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**Efficiency and Productivity Measurement: Basic Concepts**

D.S. Prasada Rao School of Economics The University of Queensland, Australia

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**Objectives for the Workshop**

Examine the conceptual framework that underpins productivity measurement Introduce three principal methods Index Numbers Data Envelopment Analysis Stochastic Frontiers Examine these techniques, relative merits, necessary assumptions and guidelines for their applications

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**Objectives for the Workshop**

Work with computer programs (we use these in the afternoon sessions) TFPIP; EXCEL DEAP FRONTIER Briefly review some case studies and real life applications Briefly review some advanced topics on Thursday and Friday morning

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Main Reference An Introduction to Efficiency and Productivity Analysis (2nd Ed.) Coelli, Rao, O’Donnell and Battese Springer, Supplemented with material from other published papers

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**Outline for today Introduction Concepts**

Production Technology Distance Functions Output and Input Oriented distance functions Techniques for Efficiency and Productivity Measurement: Index Number methods

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**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

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**Efficiency: Productivity:**

How much more can we produce with a given level of inputs? How much input reduction is possible to produce a given level of observed output? 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

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**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?

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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:

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**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 Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**Output and Input sets P(x) = {q: x can produce q} = {q : (x,q) S}**

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 Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**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. Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**Output Distance Function**

y1A y2A B C A y1 y2 P(x) PPC-P(x) The value of the distance function for the firm using input level x to produce the outputs, defined by the point A, is equal to the ratio =0A/0B. The value of the distance function for the point, A, which defines the production point where firm A uses x1A of input 1 and x2A of input 2, to produce the output vector q, is equal to the ratio =0A/0B. Output oriented TE = Do Input oriented TE = 1/Di 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)

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**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. Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**Input Distance Function**

The value of the distance function for the firm using input level x to produce the outputs, defined by the point A, is equal to the ratio =0A/0B. The value of the distance function for the point, A, which defines the production point where firm A uses x1A of input 1 and x2A of input 2, to produce the output vector q, is equal to the ratio =0A/0B. Output oriented TE = Do Input oriented TE = 1/Di 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

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**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 Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**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 ….

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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

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**Profit maximisation Iso-profit line: q q = π/p + (w/p)x frontier**

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**Cost minimisation The firm must produce output, q0**

Minimum cost is defined as: x1 Cost min Iso-cost line: x1 = c/w1 – (w2/w1)x2 Isoquant (q=q0) x2

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**Revenue maximisation The firm has input allocation, x0**

Maximum revenue is defined as: y1 Revenue max Iso-revenue line: y1 = r/p1 – (p2/p1)y2 PPC (x=x0) y2

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**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.

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Production function Marginal product Production elasticity Scale elasticity

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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).

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Returns to scale q DRS CRS IRS x

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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

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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. Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**Productivity and Efficiency Concepts**

technical efficiency scale efficiency allocative efficiency cost efficiency revenue efficiency total factor productivity (TFP) Brief overview of empirical methods

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**Technical Efficiency q Frontier B Output orientation: TEO=DA/DB**

Input orientation: TEI=EC/EA C E Firms are on that frontier, if they are technically efficient, or beneath the frontier if they are not technically efficient. In this figure, firm A is technically inefficient, while firms B and C are technically efficient. One can define input-orientated or output-orientated technical efficiency (TE) scores. The output orientated TE score of firm A is TEO=DA/DB. This is approximately 0.80, indicating that the firm is producing 20% below its potential output, given the inputs it has available. The input orientated score is TEI=EC/EA. It is also approximately 0.80, suggesting it can produce the same output with 20% less inputs. These scores take a value between zero and one – a value of one indicates full efficiency. TEO and TEI are usually similar, and will be identical if the technology exhibits constant returns to scale (CRS). A CRS technology is (roughly) one in which an X% increase in all inputs causes output to increase by X%. These TE measures can be generalised to multi-input (and multi-outputs) by looking at proportional changes – which we describe later. The feasible production set is the set of all input-output combinations that are feasible. This set consists of all (non-negative) points on or below the production frontier. A D x

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**Scale Efficiency CRS Frontier q VRS Frontier TEVRS=DB/DA TECRS = DC/DA**

SE=DC/DB = TECRS/TEVRS C D Firms are on that frontier, if they are technically efficient, or beneath the frontier if they are not technically efficient. In this figure, firm A is technically inefficient, while firms B and C are technically efficient. One can define input-orientated or output-orientated technical efficiency (TE) scores. The output orientated TE score of firm A is TEO=DA/DB. This is approximately 0.80, indicating that the firm is producing 20% below its potential output, given the inputs it has available. The input orientated score is TEI=EC/EA. It is also approximately 0.80, suggesting it can produce the same output with 20% less inputs. These scores take a value between zero and one – a value of one indicates full efficiency. TEO and TEI are usually similar, and will be identical if the technology exhibits constant returns to scale (CRS). A CRS technology is (roughly) one in which an X% increase in all inputs causes output to increase by X%. These TE measures can be generalised to multi-input (and multi-outputs) by looking at proportional changes – which we describe later. The feasible production set is the set of all input-output combinations that are feasible. This set consists of all (non-negative) points on or below the production frontier. A B x

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**Allocative Efficiency**

We now return to our shirt factory example (see data on slide 5) to discuss allocative efficiency. In this figure we have plotted the isoquant corresponding to 200 shirts. The isoquant is a horizontal slice through the 3D production frontier – showing the various (minimum) input bundles which can produce 200 shirts. All points on or to the north-west of this isoquant are technically feasible We have plotted the example data so both firms happen to operate on the isoquant – hence we assume no technical inefficiency in these 2 firms. The isocost lines link points of equal cost, given the assumed prices. Firm A is (approx.) allocatively efficient – because it cannot change its input mix to attain a lower isocost line. Firm B has allocative inefficiency – it could change the input mix to that of point A and save $60. One possible allocative efficiency score for Firm B is 360/420=0.86, which indicates that it can reduce costs by 14%. AE=360/420=0.86

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**Allocative Efficiency (2)**

Consider now a new firm, firm C, which uses four machines and two workers to produce 200 shirts per day. This firm is both technically inefficient (it is inside the isoquant) and allocatively inefficient (it is not using the minimum cost input mix). Its costs are $560, which are $200 higher than firm A. We could say it has a cost efficiency (CE) score of 560/360= It could reduce costs by 36%. How can we divide this score into technical and allocative components? Farrell (1957) suggested measuring technical efficiency by holding the input mix fixed, and proportionally shrinking all inputs until the frontier was reached. If we do this to firm C we obtain the point D. This point uses (approximately) 2.8 machines and 1.5 workers, implying a cost of $400. Hence, technical efficiency (TE) = 400/560 = 0.71, and allocative efficiency (AE) = 360/400 = 0.9. TE=400/560=0.71 AE=360/400=0.9 CE=360/560=0.64

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**Output orientated efficiency**

shirts C B D iso-revenue line A PPC TEO=0A/0B AEO=0B/0C RE=0A/0C =TEO×AEO Consider the case where our clothing factories now produce two outputs: shirts (y1) and trousers (y2), using capital and labour. For any particular input vector (eg. two machines and three workers) we can draw a production possibility curve like that depicted in the figure. All points on or below the curve are technically feasible. The isorevenue line links points of equal revenue – its slope reflects the relative output prices. Firm A is technically inefficient and (output-mix) allocatively inefficient. Firm B is technically efficient and (output-mix) allocatively inefficient. Firm D is technically efficient and (output-mix) allocatively efficient. Hence, it defines the point of maximum revenue (for the given input quantities) - it is revenue efficient (RE). Our efficiency measures are very similar to the input-orientated case. Consider firm A in Figure 6. A measure of output-orientated technical efficiency is the ratio TEO = 0A/0B, which is the amount by which it can radially expand both outputs, given the input vector it possesses. The (output-mix) allocative efficiency is AEO = 0B/0C, which reflects the revenue change when we move from point B to point C. The revenue efficiency (RE) is the product of these two measures RE = (0A/0C) = (0A/0B)(0B/0C) = TEOAEO, which reflects the revenue change from point A to point C. Again, we note that all of these three measures are bounded by zero and one. Note that our radial technical efficiency measures are units invariant. That is, changing the units of measurement (e.g., measuring quantity of labour in person hours instead of person years) does not change the value of the efficiency measure. trousers

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**Productivity? productivity = output/input**

What to do if we have more than one input and/or output? partial productivity measures aggregation Partial productivity examples: electricity: customers/employees and kwh/employees agriculture: yield/hectare Aggregation is by index numbers - using either prices (PIN) or shadow prices (SFA and DEA) - more on this later

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Example Two firms producing t-shirts using labour and capital (machines). The partial productivity ratios conflict. Each firm produces 200 t-shirts per day We ignore other possible inputs to keep the example simple

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**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. Here we suggest a linear aggregation function - there are many other options. Note that this TFP measure is equal to the inverse of the unit cost measures, which are $1.80 and $2.10, respectively. What do we do if the firms face different prices? – we could select one set of prices (toss a coin?) or we could use both sets of prices - eg. use a Tornqvist or Fisher index – see session 5. What do we do if we have no price information? – we could use the shadow prices implicit in an estimated production technology - eg. SFA or DEA – more on this in later sessions. Having no price information is a common issue in trying to construct aggregate output indices in public service sectors, such as police, education and health.

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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 To be able to define these three types of efficiency - we must first define a production frontier We need good data and advanced methods (eg. SFA or DEA) to estimate the technology Environmental differences (eg. population density in electricity distribution) may also contribute.

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**TFP growth components technical change (TC)**

technical efficiency change (TEC) scale efficiency change (SEC) allocative efficiency change (AEC) The decomposition of TFP growth is discussed in sessions 6-8. Again, environmental changes may also play a role.

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**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. Fare et al 1994, OECD and MPI (regional concept) Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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**Overview of Methods index numbers (IN) data envelopment analysis (DEA)**

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 The discussion to date has assumed that we know the true production technology – in reality we must estimate it using sample data. There are essentially three main methods: The PIN methods, such as Tornqvist and Fisher indices, are used to measure TFP differences between firms and between time periods. If we wish to decompose these differences into technical efficiency, allocative efficiency, scale efficiency and (if applicable) technical change effects, then we need to use DEA or SFA. DEA is a non-parametric method, which uses linear programming methods to construct a piecewise-linear frontier over the sample data. If one has panel data, one can construct a frontier in each year and then calculate and decompose TFP change using Malmquist index formulae. SFA methods are parametric methods, which use econometric methods to estimate the frontier. These may be production, cost or distance functions. If one has panel data, one can estimate a panel data model and then calculate and decompose TFP change using Malmquist index formulae.

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**Relative merits of Index Numbers**

Advantages: only need 2 observations transparent and reproducible easy to calculate Disadvantages: need price information cannot decompose SFA and DEA need a large sample of firms to get decent results.

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**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 This is an incomplete list. DEA and SFA are discussed in the next two sessions. Look at the list of remaining sessions again

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