SECTION A

1.         Let

A =  [2

−2

−1

2

−2].

(i) Evaluate the determinant ofA.

(ii) Determine whether A is orthogonal.

2.          Consider the following form for the production function of an economy Q = F(K, L, t)

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

IfK 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:

Q = eTt K a LF                      (T, a, F > 0).

3.         Show that the function

 x2  + 2xy  2y2   3x + 4y +1

is concave.

Find its global maximum value.

4.         Consider the difference equation      yt+1 + ayt  = b

where  a   and   b   are   constants  with   a 士 1 .  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

V(p1 , p2 , p3 , m) = p1(a)pp3(Y)

where pi  and m denote the price ofthe i-th good and the consumer’s income respectively and  a,  andY 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  −  = 3t .

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

7.         Find the general solution of the difference equation

yt+2  − 5yt+1  + 6yt  = 3 + 2t .

8.         Let

3    −1

4    −2

Find the eigenvalues and eigenvectors ofA and use your results to write down (i) the general solution of the difference equation y(t + 1) = Ay(t) ,             (ii) the general solution of the differential equation dy/dt = Ay(t).

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 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 .

Let

v1  = [0], v2  = [1], v3  = [1].

Show that

(i)  v1 , v2 , v3  are linearly independent;

(ii) any 3-vector can be expressed as a linear combination of v1 , v2 , v3 . How do you think results (i) and (ii) can be generalised?

10.       (a) 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 .

For a homogeneous utility function of 2 variables, 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.

(b) Suppose the production function in an economy takes the form

Q = F(K, L)

where Q, K and L denote aggregate output, capital and labour. Suppose further that F(K,L) is homogeneous of degree 1. Show that the production function can be written in the form

Q = L0(k)

where  0 is a function of a single variable and k = K / L .

In the case where

F(K, L) = (K−2  + L−2) −1/2

show that F(K, L) is homogeneous of degree 1 and find the function  0 .

11.       A firm has a Cobb-Douglas production function Q = Ka Lb

where Q, K and L denote output, capital and labour respectively and where a and b  are positive constants. Determine how the returns to scale depend on a and b and find the marginal products of capital and labour.

Show that each isoquant is negatively sloped, convex and has the axes as asymptotes. Sketch the isoquant diagram.

When the prices of capital and labour are r and w respectively, find the conditional input demand functions and hence show that the minimum cost of producing output level q is

(a + b)() () q  .

12.       (a) A consumer has the utility function

U(x1 , x2 ) = 40   + x2              (x1 > 0, x2  > 0)

where xi   denotes the consumption of the ith good. Let the prices of goods 1 and 2 be 4 and 1 respectively and suppose the consumer’s income is m. Find the quantities of the two goods demanded if (i) m = 300 , (ii) m = 50 .

(b) Solve the problem

minimise (x 1)2  + (y 1)2   subject to 2x + y  a ,

if (i) a = 4 , (ii) a = 2 .

13.       (a) State Taylor’s theorem for the expansion of a smooth function about the point x = a . Taking   f(x) = x1/3 , find an approximation for   91/3 as a sum of fractions using the first three terms of the Taylor expansion of  f(x) about x = 8. By considering the expression for the remainder term in this case, show that the absolute value of the error is less than 1/4000.

(b) Evaluate

∫冗/2teit dt

and hence evaluate

冗/2t sin t dt .

14.       Suppose the non-linear system

 = f(x, y),       = g(x, y)

has a fixed point at (x*,y*) . Derive the linearisation of the system about (x*,y*) .          Under what circumstances are the solution paths of the non-linear system close to those of the linearisation?

For the following system

 = x(3 − 2x − y),

 = y(3 − 2y − x)