Most indexes used for computation of purchasing power parities (*PPP*) are defined in this way: Suppose there are countries and 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 in turn is defined as a weighted average of prices of the commodity in different countries deflated by respected *PPP*s. This leads to an system of equations in *m* unknown *PPP*s and *N* unknown *P*s. 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:

- 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. - The weights in
*PPP*equations and the weights in*P*equations cannot be independent. - The main tool for proving such results is (nonlinear)
*Perron*–*Frobenius*theorems.