Topic 2

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


Terminology

• Productivity• Technical efficiency• Allocative efficiency• Cost efficiency• Technical change• Scale efficiency• Constant returns to scale ...


Terminology

• Point of (technically) optimal scale• Total factor productivity (TFP)• Production frontier• Feasible production set• Output-mix allocative efficiency• Revenue efficiency


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


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


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


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


• 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


• 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


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


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


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’


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


• 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


• 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


• 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


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


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


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


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


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

Topic 2

Technology

Transcript of Topic 2