# Contributions to percent change

The real Gross Domestic Product (GDP) by industry series estimates by province and territory are based on the chain Fisher formulaNote 1, which is not additive. Non-additivity of the real series comes both from chaining and from the Fisher formula itself. Chaining destroys the additive consistency of accounting equations and the Fisher formula does not have the additivity property. The fact that the real aggregates are not additive makes it more difficult to measure the contribution of an individual aggregate, sector or industry to the total economy, as the sum of the underlying components does not add to the total. This problem of additivity increases the further the distance from the reference year and the more relative prices change.

There are a variety of ways to overcome this additivity issue. For an analysis of industry shares, current values (or current prices) may be a sufficient or even more desirable alternative, because they reflect the economic structure at the prices of the period under study. For growth analysis, however, current values are not always appropriate as they combine both changes in prices and in volumes. A useful complementary measure is provided by the Contributions to Percentage Change (CPC) formula, which presents a strictly additive decomposition of the variations in the aggregate. The formula re-weights the contributions of the detailed series in such a way that they become strictly additive to the total variation of the aggregate.

The CPC formula is a function of GDP at current values, the Paasche and Laspeyres volumes of GDP and the Fisher volume index.

$\Delta {%}_{i,t-1\to t}=\frac{100×\left(\sum _{i}{\mathrm{CGDP}}_{t}^{i}}{\sum _{i}{\mathrm{CGDP}}_{t-1}^{i}}\right)×\left({\mathrm{LGDP}}_{t}^{i}-{\mathrm{CGDP}}_{t-1}^{i}\right)+{\mathrm{FV}}_{t}\left({\mathrm{CGDP}}_{t}^{i}-{\mathrm{PGDP}}_{t-1}^{i}\right)}{\sum _{i}{\mathrm{CGDP}}_{t}^{i}+{\mathrm{FV}}_{t}\sum _{i}{\mathrm{PGDP}}_{t-1}^{i}}$

Where:

CGDPi: GDP at current values for industry i at time t and t-1

LGDPi: Laspeyres GDP volume for industry i at time t

PGDPi: Paasche GDP volume for industry i at time t-1

FV: Fisher volume index at time t

This formula is the basis of the CPC series published by the Provincial and Territorial GDP by Industry program. It allows measuring the contribution of an individual industry or aggregate to the percentage change in total GDP in "real" terms.

The CPC statistic applies only to a single period. It should not be used to conduct a period over period growth rate analysis. Furthermore, the contributions to percentage change should not be interpreted as proportions.

Table 1 provides an example of contributions to percentage change in the context of a Fisher index. As noted above, the sum of the detailed industries does not add up to the total index for columns t and t-1, making it difficult to relate the individual industry growth rates to the aggregate growth rate. The CPC, however, provides a completely additive measure of the contribution of each industry to the aggregate growth rate.

Table 1: Example of contributions to percent change for a Fisher index
Table summary
This table displays the results of Example of contributions to percent change for a Fisher index. The information is grouped by Industry (appearing as row headers), t-1, t, Growth rate and CPC (appearing as column headers).
Industry t-1 t Growth rate CPC
A 108 144 33.33% 1.98
B 224 238 6.25% 1.66
C 525 540 2.86% 1.49
D 150 162 8.00% 1.22
Total (Fisher) 1003.7 1067.4 6.35% 6.35
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