*This is the first of a series of papers concerned with PPP measures. The papers are in varying presentational styles and also reflects may growing understanding of the issues involved, and my improving presentation of them.*See **Measuring PPP-adjusted GDP Index** for the other papers. *This paper, written in September 2003, is a simple mathematical exposition.*

**Keywords**: Statistics;

This paper is presents a simple proposition:

*Where the international prices of exported goods are less than the purchasing power parity prices of the same consumption goods the purchasing power parity adjusted GDP measured on the production side will be less than the purchasing power parity adjusted GDP measured on the expenditure side for those countries which are net exporters of the good.*

The theorem, and its converse which applies to net importers, is important where there is a high level of international dumping, as occurs in regard to agricultural products.

The theorem is proved in the case of a single export (which may be a composite commodity).

Some Definitions

We take

Y(i) = the market value of good i, as produced in a particular country.

X(i) = the market vale of good i, as expended (purchased) in a particular country,

N(I) = the net exports of god I = Y(i)- X(i)

p(i) = the local price of good I and

P(i) = the PPP price which is applied in purchasing power parity comparisons to put all countries on the same price level, and which is measured in some standard currency say $US.

e = the exchange rate between the domestic currency and the standard currency.

For the record, the domestic value of any good is converted to the PPP level by multiply it by P(i)/p(i), except that (and it is this where the anomaly which gives the theorem arises) in expenditure side comparisons exports and imports are adjusted by the direct conversion of the value into the standard currency (e.g. the US dollar) at the standard exchange rate. So the conversion is by multiplying by e.

**Proof**

GDP adjusted for purchasing power parity on the __expenditure__ side (the usual way) is defined by

GDP(X) = ΣX*P/p + ΣN*e

GDP adjusted for purchasing power parity on the production side is defined by

GDP(Y) = ΣY*P/p

= Σ(X+N)*P/p

= ΣX*P/p + ΣN*P/p

= ΣX*P/p + ΣN*e + ΣN*(P/p-e)

= GDP(X) + ΣN*(P-p*e)/p

Thus if P(i) > p(i)*e where N > 0

Then

GDP(Y) > GDP(X)

QED.

This covers the case of a country which charges the same price for a good whether domestically purchased or exported. If the country itself dumps then the theorem involves treating the produced commodity as two separate items, in which case the āpā in the proof equations refers to the price involved in exporting.