Hi!!
The inverse demand function is the inverse function for the demand,
that is if demand (Q) function is defined by Q = f(P), where P is the
price; then the inverse demand function is the inverse function of
f(P) and it is given by P = f^1(Q), it defines P in function of Q.
See the folowing page for further reference:
http://www.econ.ucsb.edu/~rabbit/econ100b/studentquestion1
The cost equation defines the cost in function of the quantity Q
produced, in other words gives us how much cost to produce certain
quantity Q of products, in this case to satisfy the demand.
Now, the solution:
a. find the firm's optimal quantity, price, and profit
(1) by using the profit and marginal profit equations
Revenue = P*Q = (120  0.5*Q)*Q =
= 120*Q  0.5*Q^2
Profit = Revenue  Cost =
= (120*Q  0.5*Q^2)  (420 + 60*Q + Q^2) =
= 60*Q  1.5*Q^2  420
Marginal Profit is the additional profit derived from the sale of one
additional unit, and it is easy calculated as the derivative of the
profit. The same is valid for all the marginal values for the purposes
of this problem.
For further references see:
"Marginal Quantities":
http://earthmath.kennesaw.edu/main_site/review_topics/marginal_quantities.htm
Then:
Marginal Profit = 60  3*Q
If profit has a maximum it occurs when its derivative is zero, since
Marginal Profit is the derivative of the profit, if profit has a
maximum it occurs when marginal profit is zero:
Marginal Profit = 60  3*Q = 0
Then
Q = 60/3 = 20 > This is the optimal quantity
Replacing this in the price function:
P = 120  0.5*Q =
= 120  0.5*20 =
= 110 > this is the optimal price.
Profit = 60*Q  1.5*Q^2  420 =
= 60*20  1.5*20^2  420 =
= 1200  600  420 =
= 180
(2) by setting MR equal to MC
MR = derivative of Revenue =
=
MC = derivative of cost =
= 60 + 2*Q
MC = MR ==> 60 + 2*Q = 120  Q ==> Q = (120  60)/3 = 20
Then
P = 120  0.5*Q = 120  10 = 110
We find the same value using both methods.

I hope that this helps you. Feel free to request for a clarification
if you need it.
Regards,
liviofloresga 
Request for Answer Clarification by
tomandsharonzga
on
11 Sep 2005 14:27 PDT
Could you please provide more detail on solving the math?
1. Are you saying that revenue equals = 120*Q  0.5*Q^2 (^2 does this
mean squared?... I'm just making sure...)
2. Are you saying that profit equals = 60*Q  1.5*Q^2  420
3. How did you solve for MP? Specifically, how did you get the 3*Q?
4. How did you solve for Price = 110. And, Profit = 180.
5. I'm lost...
Can you help?
Thanks!
Tom

Request for Answer Clarification by
tomandsharonzga
on
11 Sep 2005 14:53 PDT
Two more questions:
1. How would you graph MR and MC?
2. Suppose the instead that the firm can sell any and all of its
output at the fixed market price P = 120. Find the firm's optimal
output?
Thanks!
Tom and Chuck (he's lost,too)...

Clarification of Answer by
liviofloresga
on
11 Sep 2005 21:29 PDT
Hi Tom!!
1. Are you saying that revenue equals = 120*Q  0.5*Q^2 (^2 does this
mean squared?... I'm just making sure...)
Revenue is the amount of money that a firm actually receives from its
activities, in this case for selling its products, and this amount is
quantity sold (Q) times the product price (P):
Revenue = P*Q = 120*Q  0.5*Q^2 (and yes, ^2 means squared).
2. Are you saying that profit equals = 60*Q  1.5*Q^2  420
Yes, profit is revenue less cost and the result of such subtraction is this.
3. How did you solve for MP? Specifically, how did you get the 3*Q?
As I told you, the marginal profit is calculated as the derivative of
the Profit formula, in this case we must to find the derivative
(Profit)' of:
Profit = 60*Q  1.5*Q^2  420
Then
Marginal Profit = (Profit)' = 60*1  1.5*2*Q  0 = 60  3*Q
Again take a look at the page "Marginal Quantities":
http://earthmath.kennesaw.edu/main_site/review_topics/marginal_quantities.htm
4. How did you solve for Price = 110. And, Profit = 180.
You want to maximize the profit, to do that you must equal the
Marginal Profit to zero, that is:
0 = 60  3*Q ==> 3*Q = 60 ==> Q = 60/3 = 20
This value of Q is the optimal quantity, if you put this value in the
Price formula you will get the optimal price:
P = 120  0.5*Q = 120  0.5*20 = 120  10 = 110
Doing the same with the Profit formula and you will get the maximum profit:
Maximum Profit = 60*Q  1.5*Q^2  420 evaluated at Q = 20, that is
Maximum Profit = 60*20  1.5*20^2  420 =
= 1200  1.5*400  420 =
= 1200  600  420 =
= 180
I hope this helps you to understand this answer. Feel free to continue
using the clarification feature if you need it.
Regards,
liviofloresga

Clarification of Answer by
liviofloresga
on
11 Sep 2005 22:34 PDT
Hi again!!
You are asking two new questions that are related to this one but
require additional work, the correct way in these cases is to open a
new question in the forum pricing it as your criterion. I will try to
answer these two additional questions for you, but if you have more
questions to do please post them at Google Answers' forum as separated
ones.
1. How would you graph MR and MC?
MR is the marginal revenue and it is the derivative of the Revenue function:
Revenue = 120*Q  0.5*Q^2 ==> MR = 120  Q
The graph of MR as a function of Q is a straight line with slope 1 and
when Q is zero MR = 120 and when Q = 120 then MR = 0.
MC is the marginal cost and it is the derivative of the Cost function:
Cost = 420 + 60*Q + Q^2 ==> MC = 60 + 2*Q
The graph of MC as a function of Q is a straight line with slope 2 and
when Q is zero MC = 60 and to get another point use Q = 30, then MC =
120.
One more useful graph in these cases is the Demand curve, that is the
graph of the Price function, in this case Price = 120  0.5*Q , again
we have a straight line where Price is 120 when Q = 0 and Price is
zero if Q = 240.
To see and approximate graph follow this link:
http://www.geocities.com/artistaflores/MC_MRgraph.jpg
2. Suppose the instead that the firm can sell any and all of its
output at the fixed market price P = 120. Find the firm's optimal
output?
If Price is fixed at $120 then:
Revenue = P*Q = 120*Q ==> MR = derivative of (120*Q) = 120
Setting MC = MR we can find the optimal output:
60 + 2*Q = 120 ==> 2*Q = 12060 ==> 2*Q = 60 ==> Q = 60/2 = 30 .
For a fixed price of $120 the firm's optimal output is Q = 30.

I hope this helps you.
Best regards,
liviofloresga
