This paper shows that the decomposition of log-change in aggregate labor productivity (ALP) devised by Balk (2013) based on Sato-Vartia indexes is inexact when applied to gross domestic product (GDP) in chained or in constant prices so that sectoral contributions do not necessarily add up to "actual" log-change in ALP. However, this paper adjusts Balk's decomposition by incorporating "relative prices" - from the "generalized exactly additive" (GEAD) decomposition of "arithmetic change" in ALP (Dumagan 2013) - and shows that the adjusted Balk decomposition is exact for GDP in chained or in constant prices like GEAD. An important finding is that relative prices could reverse the signs of sectoral contributions from Balk's inexact decomposition. Hence, results from related decompositions of log-change in ALP, e.g., those based on the Tornqvist framework, that do not explicitly recognize relative prices could be misleading and, therefore, may need reconsideration.
relative prices productivity change decomposition index number theory