Divya and Emily have to divide a bag of 200 pistachios between them. They derive enjoyment from pistachios according to the following hedonic utility functions: uD(x) = √x for Divya; uE (x) = 2√x for Emily (where x is the number of pistachios eaten). (a) Consider all possible ways of dividing the pistachios. Treating the utilities given above as the two agents" "levels," which division is best according to the utilitarian criterion? (b) Which division is best according to the leximin criterion? (c) Which is best according to the Nash criterion?
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- 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.Suppose that U(f,c) = f + 8c^(1/2)is a utility function that describes Amelia’s preferences over two goods: fish(f)and custard (c). For the following, think of fish as the good graphed on the horizontal axis.a. Derive an expression for her marginal utility (Uf)from a small increase in f holding c fixed. Also find themarginal utility for custard (Uc).b. What is Amelia’s marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitivedescription of what MRS represents. If Amelia has 4 units of custard, holding her utility constant, howmany units of custard would she be willing to give up in order to get one more unit of fish?c. Graph Amelia’s indifference curve for a utility level of 40. Be sure to specify at least 3 bundles of goodson the indifference curve.d. Does the fact that Amelia’s indifference curve intersects with the custard axis violate any of the 5properties of indifference curves? Briefly support your answer.e. Give another utility function that represents…Problem 1 Last week, neither Alistair nor Bodana purchased blueberries (x). Both consumers had $30 to spend on blueberries and/or raspberries (y), the price of the two fruits was $3 per pack and both consumers bought raspberries only. Alistair's utility for blueberries and raspberries is UA(x,y) = x+2y. Bodana's utility for blueberries and raspberries is UB(x,y) = x +12ln(y). a) In two separate graphs, illustrate Alistair's and Bodana's optimal choices. Your graphs must illustrate the consumers' budget constraints and their indifference curves at the optimal bundle. b) At their optimal bundle, what is Alistair's marginal rate of substitution of raspberries for blueberries (- MUx/MUy)? At their optimal bundle, what is Bodana's marginal rate of substitution? This week, blueberries cost $3 per pack just like last week, however raspberries now cost $4 per pack. c) How many packs of blueberries and raspberries will Alistair buy this week? Clearly justify your answer. d) Using the…
- A child is given $4.80 of pocket money to be spent on either hard candies or chocolates. Chocolates cost 40 cents and hard candies $1.20 each. The marginal utilities derived from each product are as shown in the following table. Number of Items MU of Chocolates MU of Hard Candies 1 2 3 4 5 6 7 8 60 50 40 30 20 10 5 150 140 120 100 80 70 50 20 Which combination would give the child the maximum utility out of spending $4.80? 6 chocolates and 2 hard candies 8 chocolates and 0 hard candies 3 chocolates and 3 hard candies 0 chocolates and 4 hard candiesuestion1:Donna and Jim are two consumers purchasing strawberries and chocolates. Jim’s utility function is U (x, y) = xy and Donna’s utility function is U (x, y) = x2y where x denotes strawberries and y denoteschocolates. Jim’s marginal utility functions are MUX=y and MUy=x while Donna’s are MUX=2xy and MUy=x2. Jim’s income is $100, and Donna’s income is $150. What is the optimal bundle for Donna if the price of strawberries is $2 and the price of chocolate is $4? What is the optimal bundle for Jim, and for Donna, when the price of strawberries rises to $3?Question2:Consider a one-period ecoSuppose Al is currently consuming five movies and two concerts per month. If his utility function is given by U = 15MC, where Mrepresents the number of movies seen and C represents the number of concerts attended, Al's total utility is equal to: 75 15 150 10 30
- Suppose there are two individuals, Jane and Keith. Jane's utility function is Uj = XY, while Keith’s is Uk = X^2 Y^2 . Where X and Y are two separate goods. a) Find the marginal utilities of the two goods for Jane and Keith. b) Sketch the marginal utilities for Jane and Keith. c) What do the marginal utilities tell us about their tastes? d) Show by implicit differentiation that, for any given combination of X and Y, Jane's indifference curve has the same slope as Keith's.Rohan’s current marginal utility from consuming peanuts is 100 utils per ounce and his marginal utility from consuming cashews is 200 utils per ounce. If peanuts cost $0.10 per ounce and cashews cost $0.50 per ounce, is Rohan maximizing his total utility from the kinds of nuts? Enter your responses as whole numbers. At the current level of peanut consumption, Rohan receives: utils per dollar.At the current level of cashew consumption, Rohan receives: utils per dollar.Therefore, Rohan maximizing his total utility because MUp/Pp is MUc/Pc.Donna and Jim are two consumers purchasing strawberries and chocolate. Jim’s utility function is ?(?,?) = ?? and Donna’s utility function is ?(?,?) = ?2? where x is strawberries and y is chocolate. Jim’s marginal utility functions are MUx=y and MUy=x while Donna’s are MUx=2xy and MUy=x2. Jim’s income is $100, andDonna’s income is $150. What is the optimal bundle for Donna if the price of strawberries is $2 and the price of chocolate is $4?
- Chuck and Diane are both currently consuming 15 apples and 3 bananas each week, which we can call bundle A. Assume that bananas are the horizontal axis good for both consumers. Let's say both Chuck and Diane consume one more banana each week (4 bananas in total), moving to their respective bundle B on their own indifference curves. In order for Chuck and Diane to remain on the same indifference curve: • Chuck must give up 3 apples, to account for the marginal utility gained from a fourth banana ● Diane must give up 5 apples, to account for the marginal utility gained from a fourth banana Based on the information above, what can we tell about Chuck and Diane's preferences? Diane's indifference curves are flatter than Chuck's indifference curves Chuck and Diane have the same MRS between the bundles with 3 and 4 bananas MRS at The absolute value of the slope of Diane's indifference curves is greater for Diane than Chuck All of the above statements are trueRafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3Your sister’s utility function over the 2 goods X and Y is defined by: U (X,Y) = XY +4Y a. If she originally consumed 10 units of X and 24 units of Y, and if the consumption of X were increased to 12 units, how much Y would she be willing to give up and maintain the initial level of satisfaction? That is, how much less Y is needed in exchange for the increase in the consumption of X if she is to be equally satisfied before and after the change occurs? b. On a carefully constructed graph (you only need to consider X and Y values up to 50), indicate your sister’s original consumption position, with X plotted on the horizontal axis and Y on the vertical axis. Draw her indifference curve through the original consumption bundle from part a. As you can verify (or have already verified), she is indifferent to the points (10,24) and (12,21). If you double the amount of each good in these two consumption bundles, you would have the bundles (20,48) and (24,42). Since the two original bundles…