Question

suppose the cost function is C= 1000+10q^2
at what output level AC is minimum
at what level AVC is minimum and what is the relation between mc,AVC and ATC
i am confused with AVC please explain properly

Answer 1

Answer
AC=C/q=1000/q+10q
Minimum of a function is found by equating first derivative equal to zero
dAC/dq=-1000/q^2+10=0
10q^2=1000
q^2=100
q=10
AC is minimum at q=10
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VC is the part of total cost function except for constant term
VC=10q^2
AVC=VC/q=10q
dAVC/dq=10
the AVC function is linear, so the AVC is minimum at zero output level where AVC=0
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MC is the change in VC or C
dC/dq=dVC/dq=20q

the MC is a change in cost or change of variable cost as the fixed cost is constant, so differentiation of a fixed or constant is zero, so there is no scope of it.
MC=AVC is at a minimum of AVC and the same for MC=VC is at a minimum of AC