# Maximizing Value and Profit Simultaneously

1. Mar 28, 2010

### nautikal

1. The problem statement, all variables and given/known data
This isn't a specific math problem but rather a business problem I would like to try and solve using calculus. Basically, a car company wants to maximize value AND profit (per car).

2. Relevant equations

$$Value = \frac{Benefit}{Price}$$ (see: http://en.wikipedia.org/wiki/Value_(marketing))

$$Profit Per Car = Price - Cost Per Car$$

Assume that Cost Per Car is $9,000 and the Benefits Per Car are$11,000.

So for example, if the car company charged $9,500 per car, the value would be 1.158 and the profit per car would be$1,500.

3. The attempt at a solution

I know how to do optimization problems when it is things like area maximization or cost minimization because there is a direct relationship between the two variables, but in this instance the relationship between profit and value is less direct.

If you change the equation for value to be: $$Value = \frac{Benefit - Price}{Price}$$
and then multiply it by the profit equation, you can find a price (\$9,950) such that "Value-Profit" is maximized*. If you assume that the company cares about value and profit per car equally, is this a mathematically valid way of solving the problem?

*Changing the value equation is necessary so that there ends up being a maximum value for "Value-Profit."

How would you solve this problem if value is twice as important as profit per car? If value is half as important as profit per car? Etc.

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