10) Suppose that the utility function of an individual can be described as U(X,Y) = 4X + 2Y. For this utility function the MRS A) is always X*Y B) is always constant C) is always X/Y D) is always X+Y E) is always X-Y.

Consider a utility function given by:

Donald likes fishing (X1) and hanging out in his hammock (X2). His utility function for these two activities is u(x1, x2) = 3X12X24. (A) Calculate MU1, the marginal utility of fishing. (B) Calculate MU2, the marginal utility of hanging out in his hammock. (C) Calculate MRS, the rate at which he is willing to substitute hanging out in his hammock for fishing. (D)Last week, Donald fished 2 hours a day, and hung out in his hammock 4 hours a day. Using your formula for MRS from (c) find his MRS last week. (E) This week, Donald is fishing eight hours a day, and hanging out in his ham­ mock two hours a day. Calculate his MRS this week. Has his MRS increased or decreased? Explain why? (F) Is Donald happier or sadder this week compared to last week? Explain.

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)?

Anna has an income of 1000 euros and utility function U(x,y)=x²y³. The prices of the two goods are P₁=5 and Py-20. a) Define the budget constraint and represent it graphically. (

Consider the following utility functions. G(x,y) = x² + 3y² H(x,y) = (x+1)¹/2 + y + 1 L(x,y) = xey U(x,y) = In(x) + y² W(x,y) = 100 x - 4x² + 3 y Z(x,y) = (x)¹/² + y Which function or functions have strictly convex indifference curves (at every point where x>0 and y>0)?

1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.

21. 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 på 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 px = 15, 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). =

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 the following utility functions: (a) u(x1, x2)=x1x2, I (b) u(x1, x2)=xx, (c) u(x1, x2) = ln x1 + ln x2, and (d) u(x1, x2)=√√x1x2, where x is the amount of good 1 and x2 is the amount of good 2. (1) (2) (3) (4) For each of the utility functions, find the marginal utility (MU₁) with respect to good 1. (5 points) For each of the utility functions, find the marginal utility (MU2) with respect to good 2. (5 points) For each of the utility functions, consider the following indifference curve: k = u(x1, x2) for some constant k. Find the marginal rate of substitution (MRS) of good 1 for good 2 at a point (x1, x2) on the indifference curve. (5 points) Show that each of the utility functions has a diminishing MRS. (5 points)