SUMMER TERM EXAMINATIONS 2017
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. State a criterion in terms of the determinant for a square matrix to have an inverse.
Find the values ofa for which the matrix
has an inverse.
2. Let
Verify that any two of the three 3-vectors u,v,w are orthogonal. Find scalars λ, μ,v such that [λu μv vw] is an orthogonal matrix.
3. Suppose
z = x ln(1+ xy) .
Use the small increments formula to find an approximation to the change in z when x changes from 1 to 1+ Δx andy changes from 1 to 1+ Δy where Δx and Δy are small.
Now suppose you are additionally given
x = 1 + t, y = t 2 .
Use the chain rule to find dz / dt when t = 1.
4. Show that the function
x2 - 2xy + 5y2 -10x + 2y
is convex.
Find its global minimum value.
5. Find the general solution of the differential equation
Also find the solutions which satisfy y=1 when t=0 and y=0 when t=1 and sketch them on the same axes.
SECTION B
6. (a) Define the terms linear combination, linear dependence and linear independence as applied to vectors.
Suppose a,b and c are three vectors in Rn . Show that a,b and c are linearly dependent if and only if there exist scalars α,β and γ, not all zero, such that
αa+βb+γc=0 .
State the generalisation of this result to the case ofk vectors in Rn .
(b) 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 .
(c) Use the results in (a) and (b) to show that ifwe have a set of more than n vectors in Rn , these vectors must be linearly dependent.
7. (a) Suppose the production function of an economy is
Q = F(K, L)
where Q, K and L denote aggregate output, capital and labour respectively.
Explain what is meant by saying (i) F(K,L) displays diminishing returns to each factor, (ii) F(K,L) displays increasing returns to scale.
Now consider the special case
F(K, L) = KαLβ (α, β > 0) .
Find a condition in terms of α and β for the production function to exhibit decreasing returns to scale. Show that if this production function exhibits decreasing returns to scale then it also exhibits diminishing returns to each input. State, with reasons, whether the converse is true.
(b) Now suppose the production function is
G(K, L, t) = ertKαLβ (α, β, r > 0)
and K and L have constant rates of growth m and n respectively. Find the rate of growth of output.
8. Explain what is meant by a homogeneous function of 2 variables of degree h. Show that the partial derivatives of such a function are homogeneous of degree h-1 .
Show that the utility function
U(x, y) = xαyβ ,
where α and β are positive constants, is homogeneous and state the degree of
homogeneity, h. Verify that the two marginal utilities are homogeneous of degree h-1 .
For this utility function, show that the slope of the indifference curves is constant along the line
y = cx
where c is a positive constant. Draw a diagram to illustrate this result.
Now show the same result is true when the utility function is a general homogeneous function of 2 variables.
9. A consumer has a utility function
where xi denotes the consumption of the i-th commodity. Show that each indifference curve is negatively sloped, convex and has two asymptotes. Find the equations of the asymptotes corresponding to the indifference curve U = c (c > 0) . Sketch the indifference curve diagram.
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.
Find the demand functions, explaining carefully each step of your argument.