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. Show that Cobb-Douglas preferences are homothetic preferences
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- Suppose there are two firms selling protein bars. Firm 1 sells 'AggieBars' with 10 grams of protein and firm 2 sells 'DavisBars' with 20 grams of protein. Consumers are distributed uniformly over their preferences for grams of protein between 10 and 20. Suppose firm 1 sells AggieBars for $2 and firm 2 sells DavisBars for $3. The 'cost to consumers of deviating from their optimal amount of protein is $0.20 per gram. a. What protein content does the marginal consumer (the consumer who is indifferent between AggieBars and DavisBars) prefer? The equation for finding the marginal consumer (when the range of product attribute values is 10) is V - p1 - tx m = V – p2 – t(10 – x m) 4 b. How would the proportion of consumers buying each product change if the cost to deviation from one's optimal amount of protein increased (was greater than $0.20 per gram)?Which one of the following statements about preferences is true? O Quasilinear tastes can be defined, such that if (x1, x2) > (Y1₁. y2), then (tx1, tx2) > (ty1₁, ty2) O Homothetic preferences have the same MRS on a ray from the origin, whereas quasilinear preferences have the same MRS on a vertical ray. The MRS for perfect complements is 0 on the vertical part of the indifference curve, -∞ on the horizontal part of the indifference curve and undefined at the kink. O If goods X and Y are perfect substitutes (with X measured on the horizontal axis) and the MRS MUX> Px, then the consumer will optimize by consuming only good Y. MUY Py O The Engel curve for a normal good will bend backwards, whereas the Engel curve for an inferior good will slope upwards. O No answer.1) Suppose that Abel and Eden spend their incomes on two goods, food (F) and clothing (C).Abel’s preferences are represented by the utility function U(F,C)=10FC, while Eden’spreferences are represented by the utility function U(F,C)= 0.20F2C2A) With food on the horizontal axis and clothing on the vertical axis, identify on a graph theset of points that give Abel the same level of utility as the bundle (10, 5). Do the same forEden on a separate graph.B) On the same two graphs, identify the set of bundles that give Abel and Eden the samelevel of utility as the bundle (15, 8).C) Do you think Abel and Eden have the same preferences or different preferences? Explain 2. Graphically show the effect of an increase in price of Coca Cola on the demand of PepsiCola 3. Assume a budget line is drawn for two commodities: X on the x-axis and Y on the y-axis. Ifthe income of the consumer is 120 Birr, the y-intercept is 3, and the slope of the budget lineis -0.5 then determined the price of commodity…
- 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 candiesThe local mall has a make-your-own sundae shop. They charge customers 35 cents for each fresh fruit topping and 25 cents for each processed topping. Barbara is going to make herself a sundae. The total utility that she receives from each quantity of topping is given by the following table: Fresh Fruit Тоpping Processed Тоpping Total Utility # of # of Units Total Utility Units 1 10 1 10 18 2 19 3 24 3 27 4 28 4 33 30 5 38 28 38 7 24 7 36 If Barbara has $12 to spend on these goods , how many fresh fruit toppings and processed toppings will she purchase to maximize utility?John and Belle consume only two goods, x and y. They have strictly convex preferences and no kinks in their indifference curves. At the initial endowment point, the ratio of John's marginal utility of x to his marginal utility of y is J and the ratio of Belle's marginal utility of x to her marginal utility of y is B, where ] B. b. C < J. c. C = J. d. C = B. e. JFor the case of two goods, give an example of a utility function U(x1, x2) that represents the preferences of a consumer who regards the two goods as perfect complements. Next, take the transformation f(U) = U³ of the your example utility function and explain if this newly gener ated function represents the original preferences. Further, provide clear arguments supporting or rejecting the claim that "f(U(x1, x2)) must be strongly increasing in (x1, x2)."Suppose that Abel and Eden spend their incomes on two goods, food (F) and clothing (C).Abel’s preferences are represented by the utility function U(F,C)=10FC, while Eden’spreferences are represented by the utility function U(F,C)= 0.20F2C2A) With food on the horizontal axis and clothing on the vertical axis, identify on a graph theset of points that give Abel the same level of utility as the bundle (10, 5). Do the same forEden on a separate graph.B) On the same two graphs, identify the set of bundles that give Abel and Eden the samelevel of utility as the bundle (15, 8).C) Do you think Abel and Eden have the same preferences or different preferences? ExplainWe claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactlyonce. Can you prove this rigorously? (Hint: what would happen if itintersected some indifference curve twice?)uestion1: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 eco16.11. Ted and Joe each consume peaches, x, and plums, y. The consumers have identical 10y7x7, MRS = 10yr^TTogether, they have 10 peaches MRSJoe utility functions, with and 10 plums. Verify whether each of the following allocations is on the contract curve: a) Ted: 8 plums and 9 peaches; Joe: 2 plums and 1 peach. b) Ted: 1 plum and 1 peach; Joe: 9 plums and 9 peaches. %3DThe following table shows the marginal utility per dollar of apples and pears for five consumers: Consumer Marginal Utility Per Dollar Apples Pears Darnell 8 5 Eleanor 7 8 Jacques 6 6 Kyoko 5 4.5 Musashi 4 4 Darnell, Eleanor, and Kyoko are not optimizing over their choice of fruit. In the following table, indicate which fruit each consumer should increase consumption of in order to achieve the optimal fruit consumption bundle. Consumer Apples Pears Darnell Eleanor KyokoSEE MORE QUESTIONS