### lecture 8

```DMOR
DEA
Variable Returns to Scale
O7
O2
O6
OR
O3
O4
O2
O1
Constant Returns to Scale
DEA example
• For each bank branch we have one output
measure and one input measure
Branch
Croydon
Dorking
Redhill
Reigate
Personal
No. of
transactions ('000) workers
125
18
44
16
80
17
23
11
Efficiency
• Inputs are changes into outputs
Branch
Croydon
Dorking
Redhill
Reigate
Personal transactions
per worker ('000)
6.94
2.75
4.71
2.09
Relative efficiency
• We can compare all branches relative to
Croydon
Branch
Relative efficiency
Croydon
Dorking
Redhill
Reigate
=6.94/6.94 = 100%
=2.75/6.94 = 40%
=4.71/6.94 = 68%
=2.09/6.94 = 30%
More outputs
Branch
Croydon
Dorking
Redhill
Reigate
Personal
No. of workers
transactions ('000) transactions ('000)
125
50
18
44
20
16
80
55
17
23
12
11
Efficiency
• We now haev two “efficiencies”:
Branch
Croydon
Dorking
Redhill
Reigate
Personal
transactions per
worker ('000)
6.94
2.75
4.71
2.09
transactions per
worker ('000)
2.78
1.25
3.24
1.09
Graphically
• Reigate
– Personal transactions per worker 2090
– Business transactions per worker 1090
– Slope 2090/1090=1.92
1.92
Relative efficiency
• Relative efficiency for Reigate
length from 0,0 to Reigate
length from 0,0 to Best
• For Reigate = 36%
• For Dorking = 43%
Relative efficiency
• Technical efficiency
– Extended efficiency due to Koopmans,
Pareto:
• A given entity is fully efficient, if no input and
no output can be improved without worsening
some other input or output.
Dominating
entities
– Relative efficiency:
• A given entity is efficient based on the
available evidence, if performance of other
entities do not indicate that no input and no
output can be improved without worsening
some other input or output.
• There is no reference to prices and weights
of inputs and outputs.
• You don’t need to establish the relation
between inputs and outputs
A desirable
direction
capital
If we know that there is a technology
which enables
• producing q0 units of output
• using L units of labor and K units of
capital according to the prodction
function:
Technical efficiency definition:
Produce a given level of output
using the minimal level of inputs
labor
capital
Then we can measure inefficiency:
• e.g. suppose that entity A produces
q0 units of outputs
• Then OA’/OA is entity A’s efficiency
A
A’
O
labor
DEA approach
E, F, G, H, I is the efficient
frontier
capital
• Production function isoquant is
not known directly
• DEA estimates it from the data
using interval-wise linear
interpolation
• Assume that firms A, B, C, D, E, F,
G, H, I all produce q0 units of output
C
E
A
B
F
D
G
H
O
I
labor
DEA efficiency
E, F, G, H, I is efficient
frontier
capital
C
E
A
B
F
D
G
A’
H
O
I
labor
• Efficiency of A according to DEA is OA’/OA
• A’ is a shadow or a phantom of A
– It is a linear combination of F and G
Adding a couple of new branches
Personal
transaction
transactions
Branch
s per
per worker
worker
('000)
('000)
Croydon
6.94
2.78
Dorking
2.75
1.25
Redhill
4.71
3.24
Reigate
2.09
1.09
A
1.23
2.92
B
4.43
2.23
C
3.32
2.81
D
3.70
2.68
E
3.34
2.96
• Branch F has 1000 personal transactions per
worker
• And 6000 business transactions per worker
Branch
Croydon
Dorking
Redhill
Reigate
A
B
C
D
E
F
G
Personal
transactions per
worker ('000)
6.94
2.75
4.71
2.09
1.23
4.43
3.32
3.70
3.34
1.00
5.00
transactions per
worker ('000)
2.78
1.25
3.24
1.09
2.92
2.23
2.81
2.68
2.96
6.00
5.00
The efficient one does not have to win
in any category
Constant returns to scale (CCR)– primal
problem
F is efficient in a
weak sense
Constant and Variable returns to scale (CRS i VRS):
decomposition into scale efficiency and pure technical
efficiency
• Primal problem
Multiplier model:
• Dual problem:
Envelopment
model:
• “Strong disposal” assumption
– Ignores presence of nonzero slack variables
– Different solutions may have nonzero slack variables or not
– Therefore one uses 2 phase of the dual problem to
maximize these variables (to see whether there exists a
solution with nonzero slack variables)
First and second phase of the dual problem may be written
together and solved in two steps
Model
Input-oriented
Output-oriented
Example: Input oriented dual problem for P5
Input oriented primal problem for P5
Results
Efficient frontier projection in input
oriented model
Efficient frontier projection in output
oriented model
Next example
Model BBC (Variable Returns to Scale):
Dual problem for DMU5
Variable Returns to
Scale
Technical efficiency for DMU5 may be reached for DMU2, which lies on the efficient
frontier
DMU 4 is weakly DEA efficient
3)
The same problem for DMU4 gives:
How to interpret weights?
• Assume that we consider an entity with efficiency less than 1
• Assume that
, the rest of the weights are zero
• Then the phantom inputs of the entity are:
• And the phantom outputs of the entity are:
```