100 A - Econ
100 A - Econ
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Date Created: 02/14/14
Problem Set 2 Econ 100 A Fall 2012 September 18 2012 Due in lecture Wednesday September 26th Please show all the necessary math and explain your answer in a concise way 1 In this problem you will have to overcome your fear of working out the algebra of utility maximization to find demand functions and perform some comparative statics exercises Remember Pierre the existentialist philosophy major that lives for cigarettes and cappuccinos Suppose that his preferences for those two goods are represented by the following utility function U X Y X Y k1 where Xrepresents cigarettes and Ycappuccinos O lt a lt 1 and k gt 0 Let px and py represent the prices of cigarettes and cappuccinos respectively and M his income a Use your preferred mathematical technique Lagrangian optimization substitution etc to solve for the demand functions for cigarettes and cappuccinos Make sure to simplify your demand functions as much as possible and check for corner solutions Hint is there any level of income such that the demand for one of the goods becomes negative b Now consider just the interior solutions ie the solutions such that each good is consumed in positive amounts i Is good X a normal or inferior good Show your answer mathematically and explain in ONE sentence what it means ii Is good X a gross complement or substitute for good Y1 What about the opposite Show your answer mathematically and explain in ONE sentence what it means C Give an economic interpretation for the parameter k Hint think of what happens when Pierre s income increases starting from zero 22 In 1933 the Swedish economist Gunnar Myrdal who later won a Nobel prize in economics and a group of his associates at Stockholm University collected a fantastically detailed historical series of prices and price indexes in Sweden from 1830 until 1930 This was published in a book called The Cost of Living in Sweden In this book you can find 100 years of prices for goods such as oat groats hard rye bread salted codfish beef reindeer meat birchwood tallow candles 1 Look for these de nitions on section 25 of the textbook You are FORBIDEN to ask your GSI 2 Taken from Bergstrom T amp Varian H 1996 Workouts in Intermediate Microeconomics W W Norton amp Co Inc 4th edition eggs sugar and coffee There are also estimates of the quantities of each good consumed by an average workingclass family in 1850 and again in 1890 The table below gives prices in 1830 1850 1890 and 1913 for our meat milk and potatoes In this time period these four staple foods accounted for about 23 of the Swedish food budget Prices of Staple Foods in Sweden3 1830 1850 1890 1913 GrainFlour 14 14 16 19 Meat 28 34 66 85 Milk 07 08 10 13 Potatoes 032 044 051 064 Based on the tables published in Myrdal s book typical consumption bundles for a working class Swedish family in 1850 and 1890 are listed below Quantities Consumed by a Typical Swedish Family4 1850 1890 Grain Flour 165 220 Meat 22 42 Milk 120 1 80 Potatoes 200 200 a Copy and complete the table below which reports the annual cost of the 1850 and 1890 bundles of staple foods at Various years prices Cost 1850 bundle 1890 bundle Cost at 1830 Prices 441 616 Cost at 1850 Prices Cost at 1890 Prices Cost at 1913 Prices 785 1137 b Calculate the Laspeyres price index for 1890 with base year 1850 c If a Swede were rich enough in 1850 to afford the 1890 bundle of staple foods in 1850 he would have to spend times as much on these foods as does the typical Swedish worker of 1850 d If a Swede in 1890 decided to purchase the same bundle of food staples that was consumed by typical 1850 workers he would spend the fraction j of the amount that the typical Swedish worker of 1890 spends on these goods 3 Prices are in Swedish KronorKg except for milk which is in Swedish Kronorl 4 Quantities are in Kgyear except for milk which is in litersyear Now let s work with some data that relate to the US economy You re going to need a computer and a spreadsheet to make these calculations Visit the website of the Bureau of Labor Statistics wwwblsgov and download annual data for the CPI for all urban consumers in US between 2002 and 2011 for the following categories all items food and beverages housing apparel transportation medical care medical care services recreation education and communication and other goods and services The data for 2002 are given below so that you can check that you found the right numbers Series ID 2002 All Items 1799 Food and Beverages 1768 Housing 1803 Apparel 1240 Transportation 1529 Medical Care 2856 Medical Care Services 2929 Recreation 1062 Education and Communication 1079 Other Goods and Services 2932 The All Items category represents the whole CPI and the other categories represent the average price of the goods that fit in that definitions Actually they don t express the price level but we are interested in the percentage variation of each category After downloading the data convert the categories price index to real prices in terms of 2010 dollars and express everything using 2002 as a basis 100 This indexes are going to give you a figure on the percentage variation of the real price of each category with respect to the base year After doing these calculations answer the following questions e Plot the indexes that you calculated in a line graph that shows how the real price of each category evolved between 2002 and 2011 If possible print it in colors and don t forget to include legends f Which category had the highest percentage variation in real terms between 2002 and 2011 How much was it g Which category had the lowest variation in real terms between 2002 and 2012 How much was it 3 Let s analyze the gasoline consumption of David a Berkeley Graduate student and how he reacts after the imposition of a tax on gasoline After paying for rent and utilities6 he has 400 per month to spend on gasoline G for his moped and other goods 0 Assume he has 5 Check the FAQ section for the CPI on the BLS website for more details 6 David doesn t need to buy food As a good graduate student he free rides on seminar lunches in different departments like economics and ARE and other social events that serve free food and drinks Cobb Douglas preferences represented by U G 0 G 01 where a 05 and the initial price of gasoline is 400 a Compute David s monthly consumption of gasoline7 at the initial price and draw a graph of his budget constraint and indifference curve at his optimal bundle Put gasoline on the horizontal axis and consider the price of other goods equal to 1 Compute David s utility level at this initial situation Label his initial indifference curve with this utility level Compute David s consumption after the govemment imposes a tax of 100 per gallon on gasoline Add his new budget constraint ad optimal bundle to your graph What is the total change in his demand for gasoline and his new level of utility as a result of the tax Label his new indifference curve with this utility level David s family are rich farmers in colorful Colorado with no knowledge about economics They feel sorry for their kid after knowing about the hardness that have been imposed by the big government of California on him and decided to help by sending some extra stipend How much money should they send to make his initial bundle just affordable at the new prices How much gasoline he consumes and how much utility he gets after he receives this extra money David on the other hand is a good son and modest in his consumption habits He also likes to demonstrate the knowledge he acquired after some years away from home in the Golden State So he asks his parents for a Hicksian compensation for the tax Now help David s family explaining to them what is a Hicksian compensation and how much money should they send to him Is it more or less than what they offered Use the initial graph to represent the effects of the compensation on David s budget constraint and his consumption after he receives the extra stipend How can you relate the findings in part c and d to the bias in the estimation of cost of living on the CPI 7 At this stage you don t need to recalculate the demand by using Lagrangian or any other math tool Just jump to the uncompensated demand function for Cobb Douglas preferences that you already know from the top of your head by now
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