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Unformatted text preview: Number 12 The given utility function is negative, but for the purpose of solving, we have : Clear @ L D Setting up the Lagrangian function through the given utility function and constraint : L = - H x1 + 3 L ^- 1- H x2 + 4 L ^- 1 + Λ H 102- 4 x1- 9 x2 L- 1 3 + x1- 1 4 + x2 + H 102- 4 x1- 9 x2 L Λ Mike ' s Lagrangian function, subject to a budget constraint Solving for the optiaml values using the Lagrangian technique. Getting the Optimal values of x1, x2, x3 through the FOC, we equate the first order partials of the Lagrangian with respect to each variable. We get : Solve @8 D @ L, x1 D 0, D @ L, x2 D 0, D @ L, Λ D < , 8 x1, x2, Λ <D :: x1 fi - 78, x2 fi 46, Λ fi 1 22 500 > , : x1 fi 12, x2 fi 6, Λ fi 1 900 >> The optimum consumption bundle for Mike ' s given utility function is the set that gave positive values. To check for the for the sufficiency condition of a maximium, we must satisfy the SOC. Checking for SOC : We get the second partial of L wrt to each variable. Through solving, we have : D @ L, x1, x1 D- 2 H 3 + x1 L 3 D @ L, x2, x2 D- 2 H 4 + x2 L 3 D @ L, x1, x2 D Setting up our Bordered Hessian Matrix and solving for the determinant of each minor to check for the sufficiency condition of the values for x1 & x2 : H1 = Det B- 4- 9- 2 H 3 + x1 L 3 F H2 <- 36 Det B- 4- 9- 4- 2 H 3 + x1 L 3- 9- 2 H 4 + x2 L 3 F H3 > 162 H 3 + x1 L 3 + 32 H 4 + x2 L 3 We have proven, through the Bordered Hessian, that H1 = 0, H2 < 0, H3 > 0. Values for x1 & x2 are indeed maximum. The value of the Lagrange mulitplier using the optimal values through substituting the x1, x2 and Lambda. L = - H 12 + 3 L ^- 1- H 6 + 4 L ^- 1- 1 6 Number 13 Clear @ L, U D Utility- maximization problem : Setting up the Lagrangian function given the utiltiy function and budget constrain in maximizing Kurt ' s utility. L = H x1 ^ 0.5 + x2 ^ 0.5 L ^ 2 + Λ H m- p1 * x1- p2 * x2 L I x1 0.5 + x2 0.5 M 2 + H m- p1 x1- p2 x2 L Λ Kurt ' s Lagrangian function Solving for the optimal consumption values using the Lagrangian technique....
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