Suppose Mary has the utility function U(x1, x2) = v(x;) + 2x2 with v such that v' > 0 and v" < 0. Suppose initially Mary chooses the optimal bundle x = 10 and x = 5. What is the new optimal x if prices for both x and x2 are cut in half (reduced by 50%)? Ox1 = 5 Ox1 = 11.7 Ox1 - 20
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- Marina decides to purchase a ring made from an alloy composed exclusively of gold (G) and titanium (T). The price of gold is $60 per gram, and the price of titanium is $30 per gram. Her total budget for the ring is $600. Her utility function is given by U(G,T) =GT. Suppose the price of titanium falls to $20 per gram. At the final basket, the optimal amount of titanium is（）grams.The preferences of a typical Californian can be represented by the following utility function: U (x1 , x2 ) = α ln(x1) + (1 − α) ln(x2) Here, x1 and x2 are the quantities of electricity and gasoline, respectively. The consumer faces prices given by p1 and p2 and has income m. Currently, the government has decided to impose a consumption restriction so that any person in the state is allowed to consume at most 50 units of electricity (x1 ≤ 50). Call this restriction a rationing constraint. (a) If α=0.25, m=100,and p1 =p2 =1, find the optimal consumption bundle of gasoline and electricity. Is the electricity rationing constraint binding (meaning does x1∗ = 50)? (b) Suppose that α = 0.75, but the other parameters are the same. What is the optimal consumption bundle? Is the rationing constraint on electricity consumption binding? (c) Now, assume that there is no rationing constraint. Assume m = 100 and p1 = p2 = 1, but α remains as a generic parameter. Solve for the optimal quantity…Consider the utility function U(x,y)=3x +6y. An agent with a budget constraint of 15x + 5y = 30 will choose which of the following bundles in order to maximize his/her utility? 2 units of x and 0 units of y 2/3 units of x and 4 units of y 1 unit of x and 3 units of y 0 units of x and 6 unit of y 1/2 unit of x and 1 unit of y
- Q1. Bob has a utility function defined over goods 1 and 2:u(x1; x2) = min{6*x1 + x2, x1 + 2*x2} Consider Bob’s preferences represented by the utility function inQuestion 1. Which of the following is correct?(A) Bob’s preferences are not homothetic.(B) Bob’s preferences are not transitive.(C) Bob’s preferences are not convex.(D) A typical indifference curve for Bob’s preferences is a straight line.(E) None of the above.Heather and Jeremiah both enjoy sunflower butter (s) and jelly (j) sandwiches, but they have varying preferences regarding the optimal ratio of the two. Heather prefers her sandwiches to have 6 spoonfuls of sunflower butter for every 3 scoops of jelly, whereas Jeremiah prefers 3 scoops of sunflower butter for every 5 scoops of jelly. They both would prefer as many sandwiches as possible, but derive no utility from sandwiches without their preferred ratio of s and j. A) Write down a utility function for both Heather and Jeremiah regarding consumption of s and j (U(s,j)) B) On the same graph, draw a set of indifference curves for both Heather and Jeremiah, labeling which is which. C) In addition to s and j, Heather and Jeremiah now need bread, b, to make sandwiches. Heather prefers to separate her sunflower butter and jelly and therefore uses 3 slices of bread per sandwich. Write down Heather’s utility function Uh(s,j,b) as a function of the three inputs.Bob lives in San Diego and loves to eat desserts. He spends his entire weekly allowance on pudding and pie. A bowl of pudding is priced at $1.00, and a piece of apple pie is priced at $4.00. At his current consumption point, Bob's marginal rate of substitution (MRS) of pudding for pie is 4. This means that Bob is willing to trade four bowls of pudding per week for one piece of pie per week. Does Bob's current bundle maximize his utility—in other words, make him as well off as possible? If not, how should he change it to maximize his utility? Bob could increase his utility by buying more pudding and less pie per week. Bob's current bundle maximizes his utility, and he should keep it unchanged. Bob could increase his utility by buying less pudding and more pie per week.
- alculate the marginal rate of substitution (MRS12) for the following utility function: U(q1,q2)= 60q1+ 0.2(q2)2 What is the value of MRS12 at bundle (4, 3)?Casper consumes cocoa and cheese. Cocoa is sold in an unusual way. There is only one supplier, and the more cocoa you buy from him, the lower the price you have to pay per unit. In fact, x2x2 units of cocoa will cost Casper √x2x2 dollars. Cheese is sold in the usual way at a price of $2 per unit. Casper's income is $10 and his utility function is u(x1,x2)=x21x2u(x1,x2)=x12x2, where x1x1 is his consumption of cheese and x2x2 is his consumption of cocoa. (1) Sketch Casper's budget set. (2) Sketch some of his indifference curves. (3) Calculate the amount of cheese and the amount of cocoa that Casper demands at these prices and this income. Do not forget to check the corners. (Hint: Write down the Lagrangean for this problem and solve the maximization programConsider a consumer whose utility function is U(q1,q2) = q1*q2. This consumer has $100 and prices of two goods are p1 = $1 and p2 = $2.a) Draw indifference curves corresponding to utility levels 1 and 4.b) Now consider utility function U(q1,q2) = (q1*q2)2. Check that it generates same indifference curve map as initial utility function and JUSTIFY this result.(c) Calculate marginal utilities of both goods and COMMENT on their properties.d) Calculate marginal ratio of substitution (MRS) at points (1,1) ; (1/2,2) ; (2,1/2) ; (2,2) ; (3,1) and (1,3) and INTERPRET results. THANKS.
- Reese thinks peanut butter and chocolate are great when separate, but when they combine they are even more epic. In other words, Reese likes to eat either peanut butter or chocolate, but when he eats them together, he gets additional satisfaction from the combination. His preference over peanut butter (x) and chocolate (y) is represented by the utility function: u(x, y) = xy + x + y Suppose that now Reese loses almost his entire income, so that he is left with only one dollar, i.e. his new income is I0 = 1. If prices are still px = 2, py = 4, what is his new optimal consumption of x and y (Hint: Remember that consumption of both goods must be weakly positive, i.e. x∗ ≥ 0 and y∗ ≥ 0) (a) x∗ = 0.5, y∗ = 0(b) x∗ = 0.25, y∗ = 0(c) x∗ = 0.75, y∗ = 0.25(d) x∗ = 0.75, y∗ = 0(e) x∗ = 0.5, y∗ = 11. Consider a consumer who chooses bundles consisting of two commodities, x and y. Suppose that, regardless of the prices px and py (which are always positive), the consumer chooses to consume x and y in a ratio of 2:1. This behavior is consistent with: a. A consumer having Cobb-Douglas utility function. b. The consumer’s utility function exhibiting perfect complements. c. The consumer’s utility function exhibiting perfect substitutes. d. Any of the above could be correct.A consumer has a perfect complements utility function, where she prefers to have one unit of H with each unit of G. Also, she has an income of $210. Assume that the price of H is $8 and the price of G is $6. What is the consumer's optimal choice for good G*? Group of answer choices 11 12 15 16
