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

ECON0006: INTRODUCTION TO MATHEMATICS FOR ECONOMICS SPECIMEN PAPER

TIME ALLOWANCE: 2 HOURS

Answer ALL FIVE questions in Section A and TWO questions from Section B. Each question in Section A carries 10 marks and each question in Section B carries 25 marks.

In cases where a student answers more questions than requested by the examination rubric, the policy of the Economics Department is that the student’s first set of answers up to the required number will be the ones that count (not the best answers). All remaining answers will be ignored.

SECTION A

1.         Find the complete solution of the system of equations

            where

.

Find also the rank of the matrix A.

2. Let

(i) Evaluate the determinant of A.

(ii) Determine whether A is orthogonal.

3.  Consider the following form for the production function of an economy

where Q, K, L and t denote output, capital, labour and time respectively.

If K and L depend only on t, find an expression for dQ/dt in terms of the partial derivatives of the production function and the time derivatives of K and L.

Now suppose that K and L have constant proportionate rates of growth m and n respectively. Find the rate of growth of output when the production function takes the following form:

4. Show that the function

is concave.

Find its global maximum value.

5. Consider the difference equation

where a and b are constants with . Find the stationary solution, the complementary solution and the general solution.

Find also the range of values of a for which

      (i) the stationary solution is stable,

      (ii) the general solution is alternating.

SECTION B

6. Define the terms linear combination, linear dependence and linear independence as applied to vectors.

Suppose a,b and c are three vectors in . Show that:

(i) if a,b and c are linearly dependent then there exist scalars α,β and γ, not all zero, such that

αa+βb+γc=0 ,

     (ii) if there exist scalars α,β and γ, not all zero, such that

   αa+βb+γc=0

     then a,b and c are linearly dependent.

      State the generalisation of this result to the case of k vectors in .

Use the generalisation to prove the following statements:

(a) Any set of vectors containing the zero vector is linearly dependent.

(b) Any set of more than n vectors in  is linearly dependent.

7. (a) Find the critical points of the function

.

Determine the nature of each critical point.

(b) Show that the function

has a critical point at  and determine whether this point is a local maximum, a local minimum, a saddle point or none of these.

8. Each week an individual with income m consumes quantities x and y of two goods whose prices are p and q. Her utility function is

which is defined for . Find the demand functions.

Now suppose the individual works for l hours per week out of an available L hours so that she has hours of free time and suppose her utility function is now

which is defined for and . Suppose m now denotes unearned income and w is the wage per hour. Assuming that , find the demand functions and labour supply. What happens if ?