Excess Capital in Agricultural Production
ABSTRACT In this article we propose a theoretical model for analyzing capital requirement in agricultural production and define excess capital thereupon. We develop a two-step method that allows endogenous regressors in the maximum likelihood estimation. The two-step procedure is also capably of recovering the parameters of time invariant variables in fixed effect models. The model and method are applied to a capital requirement study using data from cash crop farms in the Netherlands. Empirical results show that excess capital widely exists on the farm. The implications of excess capital are further demonstrated with a production frontier analysis.
Excess Capital in Agricultural Production
Michigan State University
Subal C. Kumbhakar
State University of New York at Binghamton
Alfons Oude Lansink
Selected Paper prepared for presentation at the American Agricultural Economics
Association Annual Meeting, Portland, OR, July 29-August 1, 2007
Copyright 2007 by Zhengfei Guan, Subal C. Kumbhakar, and Alfons Oude Lansink. All rights
reserved. Readers may make verbatim copies of this document for non-commercial purposes by
any means, provided that this copyright notice appears on all such copies.
Excess Capital in Agricultural Production
In this article we propose a theoretical model for analyzing capital requirement in agricultural
production and define excess capital thereupon. We develop a two-step method that allows
endogenous regressors in the maximum likelihood estimation. The two-step procedure is also
capably of recovering the parameters of time invariant variables in fixed effect models. The
model and method are applied to a capital requirement study using data from cash crop farms in
the Netherlands. Empirical results show that excess capital widely exists on the farm. The
implications of excess capital are further demonstrated with a production frontier analysis.
Keywords: Agricultural production, capital requirement, endogeneity, excess capital, fixed
effect, maximum likelihood estimation, stochastic frontier
In empirical analysis of production, factor demand functions are usually derived under the
assumptions of profit maximization or cost minimization. The factor demand functions derived
under these behavioral assumptions indicate how much inputs a producer should use in order to
maximize profit or minimize cost given prices and the state of technology. In practice, however,
the actual usage of inputs can be higher or lower than the optimal amount. The amount of an
input used in actual production depends on various factors and can be studied directly from a
technical perspective with an input requirement function. The input requirement function shows
the minimum amount of an input that is required to produce a given level of output, given other
inputs and the technology. This technical approach to studying input requirement is desirable for
several reasons. First, empirical studies often reject the behavioral assumptions (Lin, Dean, and
Moore 1974; Ray and Bhadra 1993; Driscoll et al. 1997; Tauer and Stefanides 1998), in which
case, imposing behavioral assumptions to derive factor demand functions would result in biased
and inconsistent parameter estimates (Pope and Chavas 1994). Second, the price information
required for deriving factor demand functions is often unavailable which makes the traditional
approaches based on profit maximization of cost minimization inapplicable. Third, a study of
input actually used or technically needed yields insights on input requirement in production. This
direct perspective of factor demand is particularly relevant for producers in making decisions
regarding input use given resource endowment, production level, production technology adopted,
and production environment, etc. The information on input requirement is also useful for policy
making on resource use.
In the existing literature, primal studies on factor requirements include Diewert (1974),
Kumbhakar and Hjalmarrson (1995, 1998), Battese, Heshmati, and Hjalmarsson (2000),
Heshmati (2001), Kumbhakar, Heshmati, and Hjalmarsson (2002), El-Gamal and Inanoglu
(2005). This literature exclusively focuses on labor requirement in the production process. Until
now, we are not aware of any study that studies capital requirement. As one of the major factors
of production, capital presents not only an important but also a more complex and interesting case
Capital is often found overused in agricultural production. In studies using both farm- and
crop-level data, Guan et al. (2005, 2006) and Guan and Oude Lansink (2006) found that capital is
overused on cash crop farms in the Netherlands. Using a nonparametric method, Guan and Oude
Lansink (2003) concluded that Dutch agriculture is over-invested in capital and that capital is
weakly disposable (i.e., it can not be disposed of costlessly when in excess). Because of weak
disposability of capital, findings in these studies suggest that producers tend to have excess
capital, which is either not used or not fully used in actual production. As capital investment is
often an irreversible decision as suggested by Pindyck (1991), excess capital tends to persist.
The presence of excess capital means more than just a failure of profit maximization or cost
minimization. It has serious implications for the econometric analysis of production. It leads to
systematic measurement error if accounting data of capital stock are used in econometric
modeling. For empirical econometric modeling of production, the “fixedness” of capital makes it
“safe” to assume that capital is exogenous. Unfortunately, the exogeneity may not be as true as it
seems, because the capital actually used in the production depends on the production levels. In
agriculture, for example, a higher output level requires more capital for harvesting, processing,
and storage of the output. This implies simultaneity of capital. In fact, the measurement error and
simultaneity come hand in hand. Excess capital serves as a reservoir of capital supply when more
capital is needed due to a higher yield; and vice versa, when less capital is used due to a low
yield, excess capital appears. Measurement error and simultaneity of independent variables are
fundamental sources of endogeneity that jeopardizes the econometric estimation, if not properly
To date, excess capital has not been explicitly explored in the literature. Somewhat related to
excess capital, the concepts of “excess capacity” is proposed in the capacity utilization literature
(Klein 1960; Fare, Grosskopf, and Kokkelenberg 1989; Morrison Paul 1999; Dupont et al. 2002;
Kirkley, Morrison Paul, and Squires 2002, 2004; Felthoven and Morrison Paul 2004). In this
literature, the “capacity output” is defined as the maximum or potential output that the existing
capital stock, in conjunction with other inputs, can produce under normal working conditions. If
the capacity output is not achieved, there exists “excess capacity”. Notice that the excess capacity
is an output-oriented concept and is an index measured with all inputs, whereas the excess capital
is input-oriented and concerns with capital stock only. As a result, excess capacity is not a proper
proxy for excess capital. Furthermore, the presence of excess capital per se would bias the
measurement of excess capacity and capacity utilization.1 This situation calls for a direct
measurement of capital requirement to define excess capital.
We propose to use the stochastic frontier approach to measure excess capital. The theoretical
basis of the frontier analysis dates back to 1950s from Koopmans (1951), Debreu (1951), and
Shephard (1953). These studies construct a frontier and defined the distance relative to the
frontier as an efficiency measure. The stochastic frontier approach originated in the works of
Meeusen and van den Broek (1977) and Aigner, Lovell, and Schmidt (1977). Kumbhakar and
Lovell (2000) gave a comprehensive overview of this literature. The Stochastic Frontier Analysis
(SFA) has been widely applied in the economics literature, mainly to measure firms’ efficiency.
This approach uses maximum likelihood estimator to estimate the frontier function and the
composed error terms (see, e.g. Kumbhakar and Lovell, 2000). The weakness of the maximum-
likelihood based Stochastic Frontier Analysis is that when independent variables are endogenous
the estimation is inconsistent, in which case the traditional Stochastic Frontier Analysis would
fall flat and a solution is needed.
To address the issues raised, this study is aimed at both theoretical and methodological
contributions to the literature. First, we define a theoretical framework of capital requirement in
agricultural production from a primal, technical perspective. Based on the theoretical model, we
further develop a concept of excess capital. In the methodological respect, we propose a two-step
approach to solve the endogeneity and the resulting inconsistency problem in the maximum
likelihood estimation and apply it to the stochastic frontier analysis. This study further analyzes
the potential impact of the presence of excess capital on empirical analysis of production.
Capital requirement and excess capital
The capital requirement in agricultural production depends on many factors. Major factors
include the type of product produced on the farm, the production level, resource endowment and
technology used, natural and geographical condition, farm organizational arrangement, the
demographical characteristics of the farmer, and other unobserved factors.
The type and mix of enterprises in farm production determine the type of buildings,
machinery, and equipment and installations to be placed on the farm. For example, sowing
machine and harvesting combines are often necessary for cereal production while other types of
planting and harvesting machines are required for potato production. And for each type of
product, a higher production level generally requires a higher capital stock.
The resource endowment of the farm and the technology adopted in production directly
affect input-output combinations and the level of capital stock required. Strategically, if a farm
has less land relative to (family) labor, the farmer may adopt a labor-intensive production
technology which would require less capital for production. The technical substitution between
capital and other inputs can also affect capital needs. Chemicals, for example, can be used for
weed control and are substitute of mechanical weeding. In some circumstances, complementarity
may exist, which means the use of one input requires the use of other inputs. An example is that
fertilizer and pesticide application often requires machinery use.
Natural and geographical conditions that affect capital requirement include climate, weather,
geographic and soil conditions, etc. Extreme weather conditions would require additional
machinery in harvesting and drying and more storage spaces. For crop production, clay soil
would require more capital use than sandy soil, as it is easier for machinery to work on loose soils
than on sticky ones.
Organizational factors that affect capital requirement include land tenure regime,and use of
contract work or outsourcing, etc. When certain operations, such as breeding, planting and soil
disinfection are outsourced, capital stock to be maintained on the farm can be substantially
reduced. The difference in land tenure may induce strategic difference in production technology
and capital investment. Other than that, a leased farm is often equipped with some basic
infrastructure, but this may not be reported in the bookkeeping due to the differences in
bookkeeping rules. Other factors, such as the demographic and personal properties of the farm
operator (e.g. education level and farming experience) may affect how efficiently the capital is
used and therefore affect the capital required in the production.
As capital requirement can vary over time, some of the capital stock may not be used due to,
for example, yearly crop rotations. In the meanwhile, a farmer may opt to maintain a high level of
capital stock on the farm simply because he is risk averse and prefers to have more capital at his
disposal, to guarantee timely sowing or harvesting in the case of adverse weather conditions. All
these cases would result in excess capital on the farm. In the next section, we propose a
theoretical model and a two-step method to study the capital requirement in agricultural
production and measure excess capital.
Model and method
The theoretical model of capital requirement can be formulated as follows:
where k is the capital stock maintained on the farm, f(.) is the amount of capital required in
production, which is a function of all the factors discussed. Y is a vector of outputs produced on
the farm; X is a vector of inputs except capital used in the production; O represents all the other
factors discussed in the preceding section.
represents excess capital. When u is zero,
e is 1
and there is no excess capital. Thus, u > 0 measures the percentage of capital in excess. Random
factors like weather and other nonsystematic elements that affect capital use are accommodated in
the model by appending a random term v. Thus the stochastic capital requirement function is
where v can take both positive and negative values. The minimum amount of capital required to
produce a certain level of Y given the technology f(.), X, O and v is:
Thus, the excess capital can be measured from
By taking logarithm of both sides of the equation (2), the stochastic capital requirement function
can be rewritten as:
We assume half-normal and normal distributions for u and v, respectively:
),, 0 (
. . .
),, 0 (
u and v are distributed independently of each other.
Based on these assumptions, the probability density function of the joint distribution of
Φ ⋅ ⎟
, and () .
and () .
are the standard normal probability density
and cumulative distribution functions, respectively (see Appendix A). With an explicit functional
form of f(.), the capital requirement function can be estimated with the maximum likelihood (ML)
estimation. Excess capital u for each observation can then be derived from the conditional
based on the conditional probability density function, ()
Endogeneity, ML estimation, and a two-step method
Since Kumbhakar and Hjalmarrson (1995) first used the stochastic frontier to study labor use in
the Swedish insurance offices, this approach has been used in several studies to model labor use
efficiency (Kumbhakar and Hjalmarsson 1998; Battese, Heshmati, and Hjalmarsson 2000;
Heshmati 2001; Kumbhakar, Heshmati, and Hjalmarsson 2002). A common assumption in this
literature is that the level of output produced with labor, an explanatory variable of labor use, is
considered exogenous when estimating the labor requirement function. Although this assumption
may not pose problems in some special cases where exogeneity does hold, theoretically this is a
strong assumption. The vast literature on production function models where output is modeled as
a function of inputs (including labor) makes the endogeneity of output a legitimate issue to be
addressed in the input requirement model.
Endogeneity poses a general problem to the maximum likelihood (ML) estimation in the
stochastic frontier analysis. The main sources of endogeneity are heterogeneity, and simultaneity
of and/or measurement errors in regressors. In least square (LS) based estimations, the
endogeneity problem may be solved with instrumental variables (IV) method. The IV method,
however, generally does not apply to ML estimation. For the ML-based stochastic frontier
analysis (SFA), the problem of endogeneity points to the very foundation of this literature.
In this study we propose a two-step method to solve the problem. In the first step the model
is estimated with least squares where endogeneity is addressed with instrumental variables. The
residuals from the first step estimation are further regressed on a constant, and additional
variables, if any, in the second step with ML estimation.
In the capital requirement model, we address the endogeneity of output Y in the first step and
employ the ML estimation to derive the excess capital in the second step. For this purpose we
assume a log-linear relationship between factors (Y, X), and O, and rewrite eq. (5) as,
ln( ) ( ,; )
( ; )
f Okf Y Xuv
=+ + +
where α andβ are vectors of parameter to be estimated. We further rewrite the model as:
The first step is to estimate the model in (8). Notice that, the model (8) now has omitted variables
O. The consequence of omitting other factors is that these factors are captured by the residuals
1e , which may cause the residuals to correlate with the regressors in the first-step model and
result in biased and inconsistent estimates (see, e.g., Pindyck and Rubinfeld 1998, pp.185). This
problem must be explicitly addressed, for which a robust estimation procedure must be used.
1e as dependent variable in (9), we estimate the effect of the O variables on capital use
as well as excess capital for each observation. The model is estimated with ML estimator based
on the joint distribution of the composed error
given in (6). As the dependent variable
1e is not observable, it is replaced by
ln( )( , ; )
kf Y X α=−
After the estimation, the excess capital component u for each observation is obtained from
The model in (8) is specified as:
)ln()ln() ln() ln()ln() ln(
where k denotes capital stock; the subscript i indexes individuals, and t indexes time periods;
subscripts j and l index inputs. The variable y denotes the output level produced on the farm; the
variable t and t2 specify a quadratic time trend. The error term is defined as
ie is the
individual effect, and model (11) is a fixed effect model. This translog model is similar to
production function models except that the capital stock and the output variables are switched.
The model in the second step regresses the residuals from (11) on other factors O, which
affect capital requirements but are not included in the first step:
e Age ContrShareSize
it u and
it v are assumed to follow half-normal and normal distribution,
respectively, as mentioned before;
DType, dummy variable for product types (0 for not being a particular product, 1 for yes),
DSoil, dummy soil type (0 for sandy soil, 1 for clay),
DTenu, dummy land tenure (0 for own land, 1 for lease),
DEdu, discrete education level (1 for primary school, 2 for non-agri education,
3 for vocational education in agriculture, 4 for higher education in agriculture),
Size, size of farm operation, in NGE (standardized Dutch Farm Unit),
Share is the share of non-arable farming operations on the farm in terms of size
Contr, the amount of contract work
Age, the age of farmer,
Di is the farm dummy.
In this model, the dummy variables DType represent the type of major product on the farm, and
its number m depends on the number of enterprises or products in the sample. DSoil is a dummy
variable for soil; DTenu is a dummy for land tenure; DEdu represents the level of education. Size
is measured in standardized Dutch Farm Unit (NGE), which is defined based on the scale,
intensity and income generating ability of the farm operations (Van den Tempel and Giesen 1992,
pp. 285-288). Share is the share of non-arable farming operations in terms of NGE for the case
study of cash crop productions in the Netherlands. Contr denotes the amount of contract work.
The variable Age of the farm operator is a proxy for experience and perhaps some other
demographic characteristics as well.
The product dummy and the share of non-arable operation distinguish the capital
requirements of different enterprises or product mixes. The farm and soil dummies capture the
impacts of natural and geographical factors. Land tenure, amount of contract work, and the size
of the farm represent the organizational factors. The education level and the age reflect the
demographic differences of farm operators. The factors used in the second step cover both factors
that affect the “standard” technical requirement of capital (e.g., from product or soil type) and
those that cause additional “non-standard” or inefficient use of capital (e.g., education or
experience). The unexplained part of the capital stock is due to white noise v and a one-sided
error term u which captures excess capital.
The rationale underlying the split of model (11) and (12) is that the former addresses the
basic input-output relations and the latter investigates the effect of farm characteristics on capital
requirement. Technically, this split also ensures the recovery of parameters of time invariant farm
characteristics (e.g. soil type) which would otherwise be impossible in the fixed effect model.
This further contributes to the literature and justifies the two-step approach.
Data and Estimation
The empirical study of capital requirement and excess capital is applied to data from the farm
accountancy data network (FADN) of the Agricultural Economics Research Institute (LEI)) in the
Netherlands. Panel data are available over the period 1990-1999 from 486 cash crop farms with a
total of 2511 observations. The panel is unbalanced and farms stay in the sample for 5 years, on
The capital requirement function in the first-step is estimated with a single output and 5 inputs.
The capital stock consists of buildings, machinery, equipment and installations. The output
measured revenues from all products. The inputs included are land (x1), labor (x2), fertilizer (x3),
pesticide (x4), and miscellaneous inputs (x5). Land was measured in hectares, and labor in quality-
corrected man-years. Miscellaneous inputs included seed, feed, energy, and services. The capital
stock, output, fertilizer, pesticide, and miscellaneous inputs were deflated to 1990 prices (prices
were obtained from the LEI/CBS2). Tornqvist price indices were calculated for capital and
miscellaneous inputs. For the second step model, 7 product types were distinguished, viz.,
cereals, root crops, mix of cereals and root crops, mix of root and other crops, open-field
vegetables, and mix of arable, horticultural and fruit production. The soil dummy takes the value
0 for sandy soil and 1 for clay soil. Land tenure distinguished own land and leased land3 for the
farm production. The education of farm operators was measured in 4 levels from low to high.
The dummy soil type is time invariant for individual farms; the dummies for product type, land
tenure and education have no or little variation over time. Other variables include the amount of
contract work, the size of the farm, and the age of the farm operator. The summary statistics of
non-dummy variables are presented in table 1.
Table 1. Summary Statistics of Cash Crop Farms in the Netherlands, 1990-1999
Variable Unit Mean Std Dev.
Capital thousand euro 229.33 180.57
Output thousand euro 224.82 175.34
Land hectare 64.92 43.63
Labor man-year 1.92 1.18
Fertilizer thousand euro 9.24 6.72
Pesticide thousand euro 16.58 12.09
Misc. thousand euro 47.46 39.01
Contract work thousand euro 10.27 7.40
Size Dutch farm unit (NGE) 114.93 79.55
Age years 49.08 10.94
Share non-arable ratio 0.09 0.12
Source: Dutch Agricultural Economics Research Institute (LEI)
Note: The statistics are per farm year, computed with 2511 observations from
486 farms; the monetary unit is in 1990 prices.
Step 1: In the estimation of the panel data model in (11) there are three issues to be addressed: i)
The heterogeneity across farms, ii) the simultaneity of output, and iii) the omission of other