1. ECON 377/477

2. Econ 377/477-Topic 3 2 Topic 3 Data Envelopment Analysis

3. Outline Introduction Efficiency measurement concepts The constant returns-to-scale model The variable returns-to-scale model Input and output orientations Modelling allocative efficiencies … Econ 377/477-Topic 3 3

4. Outline Adjusting for the environment Non-discretionary variables and input congestion Exercises Estimating technical and allocative efficiency using DEAP Econ 377/477-Topic 3 4

7. Motivation 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 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 Econ 377/477-Topic 3 6

9. Stochastic frontier analysis (SFA)Econ 377/477-Topic 3 7

12. x /q S 2 P · A · Q · R ¢ Q · ¢ S ¢ 0 A x /q 1 Technical and allocative efficiencies TEi=0Q/0P AEi=0R/0Q EEi=0R/0P Econ 377/477-Topic 3 10

13. Technical and allocative efficiencies 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 The technical efficiency measure (0Q/0P) must lie between zero and one At the value of one, a firm is fully efficient Econ 377/477-Topic 3 11

14. Technical and allocative efficiencies Given an input-price ratio, the allocative efficiency is 0R/0Q Reduction in cost if production is at Q’ Point Q is technically efficient but allocatively inefficient Economic efficiency is: EEi= 0R/0P That is, EE = TE * AE The efficient isoquant must be estimated using sample data Econ 377/477-Topic 3 12

15. Non-parametric estimation: the piecewise linear convex isoquant S x2/q : firms No observed point should lie to the left or below line SS Frontier S x1/q 0 Econ 377/477-Topic 3 13

16. Output-orientated measures Aigner and Chu (1968) is the seminal paper on non-parametric output measures Recall from Topic 2 that to specify a parametric frontier production function in input-output space, a Cobb-Douglas function is: ln(qi) = f(ln(xi), ) - ui where qi is the output of the i-th firm, xi is an input vector and ui is a non-negative variable representing inefficiency Econ 377/477-Topic 3 14

17. Output-orientated measures Technical efficiency is calculated as: TEi = qi/f(ln(xi),) = exp (-ui) 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 The non-parametric approach does not account for noise Econ 377/477-Topic 3 15

18. Technical and allocative efficiencies 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 Changing the units of measurement will not change the value of the efficiency measure Econ 377/477-Topic 3 16

19. Standard DEA models DEA is a non-parametric mathematical programming approach to frontier estimation Charnes, Cooper and Rhodes (1978) coined the term, data envelopment analysis We begin a description of its use assuming an input orientation and constant returns to scale Banker, Charnes and Cooper (1984) proposed variable returns to scale models that are discussed below Econ 377/477-Topic 3 17

20. Standard DEA models For DEA, we need data on the input and output quantities of each firm Linear programming is used to construct a non-parametric piecewise surface over the data TE is the distance of each firm below this surface Econ 377/477-Topic 3 18

21. Standard DEA models Remember that in the input-orientated model, we look at the amount by which inputs can be proportionally reduced, with outputs fixed And for the output-orientated model, we look at the amount by which outputs can be proportionally increased with inputs fixed Econ 377/477-Topic 3 19

23. N inputs (N×I input matrix, X)

24. M outputs (M×I output matrix, Q)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 Econ 377/477-Topic 3 20

25. CRS models We introduce DEA via the ratio form For each firm we would like to obtain a measure of the ratio of all outputs over all inputs, such as: (uqi/vxi) where u is an M×1 vector of output weights and v is an N×1 vector of input weights We wish to obtain the optimal weights by solving the linear programming (LP) problem on the next slide The LP problem must be solved I times Econ 377/477-Topic 3 21

26. CRS models Infinite number of solutions maxu,v (uqi/vxi) stuqj/vxj  1j = 1,2,...,I u, v  0 Impose the constraint vxi= 1 max, (qi) st xi = 1 qj - xj  0j = 1,2,...,I ,  0 Multiplier form of the DEA model = M×1 vector of output weights  = N×1 vector of input weights Identical TE scores: TEi= qi/xi Econ 377/477-Topic 3 22

27. Input-orientated CRS DEA Model Using duality, the input-orientated CRS model is: min, st -qi + Q 0 xi - X 0  0 xi is a N×1 vector of inputs of the i-th firm qi is M×1 vector of outputs of the i-th firm X is a N×I input matrix Q is M×I output matrix θis a scalar (used to estimate TE)  is a I×1 vector of constants Econ 377/477-Topic 3 23

28. Input-orientated CRS DEA Model The problem for the i-th firm is radially to contract the input vector xi as much as possible The inner boundary is a piecewise linear isoquant determined by the observed data points The radial contraction of the input vector xi produces a projected point (X, Q) on the surface of this technology The diagram on the next slide shows the radial contraction for two inefficient firms, A and B Econ 377/477-Topic 3 24

29. Non-parametric estimation: the piecewise linear convex isoquant S x2/q A A B C B D S x1/q 0 Econ 377/477-Topic 3 25

30. Input-orientated CRS DEA Model The projected points are a linear combination of the observed data points Firm A can reduce use of inputs x1 and x2, and move to point A without reducing output Firm B can reduce use of inputs x1 and x2, and move to point B without reducing output The constraints ensure that this projected point cannot lie outside the feasible set Econ 377/477-Topic 3 26

32. Is point A an efficient point?

33. We could reduce the amount of input x2 (by the amount CA) and still produce the same output

34. Input slacks will be equal to zero if and only if xi -X= 0 (for the given optimal values of  and )

35. For output-orientated models, output slacks will be equal to zero if and only if Q- qi= 0Econ 377/477-Topic 3 27

36. A simple numerical example Data for a simple numerical example are presented below, and a graph of the input-output ratios is shown on the next slide Econ 377/477-Topic 3 28

37. A simple numerical example 5 S x2/q : projected points for inefficient firms 1 4 3 1 4 2 3 2 3 4 1 S 5 3 x1/q 0 1 2 4 5 Econ 377/477-Topic 3 29

38. LP for firm number 3 The DEA frontier results from running five LP problems – one for each firm We could rewrite the minimisation problem for firm 3, for example, as: min, st -q3 + (q11 + q22 + q33 + q44 + q55)  0 x13 - (x111 + x122 + x133 + x144 + x155)  0 x23 - (x211 + x222 + x233 + x244 + x255)  0  0 where  = (1, 2, 3, 4, 5) Econ 377/477-Topic 3 30

39. LP for firm number 3 The values of  and  that provide a minimum value for  are listed in row 3 of the table on the next slide (in pink) That is, firm 3 could possibly reduce the consumption of all inputs by 16.7 per cent without reducing output This implies production at point 3 in the diagram on the previous slide but one Econ 377/477-Topic 3 31

40. CRS Input-orientated DEA results Econ 377/477-Topic 3 32

41. LP for firm number 1 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) Firm 1 has a TE score of 0.5 This means it has the potential to reduce input usage by 50 per cent It also has a (non-radial) input slack of 0.5 units of x2 (per unit of q) Econ 377/477-Topic 3 33

42. Peers for firm 3 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 These firms are termed its peers Peers define the relevant part of the frontier, namely the efficient production for individual firms λ values are used to measure the weights of peers 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 Econ 377/477-Topic 3 34

43. Peers for firm 3 x2/q S 1 4 3 Peer for firm 3 1 4 2 3 3 2 Peer for firm 3 4 1 S 5 x1/q 0 1 2 3 4 5 Econ 377/477-Topic 3 35

44. Peers for firms 1,2 and 5 The projected point 1 for firm 1 in the diagram on the next slide lies on the vertical line from point 2 Thus, firm 2 is its sole peer Because firms 2 and 5 lie on the frontier, they are technically efficient They are therefore their own peers Hence, λ2 = 1.0 for firm 2 and λ5 = 1.0 for firm 5 Econ 377/477-Topic 3 36

45. Peers for firms 1, 2 and 5 x2/q S 1 4 3 Peer for firm 1 and own peer 1 4 2 3 3 2 Own peer 4 1 S 5 x1/q 0 1 2 3 4 5 Econ 377/477-Topic 3 37

46. Targets for firm 3 Targets are defined by the coordinates of the efficient points These targets provide the amount of inputs that should be used (or output produced for an output-orientated model) The targets of firm 3 are the coordinates of the efficient projection point 3, equal to 0.833(2,2), = (1.666,1.666) 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 Econ 377/477-Topic 3 38

47. Targets for firm 3 x2/q S 1 4 3 1 4 2 3 3 2 1.666 4 1 S 5 x1/q 0 1 2 3 4 5 1.666 Econ 377/477-Topic 3 39

48. Targets for firm 1 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 This would result in targets of (x1/q = 1, x2/q = 2) These targets are the coordinates of the efficient firm 2, at point 2, equal to 1.0(1,2), = (1,2) 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 Econ 377/477-Topic 3 40

49. Targets for firm 1 x2/q S 1 4 3 1 4 2 3 3 2 2.0 4 1 S 5 x1/q 0 1 2 3 4 5 1.0 Econ 377/477-Topic 3 41

51. DEAP – DOS-based DEA program

52. Onfront, Warwick DEA – Windows-based specialist DEA programsWe use DEAP Version 2.1 Econ 377/477-Topic 3 42

54. Copy the data into a data file for the DEAP program

55. Create an instruction file for DEAP program

56. Execute the DEAP program to produce the output fileRead pages 252-253 of CROB Econ 377/477-Topic 3 43

58. The file must be saved as .dta

59. Use either

60. Excel (save as Tab delimited)

61. NotepadEcon 377/477-Topic 3 44

62. The instruction file Edit the instruction file (CROB, page 169) Right click and open with notepad eg1.dta DATA FILE NAME eg1.out OUTPUT FILE NAME 5 NUMBER OF FIRMS 1 NUMBER OF TIME PERIODS 1 NUMBER OF OUTPUTS 2 NUMBER OF INPUTS 0 0=INPUT AND 1=OUTPUT ORIENTATED 0 0=CRS AND 1=VRS 0 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST- DEA, 3=DEA(1-STAGE), 4=DEA(2-STAGE) Econ 377/477-Topic 3 45

63. Output file Efficiency summary Summary of output slacks Summary of input slacks Summary of peers Summary of peer weights Peer count summary Summary of output targets Summary of input targets Firm by firm results Econ 377/477-Topic 3 46

64. The VRS DEA model The CRS assumption is only appropriate when all firms are operating at an optimal scale 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) The use of VRS specification permits the calculation of TE devoid of these SE effects SE can be calculated by estimating both the CRS and VRS models and looking at the difference in scores Econ 377/477-Topic 3 47

65. The VRS DEA model Add a convexity constraint to the CRS model: min, st -qi + Q 0 xi - X 0 I1= 1  0 where I1 is an I×1 vector of ones Econ 377/477-Topic 3 48

66. The VRS DEA model A convex hull of intersecting planes is formed that envelops the data points more tightly than the CRS conical hull It thus provides TE scores that are greater than or equal to those obtained using the CRS model The convexity constraint, I1= 1, ensures that an inefficient firm is only benchmarked against firms of a similar size Econ 377/477-Topic 3 49

67. The VRS DEA model 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 This situation, where TECRS = TEVRSSE, is represented in the diagram on the next slide using a one-input one-output example 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 Econ 377/477-Topic 3 50

68. Scale efficiency measurement in DEA q CRS frontier R TECRS = APC/AP TEVRS = APV/AP SE = APC/APV PC PV A P VRS frontier 0 x Econ 377/477-Topic 3 51

69. The VRS DEA model: returns to scale 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 We can achieve this aim by replacing I1= 1 with I1≤ 1 The NIRS curve is shown on the next slide Econ 377/477-Topic 3 52

70. The VRS DEA model: returns to scale The nature of the scale inefficiencies can be determined by checking whether the NIRS TE score is equal to the VRS TE score This is shown on the next slide They are unequal at point P, and so IRS exist They are equal at point G, and so DRS apply Econ 377/477-Topic 3 53

72. unequal at PIRSPC PV A P VRS frontier 0 x Econ 377/477-Topic 3 54

73. Scale efficiency example Consider an example of five firms producing a single output using a single input The data set is: Firm qx 1 1 2 2 2 4 3 3 3 4 5 5 5 5 6 55 Econ 377/477-Topic 3

74. Scale efficiency example CRS frontier 5 VRS frontier 4 q 3 2 1 Econ 377/477-Topic 3 56

75. Scale efficiency example: results Firm CRS TE VRSTE Scale 1 0.500 1.000 0.500 irs 2 0.500 0.625 0.800 irs 3 1.000 1.000 1.000 - 4 0.800 0.900 0.889 drs 5 0.833 1.000 0.833 drs mean 0.727 0.905 0.804 Econ 377/477-Topic 3 57

76. Using DEAP qx 1 2 2 4 3 3 4 5 5 6 eg2.dta DATA FILE NAME eg2.out OUTPUT FILE NAME 5 NUMBER OF FIRMS 1 NUMBER OF TIME PERIODS 1 NUMBER OF OUTPUTS 1 NUMBER OF INPUTS 0 0=INPUT AND 1=OUTPUT ORIENTATED 1 0=CRS AND 1=VRS 0 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA, 3=DEA(1-STAGE), 4=DEA(2-STAGE) Econ 377/477-Topic 3 58

77. DEAP output file: components Efficiency summary Summary of output slacks Summary of input slacks Summary of peers Summary of peer weights Peer count summary Summary of output targets Summary of input targets Firm by firm results Econ 377/477-Topic 3 59

78. Output-orientated DEA models Outputs are proportionally expanded, with inputs held fixed The same frontier is produced as for the input-orientated model The TE scores are identical under CRS – but can differ under VRS Selection of orientation depends on which set (outputs or inputs) the firm has most control over Econ 377/477-Topic 3 60

79. Output-orientated DEA models For the output-orientated VRS model: max , st -qi + Q 0 xi - X 0 I1= 1  0 where 1  < , and - 1 is the proportional increase in outputs that could be achieved by the i-th firm with input quantities held constant Note that 1/ defines a TE score, which varies between zero and one (this is the output-orientated TE score reported by DEAP) min, st -qi + Q0 xi - X 0 I1= 1  0 Equivalent input-orientated VRS model Econ 377/477-Topic 3 61

80. Output-orientated DEA models A two-output example of an output-orientated DEA is represented by a piecewise linear production possibility curve on the next slide Sections of the curve at right angles to the axes result in output slack 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 Econ 377/477-Topic 3 62

81. Output-orientated DEA models q2 P A P Q 0 q1 Econ 377/477-Topic 3 63

82. Output-orientated DEA models Output- and input-orientated DEA models will estimate exactly the same frontier Therefore, they will identify the same set of firms as being efficient It is only the efficiency measures associated with the inefficient firms that may differ between the two methods They differ when constant returns to scale do not prevail Econ 377/477-Topic 3 64

84. economic efficiency, known as cost efficiency (CE) for cost minimisation and revenue efficiency (RE) for revenue maximisationEcon 377/477-Topic 3 65

86. cost efficiency (CE) modelAllocative efficiency (AE) is then calculated as AE = CE / TE Econ 377/477-Topic 3 66

87. Cost minimisation Solve the cost-minimisation DEA: min,xi*wixi* st -qi + Q 0 xi* - X 0 I1= 1  0 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 min, st -qi + Q0 xi - X 0 I1= 1  0 Original input-orientated VRS model Econ 377/477-Topic 3 67

88. Cost minimisation The cost efficiency of the i-th firm is: CE = wi/xi* / wi/xi That is, CE is a ratio of the minimum cost to the observed cost for the i-th firm The AE, TE and CE measures can take values ranging from 0 to 1 where a value of 1 indicates full efficiency Note that this procedure implicitly includes any slacks into the allocative efficiency measure in that slacks represent inappropriate input mixes Econ 377/477-Topic 3 68

89. Revenue maximisation Solve the revenue-maximisation problem: max ,qi*piqi* st -qi* + Q 0 xi - X 0 I1= 1  0 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 max , st -qi + Q 0 xi - X 0 I1= 1  0 Original output-orientated VRS model Econ 377/477-Topic 3 69

90. Revenue maximisation The revenue efficiency of the i-th firm is: RE = pi/qi / pi/qi* That is, RE is a ratio of the observed revenue for the i-th firm to the maximum revenue The AE, TE and RE measures can take values ranging from 0 to 1, where a value of 1 indicates full efficiency Econ 377/477-Topic 3 70

91. CRS cost-efficiency DEA example Data are used for a two-input one-output, input-orientated DEA example, shown in the table on the next slide All firms are assumed to face the same prices, which are 1 and 3 for inputs 1 and 2, respectively The solution of the problem is shown in the figure on the following slide Firm 5 is the only cost-efficient firm, and all others have some allocative inefficiency Econ 377/477-Topic 3 71

92. CRS cost efficiency DEA example: results Econ 377/477-Topic 3 72

93. CRS cost-efficiency DEA example x2/q 6 1 5 4 3 1 2 3 3 2 4 4 3 1 5 0 1 4 2 3 5 6 x1/q 73 Econ 377/477-Topic 3

94. How to use DEAP The data file comprises output in the first column, input quantities in the next two columns and input prices in the final two columns: 1 2 5 1 3 2 2 4 1 3 3 6 6 1 3 1 3 2 1 3 2 6 2 1 3 Econ 377/477-Topic 3 74

95. The instruction file eg3.dta DATA FILE NAME eg3.out OUTPUT FILE NAME 5 NUMBER OF FIRMS 1 NUMBER OF TIME PERIODS 1 NUMBER OF OUTPUTS 2 NUMBER OF INPUTS 0 0=INPUT AND 1=OUTPUT ORIENTATED 0 0=CRS AND 1=VRS 1 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA, 3=DEA(1-STAGE), 4=DEA(2-STAGE) Econ 377/477-Topic 3 75

96. Non-discretionary variables In input- (output-) orientated DEA models, all inputs (outputs) can be readily reduced (expanded) This is not the case all the time, such as changing labour and materials in the short run but not capital We wish to cater for this variation in ability to account for different abilities to alter input usage Econ 377/477-Topic 3 76

97. Non-discretionary variables We can formulate a model in which we seek radial reduction in the inputs over which the manager has discretionary control Inputs can be divided into discretionary and non-discretionary sets, as shown on the next slide The discretionary and non-discretionary input sets are denoted by XD and XND, respectively Econ 377/477-Topic 3 77

98. Non-discretionary variables The VRS DEA problem with discretionary and non-discretionary variables can be written as: min, st -qi + Q 0 xiD - XD 0 xiND - XND 0 I1= 1  0 min, st -qi + Q0 xi - X 0 I1= 1  0 Original input-orientated VRS model Econ 377/477-Topic 3 78

100. Location characteristics

101. Labour union power

102. Government regulationsThere are four main methods to accommodate them in a DEA analysis Econ 377/477-Topic 3 79

103. Accounting for the environment: Method 1 If the values of the environmental variables can be ordered, use the method of Banker and Morey (1986) 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 Econ 377/477-Topic 3 80

105. Divide the sample accordingly

106. Project all observed data points onto their respective frontiers

107. Solve DEA separately and compare the mean technical efficiencyEcon 377/477-Topic 3 81

108. Disadvantages of Methods 1 and 2 The comparison set is reduced Results might be misleading as many firms might be found efficient, thus reducing the discriminating power of the analysis Only one environmental variable can be considered The environmental variable must be categorical Prior judgment is required Econ 377/477-Topic 3 82

110. a non-discretionary neutral variableEcon 377/477-Topic 3 83

111. Accounting for the environment: Method 3.1 The standard non-discretionary variable is assumed to have a positive effect on efficiency: min, st -qi + Q 0 xi - X 0 zi - Z 0 I1= 1  0 min, st -qi + Q0 xi - X 0 I1= 1  0 Original input-orientated VRS model Adding this constraint ensures the i-th firm is only compared with a (theoretical) frontier firm that has an environment that is no better Econ 377/477-Topic 3 84

113. It assumes that environmental variables can be reduced

114. But firms can not alter their environment

115. By removing  from the additional constraint, the environmental variables are not included in the calculation of the efficiency scoresEcon 377/477-Topic 3 85

116. Accounting for the environment: Method 3.1 This next formulation is for an environmental variable with a negative impact The standard non-discretionary variable is assumed to have a negative effect on efficiency: min, st -qi + Q 0 xi - X 0 -zi + Z 0 I1= 1  0 Note the changes in signs Econ 377/477-Topic 3 86

118. This approach avoids having to pre-specify the direction of influence

119. But it greatly reduces the reference set for each firm, inflating the efficiency scoresEcon 377/477-Topic 3 87

120. Accounting for the environment: Method 3.2 Inclusion of an environmental variable as a non-discretionary neutral variable: min, st -qi + Q 0 xi - X 0 -zi + Z= 0 I1= 1  0 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 Econ 377/477-Topic 3 88

122. prior expectations about the direction of the effect

123. the inability to test for statistical significance

124. the inability to include categorical variables

125. the inability to include variables with negative valuesEcon 377/477-Topic 3 89

127. Advantages of the two-stage method It can accommodate more than one variable It can accommodate both continuous and categorical variables It does not make prior assumptions regarding the direction of the influence of the categorical variable A hypothesis test can be conducted to see if the variable has a significant influence upon efficiency It is easy to calculate The method is simple and transparent Econ 377/477-Topic 3 91

128. Disadvantages of the two-stage method A significant proportion of the efficiency scores might equal unity and OLS might estimate scores greater than unity This problem can be overcome by using the Tobit regression method to account for truncated data If the variables used in the first stage are highly correlated with second-stage variables, results are likely to be biased Econ 377/477-Topic 3 92

129. Input congestion Input congestion implies that isoquants ‘bend backwards’ – negative marginal product Excess input use can also be due to constraints that are beyond the control of the firm Standard DEA assumes strong disposability of inputs (and outputs) That is, P(x) satisfies strong disposability in inputs: if q can be produced from x, it can be produced from any x* ≥ x DEA that accounts for input congestion relaxes this strong disposability assumption Econ 377/477-Topic 3 93

130. Input congestion 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: min,, st -qi + Q 0 xi - X = 0 I1= 1  0, 0 <   1 min, st -qi + Q0 xi - X 0 I1= 1  0 Original input-orientated VRS model Econ 377/477-Topic 3 94

132. Scale inefficiency

133. Congestion inefficiency

134. Three DEA models are solved for this purpose: CRS assuming strong disposability; VRS assuming strong disposability; and VRS assuming weak disposabilityEcon 377/477-Topic 3 95

135. Input congestion x2 SS SW Input congestion efficiency A ICE = 0PS/0PW P TES = (0PS/0PW)(0PW/0P) TES = 0PS/0P PW PS TES = ICETEW S 0 x1 96 Econ 377/477-Topic 3

136. Overview of DEA Constant returns to scale Variable returns to scale Input-orientation Output-orientation Technical efficiency Scale efficiency Cost efficiency Allocative efficiency Econ 377/477-Topic 3 97

137. Overview of DEA Using DEAP Creating data file Creating instruction file Reading and interpreting output file Adjusting for environmental variables Limitations Econ 377/477-Topic 3 98

138. Overview of DEA: limitations Measurement error and other noise may influence the shape and position of the frontier Outliers may influence the results The exclusion of an important input or output can result in biased results Econ 377/477-Topic 3 99

140. they say nothing about the efficiency of one sample relative to the otherEcon 377/477-Topic 3 100

141. Overview of DEA: limitations The addition of an extra firm in a DEA analysis cannot result in an increase in the TE scores of the existing firms The addition of an extra input or output in a DEA model cannot result in a reduction in the TE scores When one has few observations and many inputs and/or outputs, many of the firms will appear on the DEA frontier Econ 377/477-Topic 3 101

142. Overview of DEA: limitations Treating inputs and/or outputs as homogeneous commodities when they are heterogeneous may bias results Not accounting for environmental differences may give misleading indications of relative managerial competence Standard DEA does not account for multi-period optimisation nor risk in management decision making Econ 377/477-Topic 3 102

143. Reading: DEA application Carrington et al. (1997), ‘Performance measurement in government service provision: The case of police services in NSW’ DEA was used Data comprised 163 police patrols in 1994/95 Outputs were offences, arrests, summons, major car accidents and kilometres travelled Inputs were officers, civilians and cars Econ 377/477-Topic 3 103