The law of one price is connected to another measure of the exchange rate—the real exchange rate. This exchange rate is a measure of the price of goods and services in one country relative to another when prices are expressed in a common currency. It is about exchanging goods, rather than money, across countries.
The real exchange rate between the United States and Europe is given as follows:
You can think of the real exchange rate as the number of units of European gross domestic product (GDP) you can get for one unit of US GDP. [***Let us check the units of the real exchange rate. The US price level over the European price level is in dollars/euros: it is the price of a unit of US real GDP divided by the price of a unit of European real GDP. The nominal exchange rate is measured in euros per dollar. Thus the units are as follows: The dollars and the euros cancel out in this expression, so the real exchange rate is just a number.***] For example, if the price level in the
United States is $1,600, the price level in Europe is EUR 400, and the price of dollars in euros is EUR 0.5, then the real exchange rate is as follows:
One unit of US GDP will get you two units of European GDP.
The real exchange rate is intimately linked to the law of one price. The easiest way to see this is to suppose that we measure US real GDP and European GDP in the same units: that is, suppose we use the same bundle of goods in each case. We know that the law of one price should hold for tradable goods—that is, goods for which arbitrage is possible and practical. If every good that went into GDP were tradable, then the law of one price would hold for every good, and the real exchange rate would equal 1. If the real exchange rate was not 1, you could make arbitrage profits by buying and selling “units of GDP.”
As before, suppose the US price level is $1,600, the European price level is EUR 400, and the nominal exchange rate (dollars per euro) is 0.5. Imagine that US GDP and European GDP measure the same bundle of (tradable) goods. Then you could take $800 and buy EUR 400. With these euros, you could buy a basket of goods in Europe. You could sell this basket in the United States for $1,600. The law of one price is violated. We would expect the following:
- Prices in the United States would increase.
- Prices in Europe would decrease.
- The nominal exchange rate would depreciate (the dollar would become less valuable).
Because arbitrage is not possible for all goods and services, we do not expect—nor do we observe—the real exchange rate to be exactly one. But this benchmark is still useful in understanding movements in the real exchange rate.
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