A
price index (
plural: “price indices” or
“price indexes”) is a normalized
average
(typically a
weighted
average) of
prices for a given class of
goods or
services in a given region, during a
given interval of time. It is a
statistic
designed to help to compare how these prices, taken as a whole,
differ between time periods or geographical locations.
Price indices have several potential uses. For particularly broad
indices, the index can be said to measure the economy's
price level or a
cost
of living. More narrow price indices can help producers with
business plans and pricing. Sometimes, they can be useful in
helping to guide investment.
Some notable price indices include:
History of early price indices
No clear consensus has emerged on who created the first price
index.
The
earliest reported research in this area came from Welshman
Rice Vaughan
who examined price level change in his 1675 book A Discourse of Coin and
Coinage. Vaughan wanted to separate the inflationary
impact of the influx of precious metals brought by Spain
from the
New World from the effect due to currency debasement. Vaughan compared
labor statutes from his own time to
similar statutes dating back to
Edward III. These statutes set wages for
certain tasks and provided a good record of the change in wage
levels. Vaughan reasoned that the market for basic labor did not
fluctuate much with time and that a basic laborers salary would
probably buy the same amount of goods in different time periods, so
that a laborer's salary acted as a basket of goods. Vaughan's
analysis indicated that price levels in England had risen six to
eightfold over the preceding century.
While Vaughan can be considered a forerunner of price index
research, his analysis did not actually involve calculating an
index. In 1707 Englishman
William
Fleetwood created perhaps the first true price index. An Oxford
student asked Fleetwood to help show how prices had changed. The
student stood to lose his fellowship since a fifteenth century
stipulation barred students with annual incomes over five pounds
from receiving a fellowship. Fleetwood, who already had an interest
in price change, had collected a large amount of price data going
back hundreds of years. Fleetwood proposed an index consisting of
averaged price relatives and used his methods to show that the
value of five pounds had changed greatly over the course of 260
years. He argued on behalf of the Oxford students and published his
findings anonymously in a volume entitled
Chronicon
Preciosum.
Formal calculation
Given a set C of goods and services, the total market value of
transactions in C in some period t would be
- \sum_{c\,\in\, C} (p_{c,t}\cdot q_{c,t})
where
- p_{c,t}\, represents the prevailing price of c in period t
- q_{c,t}\, represents the quantity of c sold in period t
If, across two periods t_0 and t_n, the same quantities of each
good or service were sold, but under different prices, then
- q_{c,t_n}=q_c=q_{c,t_0}\, \forall c
and
- P=\frac{\sum (p_{c,t_n}\cdot q_c)}{\sum (p_{c,t_0}\cdot
q_c)}
would be a reasonable
measure of the
price of the set in one period relative to that in the other, and
would provide an
index measuring
relative prices overall, weighted by quantities sold.
Of course, for any practical purpose, quantities purchased are
rarely if ever identical across any two periods. As such, this is
not a very practical index formula.
One might be tempted to modify the formula slightly to
- P=\frac{\sum (p_{c,t_n}\cdot q_{c,t_n})}{\sum (p_{c,t_0}\cdot
q_{c,t_0})}
This new index, however, doesn't do anything to distinguish growth
or reduction in quantities sold from price changes. To see that
this is so, consider what happens if all the prices double between
t_0 and t_n while quantities stay the same: P will double. Now
consider what happens if all the
quantities double between
t_0 and t_n while all the
prices stay the same: P will
double. In either case the change in P is identical. As such, P is
as much a
quantity index as it is a
price
index.
Various indices have been constructed in an attempt to compensate
for this difficulty.
Paasche and Laspeyres price indices
The two most basic formulas used to calculate price indices are the
Paasche index (after the German economist
Hermann Paasche ) and the
Laspeyres
index (after the German economist
Etienne Laspeyres ).
The Paasche index is computed as
- P_P=\frac{\sum (p_{c,t_n}\cdot q_{c,t_n})}{\sum (p_{c,t_0}\cdot
q_{c,t_n})}
while the Laspeyres index is computed as
- P_L=\frac{\sum (p_{c,t_n}\cdot q_{c,t_0})}{\sum (p_{c,t_0}\cdot
q_{c,t_0})}
where P is the change in price level, t_0 is the base period
(usually the first year), and t_n the period for which the index is
computed.
Note that the only difference in the formulas is that the former
uses period n quantities, whereas the latter uses base period
(period 0) quantities.
When applied to bundles of individual consumers, a Laspeyres index
of 1 would state that an agent in the current period can afford to
buy the same bundle as he consumed in the previous period, given
that income has not changed; a Paasche index of 1 would state that
an agent could have consumed the same bundle in the base period as
she is consuming in the current period, given that income has not
changed.
Hence, one may think of the Paasche index as one where the
numeraire is the bundle of goods using base year
prices but current quantities. Similarly, the Laspeyres index can
be thought of as a price index taking the bundle of goods using
current prices and current quantities as the numeraire.
The Laspeyres index systematically overstates inflation, while the
Paasche index understates it, because the indices do not account
for the fact that consumers typically react to price changes by
changing the quantities that they buy. For example, if prices go up
for good c then,
ceteris
paribus, quantities of that good should go down.
Fisher index and Marshall-Edgeworth index
A third index, the
Marshall-Edgeworth index (named
for economists
Alfred Marshall and
Francis Ysidro Edgeworth),
tries to overcome these problems of under- and overstatement by
using the arithmethic means of the quantities:
- P_{ME}=\frac{\sum [p_{c,t_n}\cdot
\frac{1}{2}\cdot(q_{c,t_0}+q_{c,t_n})]}{\sum [p_{c,t_0}\cdot
\frac{1}{2}\cdot(q_{c,t_0}+q_{c,t_n})]}=\frac{\sum [p_{c,t_n}\cdot
(q_{c,t_0}+q_{c,t_n})]}{\sum [p_{c,t_0}\cdot
(q_{c,t_0}+q_{c,t_n})]}
A fourth, the
Fisher index (after the American
economist
Irving Fisher), is
calculated as the
geometric mean of
P_P and P_L:
- P_F = \sqrt{P_P\cdot P_L}
Fisher's index is also known as the “ideal” price index.
However, there is no guarantee with either the Marshall-Edgeworth
index or the Fisher index that the overstatement and understatement
will thus exactly one cancel the other.
While these indices were introduced to provide overall
measurement of relative prices, there is
ultimately no way of measuring the imperfections of any of these
indices (Paasche, Laspeyres, Fisher, or Marshall-Edgeworth) against
reality.
Practical measurement considerations
Normalizing index numbers
Price indices are represented as
index
numbers, number values that indicate relative change but not
absolute values (i.e. one price index value can be compared to
another or a base, but the number alone has no meaning). Price
indices generally select a base year and make that index value
equal to 100. You then express every other year as a percentage of
that base year. In our example above, let's take 2000 as our base
year. The value of our index will be 100. The price
- 2000: original index value was $2.50; $2.50/$2.50 = 100%, so
our new index value is 100
- 2001: original index value was $2.60; $2.60/$2.50 = 104%, so
our new index value is 104
- 2002: original index value was $2.70; $2.70/$2.50 = 108%, so
our new index value is 108
- 2003: original index value was $2.80; $2.80/$2.50 = 112%, so
our new index value is 112
When an index has been normalized in this manner, the meaning of
the number 108, for instance, is that the total cost for the basket
of goods is 4% more in 2001, 8% more in 2002 and 12% more in 2003
than in the base year (in this case, year 2000).
Relative ease of calculating the Laspeyres index
As can be seen from the definitions above, if one already has price
and quantity data (or, alternatively, price and expenditure data)
for the base period, then calculating the Laspeyres index for a new
period requires only new price data. In contrast, calculating many
other indices (e.g., the Paasche index) for a new period requires
both new price data and new quantity data (or, alternatively, both
new price data and new expenditure data) for each new period.
Collecting only new price data is often easier than collecting both
new price data and new quantity data, so calculating the Laspeyres
index for a new period tends to require less time and effort than
calculating these other indices for a new period.
Calculating indices from expenditure data
Sometimes, especially for aggregate data, expenditure data is more
readily available than quantity data. For these cases, we can
formulate the indices in terms of relative prices and base year
expenditures, rather than quantities.
Here is a reformulation for the Laspeyres index:
Let E_{c,t_0} be the total expenditure on good c in the base
period, then (by definition) we haveE_{c,t_0} = p_{c,t_0}\cdot
q_{c,t_0}and therefore also\frac{E_{c,t_0}}{p_{c,t_0}} =
q_{c,t_0}.We can substitute these values into our Laspeyres formula
as follows:P_L=\frac{\sum (p_{c,t_n}\cdot q_{c,t_0})}{\sum
(p_{c,t_0}\cdot q_{c,t_0})}=\frac{\sum (p_{c,t_n}\cdot
\frac{E_{c,t_0}}{p_{c,t_0}})}{\sum E_{c,t_0}}=\frac{\sum
(\frac{p_{c,t_n}}{p_{c,t_0}} \cdot E_{c,t_0})}{\sum
E_{c,t_0}}
A similar transformation can be made for any index.
Chained vs non-chained calculations
So far, in our discussion, we have always had our price indices
relative to some fixed base period. An alternative is to take the
base period for each time period to be the immediately preceding
time period. This can be done with any of the above indices, but
here's an example with the Laspeyres index, where t_n is the period
for which we wish to calculate the index and t_0 is a reference
period that anchors the value of the series:
P_{t_n}=\frac{\sum (p_{c,t_1}\cdot q_{c,t_0})}{\sum (p_{c,t_0}\cdot
q_{c,t_0})}\times\frac{\sum (p_{c,t_2}\cdot q_{c,t_1})}{\sum
(p_{c,t_1}\cdot q_{c,t_1})}\times\cdots\times\frac{\sum
(p_{c,t_n}\cdot q_{c,t_{n-1}})}{\sum (p_{c,t_{n-1}}\cdot
q_{c,t_{n-1}})}
Each term
- \frac{\sum (p_{c,t_n}\cdot q_{c,t_{n-1}})}{\sum
(p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}
answers the question "by what factor have prices increased between
period t_{n-1} and period t_n". When you multiply these all
together, you get the answer to the question "by what factor have
prices increased since period t_0.
Nonetheless, note that, when chain indices are in use, the numbers
cannot be said to be "in period t_0" prices.
Index number theory
Price index formulas can be evaluated in terms of their
mathematical properties
per se. Several different tests of
such properties have been proposed in index number theory
literature. W.E. Diewert summarized past research in a list of nine
such tests for a price index I(P_{t_0}, P_{t_m}, Q_{t_0}, Q_{t_m}),
where P_0 and P_n are vectors giving prices for a base period and a
reference period while Q_{t_0} and Q_{t_m} give quantities for
these periods.
- Identity test:
- :I(p_{t_m},p_{t_m},\alpha \cdot q_{t_m},\beta\cdot
q_{t_n})=1~~\forall (\alpha ,\beta )\in (0,\infty )^2
- :The identity test basically means that if prices remain the
same and quantities remain in the same proportion to each other
(each quantity of an item is multiplied by the same factor of
either \alpha, for the first period, or \beta, for the later
period) then the index value will be one.
- Proportionality test:
- :I(p_{t_m},\alpha \cdot p_{t_m},q_{t_m},q_{t_n})=\alpha \cdot
I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})
- :If each price in the original period increases by a factor α
then the index should increase by the factor α.
- Invariance to changes in scale test:
- :I(\alpha \cdot p_{t_m},\alpha \cdot p_{t_n},\beta \cdot
q_{t_m}, \gamma \cdot
q_{t_n})=I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})~~\forall
(\alpha,\beta,\gamma)\in(0,\infty )^3
- :The price index should not change if the prices in both
periods are increased by a factor and the quantities in both
periods are increased by another factor. In other words, the
magnitude of the values of quantities and prices should not affect
the price index.
- Commensurability test:
- :The index should not be affected by the choice of units used
to measure prices and quantities.
- Symmetric treatment of time (or, in parity measures, symmetric
treatment of place):
-
:I(p_{t_n},p_{t_m},q_{t_n},q_{t_m})=\frac{1}{I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})}
- :Reversing the order of the time periods should produce a
reciprocal index value. If the index is calculated from the most
recent time period to the earlier time period, it should be the
reciprocal of the index found going from the earlier period to the
more recent.
- Symmetric treatment of commodities:
- :All commodities should have a symmetric effect on the index.
Different permutations of the same set
of vectors should not change the index.
- Monotonicity test:
- :I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \le
I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\Leftarrow~~p_{t_n} \le
p_{t_r}
- :A price index for lower later prices should be lower than a
price index with higher later period prices.
- Mean value test:
- :The overall price relative implied by the price index should
be between the smallest and largest price relatives for all
commodities.
- Circularity test:
- :I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \cdot
I(p_{t_n},p_{t_r},q_{t_n},q_{t_r})=I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\Leftarrow~~t_m
\le t_n \le t_r
- :Given three ordered periods t_m, t_n, t_r, the price index for
periods t_m and t_n times the price index for periods t_n and t_r
should be equivalent to the price index for periods t_m and
t_r.
Quality change
Price indices often capture changes in price and quantities for
goods and services, but they often fail to account for improvements
(or often deteriorations) in the quality of goods and services.
Statistical agencies generally use
matched-model price
indices, where one model of a particular good is priced at the same
store at regular time intervals. The matched-model method becomes
problematic when statistical agencies try to use this method on
goods and services with rapid turnover in quality features. For
instance, computers rapidly improve and a specific model may
quickly become obsolete. Statisticians constructing matched-model
price indices must decide how to compare the price of the obsolete
item originally used in the index with the new and improved item
that replaces it. Statistical agencies use several different
methods to make such price comparisons.
The problem discussed above can be represented as attempting to
bridge the gap between the price for the old item in time t,
P(M)_{t}, with the price of the new item in the later time period,
P(N)_{t+1}.
- The overlap method uses prices collected for both
items in both time periods, t and t+1. The price relative
{P(N)_{t+1}}/{P(N)_{t}} is used.
- The direct comparison method assumes that the
difference in the price of the two items is not due to quality
change, so the entire price difference is used in the index.
P(N)_{t+1}/P(M)_t is used as the price relative.
- The link-to-show-no-change assumes the opposite of the
direct comparison method; it assumes that the entire difference
between the two items is due to the change in quality. The price
relative based on link-to-show-no-change is 1.
- The deletion method simply leaves the price relative
for the changing item out of the price index. This is equivalent to
using the average of other price relatives in the index as the
price relative for the changing item. Similarly, class
mean imputation uses the average price relative for items with
similar characteristics (physical, geographic, economic, etc.) to M
and N.
See also
Notes
- Chance, 108.
- Chance, 108-109
- Statistics New Zealand; Glossary of Common Terms,
“Paasche Index”
- Statistics New Zealand; Glossary of Common Terms,
“Laspeyres Index”
- Diewert (1993), 75-76.
- Triplett (2004), 12.
- Triplett (2004), 18.
- Triplett (2004), 34.
- Triplett (2004), 24-26.
References
- Chance, W.A. “A Note on the Origins of Index Numbers“, The
Review of Economics and Statistics, Vol. 48, No. 1. (Feb.,
1966), pp. 108-10. Subscription URL
- Diewert, W.E. Chapter 5: “Index Numbers” in Essays in Index
Number Theory. eds W.E. Diewert and A.O. Nakamura. Vol 1.
Elsevier Science Publishers: 1993. ( Also
online.)
- McCulloch, James Huston. Money and Inflation: A Monetarist
Approach 2e, Harcourt Brace Jovanovich / Academic Press,
1982.
- Triplett, Jack E. “Economic Theory and BEA's Alternative Quantity and Price
indices”, Survey of Current Business April 1992.
- Triplett, Jack E.
[http://www.oecd.org/dataoecd/37/31/33789552.pdf Handbook on
Hedonic Indexes and Quality Adjustments in Price Indexes: Special
Application to Information Technology Products. OECD
Directorate for Science, Technology and Industry working paper.
October 2004.
- U.S. Department of Labor BLS “Producer Price
Index Frequently Asked Questions”.
- Vaughan, Rice. A
Discourse of Coin and Coinage (1675). (Also online
by chapter.)
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