### Reconciling Gross Output TFP Growth with Value Added TFP Growth

Erwin Diewert
unpublished
This article obtains relatively simple exact expressions that relate value added total factor productivity growth (TFP) in a value added framework to the corresponding measures of TFP growth in a gross output framework when Laspeyres or Paasche indexes are used to aggregate outputs and inputs. Basically, as the input base becomes smaller, the corresponding estimates of TFP growth become larger. A fairly simple approximate relationship between Fisher indexes of gross output TFP growth and the
more » ... responding Fisher index of value added TFP growth is also derived. The methodology developed in this article has a number of applications. SCHREYER (2001:26) DEVELOPED AN approximate formula to relate total factor productivity growth (or multifactor productivity growth) in a gross output model of production to TFP (or MFP) growth in a value added setting. In this article, we take another look at this issue 2 and develop an exact relationship between the two measures when Laspeyres (or Paasche) output and input indexes are used to compute aggregate growth rates of inputs and outputs. We develop rules relating the two productivity concepts that are simpler than the existing rules that have been developed in the literature. Sections 1 and 2 discuss the construction of the aggregate outputs and inputs using the Laspeyres and Paasche formulae respectively, while Section 3 uses the Fisher (1922) ideal index number formula to aggregate inputs and outputs. Section 4 concludes. The Laspeyres Case We first consider the situation where Laspey-res indexes are used to aggregate outputs and inputs. For simplicity, consider a situation where we want to compute gross output and value added productivity growth rates for a production unit that produces gross output at prices , uses intermediate input at prices and uses primary input at prices for t = 0,1. 3 2 See Balk (2009) for a comprehensive discussion on this topic. We obtain approximately the same result as Balk obtains but our method of derivation is much simpler. 3 In the case where there are many inputs and ouputs, the output and input quantity ratio aggregates, / , / and / the output and input quantity ratio aggregates can be interpreted as Laspeyres indexes of the micro quantities. The corresponding aggregate price ratios are of course Paasche price indexes. q Y t 0 > p Y t 0 > q M t 0 > p M t 0 > q X t 0 > p X t 0 > q Y 1 0 > q Y 0 0 > q M 1 0 > q M 0 0 > q X 1 0 > q X 0 0 >