11
A Net Profit Approach to Productivity A Net Profit Approach to Productivity Measurement, with an Application to Italy Measurement, with an Application to Italy
byby Carlo MilanaCarlo MilanaIstituto di Studi e Analisi EconomicaIstituto di Studi e Analisi Economica, Rome, Italy, Rome, Italy
This presentation has been prepared for the OECD Workshop on productivity This presentation has been prepared for the OECD Workshop on productivity measurement, 16-18 October, 2006, Bern, Switzerland.measurement, 16-18 October, 2006, Bern, Switzerland.
22
ContentsContents
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
33
Unit cost of productionUnit cost of productionwith constant returns to scalewith constant returns to scale
C(w0,y0) C(w0,y1)
y
C(w1,y0)C(w0,y0)
C(w1,y1)C(w0,y1)= = c(w1)
c(w0)___
C(w,y) = c(w) · y
C(w,y)
y
A
BC(w1,y0) C(w1,y1)
D
E
y0 y1
F G
Inverse of MFP
Invariant index number (with respect to y)
Output
Average cost
C(w1,y0)C(w0,y0)
C(w1,y1)C(w0,y1)
Invariant index number (with respect to y)
44
Unit costs of production withUnit costs of production withnon-constant returns to non-constant returns to scalescale
y
C(w,y) = c(w) · g(y)
C(w,y)
y
y0 y1
C(w1,y1)
C(w0,y1)
C(w0,y0)
C(w1,y0)
C(w1,y0)C(w0,y0)
C(w1,y1)C(w0,y1)
>
Inverse of MFP
Effects on unit cost from diseconomies of scale
Non-invariant index number(with respect to y)
Average cost
Output
55
““Superlative” index numbersSuperlative” index numbersThe Translog-TThe Translog-Törnqvist caseörnqvist case
withwith C(w,y) = c(w) · g(y)
Diewert (1976)Diewert (1976) has shown that if the cost function has a has shown that if the cost function has a Translog Translog functional formfunctional form, , yy affects only the affects only the first-order terms in first-order terms in ww, then, then
)(2
1
0
1
2/1100
2/110110
)(,
)(,ii ss
i
ii
Tr
Tr
w
w
yywC
yywC
Caves, Christensen, and Diewert (1982) have shown that
)(2
1
0
12
1
10
11
00
0110
),(
),(
),(
),(ii ss
i
ii
Tr
Tr
Tr
Tr
w
w
ywC
ywC
ywC
ywC
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.
Törnqvist index number
66
Unit cost of production withUnit cost of production withnon-constant returns to non-constant returns to
scalescale
y
Non-invariant index number
C(w,y)
C(w,y)
y
Diseconomies of scale
C(w1,y1)
C(w0,y1)
C(w0,y0)
C(w1,y0)
C(w1,y0)C(w0,y0)
C(w1,y1)C(w0,y1)
>
Inverse of MFP
y0 y1
Average cost
Output
77
““Superlative” index Superlative” index numbersnumbers
The Translog-TThe Translog-Törnqvist caseörnqvist casewith the general case ofwith the general case of C(w,y)
Diewert (1976)Diewert (1976) has shown that if the cost function has a has shown that if the cost function has a Translog Translog
functional formfunctional form, , yy affects only the affects only the first-order terms in first-order terms in ww, then, then
)(2
1
0
1
2/1100
2/110110
)(,
)(,ii ss
i
ii
Tr
Tr
w
w
yywC
yywC
Caves, Christensen, and Diewert (1982) have shown that
)(2
1
0
12
1
10
11
00
0110
),(
),(
),(
),(ii ss
i
ii
Tr
Tr
Tr
Tr
w
w
ywC
ywC
ywC
ywC
Moreover, if the Translog cost function has also the second-order terms
in w affected by y, then (see, Milana, 2005): )(
2
1
0
1)1(
10
11
00
0110
),(
),(
),(
),(ii ss
i
ii
Tr
Tr
Tr
Tr
w
w
ywC
ywC
ywC
ywC
Törnqvist index number
88
Homothetic caseHomothetic case
Laspeyres
Paasche
In the homothetic case we always have
Laspeyres)(
)(
),(
),(index type-Laspeyres True""
0
1
00
01
wc
wc
ywC
ywC
)(
)(
),(
),(index type-Paasche True""Paasche
0
1
10
11
wc
wc
ywC
ywC
Ideal Fisher
The ratio falls into the interval between Paasche and Laspeyres
c(w1)/c(w0)index numbers. The ideal Fisher is just one of the points belonging to this interval. The “true” index may be equal to
.10 with )()(
)( 10
1
PLwc
wc
99
General non-homothetic General non-homothetic casecase
• 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.
1010
General non-homothetic General non-homothetic casecase
•In the nonhomothetic case, we might have the following reverse position
Laspeyres PaascheIdeal Fisher
),(
),( type-Paasche True""
10
11
ywC
ywC
Tr
Tr
• ••
2
1
10
11
00
01
),(
),(
),(
),(
ywC
ywC
ywC
ywC
Tr
Tr
Tr
Tr
) transloga of case in theTornqvist ),(
),(
),(
),(
type-Paasche and -Laspeyres True"" theofmean Geometric
2
1
10
11
10
11
TrTr
Tr
Tr
Tr CywC
ywC
ywC
ywC
numbers!index true""
two theofmean geometric theto
Laspeyres) (and Paaschen closer tha
be toexpected isFisher Ideal The
),(
),( typeLaspeyres True""
00
01
ywC
ywC
Tr
Tr
1111
Since a geometric average of two non-invariant economic index Since a geometric average of two non-invariant economic index numbers is generally non-invariant with respect to reference numbers is generally non-invariant with respect to reference variables, variables, the “superlative” index numbers are also non-invariant the “superlative” index numbers are also non-invariant in the non-homothetic case.in the non-homothetic case.
While the price economic index number is linearly homogeneous While the price economic index number is linearly homogeneous by construction, in general the corresponding by construction, in general the corresponding quantity index quantity index number fails to satisfy the linear homogeneity requirements in the number fails to satisfy the linear homogeneity requirements in the non-homothetic case.non-homothetic case. (see, for example, (see, for example, Samuelson and Swamy, Samuelson and Swamy, 1974, Diewert, 1983, p. 1791974, Diewert, 1983, p. 179).).
Samuelson and Swamy (1974, p. 576)Samuelson and Swamy (1974, p. 576) observed that, in the observed that, in the general non-homothetic case, the corresponding quantity index general non-homothetic case, the corresponding quantity index obtained implicitly by deflating the nominal cost by means of the obtained implicitly by deflating the nominal cost by means of the economic price index fails to satisfy the requirements of the linear economic price index fails to satisfy the requirements of the linear homogeneity test. homogeneity test.
Samuelson and Swamy (1974, p. 570) noted: Samuelson and Swamy (1974, p. 570) noted: “[t]he invariance of “[t]he invariance of
the price index is seen to imply and to be implied by the the price index is seen to imply and to be implied by the invariance of the quantity index from its reference price base”.invariance of the quantity index from its reference price base”.
General non-homothetic caseGeneral non-homothetic case
1212
Empirical evidence (I)Empirical evidence (I)
Table 1. Alternative Measures of TFP Changes Based on Different Cost Functions (in percentage)
All industries in the Italian economy
YearYear
Implicit Implicit LaspeyresLaspeyres
(direct (direct Paasche)Paasche)
(1)(1)
ImplicitImplicitKonüs-Konüs-
ByushgensByushgens(ideal Fisher)(ideal Fisher)
(2)(2)
ImplicitImplicitGeneralizedGeneralized
LeontiefLeontief(3)(3)
Implicit Implicit PaaschePaasche(direct (direct
Laspeyres)Laspeyres)(4)(4)
Direct Direct Paasche/Direct Paasche/Direct
Laspeyres Laspeyres ratioratio
(5) = (1)/(4)(5) = (1)/(4)
Difference Difference betweenbetween
direct Paaschedirect Paascheand direct and direct LaspeyresLaspeyres
(6) = (1) - (4)(6) = (1) - (4)
19711971 0.650.65 0.470.47 0.480.48 0.300.30 2.202.20 0.350.35
19721972 -1.33-1.33 -1.49-1.49 -1.8-1.8 -1.64-1.64 0.820.82 0.300.30
19731973 2.932.93 2.862.86 2.862.86 2.782.78 1.051.05 0.150.15
19741974 1.951.95 1.791.79 1.781.78 1.641.64 1.191.19 0.320.32
19751975 -3.30-3.30 -3.45-3.45 -3.44-3.44 -3.61-3.61 0.910.91 0.310.31
19761976 1.511.51 1.461.46 1.461.46 1.411.41 1.071.07 0.110.11
19771977 -0.61-0.61 -0.65-0.65 -0.65-0.65 -0.68-0.68 0.890.89 0.070.07
19781978 -0.06-0.06 -0.12-0.12 -0.12-0.12 -0.17-0.17 0.340.34 0.110.11
19791979 -0.82-0.82 -0.93-0.93 -0.93-0.93 -1.05-1.05 0.780.78 0.230.23
19801980 0.580.58 0.350.35 0.350.35 0.120.12 4.864.86 0.460.46
19811981 -1.46-1.46 -1.50-1.50 -1.50-1.50 -1.54-1.54 0.940.94 0.090.09
19821982 -0.70-0.70 -0.71-0.71 -0.71-0.71 -0.72-0.72 0.970.97 0.020.02
19831983 0.170.17 0.140.14 0.140.14 0.120.12 1.351.35 0.040.04
19841984 0.220.22 0.210.21 0.210.21 0.190.19 1.151.15 0.030.03
19851985 1.681.68 1.661.66 1.661.66 1.631.63 1.031.03 0.050.05
19861986 0.600.60 0.640.64 0.640.64 0.680.68 0.880.88 -0.08-0.08
19871987 0.560.56 0.490.49 0.490.49 0.430.43 1.321.32 0.140.14
19881988 1.001.00 0.980.98 0.980.98 0.950.95 1.051.05 0.050.05
Strongnonhomotheticchanges
1313
Empirical evidence (I)Empirical evidence (I)
Table 1. (Continued) Alternative Measures of TFP Changes Based on Different Cost Functions (in percentage) All industries in the Italian economy
YearYear
Implicit Implicit LaspeyresLaspeyres
(direct (direct Paasche)Paasche)
(1)(1)
ImplicitImplicitKonüs-Konüs-
ByushgensByushgens(ideal Fisher)(ideal Fisher)
(2)(2)
ImplicitImplicitGeneralizedGeneralized
LeontiefLeontief(3)(3)
Implicit Implicit PaaschePaasche(direct (direct
Laspeyres)Laspeyres)(4)(4)
Direct Direct Paasche/Direct Paasche/Direct
Laspeyres Laspeyres ratioratio
(5) = (1)/(4)(5) = (1)/(4)
Difference Difference betweenbetween
direct Paaschedirect Paascheand direct and direct LaspeyresLaspeyres
(6) = (1) - (4)(6) = (1) - (4)
19891989 0.290.29 0.260.26 0.260.26 0.240.24 1.231.23 0.050.05
19901990 -0.32-0.32 -0.35-0.35 -0.35-0.35 -0.38-0.38 0.830.83 0.060.06
19911991 -0.34-0.34 -0.31-0.31 -0.31-0.31 -0.28-0.28 1.231.23 -0.06-0.06
19921992 0.930.93 0.890.89 0.880.88 0.840.84 1.111.11 0.090.09
19931993 0.940.94 0.940.94 0.940.94 0.940.94 1.001.00 0.000.00
19941994 1.651.65 1.641.64 1.641.64 1.631.63 1.011.01 0.020.02
19951995 1.201.20 1.201.20 1.201.20 1.211.21 0.990.99 -0.02-0.02
19961996 -0.26-0.26 -0.26-0.26 -0.26-0.26 -0.26-0.26 1.001.00 0.000.00
19971997 0.540.54 0.520.52 0.520.52 0.500.50 1.071.07 0.030.03
19981998 -0.29-0.29 -0.30-0.30 -0.30-0.30 -0.30-0.30 0.970.97 0.010.01
19991999 -0.08-0.08 -0.09-0.09 -0.09-0.09 -0.10-0.10 0.790.79 0.020.02
20002000 0.730.73 0.630.63 0.620.62 0.530.53 1.361.36 0.190.19
20012001 -0.31-0.31 -0.31-0.31 -0.31-0.31 -0.31-0.31 0.980.98 0.010.01
20022002 -0.34-0.34 -0.34-0.34 -0.34-0.34 -0.35-0.35 0.960.96 0.010.01
20032003 -0.42-0.42 -0.42-0.42 -0.42-0.42 -0.42-0.42 0.990.99 0.000.00
1414
The Net Profit Approach (I)The Net Profit Approach (I)
The basic idea is to find an unrestricted function where The basic idea is to find an unrestricted function where there are no reference variables.there are no reference variables.
We build on We build on the seminal research of the seminal research of Diewert and Morrison Diewert and Morrison (1986)(1986) and and Kohli (1990)Kohli (1990), who used the , who used the restricted revenue restricted revenue functionfunction to measure the terms-of-trade component of to measure the terms-of-trade component of welfare change.welfare change.
We base our developments on the We base our developments on the theory of profit functionstheory of profit functions. .
(See(See Lawrence J. Lau, “Profit Functions of Technologies with Lawrence J. Lau, “Profit Functions of Technologies with Multiple Inputs and Outputs”, Multiple Inputs and Outputs”, Review of Economics and Review of Economics and Statistics, Statistics, August 1972, Vol. 54, no. 3, pp. 281-289.)August 1972, Vol. 54, no. 3, pp. 281-289.)
1515
The Net Profit Approach (I)The Net Profit Approach (I)
The function should exhibit some desirable The function should exhibit some desirable properties, such asproperties, such as differentiability, homogeneity, differentiability, homogeneity, and separability with respect to other variablesand separability with respect to other variables..
A possible candidate is theA possible candidate is the net profit function net profit function ΠΠtt((p,wp,w)) which can be considered as a which can be considered as a transformation function in the space of output and transformation function in the space of output and input prices for a given profit value. It is dual to input prices for a given profit value. It is dual to thethe transformation function transformation function TTtt[[yy,(-,(-xx)])] defined in the defined in the space of output and input quantities.space of output and input quantities.
1616
The Net Profit Approach (II)The Net Profit Approach (II)
Let’s start with the simplest model of Let’s start with the simplest model of one output (one output (yy)) and and one input (one input (xx)) of a price-taking firm producing under constant returns to of a price-taking firm producing under constant returns to scale and facing the scale and facing the output price output price ((pp)) and the and the input price (input price (ww)) in perfectly competitive markets in perfectly competitive markets.. Productivity ( Productivity (TFPTFP) is defined as) is defined as
x
yTFP
1717
The Net Profit Approach (III)The Net Profit Approach (III)The aim is to provide a The aim is to provide a measure of the relative rate of technical measure of the relative rate of technical change change (productivity net of scale effect). (productivity net of scale effect).
Under constant returns to scale (no scale effect) and perfect Under constant returns to scale (no scale effect) and perfect competition, in a competition, in a one-output, one-input model of productionone-output, one-input model of production, the , the relative rate ofrelative rate of productivity or technical change (productivity or technical change (TFPGTFPG00)) between between tt=0=0 and and tt=1,=1, as seen from the perspective of situation as seen from the perspective of situation tt=0,=0, isis
Similarly, we could define the relative rate of change in Similarly, we could define the relative rate of change in TFPTFP with with respect to the comparison situation respect to the comparison situation tt=1=1..
10
1001
0
0
0
0
1
10 /
xy
xyxy
x
y
x
y
x
yTFPG
1818
The Net Profit Approach (IV)The Net Profit Approach (IV)
1
1
1
1
10
1001
10
1001
0
0
0
0
1
10 //
x
y
p
w
xy
xyxy
xy
xyxy
x
y
x
y
x
yTFPG
1
1
10
1001
p
w
yy
xyxy
)]~,1(~
)~,1(~
[ 1011 ww
),(1
),(1 110
01111
111
1
0
0
1
1
wpyp
wpypp
w
y
x
y
x
where k is the degree of RS. With CRS: k=1
x
y
p
w
k
1In general,
k is generally unknwon
However,x
y
p
w
*
with w* = w / k
if w* rather than wis observed.
Normalized net profit function
1919
The Net Profit Approach (V)The Net Profit Approach (V)
)],(1
),(1
[ 11001
11111
0 wpyp
wpyp
TFPG
)],(1
),(1
[ 00000
11111
wpyp
wpyp
)],(1
),(1
[ 00000
11001
wpyp
wpyp
1
1
0
0
0
0000000111111 ]/)(/)[(
p
w
p
w
y
xypxwypypxwyp
Normalized net profit change Laspeyres-type relativeprice change component
2020
The Net Profit Approach (VI)The Net Profit Approach (VI)
IfIf
then
1
1
0
0
1
1
10
1
0
0
10
01,0
~~
~
~~
~
p
w
p
w
y
x
y
xP
and 1,0000000111111 ]/)(/)[(change l technicaRelative Pypxwypypxwyp
Normalized net profit changeRelative price change component
2 2
12
221211 wwpt and tt
ttt
pywp
yww
1),(
1)~,1()~,1(
~
2121
Some empirical results (I)Some empirical results (I)Figure 4. Measures of effects of TFP growth on real factor prices,
based on the GL and KB cost functions
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
2.0%All industries in the Italian economy
2000 2001 2002 2003
E
M
S LNK IC
TFP
E M S
L
NK ICT TFP E M
S
L
NK ICT TFP E M S
L
NK
ICT
TFP
E = Energy inputs
NK = Non-ICT fixed capital
ICT = ICT fixed capital
M = Relative change in real prices of non energy materials
L = Labour
S = Service inputs
M = Materials
TFP = Total factor productivity
Cost-based measure of TFPG and its components
Figure 5. Measures of effects of TFP growth on real factor prices, based on the GL and KB profit functions
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
2.0%
2000 2001 2002 2003
TFP
LNK ICT
E
M
M
S TFPE M S
L
NK ICT E MS
L
NK ICT TFP TFPE M S
L
ICTNK
All industries in the Italian Economy
NK = Non-ICT fixed capital
ICT = ICT fixed capital
L = Labour
E = Energy inputs
M = Materials
S = Service inputs
TFP = Total factor productivity
Profit-based measure of TFPG and its components
2222
Empirical resultsEmpirical resultsMain conclusions (I)Main conclusions (I)
Italy has had some special Italy has had some special reasons to be concerned about productivityreasons to be concerned about productivity of the economy. The high public debt and the unresolved north-south regional divide require sustained growth in production. 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 activitiesMany factors seem to constrain economic activities, including highly regulated markets and protective institutional setting in favour of incombents. , 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, 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 decreasing returns to scale may be more dominant in Italy . .
2323
Empirical resultsEmpirical results
Main conclusions (II)Main conclusions (II)
• One important element of productivity growth in Italy is One important element of productivity growth in Italy is technicaltechnical
changechange (TFP net of the scale economies or diseconomies). (TFP net of the scale economies or diseconomies). • Scale diseconomiesScale diseconomies seem to affect the internal structure of production.seem to affect the internal structure of production.
• Non-homotheticityNon-homotheticity appear to prevail over the whole period 1970- 2003, appear to prevail over the whole period 1970- 2003,
except three years. except three years.
• Non-constant returns to scale are not neutral, thus bringing aboutNon-constant returns to scale are not neutral, thus bringing about
rather strong rather strong asymmetric changes in the composition of production andasymmetric changes in the composition of production and
in the use of factor inputsin the use of factor inputs. .
2424
Empirical resultsEmpirical results
Main conclusions(III)Main conclusions(III)
• 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
Towards “TFP growth accounting”