Malmquist Total Factor Productivity Index for Modeling Dairy Sector

Seminar II
Presented By:
Arnab Roy ,UAS ,GKVK
PALB 7003
2

Flow of Presentation
3
 Significance of dairy sector
 Efficiency Measurement Approach
 Data envelopment analysis
 Malmquist Productivity Index
 Case Study
 Govt. initiatives
 Conclusion
 References

Significance of Dairy Sector
4
India ranks first among the world’s milk producing Nations
17 mt (1950-51) 176.4 mt(2017-18)6.65 %
130 gram
374 gram
per capita availability
 48 % milk is consumed at the producer level
 52 % of the milk is marketable surplus

5
Fig 1: Growth of per capita consumption of dairy products

Fig 2: Indian Dairy products consumption
6Source: NDDB,2015

7Source: NAP for Dairy Development ,Vision-2022
Fig 3: Milk Processing capacity of Different organization in 2016-17( in LLPD)

Table 1:Per capita monthly consumption expenditure on broad groups of
items(in Rs.)
8
NSS Round Rural/Urban
Milk and
Milk
Products
Meat, Egg,
Fish Total Food
57th (July 2001- June2002)
Rural 41.91 (15.17) 16.72 276.35
Urban
75.82(18.85)
25.83 402.31
Rural 45.34(15.51) 18.31 292.27
58th (July 2002- -Dec.2002)
Urban 78.19(18.19 ) 27.07 429.79
59th (Jan. 2003- Dec.2003) Rural 44.69(14.90 ) 17.93 299.86
Urban
80.03(18.65 )
27.18 429.12
60th (Jan. 2004- June2004)
Rural
47.60(15.63 )
18.60 304.60
Urban 82.98( 18.80) 27.84 441.48
61st (July 2004- June2005)
Rural 47.31(15.31 ) 18.60 307.60
Urban
83.30(18.62 )
28.47 447.41
66th (July 2009- June2010)
Rural
79.78(16.04 )
32.47 497.25
Urban
139.29(19.16 )
48.22 726.82
68th (July 2011- June2012)
Rural 116.38(18.72 ) 46.04 621.56
Urban 187.14( 20.26) 67.18 923.71
Source : National Sample Survey Organisation, GoI (2012-13)
Note: Per cent value of total food in parenthesis

9
Efficiency
Measurement
Parametric Techniques
SFA
Distribution Free
Approach
Non-Parametric
Techniques
DEA
Free Disposal Hull
(FDH)
Mesaurement of Efficiency

DATA ENVELOPMENT ANALYSIS
10
 It is mainly based on the earlier concept of Frontier
Analysis (Farrel, 1957)
 It assesses the relative efficiency scores of a particular
set of Decision-Making-Units (DMU)
 Efficiency scores of DMUs which are equal to 1 are called
“efficient” and different to 1 are called “inefficient”
 Most effective DMUs is called “efficiency frontier”

Mathematical Structure of DEA
11
jkuv
i
xu
yv
ts
xu
yv
jk
m
j
jij
ki
s
k
k
m
j
jpj
kp
s
k
k
,0,
,1.
max
1
1
1
1










jkuv
ixuyv
xuts
yv
jk
m
j
jijki
s
k
k
m
j
jpj
kp
s
k
k
,0,
,0
1.
max
11
1
1









where: p is the unit being evaluated; s represents the number of outputs; m represents the
number of inputs; yki is the amount of output k provided by unit i; xji is the amount of
input j used by unit i; vk and uj are the weights given to output k and input j, respectively.

Fig 4: Input Oriented TE measure and Scale Efficiency 12
Where, Y = Output ; X = Input

CCR Ratio Model (Input-Oriented)
13
i
kyy
jxxts
i
kpki
i
i
jpji
i
i





0
.
min




where:  represents the efficiency score of unit p; s represent the dual variables that
identify the benchmarks for inefficient units.

CCR Ratio Model (Output-Oriented)
14
i
kyy
jxxts
i
kpki
i
i
jpji
i
i





0
.
max




Efficiency =

1

CCR Model: Illustration
15
DMU Input 1 Input 2 Output 1 Output 2 Output 3
1 5 14 9 4 16
2 8 15 5 7 10
3 7 12 4 9 13
0,,,,
01271394
01581075
01451649
1145
..
1649
21321
21321
21321
21321
21
321






uuvvv
uuvvv
uuvvv
uuvvv
uu
ts
vvvMaximize

CCR Model: Illustration Results
16
DMU 1 and DMU 3 are efficient
(efficiency of 1.00 with no slacks)
DMU 2 is inefficient (efficiency < 1.00)
DMU 2 can utilize DMU 1 and DMU 3
as benchmarks for improvement

Graphical Illustration of DEA
17
To illustrate, consider seven DMUs
which each have one input and one
output:
 L1 = (2,2),
 L2 = (3,5),
 L3 = (6,7),
 L4 = (9,8),
 L5 = (5,3),
 L6 = (4,1),
 L7 = (10,7)
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9 10 11
Input
Output

Table 1 : Hypothetical Example of DMU
18
Company A B C D E F G H
Employees 4 3 3 2 8 6 5 5
Output 3 2 3 1 5 3 4 2
Output/Employee 0.75 0.667 1 0.5 0.625 0.5 0.8 0.4

0
Output
Employees
C
Efficiency Frontier
Figure 5 : Regression Line and Efficiency Frontier
Regression Line
D
B
A
G
H
F
E

Table 2 : Efficiency scores of Different DMU
20
Company A B C D E F G H
Efficiency 0.75 0.667 1 0.5 0.625 0.5 0.8 0.4
1
ofemployeeperSales
anotherofemployeeperSales
0 
C
1 = C > G > A> B > E > D = F > H = 0.4

BCC Model of DEA
21
 CCR model considers constant returns to
scale (CRS) whereas the BCC model
considers variable returns to scale (VRS)
i
kyy
jxxts
i
i
i
kpki
i
i
jpji
i
i







0
1
.
min






Malmquist Productivity Index
22
• Malmquist Productivity index makes use of distance
functions to measure productivity change.
• It can be defined using input or output orientated
distance functions.
• This approach was first proposed in Caves, Christensen
and Diewert (1982).
• We just look at the Output-orientated Malquist
productivity Index (MPI).

MATHEMATICAL FORMULATION OF MALMQUIST
PRODUCTIVITY INDEX
Let us take the two time periods t1 and t2.
•Period t1, a firm uses input x1to produce output y1
•Period t2, the same firm uses input x 2 to produce output y2.
Let us define the production set at time t, as St = { (xt , y t ): xt
can produce yt }
Where xt is an input vector and yt is an output vector such
that 𝑥 𝑡 𝜖 𝑅+
𝑁 𝑎𝑛𝑑 𝑦 𝑡 𝜖 𝑅+
𝑀 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡.
This distance function is defined as the inverse of Farrel‟s
(1957) technical efficiency measure
𝐷 𝑡1 (𝑥 𝑡1 , 𝑦 𝑡1 ) = (𝑆𝑢𝑝 {𝜃𝜖 𝑅 𝑥 𝑡1 , 𝜃𝑦 𝑡1 𝜖 𝑆𝑡1 }) −1

24
• Using period s-technology:
• Using period t-technology:
 
 
 ,
,
, ,
,
s
s o t t
o s t s t s
o s s
d
m
d

q x
q q x x
q x
   
 ss
t
o
tt
t
o
tsts
t
o
d
d
m
xq
xq
xxqq
,
,
,,, 

Since there are two possible MFP measures, based on period s
and period t technology, the MFP is defined as the geometric
average of the two:
25
     
 
 
 
 
0.5
0.5
, , , , , , , , ,
, ,
, ,
s t
o s t s t o s t s t o s t s t
s t
o t t o t t
s t
o s s o s s
m m m
d d
d d
   
 
  
  
q q x x q q x x q q x x
x q x q
x q x q

Malmquist Productivity Index - Properties
26
1. It can be decomposed into efficiency change and
technical change:
   
 
 
 
 
 
5.0
,
,
,
,
,
,
,,,









ss
t
o
ss
s
o
tt
t
o
tt
s
o
ss
s
o
tt
t
o
sstso
d
d
d
d
d
d
m
qx
qx
qx
qx
qx
qx
xxqq
2. Malmquist productivity index is the same as the Hicks
Moorsteen index if the technology exhibits global
constant returns to scale and inverse homotheticity.

27
x
q
0
E
D

 

qs
qa
qc
qt
qb
xs xt
Frontier in
period s
Frontier in
period t
Efficiency change =
as
ct
qq
qq
/
/ 5.0
/
/
/
/







bs
as
ct
bt
qq
qq
qq
qq
Technical change =

28
3. The input-orientated Malmquist productivity index is given
by:
4.Output-orientated and input-orientated Malmquist indexes
coincide if the technology exhibits constant returns to scale.
5. The Malmquist Productivity index does not adequately
account for scale change.
6. The Malmquist productivity index does not satisfy transitivity
property. So we need to use the EKS method to make them
transitive.
Cont..
 
 
 
 
0.5
, ,
, ,
s s t t s s
s t t t t t
d q x d q x
TFPC
d q x d q x
 
  
 

Components approach of Malmquist
Productivity Index
29
The last approach is to measure productivity change by
identifying various sources of productivity growth:
1. Efficiency change
2. Technical change
3. Scale efficiency change
4. Output and input mix effect
Then Productivity change is measured as the product of
the four changes above. The resulting index is:
   
 
 
 
5.0
,
,*
,*
,*
,*
,,,









ss
t
o
tt
t
o
ss
s
o
tt
s
o
tsts
ts
d
d
d
d
TFPC
qx
qx
qx
qx
qqxx

Malmquist Productivity Index Using DEA
30
[do
t
(qt,xt)]-1
=max, ,
st -qit +Qt0,
xit -Xt0,
0,
[do
s
(qs,xs)]-1
=max, ,
st -qis +Qs0,
xis –Xs0,
0,
[do
t
(qs,xt)]-1
=max, ,
st -qis +Qt0,
xis -Xt0,
0,
[do
s
(qt,xt)]-1
=max, ,
st -qit +Qs0,
xit –Xs0,
0,

Calculation of MPI using DEA
31
Listing of Instruction File, EG4-INS.TXT
___________________________________________________________________
eg4-dta.txt DATA FILE NAME
eg4-out.txt OUTPUT FILE NAME
5 NUMBER OF FIRMS
3 NUMBER OF TIME PERIODS
1 NUMBER OF OUTPUTS
1 NUMBER OF INPUTS
1 0=INPUT AND 1=OUTPUT ORIENTATED
0 0=CRS AND 1=VRS
2 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA,
3=DEA(1-STAGE), 4=DEA(2-STAGE)
___________________________________________________________________

Technical efficiency and total factor productivity growth in the
Dairy cooperatives unions in India
-K. Rajendran and Samarendu Mohant(2004)
Journal of Food Distribution Research
32
Case Study -I

Table 3. Performance of Dairy Co-operatives Organized through
Operation Flood.
33
Regions Anand-
Pattern
DCS
Producer
members
(000)
Processing
capacity
(000 lbs)
Average
procurement
(000kgs/day)
Average an-
nual market-
ing (000 lbs)
Artificial
insemina-
tion centers
(DCS)
Mobile
veterinary
clinics
Northern 22,166 1,343 4,630 1,451 1,957 3,365 151
region
Western 20,854 3,140 9,375 4,984 3,262 5,584 328
region
Southern 20,886 4,241 5,504 3,546 3,341 5,711 242
region
Eastern 5,065 268 1,536 304 833 1,520 31
region
Total 69,868 8,992 21,045 10,285 9,393 16,180 752
Source: Dairy India (2014).

Table 4. Technical Efficiency Levels of the Dairy cooperative (2004-2008)
34
Dairy
Cooperative
Year
2004 2005 2006 2007 2008 mean
West Bengal 0.858 1.000 1.000 1.000 1.000 0.972
Karnataka 0.843 0.898 0.825 0.814 0.929 0.862
Kerala 0.781 1.000 1.000 0.812 0.933 0.905
Odisha 1.000 0.792 0.919 0.779 0.928 0.884
Haryana 1.000 0.984 1.000 0.986 1.000 0.994
Maharastra 0.848 0.868 0.824 0.790 0.937 0.853
Gujrath 0.830 0.976 0.920 0.966 0.928 0.924
Andhra Pradesh 0.832 0.927 0.894 0.861 0.929 0.889
Rajasthan 0.954 0.995 0.982 0.842 0.930 0.941
UP 1.000 1.000 1.000 0.825 0.929 0.951
Punjab 1.000 0.644 0.733 1.000 0.932 0.862
Source: NDDB

Table 5. Average Total Factor Productivity Growth Components
35
Dairy
Cooperative
Technical
efficiency
Change
Technical
change
Pure
efficiency
change
Scale
efficiency
change
Total factor
productivity
change
West Bengal 1.039 1.092 1.000 1.039 1.135
Karnataka 1.025 0.985 1.009 1.016 1.009
Kerala 1.045 0.992 1.037 1.008 1.037
Odisha 0.982 0.900 0.982 1.000 0.884
Haryana 1.000 1.009 1.000 1.000 1.009
Maharastra 1.025 0.953 0.999 1.026 0.977
Gujrath 1.028 0.968 1.014 1.014 0.996
Andhra
Pradesh 1.028 0.960 1.006 1.022 0.987
Rajasthan 0.994 0.990 1.003 0.991 0.983
UP 0.982 0.958 0.988 0.994 0.941
Punjab 1.006 0.922 1.007 0.999 0.928
Source: Author’s calculation

Case Study-II
Performance Analysis of Food Processing Industries in
Punjab using Data Envelopment Analysis
- Rohin Malhotra , International Journal of Economics &
Management Sciences(2018)
36
 This paper has analysed development and financial
performance, with special reference to working capital
management industries
 It prioritises these industries for development based on the
performance criteria

Table 6: Dominance table of Processed products
Industry
Characteristics/
Year
Number of
factories (no.)
Number of
workers (no.)
Invested
capital
Total
output
Total
inputs
Profit
Slaughtering, preparation
and
preservation of meat
1980-81 17.33 10.95 17.56 33.33 34.9 22
1990-91 13.49 12.93 18.94 37.98 41.17 20.18
2000-01 5.39 7.09 14.06 24.07 26.94 8.51
2010-11 10.95 12.12 24.05 25.5 27.52 1.49
2015-16 7.89 12.76 14.47 23.02 20.17 160.23
Manufacture of dairy
product
1980-81 1.47 3.5 16.17 11.89 11.28 26.16
1990-91 1.41 6 17.51 13.83 11.57 48.81
2000-01 3.03 8.23 12.61 23.49 18.78 88.96
2010-11 2.05 6.6 5.88 12.99 13.41 1.21
2015-16 2.14 8.77 11.86 14.82 15.36 1.82
Manufacture of other
food products
1980-81 6.13 24.72 16.2 8.54 6.87 28.71
1990-91 3.61 12.65 22.21 10.08 9.17 6.7
2000-01 5.8 16.69 31.19 11.94 11.19 -1.05
2010-11 3.23 11.21 11.67 9.89 9.61 -0.95
2015-16 2.79 12.58 14.84 9.37 9.13 -15.99
Manufacture of beverages 1980-81 2.27 3.99 5.42 3.49 3.29 2.55
1990-91 3.37 6.93 7.67 7.11 5.85 28.67
2000-01 3.57 7.51 11.61 7.51 7.28 3.71
2010-11 3.91 8.98 12.25 17.51 13.45 97.22
2015-16 3.14 7.95 10.81 11.93 11.85 5.86
Source: Authors calculation(2013)

Table 7 : Total factor productivity change and various efficiency change
38
Industry
Technical
efficiency
Technological
change
Pure
technical
Scale
efficiency
Total factor
productivity
change
efficiency
change change change
Meat 1.008 1.21 1.001 1.006 1.219
Dairy 0.903 1.162 0.951 0.949 1.049
Grain, Starch 0.873 1.049 0.93 0.939 0.916
Other Food
Products 0.873 1.17 0.874 0.999 1.021
Beverages 0.946 1.184 1 0.946 1.12
Mean 0.919 1.154 0.95 0.967 1.06
Source: Malhotra (2013)

Table 8 : Malmquist index summary of firm means
39
Period Period
1980-81 to 1997-98 1998-99 to 2015-16
Industry Technical Technological Pure Scale Total Factor Technical Technological Pure Scale Total Factor
Efficiency Change Technical Efficiency Productivity Efficiency Change Technical Efficiency Productivity
Change Efficiency Change Change Change Efficiency Change Change
Change Change
Meat 0.909 1.72 0.975 0.932 1.563 1 1.099 1 1 1.099
Dairy 1 2.072 1 1 1.437 0.943 1.005 0.905 1.042 0.948
Grain,
Starch
0.641 1.346 0.684 0.938 0.863 0.874 0.972 1.007 0.868 0.85
Other Food 0.773 1.713 1 0.773 1.324 1.009 1.014 1.012 0.998 1.023
Products
Beverages 0.896 1.593 0.828 1.082 1.427 0.905 1.083 1.032 0.877 0.979
Mean 0.834 1.676 0.888 0.939 1.398 0.945 1.034 0.99 0.954 0.976
Source: Authors calculation(2013)

Fig 6 : Growth Rates in Percentage of Different Industries40

Advantages of MPI
41
 Its unique ability to measure the efficiency of
multiple-input & output of DMUs without
assigning prior weight to the input and output
 Non imposing apriori parametric restrictions on
the underlying technology
 Does not have as many restrictive assumptions
as parametric statistical inference.

Government initiatives for Dairy Development
42
 Dairy Processing and Infrastructure Development Fund (DIDF)
 National Programme for Dairy Development(NPDD)
 National Dairy Plan (Phase-I)
 Dairy Entrepreneurship Development Scheme(DEDS)
 Support to Dairy Cooperatives
 National Action Plan (NAP) ( 2021-22 and 2023-24)

43
 The analysis of input slacks in the dairy processing industry
suggests that the industry is labour intensive
 The industry needs to modernize its production system to
improve the capacity utilization of factor inputs, mainly of raw
material, capital and energy
 Majority of farmers use available technology sub-optimally
and produce less than potential output.
 Strong and effective linkage of farms to market would provide
incentives towards increasing their efficiency in production
Conclusion

References:
44
1. CHARNES, R., COOPER, W., RHODES, E., 1978, Measuring the efficiency of
decision making units. European Journal of Operational Research 2: 429-444.
2. FABIO A. MADAU & ROBERTO FURESI & PIETRO PULINA.,2017, Technical
efficiency and total factor productivity changes in European dairy farm sectors.
Agricultural and Food Economics, Springer;Italian Society of Agricultural Economics
(SIDEA),5(1): 1-14,
3. KARMAKAR, K. G., & BANERJEE, G. D., 2006, Opportunities and challenges in the
Indian dairy industry. Technical Digest, 2006: 24‐27.
4. MALMQUIST, S.,1953, Index numbers and indifference curves”, Trabajos de
Estatistica, 4(1): 209-242.
5. MALHOTRA, R .,2018,Performance Analysis of Food Processing Industries in Punjab
using Data Envelopment Analysis. Int J Econ Manag Sci 7: 550.

Malmquist Total Factor Productivity Index for Modeling Dairy Sector

Malmquist Total Factor Productivity Index for Modeling Dairy Sector

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