Student ID ________________
Semester One Assessment, 2019
Faculty / Dept: Economics
Subject Number ECON30010
Subject Name Microeconomics
Writing time 2 hrs
Reading 15 minutes
Open Book status 10 double-sided pages
Number of pages (including this page) 5
Authorised Materials:
10 double-sided sheets of paper with any information on them, brought by students. Can be
handwritten, printed or photocopied.
Instructions to Students:
Instructions to Invigilators:
Calculators are not allowed
Paper to be held by Baillieu Library: yes ____ No __x__
Extra Materials required (please supply)
Graph paper _____ Multiple Choice form ____
Problem 1 [22pt]
Let Ann’s utility function be uA(q1, q2) = (q1)
1/3 + q2, prices are p1 and p2 for
goods q1 and q2, and her income is Y .
(a) [1pt] Write Ann’s expenditure minimisation problem.
(b) [6pt] Write down the Lagrangian and derive first-order conditions and ex-
pressions for q1 and q2.
(c) [3pt] What conditions should be imposed so that (b) gives you the solution
to Ann’s problem? Do not simplify the expressions.
(d) [4pt] Suppose that the condition(s) you state in (c) do not hold. What is
the solution to Ann’s optimisation problem in that case?
Consider Ben, with utility function
uB(q1, q2) =
{
q
1/3
1 + q2 if q1 < 1
q1 + q2 if q1 ≥ 1
(e) [8pt] Find Ben’s Hicksian demand.
Problem 2 [28pt]
Let x be the amount of money subject 1 receives and y be the amount of money
subject 2 receives.
Subject 1 may have “standard” preferences (denoted by S), defined as
uS1 (x, y) = x,
or Fehr-Schmidt preferences (denoted by FS), defined as
uFS1 (x, y) = x− α|x− y|,
where |x− y| denotes the absolute value of (x− y) and α > 0. That is, the utility
of subject 1 with FS preferences depends on both subject 1’s money x as well as
subject 2’s money y.
You are asked to design an experiment that would determine whether subject 1
has S or FS preferences. In your design, you need to find two bundles (x1, y1) >> 0
and (x2, y2) >> 0 such that subject 1 with S-preferences would necessarily choose
(x1, y1), while subject 1 with FS-preferences would necessarily choose (x2, y2).
EXAMINATION CONTINUES ON THE NEXT PAGE
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(a) [6pt] Find two such bundles, or explain why it is impossible.
(b) [5pt] Suppose that the definition of Fehr-Schmidt preferences also imposes
the condition that α ∈ [0.01, 1]. Find two bundles (x1, y1) and (x2, y2) that dis-
tinguish between S- and FS-preferences as described above or explain why it is
impossible.
Suppose now that subject 1 has FS preferences with parameter α = 1 and
subject 2 has S preferences. Suppose that two subjects play a game in which
they simultaneously decide whether to accept the bundle (1,2) or reject it. If both
accept (1,2), they consume it (subject 1 consumes $1 and subject 2 consumes $2).
If at least one of them reject it, they consume (0,0).
(c) [3pt] Write this game in a normal form (that is, as a matrix)
(d) [8pt] Find all mixed-strategy Nash equilibria of this game (including those
where subjects play degenerate mixed strategies, i.e. pure strategies). Be sure to
double-check that you have found all such strategies.
With three subjects, FS preferences (with parameter α = 1) are defined as
follows:
uFS1 (x, y, z) = x−
∣∣∣∣x− y + z2
∣∣∣∣ , (1)
where x, y, z are quantities consumed by subjects 1, 2 and 3 respectively.
Consider next the game with three subjects; subject 1 has FS preferences as
defined in (1) and subjects 2 and 3 have S preferences (that is, u2
S(x, y, z) = y and
u2
S(x, y, z) = z).
The game is similar to that in (d): if all subjects accept bundle (1, 2, 2), then
they consume it (that is, subject 1 consumes $1 and subjects 2 and 3 consume $2
each); if at least one of them rejects, then all consume (0, 0, 0).
(e) [6pt] Find all pure-strategy Nash equilibria.
EXAMINATION CONTINUES ON THE NEXT PAGE
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Problem 3 [10pt]
Consider a game with two players.
Definition: Strategy si is weakly dominated if there exists a strategy s
′
i such
that:
1. ui(s
′
i, s−i) ≥ ui(si, s−i) for any strategy of the opponent s−i and
2. ui(s
′
i, s−i) > ui(si, s−i) for at least one strategy of the opponent s−i.
Construct an example where two players both play their weakly dominated
strategies in a pure-strategy Nash equilibrium or explain why it is impossible.
Problem 4 [20pt]
Consider a political competition game similar to the one discussed in class, where
parties A and B simultaneously choose their position along the [0, 1] interval.
Unlike the model in class, one-half of the population occupies [0, 1/4] and the
other half occupies [1/3, 1]; no one occupies [1/4, 1/3]. Assume that the election
outcome is deterministic: the party that has more support wins elections with
certainty; if parties have equal support, then they each win with probability 1/2.
These assumptions are retained for the entire problem.
(a) [6pt] Suppose that parties only care about winning. Find all pure-strategy
Nash equilibria of this game.
(b) [2pt] Find one mixed-strategy Nash equilibrium of this game.
Suppose now that party A cares about the position of the winning party and its
ideal position is 0. That is, A’s utility function is uA = −x, where x is the position
of the winning party. Party B only cares about winning.
(c) [6pt] Find one pure-strategy Nash equilibrium where parties do not locate
at the same point or explain why it is impossible.
(d) [6pt] Find one pure-strategy Nash equilibrium where parties do locate at the
same point or explain why it is impossible.
EXAMINATION CONTINUES ON THE NEXT PAGE
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Problem 5 [20pt]
Consider a congestion game similar to the one in class, with infinitely many drivers
choosing which route from A to B to take, on the following road network:
A B
L
2 min
x
l
3 min
1− x
H
{
2 if y ≤ 0.5
1.5 + y if y > 0.5
min
y
h{
1 if 1− y ≤ 0.7
0.3 + (1− y) if 1− y > 0.7 min
1− y
C
D
c min s
Numbers on the edges show the travel time (in minutes) on each road; x, 1 −
x, y, 1−y are fractions of drivers on each road. Unlike the model in class, the travel
time on highways h and H increases only after a certain threshold is reached (when
more than 0.5 mass of drivers travel on H and more than 0.7 mass of drivers travel
on h).
In parts (a)–(b), let c = 1.
(a) [6pt] Find one pure-strategy Nash equilibrium of this game.
(b) [7pt] Find all pure-strategy Nash equilibria. Explain why no other pure-
strategy Nash equilibria exist.
(c) [7pt] Let c = 0.3. Find one pure-strategy Nash equilibrium of the new game.
END OF EXAMINATION
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