Basic Efficiency Measure (see Ch 6, CRB; Ch 2 On Front)
Reference Technologies
Output Set
P(x)
y1
P(x)
y1
Input Set
L(y)
x2
L(y)
x1
Graph
GR
y
GR
x
Input Based Measures (Farrell, 1957)
Technical Effic.
x2
P
Q
O
Technical Effic.:
Fi(y,x) = min{λ: λx  L(y)}
L(y)
x1
For Obs P:
Fi(y,x) = OQ/OB < 1
Estimating Effic. with DEA (activity analysis)
1. Construct technology from data
2. Estimate efficiency relative to tech for each observation
Notation
k = 1,…,K
indexes observations or DMU’s
m = 1,…, M
indexes different types of outputs
n = 1,…,N
indexes different types of inputs
ykm
Individual output and input data for
xkn
observation k
ym
Elements of technology created from out
xn
sample data
zk
Intensity (activity) variables
Constructing the Reference Technology
x   N , y   M
1. L(y) = { x = (x1,…,xN): x can produce y}
2. P(x) = { y = (y1,…,yM): y producible from x}
3. GR = {(x,y) = (x1,…,xN , y1,…,yM): x can produce y}
Activity Analysis Constraints Used to Construct Tech
K
outputs:

k 1
zkykm  ym, m=1,.., M
K

inputs:
zkxkn  xn, n=1,.., N
k 1
zk  0, k=1,…, K
intensity vars:
Additional Constraints to Change Returns to Scale
K
NIRS

zk  1,
k 1
K
VRS

zk = 1,
k 1
Production Properties and DEA
Disposability
1. Strong (free) disposability of outputs (SDO):
if y  y0, y0  P(x)  y  P(x)
y2
y
y0
P(x)
y1
impose in DEA through inequalities:
K

zkykm  ym, m=1,.., M
k 1
2. Strong (free) disposability of inputs (SDI):
if x  x0, x0  L(y)  x  L(y)
L(y)
x2
x
0
x
x1
K
DEA restrictions:

zkxkn  xn, n=1,.., N
k 1
Convexity
Data:
Obs
x1
x2
y
A
3
1
1
B
1
3
1
x2
3
L(1)
B
1/2B +1/2A
2
A
1
1
2
3
x1
intensity variables
‘connect’ feasible points (dashed lines)
zAxA1 + zBxB1  x1
zAxA2 + zBxB2  x2
EXAMPLE – Input set L(y)
Obs(k)
y
x1
x2
A
1
3
1
B
1
1
3
x2
3
B
2
A
1
O
1
2
3
x1
L(1) = {(x1,x2):
zA1+ zB1  1
output
zA3+ zB1  x1
input x1
zA1+ zB3  x2 input x2
zA, zB1  0 } intensity variables
Example, cont.
Role of Intensity Variables in Constructing Technologies
“Dot Connectors” – creates convex combinations of observed data
EG: from Table 1 + L(1)
let zA = 1, zB = 0
then 3  x1
1  x2 (Strong disposability)
x2
3
B
2
A
1
O
1
let zA = 0, zB = 1
then 1  x1
3  x2
2
3
x1
x2
B
3
2
A
1
O
1
2
3
x1
let zA = 1/2, zB = 1/2
then 3/2+1/2  x1
½+3/2  x2
x2
3
L(1)
B
2
A
1
O
1
and so on.
2
3
x1
Technical Efficiency-example for obs:
Fi(3,1) = min λ
s.t.
zA1+ zB1  1
output
zA3+ zB1  λ3
x1
for A = 3
zA1+ zB3  λ1
x2
for A = 1
zA, zB1  0
x2
3
B
L(1)
2
A
1
O
1
2
Solution Fi(3,1) = 1
zA = 1, zB = 0
11
3  1*3
1  1*1
zA, zB  0
3
x1
Estimating Technical Efficiency with Activity Analysis
Example 1: Input Tech Eff: Fi(yk,xk) = min{λ: λxk  L(yk)}
k
y
x1
x2
A
1
2
1
B
1
1
2
C
1
2
2
x2
3
2
1
O
1
2
Write out technology
(assume CRS, SDI)
L(1| c, s) = {( x1,x2):
3
x1
Write out Linear Programming Problem for obs c:
Output Based Efficiency:
Technical Effic.
y2
P(xA)
B
A
O
y1
Tech Efficiency
Fo(y, x) = max{θ: θy  P(x)}
Fo(yA,xA) = OB/OA
if y  P(x)  Fo(y, x)  1
Example 2: Output Tech Eff: Fo(yk,xk) = max{θ: θyk  P(xk)}
k
x
y1
y2
D
1
2
1
E
1
1
2
F
1
1
1
y2
3
2
1
O
1
2
3
Write out technology: P(1| c, s)
P(1| c, s) = {(y1,y2):
Write out linear problem for obs F:
y1
Scale Efficiency
y
F’
F
E’
1
D’
D
C
B
O
C’
Scale Eff:
B’
GRc
E
GRv
A
A’
x
SE = Fc/Fv
Input based (see obs A)
SEi = Fi(y,x|c)/Fi(y,x|v)
for obs A: SE iA = (OC’/OA’)/(OB’/OA’) = OC’/OB’
Output based (see obs D)
SEo = Fo(y,x|c)/Fo(y,x|v)
for obs D: SE oD = (OF’/OD’)/(OE’/OD’) = OF’/OE’