# UCLA STATS 100C - Midterm1_2015_Solution (7 pages)

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## Midterm1_2015_Solution

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

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UCLA Economics 11 Fall 2015 Professor Mazzocco MIDTERM 1 Version 1 NAME ID TA Part I Multiple Choice Questions 3 5 points each 1 If all prices and income double the Marshallian demands for goods X and Y will a both increase since the consumer has more income b both decrease since the two goods are now more expensive c remain unchanged because the prices and income increase by the same proportion Answer d the demand for X might increase or decrease depending on the preference of the consumer 2 If Angelina s utility function is given by U x y min 2x 4y and she is considering the two consumption bundles A x 100 y 25 and B x 50 y 50 then we can say that the consumer a is indifferent between A and B Answer b prefers A relative to B c prefers B relative to A d can t compare A and B because preferences are not complete 3 Alice s utility function is such that she is always indifferent between 1 unit of x and 3 units of y Suppose initially px 2 py 1 and I 10 Alice chooses x and y to maximize her utility Then the government imposes a 100 sale tax on x the price of x doubles How much MORE does the person have to earn in order to achieve the same level of utility as before Find the increase in income required to achieve the same utility level 1 a 0 b 5 Answer c 10 d 18 4 Suppose the good x is graphed on the x axis and the good y is graphed on the y axis The slope of the budget constraint is a px py b ppxy Answer c ppxy d py px 5 Suppose that the price of champagne is pc 70 and that the price of cheap wine from Trader Joe s is pt 10 and Alessio has income I 210 Alessio s M RS of cheap wine Uc is equal to 8 for any consumption bundle for each point for champagne that is M M Ut of the indifference curves If he spends his income optimally then Alessio will a Buy only champagne Answer b Buy 8 units of champagne and 2 units of wine c Buy a 4 1 ratio of champagne to wine we don t know exactly how much he can afford d There is not enough information to answer the question 6 Which of the following functions is not a homothetic function a U x y x2 y 2 b U x y x y 1 c U x y x y 1 d U x y xy x y Answer 7 Which of the following utility functions represent the same preferences as U x y xy a U x y 10 xy 2 b U x y xy c U x y ln x ln y d All of the above represent the same preferences Answer 8 If the utility function is U x y 6x 3y then the bundle 3 2 provides the same utility as a 2 4 Answer b 5 0 c 1 5 d None of the above 9 If an individual s utility function is given by U x y y ln x2 then MRS a Depends only on y b Depends only on x c Depends on x and y Answer d Is not determined 10 If Marlon s utility function for goods X and Y is given by U X Y minX 3Y the indirect utility function will be given by 3I p x py I b V X Y 3px py I c V X Y px 3py 3I d V X Y Answer 3px py a V X Y 3 Part II Essay Questions Question 1 36 Points 1 Arnold has the following utility function U x y xy x a For a given income I and prices px and py find Arnold s Marshallian demands for x and y 7 Points b Find the indirect utility function 5 Points c Suppose Arnold has 20 dollars to spend on X and Y and the prices of these goods are px 1 and py 4 What is his optimal consumption bundle What is his utility at that consumption bundle 5 Points d The government wants to increase Arnold s welfare It decides to give Arnold a cash transfer of T 8 additional dollars The prices stay at px 1 and py 4 What will be the new optimal consumption bundle and his new utility level 5 Points e Suppose that instead of making the cash transfer the government subsidizes Arnold by providing a price subsidy for the consumption of the good X The government chooses the subsidy so that the total amount paid in subsidies to Arnold is exactly equal to T 8 Find the value of the subsidy What is his new consumption bundle By how much will his utility increase under this arrangement 7 Points f Which program should the government use if the goal is to maximize Arnold s utility the transfer or the subsidy program 5 Points 4 Question 2 29 Points 1 Pearl is a student who cares only about two goods books B and food F Her utility function is given by u B F B 0 5 F 0 5 and the prices of books and food are pB and pF a Find Pearl s Hicksian demand functions for B and F 8 Points b Find her expenditure function 8 Points c Suppose that pB 1 pF 4 and that Pearl want to achieve the utility level U 10 What is the optimal consumption bundle that allows Pearl to achieve that utility level What is Pearl s expenditure at the optimum 5 Points d Now suppose the price of books rises to p0B 4 Pearl wants absolutely to stay at the same utility level U 10 How much income does she needs to stay at U 10 now that the price of books has increased to p0B 4 Hint remember the economic definition of the expenditure function 8 Points 5 Question 1 Solution a Taking a strictly increasing transformation u x y ln x ln y 1 From the tangency condition px py ux uy we get px py y 1 x x y 1 py px Plugging this into budget constraint y 1 py py y I Thus y I py I py x 2py 2px b The indirect utility function is V I py I py I py 2 1 2px 2py 4px py c x 12 y 2 V 36 0 d Now the income is I I T New consumptions are x I T py I T py 16 y 3 2px 2py And new utility is V I T py 2 64 4px py I py e The consumptions are x 2 p and y x The total amount of subsidies must satisfy I py 2py And the utility is V x T Therefore 2 px 2px T T 0 4 I py I py 2I 6 I py 2 4 px py And x 20 y …

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