Price index

The price of an individual good can obviously change through time. What economists find, however, is that prices do not change altogether independently. On the contrary, the prices of whole classes of goods will often increase (or decrease) more or less simultaneously; even if a few of the goods in the class decrease (or increase) in price, for instance, it is often reasonable to say that the prices, taken as a whole, have gone up (or gone down). A price index is a numerical time series measure designed to help compare how the prices of some class of goods, taken as a whole, change over time. If the class of goods in question is particularly broad (e.g. "all the goods in the economy", or "the goods that a typical urban consumer needs to by"), then the price index is said to measure the economy's overall price level.

Almost all price indices are based on ratios. Thus

${\frac {\text{value of index at some time}}{\text{value of index at base time}}}$

should indicate by what factor prices have increased or decreased between the two periods. For instance, if you have an index where the value for 2006 is 100 and the value for 2007 is 105, then the quotient is 105/100 = 1.05 = 105%. This means that the overall price level has increase by 5% between 2006 and 2007.

Note that the individual values of the index, taken separately, don't represent anything about reality. Thus if you know that the value of a price index for 2006 is 100 but you do not know the value at any other time, then you don't actually know anything about the state of prices. It's only when you can compare the ratio between two index values that an index shows anything real, namely by what factor prices have increased.

Because only the ratio matters, when you take an index and multiply all its particular index values by the same constant factor, then you get an equivalent index. In the above example, for instance, it gives you the exact same information to say that the value at 2006 is $2*100=200$ and the value at 2007 is $2*105=210$.

Stated technically, but still very generally, a price index is a function $I(P,Q,n)$, where $P$ is information about the price of various goods at various times, $Q$ is information about the quantity sold of various goods at various times, and $n$ indicates the time period (e.g., the year, or the quarter) for which one wishes to compute the index.

In typical cases, not all prices and quantities of purchases can be recorded, so a representative sample is used instead. Price indices can be constructed for goods or services individually, by category, or across an entire economy. Economy-wide price indices are used to measure inflation. Price indices can also be used to help measure other economic statistics such as Gross Domestic Product.

Conceptually, price indices are often thought of as tracking the price of a basket of goods, such as a set of goods purchased by a typical consumer. When making such a price index based on such a market basket a certain set of items is chosen. The change in the total cost of the goods is then tracked by the price index as the prices of the items in the basket change. While intuitively easy, the fixed market basket approach can suffer from substitution bias. For example, in a consumer price index, substitution bias would occur if consumers stopped buying apples and only bought oranges because the cost of apples had risen and oranges were good substitutes. A price index based on a fixed basket of goods would overstate price change since it would reflect the rising cost of apples even though they were substituted out of the real world market basket. Instead of using a fixed market basket, some price indices attempt to reflect a cost of living concept. In a cost of living index, the standard of living for an initial period is gauged. The price index reflects the change in the amount of money required to maintain that standard of living in later periods. The cost of living index shows the change in the amount of money required to attain a certain level of utility.

A price index can also be thought of as measuring an average change in prices. In price index formulas price changes are represented by price relatives. A price relative is found by dividing a later period price by a base period price. Different price index formulas use different methods for weighting and average price relatives. A Jevons Index uses the unweighted geometric average of price relatives while other indices use weights and arithmetic averages.

While price index formulas all use price and quantity data, they amalgamate this data in different ways. A simple price index can be constructed using various combinations of base period prices ($p_{0}$),later period prices ($p_{t}$), base period quantities ($q_{0}$, and later period quantities ($q_{t}$). Price index formulas can be framed as comparing expenditures (An expenditure is a price times a quantity) or taking a weighted average of price relatives.

Notable price indices are

• Consumer price index
• Producer price index
• Personal consumption expenditures price index

The GDP deflator does not assume a fixed market basket of goods and services.

Most price indices are normalized to a value of 100 in the base year(s), to indicate the percentage level of the price index in each year relative to the base year. So a price-index value of 110 for a given year means that the price index is 10 percent higher in that year than the base year.

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 Englishman 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 Vaughan's fellow Englishman William Fleetwood probably 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

$\sum _{c\,\in \,C}(p_{c,t}\cdot q_{c,t})$

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 the inflation rate when taking the numeraire as the bundle of goods using previous prices but current quantities. Similarly, the Laspeyres index can be though 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:

$P_{ME}={\frac {\sum [p_{c,t_{n}}\cdot {\frac {1}{2}}(q_{c,t_{0}}+q_{c,t_{n}})]}{\sum [p_{c,t_{0}}\cdot {\frac {1}{2}}(q_{c,t_{0}}+q_{c,t_{n}})]}}={\frac {\sum [p_{c,t_{n}}\cdot (q_{c,t_{0}}+q_{c,t_{n}})]}{\sum [p_{c,t_{0}}\cdot (q_{c,t_{0}}+q_{c,t_{n}})]}}$

A fourth, the Fisher index (after the American economist Irving Fisher), is calculated as the geometric mean of $P_{P}$ and $P_{L}$:

$P_{F}={\sqrt {P_{P}\cdot P_{L}}}$

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

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 indexes (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 indexes 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 have $E_{c,t_{0}}=p_{c,t_{0}}\cdot q_{c,t_{0}}$ 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 preceeding time period. This can be done with any of the above indexes, 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:

$P_{t_{n}}={\frac {\sum (p_{c,t_{0}}\cdot q_{c,t_{1}})}{\sum (p_{c,t_{0}}\cdot q_{c,t_{0}})}}\times {\frac {\sum (p_{c,t_{1}}\cdot q_{c,t_{2}})}{\sum (p_{c,t_{1}}\cdot q_{c,t_{1}})}}\times \cdots \times {\frac {\sum (p_{c,t_{n}}\cdot q_{c,t_{n-1}})}{\sum (p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}}$

Note that, when chain indexes 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 there 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.

1. 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.

2. 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.

3. 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.

4. Commensurablity test:

   The index should not be affected by the choice of units used to measure prices and quantities.

5. 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.

6. 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.

7. Monotonicity test:

   $I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})\leq I(p_{t_{m}},p_{t_{r}},q_{t_{m}},q_{t_{r}})~~\Leftarrow ~~p_{t_{n}}\leq p_{t_{r}}$

   A price index for lower later prices should be lower than a price index with higher later period prices.

8. Mean value test:

   The overall price relative implied by the price index should be between the smallest and largest price relatives for all commodities.

9. Circularity test:

   $I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})\cdot I(p_{t_{n}},p_{t_{r}},q_{t_{n}},q_{t_{r}})=I(p_{t_{m}},p_{t_{r}},q_{t_{m}},q_{t_{r}})~~\Leftarrow ~~t_{m}\leq t_{n}\leq t_{r}$

   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 $t_{r}$.

See also

• Aggregation problem
• Inflation
• GDP deflator
• Etienne Laspeyres
• Hermann Paasche
• Irving Fisher
• Real versus nominal value

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.
• U.S. Department of Labor BLS “Producer Price Index Frequently Asked Questions”.
• Vaughn, Rice. A Discourse of Coin and Coinage (1675). (Also online by chapter.)

External links

• PPI data from BLS