The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity Indices

Abstract

This chapter is concerned with the relation between measures of output and labour productivity change for individual production units and for aggregates such as industries, sectors, or economies. In the framework of discrete time periods several useful, symmetric expressions are derived and confronted with results from the literature.

Appendices

Appendix A: The Tang and Wang Method

In the National Accounts it is usual to measure (aggregate) output as (aggregate) real value added, which is calculated as nominal value added deflated by an appropriate price index relative to some reference period b. The formal definition is given by expression (6.1). Recall that there are price and quantity indices such that revenue ratios and intermediate input cost ratios can be split into price and quantity components; see expressions (6.2) and (6.3). Real gross output is then defined by $Y^{k}(t,b) \equiv R^{kb}Q_{R}^{k}(t,b)$, real intermediate input use by $X_{\mathit {EMS}}^{k}(t,b) \equiv C_{\mathit {EMS}}^{kb}Q_{\mathit {EMS}}^{k}(t,b)$, the relative gross output price index by $\tilde {P}_{R}^{k}(t,b) \equiv P_{R}^{k}(t,b)/P_{\mathit {VA}}^{\mathcal {K}}(t,b)$, and the relative intermediate input price index by $\tilde {P}_{\mathit {EMS}}^{k}(t,b) \equiv P_{\mathit {EMS}}^{k}(t,b)/P_{\mathit {VA}}^{\mathcal {K}}(t,b)$. After substituting all these definitions into expression (6.1) it appears that aggregate output can be written as

$$\displaystyle \begin{aligned} \mathit{RVA}^{\mathcal{K}}(t,b) = \sum_{k \in \mathcal{K}} \left(\tilde{P}_{R}^{k}(t,b)Y^{k}(t,b) - \tilde{P}_{\mathit{EMS}}^{k}(t,b)X_{\mathit{EMS}}^{k}(t,b) \right). \end{aligned} $$

(6.31)

This is an important building block for what follows.

Aggregate output change, going from an earlier period 0 to a later period 1, is naturally measured by $\mathit {RVA}^{\mathcal {K}}(1,b) - \mathit {RVA}^{\mathcal {K}}(0,b)$. This can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b) =} \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}\left(\tilde{P}_{R}^{k}(1,b)Y^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)Y^{k}(0,b)\right) - \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b)X_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)X_{\mathit{EMS}}^{k}(0,b)\right). \end{array} \end{aligned} $$

(6.32)

The two parts at the right-hand side of this expression can be decomposed according to the Laspeyres, forward-looking perspective, yielding

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b) = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}\tilde{P}_{R}^{k}(0,b)\left(Y^{k}(1,b) - Y^{k}(0,b)\right) \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}Y^{k}(0,b)\left(\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)\right) \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}\left(\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)\right)\Big(Y^{k}(1,b) - Y^{k}(0,b)\Big) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\tilde{P}_{\mathit{EMS}}^{k}(0,b)\left(X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)\right) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}X_{\mathit{EMS}}^{k}(0,b)\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right)\Big(X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)\Big).{} \end{array} \end{aligned} $$

(6.33)

Switching to relative changes (forward-looking growth rates) delivers

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\frac{\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{R}^{k}(0,b)Y^{k}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(0,b)} \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{R}^{k}(0,b)Y^{k}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } \frac{\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)}{\tilde{P}_{R}^{k}(0,b)} \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{R}^{k}(0,b)Y^{k}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } \frac{\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)}{\tilde{P}_{R}^{k}(0,b)} \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(0,b)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{\mathit{EMS}}^{k}(0,b)X_{\mathit{EMS}}^{k}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } \frac{X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)}{X_{\mathit{EMS}}^{k}(0,b)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{\mathit{EMS}}^{k}(0,b)X_{\mathit{EMS}}^{k}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } \frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)}{\tilde{P}_{\mathit{EMS}}^{k}(0,b)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{\mathit{EMS}}^{k}(0,b)X_{\mathit{EMS}}^{k}(0,b)}{\mathit{RVA}^{\mathcal{K}}(0,b) } \frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)}{\tilde{P}_{\mathit{EMS}}^{k}(0,b)} \frac{X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)}{X_{\mathit{EMS}}^{k}(0,b)}. {} \end{array} \end{aligned} $$

(6.34)

Notice that, by using the definitions of $\tilde {P}_{R}^{k}(0,b)$, Y ^k(0, b), $\mathit {RVA}^{\mathcal {K}}(0,b)$, and expression (6.2), it appears that $\tilde {P}_{R}^{k}(0,b)Y^{k}(0,b)/\mathit {RVA}^{\mathcal {K}}(0,b) = R^{k0}/\mathit {VA}^{\mathcal {K}0}$, which is the base period share of nominal revenue of unit k in aggregate nominal value added. Similarly it appears that $\tilde {P}_{\mathit {EMS}}^{k}(0,b)X_{\mathit {EMS}}^{k}(0,b)/\mathit {RVA}^{\mathcal {K}}(0,b)$ $= C_{\mathit {EMS}}^{k0}/\mathit {VA}^{\mathcal {K}0}$, which is the base period share of nominal intermediate input cost of unit k in aggregate nominal value added. Hence, these shares are independent of the reference period b. This, however, does not hold for the growth rates, contrary to the assertion of TW. Notice that, for instance,

$$\displaystyle \begin{aligned} \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(0,b)} = \frac{Q_R^k(1,b) - Q_R^k(0,b)}{Q_R^k(0,b)}, \end{aligned}$$

which in general stays dependent on reference period b data.

The right-hand side of expression (6.34) consists of six terms: the first and the fourth give the aggregate effect of quantity change, the second and the fifth give the aggregate effect of relative price change, and the third and the sixth give the aggregate effect of the interaction of quantity and price change. The entire expression corresponds to expression (2) of Tang and Wang (2015). As noticed in the main text, this method can be extended to aggregate (simple) labour productivity change, but also to aggregate TFP change.

Instead of decomposing aggregate output change according to the Laspeyres-perspective, as in expression (6.33), one can use the Paasche, backward-looking perspective. Then

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b) = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}\tilde{P}_{R}^{k}(1,b)\left(Y^{k}(1,b) - Y^{k}(0,b)\right) \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}Y^{k}(1,b)\left(\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)\right) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\left(\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)\right)\Big(Y^{k}(1,b) - Y^{k}(0,b)\Big) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\tilde{P}_{\mathit{EMS}}^{k}(1,b)\left(X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)\right) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}X_{\mathit{EMS}}^{k}(1,b)\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right) \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right)\Big(X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)\Big). {} \end{array} \end{aligned} $$

(6.35)

Notice that the signs on the interaction terms in expression (6.35) differ from those in expression (6.33). Switching to backward-looking relative changes delivers

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\frac{\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b)}{\mathit{RVA}^{\mathcal{K}}(1,b) } = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{R}^{k}(1,b)Y^{k}(1,b)}{\mathit{RVA}^{\mathcal{K}}(1,b) } \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(1,b)} \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{R}^{k}(1,b)Y^{k}(1,b)}{\mathit{RVA}^{\mathcal{K}}(1,b)} \frac{\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)}{\tilde{P}_{R}^{k}(1,b)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{R}^{k}(1,b)Y^{k}(1,b)}{\mathit{RVA}^{\mathcal{K}}(1,b) } \frac{\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)}{\tilde{P}_{R}^{k}(1,b)} \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(1,b)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b)X_{\mathit{EMS}}^{k}(1,b)}{\mathit{RVA}^{\mathcal{K}}(1,b) } \frac{X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)}{X_{\mathit{EMS}}^{k}(1,b)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b)X_{\mathit{EMS}}^{k}(1,b)}{\mathit{RVA}^{\mathcal{K}}(1,b) } \frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)}{\tilde{P}_{\mathit{EMS}}^{k}(1,b)} \\ & &\displaystyle \qquad + \!\sum_{k \in \mathcal{K}}\!\frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b)X_{\mathit{EMS}}^{k}(1,b)}{\mathit{RVA}^{\mathcal{K}}(1,b) } \frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b) {-} \tilde{P}_{\mathit{EMS}}^{k}(0,b)}{\tilde{P}_{\mathit{EMS}}^{k}(1,b)}\frac{X_{\mathit{EMS}}^{k}(1,b) {-} X_{\mathit{EMS}}^{k}(0,b)}{X_{\mathit{EMS}}^{k}(1,b)}. {} \end{array} \end{aligned} $$

(6.36)

Now $\tilde {P}_{R}^{k}(1,b)Y^{k}(1,b)/\mathit {RVA}^{\mathcal {K}}(1,b) = R^{k1}/\mathit {VA}^{\mathcal {K}1}$, which is the comparison period share of nominal revenue of unit k in aggregate nominal value added, and $\tilde {P}_{\mathit {EMS}}^{k}(1,b)X_{\mathit {EMS}}^{k}(1,b)/\mathit {RVA}^{\mathcal {K}}(1,b) = C_{\mathit {EMS}}^{k1}/\mathit {VA}^{\mathcal {K}1}$, which is the comparison period share of nominal intermediate input cost of unit k in aggregate nominal value added. The right-hand side of expression (6.36) consists of six terms: the first and the fourth give the aggregate effect of quantity change, the second and the fifth give the aggregate effect of relative price change, and the third and the sixth give the aggregate effect of the interaction of quantity and price change.

Forward- and backward-looking relative changes are related in a straightforward way. For instance,

$$\displaystyle \begin{aligned} \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(1,b)} = 1 - \left(1 + \frac{Y^{k}(1,b) - Y^{k}(0,b)}{Y^{k}(0,b)}\right)^{-1}. \end{aligned}$$

The decompositions in expressions (6.33) and (6.35), (6.34) and (6.36), respectively, have the same structure, and it is by and large a matter of taste and convenience which of the two is preferred. Therefore, as another alternative the simple arithmetic mean of the Laspeyres- and Paasche-perspective decompositions, expressions (6.33) and (6.35), could be chosen. The resulting decomposition is named after Bennet and has the additional feature that the two interaction terms cancel:

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b) = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{K}}(1/2)\left(\tilde{P}_{R}^{k}(1,b) + \tilde{P}_{R}^{k}(0,b)\right)\Big(Y^{k}(1,b) - Y^{k}(0,b)\Big) \\ & &\displaystyle \qquad + \sum_{k \in \mathcal{K}}(1/2)\left(\tilde{P}_{R}^{k}(1,b) - \tilde{P}_{R}^{k}(0,b)\right)\Big(Y^{k}(1,b) + Y^{k}(0,b)\Big) \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}(1/2)\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) + \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right)\Big(X_{\mathit{EMS}}^{k}(1,b) - X_{\mathit{EMS}}^{k}(0,b)\Big)\\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}(1/2)\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) - \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right)\Big(X_{\mathit{EMS}}^{k}(1,b) + X_{\mathit{EMS}}^{k}(0,b)\Big).{} \end{array} \end{aligned} $$

(6.37)

Switching to forward- or backward-looking relative changes would destroy the symmetry, as one easily checks. The symmetry can be retained when differences are related to logarithmic means, as in

$$\displaystyle \begin{aligned} \frac{\mathit{RVA}^{\mathcal{K}}(1,b) - \mathit{RVA}^{\mathcal{K}}(0,b)} {\mathit{LM}(\mathit{RVA}^{\mathcal{K}}(1,b),\mathit{RVA}^{\mathcal{K}}(0,b))} = \ln(\mathit{RVA}^{\mathcal{K}}(1,b) / \mathit{RVA}^{\mathcal{K}}(0,b)). \end{aligned}$$

This delivers

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln(\mathit{RVA}^{\mathcal{K}}(1,b) / \mathit{RVA}^{\mathcal{K}}(0,b)) = } \\ & &\displaystyle \sum_{k \in \mathcal{K}}\frac{\left(\tilde{P}_{R}^{k}(1,b) + \tilde{P}_{R}^{k}(0,b)\right)\mathit{LM}(Y^{k}(1,b),Y^{k}(0,b))} {2\mathit{LM}(\mathit{RVA}^{\mathcal{K}}(1,b),\mathit{RVA}^{\mathcal{K}}(0,b))}\ln\left(\frac{Y^{k}(1,b)}{Y^{k}(0,b)}\right) \\ & &\displaystyle + \sum_{k \in \mathcal{K}}\frac{\mathit{LM}(\tilde{P}_{R}^{k}(1,b),\tilde{P}_{R}^{k}(0,b))\left(Y^{k}(1,b) + Y^{k}(0,b)\right)} {2\mathit{LM}(\mathit{RVA}^{\mathcal{K}}(1,b),\mathit{RVA}^{\mathcal{K}}(0,b))}\ln\left(\frac{\tilde{P}_{R}^{k}(1,b)}{\tilde{P}_{R}^{k}(0,b)}\right)\\ & &\displaystyle - \sum_{k \in \mathcal{K}}\frac{\left(\tilde{P}_{\mathit{EMS}}^{k}(1,b) + \tilde{P}_{\mathit{EMS}}^{k}(0,b)\right) \mathit{LM}(X_{\mathit{EMS}}^{k}(1,b),X_{\mathit{EMS}}^{k}(0,b))} {2\mathit{LM}(\mathit{RVA}^{\mathcal{K}}(1,b),\mathit{RVA}^{\mathcal{K}}(0,b))}\ln\left(\frac{X_{\mathit{EMS}}^{k}(1,b)}{X_{\mathit{EMS}}^{k}(0,b)}\right)\\ & &\displaystyle - \sum_{k \in \mathcal{K}}\frac{\mathit{LM}(\tilde{P}_{\mathit{EMS}}^{k}(1,b),\tilde{P}_{\mathit{EMS}}^{k}(0,b))\left(X_{\mathit{EMS}}^{k}(1,b) {+} X_{\mathit{EMS}}^{k}(0,b)\right)}{2\mathit{LM}(\mathit{RVA}^{\mathcal{K}}(1,b),\mathit{RVA}^{\mathcal{K}}(0,b))} \ln\left(\frac{\tilde{P}_{\mathit{EMS}}^{k}(1,b)}{\tilde{P}_{\mathit{EMS}}^{k}(0,b)}\right).{} \end{array} \end{aligned} $$

(6.38)

What do we see here? At the left-hand side we have the logarithmic difference of aggregate real value added. If small then this approximates a forward-looking percentage change or the negative of a backward-looking percentage change. The right-hand side decomposes this into four terms. Each term is a weighted sum of subaggregate changes. The first term gives the total effect of real gross output quantity change, where each individual term is weighted by the share of mean nominal revenue in mean nominal aggregate value added. The second term gives the total effect of relative gross output price change, where each individual term is also weighted by the share of mean nominal revenue in mean nominal aggregate value added. The third term gives the total effect of real intermediate input quantity change, where each individual term is weighted by the share of mean nominal intermediate input cost in mean nominal aggregate value added. The fourth term gives the total effect of relative intermediate input price change, where each individual term is also weighted by the share of mean nominal intermediate input cost in mean nominal aggregate value added.

Though symmetric with respect to time, the weights in the first and second term, and those in the third and fourth term, are not equal, due to different combinations of arithmetic and logarithmic means. This is not completely satisfactory. Another disadvantage of this decomposition is that aggregate real value added change is not written as a weighted sum or product of subaggregate real value added changes.

Appendix B: Dumagan’s Decomposition

Dumagan (2013a) considered another decomposition of aggregate simple labour productivity. As in expression (6.1), aggregate real value added is defined as

$$\displaystyle \begin{aligned} \mathit{RVA}^{\mathcal{K}}(t,b) \equiv \mathit{VA}^{\mathcal{K}t}/P_{\mathit{VA}}^{\mathcal{K}}(t,b), \end{aligned} $$

(6.39)

where $P_{\mathit {VA}}^{\mathcal {K}}(t,b)$ is an appropriate price index relative to some base period b. Similarly, every production unit’s real value added is defined as

$$\displaystyle \begin{aligned} \mathit{RVA}^{k}(t,b) \equiv \mathit{VA}^{kt}/P_{\mathit{VA}}^{k}(t,b) \: (k \in \mathcal{K}), \end{aligned} $$

(6.40)

where $P_{\mathit {VA}}^{k}(t,b)$ are also appropriate price indices relative to some base period b; there is no assumption about any relation between the aggregate deflator $P_{\mathit {VA}}^{\mathcal {K}}(t,b)$ and the unit-specific deflators $P_{\mathit {VA}}^{k}(t,b)$ $(k \in \mathcal {K})$; they may or may not exhibit the same functional form.

Define for any production unit k its relative value added price index as $\tilde {P}_{\mathit {VA}}^{k}(t,b) \equiv P_{\mathit {VA}}^{k}(t,b)/P_{\mathit {VA}}^{\mathcal {K}}(t,b)$ and its labour share as $\tilde {L}^{kt} \equiv L^{kt}/L^{\mathcal {K}t}$. Since aggregate value added is the sum of unit-specific value added, $\mathit {VA}^{\mathcal {K}t} = \sum _{k \in \mathcal {K}}\mathit {VA}^{kt}$, it readily appears, by substituting all the foregoing definitions, that aggregate simple labour productivity can be written as

$$\displaystyle \begin{aligned} \frac{\mathit{RVA}^{\mathcal{K}}(t,b)}{L^{\mathcal{K}t}} = \sum_{k \in \mathcal{K}}\tilde{P}_{\mathit{VA}}^{k}(t,b)\tilde{L}^{kt}\frac{\mathit{RVA}^{k}(t,b) }{L^{kt}}. \end{aligned} $$

(6.41)

We see here three factors: relative price index, labour share, and unit-specific simple labour productivity. Next, aggregate simple labour productivity growth, again defined as

$$\displaystyle \begin{aligned} \frac{\mathit{RVA}^{\mathcal{K}}(1,b)/L^{\mathcal{K}1} - \mathit{RVA}^{\mathcal{K}}(0,b)/L^{\mathcal{K}0}} {\mathit{RVA}^{\mathcal{K}}(0,b)/L^{\mathcal{K}0}}, \end{aligned}$$

was decomposed according to the Laspeyres, forward-looking perspective, into three factors, respectively called pure productivity effect, Denison effect, and Baumol effect. This final expression, called GEAD, goes back to Tang and Wang (2004). However, this decomposition is not unique and the distinction between the two effects appears to be rather artificial. This was discussed in Section 5.5.2.

Next, Dumagan considered the special case where the relation between aggregate and unit-specific deflators is given by

$$\displaystyle \begin{aligned} P_{\mathit{VA}}^{\mathcal{K}}(t,b) = \left(\sum_{k \in \mathcal{K}}\frac{\mathit{VA}^{kt}}{\mathit{VA}^{\mathcal{K}t}}\left(P_{\mathit{VA}}^{k}(t,b)\right)^{-1}\right)^{-1}. \end{aligned} $$

(6.42)

Thus the aggregate deflator is a two-stage Paasche index of the unit-specific deflators (see Chapter 2, Appendix A). Rewriting this expression leads to $\mathit {RVA}^{\mathcal {K}}(t,b) = \sum _{k \in \mathcal {K}}\mathit {RVA}^{k}(t,b)$, and then to

$$\displaystyle \begin{aligned} \frac{\mathit{RVA}^{\mathcal{K}}(t,b)}{L^{\mathcal{K}t}} = \sum_{k \in \mathcal{K}}\tilde{L}^{kt}\frac{\mathit{RVA}^{k}(t,b) }{L^{kt}}, \end{aligned} $$

(6.43)

which is a simplification of expression (6.41) in the sense that there are but two factors left: labour share and unit-specific simple labour productivity. Aggregate simple labour productivity growth can then again be decomposed in a number of ways, one of which was called TRAD. The paper contains interesting empirical comparisons of the GEAD and TRAD decompositions. Another empirical comparison was provided by De Avillez (2012).

Finally, notice that expression (6.43) also materializes when for aggregate and unit-specific value added the same deflators are used; that is when $P_{\mathit {VA}}^{\mathcal {K}}(t,b) = P_{\mathit {VA}}^{k}(t,b)$ $(k \in \mathcal {K})$.

Appendix C: Proof of Expression (6.13)

Our point of departure is the identity

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}}\frac{\mathit{VA}^{k1}}{\mathit{VA}^{\mathcal{K}1}} - \sum_{k \in \mathcal{K}}\frac{\mathit{VA}^{k0}}{\mathit{VA}^{\mathcal{K}0}}= 0. \end{aligned} $$

(6.44)

Using the definition of value added, this can be rewritten as

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}}\frac{R^{k1}-C^{k1}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}1}} - \sum_{k \in \mathcal{K}}\frac{R^{k0}-C^{k0}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}0}}= 0, \end{aligned} $$

(6.45)

which can be rearranged to

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}}\left(\frac{R^{k1}}{\mathit{VA}^{\mathcal{K}1}} - \frac{R^{k0}}{\mathit{VA}^{\mathcal{K}0}}\right) - \sum_{k \in \mathcal{K}}\left(\frac{C^{k1}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}1}} - \frac{C^{k0}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}0}}\right) = 0, \end{aligned} $$

(6.46)

As we know, the logarithmic mean transforms differences into ratios. Thus the last expression can be rewritten as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\sum_{k \in \mathcal{K}}\mathit{LM}\left(\frac{R^{k1}}{\mathit{VA}^{\mathcal{K}1}},\frac{R^{k0}}{\mathit{VA}^{\mathcal{K}0}}\right) \ln\left(\frac{R^{k1}/\mathit{VA}^{\mathcal{K}1}}{R^{k0}/\mathit{VA}^{\mathcal{K}0}}\right)} \\ & &\displaystyle \qquad - \sum_{k \in \mathcal{K}}\mathit{LM}\left(\frac{C^{k1}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}1}},\frac{C^{k0}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}0}}\right) \ln\left(\frac{C^{k1}_{\mathit{EMS}}/\mathit{VA}^{\mathcal{K}1}}{C^{k0}_{\mathit{EMS}}/\mathit{VA}^{\mathcal{K}0}}\right) = 0. \end{array} \end{aligned} $$

(6.47)

This, finally, can be rearranged as

$$\displaystyle \begin{aligned} \ln\left(\frac{\mathit{VA}^{\mathcal{K}1}}{\mathit{VA}^{\mathcal{K}0}}\right) = \sum_{k \in \mathcal{K}}\xi^{k}_{R}(1,0)\ln\left(\frac{R^{k1}}{R^{k0}}\right) - \sum_{k \in \mathcal{K}}\xi^{k}_{\mathit{EMS}}(1,0)\ln\left(\frac{C^{k1}_{\mathit{EMS}}}{C^{k0}_{\mathit{EMS}}}\right), \end{aligned} $$

(6.48)

where

$$\displaystyle \begin{aligned} \xi^{k}_{R}(1,0) \equiv \frac{\mathit{LM}\left(\frac{R^{k1}}{\mathit{VA}^{\mathcal{K}1}},\frac{R^{k0}}{\mathit{VA}^{\mathcal{K}0}}\right)} {\sum_{k \in \mathcal{K}}\mathit{LM}\left(\frac{R^{k1}}{\mathit{VA}^{\mathcal{K}1}},\frac{R^{k0}}{\mathit{VA}^{\mathcal{K}0}}\right) - \sum_{k \in \mathcal{K}}\mathit{LM}\left(\frac{C^{k1}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}1}},\frac{C^{k0}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}0}}\right)} \: (k \in \mathcal{K}), \end{aligned}$$

and

$$\displaystyle \begin{aligned} \xi^{k}_{\mathit{EMS}}(1,0) \equiv \frac{\mathit{LM}\left(\dfrac{C^{k1}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}1}},\dfrac{C^{k0}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}0}}\right)} {\sum_{k \in \mathcal{K}}\mathit{LM}\left(\dfrac{R^{k1}}{\mathit{VA}^{\mathcal{K}1}},\dfrac{R^{k0}}{\mathit{VA}^{\mathcal{K}0}}\right) - \sum_{k \in \mathcal{K}}\mathit{LM}\left(\dfrac{C^{k1}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}1}},\dfrac{C^{k0}_{\mathit{EMS}}}{\mathit{VA}^{\mathcal{K}0}}\right)} \: (k \in \mathcal{K}). \end{aligned}$$

Replacing revenue and intermediate inputs cost ratios by products of price and quantity indices, we obtain

$$\displaystyle \begin{aligned} \ln\left(\frac{\mathit{VA}^{\mathcal{K}1}}{\mathit{VA}^{\mathcal{K}0}}\right) = \end{aligned} $$

(6.49)

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}}\xi^{k}_{R}(1,0)\ln(P_R^k(1,0)Q_R^k(1,0)) - \sum_{k \in \mathcal{K}}\xi^{k}_{\mathit{EMS}}(1,0)\ln(P_{\mathit{EMS}}^k(1,0)Q_{\mathit{EMS}}^k(1,0)) = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}}\left(\xi^{k}_{R}(1,0)\ln P_R^k(1,0) - \xi^{k}_{\mathit{EMS}}(1,0)\ln P_{\mathit{EMS}}^k(1,0)\right) + \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}}\left(\xi^{k}_{R}(1,0)\ln Q_R^k(1,0) - \xi^{k}_{\mathit{EMS}}(1,0)\ln Q_{\mathit{EMS}}^k(1,0)\right), \end{aligned}$$

from which expression (6.13) immediately follows. The price and quantity components after the second equality sign in expression (6.49) generalize the conventional Sato-Vartia indices (see Appendix A of Chap. 2).