# UCLA STATS 100C - Midterm1_2014_Solution (10 pages)

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## Midterm1_2014_Solution

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- School:
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- Stats 100c - Linear Models

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UCLA Economics 11 Fall 2014 Professor Mazzocco MIDTERM 1 Version 1 NAME ID TA Part I Multiple Choice Questions 3 5 points each 1 John likes 2 sugar packets S in his coffee C Which utility function would best represent his preferences a U S C min 1 2 S C b U S C min S 2C c U S C min 2S 4C d All of the above Solution d 2 Robert s preferences are described by the following utility function U x y 2x y If px 2py then the optimal consumption for good x is given by a x I px b x I 2px c x 0 d x I py Solution c 1 3 Suppose we have the following equations for the M RS of a utility function U x y Which of the following corresponds to a homothetic utility function a M RS x y xy x2 y c M RS x y 2 x y b M RS x y d M RS x y x2 y 2 xy Solution d 4 Suppose a consumer s utility function U x y is given by max 2x y Suppose you know that when the price of x is 1 the consumer s demand for x is 4 What is the consumer s income a 2 b 4 c 8 d None of the above Solution b 5 Pizza and pepsi are complements Suppose that at current consumption level of 4 pizzas and 3 pepsis an individual s marginal utility of consuming an extra pizza is 12 whereas the marginal utility of consuming an extra pepsi is 3 Then the MRS of pepsi for pizza that is the number of pizzas the individual is willing to give up to get one more pepsi is a 3 b 4 c 1 3 d 1 4 Solution d 2 6 Andy is indifferent between consuming M units of x or N units of y Which of the following utility functions of x and y can represent Andy s preference a U x y Mx Ny b U x y ln N x M y c All of the above d None of the above Solution b If Andy views M units of x to be the same thing as N units of y then they are perfect substitutes The usual form for perfect substitutes is U x y N x M y And since b is a monotonic transformation of N x M y b represents the same preferences Note that this is the correct form of the utility function since U M 0 U 0 N ln N M One can also check this by calculating M RSx y N M 7 If the preferences of an individual are represented by the utility function U x y x y and the consumption bundle x1 y1 lies on a higher indifference curve than x2 y2 it must be that a x1 x2 and y1 y2 b x1 x2 and y1 y2 c x1 x2 and y1 y2 d Any of these can happen Solution d 3 8 If an individual s utility function for good x and y is given by U x y min 4x y the indirect utility function will be given by I 4px py 4I b V px py I px py 4I c V px py I px 4py I d V px py I px 4py a V px py I Solution c 9 If an individuals utility function is given by U x y x0 5 y 0 5 and I 20 px 2 py 1 his or her preferred consumption bundle will be a x 10 y 20 b x 5 y 10 c x 4 y 8 d x 20 y 5 Solution b 10 Which of the following demand functions is not homogeneous of degree zero in px py and I a x I px 3py b x I 0 5 p 0 3 p 0 2 x y c x Ipy p2x d x I px py Solution d 4 Part II Essay Questions Question 1 35 Points 1 Jane has the following utility function U x y xa y 1 a with 0 a 1 Prices are px and py and income is I a Set up the budget constraint for Jane and use the Lagrangian multiplier method to solve for the optimal consumption of x and y 7 points Solution The budget constraint is I px x py y Therefore the Lagrangian for the utility maximization problem is given by L x y xa y 1 a I px x py y Taking first order conditions FOC s axa 1 y 1 a px 0 1 1 a xa y a py 0 2 I px x py y 0 3 L x y x L x y y L x y 1 2 and 3 can be re written as axa 1 y 1 a px 4 1 a xa y a py 5 I p x x py y 6 Dividing 4 by 5 px a y 1 ax py Rearranging to isolate y y 1 a px x a py 7 6 and 7 now describe a system of two equations and two variables Sub 5 stituting 7 into 6 we can find the optimal consumption of x 1 a px x a py 1 a I px px x a 1 I p x x a aI x gx px py I px I px x py Substituting x into 7 we can find y 1 a px x a py 1 a px aI y a py px 1 a I y gy px py I py y b Find the indirect utility function for a 1 3 7 points Solution V px py I U gx px py I gy px py I a 1 a aI 1 a I px py a 1 a a 1 a V px py I I px py So if a 1 3 then V px py I 6 1 3px 1 3 2 3py 2 3 I For the rest of the question use px 10 py 5 I 300 and a 1 3 c What is the optimal choice of x and y given those prices income and a 5 points Solution 1 3 300 10 10 2 3 300 y gy 10 5 300 40 5 1 3 2 3 1 2 V 10 5 300 300 25 2 3 10 3 5 x gx 10 5 300 The government wishes to raise taxes by using one of the following two taxes i a tax on Jane s income of T 100 dollars with the tax the new income is I T or a sale tax on each unit purchased of the good y the new price of y is therefore py d Using the prices income and a given above find the sale tax that makes the tax revenues collected from Jane using the sale tax equal to the tax revenues collected from Jane using the income tax 8 points Solution The following is the general …

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