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 arithmetic means of the quantities: Marshall price index = ∑pnqo+∑pnqn ∑poqo+∑poqn * 100 Fisher’s index number:
A fourth, the Fisher index (after the American economist Irving Fisher), is calculated as the geometric mean of PP and PL: * 100
Fisher's index is also known as the “ideal” price index. As a measure of the inflation factor (one plus the inflation rate), a Fisher Price Index is the square root of the product of Laspeyres and Paasche Price Indexes. As a measure of the growth factor (one plus the rate of growth), a Fisher Quantity Index is the square root of the product of Laspeyres and Paasche Quantity Indexes.
However, there is no guarantee with either the Marshall–Edgeworth index or the Fisher index that the overstatement and understatement will exactly 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. Example:
| ∑ Poqo=623.70
Marshall price index = ∑pnqo+∑pnqn ∑poqo+∑poqn * 100 =524.75+600.15/623.70+704.2
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