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Productivity and Efficiency � Analysis

Atakelty Hailu

SARE/UWA

Outline •  Productivity concepts •  Nonparametric approaches and DEA

–  DEA chapter handout from Coelli et al. (2005)

•  Parametric approaches – Stochastic Frontier Analysis (next lecture)

What is Productivity?

•  What is productivity? –  Rate of output per input? –  What output? What input?

•  What is efficiency? •  What is technical change? •  What are the components of

productivity change?

Productivity •  Productivity is about how much we get out of the

resources we utilize in a production activity •  Productivity is commonly defined as the ratio of

output to input –  Example:

•  Production per employee •  Output per hectare (Yield) •  Aggregate output to aggregate input

•  Productivity might be defined in absolute terms or relative to previous productivity level (i.e. benchmarked against base or reference period) or relative to the productivity of another producer (i.e. indexed or benchmarked against another observation)

Single vs Total Factor Productivity •  Productivity measures that relate output to a

single output are single or partial productivity measures (SFP): –  Example:

•  GDP to labour force ratio •  Sales per employee •  Crop yield (output per hectare)

–  These suffer from imputation problems as they do not take into account changes in other inputs

•  Labour productivity might improve (deteriorate) while the productivity of capital deteriorates (improves)

•  Total (multifactor) factor productivity (TFP) is a more appropriate measure of productivity

Total Factor Productivity (TFP)

•  TFP relates aggregate output to all inputs –  TFP = Aggregate output/Aggregate input –  It is easier to measure TFP relative to another

TFP

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛

1986

2005

1986

2005

1986

2005

1986

2005

Inputs Inputs

Outputs Outputs

Inputs Outputs

TFP TFP

•  The TFP growth (TFPG) can be computed as the difference in the growth rates of aggregate outputs and aggregate inputs

TFP … •  The TFP growth (TFPG) can be computed as the

difference in the growth rates of aggregate outputs and aggregate inputs

•  These growth rates can be computed using index number procedures such as Tornqvist or Fisher Ideal Index

•  Tornqvist – growth rate is share weighted sum of individual quantity growth rates, with quantity growth rates calculated as log of ratios

What are the sources of productivity growth?

Understanding its sources •  Productivity change results from sources that have

different managerial and policy implications –  Are we observing the effect of technology?

•  R &D, R &D spillovers •  Where would spending be more effective

–  Are there economies/diseconomies of scale? •  Can we benefit from the re-organization of production, e.g.

small scale versus mechanized agriculture, centralization/ decentralization of services

–  Is there inefficiency? •  Can we benefit from extension and the spread of best practices?

Or changing incentive structures?

•  Understanding what is driving productivity change is at least as important as measuring it

Components of productivity growth

•  TFP growth can result from a combination of at least any of these: –  Technical change (TC) –  Scale efficiency change (SEC) –  Efficiency change (EC) –  Changes in output/input mix (OM/IM)

Orientation of productivity measures

•  A productivity measure might or might not have an orientation –  Example: non-oriented is one defined as a ratio of

output index to input index •  Oriented-productivity measures

–  Output-oriented or output-based (output/ revenue enhancement concept): focus on potential changes in output for a given set of inputs

–  Input-oriented or Input-based (input/cost saving concept): focus on potential changes in output for a given set of inputs

Orientation of productivity measures

•  How do output-oriented measures compare with input-based ones?

Output y F(X)

Output orientation •G

I •

Z• Input orientation

Orientation of productivity measures

•  When does the orientation not matter? That is, when are the results identical?

Output y F(X)

Output orientation •G

I •

Z• Input orientation

Orientation of productivity measures

•  Directional measures – simultaneous output enhancement and input saving.

Output y F(X)

•G

I •

Z•

Technical change •  Technical change as measured in terms of output enhancement

(TCy) and input savings (TCx) –  TC allows the production of more output using the same

inputs (G to I), saving of inputs while producing the same output (G to Z), or a combination of both (landing on the arc from Z to I)

Input x

Output y

Fs(X)

Ft(X)

TCyTCx •G

I •

Z•

Technical change •  A shift in the technology can involve both

technical progress and regress –  Example: an improved crop variety that is less

resistant to drought or water stress

rainfall

Crop yield

Fs(X)

Ft(X)

•G

Local technical regress

Local technical progress

Technical change •  Technical change as measured in terms of output

enhancement (TCy) and input savings (TCx)

Input x1

Input x2

TCx

ISOQs(y)

ISOQt(y)

Characterizing Technical Change •  How does TC appear?

–  Embodied versus disembodied TC •  Embodied – TC coming as a new input leading to a

different structure in the technology (difficult to study) •  TC generally assumed to be disembodied

•  How TC affects relative input use –  Neutral vs non-neutral TC

•  Hicks neutral TC – not affecting the MRTS •  Cost-neutral - not affecting cost shares

•  Input augmentation effects – TC affecting the effectiveness of some inputs

•  See Chambers (1988)

Scale efficiency change (SEC) •  For any given technology, movement along the frontier will

involve scale effects if the technology is not characterized by constant returns to scale (CRS) –  Example: scaling up production from E to G involves scale

effects (diseconomies) of DG/DF ( < 1)

Input x

Output y

F0(x)

•E

F •

•G

A D O

Scale efficiency change (SEC) •  W is point of optimal scale – movements away from W reduce

scale efficiency (SE) General formula: SE = (actual output/input ratio) ÷ (output/input ratio at W) SE at E = (output rate at E) ÷ (output rate at W)

= (AE/OA)/(AJ/OA) = AE/AJ

SE at G = (DG/OD)/(DK/OD) = DG/DK SEC for move from E to G = (DG/DK)/(AE/AJ) = DG/DF < 1

Input x

Output y F (x)

W•

•G•E

A D

J F K

Efficiency •  Three concepts of efficiency:

–  Technical efficiency (TE) – defined relative to the best practice or the best output (least input) that is possible given the technology

•  The radial measure defined by (Farrell 1957) used generally

•  But growing interest in non-radial (directional) measures –  Allocative efficiency (AE) or price efficiency –

compares the mix of inputs and/or outputs to the best possible given prevailing prices

–  Economic efficiency(EE): combines TE and AE •  EE = TE.AE

–  TE is an upper bound for economic efficiency – when the firm is evaluated at the most favourable prices

•  Actual production is at A, isocost line for minimum cost is through D and C

Input-oriented TE, AE and EE

OA OD

OB OD

OA OB

A •

B•

ISOQ(y)

C •

• D

-p1/p2

So far •  Defined productivity and productivity

change •  Identified some of the key components of

productivity change •  But, how easily we can measure these

components depends on our data and the methods we use

Choice of approach to productivity measurement

What you have: Data

•

• •

•

• • •

•

•• •

Price/value data?

+ (Economic) theory

Approaches to productivity measurement

•  Different dimensions along which approaches can be classified –  Assumptions regarding efficiency

•  Allowing for inefficiency or not •  Frontier vs nonfrontier approaches

–  Representation of the technology •  Parametric (functional form) or •  Nonparametric

–  Producer behavioral assumptions •  Primal – focus on technology without making use of the

implications of producer optimizing behaviour •  Dual – incorporate implications of optimizing behaviour

–  Type of frontier •  Deterministic frontiers •  Stochastic frontiers

Frontier vs Nonfrontier approaches •  Traditionally in economics: constant returns to scale

and full efficiency were standard assumptions. –  Simplified analysis

•  Further, usually a scalar output used (aggregate output index) –  Productivity change and technical change were synonymous

•  Nonfrontier approaches have greately expanded since Farrell’s (1957) pioneering work allowing for technical and allocative inefficiency

•  Integrated approaches – now more common –  Pioneering work by Nishimuzi and Page (1982) where they

used a productivity measure that they decomposed into technical change and efficiency change

–  Many more similar studies since then

Parametric vs nonparametric •  The technology is a set:

T = {(x,y) | x can produce y} Equivalently, the technology can be represented using the input

requirement set L(y) = {x | (x,y) is element of T}

Or the output possibility set P(x) = {y | (x,y) is element of T}

•  Parametric forms represent the technology with a parameterized function, e.g. production function y = F(x)

T = {(x,y) | y ≤ F(x)}

•  Nonparametric forms use set representation (e.g. data envelopment analysis/DEA) or no representation of the technology (index number methods)

Primal vs. dual approaches •  Dual – incorporate information contained in

prices •  Parametric

–  Cost functions, profit functions, revenue functions, … •  Nonparametric

–  methods using revealed preference axioms (e.g. WAPM) –  index number procedures (economic approach)

•  Primal: –  Single output

•  Production functions –  Multi-output

•  Parametric –  distance functions (radial and directional)

•  Nonparametric –  data envelopment analysis (DEA)

Deterministic vs. stochastic frontiers •  Deterministic – all deviations from the frontier are

attributed to inefficiency (e.g. DEA, non-stochastic distance functions) –  mathematical programming used to estimate efficiency or to

compute parameters for estimated frontier •  Stochastic (SFA) – deviations attributed to

inefficiency and stochastic variations in the frontier

•E •D

•F •A

•B •C

determintic

stochastic

Nonparametric approaches � to productivity measurement

Atakelty Hailu

SARE/UWA

Set representations •  There have been both primal and dual

nonparametric approaches that depend on a set representation of the underlying technology

•  The primal (data envelopment analysis – DEA) has its roots in economics but was popularized in the management science literature

•  The dual nonparametric approach utilizes revealed preferences to ‘trace out’ the boundary of the technology

Nonparametric bounds to T •  Economics work on nonparametric bounds began

with Afriat (1972) and Hanoch and Rothschild (1972)

•  Varian (1984) builds on that and shows how nonparametric bounds for the underlying technology can be constructed if the observed data are consistent with the Weak Axiom of Profit Maximization (WAPM)

•  Banker and Maindiratta (1988) extend Varian's approach to cases where, because of technical or allocative inefficiency, the data may not be rationalizable in the sense of Varian.

•  They define the tightest inner- and outer-bounds to the underlying technology

Nonparametric bounds to T •  Let x ∈ℜ+N, r ∈ℜ+N, y ∈ℜ+M, p ∈ℜ+M denote

vectors of inputs, input prices, outputs, and output prices, respectively.

•  Start with observed input and output quantity and price data for a set S of observations, with the purpose of constructing nonparametric bounds for the underlying production technology.

•  This requires specifying the minimum requirements that a set T ⊆ ℜ+ N × ℜ+ M must satisfy to qualify as a production possibility set representing the technology underlying the set of observations in S.

Admissible technologies •  A1) T is closed and convex. •  A2) For all s ∈ S, (xs, ys) ∈T. •  A3) Y rationalizes the subset of observations E={t: Δt=0} ⊆ S, where the criterion function Δ is defined by Δt = max {(ptys- wtxs)-(ptyt-wtxt); s, t ∈ S } ≥ 0. Alternatively, we have (ptyt-wtxt) ≥ (ptys-wtxs), for all (ys, xs) ∈ T and for all t ∈ E.

•  A4) If (y, x) ∈Y and y ≥ y', x' ≥ x, then (y', x') ∈Y.

Admissible technologies •  Closure and convexity (A1) are basic regularity

conditions that are customarily imposed on the production possibility set.

•  A2 ensures that the constructed technology includes (or supports as feasible) all the empirically observed input-output combinations in the sample.

•  A3 requires that the production possibility set rationalize the subset E of observations passing the WAPM test.

•  A4 is the monotonicity conditions (or free disposability).

Inner and outer bounds •  All admissible production sets T satisfying

conditions A1 to A4 are bounded by EYI EYO (i.e. EYI ⊆ T ⊆ EYO)

( ) ⎭ ⎬ ⎫

⎩ ⎨ ⎧

∈≥=≥≤= ∑ ∑∑ ∈ ∈ ∈

S S SS

t;,,;1z,z,z,EYI)1( t t

ttt

tt 0zxyxxyyxy

( ){ }0xvxwypxwypxv ≥∈−≤−= ,;t;,EYO)2( tttttt E

•  These bounds are illustrated in the next slide using observed data for points A, B, C, and D with A, C and D being passing the WAMP test

Illustration of Outer and Inner Technology Bounds

Inner and outer bounds •  The outer bound defines the technology set as the

intersection of the half spaces defined by the isoprofit lines (The Supporting Hyperplane Theorem)

•  The inner bound is the variable returns to scale (VRS) version of the data envelopment analysis (DEA) model – the smallest convex hull containing all the data

•  The inner and outer bounds will provide different efficiency scores, with those computed against the outer bound being the lower bound to the true efficiency scores

•  In practice, the dual frontier is rarely used (but see Chavas and Cox, and Hailu (AJAE 2001)).

•  Application of the DEA has expanded enormously

DEA •  Another name for this is activity analysis –

as it constructs a piece-wise linear frontier by combining observed input-output combinations

X 1

W h

T h

X 1

ISOQ(y)

DEA •  DEA scores are computed for each observation by

projecting the observation onto the frontier – weights are adjusted so that the efficiency score is minimized

•  Observed points: R, S, T and W •  R’s technical efficiency is OP/OR

W h

T h

X 1

ISOQ(y)

S•

DEA •  DEA scores are computed for each observation by

projecting the observation onto the frontier – weights are adjusted so that the efficiency score is minimized

•  Observed points: R, S, T and W •  R’s technical efficiency is OP/OR

W h

T h

X 1

ISOQ(y)

S•

DEA •  Linear programming is used to compute efficiency

scores; LP used as many times as there are observations.

•  Input oriented TE:

TEx(x,y) = maxθ θ |y ≤ z tyt

t∈S ∑ ,

x θ ≥ ztxt, t∈S ∑ zt =1,

t∈S ∑ y,x,z ≥0,t ∈ S

⎧ ⎨ ⎩

⎫ ⎬ ⎭

•  Output oriented TE:

⎭ ⎬ ⎫

⎩ ⎨ ⎧

∈≥=≥≤= ∑∑∑ ∈ ∈ ∈

S SSS

t,,,1z,z,z|max),(TE t

tt y 0,zxyxxy

y δ

δδyx

DEA •  The z variables are activity levels assigned to

peers in the construction of the best practice frontier

•  Restrictions on the Z vector impose returns to scale restrictions, –  Z add up to 1 – variable returns to scale (VRS) –  Z are free – constant returns to scale (a cone technology)

(CRS) –  Z sums up to not greater than 1 – non-increasing returns

to scale (NIRS)

• •

•

• •

• CR S

VRS

DEA and RTS versions •  Variable returns to scale (VRS)

–  Observations are compared against observations of similar size (similar peers)

–  This is because the projected point is a convex combination of observed points

–  ‘Conservative’ in its assessment of inefficiency levels •  Constant returns to scale (a cone technology) (CRS)

–  Assumes production can be scaled up or down easily –  Projected point is a linear combination of observed points –  Observations can be compared against any observations of any size –  ‘liberal’ in its assessment of inefficiency (can generate big inefficiency

estimates) –  Can confuse scale inefficiency with technical inefficiency – not good if

observations are not operating at optimal scales •  Non-increasing returns to scale (NIRS)

–  Allows comparison with smaller observations but not with larger ones –  In other words, assumes production can be scaled down but not up –  Similar to CRS on the down side, and to VRS on the up side (from the

optimal scale point)

Exercise •  Draw the NIRS frontier for these data

• • •

• •

•

CRS

VRS

Scale Efficiency (SE) from DEA •  Nature of RTS and relationship to efficiency scores

–  Scores from VRS at least as big as those from NIRS which are in turn at least as big as those from a CRS

–  TECRS ≤ TENIRS ≤ TEVRS •  SE = TECRS ÷ TEVRS •  How do we know if the observation is operating in

the area of increasing or decreasing returns to scale? –  Increasing returns if TENIRS ≠ TEVRS –  Decreasing returns if TENIRS = TEVRS –  See Coelli et al (2005) DEA handout for examples (p.174)

Variations on DEA model •  Discretionary vs. Non-Discretionary Variables

–  If a variable is non-discretionary, then the producer cannot be assumed to be able to change it in the same way as other variables, e.g. fixed quantity of an input

•  Exclude variable from contraction or expansion (in the measurement of efficiency)

•  Environmental variables that affect efficiency but are not under the control of the manger (e.g. location, etc.)

–  Solution depends on type of variable and whether it has natural ordering, sample size, etc.

–  In some cases, the environmental variable can be included as a non- discretionary input, output or neutral variable in the DEA model

Variations on DEA model… •  Input congestion – strong or free disposability of

input not appropriate if there is a ‘backward bending’ isoquant type effect –  Use weak disposability formulations (equality rather than

inequality restrictions in DEA formulation) –  Can have an associated measure of input congestion

efficiency (ICE) –  Used for bad outputs as well –  Approach can be controversial (e.g. Hailu (2003, AJAE))

Bootstrapped DEA •  DEA scores are highly sensitive to sampling and

data errors •  The sampling distribution of efficiency estimates has

been the subject of recent research, resulting –  Analytical asymptotic analysis (only for single input/single

output cases) –  Bootstrapping techniques

•  Resampling is used to provide distributions of DEA efficiency scores

•  Details of bootstrapping technique tailored for the DEA case are discussed in Simar and Wilson (2000) and their earlier papers

•  Bootstrapping only for sampling errors not for noise and misspecification errors

Intertemporal analysis in DEA •  How to handle data from different periods •  Alternatives

–  Intertemporal frontier – treat all observations as a cross- section and then interpret efficiency scores as productivity scores

–  Window analsys – window of years used as a cross-section e.g. a window that is 3 years wide

–  If enough data is available, construct frontiers year by year –  See Cooper et al (2000)

Understanding sensitivity/sensibility of DEA scores

•  Sample size versus inputs and outputs (curse of dimensionality)

•  Choice of orientation might be important, especially if there is no variability along either inputs or outputs

References •  Afriat, S. "Efficiency Estimates of Production

Functions." Int. Econ. Rev. 13(October 1972):67-77. •  Banker, R.D, A. Charnes, and W.W. Cooper. "Some

Models for Estimating Technical and Scale Inefficiency in Data Envelopment Analysis." Manage. Sci. 30(September 1984):1078-92.

•  Banker, R.D. and A. Maindiratta. "Nonparametric Analysis of Technical and Allocative Efficiencies in Production." Econometrica. 56 (November 1988): 1315-32.

•  Chavas, J.P. and T.L. Cox. "Production Analysis: A Nonparametric Time Series Application to U.S. Agriculture." J. Agr. Econ. 48(September 1997): 330-48.

•  Cherchye, L. and T. Post. 2003. “Methodlogical Advances in DEA: A survey and an application for the Dutch electricity sector”, Statistica Neerlandica, 57, 410-238.

References … •  Hailu A and Veeman TS (2001) Nonparametric Productivity

Analysis with Undesirable Outputs: An Application to the Canadian Pulp and Paper Industry. American Journal of Agricultural Economics 83: 605-616.

•  Hanoch, G., and M. Rothschild. "Testing the Assumption of Production Theory: A Nonparametric Approach." J. Polit. Economy. 80(March 1972):256-75.

•  Simar, L. 2003. “Detecting Outliers in Frontier Models: A Simple Approach”, Journal of Productivity Analysis, 20, 391– 424, 2003

•  Simar, L. and P. W. Wilson. 2000. “Statistical Inference in Nonparametric Frontier Models: The State of the Art”, Journal of Productivity Analysis, 13, 49–78.

•  Varian, H.R. "The Nonparametric Approach to Production Analysis." Econometrica. 52(May 1984):579-97.

Software •  Frontier (by Tim Coelli) can estimate stochastic frontiers. It is a

free software downloadable on the web. •  R is a powerful open source statistical/mathematical software. It

is available at http://www.r-project.org. It is available for both Windows and Unix-like platforms. Start by reading the “An Introduction to R”. Packages other than the base can be downloaded; download the lpSolve and micEcon (microeconomic) packages.

–  Frontier R package –  Apear R package (obtain from instructor) –  Benchmarking R package

•  GAMS – you can get a demo version for free. Read the tutorial and try to find out how you would solve several optimization problems in a loop. And how to write data to files.