K.P. Kalirajan, R.T. Shand Agricultural Economics 24 2001 297–306 301
Fig. 2. Paths of efficiency with changing technical and allocative efficiencies with technological change over time.
programmes to identify best technical practices, when, as discussed in the Bayesian learning models, farm-
ers are left to discover these in a learning-by-doing process andor through interaction between farmers in
this process. In practice, this latter situation has been the norm in agriculture, and for this study, it has led
to our third hypothesis which is that the time path will be the second one, i.e. ii in Fig. 1.
As discussed by Silverberg 1991, technologies develop considerably during the diffusion process.
Following Atkinson and Stiglitz 1969, it may be argued that a major component of the technological
change emanates from learning on existing technolo- gies. One obvious reason is learning by doing and
learning by using, on the part of researchers and farmers, respectively Rosenberg, 1982. Thus, in the
present study, the possibility of technological change occurring over the time period in which technical
efficiency is being measured cannot be ruled out. In such a case, technical efficiency is being measured
against a moving technical frontier see Fig. 2. Over time a farmer may shift from points A
1
on A
1
A
′ 1
to A
20
A
′ 20
, while his true frontier may also be shift- ing from F
1
F
′ 1
to F
20
F
′ 20
, and with it, the optimum point of economic efficiency from E
1
to E
20
. Such situations would require different interpretations of
efficiency improvement, and require decomposition of the changes due to the shifts in the farm’s efficiency
and those due to technological change.
4. Data and estimation
The data are drawn from a cost of cultivation project conducted by the Tamil Nadu Agricultural
University. For empirical analysis, data from irrigated
302 K.P. Kalirajan, R.T. Shand Agricultural Economics 24 2001 297–306
and non-irrigated paddy farms from North Arcot dis- trict, India, were used. Within the region, farms with
reasonably homogeneous land and equipment were sampled. The data covers the 10-year period from
1973 to 1982, and each year has two seasons of similar crops which give a total number of twenty periods.
1
Sample farmers were following the high yielding seed technology which is otherwise called the ‘Green
Revolution’ technology. They were cultivating the high yielding variety of paddy, IR20, since 1970. In
each crop period, longitudinal data from 25 irrigated and 25 non-irrigated farmers were selected for the
survey.
The following farm-specific stochastic Cobb- Douglas production frontier was separately estimated
for each crop season:
2
ln y
it
= α
+ X
α
jt
ln x
ijt
+ β
1t
ln x
4t
+ u
it
+ ν
it
, i =
1, 2, . . . , n observations, t =
1, 2, . . . , T periods 1
where y is the observed paddy output in kg, x
1
the pre-harvest labour in man-days, x
2
the fertiliser in kg, x
3
animal power pair-days, x
4
the area operated in acres, multiplied by a soil fertility index. This is
considered here as a fixed input, e
u
the firm-specific technical efficiency defined above, u the non-positive
random variable representing the combined effects of farm-specific characteristics that influence technical
efficiency and ν is a statistical random variable.
It is assumed that u follows a truncated normal distribution with mean µ and variance σ
2 u
and ν fol- lows a normal distribution N 0, σ
2 ν
. The maximum likelihood methods of estimation is used to obtain
estimates of the parameters of the maximum possible
1
Published data on later years are at aggregated level and firm-level data on inputs and outputs are not available. Permission
from the Ministry of Agriculture to obtain firm-level data from the Tamil Nadu Agricultural University could not be obtained.
2
Alternative functional forms such as translog and quadratic were tried, but because of high ¯
R
2
values and the number of significant variables, the Cobb–Douglas form was preferred for
further analysis. In addition, in the translog form, all null hypothe- ses of linear and non-linear separabilities could not be rejected at
the 5 level. Apparently, complete global separability could not be rejected. Thus, the Cobb–Douglas function can be considered
as an appropriate model describing the technology for the given data set.
Table 1 Calculated mean technical efficiencies of sample farmers from
North Arcot, Tamil Nadu, 1973–1982 Sl. no.
Crop season TE
Year Season no.
Irrigated Non-irrigated
1 1973
1 0.6782
0.6315 2
2 0.6810
0.6280 3
1974 1
0.6796 0.6246
4 2
0.6820 0.6300
5 1975
1 0.6853
0.6352 6
2 0.6726
0.6285 7
1976 1
0.6792 0.6310
8 2
0.6817 0.6389
9 1977
1 0.6856
0.6410 10
2 0.6921
0.6420 11
1978 1
0.6997 0.6435
12 2
0.7003 0.6460
13 1979
1 0.7126
0.6482 14
2 0.7285
0.6501 15
1980 1
0.7392 0.6513
16 2
0.7452 0.6533
17 1981
1 0.7486
0.6582 18
2 0.7488
0.6587 19
1982 1
0.7492 0.6587
20 2
0.7498 0.6591
output frontier of Eq. 1 and mean technical effi- ciencies Kalirajan and Shand, 1994.
3
The mean technical efficiency for each crop period is calculated
as follows Table 1: Ee
u
i
= 8 σ
u
− λ
σ
u
8 −λ
σ
u −
1
exp−λ + σ
2 u
where λ = σ
u
σ
ν
and 8 is the standard normal dis- tribution function evaluated at the point given in the
brackets. Allocative efficiency for the output of each farmer
for each crop season is derived first by simultaneously solving the firm-specific observed production function
and the marginal productivity conditions yielding the optimum output and variable inputs, and second, by
calculating the ratio of observed profit to the optimum
3
The mean parameter estimates of the frontier production func- tion for the sample participants in the irrigated and non-irrigated
environments, respectively, are: 1 lnpaddy=3.1628+0.2432 lnlabor+0.1712
lnfertiliser+ln 0.0804 lnanimal
power+ 0.5301 lnland 2 lnpaddy=2.4627+0.2263 lnlabor+0.1328
lnfertiliser+0.1269 lnanimal power+0.5185 lnland.
K.P. Kalirajan, R.T. Shand Agricultural Economics 24 2001 297–306 303
maximum profit: α
1
ln x
1
+ α
2
ln x
2
+ α
3
ln x
3
− ln y
= − β
1
ln x
4
− α
− u
ln x
1
− ln y = ln α
1
− ln p
1
− ln p
y
ln x
2
− ln y = ln α
2
− ln p
2
− ln p
y
ln x
3
− ln y = ln α
3
− ln p
3
− ln p
y
2
There are four equations in four unknowns x
1
, x
2
, x
3
and y; the production parameters α , α
1
, α
2
, α
3
, u and β
1
are MLE estimates of Eq. 1. The calculated optimal output y
∗∗
, along with the concerned opti- mal inputs x
∗ 1
, x
∗ 2
, x
∗ 3
, and their relevant prices are used to work out the maximum profit. The ratio of
the observed to the above optimum profit for each observation is calculated for each crop season. A
simple average of these ratios for each crop season is worked out which then serves as a measure of
allocative efficiency for that particular crop season Table 2.
Now, economic efficiency is calculated. First, the following equations representing the frontier produc-
tion function and the marginal productivity conditions are simultaneously solved:
Table 2 Calculated mean allocative efficiencies of sample farmers from
North Arcot, Tamil Nadu, 1973–1982 Sl. no.
Crop season AE
Year Season no.
Irrigated Non-irrigated
1 1973
1 0.7526
0.7610 2
2 0.7625
0.7680 3
1974 1
0.7820 0.7815
4 2
0.7650 0.7520
5 1975
1 0.7852
0.7785 6
2 0.7980
0.7910 7
1976 1
0.8005 0.8120
8 2
0.8056 0.8156
9 1977
1 0.8120
0.8200 10
2 0.8178
0.8215 11
1978 1
0.8250 0.8270
12 2
0.8289 0.8305
13 1979
1 0.8325
0.8340 14
2 0.8410
0.8400 15
1980 1
0.8592 0.8485
16 2
0.9001 0.8920
17 1981
1 0.9025
0.9010 18
2 0.9185
0.9125 19
1982 1
0.9310 0.9420
20 2
0.9526 0.9510
Table 3 Calculated mean economic efficiencies of sample farmers from
North Arcot, Tamil Nadu, 1973–1982 Sl. no.
Crop season EE
Year Season no.
Irrigated Non-irrigated
1 1973
1 0.5008
0.4703 2
2 0.5072
0.4736 3
1974 1
0.5206 0.4792
4 2
0.5162 0.4636
5 1975
1 0.5216
0.4812 6
2 0.5269
0.4896 7
1976 1
0.5381 0.5013
8 2
0.5402 0.5185
9 1977
1 0.5486
0.5204 10
2 0.5530
0.5296 11
1978 1
0.5623 0.5318
12 2
0.5765 0.5386
13 1979
1 0.5828
0.5397 14
2 0.6010
0.5401 15
1980 1
0.6125 0.5462
16 2
0.6532 0.5737
17 1981
1 0.6612
0.5816 18
2 0.6785
0.5936 19
1982 1
0.6818 0.6182
20 2
0.6987 0.6206
α
1
ln x
1
+ α
2
ln x
2
+ α
3
ln x
3
− ln y
= − β
1
ln x
4
− α
ln x
1
− ln y = ln α
1
− ln p
1
− ln p
y
ln x
2
− ln y = ln α
2
− ln p
2
− ln p
y
ln x
3
− ln y = ln α
3
− ln p
3
− ln p
y
3
These equations are similar to those in System 2. The only difference is that in the first equation in System
3, u=0, which implies that it represents the potential of the technology and not the actually realised pro-
duction function as in Eq. 2. The solution to Eq. 3 gives the optimal output and inputs evaluated at the
frontier using the price information. Then the optimal net returns or profit is calculated. Finally, the ratio of
the realised profit to the above optimal profit evalu- ated at the frontier is calculated to provide a measure
of economic efficiency Table 3.
5. Results
