11

Alternative Measures of Productivity in a Alternative Measures of Productivity in a Changing Structure of ProductionChanging Structure of Production

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 workshop of the EUKLEMS project This presentation has been prepared for the workshop of the EUKLEMS project organized in Brussels, on the 16 and 17 March 2007.organized in Brussels, on the 16 and 17 March 2007.

22

OverviewOverview

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

I. Defining the problem: Building blocks of economic theory of index numbers

VI. Conclusion

33

Special reference will be made to Afriat’s Special reference will be made to Afriat’s theory of economic index numberstheory of economic index numbers

Afriat, S.N. (1956), “Theory of Economic Index Numbers”, Research Afriat, S.N. (1956), “Theory of Economic Index Numbers”, Research Report, Department of Applied Economics, University of Cambridge, Report, Department of Applied Economics, University of Cambridge, UK.UK.

____ (1967), “The Construction of a Utility Function from Expenditure ____ (1967), “The Construction of a Utility Function from Expenditure Data”, Data”, International Economic Review International Economic Review 8(1): 67-71.8(1): 67-71.

____ (1972), “The Theory of International Comparisons of Real Income ____ (1972), “The Theory of International Comparisons of Real Income and Prices”, in D.J. Daly (ed.by), and Prices”, in D.J. Daly (ed.by), International Comparisons of Prices International Comparisons of Prices and Output, and Output, NBER, Studies in Income and Wealth Volume 37, New NBER, Studies in Income and Wealth Volume 37, New York, 1972, Ch. 1, pp. 13-84.York, 1972, Ch. 1, pp. 13-84.

____ (1977), ____ (1977), The Price Index, The Price Index, Cambridge, UK, Cambridge University Cambridge, UK, Cambridge University Press.Press.

____ (1981), “On the Constructability of Consistent Price Indices ____ (1981), “On the Constructability of Consistent Price Indices Between Several Periods”, in A. Deaton (ed. by), Between Several Periods”, in A. Deaton (ed. by), Essays in the Essays in the Theory and Measurement of Consumer BehaviourTheory and Measurement of Consumer Behaviour. Cambridge . Cambridge University Press, pp. 133-61.University Press, pp. 133-61.

44

References to recent References to recent contributionscontributions

Sydney N. Afriat (2005), Sydney N. Afriat (2005), The Price Index and Its Extensions—A Chapter The Price Index and Its Extensions—A Chapter in Economic Measurement, in Economic Measurement, Forward by Angus DeatonForward by Angus Deaton, , London and London and New York, Routledge.New York, Routledge.

Carlo Milana (2005), “The Theory of Exact and Superlative Index Carlo Milana (2005), “The Theory of Exact and Superlative Index Numbers Revisited”, EUKLEMS Working Paper no. 3.Numbers Revisited”, EUKLEMS Working Paper no. 3.

Afriat, S.N. and C. Milana (2007a), “The Super Price Index: Irving Fisher Afriat, S.N. and C. Milana (2007a), “The Super Price Index: Irving Fisher and After”, University of Siena, Quaderno no. 492.and After”, University of Siena, Quaderno no. 492.

Afriat, S.N. and C. Milana (2007b), “Price-Level Computation Method”, Afriat, S.N. and C. Milana (2007b), “Price-Level Computation Method”, paper prepared for the EUKLEMS project.paper prepared for the EUKLEMS project.

Afriat, S.N. and C. Milana (2007c), “Price-Level Computation: An Afriat, S.N. and C. Milana (2007c), “Price-Level Computation: An Illustration”, paper prepared for the EUKLEMS project. Illustration”, paper prepared for the EUKLEMS project.

55

Building blocks of economic index numbers (1) Building blocks of economic index numbers (1)

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 notion of “exact” index numbersThe notion of “exact” index numbers: : Byushgens (1925), KonByushgens (1925), Konüs-üs-Byushgens (1926), Byushgens (1926), Samuelson (1947)(1984).Samuelson (1947)(1984).

The index number problem: the “impossibility theorem” on Fisher’s The index number problem: the “impossibility theorem” on Fisher’s (1922)-Frisch (1930) test criteria(1922)-Frisch (1930) test criteria:: ((ii) linear homogeneity; () linear homogeneity; (iiii) time-) time-reversal test; (reversal test; (iiiiii) circularity or transitivity test; () circularity or transitivity test; (iviv) dimensional test; () dimensional test; (vv) ) (weak) factor reversal test(weak) factor reversal test: : Samuelson (1974),Samuelson (1974), Samuelson and Swamy Samuelson and Swamy (1974, pp. 571-575). (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).

66

At least in theory, following Afriat (1972) and At least in theory, following Afriat (1972) and Samuleson and Swamy (1974),Samuleson and Swamy (1974),

No “smoothing out” inconsistency problemsNo “smoothing out” inconsistency problems with with transitivity requirements, for example by transitivity requirements, for example by applying traditional methods like the EKS and applying traditional methods like the EKS and CDD, is needed with economic index numbers in CDD, is needed with economic index numbers in the canonical homothetic case.the canonical homothetic case.

What is needed is a preliminary What is needed is a preliminary test for test for consistencyconsistency of data with homotheticity of data with homotheticity assumptions.assumptions.

77

Building blocks of economic index numbers (2) Building blocks of economic index numbers (2)

What if the chosen index number formula does not respect What if the chosen index number formula does not respect the circularity or transitivity test?the circularity or transitivity test?

Violation of this test could be due to at least one of the following Violation of this test could be due to at least one of the following reasons:reasons:

(i)(i) The observed data cannot be rationalized by an underlying The observed data cannot be rationalized by an underlying homothetic utility or technology function homothetic utility or technology function (non-homothetic case)(non-homothetic case);;

(ii)(ii) The economic agents are not optimizing their choices The economic agents are not optimizing their choices (inefficiency case)(inefficiency case);;

(iii)(iii) The chosen index number formula is not exact for the The chosen index number formula is not exact for the underlying utility or technology function underlying utility or technology function (wrong-formula case)(wrong-formula case). . As for the reason (i), a consistency test could be devised in order to determine 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 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 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” passed, then we can still determine upper and lower bounds for the “true” unknown economic index. unknown economic index.

88

Building blocks of economic index numbers (3) Building blocks of economic index numbers (3)

Since the “true” index number (when this exists) is generally unknown, any Since the “true” index number (when this exists) is generally unknown, any index number formula should be abandoned since the unknown value may index number formula should be abandoned since the unknown value may be any point between the Laspeyres (L) and Paasche (P) indexes.be any point between the Laspeyres (L) and Paasche (P) indexes.

This conclusion was already reached by Afriat (1977, p. 112):This conclusion was already reached by Afriat (1977, p. 112):““Any ambiguity about “true index” […] shows no effect in the unambiguous 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 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; 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 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 being a better approximation to “the true index” than others. There is no proper distinction of “constant utility” indices, since all these points have proper distinction of “constant utility” indices, since all these points have that distinctionthat 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.“Index Formula: Any point in the LP-interval, if any.”

99

The “true” index numberThe “true” index number

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,

When P is a Laspeyres index, whereas when P is a Paasche index. Finally, when P is a Fisher “ideal” index.

Afriat’s (2005, p. xxiii) index formula was obtained in an explicit form by Milana (2005) as follows:

1010

Building blocks of economic index numbers (4) Building blocks of economic index numbers (4)

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), who 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.

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).

Samuelson and Swamy (1974, p. 585)Samuelson and Swamy (1974, p. 585) found that, in the general found that, in the general non-homothetic case where the Afriat factorization conditions are violated,non-homothetic case where the Afriat factorization conditions are violated, ““[…] […] It is evident that the Ideal [Fisher] index cannot give high-powered approximation It is evident that the Ideal [Fisher] index cannot give high-powered approximation to the true index in the general, nonhomothetic caseto the true index in the general, nonhomothetic case . A simple example will illustrate . A simple example will illustrate the degree of this failure […]. “Even if (P1,P0,Pthe degree of this failure […]. “Even if (P1,P0,Pαα) and (Q1,Q0) are ‘sufficiently close ) and (Q1,Q0) are ‘sufficiently close together,’ it is not true that the Laspeyres and Paasche indexes provide two-sided 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 Laspeyresbounds for the true index. In this example, the true index lies outside the Laspeyres-Paasche interval!”-Paasche interval!”

1111

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

1212

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.

1313

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

1414

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 be linearly homogeneous. economic price index fails to be linearly homogeneous.

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

1515

Empirical evidenceEmpirical evidence

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

1616

Empirical evidenceEmpirical evidence

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

1717

The upper bound of the aggregate input The upper bound of the aggregate input price index with two observationsprice index with two observations

A

B

Factor input 2

Factor input 1O

1818

Extending a bilateral comparison to a Extending a bilateral comparison to a multilateral context (Afriat, 1981)multilateral context (Afriat, 1981)

A

C

B

Factor input 2

Factor input 1O

1919

Afriat’s upper and lower boundsAfriat’s upper and lower boundsLet

represent the Laspeyres price index number where the subscripts i and j denote the base and the comparison situations, respectively. Then

is the Paasche index number

(1)

(2)

Then, let us define the chained Laspeyres and chained Paasche indexes between the reference and current situations, denoted respectively with s and t, respectively as

(3)

(4)

ktijsikijktsij

...... max

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)

(6)

2020

Related contributionsRelated contributionsAfriat’s Afriat’s minimum (maximum) path chained Laspeyers (Paasche) index numbersminimum (maximum) path chained Laspeyers (Paasche) index numbers are are very close to the following contributions combined together:very close to the following contributions combined together:

Dale W. Jorgenson and Zvi Griliches (1967)Dale W. Jorgenson and Zvi Griliches (1967), “The Explanation of Productivity Change”, , “The Explanation of Productivity Change”, Review of Economic Review of Economic Studies Studies 34: 249-283.34: 249-283.

Dale W. Jorgenson and Zvi Griliches (1971)Dale W. Jorgenson and Zvi Griliches (1971), “Divisia Index Numbers and Productivity Measurement”, , “Divisia Index Numbers and Productivity Measurement”, Review of Review of Income and WealthIncome and Wealth , pp. 227-229, who wrote: “The main advantage of a , pp. 227-229, who wrote: “The main advantage of a chain indexchain index is the reduction of errors is the reduction of errors of approximation as the economy moves from one production configuration to another. […] of approximation as the economy moves from one production configuration to another. […] The The Laspeyres approximationLaspeyres approximation to the Divisia index of total factor productivity was employed in our original to the Divisia index of total factor productivity was employed in our original study of productivity change (1967)”study of productivity change (1967)”

Samuelson and Swamy (1974, pp. 576)Samuelson and Swamy (1974, pp. 576) where it is stated that “Fisher missed the point made in where it is stated that “Fisher missed the point made in Samuelson (1947, p. 151) that Samuelson (1947, p. 151) that knowledge of a third situationknowledge of a third situation can add information relevant to can add information relevant to the comparison of two given situations”the comparison of two given situations”

Kruskal's Kruskal's Minimum Spanning TreeMinimum Spanning Tree recently proposed. See recently proposed. SeeHill, Robert J. (1999a)Hill, Robert J. (1999a), "Comparing Price Levels across Countries Using Minimum Spanning Trees", Review of , "Comparing Price Levels across Countries Using Minimum Spanning Trees", Review of Economics and Statistics 81: 135-142. Economics and Statistics 81: 135-142.

Hill, Robert J. (1999b)Hill, Robert J. (1999b), "International Comparisons using Spanning Trees", in A. Heston and , "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 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.and Wealth, Volume 61, NBER, Chicago: The University of Chicago Press.

Hill, Robert J. (2001)Hill, Robert J. (2001), "Linking Countries and Regions using Chaining Methods and Spanning , "Linking Countries and Regions using Chaining Methods and Spanning Trees", prepared for the Joint World Bank-OECD Seminar on Purchasing Power Parities 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.Recent Advances in Methods and Applications, Washington, D.C., 30th January-2nd February, 2001.

2121

Laspeyres and Paasche matrices withLaspeyres and Paasche matrices withAfriat’s tight upper and lower bound matricesAfriat’s tight upper and lower bound matrices

1

1

1

1

434241

343231

242321

141312

LLL

LLL

LLL

LLL

L

1111

11

11

111

1

1111

342414

432313

423212

413121

LLL

LLL

LLL

LLL

P

1)()(

1)(

)(1

1

433243213243

34322132

34232321

342312231212

LLLLLL

LLLL

LLLL

LLLLLL

M

1111

11

11

111

1

1111

343423342312

43232312

32433212

213243213221

LLLLLL

LLLL

LLLL

LLLLLL

H

So that we have the following range of possible values:

rsrsrsrs LMHP measure True

2222

An example from An example from S. Dowrick and John S. Dowrick and John Quiggin, “True Measures of GDP and Quiggin, “True Measures of GDP and Convergence”, Convergence”, American Economic ReviewAmerican Economic Review, , March 1997, pp. 41-64March 1997, pp. 41-64

Constant price and Afriat’s bounds on per capita GDP ratios, Constant price and Afriat’s bounds on per capita GDP ratios, 19801980

__________________________________________________________________________________________________________________________________________________________________________________________

L: matrix of Laspeyres ratios (log values)L: matrix of Laspeyres ratios (log values)

Canada US Norway Luxembourg …Canada US Norway Luxembourg …

1.1. Canada Canada 0.000 0.000 -0.009-0.009 -0.032 -0.032 -0.063-0.063 … …2.2. USUS 0.0170.017 0.000 0.017 0.000 0.017 -0.062-0.062 … … 3.3. Norway Norway 0.118 0.120 0.000 0.0340.118 0.120 0.000 0.034 … …4.4. LuxembourgLuxembourg 0.1830.183 0.1320.132 0.114 0.000 0.114 0.000 … …5.5. … … … … … …… … … … … …

M: Afriat’s minimum path matrix/true indexes (log values)M: Afriat’s minimum path matrix/true indexes (log values)

Canada US Norway Luxembourg …Canada US Norway Luxembourg …

1.1. CanadaCanada 0.000 -0.009 -0.0550.000 -0.009 -0.055 -0.097-0.097 … …2.2. US US 0.017 0.000 -0.044 -0.0860.017 0.000 -0.044 -0.086 … … 3.3. Norway Norway 0.118 0.109 0.000 0.0220.118 0.109 0.000 0.022 … …4.4. LuxembourgLuxembourg 0.1480.148 0.132 0.088 0.0000.132 0.088 0.000 … …5.5. … … … … … …… … … … … …

Column value means upper boundColumn value means upper bound(-) row value means lower bound(-) row value means lower bound

Upper bound

Lower bound

2323

ConclusionConclusion

Main steps towards the construction of Afriat’s tight bounds of the unknown “true” index:

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 1: Construct/revise an appropriate economic-theoretic model of demand (supply) behaviour

Step 5: Derive Afriat’s tight bounds from the results of Step 4.