Transcript Company Meeting Title

Alternative Measures of Productivity in a
Changing Structure of Production
by Carlo Milana
Istituto di Studi e Analisi Economica, Rome, Italy

This presentation has been prepared for the workshop of the EUKLEMS project
organized in Brussels, on the 16 and 17 March 2007.
1
Overview
I. Defining the problem: Building blocks of economic theory of
index numbers
II. Measurement problems with non-invariant index numbers
III. Empirical evidence in Italy
IV. Upper and lower bounds in the case of only two observations
V. Afriat’s tight bounds in a multilateral context
VI. Conclusion
2
Special reference will be made to Afriat’s theory
of economic index numbers
Afriat, S.N. (1956), “Theory of Economic Index Numbers”, Research Report,
Department of Applied Economics, University of Cambridge, UK.
____ (1967), “The Construction of a Utility Function from Expenditure Data”,
International Economic Review 8(1): 67-71.
____ (1972), “The Theory of International Comparisons of Real Income and
Prices”, in D.J. Daly (ed.by), International Comparisons of Prices and Output,
NBER, Studies in Income and Wealth Volume 37, New York, 1972, Ch. 1, pp.
13-84.
____ (1977), The Price Index, Cambridge, UK, Cambridge University Press.
____ (1981), “On the Constructability of Consistent Price Indices Between
Several Periods”, in A. Deaton (ed. by), Essays in the Theory and
Measurement of Consumer Behaviour. Cambridge University Press, pp. 13361.
3
References to recent contributions
Sydney N. Afriat (2005), The Price Index and Its Extensions—A Chapter in
Economic Measurement, Forward by Angus Deaton, London and New York,
Routledge.
Carlo Milana (2005), “The Theory of Exact and Superlative Index Numbers
Revisited”, EUKLEMS Working Paper no. 3.
Afriat, S.N. and C. Milana (2007a), “The Super Price Index: Irving Fisher and
After”, University of Siena, Quaderno no. 492.
Afriat, S.N. and C. Milana (2007b), “Price-Level Computation Method”, paper
prepared for the EUKLEMS project.
Afriat, S.N. and C. Milana (2007c), “Price-Level Computation: An Illustration”,
paper prepared for the EUKLEMS project.
4
Building blocks of economic index numbers (1)
The notion of “exact” index numbers: Byushgens (1925), KonüsByushgens (1926), Samuelson (1947)(1984).
The factorization theorem and the invariance (existence) of aggregating
index number: Shephard (1953) in the economic theory of production and
Afriat (1956)(1972) in the economic theory of consumption.
The index number problem: the “impossibility theorem” on Fisher’s
(1922)-Frisch (1930) test criteria: (i) linear homogeneity; (ii) timereversal test; (iii) circularity or transitivity test; (iv) dimensional test; (v)
(weak) factor reversal test: Samuelson (1974), Samuelson and Swamy
(1974, pp. 571-575).
Making the “impossible” possible : Irrespective of its functional
form, the “exact” index number for the existing “true” index always
satisfies all Fisher-Frisch test criteria (including the transtivity or
circularity test): Samuelson and Swamy (1974, pp. 571-575).
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At least in theory, following Afriat (1972) and
Samuleson and Swamy (1974),
No “smoothing out” inconsistency problems with
transitivity requirements, for example by applying
traditional methods like the EKS and CDD, is needed
with economic index numbers in the canonical
homothetic case.
What is needed is a preliminary test for consistency of
data with homotheticity assumptions.
6
Building blocks of economic index numbers (2)
What if the chosen index number formula does not respect the
circularity or transitivity test?
Violation of this test could be due to at least one of the following reasons:
(i) The observed data cannot be rationalized by an underlying homothetic
utility or technology function (non-homothetic case);
(ii) The economic agents are not optimizing their choices (inefficiency
case);
(iii) The chosen index number formula is not exact for the underlying utility
or technology function (wrong-formula case).
As for the reason (i), a consistency test could be devised in order to determine
whether the data can in fact be rationalized by an optimizing behaviour governed
by a canonical utility or technology function. If the data consistency test is
passed, then we can still determine upper and lower bounds for the “true”
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unknown economic index.
Building blocks of economic index numbers (3)
Since the “true” index number (when this exists) is generally unknown, any
index number formula should be abandoned since the unknown value may
be any point between the Laspeyres (L) and Paasche (P) indexes.
This conclusion was already reached by Afriat (1977, p. 112):
“Any ambiguity about “true index” […] shows no effect in the unambiguous
result that the set of values is in any case identical with the Paasche-Laspeyres
Interval. The “true” points are just the points in that interval and no others;
and none is more true than another. There is no sense to a point in the interval
being a better approximation to “the true index” than others. There is no
proper distinction of “constant utility” indices, since all these points have
that distinction”.
And, again, Afriat (2005, p. xxiii) states:
“Let us call the LP interval the closed interval with L and P as upper and lower
limits, so the LP-inequality is the condition for this to be non-empty. While every
true index is recognized to belong to this interval, it can still be asked what points
in this interval are true? The answer is all of them, all equally true, no one more
true than another. When I submitted this theorem to someone notorious in this
subject area it was received with complete disbelief.
“Here is a formula to add to Fisher’s collection, a bit different from the others.
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“Index Formula: Any point in the LP-interval, if any.”
The “true” index number
Afriat’s (2005, p. xxiii) index formula was obtained in an explicit form by
Milana (2005) as follows:
where
is a parameter that is related to a combination of the residual
of a first-order approximation to the underlying economic functions
at the two observation points, while
In the homothetic case,
P is a Laspeyres index, whereas when
When
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P is a Fisher “ideal” index.
P is a Paasche index. Finally, when
Building blocks of economic index numbers (4)
In addition to the foregoing results, Milana (2005) found that Diewert’s and Fisher’s
superlative indexes cannot understandably be defined as, in some way,
approximating the actual but unknown true index number up to the second order
in order to admit them as “superlative” in the language of Diewert (1976, 1978).
The Törnqvist index (corresponding to formula with code number 123 in Fisher,
1922, p. 473) is seen as the most superlative by Caves, Christensen, and Diewert
(1982, p. 41), whereas it was not deemed "superlative" by Fisher (1922, p. 247), wh
classified it, in a descending order of merit, below the classes of "excellent" and
"superlative" index numbers, with the last group ranked at the top position.
Samuelson and Swamy (1974, p. 585) found that, in the general
non-homothetic case where the Afriat factorization conditions are violated,
“[…] It is evident that the Ideal [Fisher] index cannot give high-powered approxima
to the true index in the general, nonhomothetic case. A simple example will illustrate
the degree of this failure […]. “Even if (P1,P0,Pα) and (Q1,Q0) are ‘sufficiently close
together,’ it is not true that the Laspeyres and Paasche indexes provide two-sided
bounds for the true index. In this example, the true index lies outside the Laspeyres
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-Paasche interval!”
Homothetic case
In the homothetic case we always have
C ( w1 , y1 ) c( w1 )
Paasche " True" Paasche - type index

C ( w0 , y 1 ) 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 )
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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
•In the nonhomothetic case, we might have the following reverse position
Geometric mean of the " True" Laspeyres - and Paasche
- type
1
 CTr ( w1 , y1 ) CTr ( w1 , y1 ) 



0
1
C
(
w
,
y
)
CTr ( w0 , y1 ) 
 Tr
1
2
 CTr ( w1 , y 0 ) CTr ( w1 , y1 )  2



0
0
C
(
w
,
y
)
CTr ( w0 , y1 ) 
Tr

 Tornqvist in the case of a translog CTr )
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!
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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 nonhomothetic case, the corresponding quantity index obtained implicitly by
deflating the nominal cost by means of the economic price index fails to be
linearly homogeneous.
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”.
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Empirical evidence
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
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Empirical evidence
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
16
The upper bound of the aggregate input
price index with two observations
Factor input 2
B
A
O
Factor input 1
17
Extending a bilateral comparison to a
multilateral context (Afriat, 1981)
Factor input 2
B
C
A
O
Factor input 1
18
Afriat’s upper and lower bounds
Let
(1)
represent the Laspeyres price index number where the subscripts i and j denote the base and
the comparison situations, respectively. Then
(2)
is the Paasche index number
Then, let us define the chained Laspeyres and chained Paasche indexes between
 

the reference and current situations,denoted
with s and t, respectively as
 max respectively
   
sij ...kt
ij...k
si
ij
kt
(3)
(4)
Afriat (1981) defined the following minimum-path chained Laspeyres and maximum
-path Paasche indexes, which are tight upper and lower bounds of the “true” index:
(5)
19
(6)
Related contributions
Afriat’s minimum (maximum) path chained Laspeyers (Paasche) index numbers are
very close to the following contributions combined together:
Dale W. Jorgenson and Zvi Griliches (1967), “The Explanation of Productivity Change”, Review of Economic
Studies 34: 249-283.
Dale W. Jorgenson and Zvi Griliches (1971), “Divisia Index Numbers and Productivity Measurement”, Review of
Income and Wealth , pp. 227-229, who wrote: “The main advantage of a chain index is the reduction of errors
of approximation as the economy moves from one production configuration to another. […]
The Laspeyres approximation to the Divisia index of total factor productivity was employed in our original
study of productivity change (1967)”
Samuelson and Swamy (1974, pp. 576) where it is stated that “Fisher missed the point made in
Samuelson (1947, p. 151) that knowledge of a third situation can add information relevant to
the comparison of two given situations”
Kruskal's Minimum Spanning Tree recently proposed. See
Hill, Robert J. (1999a), "Comparing Price Levels across Countries Using Minimum Spanning Trees", Review of
Economics and Statistics 81: 135-142.
Hill, Robert J. (1999b), "International Comparisons using Spanning Trees", in A. Heston and
R.E. Lipsey (eds.), International and Interarea Comparisons of Income, Output, and Prices, Studies in Income
and Wealth, Volume 61, NBER, Chicago: The University of Chicago Press.
Hill, Robert J. (2001), "Linking Countries and Regions using Chaining Methods and Spanning
Trees", prepared for the Joint World Bank-OECD Seminar on Purchasing Power Parities
-Recent Advances in Methods and Applications, Washington, D.C., 30th January-2nd February, 2001.
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Laspeyres and Paasche matrices with
Afriat’s tight upper and lower bound matrices

 1

 1
L
P   12
1

 L13
 1
L
 14
 1 L12 L13 L14 
L
1 L23 L24 
21

L
 L31 L32 1 L34 


 L41 L42 L43 1 
1
L12


L21
1
M 
 ( L32 L21 )
L32

( L43 L32 L21 ) ( L43 L32 )
L12 L23
L23
1
L43
1
L21
1
1
L23
1
L24
L12 L23 L34 
( L23 L34 ) 
L34 

1

So that we have the following range of
possible values:
1
L31
1
L32
1
1
L34
1 
L41 

1 
L42 
1 

L43 

1 

1
1
1 

 1
L21 L32 L21 L43 L32 L21 


1
1 
 1
1
 L12
L32
L43 L32 
H 
1
1
1 
1


L23
L43 
 L12 L23
 1

1
1
1 
L L L L L
 12 23 34 23 34 L34

Prs  H rs  True measure  M rs  Lrs
21
An example from S. Dowrick and John Quiggin,
“True Measures of GDP and Convergence”,
American Economic Review, March 1997, pp. 41-64
Constant price and Afriat’s bounds on per capita GDP ratios, 1980
_____________________________________________________________________________________________
L: matrix of Laspeyres ratios (log values)
Canada
1.
2.
3.
4.
5.
Canada
US
Norway
Luxembourg
…
0.000
0.017
0.118
0.183
…
US
-0.009
0.000
0.120
0.132
…
Norway
Luxembourg
-0.032
0.017
0.000
0.114
…
-0.063
-0.062
0.034
0.000
…
…
…
…
…
…
…
Lower bound
M: Afriat’s minimum path matrix/true indexes (log values)
Upper bound
Canada
1.
2.
3.
4.
5.
Canada
US
Norway
Luxembourg
…
0.000
0.017
0.118
0.148
…
Column value means upper bound
(-) row value means lower bound
US
-0.009
0.000
0.109
0.132
…
Norway
Luxembourg
-0.055
-0.044
0.000
0.088
…
-0.097
-0.086
0.022
0.000
…
…
…
…
…
…
…
22
Conclusion
Main steps towards the construction of Afriat’s tight bounds of the
unknown “true” index:
Step 1: Construct/revise an appropriate economic-theoretic model of
demand (supply) behaviour
Step 2: Test consistency of the data with a well behaved utility or
technology function (if the test is passed, then go to step 3;
if not, then go to step 1).
Step 3: Construct the matrix of Laspeyres and Paasche indexes.
Step 4: Devise an efficient algorithm to derive the matrix of minimum
(maximum) path of chained Laspeyres (Paasche) indexes.
Step 5: Derive Afriat’s tight bounds from the results of Step 4.
23