it is shown that the
2
expected land price E(pI,t) equals pI ,0 e (   / 2) dt . Given the parameter values for  and  in the
corrected base run as determined in Section 2.3 ( = 0.124 and  = 0.476), it can be verified that
the expected growth in land prices is 1.0% yr-1. As land prices are expected to grow over time,
land will first be bought, a maximum farm size will be attained, and then the land will be sold
until the farm is closed (i.e. when At = 0). The value of the farm is determined over the period in
time where At > 0.
The uncorrected and corrected base run as well as the actual situation in 1984 and 1996 for
an average beef cattle ranch in the Atlantic Zone of Costa Rica, are given in Table 2. The
uncorrected base run ( = 0.131 and  = 0.496) and the corrected base run ( = 0.124 and  =
0.476) are generated for the year 1995, and refer to the situation in which the maximum farm
size is attained at a specific moment in time t as the cattle rancher anticipates on the expected
development in land prices over time.9 The actual situation refers to the situation in 1984
(DGEC, 1987) and to the situation in 1996 (Roebeling et al., 1998).10
Table 2. (Un-)corrected base run and actual situation in 1984 and 1996 for an average beef
cattle ranch in the Atlantic Zone of Costa Rica.
Unit
Value of the farm
Resource use
Land
Cattle
Labor
Beef production
Resource use intensity
Stocking rate
Labor intensity
Notes:
106 US$
Ha yr-1
AU yr-1
LD yr-1
Tons yr-1
AU ha-1 yr-1
LD ha-1 yr-1
Base run
Uncorrected Corrected
1.94
1.93
Actual situation
19841
19962
-
199.6
285.5
574.7
33.2
173.1
247.7
498.6
28.8
214.5
232.0
566.0
-
170.4
228.9
570.0
-
1.4
2.9
1.4
2.9
1.1
2.6
1.3
3.3
1
Actual situation 1984 obtained from DGEC (1987).
2
Actual situation 1996 obtained from Roebeling et al. (1998).
Comparison of the uncorrected and corrected base run shows that the value of the farm is about
0.6% overestimated when the development in land prices is not corrected for fluctuations in
interest rates. Resource use, on the other hand, is overestimated with no less than 15% in case the
development in land prices is not corrected for fluctuations in interest rates. Resource use
intensity indicators are equal for the uncorrected and corrected base run, as the same constant
returns to scale production function is used in both base runs.
The corrected base run indicates that an average cattle ranch in the AZ of Costa Rica
encompasses about 173 ha of pasture land and 248 animal units (AU) of 400 kg live weight,
which implies a stocking rate of about 1.4 AU ha-1 yr-1. Annual labor requirements for cattle and
9
We consider a ‘starting’ cattle rancher with an initial land stock A0 of 1 ha of poorly drained land.
In both cases, beef cattle ranches are defined to have more than 50 ha of pasture and more than 50 animal units
of beef cattle of 400 kg live weight.
10
14
pasture maintenance are about 500 labor days (LD), which implies a labor use intensity of 2.9
LD ha-1 yr-1. Annual beef production is almost 30 tons yr-1, which implies an average beef
production of 116 kg AU-1 yr-1. Finally, over the concerned infinite time horizon the value of the
farm is 1.93*106 US$.
Comparison of the corrected base run and the actual situation shows that land use is in
between the 1986 and 1996 situation, while cattle and labor use are up to 25% overestimated.
This deviation can be explained by differences in beef, input and land prices, as well as interest
rates for the concerned base run and reference years. Resource use intensity indicators in the
corrected base run situation are 5% to 30% overestimated compared to the actual situation in
1996 and, especially, 1986. This overestimation can be explained by the production function
parameter estimates that are based on 1995 data of the top ten best cattle ranches in the Atlantic
Zone as well as experimental field data (Hengsdijk et al., 2000). This is confirmed by the fact
that differences are larger between 1985 and 1995 than between 1996 and 1995 as a result of
technological progress.
4.2
Policy simulations: Uncertainty in land prices and interest rate subsidies
In this Section the stochastic reversible investment model, as presented in Section 3, is used to
evaluate the effect of varying levels of interest rate subsidy and uncertainty in land prices on the
maximum farm size and value of the farm of an average beef cattle ranch in the Atlantic Zone of
Costa Rica. This analysis is performed on the basis of the corrected base run.
Table 3 shows the maximum farm size and value of the farm for various levels of interest
rate r and standard deviation in land prices . The corrected base run situation is given in the
shaded upper left corner of Table 3, where r = 6.9% yr-1 and  = 0.476. Simulations include a
gradual reduction of the interest rate subsidy (on average the interest rate subsidy on livestock
credit was about 0.7% yr-1 over the period 1985 – 1996) as well as a gradual reduction of the
standard deviation in land prices ( will be reduced by 20%). Numerical solutions of a
discretisized version of the model for a large number of stochastic land price paths, leads to
similar results and response reactions. Moreover, numerical exercises with a constant rate of
growth in land prices (i.e. with constant  – 2/2) show that the (average) value of the farm and
maximum farm size attained over time is reduced for lower levels of .
Table 3. Farm size and value of the farm for various levels of interest rate (r) and standard
deviation in land price ().
r \ 
6.9
7.1
7.3
7.5
0.48
173
163
153
143
Farm size (in ha)
0.45 0.43 0.40
111
82
66
103
77
62
96
71
57
90
66
53
0.38
56
52
49
45
Value of the farm (in 106 US$)
0.48
0.45
0.43
0.40
0.38
1.931 1.275 0.875 0.628 0.464
1.924 1.268 0.870 0.624 0.462
1.921 1.263 0.866 0.622 0.460
1.920 1.260 0.864 0.620 0.458
The maximum farm size attained over time is relatively sensitive with respect to interest rate
subsidies and, especially, fluctuations in land prices. Abolition of the interest rate subsidy would
lead to a 17% reduction in farm size, as investments are reduced due to a decrease in the
marginal production value of land. This effect is even stronger for lower values of . Reduction
of the standard deviation in land prices by one fifth would lead to a 68% reduction in farm size,
15
as investments are reduced due to more rapidly increasing land prices over time.11 This effect is
stronger for higher values of r.
The value of the farm is relatively insensitive with respect to interest rate subsidies, and
relatively sensitive with respect to fluctuations in land prices. Eradication of the interest rate
subsidy would lead to a 0.6% reduction in the value of the farm, due to a decrease in the
marginal production value of land as well as a decrease in the present value of expected rents
accruing to the adjustment technology. This effect is stronger for lower values of . Reduction of
the standard deviation in land prices by one fifth would lead to a 76% reduction in the value of
the farm, due to increased capital costs related to the ownership of land as well as a decrease in
the present value of expected rents accruing to the adjustment technology.12 This effect stronger
for higher values of r.
5
Conclusions
Analytical results indicate that the optimal rate of investment is dependent on the price of land at
a specific moment in time. Consequently, at each point in time the optimal rate of investment is
determined through comparison of the marginal value of an extra unit of land, and the marginal
investment costs of this extra unit of land. Furthermore, analytical results indicate that the value
of the farm increases with the variance in land prices. Fluctuations in land prices give rise to
speculative returns from land, in the form of rents accruing to the adjustment technology. Finally,
analytical results indicate that the maximum farm size attained over time will be larger, the larger
the variance in land prices. A smaller variance in land prices leads to an increased exponential
growth in expected land prices over time (given that land prices are modeled as a geometric
Brownian motion), and the farmer starts selling his land earlier in time.
Land price data for the Atlantic Zone of Costa Rica indicate that there is limited evidence
for Kaimowitz’s (1996) statement that land prices in Central America have risen considerably
over the last couple of decades. Over the period 1980 to 1991, expected land prices in the
Atlantic Zone have risen with less than 0.8% yr-1. Fluctuations in real land prices are relatively
large. Relative changes in land prices show a standard deviation of no less than 50% yr-1 when
the development in land prices is not corrected for fluctuations in interest rates, and about 48%
yr-1 when the development in land prices is corrected for fluctuations in interest rates. This result
is in line with Faminow (1998), who shows that real land prices in the Brazilian Amazon have
not risen over the last couple of decades, although fluctuations in land prices were large. Interest
rate data for Costa Rica confirm that interest rate subsidies have been supplied to agricultural
credit in general, and livestock credit in particular. Over the period 1984 to 1996, interest rates
charged on agricultural and livestock credit were, respectively, 0.3% and 0.7% below the
national average.
Numerical results for an average cattle rancher in the Atlantic Zone of Costa Rica indicate
that the value of the farm and resource are, respectively, 0.6% and 15% overestimated in case the
development in land prices is not corrected for fluctuations in interest rates. Policy simulations
show that interest rate subsidies and, especially fluctuations in land prices, determine to a large
extent the maximum farm size that is attained over time, as the cattle rancher anticipates on the
expected development in land prices over time.
Abolition of the interest rate subsidy would lead to a 17% reduction in farm size, while a
one fifth reduction in the standard deviation in land prices would lead to a 68% reduction in farm
size. Consequences for the value of the farm are relatively small with respect to interest rate
subsidies and relatively large with respect to fluctuations in land prices. Abolition of the interest
In case 2 smaller, then increases in pI,t are bigger than decreases in pI,t, and therefore pI,t rises more rapidly
over time. Consequently, the hacienda owner starts selling his land earlier in time.
12
Investment costs increase with the expected increase in land prices.
11
16
rate subsidy would lead to a 0.6% reduction in the value of the farm, while a one fifth reduction
in the standard deviation in land prices would lead to a 76% reduction in the value of the farm.
In this study it is shown that uncertainty in land prices is a sufficient condition for land
speculation, as it gives rise to speculative returns from land in the form of rents accruing to the
adjustment technology. For the numerical example of an average cattle rancher in the Atlantic
Zone of Costa Rica, it is shown that results are overestimated when the development in land
prices is not corrected for fluctuations in interest rates. Moreover, it is shown that lower
fluctuations in land prices lead to a reduction in speculative returns from land, as well as a
reduction in the maximum farm size attained over time. This effect is further reinforced in
combination with lower interest rate subsidies. Consequently, it can be concluded that variability
in land prices alone is a sufficient condition for land speculation and subsequent inflated levels of
deforestation of agrarian frontier areas by cattle ranchers in the Atlantic Zone of Costa Rica. This
deforestation is further promoted by subsidized livestock credit. While the role of subsidized
livestock credit in deforestation has been widely acknowledged (Ledec, 1992; Kaimowitz, 1996),
the role of land speculation has, so far, only be related to deforestation in case it was proven that
land prices tended to rise over time (Kaimowitz, 1996; Faminow, 1998).
Future research needs to address a number of topics that were not dealt with in this study.
First, the partial equilibrium approach implies that prices are not determined by aggregate supply
and demand. Second, it might be interesting to model real land prices as a mixed geometric
Brownian and Poisson jump process, given the occurrence of a jump in real land prices in the
Atlantic Zone of Costa Rica around the year 1978.
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