Page 280 - Krugmans Economics for AP Text Book_Neat

you do with it? You put it in the bank; in effect, you are lending the $1,000 to the bank,
which in turn lends it out to its customers who wish to borrow. At the end of a year, you
will get more than $1,000 back—you will receive the $1,000 plus the interest earned.
All of this means that having $1,000 today is worth more than having $1,000 a year
from now. As any borrower and lender know, this is what allows a lender to charge a
borrower interest on a loan: borrowers are willing to pay interest in order to have
money today rather than waiting until they acquire that money later on. Most interest
rates are stated as the percentage of the borrowed amount that must be paid to the
lender for each year of the loan. Whether money is actually borrowed for 1 month or 10
years, and regardless of the amount, the same principle applies: money in your pocket
today is worth more than money in your pocket tomorrow. To keep things simple in
the discussions that follow, we’ll restrict ourselves to examples of 1-year loans of $1.
Because the value of money depends on when it is paid or received, you can’t evaluate
a project by simply adding up the costs and benefits when those costs and benefits ar-
rive at different times. You must take time into account when evaluating the project be-
cause $1 that is paid to you today is worth more than $1 that is paid to you a year from
now. Similarly, $1 that you must pay today is more burdensome than $1 that you must
pay next year. Fortunately, there is a simple way to adjust for these complications so that
we can correctly compare the value of dollars received and paid out at different times.
Next we’ll see how the interest rate can be used to convert future benefits and costs
into what economists call present values. By using present values when evaluating a proj-
ect, you can evaluate a project as if all relevant costs and benefits were occurring today
rather than at different times. This allows people to “factor out” the complications cre-
ated by time. We’ll start by defining the concept of present value.

Defining Present Value

The key to the concept of present value is to understand that you can use the interest rate
to compare the value of a dollar realized today with the value of a dollar realized later. Why
the interest rate? Because the interest rate correctly measures the cost to you of delaying
the receipt of a dollar of benefit and, correspondingly, the benefit to you of de-
laying the payment of a dollar of cost. Let’s illustrate this with some examples.
Suppose that you are evaluating whether or not to take a job in which
your employer promises to pay you a bonus at the end of the first year.
What is the value to you today of $1 of bonus money to be paid one year in
the future? A slightly different way of asking the same question: what
amount would you be willing to accept today as a substitute for receiving
$1 one year from now?
To answer this question, begin by observing that you need less than $1 today in
Jorg Greuel/Getty Images that the bank can then lend it out to its borrowers. This turns any amount you have
order to be assured of having $1 one year from now. Why? Because any money that
you have today can be lent out at interest—say, by depositing it in a bank account so

today into a greater sum at the end of the year.
Let’s work this out mathematically. We’ll use the symbol r to represent the interest
rate, expressed in decimal terms—that is, if the interest rate is 10%, then r = 0.10. If you
lend out $X, at the end of a year you will receive your $X back, plus the interest on your
$X, which is $X × r. Thus, at the end of the year you will receive:

(24-1) Amount received one year from now as a result of lending $X today =
$X + $X × r = $X × (1 + r)

The next step is to find out how much you would have to lend out today to have $1 a
year from now. To do that, we just need to set Equation 24-1 equal to $1 and solve for
$X. That is, we solve the following equation for $X:

(24-2) Condition satisfied when $1 is received one year from now as a result of
lending $X today: $X × (1 + r) = $1
238 section 5 The Financial Sector