Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

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 scientiÖc 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 - Annís Savings Account (2 marks)

Ann starts a savings account and deposits $2000 in the Örst 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 Önite, 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?

(2.2) How do we deÖne the DMís 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 deÖne the DMís 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 - Reáexiveness

2 - Symmetry

3 - Transitivity

4 - Completeness

5 - Irreáexivity

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 aP b, then it is not the case that bP a.

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 aP b and bP a, then a = b. An equivalent way of making this deÖnition would be to say that P is antisymmetric if and only if whenever a; b 2 X, with a = b, then either it is not the case that aP b or it is not the case that bP a.

8 - Acyclicity. A binary relation P on a set X is said to be acyclic if and only if whenever x1P x2, x2P x3, x3P x4, ... , xn-1Pxn for some positive n and x1; x2; ... ; xn 2 X, then x1 = 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 reáexive, 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 reáexive, antisymmetric and transitive at the same time.

Problem 3 (2 marks)

Suppose that a non-empty X is Önite and % is a complete and transitive relation on X.

Prove that C  (B; %) is non-empty for every budget ? = B  X, where

C  (B; %) = fx 2 B j x % y, for all y 2 Bg