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```Date: 10/15/2003 at 05:00:24
From: Matthew
Subject: Optimisation

A firm with a total income of \$50,000 per year currently spends
expenditure, total income increases by twenty percent. By maximising
the total residue of income less advertising expediture, find the
optimum level of advertising for the firm.

(Hint: d/dx a^x = a^x ln a.)

I don't understand where to start the question.
```

```
Date: 10/15/2003 at 07:55:28
From: Doctor Luis
Subject: Re: Optimisation

Hi Matthew,

Currently, the firm has a total income of \$50,000/year, and it spends
\$10,000/year on advertising, leaving us with a residue profit of
\$40,000/year.

Now, if advertising was doubled to \$20,000/year, its income would
increase 20% to \$60,000/year.  The residue from this would still be
\$40,000/year.  Since this gives us the same net result, you gain
nothing from having spent all that money doubling the advertising
expenditure.  But is there a happy medium?

income, the residue R(x) as a function of the level of advertising x
is

R(x) = (\$50,000/year)*(1.20)^x - (\$10,000/year)*(2)^x

A(x) = (\$10,000/year)*(2)^x

and the income grows as

I(x) = (\$50,000/year)*(1.20)^x

Thus, we are comparing the growth of two exponential functions.  2^x
grows faster than 1.2^x, but the income started with a higher
coefficient.  Eventually, A(0)*2^x catches up to I(0)*1.2^x and it
becomes counterproductive to keep increasing the advertising budget.
We have already seen R(0)=\$40,000/yr, and R(1)=\$40,000/yr, so we can
guess that by the time x=1, most of the gains from advertising have
been cancelled by the cost of advertising itself.

Therefore, our happy medium is somewhere between x=0 and x=1.

Here's the plot of R(x),

Clearly, the residue profit maxes out somewhere in between advertising
levels x = 0 and x = 1, just as we expected.  Now you can just use
calculus to find the optimal value of x.

I hope this helped!

Let us know if you have any more questions.

- Doctor Luis, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
High School Calculus

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