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Chapter 3: Marginal Analysis for Optimal Decisions
Chapter 3:
MARGINAL ANALYSIS FOR OPTIMAL DECISIONS
Essential Concepts
1. Formulating an optimization problem involves specifying three things: (1) the objective function to be
either maximized or minimized, (2) the activities or choice variables that determine the value of the
objective function, and (3) any constraints that may restrict the range of values that the choice
variables may take.
2. Marginal analysis involves changing the value of a choice variable by a small amount to see if the
3. Net benefit from an activity (NB) is the difference between total benefit (TB) and total cost (TC ) for
4. The choice variables determine the value of the objective function. Choice variables can be either
5. Marginal benefit (MB) is the change in total benefit caused by an incremental change in the level of
activity. Marginal cost (MC) is the change in total cost caused by an incremental change in the level
of activity. An “incremental change” in activity is a small positive or negative change in activity,
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6. Because “marginal” variables measure rates of change in corresponding “total” variables, marginal
benefit and marginal cost are also slopes of total benefit and total cost curves, respectively. Marginal
benefit (cost) of a particular unit of activity is measured by the slope of the line tangent to the total
benefit (total cost) curve at that point of activity.
7. If, at a given level of activity, a small increase or decrease in activity causes net benefit to increase,
then this level of activity is not optimal. The activity must then be increased (if marginal benefit
8. When a manager faces an unconstrained maximization problem and must choose among discrete
levels of an activity, the manager should increase the activity if
MB >MC
and decrease the activity if
Chapter 3: Marginal Analysis for Optimal Decisions
9. Sunk costs are costs that have previously been paid and cannot be recovered. Fixed costs are costs that
are constant and must be paid no matter what level of activity is chosen. Average (or unit) cost is the
10. The ratio of marginal benefit divided by the price of an activity (MB/P) tells the decision maker the
additional benefit of that activity per additional dollar spent on that activity, sometimes referred to
11. To maximize or minimize an objective function subject to a constraint, the ratios of the marginal
benefit to price must be equal for all activities,
Answers to Applied Problems
1. a. One way of reducing traffic deaths is to reduce speed. While it may be possible to eliminate all
traffic deaths by allowing motorists to drive no faster than 15 MPH in cars equipped with driver
and passenger air bags, most American drivers would not view a 15 MPH speed limit as optimal.
b. If it costs nothing to eliminate pollution (i.e. MC = 0), then the optimal level of pollution would
indeed be zero. When the marginal cost of pollution abatement is greater than zero, as it is for
virtually every type of pollution, the optimal level of pollution occurs at that level of pollution for
which the marginal benefit to society of eliminating more pollution just equals the marginal cost
of eliminating more pollution. In fact, it is possible to have too little pollution if pollution
abatement activities have been undertaken such that the marginal cost of abatement exceeds the
marginal benefit.
d. See answer to part c.
2. Appalachian Coal Mining should minimize net cost by choosing that level of pollution (P) where the
3. The second partner is basing his objection to the move on costs that are sunk. The money spent on
office stationary, business cards, and a sign that cannot be moved to the new office are not marginal
4. a. 2
b. $500 (= $25 20 radios not stolen due to hiring 1 guard)
c. 4
5. a. The following graph illustrates such a situation. Clean-up activity is plotted along the horizontal
axis and marginal benefits and costs along the vertical. For any amount of clean up greater than
6. “Never give up”: You should give up an activity when MC > MB for extra units of the activity.
“Anything worth doing is worth doing well”: How “well” you choose to do something should be
7. a. With a payroll of $160,000, the manager should hire five people with high school diplomas and
two people with bachelor's degrees. This choice maximizes the number of customers served
because the last dollar spent on each type of employee yields the same addition to the number of
8. a. Q* = 6,000 wine decanters
MR6,000 = $70
b. TR6,000 = 70 6,000 = $420,000
Chapter 3: Marginal Analysis for Optimal Decisions
9. a. 5
b. 4
c. 3
Answers to Mathematical Exercises
1. a. MB = 170 – 2x; MC = – 10 + 4x
2. a. MB = 100 – 4x; MC = x2 – 12x + 52
b. NB = 100x – 2x2 – (1/3)x3 + 6x2 – 52x – 80
3. Z = 3x + xy + y + λ(70 – 4x – 2y)
¶Z
¶x
= 3 + y – λ4 = 0
Chapter 3: Marginal Analysis for Optimal Decisions
4. Z = 6x + 3y + λ(288– xy)
¶Z
¶x
= 6 – λy = 0
5. a. Marginal benefit is the derivative of TB with respect to A:
MB =dTB
dA =8-0.008A
. At point C,
MB = 8 – 0.008(200) = $6.40. The rest of the points can be similarly verified.
