The use of multiplicative decompositions in economic analysis
Multiplicative decompositions are often used in economics, although they may not always be thought of as such. They provide a way to break down a variable, ‘y’, into two or more components in order to study the relative contributions of those components to changes in ‘y’ over time. They are a form of non-parametric economic modelling.
There can be any finite number of components, n, in a multiplicative decomposition. Such a decomposition for variable y takes the form in equation (1):
This can be rewritten in the form:
where the v’s are the n+1 ratio variables in equation (1).
A simple example would be the decomposition of an aggregate monthly wage bill, W, into components, given the number of employees during the month, L, and the total number of hours worked by all employees during the month, H:
where h is the average number of hours worked per employee and w is the average monthly wage rate per hour.
Useful properties of a multiplicative decomposition
Working with the example in equation (3), it is easily seen that:
In other words, the relative change in W in period t compared to period t-n is exactly equal to the product of the corresponding relative changes in the three decomposition variables. So if there was an increase in the number of employees of 1.5%, a decrease in the average hours worked of employees of -1% and a 2% increase in the average wage rate then the percentage change in the wage bill would be 2.5%:
Of course, rounding may not always make this relationship appear to be exactly true.
If one prefers, the relationship between changes in W and changes in its three components can be expressed in an additive form by using the logarithmic transformation:
Since 100 times the logarithmic change is a close approximation to the percentage change for small changes,1 equation (6) says that the percentage change in the wage bill is approximately equal to the sum of the percentage changes in L, h and w. In some applications this additive property is very useful.
More multiplicative decomposition examples
Lots of practical examples of the multiplicative decomposition can be found in economic analysis. In particular, numerous examples can be found where the decomposition involves only two components. One is the equation expressing real GDP as the product of labour productivity and total hours worked in the economy:
The first term in this decomposition, real GDP per hour worked in the economy, is interpreted as labour productivity. Statistics Canada releases quarterly statistics for labour productivity that are based on precisely this multiplicative decomposition in its tables 36-10-0206-01 and 36-10-0207-01.
Another example is the equation expressing household saving as the product of the saving rate and household disposable income:
During periods when household saving is not moving in step with household income, changes in the saving rate can be studied in relation to interest rates, unemployment rates and consumer confidence measures. Statistics Canada publishes data for household disposable income, household saving and the saving rate in its table 36-10-0112-01.
Yet another example is the equation relating greenhouse gas emissions (GHG) to economic activity.
Trends in the GHG/GDP ratio reveal how, in aggregate, technologies and output composition are changing through time that tend to raise or lower the GHG-intensity of Canadian output.
Fisher’s well known equation of exchange2 implies yet another multiplicative decomposition:
where M is the money supply, P is the price level, V is the velocity of money circulation and T/M is the ratio of transaction volume to the money supply. This equation can be used to decompose and interpret changes in the money supply over time.
The Bank of Canada did a simple multiplicative decomposition of real GDP growth into components for population growth and GDP per capita changes in Chart 13 its October 2024 Monetary Policy Review.
As a final example, consider the six-part decomposition in my paper “Accounting for the decline of Canada’s real GDP per capita since mid 2022.”
Cautions
Of course, one can also construct nonsensical multiplicative decompositions. For example, consider the example in equation (11):
In this formulation Canada’s real GDP is related to the intensity of auroras in the upper atmosphere via the Kp index.3 Over the last 25 years this index decreased markedly during the recessions of 2008-2009 and 2020 as shown in Chart 1. If one believes these recessions are causally related to variations in aurora intensity, one might want to distinguish this factor separately from others influencing changes in GDP by adopting the multiplicative decomposition (11).
But of course no one believes aurora intensity is causally related to GDP. This is a misleading formulation.
The point here is that multiplicative decompositions only have useful meaning to the degree that we believe them to be sensible. These decompositions do not test whether an hypothesis is reasonable. Rather, they assume an hypothesis is reasonable.
Another caution to keep in mind is that while a particular multiplicative decomposition may indeed seem reasonable, it might embed a lot of hidden complications. In the decomposition in equation (7), for example, where GDP is related to total hours worked and aggregate labour productivity, changes in the latter component might often be unrelated to changes in what we normally think as ‘productiveness’ or ‘efficiency’. They might, rather, just reflect shifts in the industrial composition of output between more and less capital-intensive industries, or temporary changes in the composition of employment between skilled and less skilled professions.4 Ratios of two aggregates must always be interpreted cautiously.
Conclusion
Multiplicative decompositions are easy to construct and can be very useful in economic analysis. They crop up frequently in binary form, while decompositions with three or more components are not seen as often. The utility of these decompositions depends crucially on the credibility of the relationship, especially since there are no statistical tests involved. When used, the analyst must be sure to explain and justify the roles of each component in the decomposition.
For example, the percentage change between 7.0 and 7.1 is 100*(7.1-7.0)/7.0 = 1.4286 and 100*log(7.1/7.0) is 1.4185.
The Purchasing Power of Money, its Determination and Relation to Credit, Interest and Crises, by Irving Fisher, assisted by Harry G. Brown (New York: Macmillan, 1922). New and Revised Edition.
According to Wikipedia, “An aurora (pl. aurorae or auroras), also commonly known as the northern lights (aurora borealis) or southern lights (aurora australis), is a natural light display in Earth's sky, predominantly seen in high-latitude regions (around the Arctic and Antarctic). Auroras display dynamic patterns of brilliant lights that appear as curtains, rays, spirals, or dynamic flickers covering the entire sky.” The SeeTheAurora.com website describes the Kp index as “… a scale used to characterize the magnitude of geomagnetic disturbances” (that is, auroras) and provides time series for this index.
The pandemic starting in March 2020 provides a good example of this kind of effect. When it began, firms laid off lower-skilled workers and retained more highly skilled and experienced workers earning higher wages. Output fell less than employment and productivity appeared to surge. But this was an unsustainable and temporary phenomenon.