topic 2
TRANSCRIPT
ECON 377/477
2
Topic 2
Productivity and Efficiency Measurement Concepts
Econ 377/477 Topic 2 3
Productivity and efficiency measurement: concepts
• Concepts and terminology• Brief description of the methods
o Data envelopment analysiso Index numberso Stochastic frontier analysis
Econ 377/477 Topic 2 4
Terminology
• Productivity• Technical efficiency• Allocative efficiency• Cost efficiency• Technical change• Scale efficiency• Constant returns to scale ...
Econ 377/477 Topic 2 5
Terminology
• Point of (technically) optimal scale• Total factor productivity (TFP)• Production frontier• Feasible production set• Output-mix allocative efficiency• Revenue efficiency
Econ 377/477 Topic 2 6
Set theoretic representation of a production technology
• Production function: a single-output technology• Production technology: a multiple-output
production process• The technology set consists of all input-output
vectors (x,q) such that x can produce q:
S = {(x,q): x can produce q}
Properties: CROB, pp 43-44
Econ 377/477 Topic 2 7
Set theoretic representation of a production technology
• Output sets: P(x) = {q: x can produce q}
= {q: (x,q) S}• Input sets: L(q) = {x: x can produce q}
= {x: (x,q) S}• Note the properties of the output set defined
by CROB
Properties: CROB, pp 43-44
Econ 377/477 Topic 2 8
Production possibility curves and revenue maximisation: a
digression• Consider a one-input two-output example, and
specify an input requirement function:
x1 = g(q1,q2)
to illustrate a production possibility curve (PPC)• The isorevenue line is the negative ratio of the
output prices• The optimal (revenue-maximising) point is the
point of tangency between this line and the PPC, shown in the diagram on the next slide
Econ 377/477 Topic 2 9
Production possibility curves and revenue maximisation: a
digression
0 q1
q2
Isorevenue line (slope = -p1/p2)
A
Optimal point
PPC(x1=x10)
Econ 377/477 Topic 2 10
Technical change and the PPC
• Technical change can favour the production of one commodity over another, illustrated on the next slide
• The outward shift of the PPC in part (a) is consistent across the output-output space
• In contrast, the outward shift of the PPC in part (b) is greater close to the horizontal axis than it is close to the vertical axis
• That is, technical change favours the production of q1 over q2
Econ 377/477 Topic 2 11
Technical change and the PPC
Neutral technical change Non-neutral technical change
q2 q2
q1q10 0
PPC(x=x10,t=1)
PPC(x=x10,t=0)
PPC(x=x10,t=1)
PPC(x=x10,t=0)
Econ 377/477 Topic 2 12
Output and input distance functions
• Distance functions are closely related to production frontiers
• Radial contractions and expansions are involved in defining distance functions
• They enable description of multi-input and multi-output production technology without the need to specify a behavioural objective
• Either output or input distance functions are usually specified, but there are also directional distance functions
Econ 377/477 Topic 2 13
Output distance functions
• The diagram on the next slide demonstrates an output distance function
• An output distance function considers a maximal proportional expansion of the output vector, given an input vector
• It is defined on the output set, P(x), as:
do(x,q) = min{δ: (q/δ)P(x)}
Econ 377/477 Topic 2 14
Output distance functions
q2
q1
C
B
A
PPC-P(x)
0 q1A
q2A
d = 0A/0B•The reciprocal is the factor by which the production of all output quantities can be increased while remaining within the feasible PPC for the given input level
•B and C would have distance function values of 1
•A would have a distance function value that is less than 1
P(x)
Econ 377/477 Topic 2 15
Input distance functions
• The diagram on the next slide demonstrates an input distance function
• An input distance function characterises the production technology as the minimal proportional contraction of the input vector, given an output vector
• It is defined on the input set, L(q), as:
di(x,q) = max{ρ: (x/ρ)L(q)}
Econ 377/477 Topic 2 16
Input distance functions
X2
X1
C
B
A
Isoq-L(q)
0 X1A
X2A
The function di(x,q) = max{:x/ )L(q)} is:
•non-decreasing in x•non-increasing in q•linearly homogeneous in x•quasi-concave in q and concave in x
If x L(q), then di (x,q) 1; ρ = 0A/0B
If both inputs and outputs are weakly disposable, then di(x,q) 1 iff do(x,q) 1
Under constant returns to scale:
di (x,q) = 1/do(x,q) for all x and q
L(q)
Econ 377/477 Topic 2 17
Efficiency measurement concepts
• References: Farrell (1957) and Debreu (1951)• Two efficiency components are:
o Technical efficiency• the ability of the firm to obtain maximal output
from given sets of inputso Allocative efficiency
• the ability of the firm to use the inputs in optimal proportions, given their respective prices
• Economic efficiency (EE) is the product of technical efficiency (TE) and allocative efficiency (AE)
Econ 377/477 Topic 2 18
Input-orientated measures
• Assume two inputs (x1and x2) and one output (q) under the assumption of constant returns to scale
• The unit isoquant of fully efficient firms is represented by SS’
• Suppose a firm uses quantities of inputs defined by point P in the graphical representation on the next slide
• Its technical efficiency is represented by the distance, QP
Econ 377/477 Topic 2 19
Input-orientated measuresX2/q
X1/q
Q’
Q
P
S’
0
S
A
A’
R
Econ 377/477 Topic 2 20
Interpreting input-orientated measures
• Technical inefficiency can be represented as the distance QP – the amount of inputs that could be proportionally reduced without reducing output
• It represents the percentage by which all inputs need to be reduced to achieve technically efficient production, the ratio QP/0P
• The technical efficiency index, 0Q/0P, lies between zero and one
• A firm is fully efficient if it has a technical efficiency index of 1
Econ 377/477 Topic 2 21
Interpreting input-orientated measures
• Given an input-price ratio, the allocative efficiency index is 0R/0Q
• No reduction in cost is possible if production is at Q’
• Point Q is technically efficient but allocatively inefficient
Econ 377/477 Topic 2 22
Interpreting input-orientated measures
• Economic efficiency is EEi = 0R/0P• That is, EE = TE*AE• We assume the production function of
the fully efficient firm is known• Note that this is not always the case!• An efficient isoquant must be estimated
using sample data
Econ 377/477 Topic 2 23
• To specify a parametric frontier production function in input-output space, consider a Cobb-Douglas function:
ln(yi) = f(ln(xi), ) - ui
where:
yi is the output of the i-th firm
xi is an input vector
ui is a non-negative variable representing inefficiency
Output-orientated measures
Econ 377/477 Topic 2 24
• Technical efficiency is calculated as:TEi = yi/f(ln(xi),) = exp (-ui)
• An output-orientated measure indicates the magnitude of the output of the i-th firm relative to the output that could be produced by the fully efficient firm using the same input vector
• This approach does not account for noise and we need to impose a functional form
Output-orientated measures
Econ 377/477 Topic 2 25
• The difference between input- and output-orientated measures can be illustrated by a simple example of one input, x, and one output, q
• The first diagram on the next slide shows the case of a decreasing-returns-to-scale technology, f(x), and an inefficient firm operating at point P
• The Farrell input-orientated measure of TE is the ratio, AB/AP, and the output-orientated measure of TE is the ratio, CP/CD
• They are only equal when there are constant returns to scale, in the second part of the diagram
Output-orientated measures
Econ 377/477 Topic 2 26
Output-orientated measures and returns to scale
q2
C
AB
P
f(x)
0
A
C
P
D
B
0
f(x)
D
(a) DRS (b) CRS
Econ 377/477 Topic 2 27
• An output-orientated measure of TE with two outputs, q1 and q2, is shown on the next slide
• Assuming CRS, we can represent the technology by a unit PPF, ZZ’, in two dimensions
• A corresponds to an inefficient firm• Output-orientated technical efficiency is the
ratio:TE = 0A/0B, = do(x,q)
where do(x,q) is the output distance function at the observed output vector of the firm associated with point A
Output-orientated measures
Econ 377/477 Topic 2 28
Efficiency measures with an output orientationq2/x1
q1/x1
B’
B
A
D’
0
REo=OA/OC
AEo=OB/OC
TEo=OA/OBC
D
Econ 377/477 Topic 2 29
• Allocative efficiency (AE) is equal to 0B/0C• Revenue efficiency (RE) is equivalent to
economic efficiency• It is defined for an output price vector p
represented by the line, DD’RE = 0A/0C, = p’q/p’q*
where q* is the revenue-efficient vector associated with the point B’
Output-orientated measures
Econ 377/477 Topic 2 30
A few points• TE has been measured along a ray from
the origin to the observed production point
• All points measured along the ray from the origin to the observed production point hold relative proportions of inputs (outputs) constant
• Changing the units of measurement will not change the value of the efficiency measure
Econ 377/477 Topic 2 31
A few points• We have demonstrated measures of
allocative efficiency from cost-minimising and revenue-maximising perspectives
• We can also do the same for the profit-maximising perspective, using DEA and stochastic frontier analysis (SFA)
• For SFA, profit efficiency is decomposed using:
• input-allocative efficiency• output-allocative efficiency• input-orientated technical efficiency
Econ 377/477 Topic 2 32
Measuring productivity and productivity change
• Measuring productivity of a firm and change in productivity is part of performance measurement
• Partial measures were discussed in Topic 1• We represent change (growth and
decrease) of productivity by a total factor productivity (TFP) index, also known as a multifactor productivity index (MFP)
Econ 377/477 Topic 2 33
Measuring productivity and productivity change
• Consider the problem of measuring productivity change for a firm from period s to period t
• The typical period used is a year• Assume that the firm makes use of the
state of knowledge, as represented by production technologies Ss and St in periods s and t
Econ 377/477 Topic 2 34
Measuring productivity and productivity change
• Suppose the firm produces outputs qs and qt using inputs xs and xt, respectively
• In some cases, we may have information on output and input prices, represented by output price vectors ps and pt, and input vectors, ws and wt, in periods s and t, respectively
Econ 377/477 Topic 2 35
Measuring productivity and productivity change
• Given these data on this firm, how do we measure productivity change?
• There are several simple and intuitive approaches we can use to derive meaningful measures of productivity change
• We consider four possible alternatives, outlined in the following slides
Econ 377/477 Topic 2 36
Approaches to measure productivity change
1. Hicks-Moorsteen approach• This approach simply uses a measure of
productivity as output growth net of growth in inputs
• If output has doubled from period s to period t, and if this output growth was achieved using only a 60 per cent growth in input use, we conclude that the firm has achieved productivity growth
Econ 377/477 Topic 2 37
Approaches to measure productivity change
1. Hicks-Moorsteen approach (continued)
• It is easy to measure and interpret, but quite difficult to identify the main sources of productivity growth
• Suppose productivity has grown by 10 per cent; do we attribute this to technical change or to improvements in efficiency?
indexquantityInput
indexquantityOutput
inputinGrowth
outputinGrowthIndexTFPHM
Econ 377/477 Topic 2 38
Approaches to measure productivity change
2. Profitability approach • This approach measures productivity
change using growth in profitability after making appropriate adjustments for movements in input prices and output prices from period s to period t
indexpriceinput//
indexpriceoutput//
/
/indexTFP
**
**
st
st
st
st
CC
RR
CC
RR
Econ 377/477 Topic 2 39
Approaches to measure productivity change
2. Profitability approach (continued) • Since the TFP measure in equation (3.21)
in the text book does not contain any price effects, the main sources of TFP change over periods s and t can be attributed to technical change and (technical, allocative and scale) efficiency changes over this period
Econ 377/477 Topic 2 40
Approaches to measure productivity change
3. CCD Approach• This approach measures productivity by
comparing the observed outputs in period s and period t with the maximum level of outputs (keeping the output mix constant)
• A commonly used index that follows this approach is the Malmquist index
Econ 377/477 Topic 2 41
Malmquist TFP index: issues
• Refer to CROB, pages 69-74, for discussion of the following issues:o Malmquist TFP and orientationo Malmquist and HM TFP indiceso Malmquist TFP and technical inefficiencyo Malmquist TFP and returns to scaleo Malmquist TFP and transitivity
Econ 377/477 Topic 2 42
Malmquist output-orientated TFP index
• Output-orientated measures of productivity focus on the maximum level of outputs that could be produced using a given input vector and a given production technology relative to the observed level of outputs:
• Assuming technical efficiency in both periods:
ssso
ssso
tstsso
d
dm
xq
xqxxqq
,
,,,,
ttsotsts
so dm xqxxqq ,,,,
Econ 377/477 Topic 2 43
Malmquist input-orientated TFP Index
• The input-orientated productivity focuses on the level of inputs necessary to produce observed output vectors qs and qt under a reference technology:
• Assuming technical efficiency in both periods:
sssi
ttsi
tstssi
d
dm
xq
xqxxqq t
,
,,,,
ttsitsts
si dm xqxxqq ,,,,
Econ 377/477 Topic 2 44
Approaches to measure productivity change
4. Component-based approach• Following this approach, various sources
of productivity growth are identified: technical change; efficiency change; change in the scale of operations; and output mix effects
• If we can measure these effects separately, then productivity change can then be measured as the product (or sum total) of all these individual effects
Econ 377/477 Topic 2 45
TFP Index: measurement by sources of productivity change
• Technical change• Technical efficiency change• Scale efficiency change• Optimal mix effect• TFP change = technical change TE
change scale efficiency change output mix effect