Applied Differentiation Problem

[Baxter_Slade]

Exercise 2: A firm has a total cost function that differs depending on whether is produces more than 4 units of output. Specifically we have:

. . . . .\(\displaystyle TC(q)\, =\, \begin{cases}6q^2\, +\, 4\, \mbox{ if }\, q\, \leq\, 4\\cq^2\, +\, f\, \mbox{ if }\, q\, >\, 4\end{cases}\)

The output is sold at a price of 24 per unit so that the firm's profit is given by:

. . . . .\(\displaystyle \pi(q)\, =\, 24q\, -\, TC(q)\)

Find the Optimal production level is C=4 and F=36

My work so far:

• I have subbed in for C and F into the Total Cost Function
• I know that Profits are Maximized when Marginal Revenue = Marginal Cost and that i find MC by differentiating TC and i find MR by differentiating TR
• I know Total Revenue = (Price)(Quantity)
• and that Profit= TR-TC

If anyone could lead me in the right direction that would be great!
Thank you in advance.

 

[stapel]

Exercise 2: A firm has a total cost function that differs depending on whether is produces more than 4 units of output. Specifically we have:

. . . . .\(\displaystyle TC(q)\, =\, \begin{cases}6q^2\, +\, 4\, \mbox{ if }\, q\, \leq\, 4\\cq^2\, +\, f\, \mbox{ if }\, q\, >\, 4\end{cases}\)

The output is sold at a price of 24 per unit so that the firm's profit is given by:

. . . . .\(\displaystyle \pi(q)\, =\, 24q\, -\, TC(q)\)

Find the Optimal production level is C=4 and F=36

Is the "is" in the last line above supposed to be "if"? Is the "C" supposed to be "c"?

My work so far:

• I have subbed in for C and F into the Total Cost Function

• I know that Profits are Maximized when Marginal Revenue = Marginal Cost and that i find MC by differentiating TC and i find MR by differentiating TR
• I know Total Revenue = (Price)(Quantity)
• and that Profit= TR-TC

What did you get? Where are you stuck? What is your question?

Please be complete. Thank you!