This paper provides an empirical investigation into the long-standing notion of biases of technical change in South African agriculture. The second order cost function is used to derive relative bias measures between labour and machinery. The results suggest that large machinery-using biases in technology have been developed with minimal labour-using biases.Farm Management, Research and Development/Tech Change/Emerging Technologies,

Thirtle, Colin G.

Townsend, Rob F.

van Zyl, Johan

English

Research Papers in Economics

Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  492
MACHINERY AND LABOUR BIASES OF TECHNICAL 
CHANGE IN SOUTH AFRICAN AGRICULTURE: A COST 
FUNCTION APPROACH 
 
R.F. Townsend
1, J. van Zyl
2 and C. Thirtle
3 
 
 
 
This paper provides an empirical investigation into the long-standing notion of biases of 
technical change in South African agriculture. The second order cost function is used to 
derive relative bias measures between labour and machinery. The results suggest that large 
machinery-using biases in technology have been developed with minimal labour-using biases. 
 
MASJIENERIE- EN ARBEIDSVOORKEURNEIGINGS IN TEGNIESE 
VERANDERING IN DIE SUID-AFRIKAANSE LANDBOU : 'N KOSTEFUNKSIEBE-
NADERING 
 
Hierdie referaat verskaf 'n empiriese ondersoek na die langstaande opvatting van 
voorkeurneigings in tegniese verandering in die Suid-Afrikaanse landbou.  Die tweede orde 
kostefunksie word gebruik om relatiewe voorkeur metings tussen arbeid en masjienerie te 
bepaal.  Die resultate impliseer dat groot masjienerie-gebruikende voorkeurneigings in 
tegnologie, met minimale arbeids-gebruikende voorkeurneigings, ontwikkel het. 
 
1. INTRODUCTION 
 
Since Hicks (1932), the notion of biases of technical change has received much 
attention in the literature (for example, Binswanger, 1974; Blackorby et al, 1976; 
Stevenson, 1980; Antle & Capalbo, 1988). According to Hicks, techniques 
designed to facilitate the substitution of other inputs for labour in production 
are ‘labour-saving technologies’. In agriculture this generally means mechanical 
technology. This paper examines technical change in South African agriculture 
with specific focus on whether technologies developed have been  machinery or 
labour-using. A cost function approach will be used in the analysis. This 
approach is described in the next section. As time series data are used in the 
analysis, the time-series properties of the variables are established to determine 
the appropriate representation of the technology variable. The Kako 
decomposition method is then used to derive biases of technical change 
                                              
1   Department of Agricultural Economics, Extension and Rural Development, University of 
Pretoria. 
2   Vice-chancellor and Principal, University of Pretoria. 
3   Department of Agricultural Economics, University of Reading and Department of 
Agricultural Economics, Extension and Rural Development, University of Pretoria. Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  493
between machinery and labour.  
 
2.  THE COST FUNCTION APPROACH 
 
The cost function is a common approach used to measure the bias of technical 
change (Binswanger, 1974; Kako, 1978; Stevenson, 1980; Archibald & Brandt, 
1991; and Machado, 1995). Allowing for the possibility of quasi-fixity in inputs, 
a temporary equilibrium framework can be formulated and the variable cost 
function can be expressed as the minimisation of 
 
  cv cv w Z Y = (,,)  ( 1) 
 
where w is the vector of variable input price, Z is a vector of quasi-fixed and 
fixed inputs and Y is the vector of output levels. A common functional form 
used in cost function analysis is the transcendental logarithmic (translog) form 
which is considered to be a flexible representation of the production system and 
underlying technology. Functions that provide second-order numerical or 
differential approximations to some unobservable underlying function are 
called flexible functional forms. 
 
The second-order translog form for m outputs, n inputs, k fixed and quasi-fixed 
inputs and an index of technology t is specified as 
 
 
ln ln ln ln
ln ln ln ln ln ln
ln ln ln ln ln ln
ln ln ln
CV Y w Z
YY ww ZZ
Yw wZ YZ
t t tY tw tZe
Yi
i
m
ii
i
n
ii
i
k
i jjj
i
m
i
m
i jjj
i
n
i
n
i jjj
i
k
i
k
ij j j
i
n
i
m
ij j j
i
k
i
n
ij j j
i
k
i
n
ti t Y t i
i
m
ti
i
n
Zt i
i
k
t
i
ii i
=+ + +
++ +
++ +
++ + + + +
∑∑∑
∑ ∑∑ ∑∑ ∑
∑ ∑∑ ∑∑ ∑
∑∑∑
αα α β
γγδ
ρρ
φφ φ φ φ
0
2
1
2
1
2
1
2
1
2
Π
 (2) 
 
and the associated input share equations are
4; 
 
  Sw Y Z t e ii i j j
i
n
ij j
i
m
ij j
j
k
it i =+ + + + + ∑∑∑ αγ ρ ρ φ ln ln ln ,         i =1,...,n (3) 
 
                                              
4  The share equations are derived as the partial derivatives of the variable cost function with 
regard to input prices. The implicit demand functions are Hicksian (output-compensating). Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  494
and the output share equations (where the share of output i is given as the 
value of output i over variable costs)
5; 
 
  Sw Y Z t e ii Y j j
i
n
ij j
i
m
ij j
j
k
it i i =+ + + + + ∑∑ ∑ αρ γ φ ln ln ln , Π          i =1,...,m (4) 
 
The national farm-level production data for the  period 1947-91 is described in 
some detail in Thirtle et al. (1993).  These data are also used in this analysis. One 
divisia aggregated output group and four variable input groups were used in 
the analysis. The inputs were labour, machinery, fertiliser and miscellaneous 
inputs. The quasi-fixed inputs used were land, capital (fixed capital in the form 
of buildings and other formal improvements) and livestock capital (stocks of 
animals). 
 
Prior to estimation of the cost function, the time-series properties of the 
variables need to be examined to avoid possible spurious regressions arising 
with trended data. Technology in equation (2) is represented by a time-trend. 
This representation has been common to many studies (Binswanger, 1974; 
Stevenson, 1980). This may lead to OLS coefficient estimates that will be 
inefficient. Significance levels will be inflated and there will be a high 
probability of concluding that there is a significant relationship among the 
variables when, in fact, no relationship exists (Clark & Youngblood, 1992). If the 
variables are, however, integrated of order one, then cointegration among the 
variables implies that technical change is neutral.  
 
3.  DETERMINING HICKS NEUTRAL TECHNICAL CHANGE 
 
Clark and Youngblood show a more general specification of the factor shares in 
equation (3) as 
 
  Sw Y Z a e ii i j j
i
n
Yj j
i
m
Zji j
j
k
it t =+ + + ++ ∑∑∑ αγ ρ ρ ln ln ln                   i=1,..,n (5) 
 
where ait is the factor specific technological change stochastic process. If ait is 
omitted due to the absence of a direct measure of technical change, then the 
error term will become (ait + et) in equation (5). The technical change bias can 
then be determined for the ith factor to be: 
 
 1)  factor  i using if Δait >0,  
                                              
5  These are derived (in a similar manner to the variable input share equations) as the 
derivatives of the cost function in respect of output levels. Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  495
  2)   neutral if Δait=0, 
  3)   factor i saving if Δait<0. 
 
The statistical properties of Sit in equation (5) can suggest different technical 
change biases: 
 
i)  Sit is stationary around a linear deterministic time-trend and all elements 
of w and Z are stationary. In this case ait=bit + eit, where t is the time-
trend and eit is white noise.  
 
ii) Both  Sit and all the elements of w and Z contain a single unit root. 
Following this, the variables may be: 
 
•  cointegrated of order (1,1). In this case the error term, and hence ait is 
stationary, thus representing neutral technical changes. 
   
•  non-cointegration exist between the variables. Thus, ait  is I(1), 
resulting in technical change bias. 
 
In order to investigate the time-series properties of the data for South Africa, the 
four-branch decision tree represented in Table 1 of Clark & Youngblood 
(1992:356) is used. Two of the four factor shares can be adequately represented 
by a trend stationary process, together with the labour, machinery, 
miscellaneous prices and the output variable. The fertiliser and machinery 
share equations and the quasi-fixed inputs follow a random walk, with fertiliser 
price being represented by a random walk with drift. As two of the four factor 
share variables are trend stationary, together with two of the factor prices, the 
use of a time-trend to capture technical change effects can be regarded as a 
reasonable approximation to the correct (unknown) duality model (Machado, 
1995). Thus, the traditional equation (3) and (4) are used for estimation. 
 
These share equations were estimated using the iterative Zellner procedure. 
The symmetry condition follows from the assumption of twice continuous 
differentiability of the cost function in prices.  Symmetry requires γij=γji. 
Similarly, symmetry is imposed on the quasi-fixed and fixed input coefficients 
(δij=δji). The theoretical requirement of linear homogeneity of the cost function 
in prices implies the following constraints: 
 Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  496
 
αφ
γγ
ρρ
i
i
n
it
i
n
ij ji
i
n
i
n
Yj
j
n
Zj
j
n
jm
im ii
==
== =
== =
==
= =
==
∑∑
∑ ∑
∑∑
10
01
00 1
11
1 1
11
;,
,
, , ,...,
  
                    ,...,  
               
 (6) 
The representation of technology with a time-trend allows the derivation of the 
appropriate indices of biased technical change. Binswanger (1974) derived a 
bias measure in his analysis of US agriculture that captured the effects of 
technical change on factor shares which can be expressed as 
 
  BY W t
SYWt
t
i
c i
(, , )
ln ( , . )
=
∂
∂
 (7) 
‘Constant price factor shares’ were used to derive indices of factor-using 
technical change bias which were derived residually as the shares that would 
have existed in the absence of factor price changes. Equation (7) can be written 
as  
 
  BY W T
XYWT
T
S
XYWT
T
t
c i
j
j
n j
(, ,)
ln (, ,) l n (, ,)
=−
= ∑
∂
∂
∂
∂ 1
 (8) 
 
Following Machado (1995) the right-hand side terms may be interprets as: 
 
(i)  the proportional variation in input use due to technical change; 
 
(ii)  the proportional variation of the (Tornqvist-Theil) index of aggregate 
input use due to technical change. 
 
Thus, this equation provides an indication of how technical change affects the 
ratio between the quantities of input i and an aggregate measure of all inputs 
over time. In this analysis, a biased measure that is equivalent to Binswanger’s 
constant price factor share indices is calculated using the Kako decomposition 
(1978). 
 
4.  THE KAKO DECOMPOSITION 
 
This section follows the approach of Kako (1978), which is used by Machado 
(1995). Under cost-minimising behaviour, the equilibrium quantity demanded 
is given as: Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  497
  XF Y W T i n i = ( , , )           = ,..., 1  (9) 
 
Differentiating this expression yields 
 
  ∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ln
ln
ln
ln
ln
ln
ln
ln
ln
ln X
X
Y
Y
X
W
W
X
T
T i
ii
j j
j
i
=+ + ∑   (10) 
 
The first term on the right-hand side represents the output effect on the change 
in demand, the second term represents the substitution effect of a change in 
demand and the last term represents the technical change effect on a change in 
demand for a particular factor of production.  The first two effects from the 
translog cost function can be represented as: 
 
 
∂
∂
∂
ρ
∂
ln
ln
ln . ln
X
Y
Y
S
RkY
i Y
i
Y
i
=+  (11) 
 
which is the output effect, where RY can be approximated by the ratio between 
output value and total cost, and  
 
 
∂
∂
∂σ ∂
ln
ln
ln ln
X
W
WS W
i
j j
jj i jj
j ∑∑ =  (12) 
 
which is the substitution effect. σij is the Allen partial elasticity of substitution 
between input i and  j. 
 
The technical change effect can then be determined residually by rearranging 
and subtracting the estimated output and substitution effects in equation (10). 
The resulting residual effect reflects the percentage variation in the use of input i 
which was caused by technical change. 
 
 
X
X
X
T
T
i
TC
i
TC
i
.
ln
ln
ln =
∂
∂
∂  (13) 
 
Indices of input use due to technical change could then be derived. These are 
then aggregated via the common Tornqvist-Theil approximation to the divisia 
index, resulting in an index of aggregate input use due to technical change, 
XTC. Finally the absolute measures of technical change bias can be derived as Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  498
  B
X
X
i
i
TC
TC =  (14) 
 
This bias measure can be interpreted as an index of relative (to all) factor use 
due to technical change. 
 
5.  ESTIMATION AND RESULTS 
 
The system of share equations (3) and (4) were estimated as a seemingly 
unrelated system, without the miscellaneous input share, to avoid singularity, 
and subject to the restrictions of symmetry and homogeneity. Thus, there are 
three input share equations, namely for labour, machinery and fertiliser, and 
one output share equation for aggregate output. The parameters derived solely 
from the share equations allow tests of all the theoretical consistency conditions. 
The theoretical properties of the variable cost function are non-decreasing in 
variable factor prices, non-increasing in quasi-fixed factors, non-decreasing in 
output, homogeneous of degree one in variable factor prices, symmetric in the 
Hessian of variable factor prices, concave in variable factor prices, convex in 
quasi-fixed factors and convex in output quantities (Khatri, 1994). Tests 
indicated that these theoretical properties were satisfied by the model. Thus, 
there is a good expectation of the cost function being well behaved in respect of 
inputs, outputs and quasi-fixed inputs. The corresponding machinery and 
labour biases of technical change were then computed.  The resulting indices of 
relative factor biases for machinery and labour are compared graphically in 
Figure 1. 
 
These indices do not vary, as the indices derived for the US by Binswanger 
(1974), and are more consistent with those derived by Machado (1995). The 
results should be treated with caution as, with all residual based tests, there 
could be some unaccounted noise which has been included in the technical 
change bias value which may distort the results. This spillover effect is likely to 
be present in the current estimates due to the important influence of the 
exogenous variables described in the previous section. Nevertheless, these 
results provide some consistency with the results derived in other studies (see 
Townsend & Thirtle, 1996), showing that technology developments have been 
more machinery using relative to labour due to technical change. This bias is 
mostly positive, especially from 1960 to 1987. The decline in 1987 coincides with 
the removal of the policy distortions favouring mechanisation.  
 Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
  499
-3
-2
-1
0
1
2
3
4
5
1948 1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990
Year
I
n
d
e
x
 
Figure 1:  Index of relative technical change bias between machinery and 
labour  
 
6. CONCLUSION 
 
This paper has attempted to provide some empirical investigation into the 
long-standing notion of biases of technical change in South African 
agriculture. The second order cost function is used to derive relative bias 
measures between labour and machinery. The results suggest that large 
machinery-using biases in technology have been developed with minimal 
labour-using biases. These biases have not contributed to alleviating the 
unemployment problem currently faced in the labour-surplus economy of 
South Africa. The biases have largely been caused by policies favouring the 
large-scale capital-intensive production model. With removal of these biased 
policies, the bias of technical change towards machinery-using technology 
should be reduced. 
 
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technological change: An application to Japanese agriculture. Journal of Agrekon, Vol 36, No 4 (December 1997)    Townsend, Van Zyl & Thirtle 
 
 
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BLACKORBY, C., LOVELL, C.A.K. & TURSBY, M.C. (1976). Extended Hicks 
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STEVENSON, R. (1980). Measuring technological bias. American Economic 
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Estimating duality models with biased technical change : A time series approach.

Extended Hicks neutral technological change.

Measuring technological bias.

Technical change and the returns to research in UK agriculture, 1953-1990. Unpublished PhD Thesis,

Testing the induced innovation hypothesis using cointegration analysis.

The theory of wages. 1st ed. London :

Total factor productivity in South African agriculture, 1947-1991. Development Southern Africa.

Machinery and labour biases of technical change in South African agriculture: A cost function approach

Machinery and labour biases of technical change in South African agriculture: A cost function approach

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