An individual with (a strictly monotone) utility function u, facing prices and income (p, I) has a Walraisian demand of æ*(p, I). Let ū = u(æ*(p, I)). What is the (minimal) expenditure for her to obtain utility ū. Intuitively (in words), why does this answer make sense?
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- Let U(x, y) = 3x + y be the utility function of a consumer, whohas a budget of I. As a function of I, find the consumer’s Walrasian demand when the prices are px = py = 1. The price of good x increases to 2, find the new Walrasian demand for the new prices px= 2 and py = 1. Decompose this change into an income and a substitution effect.If Sally's utility function is U = 12 (91) 05 + 92. what is her Engel curve for q,? Let the price of q, be p1, let the price of q, be p2, and let income be Y. Sally's Engel curve for good q2 is Y=|: (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the _ character.)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.
- Irene spends all of her income M on soda and chips. The price of soda is 2 per unit. Irene’s utility function is U (s, c) = 5 ln(1 + s) + ln(1 + c). (a) Derive Irene’s demand function for chips (as a function of her income and the price of chips).Johanna has / =$181 to spend on goods x1 and x2. Her utility function is given by U (a1, 22) = 2 min{ax1, bx,} where a =3 and b =5. Good x1 costs p1 =$10 per unit, and good x2 costs p2 =$9 per unit. How much of good x, does Johanna consume? Enter a numerical value below. You may round to the second decimal if necessary. (Partial quantities are allowed.)A consumer has preferences over two goods, denominated by x and y, given by the utility function U(x,y) = min{αx,y} with α > 0. The prices of the good are px = 2 and py = 5. The consumer has an income of I > 0. (a) Provide an intuition for this utility function. Specifically, are these goods substitutes or complements? If x is bicycle tires and y is bicycle frames, what is the value for α? (b) For what values of α will the consumer demand (i.e., Walrasian demand) more x than y. (c) For what values of α will the consumer spend more on x than on y (given her Walrasian demands).
- Consider a person who consumes two goods, x and y, and has a utility function given by U(x, y) = In(x)+y. This person has an income of $100 and faces a price of $0.50 for good x and $1 for good y. Price of x then rises to $0.60. Solve for the compensating variation (CV) and equivalent variation (EV) of this price change. Show your work.If Philip's utility function is U=4 (41) 05+42. 0.5 what are his demand functions for the two goods? Let the price of q, be p,, let the price of q, be p2, and let income be Y. Philip's demand for q, as a function of p, and p, is 91 and his demand for good q, is (Properly format your expressions using the tools in the palette. Hover over tools to see keyboard shortcuts. Eg, a subscript oJoyce drinks both coffee (x) and tea (y). Her preferences over these two goods can be represented by the utility function U(x,y) =x + 3y^1/2 where x represents the number of pounds of coffee and y represents the number of cups of tea. a) Given her preferences, find her demand functions for coffee (x) and tea (y). b) Suppose that the price of a pound of coffee is $4 and that she has $56 to spend on coffee and tea. Write her demand curve for tea. Illustrate her demand curve. c) Suppose that the price of a cup of tea is $1 (the price of coffee and income remain $4 and $56, resp.). Use your demand functions to find her best bundle. In an indifference curve diagram illustrate her best bundle at these prices. For the remainder of the question, assume that her income rises to $60 and that the prices of coffee and tea are unchanged at = $4 and =1.
- Mary has the following utility function: u(x, y) = 3 ln(x) + 2y. Her income is given by I = 10 and the prices originally are p = 1 and py = 2 (b) How much of each good is Mary currently consuming?Utility maximization with a budget constraint. A hypothetical consumer spends all tgheir income on ramen noodles (N) and wild rice (W). N is the quantity of noodles; W is the quantity of wild rice. Their income is $1,600 per month. the price of noodles is $2 per package and the price of wild rice is $20 per pound. The utility function is U=sqrt(N*W). the MRS = -N/W. The budget constraint is: 1,600 = 2*N + 20*W Graph Qty of noodles (N) on vertical axis and Qty of wild rice (W) on horizontal axis. SOLVE: a. Graph the budget constraint. label all points. What is the slope of the budget constraint? b. Find the optimal quantities of noodles(# of packages) and the wild rice (# of pounds) given the budget constraint. graph these optimal quantities. draw your indifference curve on the same graph. c. Show on your graph what happens when the price of wild rice increases to $40 per pound. Find your new optimal quantities of noodles and wild rice. label all points on graph. label the…A consumer is in equilibrium and is spending income in such a way that the marginal utility of product X is 40 units and that of Y is 32 units. If the unit price of X is $5, then the price of Y must be:
