Question regarding real exchange rate: q(f/d) = S(f/d) * (P(d)/P(f))
It says that an increasing real exchange rate q(f/d) means that it is becoming less expensive, in real terms, to shop in the foreign country.
The only way for q(f/d) to increase would be a higher domestic price index which means higher inflation. But if higher inflation means a depreciating currency, wouldn’t that make it more expensive for people in the domestic country to shop for foreign goods?
This simply means that appreciation of exchange rate leads to increase in importation in the short run. In the long run, because the inflow of foreign goods will be at a lower price, the domestic producers are forced to lower their price as a result of intence competition, and thus a fall in inflation and price level.
However, this strategy will deter growth as import is a leakage to the economy, and thus economic growth will slow down.
The real exchange rate (rearranging your formula) is
P(d) / [(P(f) / S(f/d)]
Thats simply the domestic price level divided by the foreign price level (in domestic currency).
An increasing RER per your formula implies that the domestic price level is increasing faster than the foreign price level (in domestic currency). Hence cheaper to shop in foreign country.
You are right, higher domestic inflation will ultimately reduce S(f/d) and bring about parity (in theory).