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Why is it that if lenders expect higher inflation in years to come, they will most likely increase $nominal$ rates of interest opposed to $real$ rates of interest?

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Lenders do not set interest rates solely based on what they would like to receive; in practice, nominal interest rates roughly reflect the expected path of the short-term rate of interest (which is set by the central bank), plus premiums for credit risk and the term of lending (term premium).

If inflation is expected to rise, the usual reaction of the central bank is to raise the policy rate. This will raise nominal interest rates. If this stabilises inflation expectations, inflation expectations may end up remaining near target, and the real rate rises. Since we cannot be sure sure what the effect on expectations will be, we do not know whether an increase in nominal rates translates into an increase in inflation expectations, or the real rate.

However, I would note that this question is somewhat ambiguous. One interpretation of the question is: why do lenders not specify the real rate when lending? (This is done in the inflation-linked bond market.)

For the developed countries, the reason why lenders do not specify lending contracts in real terms is straightforward: almost no private sector entity is willing to borrow at a fixed real rate. Since you have no guarantee that your income will rise with inflation, you have no idea whether you can repay the loan. In practice, we see that the only major borrower in the inflation-linked market is the central government (which has no worries about inflation). Therefore, lenders have little choice but to set nominal interest rates in contracts.

It should be noted that some countries with high inflation feature widespread indexation of contracts to inflation. I am not familiar with such countries, but it is possible that lending contracts could be set in real terms in such places.

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Better look at the Fisher rule (Fisher equation). nominal interest rate (i) = real interest rate (r) + inflation (pi) (i = r + pi)

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