Suppose a consumer with income I has preferences over consumption bu XY. Suppose th be represented by the utility function U(X,Y) px, is 1 and the price of Y, py, is 2. At the consumer's optimal consum (X*, Y*), what must the value of the ratio be? (a) 10 X* =
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- A consumer consumes pizza and soda always in the fixed proportion of 3 slices of pizza (x) with 2 cups of soda (y). Any pizza or soda not consumed together with the proportional amount of the other is useless to her. (a) Write down a utility function u(x,y) that represents her preferences. (b) If she has I = $60 budget and px = 4, py = 4, what is her optimal bundle? (c) In (b), If the price of soda increases to $6, how much money should she be compensated so that she achieves the same level of utility before the price increase? (d) Returning to (b), If the price of soda increases to $6, how much should the price of pizza be reduced so that she achieves the same level of utility before the price hike? (e) In (a), what values of a and b she is indifferent between the bundle "30 slices of pizza and a cups of soda" and the bundle "b slices of pizza and 40 cups of soda". Find at least two (a,b) pairs to get full points.Consider a consumer whose preferences can be represented by the utility function u(x, y) = x + y (a) Originally, px = 1, py = 2 and m = 1. What bundle does the consumer choose, and what is his utility from this bundle? (b) The price of good x then rises to 3. What bundle does the consumer choose after the price change, and what is his utility from this bundle? (c) Calculate the compensating variation. (Hint: at the new price ratio, what good will he spend his income on?) (d) Calculate the equivalent variation.If the utility function (U) between food (F) and clothing (C) can be represented as U = F*C, then which combination of food and clothing will a rational consumer choose? a. (F, C) = (9, 25) O b. (F, C) = (25, 4) O c. (F, C) = (4, 16) d. (F, C) = (4, 25)
- Question 1 Consider a consumer with a utility function of U(X,,X2, ... , Xn) = II-1 X , a; > 0, i = 1,2,3, ...,n. P; is a price of 1 unit of the i-th good. Assuming that the consumer has the total budget of $M, derive his optimal consumption bundles in the following procedure. (a) Set up the consumer's utility maximization program with a Lagrangean multiplier. (b) What is an economic intuition of the Lagrangean multiplier? (c) Assuming an interior solution, get all the FOCS. (d) Find an optimal consumption bundle while specifying each step required to get the solution. (e) Characterize the maximized utility as a function of $M and the parameters. What does it look like? (f) Find an exact value of the Lagrangean multiplier.(2) Consider the utility function u(x, y) = Vx+ay. Your budget constraint is P1x+p2y = М. (a) Find the marginal utilities for these goods. (b) Show that these preferences are quasilinicar. (c) Solve the consumer's optimization problem.A utility function is given by the equation U = 20xe¬0.1x, where x is the number of glasses of wine consumed. (a) Show that this utility function has a maximum value and calculate the maximum utility. (b) Describe how marginal utility changes for glasses of wine consumed after the maximum utility is reached. Do you consider this reasonable? Give an explanation.
- For each of the following utility functions get the marginal utility of consumption of each of the goods. 1/2 1/2 (a) u(a,b) = a b (b) u(x, y) = x³/4y1/4 (c) u(a, b) = n(a) + In(x) (d) u(x, y) = ln(x) + ln(y) (e) u(a, b) = 2xa +XbA consumer is maximising her utility function: U(r. y) = (r05 +y0.5)2, subject to the budget constraint 4x + 2y = 108. (a) Set up the Lagrangian function of this utility maximisation problem and first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the multiplier. (c) What are the utility maximising amounts of x and y if the budget constrain to x+ 2y = 36? How would change? Explain your reasoning. (Hint: Yc need to calculate A, rather comment on how it would change and why.)If Jackson's preferences over consumption bundles can be represented by the utility function U(X, Y) = 2X + Y, what will Jackson's optimal bundle, (X*,Y*), be when the price of X is px = 3, the price of Y is py = 1, and Jackson's income is I = 150? (a) (110,0) (b) (0, 150) (c) (0,80) (d) (50, 0) (e) (30,60)
- Suppose that you go to a restaurant with your friends. a) One of your friends always eat exactly one chicken wrap (C) and drinks two cups of ayran (A). He never changes this combination of food choice. (i) Write the utility function of your friend to represent his preferences. (ii) Draw couple of indifference curves of your friend on graph with appropriate labels. (iii) Determine the demand function of your friend for each product as a function of prices and income. Denote prices as Pc, Pa; income as M. b) Suppose that one of your friends does not care if she consumes spaghetti(S or noodle(N).She wants to consume either two-dish spaghetti or two-dish noodle or any combination of both which adds up to two dishes. (i) Write the utility function of your friend to represent his preferences. (ii) Draw couple of indifference curves of your friend on graph with appropriate labels. (iii) Denote prices as Pn, Ps; income as M. If Pn < Ps, then determine the demand…(c) If the preferences are concave, will the consumer ever consume both of the goods together?Ice cream and cakes are perfect substitutes for a child, and 2 units of ice cream is always worth 3 units of cakes (however many ice creams or cakes she might have, she would be willing to give up 2 ice creams to get 3 more cakes to keep the same utility level) . (a) Write down a utility function u(x,y) that represents the child's preferences, where x is the number of ice creams and y is the number of cakes she has. (b) If the prices are px= 8 and py =5, and she has $140 to spend on the two goods this summer, what is her optimal bundle? (c) If the price of ice creams decreases slightly, down to px = 7, what happens to her optimal bundle in this case? Did it change "slightly" compared to (b)?