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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10

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16

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

derivative rd30)4

derivative nd24)3

derivativest 141200)()2

function primative21200)1 2

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19

Primitive function and 1st & 2nd derivatives

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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

,0 If

maximum relative a is then

,0 If

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23

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maximum 05)

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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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27

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28

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max0 min0 )8

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29

9.4 Imperfect Competition, Example 4, p. 240

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30

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31

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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35

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36

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38

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39

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40

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41

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42

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43

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9.5 Maclaurin Series of a Polynomial Function

44

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46

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47

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48

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50

9.6-3 Nth-derivative test

odd is N ifpoint inflectionan

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51

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52

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53

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