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## Purchasing Power Parity and Real Exchange Rates March 13, 2011

Posted by tomflesher in Macro, Teaching.
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In an earlier post, I talked about purchasing power through the lens of Beerflation. (Again, hat tip to James at the Supine Bovine.) There, purchasing power was used to compare the relative values of the minimum wage in the past to the minimum wage now. Another way we can use purchasing power is to compare whether current prices in one country are the same as prices in the local currency of another country.

That requires some understanding of foreign exchange rates. The convention is to list the exchange rate as a fraction with the foreign country’s value in the numerator and the local currency in the denominator. For example, as of today, one dollar would buy about .72 euro, so the exchange rate in the US would be listed as €0.72/$1. One dollar would buy about 81.89 yen, so the exchange rate there is ¥81.89/$1.1 If we wanted to list the exchange rate of yen for euro, we could work it out using the US exchange rate to be ¥81.89/€0.72, or, dividing out, about ¥113.74/€1. Since we’re comparing currency instead of goods, we call this a nominal exchange rate.

If we stop to think about prices, this should mean that$1 worth of some random good that’s of uniform quality worldwide should be the same as €0.72 or ¥81.89 worth of that same good. Think, for example, about white sugar. Currently, Domino Sugar is listed on Amazon.com at$8.99 for a ten-pound bag. If my purchasing power is the same worldwide, then I should be able to get that same ten-pound bag for around (8.99*0.72) = €6.47 or for around (8.99*81.89) = ¥736.19. If I can, then my money is worth the same around the world – prices are all the same after I change my money. This is called purchasing power parity.

If purchasing power parity holds, then the price of a good in US dollars is the same, after conversion, as the price of a good in euro. That means the exchange rate of those currencies should be the same as the ratio of those prices. In symbols, we could define e as the nominal exchange rate, P as the local price, and P* as the foreign price. (This is the convention that Greg Mankiw uses in Brief Principles of Macroeconomics.) Then,

$e = \frac{P\star}{P}$

A real exchange rate, like all real variables, relates the prices of goods to each other. The classic example of a real price is to compare the price of a good you wish to buy with the number of hours you’d need to work to get it. If I make $22 per hour, and the price of Daron Acemoglu’s Introduction to Modern Economic Growth is$61.38 (currently), then regardless of the inflation rate or where our prices are pegged, the price of the book for me is 2.79 hours of work. In order to avoid having individual prices, I could peg the price not to hourly wages (which differ from person to person) but to some staple good. Given a \$1.95 can of black beans, Daron’s book is worth 31.48 cans of beans, and I make around 11.28 cans of beans per hour. If we divide those out, we’ll get 31.48/11.28 = 2.79 hours per copy of Introduction to Modern Economic Growth.

Neat, huh?

So, if a real price relates the price of goods to each other, a real exchange rate relates local prices to each other. In general, this is done using what’s called a basket of goods that’s supposed to represent what a typical consumer buys in a year. Here, for simplicity, we’ll stick with one good.

If purchasing power parity holds, then the ratio of those prices should be 1/1 after we correct for the foreign exchange rate. So, if we define RER as the real exchange rate between two countries, then

$RER = e \times P \times \frac{1}{P\star}$

Or, in other words, prices are the same after you exchange your money. That is, with purchasing power parity, the real exchange rate is 1. More simply, if purchasing power parity holds, then prices should be the same whether you change your money into a foreign currency or not.

1 I’m rounding all prices to the nearest hundredth for simplicity.

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