ECNHIE Summer 223 3] Donnie knows that his preferences over pizza {3}] and french fries [f] are given by the follmving utility function: U [:3va = 4?[{10s{2a}+10s{f}}3] + 17 Propose a simpler function that represents Bonnie's preferences {give a formula] and explain in no more than three sentences why your simpler function represents the same underlying preferences. Consumer Choice 8:: Comparative Statics 4] Basil wants to buy owers for his wife Sybil. The cheapest owershop in town sells daisies [d] for pa = $3 and tulips [t] for pg = $2 and Basil has only as = Eli] that he's willing to spend. Suppose Basil knows that his wife's preferences over daisies and tulips are given by the utility function U (at) = said and he wants to make Sybil as happy as possible, given what he's willing to spend. Given this information. {a} 'Write dmvn an equation for Basil's budget constraint and graph Basil's budget set, with tulips {t} on the horizontal axis. Label the axes, intercept, and slope of the budget curve, and give a [onesentence] economic interpretation of the slope of the budget curve. [b] Set up the Lagrangian and derive the firstorder conditions that Basil's optimal choice of owers must satisfy. {c} Formulate an expression for Basil's Marginal Rate of Substitution [MRS] and evaluate it at the bundle (at, t] = {3, 2]. In one sentence, give an economic interpretation of this number. [d] Find the optimal bundle of owers that Basil should purchase. 5] Continuing from Problem :1, suppose Basil's income rises so that he has as = 312 available to spend, while facing the same prices 33,: = $3 and 3?; = $2. {a} Find the optimal bundle that Basil should purchase given his new level of income. [b] Based on your answer to Part {a} and to Problem 3, Part {d}, compute the income elasticity of Basil's demand for tulips {t}. {c} Formulate an equation for the Engel curve expressing Basil's demand for daisies {d} as a function of income {m} and graph the curve. ECNHlQ Summer 223 Problem Set 1: Consumer Theory Instructions | This set of problems covers consumer theory: how individuals choose which goods to buy [and in what quantities} and how these choices respond to various changes. a This problem set is due on May 30th at the beginning of class. | Be sure to shmv all of your work. Correct solutions without work shown will not receive credit. Incorrect solutions with work shown may be eligible for partial credit. | Please scan your solutions and submit them on Canvas as a single PDF file. Make sure your scanned solutions are readable! Consumer Budgets 1] Brett is buying cheeseburgers [c] and drinks {d} for breakfast with his friends. The price of cheeseburgers at Brett's favorite burger place is 33,, = 33 and the price of large drinks is 33:! = $2. Suppose that Brett has $31 to spend. {a} Graph Brett's budget set with d on the horizontal axis and c on the vertical axis. Label axes, the slope of each segment of the budget curve, the points [if any} where the slope changes, and intercepts on both axes. [b] Xow suppose the restaurant offers a bulk discount on cheeseburgers. If Brett buys more than 5 cheeseburgers, the price is discounted by 5H???- and if he buys 5 cheeseburgers or less, the original price pa = $3 applies. Graph Brett's new budget set reflecting the discount. Label axes, the slope of each segment of the budget curve, the points {if any} where the slope changes, and intercepts on both axes. Utility 3.: Preferences 2] For each of the following scenarios, write down a utility function that describes the consumers preferences and sketch the oorresponding indifference curve. {a} Leo loves pineapple pizza so much that he's always willing to give 11p 3 slices of cheese pizza for 1 slice of pineapple pizza, no matter hmv much pizza of either kind he already has. [b] Mikey always drinks soda when he eats pizza. Specifically, Mikey always drinks exactly one can of soda for every three slices of pizza he eats, and he gains no benefit from eating additional pizza if he doesn't have the right amount of soda to go with it [nor does he benet from drinking additional soda without pizza to eat with it]. Moreover, when he consumes Fi slices of pizza and '2 sodas, his utility is 2i]. ECNEIIE Summer H123 B] ISontinuing from Problem 5, suppose Basil keeps his new income of m = $121 but the price of tulips rises so that Basil now faces prices pa = $3 and 331 = 33. {a} Find the optimal bundle that Basil should purchase given the new prices he faces. (b) Based on your answer to Part (a) and to Problem 4.. Part [a], compute the price elasticity of Basil's demand for tulips (t) and the crossprice elasticity of Basil's demand for daisies (d)- {c} For both tulips [t] and daisies {d} briefly explain whether the income effect increases or decreases Basil's demand when the price of tulips changes and whether the substitution effect increases or decreases Basil's demand. 1What if anything can you say about the relative size of the effects? 7] For each of the following utility functions~ formulate an expression for the optimal quantity of xylophones (I) and the optimal quantity of yoyos [y] as functions of the income m. and the prices p, and par. {a} U (ea) = minfe-y} lib} U (I, a] = 33 + 5y