Suppose two transactions are conducted. In the first transaction, one loaf of bread is exchanged for $2. In the second transaction, one liter of milk is exchanged for $1. The price, or the rate of exchange, in the first transaction is $2/one loaf of bread. The price in the second transaction is $1/one liter of milk. In order to calculate the average price, we must add these two ratios and divide them by two; however, it is conceptually meaningless to add $2/one loaf of bread to $1/one liter of milk.
It is interesting to note that in the commodity markets, prices are quoted as dollars/barrel of oil, dollars/ounce of gold, dollars/tonne of copper, etc. Obviously it wouldn't make much sense to establish an average of these prices. Likewise, it doesn't make much sense to establish an average of the exchange rates dollar/sterling, dollar/yen, etc.
On this Rothbard wrote, "Thus, any concept of average price level involves adding or multiplying quantities of completely different units of goods, such as butter, hats, sugar, etc., and is therefore meaningless and illegitimate. Even pounds of sugar and pounds of butter cannot be added together, because they are two different goods and their valuation is completely different" (Man, Economy, and State, p. 734).
