ECM302 Coursework I
Each student needs to submit their own coursework solution. The solution
has to be handed in by the 19 November, 16.00 electronically via moodle.
Question 1 [12 marks]
Consider a consumer with preferences represented by the utility function
u(x1, x2, x3) = x1 +
√
x2x3
a) Give an example of a different utility function that represents the same
preferences as u.
[3 marks]
b) Let I = 24 be the income and p1 = 2, p2 = 1, p3 = 1 the prices. Calculate
the demand for good 1, good 2 and for good 3.
[9 marks]
Question 2 [18 marks]
Consider a representative consumer with preferences represented by the util-
ity function
u(x1, x2) =
√
1 + x1 + x2.
Assume that the price for good 2 is normalized to p2 = 1.
a) Calculate the demand for good 1 as a function of its price p and income
I.
[6 marks]
1
b) Now in addition to the consumer, consider a representative firm that
produces good 1 and has a cost function c(y1) =
1
2
· y21. Calculate the
supply as a function of the price for the good p.
[3 marks]
c) Assume that I = 4. Draw the demand and supply graph for good 1.
Find the equilibrium price and quantity of good 1.
[4 marks]
d) Calculate the producer and consumer surplus in the equilibrium you have
calculated in c).
[5 marks]
Question 3 [18 marks]
Consider an economy with one consumer, one consumption good and one
firm. The consumer generates his income from labour. The consumer can
divide one unit of time between leisure and labour `. There is complemen-
tarity between leisure and consumption and his utility is given by
u(x, `) =
(√
x +
√
1 `
)2
The firm has a production technology that only uses one input factor,
labour, and which is given by
f(L) =
√
L.
a) Let p be the price of the consumption good and let w be the wage rate.
Formulate the consumer’s utility maximisation problem and derive his
demand for the consumption good and his labour supply as a function
of p and w.
2
[6 marks]
b) Does the production function exhibit decreasing, increasing or constant
returns to scale
[3 marks]
c) Calculate the supply of the consumption good and the demand for labour
as a function of p and w.
[4 marks]
d) Find prices p and w, consumption level x and labour ` such that both
the output market and the labour market are simultaneously in equi-
librium.
[5 marks]
Question 4 [16 marks]
Consider the strategic form game depicted below. Each of two countries must
simultaneously decide on a course of action. Country 1 must decide whether
to keep its weapons or to destroy them. Country 2 must decide whether to
spy on country 1 or not. It would be an international scandal for country 1
if country 2 could prove that country 1 was keeping its weapons. The payoff
matrix is as follows.
Spy Don’t spy
Keep ( 1, 1) (1, 1)
Destroy (0, 2) (0, 2)
(a) Does either player have a strictly dominant strategy Explain your
answer!
3
[3 marks]
(b) Does either player have a weakly dominant strategy Explain your
answer!
[3 marks]
(c) Find a Nash equilibrium in which a weakly dominated strategy is used.
[5 marks]
(d) What assumption on the beliefs of a player would justify him choosing
a weakly dominated strategy Is this a realistic assumption in reality
[5 marks]
Question 5 [20 points]
a) Define the concept of a rationalisable strategy.
[5 marks]
b) Find the set of rationalisable strategies and all Nash equilibria for the
following game. Explain your reasoning.
12 L M
A 2, 7 3, 3
B 4, 2 2, 4
C 1, 7 2, 5
[5 marks]
c) Find the set of rationalisable strategies and all pure-strategy Nash equi-
libria for the following game [For this game you don’t need to calculate
the mixed-strategy Nash equilibria]. Explain your reasoning.
4
12 X Y Z
A 3, 0 0, 3 1, 2
B 0, 4 3, 1 1, 3
C 2, 5 2, 5 1, 5
[5 marks]
d) Find the set of rationalisable strategies and all Nash equilibria for the
following game. Explain your reasoning.
12 X Y Z
A 4, 5 3, 0 4, 1
B 5, 0 4, 5 1, 2
C 2, 9 1, 4 3, 3
[5 marks]
Question 6 [16 marks]
There are two players. Each of the players has an ”account” with an initial
balance of 0. The players alternate: At each stage, one of the players has the
right to stop the game. If a player chooses not to stop the game, then his
account is debited by 1 and the opponent’s is credited by 3. The game lasts
for 200 stages. If both players choose not to stop the game for 100 turns, the
game ends and each player receives the balance in his account (which is 200;
check this in order to verify that you understand the game).
a) What are the subgame-perfect equilibria of this game
[8 marks]
5
b) Now suppose the game can potentially go on for ever and is stopped
randomly: After each round in which the game has not been stopped
by the player, we toss a fair coin and stop the game if the outcome is
”heads” and continue if the outcome is ”tails”. Find a subgame perfect
equilibrium.
[8 marks]
6
