Definitions

# Price index

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. Rice 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 eight fold 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 perhaps created 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
$sum_\left\{c,in, C\right\} \left(p_\left\{c,t\right\}cdot q_\left\{c,t\right\}\right)$
where
$p_\left\{c,t\right\},$ represents the prevailing price of $c$ in period $t$
$q_\left\{c,t\right\},$ 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
$q_\left\{c,t_n\right\}=q_c=q_\left\{c,t_0\right\}, forall c$
and
$P=frac\left\{sum \left(p_\left\{c,t_n\right\}cdot q_c\right)\right\}\left\{sum \left(p_\left\{c,t_0\right\}cdot q_c\right)\right\}$
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

$P=frac\left\{sum \left(p_\left\{c,t_n\right\}cdot q_\left\{c,t_n\right\}\right)\right\}\left\{sum \left(p_\left\{c,t_0\right\}cdot q_\left\{c,t_0\right\}\right)\right\}$

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

$P_P=frac\left\{sum \left(p_\left\{c,t_n\right\}cdot q_\left\{c,t_n\right\}\right)\right\}\left\{sum \left(p_\left\{c,t_0\right\}cdot q_\left\{c,t_n\right\}\right)\right\}$
while the Laspeyres index is computed as
$P_L=frac\left\{sum \left(p_\left\{c,t_n\right\}cdot q_\left\{c,t_0\right\}\right)\right\}\left\{sum \left(p_\left\{c,t_0\right\}cdot q_\left\{c,t_0\right\}\right)\right\}$
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 the inflation rate when taking the numeraire as the bundle of goods using base year prices but current quantities. Similarly, the Laspeyres index can be thought of as the inflation rate when the numeraire is given by the bundle of goods using current prices and current quantities.

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:

$P_\left\{ME\right\}=frac\left\{sum \left[p_\left\{c,t_n\right\}cdot frac\left\{1\right\}\left\{2\right\}cdot\left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right]\right\}\left\{sum \left[p_\left\{c,t_0\right\}cdot frac\left\{1\right\}\left\{2\right\}cdot\left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right]\right\}=frac\left\{sum \left[p_\left\{c,t_n\right\}cdot \left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right]\right\}\left\{sum \left[p_\left\{c,t_0\right\}cdot \left(q_\left\{c,t_0\right\}+q_\left\{c,t_n\right\}\right)\right]\right\}$

A fourth, the Fisher index (after the American economist Irving Fisher), is calculated as the geometric mean of $P_P$ and $P_L$:

$P_F = sqrt\left\{P_Pcdot P_L\right\}$

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.

The results of these two methods are likely to be very similar, but it can be shown that a theoretically correct approach would be to take a weighted average of the two, with the Fisher result being given twice the weight of the Marshall-Edgeworth result. (Consider chaining into infinitesimally small time periods. Integral of exp(t) from t=0 to t=n is approximately equal to (2/3)(exp(n/2)+(1/3)((exp(n)+1)/2).)

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.

### 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 $E_\left\{c,t_0\right\}$ be the total expenditure on good c in the base period, then (by definition) we have $E_\left\{c,t_0\right\} = p_\left\{c,t_0\right\}cdot q_\left\{c,t_0\right\}$ and therefore also $frac\left\{E_\left\{c,t_0\right\}\right\}\left\{p_\left\{c,t_0\right\}\right\} = q_\left\{c,t_0\right\}$. 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

$frac\left\{sum \left(p_\left\{c,t_n\right\}cdot q_\left\{c,t_\left\{n-1\right\}\right\}\right)\right\}\left\{sum \left(p_\left\{c,t_\left\{n-1\right\}\right\}cdot q_\left\{c,t_\left\{n-1\right\}\right\}\right)\right\}$

answers the question "by what factor have prices increased between period $t_\left\{n-1\right\}$ 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 $I\left(P_\left\{t_0\right\}, P_\left\{t_m\right\}, Q_\left\{t_0\right\}, Q_\left\{t_m\right\}\right)$, where $P_0$ and $P_n$ are vectors giving prices for a base period and a reference period while $Q_\left\{t_0\right\}$ and $Q_\left\{t_m\right\}$ give quantities for these periods.

1. Identity test:
2. :$I\left(p_\left\{t_m\right\},p_\left\{t_m\right\},alpha cdot q_\left\{t_m\right\},betacdot q_\left\{t_n\right\}\right)=1~~forall \left(alpha ,beta \right)in \left(0,infty \right)^2$
3. :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.
4. Proportionality test:
5. :$I\left(p_\left\{t_m\right\},alpha cdot p_\left\{t_m\right\},q_\left\{t_m\right\},q_\left\{t_n\right\}\right)=alpha cdot I\left(p_\left\{t_m\right\},p_\left\{t_n\right\},q_\left\{t_m\right\},q_\left\{t_n\right\}\right)$
6. :If each price in the original period increases by a factor α then the index should increase by the factor α.
7. Invariance to changes in scale test:
8. :$I\left(alpha cdot p_\left\{t_m\right\},alpha cdot p_\left\{t_n\right\},beta cdot q_\left\{t_m\right\}, gamma cdot q_\left\{t_n\right\}\right)=I\left(p_\left\{t_m\right\},p_\left\{t_n\right\},q_\left\{t_m\right\},q_\left\{t_n\right\}\right)~~forall \left(alpha,beta,gamma\right)in\left(0,infty \right)^3$
9. :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.
10. Commensurability test:
11. :The index should not be affected by the choice of units used to measure prices and quantities.
12. Symmetric treatment of time (or, in parity measures, symmetric treatment of place):
13. :$I\left(p_\left\{t_n\right\},p_\left\{t_m\right\},q_\left\{t_n\right\},q_\left\{t_m\right\}\right)=frac\left\{1\right\}\left\{I\left(p_\left\{t_m\right\},p_\left\{t_n\right\},q_\left\{t_m\right\},q_\left\{t_n\right\}\right)\right\}$
14. :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.
15. Symmetric treatment of commodities:
16. :All commodities should have a symmetric effect on the index. Different permutations of the same set of vectors should not change the index.
17. Monotonicity test:
18. :$I\left(p_\left\{t_m\right\},p_\left\{t_n\right\},q_\left\{t_m\right\},q_\left\{t_n\right\}\right) le I\left(p_\left\{t_m\right\},p_\left\{t_r\right\},q_\left\{t_m\right\},q_\left\{t_r\right\}\right)~~Leftarrow~~p_\left\{t_n\right\} le p_\left\{t_r\right\}$
19. :A price index for lower later prices should be lower than a price index with higher later period prices.
20. Mean value test:
21. :The overall price relative implied by the price index should be between the smallest and largest price relatives for all commodities.
22. Circularity test:
23. :$I\left(p_\left\{t_m\right\},p_\left\{t_n\right\},q_\left\{t_m\right\},q_\left\{t_n\right\}\right) cdot I\left(p_\left\{t_n\right\},p_\left\{t_r\right\},q_\left\{t_n\right\},q_\left\{t_r\right\}\right)=I\left(p_\left\{t_m\right\},p_\left\{t_r\right\},q_\left\{t_m\right\},q_\left\{t_r\right\}\right)~~Leftarrow~~t_m le t_n le t_r$
24. :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, $P\left(M\right)_\left\{t\right\}$, with the price of the new item in the later time period, $P\left(N\right)_\left\{t+1\right\}$.

• The "overlap method" uses prices collected for both items in both time periods, t and t+1. The price relative $\left\{P\left(N\right)_\left\{t+1\right\}\right\}$/$\left\{P\left(N\right)_\left\{t\right\}\right\}$ 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. $P\left(N\right)_\left\{t+1\right\}$/$P\left(M\right)_t$ 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 do 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.