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 (2nd 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? Productivity: • We wish to measure the level of output per unit of input and compare it with other firms • Partial productivity measures – output per 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., qP(x)), then do(x,q) 1 and the distance is equal to 1 only if q is on the frontier. 12 Output Distance Function y2 y2A A B C PPC-P(x) P(x) 0 y1A y1 Do(x,y) The value of the distance function is equal to the ratio =0A/0B. Output-oriented Technical Efficiency Measure: TE = 0A/0B = 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., xL(q)), then di(x,q) 1 and the distance is equal to 1 only if x is on the frontier. 14 Input Distance Function x2 A x2A L(y) B Isoq-L(y) C 0 x1A x1 Di(x,y The value of the distance function is equal to the ratio =0A/0B. 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: S = {(x,q) : x can produce q} • Maximum profit is defined as: (p, w) max {(pq wx) : (x, q) S} q ,x 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, q0 • Minimum cost is defined as: c(q, w) min {wx : (x, q) S} x1 x Cost min Iso-cost line: x1 = c/w1 – (w2/w1)x2 Isoquant (q=q0) x2 20 Revenue maximisation • The firm has input allocation, x0 • Maximum revenue is defined as: r(p, x) max {pq : (x, q) S} q y1 Revenue max Iso-revenue line: y1 = r/p1 – (p2/p1)y2 PPC (x=x0) y2 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 q f (x) MPn f (x) xn f (x) xn En xn q N En n 1 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: M S c(w, q m 1 m ) c ( w, q) c( w , q) • 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 labour B isocost ($420) A isoquant (y=200) 0 capital isocost ($360) AE=360/420=0.86 31 Allocative Efficiency (2) isocost ($560) labour isocost ($400) A D E C isoquant (y=200) TE=400/560=0.71 0 isocost ($360) capital AE=360/400=0.9 CE=360/560=0.64 32 Output orientated efficiency shirts C B A D iso-revenue line PPC TEO=0A/0B AEO=0B/0C RE=0A/0C 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. firm labour capital output (x1) (x2) (q) A 2 2 200 B 4 1 200 q/x1 q/x2 100 50 100 200 35 Total factor productivity (TFP) • Use an aggregate measure of input: TFP = y/(a1x1+a2x2) • 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