TC Function Help...Urgent Watch

stem01
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Hey

Im really stuck on this question and i need some guidance, could someone please point me in the right direction?

The toal cost function of a firm is given by

TC = 400 + 37.5Q + 0.75Q^2 + 0.005Q^3

The firm is able to sell any volume of output at a price of £25 per unit. How much output should the firm produce so as to maximise profits? Be sure you check you have found a maximum.


TC = 400 + 37.5Q + 0.75Q^2 + 0.005Q^3

= 37.5 + 1.5Q + 0.015Q^2

= -1.5 + 0.030Q

Q = 1.5/0.030

Q = 50

Thanks
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MeAndBubbles
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All you did was prove that the TC function was convex by finding a postitive second derivative.

What you should include is MR = MC, and another equation too, can you find the other equation?

otherwise you'll be short on equations and too many unkowns.
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stem01
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So like this?

TC = 400 + 37.5Q + 0.75Q^2 + 0.005Q^3

= 37.5 + 1.5Q + 0.015Q^2

= -1.5 + 0.030Q

25 = -1.5 + 0.030Q

25 + 1.5 = 0.030Q

26.5 = 0.030Q

26.5/0.030 = Q

833.33 = Q
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chidona
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(Original post by stem01)
So like this?

TC = 400 + 37.5Q + 0.75Q^2 + 0.005Q^3

MC= 37.5 + 1.5Q + 0.015Q^2

= -1.5 + 0.030Q
Not sure why you differentiated the marginal cost. Since profit maximisation is at MR = MC, you only to differentiate TC once to get MC.

And you know that MR = 50Q, since the firm's demand curve is P=25Q. Thus TR = 25Q^2 and MR = 50Q. From there on it's a simple equation.
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stem01
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Hello

Im lost on what i do next. I know:

TR = 25Q^2
MR = 50Q
P = 25

Do i just have to put 50Q back into the the formula to get my final answer?

TC = 400 + 37.5(50) + 0.75(50)^2 + 0.005(50)^3


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chidona
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(Original post by stem01)
Hello

Im lost on what i do next. I know:

TR = 25Q^2
MR = 50Q
P = 25

Do i just have to put 50Q back into the the formula to get my final answer?

TC = 400 + 37.5(50) + 0.75(50)^2 + 0.005(50)^3


Thanks
Prof max is at the point where MR = MC

MR = 50Q
MC = 37.5 + 1.5Q + 0.015Q^2

Thus:

50Q = 37.5 + 1.5Q + 0.015Q^2

Now solve for Q, that's the profit maximising quantity.
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stem01
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so i would divide 50 where Q is in the formula to get 37.5?
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chidona
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(Original post by stem01)
so i would divide 50 where Q is in the formula to get 37.5?
It's a quadratic equation.
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Chrrye
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(Original post by stem01)
The firm is able to sell any volume of output at a price of £25 per unit. How much output should the firm produce so as to maximise profits? Be sure you check you have found a maximum.[/B]
I see that and I'm thinking TR=25Q, MR=25=P
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chidona
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(Original post by Chrrye)
I see that and I'm thinking TR=25Q, MR=25=P
Gosh, you're right. Christ, don't know what I was thinking, had the correct graph visualised in my head, but chucked an extra Q in there for some reason.

Sorry for that OP, simplifies your working a tad (it's still a quadratic, however).
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Chrrye
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(Original post by chidona)
Gosh, you're right. Christ, don't know what I was thinking, had the correct graph visualised in my head, but chucked an extra Q in there for some reason.

Sorry for that OP, simplifies your working a tad (it's still a quadratic, however).
Simpler than that. The cost is going up at 37.5Q + some

Theres only going to be one outcome to this...
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chidona
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(Original post by Chrrye)
Simpler than that. The cost is going up at 37.5Q + some

Theres only going to be one outcome to this...
Apologies if I'm misinterpreting you, but...

TC = 400 + 37.5Q + 0.75Q^2 + 0.005Q^3

MC = 37.5 + 1.5Q + 0.015Q^2

TR = 25Q

MR = 25

Prof. max is when MR = MC

25 = 37.5 + 1.5Q + 0.015Q^2

0.015Q^2 + 1.5Q + 12.5 = 0

Surely? I get the same equation by going through the method of working out the profit function. Even though it's yielding me two negative results, which is always heartening (knew I should have done some practice sooner).
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Chrrye
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(Original post by chidona)
Apologies if I'm misinterpreting you, but...

TC = 400 + 37.5Q + 0.75Q^2 + 0.005Q^3

MC = 37.5 + 1.5Q + 0.015Q^2

TR = 25Q

MR = 25

Prof. max is when MR = MC

25 = 37.5 + 1.5Q + 0.015Q^2

0.015Q^2 + 1.5Q + 12.5 = 0

Surely? I get the same equation by going through the method of working out the profit function. Even though it's yielding me two negative results, which is always heartening (knew I should have done some practice sooner).
Yar, as in profit is maximised at negative Q value

Theyr not going to produce anything as MC >MR for positive Q values
therefore OUTPUT=Q=0 / would not enter

That would be the answer to the question but I'm not sure why theyd ask it. Seems a bit pointless.
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chidona
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(Original post by Chrrye)
Yar, as in profit is maximised at negative Q value

Theyr not going to produce anything as MC >MR for positive Q values
therefore OUTPUT=Q=0 / would not enter

That would be the answer to the question but I'm not sure why theyd ask it. Seems a bit pointless.
Ah, yes, same wavelength now. I'd be peeved if they asked that to me in an exam, really would =/
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