Department of Economics
University or Minnesota
Spring 2021 (Remote)
Problem Set 1
Due: Wednesday, February 8th
This problem set is due on Monday, February 8th at 11:59pm (Central Time). The
maximum score is 110 points. Be sure to review the Syllabus for details about homework assignments
and their grading. Feel free to contact me or your TA if you have specific questions about the
assignment.
Note you only have to submit questions 1 to 4. Some exercises have several parts, and each
part may conceal more than one task. Be sure to answer these questions thoroughly for full credit.
Further material intended for practice is provided at the final pages.
Present all final answers neatly on the space provided, or provide an organized document (you
do not need write/type the questions, however do not forget to label your answers). Show any
relevant calculations neatly. Please remember to attribute credit to your colleagues due to any help
received and collaboration.
Name:
ID number:
Section Number:
Names (and section numbers) of classmates you worked with:
1
University of Minnesota ECON 3101 – PS1
- (POINTS: 20)
Nicholson and Snyder 3.4 - (POINTS: 20)
Nicholson and Snyder 4.1
2
University of Minnesota ECON 3101 – PS1 - Consider the following optimization problem: (POINTS: 40)
max
x,y
−(x
2
+y
2
)
s.t.
3
2
(y+2) ≤ x (Ineq1)
s.t.
1.5+y
2
≤ x (Ineq2)
(a) What is the solution of the unconstrained problem? (Points: 4/40)
Answer:
■
3
University of Minnesota ECON 3101 – PS1
(b) At this part, you should consider each restriction at a time:
i. What is the solution of the problem restricted only by (Ineq1)? (Points: 5/40)
Answer:
■
ii. And only by (Ineq2)? (Points: 5/40)
Answer:
■
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University of Minnesota ECON 3101 – PS1
(c) What is the solution of the problem when both constraints hold in equality, i.e., (Points:
6/40)
max
x,y
−(x
2
+y
2
)
s.t.
3
2
(y+2) = x (Eq1)
s.t.
1.5+y
2
= x (Eq2)
Answer:
■
5
University of Minnesota ECON 3101 – PS1
(d) Use the method of slack variables presented in lecture to find the solution for the initial
problem. Explain why the solution is different from the one you found in the previous
item. (Points: 12/40)
Answer:
■
6
University of Minnesota ECON 3101 – PS1
(e) Consider below the modified version of the initial problem:
max
x,y
−(x
2
+y
2
)
s.t.
3
2
(y+2) ≥ x (Ineq1’)
s.t.
1.5+y
2
≥ x (Ineq2’)
i. What is the solution for this modified problem? (Points: 5/40)
Answer:
■
ii. What is the role of the constraints now? (Points: 3/40)
Answer:
■
7
University of Minnesota ECON 3101 – PS1 - (POINTS: 30)
Consider an agent with preferences represented by the following utility function:
u(x, y) = xy
(a) For each pair of bundles below, calculate the utility and indicate which one is preferred
or if the agent is indifferent between them: (Points: 4/30)
a = (0,1) b = (15,0) a is to b because u(a) u(b)
c = (1,1) d = (2,2) c is to d because u(c) u(d)
e = (4,1) f = (12,
1/4) e is to f because u(e) u( f )
g = (3,3) h = (1,8) g is to h because u(g) u(h)
(b) Considering the bundles from the previous item, order them from the least preferred to
the most preferred. (Points: 3/30)
Tip: If two bundles are indifferent, the order is irrelevant.
Answer:
■
(c) Consider a bundle i = (6, y). How much of good y this bundle should contain if the
consumer is indifferent between it and the bundle f (from part (a))? (Points: 4/30)
Answer:
■
8
University of Minnesota ECON 3101 – PS1
(d) Graph and label the bundles a − i. Graph the indifference curve that go through each
point and specify its utility level. (Points: 9/30)
Answer:
■
9
University of Minnesota ECON 3101 – PS1
(e) Give an example of another utility function that represents the same preferences that
u(x, y) represents. Explain why your example works. (Points: 10/30)
Answer:
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University of Minnesota ECON 3101 – PS1
Additional Practice
The questions below are meant to provide additional practice, but the student does not need to
submit them.
• Nicholson and Snyder: 3.1; 3.9; 4.2; 4.4
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University of Minnesota ECON 3101 – PS1
• Consider preference relations defined according to the following rule:
(x1, y1) ⪰ (x2, y2) when (x1 ≥ x2 and y1 ≥ y2)
holding with strictly inequality, i.e., strictly preference, if at least one inequality is strict:
(x1, y1) ≻ (x2, y2) when ((x1 > x2 and y1 ≥ y2) or (x1 ≥ x2 and y1 > y2))
Explain why this preference relation is not complete (you can provide an example that does
not satisfy the completeness assumption).
12
University of Minnesota ECON 3101 – PS1
• Consider the following prices and income:
px = $2 and py = $5
I = $140
Let A denote the bundle (50,8).
– Draw the budget constraint given prices and income above (and all the relevant points
including bundle A). What is the equation for the budget line?
Answer:
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University of Minnesota ECON 3101 – PS1
– Consider a change in prices:
px = $1 and py = $7
∗ Draw the new budget constraint (and the relevant points) and present the equation
for the new budget line.
Answer:
■
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University of Minnesota ECON 3101 – PS1
∗ Assume an agent has strongly monotone preferences and argue why A will not be
an optimal choice under the new prices.
Answer:
■
– Consider now prices
px = $4 and py = $2
∗ Draw the budget constraints for I = $140. Explain why the consumer cannot by
bundle A.
Answer:
■
∗ How much income the consumer need so they are able to afford bundle A under the
new prices? Draw this new budget constrain and explain how it can be obtained
from the budget constraint in the previous item.
Answer:
■
15
University of Minnesota ECON 3101 – PS1
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16
University of Minnesota ECON 3101 – PS1
– Consider now a different situation. Rather than having income I = $140, the consumer
has an endowment (x, y) = (50,8). For the original prices (px = $2, py = $5), the endowment is worth $140 (i.e., = $2×50+$5×8).
∗ Draw the consumer’s budget constraint (and the relevant points).
∗ Now consider the same prices from item (b) and draw the new budget constraint.
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17
University of Minnesota ECON 3101 – PS1
∗ Mark bundle A in your graph above and explain why the consumer is able to purchase
it for both cases.
Tip: If a consumer has an endowment, that is, quantities of each good, e.g., (x¯, y¯),
their disposable income is pxx¯+ pyy¯.
Answer:
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