ECON8025 - Homework 1 - Problems
Solve all questions and items below. Type or write the solu-
tion (use 12, 13 or 14 font, or write clear and big handwritten
answers). If possible, use LaTeX or some scienti…c editor. Feel
free to add (professional-looking) graphs if you believe the graphs
improve our economic understanding. Do not write unnecessary
comments. Proof all your claims. Show all the steps in the calcula-
tions. Avoid using decimal expansions; instead write your answers
as fractions if necessary.
Notation: :A means "not A".
Problem 1 - Ann’s Savings Account (2 marks)
Ann starts a savings account and deposits $2000 in the …rst day of every
year, for ten years, never withdrawing any money. How much will she have
in the end of the tenth year? Assume that the savings account pays 3% per
year of interest. Use compound interests, of course.
Problem 2 - Relations and Preferences (7 marks total, 1 mark
for each item)
Consider a decision-maker (DM) having to choose a single deterministic
alternative in a …nite, non-empty set X. Suppose that the DM has a week-
preference relation R on set X; that is, a transitive and complete relation on
X.
(2.1) What is a (binary) relation on X?
1
(2.2) How do we de…ne the DM’s indi¤erence relation (denoted ) de-
rived from relation R?
(2.3) What are the main properties of the indi¤erence relation?
(2.4) What are the properties that the indi¤erence relation does not
satisfy?
(2.5) How do we de…ne the DM’s strict preference (denoted S)?
(2.6) What are the main properties of the strict preference relation?
(2.7)What are the properties that the strict preference relation does not
satisfy?
HINT: Consider only the following possible properties of rela-
tions:
1 - Re‡exiveness
2 - Symmetry
3 - Transitivity
4 - Completeness
5 - Irre‡exivity
6 - Asymmetry. A binary relation P on a set X is said to be asymmetric
if and only if for every pair of elements a; b 2 X, if aPb, then it is not the
case that bPa.
7 - Antisymmetry (which is less strong than asymmetry). A binary re-
lation P on a set X is said to be antisymmetric if and only if for every
pair of elements a; b 2 X, if aPb and bPa, then a = b. An equivalent way of
making this de…nition would be to say that P is antisymmetric if and only
if whenever a; b 2 X, with a 6= b, then either it is not the case that aPb or it
is not the case that bPa.
8 - Acyclicity. A binary relation P on a set X is said to be acyclic if
2
and only if whenever x1Px2, x2Px3, x3Px4, , xn1Pxn, for some positive
n and x1; x2; ; xn 2 X, then x1 6= xn.
9 - Equivalence. A binary relation P on a set X is said to be an equiva-
lence relation if and only if it is re‡exive, symmetric and transitive at the
same time.
10 - Partial Order. A binary relation P on a set X is said to be a partial
order if and only if it is re‡exive, antisymmetric and transitive at the same
time.
Problem 3 (2 marks)
Suppose that a non-empty X is …nite and % is a complete and transitive
relation on X.
Prove that C(B;%) is non-empty for every budget ? 6= B X, where
C(B;%) = fx 2 B j x % y, for all y 2 Bg
THE END