ECON1010: MATHEMATICS FOR ECONOMICS 2018
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SUMMER TERM EXAMINATIONS 2018
ECON1010: MATHEMATICS FOR ECONOMICS
SECTION A
1. 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) 一 x2x3
is (i) positive definite, (ii) positive semidefinite but not positive definite, (iii) neither of these.
2. 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 ofoutput.
3. Find the gradient vector and the Hessian matrix of the function 3x2 一 6xy + y4 + y2 .
Verify that (1,1,一1) is a critical point and determine whether it is a local maximum, local minimum or neither.
4. A consumer has a utility function
U(x1 , x2 ) = x1(3)x2
where x i denotes the consumption of the i-th commodity.
If the price of the i-th commodity is p i 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.
5. Determine whether the function
x1(3)x2 (x1 > 0, x2 > 0)
is (i) quasi-concave, (ii) concave.
6. Use De Moivre’s theorem to evaluate ( 3 + i)10
giving your answer in the form a + bi where a, b 从 R .
7. Find the general solution of the differential equation
d 2 y dy
2 一 4 + 3y = 2t .
dt dt
8. Find the characteristic polynomial of the matrix
「0 1]
| | .
Hence show the matrix is not diagonalisable.
SECTION B
9. (a) Define the term echelon matrix and say what the 4 types of echelon matrix are. Hence show that the number of solutions of the simultaneous linear equations Ax = b is 0, 1 or infinity where A is an m〉n matrix, x is a vector in Rn and b is a vector in Rm .
(b) Given that the equation
ABx = d ,
where
「1 1 0] 「 1 4 1 ] 「1]
| | | | | |
一13 一(一)1(2) 一
has a solution, find k.
10. 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.
11. Consider the differential equation
dy
(*)
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 of a for which the stationary solution is stable.
Now consider the following two difference equations:
(i) yt + ayt = b , (ii) Dyt + ayt = b
where yt = 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.
12. Suppose that, for a given parameter t, the variables x and y are chosen to maximise f (x, y, t) subject to the constraint g(x, y, t) = 0 and denote the maximal value by v(t) . Show that
?L
v,(t) =
?t
where L denotes the Lagrangian
L(x, y, 入, t) = f (x, y, t) 一 入g(x, y, t)
with x, y and 入held at their optimal values.
State the extension of this result to the case where there are r parameters and n variables. Use the extension to derive Roy’s identity.
Suppose the indirect utility function is given by
(p1 + p2 )m
where pi and m denote the price of the i-th commodity and the consumer’s income respectively. Use Roy’s identity to find the demand functions.
13. (a) Find the general solution of the difference equation x(t +1) = Ax(t) + b(t) , where
「 1 1/ 3] 「2 + t]
A = |L4 / 3 1 」| , b(t) = |L 3t 」| .
(b) Find the general solution of the differential equation = Ax(t) + b(t) , where
dt
「一 2 一 3] 「e一t ]
A = | | , b(t) = | | .
L 1 一 6」 L 0 」
14. Suppose the non-linear system
dx/ dt = f (x, y), dy / dt = 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?
In a model of pollution, capital K and pollution P satisfy the pair of differential equations
dK / dt = K(sKa一1 一 6), dP/ dt = Kb 一yP
where 0 < s < 1, 0 < a< 1, 6 > 0,y > 0 and b > 1.
Find the fixed point (K*,P*)satisfying K* > 0 and P* > 0 and show it is locally stable for the non-linear system.
Find an explicit expression for K(t) when K(0) = K0 > 0 , and find its limit as t ) 的 .
2022-08-20