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A. Ozkardas Assignment 1 ECON 211

ECON 211 - Introduction to Mathematical Economics

Assignment 1 - Spring 2021

Due date: May 30th, 11:59pm

Total marks: 40

Instructions

� Answer all questions, making sure to fully explain your answers.

� You may create your graphs using graphing software or sketch by hand. However if you choose to sketch by hand, make sure to include some numbers on the axes and draw with enough precision so that the marker can see that your graph is drawn correctly.

� It is permitted to consult with other students regarding the assignment questions. However, the �nal work submitted must be your own. It is considered cheating for students to submit identical (or nearly identi- cal) assignments. Any students who do this will be subject to disciplinary action as described in University Policy 71.

� When you submit your assignment, please upload in a single �le or one �le per question. If you upload one �le per question, please label your �les according to the question number. Also please ensure that your images are submitted so that they display upright and not sideways.

� Please submit your �les in one of the following formats: pdf, jpeg, jpg, png, doc, docx. Please do not submit any HEIC �les. If you cannot submit in one of these formats, please contact the instructor.

1. [6 marks] Prove that for subset X, Y and Z of a given universal set U

(a) [1 marks] X ∩Y = X ∪Y (b) [1 marks] X ⊆ Y implies Y ⊆ X (c) [1 marks] X ⊆ Y implies X ∪ (Y −X) = Y (d) [1 marks] X ∩Y = ∅ implies Y ∩X = Y (e) [1 marks] (X −Y )−Z = X − (Y ∪Z) (f) [1 marks] X − (Y ∪Z) = (X −Y )∩ (X −Z)

2. [6 marks] Let X = Rn. An epsilon neighborhood of a ∈ X is de�ned by

N�(a) = {x ∈ X : d(a,x) < �}

where � is any positive real number.

(a) [3 marks] As precisely as you can illustrate N�(a) for a = 2 and � = 1/2

(b) [3 marks] As precisely as you can illustrate N�(a) for a = (2,2) and � = 1

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A. Ozkardas Assignment 1 ECON 211

3. [8 marks] Let X = Rn, and let A be a non-empty subset of X. a is a boundary point of A if every neighborhood of a contains at least one point in A and at least one point in Ac. a is an interior point of A if there exists a neighborhood of a that lies entirely inside A. A set is said to be closed if it contains all of its boundary points. A set is said to be open if every point in the set is an interior point for that set.

(a) [2 marks] For the set in question 2(a), is the point 5/2 a boundary point?

(b) [3 marks] For the set in question 2(b), is the point (3/2, 3/2) an interior point?

(c) [3 marks] Is the set in 2(b) open?

4. [3 marks] A subset of R has a maximum if it contains its supremum. This supremum is then the maximum of the set. Give examples of subsets of R (including bounded subsets) that do and do not have a maximum.

5. [5 marks] Find the convex combinations of the following pairs of points and, show part (a) graphically:

(a) [3 marks] (−1,1) and (3,4) (b) [2 marks] (−2,0,1) and (1,−2,2)

6. [6 marks] Given the strictly quasiconcave function y = f(x1,x2), sketch a typical level set in each of the following cases:

(a) [2 marks] The function is increasing in x1 and decreasing in x2.

(b) [2 marks] The function is decreasing in x1 and increasing in x2.

(c) [2 marks] The function is decreasing in both variables. (Hint: First determine which way the curve of the level set must slope, then identify the area that gives the better set, and then �nd how the curvature must look to make the better set convex.)

7. [6 marks] Given the strictly quasiconvex function y = f(x1,x2), sketch a typical level set in each of the following cases:

(a) [2 marks] The function is increasing in x1 and decreasing in x2.

(b) [2 marks] The function is decreasing in x1 and increasing in x2.

(c) [2 marks] The function is decreasing in both variables.

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