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## Nominal and real interest rates

One of the most simple but crucial aspects of learning economics is understanding the relationship between nominal and real interest rates. When you see interest rates quoted for savings accounts or loans, they are usually nominal interest rates, expressed in terms of currency. Eg if you borrow £1000 for one year at 4%, then you pay back £1040, you are paying in interest 4% of the nominal value of the amount you borrowed.

However that does not tell you how the £1040 you repay compares to the £1000 you borrowed in terms of a basket of goods. If inflation was 3.5% then £1040 in a year’s time will be worth $\frac{1040}{1.035}=1004.83$ ie £1004.83 today. So you are actually only really paying $\frac{1004.83}{1000}=1.00483$ ie 0.483% interest in real terms.

This is a concept that most people in the real world won’t think about. They will just think about the nominal rate of interest quoted and not take account of how this will be eroded by inflation.

The relationship between real interest rates, nominal interest rates and inflation is

$(1+r_t) = \frac{(1 + i_t)}{1 + \pi^e_{t+1}}$.

This means that the real interest rate depends on the nominal interest rate and the expected rate of inflation next year. The real interest rate is an ex ante interest rate, because it is based on expectations of inflation. This means at the time you are making a decision (do I save/borrow at this nominal rate of interest?) you have to base your decision on what your expectations of inflation are. A year later, when you know what inflation actually was, you can find out the ex post real interest rate (ie what the real interest rate actually was, regardless of what the ex ante real interest rate suggested it would be).

You can make an approximation to this.

$(1+r_t) = \frac{(1 + i_t)}{1 + \pi^e_{t+1}} \Rightarrow (1 + r_t) (1 + \pi^e_{t+1}) = (1+i_t)$, when you expand the brackets you get:
$1+r_t + \pi^e_{t+1} + r_t \pi^e_{t+1} = 1 + i_t \Rightarrow r_t + \pi^e_{t+1} + r_t \pi^e_{t+1} = i_t$.

As the multiple of real interest rate and expected inflation $r_t \pi^e_{t+1}$ is likely to be small, we can approximate to:

$r_t + \pi^e_{t+1} \approx i_t$ or $r_t \approx i_t - \pi^e_{t+1}$.

Real interest rates are relevant when we are thinking about what something is worth in terms of a basket of goods, so for instance firms that are making decisions about whether or not to make an investment, will think in terms of real interest rates. The real interest rate is the relative price of current consumption compared to future consumption. It tells you how much consumption you gain in the future by sacrificing consumption now.

The IS relation uses the real interest rate: $Y=C(Y-T) + I(Y,r) + G + NX$. As $r \approx i - \pi^e$ we can roughly say $Y=C(Y-T) + I(Y,i - \pi^e) + G + NX$. This is how expected inflation influences the IS curve. A rise in expected inflation will mean that a higher nominal interest rate is needed for every level of output. So a rise in expected inflation will shift the IS curve upwards.

Nominal interest rates are relevant when we are thinking about money markets. When investors are deciding whether to hold money (that pays 0% interest) or some illiquid interest bearing asset like bonds (that pays i% interest) then they will think in terms of nominal interest rates.

The LM relation uses the nominal interest rate: $\frac{M}{P}=YL(i)$.