What will happen to the trade balance and the real exchange rate of a small open economy when government purchases increase, as during a war? Does your answer depend on whether this is a local war or a world war?

Answer: When a government purchase increases during a war, be it a local war or a world war. it means that it's savings has reduced, therefore the trade balance will fall. And if the purchase is done to import more goods into the country, the trade balance becomes negative, leading to a deficit.

The exchange rate of the currency will reduce because the country the government is making more currency to be available and surplus, by increasing it's purchase. When they is excess currency in the world market, the currency reduces it value. In a world war, or local war, the exchange rate may not actually reduce because, it will be difficult for the country to have enough money to make its currency to be available in the world market.