Suppose that the total benefit and total cost from a continuous activity are, re

Question

Suppose that the total benefit and total cost from a continuous activity are, respectively, given by the following equations:

B(Q) = 100 + 36Q – 4Q2 and C(Q) =80 + 12Q.

(Note: MB(Q) = 36 – 8Q and MC(Q) = 12.)

Instructions: Use a negative sign (-) where appropriate.

a. Write out the equation for the net benefits.

N(Q) =  + Q + Q2

b. What are the net benefits when Q = 1? Q = 5?

Net benefits when Q = 1:
Net benefits when Q = 5:

c. Write out the equation for the marginal net benefits.

MNB(Q) =  + Q

d. What are the marginal net benefits when Q = 1? Q = 5?

Marginal net benefits when Q = 1:
Marginal net benefits when Q = 5:

e. What level of Q maximizes net benefits?



f. At the value of Q that maximizes net benefits, what is the value of marginal net benefits?

Explanation / Answer

(a)

Net benefit = Total benefits - Total costs

N(Q) = B(Q) - C(Q)

= 100 + 36Q - 4Q2 - 80 - 12Q

= 20 + 24Q - 4Q2

(b)

When Q = 1,

N(Q) = 20 + 24 - 4 = 40

When Q = 5,

M(Q) = 20 + (24 x 5) - (4 x 25) = 40

(c)

Marginal net benefit, MNB(Q) = dN(Q) / dQ = 24 - 8Q

(d)

When Q = 1,

MNB(Q) = 24 - 8 = 16

When Q = 5,

MNB(Q) = 24 - (8 x 5) = - 16

(e)

Net benefit is maximized when dN(Q) / dQ = 0

Or,

24 - 8Q = 0

24 = 8Q

Q = 3

(f)

When Q = 3,

MNB(Q) = 24 - (3 x 8) = 0

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