On Perfect Aggregation in the National Output Deflator and Generalized Rybczynski Theorems

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Fisher and Shell [1972] attempted to place the theory of output deflation on a production-theoretic foundation in the same way as the theory of the cost-ofliving index is based on a consumer-theoretic foundation. It developed a production-theoretic output deflator to which actually computed indices, such as the GNP deflator, serve (sometimes) as bounding approximations. The present paper continues that work. In it, I inquire as to the possibility of building up the production-theoretic deflator from subindices each of which is a deflator for a particular sector or industry. In what sense, if any, is it possible, for example, to talk of the steel industry's contribution to inflation? Clearly, such a statement would be well defined if the full output deflator could be expressed in terms of specific industry deflators; the question is whether this is possible. Obviously, such questions are trivial when considering the Paasche approximation to national output deflation, the GNP deflator. Paasche or Laspeyres indices can be expressed in terms of subindices in a simple, arithmetic way. Yet to be content with this answer is, in effect, to ignore the question. The steel industry deflator in the Paasche approximation is a bound on the productiontheoretic deflator for the steel industry. The full Paasche approximation is a similar bound on the production-theoretic deflator for the economy as a whole. Yet if the economy-wide, production-theoretic deflator is not simply a function of production-theoretic deflators for separate industries, it is not at all plain that much interest should attach to the percent of Paasche-approximated deflation accounted for arithmetically by the Paasche deflator for the steel industry. More generally, someone attempting to construct deflators for a particular sector or industry will do so without regard for prices outside that sector. If this is done for every sector of the economy, can the results then be used directly to construct the national output deflator? If not, then what can one say concerning the errors that are likely to be committed in proceeding on a sector-by-sector basis? What approximation theorems can be proved? The present paper is only the beginning of research into such questions. It