True or False?
The statement is true. When the exchange rate is fixed, equilibrium in the small country depends on the intersection of its IS curve with the real interest rate line. Monetary shocks will not shift the LM curve because the small country's central bank will accommodate them with money supply changes as it purchases and sells foreign exchange reserves to keep the exchange rate from changing. Monetary shocks in the big country will affect the real interest rate, causing both countries' outputs to change in the same direction as they move along their respective IS curves. Since monetary shocks do not affect the real exchange rate and the nominal exchange rate is fixed, the two countries' price levels will move together in response to big-country monetary shocks.
The two crucial things to keep in mind are that equilibrium in the small country is determined by its IS curve in combination with the world real interest rate. Since the LM curve is endogenous, it automatically adjusts to cross through the intersection of the IS curve and the real interest rate line. These adjustments automatically compensate for exogenous monetary shocks affecting the curve.
The second thing to remember is that big-country monetary shocks affect the world real interest rate in the short run, causing output in both countries to expand along their IS curves, but have no long-run effect on outputs, the world real interest rate and the real exchange rate. This is because they have no long-run effect on the level of either country's IS curve or the full-employment level of output. Since the real and nominal exchange rates are both unaffected in the long run the price levels in the two countries must move in unison.