ECON7030 Microeconomic Analysis

Problem Set 1: 100 Marks

Answer all questions. Justify all your answers. Every graph (figure or diagram) in your

answers must be well-labelled. For functions that intersect the axes, the intercept values

must be clearly identified. All quantities of goods can be treated as continuous variables

unless explicitly stated otherwise. Show your work. Marks are as indicated.

Question 1 True or False (25 marks)

True or False? Justify your answers. A correct answer without justification will receive no

credit.

(a) (5 marks) Suppose the utility function U (x, y) = (x + 1)2 (y + 1)2 represents a con-

sumer’s preferences over the domain x, y ≥ 0. Consider the function V (x, y) = ln(x +

1) + ln(y + 1). Determine whether V represents the same preferences as U . To do so,

you must explicitly identify a function f such that V = f (U ) and verify whether f is

strictly increasing over the relevant domain.

(b) (5 marks) Consider three bundles A, B, and C in a consumption set. Suppose we only

know that A ≿ B, B ≿ C, and C ≿ A. This information is sufficient to establish that

the preference relation ≿ is both complete and transitive.

(c) (5 marks) Two friends, Ana and Bruno, walk into a clothing store with completely

different preferences and different incomes. After browsing independently, they each

find an outfit, a shirt and pair of pants, and they both purchase the outfit. A classmate

observes this and concludes: “Ana and Bruno must have had the same MRS for that

bundle of shirt and pants.” Is the classmate correct?

(d) (5 marks) Indifference curves cannot cross the horizontal axis; a bundle on the hori-

zontal axis implies zero consumption of one good, so the consumer must be at a corner

that is never optimal.

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(e) (5 marks) Suppose U (x, y) is a differentiable utility function representing a consumer’s

preferences, and let V (x, y) = f (U (x, y)) where f : R → R is strictly increasing. Then

the marginal utilities satisfy M UxU = M UxV and M UyU = M UyV . As a consequence, the

MRS is the same for U and V .

Question 2 A “Bad” Good (25 marks)

Anna consumes two goods: leisure hours (good x) and pollution exposure (good y). Anna

likes leisure but dislikes pollution — that is, pollution is a “bad”. Her preferences are

represented by the utility function:

U (x, y) = x − 21 y 2 ,

where x ≥ 0 and y ≥ 0.

(a) (5 marks) Derive the Marginal Rate of Substitution (MRS) for Anna. What does

the sign of the MRS tell you about the shape of her indifference curves? Is the MRS

diminishing, constant, or increasing in y along an indifference curve?

(b) (5 marks) In a well-labelled diagram with x on the horizontal axis and y on the vertical

axis, sketch two indifference curves for Anna. Indicate clearly the direction in which

utility is increasing. Choose a bundle E on one of the indifference curves and shade the

region of bundles that Anna strictly prefers to E.

(c) (5 marks) Are Anna’s preferences monotone? Are they strongly monotone? Justify

your answer carefully.

(d) (5 marks) Are Anna’s preferences convex? Prove your answer using the definition of

convexity.

(e) (5 marks) Suppose Anna faces prices Px = 1 and Py = p > 0, and has income M > 0.

Anna cannot choose her level of pollution freely in the market; instead, she is forced to

consume a fixed quantity ȳ of pollution (e.g., determined by her neighbourhood). Given

this constraint, what quantity of leisure x will she choose, and what is her resulting

utility? How does her utility change as ȳ increases? Interpret.

Question 3 Utility Maximisation, Expenditure Minimisation, and

Duality (25 marks)

Consider a consumer with Cobb-Douglas utility:

U (x, y) = x1/3 y 2/3 .

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The consumer faces prices Px , Py > 0 and has income M > 0.

(a) (5 marks) Find the Marshallian (uncompensated) demand functions x∗ (Px , Py , M ) and

y ∗ (Px , Py , M ).

(b) (5 marks) Derive the expenditure function e(Px , Py , ū) and the Hicksian (compensated)

demand functions xh and y h .

(c) (5 marks) Suppose Px = 1, Py = 2, and M = 12. Find the utility-maximising bundle

A and the corresponding utility level ū.

(d) (10 marks) Now suppose the price of y rises to Py′ = 8 (with Px = 1 and M = 12

unchanged).

(i) Find the new utility-maximising bundle B.

(ii) Find the expenditure-minimising bundle C that achieves the original utility level

ū from part (c) at the new prices, and state the minimum expenditure required.

(iii) In a single well-labelled graph, draw the original budget line, the compensated

budget line (corresponding to the minimum expenditure in (ii)), and sketch the

indifference curves passing through bundles A and C. Label all intercepts and

bundles clearly.

Question 4 Perfect Substitutes (25 marks)

Sam consumes two goods: Oat Milk (x) and Almond Milk (y). Sam views these as perfect

substitutes and is willing to exchange α units of Almond Milk for 1 unit of Oat Milk at any

consumption bundle, where α > 1. The price of Oat Milk is Px and the price of Almond

Milk is Py . Sam has income M > 0.

(a) (4 marks) Propose a utility function that correctly represents Sam’s preferences. Jus-

tify your choice by explaining how it captures the substitution rate between the two

goods.

(b) (5 marks) Draw Sam’s indifference map in a well-labelled diagram (put x on the

horizontal axis). Indicate the direction in which utility increases and label the slope of

the indifference curves. Does Sam exhibit a diminishing MRS?

(c) (10 marks) Using U (x, y) = αx + y with α = 3 and M = 12, answer the following:

(i) Suppose Px = 2 and Py = 1. Find Sam’s optimal bundle. In a well-labelled dia-

gram, draw his budget line and the indifference curve passing through the optimal

bundle.

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(ii) Now suppose Px = 6 and Py = 1 (with M = 12 unchanged). Find Sam’s new

optimal bundle. Draw the new budget line and optimal indifference curve in the

same diagram.

(iii) Explain in plain economic language why the optimal bundle changed from part (i)

to part (ii). Your explanation should refer to the relative slopes of the budget line

and indifference curves.

(d) (6 marks) Returning to general parameters (α > 1, Px , Py , M ), suppose Sam’s income

rises by ∆M > 0 while prices remain fixed in the region where Px /Py < α (so that the

x-axis corner is optimal). How does his demand for each good respond to this income

change? What does this imply about the income elasticity of demand for x and y? Now

contrast with the income elasticity in the region Px /Py > α. What do these results say

about the “normality” of each good?

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