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Open Access 27-09-2024 | Special Issue Paper

A decomposition of economic growth decompositions

Author: Jan Oosterhaven

Published in: The Annals of Regional Science | Issue 4/2024

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Abstract

This paper critically compares the ability of all three decomposition techniques that explain economic growth and its variation between regions and nations. Old time shift-and-share analysis (S&S) presumes that industry mix and regional competitiveness are all important. Structural decomposition analysis (SDA) presumes that final demand growth and input–output coefficient changes are all that matter. Growth accounting (GA) presumes the same for the growth of production factors and technological progress. This paper concludes that an econometric estimation of a GA equation without its residual factor productivity growth component, but with industry mix and demand components from S&Ss and SDAs, respectively, offers the best approach to explain longer term economic growth and variations therein.
Notes

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1 Introduction: Three growth decomposition methods

The explanation of economic growth and its variation between regions and nations is one of the most important theoretical and empirical issues in economics (see Johansson 1998, for a fine overview). In empirical analyses, two quite different approaches are used, namely deterministic/decomposition methods and stochastic/econometric methods. In deterministic approaches, total growth or the growth difference between regions or nations is decomposed into and fully attributed to different components according to some formula or theory. In stochastic approaches, a set of possible explanatory variables is econometrically tested to determine which variables contribute most to the explanation of growth or growth differences and which don´t; always leaving an unexplained residual. This contribution aims at a comparison of the three decomposition approaches that are used in analyses of economic growth.
The probably oldest method is shift-and-share analysis (S&S, Creamer 1942, popularized by Dunn in Perloff et al. 1960). S&S concentrates on the role of spatial differences in the industry mix in explaining differences in economic growth. Over the years, S&S received a lot of criticism, most notably by Richardson (1978), which may be one of the reasons for its diminished use nowadays. The second oldest approach is based on Leontief´s demand-driven input–output (IO) model and is known as structural decomposition analysis (SDA, Leontief 1941, see Rose & Casler 1996, for an excellent older overview). SDA attributes the growth of value added or employment to the growth of final demand and to changes in the IO model´s coefficients. This approach is still used a lot. The third somewhat newer approach is growth accounting (GA, Solow 1957). GA is based in production function theory and is mainly used as a means to measure and decompose factor productivity growth (see Jorgenson & Griliches 1967, for a seminal contribution). When used to analyse economic growth, GA exclusively attributes it to the growth of the supply of capital and labour and to technological progress.1
All three decomposition approaches have been applied to a host of different problems. Here, we primarily aim at an evaluation from the perspective of explaining economic growth and the spatial variation therein. For that purpose, we will emphasize the fundamental differences between the three approaches, and ignore combinations of them. While doing so, we will especially discus their theoretical foundations, if present, and their possibilities to estimate the statistical significance of the components proposed. We conclude that an econometric estimation of an extended growth accounting equation offers the best approach to explaining, especially, long run economic growth. Such an econometric estimation should exclude the residual, total factor productivity component of GA, whereas it should include individual components from a S&S and/or a SDA, alongside other explanatory variables.

2 Shift-and-share analysis: impact of industry mix

We start our overview with the oldest decomposition technique, which is based on the notion that the local mix of industries is important in explaining the difference between a region/nation and some other geographical unit with which one wants to compare that region/nation. The most simple shift-and-share analysis (S&S) decomposes the following identity:
$$v^{r} - v^{n} \equiv \sum\nolimits_{i} {s_{i}^{r} v_{i}^{r} } - \sum\nolimits_{i} {s_{i}^{n} } v_{i}^{n} ,\;{\text{with}}\;\sum\nolimits_{i} {s_{i}^{r} } = \sum\nolimits_{i} {s_{i}^{n} } = 1,$$
(1)
where v = variable of interest (e.g. total GDP growth, total job growth, average wage level, total energy use, total CO2 emissions) for some unit r (e.g. region) that is to be compared with some norm n (e.g. nation), and that is aggregated over some index i (e.g. industry), and where \(s_{i}^{r} \equiv v_{i}^{r} /\sum\nolimits_{i} {v_{i}^{r} }\) = share of i in r as regards v, and \(s_{i}^{n}\) = the analogous share of i in n.2 From (1), it follows that S&S may be used to analyse a multitude of issues.3 Here, we only discuss its oldest and most frequently used application to regional growth.
Table 1 shows the five ways in which (1) may be decomposed. The first decomposition shows the classical S&S of regional economic growth \(v^{r}\) into its share in national growth \(v^{n}\), plus a proportional shift due to a different industry mix \(\sum\nolimits_{i} {\left( {s_{i}^{r} - s_{i}^{n} } \right)v_{i}^{r} }\), plus a residual differential shift \(\sum\nolimits_{i} {s_{i}^{n} } \left( {v_{i}^{r} - v_{i}^{n} } \right)\) (the italics indicate the origin of the term shift-and-share). The last component gives an indication of the impact of regional competitiveness, as it measures whether the weighted average regional industry grows faster or slower than its national counterpart. Capello (2007) further clarifies that the industry mix component will primarily be related to demand-side growth factors, whereas the residual competitiveness component will primarily be related to supply-side factors.
Table 1
Possible S&S decompositions of the regional/national growth differential, vr–vn *
No.
Industry mix component, with:
Competitiveness component, with:
Specialization component
Regional growth rates
National growth rates
Regional industry shares
National industry shares
\(\sum\nolimits_{i} {\left( {s_{i}^{r} - s_{i}^{n} } \right)v_{i}^{r} }\)
\(\sum\nolimits_{i} {\left( {s_{i}^{r} - s_{i}^{n} } \right)v_{i}^{n} }\)
\(\sum\nolimits_{i} {s_{i}^{r} \left( {v_{i}^{r} - v_{i}^{n} } \right)}\)
\(\sum\nolimits_{i} {s_{i}^{n} } \left( {v_{i}^{r} - v_{i}^{n} } \right)\)
\(\sum\nolimits_{i} {\left( {s_{i}^{r} - s_{i}^{n} } \right)} \left( {v_{i}^{r} - v_{i}^{n} } \right)\)
1.
+
  
+
 
2.
 
+
+
  
3.**
½
½
½
½
 
4.
+
 
+
 
5.
 
+
 
+
+
*Where the s indicate industry shares and the v industry growth rates
**This decomposition results from taking the average of decomposition 1 and 2 as well as that of 4 and 5
Oosterhaven & van Loon (1979)
Both the industry mix and the competitiveness component may be measured—respectively weighted—differently, as is evident from a comparison of the first and second decomposition in Table 1. Taking the average of the first two decompositions delivers the third decomposition. Taking an average is the typical solution of SDA and GA to the problem of choosing between components measured in base year terms and those measured in end year terms (see below). In shift-and-share analysis, however, taking the average is not the preferred choice.4
When the research interest is in comparing different regions, each component needs to be measured/weighted in the same way for all regions. This argument makes the first three and, especially, the fourth decomposition unacceptable for interregional comparisons. Luckily, there is a fifth decomposition that measures/weighs both the industry mix component and the competitiveness component in the same way, such that they can be compared between regions. To reach this result, a third component has to be added (see the last row of Table 1).
This third, interacting differences component is theoretically interesting on its own account, as it measures whether the industries in which a region is specialized have a high or low score on the variable of interest, i.e. in our case whether they grow faster or slower than their national counterpart. This third component thus measures the impact of regional specialization, i.e. it measures so-called localization economies or diseconomies when it proves to be negative (see Oosterhaven & Broersma 2008, for the difference with cluster, urbanization and agglomeration economies, see Johansson 1998, for a further discussion). For this additional reason, the fifth decomposition should even be considered to represent the preferred decomposition when the research interest only regards a single region.5
In case of regional wage differences (Oosterhaven & van Loon 1979) and regional labour productivity differences (Oosterhaven & Broersma 2007) specialization clearly pays off, in the sense that industries in which a region is specialized have higher levels of labour productivity and pay higher wages than their national counterparts, indicating positive localization economies. In case of employment growth and value added growth, however, the specialization component proved to be negative for all Dutch regions, which was interpreted as representing diminishing returns to these positive localization economies. The same result was found for earlier periods, for different regional and different sectoral classifications (WWR 1980; Oosterhaven & Stol 1985).
In these studies, the industry mix component proved to exhibit a stable regional pattern over eight periods between 1951 and 1983, with a slowly diminishing importance, starting with “explaining” a halve and ending with “explaining” only a quarter of the regional differences in job growth. The residual competitiveness component, on the other hand, gained importance, but with an unstable regional pattern with sometimes changing signs between subperiods. Interestingly, the changing of signs appeared to be related to changes in national economic growth. Core regions showed a relative slowdown of their residual growth during periods of national growth, probably due to local congestion and supply shortages, whereas peripheral regions reduced part of their economic arrears during periods of national growth, probably due to picking up part of the core regions´ choked off growth.
The most attractive properties of S&S are its versatility and the limited amount of aggregate sectoral data needed. The most mentioned objections against S&S are (1) its lack of a theoretical foundation (Richardson 1978), and (2) the impossibility to determine the statistical significance of its components (Stillwell 1969; Chalmers & Beckhelm 1976; Stevens & Moore 1980). Besides, it was noted already early on that (3) the industry mix component is sensitive to sectoral aggregation, being more important at lower levels of aggregation, while (4) its size is underestimated due to ignored interindustry interdependences (Mackay 1968).
The lack of a theoretical foundation, however, may be turned into an advantage of S&S if the unstable residual component is dropped, while the industry mix and the specialization component are used as regular, but composite variables in an econometric estimation of the LHS of (1). This, in fact, simultaneously solves the important second objection, as it provides a measure of the statistical significance of these two components in explaining the LHS of (1). In the case of Dutch regional labour productivity levels and growth rates, using this econometric approach, Broersma & Oosterhaven (2009) find that both the industry mix and the specialization component from their S&S are highly significant, along with the regional capital/labour ratio, a regional diversity index, and the own and the neighbouring regions´ job density as indicators of agglomeration economies or diseconomies (see Johansson & Quigley 2004, for a further discussion).

3 Structural decomposition analysis: a demand-side story

Next, we consider structural decomposition analysis (SDA), which is based on the demand-driven IO model (Leontief 1941). In this model,6 exogenous final demand for products from industry i (\(y_{i} \in {\mathbf{y}}\)) and endogenous intermediate demand for products from i by all industries j (\(\sum\nolimits_{j} {z_{ij} } \in {\mathbf{Z}}\,{\mathbf{i}}\)) together determine the supply of output by industry i (\(x_{i} \in {\mathbf{x}} = {\mathbf{Z}}\,{\mathbf{i}} + {\mathbf{y}}\)). In addition, intermediate demand for products from industry i by industry j, without economies of scale, is linearly determined by the size of industry j´s total output (\(z_{ij} = a_{ij} x_{j} \in {\mathbf{Z}} = {\mathbf{A}}\,{\hat{\mathbf{x}}}\)), which implies full complementarity of all inputs. The solution of the basic Leontief model reads as follows:
$$x_{i} = \sum\nolimits_{j} {l_{ij} y_{j} \in {\mathbf{x}} = {\mathbf{Ly}} = ({\mathbf{I}} - {\mathbf{A}})^{ - 1} {\mathbf{y}}}$$
(2)
in which \(l_{ij}\) are the elements of the so-called Leontief inverse \(\left( {{\mathbf{I}} - {\mathbf{A}}} \right)^{ - 1}\), indicating the demand-driven direct plus indirect impact on total output of industry i of any change in exogenous final demand for products from industry j.
The most simple SDA splits up the absolute growth of output by industry i (\(\Delta x_{i} \in \Delta {\mathbf{x}}\)):
$$\Delta {\mathbf{x}} \equiv {\mathbf{x}}_{1} - {\mathbf{x}}_{0} = \left( {{\mathbf{I}} - {\mathbf{A}}_{1} } \right)^{ - 1} {\mathbf{y}}_{1} - \left( {{\mathbf{I}} - {\mathbf{A}}_{0} } \right)^{ - 1} {\mathbf{y}}_{0} = {\mathbf{L}}_{1} {\mathbf{y}}_{1} - {\mathbf{L}}_{0} {\mathbf{y}}_{0} ,$$
(3)
An SDA of (3) represents a comparative static analysis that sequentially looks at the impact on the variable of interest of changes in each set of parameters, holding the other sets of parameters constant. Note that SDA may be used to decompose any first order difference in a matrix equation, such as the difference between national and regional embodied CO2 emissions, or the growth of energy use (see Hoekstra and Bergh 2002, for an overview such SDAs). However, here we only discuss its most common application, namely to long run economic growth.
Just like the decomposition of (1), there are five comparable decompositions of (3) (see Table 2). Skolka (1989) presents four of them, while Decomposition 4 is added by Oosterhaven & van der Linden (1997). In choosing between the first two decompositions, neither Skolka (1989), nor Dietzenbacher et al (2004) nor Miller & Blair (2009) see any preference, which is why they all prefer taking the average, i.e. the third decomposition. This choice neglects the interaction component \(\Delta {\mathbf{L}}\,\Delta {\mathbf{y}}\) in the equally logical fifth decomposition in Table 2. However, in this case, this is not a real loss, as the interaction component is empirically found to be rather small (Uno 1989), while it is theoretically considered to have no clear economic interpretation (Skolka 1989; Miller & Blair 2009). Here SDA clearly deviates from S&S.7
Table 2
Possible structural IO decompositions of industry output growth, \(\Delta {\mathbf{x}}\)*
No.
Leontief-inverse change, with:
Final demand change, with:
Interaction component
base year y
end year y
base year L
end year L
\(\Delta {\mathbf{L}}\,{\mathbf{y}}_{0}\)
\(\Delta {\mathbf{L}}\,{\mathbf{y}}_{1}\)
\({\mathbf{L}}_{0} \,\Delta {\mathbf{y}}\)
\({\mathbf{L}}_{1} \,\Delta {\mathbf{y}}\)
\(\Delta {\mathbf{L}}\,\Delta {\mathbf{y}}\)
1.
+
  
+
 
2.
 
+
+
  
3.**
½
½
½
½
 
4.
 
+
 
+
5.
+
 
+
 
+
*Where x represents a vector of industry output, f a vector of final demand for industry output, and L the Leontief inverse (IA)−1, with A being the matrix of interindustry coefficients
**This decomposition results from taking the average of decomposition 1 and 2 as well as of 4 and 5
Departing from the most simple IO model used in (3), many, more sophisticated variants with an increasing number of components have been developed (see Rose & Casler 1996, and Miller & Blair 2009, for overviews). To increase the understanding of the type of outcomes that a SDA may generate, we showcase the decomposition, not of output growth, but of GDP growth in the EU by Oosterhaven & van der Linden (1997), as it includes most of the individual components proposed in the literature. They use an interregional IO model with I industries, R regions and Q types of final demand. Specific to their approach is that each of the interregional intermediate input coefficients \(a_{ij}^{rs}\) and each of the interregional final demand input coefficients \(b_{iq}^{rs}\) (often called bridge coefficients) is written as the product of a technical/preference coefficient and a trade origin ratio.
The solution of their interregional IO model reads as follows:
$${\mathbf{v}} = {\hat{\mathbf{c}}}\left( {{\mathbf{I}} - {\mathbf{M}}^{a} \otimes {\mathbf{A}}} \right)^{ - 1} \left( {{\mathbf{M}}^{f} \otimes {\mathbf{F}}} \right){\mathbf{y}} = {\hat{\mathbf{c}}}\;{\mathbf{L}}\;{\mathbf{B}}\;{\mathbf{y}}$$
(4)
In (4), \(\otimes\) = the Hadamar product (i.e. cell-by-cell matrix multiplication), and going backwards along the causal chain of their IO model: \(y_{ \cdot q}^{ \cdot s} = \sum\nolimits_{i}^{r} {y_{iq}^{rs} } \in {\mathbf{y}}\) = a QR column with macroeconomic levels of final demand of type q per region s, \(f_{iq}^{ \cdot s} \in {\mathbf{F}}\) = an IR x QR block matrix, with R mutually identical I x QR matrices with final demand preference coefficients, indicating the total use of product i from all over the world per unit of final demand of type q in region s, \(m_{iq}^{rs} \in \,\,{\mathbf{M}}^{f}\) = an IR x QR matrix with cell-specific trade origin ratios, indicating which fraction of that final demand originates from region r, \(a_{ij}^{ \cdot s} \in {\mathbf{A}}\) = an IR x IR block matrix, with R mutually identical I x IR matrices with technical coefficients, indicating the total use of product i from all over the world per unit of output of industry j in region s, \(m_{ij}^{rs} \in {\mathbf{M}}^{a}\) = an IR x IR matrix with cell-specific trade origin ratios, indicating which fraction of that intermediate demand originates from region r, \({\hat{\mathbf{c}}}\) = an IR x IR diagonal matrix with gross value added coefficients on its diagonal, and v = an IR column with gross value added per industry, per region.
The decomposition of the changes in the last part of (4) reads as follows:
$$\Delta {\mathbf{v}} = 0.50\;\Delta {\hat{\mathbf{c}}}\left( {{\mathbf{L}}_{0} {\mathbf{B}}_{0} {\mathbf{y}}_{0} + {\mathbf{L}}_{1} {\mathbf{B}}_{1} {\mathbf{y}}_{1} } \right) +$$
(5a)
$$0.25\left[ {{\hat{\mathbf{c}}}_{0} \;{\mathbf{L}}_{1} \;\Delta {\mathbf{M}}^{a} \otimes \left( {{\mathbf{A}}_{0} + {\mathbf{A}}_{1} } \right){\mathbf{L}}_{0} {\mathbf{B}}_{1} {\mathbf{y}}_{1} + {\hat{\mathbf{c}}}\;_{1} {\mathbf{L}}_{1} \;\Delta {\mathbf{M}}^{a} \otimes \left( {{\mathbf{A}}_{0} + {\mathbf{A}}_{1} } \right){\mathbf{L}}_{0} {\mathbf{B}}_{0} {\mathbf{y}}_{0} } \right] +$$
(5b)
$$0.25\left[ {{\hat{\mathbf{c}}}_{0} \;{\mathbf{L}}_{1} \left( {{\mathbf{M}}_{0}^{a} + {\mathbf{M}}_{1}^{a} } \right) \otimes \Delta {\mathbf{A}}\;{\mathbf{L}}_{0} {\mathbf{B}}_{1} {\mathbf{y}}_{1} + {\hat{\mathbf{c}}}_{1} {\mathbf{L}}_{1} \left( {{\mathbf{M}}_{0}^{a} + {\mathbf{M}}_{1}^{a} } \right) \otimes \Delta {\mathbf{A}}\;{\mathbf{L}}_{0} {\mathbf{B}}_{0} {\mathbf{y}}_{0} } \right] +$$
(5c)
$$0.25\left[ {{\hat{\mathbf{c}}}_{0} \;{\mathbf{L}}_{0} \;\Delta {\mathbf{M}}^{f} \otimes \left( {{\mathbf{F}}_{0} + {\mathbf{F}}_{1} } \right){\mathbf{y}}_{1} + {\hat{\mathbf{c}}}_{1} \;{\mathbf{L}}_{1} \;\Delta {\mathbf{M}}^{f} \otimes \left( {{\mathbf{F}}_{0} + {\mathbf{F}}_{1} } \right){\mathbf{y}}_{0} } \right] +$$
(5d)
$$0.25\left[ {{\hat{\mathbf{c}}}_{0} \;{\mathbf{L}}_{0} \left( {{\mathbf{M}}_{0}^{f} + {\mathbf{M}}_{1}^{f} } \right) \otimes \Delta {\mathbf{F}}\;{\mathbf{y}}_{1} + {\hat{\mathbf{c}}}_{1} \;{\mathbf{L}}_{1} \left( {{\mathbf{M}}_{0}^{f} + {\mathbf{M}}_{1}^{f} } \right) \otimes \Delta {\mathbf{F}}\;{\mathbf{y}}_{0} } \right] +$$
(5e)
$$0.50\;\left( {{\hat{\mathbf{c}}}\,_{0} {\mathbf{L}}_{0} {\mathbf{B}}_{0} + {\hat{\mathbf{c}}}\,_{1} {\mathbf{L}}_{1} {\mathbf{B}}_{1} } \right)\;\Delta {\mathbf{y}}.$$
(5f)
In (5), the average is taken of two so-called polar decompositions, which is most clear in (5a) and (5f). The above polar decomposition, however, represents only one of many possible decompositions. Dietzenbacher & Los (1998), also ignoring interaction components, show that the number of possible basic decompositions equals the faculty of the number of components (n). They, luckily, also show that taking the average of two polar decompositions, as done in (5), results in empirical outcomes that are very close to the average of all n! possible basic decompositions.
When used to analyse economic growth, SDA is usually applied to longer time periods, and practically always reports that changes in the level of final demand constitute by far the most important component. Feldman et al. (1987), following the seminal study of Anne Carter (1970) with more recent and more detailed IO data, analyse a decomposition of output growth with \({\mathbf{x}} = {\mathbf{LB}}\,{\mathbf{y}}\) for the USA over the period 1963–1978. They find that changes in y are far more important than changes in either L or B, for some 80% of the 400 American industries distinguished. Coefficient changes were only important in case of the fastest and the slowest growing industries (see Fujimagari 1989, for very comparable results for Canada). From those outcomes, they conclude that the best growth policy is a good macroeconomic policy. The question is whether that conclusion is justified.
Applying (5) to their EU intercounty input–output tables (IOTs) for 1975–1985, Oosterhaven & van der Linden (1997) also report final demand growth, especially of household consumption, to be by far the most important component for all eight countries and for almost all of the 25 industries distinguished. The combined effect of the five types of coefficient changes in (5) proved to be rather small and predominantly negative, which was mainly caused by a systematic decline in value added coefficients in (5a), indicating more roundabout production processes with longer supply chains incorporating more non-EU value added. At the industry level for individual countries, however, they did find larger impacts of different types of coefficient changes, which leads them to conclude that sector policies may be more important for economic growth than indicated by Feldman et al. (1987), also because the economically much more open EU countries have less scope for macroeconomic policies than the economically more closed USA. Again the question is whether that conclusion is justified.
Finally, we consider SDA results for the third large international trading unit, i.e. China. Andreosso-O´Callaghan & Yue (2002) also find that the growth of total final demand, and specifically the export growth of ´high-tech´ industries, constitutes by far the largest contribution to Chinese output growth for 1987–1997. They, however, do not make a distinction between ordinary exports and processing exports that add only limited amounts of domestic value added to mainly imported materials. This distinction is important as processing exports—in contrast to ordinary exports—hardly have any indirect impacts on domestic value added. Pei et al. (2012), using Chinese IOTs with both kinds of exports separated for 2002–2007, conclude that the contribution of exports to domestic value added is overestimated with 32% if the two types of exports are aggregated, while the contribution of exports to the value added of the ´high-tech´ telecommunication industry is even overestimated with 63%. Still, they too report that the growth of domestic final demand “explains” as much as 70% of Chinese GDP growth, whereas changes in coefficients “explain” only minus 5%. The remainder of about 35% is “explained” by the growth of both types of exports.
In the above paragraph and earlier, the word explained has been put between quotation marks. The phrase “deterministically attributed to” would have been more correct, be it more cumbersome. As opposed to S&S, SDA is seldom criticized. The main critique (Rose & Casler 1996; Dietzenbacher & Los 1998; Miller & Blair 2009) regards (1) the non-uniqueness of each decomposition and (2) the weak theoretical foundation for taking averages. However, just like S&S, SDA also needs to be criticized because of (3) the impossibility to determine the statistical significance of its components and (4) its sensitivity to sectoral aggregation. As opposed to S&S, however, SDA does have a theoretical foundation, namely the demand-driven IO quantity model. In case of S&S the lack of a theoretical foundation and the related presence of a residual component can be turned into an advantage that solves the problem of establishing the statistical significance of its non-residual components.
In contrast, having a theoretical foundation may easily be considered to represent the weakest aspect of SDA, for two reasons. First, as opposed to S&S, and precisely because of its specific theoretical foundation, SDA does not have a residual component that, by dropping it in an econometric estimation, may be used to establish the statistical significance of the other components. Second, depending upon the type of application, the assumed causality of the underlying demand-driven IO quantity model may represent a major problem.
This second weaknes is, especially, a problem in the largest area of SDA applications, i.e. the decomposition of industry output growth and GDP growth. In case of short run, year-to-year economic fluctuations, especially when the economy operates below full capacity, the Leontief model more or less adequately captures the, under those conditions dominant demand-side causes of short run economic growth and decline (see Oosterhaven 2022). In case of the more often analysed, longer run changes over five or more years, however, SDA unjustly ignores the impact of changes on the supply-side of the economy, such as the growth of labour supply, the growth of the capital stock and technological progress.

4 Growth accounting: a supply-side story with a residual

In contrast, both neoclassical growth theory and new growth theory (Solow 1999; Sengupta 1998), as well as empirical analyses of long run economic growth (e.g. Durlauf et al. 1996) only look at supply-side factors to explain growth differences between regions and nations. The third decomposition approach, growth accounting (GA) perfectly fits into this last mentioned literature, as it ignores the demand side entirely and decomposes the growth of industry output and GDP exclusively into the contributions of supply-side components.
GA may be based in production theory (Diewert 1976, Caves et al. 1982). Using a translog function of production possibility frontiers, and assuming competitive factor markets, full input utilization and constant returns to scale, the relative growth of multi-factor productivity of industry j (\(\Delta \ln R_{j}\)) is estimated as the residual of the relative growth of the total output of industry j (\(\Delta \ln x_{j}\)) and the relative growth of its inputs, i.e. its use of capital \(k_{j}\), labour \(l_{j}\) and intermediate inputs \(z_{j}\), weighted with their respective cost shares wqj (Timmer et al. 2010, ch. 2):
$$\Delta \ln R_{j} = \Delta \ln x_{j} - w_{kj} \;\Delta \ln k_{j} - w_{lj} \Delta \;\ln l_{j} - w_{zj} \Delta \;\ln z_{j}$$
(6a)
$${\text{with}}\;w_{kj} = {{p_{kj} k_{j} } \mathord{\left/ {\vphantom {{p_{kj} k_{j} } {p_{j} x_{j} }}} \right. \kern-0pt} {p_{j} x_{j} }},w_{lj} = {{p_{lj} l_{j} } \mathord{\left/ {\vphantom {{p_{lj} l_{j} } {p_{j} x_{j} }}} \right. \kern-0pt} {p_{j} x_{j} }},w_{zj} = {{p_{zj} z_{j} } \mathord{\left/ {\vphantom {{p_{zj} z_{j} } {p_{j} x_{j} }}} \right. \kern-0pt} {p_{j} x_{j} }},\;{\text{and with}}\;w_{kj} + w_{lj} + w_{zj} = 1,$$
(6b)
wherein: R = level of multi-factor productivity, w = respective weights and p = respective prices.
In empirical applications, capital, labour and total intermediate inputs are often split up further, mostly by means of data from IOTs or supply-use tables (SUTs), while the weights are mostly calculated as the average of the begin year weight and the end year weight, as in the well-known and often used EU KLEMS database (see Timmer et al. 2010, ch. 3). In fact, (6) may be calculated directly by means of two IOTs or two SUTs. This results in decompositions of aggregate factor productivity changes that also attribute part of its growth to changes in industry mix of final demand and changes in IO coefficients (e.g. Wolff 1985, Casler & Gallatin 1997, see Kuroda & Nomura 2004, for a fine application to Japan).
The use of the same IO data seems to suggest that SDA and GA are just two extreme cases of a single, integrated approach. The IO model, however, assumes that the demand for outputs drives production while supply follows, as opposed to GA that assumes that the supply of inputs drives production whereas demand follows. This fundamental difference is most telling in the role that investment plays in both decomposition methods. In SDA, it are the year-to-year fluctuations in the demand for investment goods that co-determine the fluctuations in total output; a mechanism known as the multiplier (Samuelson 1939; Puu 1986). In GA, it is the level of investments that co-determines the longer run growth in the supply of capital and therewith the longer run growth of output; a mechanism known as the accelerator (Samuelson 1939; Puu 1986).
The primary field of the application of GA, as opposed to SDA, is that of estimating residual total factor productivity growth. Comparing the USA and Europe, van Ark et al. (2008, see also Timmer et al. 2010) e.g. show that Europe was catching up in labour productivity until about 1995, after which it experienced a slowdown, whereas the USA significantly accelerated its productivity growth, at least until 2006. At the detailed industry level, traditional manufacturing no longer acted as the productivity engine of Europe, probably due to exhausted catching up possibilities, while Europe´s industries lagged in participating in the new knowledge economy, lagged in investing in information and communication technology, and lagged in keeping up their multi-factor productivity growth. These differences, especially, led to an increasing gap in the productivity of European trade and business services, of course, with variations from industry to industry and from country to country. Note that demand is not mentioned in this analysis.
Also in the case of China, GA tells a story that is completely different from that of SDA, where growth of final demand is the dominant “explanation”. Wu (2016) decomposes China´s 9.16% annual GDP growth over the period 1980–2000 into 6.61% due to the growth of capital, 1.32% due to the growth of labour and 1.24% due to the growth of total factor productivity (TFP). Of the 1.32% due to labour growth, 75% is attributed to quality improvement and 25% to the growth of hours worked. Of the 1.24% due to TFP growth, 70% is attributed to TFP growth at the industry level and 30% to the reallocation of capital and labour between industries. Differences in the contribution of the individual industries to these aggregate results are mainly explained by industry differences in market structures and policy interventions, running from being essentially centrally planned to being open to world competition.
Comparable to SDA, the above type of GA analyses also suffer from their deterministic nature. Also in the case of GA the statistical significance of the components cannot be tested. GA simply believes the assumptions of the production function model underlying (6), just as SDA simply believes in the IO model, be it (3) or (4). In the case of GA, solely by assumption, demand does not play a role, whereas in the case of SDA solely by assumption supply does not pay a role. Note that looking at the supply-side only, is as one-sided as looking at the demand-side only. As said before, the latter may be more or less acceptable when analysing short run, year-to-year changes in economic growth, whereas looking at the supply-side is more appropriate when analysing longer run economic growth, however, both approaches remain one-sided (see Johansson and Westin 1987, for a more balanced approach).

5 Conclusion: econometric growth accounting

Luckily, in the case of GA there is no reason to restrict the analysis to supply-side factors only. The reason is that GA contains a residual component (i.e. \(\Delta \ln R_{j}\) in Eq. 6a), just like shift-and-share analysis. And comparable to S&S, dropping the residual component allows for an econometric estimation of industry output growth (i.e. \(\Delta \ln x_{j}\)) or GDP growth (i.e. \(\Delta \ln \left( {GDP} \right) = \Delta \ln \left( {\Sigma_{j} \,c_{j} x_{j} } \right)\)). Dropping the residual component thus delivers an estimate of the statistical significance of the, in that case, estimated instead of predetermined weights of the remaining components.
Such an econometric GA approach, furthermore, enables the analyst to estimate the real importance (if any) of demand-side factors in explaining shorter or longer run economic growth, namely by adding components from either a S&S and/or a SDA as composite explanatory variables. Such components may, for example be used to test the significance of industry mix changes, be it in total output as measured by means of a S&S or in final demand as measured by a SDA. Additionally, they may e.g. be used to test the significance of the total of all changes in intermediate input coefficients from a SDA or the significance of cluster or localisation economies as measured by the specialization component from a S&S.
In this way, the data will decide which factors are most important in explaining economic growth, instead of making the assumption that one knows the answer beforehand, as is done in all three decomposition analyses. Up till now only Broersma and Oosterhaven (2009) have done an econometric analysis, in their case of labour productivity growth, that uses S&S components as explanatory variables, but unfortunately they have not done that using a GA approach without its residual component, as advocated here. Hence, empirical tests of the dominance of supply-side factors versus demand-side factors versus industry mix factors in short run versus long run economic growth remain to be done.

Acknowledgements

The author thanks several anonymous referees for useful comments, which have improved this article.
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Footnotes
1
On August 14, 2024 “shift and share analysis” scored about 1,280 hits on Google Scholar, the combination of “structural decomposition analysis” and “input–output” scored about 7,190 hits, and the combination of "growth accounting" and "production function" scored about 21,300 hits.
 
2
Note that the definition of the share sometimes needs to be adapted to the definition of the variable, as will be the case when decomposing a ratio, such as labour productivity growth (see Oosterhaven & Broersma 2007).
 
3
In international economics, with v = export growth, r = some country, n = all of the world, and i = products, S&S is known as constant market share analysis (see Jepma 1986, for an overview and several applications).
 
4
In fact, S&S may be considered as a special case of index decomposition analysis (IDA, see Hoekstra & van den Berg 2003, for an overview). The difference is that IDA links the variable of interest (e.g. energy use or job growth) explicitly to the output by industry, which results in at least one extra component related to the change in the corresponding coefficient.
 
5
The fifth decomposition also proved to be superior from an empirical point of view in case of regional wage differences in The Netherlands, as its three components were mutually uncorrelated, as opposed to the components of the other decompositions (Oosterhaven & van Loon 1979).
 
6
Bold capitals = matrices, bold lower cases = vectors, italicized lower cases = scalars, headed bold lower cases = diagonal matrices, and i = unity vector filled with ones. Hence, the unity matrix \({\mathbf{I}} = {\hat{\mathbf{i}}}\). Finally, \(\in\) indicates that the left hand scalar is the typical element of the right hand vector or matrix.
 
7
De Boer & Rodrigues (2020), in an excellent overview, show how both index decomposition analysis (IDA) and SDA may benefit from the outcomes of index number theory, which also divides the (price-quantity) interaction term over the price and quantity indices. They furthermore present the corresponding multiplicative IDAs and SDAs where geometric averages are taken instead of the arithmetic average taken in the additive, third decomposition of Table 2.
 
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Metadata
Title
A decomposition of economic growth decompositions
Author
Jan Oosterhaven
Publication date
27-09-2024
Publisher
Springer Berlin Heidelberg
Published in
The Annals of Regional Science / Issue 4/2024
Print ISSN: 0570-1864
Electronic ISSN: 1432-0592
DOI
https://doi.org/10.1007/s00168-024-01309-7