represents the contribution to productivity and welfare of changes in aggregate technology.

The Basu and Fernald decomposition can be extended by disaggregating R� into a within sectors and between sectors component. This is useful in assessing whether the gain from reallocation (if any) occur because resources are reallocated across industries or within industries across ...rms. Basu and Fernald used industry level data in their empirical exercise so they could at best evaluate the between component. (We say "at best" because if there are within-industry reallocation terms, then Basu and Fernald’s estimation using industry-level data would not give a consistent estimate of even the average industry markup, <u>�.</u> In general, one can estimate <u>�</u> correctly only by taking the average of ...rm-level markups, estimated using ...rm-level data.) If one uses ...rm-level data, one can discuss the relative importance of the within and between components. RM can also be decomposed into a within and between component, but there is a residual term.

Let P<sup>Y J</sup> Y<sup>J</sup> =<sup>P</sup> i�J<sup>P Y</sup> i<sup>Y</sup> i<sup>Jbe the total value added produced in industry J, wJ= P Y JYJ=P Y Y</sup> the share of industry J out of aggregate output and wi<sup>J</sup> = Pi<sup>YY</sup> i<sup>J=P Y JYJtheshareofvalue</sup> added of ...rm i in industry J. Denote with Qi a ...rm gross output and with Pi<sup>Q</sup> its price. Then wi<sup>QJ</sup> = Pi<sup>QQ</sup> i<sup>J=P QJQJ,whereP QJQJ= P</sup> i�J<sup>P Q</sup> ii<sup>QJ;representsthe...rmshareofindustrygross</sup> output. Finally, the primary inputs growth in industry J is d log X<sup>J</sup> = s<sup>J</sup> K<sup>d log KJ+ sJ</sup> L<sup>d log LJ,</sup> <u>Pi�J</u><sup>PK;iKi</sup> <u>Pi�J</u><sup>PL;iLi</sup> where s<sup>J</sup> K<sup>=</sup> P<sup>Y J</sup> Y<sup>J</sup> and s<sup>J</sup> L<sup>=</sup> P<sup>Y J</sup> Y<sup>J</sup> . De...ne R�<sup>JandR</sup> �<sup>Jastheindustryequivalentofthe</sup> reallocation terms R� and RM when aggregating over industry J rather than the entire economy, i.e. R�<sup>J=P</sup> i�J<sup>w</sup> i<sup>J</sup> ��i � <u>�</u><sup>~~J~~�</sup> d log Xi and RM<sup>J=P</sup> i�J<sup>w</sup> i<sup>J</sup> ��i � <u>�</u><sup>~~J~~�</sup> 1�s<sup>Q</sup> Ms<sup>Q</sup> M;i;i<sup>(d log Mi �d log Qi),where</sup> <u>�</u><sup>~~J~~</sup> =<sup>P</sup> i�J<sup>w</sup> i<sup>J�</sup> i<sup>.Wecandecomposethereallocationtermforprimaryinputs,R�;intoawithin</sup> and a between component (denoted by superscripts W and B, respectively) as follows: where R�<sup>W= P</sup> J<sup>K</sup> =1<sup>wJR</sup> �<sup>Jand R</sup> �<sup>B= P</sup> J<sup>K</sup> =1<sup>wJ �</sup> <u>�</u><sup>~~J~~</sup> � <u>��</u> d log X<sup>J</sup> :Note that the between component can be calculated on the basis of industry data only.

The decomposition for the reallocation term for materials, RM , is instead: where RM<sup>W= PK</sup> J=1<sup>wJR</sup> M<sup>J; R</sup> M<sup>B= PK</sup> J=1<sup>wJ �</sup> <u>�</u><sup>~~J~~</sup> � <u>��</u> (d log Q<sup>J</sup> �d log Y<sup>J</sup> ) and Rw�wQ =<sup>PK</sup> J=1<sup>wJ �</sup> <u>�</u><sup>~~J~~</sup> � <u>��</u> ( wi<sup>J�w</sup> i<sup>QJ</sup> )d log Qi: In the between component, d log Y<sup>J</sup> =<sup>P</sup> i�J<sup>w</sup> i<sup>Jd log YiistheDivisiaindex</sup> of industry value added,. d log Q<sup>J</sup> =<sup>P</sup> i�J<sup>w</sup> i<sup>QJ</sup> d log Qi is the Divisia index of gross output (using wi<sup>QJ</sup> as weights). The residual term, Rw�wQ, re‡ects the di¤erence between value added weights and gross output weights in aggregating ...rm level gross output within an industry.

# 7 Econometric Framework

The modi...ed Solow productivity residual can be essentially calculated from the data and requires no estimation if the distributional shares are observable (or if we observe the labor share and 22