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Unformatted text preview: Deﬁning Changes in Productivity
Measuring Technological Change Measuring Changes in Productiity: XXII
Charles B. Moss1
1 University of Florida November 15, 2011 Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change 1 Deﬁning Changes in Productivity 2 Measuring Technological Change
Total Factor Productivity and Index Number Theory
Distance Functions Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Deﬁning Changes in Productivity y1 y1
y1 Yx
Yx p2 y2 y2
Charles B. Moss p1 y2 Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change In this ﬁgure, we assume that the inputs are constant at x ,
but the total level of outputs has increased from (y1 , y2 ) to
(y1 , y2 ). Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Change in Productivity  Reduced Inputs x1
x1
x1 x2 x2
Charles B. Moss x2
Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change In both cases, most would agree that a technical change has
taken place.
Further, most would agree that the technical change has
increased the economic well being of society.
We now have more stuﬀ for the same level of inputs. However, there are some issues that need to be raised. Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change First, one could raise the question about embodied versus
disembodied technical change.
This debate regards whether there has been an increase in
knowledge, or whether there has been an increase in the
quality of inputs.
If the increase in output has been associated with an increase
in quality of an input, is it technical change?
For example, a large portion of the gains to research literature
can be traced to Griliches discussion of hybrid corn.
In this case, the increase in technology was associated with the
improvement in an input – seed.
More recently, some of the most recently observed increases in
productivity may be traced to genetically modiﬁed organisms
(GMOs).
Under most concepts of productivity, these increases do not
represent changes in productivity in agriculture, but can be
traced to changes in the input bundle.
Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change A second area of concern is whether the changes in
technology are neutral with regard to the input bundle.
Going back to the ﬁgure, the increase in technology implies
relatively more x1 it is biased toward x1 . x1
x1
x1 x2 x2
Charles B. Moss x2
Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Continued
In the hybrid corn example, additional fertilizer complemented
the use of hybrid corn.
In the classic discussion, Hicks developed the notion of labor or
capital augmenting technological development. Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions Measuring Technological Change
In the single inputsingle output analysis, one could directly
measure technical change:
y
y
y = f ( x , t ) ⇒ θ ( t ) = ⇒ t = θ −1
(1)
x
x
Several factors should be considered.
We know that proﬁtmaximizing behavior changes the point of
production even in the univariate case.
We know that the decision maker chooses to produce where
the marginal value product equals the price of the input.
Thus, if either the price of the input or the price of the output
has changed, the ratio of outputs to inputs will change. Even in the single variable case, we would wonder about
excluded factors, things beyond the decision makers control. Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions Extending the analysis to the multivariate world, we begin by
examining the productproduct relationship in the ﬁrst graph.
Note
θ (t ) =
Y (x )
p 1 y1 + p 2 y 2
=
Y (x )
p 1 y1 + p 2 y 2 (2) could be used as one measure of technical change.
Similarly, in the inputinput relationship:
θ (t ) =
V (y )
w 1 x1 + w 2 x2
=
V (y )
w 1 x1 + w 2 x2 Charles B. Moss Measuring Changes in Productiity: XXII (3) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions Note ﬁrst that each of these formulations is based implicitly
on Shephard duality:
θ (t ) = = Y (x )
π (p , w , t = 1r )
=
Y (x )
π (p , w , t = 0)
p1 y1 (p , w , t = 1) + p2 y2 (p , w , t = 1)
p2 y1 (p , w , t = 0) + p2 y2 (p , w , t = 0) V (y )
c (w , y , t = 1)
θ (t ) =
=
V (y )
c (w , y , t = 0)
=
w1 x1 (w , y , t = 1) + w2 x2 (w , y , t = 1)
w1 x1 (w , y , t = 0) + w2 x2 (w , y , t = 0) Charles B. Moss Measuring Changes in Productiity: XXII (4) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions More formally,
c (w , y , t ) = min w x : x ∈ V (y , t )
x >0
⇔ V (y , t ) = x : w x ≥ c (w , y , t ) , w > 0
∗ (5) Thus, by gross simpliﬁcation, we could envision a cost function
1
c (w , y , t ) = α0 + α w + w Aw + β +
2
1
y By + w Γy + θ (w , y , t )
2
x (w , y , t ) = α + AW + Γy + ∇w θ (w , y , t )
with θ (w , y , t ) being a measure of technical change.
Charles B. Moss Measuring Changes in Productiity: XXII (6) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions This formulation allows us to discuss several key features of
technology measurement.
First, in the grossest sense, technological change tends to be a
measurement of factors that we don’t understand.
From the preceding equation, what is the diﬀerence between
technology and a residual?
One approach is to proxy technical change with a simple time
trend, t .
Alternatively, several studies have used other proxy variables
such as spending on agricultural research. This formulation allows the researcher to examine the
neutrality of technical change
∂ c (w , y , t )
xi ( w , y , t )
xi ( w , y )
∂ wi
=
=
∂ c (w , y , t )
xj ( w , y , t )
xj ( w , y )
∂ wj
Charles B. Moss Measuring Changes in Productiity: XXII (7) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions Finally, it is possible to envision adjusting this measure for
diﬀerences in input quality.
For example, if the quality of one variable increases over time,
then we could adjust the price of that variable upward to
account for the increase in quality. Charles B. Moss Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions Total Factor Productivity and Index Number Theory
The index number approach can be looked at as an extension
of the single inputsingle product scenario above
dy
∂ f (x , t ) dx
∂ f (x , t )
=
+
dt
∂x
dt
∂t
Replacing diﬀerentiation with log diﬀerences
y = f (x , t ) ⇒ d ln [y ] ∂ ln [y ] d ln [xi ]
=
+ T (x , t )
dt
∂ ln [xi ] dt
i
d ln [xi ]
=
i
+ T (x , t )
dt
i
wi xi d ln [xi ]
=
+ T (x , t )
dt
py
i
d ln [y ] wi xi d ln [xi ]
⇒ T (x , t ) =
−
dt
dt
py
i Charles B. Moss Measuring Changes in Productiity: XXII (8) (9) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions This formulation is sometimes approximated as the
TornqvistTheil measure T (x , t ) = ln [yt ]−ln [yt −1 ]− N
1
i =1 2 [Vit + Vi ,t −1 ] [ln [xit ] − ln [xi ,t −1 ]]
(10) Working backward
d ln [y ] = d ln [x ] + T (x , t )
⇒ d ln [y ] − d ln [x ] = T (x , t )
⇒ d ln y x Qy
⇒ d ln
Qx
Charles B. Moss = T (x , t ) = T (x , t ) Measuring Changes in Productiity: XXII (11) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions In the TornqvistTheil index, the indices were implicitly Divisia
output and input indices
Qy =
i w i xi
p i yi
, Qx =
p j yj
w j xj
i
j (12) j The linkage in this case is the deﬁnition of total factor
productivity TFP = y
y
˙
˙
⇒ TFP = =
x
x
˙ Charles B. Moss ∂ f (y , x , t ) dyi
∂ yi
dt
i ∂ f (y , x , t ) dxi
∂ xi
dt Measuring Changes in Productiity: XXII (13) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions Distance Functions x1
x1
x1 x2 x2
Charles B. Moss x2
Measuring Changes in Productiity: XXII Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions These concepts actually deﬁne a distance function measure of
productivity growth.
Deﬁne a measure θ for a new technology based on the old
technology as
Dθ (x , y ) = min [θ : F (x θ) ≥ y ]
θ Charles B. Moss Measuring Changes in Productiity: XXII (14) Deﬁning Changes in Productivity
Measuring Technological Change Total Factor Productivity and Index Number Theory
Distance Functions x1
x1 x1 x2 x2
Charles B. Moss x2 Measuring Changes in Productiity: XXII ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.
 Fall '09
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