microeconomics salvatore chapter 2
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Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 1 1
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 2
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 3
0
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0 1 2 3 4 5 6 7
Q
TR
Expressing Economic Relationships
Equations: TR = 100Q - 10Q2
Tables:
Graphs:
Q 0 1 2 3 4 5 6TR 0 90 160 210 240 250 240
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 4
Total, Average, and Marginal Revenue
TR = PQ
AR = TR/Q
MR = TR/Q
Q TR AR MR0 0 - -1 90 90 902 160 80 703 210 70 504 240 60 305 250 50 106 240 40 -10
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 5
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 6
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 7
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0 1 2 3 4 5 6 7
Q
TR
-40
-20
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0 1 2 3 4 5 6 7
Q
AR, MR
Total Revenue
Average andMarginal Revenue
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 8
Total, Average, andMarginal Cost
Q TC AC MC0 20 - -1 140 140 1202 160 80 203 180 60 204 240 60 605 480 96 240
AC = TC/Q
MC = TC/Q
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 9
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 10
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 11
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 12
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 13
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 14
Profit Maximization
Q TR TC Profit0 0 20 -201 90 140 -502 160 160 03 210 180 304 240 240 05 250 480 -230
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 15
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Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 17
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 18
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 19
New Management Tools
• Benchmarking
• Total Quality Management
• Reengineering
• The Learning Organization
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 20
Other Management Tools
• Broadbanding
• Direct Business Model
• Networking
• Performance Management
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 21
Other Management Tools
• Pricing Power
• Small-World Model
• Strategic Development
• Virtual Integration
• Virtual Management
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 22
Chapter 2 Appendix
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 23
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.
0limX
dY Y
dX X
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 24
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 25
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 26
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
0dY
dX
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 27
Rules of Differentiation
Power Function Rule: The derivative of a power function, where a and b are constants, is defined as follows.
( ) bY f X aX
1bdYb aX
dX
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 28
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
dY dU dV
dX dX dX
Y U V
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 29
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
dY dV dUU V
dX dX dX
Y U V
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 30
Rules of Differentiation
Quotient Rule: The derivative of the ratio of two functions, U and V, is defined as follows.
( )U g X ( )V h X UY
V
2
dU dVV UdY dX dXdX V
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 31
Rules of Differentiation
Chain Rule: The derivative of a function that is a function of X is defined as follows.
( )U g X( )Y f U
dY dY dU
dX dU dX
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 32
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 33
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.
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 34
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.
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 35
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 36
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 37
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 38
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)
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 39
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 40
Quadratic Formula
• Write the equation in the following form:aX2 + bX + c = 0
• The solutions have the following form:2b b 4ac
2a
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 41
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 42
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 43
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 44
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 45
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 46
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 47
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 48
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
Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 49
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
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