Find the real exchange rate

Question

If the nominal exchange rate increases by 5%, while domestic inflation is 2% and foreign inflation is 3% then the real exchange rate changed by?

Answer 1

Let's say that one old Galleon (valid currency in the world of Harry Potter) is worth $x$ old dollars. Then change happens, and afterwards one old Galleon is worth the same as $1.02$ new Galleons, and one old dollar is worth $1.03$ new dollars. Also, the new exchange rate gives you $1.05\cdot x$ new dollars for each new Galleon.

Converting the new exchange rate to the old currencies, we get that the value of $1\over 1.02$ old galleons now give you $1.05\cdot x \over 1.03$ old dollars, or that one old galleon is now worth $${1.02\cdot 1.05\over 1.03}\cdot x=1.0398\cdot x {\rm \ old\ dollars}$$ In other words, the real exchange rate has increased by $3.98$ %.

We can get a good approximate answer here by noting that for small values of $a,b,c$, we have $${(1+a)(1+b)\over 1+c}\approx 1+(a+b-c)$$ which gives us that the real exchange rate increases by approximately $2+5-3=4$ %.

To verify the approximative formula, take logarithms of both sides, simplify the left hand side, and use $\log(1+x)\approx x$, which is valid for small $x$.