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3. (8 marks) Regarding the optimization of f(x) subject to the constraint g(x) x(n) are choice va...
A consumer must maximize utility, U-f(x.y), subject to the constraint that she spends all her inc...
A consumer must maximize utility, U-f(x.y), subject to the constraint that she spends all her income, M on purchasing two goods x, v. The unit prices of the goods, px and py respectively, are market determined and hence exogenous. (i) State the objective function, constraint, and choice variables of this problem (3 marks) (ii) Obtain the Lagrangean for this problem, using λ to represent the Lagrange multiplier. (3 marks) (i) Obtain the first order conditions of this problem in terms...
4. Problems 2.4 Suppose you would like to maximizefxy)-xy, subject to the constraint that x and...
4. Problems 2.4 Suppose you would like to maximizefxy)-xy, subject to the constraint that x and y are constrained to sum to 1. That is, xy 1 Instead of working through this maximization problem, however, you could also work through the dual problem to this, which is: Minimize x+ y subject to xy = 0.25 Hint: Assume x and y take only positive values The Lagrangian for this constrained minimization in the dual problem is Which of the following are...
8.(15 POINTS) Consider the following optimization problem: Max xi + subject to : 5xí +60192 +...
8.(15 POINTS) Consider the following optimization problem: Max xi + subject to : 5xí +60192 + 5x3 = 1 and 21 > 0,22 > 0. where 2 and 32 are choice variables. (a) Write the Lagrangean and the Kuhn-Tucker conditions. (6) State and verify the second order condition. Distinguish between sufficient and necessary condi tions. (c) Is the constraint qualification condition satisfied? Show clearly why or why not. (d) Solve the Kuhn-Tucker conditions for the optimal choice: x1, x, and...
1. Consider the constrained optimization problem: min f(x,x2) - (x-3)2 (x2 -3)2 Subject to Is this...
1. Consider the constrained optimization problem: min f(x,x2) - (x-3)2 (x2 -3)2 Subject to Is this problem convex? Justify your answer Form the Lagrangian function. a. b. Check the necessary and sufficient conditions for candidate local minimum points. Note that equality constraint for a feasible point is always an active constraint c. d. Is the solution you found in part (c) a global minimum? Explain your answer
S An individual has a utility function as follows subject to the budget constraint; 6r+2y 110 i) Write down the Lagrangian function for this individual. (2 marks) (6 marks) Using Cramer's rule...
S An individual has a utility function as follows subject to the budget constraint; 6r+2y 110 i) Write down the Lagrangian function for this individual. (2 marks) (6 marks) Using Cramer's rule, solve for x, y and 2. ii) Using Hessian matrix, check the second-order sufficient condition to verify that the utility of this individual is at maximum. (3 marks)
S An individual has a utility function as follows subject to the budget constraint; 6r+2y 110 i) Write down the...
3. Problems 2.3 Suppose that f(x,y)=xy, with the constraint that x and y are constrained to...
3. Problems 2.3 Suppose that f(x,y)=xy, with the constraint that x and y are constrained to sum to 1. That is, x +y = 1 Given this constraint, which of the following functions of x is equivalent to the original functionfx,y)=xy? f (x) = x2 f (x) = 1-r f (x) -x-x2 f(x) = x + x2 using the first order condition that f . (x) = 0, the value of x that maximizes f(x) (andfx,y)) is x- corresponding value...
A consumer must maximize utility, U-for.y), subject to the constraint that she spends all her inc...
A consumer must maximize utility, U-for.y), subject to the constraint that she spends all her income, M on purchasing two goods x, y. The unit prices of the goods, p, and py respectively, are market determined and hence exogenous (3 marks) (3 marks) rKS rice marks) (i e1 (2 marks) 0.8,0.2 (d) Let the utility function be U -5x ф Solve the maximization problem in this case (that is obtain x*, y*, 8y0.z and unit prices pr - p- 1...
Consider the optimization problem 5-6 5-6 F=(X-I)2 + (X Minimize: Subject to: 2-1) X +X-0.5s 0...
Consider the optimization problem 5-6 5-6 F=(X-I)2 + (X Minimize: Subject to: 2-1) X +X-0.5s 0 a. Write the expression for the augmented Lagrangian using r'p = 1. b. Beginning with λ 1 0 and λ2-0 , perform three iterations of the ALM method. c. Repeat part (b), beginning with λ 1-1 and λ2-1 d. Repeat part (b), beginning with λι--I and λ2--1
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x =...
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
a) Solve the following problem using Lagrange multiplier method. Minimize fCX)-x1+ x2+X 4. subject to: x2+x-3...
a) Solve the following problem using Lagrange multiplier method. Minimize fCX)-x1+ x2+X 4. subject to: x2+x-3 X1+3x2+ 2x)- 7 (1) (2) (Note: Please do not check the second order sufficiency conditions) b) If the right side of the above constraint (1) is changed to 3.4, using sensitivity analysis find the approximate new minimum value of fX).
a) Solve the following problem using Lagrange multiplier method. Minimize fCX)-x1+ x2+X 4. subject to: x2+x-3 X1+3x2+ 2x)- 7 (1) (2) (Note: Please do...
A consumer must maximize utility, U-f(x.y), subject to the constraint that she spends all her income, M on purchasing two goods x, v. The unit prices of the goods, px and py respectively, are market determined and hence exogenous. (i) State the objective function, constraint, and choice variables of this problem (3 marks) (ii) Obtain the Lagrangean for this problem, using λ to represent the Lagrange multiplier. (3 marks) (i) Obtain the first order conditions of this problem in terms...
4. Problems 2.4 Suppose you would like to maximizefxy)-xy, subject to the constraint that x and y are constrained to sum to 1. That is, xy 1 Instead of working through this maximization problem, however, you could also work through the dual problem to this, which is: Minimize x+ y subject to xy = 0.25 Hint: Assume x and y take only positive values The Lagrangian for this constrained minimization in the dual problem is Which of the following are...
8.(15 POINTS) Consider the following optimization problem: Max xi + subject to : 5xí +60192 + 5x3 = 1 and 21 > 0,22 > 0. where 2 and 32 are choice variables. (a) Write the Lagrangean and the Kuhn-Tucker conditions. (6) State and verify the second order condition. Distinguish between sufficient and necessary condi tions. (c) Is the constraint qualification condition satisfied? Show clearly why or why not. (d) Solve the Kuhn-Tucker conditions for the optimal choice: x1, x, and...
1. Consider the constrained optimization problem: min f(x,x2) - (x-3)2 (x2 -3)2 Subject to Is this problem convex? Justify your answer Form the Lagrangian function. a. b. Check the necessary and sufficient conditions for candidate local minimum points. Note that equality constraint for a feasible point is always an active constraint c. d. Is the solution you found in part (c) a global minimum? Explain your answer
S An individual has a utility function as follows subject to the budget constraint; 6r+2y 110 i) Write down the Lagrangian function for this individual. (2 marks) (6 marks) Using Cramer's rule, solve for x, y and 2. ii) Using Hessian matrix, check the second-order sufficient condition to verify that the utility of this individual is at maximum. (3 marks)
S An individual has a utility function as follows subject to the budget constraint; 6r+2y 110 i) Write down the...
3. Problems 2.3 Suppose that f(x,y)=xy, with the constraint that x and y are constrained to sum to 1. That is, x +y = 1 Given this constraint, which of the following functions of x is equivalent to the original functionfx,y)=xy? f (x) = x2 f (x) = 1-r f (x) -x-x2 f(x) = x + x2 using the first order condition that f . (x) = 0, the value of x that maximizes f(x) (andfx,y)) is x- corresponding value...
A consumer must maximize utility, U-for.y), subject to the constraint that she spends all her income, M on purchasing two goods x, y. The unit prices of the goods, p, and py respectively, are market determined and hence exogenous (3 marks) (3 marks) rKS rice marks) (i e1 (2 marks) 0.8,0.2 (d) Let the utility function be U -5x ф Solve the maximization problem in this case (that is obtain x*, y*, 8y0.z and unit prices pr - p- 1...
Consider the optimization problem 5-6 5-6 F=(X-I)2 + (X Minimize: Subject to: 2-1) X +X-0.5s 0 a. Write the expression for the augmented Lagrangian using r'p = 1. b. Beginning with λ 1 0 and λ2-0 , perform three iterations of the ALM method. c. Repeat part (b), beginning with λ 1-1 and λ2-1 d. Repeat part (b), beginning with λι--I and λ2--1
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
a) Solve the following problem using Lagrange multiplier method. Minimize fCX)-x1+ x2+X 4. subject to: x2+x-3 X1+3x2+ 2x)- 7 (1) (2) (Note: Please do not check the second order sufficiency conditions) b) If the right side of the above constraint (1) is changed to 3.4, using sensitivity analysis find the approximate new minimum value of fX).
a) Solve the following problem using Lagrange multiplier method. Minimize fCX)-x1+ x2+X 4. subject to: x2+x-3 X1+3x2+ 2x)- 7 (1) (2) (Note: Please do...
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