We measure inflation by choosing a unit of value which we assume constant in time and describing "real value" in those units. In the nineteenth century the unit of real value was an ounce of gold, but today it is a "market basket". (The US Bureau of Labor Statistics actually measures the inflation rate by sending workers with a fixed shopping list to stores and recording how much they pay.)
The inflation rate is a measure of how the price in dollars of a market basket of goods changes. More precisely, if
one basket costs m dollars today
andone basket costs m (1+a) dollars in a year,
then the inflation rate is a. For example, to say that the inflation rate is 3% means that stuff which costs $10 today will cost $10.30 in a year. Note thatone dollar today buys 1/m market baskets today
butin one year one dollar will buy 1/(m(1+a)) market baskets.
An investment of P dollars today at an annual interest rate of r grows to P(1+r) dollars in a year. The real growth rate of the investment is the growth rate measured in market baskets. It is computed as follows:
P dollars today buys P/m market baskets today.
ButP dollars to day grows to P(1+r) dollars in one year,
andin one year P(1+r) dollars will buy P(1+r)/(m(1+a)) market baskets.
In other words:an investment of qold=P/m market baskets today grows to qnew=P(1+r)/(m(1+a)) market baskets in one year.
The real growth rate is the number g such that(1+g)qold= qnew.
By the calculation on page 817 the real growth rate g is given byg = (qnew-qold)/qold = (r-a)/(1+a).
For example, if an investment pays interest of 4.53% and the inflation rate is 3.1%, the real growth rate isg= (0.0453-0.031)/1.031 = 0.01387=1.387%
The problems often assume a constant inflation rate. This is a reasonable assumption for predicting the future, but, as can be seen from table 21.5 on page 816, it is not true. For example, according to the table the price of a market basket was