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Estimation of the lifespan of agricultural tractor using a diffusion model at the aggregate level

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R. Muñoz, and J. Llanos. 2012. Estimation of the lifespan of agricultural tractor using a diffusion model at the aggregate level. Cien. Inv. Agr. 39(3): 557-562. The adoption of agricultural tractors by farms has resulted in different combinations, relationships and established ways of production of localities, regions and countries. The dynamics of the tractorization process are defined by the balance of the units that enter the fleet and those that are removed at the end of their useful life. This study uses a diffusion pattern and aggregated adoption of the agricultural tractor, the parameters of which were determined using a nonlinear estimation procedure and country-level data for Chile. Using the predictive power of the model, a longitudinal series of tractor fleets was estimated, and then the balance was calculated considering the units that were discharged. To more accurately determine the average lifespan of a tractor, successive iterations were performed to reduce differences with respect to a control variable for the number of tractors registered. The estimated average lifespan for agricultural tractors at the aggregate level is 22 years. This type of estimation procedure can be used not only for descriptive purposes, as in this case, but also for predictive purposes.
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Cien. Inv. Agr. 39(3):557-562. 2012
www.rcia.uc.cl
agricultural economics
Received January 29, 2012. Accepted July 11, 2012.
Corresponding author: ricardo.munozc@mail.udp.cl
Introduction
The adoption of tractors on farms has resulted in
different combinations, relationships and estab-
lished ways of production of localities, regions and
countries. Many researchers have examined the
agricultural tractor from different perspectives,
including the choice between tractor and animal
traction (Olmstead and Rhode, 2001); the factors
affecting economic evaluation of agricultural
tractors (Hetz et al.,1998); the costs, in terms
of opportunity, time savings and improvements
in the implementation of technology adoption
(Duffy and Silberberg, 2006); the demand
projection of tractors (Unakitan and Akdemir,
2007); the positive growth rate of tractors and
its relationship to technological change (Hus-
sain and Ali, 1989; Patel and Madalsa, 1996);
the use of tractors as an indicator of prosperity
or progress in the agricultural sector (Unakitan
research note
Estimation of the lifespan of agricultural tractor using a diffusion
model at the aggregate level
Ricardo Muñoz
1
, and José Llanos
2
1
Facultad de Economía y Empresa, Universidad Diego Portales, Ave. Manuel Rodríguez Sur 415. Santiago,
Chile.
2
Universidad de Santiago de Chile. Ave. Ecuador 3769. Santiago, Chile.
Abstract
R. Muñoz, and J. Llanos. 2012. Estimation of the lifespan of agricultural tractor using
a diffusion model at the aggregate level. Cien. Inv. Agr. 39(3): 557-562. The adoption
of agricultural tractors by farms has resulted in different combinations, relationships and
established ways of production of localities, regions and countries. The dynamics of the
tractorization process are dened by the balance of the units that enter the eet and those that
are removed at the end of their useful life. This study uses a diffusion pattern and aggregated
adoption of the agricultural tractor, the parameters of which were determined using a nonlinear
estimation procedure and country-level data for Chile. Using the predictive power of the
model, a longitudinal series of tractor eets was estimated, and then the balance was calculated
considering the units that were discharged. To more accurately determine the average lifespan
of a tractor, successive iterations were performed to reduce differences with respect to a control
variable for the number of tractors registered. The estimated average lifespan for agricultural
tractors at the aggregate level is 22 years. This type of estimation procedure can be used not
only for descriptive purposes, as in this case, but also for predictive purposes.
Key words: Aggregate diffusion, agricultural mechanization, lifespan, technology adoption,
tractor, tractorization.
ciencia e investigación agraria558
and Akdemir, 2007); valuation and depreciation
models for tractors (Guadalajara-Olmeda and
Fenollosa-Ribera, 2010) and the relationship
between tractor stock and adoption and diffusion
patterns (Muñoz et al., 2011). In these studies,
the useful life or lifespan of the tractor param-
eter plays an important role in the analysis. This
parameter is often given or assumed, and inu-
ences the estimation of important technical and
economic variables (Hetz et al.,1998; Unakitan
and Akdemir, 2007; Guadalajara-Olmeda and
Fenollosa-Ribera, 2010; Muñoz et al. 2011).
Most aggregate adoption models are analytically
derived from the behavior of the diffusion process
over time, with the analytical characteristics of
that behavior being tied to the nature of the model
(Rao and Kishore, 2010). Aggregate adoption
models have their origin in the growth approach
(Baptista, 1999), which has been adapted for various
applications (Mahajan et al., 2000; Singh, 2008,
Tseng and Hu, 2009) including tractors (Muñoz
et al., 2011). The results show a temporal pattern
of adoption, the graph of which is an S-shape or
sigmoidal curve.
Muñoz et al. (2011) proposed a descriptive model
based on tractorization in Chile that can be used
as a space-time reference structure for the diffu-
sion and adoption of technology in place studies
of technical change, social development and eco-
nomic development in a country. The same study
estimated the tractorization plan at an aggregate
level, based on the logistic growth paradigm.
The objective of this study was to estimate the
average lifespan of a tractor at the aggregate level
using an aggregate diffusion model and the bal-
ance of the tractor eet, based on the information
described by Muñoz et al. (2011). This model
can be used to describe the process and also to
estimate a longitudinal series of the tractor eet.
Then, when the tractor’s estimated useful life is
known, through the estimation of discharged
units, a eet balance is achievable. The model
can then be used to re-evaluate the value given
as the average lifespan of the tractor, through
successive iterations, to reduce the differences
with respect to a given control variable, such as
the annual series of tractors registered.
Materials and methods
The balance of the tractor eet was taken from
the model developed by Muñoz et al. (2011).
To estimate the tractor eet at any given time,
the following expression was used:
R
t
T
N
t
T
T
t
P
T
t
P =
1
(1)
The balance at time t (year) is given by the fol-
lowing expression:
R
t
N
t
T
t
T
t
TTPP =
1
where:
T
t
P
= Number of tractors at time t
T
t
P
1
= Number of tractors at time t-1
N
t
T
= Number of tractors to be added to the eet
at time t
R
t
T
= Number of tractors that leave the eet at
time t.
Aggregate adoption model
To determine the growth rate of the tractoriza-
tion process or the distribution and adoption of
the tractor, Muñoz et al. (2011) used a nonlinear
procedure, and based on the Marquardt method
(Rawlings et al., 1998), the following expression
was obtained:
(2)
559
VOLUME 39 Nº3 SEPTEMBER – DECEMBER 2012
Data source
The number of registered tractors was obtained
from the Registro Nacional de Vehículos Motor-
izados (RNVM, Chile), which provides informa-
tion from 1985 on. This series included units
between 1987 and 1997, selected to exclude the
cumulative number observed in the initial years
and to consider the agricultural census of 1997
(n=49,967 tractors), because these were the latest
data used in developing equation (2).
Estimates
Muñoz et al. (2011) indicated that the number
of tractors discharged from the eet (T
R
t
) can be
estimated by knowing the average lifespan of a
tractor, while knowing the number of tractors to
be incorporated (T
N
t
) requires knowing the number
of annual registrations of new tractors or units
sold in the market. The difculty in determining
T
R
t
lies in the fact that differentnal gures will
be obtained depending on the selected lifespan,
while for T
N
t
, rigorous statistics kept by the ve-
hicle registration or import entities or vendors are
required. The agricultural census data (Dirección
General de Estastica, 1933; Dirección General
de Estadística, 1938; SNEC, 1955; Dirección de
Estadística y Censos, 1969; INE, 1976; INE,
1997) offer a way to overcome the difculties
mentioned because they represent the actual
number of tractors (P
T
t
). Therefore, the choice
was made to use the growth curve described by
expression (2) for the period between 1930 and
1997. The graph is shown in Figure 1.
Afterward and also at the aggregate level, the tractor
eet (P
T
t
) was assessed: P
T
t-1
was determined, and
equation (1) was applied to estimate the number
of tractors that were added to the eet, assum-
ing in the equation an average lifespan ranging
from 10 to 30 years. Then, the variance between
the estimated number and the series values was
determined from the number of tractors registered.
The deviation of the estimate was expressed as
absolute error (%).
Results and discussion
Based on the procedure used in this study, an
average useful life of 22 years yielded the lowest
average absolute error (19.4%). The absolute error
decreased from 30.4 to 19.4% as the average life
increased from 10 and 22 years, and the absolute
error increased from 19.4 to 26.2% as the average
life increased from 22 and 30 years (Figure 2).
0
10.000
20.000
30.000
40.000
50.000
60.000
1930
1933
1936
1939
1942
1945
1948
1951
1954
1957
1960
1963
1966
1969
1972
1975
1978
1981
1984
1987
1990
1993
1996
Tractors (units)
P
T
=54,799.2/(1+EXP(-(3.48444+0.0849665*t))) r
2
=0.998
Figure 1. Growth curve of agricultural tractor.
ciencia e investigación agraria560
The average tractor lifespan result obtained in this
study is an estimate of the average time that a trac-
tor is kept in operation at the aggregate level, and
obviously it is somewhat higher than that used to
calculate economic and technical indexes for tractors
in Chile (Hetz et al., 1998). However, it should be
kept in mind that the success of the procedure used
in this study depends on the quality with which the
series of tractors enrolled in the RNVM is built each
year. With respect to this point, it was observed that
for the same year of registration, models manufac-
tured in different years were registered; therefore,
the number of units registered in a given year and
the number of units manufactured in the same year
were not coincident. This creates a dilemma con-
cerning when it is correct to register the inscription.
This also introduces variation into the data series,
which depends on the thoroughness (of the owners
and the entity or entities that regulate the process)
with which the units are annually registered in the
RNVM, a factor that is beyond the scope of the
method used in this study.
On the other hand, the annual growth of the eet,
estimated using equation (2), is based on the nature
of the data that were used to develop the model,
which were derived from agricultural censuses that,
despite determining the actual stock of tractors in the
country, is subject (according to the characteristics
of the logistic model) to estimates of interpolation
areas corresponding to the times between cen-
suses, because it is an accumulated function that
also generates a density function. In a recent study
(Tomantschger et al., 2011) estimated the probability
density function of the lifespan of the engine for a
specic tractor model (until an overhaul is needed)
may not only be used as a proxy for estimating the
tractor lifespan but may also offer a way for further
studies to integrate this approach to estimating trac-
tor lifespan at the level of the tractor model cluster
within the farm tractor aggregate total.
The proposed model for estimating the fleet
of tractors at the aggregate level (the country
level), which gives the number of tractors to be
incorporated into the stock, T
N
t
, estimated using
equation (1) and based on equation (2), can be
useful for market predictions.
The procedure presented provides an option for
integrating a eet balance equation and a model
that describes its evolution, but it does not explain
the factors that accounted for the evolution, which
is a subject that could be studied further.
0
5
10
15
20
25
30
35
40
45
50
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Lifespan (years)
Variation (%)
Figure 2. Average variation (%) between the number of tractors added to the eet and
the number of tractors enrolled annually in the RNVM, assuming a useful life between
10 and 30 years.
561
VOLUME 39 Nº3 SEPTEMBER – DECEMBER 2012
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Resumen
R. Muñoz y J. Llanos. 2012. Estimación de la vida útil del tractor agrícola mediante el uso de
un modelo de difusión a nivel agregado. Cien. Inv. Agr. 39(3): 557-562. El tractor ha sido icono de
cambio tecnológico en la agricultura, su adopción al interior de la explotación agraria ha signicado
diversas combinaciones, relaciones y formas establecidas de producción de una localidad, región
y país. La dinámica del proceso tractorización se encuentra denido por el balance dado por las
unidades que ingresan al parque y por aquellas que son resciladas por efecto de término de su vida
útil. Este trabajo hace uso del patrón de difusión y adopción agregada del tractor agrícola, cuyos
parámetros se determinaron mediante un procedimiento de estimación no lineal, con el empleo de
datos a nivel país, en Chile. Utilizando la capacidad de pronóstico del modelo, se estimó una serie
longitudinal del parque de tractores y, seguidamente, su balance mediante el cálculo de aquellas
unidades que son dadas de baja. Buscando determinar con mayor exactitud la vida útil promedio del
tractor, se realizaron sucesivas iteraciones, con objeto de reducir desviaciones cuando se confrontó
con una variable control, dada por una serie de tractores inscritos. La vida útil promedio estimada
para el tractor agrícola a nivel agregado sería de 22 años. Este procedimiento permitiría ser usado no
sólo para nes descriptivos, como en este caso, sino también para propósitos predictivos.
Palabras clave: Adopción tecnológica, difusión agregada, mecanización agrícola, tractor
agrícola, tractorización, vida útil.
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