What is the relationship between average cost and marginal cost in production?

[pattiecake]

Suppose the total cost (in dollars) of manufacturing x units of a certain commodity is C(x)= 3x^2+ 18x + 243. At what level of production is the average cost per unit the smallest? At what level of production is the average cost per unit equal to the marginal cost?

So I thought if you want the average cost per unit to be the smallest- all you have to do is take the derivitave of C(x)= 3x^2+ 18x + 243, then solve for the minima which is at x=-3. But in this case, it doesn't make sense for the x value to be negative.

But for the second part, i lack knowledge in the area of economics, so I'm not sure what value they want when they ask for "what level of production is the average cost per unit equal to the marginal cost". Are they talking about inflection points? Help!

 

[Ebolamonk3y]

differential cost

n : the increase or decrease in costs as a result of one more or one less unit of output [syn: marginal cost, incremental cost]

 

[Ebolamonk3y]

marginal cost, from my damn economics teacher pounding into my head, is the one extra unit of whatever and its effects... But really, from that definition I have nothing to work with mathamatically, just parroting the book.. Damn teacher... Hope you can find what you need...


If you are studying to become an economist, this can be much of a problem...

 

[philosophking]

This makes me happy to be a double major in math and economics :)

Average cost is the cost divided by the quantity produced, or in your notation, C(x)/x . Do you see why? x is the number of units produced and C(x) is the cost. Thus you get C(x)/x = 3x + 18 + 243/x. Now to get this minimized you have to differentiate, then set it equal to zero. This shouldn't be too hard. You get 9 as an answer, but work it out for yourself.

Marginal cost is definitely your cost function differentiated, so I also don't see how x could be negative... could you have maybe copied down the problem wrong? For instance, if it was -18x instead of +18x, you'd get the same answer for your first question, and the other two questions would make more sense.

 

[philosophking]

*shakes head* nevermind, I forgot the exact wording of the question before. Marginal cost is the derivative of C(x). Thus, to get the second question correct, you simply set the equation we got in the first part equal to C'(x). Solving for x, you should get an answer of 9 once again.

Translated to economic terms, this makes perfect sense. If you ever take a college microeconomics course, you'll learn that the marginal cost curve always intersects the average total cost curve at its minimum.

 

[Ebolamonk3y]

Heh... someone needs to review their stuff again... :( sorry I was no help.