Solve for the optimal (c,l) pair in the following problem
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Solve for the optimal (c,l) pair in the following problem
1+0 max C ед subject to...
2. Consumption-Savings Decision: The Household's decision problem is: 1- 1- max - C1,C2,8 1-7."1-7 s.t. Ci+s=(<)yi...
2. Consumption-Savings Decision: The Household's decision problem is: 1- 1- max - C1,C2,8 1-7."1-7 s.t. Ci+s=(<)yi C2 = (*)(1+r)s + y2 where ci and c2 are consumption in periods 1 and 2 respectively; yi and Y2 are income in periods 1 and 2 respectively; s is savings; r is the interest rate on savings.y is a parameter controlling the concavity of the utility function, and will determine intertemporal substitution of consumption.4 We assume that y> 1; so utility is increasing...
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function...
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function u(c) is strictly concave, increasing, C1, and as- suming interiority of the optimal solution, what is the problem the consumer is solving? What are the FOCs for this problem using an "unconstrained" ap- proach (i.e., variable substitution in "primal" problem)? (b) Do optimal solutions for all goods satisfy "MRS" "price ratio" condition (i.e., MRSy(c) for all (V) i j)? If so, explain why. If...
Solve the following Utility Maximization Problem for x* and y* that Max U(x,y)= ln(x) subject to...
Solve the following Utility Maximization Problem for x* and y* that Max U(x,y)= ln(x) subject to Pxx + pyy = I ----.-.(2) where In denotes the natural logarithm (base e) and x and y>0. a) (25 points) by Substitution and show that your values of x* and y* max U (x*,y*). Problem 1. b) (20 points) by the Lagrange Multiplier Method
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a)...
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.
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...
L I L I JUNULUI! SM 4. Consider the utility maximization problem max U(x,y) = (x...
L I L I JUNULUI! SM 4. Consider the utility maximization problem max U(x,y) = (x + y s.t. x+4y= 100. (a) Using the Lagrange method, find the quantities demanded of the two goods. (1) Suppose income increases from 100 to 101. What is the exact increase in the optimal value of U(x,y)? Compare with the value found in (a) for the Lagrange multiplier (c) Suppose we change the budget constraint to px + 4y=m, but keep the same utility...
1. Consider the profit maximization problem for the firm. Say the pro- duction function f(k, n)...
1. Consider the profit maximization problem for the firm. Say the pro- duction function f(k, n) is strictly concave jointly in all arguments and once continuously differentiable (C), where k (resp, n) denotes the input of capital (resp, labor) (a) Assume a maximum exists for the firms profit maximization problem. What are the first order conditions (FOCs) to characterize all optimal solutions for factor demands for capital and labor in the general problem? (b) Say f(k, n) = (ak" +...
ONLY ANSWER 5 and 9. Rating will be provided Exercises for Section 6.2 In Exercises 1-12 use the separation of variables method to solve the heat equation (a, t)auz(t<<l,t>0, subject t...
ONLY ANSWER 5 and 9. Rating will be provided
Exercises for Section 6.2 In Exercises 1-12 use the separation of variables method to solve the heat equation (a, t)auz(t<<l,t>0, subject to the following boundary conditions and the following initial conditions: a = V2, l = 2, u(0,t) = u(2,t)=0, and 5. 20, 0r< 1 0, a(x, 0) = rS 2. 1 1 = π, u(z, 0) = π-z, u(0, t) = uz(mt) = 0. 9.
Exercises for Section 6.2 In...
A firm has a Cobb-Douglas production function q = AKL, where K denotes capital, L is...
A firm has a Cobb-Douglas production function q = AKL, where K denotes capital, L is labor, and A, a, b, are constants. ginal returns to labor in the short run if its production function is 1. Sketch an isoquant line, write a mathematical formula for its slope, and provide an interpretation for its meaning. 2. On a separate graph, draw an isocost line, write a mathematical formula for its slope, and provide an interpretation for its meaning. 3. On...
Problem 1: Consider the following linear optimization problem: max 1 +22x;3 subject to x1 + x2...
Problem 1: Consider the following linear optimization problem: max 1 +22x;3 subject to x1 + x2 +r3 10 2x1 -r2 2-4 i20, -1,2,3 a) Bring the problem to a standard form (b) Show that the point (2,8,0)T is optimal by the optimality condition of the linear program- ming. Is it an extreme point? Provide arguments for your answers (c) Determine at least one other point different than (2,8,0)T, which is an extreme point of the constraint set 1) (d) Find...
2. Consumption-Savings Decision: The Household's decision problem is: 1- 1- max - C1,C2,8 1-7."1-7 s.t. Ci+s=(<)yi C2 = (*)(1+r)s + y2 where ci and c2 are consumption in periods 1 and 2 respectively; yi and Y2 are income in periods 1 and 2 respectively; s is savings; r is the interest rate on savings.y is a parameter controlling the concavity of the utility function, and will determine intertemporal substitution of consumption.4 We assume that y> 1; so utility is increasing...
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function u(c) is strictly concave, increasing, C1, and as- suming interiority of the optimal solution, what is the problem the consumer is solving? What are the FOCs for this problem using an "unconstrained" ap- proach (i.e., variable substitution in "primal" problem)? (b) Do optimal solutions for all goods satisfy "MRS" "price ratio" condition (i.e., MRSy(c) for all (V) i j)? If so, explain why. If...
Solve the following Utility Maximization Problem for x* and y* that Max U(x,y)= ln(x) subject to Pxx + pyy = I ----.-.(2) where In denotes the natural logarithm (base e) and x and y>0. a) (25 points) by Substitution and show that your values of x* and y* max U (x*,y*). Problem 1. b) (20 points) by the Lagrange Multiplier Method
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.
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...
L I L I JUNULUI! SM 4. Consider the utility maximization problem max U(x,y) = (x + y s.t. x+4y= 100. (a) Using the Lagrange method, find the quantities demanded of the two goods. (1) Suppose income increases from 100 to 101. What is the exact increase in the optimal value of U(x,y)? Compare with the value found in (a) for the Lagrange multiplier (c) Suppose we change the budget constraint to px + 4y=m, but keep the same utility...
1. Consider the profit maximization problem for the firm. Say the pro- duction function f(k, n) is strictly concave jointly in all arguments and once continuously differentiable (C), where k (resp, n) denotes the input of capital (resp, labor) (a) Assume a maximum exists for the firms profit maximization problem. What are the first order conditions (FOCs) to characterize all optimal solutions for factor demands for capital and labor in the general problem? (b) Say f(k, n) = (ak" +...
ONLY ANSWER 5 and 9. Rating will be provided
Exercises for Section 6.2 In Exercises 1-12 use the separation of variables method to solve the heat equation (a, t)auz(t<<l,t>0, subject to the following boundary conditions and the following initial conditions: a = V2, l = 2, u(0,t) = u(2,t)=0, and 5. 20, 0r< 1 0, a(x, 0) = rS 2. 1 1 = π, u(z, 0) = π-z, u(0, t) = uz(mt) = 0. 9.
Exercises for Section 6.2 In...
A firm has a Cobb-Douglas production function q = AKL, where K denotes capital, L is labor, and A, a, b, are constants. ginal returns to labor in the short run if its production function is 1. Sketch an isoquant line, write a mathematical formula for its slope, and provide an interpretation for its meaning. 2. On a separate graph, draw an isocost line, write a mathematical formula for its slope, and provide an interpretation for its meaning. 3. On...
Problem 1: Consider the following linear optimization problem: max 1 +22x;3 subject to x1 + x2 +r3 10 2x1 -r2 2-4 i20, -1,2,3 a) Bring the problem to a standard form (b) Show that the point (2,8,0)T is optimal by the optimality condition of the linear program- ming. Is it an extreme point? Provide arguments for your answers (c) Determine at least one other point different than (2,8,0)T, which is an extreme point of the constraint set 1) (d) Find...
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