Let me try to make the intuition of my previous post simpler and clearer.
Suppose a country wants to invest more, because the profitability of investment has increased. It wants to increase the stock of kapital (machines) in its factories. There are three ways it could do this:
1 Divert some of its own production away from producing consumer goods towards producing investment goods, and consume less (save more).
2 Divert some of its own production away from producing consumer goods towards producing investment goods, but keep consumption (and saving) the same by importing consumer goods.
3 Import more investment goods, keep its own production of consumer and investment goods the same, and keep consumption (and saving) the same.
A closed economy can only do 1. And this will require an increase in the rate of interest (or something) to persuade households to postpone consumption. And it will require an increase in the price of investment good relative to consumption goods, to persuade firms to switch from producing consumption to investment goods.
An open economy can also do either 2 or 3. In both 2 and 3 there will be what economists call "capital flows" (borrowing from abroad, a surplus on the capital account of the balance of payments, matching a deficit on the current account). But only 3 has what I will call "kapital flows".
And the difference matters, because when we talk about "perfect capital mobility" in international finance/macroeconomics (and draw a horizontal BP curve in the Mundell-Fleming model and say a small open economy has the real interest rate pinned down by arbitrage at the world rate we are talking about capital flows (international borrowing and lending) and not kapital flows (importing or exporting machines).
If I understand Paul Krugman's model correctly, he assumes that 3 is impossible. A country cannot import investment goods, it can only import consumer goods. It must be self-sufficient in investment goods. Investment goods are non-traded goods. And he assumes that each country produces a different consumption good. The home country can only produce apples; the foreign country can only produce bananas. So if an open economy does 2, the price of apples must rise relative to the price of bananas, to persuade people to consume fewer apples and more bananas. And when the demand for increased investment goes back to normal, the price of apples will fall again, relative to the price of bananas. And if people expect this, the real interest rate in terms of apples (nominal interest minus expected apple inflation rate) must rise relative to the real interest rate in terms of bananas. And yet, if I understand Paul's model correctly, each country only produces one good, that can either be consumed or invested, so we have the simple "Neoclassical" production function Y=F(K,L)=C+I and I=dK/dt, so the price of the investment good is always the same as the price of the consumption good, because firms can switch from producing consumption to investment goods, at exactly the same cost of each, so the Marginal Product of Kapital is always equal to the real rate of interest in terms of the consumption good. (Otherwise, instead of r=MPK, we would instead have r=(MPK/Pk) + (dPk/dt)/Pk where Pk is the price of the investment good in terms of the consumption good, and the second term is the expected growth rate of that price.)
(Yes, I am reading things into Paul's post, to try to reveal his implicit assumptions, and I may be wrong on this.)
And I am saying there is a theoretical problem with this. Because it is hard to understand why the consumption good can be traded but the investment good cannot, if the two goods are the same good. Hard, but not impossible, because we could assume that each country can only use a very special type of investment good that can only be produced at home, and cannot be imported. (You can only invest in apple orchards by planting apples, which can't be grown in foreign climes.) But this does not seem generally plausible.
And I am saying that 3 is more likely, because investment goods can and do get imported. Or we could imagine 3', where the country exports fewer investment goods, so it can invest more at home, which amounts to the same thing. And if 3 is what happens, the world price of investment goods might rise, and world interest rates will rise, if the home country is a large country. But if the home country is small, relative to the rest of the world, these will not happen. And there is no reason why the apple rate of interest should rise relative to the banana rate of interest, because the mix of consumption need not change. (If the shock were a change in home country consumption and saving, rather than investment, this would cause relative interest rates to change.)
On the other hand, if the increased investment were an increase in investment in housing, I think 2 would be more plausible than 3, and Paul's model would be better. Because most of the work to build a new house must be done on site, even though the lumber and materials can be imported (I think that's right). But even then, there is a problem with any model of kapital where the price of kapital goods never changes relative to the price of consumer goods. Especially if we are talking about houses, recently.
So maybe I should do the classic (and perfectly legit) economist's wimp-out and say: "It depends". But at least I've said something about what it depends on.