A firms sells in a highly competitive market. In which the going price
is $15 per unit and its cost equation is c = 25+ 0.25q^2(it is 0.25q
suared. What is the profit at the profit maximizing level? Suppose
fixed cost rise to $75 how will this affect the level of output?

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Request for Question Clarification by neurogeek-ga on 04 Oct 2005 16:46 PDT

Hi econ_student,

I will be glad to help you work through solving this problem.    

Can you help me by clarifying where you get stuck?  For example, can
you write the equation for net profit?  Do you know what to do next
with that equation?

Also, is the 0.25 squared along with the q?  If it is, you can rewrite and regroup:
(0.25*q)^2 = 0.25 * 0.25 * q * q = 0.0625 * q^2


I should be around for the next 3 hours, and also around the same time
tomorrow.    Other researchers who want to answer or chime in should
feel free.


--neurogeek

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Clarification of Question by econ_student-ga on 05 Oct 2005 06:46 PDT

To clarify. the q alone is squared. The square sign is after the q.

I suspect this is a student trying to get help to do homework?

Yes it is. I am trying to understand this particular area. As it may
be the focus for exam

Hi,

If that is the intent, I can give you directions how to solve it. This
chapter is useful from concept point of view
http://www.howardcc.edu/social_science/micropdf/unit-6.jb.pdf

Essentially, the firm would make profits as long as the incremental
revenue is more than the incremental cost. In a purely competitive
market, there are infinite buyers and sellers and hence price is
stable. The incremental revenue/ marginal revenue (MR) is equal to
unit price.

You have the cost equation, just find out derivate, which will give
incremental cost/ marginal cost (MC). The firm should produce till the
time MR=MC. Once you have that value of q, you can calculate cost of
producing, revenues from producing and hence, the profit.

The level of output depends on marginal revenue and marginal cost. If
fixed cost increases, how will the marginal revenue /cost change?
Think about it and post as comment.

If you read this article and follow the procedure, I can help you in
validating your results.

The cost equation is C=25 +0.25Q^2. (The q alone is squared.)

25 is the fixed cost

.25 is the variable cost

q squared will be all the other costs in producing the number of output. 

Therefor for example to produce 100 units it will cost:

25 + .25(100)^2
 25 +.25 (10000)
25 + 2500
2525

These sell at $15 per unit hence sales will be 15 x 100 = 1500

Then the total profit or loss is 1500 -2525
Net loss=$1025

That is where I feel I am going wrong. Am i calculating something
wrong in the equation?

Try 20,25,30,35 for q.  You will find there is a curve in which 30 is
the peak at $200 profit. ($175,$193.75,$200, and $193.75
respectively).  I think stating with a high q threw you off.  At fixed
costs of $75 q=30 is still the peak.  This can easily be plotted in MS
Excel.

Hope this helps.

If you have to know how to solve the problem for a test, make sure you
check what method your professor wants you to use to solve the
question.

possible methods:

1) you have to know that in order to maximize profits the firm will
have to choose the quantity level where the marginal cost is equal to
the marginal revenue.
You have to know that if the firm is in a competitive market MR is
equal to the price.
You have to know that the equation for margiinal cost can be obtained
from the equation for the total cost by taking the first derivative:

TC=25+0.25q^2
MC= 0.5q

you havw to know how to solve the following equation:

MR=15=0.5q=MC

q*=15/0.5=15*2=30

2) Your professor does not require to know how to obtain the MC
equation. He/she wants yo to know that the firm should increase output
as long as MR is above MC and he/she but still wants you to figure out
marginal cost by trying with different numbers:

marginal cost= increase in cost due to a unit increase in quantity
can be estimated by taking the ratio between the increase in cost and
the increase in q.

example if q=0 C=25
if q= 10 then C=25+25=50 marginal cost is 50-25/10-0=2.5 
if q=20 then c=25+100=125  marginal cost is 125-50/20-10=7.5
if q=30 then C=250 MC is 250-125/10=12.5
if q=40 then C=425 MC is 17.5 
the firm should produce 30 as going from 30 to 40 units the MR is not
above MC any more.

3) your professor allows you to find an answer by simply plotting the
profit curve and graphically observing where it peaks.

Profit equation= 15*q-25-0.25q^2
when you plot this always start with a q=0 and then increase by 5s or 10s.

Good luck!

I must offer thank you to all you has help me with this. The derivate
is something I need to research and understand how you arrived at MC =
0.5q. Once i get that then I know Marginal revenue must equal to
marginal cost. Therefor the firm must sell 30 outputs, to recieve $15.

You all just made this problem become very simple. Thank you to all
you skilled researchers and all you valuable input.