# Profit maximizing supply

1. Feb 12, 2012

### Cinitiator

1. The problem statement, all variables and given/known data
I need to find the profit maximizing production output given any random production function, an output quantity, and a unit price. In fact, I need to find a function which would output a profit maximizing production quantity.

2. Relevant equations
I have a relevant picture:

3. The attempt at a solution

We will assume that production cost function at a given quantity is C(q) = q^2 due to the diminishing returns of the production process, and consequently exponentially growing costs.
The total revenues function is R(p,q) = p*q
The total profit function is therefore P(p,q) = R(p,q) - C(q) = p*q-q^2 = q(p-q)

I assumed a constant price, and graphed P(q) as a constant function of profit at a given quantity q sold, and could find the profit maximizing quantity graphically for any price I wanted. For example, for the price of 4 dollars per unit, the profit maximizing quantity was 2.
I simply replaced p with 4 in the total profits equation, so that it would be P(q)= q*(4-q).

Now, knowing this, I need to find a function which would output the profit maximizing output at a given price (say, S(p, q), which would output the profit maximizing supply). How do I do that? I would assume I would need to use a second partial derivative test and so on and so forth, but I don't know how exactly to apply it here.

I also need to learn to be able to do that for any given production cost function, so any explanations would be greatly appreciated.

Last edited: Feb 12, 2012