How Do You Determine the Profit-Maximizing Output in Economics?

[Cinitiator]

Homework Statement

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.

Homework Equations

I have a relevant picture:

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.

 

[nate808]

Hello,

I can offer some guidance on how to approach this problem. To find the profit maximizing production output, we need to first understand the relationship between production quantity and profit. This can be done by taking the derivative of the profit function, P(q), with respect to quantity, q. This will give us the marginal profit function, which represents the change in profit for a unit change in quantity.

Next, we need to set this marginal profit function equal to zero and solve for q. This will give us the quantity at which profit is maximized. This is known as the first order condition for profit maximization.

However, it is important to note that this only gives us a critical point, which could be a maximum, minimum, or an inflection point. To determine if it is a maximum, we need to take the second derivative of the profit function and evaluate it at the critical point. If the second derivative is negative, then the critical point is a maximum and represents the profit maximizing quantity. If the second derivative is positive, then the critical point is a minimum and does not represent the profit maximizing quantity.

Now, to find a function that outputs the profit maximizing quantity at a given price, we can use the inverse function of the marginal profit function. This will give us a function S(p) that represents the profit maximizing supply at a given price.

As for using this approach for any given production cost function, the steps remain the same. We first need to find the marginal profit function by taking the derivative of the profit function, set it equal to zero to find the critical point, and then use the second derivative test to determine if it is a maximum or not.

I hope this helps in solving your problem. Good luck!