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Index Numbers: Laspeyres, Paasche, and Fisher Price and Quantity Indexes
In defining an index number, always start with a ratio of sums. Each component of each sum is a price times a quantity. The number of such price/quantity products is equal to the number of goods or services included in the index.
The next step is to insert dates for the “?” marks.
If you want a Price Index, make sure only the prices in the numerator and denominator correspond to different dates. Quantities have the same date: earlier in both numerator and denominator, for a Laspeyres Index; later, for a Paasche Index. Therefore:
Laspeyres Price Index: Paasche Price Index:
If you want a Quantity Index, make sure only the quantities in the numerator and denominator correspond to different time periods. Prices have the same date: earlier in both numerator and denominator, for a Laspeyres Index; later, for a Paasche Index.
Laspeyres Quantity Index: Paasche Quantity Index:
In general, Laspeyres and Paasche Price Indexes or Inflation Factors will differ. Their geometric averages are called Fisher Indexes (after Irving Fisher).
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. Question: Why are Fisher Indexes useful averages of Laspeyres and Paasche Indexes?
Answer: The advantage of the geometric averages is revealed by considering the ratio of nominal values where $G is the rate of change of nominal GDP:
Note that both prices and quantities are different, comparing the numerator and the denominator. If the ratio were 2, for example, value has doubled (). A
In defining an index number, always start with a ratio of sums. Each component of each sum is a price times a quantity. The number of such price/quantity products is equal to the number of goods or services included in the index.
The next step is to insert dates for the “?” marks.
If you want a Price Index, make sure only the prices in the numerator and denominator correspond to different dates. Quantities have the same date: earlier in both numerator and denominator, for a Laspeyres Index; later, for a Paasche Index. Therefore:
Laspeyres Price Index: Paasche Price Index:
If you want a Quantity Index, make sure only the quantities in the numerator and denominator correspond to different time periods. Prices have the same date: earlier in both numerator and denominator, for a Laspeyres Index; later, for a Paasche Index.
Laspeyres Quantity Index: Paasche Quantity Index:
In general, Laspeyres and Paasche Price Indexes or Inflation Factors will differ. Their geometric averages are called Fisher Indexes (after Irving Fisher).
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. Question: Why are Fisher Indexes useful averages of Laspeyres and Paasche Indexes?
Answer: The advantage of the geometric averages is revealed by considering the ratio of nominal values where $G is the rate of change of nominal GDP:
Note that both prices and quantities are different, comparing the numerator and the denominator. If the ratio were 2, for example, value has doubled (). A