Efficiency and Productivity Measurement Basic Concepts D S

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 2

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 3

Main Reference An Introduction to Efficiency and Productivity Analysis (2 nd 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 6

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? (i) Productivity: • We wish to measure the level of output per unit of input and compare it with other firms • Partial productivity measures – output 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 7

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: 9

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 10

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 11

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. 12

Output Distance Function y 2 A A B C PPC-P(x) 0 y 1 A y 1 Do(x, y) The value of the distance function is equal to the ratio =0 A/0 B. Output-oriented Technical Efficiency Measure: TE = 0 A/0 B = do(x, q) 13

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. 14

Input Distance Function Di(x, y The value of the distance function is equal to the ratio =0 A/0 B. Technical Efficiency = TE = 1/di(x, q) = OB/OA 15

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 16

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 …. 17

Profit maximisation • Firms produce a vector of M outputs (q) using a vector of K inputs (x) • The production technology (set) is: • Maximum profit is defined as: where p is a vector of M output prices and w is a vector of K input prices 18

Profit maximisation q Iso-profit line: q = π/p + (w/p)x frontie r Profit max x 19

Cost minimisation • The firm must produce output, q 0 • Minimum cost is defined as: x 1 Cost min Iso-cost line: x 1 = c/w 1 – (w 2/w 1)x 2 Isoquant (q=q 0) x 2 20

Revenue maximisation • The firm has input allocation, x 0 • Maximum revenue is defined as: y 1 Revenue max Iso-revenue line: y 1 = r/p 1 – (p 2/p 1)y 2 PPC (x=x 0) y 2 21

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. 22

Production function Marginal product Production elasticity Scale elasticity 23

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). 24

Returns to scale q DRS CRS IRS x 25

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: • 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 26

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. 27

Productivity and Efficiency Concepts • Concepts – technical efficiency – scale efficiency – allocative efficiency – cost efficiency – revenue efficiency – total factor productivity (TFP) • Brief overview of empirical methods 28

Technical Efficiency q Frontier B E C A D Output orientation: TEO=DA/DB Input orientation: TEI=EC/EA x 29

Scale Efficiency CRS Frontier q VRS Frontier TEVRS=DB/DA D C A B TECRS = DC/DA SE=DC/DB = TECRS/TEVRS x 30

Allocative Efficiency AE=360/420=0. 86 31

Allocative Efficiency (2) TE=400/560=0. 71 AE=360/400=0. 9 CE=360/560=0. 64 32

Output orientated efficiency shirts C B A D iso-revenue line PPC TEO=0 A/0 B AEO=0 B/0 C RE=0 A/0 C 0 trousers =TEO×AEO 33

Productivity? • productivity = output/input • What to do if we have more than one input and/or output? – partial productivity measures – aggregation 34

Example • Two firms producing t-shirts using labour and capital (machines). • The partial productivity ratios conflict. 35

Total factor productivity (TFP) • Use an aggregate measure of input: TFP = y/(a 1 x 1+a 2 x 2) • 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. 36

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 37

TFP growth components • • technical change (TC) technical efficiency change (TEC) scale efficiency change (SEC) allocative efficiency change (AEC) 38

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. 39

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 40

Relative merits of Index Numbers • Advantages: – only need 2 observations – transparent and reproducible – easy to calculate • Disadvantages: – need price information – cannot decompose 41

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 42