Suppose that instead of consuming all of your earnings this
year you put $1000 in your local bank in a savings account on
which you earn interest at the rate of 5 percent per year. The
bank then loans this money to another person who uses it to purchase
a stereo set. At the end of the year, you can withdraw $1050---your
initial $1000 plus $50 interest. We can think of this $50 as a
payment to you for not consuming $1000 of your income during the year.
The individual who bought the stereo pays the bank back at the end
of the year with interest at, say, 10 percent. He pays $100 for the
privilege of spending $1000 more than he earned during the year. The
bank uses the $50 excess of interest received over interest paid to
cover the cost of managing your account, processing his loan and
providing a reserve to offset the proportion of its consumer loans
that turn out to be non-collectable.
The interest rate can thus be thought of as the price of spending money
today rather than spending it in the future.
In every society it costs more to have things now than in the future
because capital is productive---income not consumed this year and added
to the capital stock will produce additional income next year.
Suppose, for example, that the flow of income produced
per unit of capital stock is 0.05. By not consuming $100 this year and
thereby increasing the capital stock by $100 we can produce an additional
$5 of income next year. Next year we could, if we wanted to, consume the
$100 we put aside this year (thereby drawing the capital stock back to its
previous level) as well as the $5 income that $100 of capital produced
during the year. The rate of interest earned would be $5/$100 or 5 percent.
Nobody will ever loan money to others at zero interest when a positive rate
of return can be had by investing in real capital such as clothes, TV sets,
further education, and so forth. Observed interest rates in the economy will
therefore always be positive. It follows from all this that today's dollar will
grow if we save it with interest until tomorrow. And tomorrow's dollar is worth
less to us today than is today's dollar.
Let A0 represent an amount of money today and suppose that the
interest rate per year is r. If we postpone expenditure of these funds until
the end of the year we will have
1. A1 = (1 + r) A0
available to spend at that time. Looking at it the other
way around, we can see that a sum A1 to be received at the end
of the year is worth only
2. A0 = A1 ⁄ (1 + r)
today. Since r is positive, A0 is clearly less than A1.
The sum A0 is called the discounted or present value of the sum
A1. A piece of paper entitling you to receive the sum A1
in one year's time is an asset whose current value is A0.
In general, an asset is a claim to future principal and/or
earnings---its market value is the amount people will pay today
for the right to receive those future funds.
3. A2 = (1 + r) A1
by the end of that year so that we can substitute Equation 1 into
Equation 3 to obtain
4. A2 = (1 + r)(1 + r) A0
= (1 + r)2 A0
In general, the accumulated value of a sum A0 set aside
now at an interest rate of r percent per year will be
5. An = (1 + r)n A0
at the end of n periods. And, from a simple manipulation of Equation 5 it
is clear that a sum An to be received in n years is worth
6. A0 = An ⁄ (1 + r)n
now. The discounted or present value of a future sum is always less than that
sum because the interest rate is always positive.
You have already learned how output and income in the economy are
flows of returns off its capital stock. The ownership of the
capital stock is represented by assets that earn income. You have
also learned that by reducing consumption this year and investing a
greater fraction of current income in additions to the capital stock,
next year's income can be increased. People thus have the opportunity,
both individually and as a group, to shift consumption between the
present and the future---to have more today and less tomorrow, or to
forego consumption today in order to increase consumption in future years.
Suppose that we lend someone a sum of money A0 to be paid
back in two years. After one year the value of that sum will be
A1 as indicated by Equation 1. The sum A1 lent out
for the second year will grow to
Time now for a test. Make sure you have worked out your own answers to the questions
before clicking on the ones provided.