MODELING
REAL REALTIONS
By Dario Gonzalez Martinez
The idea now is to draw a mathematic model to
represent and predict the behavior of a quantity that is related to another. To make this real, we will base our
discussion on real data. We will
consider the data based on the first class letter postage for the US Mail from
1919 to 2008, and we will model this problem to predict future rates.
Here is the data
Year 
Rate (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 
2004 
39 
2006 
41 
2008 
42 
We can elaborate a graph to show the variation
of the rate through the years. Figure 1
below shows this relation:

Figure 1 
Using a graph always can help us to determine
a curve that could fit for our problem. In
other words, the idea is to find a function that relates years and rates such
that the relation expressed by the function describes acceptably the real
relation between years and rates.
For our analysis, consider the following
table:
Year 
Rate (in cents) 
Rate’s average Increment per year 
1919 
2 
0.076923077 
1932 
3 
0.038461538 
1958 
4 
0.2 
1963 
5 
0.2 
1968 
6 
0.666666667 
1971 
8 
0.666666667 
1974 
10 
3 
1975 
13 
0.666666667 
1978 
15 
1.666666667 
1981 
20 
0.5 
1985 
22 
1 
1988 
25 
1.333333333 
1991 
29 
1 
1994 
32 
0.333333333 
1997 
33 
0.5 
1999 
34 
1 
2002 
37 
1 
2004 
39 
1 
2006 
41 
0.5 
2008 
42 
The third column that I added represents the
rate’s average increment per year, that is, each of these cells is the rate’s
average increment per year from the year in the same row to the year in the
next row. For example, the result of the
first cell in the third column can be obtained by
If we observe this column, we can appreciate that the rate
increase relatively fast from 1919 to 1988 with a great explosion about 1958. It is possible appreciate that the population
in the United State start increasing from 1956 to 1959 by looking the following
resource on the web:
http://www.census.gov/popest/archives/1990s/popclockest.txt
This could explain why the first class letter
postage cost increase fast from 1958. Since
the population was increasing, the demand for letter postage increases too,
which in turn could have increased the rate.
DEDUCTION
OF A SUITABLE MODEL
Although it seems highly improbable the first class
letter postage cost increase indefinitely since nobody will pay a high price to
send a letter, the rate is still relatively small, we
can use an exponential model to predict the cost in a near future.
An exponential model is:
Where a
and b are the parameters that
we need to find. To this end, we will
take natural logarithm in both sides of the above relation.
We just need to find the values of a and ln(b) in a linear
regression, and then we will obtain the values for the parameters of our
exponential model. By using the
following statistic formulas
The following table made in Excel did the work to calculate these
relations for us:
VALUES OF PARAMTERS 

Thus, our exponential model will be
Where x represents the number of years from 1919, and y is the
rates of the first class letter postage (theoretical). Let’s see how well our model fit. Figure 2 shows the real graph and our
theoretical model to predict the rate:

Figure 2 
This model could be a good approximation to
the real relation between years and rates.
Suppose that we want to know when the cost
will be 1 dollar (100 cents). According our
model, we will have:
This result suggests that first class letter postage will cost 1
dollar about 2028.