A company produces and sells a consumer product and is able to control the demand for the product by varying the selling price. The approximate relationship between price and demand is 50 units.
p = 38 + (2,700 / D) - (5,000 / D2)
Marginal (variable) cost (MC) = 40
(a) Profit is maximized by equality of Marginal revenue (MR) and MC.
Total revenue (TR) = p x D = 38D + 2,700 - (5,000 / D)
MR = dTR / dD = 38 + (5,000 / D2)
Equating MR with MC,
38 + (5,000 / D2) = 40
5,000 / D2 = 2
D2 = 2,500
Taking positive square root on each side,
D = 50
(b) When D = 50, from demand function we get
p = 38 + (2,700 / 50) - (5,000 / 2,500) = 38 + 54 - 2 = $90 (Profit-maximizing price)
Profit (\pi) ($) = Total Revenue - Total Costs = TR - (Fixed cost + Total variable cost) = (p x D) - (1,000 + 40D)
= 38D + 2,700 - (5,000 / D) - 1,000 - 40D
= 1,700 - 2D - (5,000 / D)
Profit is maximized when d\pi/dD = 0 and d2\pi/dD2 < 0.
First order condition: d\pi/dD = - 2 + (5,000 / D2)
Second order condition: d2\pi/dD2 = d/dD(d\pi/dD) = - 2 x (5,000 / D3) = - 10,000 / D3
Since D > 0, (- 10,000 / D3) < 0, which proves that profit is maximized when company produces = 50 units.
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