1. Choice: Consider a consumer with a utility function U(x, y) = ln(x + y). (a) Find the quantity demanded for both goods if p, = 5, P, = 3, and m = 40
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- 1. 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,(Review Question) Sarah consumes both wine ) and cheese (y) and considers them perfect substitutes. She always gets twice as much utility from consuming wine as she does cheese. What equation would represent her utility function? U-(2xy) U=x+2y U=2x+y U=min(x,2y)Q: A consumer’s preferences for food (F) and clothing (C) are given by U(F,C) = F0.2C0.8. The price of food is $4, and the price of clothing is $8. The consumer has an income of $3600. a) What is the utility maximizing choice of food and clothing? b) How would the utility maximizing choice change if price of clothing increased to $14? c) Given the answers to the previous parts plot a linear approximation to the demand function for clothing.
- 2) A consumer’s utility function is: U(x,y) = sqrt(9xy). The price of each good is: (PX,PY) = (15,20). The monthly budget is: M = 600, answer the following:a. What is the optimal bundle (X, Y) that maximizes the consumer’s utility?b. What is the optimal utility given your answer from part a?c. If the desired level of utility the consumer is U = 400, what would be the optimal bundle (X, Y) to minimize consumer expenses?d. Given U = 400 from part c, what would be the new optimal bundle for both goods if PX increases to 25?e. Given the price increase in good X from part d, what would be the optimal bundle for this consumer wishing to maximize its utility with an income, M = 600?f.: Graph the budget line and utility curve from part a.2) A consumer’s utility function is: U(x,y) = sqrt(9xy). The price of each good is: (PX,PY) = (15,20). The monthly budget is: M = 600, answer the following:a. What is the optimal bundle (X, Y) that maximizes the consumer’s utility?b. What is the optimal utility given your answer from part a?c. If the desired level of utility the consumer is U = 400, what would be the optimal bundle (X, Y) to minimize consumer expenses?d. Given U = 400 from part c, what would be the new optimal bundle for both goods if PX increases to 25?e. Given the price increase in good X from part d, what would be the optimal bundle for this consumer wishing to maximize its utility with an income, M = 600?f.: Graph the budget line and utility curve from part a i need help on part c21. 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). =
- 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.A consumer has utility (see image) on ice creams (x) and cakes (y). (a) Are the indifference curves bowed towards the origin? (b) Derive his demand function (as a function of prices px, py and budget I) for ice cream (x). (c)(Looking at the demand function you found in (b), Is ice cream a normal good? Are ice cream and cakes substitutes or complements? Calculate the income elasticity of market demand at the point px = 2, py = 1 and I = 12.Question 7 A consumer has a utility function 100-(x₁-4)4 -(x2-3)4 and a budget constraint of 9x1+3x2 <200. How much of good 2 do they consume?
- 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.(A) Suppose an individual who derives utility u(x, y) from consuming a units of the good X and y units of the good Y. Prices of goods X and Y are respectively p = 1 and Py = p. Furthermore, the individual has a budget m. What is the demand for Y if the utility function is: u(x, y) = min{x, ay}. (a) y(p, m) = am (b) y(p, m) = = (c) y(p, m)= (d) y(p, m) = = am a+p m a+p m 1+p (B) Same setup as (A) with u(x, y) = ax + y and ap Pr. The quantity demanded of good X is q. Which of the following is true ? (a) The substitution effect increases q but the income effect decreases q and over- come the substitution effect. Overall q decreases. (b) The substitution effect decreases q but the income effect increases q come the substitution effect. Overall q increases. and over- (c) Both the income and substitution effects increase q. Overall q increases. (d) Both the income and substitution effects decrease q. Overall q decreases. (D) Suppose an inferior good X (not Giffen). Its price increases from…The utility function of a consumer is u = √x + 2y. (a) Calculate how much MRS of that consumer when x = 4. (b) If the consumer currently consumes 4 units of goods x and the price ratio of Px / Py = 1/4, then the consumer should increase or decrease the consumption of goods x?