Economists are a conservative lot, but sometimes even
they overestimate the cost of a long-run project. Recently I saw an economist’s reckoning of the
amount of aid that North Koreas would need to build a prosperous economy. The economist thought that this would take
$30 billion per year for 10 years. So the
total cost would be $300 billion.
Checkbooks, please.
The economist is using “current value”—the value of a
sum in the moment that it is received. As I’ll explain, it is more common to
use “present value”—the value today of a sum to be received in the future.
For example, suppose that the interest rate is
10%. Then the present value of $1 to be
received next year is 91 cents, since we can put 91 cents in a savings account
today and receive $1 next year (the principal of 91 cents plus 9 cents in
interest). The current value is $1, since we would receive next year a dollar
if we withdrew our funds then.
The advantage of present value over current value is
that it enables us to compare the values of sums received at different
times. Yes, the current value of $30
billion per year for 10 years is $300 billion; but this tells us little,
because we cannot compare this sum sensibly to, say, the sum of $25 billion per
year for 15 years. It is useless to compare
the current values of these two sums, because each sum involves different
periods. The obvious thing to do is to
express each of the two sums in terms of the same period—today. That’s what present value does.
Let’s return to our example of a $1 received next year
when the interest rate is 10%. We’ve already seen that the present value of
this dollar is 91 cents. To apply present value more generally, let’s express
it as a formula. In our example, the present
value of a dollar to be received next year is PV = $1/(1+.1), since PV*(1+.1)
= $1. More generally, PV = S/(1+i), where S is the amount to be received next year ($1 in our example) and i is the interest rate (10%).
We can extend the concept of present value to 10 years
or to any other period of years. A
two-year example will show how this works. Suppose again that the interest rate
is 10%. We want the present value of $1
to be received after two years. Suppose
that we put the amount PV in the bank
today. After one year, our savings account will hold the principal and
interest, or PV*(1.1). But this time, rather than withdraw this
amount, we leave it in the bank for another year. After the second year. the
savings account will have [PV(1.1)]*(1.1)
= $1. Solving, we find that PV = $1/(1.1)^2 = $.84. In other words, if we put 84 cents in the bank
today, we will have $1 after two years.
Thus the present value of $1 to be received after two years is 84 cents.
Generally, the present value of a sum S to be received after n years when the interest rate is i is PV
= S/(1+i)^n.
Your
tax dollar at work
Now let’s return to the North Korean project. Its present value reflects 10 annual payments
of aid, each with a current value of $30 billion. To estimate this value, we
will need the amount of money that, put in the bank today, would yield $30
billion after one year; plus the amount
of money that, put in the bank today, would yield $30 billion after two years;
and so forth. The present value of the project is PV = $30B/(1+i)^1 + $30B/(1+i)^2 + … + $30B/(1+i)^10. The B denotes
a billion.
We still need the interest rate. For barroom calculation, a reasonable interest
rate might be 3%, since this might roughly approximate the annual growth of
real (that is, physical or human) capital over the very long run. The idea is this: To borrow money for
expanding their factories, producers are willing to pay an interest rate as
high as 3%, since they can pay off the interest out of profits (expressed in
terms of the new capital). Anyway, using this interest rate, the present value
of the North Korea project turns out to be $256 billion. Not pocket change, but
not $300 billion either.
In his proposal, the economist sums up the current
values of $30 billion in each of 10 years to arrive at an estimate of $300
billion. Most economists would argue
that this overstates the true cost of the project, since its cost today is less than $300 billion. After all, if we put $256 billion in the bank today when the interest rate is 3%, then we can pay out $30 billion (in current value) each year. Or, to put it another way, if we put $300 billion in the bank today, then we can pay out each year, in addition to the $30 billion of "principal," the interest that has accumulated on it. The sum of those payments over 10 years will exceed $300
billion.
Note, incidentally, that present value falls when the
interest rate rises—and rises when the interest rate falls. If the interest rate falls to 0%, then present value will equal current value, since there is no interest. Always worth bearing in mind for a pop quiz.
For simplicity, I assume no inflation. Actually, I assume a lot of things; but for
a Valentine’s Day post, the aphorism KISS seems apropos. –Leon Taylor
tayloralmaty@gmail.com
Good
reading
Frederic S. Mishkin.
The economics of money, banking,
and financial institutions. Twelfth
edition. New York: Pearson. 2019.
