For the equations linking real and nominal exchange rates, it is hard to keep track of which end is up. This exercise is meant to help with that.

First, in Mankiw’s textbook, the nominal exchange rate is expressed as the number of units of foreign currency that can be purchased with one unit of domestic currency. The real exchange rate E is then the effective price of domestic goods compared to foreign goods, and the effective price of domestic goods for foreigners is eP, where P is the domestic price and e is the nominal exchange rate. So

E = eP/P*

where P* is the foreign price. This equation can also be solved for the nominal exchange rate, yielding

e = EP*/P.

Here the intuition is that the higher the price in the foreign country, the more units of its currency we should get, for things to be fair in the sense of having a reasonable real exchange rate.

Normally different countries use different systems for their price indices, that are not comparable. In the exercise below, the price indices listed have been carefully constructed to be comparable between the two countries. For example, looking at the price of Big Macs is a simple example of a price index that is constructed to be comparable across countries. In this case, imagine that there is a basket of things that you could buy in either country, and the price index tells the number of units of the relevant currency needed to buy that basket.

"Purchasing Power Parity" means the same thing the real exchange rate being equal to 1: E=1. But the real-world principle associated with Purchasing Power Parity is that the real exchange rate seldom wanders too far from 1. All of the examples in the tables below have a real exchange rate between .5 and 2 inclusive.

Please fill in the missing number in each row using the equations above. (There is another exercise based on annual percent changes below these questions and answers.)

Here are the answers:

The equation E = eP/P* can be transformed into percent change form as

%ΔE = %Δe + %ΔP - %ΔP*

Dividing by the number of elapsed years yields the equation

(%ΔE/yr) = (%Δe/yr) + (%ΔP/yr) - (%ΔP*/yr)

The percent change in a price per year is the same thing as inflation. Let’s call the domestic inflation rate π and the foreign inflation rate π*. Then

(%ΔE/yr) = (%Δe/yr) + π - π*

To keep things straight, think of the percent change in the real exchange rate, which is how expensive things look in the U.S. to foreigners, as coming partly from the percent change in the nominal exchange rate, and partly from our prices going up faster than theirs.

Rearranging this equation to solve for the annual rate of change in the *nominal* exchange rate yields a closely related equation:

(%Δe/yr) = (%ΔE/yr) + π* - π

Here think of the intuition as saying that if the foreign prices are going up faster than the domestic prices, we should get more units of the foreign currency to be fair in the sense of having a reasonable real exchange rate not too different from where it is now.

The exercise below is based on these equations about percent changes per year in real and nominal exchange rates and rates of inflation in the two countries. Since the real exchange rate E seldom gets too far from 1, over long time periods, its rate of change is likely to be quite small. The exercise below reflects that.

Here are the answers: