chiang ch9
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1
• 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
2
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)
3
9.2 Relative Maximum and Minimum: First-Derivative Test9.2-1 Relative versus absolute extremum9.2-2 First-derivative test
5
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.)
A f '(x*) = 0
x*
y
6
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.)
x
Bf '(x*)=0
y
x*
7
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)
D f '(x*) = 0
x*
y
x
8
9.2 Example 1 p. 225
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9.2 Example 2 p. 226
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9.2 Smile test
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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
17
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
18
9.3 Example
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19
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20
Example 1, p. 228
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21
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)
22
9.3-3 The smile test
minimum relative a is then
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24
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
25
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) >0, the 2nd order condition or relative maximum if f "(x0)<0.
26
9.4-2 Profit function: Example 3, p. 238
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28
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9.4 Imperfect Competition, Example 4, p. 240
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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.
32
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
33
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36
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40
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44
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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
54
9.6-1 Taylor expansion and relative extremum
55
9.6-2 Some specific cases
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