Suppose the total benefit derived from a given decision, Q, is B(Q) = 20Q - 2Q^2

Question

Suppose the total benefit derived from a given decision, Q, is B(Q) = 20Q - 2Q^2 and the corresponding total cost is C(Q) = 4 + 2Q^2, so that MB(O) = 20 - 4Q and MC(Q) = 4Q. Instruction: Use a negative sign (-) where appropriate. What is total benefit when Q = 2? Q = 10? When Q = 2: __________ When Q = 10: __________ What is marginal benefit when Q = 2? Q = 10? When Q = 2: __________ When Q = 10: ______________ What level of Q maximizes total benefit? ___________ What is total cost when Q = 2? Q = 10? When Q = 2: ________ When Q = 10: _________ What is marginal cost when Q = 2? Q = 10? When Q = 2:__________ When Q = 10: _____________ What level of Q minimizes total cost? __________ What level of Q maximizes net benefits? ___________

Explanation / Answer

Solution:

a) Total benefit B(Q) = 20Q-2Q^2
   when Q = 2 is B(2) = 20(2) – 2*(2)^2 = 32,
       Q = 10 is B(10) = 20(10) – 2*(10)^2 = 0

b) Marginal benefit MB(Q)=20-4Q
   when Q = 2 is MB(2) = 20 – 4(2) =12,
       Q = 10 is MB(10) = 20 – 4(10) =-20

c) Total benefits are maximized when MB(Q)=0, or 20-4Q=0. Some algebra leads to Q = 20/4=5 as the level of output that maximizes total benefits.

d) Total cost C(Q) = 4 + 2Q^2
   when Q = 2 is C(2) = 4 + 2*(2)^2 = 12,
       Q = 2 is C(10) = 4 + 2*(10)^2 = 204

e) Marginal cost MC(Q)=4Q
   when Q = 2 is MC(Q) = 4(2) = 8,
       Q = 10 is MC(Q) = 4(10) = 40

f) Total costs are minimized when Q=0

g) Net benefits are maximized when MNB(Q) = MB(Q)-MC(Q) = 0, or 20 – 4Q – 4Q = 0.
Some algebra leads to Q = 20/8 = 2.5 as the level of output that maximizes net benefits.

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