Question

3) A sticky goo oozes mysteriously from the rare wazoo tree, which grows only on the farm of Wolf Molder, just outside of Pullman, Washington. This goo, when smeared on the face, results in a tightening of the skin and the elimination of fine lines. Wolf bottles the goo at a cost of $2 per bottle and sells it to Donna Scali at a wholesale price of $w per bottle. Donna sells the goo to the general public over the Internet under the name “Youth Goo” at a price of $P per bottle. The retail demand for Youth Goo is given by P = 60 - .01Q.
(a) Write Donna Scali’s profit as a function of the number of bottles of Youth Goo she sells over the Internet and the wholesale price, πD(Q;w).
Write an equation characterizing Donna’s profit-maximizing choice of output as a function of the wholesale price w.
(b) What is Wolf’s profit as a function of the number of bottles of Youth Goo he sells to Donna, πW(Q)?
What is Wolf’s profit-maximizing choice of output?

Answer 1

(a) $πD(Q;w) = (60 - 0.01Q)*Q - w*Q = (60 - w)Q - 0.01Q^2.$

(b) $πW(Q) = w*Q - 2*Q = (w - 2)*Q.$

The profit is maximized when πD'(Q) = 0, so:

$60 - w - 0.02Q = 0,$

Q = 3,000 - 50w.