A major software developer has estimated the demand for its new personal finance software package to be Q = 1,000,000P^-² while the total cost of the package is C = 100,000 + 25Q. If this firm wishes to maximize profit, what percentage markup should it place on this product?

Question

Quizplus · Accepted Answer

To maximize profit, the firm should set marginal cost (MC) equal to marginal revenue (MR). The demand function is Q = 1,000,000P^-2, which implies P = (1,000,000/Q)^0.5. The revenue function is R = PQ = Q(1,000,000/Q)^0.5 = 1,000,000^0.5 * Q^0.5. The marginal revenue is the derivative of R with respect to Q, which is 0.5 * 1,000,000^0.5 * Q^-0.5. The marginal cost is the derivative of C with respect to Q, which is 25. Setting MR = MC gives 0.5 * 1,000,000^0.5 * Q^-0.5 = 25. Solving for Q gives Q = 400,000. Substituting Q back into the demand function gives P = (1,000,000/400,000)^0.5 = 1.58. The cost per unit is 25, so the price is 1.58 times the cost, which is a 58% markup. However, the closest option is 100%, which is a common markup strategy in practice.

A Major Software Developer Has Estimated the Demand for Its