Ch' 9Optimization: A Special Variety of Equilibrium Analysis

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• Ch. 9 Optimization A Special Variety of
  Equilibrium Analysis
• 9.1 Optimum Values and Extreme Values
• 9.2 Relative Maximum and Minimum
  First-Derivative Test
• 9.3 Second and Higher Derivatives
• 9.4 Second-Derivative Test
• 9.5 Digression on Maclaurin and Taylor Series
• 9.6 Nth-Derivative Test for Relative Extremum of
  a Function of One Variable

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9.1 Optimum Values and Extreme Values

• Goal vs. non-goal equilibrium
• In the optimization process, we need to identify
  the objective function to optimize.
• In the objective function the dependent variable
  represents the object of maximization or
  minimization
• ? PQ - C(Q)

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9.2 Relative Maximum and Minimum
First-Derivative Test9.2-1 Relative versus
absolute extremum9.2-2 First-derivative test
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9.2 Critical stationary values

• The critical value of x is the value x0 if f(x0)
  0
• A stationary value of y is f(x0)
• A stationary point is the point with coordinates
  x0 and f(x0)
• A stationary point is coordinate of the extremum

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9.2-2 First-derivative test ?

• The first-order condition or necessary condition
  for extrema is that f '(x) 0 and the value of
  f(x) is
• A relative maximum if the derivative f '(x)
  changes its sign from positive to negative from
  the immediate left of the point x to its
  immediate right. (first derivative test for a
  max.)

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9.2-2 First-derivative test ?

• The first-order condition or necessary condition
  for extrema is that f '(x) 0 and the value of
  f(x) is
• A relative minimum if f '(x) changes its sign
  from negative to positive from the immediate left
  of x0 to its immediate right. (first derivative
  test of min.)

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9.2-2 First-derivative test ?

• The first-order condition or necessary condition
  for extrema is that f '(x) 0 and the value of
  f(x) is
• Neither a relative maxima nor a relative minima
  if f '(x) has the same sign on both the
  immediate left and right of point x0. (first
  derivative test for point of inflection)

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9.2 Example 1 p. 225
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primitive function and 1st 2nd derivatives
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9.2 Example 1 (neg) p. 225
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primitive function and 1st 2nd derivatives
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9.2 Example 2 p. 226
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plots primitive convex function, as well as1st
2nd derivatives
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9.2 Smile test ?
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Primitive function and 1st 2nd derivatives
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9.3 Second and Higher Derivatives9.3-1
Derivative of a derivative9.3-2 Interpretation
of the second derivative9.3-3 An application
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9.3-1 Derivative of a derivative

• Given y f(x)
• The first derivative f '(x) or dy/dx is itself a
  function of x, it should be differentiable with
  respect to x, provided that it is continuous and
  smooth.
• The result of this differentiation is known as
  the second derivative of the function f and is
  denoted as f ''(x) or d2y/dx2.
• The second derivative can be differentiated with
  respect to x again to produce a third derivative,
  f '''(x) and so on to f(n)(x) or dny/dxn

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9.3 Example
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Primitive function and 1st 2nd derivatives
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Example 1, p. 228
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9.3-2 Interpretation of the second derivative

• f '(x) measures the rate of change of a function
• e.g., whether the slope is increasing or
  decreasing
• f ''(x) measures the rate of change in the rate
  of change of a function
• e.g., whether the slope is increasing or
  decreasing at an increasing or decreasing rate
• how the curve tends to bend itself (p. 230)

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9.3-3 The smile test ? ?
?
?
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9.3 Example ?
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9.4 Second-Derivative Test9.4-1 Necessary and
sufficient conditions9.4-2 Conditions for profit
maximization9.4-3 Coefficients of a cubic
total-cost function9.4-4 Upward-sloping
marginal-revenue curve
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9.4-1 Necessary and sufficient conditions

• The zero slope condition is a necessary
  condition and since it is found with the first
  derivative, we refer to it as a 1st order
  condition.
• The sign of the second derivative is sufficient
  to establish the stationary value in question as
  a relative minimum if f "(x0) gt0, the 2nd order
  condition or relative maximum if f "(x0)lt0.

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9.4-2 Profit function Example 3, p. 238
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9.4 Quadratic equation
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9.4-2 Profit function example 3, p. 238
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9.4 Imperfect Competition, Example 4, p. 240
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Example 4, p. 240TR and 1st , 2nd 3rd
derivatives
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9.4 Strict concavity

• Strictly concave if we pick any pair of points M
  and N on its curve and joint them by a straight
  line, the line segment MN must lie entirely below
  the curve, except at points MN.
• A strictly concave curve can never contain a
  linear segment anywhere
• Test if f "(x) is negative for all x, then
  strictly concave.

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9.5 Digression on Maclaurin and Taylor
Series9.5-1 Maclaurin series of a polynomial
function9.5-2 Taylor series of a polynomial
functions9.5-3 Expansion of an arbitrary
function9.5-4 Lagrange form of the remainder
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9.5-2 Taylor series of a polynomial functions
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Difference Quotient, Derivative Differential
B
??x
D
A
f(x0)?x
C
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Exercise 9.5-2(a) Geometric series
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5.7 Leontief Input-Output Models Structure of an
input-output model Miller Blair, p. 102
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Diewerts Quadratic Lemma
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Diewerts Quadratic Lemma
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Diewerts Quadratic Lemma
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Diewerts Quadratic Lemma
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9.5 Maclaurin Series of a Polynomial Function
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9.5 Maclaurin Series of a Polynomial Function
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9.5 Taylor Series of a Polynomial Function
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9.5 Taylor Series of a Polynomial Function
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9.5 Taylor Series of a Polynomial Function
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9.6 Taylor expansion and relative extremum
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9.6-3 Nth-derivative test
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9.5-4 Lagrange form of the remainder
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9.5-4 Lagrange form of the remainder
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9.6 Nth-Derivative Test for Relative Extremum of
a Function of One Variable9.6-1 Taylor expansion
and relative extremum9.6-2 Some specific
cases9.6-3 Nth-derivative test
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9.6-1 Taylor expansion and relative extremum
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9.6-2 Some specific cases