ECON60081
2 hours

THE UNIVERSITY OF MANCHESTER

MATHEMATICAL METHODS FOR ECONOMIC ANALYSIS

SAMPLE EXAM
Date:
Time:

Answer ALL Questions in Section A and ALL Questions in Section B.
Each Section contributes to 50% of the total marks.
Please enter your answers in the Answer Book

THIS PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM

Electronic calculators may be used provided that they cannot store text.

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P.T.O.

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ECON60081

SECTION A

Answer ALL Questions. Each question carries equal weight.

Question 1. Consider the problem Ax = b where
⎛
⎞
⎞
⎛
⎛ ⎞
2 1 3
1
1
A = ⎝ 3 2 5 ⎠  x = ⎝ 2 ⎠  b = ⎝ 3 ⎠ 
1 1 2
3
2

Determine the degrees of freedom and the number of redundant equations of this system. Further, determine the solution(s) if solutions exists.
[10 marks]
Question 2. Give formal definitions for the following: (a) a convex function, (b) a strictly con­ vex set, (c) a differentiable function. Further, give an example of a concave function that is not differentiable. [10 marks]
Question 3. Find the solution of the following differential equation
 = 1 + 3 − 2 
˙

where (0) = 5.

[10 marks]

Question 4. Find the general solution of the following second order differential equation
 + 4 + 10 = 
¨
˙
[10 marks]
Question 5. Consider the following system of nonlinear difference equations
½
1+1 = 31 − 2 
2
2+1 = 2 + 1 
Find the equilibria and classify them as sink, source or saddle.

[10 marks]

Continued

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ECON60081

SECTION B

Answer ALL Questions.

Question 6. Assume  ≥ 4. For   −1 and   −1, solve the utility maximization problem
(a) max ( ) = 1 ln(1 + ) + 1 ln(1 + ) subject to the constraint 2 + 3 = .[15 marks]
2
4
(b) Let (∗ ()  ∗ ()) be the solution to the problem in part (a) as a function of .