Reinsdorf-Balk Transformationand Additive Decomposition of Indexes

Ki-Hong Choi(NPRI) and Hak K. Pyo(SNU)

I. Introduction

II. Reinsdorf-Balk Transformation

III. Additive Decompositions of Fisher Index

IV. Additive Decomposition of Chain Indexes

V. Numerical Illustrations (skip)

VI. Concluding Remarks

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I. Introduction

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arithmetic mean index

geometric mean index

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decomposition in %-change

decomposition in log-change

• Arithmetic mean index and decomposition of growth rate

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• Geometric mean index and decomposition of growth rate

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II. Reinsdorf-Balk Transformation

• Literature

Reinsdorf(1997, 2002), Balk(1999, 2004)

Kohli(2007), “Truly remarkable”

• Some need for improvements

– Reinsdorf’s proof is hard to follow.– Though Balk’s proof is easy and clear, there

is a better alternative.

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Balk’s Identity: from A index to G index

Reinsdorf’s Identity: from G index to A index

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III. Additive Decompositions of Fisher III. Additive Decompositions of Fisher IndexIndex

• Conventional Decompositions

– Decomposition in % changes

van IJzeren(1952)=Dumagan(2002)=Ehemann

et al(2002)

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– Decomposition in log-changes

Reinsdorf(1997), Reinsdorf et al(2002), Balk(2004)

Less satisfactory decomposition by Vartia(1976)

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• New Decompositions Decomposition in % changes

Reinsdorf et al.(2002), applying Reinsdorf identity to the geometric mean version of Fisher index.

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– Decomposition in log-changes

• Applying Balk’s identity to the van IJzeren form of Fisher index

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IV. Additive Decomposition of Chain Indexes

• Difficulty with additive decomposition of cumulative (compound) growth rates of chain indexes

– Cumulative growth rates is decomposable in log-changes

– But decomposition in log-change is not good since it deviates from %-changes

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• R-B transformation gives solution to this dilemma

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VI. Concluding remarks

• R-B transformation is interesting since it makes arithmetic mean and geometric mean indexes interchangeable.

• R-B transformation is useful since it can provide an additive decomposition of cumulative growth rates by chain indexes.

Thank you!