2. Let U=U(x,y) be the utility function of the agent. x and y_represent the goods. Assume positive marginal utilities. Let Px, Py be the prices and I the income. a. State the agent's utility maximization problem. b. Present the Lagrangian function. c. Derive and interpret the first order condition. In your analysis, you must include the Lagrange multiplier. d. Present a graphical interpretation of the optimality condition (include indifferent curves and budget sets). 3 دیا e. Present the second-order condition and describe the conditions under which one could secure a maximum. f. Solve the previous questions but assuming: (i) U(x,y)=xy where a+b<1, (ii) U(x,y)=ax+by, (iii) U(x,y)=Min{x,y}. ax. apx g. Now, using the general formulation U=U(x,y), express the comparative-static derivative as the sum of income and substitution effects.

2. Let U=U(x,y) be the utility function of the agent. x and y_represent the goods. Assume positive marginal utilities. Let Px, Py be the prices and I the income. a. State the agent's utility maximization problem. b. Present the Lagrangian function. c. Derive and interpret the first order condition. In your analysis, you must include the Lagrange multiplier. d. Present a graphical interpretation of the optimality condition (include indifferent curves and budget sets). 3 دیا e. Present the second-order condition and describe the conditions under which one could secure a maximum. f. Solve the previous questions but assuming: (i) U(x,y)=xy where a+b<1, (ii) U(x,y)=ax+by, (iii) U(x,y)=Min{x,y}. ax. apx g. Now, using the general formulation U=U(x,y), express the comparative-static derivative as the sum of income and substitution effects.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.13P

See similar textbooks

Similar questions

An agent has utility u(x1, x2) = (x-11+x-12)-1for goods x1 and x2. The prices of the goods are p1 and p2. The agent has income m. a) Show preferences are convex. You can do this graphically or by showing that MRS is decreasing in x1. b) Solve for the agent’s optimal choice of (x1, x2). c) Show the agent’s indirect utility function is given by:V=m(p1/21 + p1/22)2
Consider the following utility function: u(x1,x2) = x1 + x2. (A): i) Restate the consumer problem. ii) Form the Lagrangian function. iii) Find the first-order-conditions (FOC). iv) Divide Lx, by Lx,- v) From iv), can you determine the type of goods 1 and 2 are? (В): Draw the indifference curves for U = X1 + x2, where U = 1 and 2. What happens to the optimal demand for good 1, when: P1 P2 1. 3. Pi = P2
U(x, y) = xayb A consumer maximises utility subject to a budget constraint M = Pxx+Pyy Where px is the price of good x, py is the price of good y and M is the budget available. a. Derive an expression for the marginal utility of x. Under what condition is the marginal utility diminishing. b. Derive an expression for the marginal utility of y. Under what condition is the marginal utility diminishing.
Mr. Dogan has 4000 TL to spend. He considers buying meat whose price is 400 TL per kg. and cheese whose price is 160 TL /kg. Given this a) What condition (equality) leads him to a utility maximization. State the condition numerically and tell clearly what each of the variables mean in that equation b) Now suppose that income of Mr. Dogan and price of meat does not change but price of cheese is down to 130 TL/kg. Do the equality and optimal basket found in a still maximize the utility? If not; what is the new condition for utility maximization (write the condition as a numerical equation defining the variables in the equation)
Donald likes fishing (X1) and hanging out in his hammock (X2). His utility function for these two activities is u(x1, x2) = 3X12X24. (A) Calculate MU1, the marginal utility of fishing. (B) Calculate MU2, the marginal utility of hanging out in his hammock. (C) Calculate MRS, the rate at which he is willing to substitute hanging out in his hammock for fishing. (D)Last week, Donald fished 2 hours a day, and hung out in his hammock 4 hours a day. Using your formula for MRS from (c) find his MRS last week. (E) This week, Donald is fishing eight hours a day, and hanging out in his ham mock two hours a day. Calculate his MRS this week. Has his MRS increased or decreased? Explain why? (F) Is Donald happier or sadder this week compared to last week? Explain.
Given the following utility function: U(Q,Q2) = 100LN(4Q1/2 + 2Q,/2) %3D a. Show that the function is cardinal. b. Show that the function is ordinal. c. Derive the demand functions for the two goods. d. For Q1 show that the law of demand holds. e. Show that Q1 is normal. f. What is the relationship between the two goods.
Consider the following utility function и(х, у) %3 3х + 4y a. Is the utility function homothetic? Explain. b. Under what condition will the consumer purchase good x alone?
LTIPLE CHOICE. Choose the one alternative that best completes 22) The magnitude of the slope of an indifference curve is: A) called the marginal rate of substitution. B) equal to the ratio of the total utility of the goods. C) always equal to the ratio of the prices of the goods. D) all of the above E) A and C only
A consumer utility function is U=(x1,x2) Where X1 is the quantity of good 1 that is bought, X2 is the quantity of good 2 that is bought. The price of good 1 is $10 and the price for good 2 is $2. If the consumer's income is $100 what will the consumer's optimal utility level be? b. Using Lagrange multilplier method. optimise the utility function x0.25 y0.25 subject to the budget constraint 24=x/10+y
Individual that consumes two goods (X and Y) and has a CES Utility Function of the form: U = 100(X^(0.75) + Y^(0.75)). Income of 1000, the price of Good X is 10 and the Price of Good Y is 20 a) Find the Marginal Rate of Substitution as a function of the quantities consumed of Good X and Good Y. b) Write out the Lagrangian for this problem. c) Solve to find the demand for Good X, the demand for Good Y, and the highest level of utility for this individual. d) Now consider an increase in the price of Good X to 20. What is the demand for Good X and Good Y? What is the Utility of the consumer following the price change? e) Considering the change in demand for each good between parts c) and d), how much is due to the substitution effect and how much is due to the income effect? f) Show your answers on a graph.
he Calculus of Utility Maximization and Expenditure Minimization -End of Appendix Problem uppose that there are two goods, X and Y. The price of X is $2 per unit, and the price of Y is $1 per unit. There are two onsumers, A and B. The utility functions for the consumers are UA(X,Y)= X05.05 UB(X,Y)= X0.8y0.2 Consumer A has an income of $100, and Consumer B has an income of $300. Using Lagrangians, solve for the optimal bundles of goods X and Y for both consumers A and B. a. The optimal bundle for consumer A is X = 25 and Y* = 50 - b. The optimal bundle for consumer B is X = 120 and Y* = 60
Consider the following utility functions: a) U= xy b) U=(xy)l3 c) U=min(x,y/2) d) U=2x + 3y e) U=x?y? + xy 1. Check if any of utility functions a) – e) are homogeneous. If so, state the degree of homogeneity of each.