Question

Toyota produces a certain aftermarket part for their vehicle line with the following estimated demand function, Q=140,000-12,000P Q is quantity demanded per year and P is price charged. Toyota was able to say their fixed costs for this product are $11,000 and variable costs are $1.75 per unit. A. Write the total revenue function. B. Determine the marginal revenue. C. Write the total cost function. D. Solve for marginal cost. E. Write an equation for total profits. At what price and output would profit be maximized? What are the total profits? F. Solve for output by solving MR=MC. Are E and F the same?

Answer 1

Q = 140,000 - 12,000P

12,000P = 140,000 - Q

P = (140,000 - Q) / 12,000

(A) Total revenue (TR) = P x Q = (140,000Q - Q²) / 12,000

(B) Marginal revenue (MR) = dTR/dQ = (140,000 - 2Q) / 12,000 = (70,000 - Q) / 6,000

(C) Total cost (TC) = Fixed cost + Total variable cost = $11,000 + $1.75Q

(D) Marginal cost (MC) = dTC/dQ = $1.75

(E) Profit (Z) = TR - TC = [(140,000Q - Q²) / 12,000] - (11,000 + 1.75Q)

Profit is maximized when dZ/dQ = 0

[(70,000 - Q) / 6,000] - 1.75 = 0

(70,000 - Q) / 6,000 = 1.75

70,000 - Q = 10,500

Q = 59,500

P = (140,000 - 59,500) / 12,000 = 80,500 / 12,000 = $6.71

Profit ($) = (P x Q) - TC = (6.71 x 59,500) - 11,000 - (1.75 x 59,500) = 399,245 - 11,000 - 104,125 = 284,120

(F) Equating MR and MC,

(70,000 - Q) / 6,000 = 1.75

70,000 - Q = 10,500

Q = 59,500

The result is the same as in part (E).