# UCLA STATS 100C - Midterm_1_2011_Solution (6 pages)

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## Midterm_1_2011_Solution

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

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UCLA Economics 11 Fall 2011 Professor Mazzocco MIDTERM 1 Version 1 NAME ID TA Part I Multiple Choice Questions Multiple Choice Questions 1 3 points Which of the following is NOT an axiom required for a utility function to represent preference a b c d Continuity Completeness Homogeneity Transitivity 2 3 points If the prices of all goods increase by the same proportion as income the quantity demanded of good x will a b c d decrease increase remain unchanged change in a way that cannot be determined from the information given 3 3 points An individual consumes only two commodities X and Y He spends all his income on these two commodities if X is a Giffen good then 1 X must be an inferior good 2 When the price of X changes the income effect dominates the substitution effect 3 It is possible that Y is an inferior good Which of the following is correct a b c d 1 2 2 3 1 3 1 2 3 4 3 points A consumer consumes 2 goods X and Y Suppose that her optimal consumption bundle is X 1 Y 2 when Px 1 Py 2 and I 5 Now Px increases to 2 other things constant and her optimal consumption bundle becomes X 0 5 Y 1 Then the total price effect for X is a 0 5 b 0 5 c 1 d Uncertain there is not enough information 5 3 points Suppose that at current consumption levels an individual s marginal utility of consuming an extra pizza is 10 whereas the marginal utility of consuming an extra beer is 1 Then the MRS of beer for pizza that is the number of pizzas the individual is willing to give up to get one more beer is a b c d 5 2 1 2 1 10 6 3 points Which of the following utility functions best represents the idea that two goods x and y are complements U x y xy a b U x y x y U x y x y c d U x y min x y 7 3 points John is shopping for fruits in a grocery store which sells two kinds of fruits X Y He only cares about the amount of fruits he eat which of the following function best represents his preference a U X Y X Y b U X Y X Y c U X Y X Y d U X Y min X Y 8 4 points Charlie has a utility function U X Y XY the price of apples pX is 1 and the price of bananas pY is 2 If Charlie s income is 200 how many units of bananas Y would he consume if he chose the bundle that maximized his utility subject to his budget constraint a 25 b 50 c 10 d 100 e 150 9 4 points If Anita s utility function over goods X Y and Z is given by U X Y Z 3X 2Y 6Z we know that a Her marginal rate of substitution between X and Y MRS x y is equal to her marginal rate of substitution between Y and Z MRS y z if she is currently consuming 2 units of X 4 units of Y and 1 unit of Z b Anita is willing to give up 2 units of Y in exchange for 3 units of X c Anita is willing to give up 2 units of Z in exchange for 6 units of X d Anita is willing to give up 9 units of Y in exchange for 3 extra units of Z 10 4 points An individual has a utility function for Dunlop tennis balls x and Prince tennis balls y of the form U x y x y The prices are such that px py His or her expenditure function is given by a E p yU b E p xU E p x p y U px p y E U c d Part II Essay Questions Question 1 37 Points Joe views goods X and Y as perfect complements in the proportion one X to two Y Thus his utility function is a For a given amount of income and prices for X and Y as well as his indirect utility function find Joe s Marshallian demand functions b Are X and Y normal or inferior goods c Find the Hicksian demand functions and the expenditure function and d Suppose Joe has 120 dollars to spend on X and Y and the prices of these goods are What is his optimal consumption bundle The government wishes to raise taxes by using one of the following two policies i a tax on Joe s income of 20 dollars 20 dollars are subtracted from Joe s income or ii a tax on each unit purchased of the good Y a tax is a negative subsidy on the price of the good Y therefore follow what we did in class with the price subsidy but instead of subtracting as we did in class add to the price e Find the consumption tax on each unit of consumption that makes the taxes the government is able to collect from Joe using the first policy equal to the total revenues obtained from Joe using the second policy f Which policy will the government choose if the objective is to maximize Joe s welfare Answer a At the optimum we must have and plugging this into our budget constraint gives us Indirect Utility b Normal goods c No matter what the prices are we must have at maximum so Therefore given a level of utility we must have d e f Policy 1 Policy 2 Both policies have the same impact on the utility of the agent Question 2 30 Points Lisa has the following utility function U X Y X Y Px 1 and Py 2 the daily income is 10 a How many units of X and Y does she consume b Suppose that the price of Y increases to 3 Find her new optimal consumption bundle c Suppose instead that the price of X increases to 4 so now Px 4 Py 2 Find her new optimal consumption bundle d Consider the new prices Px 4 Py 2 If her income changes in a way that she can achieve exactly the level of utility she was able to achieve in part a How many units of X and Y would she buy e Find the income and substitution effect for X and Y when the price of X increases from 1 to 4 with the price of Y staying at 2 unchanged Hint observe that in part d you have computed optimal consumption holding the utility constant Answer a X 10 Y 0 b X 10 Y 0 c X 0 Y 5 d Since px py E p yU 2 10 20 X 0 Y 10 e Income effect X 0 0 0 Y 5 10 5 Substitution effect X 0 10 10 Y 10 0 10

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