Simon Fraser University
ECON 804
Spring 2012
S. Lu
Midterm Examination
February 28, in class
Duration: 105 minutes
Total: 60 points
Each problem is worth 20 points and designed for 30 minutes. (You have 15 extra minutes.)
Make sure you save time for 3c and 3d: they should be easier than 2b through 3b!
1. Two roommates share an apartment. The amount of dirt in the apartment at the start of period
, denoted , follows:
, where
, and
is the
amount of cleaning done by roommate in period . (If
, then
.)
The roommates share a common discount factor
is:
, where
.
, and roommate ’s utility in period
a) [1] Interpret in this model. Is it necessarily the same for the two agents for whom
cleaning is equally costly?
b) [5] Suppose the game is played as a one-shot game in period 0, and
.
Show that there is a unique Nash equilibrium, and that players’ strategies are interior.
What is
in equilibrium? If
, what is as a function of and ?
c) [9] Suppose the game is played indefinitely. Find the first-order conditions governing
Markov perfect equilibrium. You do not need to solve these conditions.
(Note: If you use a Bellman equation, make sure to get rid of the value functions. Also,
express everything in terms of a single period’s quantities.)
d) [3] Compare the first order conditions from parts b and c and explain the economic
intuition behind the difference(s).
e) [2] How would your answer to part c change if there were more than two roommates?
Explain briefly.
2. We learned two versions of the Median Voter Theorem, one assuming single-peaked
preferences and one assuming single crossing.
a) [5] State the similarities and differences between the results. Which result is stronger?
b) [10] Show by example that the assumptions of the weaker theorem do not imply the
conclusion of the stronger theorem. [Hint: Try 3 agents and 4 alternatives.]
c) [5] Show that it is impossible to answer part b with an example with only 3 alternatives
and strict preferences.
3. The United Nations Security Council has 15 seats, with 5 permanent members that each have
veto power (on non-procedural matters). Let be the set of all 15 members, and let be the
set of permanent members. Nine affirmative votes are required to pass a resolution.
Disregarding the possibility of abstention, voting on the Council can be characterized by the
following characteristic function game, where
is an arbitrary set of countries:
a) [4] Find the core of . Show your work.
b) [6] Find the Shapley value of . Show your work. No need to simplify.
c) [6] Let
be the SWF corresponding to the Security Council’s voting rule. May’s
theorem requires four axioms (and that there be two alternatives).
i. State the one(s), if any, that is/are satisfied by .
ii. State the one(s), if any, that is/are violated by .
In each case, provide both the name and the statement of the axiom. You do not need to
prove your answers, but please briefly justify each.
d) [4] Show that each of the four axioms you provided in part d is necessary for May’s
theorem. That is, for each of them, give a SWF that violates it but satisfies the three
others. Assume that agents can be indifferent between a motion passing and failing (the
SWF can have indifferences as well). You do not need to prove your answers.