Question

Jocelyn’s Jump Ropes Company is a profit-maximizing firm with a long run total cost function of, LTC(q) =q ^3-10 q^2- 50q. What is the lowest price that this will produce positive output in the long run?

Answer 1

The lowest price would be the minimum of LTC.

LTC would be the minimum if its derivate is 0.

LTC = q^3 – 10q^2 – 50q

Derivative of LTC = {3q^(3 – 1)} – {(10 × 2)q^(2 – 1)} – {50q^(1 – 1)}

0 = 3q^2 – 20q – 50

Now by solving the equation, (q = 8.60 or -1.93).

Since q can’t be negative, the value of q would be 8.60.

Now by putting this value in LTC, we would get price.

LTC = q^3 – 10q^2 – 50q

         = 8.60^3 – 10(8.60)^2 – 50(8.60)

         = 636.056 – 739.60 – 430

         = - 533.54

The price becomes negative, since q is positive.

Answer: - 533.54