**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**

__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---->**

*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---->**

**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.**

*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:**

I**f you have any questions, comments, suggestions,
criticisms or cash donations:**