Associated Topics || Dr. Math Home || Search Dr. Math

### Setting Interest Rate to Maximize Profit

```
Date: 7/25/96 at 10:50:32
From: Anonymous
Subject: Setting Interest Rate to Maximize Profit

A loan company is limited to charging maximum 18 percent interest on a
loan.  The amount of money available for loans is proportional to the
interest rate the company will pay its investors.  Assuming the
company can lend all the money invested with it, what interest rate
should it pay its investors to maximize its profits?

I've assigned variables:

P: profit
x: amount available for loans
y: amount invested
18/100: interest rate on money loaned out
a: interest rate paid to investors

Since amount of money available for loans is proportional to a,
I get the following:

x=y-a*y

And we know that:

P=18/100*x

So I tried substituting the first equation into the second:

P=18/100(y-a*y).

Now I think I need to differentiate P and set it equal to zero to find
maximum profit, but with respect to which variable?  Is there a rule
dealing with 2?

I really appreciate your help.  Thanks.
```

```
Date: 7/29/96 at 13:38:43
From: Doctor Erich
Subject: Re: Setting Interest Rate to Maximize Profit

Lisa,

Wow! Nice job on the problem so far. You're making my job easy!
You're almost to the end of the problem and you're right that you have
to differentiate P and set it equal to zero. Since you want to figure
out the profit maximizing interest rate to pay investors, you'd want
to differentiate P with respect to a. Generally, whenever you want to
find out the effect that one variable, y, has on another, x, you want
to differentiate x with respect to y. That will tell you the effect a
change in y will have on x.

The one other thing you want to remember when doing these problems is
that profit isn't just the money you make.  Profit is the your
revenue, the money you make, minus your costs, how much it costs you.
So in this case your definition of profit only takes the revenue into
account and not the costs. So maybe try defining P=18/100x - ax,
or something like that, which is basically the interest you made
investing the money minus the interest you had to pay your investors
you lent the money to you in the first place. When you differentiate,
you should be able to divide y out of the equation so you'll just have
a = something.

Aside from that, the rest of your analysis looks very solid... Nice
job!

-Doctor Erich,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Interest

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search