Efficiency and Productivity

1
Efficiency and Productivity

• Ming-Miin YU
• OCT.6,2002

2
Outline

• Introduction
• Economics
• Basic Concept
• Methodologies 1) Index Number2) Least Square3)
  DEA4) SFA
• Conclusion

3
Introduction

• What is Productivity?
• What is Efficiency?
• Productivity
• Technical Efficiency
• Production Frontier
• Feasible Production Set
• Scale Economies
• Technical Change

4
Productivity Growth

• Factors of Productivity Growth From One Year to
  Next Year
• Efficiency Improvement
• Technical Change
• Scale Economies
• Combination of Three Factors

5
Economics

• Microeconomics Vs. Macroeconomics
• Production Economics (PE) Vs. Consumption
  Economics(CE)
• PE)Allocate resources such that Max. Profit
  or Min.costCE)Allocate income such that Max.
  utility or Min.expenditureHere we deal only
  with PE

6
Basic Concept

• Production Technology
  gods
  input
  bads
• Productivity output / input
  

7
Production Frontier
8
Feasible Production Set

• represent a firms production technology by f(x).
• All set on or below f(x) are feasible,while set
  above f(x) are infeasible inpresent technology.
• f(x) is called production frontier.

9
Allocative Efficiency
10
Overall Efficiency

• one output y was produced by a firm,which
  utilized two inputs,x1 and x2.
• AA is frontier,for firm D, technical
  efficiencyOC/OD
• If prices of inputs are known,we candefine
  allocate efficiency by OB/OC(PP is isocost)
• Overall eff.TEAE(OC/OD)(OB/OC)OB/OD

11
Scale Effect

• Distinction between techeff.productivity
• Both B and C areon frontier,henceboth are
  efficient.
• But productivity ofC is greater than B (due to
  scale effect).

12
Methodologies

• 1.Index number methods
• 2.Least-square methods
• 3.Data Envelopment Analysis (DEA)
• 4.Stochastic Frontier Approach (SFP)

13
Frontier and Non-FrontierParametric and
Non-Parametric
14
Productivity Change

• a ratio type Productivity Index
• Py/x
• Partial factor productivity vs.Total factor
  productivity
• Paggregrate Y/aggregrate X
  Paggregrate Y/x1
• productivity change PcYt/Ys/Yt/Xs

15
Decomposition of a Simple TFP Index

• TFP Growth
• Real Product in Period t
• Homogeneous of degree

16
Decomposition of a Simple TFP Index

• Change in Technical Efficiency
• Technical Change
• Effect of a Change in the Scale of Operations
• Scale of Operations
• Returns to Scale Parameter

17
Output and Input Quantity Index

• utilizing only one input and produce one output?
• multiple inputs and multiple outputs
• Aggregrate Output output quantity index
• Aggregrate Input input quantity index

18
Quantity Index
19
Properties of Index Numbers

• Test Approach1).Positivity.2).Continuity3).Prop
  ortionality4).Commensurability 5).Time-reversal
  test.6).Mean-value test7).Factor-reversal
  test.8).Circularity test(transitivity).

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Tornqvist and CCD Index

• Tornqvist index satisfies all properties list
  above except (7)(8)
• Index should satisfies transitivity, that is
  IstIsr Irt(otherwise, cant be compared
• Caves,Christensen and Diewert(CCD,1982) convert
  Tornqvist indices into CCD indices as below,which
  satisfies transitivity

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CCD Input Quantity Index
22
CCD Output Quantity Index
23
CCD Productivity Index
24
Example Tornqvist Index

• Data for an artificial freight company
• Input(quantity)year labor capital other1996
  145 67 391997 166 75 391998
  162 78 431999 178 89
  422000 177 93 51

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Input Price

• Input(price)year labor capital other1996
  39 100 1001997 41 110
  971998 42 114 1031999 46 121
  1192000 46 142 122

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Output Quantity and Price

• output(quantity) Price year O1
  O2 O1 O2 1996 471
  293 27 181997 472 290
  28 17 1998 477 278 34
  171999 533 277 32 202000 567
  289 34 23

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Tornqvist index number

• Tornqvist index numberobsn output input
  TFP1 1.0000 1.0000 1.00002
  0.9986 1.1007 0.90733 0.9960
  1.1333 0.87884 1.0833 1.2340
  0.87735 1.1468 1.3122 0.8740
  TFP down 13

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The Total Productivity of Canadian Railways

• Tretheway,et.al (1997) The Total Productivity
  of Canadian Railways
• The paper measured TFP of CanadianNational(CN)Rai
  lway Canadian Pacific Rail(CP) by
  usingTornqvist TFP Index Number Method.
• Input labor,fuelenergy,waystructure,
  equipment,land,materialother.
• Outputton-kmpass-km.

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The Result

• The Result showed
• for CN,TFP growth rate ()
  1956-81 1981-91 11956-91 3.4
  3.0 3.3
• for CP,TFP growth rate ()
  1956-81 1981-91 11956-91 3.4
  2.7 3.2

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2.Least-square methods

• a firms production technology by yf(x)
• In practice, production function is never known.
• One can estimate production functionby using
  Econometrics methods.

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Specify a Functional Form

• Before estimate,one should specify a functional
  form.
• The most famous functional form probably is
  Cobb-Douglas Productionfunction.
• The C-D function proposed in 1928 wasyAx1ax21-a
  x1labor,x2capital

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Production Stages

• Neoclassical Production Function
• 3 stagesstage 10(MPPAPP)stage
  2(MPPAPP)(MPP0)stage 3(MPPlt0)

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Restristions of CD Function

• Because of its simplicity,C-D fn. has beenwidely
  used.
• However,it can not represent 3-stage production
  function.
• Besides,it imposed some assumption(such
  aselasticity of substitution1) to function.
• Many economists devoted to find a flexible
  functional form both from Primal and
  Dual.Noteflexible--no priori assumption.

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Production and Cost Functions

• Primal---estimate production fns.Dual-----
  estimate cost fns.
• Specify function form (such as Translog,
  Quadratic,Leontif,etc.)and estimate it to get
  parameters.
• For example,we specify Translog cost fn.as
  follows.

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Translog Cost Function

• lnCa0aLlnPLaElnPEaFlnPFßYlnYßQlnQ?TTdKlnK
  (1/2)aLL(lnPL)2aLElnPLlnPEaLFlnPLlnPF(1/2)aE
  E(lnPE)2aEFlnPElnPF(1/2)aFF(lnPF)2?LYlnPLlnY?
  EYlnPElnY?FYlnPFlnY?LQlnPLlnQ?EQlnPElnQ?FQlnP
  FlnQ?LTlnPLT?ETlnPET?FTlnPFT?LKlnPLlnPk?EKl
  nPElnPk?FKlnPFlnPk (1/2) ßYY(lnY)2 (1/2)
  ßQQ(lnQ)2 (1/2) ?TT(T)2 (1/2) d KK(lnK)2
  tYTTlnYµYKlnYlnK?YQlnYlnQ?QTTlnQpQKlnQlnK?T
  KTlnK

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Properities

• Where,Ccost Y,QoutputPL,PE,PFinputTtime
  trendKfixed factor.
• A well-behavior cost fn.satisfy
  someproperties,such as homogenous in degree
  1,i.e. ?C(y,p)C(y, ?p)therefore, one get
  aLaEaF1, aLLaLEaLF0aEE aEL
  aEF0, aFFaFL aFE0 ?LY?EY?FY0,
  ?LQ?EQ?FQ0 ?LT?ET?FT0, ?LK?EK?FK0

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Shephards lemma

• By using Shephards lemma,one getcost share
  eqs

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Parameter Estimation

• Estimating the cost share eqs.and translog cost
  fn.silmutaneously,we getparameters.(using
  computer program)
• Then,we can compute1)Partial elasticities of
  substitution.2)Prices elasticities of factor
  demand.3)Economies of density.4)Productivity
  growth.

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3.DEA

• The two methods introduced above
  arenon-frontier.
• Now,we turn the topic to Frontier.
• The Frontier methods include1)Data Envelopment
  Analysis(DEA)2)Stochastic Frontier Approach(SFA)
• DEA involve mathematical programmingwhile SFA
  using econometric methods.

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Distance Function

• Farrell (1957)proposedEconomic
  eff.(EE)Technical eff.(TE)Allocative eff.(AE)
• TEgiven input,max.output,or given
  output,min.input.
• AEgiven input prices,optimal cost,or given
  output prices,optimal revenue.

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Input Oriented DEA

• Concept(input orientated)
  TEiOQ/OP
  AEiOR/OQ
  EEiTEiAEi
  OR/OP

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Output Orienrted DEA

• Concept(output orientated)
  TEoOA/OB
  AEoOB/OC
  EEoTEoAEo
  OA/OC

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Constant Returns to Scale

• For DMU Passume CRS,TEiAB/APTEoCP/CDTEiTEo

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Variable Returns to Scale

• While in case of VRS,TEi is not equal toTEo
• D
• A B P
• D

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DEA Ratio Form

• Primal Form
• 
  
• 
  for j 1,2,,n
• 
  
• ur?0 for r1,2,,s
• vi?0 for i1,2,,m
• Revised Form

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Envelopment Form

• Dual form and Technical Efficiency

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Pure Technical Efficiency

• SE TE / PTE
• OE TE AE PTE SE AE

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Overall Efficiency
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Decomposition of Technical Efficiency

• ????( )MN/MA
• ??????( )MB/MA
• ????( )MN/MB

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Implement

• Where thita is eff.score,Y,X are output and
  input of all DMU,yi,xi are output and input of
  i-th DMU,

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Example Data for CRS DEA

• An exampleAssume CRS,5 firms,1 output,2
  inputsFirm y x1 x2 x1/y x2/y

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For Firm 3

• DEA frontier is the result of running 5LP
  problems--one for each firm.

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CRS Input-Orientated DEA Results

• Result

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CRS Input-Orientated DEA Example

• Result

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Discussion

• Discussion1).TE of firm 3 is 0.833,i.e.it could
  reduce all inputs(x1,x2) by 16.7.2).Reduction
  of inputs, firm 3 project to 3 ,on a line
  joining 2 and 5,3). The line joining 2 and 5 is
  called Frontier ,firm 2 and 5 are referred
  to as peers or targets

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Malmquist Productivity

• Yc
  Efficiency Change
• yt E
• Yb
• Ya Technical
  Change
• ys D

Frontier in period t
Frontier in period s
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Malmquist Productivity

• set of production possibilities
• Shephard output distance function

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Decomposition of Malmquist Productivity

• The productivity change taking the technology of
  the period t as reference
• The productivity change taking the technology of
  the period t1 as reference

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Malmquist productivity change index

• Malmquist productivity change index
• efficiency change
• technical change

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4.Stochastic Frontier

• CD Frontieryexp(xß)
• yi exp(xßvi)
• 1)if vigt0,above frontier
• 2)if vilt0,underfrontier

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4.Stochastic Frontier

• Deterministic frontierAll observations must lie
  on or belowthe frontier.
• Stochastic frontierIf observations can be above
  the frontier due to random events.

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4.Stochastic Frontier

• Basic Stochastic Frontier modelor
  ln(y)xßv-u
• Where youtputs xinputs f(x, ß) is
  the deterministic parts b are parameters to be
  estimated v are random error u are
  non-negative random variables which
  represents inefficiencies.

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4.Stochastic Frontier
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4.Stochastic Frontier

• How to estimate exp(ui)?
• Specify some probability distribution form,such
  as truncated normal.
• Let p.d.f. of ui be g(x) and p.d.f. of vi
  be h(x),thenjoint p.d.f. of ui and vi will be
  g(x)h(x)
• estimate by using maximum likelihood,one can
  find tech. ineff.

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Conclusion Classification of four methods
Parametric Non-parametric
Frontier S.F.A. DEA
Non- frontier L.S. INDEX
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Methods and Its Properties
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Conclusion(Index)

• Tornqvist indicesused to compare TFP across
  time.
• CCD indices-used to compare TFP across
  DMUs.-or to compare TFP across DMUs and time.

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Conclusion(Index)

• Advantages over econometrics-only need two
  observations(minimum).-easy to calculate.-does
  not assume smooth tech.progress
• Disadvantages-requires both price and quantity
  information-cannot investigate biases in tech.
  change and scale effects.

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Conclusion(L.S.)

• Using econometric methods
• Primal methods(estimate production fn.Dual
  methods(estimate cost fn.)
• Can measure technological change.

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Conclusion(DEA)

• Math-programming methods.
• Estimation of non-parametric frontiers.
• Used to compare eff. Across DMUs.
• Do not need price data.
• If price data are available,then one
  cancalculate allocative eff.

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Conclusion(SFA)

• Econometric estimation of parametricfrontiers.
• Advantages over DEA-accounts for noise-can
  conduct hypothesis test.
• Disadvantages-need to specify a functional
  form.-more difficult to accommodate multiple
  outputs.