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Linking the Chain: A New Method of Inflation Adjustment
by Mark Lovewell
In 2003, Statistics Canada refined its method for calculating real GDP. Its former method, explained in
Figure 9.3 of the text, stressed reference-year prices in weighting the outputs of various products. This
method created the potential for bias over time, especially when there was a range of price changes, with
some prices rising much more than others.1
The Basic Method
To review Statistics Canada's former calculation method, let's take two years, 2006 and 2007, where the
first represents the reference year. Recall that quantities of individual products appearing in GDP are
valued both at current-year and reference-year prices. The total valuation in current-year prices is divided
by the total valuation in reference-year prices to give the relevant value of the GDP deflator. For 2006, the
reference year, the GDP deflator equals 100. For 2007, the calculation results in a ratio, with individual
quantities for 2007 appearing both in the numerator and denominator. In the numerator, these quantities
are valued at 2007 prices, and in the denominator at 2006 prices. Once the GDP deflator is derived, real
GDP is found through the following formula:
real GDP = nominal GDP/GDP deflator (in hundredths)
Note the role played by prices from the reference year (in this case 2006) in these calculations. As long as
relative prices in the economy stay approximately similar, using reference-year prices to weight quantities
in the denominators of each of the GDP deflator ratios is not a significant problem. But if some prices
change far more rapidly than others, or if some prices are even falling, then the weighting based on
reference-year prices diverges considerably from the way quantities would be weighted using current-year
prices.
Chain Volume Index
Statistics Canada's new method minimizes this potential divergence. The method employs a concept
known as a chain volume index, which refers to the way a series of price calculations, each involving
volumes of output from two consecutive years, are linked together to form a continuous chain. To
understand this approach, let's focus on an economy's GDP in the same two years as before, 2006 and
2007. Calculating the change in real GDP using a chain volume index involves four basic steps. In the first
step, the individual quantities that appear in GDP in each of the two years are both valued at 2006 prices
to give two aggregate dollar amounts – one for total output in 2006 and one for total output in 2007. The
first step is completed by the calculation of the percentage increase in total value, expressed in terms of
2006 prices, between these two years. In the second step, the calculations of the two aggregate dollar
amounts are repeated, but now with 2007 rather than 2006 prices. The results of steps one and two are a
pair of estimates of the percentage change in total value between the two years, one using 2006 prices
and one using 2007 prices. The third step involves taking the average of these two percentage changes,
which gives a final estimate of the change in real GDP. In the fourth step, the 2007 estimate of real GDP is
employed to derive the GDP deflator for 2007. This is done by rearranging the equation for the GDP
deflator:
GDP deflator (in hundredths) = nominal GDP/real GDP
The set of calculations outlined above is repeated for each pair of consecutive years, with the real GDP
found for 2007 representing the starting point for the calculations involving 2007 and 2008. The result is a
series of real GDP values much more closely tied to the underlying price changes taking place in the
economy than the estimates found using Statistics Canada's former calculation method.
Calculations Compared
The differences between the two methods can be highlighted with a numerical example. Consider a
hypothetical economy whose output consists of just two items – hamburgers and sodas. Figure A shows
output and prices in 2006 and 2007. While 20 hamburgers and 50 sodas are produced in 2006, with prices
of $4 and $1.50 respectively, in 2007 output increases to 40 hamburgers and 60 sodas, with prices of $10
and $1.75.
Figure A
2006
2007
Output and Prices in a Two-Product Economy
Hamburger Output
Hamburger Price
20
$4
40
$10
Soda Output
50
60
Soda Price
$1.50
$1.75
Basic Method
Let's find real GDP using Statistics Canada's former calculation method, extending the approach used for a
one-product economy in Figure 9.3 of the text. We'll again presume that 2006 is the reference year, so
real GDP in 2006 is the same as nominal GDP, because the GDP deflator, by definition, equals 100. This
nominal GDP value is found using the following equation:
2006 Nominal GDP = 2006 quantities at 2006 prices
= (20 hamburgers x $4) + (50 sodas x $1.50)
= $155
The 2007 real GDP is derived first by calculating the GDP deflator, which equals current-year output
valued at current-year prices divided by current-year output valued at reference-year prices, with this
fraction multiplied by 100:
2007 GDP deflator = (2007 quantities at 2007 prices)/(2007 quantities at 2006 prices) x 100
= (40 x $10) + (60 x $1.75) x 100
(40 x $4) + (60 x $1.50)
= $505 x 100 = 202
$250
The 2007 real GDP of $250 is then found by dividing the year's nominal GDP [$505 = (40 hamburgers x
$10) + (60 sodas x $1.75)] by the GDP deflator expressed in hundredths (= 2.02). Based on this
calculation method, the growth in this economy's real GDP between 2006 and 2007 is 61 [= (($250 $155))/$155)] percent.
Chain Volume Index
Now let's use the new method based on the chain volume index. Figure B shows the first step. We find the
total value of output in both years using 2006 prices, then calculate the percentage increase in this total
value. In this case, the increase in total value is from $155 to $250, or a change of $95 [= ($250 $155)], giving a percentage increase of 61 [= ($250 - $155)/$155] percent. Note that this first step, in
which 2007 output is valued at 2006 prices, gives the same result as Statistics Canada's former
calculation method. That's because the two procedures are synonymous, with current-year output in each
method valued at 2006 prices.
Figure B
(1)
Product
2006 and 2007 Output Using 2006 Prices
(2)
(3)
(4)
2006 Output
2006 Price
2006 Output
at 2006 Price
(2) x (3)
Hamburgers
20
$4
$ 80
Sodas
50
$1.50
75
$155
(5)
2007 Output
(6)
2006 Price
40
60
$4
$1.50
(7)
2006 Output at
2006 Price
(5) x (6)
$160
90
$250
Now let's complete the second step of finding the total value of output in both years using 2007 prices.
The calculations are shown in Figure C. The increase in the value of output between the two years is from
$287.50 to $505, or a change of $217.5 [= ($505 - $287.50)], giving a percentage increase of 76 [=
($505 - $287.50)/$287.50] percent.
Figure C
(1)
Product
2006 and 2007 Output Using 2007 Prices
(2)
(3)
(4)
2006 Output
2007 Price
2006 Output
at 2007 Price
(2) x (3)
Hamburgers
20
$10
$200.00
Sodas
50
$1.75
87.50
$287.50
(5)
2007 Output
(6)
2007 Price
40
60
$10
$1.75
(7)
2006 Output at
2007 Price
(5) x (6)
$400
105
$505
In step three, the average of the two estimated percentage changes in total value is found to be 68.5 [=
(61 + 76)/2] percent. The estimate of real GDP in 2007, using this calculated percentage change as well
as the 2006 real GDP value of $155, is therefore $261.18 [= ($155 x 1.685)/100], which differs quite
markedly from the real GDP estimate ($250) found using the former calculation method. Finally, in step
four, the 2007 value of the GDP deflator is also found to be considerably different than before:
GDP deflator (in hundredths) = nominal GDP/real GDP = $505/$261.18 = 1.93
The Divergence Explained
In common sense terms, the reason for the differences in the results found using the two calculation
methods is that Statistics Canada's former method included only one half of the estimate – that part
involving step one's highlighting of 2006 prices – now incorporated in the chain volume index. In our
hypothetical example, the price of hamburgers has risen so much (from $4 to $10), that the chain volume
index, which also incorporates the 2007 price of hamburgers in its weighting, does a better job of
capturing the relatively large increase in hamburger production (from 20 to 40) occurring between 2006
and 2007. Therefore the chain volume index gives a higher real GDP estimate than the one found with
Statistics Canada's former approach. If hamburger prices and outputs were both to continue to increase
significantly, this trend would be emphasized over time as the calculations using the chain volume index
would be repeated for each pair of consecutive years. In the contemporary Canadian economy, the range
of price changes for different products can be quite large over time, so Statistics Canada's new approach
can be seen as a significant improvement over its previous inflation adjustment process.
Notes
1. Karen Wilson, The Introduction of Chain Volume Indexes (Ottawa: Statistics Canada, 2003), pp. 1-2.
If directed to do so by your instructor, the following questions can be answered online and emailed to
him/her or yourself.
In the same hypothetical economy as highlighted in the article, output in 2008 increases to 50
1 hamburgers
and 120 sodas. Meanwhile, the price of hamburgers drops to $8 and the price of sodas
rising to $4. Find real GDP and the GDP deflator for 2007 and 2008, presuming that 2007 is the
reference year, using Statistics Canada's former calculation method.
Using the illustration provided in question 1, now find real GDP and the GDP deflator in 2008 using
2 Statistics
Canada's new calculation method. Explain the divergence in your answers.