Chapter 3: Assignment
Technical Questions: 5, 8, 11, 12
Applied Questions: 4,7, 8, 9
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CHAPTER 3:
MARGINAL ANALYSIS FOR
OPTIMAL DECISIONS
I. DEFINITIONS.
A. Objective function: The function that the
decision maker seeks to maximize or
minimize.
 Examples:
a) Consumers attempt to maximize
satisfaction.
b) Managers of a business firm attempt
to maximize profits.
c) Production manager might attempt to
minimize costs for a given level of
production.
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B. Activities or choice variables.
1) Choice variables are variables that
determine the value of the objective
function.
2) Examples
a) Objective function
Q = f (L, K)
1. L and K are the choice variables.
2. Managers will choose the
combination of L and K that
minimizes cost.
b) Objective function
U = f (X, Y)
Consumers will choose the amount of
X and Y that maximizes satisfaction.
c) π = f (Q)
3) Discrete Choice Variable: A variable that
can take only specific integer values.
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4) Continuous Choice Variables: A variable
that can take on any value between two
end points.
C. Marginal Analysis:
An analytical tool for solving optimization
problems that involves changing the value of
the choice variables by a small amount to see
if the objective function can be further
increased or decreased.
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II. UNCONSTRAINED MAXIMIZATION:
A. Defined: The decision maker can choose the
level of the activity from an unrestricted set
of values.
B. Example: With a discrete choice variable.
1) NB= TB-TC
MB =
ΔTB
ΔA
ΔTC
MC = ΔA
2) Rule: Increase activity as long as
MB > MC. That will ensure reaching the
optimal level of the activity.
3) Hypothetical Example: Table 3.2
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C. Example: With a continuous choice variable.
1) Rule: To maximize total benefit, go to the
point where marginal benefit equals
marginal cost.
2) Hypothetical Example: Figure 3.1 & 3.2
3) Economic Example: Applied Question 8
D. Irrelevance of sunk costs or fixed costs.
Sunk Costs: Costs that have previously been
paid and cannot be recovered.
Fixed Costs: Costs are constant and must be
paid no matter what the level of the activity is.
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III.
CONSTRAINED OPTIMIZATION.
A. Defined: A decision maker (manager) must
choose values for the choice variables from a
restricted set of values.
1) Budgets limit the amount of labor & capital
a manager can purchase
2) Consumers face budget constraints
B. Rule
MBA
PA
=
MB B
PB
=
MBc
Pc
MBn
=….= Pn
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C. Constrained Maximization Problem
1) Example: Optimal level of advertising
expenditures on radio and TV
advertisements.
2) Ptv ad = $400 Pradio ad = $300
3) Budget constraint = $2000
# of ads MBtv MBtv/Ptv
1
2
3
4
5
6
400
300
280
260
240
200
MB radio MB radio/P radio
1
0.75
0.70
0.65
0.60
0.50
360
270
240
225
150
120
Optimal Combination:
MB TV ad MB Radio ad
PTV ad = PRadio ad
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1.20
0.90
0.80
0.75
0.50
0.40
4) How should manager reallocate budget if:
MB TV ad
MB Radio ad
a) If PTV ad > PRadio ad
MB TV ad
MB Radio ad
b) If PTV ad < PRadio ad
D. Constrained Minimization Problem. Minimize
total cost function subject to a constraint.
1. Manager must minimize the total cost of
3000 units of benefits involving two activities;
(A and B)
MB of last unit of A = 30 PA = 5
MB of last unit of B = 60 PB = 20
MBa
Pa = 30/5
MBb
Pb = 60/20
How should the budget be reallocated?
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