Production Theory

Production is an activity to create, produce, and make. Production activities

cannot be carried out if there are no raw materials to support the production
process. Production activities can be carried out if there are elements of labor,
natural resources, capital in all its forms, and skills. So, all elements that support
the effort to create value or increase the value of goods are called production
factors (Rosyid, 2009). Production theory explains how to use input to produce a
certain amount of output in a production process. The relationship between input
and output as explained in production theory will be discussed further using the
production function.

From a production function, it will be known how the addition of a certain amount
of input proportionally will be able to produce a certain amount of output. The
agricultural-based production system applies the concept of input or output and
the relationship between the two in accordance with the concept and concept of
production theory. Production factors in farming are elements needed in farming.
Production factors are inputs in the agricultural production process. The
agricultural production process is a process that combines agricultural production
factors to produce agricultural products (output). Production factors do indeed
determine the size of the production obtained (Soekartawi, 2011).

Production Function
According to Soekartawi (2002) the production function is a physical relationship
between factors or production inputs ( input ) and production ( output ). The
production factors in question are land, fertilizer, labor, capital to buy seeds,
fertilizer, medicines, pay for labor and management aspects that affect the size of
the production obtained. The relationship between input and output can be written
mathematically as follows.

Y = f(X 1,X 2,X3,...,Xm)

Where:
Y = production (output)
X 1,X 2,X3,...,Xm = production factors (input)
Production function analysis is carried out to find out information on the use of
limited production factors to produce maximum production, but in practice the use
of limited production factors is influenced by factors beyond human control such
as climate change, pest and disease attacks. These external factors are known as
uncertainty and risk factors. The magnitude of the uncertainty factor will
determine the magnitude of the risk that will be faced. The production function
used to analyze production in conditions of uncertainty and risk is a deterministic
production function that uses unchanged variables in its analysis. This function
will produce relatively good estimates with a production function form that is in
accordance with the known problems so that the function can be used to determine
a good combination of inputs and determine the magnitude of the influence of
input on output (Soekartawi, 2002). According to Khusaini (2013) in adding
production factors there are three stages, namely
1. Stage I, at this stage the output elasticity value is positive and greater than one.
The total production curve (TP) shows an increase accompanied by an increase
in the marginal product curve (MP) until it reaches a maximum and then
decreases due to the addition of input, while the average product curve (AP)
continues to increase, but has not reached maximum profit.
2. Stage II, at this stage the output elasticity value is equal to 1, the TP curve
continues to increase until it reaches a maximum. The PM curve continues to
decrease until it becomes zero when the TP curve reaches a maximum, while
the AP curve reaches a maximum when it intersects the PM curve which causes
the amount of marginal product to be equal to the amount of average product.
This stage 2 will achieve maximum profit.
3. Stage III, output elasticity is negative because the addition of input causes the
total product to decrease and is followed by a decrease in marginal product and
average product, so that the addition of input will reduce profits. The
relationship between the TP, MP and AP curves can be seen in Figure 1.3.
Figure 1.3 Relationship between TP, MP and AP curves (Khusaini, 2013).
Cobb-Douglas Production Function
The Cobb-Douglas production function is a function or equation that involves two
or more variables. In this case, one variable is called the dependent variable (the
variable that is influenced) and the other variable is called the independent
variable (the variable that influences or the variable that explains). Production
function analysis is often used in empirical research because it is used to find out
information about how limited resources such as land, capital, and labor can be
managed well so that maximum production can be obtained (Rahim, 2012). The
Cobb-Douglas production function was introduced by Charles W. Cobb and Paul
H. Douglas, systematically the Cobb-Douglas production function is written as
follows:
Y = αX1β1, X2β 2,... , Xiβi,...,Xnβneu
If the Cobb-Douglas production function is expressed by the relationship between
Y and X, then it can be:
Y = f( X1, X2,..., Xi,..., Xn)
To facilitate the estimation and estimation of the parameters, they are changed into
the form of double natural logarithm (ln) so that they become multiple linear
forms which are then analyzed using the ordinary least square method which is
formulated as follows:
LnY = Lnα + β1LnX1 + β2LnX2 + ,... ,+ βnLnXn
Information:
Y = variable being explained
X = explanatory variable
α = intercept/constant
β = regression coefficient
u = error (disturbance term)
e = natural logarithm (Rahim, 2012).
The Cobb Douglas production function is more widely used by writers because it
has the following advantages (Rahim, 2012):

 Solving the Cobb-Douglas function is relatively easier compared to other

functions, because the Cobb-Douglas function can easily be transferred to a
linear form by logarithmizing.
 The results of the estimation using the Cobb-Douglas function will produce a
regression coefficient which also shows the magnitude of elasticity.
 The total elasticity also shows the level of business scale (return of scale)
which is useful for knowing whether the activities of a business follow the
rules of increasing business scale, constant business scale or decreasing
business scale.
 The intercept coefficient of the Cobb Douglas function is a production
efficiency index that directly describes the efficiency of input use in producing
output from the production system being studied.
 The coefficients of the Cobb Douglas function directly describe the production
elasticity of each input used and considered for study in the Cobb Douglas
production function.

Stochastic Frontier Function

Frontier production is not much different from the production function in general,
but the stochastic frontier production function is widely used to explain the
concept of measuring efficiency, and also emphasizes the maximum output
conditions that can be produced (Coelli et al., 1998). The frontier production
function is derived by connecting the maximum output points for each level of
input use, so that it can represent the most technically efficient input-output
combination. According to Doll and Orazem (1984) the frontier production
function is the most practical production function that describes the potential
production of variations in the combination of production factors at a certain level
of technology and knowledge. In general, the measurement of the frontier
production function is distinguished in 4 ways, namely: (1) deterministic
nonparametric frontier, (2) deterministic parametric frontier, (3) deterministic
statistical frontier, and (4) stochastic statistical frontier (stochastic frontier). The
stochastic frontier production function can be written mathematically as follows:

ln (yi) = xi β + ( v i – u i ), i=1, 2, 3,…n

Information:
ln(y i ) = logarithm of the i-th farmer's output

xi = input used by farmer i

β = parameter to be estimated

vi = random effect, which is related to external factors

affect production

ui = random variables describing technical inefficiency

Aigner et al in (Murniati, 2014) assume that vi is a normal random variable that is

independently and identically distributed (iid) with a mean of zero and constant
variance, while ui is an independent variable that is assumed to be independently
and identically distributed exponentially or half-normally. If two deviations (vi and

ui ) are assumed to be independent of each other and independent of production

input (xi) and specific distribution assumptions (normal and half-normal
respectively) are installed, then the likelihood function (maximum likelihood
estimators) can be estimated. The stochastic frontier production function model
with two error terms used to estimate the technical efficiency of farming can be
expressed as follows:

Yi* = f (Xi; β) + єi

Information:

Yi = output or output produced by the ith observation

Xi = input vector used by the i-th observation

β = vector of unknown parameter coefficients.

Єi = error term of the i-th observation

The stochastic frontier model is also called the composed error model because the
error term consists of two components, namely:

Єi = v i – u i I = 1,2,.. ,n
There are two error components in the stochastic frontier production function,
namely, first, error due to external factors which are assumed to follow a normal
2
distribution (vi ≈ 0 ,σ v ). Second, errors due to internal factors that can be
controlled by farmers, for example the ability of farmers to manage their farming,
2
this component is assumed to be half normal distribution (ui ≈ N ( 0,σ u ) or
asymmetrically distributed (ui > 0). Production is said to be efficient if the
production achieved is the same as its potential or ui = 0, conversely if the
production achieved is below its potential, then ui > 0. The stochastic frontier
production function can be estimated using two methods, namely the Ordinary
Least Square (OLS) method and the Maximum Likelihood Estimation (MLE)
method. Ordinary Least Square (OLS) is an estimation method commonly used in
linear regression analysis, both simple and multiple. This method is the most
frequently used method. The main characteristic of OLS is the existence of a
"Line of Best Fit" which can be interpreted as the sum of the squares of the
deviations of observations or observation points from the minimum regression
line. Maximum Likelihood Estimation (MLE) is a technique used to find a certain
point to maximize a function, this technique is widely used in estimating a data
distribution parameters and remain dominant in the development of new tests.