More Technical Efficiency and Data Envelopment Analysis (DEA)

Transcription

1 More Technical Efficiency and Data Envelopment Analysis (DEA) based on OnFront, Reference Guide Coelli, et al., Introduction to Efficiency and Productivity Springer, 2005, Ch 6. April 22, 2008

2 Technology: Set Representations Equivalent Technology Sets: describe the relationship between all feasible inputs x = (x 1,..., x N ) and outputs y = (y 1,..., y M ) y x2 y2 GR L(y) P(x) x x1 y1 GR={(x,y):x can produce y} L(y)={x: (x,y) is in GR} P(x)={y: (x,y) is in GR} If (x,y) is in GR, then x is in L(y) and y is in P(x)

3 Technical Efficiency y x2 y2 GR (x,y) (x1,x2) L(y) (y1,y2) P(x) x x1 y1 Fi(y,x)=min{lambda: lambda x is in L(y)} Fo(x,y)=max{theta:theta y is in P(x)}

4 Estimating technical efficiency with DEA DEA uses linear programming to construct the technology and best practice frontier from the data in your sample Simultaneously, it estimates the distance to the best practice frontier for each observation There is a separate linear programming problem for each observation (DMU)

5 DEA Overview DEA MODEL Technology Efficiency Measure Returns to Scale Disposability Orientation Inputs and Outputs

6 Formalities k = 1,..., K, are the observations or decision making units (DMUs) x k = (x k1, x k2,..., x kn ) are the inputs 1 to N for observation (DMU) k y k = (y k1, y k2,..., y km ) are the outputs 1 to M for observation (DMU) k z = (z 1, z 2,..., z K ) are the intensity variables ( dot connectors ), which will be used to construct the best practice frontier Input technical efficiency for each k=1,...,k, is defined as follows F i (y k, x k ) = min λ,z {λ : λx k L(y k )} (1)

7 DEA (linear programming) problem for input technical efficiency, F i (y k, x k ) For each observation k solve: F i (y k, x k ) = min{λ λ,z (2) subject to K z k y km y km, m = 1,..., M k=1 K z k x kn λx kn, n = 1,..., N k=1 z k 0, k = 1,..., K. REMARK: The inequalities represent the technology, in this case L(y k ).

8 How DEA constructs the technology sets, eg: L(y) Recall: L(y) = {x = (x 1,..., x N ): x can produce y=(y 1,..., y M )}, in DEA, for observation k: L(y k ) = {(x 1,..., x N ) : (3) K z k y km y km, m = 1,..., M k=1 K z k x kn x n, n = 1,..., N k=1 z k 0, k = 1,..., K}. REMARKS: The inequalities allow for free (strong) disposability of both inputs and outputs. Note that we are looking for all possible x s that satisfy the constraints (x n rather than x kn on the RHS).

9 Now plot and use the following data to write out the constraints for the input set: capital labor widgets x 1 x 2 y Firm A Firm B Firm C

10 L(50) = {(x 1, x 2 ) : (4) z A 30 + z B 40 + z C 40 x 1 (capital) z A 40 + z B 20 + z C 30 x 2 (labor) z A 50 + z B 50 + z C 50 50(widgets) z A, z B, z C 0

11 L(50) creates additional observations of capital and labor by allowing the z s to take any values greater than or equal to zero. What amount of capital and labor do you get if: z A = z B = 1/2, z C = 0

12 L(50) creates additional observations of capital and labor by allowing the z s to take any values greater than or equal to zero. What amount of capital and labor do you get if: z A = z B = 1/2, z C = 0 capital 35, labor 30

13 L(50) creates additional observations of capital and labor by allowing the z s to take any values greater than or equal to zero. What amount of capital and labor do you get if: z A = z B = 1/2, z C = 0 capital 35, labor 30 z A = z B = 0, z C = 1

14 L(50) creates additional observations of capital and labor by allowing the z s to take any values greater than or equal to zero. What amount of capital and labor do you get if: z A = z B = 1/2, z C = 0 capital 35, labor 30 z A = z B = 0, z C = 1 capital 40, labor 30

15 L(50) creates additional observations of capital and labor by allowing the z s to take any values greater than or equal to zero. What amount of capital and labor do you get if: z A = z B = 1/2, z C = 0 capital 35, labor 30 z A = z B = 0, z C = 1 capital 40, labor 30 z A = z B = z C = 1/3

16 L(50) creates additional observations of capital and labor by allowing the z s to take any values greater than or equal to zero. What amount of capital and labor do you get if: z A = z B = 1/2, z C = 0 capital 35, labor 30 z A = z B = 0, z C = 1 capital 40, labor 30 z A = z B = z C = 1/3 capital 36.67, labor 30

17 Labor (x2) 40- A L(50) 30- C 20- B 0 l l l Capital (x1)

18 Now write out the the problem to solve for technical efficiency for Firm C.

19 F i (y C, x C ) = min λ : (5) z A 30 + z B 40 + z C 40 λ40(capital) z A 40 + z B 20 + z C 30 λ30(labor) z A 50 + z B 50 + z C 50 50(widgets) z A, z B, z C 0 solution: F i (y C, x C ) =.91, za =.36, z B =.64, z C = 0 How much capital and labor would firm C use if it were efficient?

20 How DEA constructs the technology sets, eg: GR GR = {(x 1,..., x N, y 1,..., y M ) : (x 1,..., x N ) can produce (y 1,..., y M )} (6) GR = {(x 1,..., x N, y 1,..., y M ) : (7) K z k y km y m, m = 1,..., M k=1 K z k x kn x n, n = 1,..., N k=1 z k 0, k = 1,..., K}. REMARKS: The inequalities allow for free (strong) disposability of both inputs and outputs. Note that we are looking for all possible x s AND y s that satisfy the constraints (x n and y n ) rather than x kn and y km on the RHS).

21 Now plot and use the following data to write out the constraints for the input-output set GR: output input y x DMU A 4 3 DMU B 1 1 DMU C 3 2

22 GR = {(x, y) : (8) z A 4 + z B 1 + z C 3 y(output) z A 3 + z B 1 + z C 2 x(input) z A, z B, z C 0}

23 What amounts of input and output do you get if: z A = z B = z C = 0 What does this look like?

24 What amounts of input and output do you get if: z A = z B = z C = 0 y 0, x 0 What does this look like?

25 What amounts of input and output do you get if: z A = z B = z C = 0 y 0, x 0 z C = 5, z B = z A = 0 What does this look like?

26 What amounts of input and output do you get if: z A = z B = z C = 0 y 0, x 0 z C = 5, z B = z A = 0 y 15, x 10 What does this look like?

27 What type of returns to scale does this technology exhibit? y 4 - A 3 - C 2 - GR 1 - B 0 l l l x

28 Returns to Scale and Scale Efficiency The DEA problems we have specified so far to estimate Farrell input and output based technical efficiency have restricted technology to satisfy: constant returns to scale strong disposability of inputs and outputs The returns to scale of technology are determined by the restrictions on the intensity variables, i.e., the z s. The disposability property arises from the inequalities on the input and output constraints.

29 Imposing Returns to Scale in DEA Constant Returns to Scale: Nonincreasing Returns to Scale: z k 0, k = 1,..., K z k 0, k = 1,..., K K k=1 z k 1 Variable Returns to Scale: z k 0, k = 1,..., K K k=1 z k = 1

30 Imposing Nonincreasing Returns to Scale: EG GR N, S = {(x, y) : (9) z A 4 + z B 1 + z C 3 y(output) z A 3 + z B 1 + z C 2 x(input) z A, z B, z C 0 z A + z B + z C 1} N stands for Nonincreasing returns to scale S stands for Strong disposability

31 What amounts of input and output do you get if: z A = z B = z C = 0 What does this look like?

32 What amounts of input and output do you get if: z A = z B = z C = 0 y 0, x 0 What does this look like?

33 What amounts of input and output do you get if: z A = z B = z C = 0 y 0, x 0 z C = 1/3, z A = z B = 0 What does this look like?

34 What amounts of input and output do you get if: z A = z B = z C = 0 y 0, x 0 z C = 1/3, z A = z B = 0 y 1, x 2/3 What does this look like?

35 y Nonincreasing Returns to Scale 4 - A 3 - C 2 - (GR N,S) 1 - B 0 l l l x

36 Imposing Variable Returns to Scale: EG GR V, S = {(x, y) : (10) z A 4 + z B 1 + z C 3 y(output) z A 3 + z B 1 + z C 2 x(input) z A, z B, z C 0 z A + z B + z C = 1} V stands for Variable returns to scale (increasing, constant and/or decreasing returns to scale) S stands for Strong disposability

37 What amounts of input and output do you get if: z A = z C = 0, z B = 1 What does this look like?

38 What amounts of input and output do you get if: z A = z C = 0, z B = 1 y 1, x 1 What does this look like?

39 What amounts of input and output do you get if: z A = z C = 0, z B = 1 y 1, x 1 z C = z B = 1/2, z A = 0 What does this look like?

40 What amounts of input and output do you get if: z A = z C = 0, z B = 1 y 1, x 1 z C = z B = 1/2, z A = 0 y 2, x 1.5 What does this look like?

41 y Variable Returns to Scale 4 - A 3 - C 2 - (GR V,S) 1 - B 0 l l l x

42 Scale efficiency (SE) Measures the gap between the boundary of the input-output set (GR) under CRS and VRS: SE = Farrell CRS technical efficiency/farrell VRS technical efficiency. y (GR C,S) 4 - A 3 - C 2 - (GR V,S) 1 - B 0 l l l x

43 Remarks: The VRS technology (GR V, S) can never be bigger than the CRS technology (GR C, S) for the same data set. We can measure the gap between the technologies in either the input or output direction.

44 Input based scale efficiency, SE i SE i = F i (y, x C, S)/F i (y, x V, S) y SEi(D)=(0E /0D )/(0B /0D ) =(0E /0B ) (GR C,S) 4 - A 3 - C 2 - (GR V,S) 1 - E B D 0 l l l E B D 3 x

45 Output based scale efficiency, SE o SE o = F o (y, x C, S)/F o (y, x V, S) y SEo(D)=(0C /0D )/(0C /0D ) =1 (GR C,S) 4 - A 3 - C C 2 - (GR V,S) 1 -D B D 0 l l l 3 x