(2) Consider the utility function u(x, y) := x'/³y²/3 for all nonnegative I and y. Given the budget constraint p,r + Pyy = m, show that the Slutsky equation which is given by %3D ´Əm 3p
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- Suppose that a consumer has a choicebetween two goods, X and Y. If the price of X is $2 and the priceof Y is $3, how much of X and Y does the consumer purchaseper period, given an income of $17 per period? Use the followinginformation about marginal utility:Units MUX MUY1 10 52 8 43 2 34 2 25 1 2Khan lives in a world with two consumption goods x and y. Her utility function is U (x, y) = √x² + y². a. If px = $3,py = $4, and her income, I, is equal to $50, what will be the quantities of x and y that Maya should buy to maximize her utility? Make sure that you write out the Lagrangian and the first-order conditions. (Hint: It may be easier to maximize U² than U). Have you found a true maximum? Explain your answer.Suppose a consumer with a utility function U(x,y) =x^(0.5)y^(0.5) and an income of $1,000.00. Considering that products X and Y are sold by the kilo and that their prices are 50.00 and 100.00 respectively, calculate: a) Maximizing amounts of x and y for the consumer b) Graphically sketch the consumer balance c) What would be the impact on the utility of this consumer, if a policy of distribution of good X was established so that each period the consumer "earns" 20 units of X at no cost? What will your new balance be?
- Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…true or false For the Cobb-Douglas utility function with α = 1/3 and β = 2/3, the indifference curve corresponding to a utility of 5 utils is B = 125A–2A maximizing consumer with preferences given by u = x^2+ y^2 allocates 60 dollars of income at pY= 3 and pX= 4. Provide an indifference curve and budget line diagram to illustrate and quantify her utility-maximizing choice a Bundle A. Next month the price of good x will fall to pX= 2. Update your diagram to show the Hicksian compensating and equivalent variations for this price change as Bundles B, C, and D.
- Utility Maximization. Suppose that an individual has a utility function given asU =√XYwhere U is the total utility, X is the quantity of good X that is consumed, and Y is the quantity ofgood Y consumed.It follows thatMUxMUy= X/Ywhere MUx is the marginal utility of X and MUy is the marginal utility of Y.The individual’s budget constraint isI = PxX + PyYwhere I is income, Px is the price of good X, and Py is the price of good Y.Suppose that Px = 2, Py = 1, and I = 100. How much of X and Y does this individual purchase?A decision maker has preferences represented by the utility function u(x,y) = xy over two goods and she is indifferent between the bundles (1,a), (b,4), (c,2) and (8,1). (a) Find the values a, b and c. (b) Brittney claims that another utility function, v(x,y) = (xy)^2 + 2xy - 5 , also represents the same preferences hence has the same exact indifference curves as u. Verify that the four bundles above are on the same indifference curve according to v, too, indeed.I need asnwers of d,e,f. Assume there is consumer, his utility function is u(x,y) =8 * x0.5+y , and his budget constraint is px*x +y = m, which implies py = 1. a.Please derive the Marshallian demand function of x. b.Please derive the indirect utility function. c. Please derive the expenditure function If originally m = 40, px=2. d. What is his original highest utility level? Now px has decreased to 1, m and py do not change. e. What is his new maximum utility level? f. Based on (c) (d) and (e), what is his compensating variation? g.Based on (c) (d) and (e), what is his equivalent variation?
- Households have utility u(C,D,I) over consumption C education D and investment I. The house-hold has income Y which it sp ends on these three go o ds at cost 1. Therefore the household's maxi-mization problem is:maxC,D,I u(C,D,I)s.t. Y = C + D + I A. Suppose that the household's utility function is u(C,D,I) = C1−φ−θDθIφ, with 0 <φ + θ ≤1, φ > 0, and θ > 0. Using this functional form, combine the first order conditionsfor C,D,I into two equations, by eliminating the Lagrange multiplier. B:You now have three equations (the budget constraint and your answer from b) andthree unknowns (C, D, and I). Use these equations to show that aggregate educationand investment are always prop ortional to aggregate income Y .Student question Time Left :00:09:43Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = 2X + Y UB(X,Y) = Min(X,Y) The initial endowments are: A: X = 5; Y = 3 B: X = 2; Y = 2 a. Illustrate the initial endowments in an Edgeworth Box. Be sure to label the Edgeworth Box carefully and accurately, and make sure the dimensions of the box are correct. Also, draw each consumer’s indifference curve that runs through the initial endowments. Is this initial endowment Pareto Efficient? b. Now suppose Consumer A gets all of both goods. Is this allocation Pareto Efficient? (You do not need to draw a new graph or illustrate this on the existing graph. Simply answer “yes” or “no.”) c. Now suppose Consumer B gets all of both goods. Is this allocation Pareto Efficient? (You do not need to draw a new graph or illustrate this on the existing graph. Simply answer “yes” or “no.”)Assume that a∈[-1,1] and θ∈(0,2). The utility function is given by u(a,θ)=a(θ-1). Which of the following action is never optimal ex-post? a. 0 b. All can be optimal ex-post c. 1 d. -1