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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:


And we know that:


So I tried substituting the first equation into the second:


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 
about this?  How can I get rid of another variable so that I'm only 
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


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 
-Doctor Erich,  The Math Forum
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Associated Topics:
High School Calculus
High School Interest

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