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Having just begun self-studying economics, I apologise if this question is too simple.

Do investments increase or decrease with increased interest rate?

According to the small, closed economy model (as far as I understand it) increased interest rates make borrowing harder and hence crowed out investment. However, in the large, open economy model it is mentioned that higher interest rates encourage both domestic and foreign investors to lend in the country, so investment can increase?

I appreciate that this boils down to a difference in foreign vs. domestic, and ultimately into the question wether investment is lending or borrowing. Anybody who could put me on the right track please?

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One quick way to reason the negative correlation is the concept of Present Value (PV). An increase of interest rate reduces the PV of cash (in-)flows occurring in subsequent periods.

in the large, open economy model it is mentioned that higher interest rates encourage both domestic and foreign investors to lend in the country, so investment can increase?

Yes, but that is because on the lender's side that leads to an arbitrage opportunity and/or a reduced cost of opportunity. Strictly speaking, high interest rates prompts lenders to lend money rather than to invest (or consume).

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History is plenty of different situations, but usually it works like this:

Low interest rates will increase demand of borrowing money and will decrease the interest of investors in lending (they will probably try to find another place with higher interest rates to invest in).

As a consequence, there will be less people offering money, so the interest rate will increase and the demand of borrowing money will decrease (because it will be more expensive). Maybe it will reach a kind of balance or maybe it will attract imvestors again (it depends on both local and international situation).

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