Chiang Ch9

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Chiang Ch9

  1. 1. <ul><li>Ch. 9 Optimization: A Special Variety of Equilibrium Analysis </li></ul><ul><li>9.1 Optimum Values and Extreme Values </li></ul><ul><li>9.2 Relative Maximum and Minimum: First-Derivative Test </li></ul><ul><li>9.3 Second and Higher Derivatives </li></ul><ul><li>9.4 Second-Derivative Test </li></ul><ul><li>9.5 Digression on Maclaurin and Taylor Series </li></ul><ul><li>9.6 N th -Derivative Test for Relative Extremum of a Function of One Variable </li></ul>
  2. 2. 9.1 Optimum Values and Extreme Values <ul><li>Goal vs. non-goal equilibrium </li></ul><ul><li>In the optimization process, we need to identify the objective function to optimize. </li></ul><ul><li>In the objective function the dependent variable represents the object of maximization or minimization </li></ul><ul><li> = PQ - C(Q) </li></ul>
  3. 3. 9.2 Relative Maximum and Minimum: First-Derivative Test 9.2-1 Relative versus absolute extremum 9.2-2 First-derivative test
  4. 4. 9.2 Critical & stationary values <ul><li>The critical value of x is the value x 0 if f’(x 0 ) = 0 </li></ul><ul><li>A stationary value of y is f(x 0 ) </li></ul><ul><li>A stationary point is the point with coordinates x 0 and f(x 0 ) </li></ul><ul><li>A stationary point is coordinate of the extremum </li></ul>
  5. 5. 9.2-2 First-derivative test  <ul><li>The first-order condition or necessary condition for extrema is that f ' (x*) = 0 and the value of f(x*) is: </li></ul><ul><li>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.) </li></ul>A f ' (x*) = 0 x* y
  6. 6. 9.2-2 First-derivative test  <ul><li>The first-order condition or necessary condition for extrema is that f ' (x*) = 0 and the value of f(x*) is: </li></ul><ul><li>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.) </li></ul>x B f ' (x*)=0 y x*
  7. 7. 9.2-2 First-derivative test  <ul><li>The first-order condition or necessary condition for extrema is that f ' (x*) = 0 and the value of f(x*) is: </li></ul><ul><li>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) </li></ul>D  f ' (x*) = 0 x* y x
  8. 8. 9.2 Example 1 p. 225
  9. 9. primitive function and 1 st & 2 nd derivatives
  10. 10. 9.2 Example 1 (neg) p. 225
  11. 11. primitive function and 1 st & 2 nd derivatives
  12. 12. 9.2 Example 2 p. 226
  13. 13. plots: primitive convex function, as well as 1 st & 2 nd derivatives
  14. 14. 9.2 Smile test 
  15. 15. Primitive function and 1 st & 2 nd derivatives
  16. 16. 9.3 Second and Higher Derivatives 9.3-1 Derivative of a derivative 9.3-2 Interpretation of the second derivative 9.3-3 An application
  17. 17. 9.3-1 Derivative of a derivative <ul><li>Given y = f(x) </li></ul><ul><li>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. </li></ul><ul><li>The result of this differentiation is known as the second derivative of the function f and is denoted as f ''(x) or d 2 y/dx 2 . </li></ul><ul><li>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 d n y/dx n </li></ul>
  18. 18. 9.3 Example
  19. 19. Primitive function and 1 st & 2 nd derivatives
  20. 20. Example 1, p. 228
  21. 21. 9.3-2 Interpretation of the second derivative <ul><li>f '(x) measures the rate of change of a function </li></ul><ul><ul><li>e.g., whether the slope is increasing or decreasing </li></ul></ul><ul><li>f ''(x) measures the rate of change in the rate of change of a function </li></ul><ul><ul><li>e.g., whether the slope is increasing or decreasing at an increasing or decreasing rate </li></ul></ul><ul><ul><li>how the curve tends to bend itself (p. 230) </li></ul></ul>
  22. 22. 9.3-3 The smile test    
  23. 23. 9.3 Example 
  24. 24. 9.4 Second-Derivative Test 9.4-1 Necessary and sufficient conditions 9.4-2 Conditions for profit maximization 9.4-3 Coefficients of a cubic total-cost function 9.4-4 Upward-sloping marginal-revenue curve
  25. 25. 9.4-1 Necessary and sufficient conditions <ul><li>The zero slope condition is a necessary condition and since it is found with the first derivative, we refer to it as a 1 st order condition. </li></ul><ul><li>The sign of the second derivative is sufficient to establish the stationary value in question as a relative minimum if f &quot; (x 0 ) >0, the 2 nd order condition or relative maximum if f &quot; (x 0 )<0. </li></ul>
  26. 26. 9.4-2 Profit function: Example 3, p. 238
  27. 27. 9.4 Quadratic equation
  28. 28. 9.4-2 Profit function: example 3, p. 238
  29. 29. 9.4 Imperfect Competition, Example 4, p. 240
  30. 30. Example 4, p. 240 TR and 1 st , 2 nd & 3 rd derivatives
  31. 31. 9.4 Strict concavity <ul><li>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. </li></ul><ul><li>A strictly concave curve can never contain a linear segment anywhere </li></ul><ul><li>Test: if f &quot;(x) is negative for all x, then strictly concave. </li></ul>
  32. 32. 9.5 Digression on Maclaurin and Taylor Series 9.5-1 Maclaurin series of a polynomial function 9.5-2 Taylor series of a polynomial functions 9.5-3 Expansion of an arbitrary function 9.5-4 Lagrange form of the remainder
  33. 33. 9.5-2 Taylor series of a polynomial functions
  34. 34.
  35. 35. Difference Quotient, Derivative & Differential f’(x 0 )  x  x A C D B f(x 0 +  x) f(x) f(x 0 ) x 0 x 0 +  x y=f(x) x  y  x f’(x)
  36. 36. Exercise 9.5-2(a): Geometric series
  37. 37.
  38. 38. Ch. 5b Linear Models and Matrix Algebra 5.5 - 5.8 5.7 Leontief Input-Output Models Structure of an input-output model Miller & Blair, p. 102
  39. 39.
  40. 40. Diewert’s Quadratic Lemma
  41. 41. Diewert’s Quadratic Lemma
  42. 42. Diewert’s Quadratic Lemma
  43. 43. Diewert’s Quadratic Lemma
  44. 44. 9.5 Maclaurin Series of a Polynomial Function
  45. 45. 9.5 Maclaurin Series of a Polynomial Function
  46. 46. 9.5 Taylor Series of a Polynomial Function
  47. 47. 9.5 Taylor Series of a Polynomial Function
  48. 48. 9.5 Taylor Series of a Polynomial Function
  49. 49. 9.6 Taylor expansion and relative extremum
  50. 50. 9.6-3 N th -derivative test
  51. 51. 9.5-4 Lagrange form of the remainder
  52. 52. 9.5-4 Lagrange form of the remainder
  53. 53. 9.6 N th -Derivative Test for Relative Extremum of a Function of One Variable 9.6-1 Taylor expansion and relative extremum 9.6-2 Some specific cases 9.6-3 N th -derivative test
  54. 54. 9.6-1 Taylor expansion and relative extremum
  55. 55. 9.6-2 Some specific cases

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