Author: Lori Alden
Audience: Advanced
placement and college economics students
Time required: About 30
minutes
NCEE Standards: 12,
15
Summary: This activity
teaches students how to do continuous compounding using the e^{x}
button on their scientific calculators. The problems at the
end of the activity vividly demonstrate how small changes in a
country's growth rate can drastically affect the standard of
living of future generations.
Reading:
Nestled among the rows
of buttons on any scientific calculator is one labeled e^{x}.
Punching it takes any value x that's displayed in your calculator's
window and raises e (a number equal to about 2.7) to that power.
The button's purpose? It
lets numbers travel through time.
Here's how it works.
Suppose you're putting $1,000 into a savings account that
pays 7% interest per year. How
much will your account be worth in 5 years?
The formula we'll use is
this:
Value
in D = Value in S ×
e^{r }^{×
(D ─
S)}
S represents the starting year of the number's voyage
through time (2005), D is
the destination year (2010), and r
represents the percentage rate at which your savings account will
grow (7% or .07). Your
account is now worth $1,000this is its Value
in S. We want to
find your account's Value in D.
Start by finding the
value of e's exponent in the equation.
r ×
(D  S) = .07 ×
(2010  2005) = .35
Now punch the e^{x} key.
e^{.35}
= 1.419
Next, multiply 1.419 by the account's Value
in S, $1,000.
$1,000 ×
1.419 = $1,419
Your account will grow to $1,419 in 5 years.
You can go back in time with the e^{x} key, too.
Suppose that your family's house has been growing in value by
4% per year over the past 20 years.
It's worth $100,000 today.
How much was it worth 20 years ago?
In
this problem, the starting year is 2005 and the destination year is
1985. Again, start by
calculating e's exponent:
r
× (D  S) = .04 ×
(1985  2005) = .8
Next,
punch the e^{x} key and multiply the result by the Value in S, $100,000.
Value in D = $100,000 ×
e^{─.8}
= $44,933
Your
house was worth $44,933 in 1985.
The e^{x} key works with any value that grows (or shrinks)
at a constant percentage rate r.
This means it can be used to calculate how population, GNP,
prices, and many other values change over time.
Problems
1.
Real GWP (gross world product) in 1 A.D. is estimated to have been
roughly $18.5 billion (in 1990 dollars).
Had the economies of the world experienced a steady growth
rate of 1% per year since then, what would real GWP be today?
If the world's population is roughly 6.5 billion, what would
real per capital GWP be today?
Solution:
Value
in 2005 = Value in 1 AD ×
e^{.01 }^{×
(2005 ─ 1)}
Value
in 2005 = $18.5 billion ×
504,965,162.6
Value
in 2005 = $9,341,855,509 billion
Per
capita value in 2005 = $9,341,855,509 billion/6.5 billion =
$1,437,208,540
=
$1.437 billion per person on the planet.
(This
amazing result illustrates the power of compoundinghad mankind not
experienced the lowgrowth dark ages, all of us would have been much
richer.)
2.
In 1999, the U.S.
per capita GDP was about $32,500.
If we experience 2% real annual growth from 1999 to 2024,
what will our per capital GDP be in 2024?
Solution:
Value
in 2024 = Value in 1999 ×
e^{.02 }^{×
(2024 ─ 1999)}
Value in 2024 = $32,500 (1.6487) = $53,583
3.
Two countries start with equal GDPs.
But Country A grows at an annual rate of 2 percent while
Country B grows at an annual rate of 2.5 percent.
After 25 years, how much larger than Country A is Country B?
Country A: Value in 2030 = Value in 2005 × e^{.02 }^{×
(2030 ─ 2005)} = Value in 2005 (1.6487)
Country B: Value in 2030 = Value in 2005 × e^{.025 }^{×
(2030 ─ 2005) }= Value in 2005 (1.8682)
Country
B in 2030/Country A in 2030 = Value in 2005 (1.8682)/Value
in 2005 (1.6487) = 1.133
Country
B is 13.3% larger.
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Lori Alden, 20057. All rights reserved. You may download
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