A Net Profit Approach to Productivity
Measurement, with an Application to Italy
by Carlo Milana
Istituto di Studi e Analisi Economica, Rome, Italy

This presentation has been prepared for the OECD Workshop on productivity
measurement, 16-18 October, 2006, Bern, Switzerland.
1
Contents
I. Measurement problems with non-invariant index numbers
II. Empirical evidence in Italy
III. Finding a better approach with the normalized profit function
IV. An application to Italy
V. Conclusion
2
Unit cost of production
with constant returns to scale
C(w,y) = c(w) · y
Average cost
C(w,y)
y
C(w1,y1)
C(w1,y0)
=
C(w0,y1) =
C(w0,y0)
Invariant index number
(with respect to y)
c(w1)
___
c(w0)
B
D
C(w1,y0)
C(w1,y1)
E
A
C(w0,y0)
C(w0,y1)
Inverse of MFP
F
y0
G
y1
y
Output
3
Unit costs of production with
non-constant returns to scale
C(w,y) = c(w) · g(y)
Average cost
C(w,y)
y
Non-invariant index number
(with respect to y)
C(w1,y0)
C(w1,y1)
>
C(w0,y0)
C(w0,y1)
C(w1,y1)
C(w1,y0)
C(w0,y1)
Effects on unit cost from
diseconomies of scale
C(w0,y0)
Inverse of MFP
y0
y1
Output
y
4
“Superlative” index numbers
The Translog-Törnqvist case
with C(w,y)
= c(w) · g(y)
Diewert (1976) has shown that if the cost function has a Translog
functional form, y affects only the first-order terms in w, then




w
CTr w , ( y y )



i
0
0 1 1/ 2
CTr w , ( y y )
w
1
0 1 1/ 2
1
i
0
i



1 0 1
( s i  si )
2
Törnqvist index number
Caves, Christensen, and Diewert (1982) have shown that
1
2
w
 CTr ( w , y ) CTr ( w , y ) 

  i 

0
0
0
1 
 CTr ( w , y ) CTr ( w , y ) 
w
1
0
1
1
1
i
0
i



1 0 1
( s i  si )
2
In the case of homothetic separability in y, this price index is a pure
price component of cost changes because, under the hypotheses
made, the non-invariance elements of the Laspeyres- and Paasche5
type economic indexes are completely offset in the geometric average
procedure.
Unit cost of production with
non-constant returns to scale
C(w,y)
Average cost
C(w,y)
y
C(w1,y0)
C(w1,y1)
>
C(w0,y0)
C(w0,y1)
Non-invariant index number
C(w1,y1)
C(w1,y0)
C(w0,y1)
Diseconomies of scale
C(w0,y0)
Inverse of MFP
y0
y1
Output
y
6
“Superlative” index numbers
The Translog-Törnqvist case
with the general case of C(w,y)
Diewert (1976) has shown that if the cost function has a Translog
functional form, y affects only the first-order terms in w, then




w
CTr w , ( y y )



i
0
0 1 1/ 2
CTr w , ( y y )
w
1
0 1 1/ 2
1
i
0
i



1 0 1
( s i  si )
2
Caves, Christensen, and Diewert (1982) have shown that
1
2
w
 CTr ( w , y ) CTr ( w , y ) 

  i 

0
0
0
1 
 CTr ( w , y ) CTr ( w , y ) 
w
1
0
1
1
1
i
0
i



1 0 1
( s i  si )
2
Törnqvist index number
Moreover, if the Translog cost function has also the second-order terms
in w affected by y, then (see, Milana, 2005):

 CTr ( w , y )   CTr ( w , y ) 

0
0  
0
1 
C
(
w
,
y
)
C
(
w
,
y
)
 Tr
  Tr
1
0
1
1
(1  )
w
  i 
w
1
i
0
i



1 0 1
( si  si )
2
7
Homothetic case
In the homothetic case we always have
C ( w1 , y1 ) c( w1 )
Paasche " True" Paasche - type index

C ( w0 , y1 ) c( w0 )
Paasche
Ideal Fisher
Laspeyres
1
" True" Laspeyres - type index
0
1
C ( w , y ) c( w )

 Laspeyres
0
0
0
C ( w , y ) c( w )
The ratio c(w1)/c(w0) falls into the interval between Paasche and Laspeyres
index numbers. The ideal Fisher is just one of the points belonging
to this interval. The “true” index may be equal to
c( w1 )

1

(
L

P
) with 0    1.
0
c( w )
8
General non-homothetic case
• In the non-homothetic case economic index numbers are non-invariant
(this is because it is not possible to disentangle univocally the mutual effects
of variables)
• If we deflate a nominal value by means of a non-invariant price index number
the resulting implicit quantity index is not in general homogeneous of degree 1
(if, for example, the elementary quantities double, in general the quantity index
does not double).
• This undesirable behaviour is related to an anomalous position of the “true”
index number with respect to the Laspeyres and Paasche index numbers.
9
General non-homothetic case
•In the nonhomothetic case, we might have the following reverse position
Geometric mean of the " True" Laspeyres - and Paasche
- type
1
1
2
 CTr ( w1 , y 0 ) CTr ( w1 , y1 )  2


0
0
0
1 
 CTr ( w , y ) CTr ( w , y ) 
 CTr ( w1 , y1 ) CTr ( w1 , y1 ) 

 Tornqvist in the case of a translog CTr )

0
1
0
1 
C
(
w
,
y
)
C
(
w
,
y
)
Tr
 Tr

CTr ( w1 , y1 )
" True" Paasche - type
CTr ( w0 , y1 )
•Laspeyres Ideal Fisher Paasche•
" True" Laspeyres type
•
CTr ( w1 , y 0 )
The Ideal Fisher is expected to be
CTr ( w 0 , y 0 )
closer tha n Paasche (and Laspeyres)
to the geometric mean of the two
" true" index numbers!
10
General non-homothetic case
Since a geometric average of two non-invariant economic index numbers is
generally non-invariant with respect to reference variables, the
“superlative” index numbers are also non-invariant in the non-homothetic
case.
While the price economic index number is linearly homogeneous by
construction, in general the corresponding quantity index number fails to
satisfy the linear homogeneity requirements in the non-homothetic case.
(see, for example, Samuelson and Swamy, 1974, Diewert, 1983, p. 179).
Samuelson and Swamy (1974, p. 576) observed that, in the general nonhomothetic case, the corresponding quantity index obtained implicitly by
deflating the nominal cost by means of the economic price index fails to
satisfy the requirements of the linear homogeneity test.
Samuelson and Swamy (1974, p. 570) noted: “[t]he invariance of the price
index is seen to imply and to be implied by the invariance of the quantity
index from its reference price base”.
11
Empirical evidence (I)
Table 1. Alternative Measures of TFP Changes Based on Different Cost Functions (in percentage)
All industries in the Italian economy
Implicit
Generalized
Leontief
(3)
Implicit
Paasche
(direct
Laspeyres)
(4)
Direct
Paasche/Direct
Laspeyres
ratio
(5) = (1)/(4)
Difference
between
direct Paasche
and direct
Laspeyres
(6) = (1) - (4)
0.47
0.48
0.30
2.20
0.35
-1.33
-1.49
-1.8
-1.64
0.82
0.30
1973
2.93
2.86
2.86
2.78
1.05
0.15
1974
1.95
1.79
1.78
1.64
1.19
0.32
1975
-3.30
-3.45
-3.44
-3.61
0.91
0.31
1976
1.51
1.46
1.46
1.41
1.07
0.11
1977
-0.61
-0.65
-0.65
-0.68
0.89
0.07
1978
-0.06
-0.12
-0.12
-0.17
0.34
0.11
1979
-0.82
-0.93
-0.93
-1.05
0.78
0.23
1980
0.58
0.35
0.35
0.12
4.86
0.46
1981
-1.46
-1.50
-1.50
-1.54
0.94
0.09
1982
-0.70
-0.71
-0.71
-0.72
0.97
0.02
1983
0.17
0.14
0.14
0.12
1.35
0.04
1984
0.22
0.21
0.21
0.19
1.15
0.03
1985
1.68
1.66
1.66
1.63
1.03
0.05
1986
0.60
0.64
0.64
0.68
0.88
-0.08
1987
0.56
0.49
0.49
0.43
1.32
0.14
1988
1.00
0.98
0.98
0.95
1.05
0.05
Year
Implicit
Laspeyres
(direct
Paasche)
(1)
Implicit
KonüsByushgens
(ideal Fisher)
(2)
1971
0.65
1972
Strong
nonhomoth
changes
12
Empirical evidence (I)
Table 1. (Continued) Alternative Measures of TFP Changes Based on Different Cost Functions (in percentage)
All industries in the Italian economy
Implicit
Generalized
Leontief
(3)
Implicit
Paasche
(direct
Laspeyres)
(4)
Direct
Paasche/Direct
Laspeyres
ratio
(5) = (1)/(4)
Difference
between
direct Paasche
and direct
Laspeyres
(6) = (1) - (4)
0.26
0.26
0.24
1.23
0.05
-0.32
-0.35
-0.35
-0.38
0.83
0.06
1991
-0.34
-0.31
-0.31
-0.28
1.23
-0.06
1992
0.93
0.89
0.88
0.84
1.11
0.09
1993
0.94
0.94
0.94
0.94
1.00
0.00
1994
1.65
1.64
1.64
1.63
1.01
0.02
1995
1.20
1.20
1.20
1.21
0.99
-0.02
1996
-0.26
-0.26
-0.26
-0.26
1.00
0.00
1997
0.54
0.52
0.52
0.50
1.07
0.03
1998
-0.29
-0.30
-0.30
-0.30
0.97
0.01
1999
-0.08
-0.09
-0.09
-0.10
0.79
0.02
2000
0.73
0.63
0.62
0.53
1.36
0.19
2001
-0.31
-0.31
-0.31
-0.31
0.98
0.01
2002
-0.34
-0.34
-0.34
-0.35
0.96
0.01
2003
-0.42
-0.42
-0.42
-0.42
0.99
0.00
Year
Implicit
Laspeyres
(direct
Paasche)
(1)
Implicit
KonüsByushgens
(ideal Fisher)
(2)
1989
0.29
1990
13
The Net Profit Approach (I)
The basic idea is to find an unrestricted function where there
are no reference variables.
We build on the seminal research of Diewert and Morrison
(1986) and Kohli (1990), who used the restricted revenue
function to measure the terms-of-trade component of welfare
change.
We base our developments on the theory of profit functions.
(See Lawrence J. Lau, “Profit Functions of Technologies with
Multiple Inputs and Outputs”, Review of Economics and
Statistics, August 1972, Vol. 54, no. 3, pp. 281-289.)
14
The Net Profit Approach (I)
The function should exhibit some desirable properties,
such as differentiability, homogeneity, and separability
with respect to other variables.
A possible candidate is the net profit function Πt(p,w)
which can be considered as a transformation function
in the space of output and input prices for a given
profit value. It is dual to the transformation function
Tt[y,(-x)] defined in the space of output and input
quantities.
15
The Net Profit Approach (II)
Let’s start with the simplest model of one output (y)
and one input (x) of a price-taking firm producing
under constant returns to scale and facing the
output price (p) and the input price (w) in perfectly
competitive markets. Productivity (TFP) is defined
as
y
TFP 
x
16
The Net Profit Approach (III)
The aim is to provide a measure of the relative rate of technical
change (productivity net of scale effect).
Under constant returns to scale (no scale effect) and perfect
competition, in a one-output, one-input model of production, the
relative rate of productivity or technical change (TFPG0) between
t=0 and t=1, as seen from the perspective of situation t=0, is
1
0
0
1 0
0 1


y
y
y
y
x

y
x
0
TFPG   1  0  / 0 
0 1
x
x
x
y
x


Similarly, we could define the relative rate of change in TFP with
respect to the comparison situation t=1.
17
The Net Profit Approach (IV)
1
0
0
1 0
0 1
1 0
0 1
1
1


y
y
y
y
x

y
x
y
x

y
x
w
y
TFPG0   1  0  / 0 

 1/ 1
0 1
0 1
y x
y x
p x
x x  x
y1 x 0  y 0 x1 w1

 1
0 1
y y
p
 x1 x 0  w1
1
1
1
1
1
   1  0   1  1 1  ( p , w )  1 0  0 ( p1 , w1 )
py
py
 y y  p
~ 1 ~ 1 ~ 0 ~1
 [ (1, w )   (1, w )]
Normalized net profit function
In general,
1 w y
 
k p x
where k is the degree
of RS. With CRS: k=1
k is generally unknwon
*
w
y
However,

p
x
with w* = w / k
if w* rather than w
is observed.
18
The Net Profit Approach (V)
TFPG 0  [
1
1
1
1
1
0
1
1

(
p
,
w
)


(
p
,
w
)]
1 1
1 0
p y
p y
[
[
1
1
1
1
1
0
0
0

(
p
,
w
)


(
p
,
w
)]
1 1
0 0
py
p y
1
1
0
1
1
0
0
0

(
p
,
w
)


(
p
,
w
)]
1 0
0 0
py
p y
0

x
 [( p1 y1  w1 x1 ) / p1 y1  ( p 0 y 0  w0 x 0 ) / p 0 y 0 ]   0
y
Normalized net profit change
 w0 w1  
 0  1 
p 
p
Laspeyres-type relative
price change component19
The Net Profit Approach (VI)
If
then

  11 p  212 w   22 w
t

1
2 2
and
1
1
~t ~
t
t
~
 (1, w)   (1, w) t   ( p, w) t
y
py
~0
~1
0


x

x1   w0 w1 
0,1
P   ~ 0 ~1  0  ~ 0 ~1  1  0  1 
p 
   y    y   p
Relative
and technica l change  [( p1 y1  w1x1 ) / p1 y1  ( p0 y 0  w0 x0 ) / p0 y 0 ]  P0,1
Normalized net profit change
Relative price
change component
20
Some empirical results (I)
Figure 4. Measures of effects of TFP growth on real factor prices,
based on the GL and KB cost functions
Cost-based measure of TFPG and its components
All industries in the Italian economy
2.0%
NK = Non-ICT fixed capital
ICT = ICT fixed capital
L = Labour
1.5%
E = Energy inputs
M = Materials
E
M = Relative
changeinputs
in real prices of non energy materials
S = Service
1.0%
TFP = Total factor productivity
TFP
L
0.5%
L
L
M
S
L
S
NK IC
E
M
S
NK ICT TFP
E
ICT
M
NK ICT TFP
E
M
S
NK
TFP
0.0%
-0.5%
-1.0%
2000
2001
2002
2003
Figure 5. Measures of effects of TFP growth on real factor prices,
based on the GL and KB profit functions
Profit-based measure of TFPG and its components
All industries in the Italian Economy
2.0%
NK = Non-ICT fixed capital
E
ICT = ICT fixed capital
L = Labour
1.5%
E = Energy inputs
M = Materials
S = Service inputs
TFP
1.0%
TFP = Total factor productivity
M
0.5%
M
L
L
NK ICT
0.0%
S
L
E
M
S
NK ICT
TFP
E
M
S
L
NK ICT TFP
E
M
S
NK ICT
TFP
-0.5%
-1.0%
2000
2001
2002
2003
21
Empirical results
Main conclusions (I)
Italy has had some special reasons to be concerned about productivity of
the economy. The high public debt and the unresolved north-south
regional divide require sustained growth in production.
Many factors seem to constrain economic activities, including highly
regulated markets and protective institutional setting in favour of
incombents.
While empirical studies have concluded that the US, for example, appear
to have constant or slightly decreasing returns to scale thank to a
relatively free capacity adjustment to the new opportunities of growth,
decreasing returns to scale may be more dominant in Italy .
22
Empirical results
Main conclusions (II)
• One important element of productivity growth in Italy is technical
change (TFP net of the scale economies or diseconomies).
• Scale diseconomies seem to affect the internal structure of production.
• Non-homotheticity appear to prevail over the whole period 1970- 2003,
except three years.
• Non-constant returns to scale are not neutral, thus bringing about
rather strong asymmetric changes in the composition of production and
in the use of factor inputs.
23
Empirical results
Main conclusions(III)
Towards “TFP growth accounting”
• The
negative trend in productivity noted recently in this country, almost
disappear with the proposed measure.
• Future steps in our nonparametric productivity measurement will be towards
the completion of “TFP growth accounting” by correcting our proposed
measure for other main components, as for example, market power, cyclical
behaviour, externalities, adjustments, technical and organizational inefficiency.
•
Volunteers joining the company are welcome!
• Critical comments are invited.
24