NYU ECON UA 10 - Demand and Revealed Preference

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help with Exercise 4 please

Intermediate Microeconomics
Homework Set 4: Demand and Revealed Preference
Due: Thursday, February 23rd at 2 pm via NYU Classes Exercise 1
Marie has preferences over consumption of sugar and spice representable by the utility function u(x1 , x2 ) = min{3x1 + 2x2 , 2x1 + 5x2 }, 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 her
entire 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 2
Consider a consumer with preferences over X = R2+ representable by the utility function
1/2 u(x1 , x2 ) = x1 + 2x1 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 good
1 is initially 1 and then doubles, compute the total change in demand for good 1, and
identify the portions of the change attributable to the substitution and income effects. 1 Exercise 3
Suppose you observe the following partial information about a consumer’s purchases of two
goods (which constitute his entire consumption). Good 1
Good 2 Year 1 Year 2 Quantity Price Quantity Price 100
100 100
100 120
? 100
80 Data on consumption of good 2 in year 2 is missing from the dataset. Also, as usual you do
not directly observe the consumer’s income. For what values of good 2 consumption in year
2 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 income
levels. 


Exercise 4
You observe the following sequence of prices and demands by a single consumer for two
goods:
(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 preference
relation.
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
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