Price index

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A price index is any single number calculated from an array of prices and quantities over a period. In typical cases, not all prices and quantities of purchases can be recorded, so a representative sample is used instead. Inflation and cost indices are calculated as price indices.

Notable price indices are

The GDP deflator differs from the consumer and producer price indexes in that it does not assume a fixed market basket of goods and services.

Most price indexes 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.

[edit] Calculating price indices

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

$q_{c,t}\,$ represents the quantity of

c

sold

If, across two periods $t 1$ and $t 2$ , the same quantities of each good or service were sold, but under different prices, then

$q_{c,t_2}=q_c=q_{c,t_1}\, \forall c$

and

$P=\frac{\sum (p_{c,t_2}\cdot q_c)}{\sum (p_{c,t_1}\cdot q_c)}$

would be a 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.

When a price index is constructed in an attempt to measure relative prices for a given set of consumers or for the economy as a whole, quantities purchased are rarely if ever identical across any two periods. And a measure

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

would confuse growth or reduction in quantities sold with price changes. Various indices have been constructed in an attempt to compensate for this difficulty.

There are two main methods to calculate price indices, 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.

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.

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. A third index, the Fisher index (after the American economist Irving Fisher), tries to overcome this problem. It 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 that the overstatement and understatement will thus exactly one cancel the other.

Ironically, 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 against reality.