A single-input/single-output firm

Although we focus in this book on indices for a multi-input/multi-output firm, by way of introduction it is useful to start with looking at a single-input/singleoutput firm. This is of course a highly artificial case, but extremely convenient to illustrate the main concepts. Thus, we consider a single firm through two time periods of equal length: a base period (t = 0) and a comparison period (t = 1). During the base period the firm uses x 0 quantity units of input to produce y 0 quantity units of output. Let the price (unit value) of the input in the base period be w 0 and the price of the output p 0. The corresponding data for the comparison period will be denoted by x l, y 1, w l, p l. All quantities and prices are assumed to be strictly positive. In this case it is quite natural to define the input price index number by
$$ {\omega ^1}/{\omega ^0}. $$


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  1. 1.
    The foregoing was inspired by Diewert (1992b, Section 2).Google Scholar
  2. 2.
    The first mention of such an index number appears, in the National Accounts context, in Copeland (1937). See Griliches (1996) on “the discovery of the residual.”Google Scholar
  3. 3.
    The Törnqvist quantity index is for instance extensively used in the U.S. Bureau of Labor Statistics productivity measurement program; see Dean, Harper and Sherwood (1996).Google Scholar
  4. 4.
    The fact that all four price or quantity indices differentially approximate each other to the first order at any point where w = w’ and x = x’,and that the Fisher and Törnqvist indices even approximate each other to the second order, as shown by Diewert (1978) and Vartia (1978), is of little practical help.Google Scholar
  5. 5.
    See, for instance, the empirical example provided by Allen and Diewert (1981). They suggest to give priority to the price index (quantity index) if there is less variation in the price relatives (quantity relatives) than in the quantity relatives (price relatives).Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  1. 1.Statistics NetherlandsThe Netherlands

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