 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:

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.