Transcript Lecture 1

Efficiency and Productivity Measurement:
Basic Concepts
D.S. Prasada Rao
School of Economics
The University of Queensland, Australia
1
Objectives for the Workshop
1. Examine the conceptual framework that
underpins productivity measurement
2. Introduce three principal methods
• Index Numbers
• Data Envelopment Analysis
• Stochastic Frontiers
Examine these techniques, relative merits,
necessary assumptions and guidelines for
their applications
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Objectives for the Workshop
3. Work with computer programs (we use
these in the afternoon sessions)
•
•
•
TFPIP; EXCEL
DEAP
FRONTIER
4. Briefly review some case studies and real
life applications
5. Briefly review some advanced topics on
Thursday and Friday morning
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Main Reference
An Introduction to Efficiency and
Productivity Analysis (2nd Ed.)
Coelli, Rao, O’Donnell and Battese
Springer, 2005
Supplemented with material from other
published papers
4
Outline for today
• Introduction
• Concepts
– Production Technology
– Distance Functions
• Output and Input Oriented distance functions
• Techniques for Efficiency and
Productivity Measurement:
– Index Number methods
5
Introduction
•
•
•
•
Performance measurement
– Productivity measures
– Benchmarking performance
• Mainly using partial productivity measures
• Cost, revenue and profit ratios
• Performance of public services and utilities
Aggregate Level
– Growth in per capita income
– Labour and total factor productivity growth
– Sectoral performance
• Labour productivity
• Share in the total economy
Industry Level
– Performance of firms and decision making units (DMUs)
– Market and non-market goods and services
– Efficiency and productivity
– Banks, credit unions, manufacturing firms, agricultural farms, schools
and universities, hospitals, aged care facilities, etc.
Need to use appropriate methodology to benchmark performance
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Efficiency:
(i) How much more can we produce with a given level of inputs?
(ii) How much input reduction is possible to produce a given level of
observed output?
(iii) How much more revenue can be generated with a given level of
inputs? Similarly how much reduction in input costs be achieved?
Productivity:
•
We wish to measure the level of output per unit of input and
compare it with other firms
•
Partial productivity measures – output per person employed;
output per hour worked; output per hectare etc.
•
Total factor productivity measures – Productivity measure which
involves all the factors of production
•
More difficult to conceptualise and measure
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Simple performance measures
• Can be misleading
• Consider two clothing factories (A and B)
• Labour productivity could be higher in firm
A – but what about use of capital and
energy and materials?
• Unit costs could be lower in firm B – but
what if they are located in different regions
and face different input prices?
8
Terminology?
• The terms productivity and efficiency relate to
similar (but not identical) things
• Productivity = output/input
• Efficiency generally relates to some form
benchmark or target
• A simple example – where for firm B
productivity rises but efficiency falls:
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Basic Framework: Production Technology
• We assume that there is a production technology that allows
transformation of a vector of inputs into a vector of
outputs
S = {(x,q): x can produce q}.
• Technology set is assumed to satisfy some basic axioms.
• It can be equivalently represented by
– Output sets
– Input sets
– Output and input distance functions
• A production function provides a relationship between the
maximum feasible output (in the single output case) for a
given set of input
• Single output/single input; single output/multiple inputs;
multi-output/multi-input
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Output and Input sets
• Output set P(x) for a given vector of inputs, x, is the set of all
possible output vectors q that can be produced by x.
P(x) = {q: x can produce q} = {q : (x,q)  S}
– P(x) satisfies a number of intuitive properties including: nothing can be
produced from x; set is closed, bounded and convex
– Boundary of P(x) is the production possibility curve
• An Input set L(q) can be similarly defined as set of all input vectors
x that can produce q.
L(q) = {x: x can produce q} = {x: (x , q)  S}
– L(q) satisfies a number of important properties that include:
closed and convex
– Boundary of L(q) is the isoquant curve
• These sets are used in defining the input and output distance
functions
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Output Distance Function
• Output distance function for two vectors x (input) and q
(output) vectors, the output distance function is defined as:
do(x,q) = min{: (q/)P(x)}
• Properties:
– Non-negative
– Non-decreasing in q; non-increasing in x
– Linearly homogeneous in q
– if q belongs to the production possibility set of x (i.e.,
qP(x)), then do(x,q)  1 and the distance is equal to 1
only if q is on the frontier.
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Output Distance Function
y2

y2A
A
B

C
PPC-P(x)
P(x)
0
y1A
y1
Do(x,y)
The value of the distance function is
equal to the ratio =0A/0B.
Output-oriented Technical Efficiency Measure:
TE = 0A/0B = do(x,q)
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Input Distance Function
• Input distance function for two vectors x (input) and q
(output) vectors is defined as:
di(x,q) = max{: (x/)L(q)}
• Properties:
– Non-negative
– Non-decreasing in x; non-increasing in q
– Linearly homogeneous in x
– if x belongs to the input set of q (i.e., xL(q)), then
di(x,q)  1 and the distance is equal to 1 only if x is on
the frontier.
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Input Distance Function
x2
A
x2A
L(y)
B
Isoq-L(y)
C
0
x1A
x1
Di(x,y
The value of the distance function is
equal to the ratio =0A/0B.
Technical Efficiency = TE = 1/di(x,q) = OB/OA
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Input and Output Distance Functions
• What is the relationship between input and
output distance functions?
• If both inputs and outputs are weakly
disposable, we can state that
di(x,q)  1 if and only if do(x,q)  1.
• If the technology exhibits global constant
returns to scale then we can state that:
di(x,q) = 1/do(x,q), for all x and q
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Objectives for the firm
• The production technology defines the
technological constraint faced by the firm
• The objective of the firm could be to maximise
profit
• Or minimise costs when outputs are fixed
• Or maximise revenue when inputs are fixed
• Or ….
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Profit maximisation
• Firms produce a vector of M outputs (q)
using a vector of K inputs (x)
• The production technology (set) is:
S = {(x,q) : x can produce q}
• Maximum profit is defined as:
 (p, w)  max {(pq  wx) : (x, q)  S}
q ,x
where p is a vector of M output prices
and w is a vector of K input prices
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Profit maximisation
q
Iso-profit line:
q = π/p + (w/p)x
frontie
r
Profit max
x
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Cost minimisation
• The firm must produce output, q0
• Minimum cost is defined as:
c(q, w)  min {wx : (x, q)  S}
x1
x
Cost min
Iso-cost line:
x1 = c/w1 – (w2/w1)x2
Isoquant (q=q0)
x2
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Revenue maximisation
• The firm has input allocation, x0
• Maximum revenue is defined as:
r(p, x)  max {pq : (x, q)  S}
q
y1
Revenue max
Iso-revenue line:
y1 = r/p1 – (p2/p1)y2
PPC (x=x0)
y2
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Short versus long run
• In the long run all things can vary
• In the short run some things are fixed
• Cost min can be viewed as profit max in
the short run when outputs are fixed
• Revenue max can be viewed as profit max
in the short run when inputs are fixed
• One can also fix a subset of inputs (e.g.,
capital) and look at short run profit max
or short run cost min, etc.
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Production function
Marginal product
Production elasticity
Scale elasticity
q  f (x)
MPn 
f (x)
xn
f (x) xn
En 
xn q
N
   En
n 1
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Returns to Scale
• A production technology exhibits constant returns
to scale (CRS) if a Z% increase in inputs results in
Z% increase in outputs (ε = 1).
• A production technology exhibits increasing
returns to scale (IRS) if a Z% increase in inputs
results in a more than Z% increase in outputs (ε >
1).
• A production technology exhibits decreasing
returns to scale (DRS) if a Z% increase in inputs
results in a less than Z% increase in outputs (ε <
1).
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Returns to scale
q
DRS
CRS
IRS
x
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Economies of scope
• Is it less costly to produce M different products in
one firm versus in M firms?
• One measure of economies of scope is:
M
S
 c(w, q
m 1
m
)  c ( w, q)
c( w , q)
• S > 0 implies economies of scope – it is better to
produce the M outputs in one firm.
• Other measures:
– product specific measures
– second derivative measures
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Efficiency Measures
• Using the distance functions defined so far, we can
define:
– Technical efficiency
– Allocative efficiency
– Economic efficiency
• A firm is said to be technically efficient if it operates on
the frontier of the production technology
• A firm is said to be allocatively efficient if it makes
efficient allocation in terms of choosing optimal input
and output combinations.
• A firm is said to be economically efficient if it is both
technically and allocatively efficient.
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Productivity and Efficiency Concepts
• Concepts
–
–
–
–
–
–
technical efficiency
scale efficiency
allocative efficiency
cost efficiency
revenue efficiency
total factor productivity (TFP)
• Brief overview of empirical methods
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Technical Efficiency
q
Frontier
B

E
C 
A
D
Output orientation: TEO=DA/DB
Input orientation:
TEI=EC/EA
x
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Scale Efficiency
CRS Frontier
q
VRS Frontier
TEVRS=DB/DA
D
C  
A
B
TECRS = DC/DA
SE=DC/DB = TECRS/TEVRS
x
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Allocative Efficiency
labour
B
isocost ($420)
A
isoquant (y=200)
0
capital
isocost ($360)
AE=360/420=0.86
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Allocative Efficiency (2)
isocost ($560)
labour
isocost ($400)

A
D

E

C
isoquant (y=200)
TE=400/560=0.71
0
isocost ($360)
capital
AE=360/400=0.9
CE=360/560=0.64
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Output orientated efficiency
shirts
C


B

A
D
iso-revenue line
PPC
TEO=0A/0B
AEO=0B/0C
RE=0A/0C
0
trousers
=TEO×AEO
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Productivity?
• productivity = output/input
• What to do if we have more than one input
and/or output?
– partial productivity measures
– aggregation
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Example
• Two firms producing t-shirts using labour
and capital (machines).
• The partial productivity ratios conflict.
firm labour capital output
(x1)
(x2)
(q)
A
2
2
200
B
4
1
200
q/x1
q/x2
100
50
100
200
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Total factor productivity (TFP)
• Use an aggregate measure of input:
TFP = y/(a1x1+a2x2)
• What should we use as the weights? – prices?
• Data: Labour wage = $80 per day and
Rental price of the machines = $100 per day
• Calculation:
TFPA = 200/(80×2+100×2) = 200/360 = 0.56
TFPB = 200/(80×4+100×1) = 200/420 = 0.48
=>A is more productive using this measure.
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TFP decomposition
• Can decompose TFP difference between 2
firms (at one point in time) into 3 types of
efficiency:
– technical efficiency;
– allocative efficiency; and
– scale efficiency.
• Need to know the technology
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TFP growth components
•
•
•
•
technical change (TC)
technical efficiency change (TEC)
scale efficiency change (SEC)
allocative efficiency change (AEC)
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How do we measure efficiency?
• Depends upon the type of data available for the
measurement purpose.
• Three types:
– Observed input and output data for a given firm over
two periods or data for a few firms at a given point of
time;
– Observed input and output data for a large sample of
firms from a given industry (cross-sectional data)
– Panel data on a cross-section of firms over time
• In the first case measurement is limited to
productivity measurement based on restrictive
assumptions.
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Overview of Methods
• index numbers (IN)
– Price and quantity index numbers used in
aggregation (eg. Tornqvist, Fisher)
• data envelopment analysis (DEA)
– non-parametric, linear programming
• stochastic frontier analysis (SFA)
– parametric, econometric
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Relative merits of Index Numbers
• Advantages:
– only need 2 observations
– transparent and reproducible
– easy to calculate
• Disadvantages:
– need price information
– cannot decompose
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Relative merits of Frontier Methods
• DEA advantages:
 no need to specify functional form or
distributional forms for errors
 easy to accommodate multiple outputs
 easy to calculate
• SFA advantages:
 attempts to account for data noise
 can conduct hypothesis tests
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