Consumer i's indirect utility function is of the form: mi mi v'(p1,P2, m') = a' + + P1 P2 where a' is a positive constant, and m' is her income. (a) Verify that v'(p1,P2, m') satisfies the monotonicity and homogeneity properties of the indirect utility function. (b) Derive consumer i's Marshallian demand functions for good 1 and good 2. (c) Derive consumer i's Hicksian demand functions for good 1 and good 2. (d) Are good 1 and good 2 complements or substitutes? Justify your answer using the substitution term in the Slutsky equation. (e) Suppose there are a total of I such consumers (that is, i = 1, 2, ..., I). Derive the aggregate (Marshallian) demand function for good 1. Let M be the aggregate income. Can this aggregate demand function be written as a function of (p, P2, M)?

Please use Kuhn Tucker 5. A consumer's utility function is: u(x) = x1+ /®2 (c) Verify the properties of the indirect utility function. (d) Calculate the expenditure function and verify its properties.

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.

Adrienne consumes three goods, x, y, and z. Her consumption preference is given by the utility function: u(x, y, z) = x*y} z#, (Pz-Py, Pz) where x, y, and z are the quantities she consumes (respectively). Let p = be the prices of the goods. Answer the following questions, and justify your answers carefully and mathematically. (a) Last year p = sumption bundle last year. (1, 10, 10) and Adrienne's income was 35. Find her optimal con- (b) This year, the prices of x and z have both risen 3.5% while the price of y has risen 7%, and Adrienne got an income raise of 3%. Find the (approximate) percentage changes of her consumption in the three goods due to these changes in prices and income. Does her raise adequately compensate her for the changed prices? What additional percentage change of her income would be needed if we were to restore Adrienne to last year's exact utility level?

A consumer’s indirect utility function is given by

Assume a consumer with the utility functionU(X,Y)=X²Y²and the typical budget constraintM= PxX+ PyY a. Derive the consumer’s demand for X and Y in terms of the parameters.b. If income of the consumer is 100, price of X is 2 and price of Y is 4, what the quantities of X and Y that gives the consumer maximum satisfaction?c. What is the consumer’s marginal rate of substitution between X and Y

Anthony seeks to maximize the following utility function u(x, y) = x'/3y2/3 subject to the budget constraint Pæa + PyY = I 1 where pr, Py, x, y, I > 0. a) Find Anthony's utility-maximizing bundle (x*, y*) as a function of pæ, Py, and I. b) Show that y* is decreasing in py and increasing in I (hint: use partial derivatives). c) What share of Anthony's income is spent on x? What share is spent on y? In other words, calculate Pa and Pu. Are these shares a function of prices? Pyy* Note: The above utility function is Cobb-Douglas, and all Cobb-Douglas functions have these share formulas for any values for the exponents. d) What is the impact of a change in pr on Anthony's utility?

Px An individual has utility function for X and Y of the form U(X,Y)= X + Y and Py expenditure function as a function of prices px, py and utility level Ū is ○ E(Px, Py, Ū) = PxŪ Ū E(Pæ, Py, Ū) Px E(Px, Py, U) E(pæ‚Py, Ū) = pyŪ = (Px + Py) Ū || The

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