2. If a consumer faces two goods x and y, and her utility function is given by U(x, y) = Cx- x,)15(y – yo)15, where x, and yo are positive constants. Please demonstrate the marginal rate of substitution between y and x (i.e. -dy/dx).
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- 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 /x1/31. For each of the three utility functions below, find the substitution effect, the income effect, and the total effect that result when prices change from p = (2,1) to p' = (2,4). Assume the consumer has income I = 20. (a) Before doing any calculation, make an educated guess about the relative magnitude of the three substitution effects and the three income effects to be found below. (b) u(x1, x2) = x1 + x2. (c) u(x1, x2) = x1x2. (d) u(x1,x2) = min {x1,x2). (e) Rank the substitution effects and the income effects found above by their magnitude. To what extent do they conform to your guess?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) = 2
- A consumer has the following utility function: Ulx, y) = xy -y, *21 where x and y represents the quantities consumed of goods X and Y. y 20 What will be the substitution and income effects for X and Yassuming that the consumer attempts to maintain the same level of utility achieved before price of Y increased (that is, when price of Y was $1)? SEx= +0.5 IEx = -0.5 SE, = -0.25 IE- = -0.25 SEx= +0.293 IE = -0.293 SEy = -0.414 IE, = +0.414 SEr= +0.25 IE SE, = -0.75 IE, = -0.75 = -0.25 SEx= +0.414 IEx = -0.414 SEy = -0.293 IE, = -0.207 Income = $3 Px= $1, Py= $23- 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?It is common for supermarkets to carry both generic (store-label) and brand-name (producer-label) varieties of sugar and other products. Many consumers view these products as perfect substitutes, meaning that consumers are always willing to substitute a constant proportion of the store brand for the producer brand. Consider a consumer who is always willing to substitute 4 pounds of a generic store brand for 2 pounds of a brand-name sugar. Do these preferences exhibit a diminishing marginal rate of substitution? Assume that this consumer has $24 of income to spend on sugar, and the price of store-brand sugar is $1 per pound and the price of producer-brand sugar is $3 per pound. How much of each type of sugar will be purchased? How would your answer change if the price of store-brand sugar was $2 per pound and the price of producer-brand sugar was $3 per pound?
- If the utility function of an individual takes the form: U = U ( x 1, x2) = (x1 + 2) 2 (x2 + 3) 3 Where U is total utility, and x1 and x2 are the quantities of two commoditiies consumed: (a) Find the marginal-utility function of each of the two commodities (b) Find the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.Give examples by drawing a graph of each of the following. Please draw your graphs neatly using a ruler to draw the axes and label your graphs completely. In each case draw ar least three indiffer- ence curves and indicate clearly the direction of increasing prefer- ence. (i) the preferences being convex but not strictly convex. (ii) the preferences being strictly increasing but not convex.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?
- 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?A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility yielded by their consumption is shown in the table below. Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively, and that the consumer has an income of $74 to spend. (a) Complete the following table by computing the marginal utility per dollar for successive units of X, Y, and Z to one or two decimal places. (b) How many units of X, Y, and Z will the consumer buy when maximizing utility and spending all income? Show this result using the utility maximization formula. (c) Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z? Product X Product Y Quantity Utility Marginal Quantity Utility Marginal Utility Utility per$ per$ 1 23 4567 42 82 118 148 170 182 182 1 23 4 5 6 7 14 26 36 44 50 54 56.4 Product Z Quantity Utility Marginal Utility per$ 1 23456 7 32 60 84 100 110 116 120 3..1. Consider a consumer who consumes two goods. The utility function is given by u(x1, x₂) /2x1+1+√√√x₂, - where x > 0 denotes the amount of good i = the price of good 1 is 3, the price of good 2 is p2 > 0, and the income is 12. 1,2 consumed. Unless stated otherwise, (1) Calculate the marginal rate of substitution (measuring the value of good 1) at the consumption plan (x₁, x₂) = (3, 4). (2) Calculate the consumption plan that is optimal for the consumer. (3) Let v(p₂) denote the utility level at the optimal consumption plan. Calculate its derivative with respect to p2, i.e., v'(P2), at p2 = 3. (4) Suppose that p2 = 1 and the utility level is 10. Calculate the compensated demand for each good.
