NYU ECON UA 10 – Demand and Revealed Preference

help with Exercise 4 pleaseIntermediate MicroeconomicsHomework Set 4: Demand and Revealed PreferenceDue: Thursday, February 23rd at 2 pm via NYU Classes Exercise 1Marie has preferences over consumption of sugar and spice representable by the utility function u(x1 , x2 ) = min{3×1 + 2×2 , 2×1 + 5×2 }, where x1 is units of sugar and x2 is units of spice.At current prices, Marie consumes 12 units of sugar and 40 units of spice and spends herentire income.1) Draw a graph showing her indifference curve through this point.2) The price of spice is $1. What is the price of sugar and what is Marie’s income? Exercise 2Consider a consumer with preferences over X = R2+ representable by the utility function1/2 u(x1 , x2 ) = x1 + 2×1 x2 + x22 .1) Is either good inferior? Is either good a Giffen good? Are the two goods gross complements or gross substitutes?2) Suppose the consumer’s income is 10 and the price of good 2 is 2. If the price of good1 is initially 1 and then doubles, compute the total change in demand for good 1, andidentify the portions of the change attributable to the substitution and income effects. 1 Exercise 3Suppose you observe the following partial information about a consumer’s purchases of twogoods (which constitute his entire consumption). Good 1Good 2 Year 1 Year 2 Quantity Price Quantity Price 100100 100100 120? 10080 Data on consumption of good 2 in year 2 is missing from the dataset. Also, as usual you donot directly observe the consumer’s income. For what values of good 2 consumption in year2 would you conclude:1) That his behavior is inconsistent with optimization of a monotone, strictly convex preference relation.2) That the data are rationalizable and good 1 is an inferior good at some prices and incomelevels. Exercise 4You observe the following sequence of prices and demands by a single consumer for twogoods:(x1 , x2 ) (p1 , p2 )(1,(2,(1,(1,(1, 1)1)2)2)8) (3, 9)(2, 12)(5.5, 8.25)(4, 6)(9, 3.375) As usual, incomes are not directly observed.1) Draw a diagram with all observed choices and the associated budget constraints.2) Prove that these data can be rationalized by a monotone, strictly convex preferencerelation.2 3) List all preference relations over the observed choices that are consistent with the data.(Do not allow the consumer to be indifferent between any pair of bundles.)4) Rank the consumption bundles (3, 9), (2.5, 11), and (28, 1) in order of preference. 3