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

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Discussion Overview

The discussion revolves around the relationship between average cost and marginal cost in production, specifically in the context of a mathematical cost function. Participants explore how to determine the level of production that minimizes average cost and where average cost equals marginal cost, touching on both mathematical and economic interpretations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a cost function C(x) and seeks to find the production level that minimizes average cost, noting confusion over negative production values.
  • Another participant defines marginal cost as the change in costs from producing one additional unit, but expresses frustration over the lack of mathematical clarity in their understanding.
  • A different participant explains that average cost is calculated as C(x)/x and suggests differentiating this expression to find its minimum, providing a specific answer of 9 while encouraging others to verify the calculations.
  • One participant asserts that marginal cost is the derivative of the cost function and suggests setting the average cost equal to marginal cost to find the relevant production level, also arriving at 9 as a solution.
  • Another participant expresses uncertainty about the problem's wording and acknowledges a need for further review of the concepts discussed.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the mathematical and economic concepts of average and marginal cost. While some participants agree on the method of differentiation, there is no consensus on the clarity of the problem statement or the interpretation of results.

Contextual Notes

There are indications of confusion regarding the definitions and calculations involved, particularly in relation to the cost function's parameters and the implications of negative production values. Some participants also note a lack of mathematical rigor in their explanations.

Who May Find This Useful

Students studying economics or mathematics, particularly those interested in cost functions and their applications in production theory.

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

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