Consider a consumer that seeks to minimize his expenditure E to achieve a given 1/4 1/4 level of utility Ū. Assume that E = p₁x₁ + p₂x₂; U = x^x^; and p₁ and p₂ are given. a) Set up the Lagrangian. b) Show the first-order conditions or minimization. c) Derive the expressions for the optimal levels of x, and x₂.
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- Take Jeremys total utility information in Exercise 6.1, and use the marginal utility approach to confirm the choice of phone minutes and round trips that maximize Jeremys utility.ASAP 1. Consider a consumer with utility u(x1,x2) = .5lnx1 + .5lnx2 (b) Now, consider an equivalent representation of the above utility functionx51 x52 Does this utility function have the "increasing difference property"? Howabout the "strict increasing difference" property?3- Assuming that the equation F(U, x, X2 = f(x1, x2,, ... Xn): ******* **** ,xn) = 0 implicitly defines a utilityfunction U 073 a) Find the expressions for 60, 6U and 6x4 " 3 6x2 6xn 6x2 6xn b) Interpret their respective economic meanings. c) Now. assume the utility function is U(x, y) = y√x. Does the consumer believe that more is better for each good? Do the consumer's preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing?
- A consumer buys Bread (B) and Milk (M) each week at price of 20 and 16 respectively. Suppose that the consumer's current consumption of Bread and Milk is giving him MUBread = 80 and MUMilk = 70. Is the consumer buying the utility-maximizing combination of bread and milk? If not, how should he reallocate his expenditures between the two goods to maximize utility? Draw diagram and explain.3) Suppose Joe's utility for lobster (L) and soda (S) can be represented as U =10.5 50.5 MUL=0.5 L -0.5 s0.5, MUs=0,5 L0.5 s-0.S. Joe walks into a restaurant with $72. Lobsters cost $18 each and sodas cost $2 each. How much lobster and soda will Joe consume if he intends to spend all his money? (There are no tax and no tips.)4. Consider the utility function U(r,g) r + Iny. (a) Find the marginal rate of substitution. MRS of this function. Interpret the result (b) Find the equation of the indifference curve for this function (c) Compare the marginal utility of r and y. How do you interpret these functions? How might a consumer choose between e and y as she tries to increase utility by, for example, consuming more when their income increases?
- 3. Suppose Mary enjoys Pepsi and Coke according to the functionU(P;C) = 4C + 5P. What does her utility function say about her MRS of Coke for Pepsi? What do her indi§erence curves look like? What type of goods are Pepsi and Coke for Mary? If Pepsi and Coke each cost $1 and Mary has $20 to spend on these products, how many units of each product should she buy in order to maximize her utility? Show this utility maximiz- ing combination combination of Pepsi and Coke on the graph. how would her consumption and utility maximizing bundle of Coke and Pepsi change if the price of Coke decreases to 50 cents. 4. Vera is an impoverished graduate student who has only $100 a month to spend on food. She has read in a government publication that she can assure an adequate diet by eating only peanut butter and carrots in the Öxed ratio of 2 pounds of peanut butter to 1 pound of carrots, so she decides to limit her diet to that regime. a) If peanut butter costs $4 per pound and carrots cost $2 per…2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (x1, x2, 13) = A (x1 – T1)ª (x2 – ã2)° (x3 – ã3)" - where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + B+y= 1 without loss of generality? Do so for the rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.Please no written by hand and no emage Consider a consumer that consumes 2 teaspoons of sugar with each cup of coffee. For each cup of coffee with sugar the consumer gains 10 utils.a) Write down the utility function that gives the total utility if the consumer consumes S teaspoons of sugar and C cups of coffee. The consumer has assigned £7 per week to be spent on drinking coffee with sugar. The current price of coffee is £0.50 per cup and each spoon of sugar costs £0.10. b) Calculate the optimal weekly consumption bundle for this consumer.c) Does the consumer view C and S as complements or substitutes?
- 5. What kind of preferences are represented by a utility function of the form u(x1, x2) = x1 + V*2? Is the utility function v(x1, x2) = xỉ + 2x1Vx2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = Vx1x2. What kind of pref- erences does it represent? Is the function v(1, x2) transformation of u(x1, x2)? Is the function w(x1, x2) = xỉx, a monotonic transformation of u(x1, x2)? = x²x2 a monotonic1. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are px and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. = (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px = 10, Py = 5 and m = 100. Suppose that px increases to p how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y > 0). = 15,EXERCISE 4 Milo spends £27 on stamps (S) and maple candy (C). His preferences for these goods are given by the following utility function: U(S, C) = min {S, 3C). Suppose that price per unit of S is £1.1 and the price per unit of C is £2.1. a) Write down Milo's budget equation and draw the corresponding budget line. Clearly label the axes and calculate the coordinates of the points of intersection of the budget line with each axis. In your graph, indicate the consumption bundle of 12 stamps and 6 maple candies. Can Milo afford this bundle? Explain. b) Find Milo's optimal consumption bundle, both algebraically and graphically. Explain your reasoning. c) If the price of a stamp increases to £1.4, how should Milo's income change for him to be as well off as before this change in prices? d) Discuss the implications of the price change from c) on Milo's optimal choice. In your discussion, include the analysis of the substitution and income effects as well as Milo's demand for stamps and/or…