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”