SUMMER TERM EXAMINATIONS 2018
ECON1004: INTRODUCTION TO MATHEMATICS FOR ECONOMICS
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. Define the rank ofa matrix.
Explain how to find the rank of (i) an echelon matrix, (ii) a general matrix.
Find the rank of
2. Define the terms positive definite and positive semidefinite as applied to quadratic forms.
Determine directly from the definitions whether the quadratic form.
q(x1, x2, x3 ) = x1(2) + x2(2) + x3(2) - x2x3
is (i) positive definite, (ii) positive semidefinite but not positive definite, (iii) neither of these.
3. Suppose the production function of an economy is
Q = F(K, L)
where Q, K and L denote aggregate output, capital and labour respectively and F(K, L) is a smooth homogeneous function of degree s where s is a positive constant. Suppose further that K and L have the same constant proportionate rate of growth m. Use the chain rule and Euler’s theorem to find the rate of growth of output.
4. Find the gradient vector and the Hessian matrix of the function
3x2 - 6xy + y4 + y2 .
Verify that (1,1,-1) is a critical point and determine whether it is a local maximum, local minimum or neither.
5. A consumer has a utility function
U(x1, x2 ) = x1(3)x2
where xi denotes the consumption ofthe i-th commodity.
If the price of the i-th commodity is pi and the consumer’s income is m, express the consumer's problem as a constrained maximisation problem.
Write down the Lagrangian for the problem and obtain the first-order conditions.
SECTION B
6. (a) Define the term echelon matrix and say what the 4 types of echelon matrix are. Hence show that the number of solutions ofthe simultaneous linear equations Ax = b is 0, 1 or infinity where A is an m×n matrix, x is a vector in R n and b is a vector in Rm .
(b) Given that the equation
ABx = d ,
where
has a solution, find k.
7. (a) For the function
f (x, y) = -3cx2 + 2cxy - y2 - 2x + 5y
where c is a constant, find the gradient vector and the Hessian matrix. For what values of c is f (x, y) concave?
When c=2, explain how you know that any critical point of f (x, y) must be a global maximum point. Hence find the global maximum.
(b) Suppose the profit of a firm is given by the smooth concave function Π(x1, x2 ) where x1 and x2 denote the output levels of the two products which the firm manufactures. Write down conditions, in terms of the first-order partial derivatives of
Π , sufficient for the profit to attain its global maximum value at (a, b) .
In the particular case where
Π(x1, x2 ) = -2x12 + 4x1x2 - 3x22 +10x1 -14x2 - 3
verify that Π is concave and find the profit-maximising output levels and the maximum profit.
8. Consider the production function
Q = (K1/ 2 + L1/ 2)2
where Q, K and L denote output, capital and labour respectively. Show that the isoquants are negatively sloped and convex.
Find the coordinates of the points where the isoquant, along which Q takes the value c, (c > 0) , meets the K and L axes. Find also the slope of this isoquant at each of these points. Sketch the isoquant diagram.
When the prices of capital and labour are r and w respectively, find the conditional input demand functions and the total cost function.
Find also the elasticity of substitution.
9. Consider the differential equation
where a and b are constants with a ≠ 0 . Find the stationary solution, the
complementary solution and the general solution.
Find also the range of values ofa for which the stationary solution is stable.
Now consider the following two difference equations:
(i) Δyt + ayt = b ,
(ii) Dyt + ayt = b
where Δyt = yt +1 — yt , Dyt = yt — yt—1 and a and b take the same values as in (*).
For each difference equation, find the stationary solution, the complementary solution and the general solution. Find also the range of values of a for which the stationary solution is stable.
For each of the following statements, say whether it is true or false, giving reasons for your answer:
(a) If the stationary solution of the differential equation (*) is stable, then the stationary solution of at least one of the difference equations (i) and (ii) must be stable.
(b) If the stationary solution of at least one of the difference equations (i) and (ii) is stable, then the stationary solution of the differential equation (*) must be stable.
(c) There is a set of values of a for which the stationary solutions of the differential equation (*) and the difference equations (i) and (ii) are all stable.