Existence and Uniqueness of PPP Indexes

Most indexes used for computation of purchasing power parities (PPP) are defined in this way: Suppose there are  j=1,...,m  countries and  i=1,...,n  commodities. PPP for jth-country is defined as a weighted average of prices of commodities in that country deflated by  world average price of the respected commodities. World average price for ith-commodity  P_i  in turn is defined as a weighted average of prices of the commodity in different countries deflated by respected PPPs. This leads to an  m+n  system of equations in m unknown PPPs and N unknown Ps. One question is when such systems have a unique positive solution. In this paper (joint with Prasada Rao) we address this question in detail. Our result can be summarised as follows:

  1. A necessary and sufficient condition for existence and uniqueness of these indexes is an intuitive condition called “connectedness” . This condition means that the set of countries cannot be divided into at least two groups with no commodity in common.
  2. The weights in PPP equations and  the weights in P equations cannot be independent.
  3. The main tool for proving such results is (nonlinear) PerronFrobenius theorems.

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