Cost Theory Problem

Can someone please answer the following problem thoroughly and step by step (show me how exactly i can answer a similar question)

Given: TC= 100 +60Q-12Q^2+Q^3

a) Find the equations of TVC, AVC, and MC functions.
b) Find the level of output at which AVC and MC are minimum, and prove that AVC and MC curves are U-shaped.
c) Find AVC and MC for the level of output at which the AVC curve is minimum.

Thank you so much for any help

Reply 1

Chrrye

a)
TC = 100 + 60Q - 12Q^2 + Q^3
Fixed Cost =100 (unrelated to quantity produced(Q) )
TVC = 60Q - 12Q^2 + Q^3
AVC = 60 - 12Q + Q^2 (TVC divided by Q)
MC = dTVC/dQ = 60 - 24Q + 3Q^2 (TVC differentiated by Q)

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b)
To find minimum differentiate and set to zero

i)
AVC = 60 - 12Q + Q^2
dAVC/dQ = -12 + 2Q = 0
Q*=6

ii)
MC = 60 - 24Q + 3Q^2
dMC/dQ = -24 +6Q = 0
Q*=4

Both functions (AVC and MC) are concave up (ushaped) as their second derivatives (wrt Q) are both positive ( d2AVC/dQ2=2 d2MC/dQ2=6)

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c)AVC curve minimum at Q*=6 (see b)i) above )
put Q=6 into the AVC and MC equations to determine "the level of output at which the AVC curve is minimum."

AVC = 60 - 12Q + Q^2
= 60 - 12(6) + (6)^2
= 60 - 72 + 36
= 24

MC = 60 - 24Q + 3Q^2
= 60 - 24(6) + 3(6)^2
= 60 - 144 + 3x(36)
= 24

Output Quantity = 24

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Worth noting that when AVC is minimised AVC=MC

Reply 2

antora

We give below short-run total cost function:TC = 100 50Q – 12Q2 Q3 where TC is total cost and Q is level of output.(i) Determine (a) total fixed cost function (b) total variable cost function(c) average variable cost function, (d) marginal cost function.(ii) Calculate TC, ATC, AVC and MC when the firm produces 10 units of output.(iii) Calculate the level of output at which AVC is minimum.plz help to solve this

Reply 3

10337

Given TC = 100 60Q – 12Q2 Q3. Finda.The equations of the TVC, AVC, and MC functions.b.The level of output at which AVC and MC are minimum, and prove that the AVC and MC curves are U-shaped.c.Find the AVC and MC for the level of output at which the AVC curve is minimum. need solution