Economics 214

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Economics 214

• Lecture 32
• Univariate Optimization

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Second-Order Conditions

• The second-order condition provides a sufficient
  condition but, as we will see, not a necessary
  condition, for characterizing a stationary point
  as a local maximum or a local minimum.

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Second-Order Conditions

• Local Maximum If the second derivative of the
  differentiable function yf(x) is negative when
  evaluated at a stationary point x (that is,
  f(x)lt0), then that stationary point represents
  a local maximum.
• Local Minimum If the second derivative of the
  differentiable function yf(x) is positive when
  evaluated at a stationary point x (that is,
  f(x)gt0), then that stationary point represents
  a local minimum.

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Second-Order Conditions for Our Examples
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Graph of Example 1
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Graph of Example 2
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Second-Order Conditions for Example 3
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Graph of Example 3
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Failure of Second-Order Condition
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Figure 9.6 Failure of the Second-Order Condition
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Stationary Point of a strictly concave function
If the function f(x) is strictly concave on the
interval (m,n) and has the stationary point x,
where mltxltn, the x is a local maximum in that
interval. If a function is strictly concave
everywhere, then it has, at most, one stationary
point, and that stationary point is a global
maximum. Note that example 1 satisfies this
condition.
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Stationary Point of a Strictly Convex Function
If the function f(x) is strictly convex on the
interval (m,n) and has the stationary point x,
where mltxltn, the x is a local minimum in that
interval. If a function is strictly convex
everywhere, then it has, at most, one stationary
point, and that stationary point is a global
minimum. Note that example 2 satisfies this
condition.
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Inflection Point
The twice-differentiable function f(x) has an
inflection point at x if and only if the sign of
the second derivative switches from negative in
some interval (m,x) to positive in some interval
(x,n), in which case the function switches from
concave to convex at x, or the sign of the
second derivative switches from positive in some
interval (m,x) to negative in some interval
(x,n), in which case the function switches from
convex to concave at x. Note that, in either
case, mltxltn.
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Example inflection point
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Graph of derivatives for example 3
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Graph of Example 3
Concave
Inflection point
convex
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Optimal Excise Tax
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Optimal excise tax rate
P
MRR
PeTe
S
Pe
D
MR
Q/T
Qe
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Optimal Timing
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Optimal Timing Continued
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Interpretation of results
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Interpretation of results
rate
r
t
0
t
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Second Order Condition