ECON 2030:EIntermediate Microeconomics II: ConsumersProblem Set # 2Till GrossDepartment of Economics, Carleton Universit...
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ECON 2030:E
Intermediate Microeconomics II: Consumers
Problem Set # 2
Till Gross
Department of Economics, Carleton University
Fall 2016
Due Date: October 5th 2016
In this problem set, as in all following ones and the exams, all unit increases refer
to an infinitesimally small increment. In other words, use derivatives to determine
marginal utility etc.
1
Expenditure Minimization (10 points)
Sally the Sleekâs preferences can be described by the utility function U (x, y) = x2 y 3 /512. Prices
are px = 2 and py = 6, she has an income of $80 to spend.
(a) If Sally initially consumed 4 units of x and 12 units of y, how much could she save if she
consumed 2 small more units of x and kept utility constant?1 (4)
¯ = 8. What is the minimum she has
(b) Sally decides that she does not need a higher level than U
to spend in order to attain that utility? (4)
(c) What is the cost of one additional (small) unit of extra utility in that case?2 (2)
2
Deriving Demand Curves (10 points)
Ariadneâs preferences can be described by the utility function U (x, y) = (3x1/2 + 6y 1/2 )2 . Elliottâs
utility function is U (x, y) = x + 3y 1/3 .
(a) What are Ariadneâs demand functions for goods x and y as a function of prices px and py and
income I? (3)
(b) Ariadne has an income of $10. Her three sisters Bianca, Clara, and Diana have the same
preferences but incomes of $20, $30, and $40, respectively. What is the aggregate demand
function of the four for good y when px = 1? (2)
1 Hint: Figure out how many units less of y she has to buy to keep utility constant. You donât yet need to solve
any optimization problem.
2 Hint: Remember that for any function E = f (U ), dE = ?f dU .
?U
1
(c) What are Elliottâs demand functions for goods x and y as a function of prices px and py and
income I? (3)
(d) Elliott has an income of $10. His brother Frank has the same preferences but an income of $20.
What is the aggregate demand function of the two for good y when px = 1?3 (2)
3
Income Changes (6 points)
Assume that there are only 2 goods, x and y, and that the price of each good is 1, i.e. px = py = 1.
(a) Amanda spends $200 on x and $300 on y when her income is $500. She spends $180 on x and
$420 on y when her income is $600. For this income change, characterize the type of each good
(necessity, luxury, or inferior). (2)
(b) Bill spends $10 on x and $6 on y when his income is $16. He spends $16 on x and $8 on y
when his income is $24. For this income change, characterize the type of each good (necessity,
luxury, or inferior). (2)
(c) Charles spends $3200 on x and $800 on y when his income is $4000. He spends $2800 on x
and $600 on y when his income is $3400. For this income change, characterize the type of each
good (necessity, luxury, or inferior). (2)
4
Price Changes (10 points)
Ariadne and Elliott have the utility functions U (x, y) = (3x1/2 + 6y 1/2 )2 and U (x, y) = x + 3y 1/3 ,
respectively. Assume that prices are initially px = py = 1.
(a) Ariadne has an income of $10. What are the total, income, and substitution effects on her
demand of y if the price of good y rises to 2? (5)
(b) Elliott has an income of $10. What are the total, income, and substitution effects on his demand
of y if the price of good y rises to 4? (5)
3 Only consider the range where both consume some of each good, i.e. ignore the parts where there are corner
solutions.
Intermediate Microeconomics II: Consumers
Problem Set # 2
Till Gross
Department of Economics, Carleton University
Fall 2016
Due Date: October 5th 2016
In this problem set, as in all following ones and the exams, all unit increases refer
to an infinitesimally small increment. In other words, use derivatives to determine
marginal utility etc.
1
Expenditure Minimization (10 points)
Sally the Sleekâs preferences can be described by the utility function U (x, y) = x2 y 3 /512. Prices
are px = 2 and py = 6, she has an income of $80 to spend.
(a) If Sally initially consumed 4 units of x and 12 units of y, how much could she save if she
consumed 2 small more units of x and kept utility constant?1 (4)
¯ = 8. What is the minimum she has
(b) Sally decides that she does not need a higher level than U
to spend in order to attain that utility? (4)
(c) What is the cost of one additional (small) unit of extra utility in that case?2 (2)
2
Deriving Demand Curves (10 points)
Ariadneâs preferences can be described by the utility function U (x, y) = (3x1/2 + 6y 1/2 )2 . Elliottâs
utility function is U (x, y) = x + 3y 1/3 .
(a) What are Ariadneâs demand functions for goods x and y as a function of prices px and py and
income I? (3)
(b) Ariadne has an income of $10. Her three sisters Bianca, Clara, and Diana have the same
preferences but incomes of $20, $30, and $40, respectively. What is the aggregate demand
function of the four for good y when px = 1? (2)
1 Hint: Figure out how many units less of y she has to buy to keep utility constant. You donât yet need to solve
any optimization problem.
2 Hint: Remember that for any function E = f (U ), dE = ?f dU .
?U
1
(c) What are Elliottâs demand functions for goods x and y as a function of prices px and py and
income I? (3)
(d) Elliott has an income of $10. His brother Frank has the same preferences but an income of $20.
What is the aggregate demand function of the two for good y when px = 1?3 (2)
3
Income Changes (6 points)
Assume that there are only 2 goods, x and y, and that the price of each good is 1, i.e. px = py = 1.
(a) Amanda spends $200 on x and $300 on y when her income is $500. She spends $180 on x and
$420 on y when her income is $600. For this income change, characterize the type of each good
(necessity, luxury, or inferior). (2)
(b) Bill spends $10 on x and $6 on y when his income is $16. He spends $16 on x and $8 on y
when his income is $24. For this income change, characterize the type of each good (necessity,
luxury, or inferior). (2)
(c) Charles spends $3200 on x and $800 on y when his income is $4000. He spends $2800 on x
and $600 on y when his income is $3400. For this income change, characterize the type of each
good (necessity, luxury, or inferior). (2)
4
Price Changes (10 points)
Ariadne and Elliott have the utility functions U (x, y) = (3x1/2 + 6y 1/2 )2 and U (x, y) = x + 3y 1/3 ,
respectively. Assume that prices are initially px = py = 1.
(a) Ariadne has an income of $10. What are the total, income, and substitution effects on her
demand of y if the price of good y rises to 2? (5)
(b) Elliott has an income of $10. What are the total, income, and substitution effects on his demand
of y if the price of good y rises to 4? (5)
3 Only consider the range where both consume some of each good, i.e. ignore the parts where there are corner
solutions.
