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- 3 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 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?4. Show that the two utility functions given below generate identical demand functions for goods X and Y: a. U(X,Y)= log(X) + log(Y) b. U(X,Y)= (XY)0.5I. You only consume two goods and your preferences are represented by utility function U(x₁, x₂) = (x0.5 + x2.5)² for x₁ > 2 and x₂ > 0. 1. Find your marginal utility for each good. Simplify your answers. Based on your answers, are your preferences strictly monotonic? Why or why not? 2. Do your preferences obey the law of diminishing marginal utility for each good? Why or why not? If not, what is the behavior of your marginal utility for each good? Use calculus and words to answer this question.
- a. Determine the demand functions of x and y in the case of a Cobb-Douglas type utility function, in the following cases: α=0.40;β=0.60 Graph the demand functions of the two goods (price as a function of quantity) assuming the individual's income is $500 - Determine what is the quantity demanded of x and y, if the price of good x is USD 1, the price of good y is USD 4, and income is USD 500 - Now, explain what happens to the quantity demanded if the prices of the goods are doubles holding income constant.1.Let x and y denote the amount of goods X and Y. Find the demand functions of X (do not need to find that of Y) when your preferences are represented by the utility function U = x^2y. Is X normal good? Can you confirm law of demand for X? What is the relationship of X with Y? Answer all of them by using the demand curve you derived. 2.Let x and y denote the amount of goods X and Y. Find the demand functions of X (do not need to find that of Y) when your preferences are represented by the utility function U = x + xy + y. Is X normal good? Can you confirm law of demand for X? What is the relationship of X with Y? Answer all of them by using the demand curve you derived 3.There are two goods F and C. Let MU and P denote marginal utility and price of each good. SupposeMUF = 3, MUC = 4, PF = 2, PC = 2.Are you maximizing your satisfaction? If not, what would you do to increase your satisfaction? Explain.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?
- 2. Ping receives a weekly allowance to purchase soda (x) and juice (y) at school. His preferences over these two goods can be represented by the utility function U(x,y) =0.5x+ 5lny where x represents the number of cans of soda and y represents the number of juice boxes. a) Given his preferences find his Marshallian demand functions for soda (x) and juice (y). b) Suppose that the price of a can of soda is $1.5 and that Ping has $30 to spend on soda and juice. Write Ping's demand curve for juice (y). Illustrate his demand curve. c) Suppose that the price of a box of juice is $1 (the price of soda and income remain $1.5 and $30, resp.). Use your demand functions to find his best bundle. In an indifference curve diagram illustrate his best bundle at these prices. For the remainder of the question assume that Ping's allowance (income) rises to $36 per week and the prices of the two goods are unchanged at Px= $1.5 and Py= $1. d) Use your demand functions to find his new best bundle. e)…Shanise has the following utility function: UX.Y) = X +Y where X is her consumption of candy bars, with price Px = $2.25, and Y is her consumption of espressos, with Py = $6.50. Derive Shanise's demand for candy bars and espresso. Shanise's demand of candy bars and espressos is O A. X= 1.0021 and Y = 0.5231. O B. X=0.330l and Y = 0.040I. O C. X=0.2461 and Y = 0.040I. O D. X= 0.8331 and Y = 0.6831. O E. X=0.0611 and Y =0.5231. Assume that Shanise's income is I = $75. How many candy bars and how many espressos will Shanise consume? Shanise will consume |candy bars and espressos. (Enter your responses rounded to two decimal places.) What is the marginal utility of income? The marginal utility of income is- (Enter your response rounded to two decimal places.)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= $2
- 6. Suppose that a fast-food junkie derives utility from three goods - softdrinks (x), 0.5 hamburgers (y), and ice cream sundaes (z) – according to the Cobb-Douglas utility function U(x, y, z)=x5y5 (1+z)05. Suppose also that the prices for these goods are given by p=0.25, p, = 1, and p. = 2 and that this consumer's income is equal to 2. a) How much will this consumer buy of each good to maximize utility? b) Show that the utility function is maximized (Use |H₂| to identify whether it is a maximum or a minimum). c) What is the maximum utility of the fast-food junkie?Suppose the weighted marginal utility for two goods x and y at a position of consumer equilibrium 70. If the price of good x is r10 and the relevant marginal utility for y is 140 what is the price of good y and the relevant marginal utility for x3. Tom's utility function is U (x; y) = 3x+4y: His budget constraint is ax + by = 1, a,b,I > 0 a) Derive the demand function of x. b) Derive the demand function of y. c) Please draw the demand curve for y. d) If the price of x increases from a to c; what would the value of the Hickis substitution effect of x be?
