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The geometric average of a Laspeyres and a Paasche index is known as a Fisher Index:
In certain cases, Laspeyres and Paasche index numbers are homologous in character and produce symmetric results.
GNP deflators are in essence Paasche indexes, showing what the market basket of goods and services consumed in the given year would have cost in the base year.
We have obtained relatively simple exact expressions for the relationship between the rate of gross output TFP growth and the corresponding rate of real value added TFP growth when the Laspeyres or Paasche index number formulae are used to aggregate inputs and outputs.
(46) Although the theoretical foundations have, to the best of our knowledge, not been established, it is possible that a superlative chained Fisher Ideal index that is the geometric mean of a chained Laspeyres and chained Paasche index would embody offsetting drifts, thereby resulting in a more reliable chained index than either the Laspeyres or Paasche index alone.
The Paasche index is defined as ([c.sub.2004] + [p.sub.2004][n.sub.2004])/([c.sub.2004] + [c.sub.1997][n.sub.2004]), which is a small number given the rapid decline in the price of computers; i.e., [p.sub.1997] is very high relative to [p.sub.2004].
(1) Paasche Index = P[I.sub.t] = [[SIGMA] [P.sub.t] x [Q.sub.t]]/[[SIGMA] [P.sub.t-1] x [Q.sub.t]]
Conversely, the Paasche index uses the prices of the current period to value current- and previous-period output:
Specifically, the goods are linked into the Paasche index using reservation prices equal to their launch price; exiting products are linked out of the Laspeyres index using a reservation price equal to their exit price.
It is possible to analyze other price-index formulas (e.g., the Paasche index) using the basic approach taken in Figure 8.
The representation of the growth of the Paasche index is necessarily cluttered because neither December of year y-1 nor month m of year y is the base period for the index (unless m happens to be December).
Conversely, quantity measures derived from a Paasche index, which is expressed as [sigma][q.sub.n][p.sub.n]/[sigma][q.sub.0][p.sub.n] and shows changes in quantities with prices at current period values, usually understate the output change.