# Nominal and Real Interest Rates

An important consideration when thinking about consumption across time is whether there is likely to be inflation. For example, suppose all you spend money on is apples, and that today apples cost $€1.00$ each. Therefore, if you have $€100$, then at today’s price of apples, that could buy 100 apples.

Now let’s think about what would happen if you saved some of your money. Suppose you can earn 10% interest on any savings. If you saved all you $€100$, therefore it would grow to $€110$ in a year. If the price of apples is unchanged, you could therefore afford 110 apples. So far, so good.

However, what would happen if the price of apples increased – say, to $€1.08$? Then with your $€110$, you could only afford $110/1.08 \approx 102$ apples. In terms of purchasing power, your savings only grew by about 2% (from 100 apples to about 102 apples).

How can we incorporate this into our model? The intertemporal budget constraint we’ve been dealing with has been
\(c_1 + {1 \over 1+r}c_2 = m_1 + {1 \over 1 + r}m_2\)
where $r$ is the *nominal* interest rate (10% in the example above). Instead of thinking of the axes of an intertemporal choice model as “dollars in the present” and “dollars in the future,” let’s think of them as “consumption in the present” and “consumption in the future.” (So, in the simple example above, it would be “apples in the present” and “apples in the future”.) Let’s assume that consumption in the present has a price of $p_1$, and consumption in the future has a price of $p_2$. Let’s also assume that we can express the income stream $(m_1, m_2)$ in real terms: for example, that $m_1$ is the amount of consumption your period-1 income can buy in period 1, and $m_2$ is the amount of consumption your period-2 income can buy in period 2. (One interpretation of this is that your wages are indexed to inflation, which is a feature of some negotiated labor contracts.) Then the intertemporal budget constraint becomes
\(p_1c_1 + {p_2 \over 1+r}c_2 = p_1m_1 + {p_2 \over 1 + r}m_2\)
(Note that all we’ve done here is to add in the prices $p_1$ and $p_2$.) Inflation is what relates $p_1$ and $p_2$: that is, if $\pi$ is the inflation rate, then
\(p_2 = (1 + \pi)p_1\)
Substituting this in for $p_2$ gives us
\(p_1c_1 + {p_1(1+\pi) \over 1+r}c_2 = p_1m_1 + {p_1(1+\pi) \over 1 + r}m_2\)
Now every term has a $p_1$, so we can cancel all those terms, leaving us with
\(c_1 + {1+\pi \over 1+r}c_2 = m_1 + {1+\pi \over 1 + r}m_2\)
Lastly, let’s define the “real” interest rate $\rho$ as:
\(1 + \rho = {1 + r \over 1 + \pi}\)
Substituting this into the budget constraint gives us
\(c_1 + {1 \over 1+\rho}c_2 = m_1 + {1 \over 1 + \rho}m_2\)
This is just the exact same budget constraint as we started out with, only with the real interest rate $\rho$ in place of the nominal interest rate $r$. In short, inflation simply has the effect of reducing the slope of the budget line, making it less advantageous to save and also less onerous to borrow.

Note that we can write $\rho$ as \(\rho = {1 + r \over 1 + \pi} - 1 = {1 + r \over 1 + \pi} - {1 + \pi \over 1 + \pi} = {r - \pi \over 1 + \pi}\) which is approximately equal to $r - \pi$ as long as $\pi$ is fairly low.