Q. If the central bank does not purchase foreign assets when output increases but instead holds the money stock constant, can it still keep the exchange rate fixed at E⁰?

Answer: No this is for the reason that the domestic interest rate would begin to rise above R*. Traders in the foreign exchange market would start to bid up the price of domestic currency in terms of foreign currency. In the deficiency of central bank intervention the exchange rate thus would fall below E⁰ so as to prevent this appreciation the central bank should sell domestic currency and buy foreign assets thus increasing the money supply and preventing any excess money demand from pushing the home interest rate above R*.