Stamps
by Emily Kennedy

 Year Price(in cents) 1919 2 1932 3 1958 4 1963 5 1968 6 1971 8 1974 10 1975 13 1978 15 1981 20 1985 22 1988 25 1991 29 1994 32 1997 33 1999 34 2002 37 2006 39

Since the beginning of the twentieth century, the price of stamps has been increasing every few years. The data to the left show the price of a stamp in various years.

Below is a graph of these data.

Looking at the graph, it's clear that there hasn't been a linear relationship between year and stamp price throughout the twentieth century.

What kind of relationship is it, then?

I thought that stamp prices would probably increase at a rate which was similar to the rate of inflation. Thus, assuming the rate of inflation is about constant (quite an assumption!), I expected the data to follow an exponential curve:
S = c·(1+R)Y
where Y is the year, S is the price of stamps (in cents),
R is the (constant) rate of inflation, and c is some other constant.

I chose this model since, if the rate of inflation were constant, and the price of stamps increased in the same way as inflation, then the price of a stamp would increase by the same proportion each year.

For instance, if the price in 1919 were P0 and inflation were constant at 4%,
I would expect the price in 1920 to be 4% more than P0 (1.04·P0),
the price in 1921 to be 4% more than that (1.04·1.04·P0), etc.

I put my data in an Excel spreadsheet and used the LOGEST command to determine an exponential regression on the data.
You can see the spreadsheet here.

The exponential regression is

S = (2.126 x 10-33)·(1.040155)Y
where Y is the year and S is the price of stamps (in cents)

The constant c (2.126 x 10-33) is extremely small because the values of Y -- the years -- are so large, greater than 1900 each!

The value of R, the rate of inflation, is 4.01%, a pretty reasonable estimate for the average inflation since 1919. So this is looking good!

I plotted this graph on top of my data to see how well it fit:

It looks like an okay fit, but why are there deviations?

The answer lies in the rate of inflation, which (of course) isn't actually constant.

During the 1960s, inflation was low. Using this website, we can determine the average rate of inflation per year.
The calculator on the site tells us overall inflation
from 1960 to 1970 was 31.1%.

To find the average inflation per year, we need to find
a value of r such that r10 = 1.311.
Taking the tenth root of both sides, we have that r ≈ 1.0274.
So the average yearly inflation in the 1960s was about 2.74%.
This is quite a bit lower than the century average of 4%, which explains why our data don't grow as quickly as our regression curve during that decade.

In the late '70s and early '80s, inflation was very high (averaging 7.2% per year). This explains why our data shoot up higher than the regression curve during this period.

From the mid-'80s until the mid-'90s, our data grow at about the same rate as the regression curve, but the data start out higher in 1985 than the regression curve does. So, during this period, inflation was about average (about 3.5%, close to the century average of 4%), but the actual price of a stamp is higher than what we would have predicted, because of the rapid inflation of the '70s and '80s.

Then, from about 1990 until today, we see that the price of a stamp is increasing less rapidly than our regression line would predict. The inflation calculator website tells us the average yearly inflation since 1990 has been about 2.7%, lower than the century average of 4%.

Since the constant R, 1.040, in our regression equation is just about equal to the average inflation during the period from 1919 to today, and the deviations of our data from the regression line closely parallel deviations in the inflation rate from the century's average inflation, it is safe to say that the price of stamps grows at a rate that is comparable to the rate of inflation.

How does this help us predict future prices of stamps?

If we assume (again, quite an assumption!) that inflation will stay around 4% per year forever, then we can use our regression line to predict when the price of a stamp will reach 74¢, or even a dollar.

To determine when the price of a stamp will be 74¢,
we can use the extrapolated graph above,
or we can solve the following for Y:

74 = (2.126 x 10-33)·(1.040155)Y

Dividing both sides by 2.126 x 10-33, we have
3.481 x 1034 = 1.040155Y

Now, taking the log (base 1.040155) of both sides, we have
Y ≈ 2020

So we can expect the price of a stamp to reach 74¢ around the year 2020.

But remember how average inflation has only been about 2.7% per year since 1990? What would happen if inflation continued at this rate, rather than 4%?

The price of a stamp would have to increase by (74 - 39) / 39 = 89.7% in order to end up at 74¢.

So we need a value of d such that 1.027d = 1.897.

Taking the log (base 1.027) of both sides gives us
d ≈ 24

So if inflation continues at only 2.7%, we can expect the price of stamps to reach 74¢ in 2006 + 24 = 2030, rather than in 2020.

Can you imagine a stamp costing a dollar?
Let's use our regression curve to determine when stamps will cost a dollar.
Hopefully it won't be for a long time!

We need to solve the following for Y:

100 = (2.126 x 10-33)·(1.040155)Y

Dividing both sides by 2.126 x 10-33, we have
4.7036 x 1034 = 1.040155Y

Now, taking the log (base 1.040155) of both sides, we have
Y ≈ 2027

So we can expect stamps to cost a dollar around the year 2027. Yikes!

But remember how average inflation has only been about 2.7% per year since 1900? What would happen if inflation continued at this rate, rather than 4%?

The price of a stamp would have to increase by (100 - 39) / 39 = 156.4% in order to end up at \$1.00.

So we need a value of d such that 1.027d = 2.564.

Taking the log (base 1.027) of both sides gives us
d ≈ 35

So if inflation continues at only 2.7%, we can expect the price of stamps to reach \$1.00 in 2006 + 35 = 2041, rather than in 2027.