8. In a 2-good model, where the goods are denoted x, and x,, the consumer's utility 1 1 function is as follows: U = x,x3. Money income available is denoted m. All of this income is spent on the two goods. The prices of the two goods are P1 and p2 respectively. (a) By minimising expenditure, subject to the utility function, find the compensated (Hicksian) demand functions; that is, demand for each good expressed in terms of U, P1 and P2. Do not check the second order conditions. (b) Substitute these conditional demand functions back into the objective function to find the expenditure function; that is, m in terms of U, P1 and p2.
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Can you please help awnser 8 b I have attached awnser to 8 a to make it easier to understand and complete.
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- Suppose that we can represent Joyce's preferences for cans of pop (the x-good) and pizza slices (y-good) with the utility function min[4x,5y]. a) Find her Marshallian Demand Functions. b) Find her Hicksian Demand Functions2. Consider the two-good model of the utility maximization program subject to a budget constraint. The utility function U of a hypothetical rational consumer and his/her budget constraint are given, respectively, by: U = x1x2, (U) B = p1x1 + p2x2, (B) where xi = the consumer’s demand for consumption good i (i = 1, 2), pi = the price of consumption good i (i = 1, 2), and B = the (exogenously given) budget of the consumer. In this maximization program, assume the following data: B = 240, p1 = 10, p2 = 2. (a) Using the Lagrangian function L, derive the first-order (necessary) conditions for a (local) maximum of the utility function. (b) Compute the optimal values of all choice variables, i.e., x*1 , x*2, and λ* , in the program, where λ signifies the Lagrange multiplier. (c) Using the information of the bordered Hessian matrix H¯ , verify the second order (sufficient) condition for a (local) maximum of the utility function. Note:- Do not provide handwritten solution. Maintain accuracy…2) For the following utility functions, using the budget constraint M = Pxx & Pyy, find the compensated (Hicksian) demand functions. Note: For a, minimize expenditure; For b, maximize utility and use the Shepard's Lemma. a) U(x,y) = x0.5 + 2y0.5 b) U (x, y) = min{2x, 5y}
- 1. Assume you spend your entire income on two goods X & Y with prices given as PX & PY, respectively. Prices and income (I) are exogenous and positive. Given that U = X2 + Y2 , derive the Marshallian demand function for good Y and evaluate the type of good. 2. Assume you spend your entire income on two goods X & Y with prices given as PX & PY, respectively. Prices and income (I) are exogenous and positive. Given that U= X2Y2 , derive the Hicksian demand function for good Y.3. Suppose that initially PX = 2, PY = 8, I = 96 and the Marshallian demand function for good Y is given by Y∗ = (0.5I/ PY)+(0.5PX/PY)− 0.5. Calculate the own price & income elasticities of demand for good Y. Interpret your computed values and say something about the type of good.4. Suppose the economy has 100 units each of goods X and Y and the utility functions of the (only) 2 individuals are: UA (XA,YA) = X0.25Y0.75, UB (XB,YB) = X0.75Y 0.25Show that pareto-improvement is possible if,…Suppose that i's preferences over goods x and y are represented by the following utility function U₁(x, y)=x45¹5. Let m denote the consumer's income, p denote the price of good x and let the price of good y equal 1. A) Find the Marshallian demand functions for goods x and y. B) Show how each of the demand function is affected by a change in the price of good x. C) Which of the goods is an inferior good?Assume you spend your entire income on two goods X & Y with prices given as Px & Py, respectively. Prices and income (I) are exogenous and positive. Given that U = X + Y, derive the Marshallian demand function for good Y and evaluate the type of good. Assume you spend your entire income on two goods X & Ywith prices given as Px & Py, respectively. Prices and income (I) are exogenous and positive. Given that U= X²Y², derive the Hicksian demand function for good Y. Suppose that initially Px = 2, P = 8, I = 96 and the Marshallian demand function for good Y is given by . Calculate the own price & income elasticities of demand for good Y. Interpret your computed values and say something about the type of good.
- Suppose an individual has preferences over goods x and y, and their expenditure minimization problem has the following expenditure function: E(px, Py, U) = (px + 3p,)U. What is the person's Hicksian demand? O(h, hy) = (2p U, pU) O(h, hy) = (U, 3U) O(h, hy) = (Up,',Up,) O(hx, hy) = (3U, U) What is the individual's indirect utility? OV = 3p! p OV = P.+3p, OV = P Py OV = P.PSuppose that i’s preferences over goods x and y are represented by the following utility function Ui(x, y)=x^0.8·y^0.2. Let m denote the consumer’s income, p denote the price of good x and let the price of good y equal 1. a) Find the Marshallian demand functions for goods x and y. b) Show how each of the demand function is affected by a change in the price of good x. c) Which of the goods is an inferior good?Bunde's preferences are given by the utility function u(x1, x2) = x, + x2. For each of the following cases, decompose the price effects into the substitution and income effects using the Hicks-Allen decompositions. (For each part, use the given template to draw the constraints, and then use your completed graph to fill in the description. Enter any points in order from left to right as they would appear on the graph.) 20 15 10 5 10 15 20 (a) Suppose m = 120, P1 = 10, and p2 = 15. The price p1 then falls to 6, keeping p2 and m fixed. Given m = 120 and p, = 15, the budget B° is drawn for p, = 10. The utility-maximizing point on this budget is at A = given the linear preferences yielding utility y° = When p, falls to 6, the new budget is B" and the utility-maximizing point is C = ), yielding utility un = . The movement from A to C is the price effect of units of good 1. Removing income incrementally until Bunde can barely afford the old utility of at the new prices yields the line which…
- 1) Suppose that a person consumes two goods, x and y, in fixed proportions. He or she always consumes 1 unit of x together with 3 units of y no matter what the relative prices are.a) What is the mathematical form for this person's utility function?b) Calculate the Marshallian demand functions for both goods for this person.c) Calculate the indirect utility function and the expenditure function for this person.d) In class we discussed why expenditure functions are concave in prices. Is the expenditure function you calculated in part (c) concave in ???A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x₁, x2) = x³x² where x₁ and ⁄2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. Derive the consumer's Marshallian demand functions. Derive the consumer's Hicksian demand functions. Derive the consumer's expenditure function. Let m = = 120, p₁ = 2, and p2 = 1. Suppose that the price of good 1 drops to p₁ = 1. Find the following Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV. Let m = 120, P₁ P₁ = 2. Find the following = 1, and p2 = 1. Suppose that the price of good 1 increases to Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV.a good is normal, then an increase in the price of the good will lead to which of the following to be true for this good? (Assume that there are only two goods, the individual's preferences lead to well-behaved preferences with strictly convex indifference curves and an interior solution for all budgets). Let SE = substitution effect, IE = income effect) (a) The magnitude of the IE for this good must be larger than the magnitude of the SE (b) The magnitude of the SE for this good must be larger than the magnitude of the IE (c) The good could be a Giffen good d) The good must be an ordinary good ( (e) None of the above