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L I L I JUNULUI! SM 4. Consider the utility maximization problem max U(x,y) = (x...
M 4. Consider the utility maximization problem max U(x,y) = x +y s.t. x + 4y...
M 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. (b) 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 + y = m, but keep the same utility...
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
9. Consider the utility maximization problem max x + y s.t. px + y =m, where...
9. Consider the utility maximization problem max x + y s.t. px + y =m, where the constants p, 9, and m are positive, and the constant a € (0,1). (a) Find the demand functions, x* (p, m) and y* (p, m). (b) Find the partial derivatives of the demand functions w.r.t. p and m, and check their signs. (c) How does the optimal expenditure on the x good vary with p?8 (d) Put a = 1/2. What are the...
Suppose the following equations represent an individual’s utility maximization problem: U(X,Y) = X0.5 + Y0.5 And the bu...
Suppose the following equations represent an individual’s utility maximization problem: U(X,Y) = X0.5 + Y0.5 And the budget constraint is: I = PxX + PyY (a) Set up the individual’s maximization problem using the Lagrange technique. (b) Find the individual’s demand function for X and Y (Derive from first order condition). (c) Find the indirect utility function. (d) Find the expenditure function. (e) Find the share of X and Y on expenditure. (f) Find the marginal utility of income.
6. Consider the following constrained maximization problem: 2 5 tu (х, у) x7y7 max х,у s.t...
6. Consider the following constrained maximization problem: 2 5 tu (х, у) x7y7 max х,у s.t Рxх + pуy < м 3, py = 4, M = 12. Answer the following questions with px a. Write down the Lagrangian function b. Derive the first order conditions c. Derive the optimality condition from those conditions d. Write the other optimality condition (since there should be two in order for us to solve for two unknowns) e. Find the optimal values for...
Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4)...
Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4) Denote the price of good X by Px, price of good Y by Py and the income of the consumer by I. a) (2 points) Write down the budget constraint for the individual. b) (4 points) Derive the marginal utilities of X and Y. c) (3 points) Derive the expression for the marginal rate of substitution of X for Y. Write down the tangency...
(5) Consider how a consumer decides how much of two goods, x and y, to consume...
(5) Consider how a consumer decides how much of two goods, x and y, to consume but races amb constraint. The consumer maximizes the following utility function s.t. the budget constraint U(x, y) = (x - rayß subject to Pxx + p,Y = 1 Where x and y stand for the two goods, Px, and Py stand for the prices of the goods X and Y, respectively, and I stands for Income (a) Write down the Lagrangian and solve for...
Optimal Consumption of good x and good y: Maximization Rule - Maximization of Utility given a...
Optimal Consumption of good x and good y: Maximization Rule - Maximization of Utility given a Budget Constraint = Marginal Utility of good x/Price of good x = Marginal Utility of good y/Price of y Calculate Consumption Bundle using the following information: Price of Good x = $5, Price of Good y = $16 and Income = $100 / 0 Quantity Consumed Total Utility Quantity Consumed Total Utility Calculate: a.) Price Elasticity of Demand =% Change in Quantity Demanded/%Change in...
Your friend has the following utility unction: U(X,Y) = 10 X + 40 Y – X2-...
Your friend has the following utility unction: U(X,Y) = 10 X + 40 Y – X2- 3Y2 Where X is her consumption of Redbox movies, with price Px = $1, and Y is her consumption of iTunes, with Py = $2. Income is 48 dollars. a. Using the Lagrangian approach, derive your friend’s demand equations for Redbox movies and iTunes. That is, find X and Y. (Hint: Substitute the budget constraint in the Lagrangian problem using the given prices and...
Question 1: Colin's utility function for goods X and Y is represented by U(XY) = X0.5Y0.5...
Question 1: Colin's utility function for goods X and Y is represented by U(XY) = X0.5Y0.5 . Assume his income is $1000 and the prices of X and Y are $50 and S100, respectively. a. Write an expression for Colin's budget constraint. b. Calculate the optimal quantities of X and Y that Colin should choose, given his budget constraint. Graph your answer. Suppose that government subsidy program lowers the price of Y from $100 per unit to $ 50 per...
M 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. (b) 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 + y = m, but keep the same utility...
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
9. Consider the utility maximization problem max x + y s.t. px + y =m, where the constants p, 9, and m are positive, and the constant a € (0,1). (a) Find the demand functions, x* (p, m) and y* (p, m). (b) Find the partial derivatives of the demand functions w.r.t. p and m, and check their signs. (c) How does the optimal expenditure on the x good vary with p?8 (d) Put a = 1/2. What are the...
6. Consider the following constrained maximization problem: 2 5 tu (х, у) x7y7 max х,у s.t Рxх + pуy < м 3, py = 4, M = 12. Answer the following questions with px a. Write down the Lagrangian function b. Derive the first order conditions c. Derive the optimality condition from those conditions d. Write the other optimality condition (since there should be two in order for us to solve for two unknowns) e. Find the optimal values for...
(5) Consider how a consumer decides how much of two goods, x and y, to consume but races amb constraint. The consumer maximizes the following utility function s.t. the budget constraint U(x, y) = (x - rayß subject to Pxx + p,Y = 1 Where x and y stand for the two goods, Px, and Py stand for the prices of the goods X and Y, respectively, and I stands for Income (a) Write down the Lagrangian and solve for...
Optimal Consumption of good x and good y: Maximization Rule - Maximization of Utility given a Budget Constraint = Marginal Utility of good x/Price of good x = Marginal Utility of good y/Price of y Calculate Consumption Bundle using the following information: Price of Good x = $5, Price of Good y = $16 and Income = $100 / 0 Quantity Consumed Total Utility Quantity Consumed Total Utility Calculate: a.) Price Elasticity of Demand =% Change in Quantity Demanded/%Change in...
Question 1: Colin's utility function for goods X and Y is represented by U(XY) = X0.5Y0.5 . Assume his income is $1000 and the prices of X and Y are $50 and S100, respectively. a. Write an expression for Colin's budget constraint. b. Calculate the optimal quantities of X and Y that Colin should choose, given his budget constraint. Graph your answer. Suppose that government subsidy program lowers the price of Y from $100 per unit to $ 50 per...
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