Liquidity Trap and Foreign Exchange Rates: A Speculation

I might be wrong, but I want to share this idea. I'm not even sure I'm original at this.

Let's imagine an open economy in a liquidity trap. In the short run, price level is sticky but not one hundred percent. Perfect capital mobility and flexible exchange rates are assumed too. Nominal interest rate is zero, but the demand for money is much shorter than the money supply at zero nominal rate. It needs to fall further, but the central banks somewhat do not let it happen. The central bank has made a pledge to price stability and it is very responsible.

To stimulate the economy, however, the central bank implements very aggressive expansionary monetary policy. Initially, the return on foreign assets is expected to rise because much more domestic currency in the FX market is expected. Due to the lack of money demand and zero lower bound, actually, the quantity of money in the market does not grow: the market is worried about the central bank's future contraction and do not use the newly printed money. As a result, the expectation of higher exchange rate does not come true. Rather, the interest rate is not low enough at zero, so the domestic currency remains over-valued. As a result, return on foreign deposit is expected to fall and it will really come true because the interest rate does not change and there is net capital inflow; the economy runs the current account deficit.

Now, the government implements a massive expansionary fiscal policy. As a result, the domestic price level and output is expected to rise and the demand for money starts rising. The central bank still secures the increased monetary stock to stay there; in other words, it makes a pledge to be responsibly irresponsible at inflation over its target. The quantity of money in the market will grow, the zero nominal interest rate eventually becomes low enough, as a result, the domestic currency is eventually expected to depreciate and it indeed does. Then people start depositing in foreign accounts. As a result, there is a rising net capital outflow, so the net export will rise. There is not only no crowding out effect but also gains in the current account from expansionary fiscal policy.