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Topic 3

  1. 1. ECON 377/477<br />
  2. 2. Econ 377/477-Topic 3<br />2<br />Topic 3<br />Data Envelopment Analysis<br />
  3. 3. Outline<br />Introduction<br />Efficiency measurement concepts<br />The constant returns-to-scale model<br />The variable returns-to-scale model<br />Input and output orientations<br />Modelling allocative efficiencies<br /> …<br />Econ 377/477-Topic 3<br />3<br />
  4. 4. Outline<br />Adjusting for the environment<br />Non-discretionary variables and input congestion<br />Exercises<br />Estimating technical and allocative efficiency using DEAP<br />Econ 377/477-Topic 3<br />4<br />
  5. 5. Motivation<br />Microeconomic theory defines:<br /><ul><li>a production function as the maximum output attainable from given sets of inputs
  6. 6. a cost function as the minimum cost of producing a particular output, given input prices</li></ul>Empirical work in all fields is dominated by OLS and its variants<br /><ul><li>fit a line of best fit through the sample data rather than over the data (production function) or under the data (cost function)</li></ul>Econ 377/477-Topic 3<br />5<br />
  7. 7. Motivation<br />A benefit of estimating frontiers (rather than the average functions) is that an average function shows the shape of the technology of an average firm while the frontier function is heavily influenced by the best performing firms, and hence reflects the technology they are using<br />The frontier represents a best-practice technology against which the efficiency of firms within the industry can be measured – the basis of most empirical work in the use of frontier in recent years<br />Econ 377/477-Topic 3<br />6<br />
  8. 8. Motivation<br />A major advantage of efficiency measures using frontiers is that they can account for more than one input (and more than one output).<br />Frontier analytical methods are:<br /><ul><li>Data envelopment analysis (DEA)
  9. 9. Stochastic frontier analysis (SFA)</li></ul>Econ 377/477-Topic 3<br />7<br />
  10. 10. Econ 377/477-Topic 3<br />8<br />Efficiency measurement concepts<br />Farrell (1957) and Debreu (1951) are two seminal papers on frontier analysis<br />Two efficiency components mentioned in earlier topics are:<br /><ul><li>technical efficiency – the ability of a firm to obtain maximal output from a given set of inputs
  11. 11. allocative efficiency – the ability of a firm to use the inputs in optimal proportions, given their prices</li></li></ul><li>Input-orientated measures<br />Assume a model with two inputs (x1and x2) and one output (q)<br />Constant returns to scale (unit isoquant) are assumed<br />We can extend the model to the use of multiple inputs and multiple outputs, and non-constant returns to scale<br />See the graphical representation on the next slide<br />Econ 377/477-Topic 3<br />9<br />
  12. 12. x<br />/q<br />S<br />2<br />P<br />·<br />A<br />·<br />Q<br />·<br />R<br />¢<br />Q<br />·<br />¢<br />S<br />¢<br />0<br />A<br />x<br />/q<br />1<br />Technical and allocative efficiencies<br />TEi=0Q/0P<br />AEi=0R/0Q<br />EEi=0R/0P<br />Econ 377/477-Topic 3<br />10<br />
  13. 13. Technical and allocative efficiencies<br />If the firm uses quantities of inputs defined by point P to produce a unit of output, then inefficiency can be represented as the distance QP, which isthe amount that inputs could be reduced without a reduction in output<br />The technical efficiency measure (0Q/0P) must lie between zero and one<br />At the value of one, a firm is fully efficient<br />Econ 377/477-Topic 3<br />11<br />
  14. 14. Technical and allocative efficiencies<br />Given an input-price ratio, the allocative efficiency is 0R/0Q<br />Reduction in cost if production is at Q’<br />Point Q is technically efficient but allocatively inefficient<br />Economic efficiency is: EEi= 0R/0P<br />That is, EE = TE * AE<br />The efficient isoquant must be estimated using sample data<br />Econ 377/477-Topic 3<br />12<br />
  15. 15. Non-parametric estimation: the piecewise linear convex isoquant<br />S<br />x2/q<br />: firms<br />No observed point should lie to the left or below line SS<br />Frontier<br />S<br />x1/q<br />0<br />Econ 377/477-Topic 3<br />13<br />
  16. 16. Output-orientated measures<br />Aigner and Chu (1968) is the seminal paper on non-parametric output measures<br />Recall from Topic 2 that to specify a parametric frontier production function in input-output space, a Cobb-Douglas function is:<br /> ln(qi) = f(ln(xi), ) - ui<br />where qi is the output of the i-th firm, xi is an input vector and ui is a non-negative variable representing inefficiency<br />Econ 377/477-Topic 3<br />14<br />
  17. 17. Output-orientated measures<br />Technical efficiency is calculated as:<br /> TEi = qi/f(ln(xi),) = exp (-ui)<br />An output-orientated measure indicates the magnitude of the output of the i-th firm relative to the output that could be produced by the fully efficient firm using the same input vector<br />The non-parametric approach does not account for noise<br />Econ 377/477-Topic 3<br />15<br />
  18. 18. Technical and allocative efficiencies<br />As also discussed in Topic 2, all efficiency measures are along the ray from the origin to the observed production point, holding relative proportions of inputs (outputs) constant<br />Changing the units of measurement will not change the value of the efficiency measure<br />Econ 377/477-Topic 3<br />16<br />
  19. 19. Standard DEA models<br />DEA is a non-parametric mathematical programming approach to frontier estimation<br />Charnes, Cooper and Rhodes (1978) coined the term, data envelopment analysis<br />We begin a description of its use assuming an input orientation and constant returns to scale<br />Banker, Charnes and Cooper (1984) proposed variable returns to scale models that are discussed below<br />Econ 377/477-Topic 3<br />17<br />
  20. 20. Standard DEA models<br />For DEA, we need data on the input and output quantities of each firm<br />Linear programming is used to construct a non-parametric piecewise surface over the data<br />TE is the distance of each firm below this surface<br />Econ 377/477-Topic 3<br />18<br />
  21. 21. Standard DEA models<br />Remember that in the input-orientated model, we look at the amount by which inputs can be proportionally reduced, with outputs fixed<br />And for the output-orientated model, we look at the amount by which outputs can be proportionally increased with inputs fixed<br />Econ 377/477-Topic 3<br />19<br />
  22. 22. CRS models<br />The following notation is used:<br /><ul><li>I decision-making units (DMUs) or firms
  23. 23. N inputs (N×I input matrix, X)
  24. 24. M outputs (M×I output matrix, Q)</li></ul>Note that the purpose of DEA is to construct a non-parametric envelopment frontier over the data points such that all observed points lie on or below the production frontier<br />Econ 377/477-Topic 3<br />20<br />
  25. 25. CRS models<br />We introduce DEA via the ratio form<br />For each firm we would like to obtain a measure of the ratio of all outputs over all inputs, such as:<br /> (uqi/vxi)<br /> where u is an M×1 vector of output weights and v is an N×1 vector of input weights<br />We wish to obtain the optimal weights by solving the linear programming (LP) problem on the next slide<br />The LP problem must be solved I times<br />Econ 377/477-Topic 3<br />21<br />
  26. 26. CRS models<br />Infinite number of solutions<br />maxu,v (uqi/vxi)<br />stuqj/vxj  1j = 1,2,...,I<br />u, v  0<br />Impose the constraint vxi= 1<br />max, (qi)<br />st xi = 1<br />qj - xj  0j = 1,2,...,I<br />,  0<br />Multiplier form of the DEA model<br />= M×1 vector of output weights<br /> = N×1 vector of input weights<br />Identical TE scores: TEi= qi/xi<br />Econ 377/477-Topic 3<br />22<br />
  27. 27. Input-orientated CRS DEA Model<br />Using duality, the input-orientated CRS model is:<br /> min,<br /> st -qi + Q 0<br />xi - X 0<br /> 0<br />xi is a N×1 vector of inputs of the i-th firm<br />qi is M×1 vector of outputs of the i-th firm<br />X is a N×I input matrix<br />Q is M×I output matrix<br />θis a scalar (used to estimate TE)<br /> is a I×1 vector of constants<br />Econ 377/477-Topic 3<br />23<br />
  28. 28. Input-orientated CRS DEA Model<br />The problem for the i-th firm is radially to contract the input vector xi as much as possible<br />The inner boundary is a piecewise linear isoquant determined by the observed data points<br />The radial contraction of the input vector xi produces a projected point (X, Q) on the surface of this technology<br />The diagram on the next slide shows the radial contraction for two inefficient firms, A and B<br />Econ 377/477-Topic 3<br />24<br />
  29. 29. Non-parametric estimation: the piecewise linear convex isoquant<br />S<br />x2/q<br />A<br />A<br />B<br />C<br />B<br />D<br />S<br />x1/q<br />0<br />Econ 377/477-Topic 3<br />25<br />
  30. 30. Input-orientated CRS DEA Model<br />The projected points are a linear combination of the observed data points<br />Firm A can reduce use of inputs x1 and x2, and move to point A without reducing output<br />Firm B can reduce use of inputs x1 and x2, and move to point B without reducing output<br />The constraints ensure that this projected point cannot lie outside the feasible set<br />Econ 377/477-Topic 3<br />26<br />
  31. 31. Slacks<br /><ul><li>The radial contraction from Ato A on the previous slide raises the question:
  32. 32. Is point A an efficient point?
  33. 33. We could reduce the amount of input x2 (by the amount CA) and still produce the same output
  34. 34. Input slacks will be equal to zero if and only if xi -X= 0 (for the given optimal values of  and )
  35. 35. For output-orientated models, output slacks will be equal to zero if and only if Q- qi= 0</li></ul>Econ 377/477-Topic 3<br />27<br />
  36. 36. A simple numerical example<br />Data for a simple numerical example are presented below, and a graph of the input-output ratios is shown on the next slide<br />Econ 377/477-Topic 3<br />28<br />
  37. 37. A simple numerical example<br />5<br />S<br />x2/q<br />: projected points for inefficient firms<br />1<br />4<br />3<br />1<br />4<br />2<br />3<br />2<br />3<br />4<br />1<br />S<br />5<br />3<br />x1/q<br />0<br />1<br />2<br />4<br />5<br />Econ 377/477-Topic 3<br />29<br />
  38. 38. LP for firm number 3<br />The DEA frontier results from running five LP problems – one for each firm<br />We could rewrite the minimisation problem for firm 3, for example, as:<br /> min,<br /> st -q3 + (q11 + q22 + q33 + q44 + q55)  0<br />x13 - (x111 + x122 + x133 + x144 + x155)  0<br />x23 - (x211 + x222 + x233 + x244 + x255)  0<br /> 0<br /> where  = (1, 2, 3, 4, 5)<br />Econ 377/477-Topic 3<br />30<br />
  39. 39. LP for firm number 3<br />The values of  and  that provide a minimum value for  are listed in row 3 of the table on the next slide (in pink)<br />That is, firm 3 could possibly reduce the consumption of all inputs by 16.7 per cent without reducing output<br />This implies production at point 3 in the diagram on the previous slide but one<br />Econ 377/477-Topic 3<br />31<br />
  40. 40. CRS Input-orientated DEA results<br />Econ 377/477-Topic 3<br />32<br />
  41. 41. LP for firm number 1<br />The values of  and  that provide a minimum value for  for firm 1 are listed in row 1 of the table on the previous slide (in light green)<br />Firm 1 has a TE score of 0.5<br />This means it has the potential to reduce input usage by 50 per cent<br />It also has a (non-radial) input slack of 0.5 units of x2 (per unit of q)<br />Econ 377/477-Topic 3<br />33<br />
  42. 42. Peers for firm 3<br />The projected point 3 from radial contraction for firm 3 in the diagram on the next slide lies on the line joining points 2 and 5<br />These firms are termed its peers<br />Peers define the relevant part of the frontier, namely the efficient production for individual firms<br />λ values are used to measure the weights of peers<br />For example, λ2 = 1.0 and λ5 = 0.5 for firm 3 indicate that firm 2 is accorded twice the weight as a peer as firm 2 in that firm 3’s projected efficiency point 3 is much closer to point 2<br />Econ 377/477-Topic 3<br />34<br />
  43. 43. Peers for firm 3<br />x2/q<br />S<br />1<br />4<br />3<br />Peer for firm 3<br />1<br />4<br />2<br />3<br />3<br />2<br />Peer for firm 3<br />4<br />1<br />S<br />5<br />x1/q<br />0<br />1<br />2<br />3<br />4<br />5<br />Econ 377/477-Topic 3<br />35<br />
  44. 44. Peers for firms 1,2 and 5<br />The projected point 1 for firm 1 in the diagram on the next slide lies on the vertical line from point 2<br />Thus, firm 2 is its sole peer<br />Because firms 2 and 5 lie on the frontier, they are technically efficient<br />They are therefore their own peers<br />Hence, λ2 = 1.0 for firm 2 and λ5 = 1.0 for firm 5<br />Econ 377/477-Topic 3<br />36<br />
  45. 45. Peers for firms 1, 2 and 5<br />x2/q<br />S<br />1<br />4<br />3<br />Peer for firm 1 and own peer<br />1<br />4<br />2<br />3<br />3<br />2<br />Own peer<br />4<br />1<br />S<br />5<br />x1/q<br />0<br />1<br />2<br />3<br />4<br />5<br />Econ 377/477-Topic 3<br />37<br />
  46. 46. Targets for firm 3<br />Targets are defined by the coordinates of the efficient points<br />These targets provide the amount of inputs that should be used (or output produced for an output-orientated model)<br />The targets of firm 3 are the coordinates of the efficient projection point 3, equal to 0.833(2,2), = (1.666,1.666)<br />Thus, the diagram on the next slide shows that firm 3 should aim to produce its 3 units of output with 3(1.666,1.666) = (5,5) units of x1 and x2<br />Econ 377/477-Topic 3<br />38<br />
  47. 47. Targets for firm 3<br />x2/q<br />S<br />1<br />4<br />3<br />1<br />4<br />2<br />3<br />3<br />2<br />1.666<br />4<br />1<br />S<br />5<br />x1/q<br />0<br />1<br />2<br />3<br />4<br />5<br />1.666<br />Econ 377/477-Topic 3<br />39<br />
  48. 48. Targets for firm 1<br />The targets for firm 1 would be to reduce the usage of both inputs by 50 per cent and also to reduce the usage of x2 by a further 0.5 units<br />This would result in targets of (x1/q = 1, x2/q = 2)<br />These targets are the coordinates of the efficient firm 2, at point 2, equal to 1.0(1,2), = (1,2)<br />Thus, the diagram on the next slide shows that firm 1 should aim to produce its 1 unit of output with 1(1,2) = (1,2) units of x1 and x2<br />Econ 377/477-Topic 3<br />40<br />
  49. 49. Targets for firm 1<br />x2/q<br />S<br />1<br />4<br />3<br />1<br />4<br />2<br />3<br />3<br />2<br />2.0<br />4<br />1<br />S<br />5<br />x1/q<br />0<br />1<br />2<br />3<br />4<br />5<br />1.0<br />Econ 377/477-Topic 3<br />41<br />
  50. 50. DEA computing<br />Programs to compute DEA include:<br /><ul><li>Excel, QSB+, SHAZAM – user programs that do calculations with an LP option
  51. 51. DEAP – DOS-based DEA program
  52. 52. Onfront, Warwick DEA – Windows-based specialist DEA programs</li></ul>We use DEAP Version 2.1<br />Econ 377/477-Topic 3<br />42<br />
  53. 53. Using DEAP software<br />The four steps involved in using DEAP are:<br /><ul><li>Create the data file
  54. 54. Copy the data into a data file for the DEAP program
  55. 55. Create an instruction file for DEAP program
  56. 56. Execute the DEAP program to produce the output file</li></ul>Read pages 252-253 of CROB<br />Econ 377/477-Topic 3<br />43<br />
  57. 57. The data file<br />This is the data file from an earlier slide<br />1 2 5<br />2 2 4<br />3 6 6<br />1 3 2<br />2 6 2<br /><ul><li>Note the ordering of the variables: q, x1, x2
  58. 58. The file must be saved as .dta
  59. 59. Use either
  60. 60. Excel (save as Tab delimited)
  61. 61. Notepad</li></ul>Econ 377/477-Topic 3<br />44<br />
  62. 62. The instruction file<br />Edit the instruction file (CROB, page 169)<br />Right click and open with notepad<br />eg1.dta DATA FILE NAME<br />eg1.out OUTPUT FILE NAME<br />5 NUMBER OF FIRMS<br />1 NUMBER OF TIME PERIODS <br />1 NUMBER OF OUTPUTS<br />2 NUMBER OF INPUTS<br />0 0=INPUT AND 1=OUTPUT ORIENTATED<br />0 0=CRS AND 1=VRS<br />0 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST- DEA, 3=DEA(1-STAGE), 4=DEA(2-STAGE)<br />Econ 377/477-Topic 3<br />45<br />
  63. 63. Output file<br />Efficiency summary<br />Summary of output slacks<br />Summary of input slacks<br />Summary of peers<br />Summary of peer weights<br />Peer count summary<br />Summary of output targets<br />Summary of input targets<br />Firm by firm results<br />Econ 377/477-Topic 3<br />46<br />
  64. 64. The VRS DEA model<br />The CRS assumption is only appropriate when all firms are operating at an optimal scale<br />The use of CRS specification when all firms are not operating at the optimal scale results in measures of TE that are confounded by scale efficiency (SE)<br />The use of VRS specification permits the calculation of TE devoid of these SE effects<br />SE can be calculated by estimating both the CRS and VRS models and looking at the difference in scores<br />Econ 377/477-Topic 3<br />47<br />
  65. 65. The VRS DEA model<br />Add a convexity constraint to the CRS model:<br /> min,<br /> st -qi + Q 0<br />xi - X 0<br />I1= 1<br /> 0<br /> where I1 is an I×1 vector of ones<br />Econ 377/477-Topic 3<br />48<br />
  66. 66. The VRS DEA model<br />A convex hull of intersecting planes is formed that envelops the data points more tightly than the CRS conical hull<br />It thus provides TE scores that are greater than or equal to those obtained using the CRS model<br />The convexity constraint, I1= 1, ensures that an inefficient firm is only benchmarked against firms of a similar size<br />Econ 377/477-Topic 3<br />49<br />
  67. 67. The VRS DEA model<br />Scale efficiency (SE) measures can be obtained for each firm by conducting both a CRS and VRS DEA and decomposing the TE scores into scale inefficiency and pure TE<br />This situation, where TECRS = TEVRSSE, is represented in the diagram on the next slide using a one-input one-output example<br />SE can be roughly interpreted as the ratio of the AP of a firm operating on the VRS frontier at Pvto the AP of a firm operating at the technically optimal scale at R<br />Econ 377/477-Topic 3<br />50<br />
  68. 68. Scale efficiency measurement in DEA<br />q<br />CRS frontier<br />R<br />TECRS = APC/AP<br />TEVRS = APV/AP<br />SE = APC/APV<br />PC<br />PV<br />A<br />P<br />VRS frontier<br />0<br />x<br />Econ 377/477-Topic 3<br />51<br />
  69. 69. The VRS DEA model: returns to scale<br />To determine whether a firm is operating in an area of increasing returns to scale (IRS) or decreasing returns to scale (DRS), we can run an additional DEA problem with non-increasing returns to scale (NIRS) imposed<br />We can achieve this aim by replacing<br /> I1= 1<br /> with<br /> I1≤ 1<br />The NIRS curve is shown on the next slide<br />Econ 377/477-Topic 3<br />52<br />
  70. 70. The VRS DEA model: returns to scale<br />The nature of the scale inefficiencies can be determined by checking whether the NIRS TE score is equal to the VRS TE score<br />This is shown on the next slide<br />They are unequal at point P, and so IRS exist<br />They are equal at point G, and so DRS apply<br />Econ 377/477-Topic 3<br />53<br />
  71. 71. Scale efficiency measurement in DEA<br />q<br />NIRS frontier<br />CRS frontier<br />G<br />R<br />VRS and NIRS are:<br /><ul><li>equal at G DRS
  72. 72. unequal at PIRS</li></ul>PC<br />PV<br />A<br />P<br />VRS frontier<br />0<br />x<br />Econ 377/477-Topic 3<br />54<br />
  73. 73. Scale efficiency example<br />Consider an example of five firms producing a single output using a single input<br />The data set is:<br /> Firm qx<br /> 1 1 2 <br /> 2 2 4 <br /> 3 3 3 <br /> 4 5 5 <br /> 5 5 6 <br />55<br />Econ 377/477-Topic 3<br />
  74. 74. Scale efficiency example<br />CRS frontier<br />5<br />VRS frontier<br />4<br />q<br />3<br />2<br />1<br />Econ 377/477-Topic 3<br />56<br />
  75. 75. Scale efficiency example: results<br />Firm CRS TE VRSTE Scale <br /> 1 0.500 1.000 0.500 irs <br /> 2 0.500 0.625 0.800 irs <br /> 3 1.000 1.000 1.000 - <br /> 4 0.800 0.900 0.889 drs <br /> 5 0.833 1.000 0.833 drs <br />mean 0.727 0.905 0.804 <br />Econ 377/477-Topic 3<br />57<br />
  76. 76. Using DEAP<br />qx<br />1 2<br />2 4<br />3 3<br />4 5<br />5 6<br />eg2.dta DATA FILE NAME<br />eg2.out OUTPUT FILE NAME<br />5 NUMBER OF FIRMS<br />1 NUMBER OF TIME PERIODS <br />1 NUMBER OF OUTPUTS<br />1 NUMBER OF INPUTS<br />0 0=INPUT AND 1=OUTPUT ORIENTATED<br />1 0=CRS AND 1=VRS<br />0 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA, 3=DEA(1-STAGE), <br /> 4=DEA(2-STAGE)<br />Econ 377/477-Topic 3<br />58<br />
  77. 77. DEAP output file: components<br />Efficiency summary<br />Summary of output slacks<br />Summary of input slacks<br />Summary of peers<br />Summary of peer weights<br />Peer count summary<br />Summary of output targets<br />Summary of input targets<br />Firm by firm results<br />Econ 377/477-Topic 3<br />59<br />
  78. 78. Output-orientated DEA models<br />Outputs are proportionally expanded, with inputs held fixed<br />The same frontier is produced as for the input-orientated model<br />The TE scores are identical under CRS – but can differ under VRS<br />Selection of orientation depends on which set (outputs or inputs) the firm has most control over<br />Econ 377/477-Topic 3<br />60<br />
  79. 79. Output-orientated DEA models<br /> For the output-orientated VRS model:<br /> max ,<br /> st -qi + Q 0<br />xi - X 0<br /> I1= 1<br />  0<br /> where 1  < , and - 1 is the proportional increase in outputs that could be achieved by the i-th firm with input quantities held constant<br />Note that 1/ defines a TE score, which varies between zero and one (this is the output-orientated TE score reported by DEAP)<br />min,<br />st -qi + Q0<br />xi - X 0<br />I1= 1<br />  0<br />Equivalent input-orientated VRS model<br />Econ 377/477-Topic 3<br />61<br />
  80. 80. Output-orientated DEA models<br />A two-output example of an output-orientated DEA is represented by a piecewise linear production possibility curve on the next slide<br />Sections of the curve at right angles to the axes result in output slack<br />Point P is projected to point P, which is on the frontier but not on the efficient frontier because production of q1 could be increased by AP without using any more inputs<br />Econ 377/477-Topic 3<br />62<br />
  81. 81. Output-orientated DEA models<br />q2<br />P<br />A<br />P<br />Q<br />0<br />q1<br />Econ 377/477-Topic 3<br />63<br />
  82. 82. Output-orientated DEA models<br />Output- and input-orientated DEA models will estimate exactly the same frontier<br />Therefore, they will identify the same set of firms as being efficient<br />It is only the efficiency measures associated with the inefficient firms that may differ between the two methods<br />They differ when constant returns to scale do not prevail<br />Econ 377/477-Topic 3<br />64<br />
  83. 83. Modelling allocative efficiencies<br />Price information and a behavioural objective such as cost minimisation or revenue maximisation enables estimation of allocative efficiencies as well as technical efficiencies<br />Two sets of linear programs are required, to measure:<br /><ul><li>technical efficiency
  84. 84. economic efficiency, known as cost efficiency (CE) for cost minimisation and revenue efficiency (RE) for revenue maximisation</li></ul>Econ 377/477-Topic 3<br />65<br />
  85. 85. Calculation of allocative efficiency in inputs<br />Input price data are required<br />The two DEA models that must be solved are the:<br /><ul><li>standard TE model
  86. 86. cost efficiency (CE) model</li></ul>Allocative efficiency (AE) is then calculated as AE = CE / TE<br />Econ 377/477-Topic 3<br />66<br />
  87. 87. Cost minimisation<br />Solve the cost-minimisation DEA:<br /> min,xi*wixi*<br /> st -qi + Q 0<br /> xi* - X 0<br />I1= 1<br /> 0<br /> where wi is a vector of input prices for the i-th firm and xi* (which is calculated by the LP) is the cost-minimising vector of input quantities for the i-th firm, given the input prices wi and the output levels qi<br />min,<br />st -qi + Q0<br />xi - X 0<br />I1= 1<br />  0<br />Original input-orientated VRS model<br />Econ 377/477-Topic 3<br />67<br />
  88. 88. Cost minimisation<br />The cost efficiency of the i-th firm is:<br /> CE = wi/xi* / wi/xi<br />That is, CE is a ratio of the minimum cost to the observed cost for the i-th firm<br />The AE, TE and CE measures can take values ranging from 0 to 1 where a value of 1 indicates full efficiency<br />Note that this procedure implicitly includes any slacks into the allocative efficiency measure in that slacks represent inappropriate input mixes<br />Econ 377/477-Topic 3<br />68<br />
  89. 89. Revenue maximisation<br />Solve the revenue-maximisation problem:<br /> max ,qi*piqi*<br /> st -qi* + Q 0<br /> xi - X 0<br /> I1= 1<br /> 0<br /> where pi is a M×1 vector of output prices for the i-th firm and qi* (which is calculated by the LP) is the revenue-maximising vector of output quantities for the i-th firm, given the output prices pi and the input levels xi<br />max ,<br />st -qi + Q 0<br /> xi - X 0<br />I1= 1<br />  0<br />Original output-orientated VRS model<br />Econ 377/477-Topic 3<br />69<br />
  90. 90. Revenue maximisation<br />The revenue efficiency of the i-th firm is:<br /> RE = pi/qi / pi/qi*<br />That is, RE is a ratio of the observed revenue for the i-th firm to the maximum revenue<br />The AE, TE and RE measures can take values ranging from 0 to 1, where a value of 1 indicates full efficiency<br />Econ 377/477-Topic 3<br />70<br />
  91. 91. CRS cost-efficiency DEA example<br />Data are used for a two-input one-output, input-orientated DEA example, shown in the table on the next slide<br />All firms are assumed to face the same prices, which are 1 and 3 for inputs 1 and 2, respectively<br />The solution of the problem is shown in the figure on the following slide<br />Firm 5 is the only cost-efficient firm, and all others have some allocative inefficiency<br />Econ 377/477-Topic 3<br />71<br />
  92. 92. CRS cost efficiency DEA example: results<br />Econ 377/477-Topic 3<br />72<br />
  93. 93. CRS cost-efficiency DEA example<br />x2/q<br />6<br />1<br />5<br />4<br />3<br />1<br />2<br />3<br />3<br />2<br />4<br />4<br />3<br />1<br />5<br />0<br />1<br />4<br />2<br />3<br />5<br />6<br />x1/q<br />73<br />Econ 377/477-Topic 3<br />
  94. 94. How to use DEAP<br />The data file comprises output in the first column, input quantities in the next two columns and input prices in the final two columns:<br /> 1 2 5 1 3<br /> 2 2 4 1 3<br /> 3 6 6 1 3<br /> 1 3 2 1 3<br /> 2 6 2 1 3<br />Econ 377/477-Topic 3<br />74<br />
  95. 95. The instruction file<br />eg3.dta DATA FILE NAME<br />eg3.out OUTPUT FILE NAME<br />5 NUMBER OF FIRMS<br />1 NUMBER OF TIME PERIODS <br />1 NUMBER OF OUTPUTS<br />2 NUMBER OF INPUTS<br />0 0=INPUT AND 1=OUTPUT ORIENTATED<br />0 0=CRS AND 1=VRS<br />1 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA, <br /> 3=DEA(1-STAGE), 4=DEA(2-STAGE)<br />Econ 377/477-Topic 3<br />75<br />
  96. 96. Non-discretionary variables<br />In input- (output-) orientated DEA models, all inputs (outputs) can be readily reduced (expanded)<br />This is not the case all the time, such as changing labour and materials in the short run but not capital<br />We wish to cater for this variation in ability to account for different abilities to alter input usage<br />Econ 377/477-Topic 3<br />76<br />
  97. 97. Non-discretionary variables<br />We can formulate a model in which we seek radial reduction in the inputs over which the manager has discretionary control<br />Inputs can be divided into discretionary and non-discretionary sets, as shown on the next slide<br />The discretionary and non-discretionary input sets are denoted by XD and XND, respectively<br />Econ 377/477-Topic 3<br />77<br />
  98. 98. Non-discretionary variables<br />The VRS DEA problem with discretionary and non-discretionary variables can be written as:<br /> min,<br /> st -qi + Q 0<br />xiD - XD 0<br /> xiND - XND 0<br />I1= 1<br /> 0<br />min,<br />st -qi + Q0<br />xi - X 0<br />I1= 1<br />  0<br />Original input-orientated VRS model<br />Econ 377/477-Topic 3<br />78<br />
  99. 99. How to account for the environment<br />The environment refers to factors that could influence the efficiency of a firm that are not under the control of the manager<br />They include:<br /><ul><li>Ownership differences
  100. 100. Location characteristics
  101. 101. Labour union power
  102. 102. Government regulations</li></ul>There are four main methods to accommodate them in a DEA analysis<br />Econ 377/477-Topic 3<br />79<br />
  103. 103. Accounting for the environment: Method 1<br />If the values of the environmental variables can be ordered, use the method of Banker and Morey (1986)<br />The efficiency of the i-th firm is compared with those firms in the sample that have a value of the environmental variable which is less than or equal to that of the i-th firm<br />Econ 377/477-Topic 3<br />80<br />
  104. 104. Accounting for the environment: Method 2<br />If there is no natural ordering of the environmental variable, choose the method by Charnes, Cooper and Rhodes (1981)<br /><ul><li>Make an arbitrary grouping (public and private)
  105. 105. Divide the sample accordingly
  106. 106. Project all observed data points onto their respective frontiers
  107. 107. Solve DEA separately and compare the mean technical efficiency</li></ul>Econ 377/477-Topic 3<br />81<br />
  108. 108. Disadvantages of Methods 1 and 2<br />The comparison set is reduced<br />Results might be misleading as many firms might be found efficient, thus reducing the discriminating power of the analysis<br />Only one environmental variable can be considered<br />The environmental variable must be categorical<br />Prior judgment is required<br />Econ 377/477-Topic 3<br />82<br />
  109. 109. Accounting for the environment: Method 3<br />A third method is to include the environmental variable(s) directly into the LP formulation of the DEA model<br />An environmental variable is included either as:<br /><ul><li>a non-discretionary input or output variable, using the non-discretionary variables approach outlined above, as shown on the next slide
  110. 110. a non-discretionary neutral variable</li></ul>Econ 377/477-Topic 3<br />83<br />
  111. 111. Accounting for the environment: Method 3.1<br />The standard non-discretionary variable is assumed to have a positive effect on efficiency:<br /> min,<br /> st -qi + Q 0<br />xi - X 0<br />zi - Z 0<br />I1= 1<br /> 0<br />min,<br />st -qi + Q0<br />xi - X 0<br />I1= 1<br />  0<br />Original input-orientated VRS model<br />Adding this constraint ensures the i-th firm is only compared with a (theoretical) frontier firm that has an environment that is no better<br />Econ 377/477-Topic 3<br />84<br />
  112. 112. Accounting for the environment: Method 3.1<br /><ul><li>This approach treats environmental variables as regular inputs
  113. 113. It assumes that environmental variables can be reduced
  114. 114. But firms can not alter their environment
  115. 115. By removing  from the additional constraint, the environmental variables are not included in the calculation of the efficiency scores</li></ul>Econ 377/477-Topic 3<br />85<br />
  116. 116. Accounting for the environment: Method 3.1<br />This next formulation is for an environmental variable with a negative impact<br />The standard non-discretionary variable is assumed to have a negative effect on efficiency:<br /> min,<br /> st -qi + Q 0<br />xi - X 0<br /> -zi + Z 0<br />I1= 1<br /> 0<br />Note the changes in signs<br />Econ 377/477-Topic 3<br />86<br />
  117. 117. Accounting for the environment: Method 3.2<br /><ul><li>If the decision maker is unsure about the direction of influence of the environmental variables, they can be included in an equality form as shown on the next slide
  118. 118. This approach avoids having to pre-specify the direction of influence
  119. 119. But it greatly reduces the reference set for each firm, inflating the efficiency scores</li></ul>Econ 377/477-Topic 3<br />87<br />
  120. 120. Accounting for the environment: Method 3.2<br />Inclusion of an environmental variable as a non-discretionary neutral variable:<br />min,<br /> st -qi + Q 0<br />xi - X 0<br /> -zi + Z= 0<br />I1= 1<br /> 0<br />Adding this constraint ensures the i-th firm is only compared with a (theoretical) frontier firm that has an environment that is no better or worse<br />Econ 377/477-Topic 3<br />88<br />
  121. 121. Including variables into a DEA model<br />Including variables into a DEA model needs to take account of:<br /><ul><li>the reduced degrees of freedom
  122. 122. prior expectations about the direction of the effect
  123. 123. the inability to test for statistical significance
  124. 124. the inability to include categorical variables
  125. 125. the inability to include variables with negative values</li></ul>Econ 377/477-Topic 3<br />89<br />
  126. 126. Accounting for the environment: Method 4, the two-stage method<br />First stage:<br /><ul><li>Solve DEA involving only the traditional inputs and outputs</li></ul>Second stage:<br /><ul><li>Regress the efficiency scores on the environmental variables</li></ul>The signs of the coefficients indicate the direction of the influence<br />Econ 377/477-Topic 3<br />90<br />
  127. 127. Advantages of the two-stage method<br />It can accommodate more than one variable<br />It can accommodate both continuous and categorical variables<br />It does not make prior assumptions regarding the direction of the influence of the categorical variable<br />A hypothesis test can be conducted to see if the variable has a significant influence upon efficiency<br />It is easy to calculate<br />The method is simple and transparent<br />Econ 377/477-Topic 3<br />91<br />
  128. 128. Disadvantages of the two-stage method<br />A significant proportion of the efficiency scores might equal unity and OLS might estimate scores greater than unity<br />This problem can be overcome by using the Tobit regression method to account for truncated data<br />If the variables used in the first stage are highly correlated with second-stage variables, results are likely to be biased<br />Econ 377/477-Topic 3<br />92<br />
  129. 129. Input congestion<br />Input congestion implies that isoquants ‘bend backwards’ – negative marginal product<br />Excess input use can also be due to constraints that are beyond the control of the firm<br />Standard DEA assumes strong disposability of inputs (and outputs)<br />That is, P(x) satisfies strong disposability in inputs: if q can be produced from x, it can be produced from any x* ≥ x<br />DEA that accounts for input congestion relaxes this strong disposability assumption<br />Econ 377/477-Topic 3<br />93<br />
  130. 130. Input congestion<br />Input congestion is accounted for in the input-orientated VRS DEA model by changing the inequalities in the input restrictions to equalities and introducing a  parameter in the input restrictions:<br /> min,,<br /> st -qi + Q 0<br />xi - X = 0<br />I1= 1<br /> 0, 0 <   1<br />min,<br />st -qi + Q0<br />xi - X 0<br />I1= 1<br />  0<br />Original input-orientated VRS model<br />Econ 377/477-Topic 3<br />94<br />
  131. 131. Input congestion<br />TE VRS DEA (weak) ≥TE VRS DEA (strong)<br />Congestion inefficiency has been removed<br />The TE measure from a CRS DEA model can be decomposed into:<br /><ul><li>Pure technical inefficiency
  132. 132. Scale inefficiency
  133. 133. Congestion inefficiency
  134. 134. Three DEA models are solved for this purpose: CRS assuming strong disposability; VRS assuming strong disposability; and VRS assuming weak disposability</li></ul>Econ 377/477-Topic 3<br />95<br />
  135. 135. Input congestion<br />x2<br />SS<br />SW<br />Input congestion efficiency<br />A<br />ICE = 0PS/0PW<br />P<br />TES = (0PS/0PW)(0PW/0P)<br />TES = 0PS/0P<br />PW<br />PS<br />TES = ICETEW<br />S<br />0<br />x1<br />96<br />Econ 377/477-Topic 3<br />
  136. 136. Overview of DEA<br />Constant returns to scale<br />Variable returns to scale<br />Input-orientation<br />Output-orientation<br />Technical efficiency<br />Scale efficiency<br />Cost efficiency<br />Allocative efficiency<br />Econ 377/477-Topic 3<br />97<br />
  137. 137. Overview of DEA<br />Using DEAP<br />Creating data file<br />Creating instruction file<br />Reading and interpreting output file<br />Adjusting for environmental variables<br />Limitations<br />Econ 377/477-Topic 3<br />98<br />
  138. 138. Overview of DEA: limitations<br />Measurement error and other noise may influence the shape and position of the frontier<br />Outliers may influence the results<br />The exclusion of an important input or output can result in biased results<br />Econ 377/477-Topic 3<br />99<br />
  139. 139. Overview of DEA: limitations<br />The efficiency scores obtained are only relative to the best firms in the sample and the inclusion of extra firms may reduce efficiency scores<br />Be careful when comparing the mean efficiency scores from two studies because:<br /><ul><li>they only reflect the dispersion of efficiencies within each sample
  140. 140. they say nothing about the efficiency of one sample relative to the other</li></ul>Econ 377/477-Topic 3<br />100<br />
  141. 141. Overview of DEA: limitations<br />The addition of an extra firm in a DEA analysis cannot result in an increase in the TE scores of the existing firms<br />The addition of an extra input or output in a DEA model cannot result in a reduction in the TE scores<br />When one has few observations and many inputs and/or outputs, many of the firms will appear on the DEA frontier<br />Econ 377/477-Topic 3<br />101<br />
  142. 142. Overview of DEA: limitations<br />Treating inputs and/or outputs as homogeneous commodities when they are heterogeneous may bias results<br />Not accounting for environmental differences may give misleading indications of relative managerial competence<br />Standard DEA does not account for multi-period optimisation nor risk in management decision making<br />Econ 377/477-Topic 3<br />102<br />
  143. 143. Reading: DEA application<br />Carrington et al. (1997), ‘Performance measurement in government service provision: The case of police services in NSW’<br />DEA was used<br />Data comprised 163 police patrols in 1994/95<br />Outputs were offences, arrests, summons, major car accidents and kilometres travelled<br />Inputs were officers, civilians and cars<br />Econ 377/477-Topic 3<br />103<br />

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