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Lesson #3 2011/10/14

Authors
Keywords
  • Economia
  • 2011
  • 6
  • 0054
  • Calculus (1: Modulo Generico)
  • 37258
  • Quantitative Finance
  • 8409

Abstract

1/25 P�i? 22333ML232 Q-Finance 2011 Calculus @ QFinance Lesson 2.3 Friday October 14th 2011 Fubini’s Theorem professor Daniele Ritelli www.unibo.it/docenti/daniele.ritelli 2/25 P�i? 22333ML232 Sufficient conditions Find max or min of f(x, y) under the constraint g(x, y) = 0 Lagrangean L(x, y, λ) = f(x, y)− λg(x, y) After solving the system f ′ x (x, y)− λg′x (x, y) = 0, f ′ y (x, y)− λg′y (x, y) = 0, g (x, y) = 0. 3/25 P�i? 22333ML232 evaluate Λ = det  L ′′ xx L ′′ xy gx L ′′ xy L ′′ yy gy gx gy 0  Λ > 0 maximum Λ < 0 minimum 4/25 P�i? 22333ML232 Let us generalize an exercise we saw yesterday: take the n variables function u(x) = u (x1, x2, . . . , x`) = x α1 1 x α2 2 · · ·xα`` = ∏` j=1 x αj j where αj > 0 for j = 1, . . . , ` Then fix a vector in Rn with positive coordinates: pi = (pi1, . . . , pi`) . Maximize u with the constraint x · pi = c ∈ R+ 4/25 P�i? 22333ML232 Let us generalize an exercise we saw yesterday: take the n variables function u(x) = u (x1, x2, . . . , x`) = x α1 1 x α2 2 · · ·xα`` = ∏` j=1 x αj j where αj > 0 for j = 1, . . . , ` Then fix a vector in Rn with positive coordinates: pi = (pi1, . . . , pi`) . Maximize u with the constraint x · pi = c ∈ R+ Form the multiplier’s theorem we have for i = 1, . . . , ` ∂u ∂xi = αi xi u = λpii (M) 4/25 P�i? 22333ML232 Let us generalize an exercise we saw yesterday: take the n variables function u(x) = u (x1, x2, . . . , x`) = x α1 1 x α2 2 · · ·xα`` = ∏` j=1 x αj j where αj > 0 for j = 1, . . . , ` Then fix a vector in Rn with positive coordinates: pi = (pi1, . . . , pi`) . Maximize u with the constraint x · pi = c ∈ R+ Form the multiplier’s theorem we have for i = 1, . . . , ` ∂u ∂xi = αi xi u = λpii (M) 5/25 P�i? 22333ML232 write (M) as: αiu = λpiixi (Ma) 6/25 P�i? 22333ML232 sum (Ma) between i = 1, . . . , ` ∑` i=1 αiu = ∑` i=1 λpiixi = λpi · x = cλ, 6

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