True or False?
The statement is true. In the case of monetary shocks, which affect the big country's LM curve only, that LM curve returns to its original level in the long-run regardless of what other countries do. In the case of real shocks in the big country, the world IS curve shifts by a given amount and stays at that level regardless of the nature of subsequent adjustments. As a result, in the long run the world interest rate must rise by the same amount whether other countries exchange rates with respect to the big country are fixed or flexible. And the big country's real exchange rate must adjust in the long-run by the amount that will shift the its IS curve to the left, and the rest-of-the world's aggregate IS curve to the right to maintain full employment everywhere. This will be independent of how the required adjustment of the LM curves takes place.
Note that the world IS curve is the sum of the individual countries' IS curves. Since real exchange rate changes simply reallocate the fixed world level of aggregate demand between the two countries, the post-shock world IS curve does not change in response to movements in the real exchange rate. World full-employment output is the sum of the full-employment output levels of the individual countries, and these output levels do not change. Any horizontal deviation of the big country's IS curve from its full employment level at a given world real interest rate must at all times be equal to the negative of the sum of the horizontal deviations of all other countries' IS curves from their full employment levels. Real goods market equilibrium in one country will thus imply real goods market equilibrium in the aggregate of all other countries.