PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 1
1
Optimization Techniques
• Methods for maximizing or minimizing
an objective function
• Examples
– Consumers maximize utility by purchasing
an optimal combination of goods
– Firms maximize profit by producing and
selling an optimal quantity of goods
– Firms minimize their cost of production by
using an optimal combination of inputs
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 2
Expressing Economic
Relationships
Equations:
Tables:
TR = 100Q - 10Q2
Q
TR
0
0
1
90
2
3
4
5
6
160 210 240 250 240
TR
300
250
Graphs:
200
150
100
50
0
0
1
2
3
4
5
6
7
Q
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 3
Total, Average, and Marginal
Revenue
TR = PQ
AR = TR/Q
MR = TR/Q
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Q
0
1
2
3
4
5
6
TR
0
90
160
210
240
250
240
AR
90
80
70
60
50
40
Copyright  2007 by Oxford University Press, Inc.
MR
90
70
50
30
10
-10
Slide 4
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 5
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 6
TR
300
250
Total Revenue
200
150
100
50
0
0
1
2
3
4
5
6
7
Q
AR, MR
120
100
Average and
Marginal Revenue
80
60
40
20
0
-20
0
1
2
3
4
5
6
7
-40
Q
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 7
Total, Average, and
Marginal Cost
AC = TC/Q
MC = TC/Q
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Q
0
1
2
3
4
5
TC AC MC
20 140 140 120
160 80 20
180 60 20
240 60 60
480 96 240
Copyright  2007 by Oxford University Press, Inc.
Slide 8
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 9
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 10
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 11
Geometric Relationships
• The slope of a tangent to a total curve
at a point is equal to the marginal value
at that point
• The slope of a ray from the origin to a
point on a total curve is equal to the
average value at that point
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 12
Geometric Relationships
• A marginal value is positive, zero, and
negative, respectively, when a total
curve slopes upward, is horizontal, and
slopes downward
• A marginal value is above, equal to, and
below an average value, respectively,
when the slope of the average curve is
positive, zero, and negative
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 13
Profit Maximization
Q
0
1
2
3
4
5
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
TR
0
90
160
210
240
250
TC Profit
20
-20
140
-50
160
0
180
30
240
0
480 -230
Copyright  2007 by Oxford University Press, Inc.
Slide 14
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 15
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 16
Steps in Optimization
• Define an objective mathematically as a
function of one or more choice variables
• Define one or more constraints on the
values of the objective function and/or
the choice variables
• Determine the values of the choice
variables that maximize or minimize the
objective function while satisfying all of
the constraints
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 17
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 18
New Management Tools
•
•
•
•
Benchmarking
Total Quality Management
Reengineering
The Learning Organization
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 19
Other Management Tools
•
•
•
•
Broadbanding
Direct Business Model
Networking
Performance Management
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 20
Other Management Tools
•
•
•
•
•
Pricing Power
Small-World Model
Strategic Development
Virtual Integration
Virtual Management
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 21
Chapter 2 Appendix
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 22
Concept of the Derivative
The derivative of Y with respect to X is
equal to the limit of the ratio Y/X as
X approaches zero.
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 23
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 24
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 25
Rules of Differentiation
Constant Function Rule: The derivative
of a constant, Y = f(X) = a, is zero for all
values of a (the constant).
Y  f (X )  a
dY
0
dX
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 26
Rules of Differentiation
Power Function Rule: The derivative of
a power function, where a and b are
constants, is defined as follows.
Y  f (X )  aX b
dY
 b  a X b 1
dX
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 27
Rules of Differentiation
Sum-and-Differences Rule: The derivative
of the sum or difference of two functions,
U and V, is defined as follows.
U  g( X )
V  h( X )
Y  U V
dY dU dV


dX dX dX
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 28
Rules of Differentiation
Product Rule: The derivative of the
product of two functions, U and V, is
defined as follows.
U  g( X )
V  h( X )
Y  U V
dY
dV
dU
U
V
dX
dX
dX
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 29
Rules of Differentiation
Quotient Rule: The derivative of the
ratio of two functions, U and V, is
defined as follows.
U  g( X )
dY

dX
V  h( X )

V dU
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
dX
 
 U dV
V
U
Y
V
dX

2
Copyright  2007 by Oxford University Press, Inc.
Slide 30
Rules of Differentiation
Chain Rule: The derivative of a function
that is a function of X is defined as follows.
Y  f (U )
U  g( X )
dY dY dU


dX dU dX
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 31
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 32
Optimization with Calculus
Find X such that dY/dX = 0
Second derivative rules:
If d2Y/dX2 > 0, then X is a minimum.
If d2Y/dX2 < 0, then X is a maximum.
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 33
Univariate Optimization
Given objective function Y = f(X)
Find X such that dY/dX = 0
Second derivative rules:
If d2Y/dX2 > 0, then X is a minimum.
If d2Y/dX2 < 0, then X is a maximum.
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 34
Example 1
• Given the following total revenue (TR)
function, determine the quantity of
output (Q) that will maximize total
revenue:
• TR = 100Q – 10Q2
• dTR/dQ = 100 – 20Q = 0
• Q* = 5 and d2TR/dQ2 = -20 < 0
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 35
Example 2
• Given the following total revenue (TR)
function, determine the quantity of
output (Q) that will maximize total
revenue:
• TR = 45Q – 0.5Q2
• dTR/dQ = 45 – Q = 0
• Q* = 45 and d2TR/dQ2 = -1 < 0
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 36
Example 3
• Given the following marginal cost
function (MC), determine the quantity of
output that will minimize MC:
• MC = 3Q2 – 16Q + 57
• dMC/dQ = 6Q - 16 = 0
• Q* = 2.67 and d2MC/dQ2 = 6 > 0
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 37
Example 4
• Given
– TR = 45Q – 0.5Q2
– TC = Q3 – 8Q2 + 57Q + 2
• Determine Q that maximizes profit (π):
– π = 45Q – 0.5Q2 – (Q3 – 8Q2 + 57Q + 2)
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 38
Example 4: Solution
• Method 1
– dπ/dQ = 45 – Q - 3Q2 + 16Q – 57 = 0
– -12 + 15Q - 3Q2 = 0
• Method 2
– MR = dTR/dQ = 45 – Q
– MC = dTC/dQ = 3Q2 - 16Q + 57
– Set MR = MC: 45 – Q = 3Q2 - 16Q + 57
• Use quadratic formula: Q* = 4
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 39
Quadratic Formula
• Write the equation in the following form:
aX2 + bX + c = 0
• The solutions have the following form:
 b  b  4ac
2a
2
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 40
Multivariate Optimization
• Objective function Y = f(X1, X2, ...,Xk)
• Find all Xi such that ∂Y/∂Xi = 0
• Partial derivative:
– ∂Y/∂Xi = dY/dXi while all Xj (where j ≠ i) are
held constant
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 41
Example 5
• Determine the values of X and Y that
maximize the following profit function:
– π = 80X – 2X2 – XY – 3Y2 + 100Y
• Solution
– ∂π/∂X = 80 – 4X – Y = 0
– ∂π/∂Y = -X – 6Y + 100 = 0
– Solve simultaneously
– X = 16.52 and Y = 13.92
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 42
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 43
Constrained Optimization
• Substitution Method
– Substitute constraints into the objective
function and then maximize the objective
function
• Lagrangian Method
– Form the Lagrangian function by adding
the Lagrangian variables and constraints to
the objective function and then maximize
the Lagrangian function
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 44
Example 6
• Use the substitution method to
maximize the following profit function:
– π = 80X – 2X2 – XY – 3Y2 + 100Y
• Subject to the following constraint:
– X + Y = 12
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 45
Example 6: Solution
• Substitute X = 12 – Y into profit:
– π = 80(12 – Y) – 2(12 – Y)2 – (12 – Y)Y – 3Y2 + 100Y
– π = – 4Y2 + 56Y + 672
• Solve as univariate function:
– dπ/dY = – 8Y + 56 = 0
– Y = 7 and X = 5
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 46
Example 7
• Use the Lagrangian method to
maximize the following profit function:
– π = 80X – 2X2 – XY – 3Y2 + 100Y
• Subject to the following constraint:
– X + Y = 12
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 47
Example 7: Solution
• Form the Lagrangian function
– L = 80X – 2X2 – XY – 3Y2 + 100Y + (X + Y – 12)
• Find the partial derivatives and solve
simultaneously
– dL/dX = 80 – 4X –Y +  = 0
– dL/dY = – X – 6Y + 100 +  = 0
– dL/d = X + Y – 12 = 0
• Solution: X = 5, Y = 7, and  = -53
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 48
Interpretation of the
Lagrangian Multiplier, 
• Lambda, , is the derivative of the
optimal value of the objective function
with respect to the constraint
– In Example 7,  = -53, so a one-unit
increase in the value of the constraint (from
-12 to -11) will cause profit to decrease by
approximately 53 units
– Actual decrease is 66.5 units
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 49