Relative purchasing power parity

Suppose that the currency of Country A is called the A$ (A-dollar) and the currency of country B is called the B$.

The theory states that if the price P in country A of a basket of commodities and services is P A-dollars, then the price Q of the same basket in country B will be C×P A-dollars, where C is a constant which does not vary over time, or, equivalently, C×P×S B-dollars, where S is the (variable) number of B-dollars required to buy one A-dollar, i.e. the exchange rate.

If (1) and (2) denote two different dates, then it follows that

${\displaystyle {\tfrac {1}{C}}={\tfrac {P(1)\times S(1)}{Q(1)}}={\tfrac {P(2)\times S(2)}{Q(2)}}}$

and hence

${\displaystyle {\tfrac {S(2)}{S(1)}}={\frac {{\big (}{\tfrac {Q(2)}{Q(1)}}{\big )}}{{\big (}{\tfrac {P(2)}{P(1)}}{\big )}}}}$

or, in words, the factor representing the movement in market exchange rates is equal to the ratio of the inflation factors (changes in price levels) of the two countries (as one would intuitively expect).

The function and the deduction of the function follow that

${\displaystyle {\tfrac {\Delta (P/Q)}{P/Q}}={\frac {\Delta P(1)}{P(1)}}-{\frac {\Delta Q(1)}{Q(1)}}=({\frac {P(2)-P(1)}{P(1)}})-({\frac {Q(2)-Q(1)}{Q(1)}})=\pi (A)-\pi (B)}$

(Using logarithmic derivation)

${\displaystyle {\text{S=P/Q}}}$

and hence

${\displaystyle {\tfrac {\Delta S(1)}{S(1)}}=\pi (A,1)-\pi (B,1)}$

The function above implies that the rate of depreciation of the nominal exchange rate equals the difference between the inflation rates of two countries.

Absolute purchasing power parity occurs when C=1, and is a special case of the above.

According to this theory, the change in the exchange rate is determined by price level changes in both countries. For example, if prices in the United States rise by 3% and prices in the European Union rise by 1% the purchasing power of the EUR should appreciate by approximately 2% compared to the purchasing power of the USD (equivalently the USD will depreciate by about 2%).

Note that it is incorrect to do the calculation by subtracting these percentages – one must use the above formula, giving ${\displaystyle {\tfrac {1.01}{1.03}}}$ = 0.98058, i.e. a 1.942% depreciation of the USD. With larger price rises, the difference between the incorrect and the correct formula becomes larger.

Unlike absolute PPP, relative PPP predicts a relationship between changes in prices and changes in exchange rates, rather than a relationship between their levels. Remember that relative PPP is derived from absolute PPP. Hence, the latter always implies the former: if absolute PPP holds, this implies that relative PPP must hold also. But the converse need not be true: relative PPP does not necessarily imply absolute PPP (if relative PPP holds, absolute PPP can hold or fail).

Commonly called absolute purchasing power parity, this theory assumes that equilibrium in the exchange rate between two currencies will force their purchasing powers to be equal. This theory is likely to hold well for commodities which are easily transportable between the two countries (such as gold, assuming this is freely transferable) but is likely to be false for other goods and services which cannot easily be transported, because the transportation costs will distort the parity.[3]

• Exchange rate
• Purchasing power parity