I give up.. I have been trying to solve this problem for hours now. I
cannot seem to understand how to derive 'q' in this problem. Any help
appreciated.
***
Suppose that a firm?s fixed proportion production function is given by
q= min (5K, 10L) and the rental rates for capital and labor are given
by v=1, w=3.
a. Calculate the firm?s long-run total, average, and marginal cost curves.
b. Suppose K is fixed at 10 in the short run. Calculate the firm?s
short-run total, average and marginal
cost curves.

***
This is what I got so far...

TC = q(v/a + w/a)=> q(1/5+3/10) = q/2
AC = TC/q => (q/2)/q ; MC = dAC/dq

Hi mastert!
You got the long-run total cost and average cost curves absolutely
right; however, there's a mistake in the formula you wrote for the
marginal cost function. The marginal cost function can be found by
taking the derivative of the TOTAL cost rather than the average cost
function, as you wrote. Therefore, we have that:

MC = dTC/dq
MC = d(q/2)/dq
MC = 1/2

Thus, the marginal cost function (which in this case turns out to be
equal to the average cost function) is constant at 1/2. So, for
question (a),

TC = q/2
AC = 1/2
MC = 1/2

b. In the short-run, we have that K=10. Therefore, the production function becomes:

q = min (5*10, 10L)
q = min (50, 10L)

Notice, then, that in the short run, production can't be higher than
50 units. So the production function can be rewritten as:

    / 10L      if L<5
q = \ 50       if L>=5

Therefore, the total cost curve becomes:

TC = 10 + q(3/10)   for q<=50

The "10" comes from the cost of 10 units of capital (which is fixed at
K=10 regardless of the number of units produced). The cost function is
not defined for q greater 50, since it's impossible to produce more
than 50 units in the short run.

Given the TC, we can now derive the AC and MC functions:

AC = TC/q = (10 + q(3/10))/q = 10/q + 3/10

MC = dTC/dq = d(10 + q(3/10))/dq = 3/10


For additional information on how to derive cost functions, you may
want to visit the following link:

The Cost Function
http://cepa.newschool.edu/het/essays/product/cost.htm#cost


I hope this helps! If you have any questions regarding my answer,
please don't hesitate to request a clarification. Otherwise I await
your rating and final comments.

Best wishes!
elmarto

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Clarification of Answer by elmarto-ga on 04 Apr 2005 06:19 PDT

Hi mastert,
I forgot to mention that, in question (b), the average and marginal
cost functions are not defined for q>50, just as the total cost
function. So the correct answer would be:

TC = 10 + q(3/10)   if q<=50
AC = 10/q + 3/10    if q<=50
MC = 3/10           if q<=50

Best regards,
elmarto

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Request for Answer Clarification by mastert-ga on 04 Apr 2005 06:45 PDT

Thanks for your answer. However, I am getting confused why production
can't be higher 50 in the short run? Also, I was assuming the problem
requires one to calculate the exact "q" this firm needs to produce.
Was I wrong? Thanks a lot!

---

Clarification of Answer by elmarto-ga on 04 Apr 2005 07:00 PDT

Hi mastert!
Production can't be higher than 50 in the short run simply because K
is fixed at 10. The production function is:

q= min (5K, 10L)

Replacing K=10, we get

q = min (50, 10L)

So, if 10L<50, then production will be equal to 10L; while if 10L>50,
then production will be equal to 50 (recall that it's the minimum
between 50 and 10L). Therefore, production can't be higher than 50 if
K is fixed at 10.

Regarding your assumption that the problem asks you to calculate the
exact q the firm needs to produce, I think you are wrong, because the
problem doesn't specify neither a budget for the firm (how much it can
spend in capital and labor) nor a selling price for the good it
produces. You need to know both what the firm can spend and what it
can earn in order to decide how much of the good the company should
produce.

Best wishes!
elmarto

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Request for Answer Clarification by mastert-ga on 04 Apr 2005 07:09 PDT

Oh I see... thanks so much!!!

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Clarification of Answer by elmarto-ga on 04 Apr 2005 07:17 PDT

I'm glad you liked my answer. Thanks for the rating!

elmarto

Great job!