If the pre-tax cost function for John’s Shoe Repair is C(q) = 100 + 10q - q2 + 1/3
q3, and it faces a specific tax of t = 10, what is its profit-maximizing condition if the
market price is p? Can you solve for a single, profit-maximizing q in terms of p?
(Hint: See Exercise 3.3 and Solved Problem 8.2.)
If the cost function function before tax is
C(q)=100+10q-"q^2" +"\\frac{1}{3}q^3"
then
Marginal Cost (MC)="\\frac{dc}{dq}" =10-2q+"q^2"
at profit max. condition:
P=MC (but is affected by tax) therefore
P+10=10-2q+"q^2"
P="q^2" -2q
By factorizing P=q(q-2)
divide by q-2
(this profit maximixing quantiy)
q="\\frac{p}{q-2}"
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