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ECON0010: MATHEMATICS FOR ECONOMICS SPECIMEN PAPER

TIME ALLOWANCE: 3 HOURS

Answer ALL EIGHT questions in Section A and THREE questions from Section B. Each question in Section A carries 5 marks and each question in Section B carries 20 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. Let

(i) Evaluate the determinant of A.

(ii) Determine whether A is orthogonal.

2. 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:

3. Show that the function

is concave.

Find its global maximum value.

4. 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.

5. State Roy’s identity.

Suppose a consumer’s indirect utility function is given by

where denote the price of the i-th good and the consumer’s income respectively and are positive constants. Use Roy’s identity to find the demand functions for the three goods.

6. Find the general solution of the differential equation

Describe the behaviour of the general solution when t is small and positive.

7. Find the general solution of the difference equation

8. Let

Find the eigenvalues and eigenvectors of A and use your results to write down

(i) the general solution of the difference equation ,

(ii) the general solution of the differential equation .

SECTION B

9. 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.

10. 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 ?

11. A consumer has a Stone-Geary utility function

where x_i denotes the consumption of the i-th commodity and b₁,b₂,c₁ and c₂ are positive

constants. The price of the i-th commodity is p_i and the consumer's income, m, is such

that

(*) .

Show that each indifference curve is negatively sloped, convex and has the lines x₁=c₁ and x₂=c₂ as asymptotes. Sketch the indifference curve pattern.

Express the consumer's problem as a constrained maximisation problem. Explain the significance of the condition (*).

Explain with the aid of a diagram how the indifference curve pattern is modified when b₁,b₂and c₂ are positive but c₁is negative.

In the case , find the demand functions which are applicable whenever .

12. (a) Suppose that x and y satisfy the following system of differential equations

Find the second-order differential equation satisfied by the complex variable . Find the general solution of this differential equation for all real values of . Hence find, for all real values of , the solutions satisfying the conditions at .