A Net Profit Approach to Productivity Measurement, with an Application to Italy by Carlo Milana Istituto di Studi e Analisi Economica, Rome, Italy

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A Net Profit Approach to Productivity
Measurement, with an Application to Italy by
Carlo MilanaIstituto 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.

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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
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Unit cost of productionwith constant returns to
scale
C(w,y) c(w) y
Average cost
___
c(w1) c(w0)


E
A
C(w0,y1)
C(w0,y0)
G
F
y
y1
y0
Output
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Unit costs of production withnon-constant
returns to scale
C(w,y) c(w) g(y)
Average cost
C(w1,y0) C(w0,y0)
C(w1,y1) C(w0,y1)
gt
C(w1,y1)
C(w1,y0)
C(w0,y1)
C(w0,y0)
Inverse of MFP
y
y1
y0
Output
5
Superlative index numbersThe
Translog-Törnqvist casewith 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

Caves, Christensen, and Diewert (1982) have
shown that
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
Paasche-type economic indexes are completely
offset in the geometric average procedure.
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Unit cost of production withnon-constant returns
to scale
C(w,y)
Average cost
Non-invariant index number
Diseconomies of scale
y1
y
y0
Output
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Superlative index numbersThe
Translog-Törnqvist casewith 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

Caves, Christensen, and Diewert (1982) have shown
that
Moreover, if the Translog cost function has also
the second-order terms in w affected by y, then
(see, Milana, 2005)
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Homothetic case
In the homothetic case we always have

Paasche
Ideal Fisher
Laspeyres
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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.

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General non-homothetic case

0
1
)
,
(
y
w
C

type
Laspeyres

True"
"
Tr
0
0
)
,
(
y
w
C
Tr
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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 non-homothetic 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
  the 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.

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Empirical evidence (I)

Table 1. Alternative Measures of TFP Changes
Based on Different Cost Functions (in
percentage) All industries in the Italian economy
Year Implicit Laspeyres (direct Paasche) (1) Implicit Konüs-Byushgens (ideal Fisher) (2) 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)
1971 0.65 0.47 0.48 0.30 2.20 0.35
1972 -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
Strong nonhomothetic changes
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Empirical evidence (I)

Table 1. (Continued) Alternative Measures of
TFP Changes Based on Different Cost Functions (in
percentage)
All industries
in the Italian economy
Year Implicit Laspeyres (direct Paasche) (1) Implicit Konüs-Byushgens (ideal Fisher) (2) 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)
1989 0.29 0.26 0.26 0.24 1.23 0.05
1990 -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
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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.)

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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 ?t(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
Tty,(-x) defined in the space of output and
input quantities.

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The Net Profit Approach (II)
Lets 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

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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 t0 and t1, as seen from the
perspective of situation t0, is Similarly, we
could define the relative rate of change in TFP
with respect to the comparison situation t1.

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The Net Profit Approach (IV)

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The Net Profit Approach (V)

Normalized net profit change
Laspeyres-type relative price change component
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The Net Profit Approach (VI)

If
and
then
Relative price change component
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Some empirical results (I)

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Empirical resultsMain 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 .

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Empirical resultsMain 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.

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Empirical resultsMain 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.
• 
  
• 
  Carlo Milana
• 
  ISAE, Rome, Italy