# UCLA STATS 100C - Midterm1_Solution (6 pages)

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## Midterm1_Solution

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- Pages:
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- School:
- University of California, Los Angeles
- Course:
- Stats 100c - Linear Models

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UCLA Economics 11 Fall 2016 Professor Mazzocco MIDTERM 1 SOLUTIONS Part I Multiple Choice Questions 3 5 points each 1 2 3 4 5 6 7 8 9 10 Version 1 B B D B C C A B A C Version 2 B C C A B A C B B D 1 Version 3 A B A C B B D B C C Part II Essay Questions Question 1 34 Points 1 Janet loves eating large salads S while drinking Odwalla juices J Drinking the juices improves the taste of salads Therefore her utility function has the following form U S J JS S Version 1 2 Version 2 3 Version 3 4 a For a given income I and prices ps and pj find Janet s Marshallian demands for S and J 8 Points Apply the tangency condition to get M RS Thus S S pj J ps pj J ps Replace in the budget constraint to get pj J pj J I So I pj 2pj I pj 2ps J S Version 1 Version 2 Version 3 J S I 2pj 2pj I 2pj 2ps I 3pj 2pj I 3pj 2ps I 4pj 2pj I 4pj 2ps b Find the indirect utility function 8 Points V pj ps I I pj I pj I pj I pj 2 2pj 2ps 2ps 4pj ps 2 Version 1 Version 2 Version 3 V I 2pj 2 4pj ps I 3pj 2 4pj ps I 4pj 2 4pj ps c Suppose Janet has 50 dollars of income version 1 and 2 for version 3 I 40 to spend on salads and juices and the prices of these goods are ps 10 and pj 5 version 1 version 2 ps 4 and pj 2 version 3 ps 6 and pj 2 What is her optimal consumption bundle What is her utility at that consumption bundle 5 Points J S V Version 1 Version 2 Version 3 4 11 8 3 7 4 18 98 48 Janet s mother is really proud of her daughter s healthy diet She therefore wants to find a way to improve her daughter s happiness utility To do this she comes up with two possible plans i The first plan is to increase Janet s income by 10 from its current value of 50 version 1 and 2 for version 3 I 40 ii The second plan is to subsidize Janet s consumption of juices by paying 20 of the price of each juice she buys d Which of the two plans should Janet s mother choose if she wants to maximize Janet s utility To obtain the points show the calculation that enables you to choose the plan 8 points Indirect utility changes in the following way for each plan i New income is I 0 I 0 1I 1 1I so the new indirect utility is V pj ps I 0 0 9I pj 2 4pj ps Replace numbers to get V Version 1 Version 2 Version 3 21 125 116 281 56 333 3 0 ii New price of juice is pj pj 0 2pj 0 8pj so the new indirect utility is 0 V pj ps I I 0 8pj 2 4 0 8pj ps Replace numbers to get V Version 1 Version 2 Version 3 21 025 117 306 56 067 So plan 1 is the best for version 1 and 3 plan 2 for version 2 e How much money does Janet s mother have to spend to implement each plan 5 points The cost of plan 1 is 10 of income so 5 for version 1 and 2 4 for version 3 The cost of plan 2 is given by the subsidy multiplied by the number of juices bought That is I 0 8pj S 0 2pj J 0 8pj ps I 8 or Version 1 Version 2 Version 3 S 5 25 5 65 4 2 4 Question 2 31 Points 1 Janet got tired of eating salads and drinking juices and she decides that from now on she will only eat steaks s and drink red wine w Her preferences for those two goods are such that she always wants to eat one steak with glasses of wine Her utility is therefore U s w min s w Version 1 2 Version 2 4 Version 3 2 The prices of steak and wine are ps and pw a Find Janet s Hicksian demand functions for s and w 7 Points Hicksian demand is always at the corner of the indifference curve therefore s w U Which returns the Hicksians hs hw Version 1 Version 2 Version 3 U U U 2 4 2 U U U b Find her expenditure function 7 Points E Version 1 U p2s pw Version 2 Version 3 ps U 4 pw U p2s pw c Suppose that ps 10 pw 5 version 1 and 2 version 3 ps 8 pw 4 How much income does Janet needs to achieve the utility level U 100 version 1 version 2 U 200 version 3 U 400 4 Points Evaluate the expenditure function to get E Version 1 Version 2 Version 3 1000 1500 3200 A new scientific article has provided evidence that excessive consumption of red meat reduces life expectancy As a consequence the government has decided to impose a sale tax of 50 on steaks 5 d If Janet wants to stay at the same utility level how much income does she need to spend after the introduction of the tax 6 Points The new tax changes the price of steak to 1 5pw The new expenditure is E Version 1 Version 2 Version 3 1250 1600 4000 e Janet realizes that she cannot afford to spend that much and decides to spend the same amount she was spending before the introduction of the tax What will her new level of utility be after the introduction of the tax 7 Points The utility level can be found by solving for the utility U that will give you the same level of expenditure with the increased price That is solving U Version 1 5 1000 U 15 2 Version 2 5 1500 U 15 4 Version 3 4 3200 12 2 Which solves to Version 1 Version 2 Version 3 U 80 187 5 320 6

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