Chapter five discusses technical efficiency, defining key terminologies and methods for estimating it, focusing on both parametric and non-parametric approaches. The stochastic frontier analysis is highlighted as a significant method that separates inefficiency from statistical noise. Ultimately, both econometric and mathematical approaches offer different advantages and disadvantages in estimating technical efficiency.

2. Chapter-Five
5. Technical Efficiency
Introduction
5.1.Definitions of terminologies
5.2. Estimation of technical efficiency
5.2.1. Parametric (econometric) approach
5.2.2. Non-parametric (mathematical) methods
5.3. The Stochastic Frontier Analysis

3. 5.1.Definitions of terminologies
The concept of efficiency emanates from the overwhelming idea
of production function.
Thus the explanation of efficiency will only be complete and clear
if we start from the concept of production and production function.
The term "production" refers to the process of transforming inputs
(such as raw materials, labor, capital, etc.) into outputs (finished
goods and services). This is a central concept in economic theory.

4. -The production function is the functional (mathematical) relationship
between inputs and output.
-It tells us the maximum quantity of output the firms can produce
given the quantities of the inputs that it might employ and the
technology used.
-For various reasons, only few firms could produce on the production
function and the vast majority of firms often produce below the
production function curve.

5.  The production function curve is a graphical representation of the production
function in economics.
 It shows the relationship between the quantity of inputs used and the
maximum amount of output that can be produced.
 The typical production function curve has the following characteristics:
 Positive slope: The curve slopes upwards from left to right, indicating that as
more inputs are used, more output is produced.
 Diminishing marginal returns: The curve becomes flatter as more inputs are
added. This means that each additional unit of input produces a smaller
increase in output, reflecting the law of diminishing returns.
 Concave shape: The curve is typically concave to the origin, meaning it is
bowed inward. This reflects the diminishing marginal returns.

6.  The mathematical expression for a simple production function is:
 Q = f(L, K)
Where:
Q is the quantity of output
L is the quantity of labor input
K is the quantity of capital input
f() is the production function
The production function curve plots the maximum output (Q) that can be produced
for different combinations of labor (L) and capital (K).

7.  Efficiency is how close a firm is to the production function curve.
 Thus, efficiency refers to how well a firm is performing in using resources to
produce outputs given available technology.
Economic efficiency is an overall performance measurement.
 It represents the efficient resource/input mix/ for any given
output that minimizes the cost of producing that level of output.

8. ▫ Technical efficiency reflects the ability of a firm to maximize output for a
given set of inputs.
-can be estimated when price information is available
▫ Allocative efficiency reflects the ability of the firm to use the inputs in optimal
proportions given their respective prices and the production technology.
- can be estimate only from data pertaining to physical quantity of inputs
and output

9.  Inefficiency is deviations of observed output from the best
production or efficient production frontier.

10. Estimation of technical efficiency
 The level of a firm’s efficiency is a measure of how far the firms
production is relative to the optimal level of output.
▫ Underlying assumption in this theoretical discussion is, the optimal
level of output from agiven set of inputs (and technology) is known.
Conceptually at least two possibilities to identify the efficient
production function.

11. 1. A theoretical production function specified by engineers
-Difficult to apply in practice
2. Empirical function based on the best results observed in practice
-Most commonly used in research and in practice
Michael Farrell was often credited for shading light on the
empirical measurement of technical efficiency.
• Idea: information is extracted from extreme observations of a body
of data to determine the best practice production frontier
▫

12.  From the computed best practice production frontier the
relative measure of technical efficiency for the individual firm
can be derived.
 Generally, the techniques fall under two distinctly opposing
approaches:
1. parametric (econometric) approach and
2. non-parametric (mathematical) methods

13. 5.2.1.The parametric approach
-It estimates a frontier function under a specific production function.
▫ The most common functional forms include the Cobb-Douglas,
Constant Elasticity of Substitution and Translog production functions.
 This implies that it impose a functional form on the production
function and make distributional assumptions about the data.
▫ This is an often quoted disadvantage of the technique.
▫ The most popular representative of parametric approach is stochastic
frontier analysis (SFA).

15. Advantage & disadvantage of parametric approaches:
 Advantage the parametric approach:
▫ it allows statistical inference and test of hypothesis
▫ Separates noise from inefficiency
 Disadvantage the parametric approach :
▫ Need to specify explicit functional form and distribution
assumption on the data.
-Possibility of misspecification

16. 5.2.2.A nonparametric approach
 is a linear programming technique and often represented by a technique is
refers to Data Envelopment Analysis (DEA).
▫ The DEA technique does not impose any assumptions about functional form
and hence it is less prone to misspecification.
▫ Further, since a non-parametric approach does not take into account random
errors, it is not subject to the problems of assuming an underlying distribution
about the error term.

17.  However, DEA cannot take account of such statistical noise means that the
efficiency estimates may be biased if the production process is largely
characterized by stochastic elements.
• Advantage : ▫ Doesn’t need to specify functional form or doesnot need
assumptions about error term
▫ Very flexible structure
• Disadvantage : ▫ Is not stochastic, so treat noise as inefficiency
▫ Sensitive to extreme observation
▫ it is not possible to estimate parameters for the model and hence impossible to
test hypothesis concerning the performance of the model.

18.  In general, the econometric and mathematical approaches differ in many ways and
none is absolutely superior to the other.
• Each of them has both advantages and disadvantages.
 A well-recognized limitation of the nonparametric approach is its deterministic
nature.
 Deterministic frontiers assume that all the deviations from the frontier are a result of
firms’inefficiency.
 ▫ Although it may suffer from specification error (if the functional form is
misspecified), econometric approach is superior to the mathematical approach in
distinguishing the effect of noise from the effects of inefficiency.

19. 5.3.The Stochastic Frontier Analysis
 The stochastic frontiers assume that part of the deviation from the
frontier is due to random events (reflecting measurement errors
and statistical noise) and the other part is due to firm specific
inefficiency
• It decomposes the error term into
▫ a two-sided random error that captures the random effects outside
the control of the firm (the decision making unit) and
▫the one-sided efficiency component.

20. Where:
Yi is the quantity of output of household i
Xi is the vector of input quantities used by household i
β is a vector of unknown parameters
• Taking the natural logarithm of both sides
▫ stochastic production function has two parts:
 the deterministic part and stochastic part.

21.  the stochastic frontier distinguishes itself from other econometric
models by partitioning the stochastic error term into two
components:
▫ the systematic random error accounting for statistical noise,and
▫ the inefficiency component.
The former is a two sided (symmetric) error component while the
latter is one sided error component

22.  ▫ If ui is 0 means that no inefficiency and the firm is
operating at its maximum possible production frontier,
given the technology and other exogenous factors
 Note: The one sided error component (ui)is assumed to
follow some particular distributions, of which the most
frequently used are:-
▫ half-normal ▫ truncated normal and
▫ exponential ▫ gamma distributions.

23.  Different distributions could potentially give rise to different
efficiency estimates and the extent to which the efficiency
scores and their ranking are sensitive to distributions is not
well documented in the literature.
 Therefore, the choice of distribution is sometimes a matter of
computational convenience.

24.  It implies that the technical efficiency of an individual farm can
be defined in terms of the ratio of the observed output to the
corresponding frontier output, given the available technology
• To make it clear, let’s rewrite the stochastic production function
(of Equation 1) as follows

25.  The frontier function (Y i*) can be derived from
the observed or stochastic function by letting
ui=0. Thus, the frontier function can be written
as
Given the equations for stochastic and frontier functions, the
technical efficiency of a given household can be written as:

26.  The one sided component reflects technical efficiency relative
to the stochastic frontier. Thus for a farm whose output lies on
the frontier, and for one whose output is below the frontier.