(Who will profit?)
Math 471
Mathematical Modeling
California Lutheran University
Spring '99
Victoria Walker
Math 471
Mathematical Modeling
California Lutheran University
Spring '99
Victoria Walker
Constants and functions necessary to determine future profit values:
Purpose: The purpose of this project is to predict how chosen factors will affect the Tobacco Industry's profits and the Health Industry's expenses. Also, to show how the Tobacco Industry's profits are affected by the chosen factors and then to see how the profits would change without these factors. Likewise, to show how the Health Industry's expenses are affected by these chosen factors and then to show how the Health Industry's expenses would look without these factors.
Constants and functions necessary to determine future profit values: In order to find the principal functions, there are a certain number of constants and functions that will contribute and make up the principal functions that will need to be found first.
L(t)=the number of lawsuits at time t.
Nl(t)=the number of laws at time t.
C(t)=the number of cessation techniques
(stop smoking techniques) at time t.
(These functions will affect both the function for the Health Industry and that of the Tobacco Industry)
L=the cost of each lawsuit for the Tobacco
Industry.
Nl=the cost of each new law for the
Tobacco Industry.
C=the cost of each cessation technique
for the Tobacco Industry.
(These values affect only the Tobacco Industry)
l=the savings of each lawsuit for the
Health Industry.
nl= the savings of each law for the
Health Industry.
c=the savings of each cessation technique
for the Health Industry.
(These last values affect only the Health Industry)
Finally the desired functions are:
E(t)=the expense function of the Health Industry.
P(t)=the profit function for the Tobacco Industry.
Model Assumptions: Using the Deterministic Model, there are a certain number of steps that must be completed in order to create an accurate model. There are a number of assumptions that need to be made and all of the parameters must be defined. The parameters of this model have already been defined in the previous section.
The year 1999 will represent the initial time. So I have set time zero to be 1999.
(The necessary assumptions)
1. The only factors that
affect the Tobacco Industry's profits and the Health Industry's expenses
are laws,
lawsuits and cessation techniques.
2. The only cessation techniques
of concern are valid FDA approved techniques. Such as the patch,
gum,
inhaler...
3. Politician's
views concerning tobacco use will not change during the time period in
question. (A politician's
views greatly affect the number of laws that are put into affect concerning
the tobacco industry and the money
that is invested to create more cessation techniques. While Bill
Clinton has been a great advocate of
increasing the number of laws and distributing money to the necessary divisions,
if Bob Dole had been
elected into office, these changes would not have occurred).
4. The three leading Tobacco
Companies will represent the Tobacco Industry as a whole. These companies
are:
R.J. Reynolds, Philip Morris, and Universal.
(There are too many companies that incorporate the Tobacco
Industry to consider all of them).
5. The top two Health Agencies
will represent the Health Industry as a whole. These agencies are:
Tenet
Healthcare Corporation and Scherer Healthcare Inc. (Again,
there are too many health agencies to take all of
them into consideration).
Determine values:(The following values were determined from information provided on the internet, see resources)
E=The Health Industry's expenses at time zero=5.99
million dollars.
P=The Tobacco Industry's profits at time zero
= 41.33 million dollars.
L=The number of lawsuits at time zero = 1980.
Nl=The number of laws at time zero=128.
C=The number of cessation techniques at time
zero = 3.
The following three functions affect both of the desired functions. Each of the functions were found in similar ways:
L(t)=(The rate
of change of the number of lawsuits)=dL/dt=(constant)L.
L(-3)=1170. L(0)=1980=L(0)e^kt---->
1980=1170e^k(-3)---->1980/1170=e^k(-3)---->1.69=e^-3k----->Ln(1.69)=-3k----->.53=-3k--->-.18=k.
So, the function is represented by:
L(t)=1170e^(-.18(t)).
Just as L(t) was found, Nl(t) and C(t) were found in the same way.
Nl(t)=(The
rate of change of the number of laws)=dNl/dt=(constant)Nl.
Nl(-35)=4. Nl(-3)=128=Nl(-35)e^kt---->
104=4e^k(-35)---->104/4=e^k(-35)---->26=e^-35k----->Ln(26)=-35k----->3.25=-35k--->-.09=k.
So, the function is represented by:
Nl(t)=4e^(-.09(t)).
C(t)=(The rate
of change of the number of cessation techniques)=dC/dt=(constant)C.
C(-8)=1. C(-1)=3=C(-8)e^kt---->3=e^k(-8)---->3=e^-8k----->Ln(3)=-8k----->1.10=-8k--->-.14=k.
So, the function is represented by:
C(t)=e(-.14(t)).
The final values that are left to be determined are the cost values for each of these factors.
Looking at the profits of our Tobacco Industry for the past ten years, I used the data as well as the following to determine the cost values that affect the Tobacco Industry:
Year
Tobacco Industry's Profits
Change
1991
20.67 million dollars
1992
32.67 million dollars
+12.00 million
1993
34.67 million dollars
- 1.33 million
1994
30.33 million dollars
- 3.67 million
1995
23.67 million dollars
- 6.67 million
1996
28.67 million dollars
+ 5.00 million
1997
35.67 million dollars
+ 7.00 million
1998
41.33 million dollars
+ 5.67 million
L=cost of the Tobacco Industry for successful lawsuits. (For this value I chose it to be equal to 1/3 of total profit decreases) = 15.8 million dollars.
Because laws and cessation techniques do not affect the Tobacco Industry for only a moment, but rather cause a loss from the moment they are enacted, there needs to be a growth function describing the dollar amount that is lost due to these laws. As more and more laws and techniques are added to the list of existing ones the cost to the Tobacco Industry is going to increase exponentially. Again differential equations will be used to determine the value of these functions.
Let the dollar value for the laws in 1991 be 22.29 million dollars (This is not an accurate figure, but rather an estimated amount)
(Using the same technique as before) Nl(t)=22.29e^.33(t+9).
Let the dollar value for the cessation techniques in 1991 be 6.48 million dollars. (Again, this is not an accurate figure) C(t)=6.48e^.35(t+9).
Using the fact that 7% of all of the Health Industry's expenses are due to treating tobacco related illnesses and the profits of the Health Industry over the last ten years, the cost values for these affective factors was determined.
Year
Health Industry's Profits
Change
1991
3.13 million dollars
1992
3.70 million dollars
+ .57 million
1993
4.93 million dollars
+ 1.23 million
1994
5.56 million dollars
+ .63 million
1995
6.67 million dollars
+ 1.11 million
1996
6.91 million dollars
+ .24 million
1997
8.41 million dollars
+ 1.50 million
1998
8.56 million dollars
+ .15 million
Here is the graph of this growth:
The x-axis represent the years beginning with 1990.
I=the dollar amount saved by the Health Industry for each lawsuit brought against the Tobacco Industry which I chose to equal .94 million dollars.
(Again, it is necessary to have functions representing the cost values for cessation techniques and new laws)
nl(t)=dollar amount saved by the Health
Industry for each new law = 5e^.15(t+9).
c(t)= dollar value saved by the Health
Industry for each cessation technique = 1.69e^.14(t+9)
Now that all of the necessary values and functions have been determined, the desired functions can be completed for the profits of the Tobacco Industry and the expenses of the Health Industry.
P(t)=-(# of lawsuits multiplied by the dollar amount that each lawsuit cost the T.I.)-(# of laws multiplied by the dollar amount that each will cost the T.I.)-(# of cessation techniques multiplied by the dollar amount that each will cost the T.I.)=-(L(t)*L)-(Nl(t)*Nl(t))-(C(t)*C(t))------>-(
E(t)=-(# of lawsuits multiplied by the dollar amount that each lawsuit saves the H.I.)-(# of laws multiplied by the dollar amount that each will save the H.I.)-(# of cessation techniques multiplied by the dollar amount that each will save the H.I.)=-(L(t)*L)-(Nl(t)*Nl(t))-(C(t)*C(t))------>-(
Now the projections can be made.
Here are the actual profits for the T.I.,
over the last ten years:
Year Tobacco Profits
1991
27,000,000
1992
39,000,000
1993
37,666,667
1994
34,000,000
1995
27,333,333
1996
32,333,333
1997
39,333,333
1998
45,000,000
Here is a graph indicating this growth:
The x-axis represents the years beginning with 1990. This graph shows nine years worth of actual profits by the Tobacco Industry.
Now that we have seen how the change in the profits of both the Tobacco Industry and the Health Industry have occurred over the last nine to ten years, we can show how they are expected to change according to the assumptions that we have made:
The following graph is a representation of the Tobacco Industry's Profits versus that of the Health Industry with the assumed factors:
The red curve indicates the Tobacco Industry's profits, and the blue line indicates the Health Industry's profits. Again, time t=zero is 1999, so this graph indicates that the Tobacco Industry's profits will begin to decline in approximately 20 years, or in 2019. The Health Industry's profits will not decrease due to the increase in these factors, but will eventually exceed the Tobacco Industry's profits.
When considering the factors, I assumed
that the factors grow exponentially due to the fact that more and more
information about the destruction of the use of tobacco is causing a great
many more responses, however, if I had assumed that the growth in these
factors was instead linear, the outcome would be altered as follows:
Here instead, the Tobacco Industry's profits begin to decline in approximately 27 years as opposed to 20 years as is represented by the red curve, while the Health Industry's profits become larger than the Tobacco Industry's profits in 2031.
Whether considering exponential growth of the affective factors or linear growth of the factors, the Tobacco Industry will eventually bite the dust provided that the assumptions were correct. A good example of this is the recent sell-out of R.J.. Reynolds to a Japanese Corporation. It was said that R.J.. Reynolds could no longer continue to profit with all of the lawsuits that are now being brought against them.
This model provides a fairly accurate account of the Tobacco Industry and Health Industry's profits if the three factors that were taken into affect are in fact the only factors that affect the profits. In reality there are many more factors that affect the Tobacco Industry's profits and the Health Industry's expenses, however there are so many that to include all in this model would not be time efficient, in fact I would be nowhere to finished if I had.
Also, by representing the entire Tobacco Industry by the three companies and the entire Health Industry by the two agencies the accuracy was again sacrificed. There is no way that all of the companies and agencies could possibly be included and still complete this model in a timely manner.
1. Daniel P. Maki and Maynard
Thompson, Mathematical Models and Applications (Englewood Cliffs:
Prentice-Hall
Inc., 1973).
2. Paul Blanchard, Robert
L. Devaney and Glen R. Hall, Differential Equations (Pacific Grove:
Cole Publishing
Co.,
1996).
3. http://www.mnbluecrosstobacco.com/toblit/factsheets/timeline.html
4. http://www.hhp.ufl.edu/hse/faculty/sdorman/tobacco/sld028.htm
5. http://www.yahoo.com/q?s=uvv&d=1y
6.
http://quote.yahoo.com/q?s=JRJR&d=2y
7.
http://quote.yahoo.com/q?s=MO&d=5y
8.
http://sec.yahoo.com/e/l/jrjr/html
9.
http://www.cdc.gov.tobacco.andth.htm
10.
http:// www.cdc.gov.tobacco.mediexp.htm
11. http://cdc.gov.tobacco.regulate.htm
12. http://stic.neu.edu
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