Dynamics: The Bottom-Up Approach

Abstract

An industry is usually an ensemble of individual firms (decision making units) which may or may not interact with each other. Similarly, an economy is an ensemble of industries. It is natural to expect that there is a relation between (aggregate) industry or economy productivity and the (disaggregate) firm- or industry-specific productivities. This chapter considers the relation between (total factor or labour) productivity measures for lower-level production units and aggregates thereof such as industries, sectors, or entire economies. In particular, this chapter contains a review of the so-called bottom-up approach, which takes an ensemble of individual production units, be it industries or enterprises, as the fundamental frame of reference. At the level of industries the various forms of shift-share analysis are reviewed. At the level of enterprises the additional features that must be taken into account are entry (birth) and exit (death) of production units.

Appendices

Appendix A: Reinsdorf’s Extension of the GR Method

The extension, proposed by Reinsdorf (2015), concerns the case where there is neither exit nor entry; that is $\mathcal {K}^{0} = \mathcal {K}^{1} = \mathcal {C}^{01}$. Typically, this is the situation of an economy consisting of a fixed set of industries. Then the GR method, expression (5.61), reduces to

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\mathit{PROD}^{1} - \mathit{PROD}^{0} = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{C}^{01}}\dfrac{\theta^{k0} + \theta^{k1}}{2}\left(\mathit{PROD}^{k1} - \mathit{PROD}^{k0}\right) \\ & &\displaystyle \qquad +\: \sum_{k \in \mathcal{C}^{01}}(\theta^{k1} - \theta^{k0})\left(\dfrac{\mathit{PROD}^{k0} + \mathit{PROD}^{k1}}{2} - a\right), {} \end{array} \end{aligned} $$

(5.88)

where a is an arbitrary scalar. A rather natural choice is a = (PROD ⁰ + PROD ¹)∕2, the overall two-period mean aggregate productivity. The growth rate of aggregate productivity is obtained by dividing both sides of Eq. (5.88) by base period aggregate productivity, PROD ⁰,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\dfrac{\mathit{PROD}^{1} - \mathit{PROD}^{0}}{\mathit{PROD}^{0}} = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{C}^{01}}\dfrac{\theta^{k0} + \theta^{k1}}{2} \dfrac{\mathit{PROD}^{k0}}{\mathit{PROD}^{0}} \dfrac{\mathit{PROD}^{k1} - \mathit{PROD}^{k0}}{\mathit{PROD}^{k0}} \\ & &\displaystyle \qquad + \: \sum_{k \in \mathcal{C}^{01}}(\theta^{k1} - \theta^{k0})\left(\dfrac{\mathit{PROD}^{k0} + \mathit{PROD}^{k1}}{2\mathit{PROD}^{0}} - \dfrac{\mathit{PROD}^{0} + \mathit{PROD}^{1}}{2\mathit{PROD}^{0} }\right), {} \end{array} \end{aligned} $$

(5.89)

which can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\frac{\mathit{PROD}^{1} - \mathit{PROD}^{0}}{\mathit{PROD}^{0}} = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{C}^{01}}\frac{1}{2} (\theta^{k1}/\theta^{k0} + 1)\theta^{k0} \frac{\mathit{PROD}^{k0}}{\mathit{PROD}^{0}} \frac{\mathit{PROD}^{k1} - \mathit{PROD}^{k0}}{\mathit{PROD}^{k0}} \\ & &\displaystyle \qquad + \: \sum_{k \in \mathcal{C}^{01}}\frac{1}{2}(\theta^{k1}/\theta^{k0} - 1) \Big[\theta^{k0}\frac{\mathit{PROD}^{k0}}{\mathit{PROD}^{0}} \left(\mathit{PROD}^{k1}/\mathit{PROD}^{k0} + 1\right) \\ & &\displaystyle \qquad - \: \theta^{k0}\left(\mathit{PROD}^{1}/\mathit{PROD}^{0} + 1\right)\Big]. {} \end{array} \end{aligned} $$

(5.90)

Like in Section 5.5.2 we consider the case of simple labour productivity; that is, for each production unit $k \in \mathcal {C}^{01}$ its productivity (level) is defined as $\mathit {PROD}^{kt} \equiv \mathit {SLPROD}_{\mathit {VA}}^{k}(t,b)$, and its weight as its labour share, $\theta ^{kt} \equiv L^{kt}/\sum _{k \in \mathcal {C}^{01}}L^{kt}$. One immediately checks that

$$\displaystyle \begin{aligned} \theta^{k0}\frac{\mathit{PROD}^{k0}}{\mathit{PROD}^{0}} = \frac{\mathit{RVA}^{k}(0,b)}{\sum_{k \in \mathcal{C}^{01}}\mathit{RVA}^{k}(0,b)} \: (k \in \mathcal{C}^{01}). \end{aligned} $$

(5.91)

If the reference period of all the deflators is selected as being b = 0, then, by definition, RVA ^k(0, 0) = VA ^k0, and the real-value-added shares reduce to nominal-value-added shares,

$$\displaystyle \begin{aligned} \theta^{k0}\frac{\mathit{PROD}^{k0}}{\mathit{PROD}^{0}} = \frac{\mathit{VA}^{k0}}{\sum_{k \in \mathcal{K}}\mathit{VA}^{k0}} \equiv s_{\mathit{VA}}^{k0} \: (k \in \mathcal{C}^{01}). \end{aligned} $$

(5.92)

Expression (5.90) then reduces to

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\dfrac{\mathit{PROD}^{1} - \mathit{PROD}^{0}}{\mathit{PROD}^{0}} = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{C}^{01}}\dfrac{1}{2} (\theta^{k1}/\theta^{k0} + 1)s_{\mathit{VA}}^{k0} \dfrac{\mathit{PROD}^{k1} - \mathit{PROD}^{k0}}{\mathit{PROD}^{k0}} \\ & &\displaystyle \qquad +\: \sum_{k \in \mathcal{C}^{01}}\dfrac{1}{2}(\theta^{k1}/\theta^{k0} - 1) \Big[s_{\mathit{VA}}^{k0} \left(\mathit{PROD}^{k1}/\mathit{PROD}^{k0} + 1\right) \\ & &\displaystyle \qquad - \: \theta^{k0}\left(\mathit{PROD}^{1}/\mathit{PROD}^{0} + 1\right)\Big], {} \end{array} \end{aligned} $$

(5.93)

which corresponds to Reinsdorf’s (2015) equations (24)–(25). The first term at the right-hand side of expression (5.93) was called the ‘within-industry productivity effect’, and the second term was called the ‘reallocation effect’. Recall that the identity holds for any functional form of the value-added based, production-unit specific deflators.

Let now the joint reference period be selected as b = 1. By definition, $\mathit {RVA}^{k}(0,1) = \mathit {VA}^{k0}/P_{\mathit {VA}}^{k}(0,1)$. For any well-defined price index P(.) there exists a price index P ^∗(.) such that 1∕P(0, 1) = P ^∗(1, 0). And if P(.) satisfies the Time Reversal Test then P ^∗(.) is identically equal to P(.). Consequently, the real-value-added shares appear to be price-updated nominal-value-added shares,

$$\displaystyle \begin{aligned} \theta^{k0}\frac{\mathit{PROD}^{k0}}{\mathit{PROD}^{0}} = \frac{\mathit{VA}^{k0}P_{\mathit{VA}}^{*k}(1,0)} {\sum_{k \in \mathcal{C}^{01}}\mathit{VA}^{k0}P_{\mathit{VA}}^{*k}(1,0)} \equiv s_{\mathit{VA}}^{k01} \: (k \in \mathcal{C}^{01}). \end{aligned} $$

(5.94)

Notice that in general these price-updated shares differ from the comparison period’s nominal-value-added shares; that is $s_{\mathit {VA}}^{k01} \neq s_{\mathit {VA}}^{k1} \equiv \mathit {VA}^{k1}/\sum _{k \in \mathcal {C}^{01}}\mathit {VA}^{k1}$ $(k \in \mathcal {C}^{01})$.

Additivity of real value added holds if and only if

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{C}^{01}}\mathit{VA}^{k0}P_{\mathit{VA}}^{*k}(1,0) = (\sum_{k \in \mathcal{C}^{01}}\mathit{VA}^{k0})P_{\mathit{VA}}^{*\mathcal{C}^{01}}(1,0), \end{aligned} $$

(5.95)

where $P_{\mathit {VA}}^{*\mathcal {C}^{01}}(.)$ is a value-added based price index appropriate for the aggregate production unit $\mathcal {C}^{01}$. Then the price-updated shares can be simplified to

$$\displaystyle \begin{aligned} s_{\mathit{VA}}^{k01} = s_{\mathit{VA}}^{k0} \frac{P_{\mathit{VA}}^{*k}(1,0) }{P_{\mathit{VA}}^{*\mathcal{C}^{01}}(1,0)} \: (k \in \mathcal{C}^{01}), \end{aligned} $$

(5.96)

and expression (5.90) reduces to

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\frac{\mathit{PROD}^{1} - \mathit{PROD}^{0}}{\mathit{PROD}^{0}} = } \\ & &\displaystyle \qquad \sum_{k \in \mathcal{C}^{01}}\frac{1}{2} (\theta^{k1}/\theta^{k0} + 1)s_{\mathit{VA}}^{k0}\dfrac{P_{\mathit{VA}}^{*k}(1,0) }{P_{\mathit{VA}}^{*\mathcal{C}^{01}}(1,0)} \dfrac{\mathit{PROD}^{k1} - \mathit{PROD}^{k0}}{\mathit{PROD}^{k0}} \\ & &\displaystyle \qquad +\sum_{k \in \mathcal{C}^{01}}\dfrac{1}{2}(\theta^{k1}/\theta^{k0} - 1) \Big[s_{\mathit{VA}}^{k0}\dfrac{P_{\mathit{VA}}^{*k}(1,0) }{P_{\mathit{VA}}^{*\mathcal{C}^{01}}(1,0)} \left(\mathit{PROD}^{k1}/\mathit{PROD}^{k0} + 1\right) \\ & &\displaystyle \qquad - \: \theta^{k0}\left(\mathit{PROD}^{1}/\mathit{PROD}^{0} + 1\right)\Big], {} \end{array} \end{aligned} $$

(5.97)

which corresponds to Reinsdorf’s (2015) equation (26). Though in the derivation of expression (5.97) the assumption of additivity of real value added appears to be required, there is no restriction on the functional form of the value-added based, production-unit specific deflators.

Superficially, expressions (5.93) and (5.97) might look like being two different decompositions of the same entity. This, however, is not the case. The decomposition in expression (5.93) is based on productivities with reference period b = 0, whereas the decomposition in expression (5.97) is based on productivities with reference period b = 1. Nevertheless, Reinsdorf (2015) proposes to take a convex combination of these two decompositions.

In the empirical illustration, on United States data concerning the years 1998–2012, a simplification of Reinsdorf’s decomposition method was compared with the GEAD-Diewert method. As expected, the differences appear to be in the industrial details.

5.12 Appendix B: Exercises on the Netherlands Manufacturing Industry, 1984–1999

This Appendix summarizes Balk and Hoogenboom-Spijker (2003). The data for this study came from the production surveys. These annual surveys contain detailed information on revenue and cost components of private firms; in particular, gross output, value added, capital input cost, labour cost, intermediate input cost, and number of employees.

The statistical unit in the production surveys is the firm (kind-of-activity unit, establishment), considered to be the actual agent in the production process, characterised by its autonomy with respect to that process and by the sale of its goods or services to the market. A firm can consist of one or more juridical units or can be part of a larger juridical unit. Firms are classified according to their main economic activity.

For 1984–1986 more data were available than for the years thereafter, since the observation threshold was changed in 1987. Prior to this year all firms with 10 or more employees were surveyed, while from 1987 all firms with at least 20 employees were surveyed. Firms with less employees were sampled. In this study bilateral comparisons were restricted to firms with 20 or more employees.

The focus is on firms of the Manufacturing industry, consisting of the 2-digit industries 15–37 of the Standard Industrial Classification (SBI; an extension of NACE Rev. 1, and corresponding with ISIC Rev. 3.1) used in the Netherlands. There were no data for SBI 36631, social job creation. The industrial classification has been changed in 1993. This caused a break in the data series and led to some difficulties in finding appropriate deflators for the years prior to 1993. Because of that change data for 1992 were classified in two ways.

Nominal gross output and value added were deflated by producer output price index numbers (PPI) for total turnover. Where available, the indices at the three-digit level of the Netherlands’ SBI were used, otherwise those at the two-digit level. To assign these sectoral price indices to firms, one must know to which industry a firm belongs. This can change through time, however. The pragmatic solution was, that per firm the industry of the comparison period was taken, unless there was no observation in that period. Then the industry of the base period was taken.

The cost of energy, materials, and services was deflated by producer input price index numbers for total expenditures at the two-digit level of the SBI classification. Since the production surveys do not contain data on the capital stock, depreciation cost was used as input variable. The nominal values of depreciation cost have not been deflated.

For each year, firms with an incomplete data record and/or zero or negative values were deleted from the database. In addition, firms with unreasonable high or low profitabilities were deleted.

The information in the data records allowed, where possible, to link the data over the years. After linking two adjacent periods, the firms which appear to be outlier in either one of the periods and the firms with incomplete data in either one of the two periods were deleted. The entry or exit status of a firm was determined from the remaining data. If a firm occured with data in a base period but not in a comparison period, then this firm was defined as an exiting firm. If a firm did not occur with data in a base period but did in a comparison period, then this firm was called an entering firm. The disadvantage of this approach is that the entry and exit sets are polluted with firms which were sampled in only one of the two periods. It would be better to define exit and entry from a business register. Register-based data allow firms to be tracked through time, because addition or removal of firms from the register usually reflects the entry and exit of firms in the ‘real world’.

The pairs of time periods studied ranged from 1984–1985 to 1998–1999. The numbers of continuing firms, entering, and exiting firms hovered about 5000, 500, and 400, respectively.

The main suite of decompositions was concerned with change of aggregate value-added based TFP levels, defined as

$$\displaystyle \begin{aligned} \mathit{PROD}^{t} \equiv \sum_{k\in \mathcal{K}^{t}}\theta^{kt} \mathit{TFPROD}_{\mathit{VA}}^{k}(t,b) = \sum_{k\in \mathcal{K}^{t}}\theta^{kt} \frac{\mathit{RVA}^{k}(t,b)}{X_{KL}^{k}(t,b)}, \end{aligned} $$

(5.98)

where real value added is defined by expression (5.23), and real primary input is defined by expression (5.25). Anytime, two adjacent periods were considered, t = 0, 1, and the reference period was chosen as b = 0. The various elements of this expression will now be discussed.

Real value added is nominal value added divided by a suitable price index number. Real primary input is nominal primary input cost divided by another suitable price index number. However, not all the theoretically necessary values and index numbers were available. Also the fact that there are entering firms in the comparison period (t = 1) necessitates an adaptation of the definitions. Obviously, for entering firms base period (t = 0) information does not exist. Therefore, to obtain commensurability with the real values of the continuing firms, comparison period value added of an entering firm was deflated by a suitable low-level PPI, and the same was carried out with capital and labour cost. Thus, operationally, the following definitions were used:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathit{TFPROD}_{\mathit{VA}}^{k}(0,0) & \equiv &\displaystyle \frac{\mathit{VA}^{k0}}{C_{KL}^{k0}} \: \mbox{for} \: k \in \mathcal{C}^{01} \cup \mathcal{X}^{0} \\ \mathit{TFPROD}_{\mathit{VA}}^{k}(1,0) & \equiv &\displaystyle \frac{\mathit{VA}^{k1}/\mathit{PPI}^{k}(1,0)} {C_{KL}^{k0}Q_{KL}^{k}(1,0)} \: \mbox{for} \: k \in \mathcal{C}^{01} \\ & \equiv &\displaystyle \frac{\mathit{VA}^{k1}/\mathit{PPI}^{k}(1,0)}{C_{KL}^{k1}/\mathit{PPI}^{k}(1,0)} \: \mbox{for} \: k \in \mathcal{N}^{1}, \end{array} \end{aligned} $$

(5.99)

where PPI ^k(1, 0) denotes a producer price index number for the lowest-level industry group to which firm k belongs. Notice that, for the continuing firms, real primary input of period 1 is calculated as nominal primary input cost of period 0 times an input quantity index. The firm-specific quantity index was defined as

$$\displaystyle \begin{aligned} Q_{KL}^{k}(1,0) \equiv \left(\frac{L^{k1}}{L^{k0}}\right)^{\alpha^{k01}} \left(\frac{C_{K}^{k1}}{C_{K}^{k0}}\right)^{1-\alpha^{k01}} \: (k \in \mathcal{C}^{01}), \end{aligned} $$

(5.100)

where L ^kt is the number of employees and $C_{K}^{kt}$ is the depreciation cost of firm k in period t. The exponents α ^k01 are defined as mean (over two periods) labour cost shares; that is,

$$\displaystyle \begin{aligned} \alpha^{k01} \equiv \frac{1}{2}\left(\frac{C_{L}^{k0}}{C_{KL}^{k0}} + \frac{C_{L}^{k1}}{C_{KL}^{k1}}\right) \: (k \in \mathcal{C}^{01}). \end{aligned} $$

(5.101)

The weights θ ^kt were defined as real input shares; that is,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \theta^{k0} & \equiv &\displaystyle \dfrac{C_{KL}^{k0}}{\sum_{k \in \mathcal{C}^{01} \cup \mathcal{X}^{0}}C_{KL}^{k0}} \: \mbox{for} \: k \in \mathcal{C}^{01} \cup \mathcal{X}^{0}\\ \theta^{k0} & \equiv &\displaystyle \dfrac{C_{KL}^{k0}Q_{KL}^{k}(1,0)} {\sum_{k \in \mathcal{C}^{01}}C_{KL}^{k0}Q_{KL}^{k}(1,0) + \sum_{k \in \mathcal{N}^{1}}C_{KL}^{k1}/\mathit{PPI}^{k}(1,0)} \: \mbox{for} \: k \in \mathcal{C}^{01} \\ & \equiv &\displaystyle \dfrac{C_{KL}^{k1}/\mathit{PPI}^{k}(1,0)} {\sum_{k \in \mathcal{C}^{01}}C_{KL}^{k0}Q_{KL}^{k}(1,0) + \sum_{k \in \mathcal{N}^{1}}C_{KL}^{k1}/\mathit{PPI}^{k}(1,0)} \: \mbox{for} \: k \in \mathcal{N}^{1}. \end{array} \end{aligned} $$

(5.102)

For each pair of adjacent years PROD ¹ −PROD ⁰ was decomposed according to the five methods discussed earlier in this chapter. Respectively,

• the BHC method, defined by expression (5.48), with a = 0;

• the dual BHC method, defined by expression (5.49), with a = 0;

• the FHK method, defined by expression (5.50), with a = PROD ⁰;

• the dual FHK method, defined by expression (5.60), with a = PROD ¹;

• the GR method, defined by expression (5.61), with a = 0;

• the GR method, defined by expression (5.61), with a = (PROD ⁰ + PROD ¹)∕2.

The decompositions were then turned into contributions to the percentage change ((PROD ¹ −PROD ⁰)∕PROD ⁰) × 100%, and averaged over time. The results are summarized in Table 5.1. It is straightforward to check that the mean of the BHC and dual BHC decompositions equals the GR decomposition (with a = 0), and that the mean of the FHK and dual FHK decompositions equals the GR decomposition (with $a = \frac {\mathit {PROD}^{0} + \mathit {PROD}^{1}}{2}$). It is also clear that decompositions where the scalar a has been set equal to 0 exhibit (in absolute value) larger contributions of the entry, exit, and between components at the expense of the within component.

Table 5.1 Netherlands Manufacturing industry, 1984–1999. Mean annual value-added based TFP change (percentage) and decomposition

The study considered a number of alternatives, the results of which are here summarized:

• Instead of computing value-added based TFP levels as arithmetic means geometric means were used. The differences appeared to be negligible.

• Variations in the definition of the exponents α ^k01 appeared to be inconsequential.

• The weights θ ^kt can be based on numbers of employees L ^kt or on real output VA ^kt∕PPI ^k(t, 0). Though the actual outcomes changed, the overall picture did not change.

• Instead of value-added based TFP levels, the analysis can be based on labour productivity levels $\mathit {RVA}^{k}(t,b)/X_{L}^{k}(t,b)$ and weights as real labour input. Again, the overall picture did not change much.

• For a number of years it was possible to compare value-added based results to gross-output based results. The differences were as expected.

• For a number of years it was also possible to supplement exit and entry as defined from the production survey data with information about entry and exit flowing from the business register. It turned out that, as result of more precise definitions, for those years the entry, exit, and between components of aggregate productivity change almost vanished. .

The study of Balk and Hoogenboom-Spijker (2003) was continued and extended by Polder et al. (2018). The continuation concerned the years 2007–2012. The extension concerned a broadening of the scope from Manufacturing to all industries; a more precise determination of the status of production units by using business register information; and special attention for the ICT intensity of the production units. The productivity measure used was simple value-added based labour productivity; the weights were based on numbers of full-time equivalent employees; and the aggregation method was the GR method, defined by expression (5.61), with a = (PROD ⁰ + PROD ¹)∕2.

As far as Manufacturing was concerned the results of both studies were consistent: aggregate productivity change turned out to be mainly due to the ‘within’ component of continuing production units.

Appendix C: Generalization of the OP Decomposition

We start by recalling some definitions. Let the aggregate (or mean) productivity level at period t be defined as the weighted arithmetic mean of the unit-specific productivity levels; that is, $\mathit {PROD}^{t} \equiv \sum _{k\in \mathcal {K}^{t}}\theta ^{kt}\mathit {PROD}^{kt}$, where the weights add up to 1, $\sum _{k \in \mathcal {K}^{t}}\theta ^{kt} = 1$.

Let $\#(\mathcal {K}^t)$ denote the number of production units in the set $\mathcal {K}^t$. Then $\overline {\mathit {PROD}}^{t} \equiv \sum _{k \in \mathcal {K}^t}\mathit {PROD}^{kt}/\#(\mathcal {K}^t)$ is the unweighted arithmetic mean of the unit-specific productivity levels, and $\bar {\theta }^{t} \equiv \sum _{k \in \mathcal {K}^t}\theta ^{kt}/\#(\mathcal {K}^t) = 1/\#(\mathcal {K}^t)$ is the unweighted arithmetic mean of the weights. The Bortkiewicz relation is

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}^t}(\theta^{kt} - \bar{\theta}^{t})(\mathit{PROD}^{kt} - \overline{\mathit{PROD}}^{t}) = \mathit{PROD}^{t} - \overline{\mathit{PROD}}^{t}. \end{aligned} $$

(5.103)

Let now the set $\mathcal {K}^t$ consist of J disjunct subsets; that is,

$$\displaystyle \begin{aligned} \mathcal{K}^t = \bigcup_{j=1}^{J}\mathcal{K}_j^t, \: \mathcal{K}_j^t \cap \mathcal{K}_{j'}^t = \emptyset \: (j \neq j'). \end{aligned}$$

Then the aggregate productivity level PROD ^t is a weighted mean of the subaggregate productivity levels PROD ^jt (j = 1, …, J),

$$\displaystyle \begin{aligned} \mathit{PROD}^{t} = \sum_{j=1}^{J}\vartheta^{jt}\mathit{PROD}^{jt}, \end{aligned} $$

(5.104)

where $\vartheta ^{jt} \equiv \sum _{k \in \mathcal {K}_j^{t}}\theta ^{kt}$ and $\mathit {PROD}^{jt} \equiv \sum _{k\in \mathcal {K}_j^{t}}\theta ^{kt}\mathit {PROD}^{kt}/\sum _{k\in \mathcal {K}_j^{t}}\theta ^{kt}$ (j = 1, …, J). Notice that $\sum _{j=1}^{J}\vartheta ^{jt} = 1$.

In the same way the overall unweighted mean $\overline {\mathit {PROD}}^{t}$ appears to be a weighted mean of the subaggregate means $\overline {\mathit {PROD}}^{jt}$ (j = 1, …, J),

$$\displaystyle \begin{aligned} \overline{\mathit{PROD}}^{t} = \sum_{j=1}^{J}\#(\mathcal{K}_j^t)\overline{\mathit{PROD}}^{jt}/ \#(\mathcal{K}^t), \end{aligned} $$

(5.105)

where $\overline {\mathit {PROD}}^{jt}\equiv \sum _{k \in \mathcal {K}_j^t}\mathit {PROD}^{kt}/\#(\mathcal {K}_j^t)$ (j = 1, …, J). Notice that $\#(\mathcal {K}^t) = \sum _{j=1}^{J}\#(\mathcal {K}_j^t)$.

By substituting expressions (5.104) and (5.105) in the Bortkiewicz relation, expression (5.103), we obtain

$$\displaystyle \begin{aligned} \mbox{covar}(\mathcal{K}^t) \equiv \sum_{k \in \mathcal{K}^t}(\theta^{kt} - \bar{\theta}^{t})(\mathit{PROD}^{kt} - \overline{\mathit{PROD}}^{t}) = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{j=1}^{J}\vartheta^{jt}\mathit{PROD}^{jt} - \sum_{j=1}^{J}\#(\mathcal{K}_j^t)\overline{\mathit{PROD}}^{jt}/ \#(\mathcal{K}^t) = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{j=1}^{J}\vartheta^{jt}\left(\mathit{PROD}^{jt} - \overline{\mathit{PROD}}^{jt}\right) + \sum_{j=1}^{J}\left(\vartheta^{jt} - \#(\mathcal{K}_j^t)/\#(\mathcal{K}^t)\right)\overline{\mathit{PROD}}^{jt} = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{j=1}^{J}\vartheta^{jt}\mbox{covar}(\mathcal{K}_j^t) + \sum_{j=1}^{J}\left(\vartheta^{jt} - \#(\mathcal{K}_j^t)/\#(\mathcal{K}^t)\right)\overline{\mathit{PROD}}^{jt}, \end{aligned} $$

(5.106)

where in the final step the Bortkiewicz relation was applied at the subset level. Thus the overall covariance between productivity levels and weights, $\mbox{covar}(\mathcal {K}^t)$, can be decomposed into two parts: a weighted mean of subset covariances (aka the within-groups effect), and a covariance of unweighted subset productivity levels and relative subset weights (aka the between-groups effect).

The decomposition in expression (5.106) is, however, not unique. The alternative is

$$\displaystyle \begin{aligned} \mbox{covar}(\mathcal{K}^t) \equiv \sum_{k \in \mathcal{K}^t}(\theta^{kt} - \bar{\theta}^{t})(\mathit{PROD}^{kt} - \overline{\mathit{PROD}}^{t}) = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{j=1}^{J}\left(\#(\mathcal{K}_j^t)/\#(\mathcal{K}^t)\right) \mbox{covar}(\mathcal{K}_j^t) + \sum_{j=1}^{J}\left(\vartheta^{jt} - \#(\mathcal{K}_j^t)/\#(\mathcal{K}^t)\right)\mathit{PROD}^{jt}, \end{aligned} $$

(5.107)

but also a mean of the two decompositions could be contemplated.

Expression (5.106) can be used to compare the overall covariance, $\mbox{covar}(\mathcal {K}^t)$, to any of the subset covariances $\mbox{covar}(\mathcal {K}_{j'}^t)$ (j′ = 1, …, J). As $\sum _{j=1}^{J}\vartheta ^{jt} = 1$, we immediately see that

$$\displaystyle \begin{aligned} \mbox{covar}(\mathcal{K}^t) - \mbox{covar}(\mathcal{K}_{j'}^t) = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{j=1}^{J}\vartheta^{jt}\left(\mbox{covar}(\mathcal{K}_j^t) - \mbox{covar}(\mathcal{K}_{j'}^t)\right) + \sum_{j=1}^{J}\left(\vartheta^{jt} - \#(\mathcal{K}_j^t)/\#(\mathcal{K}^t)\right)\overline{\mathit{PROD}}^{jt}. \end{aligned} $$

(5.108)

And, as $\sum _{j=1}^{J}\left (\vartheta ^{jt} - \#(\mathcal {K}_j^t)/\#(\mathcal {K}^t)\right ) = 0$, in the between-groups component of expression (5.108) the unweighted subset productivity levels $\overline {\mathit {PROD}}^{jt}$ may be replaced by relative levels $\overline {\mathit {PROD}}^{jt} - \overline {\mathit {PROD}}^{j't}$. The resulting expression plays a big role in the analysis of Maliranta and Määttänen (2015); there called “augmented OP decomposition method”.

Finally, by substituting expression (5.106) in expression (5.83), and using again the fact that $\sum _{j=1}^{J}\left (\vartheta ^{jt} - \#(\mathcal {K}_j^t)/\#(\mathcal {K}^t)\right ) = 0$, we obtain

$$\displaystyle \begin{aligned} \mathit{PROD}^{t} = \overline{\mathit{PROD}}^{t} + \sum_{j=1}^{J}\vartheta^{jt}\mbox{covar}(\mathcal{K}_j^t) \end{aligned} $$

(5.109)

$$\displaystyle \begin{aligned} + \: \sum_{j=1}^{J}\left(\vartheta^{jt} - \#(\mathcal{K}_j^t)/\#(\mathcal{K}^t)\right)\left(\overline{\mathit{PROD}}^{jt} - \overline{\mathit{PROD}}^{t}\right). \end{aligned}$$

This could be seen as a generalization of the original OP decomposition. The reallocation term is split into two terms, measuring intra-subset and inter-subset reallocation, respectively.