Chiang_Ch9 - Ch.9 Optimization:ASpecial...

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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  N th -Derivative Test for Relative  Extremum of a Function of One Variable
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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 2200 π  = PQ - C(Q)
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3 9.2 Relative Maximum and Minimum:   First-Derivative Test 9.2-1 Relative versus absolute extremum 9.2-2 First-derivative test
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4 The critical value of x is the value x 0  if  f’(x 0 ) = 0 A stationary value of y is f(x 0 • A stationary point is the point with  coordinates x 0  and f(x 0 )   • A stationary point is coordinate of the  extremum
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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
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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 x 0  to its immediate right. (first derivative test of  min.) x B f ' (x*)=0 y x*
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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 x 0 . (first derivative test for point of  inflection) D f ' (x*) = 0 x* y x
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8 9.2 Example 1 p. 225 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 + - = = - = = = = + - = = = = = = = = = = - - = + - = = + - = + + - = = , 75 . 6 ) 5 . 6 ( ' 0 ) 6 ( ' 25 . 5 ) 5 . 5 ( ' min 625 . 9 ) 5 . 6 ( 8 ) 6 ( 375 . 9 ) 5 . 5 ( 1 , 25 . 5 ) 5 . 2 ( ' 0 ) 2 ( ' 75 . 6 ) 5 . 1 ( ' max 625 . 38 5 . 2 40 ) 2 ( 375 . 38 ) 5 . 1 ( right left extrema 6 ; 2 0 6 2 0 12 8 3 ) ( ' derivative st 1 0 36 24 3 ) ( ' function primative 8 36 12 ) ( * * * 2 * 1 2 2 2 3 f f f f f f f f f f f f x f y x x x x x x x f x x x f x x x x f y i
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9 primitive function and 1 st nd derivatives ( 29 ( 29 4 0 24 6 ) ( ) 3 6 2 0 12 8 ) ( 36 24 3 ) ( ) 2 8 36 12 ) ( ) 1 2 2 2 3 = = - = = = = + - = + - = + + - = x x x f x x x x x f x x x f x x x x f
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This note was uploaded on 11/09/2011 for the course ECON 101 taught by Professor Richards during the Spring '11 term at Cambrian College.

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Chiang_Ch9 - Ch.9 Optimization:ASpecial...

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