Sample Problems from Class (if I had had time):
6. Fill in the blanks in the following table to answer the questions below.
a.
A
TB
TC
NB
MB
MC
0
$ ____
$0
$ ____
1
____
50
50
$100
$ ____
2
175
____
____
____
60
3
____
____
70
70
____
4
314
241
____
____
____
5
381
____
72
____
68
6
____
380
____
66
____
What is the optimal level of activity in the table above?
b. What is the value of net benefit at the optimal level of activity? Can total benefit be
increased by moving to any other level of A? Explain. Can total cost be decreased by
moving to any other level of A? Explain.
c. Using the numerical values in the table, comment on the statement, “The optimal level of
activity occurs where the difference between marginal benefit and marginal cost is
minimized.”
8. You are interviewing three people for one sales job. On the basis of your experience and
insight, you believe Jack can sell 64 widgets per day, Jill can sell 50 widgets per day, and
John can sell 100 widgets per day. The daily salary each person is asking is as follows: Jack,
$100; Jill, $100; and John, $300. How would you rank the three applicants?
10. A decision maker is choosing the levels of two activities, X and Y, so as to maximize total
benefits under a given budget. The prices and marginal benefits of the last units of X and Y
are denoted P , P , MB , and MB .
X
Y
X
Y
a. If P = $30, P = $40, MB = 420, and MB = 640, what should the decision maker do?
X
Y
X
Y
b. If P = $15, P = $17.50, MB = 750, and MB = 840, what should the decision maker do?
X
Y
X
Y
c. If P = $56, P = $14, MB = 2,800, and MB = 1,050, how many units of Y can be obtained if X
is reduced by one unit? How much will benefits increase if this exchange is made?
X
Y
X
Y
d. If the substitution in part c continues until the point of equilibrium is reached, and MB
rises to 3,360, then what will MB be in equilibrium?
X
Y
12. Suppose a firm is considering two different activities, X and Y, which yield the total benefits
presented in the schedule below. The price of X is $40 per unit, and the price of Y is $50 per
unit.
Total
benefit of
activity X
(TBX)
$ 0
Total
benefit of
activity Y
(TBY)
$ 0
1
1,200
1,300
2
2,160
2,550
3
3,040
3,750
4
3,840
4,750
5
4,560
5,650
Level of
activity
0
a.
The firm places a budget constraint of $360 on expenditures on activities X and Y. What
are the levels of X and Y that maximize total benefit subject to the budget constraint?
b. What is the total benefit associated with the optimal levels of X and Y in part a?
c.
Now let the budget constraint decrease to $230. What are the optimal levels of X and Y
now? What is the total benefit when the budget constraint is $230?
Answers:
6. Your table should look like this:
A
TB
TC
NB
MB
MC
0
$0
$0
$0
1
100
50
50
$100
$ 50
2
175
110
65
75
60
3
245
175
70
70
65
4
314
241
73
69
66
5
381
309
72
67
68
6
447
380
67
66
71
a. A* = 4
b. NB* = $73. Yes, by increasing A to 6 units of activity, total benefit will continue to
increase. Although A = 4 maximized net benefit, it does not generally maximize total
benefit. Yes, by decreasing A to zero units of activity, total cost can be decreased to
zero. Minimizing total cost is clearly not optimal.
c. For unconstrained optimization problems with discrete activity levels, it may not be
possible to adjust A to precisely the level where MB= MC. However, adjusting A to the
point where the difference between MB and MC is minimized may, in some instances,
fail to achieve the maximum net benefit. This problem is one such example. At 5 units
of activity the difference between MB and MC is just $1 ($67 – $68). At the optimal
level, A = 4, the difference between MB and MC is $3 ($69 – $66). The correct rule to
follow for discrete choice variables is to increase A until the last (or highest) level of
activity is reached for which MB is greater than MC.
8. Compare the marginal benefit per dollar for each applicant: For Jack, MB/P = 64/100 = 0.64;
for Jill, MB/P = 50/100 = 0.50; for John, MB/P = 100/300 = 0.33. Thus Jack ranks first, Jill
ranks second, and John comes in last.
10. Answers are found by comparing the ratios of the marginal benefit divided by price for the
two activities X and Y.
a.
MBX 420
640 MBY
=
= 14 < 16 =
=
PX
30
40
PY
Use more Y and use less X, while keeping total expenditure on X and Y constant.
b.
MBX 750
MBY
840
=
= 50 > 48 =
=
PX
15
17.50
PY
Use more X and use less Y, while keeping total expenditure on X and Y constant.
c. A 1-unit reduction in X frees up $56, which can then be used to purchase 4 units of Y.
Total benefits will increase by 1,400 (= –2,800 + 4,200), which is the sum of the lost
benefit from one less X (= 2,800) and the gain in benefit from adding 4 units of Y (= 4 
1,050).
d. In equilibrium 3,360/56 = MBY/14, so MBY must equal 840.
12. a. The combination 4X, 4Y maximizes total benefit subject to a budget constraint of $360.
b. Total benefit of 4X, 4Y is $8,590 (= $3,840 + $4,750).
c. The combination 2X, 3Y is optimal when the budget constraint is $230. Total benefit of
2X, 3Y is $5,910 (= $2,160 + $3,750).