Mathematics for Business Administration:
                                    Multivariable Optimization
                                             Universidad de Murcia
                                         Mar´ıa Pilar Mart´ınez-Garc´ıa
                                       Universidad de Murcia      Mathematics for Business Administration: Multivariable Optimization
           Chapter Four: Constrained Optimization
                        Useful links
               Review problems for Chapter 4
             Multiple choice questions Chapter 4
        Chapter Four: Constrained Optimization. The
                 Lagrange Multiplier Method
                                Chapter Four: Constrained Optimization. The Lagrange Multiplier Method
              Chapter Four: Constrained Optimization
                               Useful links
                   Review problems for Chapter 4
                Multiple choice questions Chapter 4
   Outline
           Introduction
           The Lagrange Multiplier Method (the two-variable case)
           The Lagrange Multiplier is a shadow price
           The Lagrange method applied to the general multivariable
           case
                                         Chapter Four: Constrained Optimization. The Lagrange Multiplier Method
                      Chapter Four: Constrained Optimization      Introduction
                                                 Useful links     The Lagrange Multiplier Method.
                               Review problems for Chapter 4      The Lagrange Multiplier is a shadow price
                          Multiple choice questions Chapter 4     TheLagrangemethodappliedtothegeneralmultivariablecase.
          consumer’s optimization problem
                                 maxU(x,y) subject to p·x+y = b.                                              (P)
          Note that:
                                      ∗    ∗
                  The point (x ,y ) that solves problem (P) is not necessarily a
                  maximum point (global or local) of the function U(x,y)
                  In this case y = b − p · x, ⇒ Max f(x) = U(x,b −px)
                  unconstrained optimization problem with one variable less
                  If the substitution method is difficult or impossible to carry
                  out in practise ⇒ Lagrange Method
                                                                  Chapter Four: Constrained Optimization. The Lagrange Multiplier Method