Question

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.)

Answer 1

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=$\frac{p}{q-2}$