Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
Problem Set: Class 2
Instructions
↪→ The problem set is meant to be done in the class, so it should take you around 90 minutes to solve it. Despite this, you are
given over 6 hours. Late submissions will not receive credit.
↪→ We encourage you to work in groups, discuss your thoughts with neighbors and ask questions to the Teaching Team
present in the workshop.
↪→ You may decide to submit your work for marking.
↪→ Despite working in groups, everyone who wants their work marked must prepare their own hand-written submission. We
will accept both scans of writing on paper and PDF submissions generated on iPads or equivalent.
↪→ Please submit only one PDF including your work.
↪→ Name your submission as ”xxx problem set 1.pdf” where xxx is your student ID.
↪→ Your answers don’t need to be fully correct to receive credit, partial correct intuitions will be rewarded.
↪→ Any use of AI is not permitted in solving the problem sets.
↪→ The submissions have to fulfill minimal readability requirements to be marked. In particular, make sure that:
– your writing is readable (you can use capital letters if this helps)
– it is clear that a given solution refers to a given exercise in the problem set
– if you made a mistake, cross out the wrong part and continue below. If possible, use a new sheet of paper.
– your answers are clear and logical
– your graphs are readable, including a description of axes and their units.
– make life of your markers easy: any doubt may be resolved to your detriment.
Good luck!
Exercise 2.1 [4 pts] Jacob is on a road trip in a desert. He and his car both need to drink: water costs $2 per gallon, and gasoline
costs $4 per gallon. Jacob brought $80 for that trip. We assume that consumption of negative quantities of water and gasoline
is impossible.
(i) [1 pt] Assigning water to the horizontal axis and gasoline to the vertical, draw Jacob’s budget set. Write down the equation
of budget constraint, and label its intercepts and kinks.
(ii) [1 pt] Suppose the local government in the desert is the only gasoline seller and it wants to generate revenue from taxation.
They impose $1 tax on each gallon of gasoline sold. Draw Jacob’s budget set, write down the equation of budget constraint,
and label its intercepts and kinks.
(iii) [1 pt] Suppose the tax raised strong concerns among the desert citizens, who claimed taxing goods necessary for survival is
immoral. During the negotiations, the government and desert citizens agreed that it is safe to assume that 10 gallons of gasoline
should be enough to survive within a certain period. Therefore, the government imposes a tax of $1 per gallon on every unit
above 10 gallons of gasoline sold. Draw Jacob’s budget set, and write down in detail the equation of budget constraint,
and label its intercepts and kinks.
(iv) [1 pt] Suppose now elections are coming, so the ruling party removes any taxes. Unfortunately, the company responsible for
the water supply made a typo in their spreadsheet, which led to ordering only 30 gallons of water. Draw Jacob’s budget set,
and write down in detail the equation of budget constraint, and label its intercepts and kinks.
Exercise 2.2 [4 pts] Suppose a consumer consumes positive amounts of goods x1 and x2, and her preferences are represented by
utility function U(x1, x2) = x1 · x2.
(i) [1pt] Consider bundle x = (1, 2). Define and draw the indifference curve at x and the set of bundles weakly preferred
to x.
(ii) [1pt] Define the Marginal Rate of Substitution, MRS12(x), and provide its interpretation. Calculate MRS12(x) at
x∗ = (1, 2). Suppose x∗ is the consumer’s optimal choice, maximizing her utility at a given set of prices and income. Derive
the ratio of prices. Do we have enough information to infer what is p1 and what is p2? If we know w = 12, do we have enough
information to infer p1 and p2? Justify in details each of your answers.
(iii) [1pt] Propose another utility function that represents the consumer’s preferences and call it V (x). Formally prove your answer
or provide detailed explanations to obtain full credit.
(iv) [1pt] CalculateMRS12(x) for at x
∗ = (1, 2) for the new utility function V suggested in the previous point. Is x∗ still an optimal
bundle given the prices from the previous point? Is it a coincidence of a rule? Prove formally or provide a detailed intuition
for the full mark.
Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
Exercise 2.3 [4 pts] Consider two goods with prices p = (p1, p2) = (3, 1), and a consumer with income w = 20.
(i) [1pt] Suppose the goods are perfect complements. Write down the utility function. Draw the set of indifference curves,
remembering to mark the direction of the increase in utility. Solve for the optimal bundle/bundles.
(ii) [1pt] Suppose the goods are perfect substitutes. Write down the utility function. Draw the set of indifference curves, remem-
bering to mark the direction of the increase in utility. Solve for the optimal bundle/bundles.
(iii) [2pts] Suppose consumer’s preferences are represented by the utility function u(x1, x2) = x1 · x2. Solve for the optimal bundle.
Hint: express x2 as a function of x1 and known objects using the budget constraint. Plug it back to the utility so that now it
depends only on x1. Does u(x1) belong to a class of functions that you are familiar with? What do we know about the extrema
of functions within this class?
Exercise 2.4 [4 pts] Consider the following maximization problem:
max
(x1,x2)>0
x
1
3
1 x
2
3
2
such that x1 + 2x2 ≤ 5
of a consumer with the utility function U(x) = x
1
3
1 x
2
3
2 facing prices (p1, p2) = (1, 2) and enjoying the income w = 5. You will solve
it in a few steps.
(i) [1pt] Can we write x1 + 2x2 = 5 instead of x1 + 2x2 ≤ 5? Formally prove or explain in detail for full credit.
(ii) [1pt] Write down the Lagrangian function.
(iii) [1pt] Write down the first-order conditions.
(iv) [1pt] Solve for optimal (x1, x2).
Exercise 2.5 [4 pts] Preference relation ⪰ defined on R2+ is said to be satiated if there exists a bundle that is preferred to any other
in the choice set. This bundle is called a bliss point.
(i) [1pt] Draw a set of indifference curves for satiated preferences. Remember to mark the direction of an increase in utility as
well as the bliss point.
(ii) [1pt] Is ⪰ monotone? Prove or give a counter-example.
(iii) [2pts] Give an example of preference relation ⪰ that is both convex and satiated. Explain your reasoning. Hint: it may be
useful to work on indifference curves.