Suppose that Jack has the utility function u (x, y) = Vx + y. The price per unit %3D of x is $1 and the price per unit of y is $4. He receives $100 that all has to be spent on two commodities x and y.
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i. Write down the Lagrangian form and solve the utility maximization problem.
ii. Assume that the income increases from 100 to $101. What is the increase in the optimal value of u(x,y)?
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- A consumer has a utility function U(X,Y ) = X1/2 Y1/2 . Prices of the goods are pX = £10 and pY = £5, respectively, and the consumer has income M = £200 to spend on X and Y. Currently, she buys 2 units of good X and spends the rest of her income on good Y.a) Determine this consumer’s current utility level. b) Computetheconsumer’smarginalutilitiesofgoodsXandY. c) Explain why the consumer’s current consumption of goods X and Y is not optimal. Should she substitute X for Y or vice versa? The consumer’s marginal rate of substitution is MRS = Y/X.d) Determine this consumer’s optimal consumption of both goods. e) By how much does the consumer’s utility increase when she consumes the optimal bundle?a. A young connoisseur has $600 to spend to build a smallwine cellar. She enjoys two vintages in particular: a 2001French Bordeaux (wF) at $40 per bottle and a less expensive 2005 California varietal wine (wC) priced at $8. Ifher utility is U (wF, wC))=wF2/3 w C1/3 ,then how much of each wine should she purchase?b. When she arrived at the wine store, this young oenologistdiscovered that the price of the French Bordeaux hadfallen to $20 a bottle because of a decrease in the valueof the euro. If the price of the California wine remainsstable at $8 per bottle, how much of each wine shouldour friend purchase to maximize utility under thesealtered conditions?c. Explain why this wine fancier is better off in part (b)than in part (a). How would you put a monetary value onthis utility increase?Suppose that the consumer maximizes his utilitysuch that the marginal utilities of goods X and Y are; MUx=80-16x MUy=60-2y Let Px=P4,Py=P2 and I=P80 Find the utility maximizing quantities X and Y
- abby is a single mom with a weekly budget of $100. She can spend her weeklyincome on food and diapers for her son. Price of food is $1 and price of a diaperis 50 cents. Her son needs at least 5 diapers a day, therefore she must purchaseat least 35 diapers weekly. Jennifer’s utility function is U (f, d) = f 0.9d0.1,where f and d represent quantity of food and diapers, respectively.Question 1 Part aFind abbys optimal consumption bundle. Make sure to draw her budgetconstraint and indifference curves and clearly demonstrate your solution.I like drinking Smirnoff and Stoli Vodka, but I really only careabout the total amount of alcohol I get out of it. They areperfect substitutes. Smirnoff is sold in liter bottles that are100 proof (that’s 50% alcohol). Stoli is sold in liter bottlesthat are 80 proof (40% alcohol). What is my utility functionfor bundles of these two vodkas? What do my indifferencecurves look like?a) You are given a utility function described as:U= U(K, L, M) = 10LogK + SLogL. + 2LogMWhere K, L and M are goods consumed. The individual has K100 to spend on the threeitems. Furthermore, you are informed that monthly salary is $500 and the prices of K, Land M are $2, $10 and $4 respectively. Calculate the consumer's optimal bundle.D.b) Provided a demand curve for Beans is as followsO = 4000 - 255PWhere Q is the quantity demanded and P is the price. Find the elasticity at K7, K10 and K15.
- a. A consumer is willing to trade 3 units of x for 1 unitof y when she has 6 units of x and 5 units of y. She isalso willing to trade in 6 units of x for 2 units of y whenshe has 12 units of x and 3 units of y. She is indifferentbetween bundle (6, 5) and bundle (12, 3). What is theutility function for goods x and y? Hint: What is theshape of the indifference curve?b. A consumer is willing to trade 4 units of x for 1 unitof y when she is consuming bundle (8, 1). She is alsowilling to trade in 1 unit of x for 2 units of y when sheis consuming bundle (4, 4). She is indifferent betweenthese two bundles. Assuming that the utility function isCobb–Douglas of the form U (x, y) = xα y β, where α andβ are positive constants, what is the utility function forthis consumer?c. Was there a redundancy of information in part (b)? Ifyes, how much is the minimum amount of informationrequired in that question to derive the utility function?If Utility =.5lnX++3Z+10+4√W then Marginal utility from W is...maddy is a single mom with a weekly budget of $100. She can spend her weeklyincome on food and diapers for her son. Price of food is $1 and price of a diaperis 50 cents. Her son needs at least 5 diapers a day, therefore she must purchaseat least 35 diapers weekly. maddy utility function is U (f, d) = f 0.9d0.1,where f and d represent quantity of food and diapers, respectively.Part aFind Jennifer’s optimal consumption bundle. Make sure to draw her budgetconstraint and indifference curves and clearly demonstrate your solution.Part bmaddy uncle Sam, who is aware of her financial troubles, offers to help herout. Specifically, he offers to pay half of her weekly diaper expenses. Howmany diapers maddy would buy now? Again, demonstrate your solution bothgraphically and analytically.
- Suppose that Omar's marginal utility for cups of coffee is constant at 5.5 utils per cup no matter how many cups he drinks. In contrast, his marginal utility per doughnut is 10 for the first doughnut he eats, 9 for the second, 8 for the third, and so on (that is, declining by 1 util per additional doughnut) In addition, suppose that coffee costs $1 per cup, doughnuts cost $1 each, and Omar has a budget that he can spend only on doughnuts, coffee, or both. How big would that budget have to be before he will spend a dollar buying a first cup of coffee? Instructions: Enter your answer as a whole numberGuerdon always puts half a sliced banana, q, , on his bowl of cereal, q, The two 14 goods are perfect complements. What is his utility function? Derive his demand curve for bananas graphically and mathematically. 12- 10- e, Guerdon's utility function (as a function of q, and q2) is 6- VA U= min/2q1.92} 4- O B. U=q, +92- 2- 0+ OC. U=29, + q92- 10 12 14 16 18 20 6 22 Bananas O D. U=0.5q1 + q2- OE. U=q, xq2- 5.00- 4.50- Guerdon's demand function for bananas (as a function of the price of bananas, P1. the price of a bowl of cereal, p2, and income, Y) is 4,00- 3.50- 91 = (p1 + 2p2) (Properly format your expression using the tools in the E 3.00 palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ character.) 2.50- * 2.00- P. 1.50- 1.00- 0.50- 0.00+ 10 12 14 Bananas tv 80 DI DD F1 F3 F4 F5 F6 F7 F8 F10 @ %23 $ & 1 3 4 7 Q W E Y U S F H. J く C V M option command command optic ーの * CO Cereal (bowls) p. S per banana B * * N AIf Utility=.5lnX+3Z+10+4VW then Marginal utility from X is...... and the Mu of Z is...... Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a b с 10; 3 3,4 5/X; 3 d 2/VW; 3