1 Marginal utility and marginal rate of substitution Suppose an individual who derives utility u(x, y) from consuming a units of good X and y units of good Y. For each of the following utility functions, derive the marginal utility of good X, the marginal utility of good Y, and the marginal rate of substitution between X and Y. (A) u(x, y) = x+by¹/3 (B) u(x, y) = (xy) (C) u(x, y) = ax + xy + by² (D) u(x, y) = ln(ax + y) 1 (E) u(x, y) = (x² + yyp) &
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PART D E
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- 1 Marginal utility and marginal rate of substitution Suppose an individual who derives utility u(x, y) from consuming a units of good X and y units of good Y. For each of the following utility functions, derive the marginal utility of good X, the marginal utility of good Y, and the marginal rate of substitution between X and Y. (A) u(x, y) = x+by¹/3 (B) u(x, y) = (xy) (C) u(x, y) = ax + xy + by² (D) u(x, y) = n(ax + y) 1 (E) u(x, y) = (x² + √y²) = 24. 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?2)Ben is choosing between cookies and soda. Denote the marginal utility of cookies is MUc, and the marginal utility of soda is MUs. Which of the following relationship is true regarding the marginal utility of substitution (MŘS) of cookie in terms of soda? A) MRS= MUc B) MRS = MUS C) MRS = MUS / MUC D) MRS = MUc / MUs
- Quantity X TUx MUX MUX/Px Quantity Y TUy MUY MUy/Py 1 75 1 96 135 60 B 72 185 50 3 234 66 4 230 45 A 4 294 60 270 5 348 54 2.Harry consumes 2 goods, X and Y and he spends N$60 per month. The price of good X is N$4 and the price of good Y is N$10. Harry's utility function is U(X, Y) = XY. (i) What is Harry's marginal rate of substitution? (5) (ii) What is the expression for Harry's budget constraint? (3) (iii) What is the slope of Harry's budget constraint? (2) (iv)Find the values of good X and Y maximizes Harry's utility. Show all your work. (6) (v) Illustrate Harry's utility maximizing combination on a clearly labelled graph. (4) (vi) Suppose the price of good X increased to N$6. Illustrate Harry's income and substitution effects on a well labelled graph. Show your work. (10) (vii) Suppose Genie is consuming two goods, coffee and beer. She spends all her income on a combination of coffee and beer where MUcoffee/Peoffee is 5 and MUbeer/Pbeer is 3. Explain why this combination is not maximizing her satisfaction. What should she do to maximize her utility? (5)1. Calculate the marginal rate of substitution for the following utility functions: (a) u(x1, x2) = ax₁ + bx2 (b) u(x1, x2) = 2√√x₁ + x₂ (c) u(x1, x2) = ln(x₁) + x2 (d) u(x1, x2) = x1x2 (e) u(x1, x2) = xx h (f) u(x₁, x₂) = x + x² 2. Marie Curie has a utility function U(x, y) = max{x, 2y}. Note that the max{} function es- sentially says that the value of the function takes the highest of the values in the set (the set is denoted as the things inside the curly braces). (a) Draw a graph, plotting the indifference curve along which U(x, y) = 10. (b) Are Marie's preferences convex?
- Find the marginal rate of substitution given the following utility functions for x and y: U(x) = 2xy U(y) = 3x1/3,2/3 O x1/3 / yA/3 O 213 x213y1/3 y2/3 / x1/3 'yA/3 /x1/3Utility is given by the function U(x,y) = x0(y + 4)0.3. a. Calculate the marginal utility of x in terms of x and y. Simplify. b. Calculate the marginal utility of y in terms of x and y. Simplify. c. Calculate the marginal rate of substitution in terms of x and y. Simplify.Compute marginal utilities and marginal rate of substitution for each of the utility functions given below. In cach case. explain whether the assumptions of diminishing marginal utility and diminishing marginal rate of substitution are satisfied or not. a) U(X,Y)= 2x-yi b) U(X,Y) = XY +3X + 5Y c) U(X,Y) = (0.3X + 0.7Y") d) U(X,Y) = In (5VX + 2VT) %3D %3D %3D
- Lee has calculated her marginal utility from buying pizza slices and soda (shown on the table in the worksheet) and wants to use these data to decide how to rationally spend the money she has. The price of pizza is $2.00 and the price of soda is $2.00 question 9 (a) given these prices and your worksheet calculations, what is the first product lee will buy? (b) would lee ever buy four slides of pizza? Why/why not? question 10Given the data in your worksheet, how many pizza slices and sodas lee should buy to maximize her total utility if she rationally spends her budget of 10.00? question 11 when the price of pizza changes, lee's MU/$P changes. If the price of pizza decreases to $1.50 and her budget decreases to $7.50, how many pzza slices and sodas lee should buy to maximize her total utility?8. Let be the utility function U(x, y) =(xi + 2ya), , where x and y are the quantities consumed of two goods. (a) Show that the functional equation U(x, y) U(2,2) defines y as a function of x in a %3D neighborhood of point (2, 2). (b) If the quantity consumed of good x increases by 1 unit, by how much does the quantity have to increase or decrease? Consumed of the good and to keep utility constant? (c) If the quantity consumed of good x increases by 1%, by how much does the quantity have to increase or decrease? Consumed of the good and to keep utility constant?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…