Consider the following demand function for a firm QD= 14,000 –2,000 P, where P =
ID: 1106099 • Letter: C
Question
Consider the following demand function for a firm QD= 14,000 –2,000 P, where P = price per unit and Q = quantity demanded per year.Assume that total costs are $3,000 when nothing is produced and increase by $.50per unit for each unit produced.
a.Write an equation for the total cost function.
b.Specify the marginal cost function.
c.Write an equation for total revenue in terms of Q.
d.Specify the marginal revenue function.
e.Write an equation for total profits in terms of Q.i.Find the level of Q that maximizes the total profit function.ii.Find the level of P that will be charged at the profit maximizing level.iii.Determine the level of total profits at the profit-maximizing level.
f.Check your answer in part e above by equating marginal revenue and marginal cost and solving for Q.g.Is this firm in a perfectly competitive industry? Why or why not?
Explanation / Answer
(a) Total cost: C ($) = 3,000 + 0.5Q
(b) Marginal cost (MC) = dC / dQ = 0.5
(c) Q = 14,000 - 2,000P
2,000P = 14,000 - Q
P = (14,000 - Q) / 2,000
Total revenue (TR) = P x Q = (14,000Q - Q2) / 2,000
(d) Marginal revenue (MR) = dTR / dQ = (14,000 - 2Q) / 2,000 = (7,000 - Q) / 1,000
(e) Profit = TR - C = [(14,000Q - Q2) / 2,000] - (3,000 + 0.5Q) = [(14,000Q - Q2) / 2,000] - 3,000 - 0.5Q
(i) Profit is maximized when d(Profit) / dQ = 0
[(14,000 - 2Q) / 2,000] - 0.5 = 0
(14,000 - 2Q) / 2,000 = 0.5
14,000 - 2Q = 1,000
2Q = 13,000
Q = 6,500
(ii) P = (14,000 - 6,500) / 2,000 = 7,500 / 2,000 = $3.75
(iii) TR ($) = 3.75 x 6,500 = 24,375 & C ($) = 3,000 + (0.5 x 6,500) = 3,000 + 3,250 = 6,250
Profit ($) = TR - C = 24,375 - 6,250 = 18,125
NOTE: First 5 parts are answered.
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