Suppose the short-run cost function of Tyrell Corporation is π = ππππ + ππππ + πππ.Β
a.Β Β Find the level/s of Β that satisfies/satisfy the first-order and second-order conditions for the minimisation
of Tyrellβs average cost (ππ). [4]
Show that ππ equals marginal cost (ππ) at the level/s of Β obtained in (a). [2
(a)
Given cost function:
"C = 5000 + 100q + 8q^2"
Average cost (AC)
"=\\frac{ C}{q}"
"=\\frac{5000}{q}+100+8q"
To minimize AC, FOC :-
"=\\frac{\\delta(AC)}{\\delta q}=0"
or, "5000(-1)q^{-2} + 8 = 0"
or, "q = 25"
(b)
SOC :-
"\\frac{d^{2}(AC)}{dq^{2}}=-5000(-2)q^{-3}"
Substituting q = 25 we get 0.64, which is > 0. Hence SOC is satisfied. AC must be minimum at q = 25.
Substituting q=25 in AC we get 500.
Now, MC:-
"\\frac{\\delta C}{\\delta q}=100+16q"
Substituting q=25 in MC we get 500.
Hence MC = AC at q=25.
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