What happens in a Mundell-Fleming IS/LM Model, foreign income down with fixed exchange rates

This post will be part of an ongoing, what happens in the Mundell Fleming Model series, in particular this post will answer: What happens in a Mundell-Fleming IS/LM Model when there is a decrease in foreign income given fixed exchange rates.  In the future I will go over the construction of the Mundell Fleming IS/LM model (which is just an open economy version of the IS/LM model) but for now I will go through the somewhat complicated problem we see when something happens in a foreign country, yet our home country keeps exchange rates constant.

First, we recognize the IS and LM lines, and the e on the Y axis represents exchange rates, which are endogenous in the model.  The horizontal line going through the model labeled FE shows the fixed exchange rates as dictated by the domestic economy.
When the income of a foreign country decreases, this enters in the equation for the IS equation.  But where does it enter, look at the equation for the IS curve below:



The only place where a foreign country can have influence is through the net exports (NX) variable.  NX shows us the amount of net exports for a given economy, so if exports are greater than imports, this number will be positive, and if imports are greater than exports, this number will be negative (think of it as imports lowering our domestic production because production is occurring overseas).  When foreign income goes down, then the amount that we are able to export to those countries goes down as well because they cannot afford as many of our goods as they were once able to. 

Since our NX term goes down, our Y goes down, and our IS curve shifts to the left.  But now we have an issue because we are no longer in equilibrium.  Either the LM curve needs to shift, or the exchange rate needs to change.  If the exchange rate could change, then we would arrive at a new equilibrium which results in a lower exchange rate, but the Y for our economy would stay the same (because the exchange rate is lower, other countries can afford more of our goods, and thus will buy more, in effect the change in the exchange rate would cancel our their decrease in income).  However, since our home economy has fixed exchange rates, this cannot occur and we need to shift our LM curve instead to get the resulting equilibrium condition.

The LM curve will shift because of the creation of arbitrage opportunities for investors.  This means that arbitrageurs will buy the home currency in the foreign-exchange markets (that are floating) and use them to buy foreign currency from the home market (where exchange rates are fixed).  This process automatically reduces the money supply, shifting the LM curve left, and will continue occurring until equilibrium at the fixed exchange rate level is reached.  The LM curve is moving left (less home currency available) because home currency available in foreign countries is going down, because it is being used to buy foreign currency from the home country's central bank.

So the takeaway from this problem is that a reduction in foreign income levels with a fixed exchange rate policy is that arbitrage will occur in the currency markets until a lower output results in the home economy, and the original fixed exchange rate is met.  If the exchange rate were allowed to float, then the LM curve wouldn't shift, and output would remain the same, only the exchange rate would lower.