Lecture 3: Optimization: One Choice Variable

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Lecture 3 OptimizationOne Choice Variable

• Necessary conditions
• Sufficient conditions
• Reference
• Jacques, Chapter 4
• Sydsaeter and Hammond, Chapter 8.

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1. Optimization Problems
Economic problems Consumers Utility
maximization Producers Profit
maximization Government Welfare maximization
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Maximization problem maxx f(x)
f(x) Objective function with a domain D x
Choice variable x Solution of the maximization
problem
A function defined on D has a maximum point at x
if f(x) ? f(x) for all x ? D. f(x) is
called the maximum value of the function.
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Minimization problem minx f(x)
f(x) Objective function with a domain D x
Choice variable x Solution of the maximization
problem
A function defined on D has a minimum point at x
if f(x) ? f(x) for all x ? D. f(x) is
called the minimum value of the function.
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• Example 3.1 Find possible maximum and minimum
  points for
• f(x) 3 (x 2)2
• g(x) ?(x 5) 100, x ? 5.

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2. Necessary Condition for Extrema
What are the maximum and minimum points of the
following functions? y 60x 0.2x2 y
x3 12x2 36x 8
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Characteristic of a maximum point
Maximum x lt x dy / dx gt 0 x gt x dy / dx lt
0 x x dy / dx 0
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Characteristic of a minimum point
Minimum x lt x dy / dx lt 0 x gt x dy / dx gt
0 x x dy / dx 0
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Theorem (First-order condition for an extremum)
Let y f(x) be a differentiable function. If the
function achieves a maximum or a minimum at the
point x x, then
dy / dx xx f(x) 0
Stationary point x Stationary value
y f(x)
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Example 3.2 Find the stationary point of the
function y 60x 0.2x2.
The first-order condition is a necessary, but not
sufficient, condition.
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Example 3.3 Find the stationary values of the
function y f(x) x3 12x2 36x 8.
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3. Finding Global Extreme Points

• Possibilities of the nature of a function f(x) at
  x c.
• f is differentiable at c and c is an interior
  point.
• f is differentiable at c and c is a boundary
  point.
• f is not differentiable at c.

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3.1 Simple Method
Consider a differentiable function f(x) in a,b.

1. Find all stationary points of f(x) in (a,b)
2. Evaluate f(x) at the end points a and d and at
   all stationary points
3. The largest function value in (b) is the global
   maximum value in a,b.
4. The smallest function value in (b) is the global
   minimum value in a,b.

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3.2 First-Derivative Test for Global Extreme
Points
Global maximum
Global minimum
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First-derivative Test

• If f(x) ? 0 for x ? c and f(x) ? 0 for x ? c,
  then x c is a global maximum point for f.
• If f(x) ? 0 for x ? c and f(x) ? 0 for x ? c,
  then x c is a global minimum point for f.

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Example 3.4 Consider the function y
60x 0.2x2. a. Find f(x). b. Find the
intervals where f increases and decreases and
determine possible extreme points and values.
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Example 3.5 y f(x) e2x 5ex 4. a. Find
f(x). b. Find the intervals where f increases
and decreases and determine possible extreme
points and values. c. Examine limx?? f(x) and
limx?-?f(x).
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3.3 Extreme Points for Concave and Convex
Functions

• Let c be a stationary point for f.
• If f is a concave function, then c is a global
  maximum point for f.
• If f is a convex function, then c is a global
  minimum point for f.

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Example 3.6 Show that f(x) ex1 x. is a
convex function and find its global minimum point.
Example 3.7 The profit function of a firm is
?(Q) -19.068 1.1976Q 0.07Q1.5. Find the
value of Q that maximizes profits.
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4. Identifying Local Extreme Points
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4.1 First-derivative Test for Local Extreme
Points
Let a lt c lt b. a. If f(x) ? 0 for a lt x lt c and
f(x) ? 0 for c lt x lt b, then x c is a local
maximum point for f. b. If f(x) ? 0 for a lt x lt
c and f(x) ? 0 for c lt x lt b, then x c is a
local minimum point for f.
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Example 3.8 y f(x) x3 12x2 36x 8. a.
Find f(x). b. Find the intervals where f
increases and decreases and determine possible
extreme points and values. c. Examine limx??
f(x) and limx?-?f(x).
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Example 3.9 Classify the stationary points of
the following functions.
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4.2 Second-Derivative Test
The nature of a stationary point Decreasing
slope ? Local maximum
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The nature of a stationary point
Increasing slope ? Local minimum
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The nature of a stationary point Point of
inflection ? Stationary slope
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Second order condition Let y f(x) be a
differentiable function and f(c) 0.
f(c) lt 0 ? Local maximum f(c) gt 0 ? Local
minimum f(c) 0 ? No conclusion
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• Example 3.10 Identify the nature of the
  stationary points of the following functions
• y 4x2 5x 10
• y x3 3x2 2
• y 0.5x4 3x3 2x2.

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4.3 Point of Inflection
a
b
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Test for inflection points
Let f be a twice differentiable function. a. If
c is an inflection point for f, then f(c)
0. b. If f(c) 0 and f changes sign around c,
then c is an inflection point for f.
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Example 3.11 y 16x 4x3 x4. dy / dx
16 12x2 4x3. At x 2, dy/dx 0. However,
the point at x 2 is neither a maximum nor a
minimum.
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Example 3.12 Find possible inflection points
for the following functions, a. f(x) x6
10x4. b. f(x) x4.
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4.4 From Local to Global
Consider a differentiable function f(x) in a,b.

1. Find all local maximum points of f(x) in a,b
2. Evaluate f(x) at the end points a and b and at
   all local maximum points
3. The largest function value in (b) is the global
   maximum value in a,b.

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5. Curve Sketching

• Find the domain of the function
• Find the x- and y- intercepts
• Locate stationary points and values
• Classify stationary points
• Locate other points of inflection, if any
• Show behavior near points where the function is
  not defined
• Show behavior as x tends to positive and negative
  infinity

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• Example 3.13 Sketch the graphs of the following
  functions by hand, analyzing all important
  features.
• a. y x3 12x
• y (x 3)?x
• y (1/x) (1/x2).

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6. Profit Maximization
Total revenue TR(q) ? Marginal revenue
MR(q) Total cost TC(q) ? Marginal cost
MC(q) Profit ?(q) TR(q) TC(q)
Principles of Economics MC MR MC
curve cuts MR curve from below.
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Calculus
First-order condition
I.e., MR MC 0
Thus the marginal condition for profit
maximization is just the first-order condition.
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Calculus
Second-order condition
At profit-maximization, the slope of the MR
curve is smaller than the slope of the MC curve.
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6.1 A Competitive Firm
Example 3.14 Given (a) perfect competition (b)
market price p (c) the total cost of a firm is
TC(q) 0.5q3 2q2 3q 2. If p 3, find the
maximum profit of the firm.