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It is clear to me that increasing the interest rate should make loaning money more expensive, therefore money more scarce, and that that would curb aggregate demand, and that's what I recall I learned.

However, it seems to me that it would decrease aggregate supply too. (As it increases the capital costs of production and furthermore requires the business to increase their profits above the new interest rate)

How can we tell that the prices should decrease in such a case?

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    $\begingroup$ you should make clear you are talking about aggregate demand and supply, if you talk just about demand and supply it makes it look like you are only talking about market for loanable funds or some other single market, I took liberty to edit your Q to make it clearer $\endgroup$
    – 1muflon1
    Jul 27, 2023 at 13:35

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Here you need to be careful to distinguish between shifts and movement along aggregate demand and supply respectively.

Increase in interest rate shifts aggregate demand (AD) to the left, since aggregate demand is derived from goods market equilibrium (IS) and money market equilibrium (LM) giving and the AD is simply a collection of equilibrium points such IS=LM given $Y^*$ and $i^*$.

If you change interest rate whole AD curve shifts either to the left (higher interest rate) or to the right (lower interest rate), because there will be less or more demand at any price level. In the picture below you see the shift to the left.

However, aggregate supply (AS) works differently. First, when it comes to AS we distinguish between long-run AS (LAS) and short-run AS (SAS). Long-run AS is vertical and completely unaffected by interest rates since nominal interest rates do not affect amount of real resources, factors and technologies society has access to. In long run prices and hence profits would just adjust so in real terms they are the same as before interest change even if nominally the numbers might differ.

SAS however does respond to interest rate although not directly but indirectly through inflation (increase in price level). Lower interest rate means higher price level (result of inflation), which means lower real wages which allow firms to higher more workers and supply higher quantity of output. However, note this is movement along SAS not shift in SAS. Firms are not willing to supply more at any price level. They are willing to supply more because price level changed (and vice versa for drop of price level).

Hence what happens to supply is that you move along the SAS. There is less output supplied but no shift in supply. Figure below showcases this.

enter image description here

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