Question 4 Marginal Cost and Marginal Benefit a. Suppose you are given a benefit

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Question 4 Marginal Cost and Marginal Benefit a. Suppose you are given a benefit equation as B(t) =-3t2 + 15t + 160 and cost equation as C(t) = t2-51 60, where t = time in hours. Find a NB equation b. Solve for MB, MC, and find a MNB expression. c. Find the optimal amount of time you should spend on the activity and the value of NB associated with that optimal time d. If you were given the following tables of benefits and costs for an activity, find the optimal level of the activity. To do this find, MB (marginal benefit), MC (marginal cost), NB (net benefit), and MNB (marginal net benefit). Explain why it is the optimal amount Equation used to solve for column values Benefits 30 50 80 120 170 210 240 260 NB Costs 20 60 90 110 120 140 170 210 MB MC MNB 4

Explanation / Answer

(a)

NB(t) = B(t) - C(t) = - 3t2 + 15t + 160 - (t2 - 5t + 60) = - 3t2 + 15t + 160 - t2 + 5t - 60

= - 4t2 + 20t + 100

(b)

MB(t) = dB(t) / dt = - 6t + 15

MC(t) = dC(t) / dt = 2t - 5

MNB(t) = dNB(t) / dt = - 8t + 20

(c)

Optimality is achieved when MB(t) = MC(t)

- 6t + 15 = 2t - 5

8t = 20

t = 2.5

NB(t) = (- 4 x 2.5 x 2.5) + (20 x 2.5) + 100 = - 25 + 50 + 100 = 125

(d)

For the table,

NB = Benefit - Cost

MB = Change in Benefit / Change in Q

MC = Change in Cost / Change in Q

MNB = Change in NB / Change in Q

Therefore:

Optimal activity is when MB = MC = 30, when Q = 7. It may be noted that MB = MC = 30 also when Q is 3 units, but that activity level is sub-optimal because MC is falling in that range. At activity level of 7 units, if output increases, marginal net benefit (MNB) is negative, so a reduction of activity will reduce the negative MNB until MNB is zero. Similarly, if output decreases, marginal net benefit (MNB) is positive, so an increase of activity will increase the MNB.

Q BENEFIT COST NB MB MC MNB 1 30 20 10 30 20 10 2 50 60 -10 20 40 -20 3 80 90 -10 30 30 0 4 120 110 10 40 20 20 5 170 120 50 50 10 40 6 210 140 70 40 20 20 7 240 170 70 30 30 0 8 260 210 50 20 40 -20

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