Economics Question

[TevashSzat]

Okay, I have a problem that I have been thinking of for some time now.

Marginal Cost as everyone knows, crosses the average total cost (ATC) and the average variable cost (AVC) at each of their minimums. The thing is, how is that mathematically proved?

I can think of a potential proof given that the total cost (TC) can be denotated as a polynomial function:

ax^n+bx^(n-1)+....+cx+d (TC)

Marginal cost is just the derivative so it would be:

anx^(n-1)+b(n-1)x^(n-2)+....+c (MC)

Average total cost is the total cost divided by the quantity, which would be x in this case so ATC would be


ax^(n-1)+bx^(n-2)+....+c+d/x (ATC)

If you set MC=ATC,

anx^(n-1)+b(n-1)x^(n-2)+....+c = ax^(n-1)+bx^(n-2)+....+c+d/x

multiply by x on both sides

anx^n+b(n-1)x^(n-1)+....+cx = ax^n+bx^(n-1)+....+cx+d

And use simple algebra, you get

a(n-1)x^n+b(n-2)x^(n-1)+....= d

which is essentially unsolveable




However, if you take the derivative of the ATC and solve for 0 to find the minimum,

a(n-1)x^(n-2)+b(n-2)x^(n-3)+....-d/(x^2) (derivative of ATC)

a(n-1)x^(n-2)+b(n-2)x^(n-3)+....-d/(x^2)=0

add by d/(x^2)

a(n-1)x^(n-2)+b(n-2)x^(n-3)+....=d/(x^2)

multiply by x^2

a(n-1)x^(n)+b(n-2)x^(n-1)+....=d

And the result is the same as the simplification as when you set MC=ATC, which proves that MC intersects ATC at the minimum of ATC. One could extend this to AVC by saying that since

TC = Variable Cost + Fixed Cost

Fixed cost is a constant so VC can be modeled as a polynomial

ax^n+bx^(n-1)+....+cx+d-alpha

where alpha is the fixed cost. Afterwards, you would use the same logic as above for ATC in order to prove the same for AVC.

Now my question is, how would this be proved for functions that were not polynomial in nature? Is it possible for total cost function models to be exponential, logarithmic, or trigonometric in nature? If so, I do not see an easy way of proving the same thing for this model.

An alternative is if you can prove the same for functions in general, but I can't see how one would approach this.

If anyone can think of the answer to this or can find it online somehow (I've tried but can't), it would be appreciated

[naut]

Why'd you want to know?

Edit: Pity I just finished lectures a couple weeks ago, I could have asked my lecturer for you.

[TevashSzat]

Well, it was just something I've found interesting to ponder. The statement seems correct just through reasoning/common sense, but it still needs to be mathematically proved to be really consider fact. The thing is, I can't seem to find anything online that shows its proof.

Edit: I actually basically typed everything I was going to say in Maple and uploaded the stuff so here is a linky for those who like pictures:

Linky

[a completely inoffensive name]

My head just exploded.

[Megas Methuselah]

My head just exploded.

Well, that is to be expected from a person like you.

I honestly tried to help, but I'm only in my first year of uni. Introductory Macroeconomics is bad enough as it is...

[Papewaio]

I thought this was the science forum, not the home science forum.

[spmetla]

I thought this was the science forum, not the home science forum.

Well it is known as the dismal science!

[Marshal Murat]

When you said polynomial functions I stopped reading.

[Papewaio]

Dictionary definition of economics btw:

Economics
E`co*nom"ics\, n. [Gr. ?, equiv. to ? ?. See Economic.]

1. The science of household affairs, or of domestic management.

2. Political economy; the science of the utilities or the useful application of wealth or material resources. See Political economy, under Political. "In politics and economics." --V. Knox.

[rasoforos]

I cant answer that.

I am sorry they put you through it tho because you ll never need it unless you follow an academic career and use to terrorize uni students...

...in the meanwhile I ll tear my BSc and MSc in economics out of shame

[TevashSzat]

Well, regarding my attempts at proving this, the fact that Marginal Cost intersects Average Total Cost and Average Variable Cost is a basic fact of microeconomics. The thing is why? It makes sense logically, but then you can just say its because it works and it makes sense so I tried to find a mathematical proof of it. Strangely, there doesn't seem to be anything online that demonstrates a proof so here is my attempt at doing it with polynomials.

The big problem is that Total Cost is not necessarily always modeled by polynomial functions. I suppose under cetain conditions, exponentials, logarithmics, trigonometric, or composite functions would all work, but I have no idea of how to prove this for those types of functions or to do so for functions in general

[Jolt]

THAT'S EASY!

Y = (C + I)/(1 - c)

There we go. Now we can all realize how easy Economics is.

[Ice]

I'd love to help, but it's been a while since I've taken calc and micro, although I remember learning exactly about this stuff.

This seems pretty straightforward, though, and I'm sure someone will be able to answer.

[Old Geezer]

Just write Obama and he'll be able to explain it clearly in less than 200 words.

[Ibrahim]

THAT'S EASY!

Y = (C + I)/(1 - c)

There we go. Now we can all realize how easy Economics is.

all that up there is in this?

GLORIOUS! (where is ichigo's sig when you need it?)