ReinsdorfBalk Transformation and Additive Decomposition of Indexes

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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

• Arithmetic mean index and decomposition of
  growth rate

arithmetic mean index
decomposition in -change

• Geometric mean index and decomposition of growth
  rate

geometric mean index
decomposition in log-change
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II. Reinsdorf-Balk Transformation

• Literature
• Reinsdorf(1997, 2002), Balk(1999, 2004)
• Kohli(2007), Truly remarkable

• Some need for improvements
• Reinsdorfs proof is hard to follow.
• Though Balks proof is easy and clear, there is a
  better alternative.

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Balks Identity from A index to G index
Reinsdorfs Identity from G index to A index
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III. Additive Decompositions of Fisher Index

• 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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Reinsdorf et al(2002), numerically identical
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• Decomposition in log-changes
• Applying Balks 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! ?